mm-4849 === Subject: A speculation about polynomial sequences and primes Dirichlet proved that a linear sequence a*n+b contains an infinite number of primes except when a and b have a common factor, in which case it contains either 0 or 1 primes. How about quadratic and higher polynomial sequences? I've looked at the cases where the answer is obvious and it seems that such sequences contain either 0 or 1 prime. I speculate:= Any polynomial sequence contains an infinite number of primes except for those sequences that contain either 0 or 1 primes. To prove this would earn immortal fame. To disprove it seems easier. Are there polynomial sequences (a*n^m+b*n^(m-1)....) which clearly contain 2 or 3 or some other finite number of primes? Dick Batchelor === Subject: Re: A speculation about polynomial sequences and primes > Dirichlet proved that a linear sequence a*n+b contains an infinite > number of primes except when a and b have a common factor, in which > case it contains either 0 or 1 primes. > How about quadratic and higher polynomial sequences? I've looked at > the cases where the answer is obvious and it seems that such sequences > contain either 0 or 1 prime. This is a well known problem. Look up the Bateman-Horn conjecture and Schinzel's conjecture. === Subject: Re: A speculation about polynomial sequences and primes > Dirichlet proved that a linear sequence a*n+b contains an infinite > number of primes except when a and b have a common factor, in which > case it contains either 0 or 1 primes. > How about quadratic and higher polynomial sequences? I've looked at > the cases where the answer is obvious and it seems that such sequences > contain either 0 or 1 prime. > I speculate:= > Any polynomial sequence contains an infinite number of primes except > for those sequences that contain either 0 or 1 primes. To prove this would earn immortal fame. To disprove it seems easier. > Are there polynomial sequences (a*n^m+b*n^(m-1)....) which clearly > contain 2 or 3 or some other finite number of primes? Dick Batchelor How about (1 + (p-1)(n-1)^2)(1 + (q-1)n^2) where p and q are distinct primes? --- J K Haugland http://www.neutreeko.net === Subject: Re: iteration equation > the operator half iterate at fixed point ... is allowed. Something for you to play with: Let a and b be positive real numbers and x be a positive real-valued variable. If f(x) = (a^c)*(x^d), where c = 1 / [sqrt(b) + 1] d = sqrt(b), then f(f(x)) = a*x^b. Thus, using this we can get explicit expressions for an iteration square root of functions such as 5x^2, 2x^3, pi*x^sqrt(3), etc. Dave L. Renfro === Subject: Re: R E L A T I V I T Y: Fitting a Square Peg into a Round Hole > The answer is simple. It disproves relativity. The measured quantum > posted speed limit of light as promulgated by Einstein's relativity. > Here is what the authors say: > We conclude that with our measurement we can put an upper limit of > 34as on a delta-t D in the nonadiabatic tunnelling regime with a gamma > ranging from 1.45 to 1.17. Quantum mechanical simulation using time- > dependent Schrodinger equation predict instantaneous ionization with > no angular delay... The measured upper limit of the tunneling delay > time is much shorter than the Buttiker-Landauer traversal time delta- > t T which for present conditions is predicted to range between 450 and > 560as... Our experimental results give a strong indication that there > is no real tunnelling delay time and we expect that this conclusion > will shed some light on the ongoing theoretical discussion on > tunneling time.http://www.sciencemag.org/cgi/content/abstract/322/5907/1525 > Let me requote that: ... There is NO REAL TUNNELING DELAY TIME ... > For the intellectually challenged, that simply means... > INSTANTANEOUS. For the Einstein groupee subset, that also means... > FASTER THAN THE SPEED OF LIGHT. Oh dear. You have no idea what you just read, do you? > You do know, don't you, that quantum mechanical tunneling through a > barrier does not imply any kind of *travel*? Part of the wavefunction > is *already* outside the barrier. > No, I guess you don't. PD- What imbecility! What idiocy! What ignorance! The wavefunction is a MATHematical construct. It is NEITHER inside or inside the barrier, and TUNNELS through, to the outside of the barrier. Thus the term 'quantum tunneling'. And it does so in And you say you know some QM. That was a spectacular display of stupidity. No wonder you like relativity. You simply don't have the cerebral muscle to handle the cognitive requirements needed for QM. Now if you say your schizophrenia got the better of you, then I'll take back the laughter. I'll merely refer you to Massachusetts General Hospital. They have a good psychiatry program. Maybe there's an experimental medication that will work for you. === Subject: Re: R E L A T I V I T Y: Fitting a Square Peg into a Round Hole posting-account=vma-PgoAAABrctSmMdefNKZ-c5S8buvP > I'll merely refer you to Massachusetts General Hospital. Did they just move you to the terminally insane ward, David? === Subject: Re: R E L A T I V I T Y: Fitting a Square Peg into a Round Hole posting-account=-Yr6sAoAAAArJwgdHLA4MBxm4oIdzdtZ InfoPath.1; .NET CLR 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.4506.2152; .NET CLR 3.5.30729; MS-RTC EA 2),gzip(gfe),gzip(gfe) (NetCache NetApp/6.0.2P1) > I'll merely refer you to Massachusetts General Hospital. Did they just move you to the terminally insane ward, David? DoNut, there is no terminally insane ward in any hospital. It is obvious you have a form of mental retardation*. As we said before, your DoNut has a hole :-) *Mental retardation is a generalized, triarchic disorder, characterized by subaverage cognitive functioning and deficits in two or more adaptive behaviors with onset before the age of 18. === Subject: Re: R E L A T I V I T Y: Fitting a Square Peg into a Round Hole > The answer is simple. It disproves relativity. The measured quantum > posted speed limit of light as promulgated by Einstein's relativity. > Here is what the authors say: > We conclude that with our measurement we can put an upper limit of > 34as on a delta-t D in the nonadiabatic tunnelling regime with a gamma > ranging from 1.45 to 1.17. Quantum mechanical simulation using time- > dependent Schrodinger equation predict instantaneous ionization with > no angular delay... The measured upper limit of the tunneling delay > time is much shorter than the Buttiker-Landauer traversal time delta- > t T which for present conditions is predicted to range between 450 and > 560as... Our experimental results give a strong indication that there > is no real tunnelling delay time and we expect that this conclusion > will shed some light on the ongoing theoretical discussion on > tunneling time.http://www.sciencemag.org/cgi/content/abstract/322/5907/1525 > Let me requote that: ... There is NO REAL TUNNELING DELAY TIME ... > For the intellectually challenged, that simply means... > INSTANTANEOUS. For the Einstein groupee subset, that also means... > FASTER THAN THE SPEED OF LIGHT. > Oh dear. You have no idea what you just read, do you? > You do know, don't you, that quantum mechanical tunneling through a > barrier does not imply any kind of *travel*? Part of the wavefunction > is *already* outside the barrier. > No, I guess you don't. > What imbecility! What idiocy! What ignorance! > The wavefunction is a MATHematical construct. It is NEITHER inside or > inside the barrier, and TUNNELS through, to the outside of the > barrier. Thus the term 'quantum tunneling'. Nice that you just made that up. Would you like a reading reference so > that you can learn some quantum mechanics? You say you like the stuff. > It might be good if you learned a little about it. It looks like your schizophrenia is morbidly malignant. Here is a simple starter course: http://en.wikipedia.org/wiki/Wave function I'm sure the first sentence is self-explanatory: A wave function or wavefunction is a mathematical tool used in quantum mechanics to describe any physical system. Keep on hallucinating. Pretty soon there will be enough evidence to get you committed involuntarily. >And it does so in > And you say you know some QM. That was a spectacular display of > stupidity. No wonder you like relativity. You simply don't have the > cerebral muscle to handle the cognitive requirements needed for QM. > Now if you say your schizophrenia got the better of you, then I'll > take back the laughter. I'll merely refer you to Massachusetts > General Hospital. They have a good psychiatry program. Maybe there's > an experimental medication that will work for you.- Hide quoted text - - Show quoted text -- Hide quoted text - - Show quoted text - It looks like your schizophrenia is morbidly malignant. Here is a simple starter course: http://en.wikipedia.org/wiki/Wave function I'm sure the first sentence is self-explanatory: A wave function or wavefunction is a mathematical tool used in quantum mechanics to describe any physical system. Keep on hallucinating. Pretty soon there will be enough evidence to get you committed involuntarily. === Subject: Re: R E L A T I V I T Y: Fitting a Square Peg into a Round Hole > The answer is simple. It disproves relativity. The measured quantum > posted speed limit of light as promulgated by Einstein's relativity. > Here is what the authors say: > We conclude that with our measurement we can put an upper limit of > 34as on a delta-t D in the nonadiabatic tunnelling regime with a gamma > ranging from 1.45 to 1.17. Quantum mechanical simulation using time- > dependent Schrodinger equation predict instantaneous ionization with > no angular delay... The measured upper limit of the tunneling delay > time is much shorter than the Buttiker-Landauer traversal time delta- > t T which for present conditions is predicted to range between 450 and > 560as... Our experimental results give a strong indication that there > is no real tunnelling delay time and we expect that this conclusion > will shed some light on the ongoing theoretical discussion on > tunneling time.http://www.sciencemag.org/cgi/content/abstract/322/5907/1525 > Let me requote that: ... There is NO REAL TUNNELING DELAY TIME ... > For the intellectually challenged, that simply means... > INSTANTANEOUS. For the Einstein groupee subset, that also means... > FASTER THAN THE SPEED OF LIGHT. > Oh dear. You have no idea what you just read, do you? > You do know, don't you, that quantum mechanical tunneling through a > barrier does not imply any kind of *travel*? Part of the wavefunction > is *already* outside the barrier. > No, I guess you don't. > What imbecility! What idiocy! What ignorance! > The wavefunction is a MATHematical construct. It is NEITHER inside or > inside the barrier, and TUNNELS through, to the outside of the > barrier. Thus the term 'quantum tunneling'. > Nice that you just made that up. Would you like a reading reference so > that you can learn some quantum mechanics? You say you like the stuff. > It might be good if you learned a little about it. > It looks like your schizophrenia is morbidly malignant. Here is a > simple starter course:http://en.wikipedia.org/wiki/Wave function > I'm sure the first sentence is self-explanatory: > A wave function or wavefunction is a mathematical tool used in > quantum mechanics to describe any physical system. > Keep on hallucinating. Pretty soon there will be enough evidence to > get you committed involuntarily. >And it does so in > And you say you know some QM. That was a spectacular display of > stupidity. No wonder you like relativity. You simply don't have the > cerebral muscle to handle the cognitive requirements needed for QM. > Now if you say your schizophrenia got the better of you, then I'll > take back the laughter. I'll merely refer you to Massachusetts > General Hospital. They have a good psychiatry program. Maybe there's > an experimental medication that will work for you.- Hide quoted text - > - Show quoted text -- Hide quoted text - > - Show quoted text - > It looks like your schizophrenia is morbidly malignant. Here is a > simple starter course:http://en.wikipedia.org/wiki/Wave function Yes, we've already noticed how you like to read the first two Would you like a reference to something better? > I'm sure the first sentence is self-explanatory: > A wave function or wavefunction is a mathematical tool used in > quantum mechanics to describe any physical system. > Keep on hallucinating. Pretty soon there will be enough evidence to > get you committed involuntarily.- Hide quoted text - - Show quoted text -- Hide quoted text - - Show quoted text - Au contraire. It appears you need to do some work on your understanding of QM. You seem to have a vague grasp of what a wave function is and are under the illusion (or hallucination) that it is something physical. Should I explain further? === Subject: Re: R E L A T I V I T Y: Fitting a Square Peg into a Round Hole > The answer is simple. It disproves relativity. The measured quantum > posted speed limit of light as promulgated by Einstein's relativity. > Here is what the authors say: > We conclude that with our measurement we can put an upper limit of > 34as on a delta-t D in the nonadiabatic tunnelling regime with a gamma > ranging from 1.45 to 1.17. Quantum mechanical simulation using time- > dependent Schrodinger equation predict instantaneous ionization with > no angular delay... The measured upper limit of the tunneling delay > time is much shorter than the Buttiker-Landauer traversal time delta- > t T which for present conditions is predicted to range between 450 and > 560as... Our experimental results give a strong indication that there > is no real tunnelling delay time and we expect that this conclusion > will shed some light on the ongoing theoretical discussion on > tunneling time.http://www.sciencemag.org/cgi/content/abstract/322/5907/1525 > Let me requote that: ... There is NO REAL TUNNELING DELAY TIME ... > For the intellectually challenged, that simply means... > INSTANTANEOUS. For the Einstein groupee subset, that also means... > FASTER THAN THE SPEED OF LIGHT. > Oh dear. You have no idea what you just read, do you? > You do know, don't you, that quantum mechanical tunneling through a > barrier does not imply any kind of *travel*? Part of the wavefunction > is *already* outside the barrier. > No, I guess you don't. > PD- > BWAH HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA > BWAH HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA > BWAH HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA > BWAH HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA > BWAH HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA > What imbecility! What idiocy! What ignorance! > The wavefunction is a MATHematical construct. It is NEITHER inside or > inside the barrier, and TUNNELS through, to the outside of the > barrier. Thus the term 'quantum tunneling'. > Nice that you just made that up. Would you like a reading reference so > that you can learn some quantum mechanics? You say you like the stuff. > It might be good if you learned a little about it. > It looks like your schizophrenia is morbidly malignant. Here is a > simple starter course:http://en.wikipedia.org/wiki/Wave function > I'm sure the first sentence is self-explanatory: > A wave function or wavefunction is a mathematical tool used in > quantum mechanics to describe any physical system. > Keep on hallucinating. Pretty soon there will be enough evidence to > get you committed involuntarily. >And it does so in > And you say you know some QM. That was a spectacular display of > stupidity. No wonder you like relativity. You simply don't have the > cerebral muscle to handle the cognitive requirements needed for QM. > Now if you say your schizophrenia got the better of you, then I'll > take back the laughter. I'll merely refer you to Massachusetts > General Hospital. They have a good psychiatry program. Maybe there's > an experimental medication that will work for you.- Hide quoted text - > - Show quoted text -- Hide quoted text - > - Show quoted text - > It looks like your schizophrenia is morbidly malignant. Here is a > simple starter course:http://en.wikipedia.org/wiki/Wave function > Yes, we've already noticed how you like to read the first two > Would you like a reference to something better? > I'm sure the first sentence is self-explanatory: > A wave function or wavefunction is a mathematical tool used in > quantum mechanics to describe any physical system. > Keep on hallucinating. Pretty soon there will be enough evidence to > get you committed involuntarily.- Hide quoted text - > - Show quoted text -- Hide quoted text - > - Show quoted text - > Au contraire. It appears you need to do some work on your > understanding of QM. You seem to have a vague grasp of what a wave > function is and are under the illusion (or hallucination) that it is > something physical. Should I explain further? Actually, the part that you made up is this: > physical construct. It is inside the barrier, and TUNNELS through, to > the outside of the barrier. Thus the term 'quantum tunneling'. > This is why I suggested some better reading than Wikipedia. Would you > like a reading reference? PD- Hide quoted text - - Show quoted text - Do not misquote me to reinforce your hallucination. This is what I said: The wavefunction is a MATHematical construct. It is NEITHER inside is inside the barrier, and TUNNELS through, to the outside of the barrier. Thus the term 'quantum tunneling'. And it does so in It is obvious that a mathematical construct such as the wavefunction is in neither location. Do you want me to explain the paragraph further. This is basic quantum mechanics. I am surprised you find it so difficult to understand. Tell me the sentence you cannot grasp so I can help you. [I know you are merely trying to cover up your stupidity. Here is what you said exactly: You do know, don't you, that quantum mechanical tunneling through a barrier does not imply any kind of *travel*? Part of the wavefunction is *already* outside the barrier. Obviously you were under the illusion that the wave function is a physical entity and that a part of it was outside the barrier. Basic misconceptions like these are common among the uneducated. === Subject: Re: R E L A T I V I T Y: Fitting a Square Peg into a Round Hole PD pretends: Now, now, where did I say that a wave function is a physical entity? Strich quotes PD: You do know, don't you, that quantum mechanical tunneling through a barrier does not imply any kind of *travel*? Part of the wavefunction is *already* outside the barrier. Strich re-educates PD: The wavefunction is a MATHematical construct. physical construct. It is inside the barrier, and TUNNELS through, to the outside of the barrier. Thus the term 'quantum tunneling'. And Strich asks: Tell me the part you don't understand, so I can help you. [Note that I have disregarded the rest of your post where there were smatterings of thought derailment and alogia.] === Subject: Re: R E L A T I V I T Y: Fitting a Square Peg into a Round Hole > PD pretends: Now, now, where did I say that a wave function is a > physical entity? > Strich quotes PD: You do know, don't you, that quantum mechanical > tunneling through a barrier does not imply any kind of *travel*? Part > of the wavefunction is *already* outside the barrier. Yes. Note there is nothing in that statement that says a wavefunction > is a physical entity. I suppose you have to read between the lines. I > suppose you can also read between the lines that electroshock therapy > is being used to keep you from successfully running for president. > Since you have denied your original assertion, then you are now agreeing with me. Good. > The wavefunction is a MATHematical construct. > physical construct. It is inside the barrier, and TUNNELS through, to > the outside of the barrier. Thus the term 'quantum tunneling'. And > Tell me what you misunderstand again. I'll allow you to deny your misunderstanding again, after you agree with my explanation again. === Subject: Re: R E L A T I V I T Y: Fitting a Square Peg into a Round Hole > PD pretends: Now, now, where did I say that a wave function is a > physical entity? > Strich quotes PD: You do know, don't you, that quantum mechanical > tunneling through a barrier does not imply any kind of *travel*? Part > of the wavefunction is *already* outside the barrier. > Yes. Note there is nothing in that statement that says a wavefunction > is a physical entity. I suppose you have to read between the lines. I > suppose you can also read between the lines that electroshock therapy > is being used to keep you from successfully running for president. > Since you have denied your original assertion, then you are now > agreeing with me. Good. > The wavefunction is a MATHematical construct. > physical construct. It is inside the barrier, and TUNNELS through, to > the outside of the barrier. Thus the term 'quantum tunneling'. And > Tell me what you misunderstand again. I'll allow you to deny your > misunderstanding again, after you agree with my explanation again. Ask them what's quantum about objective real time measurements? If the tunneling can be objectively and peer replicated as having a > real time FTL measurement, then where's this quantum part getting > involved? ~ BG- Hide quoted text - - Show quoted text - PD's schizophrenia prevents him from grasping that reality that objective real time measurements are consistent with faster-than-light transmission of the electron. He will continue to deny reality to conform to his delusion of a luminal speed limit as originally postulated, without evidence, by Einstein. === Subject: Re: R E L A T I V I T Y: Fitting a Square Peg into a Round Hole > PD pretends: Now, now, where did I say that a wave function is a > physical entity? > Strich quotes PD: You do know, don't you, that quantum mechanical > tunneling through a barrier does not imply any kind of *travel*? Part > of the wavefunction is *already* outside the barrier. Yes. Note there is nothing in that statement that says a wavefunction > is a physical entity. Since you have denied your original assertion, then you are now agreeing with me. Good. > The wavefunction is a MATHematical construct. > physical construct. It is inside the barrier, and TUNNELS through, to > the outside of the barrier. Thus the term 'quantum tunneling'. And This part above is the part you made up. Tell me what you misunderstand again. I'll allow you to deny your misunderstanding after you agree with my explanation again. === Subject: Re: R E L A T I V I T Y: Fitting a Square Peg into a Round Hole > PD pretends: Now, now, where did I say that a wave function is a > physical entity? > Strich quotes PD: You do know, don't you, that quantum mechanical > tunneling through a barrier does not imply any kind of *travel*? Part > of the wavefunction is *already* outside the barrier. > Yes. Note there is nothing in that statement that says a wavefunction > is a physical entity. > Since you have denied your original assertion, then you are now > agreeing with me. Good. Where did I ever assert a wavefunction is a physical entity? > I'm not saying it is or is not. I'm asking you where I asserted it. > The wavefunction is a MATHematical construct. > physical construct. It is inside the barrier, and TUNNELS through, to > the outside of the barrier. Thus the term 'quantum tunneling'. And > This part above is the part you made up. > Tell me what you misunderstand again. I'll allow you to deny your > misunderstanding after you agree with my explanation again. I understand perfectly what you said in the content that you made up. > Understanding it does not indicate assent. What you made up is wrong, > and not what quantum mechanics says. You say you like quantum > mechanics. Would you like a reading reference so you can see what > quantum mechanics says, or are you just happy making stuff up about > it? PD- Well well, if it is not schizo unable to make up his mind. He is too fear stricken to identify my wrong statement knowing he'd have to make another retraction. === Subject: Re: R E L A T I V I T Y: Fitting a Square Peg into a Round Hole > 'quantum tunneling' at FTL has been accomplished many times and > objectively measured as such, so it's hardly unreal. Tunneling has never been measured to be FTL. Try again. ~ BG === Subject: Re: R E L A T I V I T Y: Fitting a Square Peg into a Round Hole > PD pretends: Now, now, where did I say that a wave function is a > physical entity? > Strich quotes PD: You do know, don't you, that quantum mechanical > tunneling through a barrier does not imply any kind of *travel*? Part > of the wavefunction is *already* outside the barrier. > Yes. Note there is nothing in that statement that says a wavefunction > is a physical entity. > Since you have denied your original assertion, then you are now > agreeing with me. Good. > Where did I ever assert a wavefunction is a physical entity? > I'm not saying it is or is not. I'm asking you where I asserted it. > The wavefunction is a MATHematical construct. > physical construct. It is inside the barrier, and TUNNELS through, to > the outside of the barrier. Thus the term 'quantum tunneling'. And > This part above is the part you made up. > Tell me what you misunderstand again. I'll allow you to deny your > misunderstanding after you agree with my explanation again. > I understand perfectly what you said in the content that you made up. > Understanding it does not indicate assent. What you made up is wrong, > and not what quantum mechanics says. You say you like quantum > mechanics. Would you like a reading reference so you can see what > quantum mechanics says, or are you just happy making stuff up about > it? > PD- > Well well, if it is not schizo unable to make up his mind. He is too > fear stricken to identify my wrong statement knowing he'd have to make > another retraction. I have no problem identifying the wrong statement. I am assuming you know grammar. You are using 'statement' in singular form. Please identify the wrong statement (singular) in the following: The wavefunction is a MATHematical construct. It is NEITHER inside or inside the barrier, and TUNNELS through, to the outside of the barrier. Thus the term 'quantum tunneling'. And it does so in If you are having a problem with decision-making, let your doctor know. Maybe he can give you another medication. === Subject: Re: R E L A T I V I T Y: Fitting a Square Peg into a Round Hole > PD pretends: Now, now, where did I say that a wave function is a > physical entity? > Strich quotes PD: You do know, don't you, that quantum mechanical > tunneling through a barrier does not imply any kind of *travel*? Part > of the wavefunction is *already* outside the barrier. > Yes. Note there is nothing in that statement that says a wavefunction > is a physical entity. > Since you have denied your original assertion, then you are now > agreeing with me. Good. > Where did I ever assert a wavefunction is a physical entity? > I'm not saying it is or is not. I'm asking you where I asserted it. > The wavefunction is a MATHematical construct. > physical construct. It is inside the barrier, and TUNNELS through, to > the outside of the barrier. Thus the term 'quantum tunneling'. And > This part above is the part you made up. > Tell me what you misunderstand again. I'll allow you to deny your > misunderstanding after you agree with my explanation again. > I understand perfectly what you said in the content that you made up. > Understanding it does not indicate assent. What you made up is wrong, > and not what quantum mechanics says. You say you like quantum > mechanics. Would you like a reading reference so you can see what > quantum mechanics says, or are you just happy making stuff up about > it? > PD- > Well well, if it is not schizo unable to make up his mind. He is too > fear stricken to identify my wrong statement knowing he'd have to make > another retraction. > I have no problem identifying the wrong statement. > I am assuming you know grammar. You are using 'statement' in singular > form. Please identify the wrong statement (singular) in the > following: > The wavefunction is a MATHematical construct. It is NEITHER inside or > inside the barrier, and TUNNELS through, to the outside of the > barrier. Thus the term 'quantum tunneling'. And it does so in > If you are having a problem with decision-making, let your doctor > know. Maybe he can give you another medication. Why can't 'quantum tunneling' take place on a interstellar basis? Why can't you calculate the answer yourself? ~ BG === Subject: Re: R E L A T I V I T Y: Fitting a Square Peg into a Round Hole > Well well, if it is not schizo unable to make up his mind. He is too > fear stricken to identify my wrong statement knowing he'd have to make > another retraction. > I have no problem identifying the wrong statement. > I am assuming you know grammar. You are using 'statement' in singular > form. Please identify the wrong statement (singular) in the > following: Well that's easy. > The wavefunction is a MATHematical construct. The following is A wrong statement (singular). > It is NEITHER inside or outside the barrier. > Bzzzt. Mathematical constructs are non-localizable. > The following is A wrong statement (singular). Bzzzt. Too obvious to require any supporting argument. Only a schizophenic would argue otherwise. > The following is A wrong statement (singular). > It is inside the barrier, and TUNNELS through, to the outside of the barrier. > Bzzzt. This is a simple definition of tunneling. Only a schizophenic would see it differently. > The following is A wrong statement (singular). > Thus the term 'quantum tunneling'. > Bzzzt. This is mere rephrasing. Only a schizophenic would quibble for the sake of argument. > The following is A wrong statement (singular). > The fact that the above are singular wrong statements may have > something to do with you making them up and you being singularly > stupid. > The fact that you see the world differently is obviously because of your singular diagnosis of schizophrenia. This should be obvious to everybody by now. === Subject: Re: R E L A T I V I T Y: Fitting a Square Peg into a > Well well, if it is not schizo unable to make up his mind. He is too > fear stricken to identify my wrong statement knowing he'd have to make > another retraction. > I have no problem identifying the wrong statement. > I am assuming you know grammar. You are using 'statement' in singular > form. Please identify the wrong statement (singular) in the > following: > Well that's easy. > The wavefunction is a MATHematical construct. > The following is A wrong statement (singular). > It is NEITHER inside or outside the barrier. > Bzzzt. Mathematical constructs are non-localizable. > The following is A wrong statement (singular). > Bzzzt. Too obvious to require any supporting argument. Only a > schizophenic would argue otherwise. > The following is A wrong statement (singular). > It is inside the barrier, and TUNNELS through, to the outside of the barrier. > Bzzzt. This is a simple definition of tunneling. Only a schizophenic > would see it differently. > The following is A wrong statement (singular). > Thus the term 'quantum tunneling'. > Bzzzt. This is mere rephrasing. Only a schizophenic would quibble > for the sake of argument. > The following is A wrong statement (singular). > The fact that the above are singular wrong statements may have > something to do with you making them up and you being singularly > stupid. > The fact that you see the world differently is obviously because of > your singular diagnosis of schizophrenia. This should be obvious to > everybody by now. Well, as I said: > Now, if you were thinking that identification of wrong statements > doesn't occur until you are *convinced* and *confess* that they are > wrong, then I'm afraid I don't have the ability to do that for beer > coasters or for cans of soup or for you. When a goofball statement is made by a jackass, having the jackass > follow those statements by saying, It's obvious, and only a > schizophrenic would think otherwise, isn't exactly compelling. PD- If you have overwhelming evidence that my statements are wrong, now is the time to bring them up. Your bluff has been called and you are obliged to show your cards. === Subject: Re: R E L A T I V I T Y: Fitting a Square Peg into a Round Hole > >Well well, if it is not schizo unable to make up his mind. He is too >fear stricken to identify my wrong statement knowing he'd have to make >another retraction. >>I have no problem identifying the wrong statement. >I am assuming you know grammar. You are using 'statement' in singular >form. Please identify the wrong statement (singular) in the >following: >>Well that's easy. >The wavefunction is a MATHematical construct. >>The following is A wrong statement (singular). >It is NEITHER inside or outside the barrier. >Bzzzt. Mathematical constructs are non-localizable. >>The following is A wrong statement (singular). >Bzzzt. Too obvious to require any supporting argument. Only a >schizophenic would argue otherwise. >>The following is A wrong statement (singular). >It is inside the barrier, and TUNNELS through, to the outside of the barrier. >Bzzzt. This is a simple definition of tunneling. Only a schizophenic >would see it differently. >>The following is A wrong statement (singular). >Thus the term 'quantum tunneling'. >Bzzzt. This is mere rephrasing. Only a schizophenic would quibble >for the sake of argument. >>The following is A wrong statement (singular). >>The fact that the above are singular wrong statements may have >>something to do with you making them up and you being singularly >>stupid. >The fact that you see the world differently is obviously because of >your singular diagnosis of schizophrenia. This should be obvious to >everybody by now. >>Well, as I said: >>Now, if you were thinking that identification of wrong statements >>doesn't occur until you are *convinced* and *confess* that they are >>wrong, then I'm afraid I don't have the ability to do that for beer >>coasters or for cans of soup or for you. >>When a goofball statement is made by a jackass, having the jackass >>follow those statements by saying, It's obvious, and only a >>schizophrenic would think otherwise, isn't exactly compelling. >>PD- > If you have overwhelming evidence that my statements are wrong, now is > the time to bring them up. Your bluff has been called and you are > obliged to show your cards. must stand without proof while science has to try to change your mind. The world has no obligation to deal with your delusions. We actually enjoy watching you rant and rave and make a fool of yourself on a daily basis. === Subject: Re: R E L A T I V I T Y: Fitting a Square Peg into a Round Hole >Well well, if it is not schizo unable to make up his mind. He is too >fear stricken to identify my wrong statement knowing he'd have to make >another retraction. >>I have no problem identifying the wrong statement. >I am assuming you know grammar. You are using 'statement' in singular >form. Please identify the wrong statement (singular) in the >following: >>Well that's easy. >The wavefunction is a MATHematical construct. >>The following is A wrong statement (singular). >It is NEITHER inside or outside the barrier. >Bzzzt. Mathematical constructs are non-localizable. >>The following is A wrong statement (singular). >Bzzzt. Too obvious to require any supporting argument. Only a >schizophenic would argue otherwise. >>The following is A wrong statement (singular). >It is inside the barrier, and TUNNELS through, to the outside of the barrier. >Bzzzt. This is a simple definition of tunneling. Only a schizophenic >would see it differently. >>The following is A wrong statement (singular). >Thus the term 'quantum tunneling'. >Bzzzt. This is mere rephrasing. Only a schizophenic would quibble >for the sake of argument. >>The following is A wrong statement (singular). >>The fact that the above are singular wrong statements may have >>something to do with you making them up and you being singularly >>stupid. >The fact that you see the world differently is obviously because of >your singular diagnosis of schizophrenia. This should be obvious to >everybody by now. >>Well, as I said: >>Now, if you were thinking that identification of wrong statements >>doesn't occur until you are *convinced* and *confess* that they are >>wrong, then I'm afraid I don't have the ability to do that for beer >>coasters or for cans of soup or for you. >>When a goofball statement is made by a jackass, having the jackass >>follow those statements by saying, It's obvious, and only a >>schizophrenic would think otherwise, isn't exactly compelling. >>PD- > If you have overwhelming evidence that my statements are wrong, now is > the time to bring them up. Your bluff has been called and you are > obliged to show your cards. must stand without proof while science has to try to change your > mind. The world has no obligation to deal with your delusions. > We actually enjoy watching you rant and rave and make a fool of > yourself on a daily basis.- Hide quoted text - - Show quoted text - Here are typical arguments with PD and Doug: Normal: A circle is round. PD: Wrong!!! Normal: Prove it... PD: Oh come now. nonsense must stand without proof while science has to try to change your mind. The world has no obligation to deal with your delusions. We actually enjoy watching you rant and rave and make a fool of yourself on a daily basis. Normal: Three is a number. PD: Wrong!!! Normal: Prove it... PD: Oh come now. nonsense must stand without proof while science has to try to change your mind. The world has no obligation to deal with your delusions. We actually enjoy watching you rant and rave and make a fool of yourself on a daily basis. Normal: Theta is a greek letter. PD: Wrong!!! Normal: Prove it. PD: Oh come now. nonsense must stand without proof while science has to try to change your mind. The world has no obligation to deal with your delusions. We actually enjoy watching you rant and rave and make a fool of yourself on a daily basis. === Subject: Re: R E L A T I V I T Y: Fitting a Square Peg into a Round Hole Let me summarize how PD's nervous breakdown started: First PD blusters: You do know, don't you, that quantum mechanical tunneling through a barrier does not imply any kind of *travel*? Part of the wavefunction is *already* outside the barrier. Strich laughed: The wavefunction is a MATHematical construct. It is construct. It is inside the barrier, and TUNNELS through, to the outside of the barrier. Thus the term 'quantum tunneling'. And it Then PD cries and denies: Wrong! Strich gives PD a chance: Correct it then. Finally PD whines and breaks down: Oh come now!!! __________________________ Teaching point: PD was confused about what a wavefunction is. He makes a wrong guess, and when this did not coincide with reality, he goes into a psychotic breakdown. Since he has no insight into his illness, he refuses to participate in his therapy. === Subject: Re: R E L A T I V I T Y: Fitting a Square Peg into a Round Hole > >Well well, if it is not schizo unable to make up his mind. He is too >fear stricken to identify my wrong statement knowing he'd have to make >another retraction. >>I have no problem identifying the wrong statement. >I am assuming you know grammar. You are using 'statement' in singular >form. Please identify the wrong statement (singular) in the >following: >>Well that's easy. >The wavefunction is a MATHematical construct. >>The following is A wrong statement (singular). >It is NEITHER inside or outside the barrier. >Bzzzt. Mathematical constructs are non-localizable. >>The following is A wrong statement (singular). >Bzzzt. Too obvious to require any supporting argument. Only a >schizophenic would argue otherwise. >>The following is A wrong statement (singular). >It is inside the barrier, and TUNNELS through, to the outside of the barrier. >Bzzzt. This is a simple definition of tunneling. Only a schizophenic >would see it differently. >>The following is A wrong statement (singular). >Thus the term 'quantum tunneling'. >Bzzzt. This is mere rephrasing. Only a schizophenic would quibble >for the sake of argument. >>The following is A wrong statement (singular). >>The fact that the above are singular wrong statements may have >>something to do with you making them up and you being singularly >>stupid. >The fact that you see the world differently is obviously because of >your singular diagnosis of schizophrenia. This should be obvious to >everybody by now. >>Well, as I said: >>Now, if you were thinking that identification of wrong statements >>doesn't occur until you are *convinced* and *confess* that they are >>wrong, then I'm afraid I don't have the ability to do that for beer >>coasters or for cans of soup or for you. >>When a goofball statement is made by a jackass, having the jackass >>follow those statements by saying, It's obvious, and only a >>schizophrenic would think otherwise, isn't exactly compelling. >>PD- >If you have overwhelming evidence that my statements are wrong, now is >the time to bring them up. Your bluff has been called and you are >obliged to show your cards. >>must stand without proof while science has to try to change your >>mind. The world has no obligation to deal with your delusions. >>We actually enjoy watching you rant and rave and make a fool of >>yourself on a daily basis.- Hide quoted text - >>- Show quoted text - > Here are typical arguments with PD and Doug: Normal: A circle is round. > PD: Wrong!!! > Normal: Prove it... > PD: Oh come now. > nonsense must stand without proof while science has to try to change > your mind. The world has no obligation to deal with your delusions. > We actually enjoy watching you rant and rave and make a fool of > yourself on a daily basis. > Normal: Three is a number. > PD: Wrong!!! > Normal: Prove it... > PD: Oh come now. > nonsense must stand without proof while science has to try to change > your mind. The world has no obligation to deal with your delusions. > We actually enjoy watching you rant and rave and make a fool of > yourself on a daily basis. > Normal: Theta is a greek letter. > PD: Wrong!!! > Normal: Prove it. > PD: Oh come now. > nonsense must stand without proof while science has to try to change > your mind. The world has no obligation to deal with your delusions. > We actually enjoy watching you rant and rave and make a fool of > yourself on a daily basis. See what happens when you skip your meds? === Subject: Re: R E L A T I V I T Y: Fitting a Square Peg into a Round Hole > What's consistent is that any response to Strich9 that doesn't conform > to the desired outcome of his attempt to manipulate, is something he > associates with a disorder. That is, if you don't answer the way I > want you to answer, there is something wrong with you. It's another > game in his short and unimaginative repertoire. PD, you need to gain more insight into your pathology. That will help you in your recovery. You cannot blame it on me or the world. That will not get you anywhere. === Subject: Re: R E L A T I V I T Y: Fitting a Square Peg into a Round Hole > Continued lack of manipulative imagination noted. No imagination is being invoked. I have merely summarized the facts. But you are welcome to hallucinate again if you wish... === Subject: Re: R E L A T I V I T Y: Fitting a Square Peg into a Round Hole > Well well, if it is not schizo unable to make up his mind. He is too > fear stricken to identify my wrong statement knowing he'd have to make > another retraction. > I have no problem identifying the wrong statement. > I am assuming you know grammar. You are using 'statement' in singular > form. Please identify the wrong statement (singular) in the > following: > Well that's easy. > The wavefunction is a MATHematical construct. > The following is A wrong statement (singular). > It is NEITHER inside or outside the barrier. > Bzzzt. Mathematical constructs are non-localizable. > The following is A wrong statement (singular). > Bzzzt. Too obvious to require any supporting argument. Only a > schizophenic would argue otherwise. > The following is A wrong statement (singular). > It is inside the barrier, and TUNNELS through, to the outside of the barrier. > Bzzzt. This is a simple definition of tunneling. Only a schizophenic > would see it differently. > The following is A wrong statement (singular). > Thus the term 'quantum tunneling'. > Bzzzt. This is mere rephrasing. Only a schizophenic would quibble > for the sake of argument. > The following is A wrong statement (singular). > The fact that the above are singular wrong statements may have > something to do with you making them up and you being singularly > stupid. > The fact that you see the world differently is obviously because of > your singular diagnosis of schizophrenia. This should be obvious to > everybody by now. > Well, as I said: > Now, if you were thinking that identification of wrong statements > doesn't occur until you are *convinced* and *confess* that they are > wrong, then I'm afraid I don't have the ability to do that for beer > coasters or for cans of soup or for you. > When a goofball statement is made by a jackass, having the jackass > follow those statements by saying, It's obvious, and only a > schizophrenic would think otherwise, isn't exactly compelling. > PD- > If you have overwhelming evidence that my statements are wrong, now is > the time to bring them up. Your bluff has been called and you are > obliged to show your cards. Oh, come now. Your transparent bluffs are getting boring. Get another hobby. Here are more examples of typical arguments with PD: Normal: A circle is round. PD: Wrong!!! Normal: Prove it... PD: Oh come now. Normal: Three is a number. PD: Wrong!!! Normal: Prove it... PD: Oh come now. Normal: Theta is a greek letter. PD: Wrong!!! Normal: Prove it. PD: Oh come now. === Subject: Re: R E L A T I V I T Y: Fitting a Square Peg into a Round Hole Looks like Dave has fully recovered his dignity, wouldn't you say? I wonder how long until the next dignity recovery break... === Subject: Re: Rant about The Creator's Glory > >> When He made this World as it is, Who did? I mean there are so many God-wannabe entities out there > that one wouldn't know which one you mean. Care to clarify? He means the FMS, of course. Only an infidel would question His greatness (particularly with a side order of garlic bread). -- Paul Hovnanian paul@hovnanian.com ---------------------------------------------------------------------- Have gnu, will travel. === Subject: Re: My Mathematical Proof of the Week (Was: Report: Rihanna Hit First, So Chris Brown Wants A Misdemeanor) > >> The Starmaker The Starmaker \ > in >> ak@ >> 207. >> 115 .17 .10 2: >> The Starmaker >> >> Gutless Umbrella Carrying Sissy >> 69.16.18 >> 6.5 0: >> Jesus forgives sinners, not >> criminals. >> If >> S = the set of all sinners, >> and >> C = the set of all criminals, >> and >> A = the intersection operator, >> and >> S A C != an Air Force base under >> Cheyenne >> Mountain, >> then >> S A C = C; >> Quantum Electro-Dynamics (QED). >> ;D >> Mathematics does not and cannot > explain > people, or where people come from. > Mathematics is just a religious > Bible > upgraded for the information > age. >> Every DNA codon contains two bits of >> information and two more redundancy >> bits for error correction. >> And we *still* get defectives like >> Starmaker. >> When I look at the percentage of people > who have a grasp of higher math to > those who don't, then...those with the > grasp of higher > math *are* the defectives ones, because > their numbers are way smaller. You're a > mistake. You're the defective one, > don't you get it? >> Like everything else, people are > distributed along a bell curve, which is > symmetrical, making > the number and distribution of the > borderline retarded people identical to > the number and distribution of high > overachiers. >> Bees, flowers, cats and fishes don't > know math and they seem to get along > fine without it. > And most people on the planet get along > without it...it's just you guys in the > sci.math class that are spinning around > in circles... >> Withough advanced math, airplanes > wouldn't fly and computers wouldn't > connect you to the > Usenet. >> What does a bee need an airplane for? >> What does the world need you for? Nothing. >> In fact, you're much like a tumor, and >> should be cut out or dosed with radiation. >> -- >> Terry Austin >> Terry Austin: like the polio vaccine, only >> with more asshole. >> -- David Bilek > You're really *into* medicine, aren't you? > You know, if you re-arrange your name Terry > Austin, you get -- Urinary Test. >> And yours becomes retarded troll. Oh, what, > that's not rearranging your name, that *is* > your name. >> That is what I mean about 'limited >> intelligence'... >> It's good that you know your own limitations. In >> fact, it's accomplishment that you know your own >> name. > Actually, I don't even know my own name. >> I believe you. You are that stupid. >> And, you're my bitch. >> That's a first, a sissy calling me a bitch! >> Is Terry a girls name? >> Anybody with a name like Terry, and a handle name >> Sissy--I suspect is a girlish person. You know, those >> people they call gay. Those people are gay because >> of a genetic defect... Is that you, ...gay? >> Gee, I haven't heard that insult before. Well, not since >> the third grade, at any rate. >> I'm sorry, I wasn't trying to be insulting.. >> Bad liar. You're not even trying. >> But then, you *are* that retarded. >> Do you believe intelligent life exist on Earth? >> Not in your house, >> Oh, you mean next door maybe? Or maybe down the block?? Around > the corner??? > Prof. Math-Is-A-Lie. >> I *never* said math is a lie, I said it's an invention by > man... You never said Math is an invention by man. You said >> Mathematics is just a religious Bible upgraded for the >> information age. Make up your mind. Which means he said that God was invented by man. His words. No > other possible way to interpret them. Even he has no idea what he's saying. === Subject: Re: My Mathematical Proof of the Week (Was: Report: Rihanna Hit First, So Chris Brown Wants A Misdemeanor) >> The Starmaker > The Starmaker \ Gutless Umbrella Carrying Sissy >> Bluuuue Rajah > in > kn ak@ > 207. > 115 .17 .10 2: >> The Starmaker > > Gutless Umbrella Carrying Sissy >>@ > 69.16.18 > 6.5 0: >> Jesus forgives sinners, not > criminals. >> If >> S = the set of all sinners, >> and >> C = the set of all criminals, >> and >> A = the intersection operator, >> and >> S A C != an Air Force base under > Cheyenne > Mountain, >> then >> S A C = C; >> Quantum Electro-Dynamics (QED). > ;D >> Mathematics does not and cannot >> explain >> people, or where people come >>from. >> Mathematics is just a religious >> Bible >> upgraded for the information >> age. >> Every DNA codon contains two bits > of > information and two more > redundancy > bits for error correction. >> And we *still* get defectives like > Starmaker. >> When I look at the percentage of >> people who have a grasp of higher >> math to >> those who don't, then...those with >> the grasp of higher >> math *are* the defectives ones, >> because >> their numbers are way smaller. You're >> a mistake. You're the defective one, >> don't you get it? >> Like everything else, people are >> distributed along a bell curve, which >> is symmetrical, making >> the number and distribution of the >> borderline retarded people identical to >> the number and distribution of high >> overachiers. >> Bees, flowers, cats and fishes don't >> know math and they seem to get along >> fine without it. >> And most people on the planet get >> along without it...it's just you guys >> in the sci.math class that are >> spinning around >> in circles... >> Withough advanced math, airplanes >> wouldn't fly and computers wouldn't >> connect you to the >> Usenet. >> What does a bee need an airplane for? >> What does the world need you for? Nothing. > In fact, you're much like a tumor, and > should be cut out or dosed with radiation. >> -- > Terry Austin >> Terry Austin: like the polio vaccine, > only with more asshole. > -- David Bilek >> You're really *into* medicine, aren't you? >> You know, if you re-arrange your name Terry >> Austin, you get -- Urinary Test. >> And yours becomes retarded troll. Oh, what, >> that's not rearranging your name, that *is* >> your name. >> That is what I mean about 'limited > intelligence'... >> It's good that you know your own limitations. In > fact, it's accomplishment that you know your own > name. >> Actually, I don't even know my own name. >> I believe you. You are that stupid. >> And, you're my bitch. > That's a first, a sissy calling me a bitch! >> Is Terry a girls name? >> Anybody with a name like Terry, and a handle name > Sissy--I suspect is a girlish person. You know, > those people they call gay. Those people are gay > because of a genetic defect... Is that you, ...gay? >> Gee, I haven't heard that insult before. Well, not > since the third grade, at any rate. >> I'm sorry, I wasn't trying to be insulting.. >> Bad liar. You're not even trying. >> But then, you *are* that retarded. >> Do you believe intelligent life exist on Earth? >> Not in your house, >> Oh, you mean next door maybe? Or maybe down the block?? >> Around the corner??? > Prof. Math-Is-A-Lie. >> I *never* said math is a lie, I said it's an invention by >> man... >> You never said Math is an invention by man. You said > Mathematics is just a religious Bible upgraded for the > information age. Make up your mind. Which means he said that God was invented by man. His words. No >> other possible way to interpret them. Even he has no idea what he's saying. > This is news to you? -- Terry Austin Terry Austin: like the polio vaccine, only with more asshole. -- David Bilek Jesus forgives sinners, not criminals. === Subject: Re: My Mathematical Proof of the Week (Was: Report: Rihanna Hit First, So Chris Brown Wants A Misdemeanor) >> The Starmaker The Starmaker \ Gutless Umbrella Carrying Sissy >> Bluuuue Rajah in >> kn ak@ >> 207. >> 115 .17 .10 2: >> The Starmaker >> >> Gutless Umbrella Carrying Sissy >@ >> 69.16.18 >> 6.5 0: >> Jesus forgives sinners, not >> criminals. >> If >> S = the set of all sinners, >> and >> C = the set of all criminals, >> and >> A = the intersection operator, >> and >> S A C != an Air Force base under >> Cheyenne >> Mountain, >> then >> S A C = C; >> Quantum Electro-Dynamics (QED). >> ;D >> Mathematics does not and cannot > explain > people, or where people come >from. > Mathematics is just a religious > Bible > upgraded for the information > age. >> Every DNA codon contains two bits >> of >> information and two more >> redundancy >> bits for error correction. >> And we *still* get defectives like >> Starmaker. >> When I look at the percentage of > people who have a grasp of higher > math to > those who don't, then...those with > the grasp of higher > math *are* the defectives ones, > because > their numbers are way smaller. You're > a mistake. You're the defective one, > don't you get it? >> Like everything else, people are > distributed along a bell curve, which > is symmetrical, making > the number and distribution of the > borderline retarded people identical to > the number and distribution of high > overachiers. >> Bees, flowers, cats and fishes don't > know math and they seem to get along > fine without it. > And most people on the planet get > along without it...it's just you guys > in the sci.math class that are > spinning around > in circles... >> Withough advanced math, airplanes > wouldn't fly and computers wouldn't > connect you to the > Usenet. >> What does a bee need an airplane for? >> What does the world need you for? Nothing. >> In fact, you're much like a tumor, and >> should be cut out or dosed with radiation. >> -- >> Terry Austin >> Terry Austin: like the polio vaccine, >> only with more asshole. >> -- David Bilek > You're really *into* medicine, aren't you? > You know, if you re-arrange your name Terry > Austin, you get -- Urinary Test. >> And yours becomes retarded troll. Oh, what, > that's not rearranging your name, that *is* > your name. >> That is what I mean about 'limited >> intelligence'... >> It's good that you know your own limitations. In >> fact, it's accomplishment that you know your own >> name. > Actually, I don't even know my own name. >> I believe you. You are that stupid. >> And, you're my bitch. >> That's a first, a sissy calling me a bitch! >> Is Terry a girls name? >> Anybody with a name like Terry, and a handle name >> Sissy--I suspect is a girlish person. You know, >> those people they call gay. Those people are gay >> because of a genetic defect... Is that you, ...gay? >> Gee, I haven't heard that insult before. Well, not >> since the third grade, at any rate. >> I'm sorry, I wasn't trying to be insulting.. >> Bad liar. You're not even trying. >> But then, you *are* that retarded. >> Do you believe intelligent life exist on Earth? >> Not in your house, >> Oh, you mean next door maybe? Or maybe down the block?? > Around the corner??? > Prof. Math-Is-A-Lie. >> I *never* said math is a lie, I said it's an invention by > man... >> You never said Math is an invention by man. You said >> Mathematics is just a religious Bible upgraded for the >> information age. Make up your mind. >> Which means he said that God was invented by man. His words. No > other possible way to interpret them. Even he has no idea what he's saying. >This is news to you? Snap out of it. === Subject: Re: My Mathematical Proof of the Week (Was: Report: Rihanna Hit First, So Chris Brown Wants A Misdemeanor) > The Starmaker The Starmaker \ > The Starmaker >> >> Gutless Umbrella Carrying Sissy >> Bluuuue Rajah > in > fj > kn ak@ > 207. > 115 .17 .10 2: >> The Starmaker > > Gutless Umbrella Carrying > Sissy >@ > 69.16.18 > 6.5 0: >> Jesus forgives sinners, not > criminals. >> If >> S = the set of all sinners, >> and >> C = the set of all criminals, >> and >> A = the intersection operator, >> and >> S A C != an Air Force base > under > Cheyenne > Mountain, >> then >> S A C = C; >> Quantum Electro-Dynamics > (QED). > ;D >> Mathematics does not and cannot >> explain >> people, or where people come >>from. >> Mathematics is just a religious >> Bible >> upgraded for the information >> age. >> Every DNA codon contains two > bits > of > information and two more > redundancy > bits for error correction. >> And we *still* get defectives like > Starmaker. >> When I look at the percentage of >> people who have a grasp of higher >> math to >> those who don't, then...those with >> the grasp of higher >> math *are* the defectives ones, >> because >> their numbers are way smaller. >> You're >> a mistake. You're the defective >> one, >> don't you get it? >> Like everything else, people are >> distributed along a bell curve, which >> is symmetrical, making >> the number and distribution of the >> borderline retarded people identical >> to the number and distribution of >> high overachiers. >> Bees, flowers, cats and fishes >> don't >> know math and they seem to get >> along >> fine without it. >> And most people on the planet get >> along without it...it's just you >> guys >> in the sci.math class that are >> spinning around >> in circles... >> Withough advanced math, airplanes >> wouldn't fly and computers wouldn't >> connect you to the >> Usenet. >> What does a bee need an airplane for? >> What does the world need you for? > Nothing. In fact, you're much like a > tumor, and should be cut out or dosed > with radiation. >> -- > Terry Austin >> Terry Austin: like the polio vaccine, > only with more asshole. > -- David Bilek >> You're really *into* medicine, aren't >> you? You know, if you re-arrange your >> name Terry Austin, you get -- Urinary >> Test. >> And yours becomes retarded troll. Oh, >> what, that's not rearranging your name, >> that *is* your name. >> That is what I mean about 'limited > intelligence'... >> It's good that you know your own limitations. > In fact, it's accomplishment that you know > your own name. >> Actually, I don't even know my own name. >> I believe you. You are that stupid. >> And, you're my bitch. > That's a first, a sissy calling me a bitch! >> Is Terry a girls name? >> Anybody with a name like Terry, and a handle name > Sissy--I suspect is a girlish person. You know, > those people they call gay. Those people are gay > because of a genetic defect... Is that you, > ...gay? >> Gee, I haven't heard that insult before. Well, not > since the third grade, at any rate. >> I'm sorry, I wasn't trying to be insulting.. >> Bad liar. You're not even trying. >> But then, you *are* that retarded. >> Do you believe intelligent life exist on Earth? >> Not in your house, >> Oh, you mean next door maybe? Or maybe down the block?? >> Around the corner??? > Prof. Math-Is-A-Lie. >> I *never* said math is a lie, I said it's an invention by >> man... >> You never said Math is an invention by man. You said > Mathematics is just a religious Bible upgraded for the > information age. Make up your mind. >> Which means he said that God was invented by man. His words. >> No other possible way to interpret them. >> Even he has no idea what he's saying. > This is news to you? Snap out of it. > Who? Out of what? -- Terry Austin Terry Austin: like the polio vaccine, only with more asshole. -- David Bilek Jesus forgives sinners, not criminals. === Subject: Re: Which is your favorite Integral Transform ? Gecko/2009021910 Firefox/3.0.7,gzip(gfe),gzip(gfe) > Honestly, > I like all types of transforms, > especially transforms which doesn't deal with > 'COMPLEX VARIABLES' 1) Laplace Transform > 2) Fourier Transform > 3) Hartley Transform > 4) Mellin Transform > 5) Hankel Transform > 6) Y-transform > 7) Meijer Transform > 8) Bochner Transform > 9) Weber transform > 10) Kontorovich-Lebedev Transform > 11) Meler-Fock Transform > 12) Hilbert Transform These are the main Integral transorms > {to my knowledge} Google is your friend; don't forget that. 1) which is your Favorite Transform? If you know any other transforms,please > do share that. so long > nimo by the by; > can you recommend > a couple of books on Non- Linear I.E & Non- Linear D.E {if PDE's too included > that would be double bonus for me } for the reference. How about the Fokyu transform? Missed that one didn't you. Hardy === Subject: Re: Which is your favorite Integral Transform ? posting-account=LChCFQoAAACR0FoxHzVn6GGERsr9zp8c Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > Honestly, > I like all types of transforms, > especially transforms which doesn't deal with > 'COMPLEX VARIABLES' > 1) Laplace Transform > 2) Fourier Transform > 3) Hartley Transform > 4) Mellin Transform > 5) Hankel Transform > 6) Y-transform > 7) Meijer Transform > 8) Bochner Transform > 9) Weber transform > 10) Kontorovich-Lebedev Transform > 11) Meler-Fock Transform > 12) Hilbert Transform > These are the main Integral transorms > {to my knowledge} > Google is your friend; don't forget that. > 1) which is your Favorite Transform? > If you know any other transforms,please > do share that. > so long > nimo > by the by; > can you recommend > a couple of books on > Non- Linear I.E & > Non- Linear D.E > {if PDE's too included > that would be double bonus for me } > for the reference. How about the Fokyu transform? Missed that one didn't you. Hardy I'm looking for that. Any other, from anyone..? === Subject: Discussion from Ask-An-Algebraist Currently, there is the following question: R is a commutative ring with identity, not an integral domain, and if I is any non-zero ideal, R/I is finite. Show R is finite. Apparently this is some unproven statement in notes written by P.L. Clark. \ There has been no resolution yet, so seeing if anyone here could help. jr === Subject: Re: Discussion from Ask-An-Algebraist posting-account=_l4K0QkAAAC09JhOoK_ZfoJKXOmr_jZf Gecko/2009030422 Ubuntu/8.04 (hardy) Firefox/3.0.7,gzip(gfe),gzip(gfe) > Currently, there is the following question: R is a commutative ring with identity, not an integral domain, and if I is \ any non-zero ideal, R/I is finite. Show R is finite. Apparently this is some unproven statement in notes written by P.L. Clark. \ There has been no resolution yet, so seeing if anyone here could help. Maybe I can make up for it. Let a be any nontrivial zero divisor in R, and let ann(a) = {r in R : ra=0} be the annihilator of a. Then ann(a) is an ideal of R (if ra=sa = 0, then (r+s)a=0; and if ra = 0 and x is in R, then xra = 0), and is nonzero (since a is a zero divisor, so there exists a nonzero b in R such that ba=0), and is not the entire ring (since 1 is not in ann(a), given that a is nontrivial). Thus, R/ann(a) is finite. The map R --> Ra given by r |-> ra is not a ring homomorphism in general, but it is an abelian group homomorphism (since (r+s)a = ra + sa). It kernel is precisely ann(a), so Ra is isomorphic, as a group, to the group R/ann(a). This group has the same underlying set as the ring R/ann(a), and therefore is finite. Thus, the ideal generated by a, Ra, is finite; since a is nontrivial, it is also a nonzero ideal of R. If Ra = R, then a is a unit; but if a unit is a zero divisor, then 1 = 0, so R = {0}, and R is finite, and we are done. Otherwise, Ra is proper nonzero ideal of R, so R/Ra is finite; and since Ra is finite it follows that R is finite. -- Arturo Magidin, sans .sig === Subject: Re: Discussion from Ask-An-Algebraist >> Currently, there is the following question: >> R is a commutative ring with identity, not an integral domain, >> and if I is any non-zero ideal, R/I is finite. Show R is finite. >> Apparently this is some unproven statement in notes written by P.L. Clark. Proposition 3 in http://math.uga.edu/~pete/factorization.pdf >> There has been no resolution yet, so seeing if anyone here could help. Maybe I can make up for it. Let a be a nontrivial zero divisor in R, and ann(a) = {r in R : ra = 0} > be the annihilator of a. Then ann(a) is an ideal of R (if ra = sa = 0, > then (r+s)a = 0; and if ra = 0 and x is in R, then xra = 0), and is > nonzero (since a is a zero divisor, so there exists a nonzero b in R > such that ba = 0), and is not the entire ring (since 1 is not in ann(a), > given that a is nontrivial). Thus, R/ann(a) is finite. The map R --> Ra given by r |-> ra is not a ring homomorphism in general, > but it is an abelian group homomorphism (since (r+s)a = ra + sa ). > Its kernel is precisely ann(a), so Ra is isomorphic, as a group, to the > group R/ann(a). This group has the same underlying set as the ring R/ann(a) > and therefore is finite. Thus, the ideal generated by a, Ra, is finite; > since a is nontrivial, it is also a nonzero ideal of R. If Ra = R, then > a is a unit; but if a unit is a zero divisor, then 1 = 0, so R = {0}, and \ > R is finite, and we are done. Otherwise, Ra is proper nonzero ideal of R, \ > so R/Ra is finite; and since Ra is finite it follows that R is finite. This is a well-known basic result in the theory of finite rings, e.g. see the first theorem in Gilmer, Robert. Zero Divisors in Commutative Rings. The American Mathematical Monthly, 93 #5 (May, 1986), pp. 382-387 http://www.jstor.org/stable/2323603 Also posted to the original AAA thread here http://at.yorku.ca/cgi-bin/bbqa?forum=ask_an_algebraist;task=show_msg;msg=11\ 45 --Bill Dubuque === Subject: Re: Discussion from Ask-An-Algebraist > Currently, there is the following question: R is a commutative ring with identity, not an integral domain, and if I is \ any non-zero ideal, R/I is finite. Show R is finite. Apparently this is some unproven statement in notes written by P.L. Clark. \ There has been no resolution yet, so seeing if anyone here could help. Surely there is more to it thatn what you write. Take R = (Z/mZ)[x], the ring of polynomials over Z/mZ, with m a composite integer. This is a commutative ring with identity, not an integral domain (since Z/mZ has zero divisors). Let I = (x), which is a nonzero ideal of R. Then R/ I is isomorphic to Z/mZ, which is finite, but R itself is not finite. -- Arturo Magidin === Subject: Re: Discussion from Ask-An-Algebraist posting-account=_l4K0QkAAAC09JhOoK_ZfoJKXOmr_jZf Gecko/2009030422 Ubuntu/8.04 (hardy) Firefox/3.0.7,gzip(gfe),gzip(gfe) > Currently, there is the following question: > R is a commutative ring with identity, not an integral domain, and if I is any non-zero ideal, R/I is finite. Show R is finite. > Apparently this is some unproven statement in notes written by P.L. Clark. There has been no resolution yet, so seeing if anyone here could help. Surely there is more to it than what you write. Oh, I see my problem... that is, R/I is finite for EVERY nonzero ideal I of R.... Sorry about that. Arturo Magidin === Subject: Re: Self-proclaimed genius discovers division by zero WAS:Small numbers for small Denialist brains posting-account=32kyHQoAAABCJbXmMdnX-j-O3VdI5K3F Gecko/20071126 Fedora/1.5.0.12-7.fc6 Firefox/1.5.0.12,gzip(gfe),gzip(gfe) > I, JohnM, am a blithering douche-nozzle. Are you now? So why are you telling Usenet about that. === Subject: Re: SUBTHREAD : fixed point sha1:yzl1W4fdaIHhSHJHvLIaDYel4Iw= I'm a bit puzzled by some of this: > Every object is part of itself. A part of an object > other than the object itself is called a proper part. An object whose only proper part is [] is called an > atom, or atomic object. Notice that [] is not considered > to be atomic, for the same reason that 1 isn't considered > to be prime in standard number theory. Definition: > x atomic <-> (Ew (wcx & ~w=x) > & Ay (ycx <-> y=x or Az (zcy <-> z=y))) So far, the axioms do not guarantee that any atom exists, > so that V could actually equal []. So lets give some > more axioms. II. Ordered pairs Ordered pairs are written as P(a,b), with P a primitive > symbol, and all pairs P(a,b) are atomic. 1. Existence: Aa (Ab (Ex (x = P(a,b)))) This is, of course, not really needed. In FOL, all function symbols are interpreted as total, so your axiom (1) above is redundant. (Put differently, in *any* theory with a two-place function symbol P, axiom (1) is a theorem.) > 2. Uniqueness: Aa (Ab (Ac (Ad > ((P(a,b) = P(c,d)) <-> (a=c & b=d))))) > 3. Atomicity: Aa (Ab (Ax ((x = P(a,b)) -> x atomic) Now I just don't get (3). I'd assume that if x is a pair, made from a and b, then both a and b are proper parts of x. Why on earth do you introduce ordered pairs, only to assume that they are atomic? **** DON'T ANSWER THIS! I REALIZED LATER WHY YOU DID THIS.**** > III. Natural numbers > Natural numbers are atoms. The primitive N is the object > whose atomic elements are exactly the naturals. These > axioms are similar to Peano's. I decided to start the > naturals with 1 rather than 0. Of course, both 1 and S > (successor) are primitives. > Definition: > n natural <-> (n atomic & ncN) > 1. 1 natural > 2. An (n natural -> S(n) natural) > 3. An (Am ((n natural & m natural) > -> (S(n)=S(m) <-> n=m))) > 4. An (n natural -> ~S(n)=1) > 5. Induction Schema: > Ax ((1cx & An ((ncx -> (n natural & S(n)cx))) -> x=N)) I suppose you *could* do it this way, but I don't see the advantage. Why not interpret N in your theory and see if you can prove the above? A natural suggestion: 1 = <0,0> (I'm using angle brackets for your pairing function P(-,-)) 2 = <1,0> = <<0,0>,0> 3 = <2,0> and so on. Now we'll hit a snag, since it's not clear how to define N itself, but here we could use an axiom: Axiom of N: (EN)( 1 c N & (Ax)( x c N -> c N ) & (AS)(( 1 c S & (Ax)( x c S -> c S ) ) -> (As)(s c S -> s c N)). I think I see where a problem may arise later on now: you still want the flat (i.e., structureless!) features to hold, so that (As)(s c S -> s c N) -> S c N. *That*'s why you defined pairs as atoms! If x and y were parts of , then it would follow that x = y. Okay, now I get why you chose the axioms as you did, but the result is a thoroughly dull, ad hoc theory as far as I'm concerned. Your opinion may differ, but what's nice about sets is that we don't *have* to add new axioms for every structure we want to define. Once we have N, it's easy to construct a whole slew of interesting mathematical structures. That won't be the case with this theory. -- Jesse F. Hughes Leaving things always seems to fix me, Running seems to ease my worried mind. -- Bad Livers, Honey, I've Found a Brand New Way === Subject: Re: SUBTHREAD : fixed point <30127591.1237067260324.JavaMail.jakarta@nitrogen.mathforum.org> <874oxq3ck1.fsf@phiwumbda.org> posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > Every object is part of itself. A part of an object > other than the object itself is called a proper part. Oops! I've forgotten about this thread, since I was obviously more concerned with the MR 0.999... thread. > II. Ordered pairs > Ordered pairs are written as P(a,b), with P a primitive > symbol, and all pairs P(a,b) are atomic. > 1. Existence: Aa (Ab (Ex (x = P(a,b)))) > This is, of course, not really needed. In FOL, all function symbols > are interpreted as total, so your axiom (1) above is redundant. (Put > differently, in *any* theory with a two-place function symbol P, axiom > (1) is a theorem.) OK then, I'll remove it. > III. Natural numbers > Natural numbers are atoms. The primitive N is the object > whose atomic elements are exactly the naturals. These > axioms are similar to Peano's. I decided to start the > naturals with 1 rather than 0. Of course, both 1 and S > (successor) are primitives. > Definition: > n natural <-> (n atomic & ncN) > 1. 1 natural > 2. An (n natural -> S(n) natural) > 3. An (Am ((n natural & m natural) > -> (S(n)=S(m) <-> n=m))) > 4. An (n natural -> ~S(n)=1) > 5. Induction Schema: > Ax ((1cx & An ((ncx -> (n natural & S(n)cx))) -> x=N)) > I suppose you *could* do it this way, but I don't see the advantage. > Why not interpret N in your theory and see if you can prove the above? > A natural suggestion: > 1 = <0,0> (I'm using angle brackets for your pairing function P(-,-)) > 2 = <1,0> = <<0,0>,0 3 = <2,0> and so on. OK, I don't see why we can't do this. > Now we'll hit a snag, since it's not clear how to define N itself, but > here we could use an axiom: > Axiom of N: > (EN)( 1 c N & (Ax)( x c N -> c N ) & > (AS)(( 1 c S & (Ax)( x c S -> c S ) ) - (As)(s c S -> s c N)). > I think I see where a problem may arise later on now: you still want > the flat (i.e., structureless!) features to hold, so that > (As)(s c S -> s c N) -> S c N. > *That*'s why you defined pairs as atoms! If x and y were parts of > , then it would follow that x = y. > Okay, now I get why you chose the axioms as you did, but the result is > a thoroughly dull, ad hoc theory as far as I'm concerned. Your > opinion may differ, but what's nice about sets is that we don't *have* > to add new axioms for every structure we want to define. The reason that I define the axioms this way is that this theory is based on tommy1729's desiderata, so I must defer to tommy1729's desires as to what he wants in a theory. The reason for the flattened mereology is that tommy1729 doesn't want to distinguish between element and subset. The only way to accomplish this is via the flattened mereology. > Once we have > N, it's easy to construct a whole slew of interesting mathematical > structures. > That won't be the case with this theory. Also, we don't want the existence of N to imply the existence of this whole slew of interesting mathematical structures, since tommy1729 is opposed to the existence of some of these structures. In particular, tommy1729 is opposed to infinite ordinals, as well as cardinals greater than aleph omega (which he calls aleph oo). Therefore, adding axioms for every structure we want to define ensures that we have more control over which structures can exist. We can add axioms that guarantee that the objects that tommy1729 does want will exist, and leave out axioms which imply that the objects that are undesirable to tommy1729 exist. So far, the axiomatization is as follows: I. Mereology 1. Extensionality: Ax (Ay (x=y <-> (Az (xcz <-> ycz)))) 2. Reflexivity: Ax (xcx) 3. Antisymmetry: Ax (Ay ((xcy & ycx) -> x=y) 4. Transitivity: Ax (Ay (Az ((xcy & ycz) -> xcz))) 5. Bottom: E!x (Ay (xcy)) 6. Top: Ex (Ay (ycx)) 7. Union: Aa (Ab (Ex (Ay (ycx <-> (yca or ycb))))) 8. Intersection: Aa (Ab (Ex (Ay (ycx <-> (yca & ycb))))) Definition: x atomic <-> (Ew (wcx & ~w=x) & Ay (ycx <-> (y=x or Az (ycz)))) II. Ordered Pairs 1. Uniqueness: Aa (Ab (Ac (Ad ((P(a,b) = P(c,d)) <-> (a=c & b=d))))) 2. Atomicity: Aa (Ab (Ax ((x = P(a,b)) -> x atomic) III. Natural Numbers 1. Axiom of N: (EN)( 1 c N & (Ax)( x c N -> c N ) & (AS)(( 1 c S & (Ax)( x c S -> c S ) ) -> (As)(s c S -> s c N)). Definition: n natural <-> ncN & n atomic My next task is to define the: IV. Real Numbers since tommy1729 does accept standard classical analysis. === Subject: Re: SUBTHREAD : fixed point sha1:lBFHLuiKJydee1mCOzW2/gpf9bU= > Therefore, adding axioms for every structure we want to > define ensures that we have more control over which > structures can exist. We can add axioms that guarantee > that the objects that tommy1729 does want will exist, and > leave out axioms which imply that the objects that are > undesirable to tommy1729 exist. Perhaps so, but this approach is also uninteresting to practicing mathematicians and logicians. There is no particular *point* in studying such an ad hoc theory. Thus, this approach will naturally be ignored. At which point, you will pretend that the problem is prejudice and bigotry or a deep belief in the absolute truth of this claim or that. You may find it an interesting exercise to provide a theory satisfying Tommy's arbitrary whims. If so, by all means, enjoy! But do not expect others to share the fascination when the resulting theory is an ad hoc tangled mess. -- Jesse F. Hughes -- A lesson in meta-honesty -- Quincy (age 7): I won't be honest next time. And that's more honesty. === Subject: Re: JSH: another one bites the dust > Moving on to the next phase where I've finished theory, worked out the > algorithm, so now I'm playing with code as I do implementation. > Could you be so kind as to post the code when you've done? I've seen > more than a few cases where someone's code does not match the algorithm > they're trying to code... I'm putting code on comp.theory only. Most up to date code is at my > math group. Google: mymathgroup A code update is coming in a couple of hours, as well as blog updates, > and updates at my math group site. Why a code *update*? The way I understood you, you had correct code from the beginning ... > James Harris === Subject: Re: JSH: another one bites the dust sha1:IIg1jp3OKyQZvgZ48s0q3JHgSGM= > > Btw, get a newsreader and a decent Usenet service. Using Google Groups > and solving all the Worlds security issues by Einsteinian levels of > mathematics, well, I got to tell ya', it's pole opposite methodologies. >> And it makes you look like a moron. Posting the identical message 4 times in 10 minutes doesn't instill >> confidence in your qualifications to lecture others on what newsreader >> they should be using. Idiot. The glitch wasn't the newsreader. So, the problem was that you don't have a decent Usenet service? > Idiot. Right. Insinuating that JSH couldn't possibly be mathematically competent because he uses a substandard Usenet portal is, on the other hand, a stellar showing of intellectual prowess. Because there is surely no better proof that JSH's math is subpar, is there? -- Jesse F. Hughes As you can see, I am unanimous in my opinion. -- Anthony A. Aiya-Oba (Poeter/Philosopher) === Subject: please clarify me here.. posting-account=LChCFQoAAACR0FoxHzVn6GGERsr9zp8c Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) Hi..., After Pixel values are Transformed [ DCT, wavelet..etc..] The next stage is 'Quantization' after quantization, entropy coding and so on; Coming to the point, => In the Quantization stage the values are 'Rounded' to nearest integers. => De-Quantization is performed over this 'Rounded Values' but not on original transformed values. =>My doubt is, we are implementing Inverse Transform on values =>which we are not transformed previously. but, still will we get the original values ? ________ Eg:- 1) 25 is transformed into 50; 2) assume that 50 is going to be quantized with some value; say 13 50/13 = 3.8461 ~= 4 (Rounded) now de-quntization 1) 4 * 13 = 52; and inverse transform on 52 but not on 50; still, can we expect 25 as the original answer. if no, then All the values which we sampled originally is not equal to the compressed values yet we can see the image or what ever it is like original file. so, what's the reason for this ? === Subject: Re: please clarify me here.. > All the values which we sampled originally is not equal to the > compressed values That's why we call it lossy compression, and why purists prefer lossless compression. > yet we can see the image or what ever it is like original file. so, what's the reason for this ? Like the original file, yes - but not identical. The reason is that the transform is designed so that distance in the transformed space is closer to perceptual distance. So the image resulting from a small rounding in the transformed data is (hopefully) difficult to distinguish from the original. There are usually cases where the transform doesn't match all that well, e.g. JPEG is notably poor at handling uniform colour regions with sharp boundaries. - Tim === Subject: Re: please clarify me here.. > Hi..., After Pixel values are > Transformed [ DCT, wavelet..etc..] The next stage is 'Quantization' > after quantization, entropy coding and so on; Coming to the point, => In the Quantization stage the values are 'Rounded' > to nearest integers. => De-Quantization is performed over this 'Rounded Values' > but not on original transformed values. =>My doubt is, we are implementing > Inverse Transform on values > =>which we are not transformed previously. but, still > will we get the original values ? Of course not. Rounding is not one-to-one. > Eg:- > 1) 25 is transformed into 50; > 2) assume that 50 is going to be quantized > with some value; say 13 > 50/13 = 3.8461 ~= 4 (Rounded) now de-quntization 1) 4 * 13 = 52; and inverse transform on 52 > but not on 50; still, can we expect 25 as the original answer. > if no, then All the values which we sampled originally > is not equal to the compressed values yet we can see the image or what ever it is > like original file. so, what's the reason for this ? The original image has lots of redundant information. Even though it is somewhat degraded, we can still see the picture. -- === Subject: Re: New Lower Limit on Higgs Boson Mass <49bd4ddc$1@news.bnb-lp.com> posting-account=5GUrzQkAAADun29oaK3p_W_saUVxxHUF Gecko/2009021910 Firefox/3.0.7,gzip(gfe),gzip(gfe) > The way you describe it, it sounds like we could have already seen the > signature but have had it dumped as noise. > Well, yes, of course. There are 300 trillion collisions in the dataset > already analyzed, and another 350 trillion in the dataset not yet > combed through. The Higgs cross-section, if there is a Higgs at all, > tells you how many Higgs bosons have been created and are sitting in > that pile somewhere. > But the art is finding distinguishing footprints. For that you have to > choose a decay channel that provides a signature that is more easily > distinguishable, which means that you are choosing only a fraction of > the Higgs bosons that are created. (And the branching ratio of decays > to that channel strongly depends on the Higgs mass.) > And even then, the signature is produced by a variety of processes > that happen way, way more frequently than Higgs boson production. But > in most of those cases, the background for that signature has a broad > and predictable spectrum for some kinematic variable (like > reconstructed mass, or missing transverse energy, or rapidity of > leading jets, or some such), while the Higgs signature should present > some excess or bump in a region of that spectrum. But of course, the > background doesn't have a completely smooth spectrum -- it also has > statistical fluctuations that look like bumps. So you do a > chemotherapy-in-reverse approach -- you cut the data some more with > some poison that kills both background and Higgs signal, but hopefully > kills the Higgs signal a little less, so that it stands out a little > more against the background. > So, if they are there at all, there are Higgs bosons scattered > throughout the data. But most of them decay into channels that are > indistinguishable from other processes, or most of them have signature > kinematic variables that are spread out enough that they don't rise > above background, or the cuts you've applied to the data have killed > most of the Higgs. The good news is that you don't have to find all > the Higgs bosons in the data. You only need to find enough of them > that you can say with statistical confidence that there are more > events in this little bunch right here than can be accounted for by > background alone. In that little bunch, you won't even be able to say > which of those are background and which are Higgs bosons, because they > still look alike. But you'd be able to say that surely some of those > are Higgs bosons, and this is enough to declare discovery. > Fun stuff. Really, really, really hard. > PD > ------------------ > with that 'philosophy' > you already found > my and yous Mother in law > in those experiments!! No, because there's no predicted production of your mother-in-law in really ??(:-) my mother in law has a nose and eyes so go ahead with your Higgs and its predictions (:-) ---------------------- > proton-antiproton collisions. There is, however, a pretty darned firm > prediction of the the production of Higgs bosons in proton-antiproton > collisions. Please read what I said. > 2 > suddenly the Higgs has mass ?? > no more > in shell out shell ??? Perhaps it would help if you knew what the terms on-shell and off- > shell in physics mean first do** you** know what is in shell and off shell ??? !! is it natures creation or an idiotic creation of mathematicians ??!! do you understand basic physics?? we will find out later !!! Y.Porat ---------------------------. > stop fooling yourself and others > and wasting your precious time > (unless you dont mind > stiing all your life and mumbling) > Y.Porat > ---------------------------- === Subject: Re: New Lower Limit on Higgs Boson Mass <49bd4ddc$1@news.bnb-lp.com> posting-account=qM-V3woAAACY9giQRDoOMhGrXbm9o2rn Gecko/2009011913 Firefox/3.0.6,gzip(gfe),gzip(gfe) > What's the next range of masses up beyond 185 GeV where a Higgs could > possibly be, according to theory? According to theory, there isn't a next range. The constraint that it > has to be less than 185 GeV comes from matching the theory against > measured data of other Standard Model processes. You mean it's not possible that Higgs could be twice as massive, for them)? > Ah okay, so I was wondering why they were so excited about finding a > single Top quark, when they had already discovered double Top quarks in > experiments previously. This is on the doorstep to Higgs. Funny how none A number of the physics blogs actually did point this out. But what > gets reported in the press isn't really designed to give people any > kind of appreciable understanding of what's going on, and even if > interviewees told them that, I'm sure it got cut. There's only so much > that you can put in a sound bite that is aimed to do little more than > garner interest. It seems that they usually go for more passive & indirect wording rather than with a direct and simple statement. I mean what's so hard about saying, the discovery of the single top-quarks is significant because it could be one of the products of a Higgs boson decaying. Yousuf Khan === Subject: Re: New Lower Limit on Higgs Boson Mass >> What's the next range of masses up beyond 185 GeV where a Higgs >> could possibly be, according to theory? >> According to theory, there isn't a next range. The constraint that it >> has to be less than 185 GeV comes from matching the theory against >> measured data of other Standard Model processes. You mean it's not possible that Higgs could be twice as massive, for > them)? > >> Ah okay, so I was wondering why they were so excited about finding >> a single Top quark, when they had already discovered double Top >> quarks in experiments previously. This is on the doorstep to Higgs. >> Funny how non > e >> A number of the physics blogs actually did point this out. But what >> gets reported in the press isn't really designed to give people any >> kind of appreciable understanding of what's going on, and even if >> interviewees told them that, I'm sure it got cut. There's only so >> much that you can put in a sound bite that is aimed to do little more >> than garner interest. It seems that they usually go for more passive & indirect wording > rather than with a direct and simple statement. I mean what's so hard > about saying, the discovery of the single top-quarks is significant > because it could be one of the products of a Higgs boson decaying. CYA. You saw how PD jumped down my throat for reporting about the NPR lady saying the Higgs is expected to be at 185 GeV, rather than in a narrow band between 180 and 183 Gev. They wouldn't want to get shouted at by a hair-splitter, would they? === Subject: Hammer falls on JSH's foot, Pell's Equation factoring solution is crap posting-account=fwSgtAkAAACFnX70ssKwbvm9_oCZVHrx Gecko/2009021910 Firefox/3.0.7,gzip(gfe),gzip(gfe) JSH describes the complete factoring algorithm here: Have a look at this gem: For integer v, which is the preferred case, the maximum range is given by: f1(1 - sqrt(1 - [1 - D](D+1)))/(D+1) < v < f1(1 + sqrt(1 - [1 - D](D +1)))/(D+1) which corresponds to v having f1 itself as a factor. How bad is this? Well, f1 can have size at most sqrt(D-1). Plugging that into these range equations yields a maximum range width of 2Dsqrt(D-1)/(D +1). For large enough D, sqrt(D-1) is approximately sqrt(D), and D+1 is approximately D, so it is asymptotic to 2D as D -> oo. The asymptotic worst case complexity for the search part of his algorithm is therefore O(D), which is no better than trial division -- it's worse, actually, as trial division only goes up to O(sqrt(D)). That's just the search part. That doesn't count the recursion to factor D-1, etc. So even if JSH's algorithm works (not sure if it it does), it is no better than trial division. If he tries to cover his bases by saying... but what about the coprime-to-f1 v range??? the answer is: a working v may not be in there!!!! (at least integer v) We are talking about *worst-case* complexity here. === Subject: Re: Hammer falls on JSH's foot, Pell's Equation factoring solution \ is crap >With this latest essentially random change to his algorithm (no way does >James actually understand what's going on--hence, all his changes now >are essentially random), he now needs to augment his proof with a >proof that if a factor exists, it can be found with a low f1, so the >search space won't blow up. James fully understands what he is doing. He conceals it by calling it brainstorming, but in reality he is using a genetic algorithm. He starts with a random set of equations and we check them to see if they work. He then makes directed changes (in the cases where he understands the issue) or undirected changes (in cases where he does not). We check the revised algorithm and the process is repeated. Evolution works with directed changes as well as undirected changes, so either way James is using a GA to improve each of his ideas. It might be an interesting project to develop James' method on a more formal basis - GAs can sometimes come up with unusual solutions that would not normally be thought of by humans. See http://ti.arc.nasa.gov/projects/esg/research/antenna.htm for some examples. rossum === Subject: Re: Hammer falls on JSH's foot, Pell's Equation factoring solution \ is crap [Tim Smith] > ... > With this latest essentially random change to his algorithm (no way > does James actually understand what's going on--hence, all his changes > now are essentially random), Oh, that's always been true, since the first time he got into the miracle factoring method game. He /used/ to pay attention to counterexamples, and then each one presented would prompt a seemingly random change in the algorithm du jour, accompanied by a gibberish explanation for why /now/ the algorithm was perfect. Lather, rinse, repeat. This time it had to wait until he implemented the thing himself. The good news is that he apparently still believes the counterexamples he himself finds ;-) And he's being fiendishly clever about it (in a perverse way) this time too, writing a program that can only /try/ to factor one of 29 fixed integers. Changing things more-or-less at random will eventually (and may already have -- I didn't check) yield a program that nails all 29 of 'em (where nails includes making up excuses for the results it gets wrong -- oh, it can't do a square or oh, the smallest prime factor can't be congruent to 1 modulo 3 ... whatever special cases it takes to discount the failures). > he now needs to augment his proof with a proof that if a factor > exists, it can be found with a low f1, so the search space won't blow > up. That's the usual outcome with these things: his random changes increase the size of the search space, effectively increasing the number of pseuo- random gcd candidates to try. Increase the search space by enough, and eventually the algorithm factors every integer people can be bothered to try. After days to months of aggressive denial, James eventually accepts (with or without public acknowledgement) that he's increased the search space so much that the algorithm is of no practical value, and slinks away -- but, as a Major Discoverer, he Will Return :-) === Subject: Re: Hammer falls on JSH's foot, Pell's Equation factoring solution \ is crap posting-account=wVv_VwoAAAAVTfUuyxLzug5SzYWCgHj1 Gecko/20081217 Firefox/2.0.0.20,gzip(gfe),gzip(gfe) > JSH describes the complete factoring algorithm here: > Have a look at this gem: For integer v, which is the preferred case, the maximum range is > given by: > f1(1 - sqrt(1 - [1 - D](D+1)))/(D+1) < v < f1(1 + sqrt(1 - [1 - D](D > +1)))/(D+1) > which corresponds to v having f1 itself as a factor. How bad is this? Well, f1 can have size at most sqrt(D-1). Plugging > that into > these range equations yields a maximum range width of 2Dsqrt(D-1)/(D > +1). For large enough D, sqrt(D-1) is approximately sqrt(D), and D+1 > is approximately D, so it is asymptotic to 2D as D -> oo. The > asymptotic worst case complexity for the search part of his algorithm > is therefore O(D), which is no better than trial division -- it's > worse, actually, as trial division only goes up to O(sqrt(D)). That's > just the search part. That doesn't count the recursion to factor D-1, > etc. So even if JSH's algorithm works (not sure if it it does), it is no > better than trial division. > Yes, but ... Since D in real applications is odd, D - 1 is divisible by f1 = 1 or f1 = 2. Both of these give a narrow range. In fact for f1, the range is basically -1 to +1; for f1 = 2, it is basically -2 to +2. But of course there are infinitely many ranges in those ranges. > If he tries to cover his bases by saying... but what about the > coprime-to-f1 v range??? the answer is: a working v may not be in > there!!!! (at least integer v) We are talking about *worst-case* > complexity here. === Subject: Re: Hammer falls on JSH's foot, Pell's Equation factoring solution \ is crap posting-account=fwSgtAkAAACFnX70ssKwbvm9_oCZVHrx Gecko/2009021910 Firefox/3.0.7,gzip(gfe),gzip(gfe) > JSH describes the complete factoring algorithm here: > Have a look at this gem: > For integer v, which is the preferred case, the maximum range is > given by: > f1(1 - sqrt(1 - [1 - D](D+1)))/(D+1) < v < f1(1 + sqrt(1 - [1 - D](D > +1)))/(D+1) > which corresponds to v having f1 itself as a factor. > How bad is this? Well, f1 can have size at most sqrt(D-1). Plugging > that into > these range equations yields a maximum range width of 2Dsqrt(D-1)/(D > +1). For large enough D, sqrt(D-1) is approximately sqrt(D), and D+1 > is approximately D, so it is asymptotic to 2D as D -> oo. The > asymptotic worst case complexity for the search part of his algorithm > is therefore O(D), which is no better than trial division -- it's > worse, actually, as trial division only goes up to O(sqrt(D)). That's > just the search part. That doesn't count the recursion to factor D-1, > etc. > So even if JSH's algorithm works (not sure if it it does), it is no > better than trial division. Yes, but ... Since D in real applications is odd, D - 1 is > divisible > by f1 = 1 or f1 = 2. Both of these give a narrow range. In fact for > f1, the range is basically -1 to +1; for f1 = 2, it is basically -2 > to +2. But of course there are infinitely many ranges in those > ranges. > However the algorithm involves trying other values for f1 and f2, see, not just f1 = 1 and f1 = 2. If those don't work (and they don't always work), then you have to try other pairs. Which means the range gets larger, which means the algorithm gets nastier. So in the worst case, where f1/f2 are near sqrt(D-1), then you have a LOT of vs to check. > If he tries to cover his bases by saying... but what about the > coprime-to-f1 v range??? the answer is: a working v may not be in > there!!!! (at least integer v) We are talking about *worst-case* > complexity here. === Subject: Re: Hammer falls on JSH's foot, Pell's Equation factoring solution \ is crap posting-account=fwSgtAkAAACFnX70ssKwbvm9_oCZVHrx Gecko/2009021910 Firefox/3.0.7,gzip(gfe),gzip(gfe) in the range which follows the rules, you will get a non-trivial > factorization, guaranteed. > So what's the point of the giant range then? How many vs are there I need to try? Why don't v = 1/2, 0, -1/2, for example factor 2^32 + 1? How do I factor 2^32 + 1 with your method? How many vs *do* I need to try? What about for 2^4096 + 1? > The huge range you mention is actually meaningless as v has to have > f 1 as a factor, and the range is too small to allow that possibility. > The range is too *small*? You mean it has to be even BIGGER? That would make the algorithm worse, not better! Also, if the range is meaningless then why do you bother mentioning it? What I don't get is why, if you algorithm works for any number, you haven't been able to post a code that factors 2^32 + 1 or something. === Subject: Help with function composition proof posting-account=_GQjkAgAAACN4wuHmYAaxLDDfT_6e68j 1.0.3705; Media Center PC 4.0; Media Center PC 3.0; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) Assume f : X ---> Y g: Y----> X gf is the identity function for all x in X We are asked to prove that f is a one-to-one correspondence. It's easy to show that f is one-to-one. I have a problem with showing f being onto as well. So far, this is what I come up with: y is in Y There exists an x in X such that x=g(y) There exists a z in Y such that fg(y) = z gf(g(y)) = g(y) = x = g(z) How do I get from here to y = f(x)? === Subject: Re: Help with function composition proof > Assume f : X ---> Y > g: Y----> X gf is the identity function for all x in X We are asked to prove that f is a one-to-one correspondence. It's > easy to show that f is one-to-one. I have a problem with showing f > being onto as well. Let X be the set of non-negative reals, let Y be the set of all reals let f(x) = x let g(y) = |y| then g(f(x)) = x for all x, but f is not onto Y. === Subject: Re: Help with function composition proof posting-account=_l4K0QkAAAC09JhOoK_ZfoJKXOmr_jZf Gecko/2009030422 Ubuntu/8.04 (hardy) Firefox/3.0.7,gzip(gfe),gzip(gfe) > Assume f : X ---> Y > g: Y----> X gf is the identity function for all x in X We are asked to prove that f is a one-to-one correspondence. It's > easy to show that f is one-to-one. I have a problem with showing f > being onto as well. Ehr, it doesn't have to be. Take X = {x}, Y = {a,b}, f:X->Y given by f(x)=a, and g:Y->X given by g (a)=g(b)=x. Then gf is the identity, but f is not onto. In general, if a composition gf is one to one then you can deduce (if we compose right to left) that f is one to one; if gf is onto, then we can conclude that g is onto. If gf is bijective, we can conclude that f is one-to-one and that g is onto, but nothing more (see the example above to see g need not be one-to-one either). -- Arturo Magidin, sans .sig === Subject: Re: Help with function composition proof posting-account=_GQjkAgAAACN4wuHmYAaxLDDfT_6e68j 1.0.3705; Media Center PC 4.0; Media Center PC 3.0; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Assume > f : X ---> Y > g: Y----> X > gf is the identity function for all x in X > We are asked to prove that f is a one-to-one correspondence. It's > easy to show that f is one-to-one. I have a problem with showing f > being onto as well. Ehr, it doesn't have to be. Take X = {x}, Y = {a,b}, f:X->Y given by f(x)=a, and g:Y->X given by g > (a)=g(b)=x. Then gf is the identity, but f is not onto. In general, if a composition gf is one to one then you can deduce (if > we compose right to left) that f is one to one; if gf is onto, then we > can conclude that g is onto. If gf is bijective, we can conclude that > f is one-to-one and that g is onto, but nothing more (see the example > above to see g need not be one-to-one either). -- > Arturo Magidin, sans .sig states A function that always maps distinct elements >onto< distinct elements is called one-to-one (usually a one-to-one correspondence) So I went looking for an onto that is not necessarily there. Your help is always greatly appreciated. === Subject: Re: Help with function composition proof > Assume > f : X ---> Y > g: Y----> X > gf is the identity function for all x in X > We are asked to prove that f is a one-to-one correspondence. It's > easy to show that f is one-to-one. I have a problem with showing f > being onto as well. > Ehr, it doesn't have to be. > Take X = {x}, Y = {a,b}, f:X->Y given by f(x)=a, and g:Y->X given by g > (a)=g(b)=x. Then gf is the identity, but f is not onto. > In general, if a composition gf is one to one then you can deduce (if > we compose right to left) that f is one to one; if gf is onto, then we > can conclude that g is onto. If gf is bijective, we can conclude that > f is one-to-one and that g is onto, but nothing more (see the example > above to see g need not be one-to-one either). > -- > Arturo Magidin, sans .sig states A function that always maps distinct elements >onto< distinct > elements is called one-to-one (usually a one-to-one correspondence) > So I went looking for an onto that is not necessarily there. Your > help is always greatly appreciated. You can however show that g o f = id implies that g is onto, can't you? === Subject: Re: Help with function composition proof posting-account=_GQjkAgAAACN4wuHmYAaxLDDfT_6e68j 1.0.3705; Media Center PC 4.0; Media Center PC 3.0; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Assume > f : X ---> Y > g: Y----> X > gf is the identity function for all x in X > We are asked to prove that f is a one-to-one correspondence. It's > easy to show that f is one-to-one. I have a problem with showing f > being onto as well. > Ehr, it doesn't have to be. > Take X = {x}, Y = {a,b}, f:X->Y given by f(x)=a, and g:Y->X given by g > (a)=g(b)=x. Then gf is the identity, but f is not onto. > In general, if a composition gf is one to one then you can deduce (if > we compose right to left) that f is one to one; if gf is onto, then we > can conclude that g is onto. If gf is bijective, we can conclude that > f is one-to-one and that g is onto, but nothing more (see the example > above to see g need not be one-to-one either). > -- > Arturo Magidin, sans .sig > states A function that always maps distinct elements >onto< distinct > elements is called one-to-one (usually a one-to-one correspondence) > So I went looking for an onto that is not necessarily there. Your > help is always greatly appreciated. You can however show that g o f = id implies that g is onto, can't you?- Hide quoted text - - Show quoted text - Yes indeed. x in X Exists y such that f(x)=y gf(x) = g(y) = x So for each x in X there exists a y in Y such that g(y) = x for your interest. === Subject: Re: average = ??? > consider a Coo function of z : f(z) consider a sequence starting somewhere at some z0. f(z0),f(f(z0)),f(f(f(z0))),... what is the average of this ? You mean how do the averages behave? > sure if the iterations go to oo , oo is the average ^^^^^^^^^^^^^^^^^. You mean the averages go to oo. > also if the iterations converge to number A , A is the average. but what if the iterations dont converge , what is the average then ? in many cases - i guess - a fixed point of f(z). but its not always a fixed point is it ?? No, let f(z) = (1 - z)^2, z0 = 0. The sequence of interates is 1, 0, 1, 0, ..., whose averages -> 1/2, which is not a fixed point of f. > or am i too pessimistic here ? limit cycles and saddle points for example lead me to believe its not always > a fixed point ... further if a function has more than one fixed point ? on the other hand , chaos does not occur in 2 dimensions , so maybe the > average is a fixed point afterall ? amy666 === Subject: Question about OEIS #: A143543 Can someone help me make sense of the formula given for the sequence A143543 \ in OEIS? It is for the number of labeled graphs on n nodes with k connected \ components. Any comments or references would be appreciated. Sam === Subject: Our curve around the Sun posting-account=5ApcPgoAAABKcgEyKsQmJVb3Rz63IGGL .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; WWTClient2),gzip(gfe),gzip(gfe) If you drop something it does not follow a curve in the Earth's geometry. But it does in the Sun's. What looks straight from one perspective is curved in another. Something that falls straight on Earth is falling around the Sun in a curve with the rest of the Earth. I believe this is Galilean Relativity where there are different appearences from different perspectives. Actually just two perspectives are required. Mitch Raemsch === Subject: Re: Our curve around the Sun posting-account=qKxGxgkAAADAPfYVCc-ZQkIzl0senr2M .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 1.1.4322; Zune 2.5),gzip(gfe),gzip(gfe) > If you drop something it does not follow a curve in the Earth's > geometry. But it does in the Sun's. What looks straight from one perspective is curved in another. Something that falls straight on Earth is falling around the Sun in a > curve with the rest of the Earth. I believe this is Galilean Relativity where there are different > appearences from different perspectives. Actually just two > perspectives are required. Mitch Raemsch If you are IN something that gets dropped, whatever the gravity, from your 'perspective' it falls straight down. If you don't look outside, you don't even know there is acceleration - until you hit something. Tom Davidson Richmond, VA === Subject: Re: Our curve around the Sun > If you drop something it does not follow a curve in the Earth's > geometry. But it does in the Sun's. What looks straight from one perspective is curved in another. Something that falls straight on Earth is falling around the Sun in a > curve with the rest of the Earth. I believe this is Galilean Relativity where there are different > appearences from different perspectives. Actually just two > perspectives are required. Mitch Raemsch If you are IN something that gets dropped, whatever the gravity, from your 'perspective' it falls straight down. If you don't look outside, you don't even know there is acceleration - until you hit something. Let us put aside looking for a moment. For the *experience* of force what matters is whether the force is opposed or not. A body in a gravitational field accelerates unless there is an opposing force. The human experience or sensation is that of an opposing force. My chair is compressing the flesh of my buttocks, it is providing an opposing force; gravity causes acceleration and is a force. In the absence of a g-field no acceleration takes place unless a force is applied to compress my cheeks. Twins are in identical spacecraft in a region of space where all else is absent. Gradually one accelerates with respect to the other. Is the acceleration the result of one being deeper within the gravity field of a third body, or because one of the pair is pushed? If one of the pair is pushed, a third body is present - possibly the exhaust gas of a rocket engine - in violation of the statement all else is absent. The twin with the compressed flesh is the one that is pushed; but observation alone, each looking at the other, cannot make that determination. === Subject: Re: Our curve around the Sun posting-account=5ApcPgoAAABKcgEyKsQmJVb3Rz63IGGL .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; WWTClient2),gzip(gfe),gzip(gfe) > If you drop something it does not follow a curve in the Earth's > geometry. But it does in the Sun's. > What looks straight from one perspective is curved in another. > Something that falls straight on Earth is falling around the Sun in a > curve with the rest of the Earth. > I believe this is Galilean Relativity where there are different > appearences from different perspectives. Actually just two > perspectives are required. > Mitch Raemsch If you are IN something that gets dropped, whatever the gravity, from > your 'perspective' it falls straight down. If you don't look outside, you don't even know there is acceleration - > until you hit something. Let us put aside looking for a moment. > For the *experience* of force what matters is whether the force is > opposed or not. A body in a gravitational field accelerates unless there is an opposing > force. The human experience or sensation is that of an opposing force. My chair is compressing the flesh of my buttocks, it is providing an > opposing force; gravity causes acceleration and is a force. In the absence > of a g-field no acceleration takes place unless a force is applied to > compress my cheeks. Twins are in identical spacecraft in a region of space where all else > is absent. Gradually one accelerates with respect to the other. > Is the acceleration the result of one being deeper within the gravity > field of a third body, or because one of the pair is pushed? > If one of the pair is pushed, a third body is present - possibly the > exhaust gas of a rocket engine - in violation of the statement all else > is absent. The twin with the compressed flesh is the one that is > pushed; but observation alone, each looking at the other, cannot > make that determination. You are accelerating right now in your chair in gravity. That is where your masses weight originates. Mitch Raemsch === Subject: Re: Our curve around the Sun > If you drop something it does not follow a curve in the Earth's > geometry. But it does in the Sun's. > What looks straight from one perspective is curved in another. > Something that falls straight on Earth is falling around the Sun in a > curve with the rest of the Earth. > I believe this is Galilean Relativity where there are different > appearences from different perspectives. Actually just two > perspectives are required. > Mitch Raemsch If you are IN something that gets dropped, whatever the gravity, from > your 'perspective' it falls straight down. If you don't look outside, you don't even know there is acceleration - > until you hit something. Tom Davidson > Richmond, VA Freefall is only detectable when it ends. The weightedness of mass allows you to establish the ending freefall motion. Mitch Raemsch === Subject: Re: Our curve around the Sun Any object that is moving with no local forces being applied to it, on or near the earth's surface experiences the Coriolis effect. This is the cause of those circular low and high pressure atmospheric areas. It also applies to objects that have been dropped, or projectiles that have been fired. An object dropped or a bullet fired vertically at either pole would be the only thing I can think of that would be expected to travel a straight line, relative to an earth based coordinate system. But, even these would follow a curved path due to the earth's orbital motion. A body in orbit is constantly falling (into a bottomless pit) but since the local vertical is constantly changing, the orbiting body never reaches the point about which it is orbiting. When God casts Lucifer into the bottomless pit, will Lucifer be in orbit about some cosmic center of gravity? Gordon >> If you drop something it does not follow a curve in the Earth's >> geometry. But it does in the Sun's. >> What looks straight from one perspective is curved in another. >> Something that falls straight on Earth is falling around the Sun in a >> curve with the rest of the Earth. >> I believe this is Galilean Relativity where there are different >> appearences from different perspectives. Actually just two >> perspectives are required. >> Mitch Raemsch >> If you are IN something that gets dropped, whatever the gravity, from >> your 'perspective' it falls straight down. >> If you don't look outside, you don't even know there is acceleration - >> until you hit something. >> Tom Davidson >> Richmond, VA Freefall is only detectable when it ends. The weightedness of mass >allows you to establish the ending freefall motion. Mitch Raemsch === Subject: Re: Our curve around the Sun or near the earth's surface experiences the Coriolis effect. This is > the cause of those circular low and high pressure atmospheric areas. The first sentence doesn't connect with the second. Air has numerous types of local forces being applied, at least one of which is vastly stronger than Coriolis acceleration. > An object dropped or a bullet fired vertically at either pole would > be the only thing I can think of that would be expected to travel a > straight line, relative to an earth based coordinate system. But, > even these would follow a curved path due to the earth's orbital > motion. Sort of the reverse. If Earth was somehow prevented from orbiting, the path would be even less straight in Earth's coordinate system. - Tim === Subject: Re: Our curve around the Sun posting-account=5ApcPgoAAABKcgEyKsQmJVb3Rz63IGGL .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; WWTClient2),gzip(gfe),gzip(gfe) > Any object that is moving with no local forces being applied to it, on > or near the earth's surface experiences the Coriolis effect. This is > the cause of those circular low and high pressure atmospheric areas. > It also applies to objects that have been dropped, or projectiles that > have been fired. An object dropped or a bullet fired vertically at either pole would be > the only thing I can think of that would be expected to travel a > straight line, relative to an earth based coordinate system. But, even > these would follow a curved path due to the earth's orbital motion. A body in orbit is constantly falling (into a bottomless pit) but > since the local vertical is constantly changing, the orbiting body > never reaches the point about which it is orbiting. When God casts Lucifer into the bottomless pit, will Lucifer be in > orbit about some cosmic center of gravity? Gordon >> If you drop something it does not follow a curve in the Earth's >> geometry. But it does in the Sun's. >> What looks straight from one perspective is curved in another. >> Something that falls straight on Earth is falling around the Sun in a >> curve with the rest of the Earth. >> I believe this is Galilean Relativity where there are different >> appearences from different perspectives. Actually just two >> perspectives are required. >> Mitch Raemsch >> If you are IN something that gets dropped, whatever the gravity, from >> your 'perspective' it falls straight down. >> If you don't look outside, you don't even know there is acceleration - >> until you hit something. >> Tom Davidson >> Richmond, VA >Freefall is only detectable when it ends. The weightedness of mass >allows you to establish the ending freefall motion. >Mitch Raemsch- Hide quoted text - - Show quoted text - Coriolis doesn't come from orbit but rotation which is a form of antigravity. Mitch Raemsch === Subject: Re: Our curve around the Sun >> Any object that is moving with no local forces being applied to it, on >> or near the earth's surface experiences the Coriolis effect. This is >> the cause of those circular low and high pressure atmospheric areas. >> It also applies to objects that have been dropped, or projectiles that >> have been fired. >> An object dropped or a bullet fired vertically at either pole would be >> the only thing I can think of that would be expected to travel a >> straight line, relative to an earth based coordinate system. But, even >> these would follow a curved path due to the earth's orbital motion. >> A body in orbit is constantly falling (into a bottomless pit) but >> since the local vertical is constantly changing, the orbiting body >> never reaches the point about which it is orbiting. >> When God casts Lucifer into the bottomless pit, will Lucifer be in >> orbit about some cosmic center of gravity? >> Gordon >> (snip) Coriolis doesn't come from orbit but rotation which is a form of >antigravity. Mitch Raemsch > The Coriolis effect applies throughout the universe. The spiral shape of galaxies is an example of this. The earth's rotation about its axis was caused by the Coriolis effect at the time the earth was being consolidated from cosmic materials. Gordon === Subject: Re: Our curve around the Sun posting-account=utFteQoAAAAcXe-3RaiWKR4V32bnT7f2 Gecko/20081217 Firefox/2.0.0.20 (.NET CLR 3.5.30729),gzip(gfe),gzip(gfe) > If you drop something it does not follow a curve in the Earth's > geometry. But it does in the Sun's. > What looks straight from one perspective is curved in another. > Something that falls straight on Earth is falling around the Sun in a > curve with the rest of the Earth. > I believe this is Galilean Relativity where there are different > appearences from different perspectives. Actually just two > perspectives are required. > Mitch Raemsch > If you are IN something that gets dropped, whatever the gravity, from > your 'perspective' it falls straight down. > If you don't look outside, you don't even know there is acceleration - > until you hit something. > Tom Davidson > Richmond, VA Freefall is only detectable when it ends. it never ends > The weightedness of mass > allows you to establish the ending freefall motion. Mitch Raemsch === Subject: Re: Our curve around the Sun > If you drop something it does not follow a curve in the Earth's > geometry. But it does in the Sun's. > What looks straight from one perspective is curved in another. > Something that falls straight on Earth is falling around the Sun in a > curve with the rest of the Earth. > I believe this is Galilean Relativity where there are different > appearences from different perspectives. Actually just two > perspectives are required. > Mitch Raemsch > If you are IN something that gets dropped, whatever the gravity, from > your 'perspective' it falls straight down. > If you don't look outside, you don't even know there is acceleration - > until you hit something. > Tom Davidson > Richmond, VA > Freefall is only detectable when it ends. it never ends > The weightedness of mass > allows you to establish the ending freefall motion. > Mitch Raemsch- Hide quoted text - - Show quoted text -- Hide quoted text - - Show quoted text - There is no singularity. === Subject: Re: Question on matrix analysis <2560804.1237373444197.JavaMail.jakarta@nitrogen.mathforum.org>, > Let A_n and B_n be two sequences of symmetric positive define matrices such > that A_n x_n = B_n x_n for some vector x_n with non-collinear components for > all n. Can be conclude that A_n is asymptotically equivalent to B_n as n goes > to infinity? That is, each entry of A_n is asymptotically equivalent to the > respectively one of B_n? I don't know what it means for a vector to have non-collinear components. I don't know what it means for a number to be asymptotically equivalent to another number. But it seems to me that it should be easy to find matrices A and B, and a vector x, such that A x = B x without A = B. I wouldn't be surprised to see examples with A and B symmetric and positive definite. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: The only angle is theta! - Wackypedia Expires: 60 days >>I never imagine it was for a moment. Marvellous how so many of you >relativists can say what things are not. You learned that from Diaper, >didn't you? >> You can call me a relativist...but not of the Einsteinian type. Very much so, he was full of bull too. Einstein: The velocity of light is not source dependent. >Einstein: We'll alter time to make lightspeed have the same c for >everyone. >Wilson: We'll uniate light speed to make it have the same c for >everyone. That's not what I said nor what unification does. >Wilson: My 'G' is not Newton's G. >Einstein: These two postulates suffice for the attainment of a simple and > consistent theory >Einstein: I'm a prominent theoretical physicist. >Wilson: xstart depends on how many points I want. >Wilson: My ellipses are beautiful, I said so. >Wilson: I'm a prominent theoretical physicist. Wilson, thank christ I got out of engineering and did physics instead. At least I now know the value of pi. >No fundamental difference at all, you are exactly like Einstein. You both > theories out of your arse faster than Superman can fly around the >Earth. >You are even proud of your far superior theories. >I have to hand it to you, Wilson, your bull is far superior to anyone's. 'Unification' is not TOTAL UNIFICATION, dopey. >> I can feel a plonk coming on.... >>You've got that right. I can always chat with Tom Davidson. >> >No point in you interrupting a conversation between Tom and I, >Wilson, and snipping his reply is just sticking your head up your >jealous arse. Tom, I'm asking a favour of you. I've published a spreadsheet and Henry Wilson says > he cannot change the initial variables. Since I can > and have no other way to find what the problem > is, could you please check to see if what he's saying > is generally the case or if he's got some quirk in his > software? http://www.androcles01.pwp.blueyonder.co.uk/WilsonMethod.xls > Sheets 1, 2 and Analysis. What I'm trying to do is show how him the correct method, >but he's a cantankerous old guy. http://www.androcles01.pwp.blueyonder.co.uk/Orbit.xls Androcles No problem, Androcles - I downloaded both spreadsheets. I am able to change the values in the >yellow cells, but I had to *double-click* to select the unprotected cells >for data entry. Ask Henry if he has tried that. Otherwise, I had no trouble. All seems to work correctly, and produce >reasonable results. BTW, Nice ellipses. I have done more than my share of numerical >simulations, and I appreciate the fact that errors propagate through >iterative procedures, so whenever explicit calculations can be performed >using an analytical formula, it is always to be preferred over the kind of \ >calculate-the-next-result-based-on-the-result-we-got-from-the-last-calcul ation >method that characterized so many numerical models. > Back in the 1980's when I was in the team analyzing atmospheric helium and >demonstrating empirically the rapid mixing at sea level, I had the >opportunity to examine a strategy for a numerical simulation of a body of >compressible gas with various variables (P, T, local velocity, specific >heat, heat input and radiation, adiabatic coefficient, etc.) that used a >modified volumetric cell array. The strategy *appeared* sound, but the >programmers had overlooked a critical detail: digital computers do not have \ >an optimal rounding algorithm. > When the last decimal digit of a number is 5, computers invariably >rounded up at that time, whereas the procedure recommended by mathematicians >and engineers (ASTM, at least) is to round terminal 5s up exactly half the >time, and down the other half of the time. The ASTM procedure in particular >instructs that *bias* in rounding may be eliminated by rounding up when the \ >digit preceding the 5 is odd, and down when that digit is even. This >eliminates what the mathematicians call propagation of round-off error. In modeling small systems this is normally not a problem, and it can be >avoided by using extended precision in the calculations. However in large \ >systems in which there are large numbers of iterations (due to high data >resolution in spatial and/or temporal dimensions) an error of one part in a \ >billion consistently *added* to one number of every twenty can produce >significant errors. When I learned that the modelers either had no appreciation for the >accumulation of round-off error, or did not consider it a serious issue, or \ >(in at least one case) considered the consequences of enduring such an error >to be preferable to delaying the publication of results when corrective >measures were implemented, I lost all faith in climate modeling. It was not long afterward that it was discovered that their model >invariably predicted overall warming of the atmosphere, even when the >calculations were run for *reduced* CO2, or when calculating *backwards* in \ >time, the average temperatures also increased monotonically with the amount \ >of time regressed. Little was said publicly about these results. It seemed >obvious to me that the ever-increasing temperatures were accumulating >round-off error from thousands of time iterations on millions of cells. I have not spoken with them since. > I hope I've said something helpful. > Tom >See, Tom Davidson is a rational scientist, whereas Woolly Wilson is just >another ing idiot. Now plonk that, sheep shagging moron. You repeated his words because he is one of the few who is too polite to call you appropriate names.....like 'stupid old pommie engineer'. >> I use perihelion....just a habit. >>Your xstart is still furthest from the focus, not closest. >> An ellipse has TWO foci, dopey old engineer.. Oh sure, and the Solar system has two suns, one of which is >invisible, right? So your xstart is closer to the one we can't see. >That plonk you feel coming on is getting ever more enticing. sure is... >> I told you, I have deliberatelty distorted my ellipses by a considerable \ >> amount >> to see the effect on brighness curves. It was negligible. You can tell me 'til the ing cows come home, your ellipses are still >half-polygons mirrored. Go back to shagging sheep where you belong. http://www.androcles01.pwp.blueyonder.co.uk/WilsonMethod.xls >On Sheet 2 you only have to change vellx to 0.000000001 to see you >don't have an ellipse no matter how many points you add. in' useless. You don't know what you are talking about. >>I never imagine it was for a moment. Marvellous how so many of you >>relativists can say what things are not. You learned that from Diaper, >>didn't you? >> You can call me a relativist...but not of the Einsteinian type. >>Very much so, he was full of bull too. >>Einstein: The velocity of light is not source dependent. >>Einstein: We'll alter time to make lightspeed have the same c for >>everyone. >>Wilson: We'll uniate light speed to make it have the same c for >>everyone. That's not what I said nor what unification does. Give us a list of what things are not, Wilson. A rotating frame is not a 'rotating frame'... hahahahhahahahaha! --Wilson There is no doppler shift in BaTh. -- Wilson http://tinyurl.com/2rk695 How would you know what you said, you crazy old bastard? >>Wilson: My 'G' is not Newton's G. >>Einstein: These two postulates suffice for the attainment of a simple and >> consistent theory >>Einstein: I'm a prominent theoretical physicist. >>Wilson: xstart depends on how many points I want. >>Wilson: My ellipses are beautiful, I said so. >>Wilson: I'm a prominent theoretical physicist. Wilson, thank christ I got out of engineering and did physics instead. > At > least I now know the value of pi. >No fundamental difference at all, you are exactly like Einstein. You both >> theories out of your arse faster than Superman can fly around the >>Earth. >>You are even proud of your far superior theories. >>I have to hand it to you, Wilson, your bull is far superior to >>anyone's. 'Unification' is not TOTAL UNIFICATION, dopey. *p..... > I can feel a plonk coming on.... >>You've got that right. I can always chat with Tom Davidson. >>No point in you interrupting a conversation between Tom and I, >>Wilson, and snipping his reply is just sticking your head up your >>jealous arse. >> Tom, I'm asking a favour of you. >> I've published a spreadsheet and Henry Wilson says >> he cannot change the initial variables. Since I can >> and have no other way to find what the problem >> is, could you please check to see if what he's saying >> is generally the case or if he's got some quirk in his >> software? >> http://www.androcles01.pwp.blueyonder.co.uk/WilsonMethod.xls >> Sheets 1, 2 and Analysis. >> What I'm trying to do is show how him the correct method, >>but he's a cantankerous old guy. >> http://www.androcles01.pwp.blueyonder.co.uk/Orbit.xls >> Androcles >> No problem, Androcles - >> I downloaded both spreadsheets. I am able to change the values in the >>yellow cells, but I had to *double-click* to select the unprotected cells >>for data entry. >> Ask Henry if he has tried that. >> Otherwise, I had no trouble. All seems to work correctly, and produce >>reasonable results. >> BTW, Nice ellipses. I have done more than my share of numerical >>simulations, and I appreciate the fact that errors propagate through >>iterative procedures, so whenever explicit calculations can be performed >>using an analytical formula, it is always to be preferred over the kind of >>calculate-the-next-result-based-on-the-result-we-got-from-the-last-calcu lation >>method that characterized so many numerical models. >> Back in the 1980's when I was in the team analyzing atmospheric helium >> and >>demonstrating empirically the rapid mixing at sea level, I had the >>opportunity to examine a strategy for a numerical simulation of a body of >>compressible gas with various variables (P, T, local velocity, specific >>heat, heat input and radiation, adiabatic coefficient, etc.) that used a >>modified volumetric cell array. The strategy *appeared* sound, but the >>programmers had overlooked a critical detail: digital computers do not >>have >>an optimal rounding algorithm. >> When the last decimal digit of a number is 5, computers invariably >>rounded up at that time, whereas the procedure recommended by >>mathematicians >>and engineers (ASTM, at least) is to round terminal 5s up exactly half >>the >>time, and down the other half of the time. The ASTM procedure in >>particular >>instructs that *bias* in rounding may be eliminated by rounding up when >>the >>digit preceding the 5 is odd, and down when that digit is even. This >>eliminates what the mathematicians call propagation of round-off error. >> In modeling small systems this is normally not a problem, and it can be >>avoided by using extended precision in the calculations. However in large >>systems in which there are large numbers of iterations (due to high data >>resolution in spatial and/or temporal dimensions) an error of one part in \ >>a >>billion consistently *added* to one number of every twenty can produce >>significant errors. >> When I learned that the modelers either had no appreciation for the >>accumulation of round-off error, or did not consider it a serious issue, >>or >>(in at least one case) considered the consequences of enduring such an >>error >>to be preferable to delaying the publication of results when corrective >>measures were implemented, I lost all faith in climate modeling. >> It was not long afterward that it was discovered that their model >>invariably predicted overall warming of the atmosphere, even when the >>calculations were run for *reduced* CO2, or when calculating *backwards* >>in >>time, the average temperatures also increased monotonically with the >>amount >>of time regressed. Little was said publicly about these results. It >>seemed >>obvious to me that the ever-increasing temperatures were accumulating >>round-off error from thousands of time iterations on millions of cells. >> I have not spoken with them since. >> I hope I've said something helpful. >> Tom >>See, Tom Davidson is a rational scientist, whereas Woolly Wilson is just >>another ing idiot. Now plonk that, sheep shagging moron. You repeated his words because he is one of the few who is too polite to > call > you appropriate names.....like 'stupid old pommie engineer'. > *pl..... >> I use perihelion....just a habit. >>Your xstart is still furthest from the focus, not closest. >> An ellipse has TWO foci, dopey old engineer.. >>Oh sure, and the Solar system has two suns, one of which is >>invisible, right? >>So your xstart is closer to the one we can't see. >>That plonk you feel coming on is getting ever more enticing. sure is... *plo... >> I told you, I have deliberatelty distorted my ellipses by a considerable > amount > to see the effect on brighness curves. It was negligible. >>You can tell me 'til the ing cows come home, your ellipses are still >>half-polygons mirrored. Go back to shagging sheep where you belong. >> http://www.androcles01.pwp.blueyonder.co.uk/WilsonMethod.xls >>On Sheet 2 you only have to change vellx to 0.000000001 to see you >>don't have an ellipse no matter how many points you add. in' useless. You don't know what you are talking about. You've had yours. *plonk* Do not reply to this generic message, it was automatically generated; you have been kill-filed, either for being boringly stupid, repetitive, unfunny, ineducable, repeatedly posting politics, religion or off-topic subjects to a sci. newsgroup, attempting free advertising or because you are a troll; any reply will go unread. Boringly stupid is the most common cause of kill-filing, but because this message is generic the other reasons have been included. You are left to decide which is most applicable to you. There is no appeal, I have despotic power over whom I will electronically admit into my home and you do not qualify as a reasonable person I would wish to converse with or even poke fun at. This should not trouble you, many of those plonked find it a blessing that they are not required to think and can persist in their bigotry or crackpot theories without challenge. You have the right to free speech, I have the right not to listen. The kill-file will be cleared annually with spring cleaning or whenever I purchase a new computer or hard drive. I hope you find this explanation is satisfactory but even if you don't, damnly my frank, I don't give a dear. Have a nice day. === Subject: Re: The only angle is theta! - Wackypedia Expires: 60 days >>No fundamental difference at all, you are exactly like Einstein. You both > theories out of your arse faster than Superman can fly around the >Earth. >You are even proud of your far superior theories. >I have to hand it to you, Wilson, your bull is far superior to >anyone's. >> 'Unification' is not TOTAL UNIFICATION, dopey. >*p..... >See, Tom Davidson is a rational scientist, whereas Woolly Wilson is just >another ing idiot. Now plonk that, sheep shagging moron. >> You repeated his words because he is one of the few who is too polite to \ >> call >> you appropriate names.....like 'stupid old pommie engineer'. > *pl..... plon..... >> I use perihelion....just a habit. >>Your xstart is still furthest from the focus, not closest. >> An ellipse has TWO foci, dopey old engineer.. >>Oh sure, and the Solar system has two suns, one of which is >invisible, right? >>So your xstart is closer to the one we can't see. >That plonk you feel coming on is getting ever more enticing. >> sure is... >*plo... >> I told you, I have deliberatelty distorted my ellipses by a considerable >> amount >> to see the effect on brighness curves. It was negligible. >>You can tell me 'til the ing cows come home, your ellipses are still >half-polygons mirrored. Go back to shagging sheep where you belong. >> http://www.androcles01.pwp.blueyonder.co.uk/WilsonMethod.xls >On Sheet 2 you only have to change vellx to 0.000000001 to see you >don't have an ellipse no matter how many points you add. in' useless. >> You don't know what you are talking about. >> > Ah bugger, I'll give him one more chance.... >You've had yours. *plonk* ing good idea. Harry Wilson. ASTC,BSc,DSc(T) www.users.bigpond.com/hewn/index.htm. ..... === Subject: an interesting dilemma posting-account=3AXGywoAAAAOVw_eePdhm2Je-nK91gvi AppleWebKit/525.19 (KHTML, like Gecko) Chrome/1.0.154.48 Safari/525.19,gzip(gfe),gzip(gfe) I've posted something like this before, but not quite in this form. I am essentially describing a theorem that is provably very long when formalized, but that can be proven very rapidly informally. Let n = 100000000000000000000000000000000000000 . Let B(S,n) be a brute-force proof-search algorithm, that, when given a sentence S and integer n as input, searches for a proof of S that is under n characters long. If it finds such a proof, it returns true; if, after searching exhaustively, it does not, it returns false. Consider the following algorithm (I give pseudocode): W(alg X) { Construct a formal sentence L = X(X) returns false; B(L,n) -> a; //if there's a proof of L in under n chars, B returns true if (a == true) return true; else return false; } The theorem is: W(W) returns false. If we run W(W), we find that if a is true, then the algorithm returns true; however, if a is true, then B(W(W) returns false) must have returned true, meaning that there exists a valid proof for the theorem W(W) returns false. Thus, a must be false, and so we know that W(W) returns false. (I think this is essentially a diagonalization argument.) However, since a is false, we know that B(L,n) did not return true, and thus no proof of the theorem was found in under n characters. You'll note, however, that we just proved the theorem informally in the above paragraph. Something about this whole proof is very suspicious to me. Does anyone have any thoughts on what's going on? If I'm right, it means that there exist very short informal proofs of theorems that require extremely long formal proofs. My big question is: Why can't we just formalize the above proof into a short proof? Obviously this is impossible, but why? -Phil === Subject: Re: an interesting dilemma Originator: tchow@lebesgue.mit.edu.mit.edu (Timothy Chow) >My big question is: Why can't we just formalize the above proof into >a short proof? Obviously this is impossible, but why? When you say proof of L, you must specify some theory T in which you are searching for proofs. In your informal argument, you make the assumption that theorems of T are true. Any reasonable theory T cannot prove that its theorems are true since this would in particular give a proof of the consistency of T within T itself. That's why you can't formalize your informal argument in T. -- Tim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences === Subject: Re: an interesting dilemma > Let B(S,n) be a brute-force proof-search algorithm, that, when given a > sentence S and integer n as input, searches for a proof of S that is > under n characters long. If it finds such a proof, it returns true; > if, after searching exhaustively, it does not, it returns false. ... > My big question is: Why can't we just formalize the above proof into > a short proof? Obviously this is impossible, but why? Phil, Very intriguing! I concur with bcrowell that it probably has to do with the strength of proof systems. For instance, more explicitly B(L, n) means There is a proof of L in under n characters IN SYSTEM B, that is, in the system embedded in B. I encourage you to look at Gregory Chaitin's work. In his work on your function B. You might wish to play with those. I know I want to go back and play with them. His web site is http://cs.umaine.edu/~chaitin/ Click on Books with LISP Software to reach pages which have the LISP software itself. -paul- === Subject: Re: an interesting dilemma Having thought about this a little more, I think the situation is as follows. The assertion you were trying to prove becomes a meaningful one if you replace the notion of formal versus informal with the idea of formal proof in one system versus formal proof in another system. It then becomes an assertion that some theorems require really long proofs in some systems, but have much shorter proofs in others. I think this assertion is true, but it's easier to prove than you thought it was, and the result isn't as interesting as you thought it was. Consider the statement that the number 10^10 exists. In the first-order theory of the reals, with all the usual elementary axioms of arithmetic (but no second-order axioms like completeness or the Archimedean principle), this is fairly easy to prove. The axioms assert that 1 exists. By closure under addition, I can prove that 1+1+1+1+1+1+1+1+1+1=10 exists. Then by closure under multiplication I can prove that 10x10x10x10x10x10x10x10x10x10 exists. In fact, the statement of the proposition is similar to the statement of the proof. There is no exponentiation operator in the system, so the proposition essentially has to name the number by some construction similar to the one given above, and then the proof simply consists of applying the axioms to verify existence of the result at each intermediate step. Now consider a weaker theory, a first-order theory of the naturals that has axioms about addition, but no axioms about multiplication. Because it's a first-order theory, I can't express statements of the form there exists a function f such that ..., and therefore I can't prove the existence of a multiplication function. In this theory, it's quite possible that the only way to even express the proposition that 10^10 exists is by writing it down as 1+1+1+...1 exists, where the ... indicates billions of repetitions of 1+. Since the statement of the proposition requires billions of symbols, its proof trivially requires at least that many symbols. === Subject: Re: Eppur si muove - Debunking Einstein's postulate about the invariance of the speed of light But what is the nature of that which you presume to think you are thinking? For one, quite incredibly, you actually think you are smarter than Einstein-- AB: But of course. He was the biggest bungler of all time. but while he was being squired round the world on strength of what other men and women of science AB: He inspired a lot of interest, certainly. But most sensible people thought he was talking rubbish, and there were lots of jokes about his theories of relativity. were finding in his thought to be valid and firmly based in solid scientific method and principle--you? Where are you? Where are those beloved photos of Mr. Arindam Banerjee posing with Indian Chiefs, the greatest men and women of science, AND Charlie Chaplin? AB: I have lots of contacts with many great Indians, who do not get publicity in the Western world. There is no point in my name-dropping, for \ you do not know them. Charlie Chaplin is long dead. I have been published \ in the Indian media, and given interviews. It is just that the Western world ignores me, as I suspect, it is insecure. I am most welcome to give talks, interviews, to whoever comes to my door. All are welcome. May all the Oprahs, etc. make a bee-line to my door! No charge, from my side, too - \ even though I do need some money to make my designs work, I won't charge for \ giving interviews. Any aspiring journalist, is most welcome. Now, if they are *all* yellow and do not want to talk to me, as their bosses do not wish \ to hear let alone publish, how I can be famous in the Western wolrld? My doors are w i i i i i de open to all journalists. Why, I am a journalist myself, I do have a diploma! You think you are so smart? AB: That will be selling myself short. Show me a picture of you with Jim Carey, Richard Feynman and Chief Dan George--then I'll be willing to consider it. Otherwise, I am moved to suppose that the only thing that can possibly inspire a person to think himself smarter than Albert Einstein is a Supreme Ignorance of Einstein as to what he was in fact thinking. AB: There is no point in trying to *convince* media-formed morons. Also, what morons think is normally immaterial, but we do live in abnormal times, \ when morons (Presidents/politicians, idiot-intellectuals, CEOs) have supreme \ power, money and influence. Which is why I am responding to you, in the expectation that your kind of thinking will be naturally exposed to the clever but unempowered. Best I can do, and thank you very much for your response, which is so useful. You think you know something concerning the phenomenon of light deflection about the limbs of the sun? You think you've got what it takes to second guess the first man who on this planet had the quality of mind to anticipate that such a thing would be the case--whatever might be the cause? That was not you, Mr. Banerjee, it was Einstein. sun, so the sun would act like a gravitational lens and thus bend light. To \ prove that, they took the positions of the stars around the sun during night \ and day during a complete eclipse. The stellar positions changed, as if the \ light was being bent by space-warping as per Einstein's predictions. This is the *only* supposed experimental proof of GR! What is actually was, was \ nothing but low grade legerdemain (sleight of hand). The sun has an extended atmosphere around it, like the Earth, and that bent the rays of light like any optical lens. So the effect from an optical lens, completely \ natural and predictable, was taken as the validation of GR. What a lot of rubbish! In *no* text on GR that I have read, was it ever remotely mentioned that the sun was acting as an optical lens. So you see, I am the \ first person to point out this bungle, and that makes me much smarter than all the rest of the jabbering relativists. But, this is only a very small fraction of my genius. To collar a scoundrel, or a media-exalted set of scoundrels, is the role of a lawkeeper of the detective type. My great achievement has to find a formula linking mass and energy, which is the correct and intuitive formula, on the lines of the grand Newtonian equations \ of motion, momentum, force, etc, merely with a re-statement of Newton's first law. With that, we will have faster thatn light interstellar travel hopefully in my own lifetime. Also, with the Hydrogen Transmission Network, \ we can solve the world's water and energy problems very satisfactorily. No \ doubt there are many vested interests, who will do their best to obstruct. But I have no doubt that the future generations, for whose interests I endeavour, will place me as the greatest genius of all time, for the works I \ have already done and published in Usenet/Internet. And not for just in the \ realm of science, too. True, I have yet to produce my best works. With the \ Grace of the Divine, I can and will reach greater heights. Not you. It was not you who won the Nobel Prize for discovery of the photoelectric effect, that was Einstein. Will you try to demean that? AB: Oh yes. Quantum theory sucks - it is like relativity, totally wrong. In those days they did not know much about antenna theory. The photoelectric effect can be easily explained with modern em theory. No self-respecting antenna engineer has the slightest use for quantum theory. A photon is just a small burst of em radiation, formed by atomic displacements of charges that make the atoms into dipoles. This was not understood, in those days when Einstein got the Nobel Prize. I found all this the hard way, when I was an antenna engineer, doing VSWR tests on radiating elements, for 8 years. Do so! Reveal yourself for precisely the sort that would! Demean the photoelectric effect, Mr. Banerjee. Refer to it as the work of a bungler. Come on! Show us who you really are, so we can really know, and describe you in terms far less uncertain and shy of forcefulness than these, if you please. AB: As I said, no antenna engineer has the slightest use for the quantum theory. Quantum and relativity are the two pillars of modern theoretical physics bull. Rather like derivates that crushed the modern economy, a lot of abstract maths has been put into the shonky basics, like putting good \ sauce over rotten fish. But just on the way outside chance that the world of science is utterly without excuse in failing till now to have noticed, the Man who is Smarter than Einstein, Mr. Arindam Banerjee--well, let's just see. Here, from Google Books and the Princeton University Press is a theory of gravitational light deflection, in a book by none other than Ignazio Ciufolini and John Archibald Wheeler. Let us see you take no more than just the ONE PAGE, namely page 120 and explain, by means of your wiser than Einstein genius, from top to bottom, exactly what is being said in those terms, so that at least a mathematician can understand what you would have to say about how WRONG all this is . . . AB: Sorry mate, I cannot oblige. I have better things to do. I studied GR \ long ago, and there was no mention of the sun acting as an optical lens. But it does have an atmosphere like the earth, and so light from the stars will bend as they pass through the solar atmosphere. Making it act like an \ optical lens. Case closed. Any lawyer from the Einstein side, please stand \ up... Show where in early GR literature there was mention of the sun acting \ as an optical lens! http://tinyurl.com/cacdcv -- JM http://jpdavid.blogspot.com/ http://bobbisoxsnatchers.blogspot.com http://vignettes-mackie.blogspot.com/ === Subject: Re: Eppur si muove - Debunking Einstein's postulate about the invariance of the speed of light <49ba1ffc$0$4046$afc38c87@news.optusnet.com.au> posting-account=WtWlpwoAAABtMc3YhLOw2GNjBkWJwBh5 SLCC1; .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.5.30729; \ .NET CLR 3.0.30618),gzip(gfe),gzip(gfe) > You think you know something concerning the phenomenon of light > deflection about the limbs of the sun? You think you've got what it > takes to second guess the first man who on this planet had the quality > of mind to anticipate that such a thing would be the case--whatever > might be the cause? That was not you, Mr. Banerjee, it was Einstein. sun, so the sun would act like a gravitational lens and thus bend light. To > prove that, they took the positions of the stars around the sun during night > and day during a complete eclipse. The stellar positions changed, as if the > light was being bent by space-warping as per Einstein's predictions. There. Now, you see. This is exactly the sort of ignorance of Einstein and the proofs of his theory which, as I prophesied in my last post was BOUND to be revealed, and now we've seen it. Whoopee!! Did I not say that there is nothing like ignorance of Einstein's thinking to cause some super-luminal star traveler like Mr. Banerjee to flatter himself to think he is a better thinker than Einstein? Oh, my yes, for it is now clear that this is exactly what's the way too uncool haps with you, my man. What you describe up there has nothing whatever to do with the method by which the eclipse experiments have been conducted, not in 1919, and not in 1973. It is not a simple matter of taking day/night exposures, as night follows the very day, as here you claim. Oof! How can you be so tremendously uninformed as to suppose these scientists could be that damned dumb--or how could you be so silly as to think that stars visible in the sky by day, would have remained in the same position to get their pictures taken at night? Oy gevalt! That might be true for the vanity of a great Star of India like you, Mr. Banerjee, to be hanging around for it, as such a camera hog, but not for the mere humble little stars of the sky. They per force move on across the ecliptic, having nothing to prove. As it happened for the Eddington expedition, they had to wait 6 months after the eclipse shots were taken to get night-time exposures of the same field of stars at the same position in the sky. Even at that, much data reduction work had to be done in order to compensate for parallax, since the earth itself had changed position by plus or minus 180 degrees, let alone so many million miles. And this was not the only variable that had to be taken into account and precisely http://ptonline.aip.org/journals/doc/PHTOAD-ft/vol_62/iss_3/37_1.shtml The parsec is defined as the distance at which a star, seen from Earth, will undergo apparent motion, due to parallax, of one arcsecond over the course of a year. No stars are within a parsec of our solar system; therefore, all stellar parallax work is subarcsecond in nature, much of it well below the arcsecond. Dyson had considerable experience in working on stellar parallaxes. In fact, both Dyson and Eddington began their careers working on problems of astrometry, including parallax and proper motion of stars and other bodies. The method used to determine the apparent shifts was to expose pictures of the star field during the eclipse and then take comparison exposures of the same star field at night, without the Sun present. Obviously, the comparison plates had to be taken at a different time of the year, because it would take some time for the Sun to move out of the Hyades star field. It was also desirable that they be taken at a time when the star field was at the same position in the sky, which meant waiting at the eclipse site until the Sun had moved far enough along the ecliptic for the star field to rise to that same height in the sky before sunrise. Similarly the same procedure was undergone for the Mauritania Expedition of 1973 by the University of Texas/Princeton team . . . http://adsabs.harvard.edu/full/1976AJ.....81..452T > This > is the *only* supposed experimental proof of GR! Omigod. I would say, You've got to be kidding but truly! Ignorance combined with arrogance need not be mistaken for a sense of humor. But shall we suppose that the terms Long baseline interferometry and Very Long Baseline Interferometry and Very Long Baseline Array mean nothing to you, with respect to the work that's being done to measure the arrival time of radio signals from distant Quasars, and the vector analysis that is done to test Einstein's equations and prove them all the more positively and accurately, without need of eclipse data? Too bad you were too enamored of your own all too obvious omniscience to click this link . . . http://tinyurl.com/cacdcv It's all there in the Ciufolini/Wheeler book on page 121--a book which was winner of the 1996 Association of American Publishers Award for Best Professional/Scholarly Book in Physics and Astronomy. On the subject of these authors . . . http://press.princeton.edu/titles/5635.html Yes, Mr. Banerjee, had you only gone clickety-click-click, it might well have saved you the further embarrassment that I will now undertake to feel for you in any case you shall as always have remained far too infallible to feel it for yourself. -- JM http://jpdavid.blogspot.com/ http://bobbisoxsnatchers.blogspot.com http://vignettes-mackie.blogspot.com === Subject: Re: Eppur si muove - Debunking Einstein's postulate about the invariance of the speed of light <49ba1ffc$0$4046$afc38c87@news.optusnet.com.au> .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) > You think you know something concerning the phenomenon of light > deflection about the limbs of the sun? You think you've got what it > takes to second guess the first man who on this planet had the quality > of mind to anticipate that such a thing would be the case--whatever > might be the cause? That was not you, Mr. Banerjee, it was Einstein. > sun, so the sun would act like a gravitational lens and thus bend light. \ To > prove that, they took the positions of the stars around the sun during night > and day during a complete eclipse. The stellar positions changed, as if the > light was being bent by space-warping as per Einstein's predictions. There. Now, you see. This is exactly the sort of ignorance of > Einstein and the proofs of his theory which, as I prophesied in my > last post was BOUND to be revealed, and now we've seen it. Whoopee!! > Did I not say that there is nothing like ignorance of Einstein's > thinking to cause some super-luminal star traveler like Mr. Banerjee > to flatter himself to think he is a better thinker than Einstein? Yes. I had the wits to see that the sun was behaving like an optical lens *because it has an extended atmosphere around it which was actually bending the light* and this point was never made by anyone before. > Oh, my yes, for it is now clear that this is exactly what's the way > too uncool haps with you, my man. What you describe up there has > nothing whatever to do with the method by which the eclipse > experiments have been conducted, not in 1919, and not in 1973. took into account the fact that the Sun has an extended atmosphere around it, that would create the lensing effect optically. They just took it that the Sun was a hot disc without the atmosphere - just as when they talk of black holes, it is just a point or something! > It is not a simple matter of taking day/night exposures, as night > follows the very day, as here you claim. Oof! How can you be so > tremendously uninformed as to suppose these scientists could be that > damned dumb--or how could you be so silly as to think that stars > visible in the sky by day, would have remained in the same position to > get their pictures taken at night? Oy gevalt! Irrelevant. Does not give any reference to the optical effect being passed of as a gravitational effect. > That might be true for the vanity of a great Star of India like you, > Mr. Banerjee, to be hanging around for it, as such a camera hog, but > not for the mere humble little stars of the sky. They per force move > on across the ecliptic, having nothing to prove. Irrelevant. > As it happened for the Eddington expedition, they had to wait 6 months > after the eclipse shots were taken to get night-time exposures of the > same field of stars at the same position in the sky. Even at that, > much data reduction work had to be done in order to compensate for > parallax, since the earth itself had changed position by plus or minus > 180 degrees, let alone so many million miles. And this was not the > only variable that had to be taken into account and precisely So much massaging of data was done. But no talk here of the sun always acting as an *optical* lens. http://ptonline.aip.org/journals/doc/PHTOAD-ft/vol 62/iss 3/37 1.shtml The parsec is defined as the distance at which a star, seen from > Earth, will undergo apparent motion, due to parallax, of one arcsecond > over the course of a year. No stars are within a parsec of our solar > system; therefore, all stellar parallax work is subarcsecond in > nature, much of it well below the arcsecond. Dyson had considerable > experience in working on stellar parallaxes. In fact, both Dyson and > Eddington began their careers working on problems of astrometry, > including parallax and proper motion of stars and other bodies. Irrelevant. No talk of the sun acting as an optical lens. The method used to determine the apparent shifts was to expose > pictures of the star field during the eclipse and then take comparison > exposures of the same star field at night, without the Sun present. > Obviously, the comparison plates had to be taken at a different time > of the year, because it would take some time for the Sun to move out > of the Hyades star field. It was also desirable that they be taken at > a time when the star field was at the same position in the sky, which > meant waiting at the eclipse site until the Sun had moved far enough > along the ecliptic for the star field to rise to that same height in > the sky before sunrise. Not irrelevant, but there is no mention of the Sun acting as an optical lens and twisting the positions of the stars as a consequence. Similarly the same procedure was undergone for the Mauritania > Expedition of 1973 by the University of Texas/Princeton team . . . http://adsabs.harvard.edu/full/1976AJ.....81..452T > This > is the *only* supposed experimental proof of GR! Indeed. All your handwaving so far has proved nothing. Omigod. I would say, You've got to be kidding but truly! Ignorance > combined with arrogance need not be mistaken for a sense of humor. But > shall we suppose that the terms Long baseline interferometry and > Very Long Baseline Interferometry and Very Long Baseline Array > mean nothing to you, with respect to the work that's being done to > measure the arrival time of radio signals from distant Quasars, and > the vector analysis that is done to test Einstein's equations and > prove them all the more positively and accurately, without need of > eclipse data? All rubbish, as GR can now be seen as based upon sleight of hand experimentation now being exploited by hordes of gimmicky careerists. Further, Hubble was supposed to show the outer limit of the limited universe, which according to the GRwallahs has limits. But Hubble has shown lots and lots of galaxies instead - the universe is thus infinite. Now this does blow the lid off the Einsteinians, but they will ignore this fact too! Too bad you were too enamored of your own all too obvious omniscience > to click this link . . . http://tinyurl.com/cacdcv It's all there in the Ciufolini/Wheeler book on page 121--a book which > was winner of the 1996 Association of American Publishers Award for > Best Professional/Scholarly Book in Physics and Astronomy. On the > subject of these authors . . . http://press.princeton.edu/titles/5635.html Yes, Mr. Banerjee, had you only gone clickety-click-click, it might > well have saved you the further embarrassment that I will now > undertake to feel for you in any case you shall as always have > remained far too infallible to feel it for yourself. I am not embarrassed in the least. I thank you for your efforts. You have shown that there was no effort done to take into account the solar atmosphere for bending the light, thus the optical effect was passed off as a gravitational effect. > -- > JMhttp://jpdavid.blogspot.com/http://bobbisoxsnatchers.blogspot.comhttp://vi\ g nettes-mackie.blogspot.com To repeat for the final time, my main point is that the sun was acting as a optical lens and not as a gravitational lens. Of course light will be bent around the sun, because of the sun's atomosphere! So the stellar positions will change, with respect to the presence and absence of the sun, because the denser medium around the sun will refract the light, like any optical lens. This small fact was never taken into account, as you yourself show. Now, please let us forget this nonsense of SR, GR, quantum theory, and get down to real physics and anti-gravity machines. Okay, I know I will have to go it alone, but please wait, dear Just Me. I will collect all my new ideas in physics, that I have published in Usenet over the years, and put it up in my website. Now that should clarify all matters to every intelligent person above the age of say 13, and with some slight knowledge of algebra. Arindam Banerjee === Subject: Re: Eppur si muove - Debunking Einstein's postulate about the invariance of the speed of light <49ba1ffc$0$4046$afc38c87@news.optusnet.com.au> posting-account=hLqQkwoAAAAEBL5vz2BJ6yN8RgDuNGiC Gecko/2009021910 Firefox/3.0.7,gzip(gfe),gzip(gfe) > AB: I have lots of contacts with many great Indians, who do not get > publicity in the Western world. There is no point in my name-dropping, for > you do not know them. Charlie Chaplin is long dead. I have been published > in the Indian media, and given interviews. It is just that the Western > world ignores me, as I suspect, it is insecure. Everybody knows who Chandrasekhar was. So no, the Western world does not ignore great Indians. === Subject: Re: Eppur si muove - Debunking Einstein's postulate about the invariance of the speed of light <49ba1ffc$0$4046$afc38c87@news.optusnet.com.au> .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) > AB: I have lots of contacts with many great Indians, who do not get > publicity in the Western world. There is no point in my name-dropping, for > you do not know them. Charlie Chaplin is long dead. I have been published > in the Indian media, and given interviews. It is just that the Western > world ignores me, as I suspect, it is insecure. Everybody knows who Chandrasekhar was. So no, the Western world does > not ignore great Indians. AB: Chandrasekhar supported Einstein's wrong theories and elaborated upon them. Naturally he was a hero. The Western world completely ignores the great and true Indians. === Subject: Re: Eppur si muove - Debunking Einstein's postulate about the invariance of the speed of light > AB: Rubbish. I was the first person in the world who said that the bending > of light was due to the Sun being an actual and very powerful optical > lens, > and not in any sense a gravitational lens. You are certainly not the first person in the world to think he was the first person in the world to think he was thinking something, like, 'first'. AB: It is evident that you could not find any evidence to contradict my statement. === Subject: Re: Pi Reward?! > Don't get excited. The contest has already been won. --------------------------------------------------- Magic of 355 \.85 113 355 \.85 113 is a 99.9999% accurate approximation of pi. 355 \.85 113 = 3.1415929203 pi = 3.1415926535 The difference = 0.0000002668 Can you find a BETTER fraction to more closely approximate the value of pi? The first one to do so will earn a reward of $500. (NOTE: The fraction you > submit must be simplified to lowest terms, so not exactly equivalent to > 355 \.85 113, and the numerator and denominator must each be whole numbers.) --------------------------------------------------- I hate to say it, but that was the silliest contest I've ever seen. $500, > for that? Had I known of it before it had been won, I would quickly have > submitted 3141592653/1000000000 A better contest would have been to find the rational number with the > smallest denominator which approximates pi better than 355/113. > (Answer: 52163/16604) > But of course even that would not have been a good contest because there > are many sites on the web, such as , which list rational approximations of pi. These are best in some sense of the word: 1414924551734738329738247544858147858542277332885695626972442560670981314986\ 9506691672282 / 4503844730213355935174142341927973749307473104071525277718510235858750658859\ 930934885345 2583585292612123503741799371724536693162758592309215546507042804821887041071\ 1184715378217 / 8223807404374805505982545258789382037718007526551264324866508001072395405541\ 432093061652 2462638108573531557346957179917590239217960734621757074378423165200755889027\ 828667966347 / 7838820560519063643657394250665654608969386815917862305705124706451059121784\ 29776709038 1836054974479725510647689548993527389332969655748170897832309780706033704730\ 363351840547 / 5844344499538241133454452165951464448227796961125941248659478826332210979677\ 82051340155 1390836834042263971543740881911060339969726032498219224774588140243334466416\ 1124766494 2770277116802650110983608806947735673089624318928851220978704022696593990012\ 04543830976 / 8818065937470113329679493791853418887658090287933456772259753679241805689735\ 8727236931 1486114487010923439768081802280660537694094665735933126168816560086231002900\ 8964609858 / 4730449332165295004170485003870568478397340929441414236122248377817988912391\ 978637967 2883328293158339505243252575585455494506263421522830054434220717727053813302\ 3631926995 / 9177919008257344711266923815240034921299919395540778011623633024615355751769\ 146128527 8890068086350737429291102897586558088192590994903619790341240244540819249942\ 97292721 / 2829796560732452970740461925011020354947624633420504606208637310206220730148\ 11147407 2415049549218201996673273966185424053471319264494263555210429258505163079088\ 802511764 / 7687341471398608451338383088551399814706664426527077874301561435292798901846\ 22574269 2519708766870192321140569030905433729864580339768223818919428148570826958940\ 89366399 / 8020482107987504608830026864816612501362094737344359443243504953258632885981\ 0867952 1330941785740160465869395804870256898598849975598948333582955798828338373120\ 29193524 / 4236519283362015880914538655660366045389962122171967732355858242286308643537\ 8543551 1421748046101286105982225788350800673331196114296728482464834490858497872996\ 9020649 / 4525564587365271529990504465041195894178295069995760214682115313139844010946\ 219150 1935433488591315810836151741896163599338930841615655476110882061415400747227\ 7028332 / 6160676134697986569221350836274093300473260661753595606101659917744334347808\ 790351 1193082813688030085795520722852881046920080676600524984733082208122107496955\ 002400 / 3797700546328735877020346486574963247066017052777459595765185006736269996414\ 94453 2750688797524236876948218089747867166237185919988219966994311289324813748398\ 002883 / 8755714380669678638267770739179047568817621812023434722665076032572363375795\ 82295 9951330324349996972399079072656582972900697623901499214864158888031123349100\ 8151 / 3167606822921152243391131884876000230092539333719130023610669494367998475171\ 4286 2650098669046767356331858533155392426680175905481550053795052842002875209969\ 89932 / 8435526057200916598879645775415210516763337730966025287736390696630235354487\ 9103 1076825233969550892368765865908909828003038267654921050616858441627304937095\ 412 / 3427641176646815049011775276237029045030147685496432567618386505328173790769\ 79 3137639235788407175404898713234042686299726163358012896518576555739156960184\ 3697 / 9987415880296038303135143736880495926136773164814361317432500563671965952109\ 797 1076825233969550892368765865908909828003038267654921050616858441627304937095\ 412 / 3427641176646815049011775276237029045030147685496432567618386505328173790769\ 79 2302111040706197660092308752799861506915228406897260598530587633454336300267\ 573 / 7327847033496376708033504910405175045553744139739933848628569992858967170313\ 64 1484605727670958753547770209820418509091518715874184972968707501997264260767\ 49 / 4725646802027466100099543579311169554934487687470687133917969822026195887740\ 6 1860617973669756402401037400166587225481270381304100668356339404289463806349\ 18 / 5922530954303353889520491286503273120694550730140202872676046121124562581254\ 3 1944325632423184186529855156425635646748794584547780858182010717153392155927 \ / 618898070760850575842472494572447081057644832073758211143353743672708863142 2982392207018702813920729840891432905197998820480044610603514736060638593446\ 1 / 9493249239715475590839587093631247333370051098400124543007348016292831482569\ 1944325632423184186529855156425635646748794584547780858182010717153392155927 \ / 618898070760850575842472494572447081057644832073758211143353743672708863142 2603363216262449527789326218955429997496864021131514091486997320458895751483 \ / 828676249063567529044972169616988198563023449367509587000395604874907398581 32787119094611837248558031163747405495413725203418841732949092763118630741 / 10436464147300283765026530561176271458491019807495916427771839933886743175 298379273798535131525332719728572890298518459346125974242546582911198657405 \ / 94977072682413831787207838871602131461977399411296295151551621929444360514 3295201947028596288310439254846240839794932515356398646004748043130980736 / 1048895356711277901969063821015688335558221143833047301605062524463671939 29491917147583240960247591908901164655618792688062443086944344719987650005 / 9387568790589005863057466740160583122932798663662869126166777409423071236 3130301571354470653764077870131237937259332565211253918906360374939804117 / 996405936898782647304956172035076438467029512998490713326277213713695724 164900375674125634546361384715002902535599950145144727098387668191176619 / 52489419812495254664107648980611897091191630834556588278785310749976215 164900375674125634546361384715002902535599950145144727098387668191176619 / 52489419812495254664107648980611897091191630834556588278785310749976215 162094809220209231929572945261185691618533462598648831135382347498624975 / 51596380273868063351018490384062290825580157976472124308141620214123854 2805566453916402616788439453817210917066487546495895963005320692551644 / 893039538627191313089158596549606265611472858084463970643690535852361 30233185886138308940516290931776778516408547913341720874132274948697707 / 9623521958390071635828036346230796341840933646502317687888165029062887 2805566453916402616788439453817210917066487546495895963005320692551644 / 893039538627191313089158596549606265611472858084463970643690535852361 2177521346974282772631896393604669345743672448382761244079068023181267 / 693126572118158504936450380734733685726205065657677981451259670539277 293386026147923240162267212967044631775227154043357087300310015070136 / 93387672591060080478325733290115946070401688377320013873967074600025 41273054646273363832008634278452307772360790026420544325632639230105 / 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262452630335382199398 / 83541266890691994833 2646693125139304345 / 842468587426513207 29543635323123417038 / 9404031197158832617 2646693125139304345 / 842468587426513207 3076704071730373588 / 979345322893700547 66627445592888887 / 21208174623389167 296756055405291469 / 94460386220409006 30246273033735921 / 9627687726852338 6134899525417045 / 1952799169684491 428224593349304 / 136308121570117 3137327371971917 / 998642318693672 139755218526789 / 44485467702853 288469374822515 / 91822653867264 8958937768937 / 2851718461558 23289027530608 / 7413127702599 1783366216531 / 567663097408 1804419559672 / 574364584667 21053343141 / 6701487259 309632196661 / 98558989278 21053343141 / 6701487259 14885392687 / 4738167652 2549491779 / 811528438 1068966896 / 340262731 245850922 / 78256779 165707065 / 52746197 5419351 / 1725033 31369698 / 9985285 1146408 / 364913 3126535 / 995207 312689 / 99532 208341 / 66317 355 / 113 31218 / 9937 355 / 113 2818 / 897 22 / 7 311 / 99 === Subject: Re: Pi Reward?! posting-account=Z9v8GgkAAAB54xGwWI4tZlTl3U6kju-g Gecko/2008052906 Firefox/3.0,gzip(gfe),gzip(gfe) On 16 mar, 03:18, Phil Carmody whose numerator and denominator are both primes, such that the > absolute value of the product of the denominator and the error > is minimised. That should help keep people away from continued > fractions, which are the magic bullet that's always rolled out > in situations like this. Phil With primes I found 211891/67447 = 3.141592657939 Following the Number Theory tradition the Quality Index must be : 1 / (Error x denominator^2) In this case: 1 / (4.3 10^(-9) x 67447^2) = 0.0505 355/113 have a Q.I = 294 Ludovicus === Subject: representing objects posting-account=M0xLOQoAAADoXlH-AVLsqNcnqbUe1dX0 AppleWebKit/525.19 (KHTML, like Gecko) Chrome/1.0.154.48 Safari/525.19,gzip(gfe),gzip(gfe) Suppose one wants to represent 29 objects in which each object is represented as a binary string of length n, which consists of k 0's and m 1's, and n = k + m. Find n, k, and m, where n has the smallest possible value. This would be (n!/(m!*k!)) >= 29 and n = m + k, therefore, ((m+k)!)/(m!*k!) >= 29 But how could this be simplified further to find two unknowns? -- aegis === Subject: Re: representing objects >Suppose one wants to represent 29 objects >in which each object is represented as a >binary string of length n, which consists of >k 0's and m 1's, and n = k + m. Find n, k, >and m, where n has the smallest possible >value. This would be (n!/(m!*k!)) >= 29 >and n = m + k, therefore, >((m+k)!)/(m!*k!) >= 29 But how could this be simplified >further to find two unknowns? Don't eliminate n since that's what you're trying to minimize. You want the least n such that (n choose k) >= 29 for some k, 0 <= k <= n. Thus, in Pascal's triangle, you want the first row which has at least one entry 29 or above. This happens for n = 7, using k=3, m=4. Alternatively, note that for fixed n, (n choose k) is maximized using k = n/2 or k = (n-1)/2 according as n is even or odd. Thus, the desired value of n is either the first even n such that (n choose (n/2)) >= 29, or the first odd n such that (n choose (n-1)/2) >= 29, whichever value of n happens first. The first even n is n = 8, using k = m = 4, and the first odd n is n = 7. using k = 3, m = 4. Thus, take the latter. quasi === Subject: Re: Compiling Undeniable Evidence of Einstein Worship posting-account=PTS84AoAAACr67p51zvy0Hlr3LkoIUcc 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022; InfoPath.2),gzip(gfe),gzip(gfe) > That is why Gisse is a college drop-out. His mind is totally fvcked > up. It does not how any degrees Einstein had. Einstein was still a > nitwit, a plagiarist, and a liar. Put a sock in it , Mother-er. Get a ing life. Become a > ing whore. Contribute to society for once in your ing life. This sounds like one fvcking mad Einstein Dingleberry. After Stockbauer throws his fit, the reality remains that Einstein was still a nitwit, still a plagiarist, and still a liar. Not even Einstein Dingleberries can change the history. === Subject: Re: Compiling Undeniable Evidence of Einstein Worship posting-account=lBRURwoAAAB_-Q_b04pGziaymfr5yRFx Gecko/2009021910 Firefox/3.0.7,gzip(gfe),gzip(gfe) > Ah. So good of you to judge. Yes it is. The whole world is certainly blessed. The spiritual man makes judgments about all things, but he himself is not subject to any man's judgment (1 Corinthians 2:15). Shubee === Subject: Re: Prime Sums In A Grid <3km6q4paigmqelbre63q8pm7q03jrt7e9o@4ax.com> <87k57gs489.fsf@nonospaz.fatphil.org> <8n77q4h5gbts3leik65mh5ipflf1cjchtf@4ax.com> <3nt7q49msu5s2eftlbqcdaato22naflq9b@4ax.com >> 25 24 23 22 19 (113) >> 21 20 18 17 13 (89) >> 16 15 14 12 2 (59) >> 11 9 8 7 6 (41) >> 10 5 4 3 1 (23) >>( 83) ( 73) ( 67) ( 61) ( 41) >>There are lots more. Nice. I assume you did a computer search, but even so, it couldn't have been > just brute force. What was your strategy? It seems now that it may be doable for all n > 1, however if that's > the case, I doubt that we will be able to prove it. > I've been playing with this some more. 64 63 62 61 60 59 58 52 (479) 57 56 55 54 53 51 50 45 (421) 49 48 47 46 44 43 42 40 (359) 41 39 38 37 36 35 34 33 (293) 32 31 30 29 28 27 26 24 (227) 25 23 22 21 20 19 18 15 (163) 16 17 14 13 12 10 8 11 (101) 9 6 1 2 4 7 5 3 (37) (293) (283) (269) (263) (257) (251) (241) (223) In this case each row is the maximum possible value: sum(1..64)-sum(1..56) = 484. Therefore the first row is 479. sum(1..64)-sum(1..48) = 904. 904-479=425. Second row is 421. sum(1..64)-sum(1..40) = 1260. 1260-900=360. Third row is 359. sum(1..64)-sum(1..32) = 1552. 1552-1259=293. Fourth row is 293. sum(1..64)-sum(1..24) = 1780. 1780-1552=228. Fifth row is 227. sum(1..64)-sum(1..16) = 1944. 1944-1779=165. Sixth row is 163. sum(1..64)-sum(1..8) = 2044. 2044-1942=102. Penultimate row is 101. sum(1..64) = 2080. 2080-2043=37. Last row is 37. Note that there's no guarantee it will come out as nicely as that: For the 3x3 case: sum(1..9)-sum(1..6) = 24 => First row 23 sum(1..9)-sum(1..3) = 15. This would suggest the second row should be 13. However in that case the third row would have to be 9 which is not prime. There is a solution with second and third rows=11. At least up to 8x8 this holds (that each row has the maximum possible sum consistent with the overall constraints). Given that it took half a second to verify it for 7x7 and fifteen minutes to verify it for 8x8, I'm not hopeful of being able to verify it for 9x9. Tim. -- God said, div D = rho, div B = 0, curl E = - @B/@t, curl H = J + @D/@t, and there was light. http://www.woodall.me.uk/ === Subject: Re: Prime Sums In A Grid <3km6q4paigmqelbre63q8pm7q03jrt7e9o@4ax.com> <87k57gs489.fsf@nonospaz.fatphil.org> <8n77q4h5gbts3leik65mh5ipflf1cjchtf@4ax.com> <3nt7q49msu5s2eftlbqcdaato22naflq9b@4ax.com> Hey it's way past time for niceness as I solved the factoring > problem. I have a mathematical proof using rather basic algebra that > shows that clearly, and I've put up test code showing a basic > demonstration for a specialized case. you no solve problem you have spoof, no proof code dosen't factor either Yet math people put up headlines around the world for quantum > computing when it factored 15. world news flash: JSH factors 15 system. In that class system they do not recognize people like me, so > they refuse to recognize major research results from me. you have 'garbage' troll, eat it. ALL they can see is later having to clap or applaud or pretend to like > my research. They have nightmares about the ignominy of seeing me get > a math prize and they will not have it. Here is your yellow ribbon for factoring 15, JSH !!! They are in their own little world. Projecting again. You are in this one. no matter where you go, there you are. Why can't Iran use my mathematical research? Or North Korea? Why > can't the US government for that matter? Or Russia? Or China? because it no workie. What happens to you when everything you do on the Internet-- > EVERYTHING--is open and transparent because no one will accept that > the system is broken so the people with the power are those who are > simply using the mathematics anyway? no one? not anybody believes you at all ? In the entire world ? North Korea will accept you, and has room for you. Mathematicians are changing your life here and now. yes, they scare me. They refuse to follow rules. Re-think the evidence now. Remember I > was published in a peer reviewed mathematical journal, you have no proof of that, they say your crapaper was withdrawn, trollbus They PULLED my paper after publication. The journal SHUT-DOWN. The > hosting university, Cameron University, SCRUBBED ALL MENTION of the > journal from its website though it had been around for ten years. you have no proof of your spoof. These people are not who you think. And right now your life depends > on getting a clue. your life depends upon getting a life. > for the journal story. The mathematicians are acting with one goal in mind: stop me from > getting rewards for my research or getting elevation in their society. use thicker heals. That's it. They've made up their minds on social class issues. you failed algebra, you will never be forgiven for that. You need to make up your mind, on the facts. your facts? > James Harris === Subject: Re: Physicists are too dumb to make a better theory; Data is curve-fitted to the old theories* posting-account=rIfu6QoAAAD5nXG3h9QEE0J3dZn1U45R Gecko/2009022621 Gentoo Firefox/3.0.6,gzip(gfe),gzip(gfe) [...] Why do you have a copy of the DSM-IV on your shelf, and why are you so acquainted with its' contents? === Subject: Einstein blocked the .81g Star Trek utopia .81h that Dorn Strich envisions. Einsteinblocked the ñ Star Trek utopia î that Dorn Strich envisions. Worse, most feel that Einstein's a genius and Strich's just insane. To a man, we hang on to our hopes and dreams .81 no matter how tenuous. Gamblers have their casinos, Christians their churches .81 Sci-Fi enthusiasts have Sci.Physics. === Subject: =?windows-1252?Q?Re=3A_Einstein_blocked_the_=93_Star_Trek_utopia_=94_th?= =?windows-1252?Q?at_Dorn_Strich_envisions=2E?= posting-account=-Yr6sAoAAAArJwgdHLA4MBxm4oIdzdtZ InfoPath.1; .NET CLR 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30; .NET CLR 3.0.04506.648; MS-RTC LM 8; .NET CLR 3.0.4506.2152; .NET CLR 3.5.30729),gzip(gfe),gzip(gfe) (NetCache NetApp/6.0.2P1) > Einstein[NonBreakingSpace]blocked the .81g Star Trek utopia .81h that Dorn Strich envisions. > Worse, most feel that Einstein's a genius and Strich's just insane. To a man, we hang on to our hopes and dreams .81 no matter how tenuous. > Gamblers have their casinos, Christians their churches > .81 Sci-Fi enthusiasts have Sci.Physics. You have that obviously reversed. You forget who has alogia written all over his theory. You forget who has schizophrenia running in his genes. And you forget who the writers of star trek base their script on. Einstein, einstein, einstein. === Subject: Re: Factoring problem solved, remaining issues resolved >'My Struggle', by James Harris. Tut, tut. A bit naughty there Marcus. You are in danger of losing the game under Godwin's Law. rossum === Subject: Re: Test run results (was Re: JSH: Factoring solution test run, Fermat primes) >Posters who post a lot in my threads have their own agenda. Drama Queen. === Subject: Re: Test run results (was Re: JSH: Factoring solution test run, Fermat primes) >My God...what the hell did he do? My implementation of the latest >factoring method described on his blog would take at most 3n+3 GCD >computations on 2^n+1. The algorithm on his blog is actually efficient, and only has one little >drawback--the numerous integers it can't factor at all. My algorithm fails to factor at a *much* higher rate.... -- Guy Macon === Subject: Re: Test run results (was Re: JSH: Factoring solution test run, Fermat primes) <49BF6F40.9FEBCB4F@tesco.net> factoring method described on his blog would take at most 3n+3 GCD > computations on 2^n+1. Having looked at it a little further, it probably comes as no surprise that the code has multiple bugs. I corrected three obvious ones, but there are a few more at least. I haven't bothered to look much further as I'm sure JSH will update with an adjusted algorithm having a whole new set of bugs in its implementation soon. - Tim === Subject: Re: Test run results (was Re: JSH: Factoring solution test run, Fermat primes) <49BF6F40.9FEBCB4F@tesco.net> not the other way around. This is evident by the fact that 2^29+1 only takes > 429 checkGCD calls, and 2^21+1 taking 31757. It should scale with the size \ > of the number, the bigger the number the more calls to the checkGCD. I haven't had time yet to properly check the code - all I did was add a counter to the checkGCD function to get a quick overview of the number of operations. If he's attempting to factor recursively then the number of calls should be expected to fluctuate greatly. - Tim === Subject: Re: Test run results (was Re: JSH: Factoring solution test run, Fermat primes) antony.clements@bigpond.com says... Using the most recent version he has put up on the web as of posting > time: > 2^29+1 takes only 429 checkGCD calls finding the factor 3, but > fails to find the small prime factor 59. > 2^21+1 is rather slower: it takes 31757 checkGCD operations but at > least gets all factors 3^2, 43, and 5919. > 2^16+1 is asserted to be prime after 3606 checkGCD operations. > > It looks quite a bit slower than trial division, and less reliable. > > - Tim This could be chalked up to 1 of 3 things, the language used (this is a > timing reference not a reference to checkGCD operations as different > languages will process the data at varying levels of efficiency), poor > algorithm design (my opinion from what you posted), and the fact that it's \ > not a final release. Going by your post Tim, it seems that James has things a little backwards. > It is the smaller numbers that should be taking less calls to the checkGCD, > not the other way around. This is evident by the fact that 2^29+1 only takes > 429 checkGCD calls, and 2^21+1 taking 31757. It should scale with the size \ > of the number, the bigger the number the more calls to the checkGCD. If it can't find every factor of every prime in the test data it's broken. > Simple as that. > Presuming you mean to say If it can't find every factor of every _number_ in the test data it's broken., your comment is a little unfair. For example, consider an algorithm was developed that could find for any N, the first prime factor larger than 10^6 (if it exists), and could do so in a resonable time. Although it is not explicitly a complete factoring algorithm it would be rediculous to dismiss it as 'broken' as with the addition of around 80,000 explicit checks for smaller prime factors, you would have an algorithm that could factorise any N in a few times the 'reasonable time'. Not, of course, that JSH's algorithm has anything to do with this. === Subject: Re: Test run results (was Re: JSH: Factoring solution test run, Fermat primes) > antony.clements@bigpond.com says... >> Using the most recent version he has put up on the web as of posting >> time: >> 2^29+1 takes only 429 checkGCD calls finding the factor 3, but >> fails to find the small prime factor 59. _number_ in the test data it's broken., your comment is a little > unfair. For example, consider an algorithm was developed that could find > for any N, the first prime factor larger than 10^6 (if it exists), and > could do so in a resonable time. Although it is not explicitly a > complete factoring algorithm it would be rediculous to dismiss it as > 'broken' as with the addition of around 80,000 explicit checks for > smaller prime factors, you would have an algorithm that could factorise > any N in a few times the 'reasonable time'. Not, of course, that JSH's algorithm has anything to do with this. As james said, his data consists of primes up to and including 2n+1, as Tim \ said, it failed to find a small prime factor (I can only assume that this was not a complete test of the algorithm because to do so would have taken a \ lot longer than a day or two), but as I have said, if the algorithm fails to \ find one or more prime factor that is within the test data, it is broken. If you are a programmer (wether you are or not is irrelevant so it's an assumption) and your boss asks you to write some code to complete a task, and you return to your boss with some code that does the task... mostly... except for 1 or two instances (hence not finding the small prime factor 59), \ one of two things will happen, your boss throws it back at you and says try \ again, or you lose your job. Broken is broken no matter how you view it. === Subject: Re: Test run results >If you are a programmer (wether you are or not is irrelevant so it's an >assumption) and your boss asks you to write some code to complete a task, >and you return to your boss with some code that does the task... mostly... \ >except for 1 or two instances (hence not finding the small prime factor 59), >one of two things will happen, your boss throws it back at you and says try \ >again, or you lose your job. Broken is broken no matter how you view it. Although I agree 100% in the case of an alleged factoring solution, in the general case sometimes broken is *not* broken no matter how you view it. For example, look at [ http://www.joelonsoftware.com/items/2008/02/19.html ] and search for either BIFF or 1904 to find the spot I am talking about. Broken? Yes. Lose your job? Only if you insist on writing 100% correct code no matter what. -- Guy Macon === Subject: Re: Test run results (was Re: JSH: Factoring solution test run, Fermat primes) posting-account=fwSgtAkAAACFnX70ssKwbvm9_oCZVHrx Gecko/2009021910 Firefox/3.0.7,gzip(gfe),gzip(gfe) > So how many operations (either on average or worst case) are > required to factor numbers on the order of 2^29? > Using the most recent version he has put up on the web as of posting > time: > 2^29+1 takes only 429 checkGCD calls finding the factor 3, but > fails to find the small prime factor 59. > 2^21+1 is rather slower: it takes 31757 checkGCD operations but at > least gets all factors 3^2, 43, and 5919. > 2^16+1 is asserted to be prime after 3606 checkGCD operations. > It looks quite a bit slower than trial division, and less reliable. My God...what the hell did he do? My implementation of the latest > factoring method described on his blog would take at most 3n+3 GCD > computations on 2^n+1. The algorithm on his blog is actually efficient, and only has one little > drawback--the numerous integers it can't factor at all. > Hmm. That raises an interesting real-math question: What numbers does it work for? What characteristics must the numbers have in order for them to factor with JSH's hammer? === Subject: Re: Test run results (was Re: JSH: Factoring solution test run, Fermat primes) posting-account=Cbgh4AoAAAAr0dt1RqLOClWCyUWii2fU Gecko/2009030422 Ubuntu/8.04 (hardy) Firefox/3.0.7,gzip(gfe),gzip(gfe) > So how many operations (either on average or worst case) are > required to factor numbers on the order of 2^29? > Using the most recent version he has put up on the web as of posting > time: > 2^29+1 takes only 429 checkGCD calls finding the factor 3, but > fails to find the small prime factor 59. > 2^21+1 is rather slower: it takes 31757 checkGCD operations but at > least gets all factors 3^2, 43, and 5919. > 2^16+1 is asserted to be prime after 3606 checkGCD operations. > It looks quite a bit slower than trial division, and less reliable. > My God...what the hell did he do? My implementation of the latest > factoring method described on his blog would take at most 3n+3 GCD > computations on 2^n+1. > The algorithm on his blog is actually efficient, and only has one little > drawback--the numerous integers it can't factor at all. Hmm. That raises an interesting real-math question: What numbers > does it work for? What characteristics must the numbers have > in order for them to factor with JSH's hammer? These rare numbers should thus be dubbed JSH nails. :) hagman === Subject: Re: Test run results (was Re: JSH: Factoring solution test run, Fermat primes) posting-account=fwSgtAkAAACFnX70ssKwbvm9_oCZVHrx Gecko/2009021910 Firefox/3.0.7,gzip(gfe),gzip(gfe) > So how many operations (either on average or worst case) are > required to factor numbers on the order of 2^29? > Using the most recent version he has put up on the web as of posting > time: > 2^29+1 takes only 429 checkGCD calls finding the factor 3, but > fails to find the small prime factor 59. > 2^21+1 is rather slower: it takes 31757 checkGCD operations but at > least gets all factors 3^2, 43, and 5919. > 2^16+1 is asserted to be prime after 3606 checkGCD operations. > It looks quite a bit slower than trial division, and less reliable. > My God...what the hell did he do? My implementation of the latest > factoring method described on his blog would take at most 3n+3 GCD > computations on 2^n+1. > The algorithm on his blog is actually efficient, and only has one little > drawback--the numerous integers it can't factor at all. > Hmm. That raises an interesting real-math question: What numbers > does it work for? What characteristics must the numbers have > in order for them to factor with JSH's hammer? These rare numbers should thus be dubbed JSH nails. :) > Sounds like a good name to me :) === Subject: Re: Test run results (was Re: JSH: Factoring solution test run, Fermat primes) > If it can't find every factor of every prime in the test data it's broken. \ > Simple as that. For some reason I had the idea that a prime number had no factors (apart from itself and 1). More study needed. === Subject: Re: Test run results (was Re: JSH: Factoring solution test run, Fermat primes) > For some reason I had the idea that a prime number had no factors (apart > from itself and 1). More study needed. A prime is only divisible by itself and 1, multiplication is the opposite of \ division so it stands to reason that it applies to multiplication as well (i'm not a mathmetician so don't take my word for it). Many odd numbers have \ prime factors (15 = 3*5 for example), but 15 is not a prime... in fact, the \ only prime has a 5 as the final digit is 5 itself. As for your comment about wether primes have factors other than itself and 1, I personally have no idea. I am making an assumption (probably falsely) because someone mentioned 2^29 - 1 which appears to follow the equation of Mersenne primes which is Mn = 2n - 1. === Subject: Re: Test run results (was Re: JSH: Factoring solution test run, Fermat primes) > For some reason I had the idea that a prime number had no factors (apart >> from itself and 1). More study needed. A prime is only divisible by itself and 1, multiplication is the opposite > of division so it stands to reason that it applies to multiplication as > well (i'm not a mathmetician so don't take my word for it). Many odd > numbers have prime factors (15 = 3*5 for example), but 15 is not a > prime... in fact, the only prime has a 5 as the final digit is 5 itself. As for your comment about wether primes have factors other than itself and \ > 1, I personally have no idea. I am making an assumption (probably falsely) \ > because someone mentioned 2^29 - 1 which appears to follow the equation of \ > Mersenne primes which is Mn = 2n - 1. disregard this post because we are talking about 2n + 1 === Subject: Re: Test run results (was Re: JSH: Factoring solution test run, Fermat primes) If it can't find every factor of every prime in the test data it's broken. > Simple as that. > For some reason I had the idea that a prime number had no factors (apart > from itself and 1). More study needed. > Damn tricky finding that '1' sometimes... -- Mike === Subject: Re: Test run results (was Re: JSH: Factoring solution test run, Fermat primes) > I'm a bit confused here. Do you mean Primes have equations, or > Primes have factors ? Brian Chandler The equation for a Mersenne prime is 2n-1, this is an equation of one particular subset of prime number, within this subset there are prime factors. This statement is also true for the equation of Fermat primes of 2^2n+1. A very very short list of some of the different types of prime numbers is: Weiferich primes Wilsone primes Wall-Sun-Sun primes Wolstenholme primes Unique primes Newman-Shanks-Williams primes Smarandache-Wellin primes Wagstaff primes Supersingular primes And the list goes on and on and on. === Subject: Re: Test run results (was Re: JSH: Factoring solution test run, Fermat primes) > Is that an equation or an expression? Equations are made up of one or more expressions, if you want to go by what \ wikipedia says, then Mn = 2n - 1. Is that more satisfactory for you? === Subject: Re: Test run results (was Re: JSH: Factoring solution test run, Fermat primes) >> Is that an equation or an expression? Equations are made up of one or more expressions, if you want to go by > what wikipedia says, then Mn = 2n - 1. Is that more satisfactory for you? Looking over the thread we are in fact talking about 2n + 1 which is not a Mersenne prime, so disregard most of my posts. === Subject: proof that separable inner product spaces have orthonormal bases posting-account=JTrNRgoAAABuGIzwXYnlAgkRTDpv8_sI .NET CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) Would appreciate it if someone can point to (e.g. online link) proof of the fact that separable inner product spaces have orthonormal bases. === Subject: Re: proof that separable inner product spaces have orthonormal bases posting-account=Cbgh4AoAAAAr0dt1RqLOClWCyUWii2fU Gecko/2009030422 Ubuntu/8.04 (hardy) Firefox/3.0.7,gzip(gfe),gzip(gfe) > Would appreciate it if someone can point to (e.g. online link) proof > of the fact that separable inner product spaces have orthonormal bases. Apply Gram-Schmidt to a dense sequence? === Subject: Lebesgue measure. Please help. Let A be a Lebesgue measurable subset of R. If m(A # A+d)=0, for every d belonging to an everywhere dense subset of A, then it must be the case that \ m(A)=0 or m(A')=0. Here m denotes Lebesgue measure, # denotes symmetric difference of sets and \ ' denotes set complement. Johnnie === Subject: Re: Lebesgue measure. Please help. <9732081.1237429340733.JavaMail.jakarta@nitrogen.mathforum.org>, > figure it out. Could someone please help me? Let A be a Lebesgue measurable subset of R. If m(A # A+d)=0, for every d > belonging to an everywhere dense subset of A, everywhere dense subset of A - do you mean this subset is dense in R? > then it must be the case that > m(A)=0 or m(A')=0. Here m denotes Lebesgue measure, # denotes symmetric difference of sets and ' > denotes set complement. Johnnie CC: Stephen J. Herschkorn === Subject: Connectedness again Let f be a nonnegative, continuous function on [0, 1] such that f vanishes at 0 and 1 but nowhere else. Conjecture: {(x, y) in [0, 1]^2: f(x) = f(y)} is connected. Ideas? I am not sure that the strict positivity of f on (0, 1) is needed. -- Stephen J. Herschkorn sjherschko@netscape.net Math Tutor on the Internet an din Central New Jersey === Subject: Re: Connectedness again Stephen J. Herschkorn a .8ecrit : > Let f be a nonnegative, continuous function on [0, 1] such that f > vanishes at 0 and 1 but nowhere else. Conjecture: > {(x, y) in [0, 1]^2: f(x) = f(y)} is connected. Ideas? I am not sure that the strict positivity of f on (0, 1) is needed. f piecewise linear with: f(1/6) = 2, f(1/5) = 1, f(1/4) = 3, f(1/3) = 2, f(1/2) = 3 f(0) = f(1) = 0 The point (x,y) = (1/6, 1/3) is an isolated point of the set {(x, y) in [0, 1]^2: f(x) = f(y)} because if x moves, f(x) decreases (local max), but if y moves, f(y) increases (local min). -- Nico. CC: Stephen J. Herschkorn === Subject: Re: Connectedness again > Stephen J. Herschkorn a .8ecrit : > Let f be a nonnegative, continuous function on [0, 1] such that >> f vanishes at 0 and 1 but nowhere else. Conjecture: >> {(x, y) in [0, 1]^2: f(x) = f(y)} is connected. Ideas? >> I am not sure that the strict positivity of f on (0, 1) is needed. > f piecewise linear with: > f(1/6) = 2, f(1/5) = 1, f(1/4) = 3, f(1/3) = 2, f(1/2) = 3 > f(0) = f(1) = 0 The point (x,y) = (1/6, 1/3) is an isolated point of the set > {(x, y) in [0, 1]^2: f(x) = f(y)} because if x moves, f(x) decreases > (local max), but if y moves, f(y) increases (local min). |>Let f be a nonnegative, continuous function on [0, 1] such that f >|>vanishes at 0 and 1 but nowhere else. Conjecture: >{(x, y) in >|[0, 1]^2: f(x) = f(y)} is connected. >... >|If f has only finitely many extrema, I think one can chug through a >|formal proof of the above argument. if f has a local maximum and a local minimum with the same value then >that gives an isolated point in the subspace. > -- Stephen J. Herschkorn sjherschko@netscape.net Math Tutor on the Internet and in Central New Jersey === Subject: Re: Connectedness again posting-account=SvltewoAAAAi7TTYrD3mAaLUHzDiF2d1 Gecko/2009021910 Firefox/3.0.7,gzip(gfe),gzip(gfe) >Let f be a nonnegative, continuous function on [0, 1] such that f >vanishes at 0 and 1 but nowhere else. Conjecture: >{(x, y) in [0, 1]^2: f(x) = f(y)} is connected. I don't think so. Say f is piecewise linear with f(0) = 0, > f(1/4) = 3, f(1/2) = 1, f(3/4) = 2, f(1) = 0. Seems like > a counterexample, although I haven't written down > the details. > Not a counter-example. Since it is too hard for me to visualize this, I write a little program to generate a image for the set of f(x) = f(y). Aside from the diagonal (0,0) -> (1,1), the set contains 12 more line segments, 6 of them forms a path from (1,0) to (0,1) and intersect the main diagonal at (1/4,1/4). 6 of them forms a loop and intersect the main diagonal on (1/2,1/2) and (3/4,3/4). For general f(.), it seems the key is to prove/disprove every member of the set f(x) = f(y) is 'connected' to the main diagonal x = y. CC: Stephen J. Herschkorn === Subject: Re: Connectedness again > > >Let f be a nonnegative, continuous function on [0, 1] such that f >vanishes at 0 and 1 but nowhere else. Conjecture: >{(x, y) in [0, 1]^2: f(x) = f(y)} is connected. > >I don't think so. Say f is piecewise linear with f(0) = 0, >>f(1/4) = 3, f(1/2) = 1, f(3/4) = 2, f(1) = 0. Seems like >>a counterexample, although I haven't written down >>the details. >> >Not a counter-example. Since it is too hard for me to visualize this, >I write a little program to generate a image >for the set of f(x) = f(y). Aside from the diagonal (0,0) -> (1,1), the set >contains 12 more line segments, 6 of them forms >a path from (1,0) to (0,1) and intersect the main >diagonal at (1/4,1/4). 6 of them forms a loop and >intersect the main diagonal on (1/2,1/2) and (3/4,3/4). For general f(.), it seems the key is to prove/disprove >every member of the set f(x) = f(y) is 'connected' to >the main diagonal x = y. > > Achille is correct; David's example is not a counter-example. Here is how I see it: Consider the motion of the line y = a by varying a. Intersection of this line with the graph of f give us points in C. Start the line at a = 0, where it intersects the graph at x = 0 and 1. This gives us the points (0, 1) and (1, 0) in C. Henceforth, I will only refer to the points of C where x <= y. Start by increasing a (i.e., by moving the line up) from 0 to 2. That generates, in C, a line segment from (0, 1) to (x1, 3/4) for some x1 < 1/4. Now decrease a to 1, generating a line segment from (x1, 3/4) to (x2, 1/2), where x2 < x1. Now increase a to 3, generating a line segment from (x2, 1/2) to (1/4, 1/4). It seems to me that in this way, one can generate a path in C from any point therein to the diagonal. Or, if you prefer, from (0, 1) to any point. If f has only finitely many extrema, I think one can chug through a formal proof of the above argument. The problem arises, for example, with functions that are nowhere differentiable over an interval. For a particular tough example, consider a path of the absolute value of a Brownian bridge. -- Stephen J. Herschkorn sjherschko@netscape.net Math Tutor on the Internet and in Central New Jersey === Subject: Re: Connectedness again Originator: jdolan@math.UUCP (James Dolan) |>Let f be a nonnegative, continuous function on [0, 1] such that f |>vanishes at 0 and 1 but nowhere else. Conjecture: >{(x, y) in |[0, 1]^2: f(x) = f(y)} is connected. ... |If f has only finitely many extrema, I think one can chug through a |formal proof of the above argument. if f has a local maximum and a local minimum with the same value then that gives an isolated point in the subspace. -- jdolan@math.ucr.edu === Subject: Re: Infinite odds > If the odds of something is one out of infinity if it happens once it > will never repeat and other result will never be exhausted. I'll give you a clue. What is 1/n as n goes to infinity? Approaching the closest thing to zero. === Subject: Re: Infinite odds posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > I'll give you a clue. What is 1/n as n goes to infinity? > Approaching the closest thing to zero. Apparently, it's the same infinitesimal that MR has mentioned several times in the .999... (or the .9 repeating thread). To MR, 1-.999... is not zero, as in standard analysis, but a nonzero infinitesimal, and it's also apparently the probability of an event such that if it happens once it will never repeat and other result will never be exhausted. === Subject: f_n(x)=x(x - 1/n). Show that f_n -> x^2 mu-almost everywhere Let f_n(x)=x(x - 1/n). Let mu be the Lebesgue measure. Domain of the sequence of functions f_n is all reals. Show that f_n -> x^2 mu-almost everywhere. An outline of my approach: Need to show f_n converges to f pointwise on R - S where S is a set with Lebesgue measure 0. Proof: Let S=Q, the rationals. Need to show f_n -> x^2 on R - Q. |f_n(x) - f| = |x^2 - x/n - x^2| = |- x/n | = |x/n| |x/n| = |x| * |1/n|. lim(n->oo) 1/n = 0 so for any epsilon > 0 we can have any x in R - Q s.t. |f_n(x) - f| < epsilon. We can conclude that f_n - > x^2 mu almost everywhere. Q.E.D. === Subject: Re: f_n(x)=x(x - 1/n). Show that f_n -> x^2 mu-almost everywhere > Let f_n(x)=x(x - 1/n). Let mu be the Lebesgue measure. > Domain of the sequence of functions f_n is all reals. > Show that f_n -> x^2 mu-almost everywhere. An outline of my approach: Need to show > f_n converges to f pointwise on R - S where S > is a set with Lebesgue measure 0. Proof: > Let S=Q, the rationals. Need to show > f_n -> x^2 on R - Q. > |f_n(x) - f| = |x^2 - x/n - x^2| = |- x/n | = |x/n| |x/n| = |x| * |1/n|. lim(n->oo) 1/n = 0 > so for any epsilon > 0 > we can have any x in R - Q s.t. |f_n(x) - f| < epsilon. > We can conclude that f_n - > x^2 mu almost everywhere. Q.E.D. Correction! We cannot simply choose any x in R - S. Any x in R - S such that x < n will work. So, we have pointwise convergence but since x is unbounded we do not have uniform convergence. f_n is mu-almost everywhere but not mu-almost uniform. === Subject: Regular Pentagon posting-account=em7BlwoAAADOhRXv8AilqN49TjuokE7m Trident/4.0; .NET CLR 2.0.50727; FDM; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) How many ways you can get .bc (Phi or Golden Ratio) from a Regular Pentagon? Types of construction allowed: Diagonals (of original Pentagon) , In- circle and Circumcircle === Subject: Re: Regular Pentagon posting-account=Cbgh4AoAAAAr0dt1RqLOClWCyUWii2fU Gecko/2009021910 Firefox/3.0.7 (.NET CLR 3.5.30729),gzip(gfe),gzip(gfe) > How many ways you can get .85 (Phi or Golden Ratio) from a Regular > Pentagon? > Types of construction allowed: Diagonals (of original Pentagon) , In- > circle and Circumcircle Since the diagonals produce a smaller pentagon (and P occurs already outside that smaller pentagon), the number of ways to find Phi is infinite === Subject: Inverse matrix of a submatrix This probably doesn't make sense, but I am wondering if there is an easier method of obtaining an inverse matrix of a matrix which is a sub-matrix of another matrix, whose inverse matrix is known. Actually I have an unitary Vandermonde matrix (a DFT matrix: http://en.wikipedia.org/wiki/Discrete_Fourier_transform#The_unitary_DFT) and \ I want to find the inverse of a sub-matrix of this matrix. Unfortunately \ that submatrix does not include the first row, so the Vandermonde property is destroyed. Any suggestion is appreciated. === Subject: Re: Inverse matrix of a submatrix posting-account=IBUqVwoAAADepmzxVr9iEYD5Z0A483SY Gecko/20070530 Fedora/1.5.0.12-1.fc5 Firefox/1.5.0.12,gzip(gfe),gzip(gfe) On 19 Mar, 05:24, Maduranga Liyanage The probability the second roll in a backgammon game is double-6 > is 1/36. What is the probability the last roll is double-6? Hmm, that's tougher and I think it's got to depend on one's endgame >strategy. Even if you ignore strategy, I think you have to enumerate >all the positions of stones from which a double-6 would win the game, >then for each one work out the chance that any game-winning numbers >would be rolled. Oh, and the possibility of being blocked by points >made by the opponent has to be figured in too. Pretty much intractable >without some significant computer assistance, I'd think. Oh, and not only endgame strategy, but doubling and acceptance. You're >less likely to double and end the game if your opponent just rolled a >double 6, right? If we eliminate the doubling rule (since I've never played for money or played with that rule), I wonder if that makes it easier. My intuition is that under those conditions, double-6 is about twice as likely as a last roll (compared to its probability as a first roll). I'm picturing that the guy who's about to win has all of his pieces in safely in his, um, whatever-it's-called -- those last 6 spots before you can, um, is it called bearing off? (Dang, I could use some terminology.) Anyway, he's got all his pieces in the last 6 squares there, and he's getting them off the board roughly two at a time. And now it's the roll that might be his last roll. If he has 1 or 2 pieces left, he can win on the spot without doubles. If he has 3 or 4 pieces left, he will only win on the spot with doubles. And, actually, the fact that it's double SIXES makes it even more likely he'll win on the spot. It seems to me that the average pair of non-doubled dice has a 50%-or-less chance of clearing off two randomly placed pieces. For example, a 4-3 won't clear off a 5-2, and vice versa. So now, I'm thinking that the odds of double sixes being the final roll is closer to four times as likely. In any case, somewhere between 2/36 and 4/36. Can someone just program their backgammon computer to play 10,000 consecutive games, so we can get some brute force numbers? === Subject: Re: a cute proposition >Consider a sequence of 3 coin tosses. >There are 8 such: HHT, HTH, etc. >And each is equiprobable, right? Let's play a game, betting $1, even money. >You select one of those sequences, then I select >one of the remaining 7. Toss a coin repeatedly, >whoever's sequence occurs first, wins. >Do you feel lucky? Step right up... To be precise: the coin is not tossed in separate >sets of 3, resetting each time neither player >wins. Rather, a 'sliding window' is employed; >if neither sequence occurs in the first 3 tosses, >check the 2nd... 4th, then 3rd.. 5th, etc. >Left as an exercise for the student. If I bet on HHH, I assume you'll bet on THH. Then, the only chance I'd have of winning (1/8) is if the first three flips are all heads. Otherwise you'll eventually win. I wouldn't be surprised to find that you can intercept any bet I might lay down. I wonder what you'd play if I took HTH. My guess is HHT, since, even if mine were about to come up, you'd still have a 50-50 chance of intercepting it (if an H rather than a T came up before mine). === Subject: Re: a cute proposition posting-account=oTDIagkAAACTxHurtPutBWvNQS8ZCNO9 Gecko/2009021910 Firefox/3.0.7,gzip(gfe),gzip(gfe) > For example, HHT beats HTH 2/3 of the time. Done by hand, verified \ by > computer. It turns out this game has been studied by Penney in 1969: http://en.wikipedia.org/wiki/Penney%27s game Book of Mathematics, pp 302-309, available on Google Books at http://tinyurl.com/dmluv5 where Gardner describes Conway's algorithm (damn you Conway! why must you be so clever!?!) to determine in general which of two n-length sequences of coin flips, A or B, are more likely to occur first. === Subject: Re: a cute proposition <87vdq63ooc.fsf@phiwumbda.org> if Player 1 chooses HHT, then >> Player 2 would choose THT, right? No, Player 2 would choose THH. (Or maybe HHH?) Definitely THH. If you draw a little diagram with vertices corresponding to 2-flip sequences and links to the next, a game consists of randomly hopping along the links, ending when a link corresponding to a player's sequence is taken. In that form it is clear that THH cuts off access to HH and therefore to HHT. Thus player 1 wins iff the flips start HH. - Tim === Subject: Re: a cute proposition <87vdq63ooc.fsf@phiwumbda.org> posting-account=oTDIagkAAACTxHurtPutBWvNQS8ZCNO9 Gecko/2009021910 Firefox/3.0.7,gzip(gfe),gzip(gfe) >> if Player 1 chooses HHT, then >> Player 2 would choose THT, right? > No, Player 2 would choose THH. (Or maybe HHH?) Definitely THH. If you draw a little diagram with vertices > corresponding to 2-flip sequences and links to the next, a game > consists of randomly hopping along the links, ending when a link > corresponding to a player's sequence is taken. In that form it is clear that THH cuts off access to HH and therefore > to HHT. Thus player 1 wins iff the flips start HH. > In this form, it is known as a De Bruijn graph; see interesting analysis at: http://research.cs.queensu.ca/~dawes/coins.html === Subject: Re: a cute proposition >> Consider a sequence of 3 coin tosses. >> There are 8 such: HHT, HTH, etc. >> And each is equiprobable, right? >> Let's play a game, betting $1, even money. >> You select one of those sequences, then I select >> one of the remaining 7. Toss a coin repeatedly, >> whoever's sequence occurs first, wins. >> Do you feel lucky? Step right up... >> To be precise: the coin is not tossed in separate >> sets of 3, resetting each time neither player >> wins. Rather, a 'sliding window' is employed; >> if neither sequence occurs in the first 3 tosses, >> check the 2nd... 4th, then 3rd.. 5th, etc. >> Left as an exercise for the student. >> -- >> Mark Here's the full table for the probability that row beats column: HHH HHT HTH HTT THH THT TTH TTT >HHH --- 1/2 2/5 2/5 1/8 5/12 3/10 1/2 >HHT 1/2 --- 2/3 2/3 1/4 5/8 1/2 7/10 >HTH 3/5 1/3 --- 1/2 1/2 1/2 3/8 7/12 >HTT 3/5 1/3 1/2 --- 1/2 1/2 3/4 7/8 >THH 7/8 3/4 1/2 1/2 --- 1/2 1/3 3/5 >THT 7/12 3/8 1/2 1/2 1/2 --- 1/3 3/5 >TTH 7/10 1/2 5/8 1/4 2/3 2/3 --- 1/2 >TTT 1/2 3/10 5/12 1/8 2/5 2/5 1/2 --- For example, HHT beats HTH 2/3 of the time. Done by hand, verified by >computer. Interesting. Here's a variant ... Suppose the 2 players select their sequences simultaneously. What's an optimal mixed strategy? quasi === Subject: Re: a cute proposition posting-account=YUVDEAoAAADy0b0DWFp7F-O3S06gZA3W Gecko/2009030423 Ubuntu/8.10 (intrepid) Firefox/3.0.7,gzip(gfe),gzip(gfe) >> Consider a sequence of 3 coin tosses. >> There are 8 such: HHT, HTH, etc. >> And each is equiprobable, right? >> Let's play a game, betting $1, even money. >> You select one of those sequences, then I select >> one of the remaining 7. Toss a coin repeatedly, >> whoever's sequence occurs first, wins. >> Do you feel lucky? Step right up... >> To be precise: the coin is not tossed in separate >> sets of 3, resetting each time neither player >> wins. Rather, a 'sliding window' is employed; >> if neither sequence occurs in the first 3 tosses, >> check the 2nd... 4th, then 3rd.. 5th, etc. >> Left as an exercise for the student. >> -- >> Mark >Here's the full table for the probability that row beats column: > HHH HHT HTH HTT THH THT TTH TTT >HHH --- 1/2 2/5 2/5 1/8 5/12 3/10 1/2 >HHT 1/2 --- 2/3 2/3 1/4 5/8 1/2 7/10 >HTH 3/5 1/3 --- 1/2 1/2 1/2 3/8 7/12 >HTT 3/5 1/3 1/2 --- 1/2 1/2 3/4 7/8 >THH 7/8 3/4 1/2 1/2 --- 1/2 1/3 3/5 >THT 7/12 3/8 1/2 1/2 1/2 --- 1/3 3/5 >TTH 7/10 1/2 5/8 1/4 2/3 2/3 --- 1/2 >TTT 1/2 3/10 5/12 1/8 2/5 2/5 1/2 --- >For example, HHT beats HTH 2/3 of the time. Done by hand, verified by >computer. Interesting. Here's a variant ... Suppose the 2 players select their sequences simultaneously. What's an > optimal mixed strategy? quasi It's helpful to convert the matrix into payoffs (subtract 1/2 and multiply by 120): --- HHH HHT HTH HTT THH THT TTH TTT HHH 0 0 -12 -12 -45 -10 -24 0 HHT 0 0 20 20 -30 15 0 24 HTH 12 -20 0 0 0 0 -15 10 HTT 12 -20 0 0 0 0 30 45 THH 45 30 0 0 0 0 -20 12 THT 10 -15 0 0 0 0 -20 12 TTH 24 0 15 -30 20 20 0 0 TTT 0 -24 -10 -45 -12 -12 0 0 So it seems the optimal strategy would be to take either HTT or THH, with equal probability. HHH and TTT are losers (meaning win chances are at most one-half) from the beginning. After removing them (for both players), HTH and THT are losers. After removing them, HHT and TTH are losers, since any strategy would be symmetric and have to average the two. That leaves HTT and THH. === Subject: close form I know how to solve the recurrence relation of T(n) - 2T(n-1) = 3 * 2^n T(0) = 1 (boundary condition) homogenous solution = T_h(n) = A * 2^n particular solution = T_p(n) = 3n (2)^n T(n) = T_h(n) + T_p(n) = 3n (2)^n + (2)^n = 2^n (3n + 1) but if it is in this form n*T(n) + n*T(n-1) - T(n-1) = 2^n T(0) = 273 (boundary condition) how do I solve it? === Subject: Re: close form sci.math,alt.math,alt.math.undergrad: > I know how to solve the recurrence relation of > T(n) - 2T(n-1) = 3 * 2^n > T(0) = 1 (boundary condition) > homogenous solution = > T_h(n) = A * 2^n > particular solution = > T_p(n) = 3n (2)^n > T(n) = T_h(n) + T_p(n) > = 3n (2)^n + (2)^n > = 2^n (3n + 1) > but if it is in this form > n*T(n) + n*T(n-1) - T(n-1) = 2^n > T(0) = 273 (boundary condition) > how do I solve it? Let S(n) = n * T(n), get a recurrence for S(n), and solve that. Brian === Subject: Extension Assume A closed subset X f:A -> Y continuous Can f be continuously extended to all of X? Of course if X is normal and Y is the reals. Does this theorem have any generalizations to a wider collection of topological spaces? === Subject: Re: Extension > Assume > A closed subset X > f:A -> Y continuous Can f be continuously extended to all of X? Of course if X is normal and Y is the reals. > Does this theorem have any generalizations > to a wider collection of topological spaces? No. Let X be a topological space and assume that each continuous function from a closed subset A into the reals can be extended to X. Let A and B be closed disjoint subsets of X and define _f_ from A U B into the reals by f(A) = {0} and f(B) = {1}. Then _f_ can be extended to X and this extension separates A from B. Therefore, X is normal. Jose Carlos Santos === Subject: Re: Extension Content-ID: <20090319084006.W27955@agora.rdrop.com> --------------------------------------------------------------------- Content-ID: <20090319084006.W27955@agora.rdrop.com >> Assume >> A closed subset X >> f:A -> Y continuous Can f be continuously extended to all of X? Of course if X is normal and Y is the reals. >> Does this theorem have any generalizations >> to a wider collection of topological spaces? > No. Let X be a topological space and assume that each continuous > function from a closed subset A into the reals can be extended to X. Let > A and B be closed disjoint subsets of X and define _f_ from A U B into > the reals by f(A) = {0} and f(B) = {1}. Then _f_ can be extended to X > and this extension separates A from B. Therefore, X is normal. That's what happens when Y is restricted to R. What can happen when Y is given greater allowance? Seems it could get complicated. Let Y = discrete {0,1}. Then if f isn't constant, f can be continuously extended to all of X only if X is disconnected. Perhaps Y should be restricted to connected spaces. === Subject: the 1 by n chess board Is that true the following problem will lead to the Fibonacci sequence? T(n) = T(n-1) + T(n-2) ############################### Consider a 1 by n chessboard. suppose that we color each square of the chessboard in either red or blue. Let T(n) represent the number of possible colored chessboards in which no 2 squares that are colored red are adjacent. Find the recurrence relation for T(n). T(n) = T(n-1) + T(n-2) ############################### Question is why it lead to fibonacci squence recurrence relation? === Subject: Re: the 1 by n chess board > Is that true the following problem will lead to the Fibonacci > sequence? T(n) = T(n-1) + T(n-2) If you mean the usual Fibonacci sequence (1,1,2,3,5,8,13,...) then you also need to properly specify the initial boundary conditions. > ############################### > Consider a 1 by n chessboard. suppose that we color each square of > the chessboard in either red or blue. Let T(n) represent the number of possible colored chessboards in which > no 2 squares that are colored red are adjacent. Find the recurrence relation for T(n). T(n) = T(n-1) + T(n-2) > ############################### Question is why it lead to fibonacci squence recurrence relation? That's a cute problem, which I presume is homework. Here's a hint: * Every legal sequence (of length N) of R and B must begin with B or RB (since it cannot begin with RR). * How many sequences (of length N) begin with B ? * How many begin with RB ? If you get stuck, show us what you have done and where you got stuck. -- --------------------------- | BBB b Barbara at LivingHistory stop co stop uk | B B aa rrr b | | BBB a a r bbb | Quidquid latine dictum sit, | B B a a r b b | altum viditur. | BBB aa a r bbb | ----------------------------- === Subject: Re: Shuttle launch tonight at 7:43 EDT >It will be brodacast at this link. >>http://www.nasa.gov/multimedia/nasatv/index.html >> Or just watch NASA-TV directly. It is included in many basic >>cable/satellite TV packages. Whichever way you watch it, NASA-TV provides >>the very best launch coverage. > In Pittsburgh, PA, neither Comcast, DirecTV, nor Verizon cable have it. And > I've asked... Only The Dish has it. So I'll be watching it on my computer via space.com. NASA TV is on channel 283 on DirecTV. It's available on the family package and on all packages above that. Jay Walsh === Subject: Re: Shuttle launch tonight at 7:43 EDT >> That said, I am going to have to try this for myself. I've also been informed Venus when bright and close, >ie cresent, can be resolved, with good naked eyes, >beyond point-like. There's no record of it in any of the traditional astronomical journals, however, and lots of people try (amateurs also make a sport of spotting Venus in daytime, and trying to catch the youngest moon after New Moon.) This is another laboratory possiblity that fails in the real world, due to contrast issues, atmospheric turbulence, etc. There has been suggestion that pre-historic man may have discerned Venus as a crescent, based on some cave drawings which seem to show two crescents in the sky, but most archaelogists believe it to be two different representations of the moon. And prehistoric man may well have had better eyesight than modern man. Brian === Subject: Re: Shuttle launch tonight at 7:43 EDT > There's no record of it in any of the traditional astronomical > journals, however, and lots of people try (amateurs also make a sport > of spotting Venus in daytime, and trying to catch the youngest moon > after New Moon.) This is another laboratory possiblity that fails in the real world, > due to contrast issues, atmospheric turbulence, etc. There has been > suggestion that pre-historic man may have discerned Venus as a > crescent, based on some cave drawings which seem to show two crescents > in the sky, but most archaelogists believe it to be two different > representations of the moon. And prehistoric man may well have had > better eyesight than modern man. > One problem is that it's so bright...if you were able to look through some sort of filter to dim it down a bit maybe you could see it as a crescent at closet approach, though I doubt it, but I can't see the crescent shape even through 10x50 binoculars at closest approach due to the glare. Even through a telescope at around 50 x the crescent isn't all that big. Pat === Subject: Re: Shuttle launch tonight at 7:43 EDT <2b83s4h26rvs69n3vn9rf5ter5mverhd9f@4ax.com> posting-account=c6T4LggAAACAqdk8Zx_zXI_oErOujh3M Gecko/20050915 Firefox/1.0.7,gzip(gfe),gzip(gfe) > There's no record of it in any of the traditional astronomical > journals, however, and lots of people try (amateurs also make a sport > of spotting Venus in daytime, and trying to catch the youngest moon > after New Moon.) > This is another laboratory possiblity that fails in the real world, > due to contrast issues, atmospheric turbulence, etc. There has been > suggestion that pre-historic man may have discerned Venus as a > crescent, based on some cave drawings which seem to show two crescents > in the sky, but most archaelogists believe it to be two different > representations of the moon. And prehistoric man may well have had > better eyesight than modern man. One problem is that it's so bright...if you were able to look through > some sort of filter to dim it down a bit maybe you could see it as a > crescent at closet approach, though I doubt it, but I can't see the > crescent shape even through 10x50 binoculars at closest approach due to > the glare. > Even through a telescope at around 50 x the crescent isn't all that big. > Pat Here's some factoids. Pat your Jupiter moon point is supported here, http://en.wikipedia.org/wiki/Galilean_moons#Visibility It makes good sense to me, I think I'll try it. Venus is ~1 arc minute in diameter, http://nssdc.gsfc.nasa.gov/planetary/factsheet/venusfact.html which is resolved with normal eye sight. I may try that with a filter, perhaps sunglasses. And finally here's one, 2nd paragraph, http://en.wikipedia.org/wiki/Uranus#Planetary_rings I found the Herchel story very interesting, maybe he had very good eyes. Ken === Subject: Re: Shuttle launch tonight at 7:43 EDT > Venus is ~1 arc minute in diameter, > http://nssdc.gsfc.nasa.gov/planetary/factsheet/venusfact.html > which is resolved with normal eye sight. > I may try that with a filter, perhaps sunglasses. > You might want to try a whole different approach....instead of looking for it before dawn or after sunset when it's going to be very bright and dazzle you, as well as low in the sky when it's closest to Earth, you might want to look for it in the daytime sky when it's almost directly overhead. It's generally visible if you know right where to look (I stumbled on it one day by accident with my 80 mm refractor, and was baffled by it till I figured out what it was due to its crescent shape) and the bright sky background would prevent the dazzle factor. > And finally here's one, 2nd paragraph, > http://en.wikipedia.org/wiki/Uranus#Planetary_rings > I found the Herchel story very interesting, maybe he > had very good eyes. > No, I've seen Uranus naked eye also. It's very dim, but you can see it. In conditions of very good seeing up here in North Dakota, I've seen up to nine stars in the Pleiades naked eye using the averted vision trick. Pat === Subject: Re: Shuttle launch tonight at 7:43 EDT >> That said, I am going to have to try this for myself. >I've also been informed Venus when bright and close, >ie cresent, can be resolved, with good naked eyes, >beyond point-like. There's no record of it in any of the traditional astronomical > journals, however, and lots of people try (amateurs also make a sport > of spotting Venus in daytime, and trying to catch the youngest moon > after New Moon.) This is another laboratory possiblity that fails in the real world, > due to contrast issues, atmospheric turbulence, etc. There has been > suggestion that pre-historic man may have discerned Venus as a > crescent, based on some cave drawings which seem to show two crescents > in the sky, but most archaelogists believe it to be two different > representations of the moon. And prehistoric man may well have had > better eyesight than modern man. > Brian The radius of Venus is about 5000 km. At its closest to Earth, it's about 50 million km away. So its angular diameter is about 1/5000 of a radian. That's pretty close to the one arc-minute that I quoted as the limit of resolution with the naked eye. So maybe there are some exceptional people who can resolve the crescent. Ken PS: Please use math and science. === Subject: Re: Shuttle launch tonight at 7:43 EDT The radius of Venus is about 5000 km. At its closest to Earth, it's > about 50 million km away. So its angular diameter is about 1/5000 of > a radian. That's pretty close to the one arc-minute that I quoted as > the limit of resolution with the naked eye. So maybe there are some > exceptional people who can resolve the crescent. > Ken > PS: Please use math and science. Actually the equatorial radius of Venus is 6,051.8 km (94.9% of the Earth's). Venus currently subtends 57 of arc and peaks about 59.22 on March 26th. These next few days ARE THE BEST to observe the Venus crescent! -Sam === Subject: Re: Shuttle launch tonight at 7:43 EDT >The radius of Venus is about 5000 km. At its closest to Earth, it's >about 50 million km away. So its angular diameter is about 1/5000 of >a radian. That's pretty close to the one arc-minute that I quoted as >the limit of resolution with the naked eye. So maybe there are some >exceptional people who can resolve the crescent. >PS: Please use math and science. The fact remains, there is no record of it ever happening. Math and science are terrific, but your formulae leave out atmospheric turbulence and contrast issues, among many other factors. The numbers say Venus should just barely, perhaps, be discernable as a crescent. In the real world, it has never happened. Perhaps an astronaut on the Space Station or Shuttle could pull it off. But on Earth, there are just too many other factors that make it impossible. As I told you in a previous post, here in the real world things are quite different. People can't make out Venus as a crescent, and people aren't going to make out a shape in the Space Station as it passes overhead. There's a reason mankind is putting telescopes in space and now using adaptive optics and laser guide stars for Earthbound telescopes. It's the Earth's atmosphere, and it has a nasty habit of doing its utmost to ruin observations of stars and planets. In particular, Venus is always either a Morning Star or Evening Star, always low in the sky so that its light has to travel through more of the atmosphere before reaching an observer on the ground. That murk usually makes for what astronomers call poor seeing, and if there has ever been a case of perfect seeing which is needed to see Venus as a crescent, it has never been recorded in the likes of Sky & Telescope or Astronomy magazine, among many others. And people do try. Have you actually tried any of these observations you claim should be possible? I have. I've been an amateur astronomer and satellite watcher for twenty-five years. How about taking your math and science outside and trying to apply it to the real world? Brian === Subject: Re: Shuttle launch tonight at 7:43 EDT > In particular, Venus is always either a Morning Star or Evening Star, > always low in the sky so that its light has to travel through more of > the atmosphere before reaching an observer on the ground. It can get up fairly high in the sky here... say 30 degrees over the horizon, but that would be halfway around its orbit from us, not at its closest approach. Mercury is the one that is a pain in the ass to observe - I've never gotten a good look at that through a telescope; it's always wobbling all over the place at any sort of magnification that would show a disc. Frankly, I'm surprised the ancients even realized it was a planet. Pat === Subject: Re: Shuttle launch tonight at 7:43 EDT <2fh3s41uejoik5278luco6ik794mlua8dq@4ax.com> posting-account=c6T4LggAAACAqdk8Zx_zXI_oErOujh3M Gecko/20050915 Firefox/1.0.7,gzip(gfe),gzip(gfe) >The radius of Venus is about 5000 km. At its closest to Earth, it's >about 50 million km away. So its angular diameter is about 1/5000 of >a radian. That's pretty close to the one arc-minute that I quoted as >the limit of resolution with the naked eye. So maybe there are some >exceptional people who can resolve the crescent. >PS: Please use math and science. The fact remains, there is no record of it ever happening. Math and > science are terrific, but your formulae leave out atmospheric > turbulence and contrast issues, among many other factors. The numbers > say Venus should just barely, perhaps, be discernable as a crescent. > In the real world, it has never happened. Perhaps an astronaut on the > Space Station or Shuttle could pull it off. But on Earth, there are > just too many other factors that make it impossible. As I told you in a previous post, here in the real world things are > quite different. People can't make out Venus as a crescent, and people > aren't going to make out a shape in the Space Station as it passes > overhead. There's a reason mankind is putting telescopes in space and > now using adaptive optics and laser guide stars for Earthbound > telescopes. It's the Earth's atmosphere, and it has a nasty habit of > doing its utmost to ruin observations of stars and planets. In particular, Venus is always either a Morning Star or Evening Star, > always low in the sky so that its light has to travel through more of > the atmosphere before reaching an observer on the ground. That murk > usually makes for what astronomers call poor seeing, and if there > has ever been a case of perfect seeing which is needed to see Venus as > a crescent, it has never been recorded in the likes of Sky & Telescope > or Astronomy magazine, among many others. And people do try. Have you actually tried any of these observations you claim should be > possible? I have. I've been an amateur astronomer and satellite > watcher for twenty-five years. How about taking your math and science > outside and trying to apply it to the real world? > Brian Ok Brian, I agree you're visually impaired relatively, is that good enuf? Science supports the claims of others. Ken === Subject: Re: Shuttle launch tonight at 7:43 EDT Ok Brian, I agree you're visually impaired relatively, is > that good enuf? Science supports the claims of others. > Why don't you give us a online citation of someone reputable who actually did see Venus as a crescent with their naked eyes, and a link to it. I smell troll, big time. Pat === Subject: Re: Shuttle launch tonight at 7:43 EDT >People with good eyesight can resolve things about one arc-minute >apart. That's 1/3500 of a radian. So if the spacecraft are directly >overhead, 350 km high, they could be resolved if they're 100 metres >apart. So I doubt that the ISS and shuttle can be resolved while >they're docked together, but they should certainly be resolvable when >they're flying in close formation. I don't like to say I told you so, but... Brian === Subject: Re: Shuttle launch tonight at 7:43 EDT <0uqdnRqwNpTBjiPUnZ2dnUVZ_tTinZ2d@posted.northdakotatelephone> posting-account=c6T4LggAAACAqdk8Zx_zXI_oErOujh3M Gecko/20050915 Firefox/1.0.7,gzip(gfe),gzip(gfe) >> I'd say it was certainly as bright as I've seen Venus, also, I'm >> informed, with good eyes, one can resolve it beyond a point. > No. Not even with basic binoculars. >> Sorry Brian, you're out classed :-). >> A professional associate did the arc visual acuity calculations >> and I found his computations to be correct. >> I'll help you out if you have any notions of what binoculars are. > Does personal experience count? >> Not much, it's science. > Have you actually tried to make out > more than a point? >> No. > I have. Not in the real world, you can't. > Wish it were so, but it isn't. :-) > Brian >> The person I refer to has good eyes and is a fellow scientist. >> I'll ask him to put a paragraph together and I'll post it, but >> Brian I really wish you'd do your own math, for your good. >> Ken > People with good eyesight can resolve things about one arc-minute > apart. That's 1/3500 of a radian. So if the spacecraft are directly > overhead, 350 km high, they could be resolved if they're 100 metres > apart. So I doubt that the ISS and shuttle can be resolved while > they're docked together, but they should certainly be resolvable when > they're flying in close formation. > Ken There's a difference between detection and resolution. Once I watched > Mir, the shuttle and a Soyuz (?) go overhead. It was 4 July several > years ago, I can't remember the year. I wasn't using any form of optics. > Had I used binoculars as you suggested I still couldn't have resolved > any of the objects. > Dan, U.S. Air Force, retired Sure you could Dan, resolution depends on the aperature of the objective lens. What size bino's do you use? Ken === Subject: Re: Shuttle launch tonight at 7:43 EDT >> I'd say it was certainly as bright as I've seen Venus, also, I'm >> informed, with good eyes, one can resolve it beyond a point. > No. Not even with basic binoculars. >> Sorry Brian, you're out classed :-). >> A professional associate did the arc visual acuity calculations >> and I found his computations to be correct. >> I'll help you out if you have any notions of what binoculars are. > Does personal experience count? >> Not much, it's science. > Have you actually tried to make out > more than a point? >> No. > I have. Not in the real world, you can't. > Wish it were so, but it isn't. :-) > Brian >> The person I refer to has good eyes and is a fellow scientist. >> I'll ask him to put a paragraph together and I'll post it, but >> Brian I really wish you'd do your own math, for your good. >> Ken > People with good eyesight can resolve things about one arc-minute > apart. That's 1/3500 of a radian. So if the spacecraft are directly > overhead, 350 km high, they could be resolved if they're 100 metres > apart. So I doubt that the ISS and shuttle can be resolved while > they're docked together, but they should certainly be resolvable when > they're flying in close formation. > Ken >> There's a difference between detection and resolution. Once I watched >> Mir, the shuttle and a Soyuz (?) go overhead. It was 4 July several >> years ago, I can't remember the year. I wasn't using any form of optics. >> Had I used binoculars as you suggested I still couldn't have resolved >> any of the objects. >> Dan, U.S. Air Force, retired Sure you could Dan, resolution depends on the aperature > of the objective lens. What size bino's do you use? > Ken I have an idea, take some binoculars out and see if YOU can see it. Dan, U.S. Air Force, retired === Subject: Re: Shuttle launch tonight at 7:43 EDT >> The person I refer to has good eyes and is a fellow scientist. >> I'll ask him to put a paragraph together and I'll post it, but >> Brian I really wish you'd do your own math, for your good. >> Ken >> People with good eyesight can resolve things about one arc-minute > apart. That's 1/3500 of a radian. So if the spacecraft are directly > overhead, 350 km high, they could be resolved if they're 100 metres > apart. So I doubt that the ISS and shuttle can be resolved while > they're docked together, but they should certainly be resolvable when > they're flying in close formation. > Ken > Which basicly demolishes your argument that you can see the ISS as more than a spot of light in the night sky. :-) Pat === Subject: Re: Shuttle launch tonight at 7:43 EDT <0uqdnRqwNpTBjiPUnZ2dnUVZ_tTinZ2d@posted.northdakotatelephone> posting-account=c6T4LggAAACAqdk8Zx_zXI_oErOujh3M Gecko/20050915 Firefox/1.0.7,gzip(gfe),gzip(gfe) >> The person I refer to has good eyes and is a fellow scientist. >> I'll ask him to put a paragraph together and I'll post it, but >> Brian I really wish you'd do your own math, for your good. >> Ken > People with good eyesight can resolve things about one arc-minute > apart. That's 1/3500 of a radian. So if the spacecraft are directly > overhead, 350 km high, they could be resolved if they're 100 metres > apart. So I doubt that the ISS and shuttle can be resolved while > they're docked together, but they should certainly be resolvable when > they're flying in close formation. > Ken Which basicly demolishes your argument that you can see the ISS as more > than a spot of light in the night sky. :-) > Pat Not really, the spec is based on 20/20 vision, (i arc minute), the ISS is 93 metres from this, http://en.wikipedia.org/wiki/International_Space_Station For instance a person with 20/10 vision could resolve 50 metres at 350 km, and therefore be able to say it's NOT a point of light, such as a star, and I would agree they have good eyesight, that was my point. http://en.wikipedia.org/wiki/Visual_acuity (it has typo near the end mixing arc minutes with arc seconds). I've read some fighters pilots, such as Yeager had extremely good vision, providing an advantage in air combat. Ken === Subject: Re: Shuttle launch tonight at 7:43 EDT > I've also been informed Venus when bright and close, > ie cresent, can be resolved, with good naked eyes, > beyond point-like. > It's too bright to make out the shape...it dazzles the eyes, so you just see a spot, not a crescent. Even with 7x50 binoculars, the crescent shape isn't obvious, even at its closest approach to Earth. First time I saw its crescent phase was through a 80mm refractor. Pat === Subject: Re: Shuttle launch tonight at 7:43 EDT > You know, you're reminding me of a guy in my Freshman year that proved to > me that it was impossible to see Long Island from the Connecticut shoreline. > Despite the fact that I had done so numerous times. While in a strictly atmosphereless situation you and your friend are of > course right, atmospheric diffraction really makes a huge difference here. That said, I am going to have to try this for myself. > I once managed to see Jupiter's four big moons naked eye on a very cold and still winter night by eclipsing Jupiter itself with a very thin twig on a tree while standing around ten feet from it. Pat === Subject: Re: Shuttle launch tonight at 7:43 EDT >> You know, you're reminding me of a guy in my Freshman year that proved to >> me that it was impossible to see Long Island from the Connecticut shoreline. >> Despite the fact that I had done so numerous times. >> While in a strictly atmosphereless situation you and your friend are of >> course right, atmospheric diffraction really makes a huge difference here. >> That said, I am going to have to try this for myself. >> I once managed to see Jupiter's four big moons naked eye on a very cold >and still winter night by eclipsing Jupiter itself with a very thin twig >on a tree while standing around ten feet from it. You were that close to Jupiter? :-) === Subject: Re: Shuttle launch tonight at 7:43 EDT > > You know, you're reminding me of a guy in my Freshman year that proved to > me that it was impossible to see Long Island from the Connecticut shoreline. > Despite the fact that I had done so numerous times. >> While in a strictly atmosphereless situation you and your friend are of > course right, atmospheric diffraction really makes a huge difference here. >> That said, I am going to have to try this for myself. > >> I once managed to see Jupiter's four big moons naked eye on a very cold >> and still winter night by eclipsing Jupiter itself with a very thin twig \ >> on a tree while standing around ten feet from it. You were that close to Jupiter? :-) I would have thought being ten feet from Jupiter would have exposed him to enough radiation to completely fry his brains. On the other hand, that would explain a lot... === Subject: Re: Shuttle launch tonight at 7:43 EDT > I once managed to see Jupiter's four big moons naked eye on a very > cold and still winter night by eclipsing Jupiter itself with a very > thin twig on a tree while standing around ten feet from it. >> You were that close to Jupiter? :-) I would have thought being ten feet from Jupiter would have exposed > him to enough radiation to completely fry his brains. On the other hand, that would explain a lot... I knew someone was going to take it like that. :-) ...though how you would figure out that you were ten feet from Jupiter is a good question given that its atmosphere doesn't have a top that's well defined. By standing around ten feet from the tree I was able to move into a position where I could get a thin twig between me and the planet, without having to worry about slight movements uneclipsing Jupiter itself. I was only able to pull this trick off once, on a extremely cold, clear, and still winter night, when we were at the center of a high pressure system, the air was almost completely moisture free, and Jupiter was at around its highest elevation on the plane of the ecliptic. I noted last night that Venus is about as far north of the sun as I've ever seen it. Pat === Subject: Re: Shuttle launch tonight at 7:43 EDT >> You know, you're reminding me of a guy in my Freshman year that proved >> to me that it was impossible to see Long Island from the Connecticut >> shoreline. Despite the fact that I had done so numerous times. >> While in a strictly atmosphereless situation you and your friend are of >> course right, atmospheric diffraction really makes a huge difference >> here. >> That said, I am going to have to try this for myself. > > I once managed to see Jupiter's four big moons naked eye on a very cold > and still winter night by eclipsing Jupiter itself with a very thin twig > on a tree while standing around ten feet from it. Cool. I recall the first time I saw them using a pair of binoculars. Really amazing. I could sort of appreciate the thrill that Galileo had. Pat -- Greg Moore Ask me about lily, an RPI based CMC. === Subject: Re: Shuttle launch tonight at 7:43 EDT >> I once managed to see Jupiter's four big moons naked eye on a very cold >> and still winter night by eclipsing Jupiter itself with a very thin twig \ >> on a tree while standing around ten feet from it. >> Cool. I recall the first time I saw them using a pair of binoculars. Really > amazing. I could sort of appreciate the thrill that Galileo had. > If it weren't for the fact that they get lost in the glare of Jupiter, they would be easy naked-eyed objects given their magnitude. Pat === Subject: Re: Shuttle launch tonight at 7:43 EDT >> It depends on the definition of resolution. The atmosphere alone >> limits resolution. >> ahh, not much, do the high school trig. > All the planets that are visible to the naked eye (with the possible exception of Uranus) are going to be larger than point sources of light, as that's why they don't twinkle like stars do due to atmospheric turbulence. But you aren't going to look up at the ISS with your naked eye and see a little tiny H shaped thing flying overhead any more than you are going to see a crescent Venus or Jupiter's Red Spot that way. Pat === Subject: Re: Shuttle launch tonight at 7:43 EDT >> I'd say it was certainly as bright as I've seen Venus, also, I'm >> informed, with good eyes, one can resolve it beyond a point. >> No. Not even with basic binoculars. > Some amateurs have gotten photos of it via telescope showing the basic shape, including some really fascinating ones showing it silhouetted against the Sun and Moon: http://www.dailyaviator.com/?m=200609 http://www.astronet.ru/db/xware/msg/1216878 Pat === Subject: Re: Shuttle launch tonight at 7:43 EDT >> No. Not even with basic binoculars. >Some amateurs have gotten photos of it via telescope showing the basic >shape, including some really fascinating ones showing it silhouetted >against the Sun and Moon: I've seen them. Spectacular! I wish I still had my telescope, but I gave it to my nephews. I've looked at ISS flying near-overhead through binoculars (not great German ones, but not cheap Walmart toys, either), still just a bright star, the shape is overwhelmed by the high contrast of bright star against black sky. Hence my point that in here the real world (as opposed to Ken's academia), people with good eyesight are not going to make out the shape of ISS flying overhead. Brian === Subject: Re: Shuttle launch tonight at 7:43 EDT > I've seen them. Spectacular! I wish I still had my telescope, but I > gave it to my nephews. > There's a whole subculture out there who believes they have seen Transient Lunar Phenomena near the Moon through their telescopes. These are small dark objects that cross the lunar disk at high speed, or glowing and moving objects that are seen near it. I too have seen these objects while observing the Moon through a telescope. These objects have a scientific name. That name is Heterocera... or as most people call them... moths. :-D One thing that did surprise me was when I was watching the sun by projecting its image on a white plate was quite often seeing birds at very high altitudes crossing the solar disk. Once a jet airliner flew across it also. Pat === Subject: Re: Shuttle launch tonight at 7:43 EDT > -> IRCC, it's supposed to be around as bright as Jupiter. > >> Pay >> Apparently you have not watched the ISS very much, if at all. I've seen it around 20 times total, and used to be a amateur astronomer, so have seen Jupiter several thousand times over the years via naked eye, binoculars, and in several different types of telescope up to a 8 Celestron. I was basing that statement on a news report from a few years back that stated that when completed, the ISS would be the brightest object in the on average is Jupiter, so ISS was supposed to be somewhere between Jupiter and Venus in brightness... but the design was cut back in size arrays that were supposed to be attached to it, so its total brightness would be less than predicted. Pat === Subject: Re: The modern mathematical concept of infinity is ... posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Well, you would get many surprises if you did *not* distinguish a and {a}. > My interest is the number of natural numbers. The number of numbers in > {4} is the same as the number of numbers in {{{4}}} or the number of > numbers in the next line: > 4 > It is always one number. And { } does not contain any number. > Therefore we do not need it (although it is useful on some occasions). You are using in in a way that is not common in set theory. Normally when > it is stated that something is in a set A, that means that that something > is an element of the set A. In that sense, there is no numbre in {{{4}}}. Concerning mathematics, the set {1, {2, 3}, {{{{4}}}}} contains the first four natural numbers. > The distinction becomes crucial when there are nested sets involved, the > set > {a, {b, c}} > has two elements, not three. > That may be true, but I am not at all intersted in that hierarchical > structure. With respect to numbers, or letters in the present case, > there is but one single answer, namely there are three letters. Yes, but they play a crucially different role. With respect to the question of how many natural numbers exist, this role is negligible. > When you try to show inconsistency, use the > language of the theory for which you want to show inconsistency. > Otherwise the only thing you show is that the language of the theory > is not compatible with the language you use yourself. > Some people play Chess others play Go others play set theory. I do not > object. But all that has no authority with respect to the > *mathematical* question: What do we know about the number of natural > numbers? Without the definition of set theory, nothing. Wrong. That stuff is basic and independent of theories. > I have proved my answer based on the fact that it is impossible to > distinguish X numbers in unary representation by less than X symbols > per number. No, you have proven that only when X is a natural number. For all of them, yes. === Subject: Re: The modern mathematical concept of infinity is ... > Concerning mathematics, the set {1, {2, 3}, {{{{4}}}}} contains the > first four natural numbers. Neither as members nor as subsets, which are the only meanings in standard set theory for contains. > The distinction becomes crucial when there are nested sets involved, > the > set > {a, {b, c}} > has two elements, not three. > That may be true, but I am not at all intersted in that hierarchical > structure. With respect to numbers, or letters in the present case, > there is but one single answer, namely there are three letters. > Yes, but they play a crucially different role. With respect to the question of how many natural numbers exist, this > role is negligible. On the contrary, it is essential. It is at the heart of what it means for an object to be an element of a set. While N is an infinite set, {N} is not. > What do we know about the number of natural > numbers? > Without the definition of set theory, nothing. Wrong. That stuff is basic and independent of theories. Not until WM can give an unambiguous definition of what he means by a natural number, and show that his definition has all the properties of the definitions given in various set theories. > I have proved my answer based on the fact that it is impossible to > distinguish X numbers in unary representation by less than X symbols > per number. > No, you have proven that only when X is a natural number. For all of them, yes. But that would require that no other sorts of numbers exist, which tosses out much of arithmetic and all of analysis with the bath water. > === Subject: Re: The modern mathematical concept of infinity is ... Nntp-Posting-Host: hera.cwi.nl ... > It is always one number. And { } does not contain any number. > Therefore we do not need it (although it is useful on some occasions). > > You are using in in a way that is not common in set theory. Normally > when it is stated that something is in a set A, that means that that > something is an element of the set A. In that sense, there is no numbre > in {{{4}}}. > > Concerning mathematics, the set {1, {2, 3}, {{{{4}}}}} contains the > first four natural numbers. Not according to standard terminology in set theory. > That may be true, but I am not at all intersted in that hierarchical > structure. With respect to numbers, or letters in the present case, > there is but one single answer, namely there are three letters. > > Yes, but they play a crucially different role. > > With respect to the question of how many natural numbers exist, this > role is negligible. The distinction starts when you state that all natural number equals N, making N an element of N. > Some people play Chess others play Go others play set theory. I do not > object. But all that has no authority with respect to the > *mathematical* question: What do we know about the number of natural > numbers? > > Without the definition of set theory, nothing. > > Wrong. That stuff is basic and independent of theories. Oh. What is the number of natural numbers? > I have proved my answer based on the fact that it is impossible to > distinguish X numbers in unary representation by less than X symbols > per number. > > No, you have proven that only when X is a natural number. > > For all of them, yes. Yes, you have proven it for all X, but not for the set of all X, i.e. N. You think so because you are thinking that all natural numbers is equal to N, which is incorrect in set theory. And so your reasoning is outside set theory and does not show inconsistency of set theory, only incompatibility with your opinion. -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: The modern mathematical concept of infinity is ... > This definition is contradicted by the fact that there is no set of X > FISONs with each FISON having less than X elements. > X being a natural number again. Why do you always omit that from your > considerations? You need that assertion for infinitely many FISONs, but > you can not prove it for that. > As I have proved that *all* natural numbers are used to distinguish > the elements of FISONs, there is no X remaining to distinguish an > actually infinite set of naturals. This statement makes no sense. Either you mean that there is no X remaining > to distinguish an actually infinite set of naturals from the FISONs. That > would be right, but irrelevant. No, it is very relevant to have distinctions for all nunmbers. >Or you mean that there is no X remaining to > distinguish the elements of an actually set from each other, but that is > nonsense, because you can not distinguish the elements of a set using a > single X. Using X and all smaller nunmbers. > Moreover, *each* natural number is used in the distinguishing > of elements from infinitely many FISONs But only finitely many, i.e., potentially infinitely many, but not actually infinitely many. > I have proved that at least aleph 0 symbols per at least one number > would be required to distinguish aleph 0 numbers. But there is no > number with aleph 0 symbols. No, you have not proven that. Of the aleph 0 natural numbers, 1 gets one > stroke, 2 two, and so on. Look, all aleph 0 numbers are distinguished > from each other and none of them has aleph 0 strokes. Or do you think > that there is a natural number that requires aleph 0 strokes? If so, > why? Because set theory says that aleph 0 is a number of numbers that is larger than every natural number. > Or do you disagree that there are not aleph 0 natural numbers that are all > finite? > I think it is the latter. In that case you ought to clearly state *that* > Not all that rigmarole about FISONs. And in that case you also have to > show how that is valid when we assume the axiom of infinity and the > definitions of set theory. Otherwise you will be unable to prove > inconsistency of set theory. > I have proved that at least aleph 0 symbols per at least one number > would be required to distinguish aleph 0 numbers. See above, where I show the opposite. Note: per has three different > meanings: > 1. by means of > 2. for each > 3. according to > adding at least makes no sense when meaning two is meant. It should be easy to understand what is meant here: In order to distinguish the following three numbers: o, oo, oooo, we nee at least one number with three symbols. In the actual case we have even used four symbols for one number. > But there is no > number with aleph 0 symbols. Because any smaller maximal number of > symbols per number is insufficient. Eh? How is the logic here? Every natural number X (as maximum number of symbols per number) is capable of distinguishing at most X numbers. For aleph 0 numbers no natural X is sufficient. === Subject: Re: The modern mathematical concept of infinity is ... > ... > This definition is contradicted by the fact that there is no set of > X > FISONs with each FISON having less than X elements. > X being a natural number again. Why do you always omit that from your > considerations? You need that assertion for infinitely many FISONs, > but > you can not prove it for that. > As I have proved that *all* natural numbers are used to distinguish > the elements of FISONs, there is no X remaining to distinguish an > actually infinite set of naturals. > This statement makes no sense. Either you mean that there is no X > remaining > to distinguish an actually infinite set of naturals from the FISONs. That > would be right, but irrelevant. No, it is very relevant to have distinctions for all nunmbers. The name aleph_0 is not used by any natural number therefore serves to distinguish the thing named from every natural number. Moreover, *each* natural number is used in the distinguishing > of elements from infinitely many FISONs But only finitely many, i.e., potentially infinitely many, but not > actually infinitely many. Wrong. aleph_0, distinguishes aleph-0 from infinitely many naturals. > Because set theory says that aleph_0 is a number of numbers that is > larger than every natural number. It is a number larger than any natural by the Cantor definition of comparisons of cardinality. Every natural number X (as maximum number of symbols per number) is > capable of distinguishing at most X numbers. For aleph_0 numbers no > natural X is sufficient. So we invent a non-natural number to do it for us, and give it an identification name aleph_0 different from the name or representation of any natural. Since all numbers, natural or otherwise, are inventions anyway, we just invent what we need. === Subject: Re: The modern mathematical concept of infinity is ... Nntp-Posting-Host: hera.cwi.nl ... > As I have proved that *all* natural numbers are used to distinguish > the elements of FISONs, there is no X remaining to distinguish an > actually infinite set of naturals. > > This statement makes no sense. Either you mean that there is no X > remaining to distinguish an actually infinite set of naturals from > the FISONs. That would be right, but irrelevant. > > No, it is very relevant to have distinctions for all nunmbers. No, it is irrelevant, the actually infinite set of naturals is distinguished from the FISONs not with some X but by being infinite, in contrast with each and every FISON. > Or you mean that there is no X remaining to > distinguish the elements of an actually set from each other, but that is > nonsense, because you can not distinguish the elements of a set using a > single X. > > Using X and all smaller nunmbers. But you have used up (in some sense) all smaller numbers already. > Moreover, *each* natural number is used in the distinguishing > of elements from infinitely many FISONs > > But only finitely many, i.e., potentially infinitely many, but not > actually infinitely many. Makes no sense in mathematics. According to a definition you gave some time ago of potentially infinite, potentially infinite sets do not exist. According to that definitions the only potentially infinite thing you could have was a set-valued function. > I have proved that at least aleph_0 symbols per at least one number > would be required to distinguish aleph_0 numbers. But there is no > number with aleph_0 symbols. > > No, you have not proven that. Of the aleph_0 natural numbers, 1 gets > one stroke, 2 two, and so on. Look, all aleph_0 numbers are distinguished > from each other and none of them has aleph_0 strokes. Or do you think > that there is a natural number that requires aleph_0 strokes? If so, > why? > > Because set theory says that aleph_0 is a number of numbers that is > larger than every natural number. That is not an answer. What natural number would require more strokes than itself given my statement above that each natural number n gets n strokes? > I have proved that at least aleph_0 symbols per at least one number > would be required to distinguish aleph_0 numbers. > > See above, where I show the opposite. Note: per has three different > meanings: > 1. by means of > 2. for each > 3. according to > adding at least makes no sense when meaning two is meant. > > It should be easy to understand what is meant here: In order to > distinguish the following three numbers: o, oo, oooo, we nee at least > one number with three symbols. In the actual case we have even used > four symbols for one number. So your use of the word per was wrong. But still not proven that there is at least one number required that uses aleph_0 strokes. > But there is no > number with aleph_0 symbols. Because any smaller maximal number of > symbols per number is insufficient. > > Eh? How is the logic here? > > Every natural number X (as maximum number of symbols per number) is > capable of distinguishing at most X numbers. For aleph_0 numbers no > natural X is sufficient. Right, for aleph_0 numbers there is no natural number X as maximum of number of symbols per number. There is no maximum number of symbols per number. But the number of symbols per number is finite. What is the problem? -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: The modern mathematical concept of infinity is ... posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > What is wrong with my logic when I use your statement that all natural > numbers equals N? You imply that N is more than the collection of its elements: > No, of course not. That number of elements is aleph 0, due to the axiom > of infinity and the definition of aleph 0. The distinctions is crucial to > understand how set theory works. As you do not make the distinctions you > apparently do not know how set theory works. > With or without set theory: The number of all natural numbers is not a > number that is larger than every natural number. This is provable > fact. Every opponent is wrong. You have not proven it. You are not able to understand it. === Subject: Re: The modern mathematical concept of infinity is ... What is wrong with my logic when I use your statement that all natural > numbers equals N? You imply that N is more than the collection of its elements: > No, of course not. That number of elements is aleph_0, due to the > axiom > of infinity and the definition of aleph_0. The distinctions is > crucial to > understand how set theory works. As you do not make the distinctions > you > apparently do not know how set theory works. > With or without set theory: The number of all natural numbers is not a > number that is larger than every natural number. This is provable > fact. Every opponent is wrong. > You have not proven it. You are not able to understand it. To be unable to understand how 2 + 2 = 5 is an advantage to those trying to do mathematics. But seems to be a necessity to those doing WM's Unmathematics. === Subject: Re: The modern mathematical concept of infinity is ... Nntp-Posting-Host: hera.cwi.nl > > What is wrong with my logic when I use your statement that all natural > numbers equals N? > > You imply that N is more than the collection of its elements: No. I do not say that. I say that with your assumed equivalence of N and all natural numbers, N is an element of N. Here is my logic again: > N is the set of all natural numbers. N has as element all > natural numbers. You equate all natural numbers with N, so N has as > element N. So according to you N is an element of N? If not, which of my statements above is actually wrong? > With or without set theory: The number of all natural numbers is not > a number that is larger than every natural number. This is provable > fact. Every opponent is wrong. > > You have not proven it. > > You are not able to understand it. You prove it by assuming that all natural numbers is the same as N (and so assuming that N is an element of itself). -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: The modern mathematical concept of infinity is ... posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Well, you would get many surprises if you did *not* distinguish a and {a}. > My interest is the number of natural numbers. The number of numbers in > {4} is the same as the number of numbers in {{{4}}} or the number of > numbers in the next line: > 4 Actually, the number of numbers in {{4}} can be zero if in is > interpreted to mean is a member of as is quite common. When talking about numbers *in mathematics*, then it is irrelevant whether a number belongs to a set as an element or as a subset. {1, {2, 3}, {{{4}}}} contains 4 numbers, namely the first four naturals. That is all that counts. > The distinction becomes crucial when there are nested sets involved, the set > {a, {b, c}} > has two elements, not three. > That may be true, but I am not at all intersted in that hierarchical > structure. Then do not pontificate on set theories, as they are specifically > intersted in the membership relation. I do not pontificate on set set theories. All I say is that a collection of natural numbers cannot surpass the largest number of that collection. If there is any theory that does not yield this result, then this theory is false. >When you try to show inconsistency, use the > language of the theory for which you want to show inconsistency. Otherwise > the only thing you show is that the language of the theory is not compatible > with the language you use yourself. > Some people play Chess others play Go others play set theory. I do not > object. But all that has no authority with respect to the > *mathematical* question: What do we know about the number of natural > numbers? Much of what WM claims to know about them is wrong. What in detail? > I have proved my answer based on the fact that it is impossible to > distinguish X numbers in unary representation by less than X symbols > per number. And that is one of the things that is wrong, unless X is required to be > a natural number, A number of natural numbers is a natural number. > as there are numbers, X, which are not natural for > which it is false. That may be. But they concern not only natural numbers. > One could add as a second condition that natural numbers > that cannot be distinguished do not belong to mathematics. Since WM's standard of distinguishability for naturals requires a unary > representation for each such natural, most of the naturals that > mathematics require are simply not available to him. Not in the real world. But without physical constraints, there is no natural number that lacks its individual unary representation. === Subject: Re: The modern mathematical concept of infinity is ... > > Well, you would get many surprises if you did *not* distinguish a and > {a}. > My interest is the number of natural numbers. The number of numbers in > {4} is the same as the number of numbers in {{{4}}} or the number of > numbers in the next line: > 4 > Actually, the number of numbers in {{4}} can be zero if in is > interpreted to mean is a member of as is quite common. When talking about numbers *in mathematics* We were talking more particularly of the number of naturals in {4} versus the number of naturals in {{{4}}}, so that one must be quite careful about the intended meaning of in. But WM is notoriously un-careful about meanings, which is a most un-mathematical, even anti-mathematical, sort of behaviour. > I am not at all intersted in that hierarchical > structure. > Then do not pontificate on set theories, as they are specifically > intersted in the membership relation. I do not pontificate on set set theories. All I say is that a > collection of natural numbers cannot surpass the largest number of > that collection. That is an unwarranted pontification, as it assumes, contrary to fact, that in every congregation of naturals there must be a largest. >If there is any theory that does not yield this > result, then this theory is false. Set theory defines naturals in such a way as to allow a congregation of them in which there is no largest. That WM's understanding cannot encompass such a definition is his loss, not ours. >When you try to show inconsistency, use the > language of the theory for which you want to show inconsistency. > Otherwise > the only thing you show is that the language of the theory is not > compatible > with the language you use yourself. > Some people play Chess others play Go others play set theory. I do not > object. But all that has no authority with respect to the > *mathematical* question: What do we know about the number of natural > numbers? > Much of what WM claims to know about them is wrong. What in detail? WM claims that there is in every congregation of naturals a largest one, when in the congregation of all of them there clearly is not. > And that is one of the things that is wrong, unless X is required to be > a natural number, > A number of natural numbers is a natural number. Not unless one requires that to be an axiom, which WM may but mathematics does not. > One could add as a second condition that natural numbers > that cannot be distinguished do not belong to mathematics. > Since WM's standard of distinguishability for naturals requires a unary > representation for each such natural, most of the naturals that > mathematics require are simply not available to him. Not in the real world. In a world limited by WM's standards, 10^googolplex is not available to WM until he has represented it in unary. > But without physical constraints, there is no > natural number that lacks its individual unary representation. Without physical constraints, one can have all the naturals, including a successor for each, in one set. WM wants to impose physical constraints on the ideas of others but exempt his own ideas from such constraints. === Subject: Re: The modern mathematical concept of infinity is ... posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > If you could think without prejudice then you would recognize: > Every finite maximal number of symbols (per number) is used to > distinguish a finite number of numbers. There is no place in > mathematics (that uses only distinguishable natural numbers) for a > number of numbers that is larger than every finite number. > No, I do not see it. I agree that every finite maximal number of symbols > per number can be used to distinguish a finite number of numbers. > It is your there is no place in mathematics which boggles the mind. Why > not? Just because that is your opinion? > If you recognize that every finite maximal number of symbols per > number can be used to distinguish a finite number of numbers. What > would you use to distinguish aleph 0 numbers? Obviously a finite > number of symbols per number is not sufficient. No, a finite maximal number of symbols per number is insufficient. That > is not the same, because a maximal number is assumed. But when I state > a finite number of symbols per number there is no maximum on the number > of symbols per number. Nevertheless, whether maximum or not, available is always a finite number whereas aleph 0 is larger than any finite number. That is my point. > As at no time you have properly defined your paths and path-bunches. > Every infinite sequence of edges of the binary tree is a path-bunch. > Every path is a path-bunch containing only one path. > Eh, is the last statement a definition? And if it is, is it not a little > bit circular? Can you not give proper definitions of paths and > path-bunches? > A path is but the geometrical representation of the infinite binary > representation of a real number. I assume that the latter is known. A > path bunch is the union of one or more paths in the binary tree. I still do not see a good definition, how do you define the union of two > infinite binary representations of real numbers? So let me attempt two > definitions: > (1) a path is an infinite sequence of elements from the set {0,1}. Yes, and with that definition it is identical to the binary expansion of a real number of the unit interval. All these real numbers (all their representations) are present, as paths, in the binary tree. > (2) a path bunch is a finite or infinite sequence of elements from > the set {0,1}. There are two definitions possible: (2a) A path bunch is the intersection of one or more paths. This makes every path a path bunch too. Every intersection of two or more paths becomes a finite path. (2b) A path bunch is the union of one or more paths. This makes the hole tree a bunch as well as every path. > Is that correct? (Because I have never seen a definition of the union of > sequences.) In principle the choice is arbitrary. Definition (2a) should be easier to treat: At every finite level of the tree the number of path bunches that can be distinguished at that level is limited by the number of nodes in the levels above this level. But the ghist of the proof can be understood by definition (1) alone: If n path can be distinguished then there must have been n - 1 nodes where they have separated (can be separated for the first time when going from top to bottom in the tree). And we need the two self-evident truths: Only paths that can be distinguished at some finite level are different paths. All nodes of the infinite binary tree are a countable set. This proof does not yield an enumeration of all paths. But is shows that there cannot be more paths than nodes. === Subject: Re: The modern mathematical concept of infinity is ... > If you recognize that every finite maximal number of symbols per > number can be used to distinguish a finite number of numbers. What > would you use to distinguish aleph_0 numbers? Obviously a finite > number of symbols per number is not sufficient. > No, a finite maximal number of symbols per number is insufficient. That > is not the same, because a maximal number is assumed. But when I state > a finite number of symbols per number there is no maximum on the number > of symbols per number. Nevertheless, whether maximum or not, available is always a finite > number whereas aleph_0 is larger than any finite number. That is my > point. And for every finite natual available, there is a larger one available. That is our point. > As at no time you have properly defined your paths and > path-bunches. > Every infinite sequence of edges of the binary tree is a > path-bunch. > Every path is a path-bunch containing only one path. > Eh, is the last statement a definition? And if it is, is it not a > little > bit circular? Can you not give proper definitions of paths and > path-bunches? > A path is but the geometrical representation of the infinite binary > representation of a real number. I assume that the latter is known. A > path bunch is the union of one or more paths in the binary tree. > I still do not see a good definition, how do you define the union of two > infinite binary representations of real numbers? So let me attempt two > definitions: > (1) a path is an infinite sequence of elements from the set {0,1}. Yes, and with that definition it is identical to the binary expansion > of a real number of the unit interval. All these real numbers (all > their representations) are present, as paths, in the binary tree. And Cantor showed unequivocally that there are to many endless binary seqeunces to be counted by naturals even if one uses all infinitely many naturals. (2) a path bunch is a finite or infinite sequence of elements from > the set {0,1}. There are two definitions possible: > (2a) A path bunch is the intersection of one or more paths. This makes > every path a path bunch too. Every intersection of two or more paths > becomes a finite path. > (2b) A path bunch is the union of one or more paths. This makes the > hole tree a bunch as well as every path. A path in a tree is a set of nodes such that (1) it contains the root node and (2) for every nonterminal node in the path there is exactly one of its child nodes in the path (like successorship). Then your (2b) is nonsense, as the union is merely a set of nodes, not a set of paths. A better definition for a 'path bunch' is the set of all paths through (containing) a given node. Then in every complete infinite binary tree every path bunch is equinumerous with the entire tree, and is, itself, uncountable. Is that correct? (Because I have never seen a definition of the union of > sequences.) In principle the choice is arbitrary. Definition (2a) should be easier > to treat: At every finite level of the tree the number of path bunches > that can be distinguished at that level is limited by the number of > nodes in the levels above this level. Not when using my definition. But the ghist of the proof can > be understood by definition (1) alone: If n path can be distinguished > then there must have been n - 1 nodes where they have separated (can > be separated for the first time when going from top to bottom in the > tree). And we need the two self-evident truths: > Only paths that can be distinguished at some finite level are > different paths. But any distinguishing between two paths actually distinguishes at the same time between two path bunches (my definition), each of which is uncountable. > All nodes of the infinite binary tree are a countable set. This proof does not yield an enumeration of all paths. But is shows > that there cannot be more paths than nodes. But as it is based on misrepresentations, shows nothing. > === Subject: Re: The modern mathematical concept of infinity is ... >> If you recognize that every finite maximal number of symbols per >> number can be used to distinguish a finite number of numbers. >> What > would you use to distinguish aleph_0 numbers? Obviously a >> finite > number of symbols per number is not sufficient. >> No, a finite maximal number of symbols per number is insufficient. >> That is not the same, because a maximal number is assumed. But when >> I state a finite number of symbols per number there is no maximum on >> the number of symbols per number. Nevertheless, whether maximum or not, available is always a finite >> number whereas aleph_0 is larger than any finite number. That is my >> point. And for every finite natual available, there is a larger one available. > That is our point. What should available mean? The point is that we DO MATHEMATICS with our natural numbers and don't stare at them in astonishment. For example, we prove that any natural can be written as a product of primes. Where in that proof (or in any text on number theory or any other mathematical subject) does it play a role whether or which naturals are available? And, if the naturals of today are different from the naturals of yesterday, why should then our proof from yesterday still be valid today? In Mueckenmathics it is not possible to prove any mathematical proposition. So whatever Mueckenmathics is (I'd say it is braindead nonsense) it has nothing to do with mathematics. Ralf -- How lucky we are that Cantor introduced curly brackets! But it was no[t] he who introduced the silly distinction between a and {a} that enables so called mathematicians to build card houses on nothing. (Prof. Dr. W. M.9fckenheim, mathematical mastermind of Augsburg University of Applied Science , in sci.math) === Subject: Re: The modern mathematical concept of infinity is ... Nntp-Posting-Host: hera.cwi.nl ... > No, a finite maximal number of symbols per number is insufficient. > That is not the same, because a maximal number is assumed. But when > I state a finite number of symbols per number there is no maximum > on the number of symbols per number. > > Nevertheless, whether maximum or not, available is always a finite > number whereas aleph_0 is larger than any finite number. That is my Yes, and your point is nonsense. The word available makes no sense. If for each natural number n, I give n n strokes, I can distinguish (clearly) all n from each other, and there are aleph_0 natural numbers. There are no two different n with the same number of strokes. Also there is no n with aleph_0 strokes, so each number has finitely many strokes. > A path is but the geometrical representation of the infinite binary > representation of a real number. I assume that the latter is known. A > path bunch is the union of one or more paths in the binary tree. > > I still do not see a good definition, how do you define the union of two > infinite binary representations of real numbers? So let me attempt two > definitions: > (1) a path is an infinite sequence of elements from the set {0,1}. > > Yes, and with that definition it is identical to the binary expansion > of a real number of the unit interval. All these real numbers (all > their representations) are present, as paths, in the binary tree. > > (2) a path bunch is a finite or infinite sequence of elements from > the set {0,1}. > > There are two definitions possible: > (2a) A path bunch is the intersection of one or more paths. This makes > every path a path bunch too. Every intersection of two or more paths > becomes a finite path. Pray define intersection of two sequences. Without such a definition I do not know what this means. > (2b) A path bunch is the union of one or more paths. This makes the > hole tree a bunch as well as every path. Pray define union of two sequences. Without such a definition I do not know what this means. But consider my definition, in it each path is a path bunch, as is the whole tree. The first because an infinite sequence (i.e. a path) is defined to be a path bunch. The second because the empty sequence would be the complete tree. > Is that correct? (Because I have never seen a definition of the union > of sequences.) > > In principle the choice is arbitrary. Definition (2a) should be easier > to treat: At every finite level of the tree the number of path bunches > that can be distinguished at that level is limited by the number of > nodes in the levels above this level. Well, in your definition you *again* used undefined terms: union and intersection of sequences. How do you *define* that? > But the ghist of the proof can > be understood by definition (1) alone: If n path can be distinguished > then there must have been n - 1 nodes where they have separated (can > be separated for the first time when going from top to bottom in the > tree). Again using undefined terminology... Do you agree with the following definitions? (3) Two paths can be distinguished from each other when not all terms agree. (4) Two paths separate from each other at step i when the i-th terms of there sequences are different and all earlier terms agree. (5) A collection of paths can be distinguished when they can be distinguished pairwise. (6) A node is a terminating sequence, i.e. a path bunch. But your statement about distinguishing n paths with the need of n-1 nodes is a bit sloppy, and at most valid for natural n. > And we need the two self-evident truths: > Only paths that can be distinguished at some finite level are > different paths. That is obvious as there is no infinite level. > All nodes of the infinite binary tree are a countable set. That is also obvious, because finite sequences can be enumerated. > This proof does not yield an enumeration of all paths. But is shows > that there cannot be more paths than nodes. No, it does not show that. It shows that for n paths to be distinguishable, where n is a natural number, you need n-1 nodes. So this only holds for a finite collection of paths. Now try to prove the same for an infinite collection of paths. -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: The modern mathematical concept of infinity is ... <49be5fe9$0$5829$426a74cc@news.free.fr> posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > WM a .8ecrit : > This is the contradiction: Every maximal number of symbols per number > limits the number of different elements in a set of unary > representations of natural numbers to this maximal number. As long as > there are only natural numbers of symbols per number allowed for, the > number of different elements in a set of unary representations of > natural numbers is limited to a natural number. > If N is assumed to have aleph 0 > n (forall n in N) elements, then > there must be at least one element with X > n (forall n in N) symbols. > Contradiction. > It is not difficult to see. Every student not yet spoilt by the > worshippers of the empty set and finished infinity does understand > that. >> Do you mean that you have used your academic authority to teach these >> idocies to students? > There is no authority required to teach and to learn the obvious. So, you confirm that you have actually taught your theory to students > in your University ? You can even try to learn it yourself. It requires only a level of intelligence that is not below the average and the decision to drop some dogmas. Here you find the lessons and exams: http://www.hs-augsburg.de/~mueckenh/GU/ If you wish I will evaluate your answers free of charge. By the way, I do not stand alone with my opinion concerning transfinite set theory as being nonsense. You seem to be unfamiliar with the historical and newer developments (in particular in France the true value of set theory had be recognized very early.) A small collection of supporters of this recognition can be found here: http://www.cs.nyu.edu/pipermail/fom/2009-March/013491.html === Subject: Re: The modern mathematical concept of infinity is ... > So, you confirm that you have actually taught your theory to students > in your University ? You can even try to learn it yourself. It requires only a level of > intelligence that is not below the average and the decision to drop > some dogmas. Here you find the lessons and exams: http://www.hs-augsburg.de/~mueckenh/GU/ If you wish I will evaluate your answers free of charge. As WM's mathemtical and logical talents are negligible, one would do better with almost anyone else as a guide. === Subject: Re: The modern mathematical concept of infinity is ... > WM a .8ecrit : > > This is the contradiction: Every maximal number of symbols per number > limits the number of different elements in a set of unary > representations of natural numbers to this maximal number. As long as > there are only natural numbers of symbols per number allowed for, the > number of different elements in a set of unary representations of > natural numbers is limited to a natural number. > If N is assumed to have aleph_0 > n (forall n in N) elements, then > there must be at least one element with X > n (forall n in N) symbols. > Contradiction. > It is not difficult to see. Every student not yet spoilt by the > worshippers of the empty set and finished infinity does understand > that. >> Do you mean that you have used your academic authority to teach these >> idocies to students? > There is no authority required to teach and to learn the obvious. > So, you confirm that you have actually taught your theory to students > in your University ? You can even try to learn it yourself. Whatever you do, do not try to learn finitist or intuitionist theory from WM as his version has it all wrong. There are logically consistent such theories available, though they often make heavy weather of math that the classical approach finds much easier. If anyone wants to try one of these, just make sure it is not WM's, try: http://en.wikipedia.org/wiki/Finitism http://en.wikipedia.org/wiki/Mathematical_constructivism http://en.wikipedia.org/wiki/Intuitionist === Subject: Re: The modern mathematical concept of infinity is ... WM a .8ecrit : >> WM a .8ecrit : > This is the contradiction: Every maximal number of symbols per number > limits the number of different elements in a set of unary > representations of natural numbers to this maximal number. As long as > there are only natural numbers of symbols per number allowed for, the > number of different elements in a set of unary representations of > natural numbers is limited to a natural number. > If N is assumed to have aleph_0 > n (forall n in N) elements, then > there must be at least one element with X > n (forall n in N) symbols. > Contradiction. > It is not difficult to see. Every student not yet spoilt by the > worshippers of the empty set and finished infinity does understand > that. >> Do you mean that you have used your academic authority to teach these >> idocies to students? > There is no authority required to teach and to learn the obvious. >> So, you confirm that you have actually taught your theory to students >> in your University ? You can even try to learn it yourself. It requires only a level of > intelligence that is not below the average and the decision to drop > some dogmas. Here you find the lessons and exams: http://www.hs-augsburg.de/~mueckenh/GU/ > If you wish I will evaluate your answers free of charge. There is a point in writing to your university about this pseudo-scientific crap published on its Web site. There is also a point about one of their professor being a crank. > By the way, I do not stand alone with my opinion concerning > transfinite set theory as being nonsense. You seem to be unfamiliar > with the historical and newer developments (in particular in France > the true value of set theory had be recognized very early.) A small > collection of supporters of this recognition can be found here: http://www.cs.nyu.edu/pipermail/fom/2009-March/013491.html Constructivism and intuitionism on one hand, and your crap, on the other hand, are completely disctinct. === Subject: Re: The modern mathematical concept of infinity is ... > WM a .8ecrit : >> WM a .8ecrit : > This is the contradiction: Every maximal number of symbols per number > limits the number of different elements in a set of unary > representations of natural numbers to this maximal number. As long as > there are only natural numbers of symbols per number allowed for, the > number of different elements in a set of unary representations of > natural numbers is limited to a natural number. > If N is assumed to have aleph_0 > n (forall n in N) elements, then > there must be at least one element with X > n (forall n in N) symbols. > Contradiction. > It is not difficult to see. Every student not yet spoilt by the > worshippers of the empty set and finished infinity does understand > that. >> Do you mean that you have used your academic authority to teach these >> idocies to students? > There is no authority required to teach and to learn the obvious. >> So, you confirm that you have actually taught your theory to students >> in your University ? You can even try to learn it yourself. It requires only a level of > intelligence that is not below the average and the decision to drop > some dogmas. Here you find the lessons and exams: http://www.hs-augsburg.de/~mueckenh/GU/ > If you wish I will evaluate your answers free of charge. There is a point in writing to your university about this > pseudo-scientific crap published on its Web site. There is also > a point about one of their professor being a crank. By the way, I do not stand alone with my opinion concerning > transfinite set theory as being nonsense. You seem to be unfamiliar > with the historical and newer developments (in particular in France > the true value of set theory had be recognized very early.) A small > collection of supporters of this recognition can be found here: http://www.cs.nyu.edu/pipermail/fom/2009-March/013491.html Constructivism and intuitionism on one hand, and your crap, on the > other hand, are completely disctinct. Amen! === Subject: Re: The modern mathematical concept of infinity is ... posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > I am glad to see that you have given up your futile > quest to find something that doesn't change that changes I have never done so. You are trying to deceive. > You need to find a contradiction without assuming that > N changes. > This is the contradiction: Every maximal > If N does not change, then N does not > have a maximal number. > Then it does neither have a fixed number aleph 0 of numbers that is > larger than every natural number. Nope, this does not follow. If N is fixed then the number > of elements in N is the supremum of N and this supremum is fixed. > By definition, the supremum is larger than every element of N. By set theory the supremum is the number of elements of the set of natural numbers. By mathematics the number of natural numbers cannot surpass every natural number. You are trying to argue by menas of matheology. The same arguing would allow a supremum of 10 for the sequence 1,1,1,... > A set of X natural numbers in unary representation cannot exist unless > there is at least one natural number represented by X symbols. You have only shown this for X a natural number. For every natural number. > Note that aleph 0 is not a natural number. Therefore natural numbers are not sufficient to make up a set of aleph 0. A set of aleph 0 natural numbers in unary representation can exist > even > though there is no natural number represented by aleph 0 symbols. No. > This is because the number of natural numbers in the set is the > supremum of the number of symbols used per number. If the set > has a largest element then this supremum is also a maximum. If the > set does not have a largest element (e.g. N) then the supremum is not > a maximum. Thus aleph 0 is a supremum, but not a maximum, so there > is no natural number represented by aleph 0 symbols. And the number of natural numbers is less than the supremum aleph 0. If set theory drops the claim that aleph 0 is the number of elements of the set of natural numbers, then we have no divergence. But then there is no room for uncountability and set theory is a twaddle about nothing. > As every natural number X of symbols is used to distinguish a finite > set of numbers, there is no natural number X of symbols capable of > distinguishing a larger number of numbers, i.e., an actually infinite > set of aleph 0 numbers. Correct, no *one* natural number can distinguish aleph 0 numbers. > However, a set of natural numbers without largest element > can do what no *one* natural number > can do. The Lord can do what no man can do. Amen. > In particular, aleph 0 natural numbers can distinguish > aleph 0 numbers. Numbers, in unary notation, are not distinguished by numbers but by symbols. With a slight modification however your last sentence is correct: In particular, aleph 0 symbols per natural number can distinguish aleph 0 numbers. Alas, no natural number has aleph 0 symbols per number. === Subject: Re: The modern mathematical concept of infinity is ... posting-account=1lE9SQkAAADFrJsDv61dh1YXcJ_ahy5I > By set theory the supremum is the number of elements of the set of > natural numbers. > By mathematics the number of natural numbers cannot surpass every > natural number. > There is the obvious point that no FISON can have more numbers than the set N. So i. N does not change and the number of numbers in N is the supremum of the sizes of the FISONs ii. N is something that does not change that changes. WM does not like i, so he uses ii. WM claims that making the assumption N does not change leads to a contradiction. However, he has shown no contradiction without also making the assumption that N changes. -William Hughes === Subject: tree relationship Why the follow tree recurrence relations T(h,k) = max number of leaves of a tree of height h, where each node has outdegree (number of children) k or less. Been told T(h,k) = T(h/2) + k How is this answer be reach? Is there any way to show this? === Subject: Applied Maths: Simple Mortgage Problem posting-account=HjewwQoAAAAVl3ujJpqMnaLyrkfbl0rs Gecko/2009021910 Firefox/3.0.7,gzip(gfe),gzip(gfe) Hi Guys, I think I have correctly worked out the answer to the following scenario, but I would appreciate confirmation. Problem: You have a mortgage with a bank with a current remaining balance of $200,000 which is currently on a fixed rate of 9% - there is currently 6 months remaining until this comes up for refixing. Current interest rates have dropped to 6%, but to break the mortgage fixed rate early there will be a $2,000 penalty fee. You want to break early because you expect 5 year mortgage rates are going up and by the time you come to refix you think the interest rate will be at 6.5%. (a) What will the cheapest option be (break fixed rate, or wait until refix date) and how do you get to this conclusion? (b) If you break what would the interest rate need to do so that the break option and stay fixed option cost the same? A: I approach this question by simply working out which option will cost the most in terms of interest over 5 years. First calculating the break option, I add the break fee ($4,000) to the balance on the mortgage ($200,000) and then calculate the interest over 5 years @ 6%. (204000/100)*6=$12,240 $12,240*5yrs = $61,200 Second option, calculate interest on $200,000 @ 9% for 6 months = $9,000 add to this the interest on $200,000 for 4 years, 6 months @ the predicted rate of 6.5% (200000/100)*6.5=13000 $13,000*4.5 = $58,500 total interest second option = $67,500 Answer to question (a) is therefore it is cheaper to break. Answer to question (b) is a little tricker. Nothing can be done to change the first 6 months, the question is what must the interest rate come down (or up) to, so that option (b) is the same amount of interest as option (a). So really we have a equation where the amount obtained in a must equal the forumula used in b, which must be solved for interest rate so that we pay $61,200 in interest. First lets subtract the $9,000 which we cant change from $61,200. $61,200 - 9,000 = 52200 $52200 = ($200,000/100) * INTEREST RATE * 4.5 rearranged for INTEREST RATE I think this is : 52200 = 2,000 * INTEREST RATE * 4.5 52200 / 2000 = INTEREST RATE * 4.5 52200 / 2000*4.5 = 5.8% Sound about right anyone? -Al === Subject: The square of a sphere by inverse 19 mathematics Cc: hope9900@verizon.net posting-account=nX7DRwoAAACLHrITBlEp88VmE0MRXW0H CLR 2.0.50727),gzip(gfe),gzip(gfe) Mathematicians Here is a new discovery of uinverse 19 mathematics just discovered, and dicotomy to current mathematics Question: What is the smallest unit proportion of a sphere, and its maximal Proportion. This answer will change understanding of quantum physics and mathematics by its roots. For simplicity 1. Smallest unit of space and a sphere is a conal unit 19, a 19 degree cone with exverse cap . This cone multiplied by itself , by proportionate progression in all dimensions, equals a sphere.The one [361-360] proportion left over is a constant value of (1 over 360), the constant for the expansion of infinity . 2. The maximal square of a sphere[sphere muultiplied by a sphere], is 19 square progression exact [ the root of a sphere is 19, like the square value of 2 is 4]. Space expands in units of 19 values at a constant infinite expansion. 3. Thus we have fixed min and max values in space proportion that because of the constant proportions in space apply to mass/space/ energy, all of which have m and M constants.The square of Mass, dare I even say it, is its maximum unit compaction of its atoms , i.e maximum possible density. Maximum speed is maximum compaction of distance, it has a value by the above proportion, it is not light, but it is the static Aether maximal speed is static because dimension is compacted to 1. No more before you call me crazy! . === Subject: Re: index of a subgroup >> does the index of a subgroup H in a group G equals the number of >> conjugacy classes of H in G? > What would this mean, if true, when H is normal in G? > >> If G is a finite group, can one say that the number of conjugacy >> classes of H in G is less or equal than the index of H in G? Yes: as Jack pointed out, the number of conjugates of H in G is equal > to the index of the normalizer of H in G, [G:N_G(H)]. Recall that N_G(H) = { g in G : gHg^{-1} = H}. Now, if xHx^{-1} = yHy^{-1}, then y^{-1}xHx^{-1}y = H, so y^{-1}x is > in N_G(H). Thus, of x is congruent to y on the left modulo N_G(H), > then x and y determine the same conjugate of H. And if x and y are > congruent on the left modulo N_G(H), so that y^{-1}x = n with n in N_G > (H), then H = nHn^{-1} = (y^{-1}x)H(y^{-1}x)^{-1}, from which you get > xHx^{-1} = yHy^{-1}. So the conjugates of H are in 1-to-1 > correspondence with the left cosets of N_G(H). Since H is contained in N_G(H), you get # Conjugates of H = [G:N_G(H)] <= [G:N_G(H)][N_G(H):H)] = [G:H], the inequality you asked for. Your answer is correct, of course, but I think that it is more simple and natural to define a surjective map from the set of left cosets onto th set of conjugacy classes, as follows: xH |-> xHy^{-1} where _y_ is some element of xH. Then xHy^{-1} is independent of the choice of _y_ and therefore the map is well defined. It is clear that the image of the map is a set of conjugacy classes and, more precisely, it's the set of all conjugacy classes. Jose Carlos Santos === Subject: Kant and Hilbert Here http://www.oup.com/uk/catalogue/?ci=9780198505358 I've just read Immanuel Kant's Critique of Pure Reason is widely taken to be the starting point of the modern period of mathematics while David Hilbert was the last great mainstream mathematician to pursue important nineteenth cnetury(sic) ideas. I don't know whether those are William Bragg Ewald's words or those of an OUP blurb writer. What are the opinions of professional mathematicians about those claims? -- Science is a differential equation. Religion is a boundary condition. --Alan Turing === Subject: Re: Kant and Hilbert > Here http://www.oup.com/uk/catalogue/?ci=9780198505358 I've just read > Immanuel Kant's Critique of Pure Reason is widely taken to be the > starting point of the modern period of mathematics while David Hilbert > was the last great mainstream mathematician to pursue important > nineteenth cnetury(sic) ideas. I don't know whether those are William Bragg Ewald's words or those of > an OUP blurb writer. What are the opinions of professional > mathematicians about those claims? I suspect that most mathematicians, if they think at all about Kant, would definitely exclude him from the modern period, and in fact from mathematics entirely. Of course, modern means different things to different people. I find the statement about Hilbert rather ridiculous: after all, the Riemann hypothesis and Catalan's conjecture are important nineteenth-century ideas. -- === Subject: Re: Kant and Hilbert posting-account=IBUqVwoAAADepmzxVr9iEYD5Z0A483SY rv:1.9.0.1) Gecko/2008070206 Firefox/3.0.1,gzip(gfe),gzip(gfe) On 19 Mar, 12:21, Frederick Williams Immanuel Kant's Critique of Pure Reason is widely taken to be the > starting point of the modern period of mathematics while David Hilbert > was the last great mainstream mathematician to pursue important > nineteenth cnetury(sic) ideas. No practising mathematician today gives a toss about Kant. === Subject: Re: Kant and Hilbert posting-account=n26igQkAAACeF9xA2Ms8cKIdBH40qzwr Gecko/20070505 Iceape/1.0.9 (Debian-1.0.13~pre080614i-0etch1),gzip(gfe),gzip(gfe) On Mar 19, 8:21 am, Frederick Williams Immanuel Kant's Critique of Pure Reason is widely taken to be the > starting point of the modern period of mathematics while David Hilbert > was the last great mainstream mathematician to pursue important > nineteenth cnetury(sic) ideas. I don't know whether those are William Bragg Ewald's words or those of > an OUP blurb writer. What are the opinions of professional > mathematicians about those claims? My amatuer opinion is that Kant does deserve great credit. For instance in some of his writing space and time are coupled so closely that the argument can be made that he is a direct precursor to unified spacetime. This is a corrective of the full scoop that we give Einstein in this department. Since there is not a single equation in Kant's critique I don't think the review does justice by remaining in the mathematical context. Kant seems more about reality as a genera which has been split out into philosophy, mathematics, and physics as modern day shards. We are caught as humans in reality and Kant does his best to address this whereas two of the shards feel quite free to dismiss this part of the problem. Also many modern works do not bother with any scholium or careful presentation anywhere near Kant's level so in their lacking in that department we might argue that these modern works are built with these older works in their foundation. How a modern Principia would differ from Newton's is of interest, and how a modern full dissection like Kant's critique would read might be important. Especially such burdens placed upon quantum theory where many quantum workers divorce themselves from philosophy is a curious ambiguity. The accumulation of information has gone to such an extent that we seem to have given up on philosophy. Yet we would not forsake logic would we? Have we? I like to celebrate the problem as an open system with some unworked and unconstructed parts. Under this paradigm something fundamental has been overlooked so there is a slender hope that the ambiguities could be lifted. I do think that this is the proper spirit to work from. - Tim -- > Science is a differential equation. > Religion is a boundary condition. > --Alan Turing === Subject: Re: Kant and Hilbert Hilbert thought that Kant was speaking about mathematics. He further thought that Kant was wrong. One reason that most profesional mathematicians ignore philosophy is that modern math taakes so long to master that no philosooopher can possibly master both math and philosoophy. Conversely, no mathmeatician has time to master both subjects. A similar break in understanding between physics and math. Up to the time \ of Hilbert, it was possible to master both subjects and many people did. Since then the two subjects have each grown so sophisticated that I doubt that there is a single person in the world who really understands both. > On Mar 19, 8:21 am, Frederick Williams \ Herehttp://www.oup.com/uk/catalogue/?ci=9780198505358I've just read >> Immanuel Kant's Critique of Pure Reason is widely taken to be the >> starting point of the modern period of mathematics while David Hilbert >> was the last great mainstream mathematician to pursue important >> nineteenth cnetury(sic) ideas. >> I don't know whether those are William Bragg Ewald's words or those of >> an OUP blurb writer. What are the opinions of professional >> mathematicians about those claims? My amatuer opinion is that Kant does deserve great credit. > For instance in some of his writing space and time are coupled so > closely that the argument can be made that he is a direct precursor to > unified spacetime. This is a corrective of the full scoop that we give > Einstein in this department. Since there is not a single equation in > Kant's critique I don't think the review does justice by remaining in > the mathematical context. Kant seems more about reality as a genera > which has been split out into philosophy, mathematics, and physics as > modern day shards. We are caught as humans in reality and Kant does > his best to address this whereas two of the shards feel quite free to > dismiss this part of the problem. Also many modern works do not bother with any scholium or careful > presentation anywhere near Kant's level so in their lacking in that > department we might argue that these modern works are built with these > older works in their foundation. How a modern Principia would differ > from Newton's is of interest, and how a modern full dissection like > Kant's critique would read might be important. Especially such burdens > placed upon quantum theory where many quantum workers divorce > themselves from philosophy is a curious ambiguity. The accumulation of information has gone to such an extent that we > seem to have given up on philosophy. Yet we would not forsake logic > would we? Have we? I like to celebrate the problem as an open system > with some unworked and unconstructed parts. Under this paradigm > something fundamental has been overlooked so there is a slender hope > that the ambiguities could be lifted. I do think that this is the > proper spirit to work from. - Tim > -- >> Science is a differential equation. >> Religion is a boundary condition. >> --Alan Turing > -- Using Opera's revolutionary e-mail client: http://www.opera.com/mail/ === Subject: Re: how to find z-values for a stick-out statistic? 5.1),gzip(gfe),gzip(gfe) > How to calculate the z-values for results of the following >experiment? Run 100 trials for which the output can be A, B, C, or D. > A, B, C, and D are hypothesised to be equally probable, > so the expected scores for each outcome is 25. Define the stick-out as the amount by which the HIGHEST SCORE >(H) exceeds the NEXT HIGHEST SCORE (N). Obviously the expected stick-out is not zero, but I'm not sure > what it is. I want to define a statistic for stick-out in two different ways. > First, define S = H-N. This can vary from 0 to 100. > Second, define S' = (H-N)/N. This can vary from 0 to infinity. Is there a way to use chi-squared to get z-values for observed > values of S and S'? Or does something else need to be done? I'd still be grateful for any help with this, even (especially) if there's a simple answer! John === Subject: Re: how to find z-values for a stick-out statistic? >> How to calculate the z-values for results of the following >> experiment? >> Run 100 trials for which the output can be A, B, C, or D. >> A, B, C, and D are hypothesised to be equally probable, >> so the expected scores for each outcome is 25. >> Define the stick-out as the amount by which the HIGHEST SCORE >> (H) exceeds the NEXT HIGHEST SCORE (N). >> Obviously the expected stick-out is not zero, but I'm not sure >> what it is. >> I want to define a statistic for stick-out in two different ways. >> First, define S = H-N. This can vary from 0 to 100. >> Second, define S' = (H-N)/N. This can vary from 0 to infinity. >> Is there a way to use chi-squared to get z-values for observed >> values of S and S'? Or does something else need to be done? I'd still be grateful for any help with this, even (especially) if > there's a simple answer! > John > Complicated probabilities can often be estimated with the Monte Carlo method: Here, choose a random integer from 1 to 4 with 25% chance each, and repeat 100 times. Compute: S = H-N and S' = (H-N)/N Then repeat. As a first approximation, you can put 10,000 values of each in \ an array, or more if you have more memory. What would you want to know about the random variables S and S', the most? David Bernier === Subject: Re: how to find z-values for a stick-out statistic? 5.1),gzip(gfe),gzip(gfe) > On Mar 15, 9:01 pm, John Nagelson > How to \ calculate the z-values for results of the following >> experiment? >> Run 100 trials for which the output can be A, B, C, or D. >> A, B, C, and D are hypothesised to be equally probable, >> so the expected scores for each outcome is 25. >> Define the stick-out as the amount by which the HIGHEST SCORE >> (H) exceeds the NEXT HIGHEST SCORE (N). >> Obviously the expected stick-out is not zero, but I'm not sure >> what it is. >> I want to define a statistic for stick-out in two different >> ways. >> First, define S = H-N. This can vary from 0 to 100. >> Second, define S' = (H-N)/N. This can vary from 0 to infinity. >> Is there a way to use chi-squared to get z-values for observed >> values of S and S'? Or does something else need to be done? > I'd still be grateful for any help with this, even (especially) > if there's a simple answer! Complicated probabilities can often be estimated with the Monte > Carlo method: Here, choose a random integer from 1 to 4 with 25% chance each, > and repeat 100 times. > Compute: > S = H-N > and > S' = (H-N)/N Then repeat. As a first approximation, you can put 10,000 values > of each in an array, or more if you have more memory. What would > you want to know about the random variables S and S', the most? program: totS1 = 0; totS2 = 0; Array[a, 4]; k = 100000; For[counter = 0, counter < k + 1, counter++, For[u = 1, u < 5, u++, a[u] = 0]; For[t = 1, t < 101, t++, r = Random[Integer, {1, 4}]; a[r] += 1]; v = {a[1], a[2], a[3], a[4]}; HH = a[Ordering[v, -2][[2]]]; NN = a[Ordering[v, -2][[1]]]; S1 = HH - NN; S2 = (HH - NN)/NN; totS1 += S1; totS2 += S2]; Print[N[totS1/k], , N[totS2/k]] which output a mean S1 of 3.83 and mean S2 of 0.151. What I'd most like to know, as well as the actual mean values of S1 and S2, is their values higher than the mean at one-tailed significance levels 0.05, 0.01, 0.001, and 10^-n up to 10^-6. John === Subject: Media. Should I use it in this case? Gecko/2009021910 Firefox/3.0.7 (.NET CLR 3.5.30729),gzip(gfe),gzip(gfe) I have a poll analysis that can be based on the following example: How important is good weather for you? And the answers are: Low, Normal, High I created crosses for the values I got with Sex. So I have: Male > Low (10%), Normal (40%) and High (50%) Female > Low (30%), Normal (50%) and High (20%) I would like to have a single value that represents this answer for Male and Female. Should I calculate the median of the three values? What should be the best approach? Miguel === Subject: Re: Media. Should I use it in this case? > I have a poll analysis that can be based on the following example: How important is good weather for you? > And the answers are: Low, Normal, High I created crosses for the values I got with Sex. So I have: Male > Low (10%), Normal (40%) and High (50%) Female > Low (30%), Normal (50%) and High (20%) I would like to have a single value that represents this answer for > Male and Female. > An astrologer likes to have a single number, date of birth + time of birth + longitude and latitude of birth that represents the personality and fate of a person. Those people are considered stupid by psychologists. > Should I calculate the median of the three values? > Now you want to have a single number that represents the importance to everybody's life??? That's even absurd for a single individual. For example, weather is not important to me unless I'm planning a mountain climb. Then the weather only on that day only at that location, an not where I live is important. If I'm a fire fighter, then the weather during the fire season is very important and the rest of the year it's unimportant. That's a situation that you want to distort into 'normal importance'? I would consider any research deliberately discarding such significant data to be worthless, as worthless as paid for correct results research. Your approach has an extreme built in bias. > What should be the best approach? > To consider yourself stupid and study astrology or consider yourself conniving and sell your soul to the corporate devil. === Subject: Re: Media. Should I use it in this case? <20090319083703.Y27955@agora.rdrop.com> Gecko/2009021910 Firefox/3.0.7 (.NET CLR 3.5.30729),gzip(gfe),gzip(gfe) > I have a poll analysis that can be based on the following example: > How important is good weather for you? > And the answers are: Low, Normal, High > I created crosses for the values I got with Sex. So I have: > Male > Low (10%), Normal (40%) and High (50%) > Female > Low (30%), Normal (50%) and High (20%) > I would like to have a single value that represents this answer for > Male and Female. An astrologer likes to have a single number, date of birth + time of > birth + longitude and latitude of birth that represents the personality > and fate of a person. Those people are considered stupid by psychologists. > Should I calculate the median of the three values? Now you want to have a single number that represents the > importance to everybody's life??? That's even absurd for > a single individual. For example, weather is not important > to me unless I'm planning a mountain climb. Then the weather > only on that day only at that location, an not where I live > is important. If I'm a fire fighter, then the weather during the fire season > is very important and the rest of the year it's unimportant. > That's a situation that you want to distort into 'normal > importance'? I would consider any research deliberately discarding > such significant data to be worthless, as worthless as paid for correct > results research. Your approach has an extreme built in bias. > What should be the best approach? To consider yourself stupid and study astrology or consider > yourself conniving and sell your soul to the corporate devil. What?!!!! The question Is weather important... was just an example ... It wasn't the fact. Basically I need to have a value that says if more people answered High then Low. But representing that with only one number. So basically to convert this percentage values to a representation of Yes/No answer and give the percentage of yes ... I am just looking for a different way to show it instead of having 3 percentages for male and 3 for female. I would like to give a similar idea of the results but with only one value for male and one for female. I think Median would be ok ... === Subject: Re: .9 repeating Nntp-Posting-Host: hera.cwi.nl > OK, then. So it's better for me to say that standard analysts find > theories such as NSA or those with adjacent points to be > didactically unsound and mathematically barren -- so that's why they > prefer standard analysis. Better? > Much better. > Though in that case, I have never knowingly met a standard analyst. > > I use the word standard analyst to refer to someone who > prefers standard analysis to nonstandard analysis. Aha. So you think that somebody who prefers standard analysis to nonstandard analysis finds that NSA is didacticaly unsound and mathematically barren? > Not on Usenet, > > Usenet's full of standard analysts. Indeed, virtually > anyone who considers MR to be a so-called crank would > qualify as a standard analyst. Why? Because the standard analyst rejects a claim that is not based on any theory at all? You are always misinterpreting. It is not the use of a non-standard theory that makes somebody a crank. It is the making of claims not based on sound theories of whatever form that makes somebody a crank. But in the case of MR, he does not even use an unsound theory, he ony proclaims that there is a real number adjacent to 0, and that is all he does do. -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: .9 repeating Nntp-Posting-Host: hera.cwi.nl ... > 3. Can I associate this theory with MR, without being > academically dishonest? > 4. Can I associate this theory with AP, without being > academically dishonest? Both would be dishonest because that would suggest that they were meaning that all along. -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: .9 repeating > ... > 3. Can I associate this theory with MR, without being > academically dishonest? > 4. Can I associate this theory with AP, without being > academically dishonest? Both would be dishonest because that would suggest that > they were meaning that all along. [Piggybacking to Lwalker:] Lwalker, why would you want to assert a stronger connection between your theory and either MR or AP than was suggested by? If your goal is to rehabilitate cranks into non-cranks by finding a way to say Look! These ideas aren't as crazy as all that, then your presenting ideas which are not theirs and which anyone can see are not theirs will do nothing for you. If your goal is merely to amuse yourself and others by looking for rigorous developments of their very informal ideas, you can do that without insisting that your developments are what they really meant, in some sense. If your goal is to lure them away from crankish behavior by gifting them with a non-crank version of their ideas and intuitions, then you seem to have missed the important step of convincing them that what you have is a non-crank version of /their/ ideas and intuitions. If you add that, though, your behavior looks to me indistinguishable from the behavior of the people you refer to as anti-cranks, other than the complaints against anti-cranks, of course. I don't mean to suggest that these are your only possible motives. These are just the ones that occur to me after some consideration. I would appreciate it if you would explain what you are trying to do, even though (I suspect) you have done so before. Jim Burns === Subject: Re: .9 repeating Nntp-Posting-Host: hera.cwi.nl ... > Suppose I want a theory in which (2) is true. If the theory > I come up with isn't rigorous, then I'm called a crank > for trying to pass off a nonrigorous theory as rigorous. If > the theory I come up with _is_ rigorous, then I'm called > academically dishonest for trying to insert something > that's not part of the original (standard) theory in order > to make a new theory. The dishonesty was your statement that it was MR's intention, i.e. hiding behind MR. -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: .9 repeating posting-account=-d2gtwoAAACIgCVyog_mrAWs-fwWRSD9 CLR 1.1.4322; InfoPath.1; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > It is .0R ending in a 1. > the end of an infinite series... meditate on this concept until you attain samadhi. anyone ever see the British series look around you? it's a parody of those educational science TV shows you used to see a few decades ago. the one on math starts with what is the largest number? a few folks make random guesses, then the announcer intones all these people are wrong. the largest number is 47,000. scientists have speculated about the existence of numbers larget than this, but at this point there is no proof of their existence. === Subject: Re: .9 repeating posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > It is .0R ending in a 1. > the end of an infinite series... Of course, infinite sequences, being order isomorphic to omega, have no end. But this doesn't mean that there can't be a set of digits that are isomorphic to something other than omega. In this case, that which is infinite can have an end after all. > anyone ever see the British series look around you? it's a parody of > those educational science TV shows you used to see a few decades ago. > the one on math starts with what is the largest number? a few folks > make random guesses, then the announcer intones all these people are > wrong. the largest number is 47,000. Ultrafinitism. There are several ultrafinitists here at sci.math. But most of them declare the largest natural number to be much larger than 47,000. Usually, their largest number > scientists have speculated about > the existence of numbers larget than this, but at this point there is > no proof of their existence. To the sci.math ultrafinitists, there is proof of the existence of the number 47,000 -- because there exist proof of the existence of the number googolplex (i.e., 10^10^1000), since there is no physical object === Subject: Re: .9 repeating <874oxzioby.fsf@phiwumbda.org> posting-account=-d2gtwoAAACIgCVyog_mrAWs-fwWRSD9 CLR 1.1.4322; InfoPath.1; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) >> We all know that in standard analysis, the only standard >> real that 0.999... can possibly equal is 1. >> It's all well and fine to play around with theories where >> 0.999... < 1, but let's not confuse this playing around with what >> cranks like Mitch are saying. It is good to be charitable in >> interpreting claims, but it is disrespectful to pretend that the >> claimant meant something else entirely. > And calling someone a crank, as Hughes just did in the quoted > paragraph (what cranks like Mitch are saying), isn't disrespectful > at all? Yes, I certainly am being disrespectful when I call Mitch a crank. > But I'm not being disrespectful in the name of charity. I am fairly > representing his claims and I'm judging them as they are. > That the only way to avoid being disrespectful is to assume that MR > is a 100% wrong crank and that we can't consider rigorous theories > in which MR isn't 100% wrong? Of course, you may consider rigorous theories in which 0.999... < 1, > though I don't personally know any of those theories. (You suggested > some theory to that effect, but I didn't read that part of your post. > Sorry.) But do you honestly believe that Mitch was speaking of some particular > theory in which 0.999... < 1? Or do you rather think that, in this > post, Mitch was speaking about the honest-to-gosh real numbers and > making a claim contrary to fact? > Clearly Hughes and I disagree on what constitutes respecting a > poster who contradicts standard analysis. I never claimed to respect Mitch. This is the main difference. You > show your so-called respect by pretending that Mitch is talking about > something else entirely. -- > When you go to class today, if your professor talks about algebraic > number theory, or misuses Galois Theory[,] I want you to carefully > notice how you feel. Hold on to that feeling so that you never forget > it. --James S. Harris, on channeling rage via Galois theory. well, if you've done much fortran 66 programming, you might know that in that version, it's entirely legal to declare 1 = 0 and from then on in the program, whenever you did something like if x==x+1 it would evaluate as true, y=2+1 would set y to 2, etc. etc. etc. they took that highly useful capacity away in fortran 77, which i assume is the reason why people don't program in fortran as much any more. === Subject: Re: .9 repeating Nntp-Posting-Host: hera.cwi.nl ... > well, if you've done much fortran 66 programming, you might know that > in that version, it's entirely legal to declare 1 = 0 and from then > on in the program, whenever you did something like if x==x+1 it > would evaluate as true, y=2+1 would set y to 2, etc. etc. etc. they > took that highly useful capacity away in fortran 77, which i assume is > the reason why people don't program in fortran as much any more. If you ever have read the fortran 66 standard you would have known that that is complete nonsense. -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: .9 repeating posting-account=U44YcwkAAAAbGXB70Qr7gA3kornmKE4i Gecko/20080922 Ubuntu/7.10 (gutsy) Firefox/2.0.0.17,gzip(gfe),gzip(gfe) > well, if you've done much fortran 66 programming, you might know that > in that version, it's entirely legal to declare 1 = 0 I don't think it is. (I'm fairly sure the last version of fortran I used was as recent as 66...) What you could do was define SUBROUTINE CHANGE(A, B) A = B END Then a quick call to CHANGE(1, 0) has the effect you are looking for. (But I suspect something like CHANGE (57, 391) would be more reliable.) Brian Chandler === Subject: Re: .9 repeating >> well, if you've done much fortran 66 programming, you might know that >> in that version, it's entirely legal to declare 1 = 0 >I don't think it is. (I'm fairly sure the last version of fortran I >used was as recent as 66...) >What you could do was define >SUBROUTINE CHANGE(A, B) >A = B >END >Then a quick call to CHANGE(1, 0) has the effect you are looking for. >(But I suspect something like CHANGE (57, 391) would be more >reliable.) That depended on the computer and compiler. Sometimes you might get a changed constant, sometimes nothing would happen (the compiler may generate a temporary variable/memory location initialized with the first constant, changed by CHANGE to the second, and subsequently ignored/discarded), and sometimes you'd get a memory access violation because the address of a read-only memory location was passed to CHANGE, which attempted to store its second parameter into the read- only first parameter address). === Subject: Re: .9 repeating Nntp-Posting-Host: hera.cwi.nl ... > I suspect that Brian Chandler nailed it -- it's not a Fortran thing > per se. It's a call by reference thing in a language that can't > express the notion of a read-only parameter. In the standard it is expressed and stated that a program where a read-only actual argument is given where the formal parameter is assigned to, is invalid. That compilers do not detect it is another matter. There are more things the standard states are invalid that compilers in general do not detect. -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: .9 repeating posting-account=nHkyWQoAAAAZj13mfknn7vPxoYn-Mvx3 CLR 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30; InfoPath.1),gzip(gfe),gzip(gfe) > ... > I suspect that Brian Chandler nailed it -- it's not a Fortran thing > per se. It's a call by reference thing in a language that can't > express the notion of a read-only parameter. In the standard it is expressed and stated that a program where a read-only > actual argument is given where the formal parameter is assigned to, is > invalid. That compilers do not detect it is another matter. There are > more things the standard states are invalid that compilers in general > do not detect. Section 8.3.2 is the relevant one and states that if a subroutine redefines the value of a formal parameter then the actual parameter must be a variable or array reference rather than an expression. A numeric literal is, of course, a special case of an expression. === Subject: Re: .9 repeating Nntp-Posting-Host: hera.cwi.nl > well, if you've done much fortran 66 programming, you might know that > in that version, it's entirely legal to declare 1 = 0 > > I don't think it is. (I'm fairly sure the last version of fortran I > used was as recent as 66...) > > What you could do was define > > SUBROUTINE CHANGE(A, B) > A = B > END > > Then a quick call to CHANGE(1, 0) has the effect you are looking for. And a quick look at the standard would have shown that that call was invalid according to the standard. That some compilers made it possible does not make it correct. And I know compilers where that did allow that call but where 1 would *not* change value (they used call by value/return where the caller did the value/return insofar as it was appropriate). -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: .9 repeating > >>well, if you've done much fortran 66 programming, you might know that >>in that version, it's entirely legal to declare 1 = 0 I don't think it is. (I'm fairly sure the last version of fortran I > used was as recent as 66...) What you could do was define SUBROUTINE CHANGE(A, B) > A = B > END Then a quick call to CHANGE(1, 0) has the effect you are looking for. (But I suspect something like CHANGE (57, 391) would be more > reliable.) Brian Chandler Oh, but it _is_ perfectly legal in SQL, where it simply means FALSE. Quite useful, though, for checking out that any synomyms ( &table_name ) needed for further processing really are present and okay in a database environment: select count(*) from &table_name where 1=0; If that doesn't generate an error then you can safely proceed. Otherwise better give up for the moment and have a look why the data aren't there. Han de Bruijn === Subject: Re: .9 repeating <16ffb$49c21c92$82a1e228$10653@news1.tudelft.nl> posting-account=nHkyWQoAAAAZj13mfknn7vPxoYn-Mvx3 CLR 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30; InfoPath.1),gzip(gfe),gzip(gfe) >>well, if you've done much fortran 66 programming, you might know that >>in that version, it's entirely legal to declare 1 = 0 > I don't think it is. (I'm fairly sure the last version of fortran I > used was as recent as 66...) > What you could do was define > SUBROUTINE CHANGE(A, B) > A = B > END > Then a quick call to CHANGE(1, 0) has the effect you are looking for. > (But I suspect something like CHANGE (57, 391) would be more > reliable.) > Brian Chandler Oh, but it is perfectly legal in SQL, where it simply means FALSE. Quite useful, though, for checking out that any synomyms ( &table name ) > needed for further processing really are present and okay in a database > environment: select count(*) from &table name where 1=0; If that doesn't generate an error then you can safely proceed. Otherwise > better give up for the moment and have a look why the data aren't there. Most computer professionals understand the difference between assignment and comparison. The assertion that had been made was that Fortran-66 supports an assignment statement in which the assignment target is a numeric literal. e.g. 1 = 0 In Fortran-66 the comparison operator for equality is .EQ. e.g. if ( 1 .eq. 0 ) goto 100 According to the Fortran-66 standard, the former is invalid. 7.1.1.1 Arithmetic Assignment Statement. An arithmetic assignment statement is of the form: v = e where v is a variable name or array element name of type other than logical and e is an arithmetic expression. I suspect that Brian Chandler nailed it -- it's not a Fortran thing per se. It's a call by reference thing in a language that can't express the notion of a read-only parameter. === Subject: Re: .9 repeating <16ffb$49c21c92$82a1e228$10653@news1.tudelft.nl> posting-account=U44YcwkAAAAbGXB70Qr7gA3kornmKE4i Gecko/20080922 Ubuntu/7.10 (gutsy) Firefox/2.0.0.17,gzip(gfe),gzip(gfe) >>well, if you've done much fortran 66 programming, you might know that >>in that version, it's entirely legal to declare 1 = 0 > I don't think it is. (I'm fairly sure the last version of fortran I > used was as recent as 66...) > What you could do was define > SUBROUTINE CHANGE(A, B) > A = B > END > Then a quick call to CHANGE(1, 0) has the effect you are looking for. > (But I suspect something like CHANGE (57, 391) would be more > reliable.) > Brian Chandler Oh, but it _is_ perfectly legal in SQL, where it simply means FALSE. What is? The sequence of symbols 1 = 0? Yes, of course -- but because it means something completely different from what it would in Fortran, where = is the assignment operator. > Quite useful, ... Isn't there a constant FALSE? Brian Chandler === Subject: Re: .9 repeating <3Aptl.81692$183.57185@newsfe09.ams2> posting-account=-d2gtwoAAACIgCVyog_mrAWs-fwWRSD9 CLR 1.1.4322; InfoPath.1; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Is below one by the infinitely nonzero small. It is the closest thing > to one. > Wrong, and it's a FAQ. > Add the infinitely small and you get one. > What's smaller than zero? > ;-) > -0 is less than 0 > In 32-bit binary arithmetic, which your computer uses, > 1000 0000 000 0000 0000 0000 0000 0000 > is less than > 0000 0000 0000 0000 0000 0000 0000 0000 >http://academic.evergreen.edu/projects/biophysics/technotes/misc/bin ... My computer is 2's complement. No such representation. In one's complement, where -0 lives, it has exactly the same size as > +0. Only a different sign. The magnitude is identical. If the comparison operator makes one larger than the other, it is a > computer format artifact and not a math artifact. 1's-complement is NOT. 0000 becomes 1111. > 2's-complement is NOT and increment. 0000 becomes 1111 + 0001, > which gives (1)0000 and sets the overflow flag. Both 1's- and 2's-complement uses signed integers, > the artefact is in over- or underflow as a result of using > a finite set of integers. -0 exists, flagged by the overflow bit. When I was working flight simulators 30 years ago a rather > annoying bug would keep popping up. We'd zoom off down > the runway when suddenly the entire world would flip upside- > down, the view being behind us for single frame. Then it > became normal again. To find the problem I had a colleague > fly (it was a Kuwait B747 sim, later stolen by the Iraqis in the > first Gulf War) and had him hold attitude while I looked at > the data. I traced the trouble to the sin() subroutine library > (we were using fixed point 16-bit DEC PDP-11s) which > had been written by the parent company in Binghamton NY > and shipped across the Atlantic. The sin() calculation was in 1's-complement (for speedy > computation), so as the plane rotated through zero pitch we > had sin(0) = 0 but at sin(1111 1111 1111 1111) the image > flipped because of overflow. The next number was > sin(1111 1111 1111 1110) and it flipped back to normal. Using 2's-complement in the subroutine cured the problem > but only after I was given special permission to alter a library > routine that had been approved by Quality Assurance. After all, > everything was working fine except for this one condition. Then later I was sent to Pittsburgh for a USAIR DC9 sim for > which we were supplying the visual display only, and again the > problem recurred. The host sim out of Binghamton was still > using the older 1's-complement sin() routine. I had no access to the host computer and was working with > an American team installing the sim that had not seen the problem > before. Naturally they insisted the problem was our visual display and > not the data they were sending; it took a week to get them to > change it and left some egg on faces. I tried not to be smug > and say I told you so! but I did chuckle when I was having > a drink after work with my own British team. > http://design.osu.edu/carlson/history/lesson13.html That's my war story.- Hide quoted text - - Show quoted text - heh. you oughta go over to the serious statistics newsgroups and ask innocently are the statistical functions in Excel reliable? i can't even come up with a war story, but i recall one particular mathematical games column in scientific american, way back when calculators were just becoming common/inexpensive. he was puzzled by the large submission of the indentical wrong solution one particular month, when he realized that that was what you would calculate, using an 8 digit calculator. friend of mine has a cell phone whose calculator goes to two decimal places accuracy, always, no matter what you do. i was baffled/ frustrated until somebody suggested it was intended for dollars and cents, despite not having any $ sign. === Subject: Re: .9 repeating > Is below one by the infinitely nonzero small. It is the closest thing > to one. > Wrong, and it's a FAQ. > Add the infinitely small and you get one. > What's smaller than zero? > ;-) > -0 is less than 0 > In 32-bit binary arithmetic, which your computer uses, > 1000 0000 000 0000 0000 0000 0000 0000 > is less than > 0000 0000 0000 0000 0000 0000 0000 0000 >http://academic.evergreen.edu/projects/biophysics/technotes/misc/bin_... My computer is 2's complement. No such representation. In one's complement, where -0 lives, it has exactly the same size as > +0. Only a different sign. The magnitude is identical. If the comparison operator makes one larger than the other, it is a > computer format artifact and not a math artifact. 1's-complement is NOT. 0000 becomes 1111. > 2's-complement is NOT and increment. 0000 becomes 1111 + 0001, > which gives (1)0000 and sets the overflow flag. Both 1's- and 2's-complement uses signed integers, > the artefact is in over- or underflow as a result of using > a finite set of integers. -0 exists, flagged by the overflow bit. When I was working flight simulators 30 years ago a rather > annoying bug would keep popping up. We'd zoom off down > the runway when suddenly the entire world would flip upside- > down, the view being behind us for single frame. Then it > became normal again. To find the problem I had a colleague > fly (it was a Kuwait B747 sim, later stolen by the Iraqis in the > first Gulf War) and had him hold attitude while I looked at > the data. I traced the trouble to the sin() subroutine library > (we were using fixed point 16-bit DEC PDP-11s) which > had been written by the parent company in Binghamton NY > and shipped across the Atlantic. The sin() calculation was in 1's-complement (for speedy > computation), so as the plane rotated through zero pitch we > had sin(0) = 0 but at sin(1111 1111 1111 1111) the image > flipped because of overflow. The next number was > sin(1111 1111 1111 1110) and it flipped back to normal. Using 2's-complement in the subroutine cured the problem > but only after I was given special permission to alter a library > routine that had been approved by Quality Assurance. After all, > everything was working fine except for this one condition. Then later I was sent to Pittsburgh for a USAIR DC9 sim for > which we were supplying the visual display only, and again the > problem recurred. The host sim out of Binghamton was still > using the older 1's-complement sin() routine. I had no access to the host computer and was working with > an American team installing the sim that had not seen the problem > before. Naturally they insisted the problem was our visual display and > not the data they were sending; it took a week to get them to > change it and left some egg on faces. I tried not to be smug > and say I told you so! but I did chuckle when I was having > a drink after work with my own British team. > http://design.osu.edu/carlson/history/lesson13.html That's my war story.- Hide quoted text - - Show quoted text - heh. you oughta go over to the serious statistics newsgroups and ask innocently are the statistical functions in Excel reliable? Is Excel reliable, never mind its stat functions? http://www.androcles01.pwp.blueyonder.co.uk/Euler.xls Note the discrepancy of 6.28361E-10. The first four digits of the mantissa match 2pi = 6.283 But then, pi is not exactly represented either, nor can it be. I have no immediate plans for this: http://tinyurl.com/da4wn8 but eventually I'll be forced into it as it becomes all the rage. i can't even come up with a war story, but i recall one particular mathematical games column in scientific american, way back when calculators were just becoming common/inexpensive. he was puzzled by the large submission of the indentical wrong solution one particular month, when he realized that that was what you would calculate, using an 8 digit calculator. friend of mine has a cell phone whose calculator goes to two decimal places accuracy, always, no matter what you do. i was baffled/ frustrated until somebody suggested it was intended for dollars and cents, despite not having any $ sign. Perhaps it was intended for pounds and pence, despite not having a £ sign. I was quite familiar with £sd based on the currency of the Roman Empire. 's' stood for shillings and 'd' stood for denarii. http://home.clara.net/brianp/money.html Having six half-crowns in one's pocket meant delivering newspapers all week including Sundays. http://www.freewebs.com/paulportcoins/1916%20HALF%20CROWN%20REV.jpg Being a young vagabond I quickly learned to embezzle excess newspapers and sell them myself, raising my income to £5. After all, I was responsible for bagging them, taking them out, delivering them, delivering kiddy comics and women's magazines and collecting the money on Saturdays for the newsagent, who had his hands full with a dozen other kids who complained of the cold and rain and didn't show up. Having complete control of my route and able to remember what paper went to every house, I was a star that gave him no trouble at all. We could all get five bob for finding him a new customer. I never did, all new customers were MINE. But I could never take a day off, MY customers that only I knew about would be contacting the newsagent asking why they had no paper and that wouldn't do at all. I had to keep them completely happy at all times. So I remained a star, out in all weather with the worst route, furthest from the shop, ploughing through mud on unmade roads. And making money. Toward the end I had a run of very good luck and managed to find him quite few new customers, collecting my five bob for each one. Years later he'd given up that business and ran a pub, and I went for a pint there and had a chat with him. He said he always knew something was wrong, his books didn't balance, but never thought it was me. It was a Fagin/Artful Dodger relationship, I was the richest kid in the neighbourhood. Oliver Twist never stood a chance. Then I turned 16, left school and went to work as an apprentice electrical fitter in the Dockyard; my income FELL to £3 / 3s / 0d. It was soon back up again, though. I was rewinding burnt-out motors for drills, washing machines and vacuum cleaners with H M. Government's copper wire. They expected me to take 12 hours to rewind a teleprinter armature, I'd do a dozen in that time. 144 hours to spend with a 40 hour week? Easiest job in the world. It was some years later that I developed a conscience and became honest. === Subject: Re: .9 repeating posting-account=-d2gtwoAAACIgCVyog_mrAWs-fwWRSD9 CLR 1.1.4322; InfoPath.1; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Is below one by the infinitely nonzero small. It is the closest thing > to one. Add the infinitely small and you get one. Mitch Raemsch your infinitely nonzero small = .000000000000 repeating an infinite series of zeros. i think you can guess what that equals. for more rigorous proof: X=.99999999.... 10*X = 9.999999999..... Ok so far? So, that means that 10*X=9+X. which leaves us with X=1. i.e., .9999999.... = 1. can't argue with 'rithmetic. don't they teach this in high school any more? === Subject: Re: .9 repeating posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > Is below one by the infinitely nonzero small. It is the closest thing > to one. > Add the infinitely small and you get one. > your infinitely nonzero small = .000000000000 repeating an infinite > series of zeros. i think you can guess what that equals. > for more rigorous proof: > X=.99999999.... > 10*X = 9.999999999..... > Ok so far? > So, that means that > 10*X=9+X. > which leaves us with X=1. i.e., .9999999.... = 1. can't argue with > 'rithmetic. Once again, z's proof works in standard analysis, but that doesn't preclude the existence of an alternate theory in which .999... < 1. Standard analysis isn't the only possible analysis that can exist! Indeed, this entire thread has been devoted to seeing whether or not there can be a theory in which one can: 1) Represent positive infinitesimals with digits. 2) Find a smallest positive infinitesimal. Rigorous theories have been found in which either 1) or 2) are true, but one in which both is true is lacking. > don't they teach this in high school any more? They teach calculus, which is based on standard analysis, in high school. But nonstandard analysis is hardly taught anywhere, HS or college. There can be perfectly legitimate, rigorous theories in which the Archimedean property fails, and there can exist nonzero infinitesimals. But first, z needs to think outside the box of standard analysis and realize that there are other types of analysis available. === Subject: Re: .9 repeating posting-account=-d2gtwoAAACIgCVyog_mrAWs-fwWRSD9 CLR 1.1.4322; InfoPath.1; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Is below one by the infinitely nonzero small. It is the closest thing > to one. > Add the infinitely small and you get one. > your infinitely nonzero small = .000000000000 repeating an infinite > series of zeros. i think you can guess what that equals. > for more rigorous proof: > X=.99999999.... > 10*X = 9.999999999..... > Ok so far? > So, that means that > 10*X=9+X. > which leaves us with X=1. i.e., .9999999.... = 1. can't argue with > 'rithmetic. Once again, z's proof works in standard analysis, but > that doesn't preclude the existence of an alternate > theory in which .999... < 1. Standard analysis isn't > the only possible analysis that can exist! Indeed, this entire thread has been devoted to seeing > whether or not there can be a theory in which one can: 1) Represent positive infinitesimals with digits. > 2) Find a smallest positive infinitesimal. Rigorous theories have been found in which either 1) or > 2) are true, but one in which both is true is lacking. > don't they teach this in high school any more? They teach calculus, which is based on standard analysis, > in high school. But nonstandard analysis is hardly taught > anywhere, HS or college. There can be perfectly legitimate, rigorous theories in > which the Archimedean property fails, and there can exist > nonzero infinitesimals. But first, z needs to think > outside the box of standard analysis and realize that > there are other types of analysis available. but how do you work in a system where the standard means of solving algebraic equations by arithmetic manipulation are not valid? === Subject: Re: .9 repeating sha1:rsEmh7IT7tXzS0vuLlQlW5yiEXI= > Indeed, this entire thread has been devoted to seeing > whether or not there can be a theory in which one can: 1) Represent positive infinitesimals with digits. > 2) Find a smallest positive infinitesimal. Rigorous theories have been found in which either 1) or > 2) are true, but one in which both is true is lacking. I'm not sure it's so hard to do both, you know, if you don't require that *every* infinitesimal is represented. Let's let R' be the set {f | f: (w + 1) -> {0,...,9}} u {1.000...0} That is, let R' be the set consisting of (w + 1) digit sequences (this is, according to some, an abuse of the word sequence, but let's continue). There's a natural ordering on R', namely the pointwise ordering. 0.000...0 < 0.000...1 < 0.000...9 and 0.999...0 < 0.999...4 < 0.999...9 There's an order-preserving embedding [0,1) -> R', defined as follows: We map x in [0,1) to the sequence (x1)(x2)(x3)...0, where xi is the i'th digit of x, with the proviso that if x has two different decimal expansions (one ending in all 9s and the other in all 0s), we use the expansion ending in all 0s. Now we make an arbitrary choice: we interpret infinite digit sequences (note: here I'm interpreting syntactic objects, not elements of R) (x1)(x2)(x3)... |-> (x1)(x2)(x3)...0 unless the sequence ends in all 9s (x1)(x2)(x3)... |-> (x1)(x2)(x3)...9 if it ends in all 9s. Now according to this, 0.999... is the largest number in R', analogous to the largest element in [0,1). There is a smallest positive number, which is 0.000...1, but I haven't assigned a digit sequence to this number, because I interpreted only omega sequences. If I extend this interpretation (like Mitch?) to the sequence 0.000...1, then this guy has a name. This structure satisfies Mitch's desiderata, more or less. But it's also completely uninteresting. I don't see why anyone would want to study it at all. I see no reason to believe that analysis done in this structure would be interesting or fruitful and I don't see that this structure matches any of my intuitions about numbers better than classical analysis. As far as interesting developments, of course, I could be wrong. Maybe there are some keen features of a structure something like this, but I certainly wouldn't invest my time on the hope that such features exist. I'm sure that this post is chock-full of errors, since I tossed it off as quickly as possible. My point is merely that satisfying (1) and (2) together are neither enough to vindicate MR as a solid mathematical thinker *nor* enough to justify the claim that theories in which (1) and (2) are true are worth pursuing. -- Jesse F. Hughes Besides, discoverers are too proud to kiss butt. Indiana Jones would never kiss some academic's ass to get published, and neither will I. --James Harris === Subject: Re: .9 repeating sha1:lTy8RhGWSdiKsYyjQju7RYopzb8= > Let's let R' be the set {f | f: (w + 1) -> {0,...,9}} u {1.000...0} Oops. Ignore the {1.000...0}, since that came from an earlier draft and merely complicated the presentation. That is, let R' be the set consisting of (w + 1) digit sequences > (this is, according to some, an abuse of the word sequence, but let's > continue). -- Jesse F. Hughes Of course, my ability to admit my mistakes and correct them is a trait that many of you seem to never have properly appreciated. -- JSH, discussing his 1463rd proof of Fermat's Last Theorem. === Subject: Re: .9 repeating posting-account=PalVlwoAAAA1TKsKLVCsNe7APQhpWkGd 1.5); .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Physician, heal thyself. - Jesus H. Christ, self-proclaimed son > of Gawd, Could you give me a verse reference for Jesus proclaiming himself the Son of God? Jonathan === Subject: Re: .9 repeating Originator: richard@cogsci.ed.ac.uk (Richard Tobin) >Could you give me a verse reference for Jesus proclaiming himself the >Son of God? Matthew 26:64, though translations vary. -- Richard -- Please remember to mention me / in tapes you leave behind. === Subject: Re: .9 repeating > Physician, heal thyself. - Jesus H. Christ, self-proclaimed son >> of Gawd, Could you give me a verse reference for Jesus proclaiming himself the > Son of God? God? Who said that? I'm sure I said Gawd. But anyway, yes, I'll give it to you just as soon as you give me a verse reference to SR says that the speed of light is c in every inertial frame. See the new improved but shortened version of Scruffy Rat, now animated with shadows, especially written for children and those unfortunate enough to be simple-minded incompetents. http://www.androcles01.pwp.blueyonder.co.uk/SR4kids/special_relativity_for_c\ h ildren.htmWanna play hardball, Doolin? http://tinyurl.com/bw55m7Your \ turn. === Subject: Re: .9 repeating <87fxhb3uhx.fsf@phiwumbda.org There may well be interesting geometries in \ which adjacency plays a > role. It simply doesn't follow that Mitch is therefore > intellectually respectable. There are systems in combinatorial geometry that have only finitely many points and do not assume any continuous embedding. The ones I am familiar with do not model adjacency, but I'm sure that could be trivially remedied. However, those also do not have any property of representation of points by arbitrary decimal sequences or hyper-sequences. - Tim === Subject: Re: .9 repeating <87fxhb3uhx.fsf@phiwumbda.org> posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > There may well be interesting geometries in which adjacency plays a > role. It simply doesn't follow that Mitch is therefore > intellectually respectable. > There are systems in combinatorial geometry that have only finitely > many points and do not assume any continuous embedding. > However, those also do not have any property of representation of > points by arbitrary decimal sequences or hyper-sequences. I mentioned finite geometries earlier in this thread. So it appears that adjacent points aren't grounds for being called crank, until we start having infinitely many such points or start labelling them with infinitesimals. Then again, finitism itself -- especially ultra finitism -- still appears to be grounds for being labeled a crank. === Subject: Re: .9 repeating <87fxhb3uhx.fsf@phiwumbda.org> crank, until we start having infinitely many such points or start > labelling them with infinitesimals. Then again, finitism itself -- especially _ultra_finitism -- still > appears to be grounds for being labeled a crank. How thick are you? It has been explained to you numerous times now that any such assertions in and of themselves *DO NOT CAUSE ANYONE TO BE CALLED A CRANK*. Ultrafinitism has been seriously discussed on at least a few occasions in sci.math, without any accusations of crankdom flying around - because the people discussing it displayed sound reasoning. It is repeated inability to recognise flaws in both their reasoning and in misuse and confusion of terminology that gets one labelled a crank. - Tim === Subject: Re: .9 repeating <87r60v3ur4.fsf@phiwumbda.org> <87prgeaodu.fsf@alatheia.dsl.inet.fi> \ <87mybi3liu.fsf@phiwumbda.org> <87k56mamcf.fsf@alatheia.dsl.inet.fi> posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > Of course, there is no suggestion that standard set theorists reject > a non-standard theory such as ZFA, just because the accept a (the) > standard theory. > Indeed there is no such suggestion. I am completely baffled by > lwalke's talk about acceptance and rejection of this or that. After > all, it's not as if those whose main preoccupation is standard set > theory go about saying things like I reject this foul circular > non-sense! Away with you johnny-come-latelys and your repugnant crank > notions!. Huh? To me, this thread so far is nothing except such rejection of such theories -- maybe not in those exact words, of course, but nonetheless a rejection. === Subject: Re: .9 repeating sha1:EHyc6G0ziR6ePMfMJKRCQVJiF/Q= >> Of course, there is no suggestion that standard set theorists reject >> a non-standard theory such as ZFA, just because the accept a (the) >> standard theory. >> Indeed there is no such suggestion. I am completely baffled by >> lwalke's talk about acceptance and rejection of this or that. After >> all, it's not as if those whose main preoccupation is standard set >> theory go about saying things like I reject this foul circular >> non-sense! Away with you johnny-come-latelys and your repugnant crank >> notions!. Huh? To me, this thread so far is nothing except such rejection > of such theories -- maybe not in those exact words, of course, > but nonetheless a rejection. You're mistaken. The folks you defend have never suggested any mathematical theory. Rather, they've made various pronouncements. These pronouncements are evidently (AP partially excepted[1]) about the honest-to-gosh reals. Mitch, for instance, has never suggested that it would be interesting to come up with an alternative to R. Rather, he has said that 0.999... < 1 is so obvious that it needs no proof. If there is anyone going around accepting certain claims as absolute truths and rejecting others, it is, of course, the cranks. No self-respecting mathematician would claim that there is no interesting mathematical theory in which 0.999... < 1. On the contrary, all one should say is that we've encountered no such (or, perhaps, only one such) theory[2]. How is it that *no one* has suggested that Katz and Katz are cranks? According to your ideas about standard and non-standard theories, a whole slew of posters should be rising up and proclaiming that Katz and Katz are morons or heretics, but it isn't happening. Why is that? Footnotes: [1] AP doesn't talk about the honest-to-gosh reals, but he *does* claim that the honest-to-gosh reals are nonsense and that his AP-reals are something like the only acceptable number system. [2] Or was it an infinite collection of very similar such theories? I'm afraid I didn't really read Katz and Katz. -- Jesse F. Hughes Of course, my ability to admit my mistakes and correct them is a trait that many of you seem to never have properly appreciated. -- JSH, discussing his 1463rd proof of Fermat's Last Theorem. === Subject: Re: .9 repeating <87r60v3ur4.fsf@phiwumbda.org> <87prgeaodu.fsf@alatheia.dsl.inet.fi> \ <87mybi3liu.fsf@phiwumbda.org> <87k56mamcf.fsf@alatheia.dsl.inet.fi> \ <873ada1ro4.fsf@phiwumbda.org> posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > Huh? To me, this thread so far is nothing except such rejection > of such theories -- maybe not in those exact words, of course, > but nonetheless a rejection. > How is it that *no one* has suggested that Katz and Katz are cranks? > According to your ideas about standard and non-standard theories, a > whole slew of posters should be rising up and proclaiming that Katz > and Katz are morons or heretics, but it isn't happening. Why is that? It's because there are two issues at stake: 1) Representing infinitesimals by digits. 2) Allowing for a smallest positive infinitesimal. MR has argued for both 1) and 2). Katz allows for 1), but not 2). Katz has an established theory in which 1) holds. No one has an established theory in which 2) holds. Therefore, declaring 1) doesn't make one a crank, but declaring 2) does make one a crank. I previously included 1) in my list of claims that only cranks make. Bader's posting of the Katz link proved me wrong about 1). No one's posted a link to a rigorous theory in which 2) holds. So I stand by my assertion that those who claim 2) are cranks, at least until someone finds an established theory in which 2) holds, which may never happen. === Subject: Re: .9 repeating lwalke3@lausd.net a .8ecrit : > Huh? To me, this thread so far is nothing except such rejection > of such theories -- maybe not in those exact words, of course, > but nonetheless a rejection. >> How is it that *no one* has suggested that Katz and Katz are cranks? >> According to your ideas about standard and non-standard theories, a >> whole slew of posters should be rising up and proclaiming that Katz >> and Katz are morons or heretics, but it isn't happening. Why is that? It's because there are two issues at stake: 1) Representing infinitesimals by digits. > 2) Allowing for a smallest positive infinitesimal. MR has argued for both 1) and 2). > Katz allows for 1), but not 2). > Katz has an established theory in which 1) holds. > No one has an established theory in which 2) holds. Oh, as you are using Surreal numbers in a funny way (you model of numbers born on day w is not even a group, as 1/w is here, but not 2/w), you should try the cut 1/w^On, which is indeed smaller than all infinitesimal numbers (but larger than up-arrow or tiny-two, of course) But ,as usual, you are completely missing the point, Troll Too bad, the thoery of improper gaps in Surreals (see Onag) is quite fun... Therefore, declaring 1) doesn't make one a crank, but > declaring 2) does make one a crank. I previously included 1) in my list of claims that only > cranks make. Bader's posting of the Katz link proved > me wrong about 1). No one's posted a link to a rigorous > theory in which 2) holds. So I stand by my assertion > that those who claim 2) are cranks, at least until > someone finds an established theory in which 2) holds, > which may never happen. === Subject: Re: .9 repeating sha1:cabItmV1lcve/WS5msbQTh5DadA= >> Huh? To me, this thread so far is nothing except such rejection >> of such theories -- maybe not in those exact words, of course, >> but nonetheless a rejection. >> How is it that *no one* has suggested that Katz and Katz are cranks? >> According to your ideas about standard and non-standard theories, a >> whole slew of posters should be rising up and proclaiming that Katz >> and Katz are morons or heretics, but it isn't happening. Why is that? It's because there are two issues at stake: 1) Representing infinitesimals by digits. > 2) Allowing for a smallest positive infinitesimal. MR has argued for both 1) and 2). > Katz allows for 1), but not 2). > Katz has an established theory in which 1) holds. > No one has an established theory in which 2) holds. Therefore, declaring 1) doesn't make one a crank, but > declaring 2) does make one a crank. Indeed it does! Declaring that (2) is true when in fact there is no obvious mathematical theory of satisfying (2) and otherwise suitably similar to our concept of real numbers is clearly a crankish thing to do. Declaring that (2) is true in some theory or other when it really *is* true in that theory is perfectly reasonable. Why do you think that declaring (2) is simply true (whatever that means) is reasonable? I confess that I don't even know what it *means* to declare that there is a least positive number, without reference to some particular theory or structure of numbers. There is a least positive natural number, there is not a least positive real number. There may or may not be a natural extension R' of R such that there is a least positive number in R'. All of these claims are sensible. Claiming that (2) is simply true is silly, just as claiming There are no non-well-founded sets or There are no infinite sets is silly. These claims are true in some set theories and false in others. What could be clearer? > I previously included 1) in my list of claims that only > cranks make. Bader's posting of the Katz link proved > me wrong about 1). No one's posted a link to a rigorous > theory in which 2) holds. So I stand by my assertion > that those who claim 2) are cranks, at least until > someone finds an established theory in which 2) holds, > which may never happen. Well, your new assertion seems fairly plausible. Now, the only question is: why don't you see that this would be a good reason to call someone a crank? If there's no theory satisfying (2), why should they claim that (2) is true? -- Just be aware that anti-Cantorians are sick of being called crackpots, and the day will soon come when the crankiest Cantorians will eat their words, and this rot will be extricated from mathematics. -- Tony Orlow, an anti-Cantorian ready to rumble === Subject: Re: .9 repeating <87r60v3ur4.fsf@phiwumbda.org> <87prgeaodu.fsf@alatheia.dsl.inet.fi> \ <87mybi3liu.fsf@phiwumbda.org> <87k56mamcf.fsf@alatheia.dsl.inet.fi> \ <87prgexuvr.fsf@phiwumbda.org> posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > I previously included 1) in my list of claims that only > cranks make. Bader's posting of the Katz link proved > me wrong about 1). No one's posted a link to a rigorous > theory in which 2) holds. So I stand by my assertion > that those who claim 2) are cranks, at least until > someone finds an established theory in which 2) holds, > which may never happen. > Well, your new assertion seems fairly plausible. Now, the only > question is: why don't you see that this would be a good reason to > call someone a crank? If there's no theory satisfying (2), why should > they claim that (2) is true? If one can really prove that no consistent theory can possibly satisfy (2)? Or, at least, if no theory that I decide is worth consider can prove that (2) is true. Then I'd accept that the claimant of (2) is wrong. I still wouldn't post in the thread that the claimant of (2) is a crank, since I don't like calling posters cranks. (This is why I always put the word crank in quotes, or even add the phrase so-called in front of the word.) When someone posts something that I reject, such as 2 + 2 = 5 or (2) (if I end up rejecting it), then I would just avoid posting in the thread. That's a key difference between me and the standard analysts, who wouldn't hesitate to call the poster a crank. > Just be aware that anti-Cantorians are sick of being called crackpots, > and the day will soon come when the crankiest Cantorians will eat > their words, and this rot will be extricated from mathematics. > -- Tony Orlow, an anti-Cantorian ready to rumble It's interesting how Hughes chose to quote TO in his sig, since it's very relevant to what we are discussing here. The day which TO predicts will soon come would be the day that someone comes up with a proof that ZFC is inconsistent, a proof that even the most reputable mathematicians accept as a proof of phi&~phi in ZFC. I disagree with TO that the day is imminent -- indeed, I suspect that there'll never be a proof of ~Con(ZFC). But if (and that's a big if ) that day comes, then the Cantorians (i.e., the standard set theorists -- I avoid use of the word Cantorian) would be forced to eat their words. Suddenly, the adherents of ZFC would be the new so-called cranks, since their theory would be the theory proved to be inconsistent. The mathematician Ed Nelson is currently working on a proof that PA (and hence ZFC) is inconsistent. Nelson is the mathematician most likely to make TO's prediction come true. === Subject: Re: .9 repeating sha1:muf/EpKTctqb/1EqtVLzEkMj61k= >> I previously included 1) in my list of claims that only >> cranks make. Bader's posting of the Katz link proved >> me wrong about 1). No one's posted a link to a rigorous >> theory in which 2) holds. So I stand by my assertion >> that those who claim 2) are cranks, at least until >> someone finds an established theory in which 2) holds, >> which may never happen. >> Well, your new assertion seems fairly plausible. Now, the only >> question is: why don't you see that this would be a good reason to >> call someone a crank? If there's no theory satisfying (2), why should >> they claim that (2) is true? If one can really prove that no consistent theory can possibly > satisfy (2)? Or, at least, if no theory that I decide is worth > consider can prove that (2) is true. Then I'd accept that the claimant of (2) is wrong. I still > wouldn't post in the thread that the claimant of (2) is a > crank, since I don't like calling posters cranks. (This is > why I always put the word crank in quotes, or even add the > phrase so-called in front of the word.) When someone posts something that I reject, such as 2 + 2 = 5 > or (2) (if I end up rejecting it), then I would just avoid > posting in the thread. That's a key difference between me and > the standard analysts, who wouldn't hesitate to call the > poster a crank. Yes, that is a difference. Thus, when MR says something that is not obviously wrong (i.e., it is *possible* that there is a theory in which it is true), you defend him. When he says something that is obviously wrong, you are silent. I just don't see this behavior as all that virtuous. One can correct someone's errors without calling them cranks (though, if you do, you will soon be tempted to call a spade a spade). >> Just be aware that anti-Cantorians are sick of being called crackpots, >> and the day will soon come when the crankiest Cantorians will eat >> their words, and this rot will be extricated from mathematics. >> -- Tony Orlow, an anti-Cantorian ready to rumble It's interesting how Hughes chose to quote TO in his sig, > since it's very relevant to what we are discussing here. I didn't choose it. My signature quotes are randomly selected. They often appear to fit the topic just because of the tendency of humans to interpret meaning in coincidence. > The day which TO predicts will soon come would be the day > that someone comes up with a proof that ZFC is inconsistent, > a proof that even the most reputable mathematicians accept > as a proof of phi&~phi in ZFC. I disagree with TO that the day is imminent -- indeed, I > suspect that there'll never be a proof of ~Con(ZFC). But > if (and that's a big if ) that day comes, then the > Cantorians (i.e., the standard set theorists -- I > avoid use of the word Cantorian) would be forced to eat > their words. Suddenly, the adherents of ZFC would be the > new so-called cranks, since their theory would be the > theory proved to be inconsistent. That simply does not follow. Suppose that I worked my entire life in ZFC and ZFC turns out to be inconsistent. It does not follow that I'm retroactively a crank. Nor that I'm a crank henceforth. Only if I continue to insist that ZFC is consistent, despite clear proof that it is not, would I become a crank. You still don't get the term crank. It isn't about individual mistaken judgments. It is about repeated fallacious reasoning or pronouncements without argument. -- I'm talking about mathematics--hard, brutal, extreme ... pushing your mind beyond the limits to understand what no one else can because they're afraid to risk it all, to lose their freaking worthless minds in the push to know. --James Harris, for the Nike Derivator === Subject: Re: .9 repeating I disagree with TO that the day is imminent -- indeed, I > suspect that there'll never be a proof of ~Con(ZFC). But > _if_ (and that's a big _if_) that day comes, then the > Cantorians (i.e., the standard set theorists -- I > avoid use of the word Cantorian) would be forced to eat > their words. Suddenly, the adherents of ZFC would be the > new so-called cranks, since their theory would be the > theory proved to be inconsistent. You mean like how Frege is considered to be a crank, since his / Grundgesetze der Arithmetik/ was shown to be inconsistent by Russell? Er, no. If those standard set theorists were to continue to insist that ZFC is consistent after it was shown to be inconsistent, then it would be reasonable to call them cranks. But they wouldn't. Instead they would start working on an alternative axiomatisation of set theory. === Subject: Re: .9 repeating <87tz5qaoj6.fsf@alatheia.dsl.inet.fi> posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > My labels standard set theorist, standard analyst, etc., > is my effort to avoid the label standard mathematician, > which previously led to an argument on how nonstandard > analysis is still standard mathematics. So a standard > analyst explicitly accepts standard analysis, as opposed to > nonstandard analysis. > What sort of acceptance is this? You'd be hard pressed to find any > standard analyst who forcefully proclaims I hereby reject > non-standard analysis except perhaps in the sense of finding it > didactically unsound or mathematically barren. OK, then. So it's better for me to say that standard analysts find theories such as NSA or those with adjacent points to be didactically unsound and mathematically barren -- so that's why they prefer standard analysis. Better? === Subject: Re: .9 repeating <87tz5qaoj6.fsf@alatheia.dsl.inet.fi OK, then. So it's better for me to say \ that standard analysts find > theories such as NSA or those with adjacent points to be > didactically unsound and mathematically barren -- so that's why they > prefer standard analysis. Better? Much better. Though in that case, I have never knowingly met a standard analyst. Not on Usenet, at a couple of universities, in correspondence, or at a conference. Quite a few haven't been interested in it, but that's a very long way from pronouncing it didactically unsound and mathematically barren. - Tim === Subject: Re: .9 repeating posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > OK, then. So it's better for me to say that standard analysts find > theories such as NSA or those with adjacent points to be > didactically unsound and mathematically barren -- so that's why they > prefer standard analysis. Better? > Much better. > Though in that case, I have never knowingly met a standard analyst. I use the word standard analyst to refer to someone who prefers standard analysis to nonstandard analysis. It refers to the majority of mathematicians -- hence the use of the word standard. Therefore, I'd be surprised if Little had never met one, to the extent that he's met any mathematicians at all. > Not on Usenet, Usenet's full of standard analysts. Indeed, virtually anyone who considers MR to be a so-called crank would qualify as a standard analyst. If one doesn't like the name standard analyst, then unfortunately this is the best name I'd come up with for the supporters of the status quo. Earlier so-called cranks have called them Cantorians, while I used to call them standard mathematicians. Until I come up with a better name for them, I find standard analyst to be suitable, since it directly refers to their adherence to standard analysis. === Subject: Re: .9 repeating sha1:w/R2QbspX4b9sREqcrHG7UQgHGA= >> OK, then. So it's better for me to say that standard analysts find >> theories such as NSA or those with adjacent points to be >> didactically unsound and mathematically barren -- so that's why they >> prefer standard analysis. Better? >> Much better. >> Though in that case, I have never knowingly met a standard analyst. I use the word standard analyst to refer to someone who > prefers standard analysis to nonstandard analysis. It > refers to the majority of mathematicians -- hence the > use of the word standard. Therefore, I'd be surprised > if Little had never met one, to the extent that he's met > any mathematicians at all. His point is that he's never met an analyst who claimed that NSA was didactically unsound or mathematically barren. There are plenty of analysts that don't care for NSA. That is, they prefer to work with classical analysis, for whatever reasons. There seem to be very few who argue that classical analysis is the one true theory and that NSA is fundamentally flawed. -- Often times, when [...] looking for a sense of adventure, I'd doodle with math equations. Often too, when pressures of regular life were really bothering me, I'd go for the adventure of fiddling with math. -- James S. Harris, a man of many mathematical adventures! === Subject: Re: .9 repeating > My labels standard set theorist, standard analyst, etc., > is my effort to avoid the label standard mathematician, > which previously led to an argument on how nonstandard > analysis is still standard mathematics. So a standard > analyst explicitly accepts standard analysis, as opposed to > nonstandard analysis. > What sort of acceptance is this? You'd be hard pressed to find any > standard analyst who forcefully proclaims I hereby reject > non-standard analysis except perhaps in the sense of finding it > didactically unsound or mathematically barren. > OK, then. So it's better for me to say that standard analysts > find theories such as NSA or those with adjacent points to be > didactically unsound and mathematically barren -- so that's > why they prefer standard analysis. Better? Lwalke3 In looking over your posts, it seems you think that ZFC is somehow unfriendly to non-standard number lines (where non-standard analysis can be \ done). If so then you are seriously mistaken. The surreal number line, for \ example, can be formulated within ZFC but it can also stand on its own as an \ axiomatic theory (see John Conway, On Numbers and Games). ZFC is definitely friendly to non-standard analysis. But 1=0.99999... is always true: 1 = 9*(1/9) = 9*0.11111... = 0.99999... . There are no decimal notations for non-standard number lines, at least that \ I have seen, so1=0.99999... cannot be applied to them. The class of surreal numbers satisfy the properties of an ordered field so 1 = 9*(1/9) is \ true on the surreal number line. The usual decimal notations don't make any \ sense in non-standard analysis. k === Subject: Re: .9 repeating posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > OK, then. So it's better for me to say that standard analysts > find theories such as NSA or those with adjacent points to be > didactically unsound and mathematically barren -- so that's > why they prefer standard analysis. Better? > But 1=0.99999... is always true: 1 = 9*(1/9) = 9*0.11111... = 0.99999... . > There are no decimal notations for non-standard number lines, at least that > I have seen, so1=0.99999... cannot be applied to them. As it turns out, Katz has indeed found a way to assign digits to nonstandard reals. One can follow the link in Bader's post to learn more. > The class of > surreal numbers satisfy the properties of an ordered field so 1 = 9*(1/9) \ is > true on the surreal number line. Katz uses the hyperreals, not the surreals, in the link. I agree that assigning digits to the surreals is probably awkward at best. === Subject: Re: .9 repeating posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2009021910 Firefox/3.0.7,gzip(gfe),gzip(gfe) > OK, then. So it's better for me to say that standard analysts > find theories such as NSA or those with adjacent points to be > didactically unsound and mathematically barren state which particular writings of this standard analyst you have in mind. Or are you just brainstorming film treatment characterizations for bad guys for your latest blockbuster The Revenge Of The Non-Standards: Plutonium's Return? MoeBlee === Subject: Re: .9 repeating posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > OK, then. So it's better for me to say that standard analysts > find theories such as NSA or those with adjacent points to be > didactically unsound and mathematically barren > state which particular writings of this standard analyst you have in > mind. The reason I speak in generalizations is that moment I mention a specific mathematician, whether an author or a sci.math poster, I'll be accused of misrepresenting that poster -- as has already occurred so many times. This request is just a trap for someone to accuse me of misrepresentation yet again. As for nonstandard analysis, there's a reason that it's labeled nonstandard -- because it's not standard. So it's almost tautological to say that standard analysts prefer standard analysis to NSA. And one of the reasons mentioned earlier that someone might prefer standard analysis to NSA is that they find the latter to be didactically unsound and mathematically barren. As for adjacent points, no rigorous theory has been made in which adjacent points are possible. And my lone attempt to come up with a rigorous theory of adjacent points -- by my modifying of the Katz theory -- was declared to be academically dishonest. Therefore, the theory of adjacent points is will always be considered a crank theory, no matter how rigorous an attempt to formulate them is. === Subject: Re: .9 repeating <87tz5qaoj6.fsf@alatheia.dsl.inet.fi As for adjacent points, no rigorous \ theory has been made > in which adjacent points are possible. And my lone > attempt to come up with a rigorous theory of adjacent > points -- by my modifying of the Katz theory -- was > declared to be academically dishonest. Nobody has claimed that coming up with any such theory was academically dishonest. The only similar claim I've seen was one I made, of you being unethical in *saying that your theory was MR's*. Being a deliberate misattribution, that *is* unethical. If that was in fact the so-called declaration of academic dishonesty to which you are referring, you are at best mistaken and at worst dishonest again, in mis-stating the claim. - Tim === Subject: Re: .9 repeating posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > As for adjacent points, no rigorous theory has been made > in which adjacent points are possible. And my lone > attempt to come up with a rigorous theory of adjacent > points -- by my modifying of the Katz theory -- was > declared to be academically dishonest. > Nobody has claimed that coming up with any such theory was > academically dishonest. The only similar claim I've seen was one I > made, of you being unethical in *saying that your theory was MR's*. > Being a deliberate misattribution, that *is* unethical. After you and others mentioned this, I admit that I was wrong to associate MR with the theory that I had derived from Katz. Of the two statements: 1) One can represent infinitesimals with digits. 2) There are adjacent infinitesimals. MR has both 1) and 2), but Katz has only 1). So I was unethical in associating the two because they disagree on 2), even though they agree on 1). So far, MR's theory is not rigorous. If I make too many changes to MR, then I would be academically dishonest in saying that it's still MR's theory. What I want to know is, what's a slight change I can make to what MR is saying, so that it will be accepted as a rigorous theory, but so that I can still associate the theory with MR? Too slight a change and the theory's not rigorous. Too big a change and it's no longer MR's theory (and then unethical to call it MR's theory). > If that was in fact the so-called declaration of academic dishonesty > to which you are referring, you are at best mistaken and at worst > dishonest again, in mis-stating the claim. === Subject: Re: .9 repeating sha1:UtpWT3jD2MyqFqI9C4negPzqq5g= > What I want to know is, what's a slight change I can > make to what MR is saying, so that it will be accepted > as a rigorous theory, but so that I can still > associate the theory with MR? Too slight a change and > the theory's not rigorous. Too big a change and it's > no longer MR's theory (and then unethical to call it > MR's theory). To be perfectly honest, just as soon as you write down some axioms (i.e., just as soon as there is a *theory*), then it is not MR's theory. It may be inspired by Mitch, but it is not Mitch's theory because Mitch has no theory. He has a couple of pronouncements. For a theory to count as Mitch's theory, it should be proposed by Mitch. -- Jesse F. Hughes Had you told it like it was, it wouldn't be like it is. -- Albert King === Subject: Re: .9 repeating posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2009021910 Firefox/3.0.7,gzip(gfe),gzip(gfe) > OK, then. So it's better for me to say that standard analysts > find theories such as NSA or those with adjacent points to be > didactically unsound and mathematically barren > state which particular writings of this standard analyst you have in > mind. The reason I speak in generalizations is that moment I > mention a specific mathematician, whether an author or a > sci.math poster, I'll be accused of misrepresenting that > poster -- as has already occurred so many times. This > request is just a trap for someone to accuse me of > misrepresentation yet again. You really are nuts; you know that, right? Look, if you won't be specific, then there's no way to evaluate your claim. And if you are specific but people have grounds to contest your claim about the mathematician, then that's fair. You can't rationally argue, My generalization is true; but I won't name a specific instance because that would expose my claim about the specific instance to possible refutation. So, I'll just continue the make the generalization, which no one can refute because I've not given them a specific to refute. So you do realize how nuts that is, right? > As for nonstandard analysis, there's a reason that it's > labeled nonstandard -- because it's not standard. So > it's almost tautological to say that standard analysts > prefer standard analysis to NSA. Hey, watch that: Don't SWITCH the argument like that. The question was not what standard analysts PREFER but rather your assertion that standard analyists won't recoginize the legitimacy of non-standard analysis (or whatever your actual words) as mathematical research. And if you won't name a specific person and his writings then we can't evaluate the BASIS on which he might hold such a view against non- standard analysis. MoeBlee === Subject: Re: .9 repeating posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > The reason I speak in generalizations is that moment I > mention a specific mathematician, whether an author or a > sci.math poster, I'll be accused of misrepresenting that > poster -- as has already occurred so many times. This > request is just a trap for someone to accuse me of > misrepresentation yet again. > Look, if you won't be specific, then there's no way to evaluate your > claim. And if you are specific but people have grounds to contest your > claim about the mathematician, then that's fair. You can't rationally > argue, My generalization is true; but I won't name a specific > instance because that would expose my claim about the specific > instance to possible refutation. So, I'll just continue the make the > generalization, which no one can refute because I've not given them a > specific to refute. > So you do realize how nuts that is, right? The problem is that there are many prejudices that people may have, but no one will ever admit. Most men who judge women solely on their looks will never admit that they are that superficial. Most women who judge men solely on their income will never admit that they are that superficial. Most anti-Semites will never admit that they hate Jews. (But maybe this is a bad example, since I don't want to invoke Godwin's law here.) Most dog lovers will never admit that they care more about their dog than who the President is. (I found this last one via a Google search.) And so on. And so I believe that most standard analysts believe that those who hold unorthodox beliefs, such as the existence of adjacent infinitesimals, are cranks. So let's take the first example. Suppose I accuse a man of judging women solely on their looks, and he replies, Of course I don't. Give an exact quote of when I ever claimed to be that superficial. Of course I'll fail, since he likely never uttered such words. But that doesn't mean that he really isn't superficial. Therefore, it is unrealistic for MoeBlee to insist that I find an exact quote of someone expressing his or her prejudices since they are rarely admitted. Whether I'm asking a anti-Semite whether they believe believers in the Jewish G-d must be k*k*s or a standard analyst whether they believe that all believers in adjacent infinitesimals must be cranks. (Once again, maybe I should replace Jews and k*k*s with blacks and n*gg*rs in order to avoid Godwin's law.) If I accuse many holders of one of these prejudices to be prejudiced, of course they will deny it and accuse me of misrepresenting their beliefs, whether it's a social or a mathematical prejudice. It's only human nature both to have and to deny such prejudices. > And if you won't name a specific person and his writings then we can't > evaluate the BASIS on which he might hold such a view against non- > standard analysis. This is a reasonable stance to have, except when some sort of prejudice is involved, since hardly anyone will ever admit to being prejudiced. Almost anyone I name will deny that they are prejudiced against so-called cranks. BTW, I have amended the list so that mere adherence to NSA isn't grounds for being called a crank, but belief in adjacent infinitesimals still is. === Subject: Re: .9 repeating sha1:/hBWaaYKWE/LtIc1hrKy96CFAIQ= > The problem is that there are many prejudices that people > may have, but no one will ever admit. > Most men who judge women solely on their looks will never > admit that they are that superficial. > Most women who judge men solely on their income will never > admit that they are that superficial. > Most anti-Semites will never admit that they hate Jews. > (But maybe this is a bad example, since I don't want to > invoke Godwin's law here.) > Most dog lovers will never admit that they care more about > their dog than who the President is. (I found this last > one via a Google search.) > And so on. And so I believe that most standard analysts > believe that those who hold unorthodox beliefs, such as > the existence of adjacent infinitesimals, are cranks. I must say you are superior! Most bigots do not admit their prejudices, but you are willing to come right out and state yours explicitly: you think that standard analysts are weak-minded bigots. That is your point in discussing prejudices, right? (By the way, have you read Mitch's non-responses to my questions? His mathematical insight is somehow lacking, no? Do you *really* believe that he's proposing an alternative mathematical theory and not just making silly, unjustified pronouncements?) -- Jesse F. Hughes Jesse: Quincy, you should trust me more. Quincy (age 4): Baba, I never trust you. And I've got good reasons. === Subject: Re: .9 repeating <8763i6xoo8.fsf@phiwumbda.org> posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > And so on. And so I believe that most standard analysts > believe that those who hold unorthodox beliefs, such as > the existence of adjacent infinitesimals, are cranks. > Most bigots do not admit their prejudices, but you are willing to come > right out and state yours explicitly: you think that standard > analysts are weak-minded bigots. I don't deny that I have such prejudices. Most humans have some sort of prejudice. > That is your point in discussing prejudices, right? I can't make the standard analysts' prejudices go away, any more than they can make my prejudices disappear. And I believe that many so-called cranks have a certain type of prejudice as well -- they're often prejudiced against the axiomatic method in the first place. Because they see that the consequences that they find undesirable (such as .999... = 1) can be derived from the axioms, they simply reject axiomatization period. I believe that if one doesn't agree with the consequences of the axioms, that person should choose different axioms, but these cranks still believe that the concept of axiomatization itself is to blame for the undesirable or unintuitive consequences. And so I can only reduce the prejudices. My hope is that by seeing that some of the cranks' claims can be made rigorous, the standard analysts would be less likely to see them as cranks, since one would think that rigorous theories are less likely to be cranky than those which lack rigor. And my hope is that by seeing that axioms can be written to prove some of their claims, the so-called cranks would be less likely to reject axiomatization. So far I have yet to succeed on any front, since overcoming prejudice is extremely difficult. > (By the way, have you read Mitch's non-responses to my questions? > His mathematical insight is somehow lacking, no? Do you *really* > believe that he's proposing an alternative mathematical theory and not > just making silly, unjustified pronouncements?) I'm not sure, since based on his line: Are you responding to me? I cannot tell. he's not even sure he's being addressed. Time will tell what MR's actual opinion of standard R is. === Subject: Re: .9 repeating sha1:MB5nhSjM9l2ExVY4XW5XOqWP26Y= > I can't make the standard analysts' prejudices go away, any > more than they can make my prejudices disappear. You haven't shown that any such prejudice exists. Your opinion that they *are* prejudiced is merely an expression of your own skewed view of the world. -- Jesse F. Hughes Hey look, Captain, next time someone wants to tie us up, let's put up a fight. --Adventures by Morse === Subject: Re: .9 repeating > OK, then. So it's better for me to say that standard analysts find > theories such as NSA or those with adjacent points to be > didactically unsound and mathematically barren -- so that's why they > prefer standard analysis. Better? Many people probably prefer standard analysis, for any number of reasons, including tradition and familiarity. It's simply not a matter of accepting or rejecting anything in any interesting sense. What's at issue are rather such questions as whether studying non-standard analysis sheds any light on questions that interest analysts, whether teaching calculus on basis of non-standard analysis is in any way beneficial, and so on. That is, it's entirely possible to find non-standard analysis, ill-founded set theory, and so on, a mathematical waste of time without any suggestion there's something incorrect or dubious in their mathematical development. -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon man nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: .9 repeating sha1:pkjDhIR0NPAVf6JjHgsH0hepDWI= >> My labels standard set theorist, standard analyst, etc., >> is my effort to avoid the label standard mathematician, >> which previously led to an argument on how nonstandard >> analysis is still standard mathematics. So a standard >> analyst explicitly accepts standard analysis, as opposed to >> nonstandard analysis. >> What sort of acceptance is this? You'd be hard pressed to find any >> standard analyst who forcefully proclaims I hereby reject >> non-standard analysis except perhaps in the sense of finding it >> didactically unsound or mathematically barren. OK, then. So it's better for me to say that standard analysts > find theories such as NSA or those with adjacent points to be > didactically unsound and mathematically barren -- so that's > why they prefer standard analysis. Better? This claim is probably closer to the truth, since there may well be some analysts who would say this. But it's utter nonsense to think that this is a prevailing opinion among real analysts. Just as it's asinine to think that, because I was working in ZFA, I thought that the axiom of foundation was mathematically barren (or vice versa). You might as well claim that number theorists don't believe there is any number between 0 and 1, or that working in the theory of rationals is didactically unsound or mathematically barren. Most mathematicians don't have this fixation with acceptance and rejection of different theories. -- So yeah, do the wrong math, and use the ring of algebraic integers wrong, without understanding its quirks and real mathematical properties, and you can think you proved Fermat's Last Theorem when you didn't. -- James S. Harris on hobbies === Subject: Re: .9 repeating <877i2m1u4m.fsf@phiwumbda.org> posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > OK, then. So it's better for me to say that standard analysts > find theories such as NSA or those with adjacent points to be > didactically unsound and mathematically barren -- so that's > why they prefer standard analysis. Better? > You might as well claim that number theorists don't believe there is > any number between 0 and 1, or that working in the theory of rationals > is didactically unsound or mathematically barren. Most > mathematicians don't have this fixation with acceptance and rejection > of different theories. But they do have this fixation with labeling certain theories as crank theories. And so far, the theory of adjacent infinitesimals is still labeled a crank theory, since they discourage any attempt to make adjacent infinitesimals into a rigorous theory. === Subject: Re: .9 repeating <87tz5qaoj6.fsf@alatheia.dsl.inet.fi> <877i2m1u4m.fsf@phiwumbda.org But they do have this fixation with labeling \ certain theories > as crank theories. No, some have a fixation with labelling certain collections of utterances as crank theories, on the basis that they *aren't actually theories*. They're disjointed pronouncements that aren't logically connected, and in a number of cases outright contradictory. It is not the assertions themselves that make any of them cranks. That is what you repeatedly misunderstand. It is their continuing inability to even recognize what it means to mathematically support their assertions. AP for example, claims that he has proved the Riemann hypothesis, except that the Riemann hypothesis is a proposition in complex analysis, not AP-number anlysis. What's more, his statement of the Riemann hypothesis is wrong, and his proof of it has numerous steps with no logical justification and a few outright errors. The latter two could be interpreted as simple error, but the volume of such errors and his repeated denial that they exist already puts him most of the way to crankhood, and his inability to distinguish the first seals the matter. Similar properties hold for MR, JSH, and WM - numerous obvious logical errors and omissions in their arguments together with an inability to demonstrate that they even know what they are claiming, and complete lack of ability to recognize and correct their own deficiencies. - Tim === Subject: Re: .9 repeating > OK, then. So it's better for me to say that standard analysts > find theories such as NSA or those with adjacent points to be > didactically unsound and mathematically barren -- so that's > why they prefer standard analysis. Better? > You might as well claim that number theorists don't believe there is > any number between 0 and 1, or that working in the theory of rationals > is didactically unsound or mathematically barren. Most > mathematicians don't have this fixation with acceptance and rejection > of different theories. > But they do have this fixation with labeling certain theories > as crank theories. And so far, the theory of adjacent > infinitesimals is still labeled a crank theory, since they > discourage any attempt to make adjacent infinitesimals into a > rigorous theory. In one of your posts you claimed this could not be done in ZFC and that it is a super hard problem. Not at all, this is really easy to do in ZFC. Here is one way it can be done: Let (R, <=) and (Z, <=) be the sets of reals and integers, respectively, under their usual orders. Define the ordered pair (r, i) in RxZ, with r in \ R and i in Z, to be an lwalke3 real and let all such pairs define the set of \ lwalke3 reals. The set of lwalke3 reals is ordered such that for any i and \ j in Z and any r and s in R: (a) (r, i) < (s, j) if r < s. (b) (r, i) < (r, j) if i < j. (c) (r, i) = (s, j) if r = s and i = j. We can identify the real number r with its lwalke3 counterpart (r, 0) and define an infinitesimal as an lwalke3 real (r, i) with i=/=0. Each lwalke3 \ real has aleph_zero infinitesimals around itself and each infinitesimal has \ a successor and a predecessor. Is this what you want? k === Subject: Re: .9 repeating posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > OK, then. So it's better for me to say that standard analysts > find theories such as NSA or those with adjacent points to be > didactically unsound and mathematically barren -- so that's > why they prefer standard analysis. Better? > You might as well claim that number theorists don't believe there is > any number between 0 and 1, or that working in the theory of rationals > is didactically unsound or mathematically barren. Most > mathematicians don't have this fixation with acceptance and rejection > of different theories. > But they do have this fixation with labeling certain theories > as crank theories. And so far, the theory of adjacent > infinitesimals is still labeled a crank theory, since they > discourage any attempt to make adjacent infinitesimals into a > rigorous theory. > In one of your posts you claimed this could not be done in ZFC and that it > is a super hard problem. Not at all, this is really easy to do in ZFC. > Here is one way it can be done: > Let (R, <=) and (Z, <=) be the sets of reals and integers, respectively, > under their usual orders. Define the ordered pair (r, i) in RxZ, with r in > R and i in Z, to be an lwalke3 real and let all such pairs define the set \ of > lwalke3 reals. The set of lwalke3 reals is ordered such that for any i and > j in Z and any r and s in R: > (a) (r, i) < (s, j) if r < s. > (b) (r, i) < (r, j) if i < j. > (c) (r, i) = (s, j) if r = s and i = j. > We can identify the real number r with its lwalke3 counterpart (r, 0) and > define an infinitesimal as an lwalke3 real (r, i) with i=/=0. Each lwalke3 > real has aleph zero infinitesimals around itself and each infinitesimal has > a successor and a predecessor. Either I or others have proposed similar concepts in various threads -- in particular, I recall someone in an AP thread that we use a certain subset of QxZ, with the lexicographic order. K h has suggested that we use RxZ with the lexicographic order. > Is this what you want? Well, let's see if it works, by asking the following questions about the theory. 1. Does 2) (i.e., the concept of adjacent infinitesimals) hold in this theory? Of course it does. K h has specifically constructed it to hold. 2. Does 1) (i.e., the idea of representing infinitesimals with digits) hold in this theory? Here's the trouble. It's not obvious that the elements (r,i) can have digits. One would think that maybe we could have (1,-1) correspond to .999...999, but some of the others could be tricky. 3. Can I associate this theory with MR, without being academically dishonest? Probably not. Associating Katz with MR was dishonest because MR argues for both 1) and 2), but only 1) is true in Katz. Similarly, associating K h with MR would be dishonest because only 2) is true, unless we can prove that 1) works as well. 4. Can I associate this theory with AP, without being academically dishonest? AP has expressed desires about his AP-reals that are even more stringent than MR's. For example, AP has mentioned the AP-real equivalent to the standard real known as the Chapernowne constant: C = .12345678910111213141516171819202122232425262728... The AP-Chapernowne constant would consist of the digits of all the other AP-reals concatenated together, as: .123456789101112131415...999...9997999...9998999...9999 in which there are as many nines in the last 999...9999 as there are in .999...9999 (which in turn has the same number of digits in .123456789...999...9999 itself)! It is this concept which, not only seems difficult in Katz or K h, but in any ZFC-based theory! This is why I questioned whether these numbers are possible in ZFC. So it may be possible to formulate MR in ZFC, but AP is a different matter. This is one reason that I'll no longer consider AP and MR to have the same theory. === Subject: Re: .9 repeating > Let (R, <=) and (Z, <=) be the sets of reals and integers, respectively, > under their usual orders. Define the ordered pair (r, i) in RxZ, with r > in > R and i in Z, to be an lwalke3 real and let all such pairs define the > set of > lwalke3 reals. The set of lwalke3 reals is ordered such that for any i > and > j in Z and any r and s in R: > (a) (r, i) < (s, j) if r < s. > (b) (r, i) < (r, j) if i < j. > (c) (r, i) = (s, j) if r = s and i = j. > We can identify the real number r with its lwalke3 counterpart (r, 0) > and > define an infinitesimal as an lwalke3 real (r, i) with i=/=0. Each > lwalke3 > real has aleph_zero infinitesimals around itself and each infinitesimal > has > a successor and a predecessor. > Here's the trouble. It's not obvious that the elements > (r,i) can have digits. Why do you want digits? In normal math digits make addition, division, etc \ very easy to do. So if you want some sort of digit notation you should first come up with definitions for such operations. For example, addition and subtraction could be defined as: Let L = (r, i) and M = (s, j) then L+M and L-M are defined as: L + M = (r + s, i + j) L - M = (r - s, i - j) There are probably a variety of different ways for defining multiplication. \ After you have established your definitions then come up with some kind of digit based sequence that is isomorphic to your definitions. k === Subject: Re: .9 repeating sha1:0JJbhnxkqd732mec2fFF0Ak8FII= >> OK, then. So it's better for me to say that standard analysts >> find theories such as NSA or those with adjacent points to be >> didactically unsound and mathematically barren -- so that's >> why they prefer standard analysis. Better? >> You might as well claim that number theorists don't believe there is >> any number between 0 and 1, or that working in the theory of rationals >> is didactically unsound or mathematically barren. Most >> mathematicians don't have this fixation with acceptance and rejection >> of different theories. But they do have this fixation with labeling certain theories > as crank theories. And so far, the theory of adjacent > infinitesimals is still labeled a crank theory, since they > discourage any attempt to make adjacent infinitesimals into a > rigorous theory. There is no theory of adjacent infinitesimals that I know of. Tell you what: gather up every utterance that Mitch has made regarding his mathematical theory and put them all together. Then tell me if you really think that this collection serves as a theory in some reasonable sense. -- Jesse F. Hughes Truth is common stuff, ready to your hand, but lies you have to make yourself, and you can't be sure they are any good until you've used them --- and then it's too late. John Steinbeck === Subject: Re: .9 repeating > But they do have this fixation with labeling certain theories > as crank theories. And so far, the theory of adjacent > infinitesimals is still labeled a crank theory, since they > discourage any attempt to make adjacent infinitesimals into a > rigorous theory. There is no theory of adjacent infinitesimals that I know of. Tell you what: gather up every utterance that Mitch has made > regarding his mathematical theory and put them all together. Then > tell me if you really think that this collection serves as a theory > in some reasonable sense. I'll help. Here is a collection of all recent utterances by BURT in sci.math, from 10 000 latest messages in the group here just a moment ago. Thread titles are marked with #, text by BURT with |. Empty lines separate quotes from different messages. The style of BURT is to quote an entire message and add a short note at the end, so the paragraphs in the quotes do belong together as shown. # The modern mathematical concept of infinity is ... | What about the inverse of infinity or how it applies to the small? | | This is the next thing to zero. # Aether as time sustaining every physical element | God created aether to sustain every fundamental immutable physical | element of nature. Instead of God sustaining He created Aether to do | the job. | Every clock is ticking away simultaneous to every other throughout the | universe. # What is the angle of a circular arc? | An infinitely small element of a circular form is in itself round; the | smallest round. | Some points are round if they come from a larger form. | Its the next thing to Zero. | Round has No Derivative in the Calculus. | What I mean is round curvature doesn't change calculus is about | change curvature. # .9 repeating | Is below one by the infinitely nonzero small. It is the closest thing | to one. | | Add the infinitely small and you get one. | If you leave out the last point on the far end you get .9 repeating. | One | would be the last point on the number line. | You can define a length that way. With one measuring rod measuring | one. | It is .0R ending in a 1. | Add .9r to than and you get one | There is only one infinitesimal that can be defined and that is | infinitely small. The rest cannot be calculated and therefore they | cannot be usable. They remain hidden. | .9r is short of one mathematically by a defined infinitesimal.. | Its less by the nonzero infinitely small. It is the closest thing. | No. .9 r is short of One by (the concept of) the infinitely small. | No. That ends in a 9. It is never raised to 1. | No. The 9 never ends. It is never raised to One. It falls short by an | infinitesimal. | You can get infinitely close to the lower limit (of zero.) | The closest thing less than One is .9 repeating. | Why don't you discover it Brian? | It would be .9 repeating with an ending 8 | Its not a fact. | | I say the closest to zero is the infinitely small nonzero | infinitesimal. | There is one thing closest. | I am talking about the next thing to zero defined as an infinitesimal | of 1/Infinity. | Other infinitesimal cannot be calculated therefore they cannot be | usable. | | There is only one usable nonzero infinitesimal or infinitely small. | By the infinitely small a quantity can be distinguished that is | different. | . | The infinitely small nonzero or next to zero is what I assert exists | in math. | One divided by infinity is nonzero. | Maybe you can find a proof yourself. | | It is too simple to require one. | I claim that there is something smallest next to zero. # .9 repeating [REPEAT POSTING] | What is your argument against it? | The concept of the closest thing to zero is a lower limit. | Are you responding to me? I cannot tell. | | The concept of the infinitly small nonzero is worth repeating. | | Added to .9 repeating and you get One. # .9 repeating | There is the closest thing to one and the closest thing to zero. | These two thiings are .9r and 1/infinity. | Apparently you have nothing. | | .9R + 1/infinity = one | | .9r is just short by the infinitely small as defined above. | .9r is different from one by the infinitely small. They are two | different quantities. | You are getting your meaning from a fallible place. There is no doubt | about .9r or the infinitely small. You just invented one. | | .9r is short of 1 by the infinitely small. And the infinitely small is | the closest thing to zero. | You can't do it. There is no such concept. It is closest to zero | with nothing closer | Right below it. The closest thing to 1 but <1. | I am using a number line system. The first point is zero and the last | is one. With something inbetween. | Zero and one are the beginning of math. That is their meaning. | How are you going to argue if I don't define 0 and 1? | One is wrong. It is short by the infinitely small. | It can be seen what the infinitely small nonzero must be through .9 | repeating. | At both ends of the number line there are two points to be defined. | Such as .9r and 1 or 0 and 1/infinity. The rest are the infinity of | points inbetween. # Was Einstein Wrong ? Article discussion... Of course he was... | If both clocks are going slow then they would be going at the same | rate. | Science needed to judge Einstein as wrong. He was not. And I am here | to validate it. # Determining the moment of inertia of a wheel | Are you looking for circular speed along the diameter? # Infinite odds | If the odds of something is one out of infinity if it happens once it | will never repeat and other result will never be exhausted. | I say nothing ever repeats itself and this is predetermined absolute | order. | And the options never exhaust when there is infinite odds. | Approaching the closest thing to zero. # wavelets | Is light a wavelet? # Einstein's birthday | What about the LASER? He is its father but it never seems to get | mentioned. I say lets celebrate the fruit he baired. | I say Einstein was right and he was the most ojective about Quantum | Mechanics. | Fathering the principle behind the LASER should be enough wouldn't you | say? # Unmoving Acceleration | SItting still in gravity is according to Einstein equivalent to a | steady state acceleration. | | You are accelerating in the aether when you are still in gravity but | it is a timeless acceleration. | Gravity is an acceleration even when it is still. | Gravity is round. | You're accelerating in the aether but you are going no where. | There is a timeless one. | I can tell you what I believe. I believe time flows or aether flows. | Believe there is more than one time similar to Stephen Hawking but | better. There is gravitational time and there is motion time. | | The symbol for time flow is + in that it goes in every direction. # How does an EM wave progagate? | Mediumless wave of electric energy. # The concept of the aether | I believe as time it is the sustainer of all things. The aether is | sustainer of the universe. The aether concept is one of sustainer of | all immutable physical elements. | | When aether changes so does energy. | That is where I disagree. The universal immutable elements are | maintained by aether. There are 14. | I disagree. | | Do you spend your time sustaining nothing doug? # Our curve around the Sun | If you drop something it does not follow a curve in the Earth's | geometry. But it does in the Sun's. | | What looks straight from one perspective is curved in another. | | Something that falls straight on Earth is falling around the Sun in a | curve with the rest of the Earth. | | I believe this is Galilean Relativity where there are different | appearences from different perspectives. Actually just two | perspectives are required. | Freefall is only detectable when it ends. The weightedness of mass | allows you to establish the ending freefall motion. | There is no singularity. | Coriolis doesn't come from orbit but rotation which is a form of | antigravity. | You are accelerating right now in your chair in gravity. That is where | your masses weight originates. === Subject: Re: .9 repeating Distribution: world posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) On Mar 18, 6:26am, mstem...@walkabout.empros.com (Michael Stemper) >The numbers proposed by AP and MR don't work in ZFC. So we need >to start from scratch with new axioms. If it took 181 pages for Suppes >to get from the axioms to real numbers, imagine how long it will take >us to get from the new axioms to the new reals! But that's the only >way to give a formal definition of the new reals that will satisfy the >standard analysts. > Conversely, if the new non-standard reals aren't based on the standard > rationals, or standard integers, or standard naturals, then it shouldn't > be surprising that it would take a lot of work to rebuild this from the > ground up. AP-reals, at least, aren't based on standard naturals. They're based on the AP-adics, a type of nonstandard natural. Therefore, it will take a lot of work to build the AP-reals. My problem is not that it'll take a lot of work, but that the standard analysts will have stopped reading my work well before I finish constructing the AP-reals. I'm not sure at this point whether MR-reals can be based in the standard rationals/naturals or not. >Moreover, long before we reach the new reals, most standard analysts >will have stopped reading the axiomatization. > There appears to be some projecting happening here. I'm not a > mathematician myself, but I'd guess that a new method of defining > numbers would be electrifying -- if it was coherent, consistent, > and truly new. AP-reals are sixteen years old -- I'm not sure whether that counts as truly new, but still relatively new compared to standard reals, which date back to the 19th century. > Just out of curiousity, have you already done a literature search > on non-standard analysis? You might want to start with Robinson's > book on the topic. If you don't feel like shelling out fifty bucks, > topic. recently one written by Katz, posted by Bader. The hard part is === Subject: Re: .9 repeating Nntp-Posting-Host: hera.cwi.nl ... > AP-reals, at least, aren't based on standard naturals. They're based > on the AP-adics, a type of nonstandard natural. Therefore, it _will_ > take a lot of work to build the AP-reals. Well, the AP-adics are based on the naturals and build well on them. Just like all p-adics (although they normally are used only with p prime, while AP uses p = 10). I may be wrong, of course, he could have changed his definitions since the last time I looked at them. -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: .9 repeating sha1:3vltTjKmekgl0MnR9GLCxmHuojo= > AP-reals are sixteen years old -- I'm not sure whether that counts > as truly new, but still relatively new compared to standard reals, > which date back to the 19th century. The AP-reals are an incoherent mess of contradictory notions and changing claims. It's silly to pretend that they are a 16-year-old mathematical theory. -- So, at this time, I'd like to assure you that I am not interested in I'll have prosecutors knocking on your doors. I have no problem with === Subject: Re: .9 repeating sha1:3MNX14FVQOxMe2DWhYkLEutNmTE= > AP-reals, at least, aren't based on standard naturals. They're based > on the AP-adics, a type of nonstandard natural. Therefore, it _will_ > take a lot of work to build the AP-reals. My problem is _not_ that it'll take a lot of work, but that the > standard analysts will have stopped reading my work well before I > finish constructing the AP-reals. You could give the AP-adics a go, then. But why bother? AP says that the only axioms needed are geometry! Don't you agree? -- Jesse F. Hughes How lucky we are to be able to hear how miserable Willie Nelson could imagine himself to be. -- Ken Tucker on Fresh Air === Subject: Re: .9 repeating <87iqm61uch.fsf@phiwumbda.org> posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > My problem is not that it'll take a lot of work, but that the > standard analysts will have stopped reading my work well before I > finish constructing the AP-reals. > You could give the AP-adics a go, then. > But why bother? AP says that the only axioms needed are geometry! > Don't you agree? I only partially agree. Just stating that the only axioms are geometry isn't rigorous enough, but AP does have a point that the AP-adics will work once we accept geometry, and not arithmetic, algebra, or set theory, as the foundation . Modern mathematicians insist that set theory (or possibly category theory) be the foundation of mathematics. AP states that he wants geometry to be the foundation. === Subject: Re: .9 repeating Nntp-Posting-Host: hera.cwi.nl ... > Now AP's theories require adjacent points -- actually, let me > use the term AP calls them, _consecutive_ points. So the new > axiom becomes: > > II.2: If A and C are two points of a straight line, then there > exists points B, B' on line AC such that no point lies between A > and B, no point lies between A and B', and at least one point D > so situated that C lies between A and D. This axiom is inconsistent. Because it states that there is a B such that there is no point between A and B, on the other hand, when we relabel B to C, the axiom states that there are points between them. -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: .9 repeating sha1:RHArJHTyBdbToYIU07vN2R7s1y4= >> My problem is not that it'll take a lot of work, but that the >> standard analysts will have stopped reading my work well before I >> finish constructing the AP-reals. >> You could give the AP-adics a go, then. >> But why bother? AP says that the only axioms needed are geometry! >> Don't you agree? I only partially agree. Just stating that the only axioms are > geometry isn't rigorous enough, but AP does have a point that > the AP-adics will work once we accept geometry, and not > arithmetic, algebra, or set theory, as the foundation . Really? You can *see* this, by his incoherent ramblings? Well, get on it then! Show us how to interpret AP-reals using only geometry! > Modern mathematicians insist that set theory (or possibly > category theory) be the foundation of mathematics. AP states > that he wants geometry to be the foundation. This is a non-argument. There is no reason we couldn't interpret a particular number theory in geometry and there is no argument against doing so. Indeed, there is *no* insistence that set theory or category theory must be the foundation for mathematics, but rather the observation that they each can serve as the foundation for (most?) existing mathematics. -- Jesse F. Hughes And I'm one of my own biggest skeptics as I had *YEARS* of wrong ideas, and attempts that failed. Worse, for some of them it took *MONTHS* before I figured out where I screwed up. -- James Harris === Subject: Re: .9 repeating <873adaza11.fsf@phiwumbda.org> posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > I only partially agree. Just stating that the only axioms are > geometry isn't rigorous enough, but AP does have a point that > the AP-adics will work once we accept geometry, and not > arithmetic, algebra, or set theory, as the foundation . > Well, get on it then! Show us how to interpret AP-reals using only > geometry! Well then, here's the geometry. Let's begin with the Euclid-Hilbert Axioms for geometry: I. Incidence II. Order III. Parallels IV. Congruence V. Continuity (I won't repeat the actual axioms in each group since anyone can Google them.) Now, here are the changes we must make to the given axioms: II.2 The original axiom is: If A and C are two points of a straight line, then there exists at least one point B lying between A and C and at least one point D so situated that C lies between A and D. Now AP's theories require adjacent points -- actually, let me use the term AP calls them, consecutive points. So the new axiom becomes: II.2: If A and C are two points of a straight line, then there exists points B, B' on line AC such that no point lies between A and B, no point lies between A and B', and at least one point D so situated that C lies between A and D. Next, we move on to: III. Parallels AP has stated that he wants his theory to emcompass all three types of geometry: Elliptic, Euclidean, and Hyperbolic. There are to be three types of numbers corresponding to the three types, namely positive AP-adics for Elliptic geometry, the AP-reals for Euclidean geometry, and negative AP-adics for Hyperbolic geometry. Therefore, I will delete axiom III.1, knowing that including III.1 or its Elliptic or Hyperbolic counterparts will result in three different numbering systems. Finally, we consider: V. Continuity This is the section I'm having the most trouble with. Clearly V.1, the Archimedean axiom, must be deleted, since AP wants there to be infinitesimal reals such as 0.000...0001. I suspect that we must delete axiom V.2 as well. But I'm hoping that eliminating V.1 and V.2 doesn't allow the remaining theory to admit a finite model, since AP evidently wants there to be infinitely many points. So this is the theory for now. Once I see that the geometry satisfies AP's desires, next I can see how to define the numbers that can adequately describe the geometry. === Subject: Re: .9 repeating sha1:8M8KhzaWPwAs49R8gB4u+ayrVBs= > So this is the theory for now. Once I see that the geometry > satisfies AP's desires, next I can see how to define the > numbers that can adequately describe the geometry. Oh. I'm a little disappointed. You haven't even stated what the set of All Possible Digit Combinations contains, much less interpreted that set in his geometry. But you said: > AP does have a point that the AP-adics will work once we accept > geometry, and not arithmetic, algebra, or set theory, as the > _foundation_ So I thought you had a bit more insight than this. -- [I want to] stand at the pinnacle of human achievement with no one else in all of history even close, no human being having faced what I have--and survived. Because when all is said and done, make no mistake, the simple truth is, I am better. --James S. Harris === Subject: Re: .9 repeating <87ljr33umh.fsf@phiwumbda.org> posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > My labels standard set theorist, standard analyst, etc., > is my effort to avoid the label standard mathematician, > which previously led to an argument on how nonstandard > analysis is still standard mathematics. So a standard > analyst explicitly accepts standard analysis, as opposed to > nonstandard analysis. > I've done some work in non-well-founded set theory (a long time ago), > as well as ZFC. Which theory do you think I accept? Because I need > to know if I'm standard or not. It's for a loan application. > (please be standard! please be standard! ... ) It appears that Aatu has already answered this -- ZFA is nonstandard, but still on a higher level of rigor than MR. === Subject: Re: .9 repeating sha1:VnV1XORTOBkk1Jt42cndqpMSqAw= >> My labels standard set theorist, standard analyst, etc., >> is my effort to avoid the label standard mathematician, >> which previously led to an argument on how nonstandard >> analysis is still standard mathematics. So a standard >> analyst explicitly accepts standard analysis, as opposed to >> nonstandard analysis. >> I've done some work in non-well-founded set theory (a long time ago), >> as well as ZFC. Which theory do you think I accept? Because I need >> to know if I'm standard or not. It's for a loan application. >> (please be standard! please be standard! ... ) It appears that Aatu has already answered this -- ZFA is > nonstandard, but still on a higher level of rigor than MR. I don't understand. You said that a standard set theorist views the axiom of infinity as set theorists would reject ZF-Infinity+~Infinity? Of course, ZF-Infinity+~Infinity is just as rigorous as ZFA, ZFC and so on, so I don't know why you bring up rigor here, when up 'til now, you've said that certain theories would be rejected no matter how rigorous they are. So, let's see if there is any mathematician anywhere who has ever rejected the theory of finite sets. Can you name one? Can you even imagine a reason to reject this perfectly good theory? You have a very odd idea about how mathematicians accept and reject various theories. -- In terms of story, this one is tops. All of the 'good' characters grow and become much more than they were when the movie began. This film celebrates friendship at it's basest level and demonstrates the triumph of good people above overwhelming odds. -- IMDB user comments are tops, at the basest level === Subject: Re: .9 repeating > You have a very odd idea about how mathematicians accept and > reject various theories. I for one can't even understand just what sort of acceptance or rejection is at issue. But there is a more fundamental problem here. Lwalke, in his excessively charitable readings of the claims of various non-standard mathematicians, effectively robs these poor people of their right to actually disagree with the standard mathematicians about anything. For, it seems, whenever someone claims something about the reals, sets, what-not, that seems to be false, this is to be taken as an indicator that they are speaking about something else entirely, to be explained by some hitherto unknown theory, or in light of these or those obscure pieces of logical arcana. -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon man nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: .9 repeating <87tz5q1unc.fsf@phiwumbda.org> <87prge8uzo.fsf@alatheia.dsl.inet.fi> posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > You have a very odd idea about how mathematicians accept and > reject various theories. > But there is a more fundamental problem here. Lwalke, in his > excessively charitable readings of the claims of various non-standard > mathematicians, effectively robs these poor people of their right to > actually disagree with the standard mathematicians about > anything. And therefore, calling those poor people cranks who are 100% wrong about mathematics is better? Calling them cranks who are 100% wrong isn't robbing them of their right to disagree with standard mathematics? === Subject: Re: .9 repeating sha1:bHpw6xUIvGuUFA8U34s/618mO9M= >> You have a very odd idea about how mathematicians accept and >> reject various theories. >> But there is a more fundamental problem here. Lwalke, in his >> excessively charitable readings of the claims of various non-standard >> mathematicians, effectively robs these poor people of their right to >> actually disagree with the standard mathematicians about >> anything. And therefore, calling those poor people cranks who are > 100% wrong about mathematics is better? Calling them cranks > who are 100% wrong isn't robbing them of their right to > disagree with standard mathematics? They have a right to disagree, just as I have a right to call them cranks when their disagreements are silly, incoherent half-thoughts. -- Jesse F. Hughes I could be a nerd. -- Quincy P. Hughes (age five) has ambitions === Subject: Re: .9 repeating sha1:32sav0E0DKyg75XZfRFCzzCIEII= > You have a very odd idea about how mathematicians accept and >> reject various theories. I for one can't even understand just what sort of acceptance or > rejection is at issue. But there is a more fundamental problem here. Lwalke, in his > excessively charitable readings of the claims of various non-standard > mathematicians, effectively robs these poor people of their right to > actually disagree with the standard mathematicians about > anything. For, it seems, whenever someone claims something about the > reals, sets, what-not, that seems to be false, this is to be taken as > an indicator that they are speaking about something else entirely, to > be explained by some hitherto unknown theory, or in light of these or > those obscure pieces of logical arcana. His excessive sense of charity shows itself in other ways, too. AP has said that his AP-reals need no theory. He has said: Here again, Lwal, what are you doing? I do not need a axiom system for AP-Reals or AP-adics. I simply define them as All Possible Digit Arrangements. Call that definition an axiom if it placates you. I do not need axioms because I already have the axioms of Geometry, and all I really need is find the Native numbers that fills each point in those three geometries. Now, Lwalker surely knows better than this. He knows that this is not a rigorous description of a mathematical theory. In fact, his entire pet project depends on the fact that he knows what counts as a mathematical theory and what doesn't. But he will not contradict AP here. He won't suggest that AP is wrong, even though he surely knows that AP's writing here is plain nonsense. He won't do this, because his sentiment towards the downtrodden cranks trumps what's left of his intellectual honesty. So, a crank's mathematical claim is never false. We just have to find the right theory to make it true. And a crank's metamathematical claim is never false either. We just have to... er... Hey! Look over there! Is that a bird? That's so cool. -- No feeling sympathy for mathematicians who start marching with signs like 'Will work for food' in the future... I will not show mercy going forward. I was trained as a soldier in the United States Army after all... We play to win. --James Harris, feel his wrath! === Subject: Re: .9 repeating > It appears that Aatu has already answered this -- ZFA is > nonstandard, but still on a higher level of rigor than MR. Standard and non-standard in this context have nothing whatever to do with level of rigour. -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon man nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: .9 repeating posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > MR would rather consider the geometry first, declare that adjacent > points are possible, then make the numbers fit the geometry -- and > conclude that there must exist a hypernatural H > I will not accept any pronouncement from you as to what MR would > rather consider. Using other people's names to serve as mouthpieces > for your own views and arguments is frankly unethical. The main argument by the standard analysts against MR is that MR isn't rigorous enough, right? All I'm trying to is find a way to make MR more rigorous so that the standard analysts will be more open towards his theory. I believe that most of what I've written in this thread is representative of what MR has expressed. MR has definitely mentioned adjacent points next to 0 and 1, with 0.999... or 0.9r being the last point before we reach 1. > Perhaps the distinction that must be made is that MR is thinking in > terms of points , thinking about the next point after 0 or the > last point before 1. To every point we assign a hyperreal, but we > don't assign to every hyperreal a point. > Again, here you're claiming knowledge of what MR is *thinking*: in > terms of points assigned to hyperreals. MR has never mentioned > hyperreals, as far back as I can remember. I admit that hyperreals are the only objects I mentioned that MR didn't mention. But Bader mentioned the Katz link, which mentions hyperreals. Without the hyperreals, it's impossible to make MR rigorous! So Little is putting me in a Catch-22, because: 1) MR claims that 0.9r < 1. 2) Standard analysts argue that MR lacks rigor. 3) I answer 2) by inserting hyperreals to make MR rigorous. 4) Little argues that inserting hyperreals into MR is academically dishonest and unethical. So without hyperreals, MR is not rigorous, yet adding them is unethical! And now I'm left wondering whether there's an ethical way to make MR rigorous -- or, for that matter, a way to make any theory other than standard theory rigorous without being academically dishonest! I sure hope I don't have to add this claim to the list of arguments made by standard analysts: Standard analysts argue that attempting to make nonstandard theories rigorous is unethical. === Subject: Re: .9 repeating posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2009021910 Firefox/3.0.7,gzip(gfe),gzip(gfe) > 1) MR claims that 0.9r < 1. > 2) Standard analysts argue that MR lacks rigor. > 3) I answer 2) by inserting hyperreals to make MR rigorous. > 4) Little argues that inserting hyperreals into MR is > academically dishonest and unethical. You have got to be one of the most obtuse persons I've ever met. Try it this way: 1. MR claims .999... < 1. 2. It is pointed out that MR is making this claim without any context or knowledge of a rigorous or even semi-formal mathematical theory. 3. You find that there is a rigorous theory in which .999... < 1. 4. It is argued that that theory is not consistent with MR's claim that there is a number immediately next to the number 1. And it is argued that though there are rigorous non-standard treatments of analysis, one can't just suppose that MR himself is talking about any of those theories when rather he simply declares .999... < 1. Therefore, it is reasonable to say MR is posting crankishly, since by all evidence he's just blurting an assertion that he has no notion of proving or supporting in a formal or even semi-formal theory, notwithstanding that, unknown to him, there may be existing formal approaches in which his assertion does hold. I don't know that I subscribe to all of the particulars there, but at least you can see that it's a much more reasonable rendering of the dialectic than yours is. MoeBlee === Subject: Re: .9 repeating MoeBlee a .8ecrit : > >> 1) MR claims that 0.9r < 1. >> 2) Standard analysts argue that MR lacks rigor. >> 3) I answer 2) by inserting hyperreals to make MR rigorous. >> 4) Little argues that inserting hyperreals into MR is >> academically dishonest and unethical. You have got to be one of the most obtuse persons I've ever met. > No, he is just trolling :-( > Try it this way: 1. MR claims .999... < 1. > 2. It is pointed out that MR is making this claim without any context > or knowledge of a rigorous or even semi-formal mathematical theory. > 3. You find that there is a rigorous theory in which .999... < 1. > 4. It is argued that that theory is not consistent with MR's claim > that there is a number immediately next to the number 1. And it is > argued that though there are rigorous non-standard treatments of > analysis, one can't just suppose that MR himself is talking about any > of those theories when rather he simply declares .999... < 1. > Therefore, it is reasonable to say MR is posting crankishly, since by > all evidence he's just blurting an assertion that he has no notion of > proving or supporting in a formal or even semi-formal theory, > notwithstanding that, unknown to him, there may be existing formal > approaches in which his assertion does hold. I don't know that I subscribe to all of the particulars there, but at > least you can see that it's a much more reasonable rendering of the > dialectic than yours is. MoeBlee === Subject: Re: .9 repeating posting-account=5ApcPgoAAABKcgEyKsQmJVb3Rz63IGGL .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; WWTClient2),gzip(gfe),gzip(gfe) > 1) MR claims that 0.9r < 1. > 2) Standard analysts argue that MR lacks rigor. > 3) I answer 2) by inserting hyperreals to make MR rigorous. > 4) Little argues that inserting hyperreals into MR is > academically dishonest and unethical. You have got to be one of the most obtuse persons I've ever met. Try it this way: 1. MR claims .999... < 1. > 2. It is pointed out that MR is making this claim without any context > or knowledge of a rigorous or even semi-formal mathematical theory. > 3. You find that there is a rigorous theory in which .999... < 1. > 4. It is argued that that theory is not consistent with MR's claim > that there is a number immediately next to the number 1. And it is > argued that though there are rigorous non-standard treatments of > analysis, one can't just suppose that MR himself is talking about any > of those theories when rather he simply declares .999... < 1. > Therefore, it is reasonable to say MR is posting crankishly, since by > all evidence he's just blurting an assertion that he has no notion of > proving or supporting in a formal or even semi-formal theory, > notwithstanding that, unknown to him, there may be existing formal > approaches in which his assertion does hold. I don't know that I subscribe to all of the particulars there, but at > least you can see that it's a much more reasonable rendering of the > dialectic than yours is. MoeBlee I claim that there is something smallest next to zero. Mitch Raemsch === Subject: Re: .9 repeating [REPEAT POSTING] sha1:neQOFUFx93FJW1Av35a6Lo8bACs= >> 1) MR claims that 0.9r < 1. >> 2) Standard analysts argue that MR lacks rigor. >> 3) I answer 2) by inserting hyperreals to make MR rigorous. >> 4) Little argues that inserting hyperreals into MR is >> academically dishonest and unethical. >> You have got to be one of the most obtuse persons I've ever met. >> Try it this way: >> 1. MR claims .999... < 1. >> 2. It is pointed out that MR is making this claim without any context >> or knowledge of a rigorous or even semi-formal mathematical theory. >> 3. You find that there is a rigorous theory in which .999... < 1. >> 4. It is argued that that theory is not consistent with MR's claim >> that there is a number immediately next to the number 1. And it is >> argued that though there are rigorous non-standard treatments of >> analysis, one can't just suppose that MR himself is talking about any >> of those theories when rather he simply declares .999... < 1. >> Therefore, it is reasonable to say MR is posting crankishly, since by >> all evidence he's just blurting an assertion that he has no notion of >> proving or supporting in a formal or even semi-formal theory, >> notwithstanding that, unknown to him, there may be existing formal >> approaches in which his assertion does hold. >> I don't know that I subscribe to all of the particulars there, but at >> least you can see that it's a much more reasonable rendering of the >> dialectic than yours is. >> MoeBlee I claim that there is something smallest next to zero. Why not clear up the point? When you claim this, do you mean that there is some mathematical theory (though not the theory of real numbers) in which this claim is true? Or is this claim indeed about the real numbers? You *do* understand the question, yes? [Sorry for the second posting in sci.math, but I decided to re-post in both groups in case Mitch doesn't read sci.math] -- Jesse F. Hughes Social castigation. Their pictures in the papers. Reporters hounding them with hard questions. And it won't end during their lifetimes. -- Oppose James S. Harris and you get post-mortem hardball interviews === Subject: Re: .9 repeating [REPEAT POSTING] <87k56mze5o.fsf_-_@phiwumbda.org> posting-account=5ApcPgoAAABKcgEyKsQmJVb3Rz63IGGL .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; WWTClient2),gzip(gfe),gzip(gfe) >> 1) MR claims that 0.9r < 1. >> 2) Standard analysts argue that MR lacks rigor. >> 3) I answer 2) by inserting hyperreals to make MR rigorous. >> 4) Little argues that inserting hyperreals into MR is >> academically dishonest and unethical. >> You have got to be one of the most obtuse persons I've ever met. >> Try it this way: >> 1. MR claims .999... < 1. >> 2. It is pointed out that MR is making this claim without any context >> or knowledge of a rigorous or even semi-formal mathematical theory. >> 3. You find that there is a rigorous theory in which .999... < 1. >> 4. It is argued that that theory is not consistent with MR's claim >> that there is a number immediately next to the number 1. And it is >> argued that though there are rigorous non-standard treatments of >> analysis, one can't just suppose that MR himself is talking about any >> of those theories when rather he simply declares .999... < 1. >> Therefore, it is reasonable to say MR is posting crankishly, since by >> all evidence he's just blurting an assertion that he has no notion of >> proving or supporting in a formal or even semi-formal theory, >> notwithstanding that, unknown to him, there may be existing formal >> approaches in which his assertion does hold. >> I don't know that I subscribe to all of the particulars there, but at >> least you can see that it's a much more reasonable rendering of the >> dialectic than yours is. >> MoeBlee > I claim that there is something smallest next to zero. Why not clear up the point? When you claim this, do you mean that there is some mathematical > theory (though not the theory of real numbers) in which this claim is > true? Or is this claim indeed about the real numbers? You *do* understand the question, yes? [Sorry for the second posting in sci.math, but I decided to re-post in > both groups in case Mitch doesn't read sci.math] -- > Jesse F. Hughes > Social castigation. Their pictures in the papers. Reporters hounding > them with hard questions. And it won't end during their lifetimes. > -- Oppose James S. Harris and you get post-mortem hardball interviews- Hide quoted text - - Show quoted text - What is your argument against it? Mitch Raemsch === Subject: Re: .9 repeating [REPEAT POSTING] > On Mar 18, 1:58 pm, Jesse F. Hughes >I claim that there is something smallest next to zero. >>Why not clear up the point? >>When you claim this, do you mean that there is some >>mathematical theory (though not the theory of real numbers) >>in which this claim is true? >>Or is this claim indeed about the real numbers? >>You *do* understand the question, yes? [...] > What is your argument against it? I would like to see Jesse's question answered, too. Perhaps the way to do this is to ask you if you disagree with any of the things we mean when we say x is a real number. It's really not a very long list. Assume that S is a set of real numbers and not empty. 1) Do you agree that, if S has any upper bound at all, then it has an upper bound which is less than or equal to all the other upper bounds (which is called a least upper bound)? -- By b is an upper bound of S I mean that b is a real number and, for every x in S, x <= b. -- By b' is a least upper bound of S I mean that b' is an upper bound of S and, for every upper bound b of S, b' <= b. (The reason we restrict S to non-empty sets is that every real number is an upper bound of the empty set by our definition, so there would be no least upper bound (no least real number).) Question 1 seems to me to be the one you are most likely to object to, although it's not at all clear that you will object. The others can be summarized as Do you agree that addition, multiplication and '<=' behave in the everyday manner? Do you agree that: 2,3) If x and y are real numbers, then x + y and x*y are real numbers. 4) There is a particular real number we'll symbolize by 0, such that, for all x, x + 0 = x. 5) There is another particular real number we'll symbolize by 1, such that, for all x, x*1 = x. 6) For each real number x, there is a particular real number we'll symbolize by -x, such that x + (-x) = 0 7) For each real number /other than 0/, there is a particular real number we'll symbolize by 1/x, such that x*(1/x) = 1. For all real numbers x, y, and z: 8) x + (y + z) = (x + y) + z 9) x*(y*z) = (x*y)*z 10) x + y = y + x 11) x*y = y*x 12) x*(y + z) = x*y + x*z. 13) If x <= y and y <= x, then x = y. 14) If x <= y and y <= z, the x <= z. 15) Either x <= y or y <= x. 16) If x <= y, then x + z <= y + z. 17) If 0 <= x and 0 <= y, then 0 <= x*y. (See Wikipedia http://en.wikipedia.org/wiki/Real_number http://en.wikipedia.org/wiki/Field_(mathematics) http://en.wikipedia.org/wiki/Total_order ) Jim Burns === Subject: Re: .9 repeating [REPEAT POSTING] <49C19EB0.3020102@osu.edu> posting-account=5ApcPgoAAABKcgEyKsQmJVb3Rz63IGGL .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; WWTClient2),gzip(gfe),gzip(gfe) > On Mar 18, 1:58 pm, Jesse F. Hughes >I claim that there is something smallest next to zero. >>Why not clear up the point? >>When you claim this, do you mean that there is some >>mathematical theory (though not the theory of real numbers) >>in which this claim is true? >>Or is this claim indeed about the real numbers? >>You *do* understand the question, yes? > [...] > What is your argument against it? I would like to see Jesse's question answered, too. > Perhaps the way to do this is to ask you if you disagree > with any of the things we mean when we say x is a real > number. It's really not a very long list. Assume that S is a set of real numbers and not empty. 1) Do you agree that, if S has any upper bound at all, > then it has an upper bound which is less than or > equal to all the other upper bounds (which is called > a least upper bound)? -- By b is an upper bound of S I mean that b is a real number > and, for every x in S, x <= b. > -- By b' is a least upper bound of S I mean that > b' is an upper bound of S and, for every upper bound b > of S, b' <= b. (The reason we restrict S to non-empty sets is that > every real number is an upper bound of the empty > set by our definition, so there would be no least > upper bound (no least real number).) Question 1 seems to me to be the one you are most likely > to object to, although it's not at all clear that > you will object. The others can be summarized as > Do you agree that addition, multiplication and > '<=' behave in the everyday manner? Do you agree that: 2,3) If x and y are real numbers, then > x + y and x*y are real numbers. 4) There is a particular real number we'll symbolize > by 0, such that, for all x, x + 0 = x. > 5) There is another particular real number we'll > symbolize by 1, such that, for all x, x*1 = x. > 6) For each real number x, there is a particular > real number we'll symbolize by -x, such that x + (-x) = 0 > 7) For each real number /other than 0/, there is > a particular real number we'll symbolize by 1/x, > such that x*(1/x) = 1. For all real numbers x, y, and z: > 8) x + (y + z) = (x + y) + z > 9) x*(y*z) = (x*y)*z > 10) x + y = y + x > 11) x*y = y*x > 12) x*(y + z) = x*y + x*z. > 13) If x <= y and y <= x, then x = y. > 14) If x <= y and y <= z, the x <= z. > 15) Either x <= y or y <= x. > 16) If x <= y, then x + z <= y + z. > 17) If 0 <= x and 0 <= y, then 0 <= x*y. (See Wikipediahttp://en.wikipedia.org/wiki/Real numberhttp://en.wikipedia.org/wiki/Field (mathematics)http://en.wikipedia.org/wiki/Total order) Jim Burns- Hide quoted text - - Show quoted text - The concept of the closest thing to zero is a lower limit. Mitch Raemsch === Subject: Re: .9 repeating [REPEAT POSTING] sha1:dbLuZ9M7+cL4gfHHwbDrIkUjhL4= >> On Mar 18, 1:58 pm, Jesse F. Hughes >>I claim that there is something smallest next to zero. >Why not clear up the point? >When you claim this, do you mean that there is some >mathematical theory (though not the theory of real numbers) >in which this claim is true? >Or is this claim indeed about the real numbers? >You *do* understand the question, yes? >> [...] >> What is your argument against it? >> I would like to see Jesse's question answered, too. >> Perhaps the way to do this is to ask you if you disagree >> with any of the things we mean when we say x is a real >> number. It's really not a very long list. >> Assume that S is a set of real numbers and not empty. >> 1) Do you agree that, if S has any upper bound at all, >> then it has an upper bound which is less than or >> equal to all the other upper bounds (which is called >> a least upper bound)? >> -- By b is an upper bound of S I mean that b is a real number >> and, for every x in S, x <= b. >> -- By b' is a least upper bound of S I mean that >> b' is an upper bound of S and, for every upper bound b >> of S, b' <= b. >> (The reason we restrict S to non-empty sets is that >> every real number is an upper bound of the empty >> set by our definition, so there would be no least >> upper bound (no least real number).) >> Question 1 seems to me to be the one you are most likely >> to object to, although it's not at all clear that >> you will object. The others can be summarized as >> Do you agree that addition, multiplication and >> '<=' behave in the everyday manner? >> Do you agree that: >> 2,3) If x and y are real numbers, then >> x + y and x*y are real numbers. >> 4) There is a particular real number we'll symbolize >> by 0, such that, for all x, x + 0 = x. >> 5) There is another particular real number we'll >> symbolize by 1, such that, for all x, x*1 = x. >> 6) For each real number x, there is a particular >> real number we'll symbolize by -x, such that x + (-x) = 0 >> 7) For each real number /other than 0/, there is >> a particular real number we'll symbolize by 1/x, >> such that x*(1/x) = 1. >> For all real numbers x, y, and z: >> 8) x + (y + z) = (x + y) + z >> 9) x*(y*z) = (x*y)*z >> 10) x + y = y + x >> 11) x*y = y*x >> 12) x*(y + z) = x*y + x*z. >> 13) If x <= y and y <= x, then x = y. >> 14) If x <= y and y <= z, the x <= z. >> 15) Either x <= y or y <= x. >> 16) If x <= y, then x + z <= y + z. >> 17) If 0 <= x and 0 <= y, then 0 <= x*y. >> (See Wikipediahttp://en.wikipedia.org/wiki/Real numberhttp://en.wikipedia.org/wiki/Field (mathematics)http://en.wikipedia.org/wiki/Total order) >> Jim Burns- Hide quoted text - >> - Show quoted text - The concept of the closest thing to zero is a lower limit. If you don't understand our questions, just say so and perhaps we can rephrase them. This response of yours is utterly beside the point. -- Jesse F. Hughes How lucky we are to be able to hear how miserable Willie Nelson could imagine himself to be. -- Ken Tucker on Fresh Air === Subject: Re: .9 repeating [REPEAT POSTING] <87fxhaxpd5.fsf@phiwumbda.org> posting-account=5ApcPgoAAABKcgEyKsQmJVb3Rz63IGGL .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; WWTClient2),gzip(gfe),gzip(gfe) >> On Mar 18, 1:58 pm, Jesse F. Hughes >>I claim that there is something smallest next to zero. >Why not clear up the point? >When you claim this, do you mean that there is some >mathematical theory (though not the theory of real numbers) >in which this claim is true? >Or is this claim indeed about the real numbers? >You *do* understand the question, yes? >> [...] >> What is your argument against it? >> I would like to see Jesse's question answered, too. >> Perhaps the way to do this is to ask you if you disagree >> with any of the things we mean when we say x is a real >> number. It's really not a very long list. >> Assume that S is a set of real numbers and not empty. >> 1) Do you agree that, if S has any upper bound at all, >> then it has an upper bound which is less than or >> equal to all the other upper bounds (which is called >> a least upper bound)? >> -- By b is an upper bound of S I mean that b is a real number >> and, for every x in S, x <= b. >> -- By b' is a least upper bound of S I mean that >> b' is an upper bound of S and, for every upper bound b >> of S, b' <= b. >> (The reason we restrict S to non-empty sets is that >> every real number is an upper bound of the empty >> set by our definition, so there would be no least >> upper bound (no least real number).) >> Question 1 seems to me to be the one you are most likely >> to object to, although it's not at all clear that >> you will object. The others can be summarized as >> Do you agree that addition, multiplication and >> '<=' behave in the everyday manner? >> Do you agree that: >> 2,3) If x and y are real numbers, then >> x + y and x*y are real numbers. >> 4) There is a particular real number we'll symbolize >> by 0, such that, for all x, x + 0 = x. >> 5) There is another particular real number we'll >> symbolize by 1, such that, for all x, x*1 = x. >> 6) For each real number x, there is a particular >> real number we'll symbolize by -x, such that x + (-x) = 0 >> 7) For each real number /other than 0/, there is >> a particular real number we'll symbolize by 1/x, >> such that x*(1/x) = 1. >> For all real numbers x, y, and z: >> 8) x + (y + z) = (x + y) + z >> 9) x*(y*z) = (x*y)*z >> 10) x + y = y + x >> 11) x*y = y*x >> 12) x*(y + z) = x*y + x*z. >> 13) If x <= y and y <= x, then x = y. >> 14) If x <= y and y <= z, the x <= z. >> 15) Either x <= y or y <= x. >> 16) If x <= y, then x + z <= y + z. >> 17) If 0 <= x and 0 <= y, then 0 <= x*y. >> (See Wikipediahttp://en.wikipedia.org/wiki/Real numberhttp://en.wikipedia.org/wiki/...) >> Jim Burns- Hide quoted text - >> - Show quoted text - > The concept of the closest thing to zero is a lower limit. If you don't understand our questions, just say so and perhaps we can > rephrase them. This response of yours is utterly beside the point. -- > Jesse F. Hughes How lucky we are to be able to hear how miserable Willie Nelson could > imagine himself to be. -- Ken Tucker on Fresh Air- Hide quoted text - - Show quoted text - Are you responding to me? I cannot tell. The concept of the infinitly small nonzero is worth repeating. Added to .9 repeating and you get One. Mitch Raemsch === Subject: Re: .9 repeating [REPEAT POSTING] >The concept of the closest thing to zero is a lower limit. >>If you don't understand our questions, just say so and perhaps we can >>rephrase them. This response of yours is utterly beside the point. > Are you responding to me? I cannot tell. The concept of the infinitly small nonzero is worth repeating. Added to .9 repeating and you get One. Do you agree or disagree that every bounded non-empty set of real numbers has a least upper bound? Jim Burns === Subject: Re: .9 repeating [REPEAT POSTING] sha1:dtQ1mcKkqo/bVUcFwqNM6agGGlE= >> Why not clear up the point? >> When you claim this, do you mean that there is some mathematical >> theory (though not the theory of real numbers) in which this claim is >> true? >> Or is this claim indeed about the real numbers? >> You *do* understand the question, yes? >> [Sorry for the second posting in sci.math, but I decided to re-post in >> both groups in case Mitch doesn't read sci.math] What is your argument against it? It rather depends on what you're claiming. Do you not understand my question? -- Often times, when [...] looking for a sense of adventure, I'd doodle with math equations. Often too, when pressures of regular life were really bothering me, I'd go for the adventure of fiddling with math. -- James S. Harris, a man of many mathematical adventures! === Subject: Re: .9 repeating sha1:45Ef+SajmNflY+bw+X0gB94jSiA= I claim that there is something smallest next to zero. > Why not clear up the point? When you claim this, do you mean that there is some mathematical theory (though not the theory of real numbers) in which this claim is true? Or is this claim indeed about the real numbers? You *do* understand the question, yes? -- But in our enthusiasm, we could not resist a radical overhaul of the system, in which all of its major weaknesses have been exposed, analyzed, and replaced with new weaknesses. -- Bruce Leverett (presumably with apologies to Ambrose Bierce) === Subject: Re: .9 repeating Originator: richard@cogsci.ed.ac.uk (Richard Tobin) >> I claim that there is something smallest next to zero. >When you claim this, do you mean that there is some mathematical >theory (though not the theory of real numbers) in which this claim is >true? Or is this claim indeed about the real numbers? Presumably he believes that numbers have some independent existence, and that his claim is true of them. The fact that it is not true of the so-called real numbers merely shows that they do not accurately model the *real* numbers. -- Richard -- Please remember to mention me / in tapes you leave behind. === Subject: Re: .9 repeating sha1:8td1qn2EELKu1glkLnZOW8gjyec= >> I claim that there is something smallest next to zero. >>When you claim this, do you mean that there is some mathematical >>theory (though not the theory of real numbers) in which this claim is >>true? >>Or is this claim indeed about the real numbers? Presumably he believes that numbers have some independent existence, > and that his claim is true of them. The fact that it is not true of > the so-called real numbers merely shows that they do not accurately > model the *real* numbers. Seems plausible, but lwalker already has taken on guessing what MR wants. I'd like to hear an answer from Mitch. -- Jesse F. Hughes I have written many words to sci.math, some of them are not even meaningless. --Ross Finlayson === Subject: Re: .9 repeating <87sklazgpv.fsf@phiwumbda.org> posting-account=9QOSvAoAAACEOWJVSDuswW7dB_0wApQO Gecko/2009030503 Fedora/3.0.7-1.fc10 Firefox/3.0.7,gzip(gfe),gzip(gfe) >> I claim that there is something smallest next to zero. >When you claim this, do you mean that there is some mathematical >theory (though not the theory of real numbers) in which this claim is >true? >Or is this claim indeed about the real numbers? Presumably he believes that numbers have some independent existence, > and that his claim is true of them. The fact that it is not true of > the so-called real numbers merely shows that they do not accurately > model the *real* numbers. In that case, he should be posting on a theology group... Maybe he's boring prophetic style might be appreciated there? -- m === Subject: Re: .9 repeating posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2009021910 Firefox/3.0.7,gzip(gfe),gzip(gfe) > I claim that there is something smallest next to zero. Okay, smallest next to zero, not to one. Mutatis mutandis, my remarks stand. MoeBlee === Subject: Re: .9 repeating sha1:pw7HQWWGT5AmsZIoSoaSDwrmfPI= > I admit that hyperreals are the only objects I mentioned > that MR didn't mention. But Bader mentioned the Katz > link, which mentions hyperreals. Without the hyperreals, > it's impossible to make MR rigorous! This impossibility claim requires a bit of justification, but let's let it pass. > So Little is putting me in a Catch-22, because: 1) MR claims that 0.9r < 1. > 2) Standard analysts argue that MR lacks rigor. > 3) I answer 2) by inserting hyperreals to make MR rigorous. > 4) Little argues that inserting hyperreals into MR is > academically dishonest and unethical. So without hyperreals, MR is not rigorous, yet adding them > is unethical! Note: you misrepresent MR by using hyperreals in any case, unless you can prove that, adapting the writings of Katz and Katz, there is a number adjacent to 1. I don't think that's so. If I'm right, then your theory clearly doesn't do what you want. -- Jesse F. Hughes And I'm one of the people who gets to be born just to clean up the crap. I'm like a super janitor. -- James S. Harris, a new kind of hero. === Subject: Re: .9 repeating <87ocvy1ugf.fsf@phiwumbda.org> posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > So without hyperreals, MR is not rigorous, yet adding them > is unethical! > Note: you misrepresent MR by using hyperreals in any case, unless you > can prove that, adapting the writings of Katz and Katz, there is a > number adjacent to 1. I don't think that's so. > If I'm right, then your theory clearly doesn't do what you want. So there are two issues at stake here: 1) Representing infinitesimals by digits. 2) Finding a smallest positive infinitesimal. Obviously, MR believes both 1) and 2) are possible. Here's what happened. I wondered whether there can be a rigorous theory in which 1) is possible. Bader found that in the Katz theory, 1) is possible. So then I went Aha, 1) is possible in a rigorous theory, so MR's claim of 1) is possible! But then I was called unethical -- because 2) doesn't hold in Katz, yet MR wants 2) to hold. My triumph over finding a rigorous theory in which 1) holds becomes short-lived because of 2). It is considered unethical for me to associate Katz with MR, because Katz and MR disagree about 2). Why did I associate MR and Katz, anyway? It was because of 1), not because of 2)! What I'd like to do is come up with a new theory by modifying Katz so that 2) holds -- which should be easier than starting with standard analysis and trying to modify it so that 1) and 2) both hold. Using Katz as a starting point, I'm already halfway to the goal. But will attempting to come up with a new theory by taking Katz and modifying it so that 2) holds be considered academically dishonest? Of course, what would be preferable is to find an established theory in which 2) already holds and just forget about Katz. But I suspect that there is no established theory with consecutive infinitesimals. And so, I can now safely say that simplying insisting 1) is not grounds for being considered a crank, but those who insist on 2) are still cranks. === Subject: Re: .9 repeating sha1:oSFm/wCOz9CQoNWYboy5VtM3aQo= > And so, I can now safely say that simplying insisting > 1) is not grounds for being considered a crank, but > those who insist on 2) are still cranks. You are plumb full of . If someone comes up with a mathematical theory satisfying (1) and (2) and claims, merely, that they have done so, then they are not cranks. If someone claims either that the reals already satisfy (1) and (2) or that their new theory is the *right* theory and the theory of R is wrong because, well, just because, then this is pretty good evidence of crankery. How hard is this to understand? -- Jesse F. Hughes If you believe there is any other truth but what is in your mind, you are deluding yourself. -- Demers Paradox === Subject: Re: .9 repeating sha1:TTw8h2TGdDcbtSPqxjl4leuGWYY= > But then I was called unethical -- because 2) doesn't > hold in Katz, yet MR wants 2) to hold. My triumph over > finding a rigorous theory in which 1) holds becomes > short-lived because of 2). It is considered unethical > for me to associate Katz with MR, because Katz and MR > disagree about 2). Why did I associate MR and Katz, anyway? It was because > of 1), not because of 2)! What I'd like to do is come up with a new theory by > modifying Katz so that 2) holds -- which should be > easier than starting with standard analysis and trying > to modify it so that 1) and 2) both hold. Using Katz as > a starting point, I'm already halfway to the goal. But > will attempting to come up with a new theory by taking > Katz and modifying it so that 2) holds be considered > academically dishonest? Look, there are two distinct claims. (a) There is a mathematical theory in which both (1) and (2) are true (and this theory is reasonable in some sense -- the Katz & Katz theories are extensions of R, for instance). (b) Mitch is proposing an alternative theory in which (1) and (2) are true. (a) is perhaps plausible, though the proof is in the pudding. (b) is just asinine overstatement. If you find a theory proving (a), it will not provide any evidence for (b). How hard is that to understand? -- Jesse F. Hughes The irony of my life is knowledge too advanced for my own planet. -- James S. Harris === Subject: Re: .9 repeating <87fxhazaia.fsf@phiwumbda.org> posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > And so, I can now safely say that simplying insisting > 1) is not grounds for being considered a crank, but > those who insist on 2) are still cranks. > If someone claims either that the reals already satisfy (1) and (2) or > that their new theory is the *right* theory and the theory of R is > wrong because, well, just because, then this is pretty good evidence > of crankery. I don't believe that the standard reals satisfy (1) or (2). Does MR? I can't be sure. Hughes has already tried asking MR but has yet to receive a response. Suppose I want a theory in which (2) is true. If the theory I come up with isn't rigorous, then I'm called a crank for trying to pass off a nonrigorous theory as rigorous. If the theory I come up with _is_ rigorous, then I'm called academically dishonest for trying to insert something that's not part of the original (standard) theory in order to make a new theory. Therefore, in no case can I ever post a theory in which (2) is true, which will be acceptable to standard analysts. I would either be a crank or dishonest. Also, notice that I don't necessarily believe that the theory of R is _wrong_ because (1) or (2) fail. I just believe that a theory in which (1) or (2) holds has the potential to be made just as rigorous as standard analysis, but I won't have the chance to try. === Subject: Re: .9 repeating sha1:TMEQZwNHiFvfW7+AQS4sVac9jy8= >> And so, I can now safely say that simplying insisting >> 1) is not grounds for being considered a crank, but >> those who insist on 2) are still cranks. >> If someone claims either that the reals already satisfy (1) and (2) or >> that their new theory is the *right* theory and the theory of R is >> wrong because, well, just because, then this is pretty good evidence >> of crankery. I don't believe that the standard reals satisfy (1) or (2). Does MR? I can't be sure. Hughes has already tried asking > MR but has yet to receive a response. Suppose I want a theory in which (2) is true. If the theory I come > up with isn't rigorous, then I'm called a crank for trying to pass > off a nonrigorous theory as rigorous. If the theory I come up with > _is_ rigorous, then I'm called academically dishonest for trying > to insert something that's not part of the original (standard) > theory in order to make a new theory. Huh? No, there's no academic dishonesty in proposing an alternative theory. Did anyone call the first proponents of non-well-founded set theory academically dishonest? (Hint: no. Another hint: They might have been called cranks if they had insisted that ZFA is right and ZFC is wrong -- even though their theory is rigorous.) > Therefore, in no case can I ever post a theory in which (2) > is true, which will be acceptable to standard analysts. I > would either be a crank or dishonest. Utter bull and you know it. > Also, notice that I don't necessarily believe that the > theory of R is _wrong_ because (1) or (2) fail. I just > believe that a theory in which (1) or (2) holds has the > potential to be made just as rigorous as standard analysis, > but I won't have the chance to try. Bull. And I, too, believe that one could come up with a rigorous theory satisfying both (1) and (2). I don't believe that it would be a very good replacement for the real numbers, but I could be wrong. Your martyr schtick is getting pretty old. -- Jesse F. Hughes You do know that after the get done with [outlawing] cigarettes, they're gonna come after guns, right? === Subject: Re: .9 repeating posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2009021910 Firefox/3.0.7,gzip(gfe),gzip(gfe) > And, again, you didn't answer my repeated question in my post: On what > basis did you arrive at your sweeping generalization that there is > some notion of 'absolute truth' regarding such things as the axiom of > infinity held uniformly among standard set theorists? Well, most standard set theories contain an Axiom of Infinity, > don't they? I've yet to see any set theory lack this axiom and > still be labeled standard. Similarly, I consider standard > analysts to accept as simply true many, if not most, of the > theorems of standard analysis. That a set theory has a certain axiom and that certain people work in that theory does not entail that all those people take that axiom as simply true, since there is a wide range of ways of regarding axioms - from the notion that axioms are merely strings of symbols from which to derive other strings of symbols to a fully platonistic notion, with all kinds of views between and aside. > My labels standard set theorist, standard analyst, etc., > is my effort to avoid the label standard mathematician, > which previously led to an argument on how nonstandard > analysis is still standard mathematics. So a standard > analyst explicitly accepts standard analysis, as opposed to > nonstandard analysis. Please tell me of a specific mathematician who works in non- constructive standard analysis and who is aware of non-standard analysis but doesn't grant that it is an alternate approach allowed by the very same set theoretic axioms as those ordinarily mentioned for standard analysis? > And let's add another concept to the list -- the idea of > adjacent points in geometry. Standard analysts -- or should > I say, standard geometers -- continue to frown upon coming > up with a system in which points can be adjacent, no matter > how rigorous the theory may be. Would you please give me an example of a rigorous theory such as you mention and a geometer who frowns on it. MoeBlee === Subject: Re: .9 repeating posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2009021910 Firefox/3.0.7,gzip(gfe),gzip(gfe) > One divided by infinity is nonzero. In what system - language, primitives, inference rules, axioms, definitions? MoeBlee === Subject: ? high dim conic posting-account=H-IscAoAAABkDNrURGSxo9jPN3MJ3a8A .NET CLR 1.0.3705; .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) Learnt in high school that one can reduce a more complicated equation describing a conic in 3d. But can one also do this for general high dimensional conic? f(x) = x*A*x'+B*x+c -> f(y) = y*D*y, where D is a diagonal matrix and y = T*x? Can one always find this coordinate transformation? === Subject: Re: ? high dim conic > Learnt in high school that one can reduce a more complicated equation > describing a conic in 3d. But can one also do this for general high > dimensional conic? f(x) = x*A*x'+B*x+c - > f(y) = y*D*y, where D is a diagonal matrix and y = T*x? Can one always find this coordinate transformation? > You'll need y = T*x + d, even in the original case. Then your answer is yes ... diagonalize a symmetric matrix, complete the square. === Subject: Re: ? high dim conic <190320091055470278%anniel@nym.alias.net.invalid> posting-account=H-IscAoAAABkDNrURGSxo9jPN3MJ3a8A .NET CLR 1.0.3705; .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) > Learnt in high school that one can reduce a more complicated equation > describing a conic in 3d. But can one also do this for general high > dimensional conic? > f(x) = x*A*x'+B*x+c > - > f(y) = y*D*y, where D is a diagonal matrix and y = T*x? > Can one always find this coordinate transformation? You'll need y = T*x + d, even in the original case. Then your answer is yes ... diagonalize a symmetric matrix, complete > the square. Is this pair of (T, d) the only one? Any theorem proves this? Intuitively, this seems to be always true, since using different coordinate system to describe the same conic of course shall not alter that conic. But this kind of statement is not a formal proof though. :( === Subject: Re: How to take 200th order derivative, and then evaluate it numerically? posting-account=_IRJ2QoAAADwlPG6Ye_nmoCOD0tIQEbK Gecko/2009021910 Firefox/3.0.7,gzip(gfe),gzip(gfe) > Hi all, > We need to take the 200th order derivative of a complicated formula, > and then evaluate it numerically. > What's the best way to handle this? > More specifically, we figured that it's impossible to do this via > finite difference, i.e. do everything completely in the numerical > domain, because the noise will be huge and error will dominate. > So a plan was to first obtain the 200th order derivative thru symbolic > calculation in Maple, Matlab, Mathematica, and then copy the huge > result and somehow make it into a form that is numerically evaluable > either in Matlab or C++. Then evaluate that huge expression > numerically. Still, that sounds daunting, the symbolic result of 200th > order derivative will be huge and we are not sure if Maple or > Mathematica can handle it. > Another possibility is to do some sort of Automatic Differentiation. > But we are new to AD. Is it possible that with the help from an AD > package, we just need to give inputs of the parameter values, and the > package will crank out the numerical result of the 200th order > derivative with almost no error? > We see that lots of free packages are in C/C++ or Fortune, which are > not convenient for fast prototyping(we are at evaluating stage, if > this idea doesn't work, we don't want to spend too much time on it). > So are there free AD packages that can go with Matlab or Mathematica > or Maple that can take 200th order differentiation and are convenient > for prototyping? Which one is the best AD package that we could try > on? If it doesn't work, then we would discard this crazy idea of > taking 200th order derivative completely. > Please shed some lights on us! > The first question is what the heck are you doing? If you have a formula, you can use a CAS like Mathematica and ask it > for the 200th derivative and see if it converges on returning a > formula. of course, that in itself might be suspect. If you are > strictly using numerical methods - toss this statement out the window > and re-evaluate what you are asking for because on the surface it > sounds nonsensical. ~A Yeah, I am sure after long time running, Mathematica will return a huge result, but then how do I plugin numerical parameter values to get numerical results? We can judge from the end numerical results whether it is good or bad idea to go this route... === Subject: Re: How to take 200th order derivative, and then evaluate it numerically? posting-account=VAPbmQoAAADgQtso5ktOaiLdCbXUvtKn 360SE),gzip(gfe),gzip(gfe) > Hi all, We need to take the 200th order derivative of a complicated formula, > and then evaluate it numerically. What's the best way to handle this? More specifically, we figured that it's impossible to do this via > finite difference, i.e. do everything completely in the numerical > domain, because the noise will be huge and error will dominate. So a plan was to first obtain the 200th order derivative thru symbolic > calculation in Maple, Matlab, Mathematica, and then copy the huge > result and somehow make it into a form that is numerically evaluable > either in Matlab or C++. Then evaluate that huge expression > numerically. Still, that sounds daunting, the symbolic result of 200th > order derivative will be huge and we are not sure if Maple or > Mathematica can handle it. Another possibility is to do some sort of Automatic Differentiation. > But we are new to AD. Is it possible that with the help from an AD > package, we just need to give inputs of the parameter values, and the > package will crank out the numerical result of the 200th order > derivative with almost no error? We see that lots of free packages are in C/C++ or Fortune, which are > not convenient for fast prototyping(we are at evaluating stage, if > this idea doesn't work, we don't want to spend too much time on it). > So are there free AD packages that can go with Matlab or Mathematica > or Maple that can take 200th order differentiation and are convenient > for prototyping? Which one is the best AD package that we could try > on? If it doesn't work, then we would discard this crazy idea of > taking 200th order derivative completely. Please shed some lights on us! > hello check this site you will find some idea http://www.mlibaba.com === Subject: Re: How to take 200th order derivative, and then evaluate it numerically? >> Hi all, >> We need to take the 200th order derivative of a complicated formula, >> and then evaluate it numerically. >My first question: why? I can see why you might want to take the 1st >derivative, or the 2nd derivative, or maybe even the 3rd or 4th derivative \ >of a function. The 200th? How are you planning to use the value you >receive from this procedure? I can see a possible reason, although done should try to find another way. If a Laplace transform, or a bilateral Laplace transform, can be computed by some procedure, but the computation cannot be extended in the complex plane, the function can be approximated by using high-order derivatives, and 200 might even be considered low for this problem. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: How to take 200th order derivative, and then evaluate it numerically? posting-account=_IRJ2QoAAADwlPG6Ye_nmoCOD0tIQEbK Gecko/2009021910 Firefox/3.0.7,gzip(gfe),gzip(gfe) >> Hi all, >> We need to take the 200th order derivative of a complicated formula, >> and then evaluate it numerically. >My first question: why? I can see why you might want to take the 1st >derivative, or the 2nd derivative, or maybe even the 3rd or 4th derivative >of a function. The 200th? How are you planning to use the value you >receive from this procedure? I can see a possible reason, although done should try > to find another way. If a Laplace transform, or a > bilateral Laplace transform, can be computed by some > procedure, but the computation cannot be extended in > the complex plane, the function can be approximated by > using high-order derivatives, and 200 might even be > considered low for this problem. -- > This address is for information only. I do not claim that these views > are those of the Statistics Department or of Purdue University. > Herman Rubin, Department of Statistics, Purdue University > hru...@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 Yeah we are looking for the 200th coefficient in the generating function... Any more thoughts? === Subject: Re: How to take 200th order derivative, and then evaluate it numerically? \ posting-account=06BQLAoAAADoC7Y4z9FWcUwGvMa7xMG9 7.4),gzip(gfe),gzip(gfe) >> Hi all, >> We need to take the 200th order derivative of a complicated formula, >> and then evaluate it numerically. >My first question: why? I can see why you might want to take the 1st >derivative, or the 2nd derivative, or maybe even the 3rd or 4th derivative >of a function. The 200th? How are you planning to use the value you >receive from this procedure? > I can see a possible reason, although done should try > to find another way. If a Laplace transform, or a > bilateral Laplace transform, can be computed by some > procedure, but the computation cannot be extended in > the complex plane, the function can be approximated by > using high-order derivatives, and 200 might even be > considered low for this problem. > -- > This address is for information only. I do not claim that these views > are those of the Statistics Department or of Purdue University. > Herman Rubin, Department of Statistics, Purdue University > hru...@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 > Yeah we are looking for the 200th coefficient in the generating > function... Any more thoughts?- Masquer le texte des messages pr.8ec.8edents - - Afficher le texte des messages pr.8ec.8edents - Bonjour, That reminds me about Dirichelet functions , chiefly functions well approximated by sums of exponentials. Of course if f(x) smooth all needed, very well approximated by sum(a[i]*exp( b[i]*x ) then (d/dx)^j o f(x) is directly obtained, Alain === Subject: Re: How to take 200th order derivative, and then evaluate it numerically? > Hi all, We need to take the 200th order derivative of a complicated formula, > and then evaluate it numerically. What's the best way to handle this? More specifically, we figured that it's impossible to do this via > finite difference, i.e. do everything completely in the numerical > domain, because the noise will be huge and error will dominate. So a plan was to first obtain the 200th order derivative thru symbolic > calculation in Maple, Matlab, Mathematica, and then copy the huge > result and somehow make it into a form that is numerically evaluable > either in Matlab or C++. Then evaluate that huge expression > numerically. Still, that sounds daunting, the symbolic result of 200th > order derivative will be huge and we are not sure if Maple or > Mathematica can handle it. Another possibility is to do some sort of Automatic Differentiation. > But we are new to AD. Is it possible that with the help from an AD > package, we just need to give inputs of the parameter values, and the > package will crank out the numerical result of the 200th order > derivative with almost no error? We see that lots of free packages are in C/C++ or Fortune, which are > not convenient for fast prototyping(we are at evaluating stage, if > this idea doesn't work, we don't want to spend too much time on it). > So are there free AD packages that can go with Matlab or Mathematica > or Maple that can take 200th order differentiation and are convenient > for prototyping? Which one is the best AD package that we could try > on? If it doesn't work, then we would discard this crazy idea of > taking 200th order derivative completely. Please shed some lights on us! > Perhaps what you might try is computing a Taylor series up to order x^200. \ There still might be some numerical difficulties, but it might not be so bad. For example, in Maple: f:= x^3*sin(x)/(x - cos(x)); Digits:= 20; 200! * coeff( series(f, x=0.3, 201), x-0.3, 200); -.90246797086526191165e446 -- === Subject: Re: How to take 200th order derivative, and then evaluate it numerically? posting-account=_IRJ2QoAAADwlPG6Ye_nmoCOD0tIQEbK Gecko/2009021910 Firefox/3.0.7,gzip(gfe),gzip(gfe) On Mar 18, 12:13pm, Robert Israel > Hi all, > We need to take the 200th order derivative of a complicated formula, > and then evaluate it numerically. > What's the best way to handle this? > More specifically, we figured that it's impossible to do this via > finite difference, i.e. do everything completely in the numerical > domain, because the noise will be huge and error will dominate. > So a plan was to first obtain the 200th order derivative thru symbolic > calculation in Maple, Matlab, Mathematica, and then copy the huge > result and somehow make it into a form that is numerically evaluable > either in Matlab or C++. Then evaluate that huge expression > numerically. Still, that sounds daunting, the symbolic result of 200th > order derivative will be huge and we are not sure if Maple or > Mathematica can handle it. > Another possibility is to do some sort of Automatic Differentiation. > But we are new to AD. Is it possible that with the help from an AD > package, we just need to give inputs of the parameter values, and the > package will crank out the numerical result of the 200th order > derivative with almost no error? > We see that lots of free packages are in C/C++ or Fortune, which are > not convenient for fast prototyping(we are at evaluating stage, if > this idea doesn't work, we don't want to spend too much time on it). > So are there free AD packages that can go with Matlab or Mathematica > or Maple that can take 200th order differentiation and are convenient > for prototyping? Which one is the best AD package that we could try > on? If it doesn't work, then we would discard this crazy idea of > taking 200th order derivative completely. > Please shed some lights on us! > Perhaps what you might try is computing a Taylor series up to order x^200. \ > There still might be some numerical difficulties, but it might not be so bad. > For example, in Maple: f:= x^3*sin(x)/(x - cos(x)); > Digits:= 20; > 200! * coeff( series(f, x=0.3, 201), x-0.3, 200); -.90246797086526191165e446 > -- > Robert Israel isr...@math.MyUniversitysInitials.ca > Department of Mathematics http://www.math.ubc.ca/~israel > University of British Columbia Vancouver, BC, Canada Our analytical expression is very complicated, with subcomponents that are specified by other dynamic systems. Not sure how we can wrap the whole thing up in Maple as a function? === Subject: Re: How to take 200th order derivative, and then evaluate it numerically? On 2009-03-19 11:43:38 -0300, excellentfeng@gmail.com said: > On Mar 18, 12:13pm, Robert Israel > Hi all, > We need to take the 200th order derivative of a complicated formula, > and then evaluate it numerically. > What's the best way to handle this? > More specifically, we figured that it's impossible to do this via > finite difference, i.e. do everything completely in the numerical > domain, because the noise will be huge and error will dominate. > So a plan was to first obtain the 200th order derivative thru symbolic > calculation in Maple, Matlab, Mathematica, and then copy the huge > result and somehow make it into a form that is numerically evaluable > either in Matlab or C++. Then evaluate that huge expression > numerically. Still, that sounds daunting, the symbolic result of 200th > order derivative will be huge and we are not sure if Maple or > Mathematica can handle it. > Another possibility is to do some sort of Automatic Differentiation. > But we are new to AD. Is it possible that with the help from an AD > package, we just need to give inputs of the parameter values, and the > package will crank out the numerical result of the 200th order > derivative with almost no error? > We see that lots of free packages are in C/C++ or Fortune, which are > not convenient for fast prototyping(we are at evaluating stage, if > this idea doesn't work, we don't want to spend too much time on it). > So are there free AD packages that can go with Matlab or Mathematica > or Maple that can take 200th order differentiation and are convenient > for prototyping? Which one is the best AD package that we could try > on? If it doesn't work, then we would discard this crazy idea of > taking 200th order derivative completely. > Please shed some lights on us! >> Perhaps what you might try is computing a Taylor series up to order x^200 > . >> There still might be some numerical difficulties, but it might not be so > bad. >> For example, in Maple: f:= x^3*sin(x)/(x - cos(x)); >> Digits:= 20; >> 200! * coeff( series(f, x=0.3, 201), x-0.3, 200); -.90246797086526191165e446 >> -- >> Robert Israel isr...@math.MyUniversitysInitial > s.ca >> Department of Mathematics http://www.math.ubc.ca/~israel >> University of British Columbia Vancouver, BC, Cana > da > Our analytical expression is very complicated, with subcomponents that > are specified by other dynamic systems. Not sure how we can wrap the whole thing up in Maple as a function? That would happen if there was one expession below some break point and another above it with an IF to choose which to use. If this were the case it would not make any sense to try for a single coefficient of a Taylor's series. (Not that the rest of your problem as stated seems to make a lot of sense. You sure would need a whole lot of confidence, and then maybe even more, in your formulation to justify this sort of computation.) === Subject: Re: How to take 200th order derivative, and then evaluate it numerically? > Hi all, We need to take the 200th order derivative of a complicated formula, > and then evaluate it numerically. > First things first: Fire the person who came up with the number 200... it should have been 199. === Subject: Re: How to take 200th order derivative, and then evaluate it numerically? > We need to take the 200th order derivative of a complicated formula, > and then evaluate it numerically. I suppose the formula represents an analytic function. If so, you're really just asking for just one term in the Taylor series f(z) = sum a_n z^n . So your question is how to evaluate just one term in the series (perhaps emphasizing that you don't need the lower-order terms first; you can liken this to the spigot algorithms for the bits of pi ). If you compute a line integral of f(z)/z^201 around the origin, all the summands a_n z^n contribute zero except the one with n=200. Divide by 2 pi, multiply by 200! (it's roughly 2^1245) and you have your derivative. So if you can compute values of f with thousand-bit precision, then you can compute this line integral numerically and you have your derivative. >My first question: why? I can see why you might want to take the 1st >derivative, or the 2nd derivative, or maybe even the 3rd or 4th derivative \ >of a function. The 200th? How are you planning to use the value you >receive from this procedure? Oh, I can jump in here: there are certainly questions that can be phrased as requiring a number which is computable (e.g. via generating functions) as the 200th derivative of a function. So I would be very interested in hearing other answers to this question. Here's an example: in order to factor an thousand-bit integer N, it would be more or less sufficient to compute the thousand binomial coefficients a_k = binomial(2^{k+1}, 2^k) modulo N (for k=1, 2, ..., 1000). So the question how hard is it to factor N ? changes to how hard is it to compute a_k mod N ? The ideal state would be to know how to do that computation in an amount of time which is a polynomial in k, but as far as I know, no one knows how to do that; I don't know how to compute the thousand bits of a_500 before the universe dies a cold death. (I don't know why it should be so hard; these a_k are just a little smaller than 4^(2^k), and the numbers 4^(2^k) mod N themselves can be computed in O(k) steps.) But consider this: these a_k are among the coefficients in the Taylor series of f(z) = 1/sqrt(1-4z). So, just as the OP, I find myself wishing I could compute high-order derivatives f^(m)(0) of my function. (The OP's m is 200; mine is, say, m=2^500 !) And indeed I gave a recipe above for computing these Taylor coefficients. There is no real difficulty for me to compute values of f(z)/z^(2^k), either: the denominator is computed with k squarings, and the numerator can be computed with k iterations of Newton's method, giving approximations of f(z) which are off by at most O(z^(-2^k)). But I have problems that the OP doesn't have, because of how long it takes to compute an operation, even a simple sum, when I am dealing with enormously long strings of digits (bits), and also because of the number of evaluations of f which I would need to make in order to compute the line integral numerically. It seems like a waste because the value I'm trying to compute is an integer, and I only need its value mod N , (i.e. 1000 bits of data) but I don't know how to get around these problems. So my question is similar to the OP's question: if I have a quick and accurate way to compute values of f, is there a quick and accurate way to compute any one term of its Taylor series? dave === Subject: Re: How to take 200th order derivative, and then evaluate it numerically? > Here's an example: in order to factor an thousand-bit integer N, > it would be more or less sufficient to compute the thousand binomial > coefficients a_k = binomial(2^{k+1}, 2^k) modulo N (for k=1, 2, ..., 1000). Dave, Intriguing. Care to elaborate? Bob === Subject: Polynomial-time integer factorization (Was: Re: How to take 200th order derivative, and then evaluate it numerically?) > Here's an example: in order to factor an thousand-bit integer N, >> it would be more or less sufficient to compute the thousand binomial >> coefficients a_k = binomial(2^{k+1}, 2^k) modulo N (for k=1, 2, ..., \ 1000). >Intriguing. Care to elaborate? This a_k is divisible by every (k+1)-bit prime (and no larger prime, and few smaller primes), so we successively compute gcd(N, a_k) (k=1,2,...) until we get something nontrivial. That will give a factor of N ; done! (Almost. It's a _nontrivial_ factor of N unless every prime divisor of N \ is of exactly the same length in bits; in that case we can play a similar game with the other Taylor coefficients that I mentioned, bisecting the intervals where the primes are known to lie until an interval is small enough to contain some but not all the prime divisors. So it might take 2000 gcd computations instead of 1000 to find a proper factor. Big deal. But the main issue that we have to have a fast way to compute central binomial coefficients mod N, with the a_k 's being the main example.) Well, you tell me: if N = 2^987 + 21, what's a_300 mod N ? (This N is a 298-digit composite number, so your answer fits in a few lines.) Unlike the OP's problem, in my case these coefficients a_k are known exactly. If I had a lot of numbers to factor, I could even pre-compute them once and then just reduce mod N each time I want to use them in this way. But from Stirling's approximation, we estimate a_k is about 4^(2^k)/(2^(k/2)) which is to say not thousands of bits but say 2^1000 bits! So it is critical for this program that we find a way to compute a_k mod N without finding a_k itself first. Note that the slightly larger number 4^(2^k) can be computed mod N with k successive squarings, so it's not magnitude alone that makes the job difficult. What's weird is that in principle I HAVE all these numbers a_k mod N : using Newton's method to compute 1/sqrt(1-4t) as a solution to 1/X^2 - (1-4t) = 0, we compute polynomials p_k with p_0 = 1 p_{k+1} = (1/2) * p_k * (3 - (p_k)^2*(1-4*t) ) mod N and then the first 2^k coefficients of p_k agree with those of the taylor series of 1/sqrt(1-4t) , and in particular a_k is just the coefficient of x^(2^k) in p_{k+1} . Check it out with your favorite CAS! (You can even choose to truncate each p_k after t^(2^k) .) It takes only O(k) ring operations in the ring (Z/NZ)[t] -- polynomial time factorization! BUT I don't know a way to pick out one coefficient of one of these polynomials without multiplying out everything term by term, and of course I can't do that with 2^1000 terms. The number of ring operations in Z/NZ itself is exponential in k so it's not really polynomial time. (I guess the terminology here is that these are SLP polynomials -- Straight Line Program or something -- meaning that their _values_ can be computed very easily; the question then becomes whether the one coefficient can be recovered with the same kind of speed with which values are computed. I don't know a way to do this.) Anyway, fast integer factorization is a hot topic, and as you can see I can reduce it to the question of computing a high-order derivative of an easily-computed function, so this is why I followed this thread with interest! dave === Subject: Re: How to take 200th order derivative, and then evaluate it numerically? >Hi all, We need to take the 200th order derivative of a complicated formula, >and then evaluate it numerically. What's the best way to handle this? More specifically, we figured that it's impossible to do this via >finite difference, i.e. do everything completely in the numerical >domain, because the noise will be huge and error will dominate. correct. maybe you use 1000 digits precision?? So a plan was to first obtain the 200th order derivative thru symbolic >calculation in Maple, Matlab, Mathematica, and then copy the huge >result and somehow make it into a form that is numerically evaluable >either in Matlab or C++. Then evaluate that huge expression >numerically. Still, that sounds daunting, the symbolic result of 200th >order derivative will be huge and we are not sure if Maple or >Mathematica can handle it. depends Another possibility is to do some sort of Automatic Differentiation. >But we are new to AD. Is it possible that with the help from an AD >package, we just need to give inputs of the parameter values, and the >package will crank out the numerical result of the 200th order >derivative with almost no error? no no. AD is very good for the first derivative, has already problems with hessians, but for higher order I never saw it. (the computational graph of a comp graph of a comp graph ... ) We see that lots of free packages are in C/C++ or Fortune, which are >not convenient for fast prototyping(we are at evaluating stage, if >this idea doesn't work, we don't want to spend too much time on it). >So are there free AD packages that can go with Matlab or Mathematica >or Maple that can take 200th order differentiation and are convenient >for prototyping? Which one is the best AD package that we could try >on? If it doesn't work, then we would discard this crazy idea of >taking 200th order derivative completely. Please shed some lights on us! > maybe your teacher wanted to shock you somehow , but there are some chances you didn't consider: if it is a rational function , perform the decomposition into partial fractions, then the formal derivative is obvious, and possibly combine back. nice exercise : the 200-th derivative of arctan(x) if it is a function with known or easy power series expansion, differentiate this one the same for a fourier expansion: attention: think about the convergence criteria what is that good for? maybe you must evaluate the remainder term of a 100-node gauss-quadrature?? (bad joke) hth peter === Subject: Re: How to take 200th order derivative, and then evaluate it numerically? posting-account=_IRJ2QoAAADwlPG6Ye_nmoCOD0tIQEbK Gecko/2009021910 Firefox/3.0.7,gzip(gfe),gzip(gfe) >More specifically, we figured that it's impossible to do this via >finite difference, i.e. do everything completely in the numerical >domain, because the noise will be huge and error will dominate. correct. maybe you use 1000 digits precision?? not sure if 1000 digits precision will save it? no no. AD is very good for the first derivative, has already problems with \ hessians, > but for higher order I never saw it. (the computational graph of a comp graph of > a comp graph ... ) Maybe we keep doing the first derivative for 200 times? Supposedly it has no error? > maybe your teacher wanted to shock you somehow , > but there are some chances you didn't consider: if it is a rational function , perform the decomposition into partial > fractions, then the formal derivative is obvious, and possibly combine > back. nice exercise : the 200-th derivative of arctan(x) if it is a function with known or easy power series expansion, > differentiate this one the same for a fourier expansion: attention: think about the convergence > criteria what is that good for? > maybe you must evaluate the remainder term of a 100-node gauss-quadrature?? > (bad joke) > Yeah, Peter. As you can see from other posts, we are trying to get the 200th coefficient of the Taylor expansion of a generating function. Any more thoughts? === Subject: Re: How to take 200th order derivative, and then evaluate it numerically? ... > ...If it doesn't work, then we would discard this crazy idea of > taking 200th order derivative completely. Sounds like a plan... -- === Subject: Re: How to take 200th order derivative, and then evaluate it numerically? posting-account=SIMyZAoAAAAhUsAAG9RTx1O9_n6inDDv rv:1.9.0.7) Gecko/2009021906 Firefox/3.0.7,gzip(gfe),gzip(gfe) > Hi all, We need to take the 200th order derivative of a complicated formula, > and then evaluate it numerically. What's the best way to handle this? More specifically, we figured that it's impossible to do this via > finite difference, i.e. do everything completely in the numerical > domain, because the noise will be huge and error will dominate. So a plan was to first obtain the 200th order derivative thru symbolic > calculation in Maple, Matlab, Mathematica, and then copy the huge > result and somehow make it into a form that is numerically evaluable > either in Matlab or C++. Then evaluate that huge expression > numerically. Still, that sounds daunting, the symbolic result of 200th > order derivative will be huge and we are not sure if Maple or > Mathematica can handle it. Another possibility is to do some sort of Automatic Differentiation. > But we are new to AD. Is it possible that with the help from an AD > package, we just need to give inputs of the parameter values, and the > package will crank out the numerical result of the 200th order > derivative with almost no error? We see that lots of free packages are in C/C++ or Fortune, which are > not convenient for fast prototyping(we are at evaluating stage, if > this idea doesn't work, we don't want to spend too much time on it). > So are there free AD packages that can go with Matlab or Mathematica > or Maple that can take 200th order differentiation and are convenient > for prototyping? Which one is the best AD package that we could try > on? If it doesn't work, then we would discard this crazy idea of > taking 200th order derivative completely. Please shed some lights on us! > How did you even get the coefficients for a 200th order polynomial? I agree with the others that say this is non sense. Any number, that is not extreme close to 1 taken to the 200 power, will either cause a numeric overflow or be rounded down to 0 unless you have a very high precision calculator. However, if the equation is a 200th order polynomial then one can compute the coefficients for any of the derivative functions easily enough in C or Fortran. Then the equation can be evaluated using Horner's method. No matlab, mathematica or maple required. I have done this before with 5 to 7th order polynomials. It would be easy enough to expand the arrays to do this with 200th order polynomials. Peter Nachtwey === Subject: Re: How to take 200th order derivative, and then evaluate it numerically? posting-account=_IRJ2QoAAADwlPG6Ye_nmoCOD0tIQEbK Gecko/2009021910 Firefox/3.0.7,gzip(gfe),gzip(gfe) > Hi all, > We need to take the 200th order derivative of a complicated formula, > and then evaluate it numerically. > What's the best way to handle this? > More specifically, we figured that it's impossible to do this via > finite difference, i.e. do everything completely in the numerical > domain, because the noise will be huge and error will dominate. > So a plan was to first obtain the 200th order derivative thru symbolic > calculation in Maple, Matlab, Mathematica, and then copy the huge > result and somehow make it into a form that is numerically evaluable > either in Matlab or C++. Then evaluate that huge expression > numerically. Still, that sounds daunting, the symbolic result of 200th > order derivative will be huge and we are not sure if Maple or > Mathematica can handle it. > Another possibility is to do some sort of Automatic Differentiation. > But we are new to AD. Is it possible that with the help from an AD > package, we just need to give inputs of the parameter values, and the > package will crank out the numerical result of the 200th order > derivative with almost no error? > We see that lots of free packages are in C/C++ or Fortune, which are > not convenient for fast prototyping(we are at evaluating stage, if > this idea doesn't work, we don't want to spend too much time on it). > So are there free AD packages that can go with Matlab or Mathematica > or Maple that can take 200th order differentiation and are convenient > for prototyping? Which one is the best AD package that we could try > on? If it doesn't work, then we would discard this crazy idea of > taking 200th order derivative completely. > Please shed some lights on us! > How did you even get the coefficients for a 200th order polynomial? I agree with the others that say this is non sense. Any number, that > is not extreme close to 1 taken to the 200 power, will either cause a > numeric overflow or be rounded down to 0 unless you have a very high > precision calculator. However, if the equation is a 200th order polynomial then one can > compute the coefficients for any of the derivative functions easily > enough in C or Fortran. Then the equation can be evaluated using > Horner's method. No matlab, mathematica or maple required. I have > done this before with 5 to 7th order polynomials. It would be easy > enough to expand the arrays to do this with 200th order polynomials. Peter Nachtwey Hi Peter, Yes, essentially we are finding the 200th coefficient of the Taylor expansion of a very complicated analytical expression. Let's say it's given by f(x), and f(x) is very complicated. We need the 200th coefficient... Any thoughts? === Subject: Re: How to take 200th order derivative, and then evaluate it numerically? posting-account=SIMyZAoAAAAhUsAAG9RTx1O9_n6inDDv rv:1.9.0.7) Gecko/2009021906 Firefox/3.0.7,gzip(gfe),gzip(gfe) > Hi all, > We need to take the 200th order derivative of a complicated formula, > and then evaluate it numerically. > What's the best way to handle this? > More specifically, we figured that it's impossible to do this via > finite difference, i.e. do everything completely in the numerical > domain, because the noise will be huge and error will dominate. > So a plan was to first obtain the 200th order derivative thru symbolic > calculation in Maple, Matlab, Mathematica, and then copy the huge > result and somehow make it into a form that is numerically evaluable > either in Matlab or C++. Then evaluate that huge expression > numerically. Still, that sounds daunting, the symbolic result of 200th > order derivative will be huge and we are not sure if Maple or > Mathematica can handle it. > Another possibility is to do some sort of Automatic Differentiation. > But we are new to AD. Is it possible that with the help from an AD > package, we just need to give inputs of the parameter values, and the > package will crank out the numerical result of the 200th order > derivative with almost no error? > We see that lots of free packages are in C/C++ or Fortune, which are > not convenient for fast prototyping(we are at evaluating stage, if > this idea doesn't work, we don't want to spend too much time on it). > So are there free AD packages that can go with Matlab or Mathematica > or Maple that can take 200th order differentiation and are convenient > for prototyping? Which one is the best AD package that we could try > on? If it doesn't work, then we would discard this crazy idea of > taking 200th order derivative completely. > Please shed some lights on us! > How did you even get the coefficients for a 200th order polynomial? > I agree with the others that say this is non sense. Any number, that > is not extreme close to 1 taken to the 200 power, will either cause a > numeric overflow or be rounded down to 0 unless you have a very high > precision calculator. > However, if the equation is a 200th order polynomial then one can > compute the coefficients for any of the derivative functions easily > enough in C or Fortran. Then the equation can be evaluated using > Horner's method. No matlab, mathematica or maple required. I have > done this before with 5 to 7th order polynomials. It would be easy > enough to expand the arrays to do this with 200th order polynomials. > Peter Nachtwey Hi Peter, Yes, essentially we are finding the 200th coefficient of the Taylor > expansion of a very complicated analytical expression. Let's say it's given by f(x), and f(x) is very complicated. We need the 200th coefficient... Any thoughts? Only the obvious which is what we did. Suppose you make a 200 x 200 array for you polynomial and put the coefficients in the first row or column like this Y(x)=c0+c1*x+c2*x^2 .. each column is reduced by one order. x x' x x''' c0 1*c1 1*2*c2 1*2*3*c3 c1 2*c2 2*3*c3 2*3*4*c4 c2 3*c3 3*4*c4 3*4*5*c5 c3 4*c4 4*5*c5 4*5*6*c6 c4 5*c5 5*6*c6 5*6*7*c7 c5 6*c6 6*7*c7 6*7*8*c8 Hopefully the row will not get messed up so bad you can't see what I am doing. Do you see the pattern? Is it time for a duh moment? This is shear brute force. This can be done in two for loops easily. By the time you get to the 199 column you will have just a constant. Of course you could just assume all the coefficients, c0..cn, are 1 and work out the multipliers only once head of time. We do this in real time systems but with lower order. Then we use Horner's method to evaluate the array at any x for any derivative. We hard code all of this. y(x) = ((...+c3*x+c2)*x+c1)*x+c0 One can easily make a general function out of this where one specifies x and the derivative. Put this data in an array where the column is the derivative number and the rows are the powers of x. A good optimizing C compiler makes light work out of this at run time. You may wish to try switching rows and columns to see which optimization is best. Peter Nachtwey === Subject: Re: How to take 200th order derivative, and then evaluate it numerically? > Hi all, We need to take the 200th order derivative of a complicated formula, > and then evaluate it numerically. What's the best way to handle this? More specifically, we figured that it's impossible to do this via > finite difference, i.e. do everything completely in the numerical > domain, because the noise will be huge and error will dominate. So a plan was to first obtain the 200th order derivative thru symbolic > calculation in Maple, Matlab, Mathematica, and then copy the huge > result and somehow make it into a form that is numerically evaluable > either in Matlab or C++. Then evaluate that huge expression > numerically. Still, that sounds daunting, the symbolic result of 200th > order derivative will be huge and we are not sure if Maple or > Mathematica can handle it. Another possibility is to do some sort of Automatic Differentiation. > But we are new to AD. Is it possible that with the help from an AD > package, we just need to give inputs of the parameter values, and the > package will crank out the numerical result of the 200th order > derivative with almost no error? We see that lots of free packages are in C/C++ or Fortune, which are > not convenient for fast prototyping(we are at evaluating stage, if > this idea doesn't work, we don't want to spend too much time on it). > So are there free AD packages that can go with Matlab or Mathematica > or Maple that can take 200th order differentiation and are convenient > for prototyping? Which one is the best AD package that we could try > on? If it doesn't work, then we would discard this crazy idea of > taking 200th order derivative completely. Please shed some lights on us! > The first question is what everyone is asking: Is this trip necessary? The next questions are based on an affirmative answer to the first: Do any useful transforms Tf (T: Fourier, Laplace,... [there must be others with useful expressions for T(derivative)]) exist? If so, look at T(f^(200)). Once you have T(f), this should be the trivial step. What about the inverse transform of T(f^(200))? How ugly is it? Just my two bits' worth. Dale === Subject: math in crosswords I've been enjoying the crosswords at http://suncrossword.com/Sun/ lately. I think they're better than the NYT's. I'm mentioning them for several reasons: 1. There's a lot (that is, usually one per puzzle) of math clues/answers which are surprisingly high-level. ISOMETRY was an answer last month. Another time, the clue was 80th prime in Roman. This caused me a certain amount of trouble, since I felt it was cheating to grind out the 80th prime. Further, there were only 3 squares because of a theme thing. One realizes quickly that the prime must end in I or IX. The crossing answer wanted it to end in X (although there was another possibility that would make it end in I, the X really wanted to be there.) I knew that pi(100) = 25, so that made me think the answer needed to be over 300. By elimination, with only three squares I was stuck. IX uses two, and I was pretty sure 500 was too big, so I didn't have enough squares to stick in CCC or CD. 401, 409, 509, are all prime. It turned out that, because of a theme-thing, it was really 4 digits long. The answer was CDIX. Anyway, the point is, that there's more math in these puzzles than any others I know of. Seriously ISOMETRY? And the clue was correct. Something like Distance preserving transformation. 2. While I'm not affiliated with the outfit that produces them, I'd like to plug them. Peter Gordon has managed to produce and deliver these puzzles for a while, but he's losing money on the deal. If he can get more people to sign up for his online puzzles (PDF or Across Lite), he says he can keep things going, which I'd really like to see. I think I paid $12 for access to 6 months worth of puzzles. I think he wants to raise rates a bit, to 20 cents per puzzle, which would be $30 to $50 per year. If you're a crossword fanatic, please visit the above website and click on the yellow box to Bring back the Sun Crossword. Again: Not affiliated, yadda, yadda. I just want to be able to keep buying these puzzles. Bart -- Cheerfully resisting change since 1959. === Subject: Re: JSH: Closing moments on first part of saga > With Fermat numbers though--suggested to me by Penny Hassett in a > comment to my math blog--it's easier, as if you have D = 2^k + 1, then > there are k+1 iterations, which is, um, yeah, polynomial time as in > LINEAR. >> Since the arithmetic operations for big numbers can't be computed in >> constant time I suggest to compute the bit complexity that takes >> this into account. Regardless of debates about time, there are a finite number of factors > of D-1. With those factors in hand, it's trivial to close the question on how > long it will take to factor D. For instance, if D-1 = 2p, where p is a prime, then you only have 2 > iterations possible. > >> I presume you use at least one of the following operations during each >> iteration: +, -, *, /. The complexity class of plus and minus is O(k). >> Multiplication can be done in O(k * log k * log log k) time (Sch.9anhage- >> Strassen algorithm) or O(n^1.585) time (Karatsuba algorithm). >> If I recall correctly java.math.BigInteger uses the Karatsuba >> algorithm for multiplication. The concept I had years ago, which I chased for years was about a > question, and the question was, can you factor one number by instead > factoring another? The answer, unequivocally, is yes. Debates about complexity or whether or not I can factor an RSA public > key are not about the pursuit of knowledge or human curiosity, but > about denial. There is a class barrier: a glass ceiling. And lies have been said about it for a while now, but there is a NEED > among certain people to feel like certain races are blocked from > intellectual accomplishments like mine. Know why? Because I'm not elected. I don't cater to a fan base. It doesn't > matter to me if you like me. You can't control me by voting me out. > You can't kindly suggest that I worry about my career! You're terrified of your daughters dreaming about me. Some of you would cut your wrists before you'd dare think of your sons > wanting to grow up to be like me. Even President Obama is a controlled threat. You voted him in. You can't vote me out. You can't vote me in. You cannot stop me. I am a discoverer. And what I am doing is what we do. I belong to the most exclusive club in the history of the world. Beyond races, or classes or your petty delusions of what it means to > be at the pinnacle of knowledge. I am the discoverer. And that is what you fear and fight with your social toys, while you > can. But remember, when the full story is told, you stole your own children > away from you, as they WILL despise you, and what you represent: the > antithesis of the human spirit. It was the human spirit which sustained me, inspired me, and made > certain that I would win. It was the soul of the world which you were fighting, not me. > James Harris > Do you think MI6 got a clue in time... or ? David Bernier === Subject: Re: JSH: Closing moments on first part of saga > You're terrified of your daughters dreaming about me. This one's an instant classic. > Some of you would cut your wrists before you'd dare think > of your sons wanting to grow up to be like me. This might be true. === Subject: Re: JSH: Closing moments on first part of saga > I've finally figured out the full proof, ... Where have I read those words before? Never mind, I'm _so_ pleased for you. -- Science is a differential equation. Religion is a boundary condition. --Alan Turing === Subject: Re: JSH: Closing moments on first part of saga sha1:g3RRcsHSkmJBP1OShsmvuKSsFrI= The concept I had years ago, which I chased for years was about a > question, and the question was, can you factor one number by instead > factoring another? The answer, unequivocally, is yes. Debates about complexity or whether or not I can factor an RSA public > key are not about the pursuit of knowledge or human curiosity, but > about denial. There is a class barrier: a glass ceiling. And lies have been said about it for a while now, but there is a NEED > among certain people to feel like certain races are blocked from > intellectual accomplishments like mine. Know why? Because I'm not elected. I don't cater to a fan base. It doesn't > matter to me if you like me. You can't control me by voting me out. > You can't kindly suggest that I worry about my career! You're terrified of your daughters dreaming about me. Some of you would cut your wrists before you'd dare think of your sons > wanting to grow up to be like me. Even President Obama is a controlled threat. You voted him in. You can't vote me out. You can't vote me in. You cannot stop me. I am a discoverer. And what I am doing is what we do. I belong to the most exclusive club in the history of the world. Beyond races, or classes or your petty delusions of what it means to > be at the pinnacle of knowledge. I am the discoverer. And that is what you fear and fight with your social toys, while you > can. But remember, when the full story is told, you stole your own children > away from you, as they WILL despise you, and what you represent: the > antithesis of the human spirit. It was the human spirit which sustained me, inspired me, and made > certain that I would win. It was the soul of the world which you were fighting, not me. > Ah. Now *that's* a fine JSH screed. I especially like the bit about the daughters. Kudos! It's been a while since you hit such top form. -- Kim liked the math I did for her and gave me quite a few groceries... likely so many groceries that they would have cost Kim about what she pays for two whole packages of cigarettes. Few people have ever rewarded me for my work as much as Kim did. -- Usenet nut === Subject: Re: JSH: Closing moments on first part of saga > With Fermat numbers though--suggested to me by > Penny Hassett in a > comment to my math blog--it's easier, as if you > have D = 2^k + 1, then > there are k+1 iterations, which is, um, yeah, > polynomial time as in > LINEAR. > Since the arithmetic operations for big numbers > can't be computed in > constant time I suggest to compute the bit > complexity that takes > this into account. Regardless of debates about time, there are a finite > number of factors > of D-1. With those factors in hand, it's trivial to close the > question on how > long it will take to factor D. For instance, if D-1 = 2p, where p is a prime, then > you only have 2 > iterations possible. I presume you use at least one of the following > operations during each > iteration: +, -, *, /. The complexity class of plus > and minus is O(k). > Multiplication can be done in O(k * log k * log log > k) time (Sch.9anhage- > Strassen algorithm) or O(n^1.585) time (Karatsuba > algorithm). > If I recall correctly java.math.BigInteger uses the > Karatsuba > algorithm for multiplication. The concept I had years ago, which I chased for years > was about a > question, and the question was, can you factor one > number by instead > factoring another? The answer, unequivocally, is yes. Debates about complexity or whether or not I can > factor an RSA public > key are not about the pursuit of knowledge or human > curiosity, but > about denial. There is a class barrier: a glass ceiling. And lies have been said about it for a while now, but > there is a NEED > among certain people to feel like certain races are > blocked from > intellectual accomplishments like mine. Know why? Because I'm not elected. I don't cater to a fan > base. It doesn't > matter to me if you like me. You can't control me by > voting me out. > You can't kindly suggest that I worry about my > career! You're terrified of your daughters dreaming about me. Some of you would cut your wrists before you'd dare > think of your sons > wanting to grow up to be like me. Even President Obama is a controlled threat. You > voted him in. You can't vote me out. You can't vote me in. You > cannot stop me. I am a discoverer. And what I am doing is what we do. I belong to the most exclusive club in the history of > the world. Beyond races, or classes or your petty delusions of > what it means to > be at the pinnacle of knowledge. I am the discoverer. And that is what you fear and fight with your social > toys, while you > can. But remember, when the full story is told, you stole > your own children > away from you, as they WILL despise you, and what you > represent: the > antithesis of the human spirit. It was the human spirit which sustained me, inspired > me, and made > certain that I would win. It was the soul of the world which you were fighting, > not me. > James Harris > I have never seen anyone reply to your posts favorably. I don't see anyone \ agreeing with you. Now, even if your solution is correct, and world-changing, nobody but you acknowledges it. So no matter what, when you \ are either too old or too busy to post on this newsgroup, your mathematical \ contributions, however great or small, will be a mere footnote. They will mean nothing. Surely even you can realize the infinitesimal impact you are \ making with your entire life. Why continue? You say eventually, the entire \ world will realize your brilliance. But when? It *Still* hasn't happened! \ Your life is finite; if something continually does not work out, you must move on to something different. Otherwise, in your final moments, I respectfully predict you will regret everything. === Subject: Re: JSH: Closing moments on first part of saga On 2009-03-19 00:43:35 -0400, JSH said: > With Fermat numbers though--suggested to me by Penny Hassett in a > comment to my math blog--it's easier, as if you have D = 2^k + 1, the > n > there are k+1 iterations, which is, um, yeah, polynomial time as in > LINEAR. Since the arithmetic operations for big numbers can't be computed in >> constant time I suggest to compute the bit complexity that takes >> this into account. Regardless of debates about time, there are a finite number of factors > of D-1. That you could possibly say that with a straight face reveals how profoundly ignorant you are about the very problem you're trying to solve. > I don't cater to a fan base. This is a damnable lie. > It doesn't matter to me if you like me. As is this. -o === Subject: Logic and tautology posting-account=5cbwMAoAAAAZoYdmWfebkP1kkdh09sZi AppleWebKit/525.27.1 (KHTML, like Gecko) Version/3.2.1 Safari/525.27.1,gzip(gfe),gzip(gfe) Marijuana has not been proven to be harmful. Therefore, marijuana is not a dangerous drug. How do you prove that the conclusion arrived at is false using a truth table? What logical connectives should i use? === Subject: Re: Logic and tautology > Marijuana has not been proven to be harmful. It's been proven harmful by a number of special government studies. In addition, it's harmful because it can get you arrested. Considering it' price, it can harm your budget. > Therefore, marijuana is not a dangerous drug. > A pound of marijuana is dangerous. > How do you prove that the conclusion arrived at is false using a truth > table? > Wrong approach. Politics is not logical. > What logical connectives should i use? You need the assumption that dangerous drugs are harmful. An equally likely assumption is that new drugs recently declared safe by the government, as based upon research done by the drug company that made the drug, are dangerous. === Subject: Re: Logic and tautology > Marijuana has not been proven to be harmful. > Therefore, marijuana is not a dangerous drug. > How do you prove that the conclusion arrived at is false using a truth > table? What logical connectives should i use? Marijuana is indeed a dangerous drug. It causes governments to wage war against their own people. -- Dave Seaman Third Circuit ignores precedent in Mumia Abu-Jamal ruling. === Subject: Logical connective or : inclusive or exclusive meaning? posting-account=5cbwMAoAAAAZoYdmWfebkP1kkdh09sZi AppleWebKit/525.27.1 (KHTML, like Gecko) Version/3.2.1 Safari/525.27.1,gzip(gfe),gzip(gfe) Either Galileo was a genius or he worked hard. Galileo was a genius. Therefore he did not work hard. If we consider the word or to have an inclusive meaning then it is easy to show why the consequent is false using a truth table. But I think in this problem or it has an exclusive meaning. Which one is it? === Subject: Re: Logical connective or : inclusive or exclusive meaning? <190320091106359109%edgar@math.ohio-state.edu.invalid> posting-account=G4HhgwkAAAB0h1X9lFcN2_L63S7FUgWk 1.0.3705; .NET CLR 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30; .NET CLR 3.0.04506.648; .NET CLR 3.0.4506.2152; .NET CLR 3.5.30729),gzip(gfe),gzip(gfe) On Mar 19, 11:06am, G. A. Edgar Either Galileo was a genius or he worked hard. > Galileo was a genius. > Therefore he did not work hard. > If we consider the word or to have an inclusive meaning then it is > easy to show why the consequent is false using a truth table. But I > think in this problem or it has an exclusive meaning. Which one is > it? Inclusive. In fact, in your example, Galileo was a genius and Galileo > also worked hard... So that reasoning above is invalid. Here's one that's OK... Either Galileo was a genius or he worked hard. > Galileo did not work hard. > Therefore, Galileo was a genius. -- > G. A. Edgar http://www.math.ohio-state.edu/~edgar/ Is the OR connective the same as the EITHER...OR connective? :) === Subject: Short mathematical problem posting-account=dha4uwoAAAAV8FU_1e-evlziX_uP-mKz Gecko/2009021910 Firefox/3.0.7 (.NET CLR 3.5.30729),gzip(gfe),gzip(gfe) I want to sell an X number of pieces of a product for a total price Z of EXACTLY $2416.67 ($2900,00-20%VAT, approx to two decimals) for a price/piece Y that has to be EXACTLY defined with two decimals and has to be the nearest to $9.50 (from $7.00 to $12.00 are OK). Recapitulating: X can be any integer number Y can be in the range 7.00-12.00, the nearest to 9.50 the better, two decimals Z is exactly 2416.67 X * Y = Z What are the X with the best Y to arrive to that Z? === Subject: Re: Short mathematical problem >I want to sell an X number of pieces of a product for a total price Z >of EXACTLY $2416.67 ($2900,00-20%VAT, approx to two decimals) for a >price/piece Y that has to be EXACTLY defined with two decimals and has >to be the nearest to $9.50 (from $7.00 to $12.00 are OK). Recapitulating: >X can be any integer number >Y can be in the range 7.00-12.00, the nearest to 9.50 the better, two >decimals >Z is exactly 2416.67 >X * Y = Z What are the X with the best Y to arrive to that Z? Let x = X y = 100*Y z = 100*Z = 241667 Then you seek positive integers x,y satisfying the constraints 700 <= y <= 1200 x*y = z and for which |y - 950| is as small as possible. However the constraints are unsatisfiable since z is prime. Thus it makes sense to relax the equality x*y = z to allow an error of e, say, on the product, where e is specified in advance. Thus for example, using e = 100, the new constraints would be 700 <= y <= 1200 z - 100 <= x*y <= z + 100 With these constraints, |abs(y - 950)| is minimized for x = 255 y = 948 quasi === Subject: Re: Short mathematical problem posting-account=G4HhgwkAAAB0h1X9lFcN2_L63S7FUgWk 1.0.3705; .NET CLR 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30; .NET CLR 3.0.04506.648; .NET CLR 3.0.4506.2152; .NET CLR 3.5.30729),gzip(gfe),gzip(gfe) > I want to sell an X number of pieces of a product for a total price Z > of EXACTLY $2416.67 ($2900,00-20%VAT, approx to two decimals) for a > price/piece Y that has to be EXACTLY defined with two decimals and has > to be the nearest to $9.50 (from $7.00 to $12.00 are OK). Recapitulating: > X can be any integer number > Y can be in the range 7.00-12.00, the nearest to 9.50 the better, two > decimals > Z is exactly 2416.67 > X * Y = Z What are the X with the best Y to arrive to that Z There is no solution to your problem as stated. Working in cents, you want integer X and Y solutions to the equation X * Y = 241667 241667 is prime; the only possible values of Y are 241667 and 1, both outside the alllowed range... === Subject: Re: Short mathematical problem > I want to sell an X number of pieces of a product for a total price Z > of EXACTLY $2416.67 ($2900,00-20%VAT, approx to two decimals) for a > price/piece Y that has to be EXACTLY defined with two decimals and has > to be the nearest to $9.50 =A0(from $7.00 to $12.00 are OK). > Recapitulating: > X can be any integer number > Y can be in the range 7.00-12.00, the nearest to 9.50 the better, two > decimals > Z is exactly 2416.67 > X * Y =3D Z > What are the X with the best Y to arrive to that Z There is no solution to your problem as stated. True. > Working in cents, you want integer X and Y solutions to the equation X * Y = 241667 241667 is prime; the only possible values of Y are 241667 and 1, both > outside the alllowed range... But there are some near misses which might interest the OP. Perhaps the nicest is X = 252, Y = 9.59, Z = X*Y = 2416.68 David === Subject: Re: Short mathematical problem <20090319124452.256$s6@newsreader.com> posting-account=dha4uwoAAAAV8FU_1e-evlziX_uP-mKz Gecko/2009021910 Firefox/3.0.7 (.NET CLR 3.5.30729),gzip(gfe),gzip(gfe) > I want to sell an X number of pieces of a product for a total price Z > of EXACTLY $2416.67 ($2900,00-20%VAT, approx to two decimals) for a > price/piece Y that has to be EXACTLY defined with two decimals and has > to be the nearest to $9.50 =A0(from $7.00 to $12.00 are OK). > Recapitulating: > X can be any integer number > Y can be in the range 7.00-12.00, the nearest to 9.50 the better, two > decimals > Z is exactly 2416.67 > X * Y =3D Z > What are the X with the best Y to arrive to that Z > There is no solution to your problem as stated. True. > Working in cents, you want integer X and Y solutions to the equation > X * Y = 241667 > 241667 is prime; the only possible values of Y are 241667 and 1, both > outside the alllowed range... But there are some near misses which might interest the OP. Perhaps the > nicest is X = 252, Y = 9.59, Z = X*Y = 2416.68 David You are right. I also tried all the possibilities in Excel and there are no possibilities. === Subject: Good books in elementary number theory posting-account=5cbwMAoAAAAZoYdmWfebkP1kkdh09sZi AppleWebKit/525.27.1 (KHTML, like Gecko) Version/3.2.1 Safari/525.27.1,gzip(gfe),gzip(gfe) One that is at undergrad level, has good exercises and solutions to it? === Subject: Re: Good books in elementary number theory One that is at undergrad level, has good exercises and solutions to it? It's difficult if not impossible to provide a good reply without knowing more. What is your mathematical background? Have you already learned how to do formal proofs, induction, etc. Have you had any abstract algebra? Do you intend to go on to be a pure/applied mathematician, computer scientist, or teacher? === Subject: representation by classes in homology. Def. please: Hi: I am looking for an actual definition of what it means for a homology class in H^2(X) to be represented by a surface. All I can tell is that it has to see with the fundamental class, and anything else. === Subject: Re: representation by classes in homology. Def. please: posting-account=9QOSvAoAAACEOWJVSDuswW7dB_0wApQO Gecko/2009030503 Fedora/3.0.7-1.fc10 Firefox/3.0.7,gzip(gfe),gzip(gfe) > Hi: > I am looking for an actual definition of what it > means for a homology class in H^2(X) to be > represented by a surface. All I can tell is that > it has to see with the fundamental class, and > anything else. One usually says that a class alpha in H 2(X) is represented by a map f : S --> X with S an oriented orientable surface with orientation class zeta if H 2(f)(zeta) = alpha. For cohomology one gets classes using Poincare duality. -- m === Subject: Re: representation by classes in homology. Def. please: > On Mar 19, 1:37pm, jewsforjesusalou > Hi: > I am looking for an actual definition of what > it > means for a homology class in H^2(X) to be > represented by a surface. All I can tell is that > it has to see with the fundamental class, and > mention > anything else. One usually says that a class alpha in H_2(X) is > represented by a map f : S --> X with S an > oriented orientable surface with orientation class > zeta if H_2(f)(zeta) = alpha. For cohomology one gets classes using Poincare > duality. -- m === Subject: Re: representation by classes in homology. Def. please: <27165894.1237488340738.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=9QOSvAoAAACEOWJVSDuswW7dB_0wApQO Gecko/2009030503 Fedora/3.0.7-1.fc10 Firefox/3.0.7,gzip(gfe),gzip(gfe) > On Mar 19, 1:37pm, jewsforjesusalou > Hi: > I am looking for an actual definition of what > it > means for a homology class in H^2(X) to be > represented by a surface. All I can tell is that > it has to see with the fundamental class, and > mention > anything else. > One usually says that a class alpha in H 2(X) is > represented by a map f : S --> X with S an > oriented orientable surface with orientation class > zeta if H 2(f)(zeta) = alpha. > Well, there is no need for the map f be an inclusion. For example, you cannot represent all classes in H 1(T) where T is a 2-torus using inclusions S^1 --> T of circles. It is usually an interesting problem to be able to decide *which* classes in homology can be so represented using an inclusion. (And when more structure is present, you can be more adventurous: for example, if X is an algebraic projective variety, one may be interested in knowing which cohomology classes of X are represented by an inclusion of an algebraic smooth subvariety; this is considerably more restrictive than admiting arbitrary submanifolds, and a much, much, much more difficult problem: it is more or less the Hodge conjecture, one of the millenium prize problems) -- m === Subject: Re: bottles vs. cartons >> Michael Press said: > Jerry Trumball said: >> I like to think that I am a gentle man, but your > forgiveness will have to come from someone other than me. >> Forgiveness is a much misunderstood concept. It benefits the > giver far more than the receiver (who may not even be aware that > he, or she, has been forgiven). >> Until you let them off the hook, you're still hooked to them. > Forgiveness is a release from a bitterness that hurts only the > embittered. Do you forgive people who have no remorse? Yes - but for my benefit, not theirs. I do not know how to manage that, and do not try. Understood. I'm not saying it's easy. I should have added that I do something for my benefit, > it is internal, but it is not forgiveness. My dogma got run over by my karma. Rich === Subject: Re: bottles vs. cartons Until you let them off the hook, you're still hooked to them. Nope, you can just tie them off somewhere and let them hang, hooked. -- It's times like these which make me glad my bank is Dial-a-Mattress === Subject: Re: bottles vs. cartons >CBFalconer said: You obviously have no knowledge of Duraglas, I've never heard the word before. I assumed, naturally enough, that >it was some kind of toughened glass. You'd be right. http://www.sha.org/bottle/glossary.htm -- It's times like these which make me glad my bank is Dial-a-Mattress === Subject: Re: bottles vs. cartons Matthew Russotto said: >>CBFalconer said: >> You obviously have no knowledge of Duraglas, >>I've never heard the word before. I assumed, naturally enough, >>that it was some kind of toughened glass. You'd be right. > http://www.sha.org/bottle/glossary.htm -- Richard Heathfield Email: -http://www. +rjh@ Google users: Usenet is a strange place - dmr 29 July 1999 === Subject: Re: bottles vs. cartons >> CBFalconer said: > You obviously have no knowledge of Duraglas, >> I've never heard the word before. I assumed, naturally enough, that >> it was some kind of toughened glass. You'd be right. > http://www.sha.org/bottle/glossary.htm In 1936, Henry W. Hess of Toledo filed for a patent for heat treating glass, especially milk bottles. His attorneys were Owen and Owen. It was granted November 21, 1939, (2,1890,737), just before Owens Illinois began making Duraglas bottles. I wonder if Hess invented Duraglas. === Subject: Re: bottles vs. cartons >possums.net says... > Milk isn't pressurized, duh. >> So? What does that have to do with squeare shape? >> And, uh, fruit juice isn't pressurized either, duh.... Fruit juice comes in the same types of containers as milk does. >> >And fruit juice is also not soft-drink. As long as you don't let it sit too long, it is. -- It's times like these which make me glad my bank is Dial-a-Mattress === Subject: Re: JSH: Factoring problem solution, proof of concept code > Perhaps to get past the 32 bit barrier those with a Linux/Unix bent could >> translate this program trivially to bc and see the median number of loops >> for those cases that don't fail. At last that would show whether the >> idea has any merit -- ignoring the fact the failures may be at present >> unpredictable/uncharacterisable. AFAIU, Java does provide arbitrary-precision integers. http://java.sun.com/javase/6/docs/api/java/math/BigInteger.html GNU bc implementation of trial division $ cat trialdiv.bc > define trialdiv(n) > { > m = sqrt(n) > for (i = 3; i <= m; i += 2) > if (n%i == 0) { print i, ; return trialdiv(n/i) } > return n > } > for (j = 1; j <= 50; ++j) > { > n = 2^j+1 > print 2^,j,+1 = ; trialdiv(n) > } $ time bc < trialdiv.bc > 2^1+1 = 3 > 2^2+1 = 5 > 2^3+1 = 3 3 > 2^4+1 = 17 > 2^5+1 = 3 11 > 2^6+1 = 5 13 > 2^7+1 = 3 43 > 2^8+1 = 257 > 2^9+1 = 3 3 3 19 > 2^10+1 = 5 5 41 > 2^11+1 = 3 683 > 2^12+1 = 17 241 > 2^13+1 = 3 2731 > 2^14+1 = 5 29 113 > 2^15+1 = 3 3 11 331 > 2^16+1 = 65537 > 2^17+1 = 3 43691 > 2^18+1 = 5 13 37 109 > 2^19+1 = 3 174763 > 2^20+1 = 17 61681 > 2^21+1 = 3 3 43 5419 > 2^22+1 = 5 397 2113 > 2^23+1 = 3 2796203 > 2^24+1 = 97 257 673 > 2^25+1 = 3 11 251 4051 > 2^26+1 = 5 53 157 1613 > 2^27+1 = 3 3 3 3 19 87211 > 2^28+1 = 17 15790321 > 2^29+1 = 3 59 3033169 > 2^30+1 = 5 5 13 41 61 1321 > 2^31+1 = 3 715827883 > 2^32+1 = 641 6700417 > 2^33+1 = 3 3 67 683 20857 > 2^34+1 = 5 137 953 26317 > 2^35+1 = 3 11 43 281 86171 > 2^36+1 = 17 241 433 38737 > 2^37+1 = 3 1777 25781083 > 2^38+1 = 5 229 457 525313 > 2^39+1 = 3 3 2731 22366891 > 2^40+1 = 257 4278255361 > 2^41+1 = 3 83 8831418697 > 2^42+1 = 5 13 29 113 1429 14449 > 2^43+1 = 3 2932031007403 > 2^44+1 = 17 353 2931542417 > 2^45+1 = 3 3 3 11 19 331 18837001 > 2^46+1 = 5 277 1013 1657 30269 > 2^47+1 = 3 283 165768537521 > 2^48+1 = 193 65537 22253377 > 2^49+1 = 3 43 4363953127297 > 2^50+1 = 5 5 5 41 101 8101 268501 real 0m18.719s > user 0m18.702s > sys 0m0.015s > JSH, When you gonna do that ? next year ? === Subject: Re: JSH: Now you can understand the parasites > The story of the AIG bonuses can give some of you perspective on the > parasitic who have lived among you for so long, as before, they could > have gotten away with it. I've talked before about the specifics of > the way the human brain works which allowed them to control so many of > you, so well, for so long so I won't go into that now. Best way to > put it is, many of you were simply being loyal, to creatures who see > that as weakness. Their aim was to divide humanity into two groups: the rulers and the > ruled. They use various psychological tools and amass wealth with which they > dominate further, while they will claim to any value you wish them to, > but they only hold a parasitic view. They do not want to earn a living. They believe they should be given > everything: your life, your blood, your hopes, your dreams. Their weakness is what you saw with AIG: they depend on you giving in > to them. If you don't, then they fall apart easily enough. The worst thing humanity ever said to itself was, we are free. You are only free to the extent that you keep your eyes open. Many of > you have been enslaved in mind for years now. The world you are beginning to see around you is a brand new one, but > don't think it gets easier. Your world does have monsters. The problem is, the monsters have > human faces. Learn to look beneath the surface, and learn to be truly free. > James Harris SO WHAT? YOU STILL CANT FACTOR ANYTHING ABOVE 15. ASK FOR A GOV BAILOUT ! === Subject: Re: Inconsistency of the usual axioms of set theory posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2009021910 Firefox/3.0.7,gzip(gfe),gzip(gfe) >>mathematicians have proven ZFC to be consistant > with ZFC. >>is ZFC consistant ? >>yes , we proved it consistant in ZFC. > You should really learn not to talk about things > you > know nothing about! It is simply not true that > we've > proved ZFC inconsistent in ZFC. > ^^^^^^^^^^^^ > I think you mean consistent. no he didnt. Yes, he did. He was disagreeing with the statement we proved [that ZFC is consistent] in ZFC. MoeBlee === Subject: Surfaces of negative curvature posting-account=DYfPiQoAAACTvlH85p9Qyo4MUDqqf-AH AppleWebKit/525.19 (KHTML, like Gecko) Chrome/1.0.154.48 Safari/525.19,gzip(gfe),gzip(gfe) Am I correct in believing that there does not exist a compact surface with (possibly non-constant) negative Gaussian curvature everywhere that can be immersed in R^3? Can anyone provide insight as to why this may or may not be the case? Saladin === Subject: Re: rules for submitting to a journal > Hi. I have this question about submitting to a journal. My > understanding is that if you submit to one journal > and get rejected, you can freely submit to another. > My question is, when the next journal says: IF THIS PAPER (OR A SIMILAR VERSION) WAS PREVIOUSLY SUBMITTED to > another > journal, or published in conference proceedings, please send us > all > pertinent correspondence at this time. does this mean I need to tell them about all the previous journals I > submitted to? I am currently trying Physics of Fluids and this > this question. Hi - I've not seen this either, but I do see the point. Case in point, one might say: some years back a paper appeared that had a fatal flaw in the experimental design. The author, an honest if misguided person, acknowledged twelve (!) anonymous reviewers. In fact, I was two of the twelve - the paper changed titles in medias res, so I didn't know enough to exclude myself. Apparently a journal was eventually found that sent the paper (a biological one) to no reviewer with decent fluid mechanical savvy! Steven Vogel === Subject: counting subgroups posting-account=ShKz4goAAAB-5fxA1JyErZXtNgWVpdct Gecko/2009030422 Ubuntu/8.10 (intrepid) Firefox/3.0.7,gzip(gfe),gzip(gfe) Let G = Z_{p^2} x Z_{p^2} and H = {x^p | x in G} (p is prime). The question is: what is number of subgroups of G with order p^3, that contain H ? The hint is: H is the only subgroup of G such that G/H ~ Z_p x Z_p. My attempt: Let A and B are different subgroups of G with order p^3, that contain H. Then it is easy to get A / B = H. I don't know hot to continue from here. How can I count these strange subgroups ? === Subject: Re: counting subgroups posting-account=_l4K0QkAAAC09JhOoK_ZfoJKXOmr_jZf Gecko/2009030422 Ubuntu/8.04 (hardy) Firefox/3.0.7,gzip(gfe),gzip(gfe) > Let G = Z {p^2} x Z {p^2} and H = {x^p | x in G} (p is prime). The question is: > what is number of subgroups of G with order p^3, that contain H ? The hint is: > H is the only subgroup of G such that G/H ~ Z p x Z p. My attempt: > Let A and B are different subgroups of G with order p^3, that contain > H. > Then it is easy to get A / B = H. I don't know hot to continue from > here. > How can I count these strange subgroups ? The subgroups of G that contain H are in one-to-one, inclusion preserving correspondence with the subgroups of G/H. If K is a subgroup of G/H, and K corresponds to the subgroup A = {g in G | gH is in K}, then #A = #K*#H. So, perhaps you might want to count the number of subgroups of suitable order in G/H instead? -- Arturo Magidin, sans .sig === Subject: 2-norm of orthogonal or permutation matrix posting-account=Qq_GEwoAAAD0Kc2k70q6xwuZdqx8sCzT 4334.5003; Windows NT 5.1; SV1; .NET CLR 2.0.50727; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) spider-mtc-td01.proxy.aol.com[400C7061] (Prism/1.2.1), HTTP/1.1 cache-mtc-ab12.proxy.aol.com[400C744C] (Traffic-Server/6.1.5 [uScM]) is it ever shown that if A is orthogonal (or a permutation matrix as a special case) that ||A||_2 = 1? if so, would you be willing to do it? or is the invariance to the 2-norm when multiplying it on a matrix or vector property all that ever needs to be known. if so, i am a type of computer with no creativity. edward === Subject: Re: 2-norm of orthogonal or permutation matrix >is it ever shown that if A is orthogonal (or a permutation matrix as a >special case) that ||A||_2 = 1? if so, would you be willing to do it? or is the invariance to the 2-norm when multiplying it on a matrix >or vector property all that ever needs to be known. if so, i am a type of computer with no creativity. no, you seem to be a bit lazy: in homework you normally are assumed to use the lecture notes/books chapters just dealt with. here the job is only inserting edward take the definition of a subordinate matrix norm use ||.||_2 and its definition take the square use the definition of orthogonality done hth peter === Subject: Re: 2-norm of orthogonal or permutation matrix >is it ever shown that if A is orthogonal (or a permutation matrix as a >special case) that ||A||_2 = 1? Yes. >if so, would you be willing to do it? No, because this is very easy. You can do it yourself. First, what's the _definition_ of the norm of a matrix A? And second, what does it mean to say that A is orthogonal? >or is the invariance to the 2-norm when multiplying it on a matrix >or vector property all that ever needs to be known. Yes. >if so, i am a type of computer with no creativity. No creativity required here, just two definitions. >edward David C. Ullrich Understanding Godel isn't about following his formal proof. That would make a mockery of everything Godel was up to. (John Jones, My talk about Godel to the post-grads. in sci.logic.) === Subject: Newbie question on definition of a subset posting-account=wzCFHwoAAACNjb4w9hAuIijU5lDzX9PN Gecko/2009021910 Firefox/3.0.7 (.NET CLR 3.5.30729),gzip(gfe),gzip(gfe) I'm confused. What's wrong with the following reasoning? Propositional logic tells me that ( p implies q ) = [ (not p) or q] so [(x in A) implies (x in B)] = (x not in A) or (x in B) now, A is-subset-of B can be written as forall x [(x in A) implies (x in B)] but this says forall x [(x not in A) or (x in B)] which means forall x [x in (complement(A) union B)] But if I draw a Venn diagram, (complement(A) union B) includes stuff that's outside (A union B). So elements outside A and B are somehow involved a definition of a subset, which doesn't match my intuition. I feel like I tied my shoelaces together. Alejo === Subject: Re: Newbie question on definition of a subset posting-account=_l4K0QkAAAC09JhOoK_ZfoJKXOmr_jZf Gecko/2009030422 Ubuntu/8.04 (hardy) Firefox/3.0.7,gzip(gfe),gzip(gfe) > I'm confused. What's wrong with the following reasoning? Propositional logic tells me that > ( p implies q ) = [ (not p) or q] > so > [(x in A) implies (x in B)] = (x not in A) or (x in B) > now, > A is-subset-of B > can be written as > forall x [(x in A) implies (x in B)] > but this says > forall x [(x not in A) or (x in B)] > which means > forall x [x in (complement(A) union B)] But if I draw a Venn diagram, (complement(A) union B) includes stuff > that's outside (A union B). So? > So elements outside A and B are somehow > involved a definition of a subset, You are not defining a set! You are describing a condition which is equivalent to A is a subset of B, which is not a set description. > which doesn't match my intuition. But A is a subset of B is not a description of a set! So of course there is no reason to think that it describes a set. If A is a subset of B, then it is certainly true that EVERYTHING is either in B, or else is not in A. And conversely, if everything is either in B or not in A, then A must be contained in B. If you take a generic Venn diagram, and you shade Complement(A) union B, you will see that there is a section that is not shaded (namely, A- B, the elements of A that are not in B). The statement For all x (x in complement(A) union B) tells you that in fact there is nothing in that unshaded area, so that there are no elements that are in A but not in B. In other words, that A is a subset of B. So it's not that the elements outside of A and B somehow involved a definition of subset, because there is no subset being defined. A is a subset of B is equivalent to an assertion that a certain set is empty (namely, A-B), as is the statement Everything is in B or in the complement of A. Arturo Magidin === Subject: Re: Newbie question on definition of a subset > I'm confused. What's wrong with the following reasoning? Propositional logic tells me that > ( p implies q ) = [ (not p) or q] > so > [(x in A) implies (x in B)] = (x not in A) or (x in B) > now, > A is-subset-of B > can be written as > forall x [(x in A) implies (x in B)] > but this says > forall x [(x not in A) or (x in B)] > which means > forall x [x in (complement(A) union B)] But if I draw a Venn diagram, (complement(A) union B) includes stuff > that's outside (A union B). So elements outside A and B are somehow > involved a definition of a subset, which doesn't match my intuition. > The statement for all x, P(x) is not the same as the set { x | P(x) }. A subset B does not mean the set A^c / B. === Subject: Re: Newbie question on definition of a subset <20090318221304.G51216@agora.rdrop.com> posting-account=wzCFHwoAAACNjb4w9hAuIijU5lDzX9PN Gecko/2009021910 Firefox/3.0.7 (.NET CLR 3.5.30729),gzip(gfe),gzip(gfe) > The statement > for all x, P(x) is a proposition. > is not the same as the set > { x | P(x) }. is a set. {x | x in A -> x in B} = {x | x in A^c or x in B} = {x | x in (A^c union B)} was the same as saying A subset B. Silly me! Alejo === Subject: 27.03.09 powers of 3. cube rt(729.) strange. Cc: MCDONewt@yahoo.co.nz posting-account=TV2szgkAAACrA1vyuh8IN_0zzgzcwogw .NET CLR 2.0.50727; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) 27.03.09 powers of 3. cube rt(729.) 16.04.08 powers of 2. don.lotto nz 19/3/09. wellington. cheers, === Subject: Re: 27.03.09 powers of 3. cube rt(729.) strange. posting-account=TV2szgkAAACrA1vyuh8IN_0zzgzcwogw .NET CLR 2.0.50727; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > 27.03.09 powers of 3. cube rt(729.) 16.04.08 powers of 2. don.lottonz19/3/09. wellington. cheers, 8 27 2197. cubes 64 81 === Subject: close form I know how to solve the recurrence relation of T(n) - 2T(n-1) = 3 * 2^n T(0) = 1 (boundary condition) homogenous solution = T_h(n) = A * 2^n particular solution = T_p(n) = 3n (2)^n T(n) = T_h(n) + T_p(n) = 3n (2)^n + (2)^n = 2^n (3n + 1) but if it is in this form n*T(n) + n*T(n-1) - T(n-1) = 2^n T(0) = 273 (boundary condition) how do I solve it? === Subject: Re: close form sci.math,alt.math,alt.math.undergrad: > I know how to solve the recurrence relation of > T(n) - 2T(n-1) = 3 * 2^n > T(0) = 1 (boundary condition) > homogenous solution = > T_h(n) = A * 2^n > particular solution = > T_p(n) = 3n (2)^n > T(n) = T_h(n) + T_p(n) > = 3n (2)^n + (2)^n > = 2^n (3n + 1) > but if it is in this form > n*T(n) + n*T(n-1) - T(n-1) = 2^n > T(0) = 273 (boundary condition) > how do I solve it? Let S(n) = n * T(n), get a recurrence for S(n), and solve that. Brian === Subject: the 1 by n chess board Is that true the following problem will lead to the Fibonacci sequence? T(n) = T(n-1) + T(n-2) ############################### Consider a 1 by n chessboard. suppose that we color each square of the chessboard in either red or blue. Let T(n) represent the number of possible colored chessboards in which no 2 squares that are colored red are adjacent. Find the recurrence relation for T(n). T(n) = T(n-1) + T(n-2) ############################### Question is why it lead to fibonacci squence recurrence relation? === Subject: Re: the 1 by n chess board > Is that true the following problem will lead to the Fibonacci > sequence? T(n) = T(n-1) + T(n-2) If you mean the usual Fibonacci sequence (1,1,2,3,5,8,13,...) then you also need to properly specify the initial boundary conditions. > ############################### > Consider a 1 by n chessboard. suppose that we color each square of > the chessboard in either red or blue. Let T(n) represent the number of possible colored chessboards in which > no 2 squares that are colored red are adjacent. Find the recurrence relation for T(n). T(n) = T(n-1) + T(n-2) > ############################### Question is why it lead to fibonacci squence recurrence relation? That's a cute problem, which I presume is homework. Here's a hint: * Every legal sequence (of length N) of R and B must begin with B or RB (since it cannot begin with RR). * How many sequences (of length N) begin with B ? * How many begin with RB ? If you get stuck, show us what you have done and where you got stuck. -- --------------------------- | BBB b Barbara at LivingHistory stop co stop uk | B B aa rrr b | | BBB a a r bbb | Quidquid latine dictum sit, | B B a a r b b | altum viditur. | BBB aa a r bbb | ----------------------------- === Subject: tree relationship Why the follow tree recurrence relations T(h,k) = max number of leaves of a tree of height h, where each node has outdegree (number of children) k or less. Been told T(h,k) = T(h/2) + k How is this answer be reach? Is there any way to show this? === Subject: Re: tree relationship >Why the follow tree recurrence relations T(h,k) = max number of leaves of a tree of height h, where each node >has outdegree (number of children) k or less. Been told T(h,k) = T(h/2) + k That's not right, unless I'm misunderstanding something. >How is this answer be reach? Is there any way to show this? It's easy to find the correct recurrence. Say you have a tree of hieght h. Then T(h-1,k) is the maximum number of nodes at depth h-1. Given that number, what's the maximum number of nodes at depth h? David C. Ullrich Understanding Godel isn't about following his formal proof. That would make a mockery of everything Godel was up to. (John Jones, My talk about Godel to the post-grads. in sci.logic.) === Subject: Re: 2-norm of orthogonal or permutation matrix posting-account=Qq_GEwoAAAD0Kc2k70q6xwuZdqx8sCzT 4334.5003; Windows NT 5.1; SV1; .NET CLR 2.0.50727; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) spider-mtc-te12.proxy.aol.com[400C708C] (Prism/1.2.1), HTTP/1.1 cache-mtc-ad07.proxy.aol.com[400C74C9] (Traffic-Server/6.1.5 [uScM]) >is it ever shown that if A is orthogonal (or a permutation matrix as a >special case) that ||A|| 2 = 1? Yes. >if so, would you be willing to do it? No, because this is very easy. You can do it yourself. First, what's the definition of the norm of a matrix A? > And second, what does it mean to say that A is orthogonal? >or is the invariance to ?the 2-norm when multiplying it on a matrix >or vector property all that ever needs to be known. Yes. >if so, i am a type of computer with no creativity. No creativity required here, just two definitions. >edward David C. Ullrich Understanding Godel isn't about following his formal proof. > That would make a mockery of everything Godel was up to. > (John Jones, My talk about Godel to the post-grads. > in sci.logic.) there. bird === Subject: Re: 2-norm of orthogonal or permutation matrix >>[...] there. No problem. Here's a hint for future reference: Any time you want to prove something about anything you really need to know the definition! Sometimes just knowing the definitions will make the problem trivial, as happened here. Even when the problem is not trivial from the definitions, you still need to know the definitions, otherwise you don't know what you're talking about. >bird David C. Ullrich Understanding Godel isn't about following his formal proof. That would make a mockery of everything Godel was up to. (John Jones, My talk about Godel to the post-grads. in sci.logic.) === Subject: Re: 2-norm of orthogonal or permutation matrix posting-account=Qq_GEwoAAAD0Kc2k70q6xwuZdqx8sCzT 4334.5003; Windows NT 5.1; SV1; .NET CLR 2.0.50727; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) spider-mtc-te12.proxy.aol.com[400C708C] (Prism/1.2.1), HTTP/1.1 cache-mtc-ad07.proxy.aol.com[400C74C9] (Traffic-Server/6.1.5 [uScM]) On Mar 19, 9:00?am, spellu...@fb04814.mathematik.tu-darmstadt.de >is it ever shown that if A is orthogonal (or a permutation matrix as a >special case) that ||A|| 2 = 1? >if so, would you be willing to do it? >or is the invariance to ?the 2-norm when multiplying it on a matrix >or vector property all that ever needs to be known. >if so, i am a type of computer with no creativity. no, you seem to be a bit lazy: in homework you normally are assumed to use > the lecture notes/books chapters just dealt with. > here the job is only inserting >edward take the definition of a subordinate matrix norm > use ||.|| 2 and its definition > take the square > use the definition of orthogonality > done hth > peter annihilated. scatterbrain === Subject: Re: a cute proposition Originator: msb@shell.vex.net (Mark Brader) James Allen: > The probability the second roll in a backgammon game is double-6 > is 1/36. What is the probability the last roll is double-6? >>. .. I wasn't expecting an exact answer, just much greater >> than 1/36. D. Gates: > What do you think of my ballpark that it's 4 times greater. It seems to be about right. > I'd still be interested in someone running a simulation of 10,000 > games between two decent players who don't use a doubling cube and > don't resign. I did a simplified calculation based on the assumption that all positions for one player where a double 6 would end the game (turns out there are 446 of them) are equally likely and that the last player is never blocked by a point made by the other player. On this basis, if I made no programming errors, the probability is 1,482/16,056 or 1 in about 10.834. Here's the code -- this is a one-off and not meant as a model of good style. #!/usr/bin/perl # For each possible position of one player's stones where a double 6 # would end the game (assuming no blockage by the other player's # made points), and a few other positions that it's not worth the # trouble of excluding. Whichever of $p, $q, $r, $s are defined are # constrained to be in ascending order. foreach my $p (1 .. 24) { position($p); foreach my $q ($p .. 24 - $p) { position($p, $q); if ($q <= 6) { foreach my $r ($q .. 12) { position($p, $q, $r); foreach my $s ($r .. 6) { position($p, $q, $r, $s); } } } } } printf Probability %d/%d = 1/%.3fn, $hits, $total, $total / $hits; # Called for each such position. Adds to $ending the number of possible # die rolls that would end the game, and adds 36 to $total if there was # at least one. sub position { my ($p, $q, $r, $s) = @_; my $ok = 0; foreach my $i (1 .. 6) { foreach my $j (1 .. 6) { if ($i == $j) { # doubles if (defined $s) { # 4 stones ++$ok if ($i >= $s); } elsif (defined $r) { # 3 stones ++$ok if ($i >= $q && 2 * $i >= $r); } elsif (defined $q) { # 2 stones ++$ok if (2 * $i >= $q || 3 * $i >= $q && $i >= $p); } else { # 1 stone ++$ok if (4 * $i >= $p); } } else { next if (defined $r); if (defined $q) { # 2 stones ++$ok if ( ($i >= $q && $j >= $p) || ($i >= $p && $j >= $q)); } else { # 1 stone ++$ok if ($i + $j > $p); } } } } if ($ok) { $hits += $ok; $total += 36; } } -- Mark Brader, Toronto | ...blind faith can ruin the eyesight-- msb@vex.net | and the perspective. --Robert Ludlum === Subject: Brain Freeze.calculate the equal of 5 % of the whole as a percentage of 75% If I were to issue 5% stock in the holding company but decided to rather sell stock in a subsidiary company that owns 75% stock in the holding company. How much stock do I have to sell in the subsiduary to respresent 5% \ in the holding company? === Subject: Re: Brain Freeze.calculate the equal of 5 % of the whole as a percentage of 75% <115083.1237714524392.JavaMail.jakarta@nitrogen.mathforum.org>, Vann > If I were to issue 5% stock in the holding company but decided to rather sell > stock in a subsidiary company that owns 75% stock in the holding company. \ How > much stock do I have to sell in the subsiduary to respresent 5% in the > holding company? Look at it this way: Suppose the company has 100 shares of stock; 5% of 100 is 5. The subsidiary has 75% of 100; 75% of 100 is 75. You want to know what percentage of 75 is 5. Answer: 5/75 x 100 = 6.67%. -- Paul Sperry Columbia, SC (USA) === Subject: Re: Brain Freeze.calculate the equal of 5 % of the whole as a percentage of 75% >If I were to issue 5% stock in the holding company but decided to rather sell stock in a subsidiary company that owns 75% stock in the holding company. How much stock do I have to sell in the subsiduary to respresent 5% \ in the holding company? 6 2/3 percent. --Lynn http://math.asu.edu/~kurtz === Subject: Updated general solution to Pell's Equation posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif AppleWebKit/525.19 (KHTML, like Gecko) Chrome/1.0.154.48 Safari/525.19,gzip(gfe),gzip(gfe) A previous posting of mine giving a general solution to Pell's Equation had a sign error. Here is a corrected, and much simpler general solution. In rationals, given x^2 - Dy^2 = 1 I have proven: y = 2[f_2*v - 1]/[(D-1) + 2f_2*v - f_2^2*v^2] and x = [f_2*v^2 - 2f_2*v + (D+1)]/[(D-1) + 2f_2*v - f_2^2*v^2] where f_1*f_2 = D-1, and the f's are non-zero integer factors, while v is nonzero but is otherwise a free variable. As Pell's Equation is normally considered in integers as a Diophantine equation note that you find rational v such that x and y are integers, which gives the 'why' of Pell's Equation. For instance, for D=2, f_1*f_2 = 1, so I have: y = 2(v - 1)/(1 + 2v - v^2) and x = (v^2 - 2v + 3)/(1 + 2v - v^2) and v = 2, gives y = 2, x = 3, and, of course, 9 - 2(4) = 1. My apologies for the prior sign error. James Harris === Subject: Re: Updated general solution to Pell's Equation >A previous posting of mine giving a general solution to Pell's >Equation had a sign error. So the previous version was wrong. So all the people who lied about it being wrong were right, eh? >Here is a corrected, and much simpler >general solution. Ah, _this_ one's right. Got it. >In rationals, given x^2 - Dy^2 = 1 I have proven: y = 2[f_2*v - 1]/[(D-1) + 2f_2*v - f_2^2*v^2] and x = [f_2*v^2 - 2f_2*v + (D+1)]/[(D-1) + 2f_2*v - f_2^2*v^2] where f_1*f_2 = D-1, and the f's are non-zero integer factors, while v >is nonzero but is otherwise a free variable. As Pell's Equation is normally considered in integers as a Diophantine >equation note that you find rational v such that x and y are integers, >which gives the 'why' of Pell's Equation. For instance, for D=2, >f_1*f_2 = 1, so I have: y = 2(v - 1)/(1 + 2v - v^2) and x = (v^2 - 2v + 3)/(1 + 2v - v^2) and v = 2, gives y = 2, x = 3, and, of course, 9 - 2(4) = 1. My apologies for the prior sign error. >James Harris David C. Ullrich Understanding Godel isn't about following his formal proof. That would make a mockery of everything Godel was up to. (John Jones, My talk about Godel to the post-grads. in sci.logic.) === Subject: Re: Updated general solution to Pell's Equation > A previous posting of mine giving a general solution to Pell's > Equation had a sign error. Here is a corrected, and much simpler > general solution. In your most recent blog, at http://mymath.blogspot.com/2009/03/dual-factorizations-with-pells-equation.h\ tml you state : However, the second condition then can be found by subtracting them, giving abs(2t(v)) < 0, so the second condition is abs(t(v)) < 0. : How can this be? This is a /proof/, remember? (Proofs don't have mistakes.) The absolute value of an integer function, used as the /denominator/ later is less than zero. Impressive. And there I was beginning to think that mathematics was easy. M -- === Subject: Re: Updated general solution to Pell's Equation >A previous posting of mine giving a general solution to Pell's > Equation had a sign error. Here is a corrected, and much simpler > general solution. > James Harris > I am not a pro but I know that Pell's equation has been around since before \ you were even born. This simple fact makes it completely improbable that it \ can be used to factor integers as a general method. If that were the case, it would have come out a long long time ago. Pell's equation has been milked \ and there is no milk left in it to be milked by new comers like you. === Subject: Problem not for nervous. y=pi*r^3 y'=? Only do not think, that it is a joke or disrespect for participants of a forum. The EXHAUSTIVE answer is necessary to me! === Subject: Re: Problem not for nervous. > y=pi*r^3 > y'=? > Only do not think, that it is a joke or disrespect > t for participants of a forum. The EXHAUSTIVE answer > is necessary to me! Derivate with respect to what ? y(r) = pi*r^3 -> dy/dr = pi*3*r^2 y(t) = pi*r(t)^3 -> dy/dt = pi*3*r^2*dr/dt .. Best wishes Torsten. === Subject: Re: Problem not for nervous. > y=pi*r^3 > y'=? y' = 3pi.r^2 r' === Subject: solving a recurrence involving partial bell polynomials Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Suppose r_1, ..., r_n are known quantities with r_1 not equal to 1. Let B_{n,k} denote a partial Bell polynomial, defined http://en.wikipedia.org/wiki/Bell_polynomials which is a function of r_1, ..., r_{n-k+1}. There is a recurrence equation I'd like some assistance with: s_n = C_n * Sum[ s_k * B_{n,k} , {k,1,n-1} ] = s_1 * B[n,1] + s_2 * B [n,2] +... + s_{n-1} B[n,n-1]. C_n is the reciprocal of r_1 - r_{1}^n, ie, C_n = 1 / (r_1 - r_{1} ^n ). s_n is then equal to some function of s_k for 1 <= k <= n-1. What I'm hoping for is a formula for s_n not involving any s_k. I shouldn't neglect to mention that s_1 = 1, an initial condition. One thing that would help is if there is a known recurrence that relates B[n+1, k] to B[n, k]. I found in the literature the following recurrence: B_{n,k} = (1/k) Sum[ Binomial[n, j] * r_{n-j} * B_{j,k-1} , {j, k-1, n-1} ]. I don't see how that recurrence helps but it might give a feel for what to expect in terms of a desired recurrence relating B[n+1,k] to B [n,k]. Ultimately, this is an effort to solve Shroeder's functional equation with formal power series. Koenings solved Schroder's equation in 1884. Shroeder's equation is given f, find functions g and constants r (which is neither 0 nor 1) such that g(f(x)) = r g(x) at least for x values in some interval. Koenings solution involves taking a limit of iterates of f which, in practice, I would think is more difficult than finding n-th derivatives of f. The recurrence relation above is deduced from differentiating Schroeder's equation (ie both sides) and evaluating this at a fixed point of f. (We're assuming that f and g are smooth and that f has a fixed point.) Then r_n is the value of the n-th derivative of f evaluated at a fixed point and s_n is the value of the n-th derivative of g evaluated at that fixed point. Then with a formula for s_n, g will be the usual (formal) Taylor expansion being the sum of (1/n!) * s_n * (x - the fixed point)^n as n ranges over the natural numbers. Furthermore what I'm looking for are conditions on f and the derivatives of f evaluated at a fixed point that guarantee that the radius of convergence of the formal power series for g is strictly greater than zero. But I don't seem to be able to do that without a formula for s_n. Perhaps I can use the given recurrence to at least say something about the asymptotic behavior of the sequence {s_n}. === Subject: Ten papers and an erratum published by Geometry & Topology Publications Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Six papers have been published by Algebraic & Geometric Topology in Issue 1 of Volume 9 (2009) and four papers and an erratum have been published by Geometry & Topology, completing Issue 3 of Volume 13 (2009). In AGT Volume 9: (1) Algebraic & Geometric Topology 9 (2009) 397-428 Lens space surgeries as A'Campo's divide knots by Yuichi Yamada URL: http://www.msp.warwick.ac.uk/agt/2009/09-01/p018.xhtml DOI: 10.2140/agt.2009.9.397 (2) Algebraic & Geometric Topology 9 (2009) 429-454 Circular thin position for knots in S^3 by Fabiola Manjarrez-Gutierrez URL: http://www.msp.warwick.ac.uk/agt/2009/09-01/p019.xhtml DOI: 10.2140/agt.2009.9.429 (3) Algebraic & Geometric Topology 9 (2009) 455-471 The Lawson homology for Fulton-MacPherson configuration spaces by Wenchuan Hu and Li Li URL: http://www.msp.warwick.ac.uk/agt/2009/09-01/p020.xhtml DOI: 10.2140/agt.2009.9.455 (4) Algebraic & Geometric Topology 9 (2009) 473-494 Symmetric topological complexity of projective and lens spaces by Jesus Gonzalez and Peter Landweber URL: http://www.msp.warwick.ac.uk/agt/2009/09-01/p021.xhtml DOI: 10.2140/agt.2009.9.473 (5) Algebraic & Geometric Topology 9 (2009) 495-535 Singular link Floer homology by Benjamin Audoux URL: http://www.msp.warwick.ac.uk/agt/2009/09-01/p022.xhtml DOI: 10.2140/agt.2009.9.495 (6) Algebraic & Geometric Topology 9 (2009) 537-548 Infinitesimal rigidity of a compact hyperbolic 4-orbifold with totally geodesic boundary by Tarik Aougab and Peter A Storm URL: http://www.msp.warwick.ac.uk/agt/2009/09-01/p023.xhtml DOI: 10.2140/agt.2009.9.537 In GT Volume 13: (7) Geometry & Topology 13 (2009) 1657-1727 Abelian subgroups of Out(F_n) by Mark Feighn and Michael Handel URL: http://www.msp.warwick.ac.uk/gt/2009/13-03/p040.xhtml DOI: 10.2140/gt.2009.13.1657 (8) Geometry & Topology 13 (2009) 1729-1773 Gromov-Witten theory of A_n-resolutions by Davesh Maulik URL: http://www.msp.warwick.ac.uk/gt/2009/13-03/p041.xhtml DOI: 10.2140/gt.2009.13.1729 (9) Geometry & Topology 13 (2009) 1775-1777 Erratum for the paper `On the chain-level intersection pairing for PL manifolds' by J E McClure URL: http://www.msp.warwick.ac.uk/gt/2009/13-03/p042.xhtml DOI: 10.2140/gt.2009.13.1775 (10) Geometry & Topology 13 (2009) 1779-1804 Virtual fundamental classes via dg-manifolds by Ionut Ciocan-Fontanine and Mikhail Kapranov URL: http://www.msp.warwick.ac.uk/gt/2009/13-03/p043.xhtml DOI: 10.2140/gt.2009.13.1779 (11) Geometry & Topology 13 (2009) 1805-1833 Geometric intersection number and analogues of the curve complex for free groups by Ilya Kapovich and Martin Lustig URL: http://www.msp.warwick.ac.uk/gt/2009/13-03/p044.xhtml DOI: 10.2140/gt.2009.13.1805 Abstracts follow (1) Lens space surgeries as A'Campo's divide knots by Yuichi Yamada It is proved that every knot in the major subfamilies of J Berge's lens space surgery (that is, knots yielding a lens space by Dehn surgery) is presented by an L-shaped (real) plane curve as a divide knot defined by A'Campo in the context of singularity theory of complex curves. For each knot given by Berge's parameters, the corresponding plane curve is constructed. The surgery coefficients are also considered. Such presentations support us to study each knot of lens space surgery itself and the relationship among the knots in the set of lens space surgeries. (2) Circular thin position for knots in S^3 by Fabiola Manjarrez-Gutierrez A regular circle-valued Morse function on the knot complement C_K = S^3 - K is a function from C_K to the circle which separates critical points and which behaves nicely in a neighborhood of the knot. Such a function induces a handle decomposition on the knot exterior E(K)= S^3 - N(K), with the property that every regular level surface contains a Seifert surface for the knot. We rearrange the handles in such a way that the regular surfaces are as ``simple as possible. To make this precise the concept of circular width for E(K) is introduced. When E(K) is endowed with a handle decomposition which realizes the circular width we will say that the knot K is in circular thin position. We use this to show that many knots have more than one nonisotopic incompressible Seifert surface. We also analyze the behavior of the circular width under some knot operations. (3) The Lawson homology for Fulton-MacPherson configuration spaces by Wenchuan Hu and Li Li In this paper, we compute the Lawson homology groups and Deligne-Beilinson cohomology groups for the Fulton-MacPherson configuration spaces. (4) Symmetric topological complexity of projective and lens spaces by Jesus Gonzalez and Peter Landweber For real projective spaces, (a) the Euclidean immersion dimension, (b) the existence of axial maps and (c) the topological complexity are known to be three facets of the same problem. But when it comes to embedding dimension, the classical work of Berrick, Feder and Gitler leaves a small indeterminacy when trying to identify the existence of Euclidean embeddings of these manifolds with the existence of symmetric axial maps. As an alternative we show that the symmetrized version of (c) captures, in a sharp way, the embedding problem. Extensions to the case of even-torsion lens spaces and complex projective spaces are discussed. (5) Singular link Floer homology by Benjamin Audoux We define a grid presentation for singular links, ie links with a finite number of rigid transverse double points. Then we use it to generalize link Floer homology to singular links. Besides the consistency of its definition, we prove that this homology is acyclic under some conditions which naturally make its Euler characteristic vanish. (6) Infinitesimal rigidity of a compact hyperbolic 4-orbifold with totally geodesic boundary by Tarik Aougab and Peter A Storm Kerckhoff and Storm conjectured that compact hyperbolic n-orbifolds with totally geodesic boundary are infinitesimally rigid when n>3. We verify this conjecture for a specific example based on the 4-dimensional hyperbolic 120-cell. (7) Abelian subgroups of Out(F_n) by Mark Feighn and Michael Handel We classify abelian subgroups of Out(F_n) up to finite index in an algorithmic and computationally friendly way. A process called disintegration is used to canonically decompose a single rotationless element $phi$ into a composition of finitely many elements and then these elements are used to generate an abelian subgroup A(phi) that contains $phi$. The main theorem is that up to finite index every abelian subgroup is realized by this construction. As an application we give an explicit description, in terms of relative train track maps and up to finite index, of all maximal rank abelian subgroups of Out(F_n) and of IA_n. (8) Gromov-Witten theory of A_n-resolutions by Davesh Maulik We give a complete solution for the reduced Gromov-Witten theory of resolved surface singularities of type A_n, for any genus, with arbitrary descendent insertions. We also present a partial evaluation of the T-equivariant relative Gromov-Witten theory of the threefold calA_n x P^1 which, under a nondegeneracy hypothesis, yields a complete solution for the theory. The results given here allow comparison of this theory with the quantum cohomology of the Hilbert scheme of points on the calA_n surfaces. We discuss generalizations to linear Hodge insertions and to surface resolutions of type D, E. As a corollary, we present a new derivation of the stationary Gromov--Witten theory of P^1. (9) Erratum for the paper `On the chain-level intersection pairing for PL manifolds' by J E McClure Greg Friedman has pointed out that there are sign errors in our paper `On the chain-level intersection pairing for PL manifolds', [Geom. Topol. 10 (2006), 1391-1424], and in particular Lemma 10.5(b) (which is a key step in the proof of the main theorem) is not correct as stated. The purpose of this note is to provide a correction. (10) Virtual fundamental classes via dg-manifolds by Ionut Ciocan-Fontanine and Mikhail Kapranov We construct virtual fundamental classes for dg--manifolds whose tangent sheaves have cohomology only in degrees 0 and 1. This condition is analogous to the existence of a perfect obstruction theory in the approach of Behrend and Fantechi [Invent. Math 128 (1997) 45-88] or Li and Tian [J. Amer. Math. Soc. 11 (1998) 119-174]. Our class is initially defined in K-theory as the class of the structure sheaf of the dg-manifold. We compare our construction with that of Behrend and Fantechi as well as with the original proposal of Kontsevich. We prove a Riemann-Roch type result for dg-manifolds which involves integration over the virtual class. We prove a localization theorem for our virtual classes. We also associate to any dg-manifold of our type a cobordism class of almost complex (smooth) manifolds. This supports the intuition that working with dg-manifolds is the correct algebro-geometric replacement of the analytic technique of deforming to transversal intersection. (11) Geometric intersection number and analogues of the curve complex for free groups by Ilya Kapovich and Martin Lustig For the free group F_N of finite rank N >= 2 we construct a canonical Bonahon-type, continuous and Out(F_N)-invariant geometric intersection form pairing cvbar(F_N) with Curr(F_N). Here cvbar(F_N) is the closure of unprojectivized Culler--Vogtmann's Outer space cv(F_N) in the equivariant Gromov--Hausdorff convergence topology (or, equivalently, in the length function topology). It is known that cvbar(F_N) consists of all very small minimal isometric actions of F_N on R-trees. The projectivization of cvbar(F_N) provides a free group analogue of Thurston's compactification of Teichmuller space. As an application, using the intersection graph determined by the intersection form, we show that several natural analogues of the curve complex in the free group context have infinite diameter. === Subject: Re: Connected implies path connected? Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > Is it true that a connected projective variety is path-connected? I > think it is, but I am not sure. Of course for manifolds, this is true > (because they are locally path-connected), but a projective variety > may have singularities. At the risk of using a thermonuclear bomb in order to kill a mosquito, you can observe that, according to Hironaka's theorem on the resolution of singularities over a field of characteristic zero, given a projective variety X there is a proper, birational map X' --> X with X a smooth algebraic variety. You can use your argument for manifolds on X' and then use the trivial fact that the image of a path-connected space is path-connected. -- m === Subject: Re: Conjecture about quotients of the modular group. Originator: bergv@math.uiuc.edu (Maarten Bergvelt) [..] > However, the congruence subgroup Gamma(2) of PSL(2,Z) is free on two > generators, so every finite simple > group is a quotient of Gamma(2), so a subquotient of PSL(2,Z). This > is closely related to Belyi's theorem. Ah, this is a cool fact! I mixed up quotients and subgroups again, as I did in the recent thread Is the Monster a congruence subgroup?. Gerard