mm-1439 >I asked what the ratio of the speed of recession to the distance means >in your model. When you said it reflects the age of the light, I thought >about how that same ratio holds for all receding galaxies and reflects >the age of the universe. You say it reflects the age of the light, but >if it applies to all galaxies near and far, then I can only conclude >that you mean all light is the same age. Or, perhaps you just don't >understand the evidence for the Big Bang. > > The correct inference to be drawn from my remarks, Tony, is that all > light is the same aging and not that all light is the same age. >> >>The same aging? That sentence grammar not. >> >> Oh, I dunno, Tony. A trifle awkward construction perhaps. I meant the >> appropriate inference is the same rate of aging and not the same age. >> >We calculate time from diastance and speed thus: >distance/speed=time: mile/(mile/hour)=hour > >The ratio of the distance of any given galaxy to its speed of recession >from us, as inferred from the redshift in its light frequencies, gives >the time it has been receding from us. For instance, a galaxy 10 billion >light years away might be receding at .7 times c, the speed of light, >which means it has been receding (assuming constant speed) for 10 >billion/0.7 years, or about 15 billion years. What leads to the >conclusion that it all started at once is that this ratio is the same in >all directions for galactic objects at all distances. All of them appear >to have begun their recession from us at the same time, some 15 billion >years ago. > >So, what does this ratio mean in your model? How do you explain it, if >not as the age of the universe? The Standard Model sticks because it >works. > > The ratio applies to the slowing of light in purely mechanical terms > for reasons related to the structure of photons transiting space. It > produces exactly the same effect as progressive recessional > longitudinal doppler except the effect is fully isotropic in nature > and implies no privileged geocentric or anthopocentric position. >> >>What is it the ratio of? What divided by what equals a measure of >>deceleration of light? Can you offer a formula, please? >> >> I can but I won't for the moment because there seems to be a select >> group of privileged characters around who seem to consider it their >> duty to arbitrarily naysay and disparage. I've already offered two >> explicit formulations: a universal predicate and an explanation for >> first and they won't even acknowledge the second. That's enough for >> the present. Two strikes and your out in my non Euclidean baseball >> world. >> >>Why do you think there is slowing of light in space? >> >> Isotropic frequency dilation. >> >> For instance, is >>light from more distant galaxies passing us at a slower speed than light >>from nearby galaxies? >> >> Sure. >> >> Can you offer some evidence, please? >> >> Isotropic frequency dilation. There are only two observable >> isotropisms in physics: the relative velocity of light and the >> cosmic red shift. But the former depends on second order >> measurements of velocity whereas the latter only depends on first >> order longitudinal recessional doppler effects relative to our >> position in space and there are no observable first order location >> relative isotropisms without the assumption of a privileged location. >> >> >But, the speed of light, whether from a near or far source, is observed >to be the same, is it not? > That's Einstein's postulate, Tony. The velocity of light has never > been measured in first order terms to the best of my knowledge. They > use second order frequency dependent cavity resonance devices to > measure c. No one has ever measured c from cosmologically remote > sources as far as I know and I'm not sure frequency resonance would > reveal degraded velocity in any event. > Besides, if the light were slowing down, >wouldn't the waves bunch up togther, causing INCREASED frequency? Maybe >you mean that light is speeding up, so that the waves get spread out >over time, but then we would find that local light was moving slower >than distant light. > No, definitely slowing. C is a constant locally and defines the upper > limit for velocity in propagating through space. Light rays can pass > through one another without change. Cosmologically remote light rays > have just slowed significantly as a function of the distance traveled. But then the waves would get bunched together, like waves approaching the shore in the ocean ans lowing due to decreased depth of the water. We would see blueshift if the light were slowing on its way toward us. Does anyone else concur? >As far as the ratio between recessional speed exhibited by redshift and >distance, that is a very well defined number in the Standard Model: the >age of the universe. I urge you to consider what, in your model, that >ratio really represents. > The problem in the standard model isn't the ratio of recessional speed > to distance but its isotropic distribution. Recessional velocity is a > first order effect and not a second order relativistic bidirectional > effect. There are no first order isotropic effects observed in physics > because that implies a privileged position in spacetime with respect > to the origin of time wherever it may have been. Not given a closed 3D surface of a 4D hypershpere, for instance. But repetition of this fact doesn't seem to have any effect. > And in my mechanical explanation for the cosmic redshift corresponds > to the optical horizon for light in space and not any supposed age of > the universe . Again, I ask, what happens to the light when it reaches that horizon, and why does the horizon seem to be visually where the implied age of the universe puts it given the speed of light? We have seen objects some 13 billion light years away, out of a calculated 13.6 billion years for the age of the universe.....coincidence? >As far as a privileged position, all you need to accept to dismiss that >nonsense is the closed curvature of our space. When you speak of the >origin, consider an analogy, since I know you love them so much. Drop a >stone into a pond, and you get a wave radiating from the point of >impact. Once that wave has begun expanding, where on that wave is the >origin? Where is the middle of the wave? Once the wave has begun its >expansion, the origin no longer exists within it. > Tony, I don't need to consider the origin of a wave as long as I know > there was one. That defines the center of the isotropic longitudinal > doppler recession assumed to cause the red shift and there is no way > to define away that isotropy or its implications by saying space is > curved or not curved in three dimensions or four or whatever. Space is > what it is in dimensional terms regardless. All you can say is that BB > created space in the process but that still leaves BB at the center of > the expansion as a privileged position and that notion was what the > Galilean revolution was intended to dispel. But the origina of the wave is instaneously outside of the wave. The wave starts at that point but immediately leaves it an no longer includes it. There is no center of the linear space occupied by the expanding wave, within that wave. -- Smiles, Tony === Subject: Re: Epistemology 201: The Science of Science >Lester Zick said: >>Lester Zick said: > >Lester Zick said: > >>I asked what the ratio of the speed of recession to the distance means >>in your model. When you said it reflects the age of the light, I thought >>about how that same ratio holds for all receding galaxies and reflects >>the age of the universe. You say it reflects the age of the light, but >>if it applies to all galaxies near and far, then I can only conclude >>that you mean all light is the same age. Or, perhaps you just don't >>understand the evidence for the Big Bang. >> >> The correct inference to be drawn from my remarks, Tony, is that all >> light is the same aging and not that all light is the same age. > >The same aging? That sentence grammar not. > > Oh, I dunno, Tony. A trifle awkward construction perhaps. I meant the > appropriate inference is the same rate of aging and not the same age. > >>We calculate time from diastance and speed thus: >>distance/speed=time: mile/(mile/hour)=hour >> >>The ratio of the distance of any given galaxy to its speed of recession >>from us, as inferred from the redshift in its light frequencies, gives >>the time it has been receding from us. For instance, a galaxy 10 billion >>light years away might be receding at .7 times c, the speed of light, >>which means it has been receding (assuming constant speed) for 10 >>billion/0.7 years, or about 15 billion years. What leads to the >>conclusion that it all started at once is that this ratio is the same in >>all directions for galactic objects at all distances. All of them appear >>to have begun their recession from us at the same time, some 15 billion >>years ago. >> >>So, what does this ratio mean in your model? How do you explain it, if >>not as the age of the universe? The Standard Model sticks because it >>works. >> >> The ratio applies to the slowing of light in purely mechanical terms >> for reasons related to the structure of photons transiting space. It >> produces exactly the same effect as progressive recessional >> longitudinal doppler except the effect is fully isotropic in nature >> and implies no privileged geocentric or anthopocentric position. > >What is it the ratio of? What divided by what equals a measure of >deceleration of light? Can you offer a formula, please? > > I can but I won't for the moment because there seems to be a select > group of privileged characters around who seem to consider it their > duty to arbitrarily naysay and disparage. I've already offered two > explicit formulations: a universal predicate and an explanation for > first and they won't even acknowledge the second. That's enough for > the present. Two strikes and your out in my non Euclidean baseball > world. > >Why do you think there is slowing of light in space? > > Isotropic frequency dilation. > > For instance, is >light from more distant galaxies passing us at a slower speed than light >from nearby galaxies? > > Sure. > > Can you offer some evidence, please? > > Isotropic frequency dilation. There are only two observable > isotropisms in physics: the relative velocity of light and the > cosmic red shift. But the former depends on second order > measurements of velocity whereas the latter only depends on first > order longitudinal recessional doppler effects relative to our > position in space and there are no observable first order location > relative isotropisms without the assumption of a privileged location. > > >>But, the speed of light, whether from a near or far source, is observed >>to be the same, is it not? >> That's Einstein's postulate, Tony. The velocity of light has never >> been measured in first order terms to the best of my knowledge. They >> use second order frequency dependent cavity resonance devices to >> measure c. No one has ever measured c from cosmologically remote >> sources as far as I know and I'm not sure frequency resonance would >> reveal degraded velocity in any event. >> Besides, if the light were slowing down, >>wouldn't the waves bunch up togther, causing INCREASED frequency? Maybe >>you mean that light is speeding up, so that the waves get spread out >>over time, but then we would find that local light was moving slower >>than distant light. >> No, definitely slowing. C is a constant locally and defines the upper >> limit for velocity in propagating through space. Light rays can pass >> through one another without change. Cosmologically remote light rays >> have just slowed significantly as a function of the distance traveled. >But then the waves would get bunched together, like waves approaching >the shore in the ocean ans lowing due to decreased depth of the water. >We would see blueshift if the light were slowing on its way toward us. >Does anyone else concur? No, Tony, I think you have it backward. >>As far as the ratio between recessional speed exhibited by redshift and >>distance, that is a very well defined number in the Standard Model: the >>age of the universe. I urge you to consider what, in your model, that >>ratio really represents. >> The problem in the standard model isn't the ratio of recessional speed >> to distance but its isotropic distribution. Recessional velocity is a >> first order effect and not a second order relativistic bidirectional >> effect. There are no first order isotropic effects observed in physics >> because that implies a privileged position in spacetime with respect >> to the origin of time wherever it may have been. >Not given a closed 3D surface of a 4D hypershpere, for instance. But >repetition of this fact doesn't seem to have any effect. Repetition shouldn't. Let me just add one further consideration. In science istropisms are taken to reflect properties of the observed and not the observer. This is what the Galilean revolution was all about. So far you haven't shown the isotropic cosmic red shift should be considered different. You just say there could be an explanation if 3D space is an object in 4D space. But that doesn't show it's so and it doesn't say there can't be alternative explanations in 3D space based on properties of the observed instead of those of the observer. >> And in my mechanical explanation for the cosmic redshift corresponds >> to the optical horizon for light in space and not any supposed age of >> the universe . >Again, I ask, what happens to the light when it reaches that horizon, It slows progressively towards zero velocity just as it has throughout its transit of space but approaches zero velocity asymptotically just as absolute zero is approached. There's nothing magical about this. >and why does the horizon seem to be visually where the implied age of >the universe puts it given the speed of light? We have seen objects some >13 billion light years away, out of a calculated 13.6 billion years for >the age of the universe.....coincidence? Because the ratio is the same whether we take it to indicate the age of the universe or the optical horizon in space. It just reflects the age of the light and not of the universe. No coincidence at all that the age of the light and the distance it's traveled should correlate when its velocity slows as a function of distance traveled. >>As far as a privileged position, all you need to accept to dismiss that >>nonsense is the closed curvature of our space. When you speak of the >>origin, consider an analogy, since I know you love them so much. Drop a >>stone into a pond, and you get a wave radiating from the point of >>impact. Once that wave has begun expanding, where on that wave is the >>origin? Where is the middle of the wave? Once the wave has begun its >>expansion, the origin no longer exists within it. >> Tony, I don't need to consider the origin of a wave as long as I know >> there was one. That defines the center of the isotropic longitudinal >> doppler recession assumed to cause the red shift and there is no way >> to define away that isotropy or its implications by saying space is >> curved or not curved in three dimensions or four or whatever. Space is >> what it is in dimensional terms regardless. All you can say is that BB >> created space in the process but that still leaves BB at the center of >> the expansion as a privileged position and that notion was what the >> Galilean revolution was intended to dispel. >But the origina of the wave is instaneously outside of the wave. The >wave starts at that point but immediately leaves it an no longer >includes it. There is no center of the linear space occupied by the >expanding wave, within that wave. There certainly would be in 4D hyperspace or wherever BB occurred. And that center would represent a privileged position in hyperspace. === Subject: Re: Epistemology 201: The Science of Science > That's Einstein's postulate, Tony. The velocity of light has never > been measured in first order terms to the best of my knowledge. They > use second order frequency dependent cavity resonance devices to > measure c. No one has ever measured c from cosmologically remote > sources as far as I know and I'm not sure frequency resonance would > reveal degraded velocity in any event. Fizzeau and Michelson measured the two way speed of light to within one part in a hundred thousand. The Doppler frequency shift does not change the speed of light in a vacuum. Bob Kolker === Subject: Re: Epistemology 201: The Science of Science >> That's Einstein's postulate, Tony. The velocity of light has never >> been measured in first order terms to the best of my knowledge. They >> use second order frequency dependent cavity resonance devices to >> measure c. No one has ever measured c from cosmologically remote >> sources as far as I know and I'm not sure frequency resonance would >> reveal degraded velocity in any event. >Fizzeau and Michelson measured the two way speed of light to within one >part in a hundred thousand. >The Doppler frequency shift does not change the speed of light in a vacuum. Of course not. === Subject: Re: Epistemology 201: The Science of Science Neil W Rickert said: >Neil W Rickert said: >> It depends on what you mean by compare. You can define cardinality >> using the Schroeder-Bernstein theorem. In that case, you do not make >> any reference to order. >Can you give an example, say, of comparing the integers and the >rationals using this method? > This involves defining cardinality in terms of mapping. Set A has > the same cardinality as set B if there is a one-to-one mapping of A > onto B. > Schroeder Bernstein shows that if there is a one-to-one mapping of A > into a subset of B, and if there is a one-to-one mapping of B into a > subset of A, then there is a one-to-one mapping of A onto B. > In the case of integers and rationals, we can map the integers into > the rationals in the obvious way. That is, the integer n maps to the > fraction n/1. To map rationals to integers, we proceed as follows: > given a positive rational, express as a fraction a/b where a and b > are integers, and the fraction is in simplest form (no common divisor > of a and b). We map this rational to the integer (2^a)*(3^b). And > we map -a/b to the negative of this. We map 0/1 to 0. > Then Schroeder-Bernstein shows how to construct a one-to-one mapping > of the integers onto the rationals. > Sorry if the above is a little technical. significant way from ordering the sets to achieve your mapping. Whereas with integers and evens we had a symmetrical mapping based on an arithmetic function, E(i)=i*2 and I(e)=e/2, here you are simply using two different arithmetic functions to map the rationals and integers. I am not sure how you would express the Cantorian diagonal approach to mapping integers onto rationals as a function to achieve 1-1 correspondence, and it's probably not possible, so in this case you are using two different functions neither of which provide 1-1 correspondence, but together show a mapping in each direction, because this achieves what Cantor's counting approach does, which as I've said I find lacking. Sure, you can use a mapping of A/B(i)=R(i)=i/1 and I(a/b)=2^a*3^b, but since this mapping is not symmetrical, it describes even less precisely the relationship between the sets. You are still using the values of the members, which are ordered cardinalities, to create functions for your mappings, so this is not an example of comparing infinite sets without ordering the set members. It just hides the ordering in terms of an arithmetic mapping. Does this seem like a valid objection? Aain, I ask, can you compare infinite sets WITHOUT ordering the members in some fashion? If not, doesn't the method of ordering say anything about the relationship between the sets? -- Smiles, Tony === Subject: Re: Epistemology 201: The Science of Science iD8DBQFCLKPOvmGe70vHPUMRAq4sAKDwvZUd1E1iM9kE8E3ARyAkBK79rQCdHZWV Dm5v0pxmcU5mc7iSO3eW9Sk= =V8pL >significant way from ordering the sets to achieve your mapping. Any time you form a mapping between the integers and another set, that will seem like ordering, simply because the integers have a natural order. >Sure, you can use a mapping of A/B(i)=R(i)=i/1 and I(a/b)=2^a*3^b, but >since this mapping is not symmetrical, it describes even less precisely >the relationship between the sets. We might say that there are two partial mappings. The proof of Schroeder-Bernstein shows how to cut an paste parts of these mappings to construct a complete mapping. === Subject: Re: Epistemology 201: The Science of Science > Neil W Rickert said: >Neil W Rickert said: >>It depends on what you mean by compare. You can define cardinality >>using the Schroeder-Bernstein theorem. In that case, you do not make >>any reference to order. >Can you give an example, say, of comparing the integers and the >rationals using this method? >>This involves defining cardinality in terms of mapping. Set A has >>the same cardinality as set B if there is a one-to-one mapping of A >>onto B. >>Schroeder Bernstein shows that if there is a one-to-one mapping of A >>into a subset of B, and if there is a one-to-one mapping of B into a >>subset of A, then there is a one-to-one mapping of A onto B. >>In the case of integers and rationals, we can map the integers into >>the rationals in the obvious way. That is, the integer n maps to the >>fraction n/1. To map rationals to integers, we proceed as follows: >>given a positive rational, express as a fraction a/b where a and b >>are integers, and the fraction is in simplest form (no common divisor >>of a and b). We map this rational to the integer (2^a)*(3^b). And >>we map -a/b to the negative of this. We map 0/1 to 0. >>Then Schroeder-Bernstein shows how to construct a one-to-one mapping >>of the integers onto the rationals. >>Sorry if the above is a little technical. > significant way from ordering the sets to achieve your mapping. Whereas > with integers and evens we had a symmetrical mapping based on an > arithmetic function, E(i)=i*2 and I(e)=e/2, here you are simply using > two different arithmetic functions to map the rationals and integers. I > am not sure how you would express the Cantorian diagonal approach to > mapping integers onto rationals as a function to achieve 1-1 > correspondence, and it's probably not possible, so in this case you are > using two different functions neither of which provide 1-1 > correspondence, but together show a mapping in each direction, because > this achieves what Cantor's counting approach does, which as I've said I > find lacking. > Sure, you can use a mapping of A/B(i)=R(i)=i/1 and I(a/b)=2^a*3^b, but > since this mapping is not symmetrical, it describes even less precisely > the relationship between the sets. You are still using the values of the > members, which are ordered cardinalities, to create functions for your > mappings, so this is not an example of comparing infinite sets without > ordering the set members. It just hides the ordering in terms of an > arithmetic mapping. Does this seem like a valid objection? > Aain, I ask, can you compare infinite sets WITHOUT ordering the members > in some fashion? If not, doesn't the method of ordering say anything > about the relationship between the sets? I'm just guessing without a comprehensive re-read of this subthread. But I think what Neil and others are really doing is compensating for the lack of a randomly accessible database of infinite numbers. With such a database, no ordering would be required. Without such a database then either (a) the set must be ordered or (b) a sequential search for a match must be done for each iteration. Thinking like a programmer: This is analogous to the matching of transactions to histories in pre-database punched card days. But I am a programmer, not a mathematician. So YMMV. -- I consider it quite possible that physics cannot be based on the field concept, i. e., on continuous structures. In that case nothing remains of my entire castle in the air, gravitation theory included, [and of] the rest of modern physics. -- Albert Einstein in a 1954 letter to Michele Besso. === Subject: Re: Epistemology 201: The Science of Science > I'm just guessing without a comprehensive re-read of this subthread. > But I think what Neil and others are really doing is compensating for > the lack of a randomly accessible database of infinite numbers. With > such a database, no ordering would be required. Without such a database > then either (a) the set must be ordered or (b) a sequential search for a > match must be done for each iteration. Thinking like a programmer: > This is analogous to the matching of transactions to histories in > pre-database punched card days. But I am a programmer, not a > mathematician. So YMMV. If you want to do mathematics think like a mathematician, not a programmer. The programmer's stock and trade consists of algorithms and their implementations on specific machines in the context of specific operating systems. The mathematicians stock and trade consistes of theorems derived from postulates. Bob Kolker === Subject: Re: Epistemology 201: The Science of Science >> I'm just guessing without a comprehensive re-read of this subthread. >> But I think what Neil and others are really doing is compensating for >> the lack of a randomly accessible database of infinite numbers. With >> such a database, no ordering would be required. Without such a >> database then either (a) the set must be ordered or (b) a sequential >> search for a match must be done for each iteration. Thinking like a >> programmer: This is analogous to the matching of transactions to >> histories in pre-database punched card days. But I am a programmer, >> not a mathematician. So YMMV. > If you want to do mathematics think like a mathematician, not a > programmer. The programmer's stock and trade consists of algorithms and > their implementations on specific machines in the context of specific > operating systems. An algorithm is an algorithm and data is data. So you and your 'mathematicians stock and trade'. > The mathematicians stock and trade consistes of theorems derived from > postulates. > Bob Kolker -- I consider it quite possible that physics cannot be based on the field concept, i. e., on continuous structures. In that case nothing remains of my entire castle in the air, gravitation theory included, [and of] the rest of modern physics. -- Albert Einstein in a 1954 letter to Michele Besso. === Subject: Re: Epistemology 201: The Science of Science > An algorithm is an algorithm and data is data. So you and your > 'mathematicians stock and trade'. And a postulate is a postulate and a theorem is is theorem. Why are you so hostile to what mathematicians do? In point of fact it is mathematical analysis that has produced the numerical methods frequently implimented as programs. Long before the first computing machine was ever designed, algorithms for solving differential equations (ordinary and partial) were developed and show to converge to the correct solutions. Algorithms for find roots of equations existed long before computers. Isaac Newton himself made the earliest contributions to the application and solution of finite difference equations. Effective methods of finding definite integrals with definite limits were developed long before computers. For example, Simposon's Rule. Computers and programming have been more of a force multiplier to mathematical algorithms than a replacement. The only area in which computers and programming techniques have constituted breakthoughs is in the area of graphics and other visual representations and real time control. Ray graphics cannot really be done effectively except with a computer. Bitmapped graphics along with dithering and aliasing tricks have produced dazzling displays and motion picture fx would be very backward without computers. Computerized realtime control has made fly-by-wire and hands off flying possible and safe. Even so, the underlying theory of these dazzling techniques were developed well in advance of and distinct from programs. Bellman and Wiener developed cypernetical applications on a purely mathematicatial basis for example. Bob Kolker === Subject: Re: Epistemology 201: The Science of Science >> An algorithm is an algorithm and data is data. So you and your >> 'mathematicians stock and trade'. > And a postulate is a postulate and a theorem is is theorem. Why are you > so hostile to what mathematicians do? > In point of fact it is mathematical analysis that has produced the > numerical methods frequently implimented as programs. Long before the > first computing machine was ever designed, algorithms for solving > differential equations (ordinary and partial) were developed and show to > converge to the correct solutions. Algorithms for find roots of > equations existed long before computers. Isaac Newton himself made the > earliest contributions to the application and solution of finite > difference equations. Effective methods of finding definite integrals > with definite limits were developed long before computers. For example, > Simposon's Rule. > Computers and programming have been more of a force multiplier to > mathematical algorithms than a replacement. The only area in which > computers and programming techniques have constituted breakthoughs is in > the area of graphics and other visual representations and real time > control. Ray graphics cannot really be done effectively except with a > computer. Bitmapped graphics along with dithering and aliasing tricks > have produced dazzling displays and motion picture fx would be very > backward without computers. Computerized realtime control has made > fly-by-wire and hands off flying possible and safe. > Even so, the underlying theory of these dazzling techniques were > developed well in advance of and distinct from programs. Bellman and > Wiener developed cypernetical applications on a purely mathematicatial > basis for example. All of the above is totally irrelevant to my illustrative example. I don't know just what you think I said. But, based on this reply, you certainly failed to understand my point. -- I consider it quite possible that physics cannot be based on the field concept, i. e., on continuous structures. In that case nothing remains of my entire castle in the air, gravitation theory included, [and of] the rest of modern physics. -- Albert Einstein in a 1954 letter to Michele Besso. === Subject: Re: Epistemology 201: The Science of Science Albert Wagner says... >> The way to explain it without analogies is to actually learn >> differential geometry. >Bull. You're hiding behind esoterica to compensate for you >lack of explanatory ability. Differential geometry is not esoterica. It's just geometry and calculus applied to curved spaces. There is nothing conceptually difficult about it. -- Daryl McCullough Ithaca, NY === Subject: Re: Epistemology 201: The Science of Science Lester Zick says... >>The way to explain it without analogies is to actually learn >>differential geometry. >And the best way to learn about it is to understand what isotropism >implies with respect to longitudinal recessional doppler effects in >terms of a privileged positions in space and time. Uh, not really, since that's not correct. The best way is to get a standard math textbook on differential geometry, or to get a physics textbook on General Relativity. -- Daryl McCullough Ithaca, NY === Subject: Re: Epistemology 201: The Science of Science On 7 Mar 2005 05:17:06 -0800, stevendaryl3016@yahoo.com (Daryl >Lester Zick says... >The way to explain it without analogies is to actually learn >differential geometry. >>And the best way to learn about it is to understand what isotropism >>implies with respect to longitudinal recessional doppler effects in >>terms of a privileged positions in space and time. >Uh, not really, since that's not correct. The best way is to get a >standard math textbook on differential geometry, or to get a physics >textbook on General Relativity. What isn't correct? You need a standard math textbook on differential geometry or a physics textbook on GR to understand the implications of longitudinal doppler and cosmic red shift isotropisms? Hardly. Those subjects are esoteric. Longitudinal doppler and cosmic red shift isotropisms are simple subjects not requiring any special expertise. === Subject: Re: Epistemology 201: The Science of Science Lester Zick says... >On 7 Mar 2005 05:17:06 -0800, stevendaryl3016@yahoo.com (Daryl >And the best way to learn about it is to understand what isotropism >implies with respect to longitudinal recessional doppler effects in >terms of a privileged positions in space and time. >>Uh, not really, since that's not correct. The best way is to get a >>standard math textbook on differential geometry, or to get a physics >>textbook on General Relativity. >What isn't correct? It isn't correct that isotropy implies anything about privileged positions in space. -- Daryl McCullough Ithaca, NY === Subject: Re: Epistemology 201: The Science of Science On 7 Mar 2005 09:08:13 -0800, stevendaryl3016@yahoo.com (Daryl >Lester Zick says... >>On 7 Mar 2005 05:17:06 -0800, stevendaryl3016@yahoo.com (Daryl >>And the best way to learn about it is to understand what isotropism >>implies with respect to longitudinal recessional doppler effects in >>terms of a privileged positions in space and time. >Uh, not really, since that's not correct. The best way is to get a >standard math textbook on differential geometry, or to get a physics >textbook on General Relativity. >>What isn't correct? >It isn't correct that isotropy implies anything about privileged >positions in space. Can you name any other first order isotropy that doesn't? === Subject: Re: Epistemology 201: The Science of Science > Can you name any other first order isotropy that doesn't? All of them. Isotropy means that space is the same in all directions. It says nothing about special or privileged points. Not a thing. Isotropy is a fancy word for the conservation of angular momentum. Bob Kolker === Subject: Re: Epistemology 201: The Science of Science >> Can you name any other first order isotropy that doesn't? >All of them. Which are what exactly, Bob? > Isotropy means that space is the same in all directions. So directions are the same in all directions? You're an idiot, Bob. > It >says nothing about special or privileged points. Not a thing. Isotropy >is a fancy word for the conservation of angular momentum. Yeah, well, whatever. === Subject: Re: Epistemology 201: The Science of Science Lester Zick says... >Of course all mechanisms are local. That doesn't make the mechanics of >those mechanisms local or the science defined by that mechanics any >the less universal because the mechanisms it describes are local. We >understand local mechanisms only because and to the extent we first >understand universal implications of the mechanics science studies. I don't understand what that has to do with Euclidean versus non-Euclidean geometry. The important feature of non-Euclidean geometry is that *locally* things are Euclidean, but if you map a large enough region of space, the local regions don't fit together in a Euclidean way. -- Daryl McCullough Ithaca, NY === Subject: Re: Epistemology 201: The Science of Science On 7 Mar 2005 05:14:29 -0800, stevendaryl3016@yahoo.com (Daryl >Lester Zick says... >>Of course all mechanisms are local. That doesn't make the mechanics of >>those mechanisms local or the science defined by that mechanics any >>the less universal because the mechanisms it describes are local. We >>understand local mechanisms only because and to the extent we first >>understand universal implications of the mechanics science studies. >I don't understand what that has to do with Euclidean versus non-Euclidean >geometry. The important feature of non-Euclidean geometry is that *locally* >things are Euclidean, but if you map a large enough region of space, the >local regions don't fit together in a Euclidean way. It's more accurate to say that particular assumptions regarding Euclidean space in local regions don't fit together in a Euclidean way. You can fit together topologies in Euclidean space any way you want in terms of mutually orthogonal directionality. That's your business. But when you try to call a topology in space space it becomes Euclid's business. Space in dimensional terms is Euclidean and/or Cartesian. The circumference of a circle is a two dimensional object in three dimensional space. It is also a one directional topology. In othe words the circumference of a circle is a 2D.1d. object just as a mobius strip is a 3D.1d object or at best a 3D.2d object depending on whether cross directional travel is possible. The problem is that dimensional geometry has to be specified in universal terms before it can be known or applied in particular or local terms. We don't know local geometries and try to piece together universal geometries from them. There is no mite geometry anymore than there is mite physics or mite mechanics. There is only mite history which science tries to reduce. But the reduction is universal whether right or wrong. In other words we only know what we know but what we know we know universally because that's what knowledge implies. It applies equally to every circumstance in the universe where comparable conditions are realized because science in the process of reduction eliminates those considerations which are particular to the circumstances at hand. What you have to consider in the case of Euclidean geometry is which if any of the circumstances included in it are particular and subject to change. And if curves are one of those particular characteristics then Euclidean dimensional concepts based on straight lines remain essential to geometry in general to the extent straight dimensionality remains necessary to the analysis of curves.So if you want to dispense with straight lines then you need to find some alternative way to analyze the properties of curves that does not depend on cardinal enumeration and rational and irrational numbers. === Subject: Re: Epistemology 201: The Science of Science Lester Zick says... >The problem is that dimensional geometry has to be specified in >universal terms before it can be known or applied in particular or >local terms. You seem to be saying it is impossible to characterize geometry in local terms, when there is a well-developed field, differential geometry, which does exactly that. >So if you want to dispense with straight lines then you need to >find some alternative way to analyze the properties of curves It's been known for a long time, first as non-Euclidean geometry, then as differential geometry. >that does not depend on cardinal >enumeration and rational and irrational numbers. What do those have to do with Euclidean versus non-Euclidean geometry? Non-Euclidean geometry uses the same notion of real numbers as Euclidean geometry. -- Daryl McCullough Ithaca, NY === Subject: Re: Epistemology 201: The Science of Science <42211d57.24122219@netnews.att.net> <4224dae7.73203090@netnews.att.net> <4226187c.86735527@netnews.att.net> <4227443c.105169290@netnews.att.net> <42279416.124155105@netnews.att.net> <38pn36F5r4jbfU1@individual.net> <38pt80F5qo75fU1@individual.net> <4228b652.4244959@netnews.att.net> <4229dd50.16844654@netnews.att.net> <422b66ee.34078265@netnews.att.net> <422c6a7b.55414584@netnews.att.net> In , on 03/07/2005 >You seem to be saying it is impossible to characterize geometry in >local terms, when there is a well-developed field, differential >geometry, which does exactly that. Differential Geometry also deals with global properties, e.g., cohomology groups. Not that I believe the OP knows what those are ;-) -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not === Subject: Re: Epistemology 201: The Science of Science Seymour J. says... >stevendaryl3016@yahoo.com (Daryl McCullough) said: >>You seem to be saying it is impossible to characterize geometry in >>local terms, when there is a well-developed field, differential >>geometry, which does exactly that. >Differential Geometry also deals with global properties, e.g., >cohomology groups. Not that I believe the OP knows what those are ;-) Err...neither do I. Whatever they are, aren't they determined by the collection of charts for the manifold? -- Daryl McCullough Ithaca, NY === Subject: Re: Epistemology 201: The Science of Science On 7 Mar 2005 07:34:45 -0800, stevendaryl3016@yahoo.com (Daryl >Lester Zick says... >>The problem is that dimensional geometry has to be specified in >>universal terms before it can be known or applied in particular or >>local terms. >You seem to be saying it is impossible to characterize geometry >in local terms, when there is a well-developed field, differential >geometry, which does exactly that. I don't say universal science can't be applied locally, which is what differential geometry does only that it has to be treated universally. >>So if you want to dispense with straight lines then you need to >>find some alternative way to analyze the properties of curves >It's been known for a long time, first as non-Euclidean geometry, then as >differential geometry. It has been known for a long time only as a superstructure on Euclidean geometry. >>that does not depend on cardinal >>enumeration and rational and irrational numbers. >What do those have to do with Euclidean versus non-Euclidean geometry? >Non-Euclidean geometry uses the same notion of real numbers as Euclidean >geometry. Cardinal enumeration and rational and irrational numbers are only possible in the context of straight lines and Euclidean geometry. === Subject: Re: Epistemology 201: The Science of Science Lester Zick says... >>You seem to be saying it is impossible to characterize geometry >>in local terms, when there is a well-developed field, differential >>geometry, which does exactly that. >I don't say universal science can't be applied locally, which is what >differential geometry does only that it has to be treated universally. I don't know what you mean by that. >So if you want to dispense with straight lines then you need to >find some alternative way to analyze the properties of curves >>It's been known for a long time, first as non-Euclidean geometry, then as >>differential geometry. >It has been known for a long time only as a superstructure on >Euclidean geometry. I'm not sure what you mean by superstructure. Non-Euclidean geometry, or differential geometry, is a generalization of Euclidean geometry. >Cardinal enumeration and rational and irrational numbers are only >possible in the context of straight lines and Euclidean geometry. That's not true. The reals and rationals can be developed independently of geometry. -- Daryl McCullough Ithaca, NY === Subject: Re: Epistemology 201: The Science of Science On 7 Mar 2005 09:37:49 -0800, stevendaryl3016@yahoo.com (Daryl >Lester Zick says... >You seem to be saying it is impossible to characterize geometry >in local terms, when there is a well-developed field, differential >geometry, which does exactly that. >>I don't say universal science can't be applied locally, which is what >>differential geometry does only that it has to be treated universally. >I don't know what you mean by that. I just mean that science has to be developed universally before it can be applied locally. Differential geometry, by which I take you to mean topology, had to be developed independently before it could be applied to particular topologies. >>So if you want to dispense with straight lines then you need to >>find some alternative way to analyze the properties of curves >It's been known for a long time, first as non-Euclidean geometry, then as >differential geometry. >>It has been known for a long time only as a superstructure on >>Euclidean geometry. >I'm not sure what you mean by superstructure. Non-Euclidean geometry, >or differential geometry, is a generalization of Euclidean geometry. I see it as the reverse. It takes Euclidean geometry to make sense of non Euclidean geometries just as it takes straight lines to make sense of curves and to make sense of curves in relation to one another. Without Euclidean straight lines you couldn't commensurate curves to one another to generalize non Euclidean geometries. >>Cardinal enumeration and rational and irrational numbers are only >>possible in the context of straight lines and Euclidean geometry. >That's not true. The reals and rationals can be developed independently >of geometry. Yes I've heard this before, Daryl, I just don't understand how. Counting requires different things to count and it requires equal differences between things counted or it won't amount to much. You can't match anything in the absence of equal differences. Now if there's another way than geometry to establish equal differences, I'm unaware of it. === Subject: Re: Epistemology 201: The Science of Science > I just mean that science has to be developed universally before it can > be applied locally. Differential geometry, by which I take you to mean > topology, had to be developed independently before it could be applied > to particular topologies. Differential geometry is NOT the general science of topology. Topology consists of defiing the open sets in a space. There is no notion of distances or measurements. Intuitively general topology (as opposed to algebraic topology) is a theory of nearness developed without co-ordinates or a metric. The set of mappings of one topological space to another in which nearness is preserved (topological homeomorphisms) is the gut of the sciences. Topology is a category in which the arrows are homeomorphisms. Google to see what is meant by the term arrows. Bob Kolker === Subject: Re: Epistemology 201: The Science of Science >> I just mean that science has to be developed universally before it can >> be applied locally. Differential geometry, by which I take you to mean >> topology, had to be developed independently before it could be applied >> to particular topologies. >Differential geometry is NOT the general science of topology. Topology >consists of defiing the open sets in a space. There is no notion of >distances or measurements. Intuitively general topology (as opposed to >algebraic topology) is a theory of nearness developed without >co-ordinates or a metric. The set of mappings of one topological space >to another in which nearness is preserved (topological homeomorphisms) >is the gut of the sciences. Topology is a category in which the arrows >are homeomorphisms. Google to see what is meant by the >term arrows. Ok, Bob, for your next trick consider getting back to the subject at hand. You have this really objectionable conversational technique of oblique replies in aid of nothing. === Subject: Re: Epistemology 201: The Science of Science >> I just mean that science has to be developed universally before it can >> be applied locally. Differential geometry, by which I take you to mean >> topology, had to be developed independently before it could be applied >> to particular topologies. > Differential geometry is NOT the general science of topology. Topology > consists of defiing the open sets in a space. There is no notion of > distances or measurements. Intuitively general topology (as opposed to > algebraic topology) is a theory of nearness developed without > co-ordinates or a metric. The set of mappings of one topological space > to another in which nearness is preserved (topological homeomorphisms) > is the gut of the sciences. Topology is a category in which the arrows > are homeomorphisms. Google to see what is meant by the > term arrows. Gee, Bob. That doesn't seem fair. You send Lester to Google for definitions but you demand that he produce them for you willy-nilly as you choose. -- I consider it quite possible that physics cannot be based on the field concept, i. e., on continuous structures. In that case nothing remains of my entire castle in the air, gravitation theory included, [and of] the rest of modern physics. -- Albert Einstein in a 1954 letter to Michele Besso. === Subject: Re: Epistemology 201: The Science of Science > I just mean that science has to be developed universally before it can > be applied locally. Differential geometry, by which I take you to mean > topology, had to be developed independently before it could be applied > to particular topologies. >> Differential geometry is NOT the general science of topology. Topology >> consists of defiing the open sets in a space. There is no notion of >> distances or measurements. Intuitively general topology (as opposed to >> algebraic topology) is a theory of nearness developed without >> co-ordinates or a metric. The set of mappings of one topological space >> to another in which nearness is preserved (topological homeomorphisms) >> is the gut of the sciences. Topology is a category in which the arrows >> are homeomorphisms. Google to see what is meant by the >> term arrows. >Gee, Bob. That doesn't seem fair. You send Lester to Google for >definitions but you demand that he produce them for you >willy-nilly as you choose. Nice one, Albert. === Subject: Re: Epistemology 201: The Science of Science > Gee, Bob. That doesn't seem fair. You send Lester to Google for > definitions but you demand that he produce them for you willy-nilly as > you choose. The Google queiries return entries in the peer reviewed literature. I asked for a reference and did not get one. I would even settle for Google reference, but I still will not get one. That is because Lester uses a private language only he can understand (if he understands it all). Bob Kolker === Subject: Re: Epistemology 201: The Science of Science > I just mean that science has to be developed universally before it can > be applied locally. Differential geometry, by which I take you to mean > topology, had to be developed independently before it could be applied > to particular topologies. Not true. See how Gauss developed differential geometry. Bob Kolker === Subject: Re: Epistemology 201: The Science of Science >> I just mean that science has to be developed universally before it can >> be applied locally. Differential geometry, by which I take you to mean >> topology, had to be developed independently before it could be applied >> to particular topologies. >Not true. See how Gauss developed differential geometry. Did he do it in pieces by analyzing your little dots, Bob? === Subject: Re: Epistemology 201: The Science of Science > Cardinal enumeration and rational and irrational numbers are only > possible in the context of straight lines and Euclidean geometry. Nonsense. The theory of real numbers can be developed in a purely algebriaic/arithmeitc manner. In fact the rigorization of real analysis consisted in decouplling the theory from geometry. Geometry both helps and hurts. Here is an example. Consider a continous function of a real variable. Can a function which is continuous everywhere but differentiable nowhere exist. If you think of a function as a curvey graph you may assume not. But Wierstrass showed by purely analytical (i.e. algebraic) methods that such functions exist. Using a purely geomtric approach this would not have become known. In the field of topology, deploying algebraic and group structures to encapsulate the spaces removes visual (and therefore geometric) intuitions from proving the theorems. Mathematical progress was made by decoupling geometry and arithmetic/algebra. This is in contrast to the Greeks who bundled arithmetic along with the geometry. So a geometry free mode of developing real and complex numbers exists and has existed from the middle of th 19-th century. Bob Kolker === Subject: Re: Epistemology 201: The Science of Science >> Cardinal enumeration and rational and irrational numbers are only >> possible in the context of straight lines and Euclidean geometry. >Nonsense. The theory of real numbers can be developed in a purely >algebriaic/arithmeitc manner. In fact the rigorization of real analysis >consisted in decouplling the theory from geometry. What I said was that the rationals, irrationals, and cardinal enumeration could not have been developed without geometry. You can make any assumptions you want regarding what you can count. But that doesn't make it happen especially when arithmetic is post hoc. Hell arithmetic in modern form doesn't even begin to happen until the renaissance and yet people have been counting for thousand of years without understanding what they were counting or exactly how or ever considering the possibility of number theory based on counting. >Geometry both helps and hurts. Here is an example. Consider a continous >function of a real variable. Can a function which is continuous >everywhere but differentiable nowhere exist. If you think of a function >as a curvey graph you may assume not. But Wierstrass showed by purely >analytical (i.e. algebraic) methods that such functions exist. Using a >purely geomtric approach this would not have become known. Does such a function have no slope and no tangent to its continuum? Tell me more. >In the field of topology, deploying algebraic and group structures to >encapsulate the spaces removes visual (and therefore geometric) >intuitions from proving the theorems. Mathematical progress was made by >decoupling geometry and arithmetic/algebra. This is in contrast to the >Greeks who bundled arithmetic along with the geometry. Well if your point is that non geometric thinking can produce valuable insights, I can hardly disagree.That wasn't and isn't my point though. >So a geometry free mode of developing real and complex numbers exists >and has existed from the middle of th 19-th century. Same reply. You're confusing productivity with the origin of seminal ideas. Different issues. I'm still waiting for someone to show me counting without assuming countable equal differences. === Subject: Re: Epistemology 201: The Science of Science > Does such a function have no slope and no tangent to its continuum? > Tell me more. No slope, no tangent anywhere. But it is continouous> Google and you will see the infinite series that defines the function. The series converges for every value of the independent variable but when you differentiate the series term by term you get a series that diverges for every value of the infinite variable. Bob Kolker === Subject: Re: Epistemology 201: The Science of Science >> Does such a function have no slope and no tangent to its continuum? >> Tell me more. >No slope, no tangent anywhere. But it is continouous> Google > and you will see the >infinite series that defines the function. The series converges for >every value of the independent variable but when you differentiate the >series term by term you get a series that diverges for every value of >the infinite variable. So how do you explain the explanation? === Subject: Re: Epistemology 201: The Science of Science > So how do you explain the explanation? Your ingnorance is invincible. If you do not know how functions can be defined by infinite series, no explanation is possible for you. You cannot cope with mathematics. That makes you a sub-human. Bob Kolker === Subject: Re: Epistemology 201: The Science of Science >> So how do you explain the explanation? >Your ingnorance is invincible. If you do not know how functions can be >defined by infinite series, no explanation is possible for you. You >cannot cope with mathematics. That makes you a sub-human. Don't get so bent out of shape, Bob. I ask a perfectly civil question and you go postal on me. If you can't answer questions, you have something in common with animals. Don't blame your shortcomings on me. === Subject: Re: Epistemology 201: The Science of Science [...] >>Geometry both helps and hurts. Here is an example. Consider a continous >>function of a real variable. Can a function which is continuous >>everywhere but differentiable nowhere exist. If you think of a function >>as a curvey graph you may assume not. But Wierstrass showed by purely >>analytical (i.e. algebraic) methods that such functions exist. Using a >>purely geomtric approach this would not have become known. > Does such a function have no slope and no tangent to its continuum? > Tell me more. [...] You got it. === Subject: Re: Epistemology 201: The Science of Science >[...] >Geometry both helps and hurts. Here is an example. Consider a continous >function of a real variable. Can a function which is continuous >everywhere but differentiable nowhere exist. If you think of a function >as a curvey graph you may assume not. But Wierstrass showed by purely >analytical (i.e. algebraic) methods that such functions exist. Using a >purely geomtric approach this would not have become known. >> Does such a function have no slope and no tangent to its continuum? >> Tell me more. >[...] >You got it. And how's the trick done? === Subject: Re: Epistemology 201: The Science of Science > And how's the trick done? Google and find out. Bob Kolker === Subject: Re: Epistemology 201: The Science of Science >> And how's the trick done? >Google and find out. I did. So? === Subject: Re: Epistemology 201: The Science of Science >And how's the trick done? >>Google and find out. > I did. So? Well, why don't you try differentiating the Weierstrass function. See what happens. (At least one the sites has done the arithmetic of graphing the function, I noticed. Interesting shape. Looks like it changes direction at every point. Of course, a computer generated graph isn't proof, but it is, as they say, highly suggestive.) === Subject: Re: Epistemology 201: The Science of Science >>And how's the trick done? >Google and find out. >> I did. So? >Well, why don't you try differentiating the Weierstrass function. Why don't you? > See >what happens. (At least one the sites has done the arithmetic of >graphing the function, I noticed. Interesting shape. Looks like it >changes direction at every point. Of course, a computer generated graph >isn't proof, but it is, as they say, highly suggestive.) I noticed the same thing, Bob. What's highly suggestive to me is that points are the only undifferentiable thing I know of. So the question occurs to me whether the function is really a continuum or just a collection of points. However, I realize you can't answer questions and being a particular fan of the Romans prefer to deal exclusively in === Subject: Re: Epistemology 201: The Science of Science > I noticed the same thing, Bob. What's highly suggestive to me is that > points are the only undifferentiable thing I know of. So the question > occurs to me whether the function is really a continuum or just a > collection of points. A function is a binary relation between the domain and co-domain such that if (x, y1) and (x, y2) are elements of the function then y1 = y2. Bob Kolker === Subject: Re: Epistemology 201: The Science of Science robert j. kolker said: > > Cardinal enumeration and rational and irrational numbers are only > possible in the context of straight lines and Euclidean geometry. > Nonsense. The theory of real numbers can be developed in a purely > algebriaic/arithmeitc manner. In fact the rigorization of real analysis > consisted in decouplling the theory from geometry. > Geometry both helps and hurts. Here is an example. Consider a continous > function of a real variable. Can a function which is continuous > everywhere but differentiable nowhere exist. If you think of a function > as a curvey graph you may assume not. But Wierstrass showed by purely > analytical (i.e. algebraic) methods that such functions exist. Using a > purely geomtric approach this would not have become known. > In the field of topology, deploying algebraic and group structures to > encapsulate the spaces removes visual (and therefore geometric) > intuitions from proving the theorems. Mathematical progress was made by > decoupling geometry and arithmetic/algebra. This is in contrast to the > Greeks who bundled arithmetic along with the geometry. > So a geometry free mode of developing real and complex numbers exists > and has existed from the middle of th 19-th century. > Bob Kolker Yes, I actually found the Peano Axiom discussion very enlightening a few systems can be derived aximatically. I am still of the opinion that geometry is very important, and perhaps more fundamental than symbolic mathemtical systems. It certainly matters to me. Every coin's got two sides. -- Smiles, Tony === Subject: Re: Epistemology 201: The Science of Science > Yes, I actually found the Peano Axiom discussion very enlightening a few > systems can be derived aximatically. I am still of the opinion that > geometry is very important, and perhaps more fundamental than symbolic > mathemtical systems. It certainly matters to me. Every coin's got two > sides. Geometry has been and most likely always will be a powerful intuition pump. But Geometry is not a logical necessity for the theory of real and complex numbers. Bob Kolker === Subject: Re: Epistemology 201: The Science of Science >> Yes, I actually found the Peano Axiom discussion very enlightening a few >> systems can be derived aximatically. I am still of the opinion that >> geometry is very important, and perhaps more fundamental than symbolic >> mathemtical systems. It certainly matters to me. Every coin's got two >> sides. >Geometry has been and most likely always will be a powerful intuition >pump. But Geometry is not a logical necessity for the theory of real and >complex numbers. That's nice. === Subject: Re: Epistemology 201: The Science of Science >I don't know what you mean by concentric with, either Concentric with means having a common center with. For example two circles having the same center (one will lie inside the other) are concentric or concentric with each other. If A is one circle and B is the other, then we say A is concentric with B. > I agree. But you can't have an isotropic isomorphic expansion without > temporal and spatial metric coincidence or diametric opposition. Could you say what that means. === Subject: Re: Epistemology 201: The Science of Science >>I don't know what you mean by concentric with, either >Concentric with means having a common center with. >For example two circles having the same center (one will lie inside the >other) are concentric or concentric with each other. If A is one circle >and B is the other, then we say A is concentric with B. >> I agree. But you can't have an isotropic isomorphic expansion without >> temporal and spatial metric coincidence or diametric opposition. >Could you say what that means. It means that coordinate systems for a temporal origin and spatial observer must coincide or be diametrically opposed in order for there to be isotropic expansion when a linear effect such as longitudinal recessional doppler is considered. If spacetime metrics for the temporal origin of BB and O are eccentric and do not coincide and are not diametrically opposed the cosmic red shift would be anisotropic. And the fact that we do have an isomorphic cosmic red shift indicates an isomorphism at work having the same form but different mechanical origin from longitudinal recessional doppler. === Subject: Re: Epistemology 201: The Science of Science > So what else in the magesterium of science and mathematics is a > metaphor? Or is it only those things mathematikers find themselves > embarrassed to explain in literal terms? Is transcendental > irrationality just a metaphor or just a plain contradiction? Neither. A transcendental real number is a real number which not the root of a polynomial with ratinal coefficients. What is so difficult about this definition that you do you comprehend it? ? Is SR a > metaphor or just a plain contradiction? It is a classical theory which comprehends mechanics and electrodynamics. Mechanics is modified to be Lorentz invariant. ? Is GR a metaphor or just a > plain contradiction? It is a theory of gravitiation which so far has not been falsified and is very well supported experimentaly. The GPS depends on timing corrections calculated using GTR. > Is pi just a metaphor for pi radians? I think we > may need to make a little list of all the metaphors mathematikers > suddenly find themselves faced with using that they didn't think were > metaphors but literally exact. Pi is the ratio of the circumfernece of a plane circle to its diameter. Pi radians is the angle subtended by an arc on the the circumference of a circle whose length is equal to the radius of the circle. Sometime the word radians is omitted when it is clear that angles are the subject of discussion. These are all standard definition. They can be found in any mathematics text book on trigonometry or geometry. Bob Kolker === Subject: Re: Epistemology 201: The Science of Science >> So what else in the magesterium of science and mathematics is a >> metaphor? Or is it only those things mathematikers find themselves >> embarrassed to explain in literal terms? Is transcendental >> irrationality just a metaphor or just a plain contradiction? >Neither. A transcendental real number is a real number which not the >root of a polynomial with ratinal coefficients. What is so difficult >about this definition that you do you comprehend it? Whether either or both definitions can be accommodated on straight lines between points. Wolf acknowledges that the real number line is a fiction for transcendentals. And I just want to know if there are other fictions, such as SR and GR, in the magesterium of mathematics and science. You say that SR and GR are predictive. And I say that just makes them useful fictions; it doesn't make them science. >? Is SR a >> metaphor or just a plain contradiction? >It is a classical theory which comprehends mechanics and >electrodynamics. Mechanics is modified to be Lorentz invariant. So it's a useful fiction which doesn't explain how anything happens. >? Is GR a metaphor or just a >> plain contradiction? >It is a theory of gravitiation which so far has not been falsified and >is very well supported experimentaly. The GPS depends on timing >corrections calculated using GTR. So it's a useful fiction because it makes the trains run on time. >> Is pi just a metaphor for pi radians? I think we >> may need to make a little list of all the metaphors mathematikers >> suddenly find themselves faced with using that they didn't think were >> metaphors but literally exact. >Pi is the ratio of the circumfernece of a plane circle to its diameter. And pi is defined on circular arcs not straight lines. >Pi radians is the angle subtended by an arc on the the circumference of >a circle whose length is equal to the radius of the circle. Sometime the >word radians is omitted when it is clear that angles are the subject of >discussion. However pi is an ambiguous term when gauge is omitted. Pi radians is a different angle from pi diameters. Simply stating that angles are involved doesn't tell us whether radians or diameters are involved. Thus used alone the term pi doesn't necessarily imply anything but the transcendental ratio 3.14159 . . . Your real argument here is that radians are a natural or implied gauge when the term pi is used. But pi is primarily defined in terms of diameters not radians; so that argument is nugatory. >These are all standard definition. They can be found in any mathematics >text book on trigonometry or geometry. And when texts refer to pi without specifying a standard of reference I would expect them to be referring to the transcendental ratio and not an angle. === Subject: Re: Epistemology 201: The Science of Science >So what else in the magesterium of science and mathematics is a >metaphor? Or is it only those things mathematikers find themselves >embarrassed to explain in literal terms? Is transcendental >irrationality just a metaphor or just a plain contradiction? >>Neither. A transcendental real number is a real number which not the >>root of a polynomial with ratinal coefficients. What is so difficult >>about this definition that you do you comprehend it? > Whether either or both definitions can be accommodated on straight > lines between points. Wolf acknowledges that the real number line is a > fiction for transcendentals. No, I don't. You've misread me again, which isn't surprising, considering your wierd notion that a number must be commensurate with a _straight_ line segment, or sone such nonsense. I didn't say the number line was a fiction, I said I thought it was a metaphor. I also said I wasn't sure what that meant, which is a terse way of saying I wasn't sure if it made sense to say it's a metaphor. I still don't know, but I'm inclined to think not. >And I just want to know if there are > other fictions, such as SR and GR, in the magesterium of mathematics > and science. You say that SR and GR are predictive. And I say that > just makes them useful fictions; it doesn't make them science. Your contrasting of science and useful fictions doesn't make much sense to me. Care to elaborate? === Subject: Re: Epistemology 201: The Science of Science >>So what else in the magesterium of science and mathematics is a >>metaphor? Or is it only those things mathematikers find themselves >>embarrassed to explain in literal terms? Is transcendental >>irrationality just a metaphor or just a plain contradiction? >Neither. A transcendental real number is a real number which not the >root of a polynomial with ratinal coefficients. What is so difficult >about this definition that you do you comprehend it? >> Whether either or both definitions can be accommodated on straight >> lines between points. Wolf acknowledges that the real number line is a >> fiction for transcendentals. >No, I don't. You've misread me again, which isn't surprising, >considering your wierd notion that a number must be commensurate with >a _straight_ line segment, or sone such nonsense. Well perhaps my reference to fiction was a trifle hyperbolic. You definitely said the real number line was a metaphor and based on collateral posts of yours on mental effects, I take your usage of the word metaphor to indicate an explanatory fiction. At least I've always objected to use of the term metaphor to describe some idea that isn't actually a metaphor because it doesn't describe something in terms which it is definitely not. So I've replaced that usage with the term fiction in my own writing. However since this usage for the term metaphor is so well established, I'm probably the only one who understands why I prefer the term fiction to metaphor. >I didn't say the number line was a fiction, I said I thought it was a >metaphor. I also said I wasn't sure what that meant, which is a terse >way of saying I wasn't sure if it made sense to say it's a metaphor. I >still don't know, but I'm inclined to think not. Well the problem with the term metaphor is that people don't really understand what they mean when they use it. My personal opinion is that based on the usage I see, when people use the term metaphor they actually mean a useful or convenient fictional device which isn't what the term metaphor actually means, but what the hell we seem stuck with it. >>And I just want to know if there are >> other fictions, such as SR and GR, in the magesterium of mathematics >> and science. You say that SR and GR are predictive. And I say that >> just makes them useful fictions; it doesn't make them science. >Your contrasting of science and useful fictions doesn't make much >sense to me. Care to elaborate? Sure. There are different ways to explain how something works. We can explain it analogically in terms of predictive models or we can simply explain how and why it works in self consistent reductive terms.That's what science does or is supposed to do. It's called mechanics. When we explain things in other than reductive terms, there is no mechanical explanation we can integrate with other mechanical explanations. If we explain things in truly metaphorical terms, the metaphorical terms may produce accurate predictions for accidental reasons not necessarily implied in the metaphor itself. These produce predictions without any mechanical insight allowing integration on any common basis with other mechanical ideas. QM falls into this category exclusively. So does astrology and so do allegories. The latter don't explain things very well before the fact but they are superb at descriptive mechanics after the fact. At least QM predicts things pretty effectively before the fact but that doesn't make it any the less a convenient fiction of no reductive explanatory significance. === Subject: Re: Epistemology 201: The Science of Science > Sure. There are different ways to explain how something works. We can > explain it analogically in terms of predictive models or we can simply > explain how and why it works in self consistent reductive terms.That's > what science does or is supposed to do. It's called mechanics. When we > explain things in other than reductive terms, So far every purely mechanical theory has failed to predict correctly. That is why we have quantum theory. The theory of heat based on classical molecular kinetics is not capable of predicting specific heats of a large number of substances. Quantum theory can. Mechanics was discarded as a fundemental theory because it failed emprically. Facts rule, Theories serve. Bob Kolker === Subject: Re: Epistemology 201: The Science of Science >> Sure. There are different ways to explain how something works. We can >> explain it analogically in terms of predictive models or we can simply >> explain how and why it works in self consistent reductive terms.That's >> what science does or is supposed to do. It's called mechanics. When we >> explain things in other than reductive terms, >So far every purely mechanical theory has failed to predict correctly. >That is why we have quantum theory. The theory of heat based on >classical molecular kinetics is not capable of predicting specific heats >of a large number of substances. Quantum theory can. >Mechanics was discarded as a fundemental theory because it failed >emprically. Facts rule, Theories serve. And you don't do much of anything that I can see, Bob, except blow smoke. === Subject: Re: Epistemology 201: The Science of Science > Whether either or both definitions can be accommodated on straight > lines between points. Real numbers can be developed completely without geometric content. For some applications it is useful (but not necessary) to associatate real numbers with points on a curve (or straight line). If the world were blind but could count real number theory could be developed. > Wolf acknowledges that the real number line is a > fiction for transcendentals. And I just want to know if there are > other fictions, such as SR and GR, in the magesterium of mathematics > and science. You say that SR and GR are predictive. And I say that > just makes them useful fictions; it doesn't make them science. That is exactly what a physical theory is. A useful fiction. It is a just so story with the virtue of being backed up by experiment or observation. The useful fictions of physics lead to useful real things of a technological nature. Nothing that Man proposes hypothetically is any more than a just-so story. There is no truth in bypothetics, only potentially falsifiable quantitative assertions. If you want useful fictions get a degree in physics. If you want Truth get a degree in theology. Bob Kolker === Subject: Re: Epistemology 201: The Science of Science >> Whether either or both definitions can be accommodated on straight >> lines between points. >Real numbers can be developed completely without geometric content. For >some applications it is useful (but not necessary) to associatate real >numbers with points on a curve (or straight line). If the world were >blind but could count real number theory could be developed. Do you admit there is no real number line? Certainly transcendentals cannot be accommodated between points on straight lines or on any one curve. So why haven't counting animals developed real number theory? >> Wolf acknowledges that the real number line is a >> fiction for transcendentals. And I just want to know if there are >> other fictions, such as SR and GR, in the magesterium of mathematics >> and science. You say that SR and GR are predictive. And I say that >> just makes them useful fictions; it doesn't make them science. >That is exactly what a physical theory is. A useful fiction. It is a >just so story with the virtue of being backed up by experiment or >observation. The useful fictions of physics lead to useful real things >of a technological nature. There are many useful fictions; not so many science. >Nothing that Man proposes hypothetically is any more than a just-so >story. There is no truth in bypothetics, only potentially falsifiable >quantitative assertions. A tale told by an idiot full of sound and fury signifying nothing? There are many fictions and many of them are useful. >If you want useful fictions get a degree in physics. If you want Truth >get a degree in theology. Frankly I prefer science. So what about the truth of my explanation omnipresent immanence without a degree in theology. === Subject: Re: Epistemology 201: The Science of Science >Whether either or both definitions can be accommodated on straight >lines between points. >>Real numbers can be developed completely without geometric content. For >>some applications it is useful (but not necessary) to associatate real >>numbers with points on a curve (or straight line). If the world were >>blind but could count real number theory could be developed. > Do you admit there is no real number line? Certainly transcendentals > cannot be accommodated between points on straight lines or on any one > curve. So why haven't counting animals developed real number theory? AHA! I think I finally undeestood what you mean: for a number to be accommodated on line, there must a geometric construction of a line of that length. Is that what you mean? It's not clear whether the construction is limited to straight edge and compass, or whether you permit the abstract construction possible in Cartesian co-ordinate space and the associated arithmetic. === Subject: Re: Epistemology 201: The Science of Science >>Whether either or both definitions can be accommodated on straight >>lines between points. >Real numbers can be developed completely without geometric content. For >some applications it is useful (but not necessary) to associatate real >numbers with points on a curve (or straight line). If the world were >blind but could count real number theory could be developed. >> Do you admit there is no real number line? Certainly transcendentals >> cannot be accommodated between points on straight lines or on any one >> curve. So why haven't counting animals developed real number theory? >AHA! >I think I finally undeestood what you mean: for a number to be >accommodated on line, there must a geometric construction of a line of >that length. >Is that what you mean? Mainly. The rationals and irrationals lie between points and can be pointed out on a straight line using right angles for that reason. Transcendentals require commensuration between curves and straight lines to be pointed out on curves. By the phrase accommodated on a straight line I basically mean a number must lie between points in space because points in space define straight line segments. Transcendentals don't lie between points in space. They lie on curves and not between points. >It's not clear whether the construction is limited to straight edge and >compass, or whether you permit the abstract construction possible in >Cartesian co-ordinate space and the associated arithmetic. I would expect reduction of the latter to the former. We don't need to walk around with a straight edge and compass to count and calculate but we do need to understand the reduction of numeric concepts and their implications as matters of science in general. === Subject: Re: Epistemology 201: The Science of Science >Whether either or both definitions can be accommodated on straight >lines between points. >> >>Real numbers can be developed completely without geometric content. For >>some applications it is useful (but not necessary) to associatate real >>numbers with points on a curve (or straight line). If the world were >>blind but could count real number theory could be developed. >Do you admit there is no real number line? Certainly transcendentals >cannot be accommodated between points on straight lines or on any one >curve. So why haven't counting animals developed real number theory? >>AHA! >>I think I finally undeestood what you mean: for a number to be >>accommodated on line, there must a geometric construction of a line of >>that length. >>Is that what you mean? > Mainly. The rationals and irrationals lie between points and can be > pointed out on a straight line using right angles for that reason. > Transcendentals require commensuration between curves and straight > lines to be pointed out on curves. By the phrase accommodated on a > straight line I basically mean a number must lie between points in > space because points in space define straight line segments. > Transcendentals don't lie between points in space. They lie on curves > and not between points. Does this mean that you think curves don't have endpoints? That's odd, since I'm sure someone taught you that between any two points you draw as many lines as you want, but only one them is a so-called straight line - the one whose measure of the distance between those points is smallest. Or do you believe that curves aren't lines? What are they, then? >>It's not clear whether the construction is limited to straight edge and >>compass, or whether you permit the abstract construction possible in >>Cartesian co-ordinate space and the associated arithmetic. > I would expect reduction of the latter to the former. We don't need to > walk around with a straight edge and compass to count and calculate > but we do need to understand the reduction of numeric concepts and > their implications as matters of science in general. Reduction to what? Geometry? I wish you'd be more explicit. === Subject: Re: Epistemology 201: The Science of Science >> >> >> >>Whether either or both definitions can be accommodated on straight >>lines between points. > >Real numbers can be developed completely without geometric content. For >some applications it is useful (but not necessary) to associatate real >numbers with points on a curve (or straight line). If the world were >blind but could count real number theory could be developed. >> >> >>Do you admit there is no real number line? Certainly transcendentals >>cannot be accommodated between points on straight lines or on any one >>curve. So why haven't counting animals developed real number theory? >AHA! >I think I finally undeestood what you mean: for a number to be >accommodated on line, there must a geometric construction of a line of >that length. >Is that what you mean? >> Mainly. The rationals and irrationals lie between points and can be >> pointed out on a straight line using right angles for that reason. >> Transcendentals require commensuration between curves and straight >> lines to be pointed out on curves. By the phrase accommodated on a >> straight line I basically mean a number must lie between points in >> space because points in space define straight line segments. >> Transcendentals don't lie between points in space. They lie on curves >> and not between points. >Does this mean that you think curves don't have endpoints? Surely those are arcs? > That's odd, >since I'm sure someone taught you that between any two points you draw >as many lines as you want, but only one them is a so-called straight >line - the one whose measure of the distance between those points is >smallest. I guess. >Or do you believe that curves aren't lines? What are they, then? Who knows. They're probably just some of Bob's little dots. >It's not clear whether the construction is limited to straight edge and >compass, or whether you permit the abstract construction possible in >Cartesian co-ordinate space and the associated arithmetic. >> I would expect reduction of the latter to the former. We don't need to >> walk around with a straight edge and compass to count and calculate >> but we do need to understand the reduction of numeric concepts and >> their implications as matters of science in general. >Reduction to what? Geometry? I wish you'd be more explicit. Well reduction to anything except outright assumptions would be nice for a change but in a pinch I'd take geometry. === Subject: Re: Epistemology 201: The Science of Science > Transcendentals don't lie between points in space. They lie on curves > and not between points. A correspondence between the real field and the euclidean straigh line (the indefinite line with out end points) can be established. The reals exist independent of any geometric entity but a correspondence can be established. The transcendental numbers lie on the line along with algebraic irrationals and rationals. All one big linearly ordered family. Bob Kolker === Subject: Re: Epistemology 201: The Science of Science >> Transcendentals don't lie between points in space. They lie on curves >> and not between points. >A correspondence between the real field and the euclidean straigh line >(the indefinite line with out end points) can be established. The reals >exist independent of any geometric entity but a correspondence can be >established. The transcendental numbers lie on the line along with >algebraic irrationals and rationals. All one big linearly ordered family. If you say so, Bob. Perhaps you could point a transcendental out on a straight line? === Subject: Re: Epistemology 201: The Science of Science > Do you admit there is no real number line? Certainly transcendentals > cannot be accommodated between points on straight lines or on any one > curve. So why haven't counting animals developed real number theory? Because, except for humans, they are too stupid to do so. Read what I said. Real analysis can be developed independently of geometry. Geometry is used because it is a really effective intuition pump, but is not logically necessary. Bob Kolker === Subject: Re: Epistemology 201: The Science of Science >> Do you admit there is no real number line? Certainly transcendentals >> cannot be accommodated between points on straight lines or on any one >> curve. So why haven't counting animals developed real number theory? >Because, except for humans, they are too stupid to do so. That's funny because they're smart enough to count but too stupid to develop real number theory. And apparently like some mathematikers they're also too stupid to answer questions. Probably why those same animals don't develop science. >Read what I said. Real analysis can be developed independently of >geometry. Geometry is used because it is a really effective intuition >pump, but is not logically necessary. Neither are you, Bob. === Subject: Re: Epistemology 201: The Science of Science > That is exactly what a physical theory is. A useful fiction. It is a > just so story with the virtue of being backed up by experiment or > observation. The useful fictions of physics lead to useful real things > of a technological nature. > Nothing that Man proposes hypothetically is any more than a just-so > story. There is no truth in bypothetics, only potentially falsifiable > quantitative assertions. Yes, Bob. That is correct as far as it goes. But some fictions predict and explain better than other fictions as the history of science makes clear. If you get emotionally and mentally stuck on a bad fiction, then although the yield in terms of technology may *seem* like a cornucopia, in fact, you may be missing out on other, even better technologies, not predicted by the commonly accepted fiction. > If you want useful fictions get a degree in physics. How would a degree in the current fiction better prepare one for a new and better fiction? -- I consider it quite possible that physics cannot be based on the field concept, i. e., on continuous structures. In that case nothing remains of my entire castle in the air, gravitation theory included, [and of] the rest of modern physics. -- Albert Einstein in a 1954 letter to Michele Besso. === Subject: Re: Triangular numbers, speculation > There seem to be quite a few of these universal quadratics in three > variables. Write T for the set of triangular numbers and S for the set of > squares. All these sets cover N, at least up to 15000: > T+T+T (conjectured by Fermat, proved by Gauss) > T+T+2T > T+T+4T > T+2T+2T > T+2T+3T > T+2T+4T > T+T+2S > T+T+4S > T+2T+2S > T+2T+3S > T+2T+4S > T+3T+S For quadratic forms, there is a neat theorem of Conway and Schneeberger. MR1803358 (2001m:11049) Conway, J. H.(1-PRIN) Universal quadratic forms and the fifteen theorem. (English. English summary) Quadratic forms and their applications (Dublin, 1999), 23--26, Contemp. Math., 272, Amer. Math. Soc., Providence, RI, 2000. 11E25 (11E20 11H55) æ Consider a positive definite quadratic form with coefficients in ${Bbb Z}$. The author calls such a form integer-valued. If it can be represented by a symmetric integer-valued matrix, then it is called integer-matrix. In this short note, the author gives a nice historical survey on the classification problem for universal integer-valued [resp. integer-matrix] positive definite forms, where universal means that these forms represent all positive integers. It is known that forms in at most three variables cannot be universal, but, for example, a sum of four squares is universal by Lagrange's theorem. A natural question is to ask whether there exists a smallest value $c$ [resp. $C$] such that whenever an integer-matrix [resp. integer-valued] form represents all positive integers $leq c$ [resp. $leq C$], then it is universal. The fifteen theorem states that $c$ exists, and that in fact $c=15$. This was proved in 1993 by the author and Schneeberger, using some computer calculations by Simons, but never published. A new and simplified proof has been given by M. Bhargava ref[in Quadratic forms and their applications (Dublin, 1999), 27--37, Contemp. Math., 272, Amer. Math. Soc., Providence, RI, 2000; MR1803359 (2001m:11050); see the following review]. As for $C$, the author mentions also that conjecturally $C=290$. {For the entire collection see MR1803355 (2001g:11003).} Reviewed by Detlev W. Hoffmann I'm not quite sure how this relates to this thread, as, e.g., T + T + T is a quadratic function but not a form. Here's the review of the Bhargava paper: MR1803359 (2001m:11050) Bhargava, Manjul(1-PRIN) On the Conway-Schneeberger fifteen theorem. (English. English summary) Quadratic forms and their applications (Dublin, 1999), 27--37, Contemp. Math., 272, Amer. Math. Soc., Providence, RI, 2000. 11E25 (11E20 11H55) The fifteen theorem of J. H. Conway and W. A. Schneeberger (1993) states that if a positive definite quadratic form whose associated symmetric matrix has integer values represents all positive integers $leq 15$, then it is universal, i.e. it represents all positive integers. The original proof, which needed some computer calculations, has never been published. In the present paper, the author presents a simplified proof of this theorem (which still needs the help of a computer due to some of the calculations involved). The main ingredient is the use of so-called escalator lattices. If $L$ is a nonuniversal lattice of rank $n$ (corresponding to a quadratic form as above), say with $a>0$ being the smallest integer not represented by the form, then an escalation of $L$ is any lattice of rank $n+1$ containing $L$ and generated by $L$ and a vector of norm $a$. An escalator lattice is one obtained by a succession of escalations of the lattice of rank $0$. The proof of the fifteen theorem can then essentially be reduced to the determination of escalator lattices and the verification of the fifteen theorem for these lattices, as every universal lattice has to contain a universal escalator lattice as sublattice. It turns out that universal escalator lattices are always of rank $4$ or $5$, and that there are none of rank $geq 6$, and that the total number of escalator lattices (of rank $leq 5$) is 1850. A study of these lattices leads to various variations of the main theorem, for example that universality is already implied if the form represents the values $1$, $2$, $3$, $5$, $6$, $7$, $10$, $14$, $15$. It also leads to a classification of universal quaternary forms (there are $204$ of them). the same volume ref[in Quadratic forms and their applications (Dublin, 1999), 23--26, Contemp. Math., 272, Amer. Math. Soc., Providence, RI, 2000; MR1803358 (2001m:11049); see the preceding review]. {For the entire collection see MR1803355 (2001g:11003).} Reviewed by Detlev W. Hoffmann -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Triangular numbers, speculation >>For integers w>=0 write T(w) = (ww+w)/2. >>Is every positive integer n a value of >>T(x) + 2T(y) +3T(z)? >>How about >>T(x) + T(y) +2T(z)? >>Both statements are true at least for n <= 15000. >> You have 3 degrees of freedom in your sums, and your polynomial is only >> degree 2. Therefore, the number of potential sums in [0,n] approaches >> some_constant * n^(3/2) as n goes to oo. >> So if there are holes, they are likely to be at the low end, and you've >> checked that already. Your speculation is probably correct. >Of course, this argument also applies to x^2 + y^2 + z^2 - but >the holes don't stay at the low end. But they do start at the low end: n=7 as Paul mentioned. Maybe I should rephrase -- ...if there are holes, some are likely to be at the low end. But it's all moot as Gauss has proved there are none, and his proof is better than my probability. --Keith Lewis klewis {at} mitre.org The above may not (yet) represent the opinions of my employer. === Subject: Projected values, annuities, payments... I've got a headache coming on I have someone I am trying to help out, but have taxed my knowledge level. This is long, so forgive me in advance. I have an Access database that I am working with to do projections of income, etc. For any given user, I have DOB, salary, COLA, interest on investments, current value, and current % contributed. Using forms, I am able to do a loop for the number of years required to get the persons salary, value, interest, balance, etc for a set year (50, 55, 60, etc.) What this does not allow is for me to run a report of all people in the system to show worth. So, into a query I go. Using this, I can calculate what their estimated salary will be at ageX using salary *(1+COLA) ^ number of years. But thats where I hit the wall. Is there even such a formula that will allow me to get 1: salary at age X 2: total payments made using 12.9% of salary 3. show yearly investment returns of X% (assume 8 for example) 4. current value at age x If I take the estimated salary at age X and multiply it by the 12.9%, should that value + starting value, and than adding the 8% of this total, get me a close number, or is that still missing the fact that the 8% is compounding over these years, adding to the yearly value. OK... no I am getting more confused, and need an aspirin. I hope there is enough info here for SOME help. Any direction would be greatly appreciated. Jeff === Subject: Re: [XPOST] A unique number for every person - can it be done? > The problem can be summarized in one sentence: > Calculate a number for every human being, company and organization on > earth, that is guaranteed to be unique till the end of time. I finally thought of a way of doing this, that should be good enough for your purposes. Step 1: Roll 128 ordinary eight-sided dice, as in the example displayed at http://uk.geocities.com/infobahn@btinternet.com/uniqnum.jpg Step 2: reading 8 as 0, write down the result as a 128-digit octal number. Step 3: convert to binary. This gives you a 384-bit random number, R. Step 4: Encode your surname (or a phonetic transliteration thereof) in ASCII. Do not use the @ symbol in your encoding. Call this encoding C. Step 5: Create the bitstring @C@R@C@ (note that R is exactly 384 bits, so this encoding is reversible, should that be necessary). Step 6: Base-64-encode this bitstring. This is your unique identifier. Step 7: Put the base 64 encoding in your sig block. I invite the statisticians amongst you to calculate the chances of a collision just for the 384-bit number (Birthday Paradox applies, of course). The purpose of encoding the surname is of course to make assurance of non-collision doubly sure. (Note: Usenet is *not* a central database or registry. Also note that surnames are generally not considered private data.) If you forget your unique identifier at any time, simply search any good Usenet archive. Tada! === Subject: Re: [XPOST] A unique number for every person - can it be done? >Also note that >surnames are generally not considered private data. Good thing, too, since some people don't have them. Greg. -- Greg Rose 232B EC8F 44C6 C853 D68F E107 E6BF CD2F 1081 A37C Qualcomm Australia: http://www.qualcomm.com.au === Subject: Re: [XPOST] A unique number for every person - can it be done? > Calculate a number for every human being, company and organization on > earth, that is guaranteed to be unique till the end of time. Your thoughts seem unecessarily complex. > Be creative, try to find data useful for the purpose. Things you may > want to use: > - Place of birth (consider not always known, names can change over time, > better go for coordinates) > - Name (First, Last / Name of company/organization) > - Name of parents (consider orphans / companies) > - Blood Type (consider companies have no blood type) > - Gender (consider companies) > - Eye color (should be constant, consider companies) That would be very limiting. Anyone who changed name, perhaps for marriage, would need a new number so their original number fails the test of being their number until the end of time. > I came up with plenty of ideas, but they were either too complicated or > creating collisions was too likely, that are not easily resolved. > Simply writing down some data and hashing it creates a decent number, > but how long will this be collision free? How big must the hash be to be > secure for thousands of years and 6 billions of people and millions of > companies/organizations? if the largest number you need is say 7 billion, then simply give everyone a number between 1 and 1000 000 000 000 000 and the problem is resolved. Should 'future history' show that the human race lasts longer than that, the upper limit is not really a limit and can increase to any number imaginable. There is no need for an individual or organisation to compute its number ... it would be in the planetary records office, or tattooed on one's arm as it was when the idea last came up. -- Gianna Stefani === Subject: Re: [XPOST] A unique number for every person - can it be done? > if the largest number you need is say 7 billion, then simply give > everyone a number between 1 and 1000 000 000 000 000 and the problem is > resolved. Should 'future history' show that the human race lasts longer > than that, the upper limit is not really a limit and can increase to > any number imaginable. And how get people their number assigned? > There is no need for an individual or organisation to compute its number > ... it would be in the planetary records office, or tattooed on one's > arm as it was when the idea last came up. Office means centralized control and this office will have a centralized database. Both was forbidden by the conceptual formulation. -- TGOS === Subject: Re: [XPOST] A unique number for every person - can it be done? <39327vF5s2j4aU1@individual.net> <1110246812.ffd699c09bb9a78fecf49771ef07a7db@meganetnews2> In message <1110246812.ffd699c09bb9a78fecf49771ef07a7db@meganetnews2>, >> if the largest number you need is say 7 billion, then simply give >> everyone a number between 1 and 1000 000 000 000 000 and the problem is >> resolved. Should 'future history' show that the human race lasts longer >> than that, the upper limit is not really a limit and can increase to >> any number imaginable. >And how get people their number assigned? >> There is no need for an individual or organisation to compute its number >> ... it would be in the planetary records office, or tattooed on one's >> arm as it was when the idea last came up. The Nazi's had this idea - but I don't think they would have been too bothered about duplicates since duplicates could be easily disposed. >Office means centralized control and this office will have a >centralized database. Both was forbidden by the conceptual >formulation. You could always implant an ethernet card into their forehead (unique MAC address). -- Jeremy Boden === Subject: Re: [XPOST] A unique number for every person - can it be done? > You could always implant an ethernet card into their forehead (unique > MAC address). And how do vendors get their MAC ID (24 BIT) assigned? Through centralized control. So again, won't work. -- TGOS === Subject: Re: [XPOST] A unique number for every person - can it be done? > Office means centralized control and this office will have a > centralized database. Both was forbidden by the conceptual > formulation. One of the first tasks when accepting a job is to debug the requirements. Centralised database is easy. Andrew Swallow === Subject: Re: [XPOST] A unique number for every person - can it be done? You said: No, I want to give anybody on the world the chance to get a unique id. I don't plan to assign it to people. They will assign it to themselves or they won't. I just try to set up the rules how he does so, to be sure, he really get's a unique one (and not one used already by someone else). And you said: 1) Every person, real person or corporate body (like company or organization), needs a number. Every means every one world wide. And now I'm confused. What on earth are you talking about? === Subject: Re: [XPOST] A unique number for every person - can it be done? > You said: > No, I want to give anybody on the world the chance to get a unique id. I > don't plan to assign it to people. They will assign it to themselves or > they won't. I just try to set up the rules how he does so, to be sure, > he really get's a unique one (and not one used already by someone else). > And you said: > 1) Every person, real person or corporate body (like company or > organization), needs a number. Every means every one world wide. > And now I'm confused. Why? Both statements don't contradict each other. > What on earth are you talking about? About a formular that allows every body in the world, who needs a unique number, that identifies him/her/it as a person or corporate body in a guaranteed unique way, for whatever reason, to get one. And that without central organization or control. Right now this is done by mail addresses or Internet domains and the so on. But these are either not constant (tomorrow someone else may own my mail address or domain) or they are centralized controlled (meaning someone in power can refuse you to get such an id and discriminate you as a person or organization) or both and often you even have to pay to get one. It can be used as a unique address in networks, as an identifier in instant messengers, as a replacement for mail addresses or phone numbers, as a producer code in commercial production (like the MAC address of NICs), as a namespace for OO classes (like Java now uses domains), as a way to create uniquely named directories; I can think of 100 cases where such an ID would be a great thing. -- TGOS === Subject: Re: [XPOST] A unique number for every person - can it be done? > You said: > > No, I want to give anybody on the world the chance to get a unique id. I > don't plan to assign it to people. They will assign it to themselves or > they won't. I just try to set up the rules how he does so, to be sure, > he really get's a unique one (and not one used already by someone else). > > And you said: > > 1) Every person, real person or corporate body (like company or > organization), needs a number. Every means every one world wide. > > And now I'm confused. > Why? Both statements don't contradict each other. > What on earth are you talking about? > About a formular that allows every body in the world, who needs a > unique number, that identifies him/her/it as a person or corporate body > in a guaranteed unique way, for whatever reason, to get one. And that > without central organization or control. Easy. Just append your geospace coords to the Julian date/time, making sure no-one else within a few feet of you is doing the same thing at the same time. ----snip---- === Subject: Re: [XPOST] A unique number for every person - can it be done? comp.programming: > Easy. Just append your geospace coords to the Julian date/time, > making sure no-one else within a few feet of you is doing the > same thing at the same time. Not too bad. But Julian date/time is very inaccurate. Counting seconds since 1970-01-01 is much better. The problem is rather not the date/time, but the geospace coords. Are geospace coords really the best coordinate system for earth we can think of? And how would you find out your geospace coords? Also the whole system is not reproducible. Unless you remember the exact time and location when you first calculated your number, you can't get it back in case you have lost it. -- TGOS === Subject: need help with your job search? Hello! Working hard? Looking for a job? Want to advance your career? I am a young professional based in Ottawa who want to help you out with your own job search. Low cost, affordable for all. Confidential. Services in French and English. Write now for more info. We can make it together! First contact by e-mail. ajobforme2005@hotmail.com === Subject: Nonnegative measurable function question... I can't figure out whether or not I have solved this problem...maybe someone can help? Let f be a non-negative Lebesgue integrable function. Prove that the function now, I will denote F(x) as int_(-oo,x) f for simplicity of notation. My proof : Since f is non-negative measurable, there exists f_n that are simple functions, increasing, and vanish off a set of finite measure, such that lim f_n = f. Thus, want to show for all eps > 0 there exists delta > 0 s.t. |x-y| < delta ==> |F(x) - F(y)| < eps Now, consider : Assume WLOG x < y. Then, |F(x) - F(y)| = | int_(-oo,x) f - int_(-oo,y) f | = | int_(-oo,x) lim f_n - (-oo,y) lim f_n | = | lim int_(-oo,x) f_n - lim int_(-oo,y) f_n| (by Monotone convergence theorem) = | lim int_(x,y) f_n | Now, I want to bound int_(x,y) f_n I want to say int_(x,y) f_n <= |x-y| M where M is the sup of f_n. f_n is a simple function so it has a sup. But, I am now worried because the n in f_n is variable. So I don't think what I am saying is correct. It is true that if I pick a specific n, say n = 1, then int_(x,y) f_1 is certainly bounded by |x-y| M for some M. But, in my whole calculation, I ended with |lim int_(x,y) f_n | Can I bound this by something? I am having trouble with a very small thing here... I'm just not quite sure if I can bound that whole thing by |x-y| M for some M. Am I reasonable in misunderstanding this or should this be totally trivial? James === Subject: Re: Nonnegative measurable function question... > I can't figure out whether or not I have solved this problem...maybe someone > can help? > Let f be a non-negative Lebesgue integrable function. Prove that the > function > F(x) = integral (from negative infinity to x) of f is continuous. > now, I will denote F(x) as int_(-oo,x) f for simplicity of notation. That's a cute problem, and a useful result: integration makes functions nicer. I have another suggestion. Think of F(x) = int_R 1_(-oo, x) f i.e. the integral over the whole real line of f times the characteristic function of (-oo,x). Now take any sequence x_n converging to x. It is enough to show F(x_n) -> F(x). That is, you have to pass the limit through the integral... what allows you to do this? === Subject: Re: Nonnegative measurable function question... > I can't figure out whether or not I have solved this problem...maybe someone > can help? If you can't figure that out, then you haven't. > Let f be a non-negative Lebesgue integrable function. Prove that the > function > now, I will denote F(x) as int_(-oo,x) f for simplicity of notation. > My proof : > Since f is non-negative measurable, there exists f_n that are simple > functions, increasing, and vanish off a set of finite measure, such that lim > f_n = f. > Thus, want to show for all eps > 0 there exists delta > 0 s.t. |x-y| < delta > ==> |F(x) - F(y)| < eps It looks like you are trying to show F is uniformly continuous. In fact this is true, but be aware of the distinction. > Now, consider : > Assume WLOG x < y. Then, > |F(x) - F(y)| = | int_(-oo,x) f - int_(-oo,y) f | = | int_(-oo,x) lim f_n - > (-oo,y) lim f_n | = | lim int_(-oo,x) f_n - lim int_(-oo,y) f_n| > (by Monotone convergence theorem) > = | lim int_(x,y) f_n | > Now, I want to bound int_(x,y) f_n > I want to say int_(x,y) f_n <= |x-y| M where M is the sup of f_n. f_n is a > simple function so it has a sup. But, I am now worried because the n in > f_n is variable. Which is a good thing to worry about. Better to start this way: Let eps > 0. Then there exists a simple function s, 0 <= s <= f, such that int_(-oo,oo) (f-s) < eps/2 ... === Subject: Module question.. Hello : I thought of this question today and couldn't show it : If M is a left R-module, and f : M ---> M is a surjective R-module homomorphism from M to itself, does this mean that M is injective? It is clear that if M is finite, then f is injective. What about in general? James === Subject: Re: Module question.. > If M is a left R-module, and f : M ---> M > is a surjective R-module homomorphism from M to itself, does this mean that > M is injective? It is clear that if M is finite, then f is injective. What > about in general? Your condition on finiteness of M can be relaxed to that M is Noetherian. (Consider the kernels of f^n, n e N). J. === Subject: Re: Module question.. Hi: A free abelian group of infinite rank is NOT Hopfian, meaning: it is isomorphic to a non-trivial quotient group of itself. Since any abelian group is a Z-module, you have here a counterexample. In your case, I think that not only M being finite you'll get that any epimorphism is actually an automorphism, but also if M is finitely generated...check this. Good luck! Tonio === Subject: Re: Module question.. Another interesting counterexample to the OP's question can be constructed for certain rings. Let V be a vector space over a field K such that V is of a countably infinite dimension over K. Let then R be the ring of K-linear endomorphisms of V. It is well known that the R-modules R and R^2 are then isomorphic (basically because V is isomorphic to Voplus V as a K-space). This allows us to construct a non-injective epimorphism of R-modules from R^2 to itself (first project to the first component and then map that isomorphically to the whole thing). Note BTW that in this case the module R^2 is clearly finitely generated, so that is not a sufficient condition. Jyrki === Subject: Gauge Integral and Henstock Lemma I'm having difficulty with Henstock's Lemma for gauge integrals as presented in Introduction to Real Analysis, DePree and Swartz. (My notation: [J],S'f will mean the integral of function f over the interval J; the notation [i=1->n]SGM t_i means the summation of t_i for i from 1 to n). Also, L(J) is the length of the interval J, and S(f,D) is the Riemann sum over the tagged interval D. The Lemma says: With f gauge integrable and for some gamma(x) such that if D is gamma-fine, |S(f,D) - [I],S'f | < eps J is a partial tagged division of I, that is, J = {(x_1, J_1), ... , (x_n, J_n)} and the J_i are pairwise disjoint except for endpoints (their union is not necessarily equal to I). Also each subinterval J_i is still gamma fine (i.e., J_i in gamma(x_i)). Then PART 1: | [i=1->n]SGM {f(x_i) L(J_i) - [J_i],S'f } | =< eps, and PART 2: [i=1->n]SGM |{f(x_i) L(J_i) - [J_i],S'f | =< 2eps I think when you go from Part 1 to Part 2, you may have to change the gamma function and make it finer if you want to achieve 2eps. (I am told I am wrong.) I think the following is an example demonstrating a need to change the gamma function: With f(x) = x, consider a partial tagged division that covers the whole interval [-12,12] (it's integral over [-12,12] is of course zero). We achieve an eps = 7 in the calculation of Part 1 of the Lemma if we use gamma(x) = (x-3.1, x+1.1) for x>=0 (x-0.6, x+3.6) for x<0 because the coarsest, worst result (closest to eps) is achieved with six uniform divisions, each 4 units wide, and tags at x = -11.5, -7.5, -3.5, 3, 7, 11. (I understand I have not proved this assertion, but I think it's non-controversial and not the cause of my confusion - famous last words). The calculation specified by Part 1 then yields |6|eps=7). I have some consolation in noticing that I hadn't proved worstness but still committed the sin of slothful reliance on intuition. Tom Adams === Subject: Re: Math Editor Recommendations > Looking for Win XP compatible recommendations, under $40. There is extensive online help for use of LaTeX and direct links to free downloads at http://www.artofproblemsolving.com/LaTeX/AoPS_L_About.php Many twelve-year-olds are using the resources there to learn LaTeX. Hope this helps! -- Karl M. Bunday P.O. Box 1456, Minnetonka MN 55345 Learn in Freedom (TM) http://learninfreedom.org/ remove .de to email === Subject: re:Math Editor Recommendations > Every time I've tried to install > Tex, I am unable to get it running and working Which distribution? Not Miktex I suppose. Which OS are you using? > I can > type all the commands, which is far faster than mousing to the icons > every other step. If this is your opinion, well you'd love Latex! Give it another try. I'm writing my Master Thesis with Latex and I'd be glad to answer your questinos and help spread Latex around the world! (if you use Windows, I know nothing about Linux, Mac, Solaris, etc). If you need support by very experienced person, you can visit your country's TUG (Tex Useres Group). Bye bye![/quote] ---------------------------------------------------------- ** SPEED ** RETENTION ** COMPLETION ** ANONYMITY ** ---------------------------------------------------------- http://www.usenet.com === Subject: Re: Math Editor Recommendations > Looking for Win XP compatible recommendations, under $40. Current > candidates are Lyx and Equation Maker (EM) v1.00. Dislike Lyx's > learning curve + complexity re Win instead of Linux. EM seems > incomplete (e.g. symbols for Reals, Integers [Q, Z]). I like EM's > ability to export to RTF, since my Word 97 can handle it. > Plan (for now) to only use generated documents for my private use on > screen, with no printing (currently) anticipated. Not interested in > graphing or calculations. Simply want online notebook for my math > hobby. I use OpenOffice almost exclusively. Every time I've tried to install Tex, I am unable to get it running and working. OpenOffice is a simple, crossplatform install. I prefer its math editor to Word because I can type all the commands, which is far faster than mousing to the icons every other step. === Subject: Re: Math Editor Recommendations > .... OpenOffice is a simple, > crossplatform install. I prefer its math editor to Word because I can > type all the commands, which is far faster than mousing to the icons > every other step. Is it well-known that a lot of straightforward algebra in word-processors such as Word doesn't need the equation editor at all? Unless you're introducing a fraction or matrix or integral every other step, you can do most algebra from the keyboard. For example, I'd always use the keyboard for a polynomial. Some of the keyboard commands for Word on a Mac (which I believe are pretty similar on a PC) are: option - for a reasonably long minus sign, command i for starting or stopping italics, command b for starting or stopping bold, command = for subscripts, command shift + for superscripts, control space-bar for escaping from such fancy things into plain type. (There are probably others which I use without thinking, and can't remember right now.) Ken Pledger. === Subject: Re: Math Editor Recommendations >> Looking for Win XP compatible recommendations, under $40. Current >> candidates are Lyx and Equation Maker (EM) v1.00. Dislike Lyx's >> learning curve + complexity re Win instead of Linux. EM seems >> incomplete (e.g. symbols for Reals, Integers [Q, Z]). I like EM's >> ability to export to RTF, since my Word 97 can handle it. >> Plan (for now) to only use generated documents for my private use on >> screen, with no printing (currently) anticipated. Not interested in >> graphing or calculations. Simply want online notebook for my math >> hobby. > I use OpenOffice almost exclusively. Every time I've tried to install > Tex, I am unable to get it running and working. OpenOffice is a simple, > crossplatform install. I prefer its math editor to Word because I can > type all the commands, which is far faster than mousing to the icons > every other step. Try texmacs at http://www.texmacs.org , the windows port is still alpha code but it is workable for me, by the way texmacs works also with a CAS package like axiom. Marc === Subject: Re: Math Editor Recommendations >Looking for Win XP compatible recommendations, under $40. Current >candidates are Lyx and Equation Maker (EM) v1.00. Dislike Lyx's >learning curve + complexity re Win instead of Linux. EM seems >incomplete (e.g. symbols for Reals, Integers [Q, Z]). I like EM's >ability to export to RTF, since my Word 97 can handle it. >Plan (for now) to only use generated documents for my private use on >screen, with no printing (currently) anticipated. Not interested in >graphing or calculations. Simply want online notebook for my math >hobby. >>I use OpenOffice almost exclusively. Every time I've tried to install >>Tex, I am unable to get it running and working. OpenOffice is a simple, >>crossplatform install. I prefer its math editor to Word because I can >>type all the commands, which is far faster than mousing to the icons >>every other step. > Try texmacs at http://www.texmacs.org , the windows port is still alpha code > but it is workable for me, by the way texmacs works also with a CAS package > like axiom. === Subject: Fuller Quickyl by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) with ESMTP id j284iPm31300 HOME <EM>| </EM>ABOUT US<EM> | </EM>SCIENTI<SPAN>FIC</SPAN> APP<EM>ROA</EM>CH <U>|</U> FAQ's | H<U>O</U>LLYWOOD SECR<I>ET</I>S |<B> </B>Privac<STRONG>y </STRONG>Policy<SPAN> an</SPAN>d Terms<U> </U>of Use

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=== Subject: Re: An exact 1-D summation challenge - 3 > GAE> Why ask Maple experts? > I still have some hope that persons like Alec Mihailovs, > Carl DeVore, or Joe Riel could force Maple to get rid > off those derivatives via an amazing trick they knew > a lot of and share with us so generously. > Let us wait and see still. Let n=0..infinity). Theorem. / /2 BesselI(0, x/2) Bondarenko(0,x) = |- 2/x + 1/2 exp(x/2) |----------------- x - (BesselI(0, x/2) + BesselI(1, x/2)) (ln(x) + gamma) 1/2 - BesselK(0, x/2) + BesselK(1, x/2)|| Pi // Corollary. Bondarenko(0,1) = (-2 + 1/2 exp(1/2) (2 BesselI(0, 1/2) - (BesselI(0, 1/2) + BesselI(1, 1/2)) gamma 1/2 - BesselK(0, 1/2) + BesselK(1, 1/2))) Pi = -0.4819628874318124580052940149095710946413914103217270690844145 7173661248044887351687347222620085333811074881980054065803 3577697969382521731895531855432170189175060403178091427218 1990040166615845008538108203821209452257764057684304970790 8965942152199060916283547633870315698120364374356662641450 9822649883972547585705110152505497991186354731145916721919 2288913473511951798866254282966345200748570156113855352857 2666907654061059022591515297855874922906802026704049631515 1782434382037157549725430825675237623765207501925128112659 5826011189803310624438027554437440248396800652448655498502 9505299014133578926931580684739653653078893877269467284583 4798970721345403584812817468735516171454949814401545241938 6655484117974010559288546305416419263508228399021920813276 4761154079440591622468664806008551358983064546321241754197 4799559222760359637037482303235622512875902417041404949782 9864353102140533397706550371851790021227600957708502930572 0576434306805039626077554153789844453292440429561113046392 77637551356 Alec Mihailovs http://math.tntech.edu/alec/ === Subject: Re: proving bicondtionals without cases? >ok, but don't you think that the fact that the case-by-case proofs are >harder > I don't think that's a fact. sorry, that should say the case-by-case proofs *aren't* harder > in some cases shows that our current system doesn't allow for >the most elegant proofs possible? > No. First, because elegant is a subjective term. We may very well > disagree on what is elegant and what is not (just as we disagree on > what is hard and what is not). Second, because we are talking about > our personal ability (or lack thereof) in finding a particular kind of > proof, not about whether or not such a proof even exists. Our ability > or lack thereof need not be a property of the system (current or > otherwise). no, what i'm saying is that i can't tell the difference between my own inability to find a particular kind of proof and the actual impossibility of that kind of proof... in the example with the subspaces, i'd like to find a 'simpler' proof that avoids cases by proceding by reversible steps--but having already spent some time looking for it and having not found it, i began to realize that i shouldn't assume that such a form of proof is possible--however this seems very strange, because EVERY biconditional proof before this one in my textbook could be done very simply using reversible steps without cases--at the very least i want to understand what makes this proof different--why has my method failed me all of a sudden? (p.s. my method is to write out the desired statement and work backwards by means of reversible steps until i reach an obvious tautology--it was starting to seem like a helpful process that i could apply to any proof whatsoever, especially because it seemed to explain the thought process along with the proof, which i desire (see for a discussion of the matter--i feel strongly that a proof should reflect the thought process behind it) > I truly don't understand what you are going for. i took some time to think about it, and i hope i've made it clearer above... i either want to figure out a way to make my proof method work in these 'trickier' proofs, OR i want to understand what makes these cases unsuitable for my method... and additionally, i'm interested in the idea of formulating a condition (biconditional completeness) which will guarantee that my proof method will work for all theorems in a system--then maybe i could use this condition to see if i am guaranteed to find a reversible proof if i look hard enough, or otherwise to show that sometimes such a proof is impossible... do you see my line of thinking now?? > -- > It's not denial. I'm just very selective about > what I accept as reality. > --- Calvin (Calvin and Hobbes) > Arturo Magidin > magidin@math.berkeley.edu === Subject: Re: proving bicondtionals without cases? one correction, i don't always work to a tautology.. if i have to prove p<=>q then i do: p <=> r1 <=> r2 ... <=> q however with something like f(x)=g(x): f(x)=g(x) <=> q(x)=r(x) (i.e. any equivalent expression) <=>... <=> x=x (or some other tautology) <=> T anyway, i think you get the idea--when the statement is symmetric, i want my proof to be symmetrical. === Subject: Re: proving bicondtionals without cases? days. My association with the Department is that of an alumnus. >one correction, i don't always work to a tautology.. >if i have to prove p<=>q then i do: ><=> r1 ><=> r2 >... ><=> q >however with something like f(x)=g(x): >f(x)=g(x) ><=> q(x)=r(x) (i.e. any equivalent expression) ><=>... ><=> x=x (or some other tautology) ><=> T >anyway, i think you get the idea--when the statement is symmetric, i >want my proof to be symmetrical. Well, that's, I guess, a way to interpret symmetrical. I think you'll find that this is hard to do in general. It is a favored method of many students (probably derived from the way equations are solved early in one's mathematical career), but it is certainly frought with dangers (making a step which is not truly reversible is the most common misstep)... Something else you might not be realizing is that you actually have hidden in many of these proofs biconditionals that are proven not symmetrically but rather as double implications. For example, let's take a simple example. Say you want to prove a trigonometric identity, e.g. tan^2 x + 1 = sec^2 x If I understand what you are saying, you would proceed as follows: tan^2 x + 1 = sec^2 x is equivalent to / sin x 2 / 1 2 | -------- | + 1 = |------ | which is equivalent to cos x / cos x/ sin^2 x 1 -------- + 1 = -------- which is equivalent to cos^2 x cos^2 x sin^2 x cos^2 x 1 ------- + -------- = -------- which is equivalent to cos^2 x cos^2 x cos^2 x sin^2 x + cos^2 x 1 ------------------ = -------- which is equivalent to cos^2 x cos^2 x 1 1 ------------- = ------------ cos^2 x cos^2 x which is a tautology, therefore the original equation is true. (Let us for the moment assume we ignore the fact that the equations only hold at values where x is not an odd multiple of pi/2). Now, how do you know that all the steps are equivalent? To justify one step, for example, you would need to justify that For every x different from zero (x/x = 1) To justify that, you would start from x/x = 1, multiply both sides by both sides by x is a reversible operation when x is not zero? I.e., you would have to prove: For every x, y, z (if x is not zero, then y=z if and only if yx =zx). And so on... Are you sure that nowhere along the line was an equivalence that required you to do the two implications separately? Now, most of mathematics is not in the form of biconditionals, but in the form of simple implications. The vast majority of theorems are, in the end, statements of the from If X then Y (though Y itself may be a compound statement, like a biconditional). Most people tend to think in terms not of biconditionals (i.e., not in terms of going from one statement to an completely equivalent other statement) but rather in terms of implications (going from a premise to a conclusion). That is why biconditionals are seldom introduced as primitive logical connectives, but are almost always derived logical connectives (defined in terms of other connectives; normally implication and conjunction). In that respect, I think you will find that most of the time the proof that best mirrors our thought processes are the double implication proofs (i.e., to prove A<->B, assume A and prove B, then assume B and prove A); this is not the case early on, because in arithmetical problems we do indeed tend to proceed along the lines you describe. Of course, that could very well be a chicken-and-egg thing: do we think that way because we do proofs that way, or do we do proofs that way because we think that way? In general, of course, it is easier to prove a weaker or smaller statement with more hypothesis than a stronger or larger statement with fewer ones. That's why we normally find it much easier to deal with A->B, and then separately with B->A, than we do to deal with A<->B. The other reason you will probably find that you cannot always find symmetric proofs is that whereas many of our axioms are often symmetric in this sense, our rules of inference are not. For example, the most common rule of inference is Modus Ponens: If we know that A->B, and we know A, then we may conclude B. This rule is not really reversible, so you will often have trouble reversing a step in which you applied modus ponens. -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: does this serie converge ? can anybody explain whether serie converges or not? SUM(from 1 to infinity)[sin(n^2)]/n === Subject: Re: does this serie converge ? >can anybody explain whether serie converges or not? Maybe. I instinctively use the same trick used to evaluate sum( sin(n)/n ) : >SUM(from 1 to infinity)[sin(n^2)]/n Let F(n) = SUM( from 1 to n ) [sin(k^2)], so that F(0) = 0 and sin( n^2 ) = F(n) - F(n-1). Then the partial sums you're looking at are SUM(from 1 to N)[sin(n^2)]/n = SUM(from 1 to N)[F(n) - F(n-1)]/n = -F(0)/1 + SUM(from 1 to N-1) F(n)[1/n - 1/(n+1)] + F(N)/N So if you can show that the numbers F(n) stay bounded as n --> infinity, then first of all F(n)/n --> 0 , so that the limit of the partial sums is the same as SUM( from 1 to infinity ) F(n) [ 1/(n(n+1)) ] (including the statement that the original series converges iff this one does). But if the F(n) are indeed bounded in magnitude by some number K, then this series does indeed converge, and in fact this one converges absolutely, because the tails of the sequence are bounded by K/N. [Caution: I'm not suggesting the original sequence converges absolutely, and in fact I'm quite sure it does not.] Now, I'm not really sure that the sequence F(n) IS bounded. Since sin(z) = (1/(2i)) ( exp(iz) - exp(-iz) ) it would be sufficient to show that the numbers G(n) = SUM( from 1 to n ) [exp( i k^2)] stay bounded. This is the special case q = exp(i) of the more general sum G(n, q) = SUM( from 1 to n ) [ q^(k^2) ] or, if you like, the special case z=0 of the even more general sum G(n, q, z) = SUM( from 1 to n ) [ q^(k^2) cos( 2k z ) ] These latter are the partial sums in the q-expansion of one of the Jacobi theta functions; those sums converge for q inside the unit disk. I can't recall the behaviour as q approaches exp(i) on the unit circle, and of course there is also the question of whether we can interchange the limits q -> exp(i) and n -> infinity , but I can report some numerical experiments from Maple using its procedure JacobiTheta3(z,q) = 1+2*sum(q^(n^2)*cos((2*n)*z),n=1..infinity) It reports numerical values of JacobiTheta3(0, 0.9*exp(I)) = 0.1952084132 + 0.6614875207 I JacobiTheta3(0, 0.99*exp(I)) = 1.810568519 - 2.205581077 I JacobiTheta3(0, 0.999*exp(I)) = 9.534898119 - 6.442655196 I JacobiTheta3(0, 0.9999*exp(I)) = 18.42194663 - 2.255218119 I JacobiTheta3(0, 0.99999*exp(I)) = 8.383891712 - 0.5121231374 I from which I let you make your own conjectures about whether the sequence F(n) I first described is bounded or not. Interpreting the values of sin(k^2) as steps in a random walk, I would guess that the terms F(n) are _not_ bounded. We can still make some progress with a weaker statement such as F(n) = O( sqrt(n) ), about which I am more confident. I don't see how to _prove_ that, but maybe someone else does. dave (PS -- the corresponding argument for SUM [sin(n)/n] is easier because I can directly sum the series SUM [ q^k ] !) === Subject: Re: does this serie converge ? >Let F(n) = SUM( from 1 to n ) [sin(k^2)] >So if you can show that the numbers F(n) stay bounded as n --> infinity, >Now, I'm not really sure that the sequence F(n) IS bounded. Let me throw out another morsel or two. I mentioned in that post another sequence, now of complex numbers: G(n) = SUM( from 1 to n ) [exp( i k^2)] The F's are just the imaginary parts of the G's. I noted that the === problem referred to in the Subject: line could be resolved by finding an upper bound on G(n), e.g. knowing that G(n) = O( sqrt(n) ) would do. I'm very curious to know the rate of growth of |G(n)| . Now, I can evaluate G( 355 ) pretty quickly, using the fact that pi is very nearly 355/133. For if we write (113/355) pi = 1 + eps then G( 355 ) = sum exp( 113/355 k^2 pi i ) exp( - eps i k^2 ) = sum ( exp( 113/355 k^2 pi i ) + O ( eps k^2 ) ) = sum exp( 113/355 k^2 pi i ) + O ( eps 355^3 ) The first part of the last line is a Gauss sum. More generally, we can let H(a,b) = sum_{ k < b } exp( a/b k^2 pi i ) and THESE are things we can evaluate : assuming gcd(a,b) = 1 we have H(a,b) = 0 when b is even, and for odd b we have ( H(a,b) )^2 equal to either 1 or +- b , IIRC. In our particular case H(113,355) = -1, but in any case the magnitude of H(a,b) is no more than sqrt(b). In this way we can more generally bound the values of G( n ) : it is never larger in magnitude than sqrt( n ) + O( n^3 | pi - n/m | ) where we of course would choose m to minimize this expression (m ~ n / pi). So whenever n is the numerator of a good approximation for pi -- good meaning that | pi - n/m | < O( 1 / n^{5/2} ) --- then we have the desired bound on the value of G(n) : it is provably no larger than O( sqrt(n) ). Unfortunately for me this isn't nearly good enough to complete the task I set out to do. There's no guarantee that there are ANY approximations which are good by the metric I just introduced, e.g. there is no rational approximation that good for the Golden Ratio, phi. We can do better for pi, but there is a limit to how well we can do; for example, I believe it's known that there are no n's for which | pi - n/m | < 1/n^{42} . But even if we can, as I suspect, find infinitely many n's for which pi has such a good approximation by a fraction n/m, that would still not come close to proving what I have conjectured is true: these values of n would have to show up in the Continued Fraction expansion for pi, and so the successive values of n must be at least as large as the successive powers of phi . These values of n are sparse enough that there's no easy way to bound the growth of G(n) in the long gaps between these key values of n. dave === Subject: Re: does this serie converge ? >can anybody explain whether serie converges or not? >SUM(from 1 to infinity)[sin(n^2)]/n That's hard to prove either way. Treating sin(n^2) as a pseudorandom value on [-1,1] you can prove it's less than SUM[1/n], but that's non-convergent so it doesn't help. Given that the expected value of the terms is zero, I would guess that it does converge. --Keith Lewis klewis {at} mitre.org The above may not (yet) represent the opinions of my employer. === Subject: Re: does this serie converge ? Keith A. Lewis escribi.97: > 23:28:09 -0800: >> can anybody explain whether serie converges or not? >> SUM(from 1 to infinity)[sin(n^2)]/n > That's hard to prove either way. Treating sin(n^2) as a pseudorandom > value > on [-1,1] you can prove it's less than SUM[1/n], but that's > non-convergent > so it doesn't help. > Given that the expected value of the terms is zero, I would guess > that it > does converge. It aparently converges to something so 0.17 Sum(sin(n^2)/n, n, 1, k) = [0.1680778870, 0.1672724962, 0.1673772760, 0.1668799456, 0.1663724485, 0.1695512632, 0.1701257725, 0.1699277381, 0.1700637883, 0.1699884680] for k = 10000, 20000, ..., 100000 -- Ignacio Larrosa Ca.96estro A Coru.96a (Espa.96a) ilarrosaQUITARMAYUSCULAS@mundo-r.com === Subject: Re: does this serie converge ? It does converge, but a proof is somewhat subtle. You need to show that the sequence (1,4,9,16,....) mod pi is uniformly distributed What this shows is that does not get long 'runs' of successive (or nearly successive) values of n where sin(n^2) does not change sign. [and thus the sum would diverge by oscillation]. Abel's criterion is insufficient. === Subject: Re: does this serie converge ? > can anybody explain whether serie converges or not? > SUM(from 1 to infinity)[sin(n^2)]/n This series converges by Abel's criterion. -- Julien Santini === Subject: Re: does this serie converge ? >>can anybody explain whether serie converges or not? >>SUM(from 1 to infinity)[sin(n^2)]/n > This series converges by Abel's criterion. Abel's test says: let u(n) be terms of a convergent series, and let v(n) be terms of a monotone convergent sequence; then the sum of u(n)*v(n) is convergent. How do you apply it here? Jose Carlos Santos === Subject: Re: does this serie converge ? > Abel's test says: let u(n) be terms of a convergent series, and let v(n) > be terms of a monotone convergent sequence; then the sum of u(n)*v(n) is > convergent. How do you apply it here? Cannot, I read sin(n)/n, sorry. -- Julien Santini === Subject: Re: does this serie converge ? >>Abel's test says: let u(n) be terms of a convergent series, and let v(n) >>be terms of a monotone convergent sequence; then the sum of u(n)*v(n) is >>convergent. How do you apply it here? > Cannot, I read sin(n)/n, sorry. Plese note that the sum of sin(n) diverges. Jose Carlos Santos === Subject: For Fermat fans Show that any positive integer which is congruent to 36 mod 40 is the sum of four squares all of which are 9 mod 40. The first solver wins a copy of the complete works of JSH. === Subject: Re: For Fermat fans > Show that any positive integer which is congruent to 36 mod 40 is the sum of > four squares all of which are 9 mod 40. > The first solver wins a copy of the complete works of JSH. 36 = x^2 + y^2 + z^2 + t^2 (mod 40) x = y = z = t = 9 (mod 40) 4*81 = 4 (mod 4). Nope, need to read as 36 = x^2 + y^2 + z^2 + t^2 (mod 40) x^2 = y^2 = z^2 = t^2 = 9 (mod 40) x, y, z, t = 3,37 (mod 40) 76 = 36 (mod 40) 76 = x^2 + y^2 + z^2 + t^2 x, y, z, t = +-3, +-37, +-77 Nope, not possible. Whew, I don't win. === Subject: Re: For Fermat fans >> Show that any positive integer which is congruent to 36 mod 40 is the sum >> of four squares all of which are 9 mod 40. >> The first solver wins a copy of the complete works of JSH. > 36 = x^2 + y^2 + z^2 + t^2 (mod 40) > x = y = z = t = 9 (mod 40) > 4*81 = 4 (mod 4). Nope, need to read as > 36 = x^2 + y^2 + z^2 + t^2 (mod 40) > x^2 = y^2 = z^2 = t^2 = 9 (mod 40) > x, y, z, t = 3,37 (mod 40) ???? 7^2 = 49 = 9 (mod 40). Try again! NB 76 = 49 + 9 + 9 + 9. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Elegance is an algorithm Iain M. Banks, _The Algebraist_ === Subject: Re: For Fermat fans > Show that any positive integer which is congruent to 36 mod 40 is the sum of > four squares all of which are 9 mod 40. > The first solver wins a copy of the complete works of JSH. JSH? Are you kidding? LOL === Subject: fixed point and sequence of functions Hi I appreciate anyone could have some clues or show me some references (a) suppose f_n converges to f. f is integrable. f_n and f are Borel measure functions. What kind of properties, can we guarantee that f_n is integrable for large n? If not, any counterexample? (b1) If F (dim =d) has a unique fixed point, say x0, any condition to force Jacobian of F(x0) to be nonsingular ? (b2) If given x1, f(x1,x2) is strictly convex and given x12, f(x1,x2) is strictly convex. Let (x010,x20) be the unique fixed point of f. Any condition to force |D^2 f(x1,x2)| to be nonsingular? Suppose both F and f have nice properties. === Subject: Re: fixed point and sequence of functions Hello > Hi > I appreciate anyone could have some clues or show me some references > (a) suppose f_n converges to f. f is integrable. f_n and f are Borel > measure functions. What kind of properties, can we guarantee that f_n > is integrable for large n? If not, any counterexample? Counterexample: f_n = 1/n. The f_n are measurable, converge to 0, which is measurable and integrable. Though, the f_n are not integrable. === Subject: Re: fixed point and sequence of functions Can we find a counterexample for the case that all f_n and f > 0 and bounded. (and uniformly converge) I am thinking something like (reverse) Dominated Convergence Thm. Charles >Hello >> Hi >> I appreciate anyone could have some clues or show me some references >> (a) suppose f_n converges to f. f is integrable. f_n and f are Borel >> measure functions. What kind of properties, can we guarantee that f_n >> is integrable for large n? If not, any counterexample? >Counterexample: f_n = 1/n. The f_n are measurable, converge to 0, which is >measurable and integrable. Though, the f_n are not integrable. === Subject: Re: fixed point and sequence of functions > Can we find a counterexample for the case that all f_n and f > 0 and > bounded. (and uniformly converge) Let f(x) = 1/(1+x^2), f_n(x) = f(x) + 1/n. All functions are bounded and > 0 on R, f_n -> f uniformly on R, but no f_n is integrable. === Subject: Re: Ilya Shambat's poetry & translation webpage > Poetry: >UAPD officers were dispatched to the UA Health Sciences Library, >>1501 N. Campbell Ave., Friday evening after a student called >>saying that a man was masturbating near the computer terminals, >>reports stated. [...] >>Officers had previously warned Shambat against trespassing and this time >>arrested him for third-degree trespassing. >>He was transported to Pima County Jail where he was refused by Pre-Trial >>Services and booked. === Subject: topological question : support Hi there, can you help me in proving the following please? let alpha be a p-form and d the exterior derivative. Then * * * * * * * * * * supp (d alpha) subset supp(alpha) * * * * * * * * * * where supp (alpha) denotes the support of alpha i.e. closure {p such that alpha(p)=/=0}. Saem === Subject: Re: topological question : support > Hi there, > can you help me in proving the following please? > let alpha be a p-form and d the exterior derivative. > Then > * * * * * * * * * * > supp (d alpha) subset supp(alpha) > * * * * * * * * * * > where supp (alpha) denotes the support of alpha i.e. > closure {p such that alpha(p)=/=0}. > Saem Consider a position x in the complement of supp(alpha). Then not only is alpha(x) = 0, there is also a neighborhood U of x for which y in U ==> alpha(y) = 0. The derivative is defined locally: if two functions agree on an open neighborhood of x, then their derivatives agree at x. Thus, since the exterior derivative of alpha, d alpha, is formed from alpha by taking derivatives of alpha, it follows that at x, d alpha is the same as d 0. But the exterior derivative of 0 is itself 0. -- Dale. === Subject: Re: topological question : support Sorry. I hit return before I added the punch line: >> Hi there, >> can you help me in proving the following please? >> let alpha be a p-form and d the exterior derivative. >> Then >> * * * * * * * * * * >> supp (d alpha) subset supp(alpha) >> * * * * * * * * * * >> where supp (alpha) denotes the support of alpha i.e. closure {p such >> that alpha(p)=/=0}. >> Saem > Consider a position x in the complement of supp(alpha). Then > not only is alpha(x) = 0, there is also a neighborhood U of x > for which y in U ==> alpha(y) = 0. > The derivative is defined locally: if two functions agree on > an open neighborhood of x, then their derivatives agree at x. > Thus, since the exterior derivative of alpha, d alpha, is formed > from alpha by taking derivatives of alpha, it follows that at > x, d alpha is the same as d 0. But the exterior derivative of > 0 is itself 0. Thus, if x is not in supp( alpha ), it is not in supp( d alpha ). Equivalently, supp(d alpha) is a subset of supp( alpha ). Dale. === Subject: Difficult absolute value problem At least, it's a problem that *I* am finding difficult (read impossible)! p(x)=[abs(x-1)+x-2] / [abs(x-1)-3x] The questions read: (a) Sketch p (b) Express p(x) as a piecewise defined function without absolute values. By using the graphics calculator I have managed to sketch p! It seems that there is a vertical asymptote at x=0.25, a right hand horizontal asymptote at y=-0.5 and a left hand horizontal asymptote at y=0. Is there a better way to determine what asymptotes exist than guessing and testing while looking at the graph on the calculator? I considered that calculating the limits as x->0-,0+,oo- and oo+ might be a good idea but then I realised that I don't know how to do this with absolute value functions! The other idea that I came up with was that there must be a vertical asymptote when the divisor = 0. That gave me x-1-3x=0 and 1-x-3x=0. Solving for x gave me *two* values, x=-0.5 and x=0.25 but the graph definitely shows only one vertical asymptote. Obviously my theory is half right but I don't understand what half or why! Even with the advantage of having the graph in front of me I have no idea how to go about expressing this function in piecewise form. It's obviously some form of reciprical function with domains x<0.25 and x>0.25 but beyond that I am completely lost! Any help at all will be appreciated - I've spent at least 4 hours (probably more) playing with this. Ivan. === Subject: Re: Difficult absolute value problem Progress at last! It finally dawned on me that if my proposition is that x>1 and I obtain x=- 0.5 when I solve the equation then it is as simple as there being no solutions in that interval! I still don't know about the horizontal asymptotes and sketching but at least I'm working on the problem and (hopefully) getting closer. Ivan. === Subject: Re: Difficult absolute value problem > At least, it's a problem that *I* am finding difficult (read impossible)! > p(x)=[abs(x-1)+x-2] / [abs(x-1)-3x] > The questions read: > (a) Sketch p > (b) Express p(x) as a piecewise defined function without absolute values. As so often is the case, it looks more difficult than it really is :-) There is really not very much to think about. The key is to get rid of the abs() functions. In your problem, you have two instances of abs(x-1). So the first step is to divide into two cases: x>1 and x<1. In each interval you will have an expression for p(x), written without abs() functions. Note that the two expressions for p(x) will not be the same. Now that the abs() is gone, you're back on safe ground again. Hope that helps. -Michael. === Subject: Re: Difficult absolute value problem >> p(x)=[abs(x-1)+x-2] / [abs(x-1)-3x] > As so often is the case, it looks more difficult than it really is :-) *smiles* or maybe it looks just as difficult as it is - see below > There is really not very much to think about. The key is to get rid of > the abs() functions. In your problem, you have two instances of > abs(x-1). So the first step is to divide into two cases: x>1 and x<1. > In each interval you will have an expression for p(x), written without > abs() functions. Note that the two expressions for p(x) will not be > the same. > Now that the abs() is gone, you're back on safe ground again. Yes, your explanation makes perfect sense and I thought I understood it! Perhaps I'm missing some understanding about abs(x) ? If x<1, (x-1) is negative so abs(x-1)= -(x-1)= 1-x In this case, p(x)=[(1-x)+x+2]/[(1-x)-3x] = 3/(1-4x) If x>, (x-1) is positive so abs(x-1)=(x-1) In this case, p(x)=[(x-1)+x+2]/[(x-1)-3x] = (2x+1)/(-2x-1)=-1! Plotting these two functions against the original shows quite a substantial difference between the two. I'm sorry if I'm being stupid about this, despite lots of practise and reading I have never really understood abs(x). Ivan. > Hope that helps. Every little bit helps, I'm just hoping that you're not throwing pearls before the swine :( > -Michael. === Subject: Pardon my stupidity! > p(x)=[abs(x-1)+x-2] / [abs(x-1)-3x] ^ This is the error! Should be +2! > If x<1, (x-1) is negative so abs(x-1)= -(x-1)= 1-x > In this case, p(x)=[(1-x)+x+2]/[(1-x)-3x] = 3/(1-4x) > If x>, (x-1) is positive so abs(x-1)=(x-1) > In this case, p(x)=[(x-1)+x+2]/[(x-1)-3x] = (2x+1)/(-2x-1)=-1! > Plotting these two functions against the original shows quite a > substantial difference between the two. It helps no end if one plots the *correct* functions! In my calculator I had typed -2 instead of +2 ... no wonder things didn't match up! > I'm sorry if I'm being stupid about this, I have been - sorry :( Ivan. === Subject: Re: Difficult absolute value problem >> p(x)=[abs(x-1)+x-2] / [abs(x-1)-3x] > If x<1, (x-1) is negative so abs(x-1)= -(x-1)= 1-x > In this case, p(x)=[(1-x)+x+2]/[(1-x)-3x] = 3/(1-4x) You've copied the function incorrectly. It should be -2, not +2, in the numerator. > If x>, (x-1) is positive so abs(x-1)=(x-1) > In this case, p(x)=[(x-1)+x+2]/[(x-1)-3x] = (2x+1)/(-2x-1)=-1! Ditto here. -Michael. === Subject: Re: geometric puzzle The result is essentially Snell's Law for refraction. One time I saw a money version of it, involving the cost of running cable under land versus under a riverbed (the latter more expensive). --OL === Subject: Geometric puzzle This puzzle is x-posted to r.p and s.m, because it is somewhat mathematical. The object is to determine the shortest possible time of travelling from point A to point B on a 2-dimensional surface. The surface consists of three regions: sand, shallow water, and rock. The travelling speed in the three regions are not the same: Running over sand, the speed is 3 miles per hour. Through the shallow water, the speed is 4 miles per hour. On the hard rock, the speed increases to 6 miles per hour. Geometrically, the point A is located at (x,y) = (0,0) and the point B is located at (8,8), all units in miles. The water region is a circular lake, centered at (4,4) and with a radius of 2 miles. The sand region is outside the lake and has y<3. The rocky area has y>3 and is of course outside the lake too. When travelling the perimeter of the lake, the speed is determined by the ground, i.e. sand or rock. -Michael. === Subject: Re: Geometric puzzle > This puzzle is x-posted to r.p and s.m, because it is somewhat mathematical. > The object is to determine the shortest possible time of travelling from > point A to point B on a 2-dimensional surface. The surface consists of three > regions: sand, shallow water, and rock. The travelling speed in the three > regions are not the same: Running over sand, the speed is 3 miles per hour. > Through the shallow water, the speed is 4 miles per hour. On the hard rock, > the speed increases to 6 miles per hour. > Geometrically, the point A is located at (x,y) = (0,0) and the point B is > located at (8,8), all units in miles. The water region is a circular lake, > centered at (4,4) and with a radius of 2 miles. The sand region is outside > the lake and has y<3. The rocky area has y>3 and is of course outside the > lake too. > When travelling the perimeter of the lake, the speed is determined by the > ground, i.e. sand or rock. Despite its simplicity, it seems to be a very difficult problem to solve (to me, rec.puzzles reader). Can it be solved mathematically at all, or does one have to resort to either intuition or some brute force approximation program to find a solution? === Subject: Re: Geometric puzzle >> The object is to determine the shortest possible time of travelling from >> point A to point B on a 2-dimensional surface. The surface consists >> of three >> regions: sand, shallow water, and rock. The travelling speed in the >> three >> regions are not the same: Running over sand, the speed is 3 miles per >> hour. >> Through the shallow water, the speed is 4 miles per hour. On the hard >> rock, >> the speed increases to 6 miles per hour. >> Geometrically, the point A is located at (x,y) = (0,0) and the point >> B is >> located at (8,8), all units in miles. The water region is a circular >> lake, >> centered at (4,4) and with a radius of 2 miles. The sand region is >> outside >> the lake and has y<3. The rocky area has y>3 and is of course outside >> the >> lake too. >> When travelling the perimeter of the lake, the speed is determined by >> the >> ground, i.e. sand or rock. > Despite its simplicity, it seems to be a very difficult problem to > solve (to me, rec.puzzles reader). Can it be solved mathematically at > all, or does one have to resort to either intuition or some brute > force approximation program to find a solution? This problem is in the realm of the calculus of variations. However, you do not need the big guns of that subject. Seems to me you just have several cases to consider (e.g., whether to enter the lake at all or not), optimize within each case, then choose the best amongts all the cases. -- Stephen J. Herschkorn sjherschko@netscape.net === Subject: Re: Geometric puzzle > This puzzle is x-posted to r.p and s.m, because it is somewhat mathematical. > The object is to determine the shortest possible time of travelling from > point A to point B on a 2-dimensional surface. The surface consists of three > regions: sand, shallow water, and rock. The travelling speed in the three > regions are not the same: Running over sand, the speed is 3 miles per hour. > Through the shallow water, the speed is 4 miles per hour. On the hard rock, > the speed increases to 6 miles per hour. > Geometrically, the point A is located at (x,y) = (0,0) and the point B is > located at (8,8), all units in miles. The water region is a circular lake, > centered at (4,4) and with a radius of 2 miles. The sand region is outside > the lake and has y<3. The rocky area has y>3 and is of course outside the > lake too. > When travelling the perimeter of the lake, the speed is determined by the > ground, i.e. sand or rock. > Despite its simplicity, it seems to be a very difficult problem to solve > (to me, rec.puzzles reader). Can it be solved mathematically at all, or > does one have to resort to either intuition or some brute force > approximation program to find a solution? Bit of both. The answer is quite 'neat', which make me think there is a clever way of doing it without too many sums. The sums are easy to set up, but look tedious to solve. I cheated and asked Excel to solve them for me. === Subject: Breakthrough by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) with ESMTP id j28BYGm30246 by support2.mathforum.org (8.12.10/8.12.10/The Math Forum, $Revision: 1.6 secondary) with ESMTP id j28BYEX2023685 H<U>O</U>ME | ABOUT US | SCIENTIFIC<I> A</I>PPROACH | FAQ's<I> </I>| HOLLYWOOD SECRET<U>S</U> | Privacy Polic<U>y</U> and Terms <FONT face=Tahoma>of</FONT> Use

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=== Subject: Q: How to solve this equation system ? I have to solve the following equation system in [0..t_E]: s = f(t) = A t^5 + B t^4 + C t^3 + D t^2 + E t + F v = f'(t) = 5A t^4 + 4B t^3 + 3C t^2 + 2D t + E a = f''(t) = 20A t^3 + 12B t^2 + 6C t + 2D j = f'''(t) = 60A t^2 + 24B t + 6C with the given boundary conditions: (t_E = unknown !!!) f(0) = s_0 (=> F = s_0) f(t_E) = s_E f'(0) = v_0 (=> E = v_0) f'(t_E) = v_E f''(0) = a_0 (=> D = a_0/2) f''(t_E) = a_E Furthermore the max. acceleration a_max in [0..t_E] is given, i.e.: f'''(t_a_max) = 0 // i.e. the acceleration has an extremum f'' (t_a_max) = a_max // at t_a_max (= unknown) The problem now is, that the time t_E is unknown, otherwise i'd know a solution (linear equation system with unknown A, B und C ...) Now my question is: How can I solve this equation system with a given a_max instead of t_E ? MTIA Thomas K.9ahler ROVEMA http://www.rovema.de Verpackungsmaschinen GmbH Tel +49 (0) 641 409-221 Industriestra¤e 1 Fax +49 (0) 641 409-212 35463 Fernwald mailto:thomas.koehler@rovema.de === Subject: Zenon paradox Do you happen to know a site where the Zenon paradox is explained? nkbjvg === Subject: Re: Zenon paradox > Do you happen to know a site where the Zenon paradox is explained? LOL! What a gas! === Subject: Re: Zenon paradox I dont know about any site but , after going through what Tom M. Apostol has written in his book 'CALCULUS' volume 1, published by Wiley, I could expalin the concept of limit of a sequence very well to my class..... vms === Subject: Re: Zenon paradox > Do you happen to know a site where the Zenon paradox is explained? Google . Its a gas. Bob Kolker === Subject: Re: Zenon paradox >> Do you happen to know a site where the Zenon paradox is explained? > Google . Its a gas. Xenon is a gas. Zenon is something else. John Briggs === Subject: Re: Zenon paradox Discussion, linux) > > Do you happen to know a site where the Zenon paradox is explained? >> Google . Its a gas. > Xenon is a gas. Zenon is something else. And Kolker is an annoying, unhelpful asshole. -- Jesse F. Hughes My baby don't allow me in the kitchen and I've come to love her decision. -- Bad Livers === Subject: Re: Zenon paradox > > Do you happen to know a site where the Zenon paradox is explained? > Google . Its a gas. > Bob Kolker No, the gas is Xenon. === Subject: Re: Zenon paradox >Do you happen to know a site where the Zenon paradox is explained? >>Google . Its a gas. >>Bob Kolker > No, the gas is Xenon. Either the joke passed you by, or I am the only one who saw the joke here, including Bob... alex === Subject: Re: Zenon paradox Wouldn't it be Zeno and not Zenon by any chance? http://mathworld.wolfram.com/ZenosParadoxes.html Paulo Matos > Do you happen to know a site where the Zenon paradox is explained? > nkbjvg === Subject: Re: Zenon paradox pmatos dixit: >Wouldn't it be Zeno and not Zenon by any chance? Actually, Zeno is the latin form. The correct ancient greek spelling using our alphabet is Zenon, like Plato is really Platon, with the letter 'o' being an omega and not an omicron. >http://mathworld.wolfram.com/ZenosParadoxes.html >Paulo Matos >> Do you happen to know a site where the Zenon paradox is explained? >> nkbjvg === Subject: Re: Zenon paradox > Wouldn't it be Zeno and not Zenon by any chance? Greek names sometimes end with nu (the Greek n). For example Platon (Plato, the philosopher), Heron (Hero, Tyrant of Syracusa) Bob Kolker === Subject: Re: An easy question on Manifolds at 06:47 PM, andrea_the_cat@yahoo.it (Andrea) said: >My question is.. what topology has TB???????? It's a local product topology but not, in general a product topology. Beyond that, the answer depends on how you define a manifold. -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not === Subject: Re: differential & exterior derivation operator >Obviously df definied above at (#) and the exterior differential of >f, also denoted by df are the same, infact df Nabla f= sum_i df/dx^i >dx^i and also the exterior differential of f is defined as df = >sum_i df/dx^i dx^i. While df= Nabla f= sum_i df/dx^i, dx^i you would not normally use either as a definition, because df/dx^i and dx^i are coordinate-dependent. >This seems a contradiction, isn't it? No. Just because df = Nabla f doesn't mean that DF = Nabla F[1] for an arbitrary differential form. In fact, Nabla is defined on tensor fields that don't correspond to differential forms at all; dT only makes sense if T is an antisymmetric tensor field, while Nabla is defined for any C^1 tensor field. [1] More precisely, that the canonical mapping from forms to antisymmetric tensors maps DF into Nabla F. -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not === Subject: FFT on vector of very large numbers I have a vector of n integers on which I need to perform an FFT. The problem is that the integers can be very large (of the scale of n^n). I understand that the two options are a) Perform the whole thing in normal floating point arithmetic and risk a very inaccurate answer b) Perform number theoretic FFT and risk it being very slow. My problem is that I know no more than that. Does anyone know of any references for how inaccurate or slow and a) and b) would be? I have found plenty of references for the accuracy of FFT but without obvious reference to very large numbers. Any help is very much appreciated. Raphael === Subject: Re: FFT on vector of very large numbers > I have a vector of n integers on which I need to perform an FFT. The > problem is that the integers can be very large (of the scale of n^n). I > understand that the two options are > a) Perform the whole thing in normal floating point arithmetic and risk > a very inaccurate answer > b) Perform number theoretic FFT and risk it being very slow. > My problem is that I know no more than that. Does anyone know of any > references for how inaccurate or slow and a) and b) would be? I have > found plenty of references for the accuracy of FFT but without obvious > reference to very large numbers. > Any help is very much appreciated. > Raphael See: P. Montgomery & R. Silverman An FFT Continuation to the P-1 Factoring Algorithm Math. Comp. Vol 54, 1990 Take your vector of integers. Reduce it modulo p_i such that p_1 = 1 mod n Do this for enough p_i such that their product exceeds 2*largest integer in your vector. Now do a convolution modulo each of the p_i and paste the results back together with the Chinese Remainder Theorem. This is quite fast. === Subject: Re: FFT on vector of very large numbers > I have a vector of n integers on which I need to perform an FFT. The > problem is that the integers can be very large (of the scale of n^n). I > understand that the two options are > a) Perform the whole thing in normal floating point arithmetic and risk > a very inaccurate answer > b) Perform number theoretic FFT and risk it being very slow. Multiplication of large numbers is often done with an FFT-based convolution algorithm. Thus, it appears that your runtime will be roughly the square of the floating-point runtime. I think FFTs are about O(n*log2(n)), right? So your runtime would be O[(n*log2(n))^2]. - Randy === Subject: Re: Minimum distance from a point to a surface defined by constraints >I have a n-dimensional point and m constraints as shown below; >Point P= (p1,p2,...pN) >Constraint 1: c1*p1+c2*p2+...+C < 0 >Constraint 2: d1*p1+d2*p2+...+D > 0 >... >Constraint m: z1*p1+z2*p2+...+Z < 0 >Note: Any Constraint can be viewed as a normalized hyperplane defined >on the n-dimensional space. > It can be argued that, m constraints define a surface (closed or not) >on >the n-dimensional space. Suppose that Point P does not satify the >constraints. I wonder, how can I find the smallest distance between >Point P to the surface determined by the constraints. >Any idea ? >Example: in 2-D > Point P= (-1,+1) > Const1: p1 > 0 > Const2: p2 < 0 >It is clear that X does not satify both Const1 and 2. For this case >the distance between Point P to an area (defined by the constraints) >can easly computed such that > Distance= sqrt(p1*p1+p2*p2) It is euclidean distance. >However this not always true. > | > | > P | > | >---------------- p1 plane > | > | An Area where all > | > | > p2 plane your feasible set is a convex polyhedron, since you have only linear inequality constraints. hence your problem bils down to a standard convex QP problem: minimize euclidena distance(P,Z)^2 subject to A*Z <= b with a matrix A and a right hand sibe b given. If by chance P satisifes the constraints you will get zero as the minimum. hth peter === Subject: Re: a string into two pieces (probability question) i have a doubt in the very wording of the problem. there is only one instance to cut the string in to 'two' pieces such that the second is one third of the first. instead.... the problem must have been stated as Two pieces are cut out of a string of given length. Find the probability that one is three times the other. i think this will give a scope to solve a proble vms === Subject: Re: a string into two pieces (probability question) > i have a doubt in the very wording of the problem. > there is only one instance to cut the string in to 'two' pieces such > that the second is one third of the first. > instead.... the problem must have been stated as > Two pieces are cut out of a string of given length. Find the > probability that one is three times the other. > i think this will give a scope to solve a proble Yes, I've reread the problem and it is ambiguous. And the OP did ask specifically about the second piece, which seems to mean that my interpretation of such that one piece is at least three times larger than the other piece is wrong. So let's interpret it that we are making two cuts. In that case there is a clearly-defined second piece. And let's also say we want the probability that the second piece is AT LEAST (not exactly equal) 3 times the length of the first. Let the first cut be at x1, uniformly distributed on [0,1]. The second cut will be at x2, uniformly distributed on [x1,1]. The question is P(x2-x1>=3*x1). x2 - x1 >= 3*x1 <=> x2 >= 4*x1. Since x2 <= 1, then 1 >= x2 >= 4*x1 <=> x1 <= 0.25. P(x2 >= 4*x1) = integ(0,1) P(x2 >= 4*x1 | x1) dx1 = integ(0,0.25) (1-4*x1)/(1-x1) dx1 I'm not sure what that is analytically. A quick and dirty numerical integration gives me 0.137. - Randy === Subject: Re: a string into two pieces (probability question) > P(x2 >= 4*x1) = integ(0,1) P(x2 >= 4*x1 | x1) dx1 > = integ(0,0.25) (1-4*x1)/(1-x1) dx1 > I'm not sure what that is analytically. A quick > and dirty numerical integration gives me 0.137. Actually, it's an easy integral, with the substitution u = 1-x1. I get P(x2 >= 4*x1) = 1 - 3*log(4/3) = 0.13695. - Randy === Subject: Does lim (1/a) x^(a-1) tend to ln(x) when a tends to 0 ? I would be very pleased if someone could help me. Is it right that lim (1/a) x^(a-1) tends to ln(x) when a tends to 0 ? (^ is for power) If not, what is the right expression to obtain ln(x) when a tends to 0 ? And how could I show this (probably using l'hopital rule if I can get the indetermination 0/0 when a=0). Elisa. ---------------------------------------------------------- ** SPEED ** RETENTION ** COMPLETION ** ANONYMITY ** ---------------------------------------------------------- http://www.usenet.com === Subject: re:Does lim (1/a) x^(a-1) tend to ln(x) when a tends to 0 ? Ok, I guess I have found, sorry for having posted this question. This is exaclty the Box Cox transfomation I hadn't realized... lim (when a tends to 0) (x^a-1)/a (this is indeterminate, 0/0) = lim x^a lnx by Hopital rule and it gives lnx. Have a nice afternoon. ---------------------------------------------------------- ** SPEED ** RETENTION ** COMPLETION ** ANONYMITY ** ---------------------------------------------------------- http://www.usenet.com === Subject: Re: Does lim (1/a) x^(a-1) tend to ln(x) when a tends to 0 ? >I would be very pleased if someone could help me. >Is it right that lim (1/a) x^(a-1) tends to ln(x) when a tends to 0 ? >(^ is for power) If not, what is the right expression to obtain ln(x) >when a tends to 0 ? >And how could I show this (probably using l'hopital rule if I can get >the indetermination 0/0 when a=0). If you already know the compound interest rule you can derive the following from it. x=(1+y/n)^n goes to e^y as n approaches oo. Let a=1/n x=(1+y*a)^(1/a) goes to e^y as a approaches 0. Solve for y y=(x^a-1)/a goes to ln(x) as a approaches 0. --Keith Lewis klewis {at} mitre.org The above may not (yet) represent the opinions of my employer. === Subject: Re: Does lim (1/a) x^(a-1) tend to ln(x) when a tends to 0 ? > Is it right that lim (1/a) x^(a-1) tends to ln(x) when a tends to 0 ? As written, this limit does not exist: the numerator tends to 1/x, the denominator tends to 0. > (^ is for power) If not, what is the right expression to obtain ln(x) > when a tends to 0 ? For fixed x > 0, (1/a)(x^a - 1) -> ln(x) as a -> 0. > And how could I show this (probably using l'hopital rule if I can get > the indetermination 0/0 when a=0). LHR is rarely a good way to understand anything. Instead, let f(a) = x^a. By definition of the derivative, (1/a)(x^a - 1) = (f(a) - f(0))/a -> f'(0) as a -> 0. Now recall f'(a) = ln(x)*x^a for any a. === Subject: Re: an true information theory > This does not apply to biology - in my view - because it depends on very > Large* logical frameworks and biology is a Small* one at best. Yah -- Godel was cool (and Hofstadter's tour de force, for all its flaws, is a most entertaining read) but he wasn't *that* cool. Applying him in places where he doesn't fit, e.g. biology, is a little like trying to apply Heisenbergian uncertainty to shuffling a deck of cards (which I've heard Tarot diviners claim). > The point about meaning is that philosophers have discussed meaning > since before Russell (i.e., Frege) and the consensus is that there is no > such thing apart from the context in which terms are employed. This > *does* apply to biology, because there is a persistent movement called - > this week at any rate - teleosemantics. This is the idea that genes > somehow encode meaning acquired by past selection. But if the > consensus view is correct, this is false. I think it is false. I was going to suggest an objection, but I realize I don't know what's meant by meaning -- in the vernacular, which this obviously isn't -- that word generally implies either a precise mapping or analysis by intelligence. My not-tiny but incomplete background in information theory has been focused toward how to explain things to laypeople, and toward efficient algorithms to do simple tasks, as opposed to proficiency in complex or formal aspects, and I recognize the limitations this imposes on my contributions to the discussion. That said, I perceive an analogue, at a high level, between a band-pass filter and the process of natural selection. Filters add information, by some definitions of information, through subtraction -- the same way a high tide mark on a beach is an encoding of information about how high the tide typically gets, and encodes this via the removal of some of the sand. Analogies are, of course, dangerous, and I cheerfully confess to ignorance of the details which might well make a filter analogy useless. But I'd love to know whether it has any merit. And on the other hand, it seems to require possibly troublesome reverse engineering to say that the *genes* encode information from the *environment* (and it's possible I'm coming close here to what you mean when you say meaning?). To use yet another analogy from my profession, if one takes a photograph of the character '2' from (say) newsprint, then digitally encodes the photograph, the resulting bitmap image might be claimed to be an encoding of the number '2', but one can't usefully use it as the *value* '2' without sophisticated decoding, where there's a lot more information in the decoder than the two bits of information you're extracting from the image. Oh, and for those who don't know that I know, I'll clarify: information-theory objections to evolution are bunkum. I'm just discussing definitions and analogies here, and trying to learn something and contribute to the discussion. So I'm not resolving anything, just suggesting competing analogies. Feel free to ignore, but I'd welcome comment. Or just point me to applicable research and I'll read it or shut up. > On information and meaning, try these: > Dretske, Fred I. 1981. Knowledge and the flow of information. Cambridge, > Mass.: MIT Press. > Platts, Mark de Bretton. 1979. Ways of meaning: an introduction to a > philosophy of language. London; Boston: Routledge & Kegan Paul. > Oxford, UK; Malden, MA: Blackwell Publishing. > On biological meaning > Millikan, Ruth Garrett. 1984. Language, thought, and other biological > categories: new foundations for realism. Cambridge, Mass.: MIT Press. > Oyama, Susan. 1985. The ontogeny of information: developmental systems > and evolution. Cambridge: Cambridge University Press. > Oyama, Susan, Paul E. Griffiths, and Russell D. Gray, eds. 2000. Cycles > of contingency: Developmental systems and evolution. Cambridge, MA: MIT > Press. > *Such terms mean something different to metaphysicians and logicians > than to the rest of us... eyelessgame === Subject: Re: an true information theory TAG_LEVEL=1000.0 QUARANTINE_LEVEL=1000.0 KILL_LEVEL=1000.0 >> >>This was a large project - if we could make mathematics an aspect of >>logic, and derive it and all elaborations of logic from set theory, >then >>we could in effect uncover the fundamental principles of the laws of >>thought. >> Well, based on their dubious assumption that thought was actually >> *logical*. >It seems to presumes that there was such a thing as a concise >description of laws of thought that governed the way every >intelligent thought would be reckoned. There probably are, but they are likely to be *biological* laws, not mathematical ones. That is a theory of thought *is* being developed by biologists, and what they are finding has no relationship to mathematics, except in the sense that one can describe the operation of neurons mathematically. > The question to consider is, >even if there is, is that some form of predicate logic? Without any qualification: No. -- The peace of God be with you. Stanley Friesen === Subject: markov chain algebra and operators Let P be an m x m probability matrix which characterizes the Markov Chain MC. Suppose now that we modify P, for instance: * increment and decrement some of the elements of a line; * remove transitions; * add transitions; and then obtain a new stochastic matrix P', associated to the Markov Chain MC'. The question is: what is the relation between the stationary distribution of MC and that of MC'? Where may I find relevant literature about this subject? Daniel Sadoc === Subject: Re: markov chain algebra and operators >Let P be an m x m probability matrix which characterizes the Markov >Chain MC. Suppose now that we modify P, for instance: >* increment and decrement some of the elements of a line; >* remove transitions; >* add transitions; >and then obtain a new stochastic matrix P', associated to the Markov >Chain MC'. >The question is: what is the relation between the stationary >distribution of MC and that of MC'? So P' is a perturbation of P (presumably a small perturbation). The stationary distributions, say p and q for MC and MC' respectively, are left eigenvectors of P and P' for eigenvalue 1. You say stationary distribution as if there's only one. I'll suppose P is irreducible (things are rather more complicated otherwise: an arbitrarily small perturbation can make a reducible Markov chain into an irreducible one). Then 1 is a simple eigenvalue of P. Let Gamma be a contour in the complex plane that encloses 1 but no other eigenvalue of P. There is a neighbourhood U of P in the space of n x n complex matrices such that if Q is in U, Q has only exactly eigenvalue inside Gamma and none on Gamma. The eigenprojection of Q for that eigenvalue can be written as F(Q) = (2 pi i)^(-1) int_Gamma (zI-Q)^(-1) dz and thus is an analytic function of Q in U. In the case where Q=P' is also a stochastic matrix, the eigenvalue must be 1, and Q is also irreducible; if v is a probability vector (i.e. nonnegative with sum 1), then q = v F(Q) is the unique stationary distribution, which is an analytic function of Q in U. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: External Direct Sums - Artinian and Semiprime I would like some help here. Let R1 and R2 be rings and let R=R1 + R2 (external direct sum) with componentwise multiplication. a) If R1 and R2 are right artinian prove that R is right Artinian. b) If R1 and R2 are semiprime, prove that R is semiprime. Proof(a): Let R1 and R2 be Right Artinian. Then there are descending chains B contains B1 contains B2 contains..... of R1 equal after finitely many steps and C contains C1 contains C2 contains..... of R2 equal after finitely many steps. So (B,C) contains (B1,C1) contains (B2,C2) contains..... that are equal after finitely many steps. Hence R contains R1 contains R2 contains R3 contains... is a descending chain equal after finitely many steps. SO R is right Artinian. I still don't have anything for part (b). James === Subject: Matrix Derivatives I have 2 questions: 1. I have a function of n matrices f(A1,A2,A3...). Ai=Bi+deltai. How can I use a Taylor series of f around Bi? 2. C is a scalar that is dependent on a matrix X. D is also a scalar, that is dependent on X. How can I derive C/D with respect to X? === Subject: Re: Matrix Derivatives 1. You can look at a function of matrix arguments as a function of the elements of these matrices. And then the complete machinery of functions of several variables goes at work... 2. (C/D)' = (CD' - C'D) / DD Happy studies: Johan E. Mebius >I have 2 questions: >1. I have a function of n matrices > f(A1,A2,A3...). Ai=Bi+deltai. > How can I use a Taylor series of f > around Bi? >2. C is a scalar that is dependent on a matrix X. > D is also a scalar, that is dependent on X. > How can I derive C/D with respect to X? === Subject: Coefficient of best-fit polynomial I have been struggling for the last few days over a problem in Luenberger's Optimization by Vector Space Methods and I cannot seem to get anywhere. Can anybody help me with this problem? Suppose randomly varying data in the form of a function x(t) is observed from t = 0 to t = T. We can predict the future data (t>T) by fitting a polynomial of degree (n-1) to the past data and using the extrapolated values of the polynomial as the estimate. The best fit polynomial, for our purposes, will be the polynomial p(T,t) that minimizes the integral of [x(t) = p(T,T)]^2 as t varies from 0 to T (so it's the closest in the L^2 norm). Show that the coefficients of p(T,t), call them a_i(T), do not need to be completely recomputed for each T but rather can be continuously updated according to a formula of the form d/dT [a_i(T)] = b_i*e(T)/T^i where the b_i are fixed constants and e(T) is the instantaneous error x(T) - p(T,T). The section that precedes this discusses the Legendre polynomials as an orthonormal basis for the space of polynomials, so I'm sure the first step is to take x(t) and project it to the space of (n-1) degree polynomials with teh first (n-1) Legendres as a basis. Then we have to move from the Legendres as a basis to the standard basis {1,t,t^2,...,t^(n-1)}. But I can't figure out where on earth that differential equation comes from. Any suggestions? KH === Subject: what are the right term and symbol to ... Please forgive me if I post my questions in a wrong newsgroup. Assuming I have a series epsilon_i, i = 0, 1, ..., n. I compute a new quantity, (epsilon_(i+1)-epsilon_i)/epsilon_i. What is the right term to call this quantity? What is the right symbol to represent it? === Subject: Re: what are the right term and symbol to ... > Please forgive me if I post my questions in a wrong newsgroup. > Assuming I have a series epsilon_i, i = 0, 1, ..., n. I compute a new > quantity, (epsilon_(i+1)-epsilon_i)/epsilon_i. What is the right term to > call this quantity? What is the right symbol to represent it? I don't see why you *have* to call it anything at all. In numerical analysis and in particular in the convergence of Newton's Method, it's called modulus of relative error between successive terms, but I am not sure how this would apply to your case. -- I. N. Galidakis http://users.forthnet.gr/ath/jgal/ ------------------------------------------ Eventually, _everything_ is understandable === Subject: Re: what are the right term and symbol to ... thank you all > Please forgive me if I post my questions in a wrong newsgroup. > Assuming I have a series epsilon_i, i = 0, 1, ..., n. I compute a new > quantity, (epsilon_(i+1)-epsilon_i)/epsilon_i. What is the right term to > call this quantity? What is the right symbol to represent it? > I don't see why you *have* to call it anything at all. > In numerical analysis and in particular in the convergence of Newton's > Method, it's called modulus of relative error between successive > terms, but I am not sure how this would apply to your case. > -- > I. N. Galidakis > http://users.forthnet.gr/ath/jgal/ > ------------------------------------------ > Eventually, _everything_ is understandable === Subject: Re: what are the right term and symbol to ... What makes you think that there is a right term' or 'right symbol'??? Give it any name you want. It is simply 1 less than the ratio of two successive terms. Why do you believe that it matters what you call it? (N.B. note that your expression is not defined for i = 0) === Subject: Re: what are the right term and symbol to ... > What makes you think that there is a right term' or 'right symbol'??? > Give it any name you want. It is simply 1 less than the ratio of two > successive terms. > Why do you believe that it matters what you call it? > (N.B. note that your expression is not defined for i = 0) I need to mention this quantity over and over again in a paper. I think a good term may help me to explain other things, and avoid confusing the readers. Can someone please sugguest a term for it? === Subject: Re: what are the right term and symbol to ... >>What makes you think that there is a right term' or 'right symbol'??? >>Give it any name you want. It is simply 1 less than the ratio of two >>successive terms. >>Why do you believe that it matters what you call it? >>(N.B. note that your expression is not defined for i = 0) > I need to mention this quantity over and over again in a paper. I think a > good term may help me to explain other things, and avoid confusing the > readers. Can someone please sugguest a term for it? You can call it whatever you want to call it, delta_i, sigma-hat_i, etc. As long as you clearly define it and it doesn't conflict with other symbols, you shouldn't have a problem. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Maple bugs: Thomas Richard: Hurrah, Maple quality improves! - Example 5 Hi Maple customers all over the world, Did you know these interesting facts? http://www.maplesoft.com/support/ Maplesoft> Maplesoft is committed to providing... hem-hem-hem... Maplesoft> the highest level of support for the products it sells. Great! So we the Maple customers must expect new free bug fixes coming soon. This is not all, however. Maplesoft> In addition, Maplesoft is also committed to delivering Maplesoft> quality services that will enable customers to ... Maplesoft> hem-hem-hem... fully hem-hem-hem... leverage the benefits Maplesoft> of the products it sells. Aha. Rejoice, brethren! So a large set of free bugs fixes to Maple 9.5.2 is coming. Herr Richard, Maple Support, Scientific Computers GmbH: [Maple's quality improves... akh-hem-hem-hem... well... at least it does not deteriorate] Laurent Bernardin, Chief Scientist and VP, Research and Development claims: http://bernardin.com/maple/index.php LB> Being strict about backwards compatibility can be frustrating LB> for our developers. A common comment I hear, is: It would LB> really be easier to make this command work better, if I would LB> not have to worry about compatibility. This is a good sign. LB> It means the issue is taken seriously at all levels. It also LB> means that we are finding ways to move Maple forward and LB> improve the system without breaking existing user code. http://www.maplesoft.com/ Maplesoft> Maplesoft technologies are hem-hem-hem-hem-hem- Maplesoft> hem...... ideal for students enrolled in any math, Maplesoft> science or engineering course. Learn how you can Maplesoft> use Maple to save hum-hum-hum-hem time, learn less Maplesoft> jeeez, we mean more and became a dropout, oh dog Maplesoft> our cats, we mean improve your grades. Here comes next example of Maple quality improvement, (in our stupid customer's language, degradation) one of many thousands. Enjoy! > simplify(3^z * (-4)^z/((-3)^z * 4^z)); 1 # a constant For the doubting Thomases: > plot((3^z * (-4)^z/((-3)^z * 4^z)), z=0..10); You see the straight line y = 1. Ready, steady, go! limit(3^z * (-4)^z/((-3)^z * 4^z), z= infinity); infinity infinity 1 -------------------- (2002) Maple 8 -------------------------- 1 -------------------- (2001) Maple 7 -------------------------- 1 -------------------- (2000) Maple 6 -------------------------- 1 -------------------- (1997) Maple V Rel 5 -------------------- limit(3^z*(-4)^z/((-3)^z)/(4^z),z = infinity) -------------------- (1995) Maple V Rel 4 -------------------- limit(3^z*(-4)^z/((-3)^z)/(4^z),z = infinity) -------------------- (1994) Maple V Rel 3 -------------------- limit(3^z*(-4)^z/((-3)^z)/(4^z),z = infinity) --------------------------------------------------------------- We clearly see improvement in quality in Maple 9.5/9.5.2. The same quality improvement (bug manifestation) with limit(3^z * (-8)^z/((-3)^z * 8^z), z= infinity); limit(5^z * (-4)^z/((-5)^z * 4^z), z= infinity); limit(6^z * (-9)^z/((-6)^z * 9^z), z= infinity); limit(9^z*(-4)^z/((-9)^z * 4^z), z= infinity); limit(9^z*(-8)^z/((-9)^z * 8^z), z= infinity); limit(10^z * (-8)^z/((-10)^z * 8^z), z= infinity); limit(10^z * (-9)^z/((-10)^z * 9^z), z= infinity); limit(11^z * (-8)^z/((-11)^z * 8^z), z= infinity); limit(11^z * (-9)^z/((-11)^z * 9^z), z= infinity); What about the competitors? ................................................................. Derive 6.1 > LIM(3^z * (-4)^z/((-3)^z * 4^z), z, inf) 1 ................................................................. Mathematica 5.1 > Limit[3^z (-4)^z/((-3)^z 4^z), z -> Infinity] 1 ................................................................. MuPAD 3.1.1 > limit(3^z * (-4)^z/((-3)^z * 4^z), z= infinity); 1 ................................................................. Buy Maple to save time, learn more and improve your grades, and, most important, pay out of your pocket for Maplesoft's staff education, Vladimir Bondarenko VM and GEMM architect Co-founder, CEO, Mathematical Director Cyber Tester, LLC http://www.cybertester.com/ http://maple.bug-list.org/ http://www.CAS-testing.org/ === Subject: Simple Random Number Generator/Function Hi All, I need to create a simple random number generator/function which when given an integer i will generate an integer n where 1 <= n <= i <= Indra === Subject: Re: Simple Random Number Generator/Function > Hi All, > I need to create a simple random number generator/function which when > given an integer i will generate an integer n where 1 <= n <= i <= > Indra Which programming language? If simplicity is all that counts, why not use the intrinsic generator of the standard library (e.g. int rand(void) ) in C? The standard generators are no _too_ bad. Hugo === Subject: Re: Simple Random Number Generator/Function <422E16BB.1F9D7F91@abouthugo.de> C, Pascal, or any languages that are easy to read will be fine. This generator will be used in DVD Authoring. There is a rand function in DVD specification, but few DVD players do not like them, that's why I would like to create one. === Subject: Re: Simple Random Number Generator/Function > C, Pascal, or any languages that are easy to read will be fine. This > generator will be used in DVD Authoring. There is a rand function in > DVD specification, but few DVD players do not like them, that's why I > would like to create one. The Standard C generator from Brian W Kernighan and Dennis M. Ritchie, The C Programming Language (Second Edition) Prentice Hall Software Series, 1988 uses x(1)=1, x(n)=(1103515245 * x(n-1) + 12345) mod 2^31 a(n)=floor(x(n)/2^16) C Program: static unsigned int next = 1; int rand( ) { next = next * 1103515245 + 12345; return ((next >>16) & 32767); } If you want different behavior for different runs, you have to replace next=1 by something taking a variable input (e.g. clock). C provides the function srand ( ... ); for this pupose. Hugo === Subject: Re: Simple Random Number Generator/Function <422E16BB.1F9D7F91@abouthugo.de> <422E1DAF.5265D967@abouthugo.de> === Subject: Re: Simple Random Number Generator/Function > Hi All, > I need to create a simple random number generator/function which when > given an integer i will generate an integer n where 1 <= n <= i <= 20source> -- John MexIT: http://johnbokma.com/mexit/ personal page: http://johnbokma.com/ Experienced programmer available: http://castleamber.com/ Happy Customers: http://castleamber.com/testimonials.html === Subject: Re: Simple Random Number Generator/Function Indra === Subject: Re: Simple Random Number Generator/Function - Cliff RNG A very interesting 'one line' Random Number Generator is the 'Cliff RNG'. It is: x(n+1) = abs(100*ln(x(n))mod 1)), where x(0) = 0.1 [note: mod 1 means 'fractional part'] This of course, gives you a fractional result between 0 and 1. Check it out on Mathworld.com at: http://mathworld.wolfram.com/CliffRandomNumberGenerator.html or in: Pickover, C. A. Computers, Randomness, Mind, and Infinity. Ch. 31 in Keys to Infinity. New York: W. H. Freeman, pp. 233-247, 1995. Very simple, easily portable. But is it random or pseudo-random? Good luck, Tomcee > Indra === Subject: Re: integer points on hyperbola >> The hyperbola y^2 - 54516*x^2 - 36036*x = 139129 seems >> quite unusual. All 'smallish' integer points have >> |y| a prime number. For example [-12, 2749],[-10, 2287], >> [-1, 397],[0, 373],[3, 859],[5, 1297],[8, 1979],[118, 27631] >> are such points. This pattern persists for y-values well >> past one million. >> Are there infinitely many such 'prime-y-value' integer >> points? >An equation y^2 - ax^2 - bx = c can be rewritten as >u^2 - av^2 = b^2 - 4c, where u = 2ax + b and v = 2y. [Gerry goes on to describe how to find more solutions. ] I guess I would view this as a question of finding elements of a fixed norm (=b^2-4ac) in a certain ring of algebraic integers (= Z[sqrt(a)] ). Writing norm(z) = z zbar we see the question comes down to understanding the factorization of b^2-4ac in this ring. There are issues about the class group and congruence classes mod 2a and so on but basically it's a finite problem, together with the introduction of units, here a free abelian group of rank 1. So there should be finitely many families of solutions of the form z = z_i eta^n where eta is a fundamental unit and n = +-1, +-2, +-3, ... In particular, within each family, the components of z_i = u_i + sqrt(a) v_i will grow approximately exponentially. Now, this is all deterministic; if I had the patience I could probably write out the finitely many families of solutions. But treating these for a moment as random integers with an approximately linearly growing number of digits, we can ask for the expected number of primes. That turns out to be infinite but very slowly growing. So I wouldn't be surprised if it got harder to find prime elements of the sequence that Jim proposed in the first place. (This is just the sort of reasoning which is used to argue that there ought to be infinitely many Mersenne primes, and so far I would say the data support the bogus assumptions being made about that case. On the other hand, it has taken quite a bit of labor so far to find forty-something elements of a supposedly infinite set!) I worked for a few minutes on the specific Diophantine question on the table. I notice that a,b,c, and b^2-4ac are between them divisible by quite a few of the low primes, suggesting that there may be easy congruences which force the y values not to be divisible by some of these primes. A sequence of smallish numbers is much more likely to contain primes if it is known in advance that its members are never divisible by 2, 3, 5, ..., p0 . So my previous claim that the primes will be sparse is not necessarily contradicted by the data showing many primes among the first few terms. dave PS -- Similar bogosity: prove that there are infinitely many numbers like 1093 and 3511; then find the third one. === Subject: Re: integer points on hyperbola > The hyperbola y^2 - 54516*x^2 - 36036*x = 139129 seems > quite unusual. All 'smallish' integer points have > |y| a prime number. For example [-12, 2749],[-10, 2287], > [-1, 397],[0, 373],[3, 859],[5, 1297],[8, 1979],[118, 27631] > are such points. This pattern persists for y-values well > past one million. > > Are there infinitely many such 'prime-y-value' integer points? >> An equation y^2 - ax^2 - bx = c can be rewritten as >> u^2 - av^2 = b^2 - 4c, where u = 2ax + b and v = 2y. > [...] I worked for a few minutes on [it]... Dario Alpern has written a nice Java application [1] for solving the general bivariate quadratic Diophantine equation a x^2 + b xy + c y^2 + dx + ey + f = 0 It has the capability to display a step-by-step solution. See also John Robertson's links [2] on Pell equations. --Bill Dubuque [1] http://www.alpertron.com.ar/QUAD.HTM [2] http://members.aol.com/jpr2718/ === Subject: Re: integer points on hyperbola >An equation y^2 - ax^2 - bx = c can be rewritten as >u^2 - av^2 = b^2 - 4c, where u = 2ax + b and v = 2y. > I guess I would view this as a question of finding elements of a > fixed norm (=b^2-4ac) in a certain ring of algebraic integers (= Z[sqrt(a)] ). Why are you talking about b^2 - 4ac, when I had b^2 - 4c? Oh, I see why - it's because I screwed up. Please, if there's anyone who took notes on my earlier post, change my b^2 - 4c to b^2 - 4ac. > But treating these for a moment as random integers with an > approximately linearly growing number of digits, we can ask for the > expected number of primes. That turns out to be infinite but very > slowly growing. Same heuristic applies to [(3/2)^n] (that's integer part of the n-th power of 3/2), but no one has any idea how to prove it's prime infinitely often. Many years ago Ron Graham came up with numbers a and b, relatively prime, such that the sequence u_1 = a, u_2 = b, u_{n+1} = u_n + u_{n-1} (which of course grows at the same rate as the Fibonacci numbers, hence, linearly growing number of digits) has no primes at all. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: integer points on hyperbola James Buddenhagen escribi.97: > The hyperbola y^2 - 54516*x^2 - 36036*x = 139129 seems > quite unusual. All 'smallish' integer points have >> y| a prime number. For example [-12, 2749],[-10, 2287], > [-1, 397],[0, 373],[3, 859],[5, 1297],[8, 1979],[118, 27631] > are such points. This pattern persists for y-values well > past one million. > Are there infinitely many such 'prime-y-value' integer > points? > Jim Buddenhagen The first composite y-value for positive x is 25159661 = 1511*16651, corresponding to x = 107756. The next integer points after [118, 27631] are: x: [221, 469, 1929, 29639, 49028, 107756] y: [51679, 109583, 450473, 6920387, 11447459, 25159661] No others with 0 < x < 5*10^5. -- Ignacio Larrosa Ca.96estro A Coru.96a (Espa.96a) ilarrosaQUITARMAYUSCULAS@mundo-r.com === Subject: Re: integer points on hyperbola > James Buddenhagen escribi.97: > The hyperbola y^2 - 54516*x^2 - 36036*x = 139129 seems > quite unusual. All 'smallish' integer points have >> y| a prime number. For example [-12, 2749],[-10, 2287], > [-1, 397],[0, 373],[3, 859],[5, 1297],[8, 1979],[118, 27631] > are such points. This pattern persists for y-values well > past one million. > Are there infinitely many such 'prime-y-value' integer > points? > Jim Buddenhagen > The first composite y-value for positive x is 25159661 = 1511*16651, > corresponding to x = 107756. > The next integer points after [118, 27631] are: > x: [221, 469, 1929, 29639, 49028, 107756] > y: [51679, 109583, 450473, 6920387, 11447459, 25159661] > No others with 0 < x < 5*10^5. > -- > Ignacio Larrosa Ca.96estro > A Coru.96a (Espa.96a) > ilarrosaQUITARMAYUSCULAS@mundo-r.com for all integer points for -10^8 < x < 10^8. There are 32 with y positve: [-20128306, 443*10608781 ] [-9158230, 89*9739*2467 ] [-1429340, 333731683 ] [-925779, 216156929 ] [-360389, 179*470089 ] [-87669, 449*45589 ] [-16576, 3870197 ] [-10674, 2492159 ] [-5707, 1332431 ] [-1570, 366497 ] [-1017, 237379 ] [-655, 152857 ] [-159, 37049 ] [-12, 2749 ] [-10, 2287 ] [-1, 397 ] [ 0, 373 ] [ 3, 859 ] [ 5, 1297 ] [ 8, 1979 ] [ 118, 27631 ] [ 221, 51679 ] [ 469, 109583 ] [ 1929, 450473 ] [ 29639, 6920387 ] [ 49028, 11447459 ] [ 107756, 1511*16651 ] [ 1065975, 281*885733 ] [ 1993784, 79*5892683 ] [ 3096246, 28817*25087 ] [ 11256669, 181*14520893 ] [ 17379536, 4057889131 ] The y-values above are factorized when composite. Summary: the curve y^2 - 54516*x^2 - 36036*x = 139129 has 20 integer points with 0 < y < 2*10^7 and all 20 have prime y-coordinate. For |x| < 10^8 and y>0 there are 32 integer points of which 23 have prime y coordinate. As Gerry Meyerson points out (see another message in this thread) there are infinitely many integer points per theory of Pell equations, but I don't know if there are more with prime y-coordinates. Jim Buddenhagen === Subject: Re: integer points on hyperbola James Buddenhagen escribi.97: > Ignacio Larrosa Ca.96estro >> James Buddenhagen escribi.97: > The hyperbola y^2 - 54516*x^2 - 36036*x = 139129 seems > quite unusual. All 'smallish' integer points have >> y| a prime number. For example [-12, 2749],[-10, 2287], > [-1, 397],[0, 373],[3, 859],[5, 1297],[8, 1979],[118, 27631] > are such points. This pattern persists for y-values well > past one million. > Are there infinitely many such 'prime-y-value' integer > points? > Jim Buddenhagen >> The first composite y-value for positive x is 25159661 = 1511*16651, >> corresponding to x = 107756. >> The next integer points after [118, 27631] are: >> x: [221, 469, 1929, 29639, 49028, 107756] >> y: [51679, 109583, 450473, 6920387, 11447459, 25159661] >> No others with 0 < x < 5*10^5. >> -- >> Ignacio Larrosa Ca.96estro >> A Coru.96a (Espa.96a) >> ilarrosaQUITARMAYUSCULAS@mundo-r.com > for all integer points for -10^8 < x < 10^8. There are > 32 with y positve: > [-20128306, 443*10608781 ] > [-9158230, 89*9739*2467 ] > [-1429340, 333731683 ] > [-925779, 216156929 ] > [-360389, 179*470089 ] > [-87669, 449*45589 ] > [-16576, 3870197 ] > [-10674, 2492159 ] > [-5707, 1332431 ] > [-1570, 366497 ] > [-1017, 237379 ] > [-655, 152857 ] > [-159, 37049 ] > [-12, 2749 ] > [-10, 2287 ] > [-1, 397 ] > [ 0, 373 ] > [ 3, 859 ] > [ 5, 1297 ] > [ 8, 1979 ] > [ 118, 27631 ] > [ 221, 51679 ] > [ 469, 109583 ] > [ 1929, 450473 ] > [ 29639, 6920387 ] > [ 49028, 11447459 ] > [ 107756, 1511*16651 ] > [ 1065975, 281*885733 ] > [ 1993784, 79*5892683 ] > [ 3096246, 28817*25087 ] > [ 11256669, 181*14520893 ] > [ 17379536, 4057889131 ] > The y-values above are factorized when composite. > Summary: the curve y^2 - 54516*x^2 - 36036*x = 139129 > has 20 integer points with 0 < y < 2*10^7 and all 20 have > prime y-coordinate. For |x| < 10^8 and y>0 there are 32 > integer points of which 23 have prime y coordinate. > As Gerry Meyerson points out (see another message in this > thread) there are infinitely many integer points per theory > of Pell equations, but I don't know if there are more with > prime y-coordinates. > Jim Buddenhagen In the 'Alpertron' (http://www.alpertron.com.ar/CUAD.HTM) you can introduce the coefficients of the equation, and get about 50 primitive solutions, with y > 0, and a second order no homogeneus recurrence relation with constant coefficients to get from each one of them a infinite sequence of derived solutions. -- Ignacio Larrosa Ca.96estro A Coru.96a (Espa.96a) ilarrosaQUITARMAYUSCULAS@mundo-r.com === Subject: Re: Possible proof of Gabriel's Theorem? Q: If f' exists everywhere need f' be continuous? A: No. I cannot think of *one* example where this is true. Again you are stating something irrelevant and skirting the main issue at hand. === Subject: Re: Possible proof of Gabriel's Theorem? > Q: If f' exists everywhere need f' be continuous? > A: No. > I cannot think of *one* example where this is true. Again you are > stating something irrelevant and skirting the main issue at hand. Surely you're not saying you can't think of a function which exists everywhere but is not continuous. If the left and right limits of f(x) as x->x0 are not the same, f is not continuous. Consider: 1. Does a step function exist everywhere? 2. Is the step function continuous? 3. Can you think of a function whose derivative is a step function? - Randy === Subject: Re: Possible proof of Gabriel's Theorem? >>Q: If f' exists everywhere need f' be continuous? >>A: No. >>I cannot think of *one* example where this is true. Again you are >>stating something irrelevant and skirting the main issue at hand. > Surely you're not saying you can't think of a function > which exists everywhere but is not continuous. > If the left and right limits of f(x) as x->x0 are not > the same, f is not continuous. > Consider: > 1. Does a step function exist everywhere? > 2. Is the step function continuous? > 3. Can you think of a function whose derivative > is a step function? That's not going to get the desired example. A step function has no antiderivative (though it has an almost-everywhere antiderivative or whatever the correct term is). One standard example of an everywhere-differentiable function whose derivative is not continuous is: | x^2 sin(1/x) , x != 0 f(x) = | | 0 , x = 0 === Subject: Re: Possible proof of Gabriel's Theorem? > Also, I _know_ that in general interchanging limits is > the hard part when you're proving almost anything in > analysis - if you just assume that that works you're > usually sweeping the entire proof under the rug. Hmm. This was a quite nice statement, I think. -- Eray === Subject: Re: Possible proof of Gabriel's Theorem? > Riemann's integral is a joke next to gabriel's average > derivative. Gabriel does not use a mesh value I don't claim to be an expert on analysis, nor do I have the time to decipher Gabriel's (pseudo?) math, but the Riemann integral does not necessarily use a mesh value. Most analysis texts these days seem to define the upper integral F as inf U(g,P) and the lower integral f as sup L(g,P). If F=f, then the function g is integrable. === Subject: Re: Possible proof of Gabriel's Theorem? Actually, they all use a mesh value. However, the riemann integral is an *approximation*. Gabriel's average tangent/derivative theorem is not an approximation - it is *natural integration*. You can take any definite integral and compute it's value using the average derivative. So you do not have the time to *decipher* gabriel's math yet you question its validity? Strange, I did not know that math needs deciphering for it is either true or false. If you have not looked at something, why do you by default cast doubt on its truth? You are simply following in the footsteps of most fools on this forum. The majority of posts including those by math professors demonstrate they (the real trolls) did not even bother to so much as read gabriel's stuff. All I have seen so far is sheer arrogance, stupidity and the utmost ignorance. Now allow me to critisize your post: > I don't claim to be an expert on analysis, nor do I have the time to > decipher Gabriel's (pseudo?) math, but the Riemann integral does not > necessarily use a mesh value. Most analysis texts these days seem to > define the upper integral F as inf U(g,P) and the lower integral f as > sup L(g,P). If F=f, then the function g is integrable. Most analysis texts are a load of rubbish and are taught by insecure individuals the likes of whom can be found on this forum. What are you trying to say? What does *integrable* mean? It's not an English word. Let's suppose you mean that g is a function and that by *integrable* you mean it can be integrated. So what is P then? Do you think that most people who read this forum know what you mean? === Subject: Re: Possible proof of Gabriel's Theorem? >Actually, they all use a mesh value. However, the riemann integral is >an *approximation*. Gabriel's average tangent/derivative theorem is not >an approximation - it is *natural integration*. You can take any >definite integral and compute it's value using the average derivative. >So you do not have the time to *decipher* gabriel's math yet you >question its validity? Strange, I did not know that math needs >deciphering for it is either true or false. If you have not looked at >something, why do you by default cast doubt on its truth? You are >simply following in the footsteps of most fools on this forum. The >majority of posts including those by math professors demonstrate they >(the real trolls) did not even bother to so much as read gabriel's >stuff. All I have seen so far is sheer arrogance, stupidity and the >utmost ignorance. >Now allow me to critisize your post: >> I don't claim to be an expert on analysis, nor do I have the time to >> decipher Gabriel's (pseudo?) math, but the Riemann integral does not >> necessarily use a mesh value. Most analysis texts these days seem to >> define the upper integral F as inf U(g,P) and the lower integral f as >> sup L(g,P). If F=f, then the function g is integrable. >Most analysis texts are a load of rubbish and are taught by insecure >individuals the likes of whom can be found on this forum. What are you >trying to say? What does *integrable* mean? It's not an English word. >Let's suppose you mean that g is a function and that by *integrable* >you mean it can be integrated. So what is P then? Do you think that >most people who read this forum know what you mean? Yes, most people reading this thread know what integrable means. The fact that you obviously don't is one of the many things that makes it all amusing enough to persuade some people to keep reading your posts. Definition: Suppose that f:[a,b] -> R is bounded. We say f is (Riemann) integrable if for every epsilon > 0 there exists delta > 0 such that if a = t_0 < ... < t_n = b, I_j = [t_{j-1}, t_j], M_j is the sup of f on I_j and m_j is the inf of f on I_j then the sum of (M_j - m_j)*(length(I_j)) is less than epsilon. You really didn't know that? If you don't know what the word integrable means then the fact that you've been replying to posts where people have been discussing the concept seems very curious. ************************ David C. Ullrich === Subject: Re: Possible proof of Gabriel's Theorem? <9mlo21djhdjkhd1pfolgu7mjj6gc45c0bk@4ax.com> > Definition: Suppose that f:[a,b] -> R is bounded. We say f is > (Riemann) integrable if for every epsilon > 0 there exists > delta > 0 such that if a = t_0 < ... < t_n = b, > I_j = [t_{j-1}, t_j], M_j is the sup of f on I_j and m_j is > the inf of f on I_j then the sum of (M_j - m_j)*(length(I_j)) > is less than epsilon. > You really didn't know that? If you don't know what the word > integrable means then the fact that you've been replying > to posts where people have been discussing the concept seems > very curious. Actually I know exactly what it is. I am questioning the relevance of his post. I am not sure he knows exactly what he is talking about. See, this is an example of you skirting the issues. Forget about all the other bozos on here! I am trying to discourage anyone who does not know what they are talking about from posting to this thread. As for the definition you provided, although I know it, I do not fully agree with it. As I stated earlier, I don't like to talk about constant functions such as f(x) = c for c some constant as being differentiable. They are not differentiable in my opinion. Now the above definition covers your ass for such functions but in truth, the ftoc fails miserably for any such function. i.e. f' = 0. The indefinite integral of 0 is some constant and its evaluation is 0 since f(x+w)-f(x) = 0 for any such constant function. So here you have a constant function which is differentiable but not *integrable*. Wow, makes a lot of sense for a learner, doesn't it? The epsilon-delta definition is a load of hogwash in my opinion. In this case of a constant functioin it is stating that f(x) = c is *not* integrable since no epsilon > 0 exists. I hinted at this earlier but seeing you were all so stupid, I simply avoided it. So why don't you try to answer the questions and stop being so arrogant! === Subject: Re: Possible proof of Gabriel's Theorem? <9mlo21djhdjkhd1pfolgu7mjj6gc45c0bk@4ax.com> Discussion, linux) >What does *integrable* mean? It's not an English word. [...] > Actually I know exactly what it is. I am questioning the relevance of > his post. I am not sure he knows exactly what he is talking about. You got a curious way of expressing yourself. -- 17:49 3/4/05: The proof is actually not hard, and it is perfect. 07:25 3/5/05: Nope. I made a mistake. 11:06 3/5/05: Maybe I screwed up[...] Otherwise, um, it's very easy to factor. 11:48 3/5/05: The answer is just that simple. -- JSH: A day in the life. === Subject: Re: Possible proof of Gabriel's Theorem? <9mlo21djhdjkhd1pfolgu7mjj6gc45c0bk@4ax.com> <87acpf1jil.fsf@phiwumbda.org> To what do I owe this pleasure? :-) Are you the famous JSH? > You got a curious way of expressing yourself. Well, considering all the nonsense and irrelevant junk that's posted on this forum, it's a miracle that I am even able to *express* myself. Ha, ha. I have a college math professor in this forum who is bent on reprimanding me and trolls from as far as France mouthing off at me! So forgive me if I am sometimes unable to express myself the way I should. Unfortunately, I know nothing about surrogate factoring so can't comment on your stuff. Ullrich and the others will be having a good laugh at this post. Some possible reactions might be: Giggle, a meeting/mating of fools... OR You said it, I didn't have to... OR See, I told you: Wells and Hughes are one and the same. Finally... OR Wells finally flipped over... The biggest and nastiest troll is Oliver! I think this miserable life form is studying psychotherapy with a major in BS(bull ). So when he gets his BS, it will be a double BS! I am targeting Ullrich because I think if he tries, he might understand it - maybe better than I do - but I am running out of hope fast. I have very little hope for all the others. Ed Hook initially seemed to be quite knowledgeable but then gave up me. Yan (from MIT) decided I was not worth the effort. Rodgers quit after he got some insight into my character and realized I wasn't such a bad guy. But he is still having a good laugh at me. :-) Oh well,.... Now I am studying for my Phd in math. Only module I need to complete now is the Analysis of 6-packs: I decided that the more I drink, the more eveyone on this forum begins to make sense. However, drinking a 6-pack will result in me losing my 6-pack and not drinking it, means I will never have a beer belly and thus shall never be awarded my Phd. Ha, ha. So it's quite a dilemma as you realize I am sure. I wonder what Ullrich looks like because he appeared to be annoyed at one post in which I vilified Chapman's personal hygiene and appearance. The last name implies germanic ancestry. Think Ullrich may be related to the great Weierstrass somewhere down the line? Ha, ha. So glad to have made your acquaintance Mr. Hughes. I wish you well in your endeavours. I am proud to be associated with you and gabriel! Here's to the revolution! Jason Wells Okay, I don't tell jokes ever but now will be the first time: HAve you heard this one? What's a BS, MS and PHd stand for ? BS - Bull MS - More PHd - Piled higher and deeper. Okay, so it's probably an old slapstick joke. I ask your forgiveness. I never was good at telling jokes (and no doubt someone will say I was never good at math either or I told you so). So please trolls, spare me the comments. I would like to keep this thread from getting any worse. otherwise I will just have to start a new one and god knows, some of you trolls will have a multiple fit! Ha, ha. === Subject: Re: Possible proof of Gabriel's Theorem? <9mlo21djhdjkhd1pfolgu7mjj6gc45c0bk@4ax.com> <87acpf1jil.fsf@phiwumbda.org> Discussion, linux) > To what do I owe this pleasure? :-) Are you the famous JSH? I'm merely his sycophant. I'm the unfamous JFH, not the famous JSH, but sometimes my .sig conveys the wrong message. -- Jesse F. Hughes Well, I guess that's what a teacher from Oklahoma State University considers proper as Ullrich has said it, and he is, in fact, a teacher at Oklahoma State University. -- James S. Harris presents a syllogism === Subject: Re: Possible proof of Gabriel's Theorem? > I'm the unfamous JFH, Are you sure you're not *in*famous? That's when you're *more* than famous, as in the *in*famous El Guapo. === Subject: Re: Possible proof of Gabriel's Theorem? to insert a few words to make the definition I gave correct: >> Definition: Suppose that f:[a,b] -> R is bounded. We say f is >> (Riemann) integrable if for every epsilon > 0 there exists >> delta > 0 such that if a = t_0 < ... < t_n = b, >> I_j = [t_{j-1}, t_j], length(I_j) < delta for all j, >> M_j is the sup of f on I_j and m_j is >> the inf of f on I_j then the sum of (M_j - m_j)*(length(I_j)) >> is less than epsilon. >> You really didn't know that? If you don't know what the word >> integrable means then the fact that you've been replying >> to posts where people have been discussing the concept seems >> very curious. >Actually I know exactly what it is. I am questioning the relevance of >his post. I am not sure he knows exactly what he is talking about. That's very curious - I don't see any reason to doubt that. >See, this is an example of you skirting the issues. Forget about all >the other bozos on here! I am trying to discourage anyone who does not >know what they are talking about from posting to this thread. >As for the definition you provided, although I know it, Curious that you didn't notice the _error_ in the version I posted then. >I do not fully >agree with it. As I stated earlier, I don't like to talk about constant >functions such as f(x) = c for c some constant as being differentiable. >They are not differentiable in my opinion. Right. Again, this is the sort of opinion that keeps some people reading your posts... >Now the above definition >covers your ass for such functions but in truth, the ftoc fails >miserably for any such function. i.e. f' = 0. The indefinite integral >of 0 is some constant and its evaluation is 0 since f(x+w)-f(x) = 0 for >any such constant function. So here you have a constant function which >is differentiable but not *integrable*. Wow, makes a lot of sense for a >learner, doesn't it? >The epsilon-delta definition is a load of hogwash in my opinion. In >this case of a constant functioin it is stating that f(x) = c is *not* >integrable since no epsilon > 0 exists. I hinted at this earlier but >seeing you were all so stupid, I simply avoided it. >So why don't you try to answer the questions and stop being so arrogant! Because you say so many things that are utterly ridiculous, at the same time exhibiting remarkable arrogance yourself. Here for example, you say that a constant function does not satisfy that definition? That's hilarious. Here's an excruciatingly detailed proof that if f = c then f satisfies the definition above: Suppose that epsilon > 0. Let delta = 1. Now assume that a = t_0 < ... < t_n = b, let I_j = [t_{j-1}, t_j], assume that length(I_j) < delta for all j, and let M_j and m_j be the sup and the inf of f on I_j, respectively. Then M_j = c and m_j = c, so sum (M_j - m_j)*(length(I_j)) = 0, and hence sum (M_j - m_j)*(length(I_j)) < epsilon. QED. ************************ David C. Ullrich === Subject: Re: Possible proof of Gabriel's Theorem? <9mlo21djhdjkhd1pfolgu7mjj6gc45c0bk@4ax.com> Ullrich: > Curious that you didn't notice the _error_ in the version I posted then. Jason: >I do not fully agree with it. As I stated earlier, I don't like to talk about constant >functions such as f(x) = c for c some constant as being differentiable. >They are not differentiable in my opinion. I don't pay much attention when you write incomprehensible junk. So you made a typo - big deal. I overlooked it because it was not important. See, this is what I mean about you: You are not interested in answering any questions, only in deviating from the main issue and knit-picking at stuff which really has nothing to do with the subject. You picked up where some twit had left off. Don't you have an opinion of your own? Show me step by step where gabriel's theorem is wrong or shut up. I meant that a delta > 0 does not exist, therefore we don't even need to look for an epsilon. since length(I_j) < delta and delta is eventually zero, no delta > 0 exists. This is a common problem with this sort of epsilon-delta analysis. Delta has to eventually be zero or else the integral is incomplete and thus not *integrable*. === Subject: Re: Possible proof of Gabriel's Theorem? [David C. Ullrich] >> Definition: Suppose that f:[a,b] -> R is bounded. We say f is >> (Riemann) integrable if for every epsilon > 0 there exists >> delta > 0 such that if a = t_0 < ... < t_n = b, >> I_j = [t_{j-1}, t_j], M_j is the sup of f on I_j and m_j is >> the inf of f on I_j then the sum of (M_j - m_j)*(length(I_j)) >> is less than epsilon. >> You really didn't know that? If you don't know what the word >> integrable means then the fact that you've been replying >> to posts where people have been discussing the concept seems >> very curious. ... [Jason] > The epsilon-delta definition is a load of hogwash in my opinion. In > this case of a constant functioin it is stating that f(x) = c is *not* > integrable since no epsilon > 0 exists. Of course every epsilon > 0 exists, so no epsilon > 0 exists is nonsensical. Maybe you meant to say that, given an epsilon > 0, no delta exists satisfying the condition? Nope, a constant function f(x)=c is especially _easy_ this way. Then M_j - m_j = c-c = 0 regardless of the partition, so the sum of (M_j - m_j)*length(I_j) is 0 regardless of the partition, which is indeed less than epsilon. It doesn't matter which delta you pick in this case; e.g., always picking delta = b-a, regardless of epsilon, is enough to show the Riemann integrability of a constant function. > I hinted at this earlier but seeing you were all so stupid, I simply > avoided it. === Subject: Re: Possible proof of Gabriel's Theorem? One more thing: The father of real analysis was Karl Weierstrass who by the way, did not complete his degree choosing rather to nurture his beer belly and skills in fencing. He was a pathetic drunk whose works were largely ignored a long time. Today, his slippery and vague ideas and concepts are taught in *real analysis*. The use of epsilon-delta arguments is highly questionable for real analysis. Most supporters of real analysis will argue in strong support of no hyperreal numbers existing in *real analysis*, yet the very idea of epsilon-delta assumes real numbers are points defined on a number line. Weierstrass did not put calculus on a firm footing or more robust ground, he confounded most of today's so-called mathematicians who are nothing more than fat, beer bellied drunks who are by nature thugs and slimebags. How can anything sound proceed from an unhealthy mind and a sack of beer? Wow, am I gonna get responses to this!! Sheeesh, I just don't want to think how many trolls are going to jump on their band wagon... Fat and ugly boys like hammick and company are sure to splatter their beer's worth without any doubt. Okay trolls, do your thing... === Subject: Re: Possible proof of Gabriel's Theorem? The more I discuss it on this forum, the more convinced I am becoming that it is true and every one here is a blundering fool! So how is that for objectivity? Now dale, do you have anything *mathematical* you care to post or do you just like posting crap for the heck of it? Now careful! If you disrespect me by posting another crappy message I simply won't respond. get it? By the way, I prefer to read only stuff pertaining to the subject. I don't care what you think of gabriel or of me or jesus christ. === Subject: Re: Possible proof of Gabriel's Theorem? > The more I discuss it on this forum, the more convinced I am becoming > that it is true and every one here is a blundering fool! Is this to be taken literally, or do I sense rhetoric? > So how is that for objectivity? Now dale, do you have anything > *mathematical* you care to post or do you just like posting crap for > the heck of it? Now careful! If you disrespect me by posting another > crappy message I simply won't respond. get it? > By the way, I prefer to read only stuff pertaining to the subject. I > don't care what you think of gabriel or of me or jesus christ. Why so rude? I would hope you don't use such a tone when you deal with people in person. I'd also hope that you will note the tone that I'm taking. It is at least civil, and I am well within reason to expect a civil tone in return. As far as disrespecting you, please note that I merely pointed out that your earlier disclaimer appeared not to be in effect any more, if indeed it was a bona fide disclaimer when it was issued. Pointing out the truth amounts to no disrespect. For the record, I have no problem with people having allegiances such as you appear to have wrt Gabriel's work; there are folks who hold to all sorts of preferences, and this one would hardly register as out of the ballpark. However, posing as a person with no opinion, and then attacking people for having opinions contrary to the one you have stiven to hide, smacks of trolling. BTW, if you actually cared about the subject, you would have found that the arguments presented by WWW, Ullrich, Chapman, and others were more correct than those given in Gabriel's notes or those that you yourself have given. For you to maintain, for instance, that Ullrich doesn't know Calculus, is patent nonsense. You might as well go to Paris and announce that they don't know how to speak French. You can be assured that most, if not all, of the contributors to these Gabriel threads have a more solid grasp of the subject matter than you do, unless you've been less than forthcoming about your own background. All things considered, the hypothesis that you are a troll is all that's really in balance. We have had our share of trolls here, some of them virtually indistinguishable from people with whom actual discussions could be had, and some who had verifiable knowledge of mathematics. In that regard, your case is not particularly challenging so far. As you may know, it's the uncertainty about trolls that makes for their value, not the vehemence of their rants. Dale === Subject: Re: Possible proof of Gabriel's Theorem? Your first response to me was hardly civil. Yet you have the gall to approach me now and imply that I am rude. My first defence on a forum such as this is to shoot first and ask questions later. Evidently you don't know much either for you stated that ...arguments presented by WWW, Ullrich, Chapman... were better than mine. What absolute rubbish! To begin, they posted no arguments except to attack, vilify and reduce the significance of my post. There is no mathematical value in any of their posts! Talk about trolls? This is troll behaviour without any shadow of a doubt! Are you so blinded by these fellows that you can't think for yourself? As for Ullrich knowing Calculus: I am not saying he does not know calculus but what I am saying is that he knows about the same as any other lecturer I have encountered with some exceptions. That is to say, what he knows does not seem to help him or me very much. He is not prepared to provide any proof or even consider gabriel's theorem. He has dismissed it without even having read any of the work. Have you read any of it? Have you understood any of it? I am not a troll - not in the computer sense or any other. I don't give up easily. Let's say I remain unconvinced by anything Ullrich has said (which is close to nothing pertaining to the subject). I have openly admitted that I am not a mathematician by occupation but I am not a fool either. Furthermore, I am not afraid of anything or anyone - no, not god himself. Do you think I would be afraid or intimidated by any of these ing fools?! Think again! I have a very healthy body and a strong mind. I know my weaknesses and strengths - this makes me better than most poeple I know. Dealing with idiots is one of my weaknesses and boy oh boy, are there a lot on this forum! Not to mention sociopaths. psychopaths and the likes of Ullrich - an academopath. === Subject: Re: Possible proof of Gabriel's Theorem? >The more I discuss it on this forum, the more convinced I am becoming >that it is true and every one here is a blundering fool! When you're convinced that _everyone_ else is wrong there's usually a more plausible explanation. >So how is that for objectivity? Now dale, do you have anything >*mathematical* you care to post or do you just like posting crap for >the heck of it? Now careful! If you disrespect me by posting another >crappy message I simply won't respond. get it? >By the way, I prefer to read only stuff pertaining to the subject. I >don't care what you think of gabriel or of me or jesus christ. ************************ David C. Ullrich === Subject: Re: Possible proof of Gabriel's Theorem? <722m21tqviiidui5s2jhauha932hn2199l@4ax.com> > Facinating. A minute ago I saw a post from _you_ admitting > that in fact f' need not be continuous: > Have just thought of an example where f' may not be continuous. Yes, I did say this. But, I am telling you that gabriel's theorem does not require f' to be continuous at x+w. I see no discrepancies here. > So far you've called me uninformed for my opinion on this > question. Above you say it makes my real analysis go down > the toilet and question whether I should be a professor. > Now it turns out that you were wrong about this (and > you _admit_ you were wrong). But there's been no apology > yet for the things you said. Actually I have been the only one that's forthright thus far. You were wrong about gabriel's definition containing two variables and you are still wrong about a lot of other things. you can be absolutely sure we disagree on most things but this doesn't say that you are particularly good at anything. So far, you have made several unsubstantiated claims: - You have said Gabriel's derivative definitioin contains more than one variable. You are wronog! - You have ignorantly stated that gabriel's theorem is a trivial consequence of the ftoc. You are wrong! Did you admit to any of this? No. I don't think so. As for complaining, I am bothered when supposedly educated men like you say things like: *You don't know what you are talking about.* I really don't care what most people on this forum say. I have little or no respect for them. Shouldn't they be apologizing? Daivd, every time you see you are wrong about something, you conveniently bring up an issue with the other posters on this forum. Again, their opinions mean nothing to me. Me a crackpot? Giggle, are you reading any of the other comments? Please tell me what on earth a contributor's opinion of me (troll, or whatever) has to do with the ing subkect I am discussing. They can all go to hell for all I care. Anyone who is rude, who attacks the character of someone else without knowing jack can in fact get stuffed! Trust me David, I don't lose any sleep over anyone's opinion - not in this forum or any other. And hey, am I the only one to make mistakes? You have all made mistakes. Please spare me the sanctimonious 'better than thou' crap. Just answer the questions as I have tried to answer your questions. So far, you have answered nothing and have made a few serious mistakes. Forget about my character: None of you on this site even have a clue of what kind of individual I am. And I would not bother with you in any other situation were it outside this forum. If you think my messages are so annoying, then why do you respond? if I have offended you so much, why on earth do you bother? Troll: I have not left any annoying messages on this forum yet I get the likes of *real trolls* like feldman and hale and whoever else frequents this forum being guilty of exactly what they accuse JSH and me. Now I have not read JSH's posts and know little about what he is trying to achieve - this is not the point. Point is you are in fact guilty of what you accuse others to be and not the other way around. Make a useful contribution or shut up I say. The choice is yours. I am responding to you because I still respect you. All the other assholes are not going to get a response from me. I asked you to take this offline with me so that we can discuss it without all the two cents worth of every tom, dick and harry in this forum. I know they are all basically very stupid. Have sent you a private email but so far you have not responded. Why don't you take this off-line with me? Too afraid I might convince you? :-) Truth is you may end up convincing me that gabriel is indeed wrong because I still have some questions. What really keeps me going is the fact neither you nor anyone else has been able to disprove or prove *anything* aside from rhetoric, ignorant posts and irrelevant troll-like comments. So let's see what you are really made of David - all talk or just another troll on sci.math... === Subject: Re: Possible proof of Gabriel's Theorem? >> Facinating. A minute ago I saw a post from _you_ admitting >> that in fact f' need not be continuous: >> Have just thought of an example where f' may not be continuous. >Yes, I did say this. But, I am telling you that gabriel's theorem does >not require f' to be continuous at x+w. I see no discrepancies here. >> So far you've called me uninformed for my opinion on this >> question. Above you say it makes my real analysis go down >> the toilet and question whether I should be a professor. >> Now it turns out that you were wrong about this (and >> you _admit_ you were wrong). But there's been no apology >> yet for the things you said. >Actually I have been the only one that's forthright thus far. You were >wrong about gabriel's definition containing two variables Nope. >and you are >still wrong about a lot of other things. you can be absolutely sure we >disagree on most things but this doesn't say that you are particularly >good at anything. So far, you have made several unsubstantiated claims: > - You have said Gabriel's derivative definitioin contains more > than one variable. You are wronog! > - You have ignorantly stated that gabriel's theorem is a > trivial consequence of the ftoc. You are wrong! Nope. _If_ we add the assumption that f' is continuous then it _is_ a trivial consequence of ftoc. As many people have pointed out, in varying degrees of detail. >Did you admit to any of this? No. I don't think so. As for complaining, >I am bothered when supposedly educated men like you say things like: >*You don't know what you are talking about.* That's a fact, that you prove with every post, whether it bothers you or not. >I really don't care what >most people on this forum say. I have little or no respect for them. >Shouldn't they be apologizing? Daivd, every time you see you are wrong >about something, Huh? It has not happened yet that I've seen I was wrong about anything that's come up between the two of us. >you conveniently bring up an issue with the other >posters on this forum. Again, their opinions mean nothing to me. >Me a crackpot? Giggle, are you reading any of the other comments? >Please tell me what on earth a contributor's opinion of me (troll, or >whatever) has to do with the ing subkect I am discussing. They can >all go to hell for all I care. Anyone who is rude, who attacks the >character of someone else without knowing jack can in fact get >stuffed! Do you read _your_ posts? People who are rude can get stuffed. >Trust me David, I don't lose any sleep over anyone's opinion - >not in this forum or any other. And hey, am I the only one to make >mistakes? You have all made mistakes. Yes. Doesn't change the fact that more or less everything you've said about all this is wrong. >Please spare me the sanctimonious >'better than thou' crap. Just answer the questions as I have tried to >answer your questions. So far, you have answered nothing and have made >a few serious mistakes. >Forget about my character: I don't recall ever saying anything about your character. >None of you on this site even have a clue of >what kind of individual I am. And I would not bother with you in any >other situation were it outside this forum. >If you think my messages are so annoying, then why do you respond? >if I have offended you so much, why on earth do you bother? If anyone's curious, this is the part that made me decide to reply to this message instead of just dropping it as hopeless. Back to Jason: Huh? What makes you think that I find your messages annoying? I haven't told _you_ to get stuffed, I've just commented on the math. In fact I don't find your messages annoying, I find them hilarious. In case you didn't know, so do a lot of people - every time you make a post here it leads to people all over the planet rolling on the floor. Honest. >Troll: I have not left any annoying messages on this forum yet I get >the likes of *real trolls* like feldman and hale and whoever else >frequents this forum being guilty of exactly what they accuse JSH and >me. Now I have not read JSH's posts and know little about what he is >trying to achieve - this is not the point. Point is you are in fact >guilty of what you accuse others to be and not the other way around. >Make a useful contribution or shut up I say. I have. The fact that you keep saying I'm wrong doesn't make me wrong. >The choice is yours. I am >responding to you because I still respect you. All the other assholes >are not going to get a response from me. I asked you to take this >offline with me so that we can discuss it without all the two cents >worth of every tom, dick and harry in this forum. I know they are all >basically very stupid. Have sent you a private email but so far you >have not responded. Why don't you take this off-line with me? Too >afraid I might convince you? :-) No. The reason is that it was clear long ago that there's no possibility anyone will ever convince you that you're wrong about more or less everything, so there would be no point to an offline discussion. As opposed to the online stuff, where one point is that it's public - when people make incorrect statements in a place like this where others come to learn about math then it's proper for others to point out the errors. Also it's all extremely amusing - wouldn't be anything funny about an offline exchange because nobody else would be able to see it. >Truth is you may end up convincing me >that gabriel is indeed wrong because I still have some questions. What >really keeps me going is the fact neither you nor anyone else has been >able to disprove or prove *anything* aside from rhetoric, ignorant >posts and irrelevant troll-like comments. I don't believe that anyone has _said_ that his theorem is false as stated. But nobody has given a proof of it, and people _suspect_ it's false, for example if there does indeed exist a non-constant differentiable f such that f'(r) = 0 for all rational r, as I suspect, then the theorem's certainly false. >So let's see what you are really made of David - all talk or just >another troll on sci.math... ************************ David C. Ullrich === Subject: Re: Possible proof of Gabriel's Theorem? <722m21tqviiidui5s2jhauha932hn2199l@4ax.com> > In fact I don't find your messages annoying, I find them hilarious. > In case you didn't know, so do a lot of people - every time you > make a post here it leads to people all over the planet rolling > on the floor. Honest. Hate to tell you that *you* are the one being laughed at. :-) Why? Well, I don't claim to know. That's why I post messages on this board soliciting information from *those who claim to know*. Now you claim to know yet not once have you been able to answer any of the questions I asked you. Your responses are hilarious and non-commital, e.g. Nope. Honest. etc. > I have. The fact that you keep saying I'm wrong doesn't make me wrong. You have been wrong two times I am aware of. The rest of your commitments say nothing so how can you say too much which is wrong? You are supposed to be a mathematician, no? Well, show me through mathematics where and why in a detailed explanation how Gabriel's theorem is wrong. You keep changing your tone! Well, what exactly are you saying? > No. The reason is that it was clear long ago that there's no > possibility anyone will ever convince you that you're wrong > about more or less everything, so there would be no point to > an offline discussion. Nothing is clear from what you have written. You have a non-comittal approach and you have made at least two mistakes which you refuse to admit. Once again David, you need to admit your mistakes and support your criticisms. You have not been able to convince me of anything because you have not said much that makes sense yet. You pick out some vague topic and then present an example which is irrelevant. As a mathematician you would need to show in detail (giving reasons) why a proof is incorrect. Just remember this is not my work and I am in fact answering questions which gabriel himself should be answering. I may not be answering correctly all the time but I am trying... My occupation is *hobby mathematician*. I openly admit this. You are supposed to be a professional?! > I don't believe that anyone has _said_ that his theorem is false > as stated. But nobody has given a proof of it, and people > _suspect_ it's false, for example if there does indeed exist > a non-constant differentiable f such that f'(r) = 0 for all > rational r, as I suspect, then the theorem's certainly false. Giggle, giggle. About 90% of the contributors on this forum have said his theorem is false! Can you *read* ?! Have you understood any of the posts? Now I am rolling in laughter. As for a proof: I attempted a proof and you have not been able to show me anywhere that this proof is incorrect. I don't know if it is indeed correct or not. I outlined my concerns which once again you failed to address. Probably because this is how you teach your classes too: you simply dismiss questions as irrelevant when you don't have answers or you are afraid to commit yourself. So David, if indeed the planet is laughing at me, I am at least accomplishing some good: laughter is good for the bones! Here's to the *good health* of all those reading my funny posts!! ha, ha. Please continue to read and improve your health! Your bill will be in the mail soon. What good are you doing? Dr. Jason Wells. Hee, hee. === Subject: Re: Possible proof of Gabriel's Theorem? <722m21tqviiidui5s2jhauha932hn2199l@4ax.com> > Hate to tell you that *you* are the one being laughed at. :-) Why? > Well, I don't claim to know. That's why I post messages on this board > soliciting information from *those who claim to know*. Now you claim to > know yet not once have you been able to answer any of the questions I > asked you. Your responses are hilarious and non-commital, e.g. OK. Let's start a vote here. I am laughing at Jason Wells. Score: Ullrich: 1. Jason: 0. > You are supposed to be a mathematician, no? Well, show me through > mathematics where and why in a detailed explanation how Gabriel's > theorem is wrong. You keep changing your tone! Well, what exactly are > you saying? He did not say that the theorem is wrong. He just pointed out that the proof is wrong. Do you get that, Jason? Or do you think that any attack on the proof is an attack on the theorem (conjecture/whatever) itself? > have not said much that makes sense yet. You pick out some vague topic > and then present an example which is irrelevant. As a mathematician you Just because you cannot understand what he is saying does not make it irrelevant. And *you* talk about being open minded... > would need to show in detail (giving reasons) why a proof is incorrect. No. It is the responsibility of the theorem 'prover' to come up with coherent definitions and arguments as to why the theorem is true. btw, you were given detailed reasons as to why the proof is incorrect. You chose to ignore them. > I don't believe that anyone has _said_ that his theorem is false > as stated. But nobody has given a proof of it, and people > _suspect_ it's false, for example if there does indeed exist > a non-constant differentiable f such that f'(r) = 0 for all > rational r, as I suspect, then the theorem's certainly false. > Giggle, giggle. About 90% of the contributors on this forum have said > his theorem is false! Can you *read* ?! Have you understood any of the > posts? The theorem as stated is false! As I can see on the webpage, there are three parts to the theorem and part ii is clearly false. I think David is talking about part i. > Now I am rolling in laughter. Me too, but at you. > As for a proof: I attempted a proof and you have not been able to show > me anywhere that this proof is incorrect. He has. Read carefully and try to understand this time. > Dr. Jason Wells. Hee, hee. I see you are getting better :-) === Subject: Re: Possible proof of Gabriel's Theorem? <722m21tqviiidui5s2jhauha932hn2199l@4ax.com> You are an imposter! As promised I will be reporting you! What a coward you are! Too afraid to post anything under your real name? > Hate to tell you that *you* are the one being laughed at. :-) Why? > Well, I don't claim to know. That's why I post messages on this board > soliciting information from *those who claim to know*. Now you claim > to > know yet not once have you been able to answer any of the questions I > asked you. Your responses are hilarious and non-commital, e.g. > OK. Let's start a vote here. > I am laughing at Jason Wells. > Score: Ullrich: 1. Jason: 0. > You are supposed to be a mathematician, no? Well, show me through > mathematics where and why in a detailed explanation how Gabriel's > theorem is wrong. You keep changing your tone! Well, what exactly are > you saying? > He did not say that the theorem is wrong. He just pointed out that the > proof is wrong. Do you get that, Jason? Or do you think that any attack > on the proof is an attack on the theorem (conjecture/whatever) itself? > have not said much that makes sense yet. You pick out some vague > topic > and then present an example which is irrelevant. As a mathematician > you > Just because you cannot understand what he is saying does not make it > irrelevant. And *you* talk about being open minded... > would need to show in detail (giving reasons) why a proof is > incorrect. > No. It is the responsibility of the theorem 'prover' to come up with > coherent definitions and arguments as to why the theorem is true. > btw, you were given detailed reasons as to why the proof is incorrect. > You chose to ignore them. > I don't believe that anyone has _said_ that his theorem is false > as stated. But nobody has given a proof of it, and people > _suspect_ it's false, for example if there does indeed exist > a non-constant differentiable f such that f'(r) = 0 for all > rational r, as I suspect, then the theorem's certainly false. > Giggle, giggle. About 90% of the contributors on this forum have said > his theorem is false! Can you *read* ?! Have you understood any of > the > posts? > The theorem as stated is false! As I can see on the webpage, there are > three parts to the theorem and part ii is clearly false. I think David > is talking about part i. > Now I am rolling in laughter. > Me too, but at you. > As for a proof: I attempted a proof and you have not been able to > show > me anywhere that this proof is incorrect. > He has. Read carefully and try to understand this time. > Dr. Jason Wells. Hee, hee. > I see you are getting better :-) === Subject: Re: Possible proof of Gabriel's Theorem? <722m21tqviiidui5s2jhauha932hn2199l@4ax.com> > You are an imposter! As promised I will be reporting you! > What a coward you are! Too afraid to post anything under your real > name? Silly boy. How hard is it to comprehend the fact that there could be more than one John Gabriel in this world? Why, there are at least 10 in the state of Texas alone. === Subject: Re: Possible proof of Gabriel's Theorem? <722m21tqviiidui5s2jhauha932hn2199l@4ax.com> > Silly boy. How hard is it to comprehend the fact that there could be > more than one John Gabriel in this world? Why, there are at least 10 in > the state of Texas alone. There are a lot of fools in this world. How hard is it for you to comprehend that you are one? === Subject: Re: Possible proof of Gabriel's Theorem? <722m21tqviiidui5s2jhauha932hn2199l@4ax.com> > Silly boy. How hard is it to comprehend the fact that there could be > more than one John Gabriel in this world? Why, there are at least 10 > in > the state of Texas alone. > There are a lot of fools in this world. How hard is it for you to > comprehend that you are one? Taking your maturity into account, a reply to this would be: It takes one to know one. Giggle Giggle.. tee hee.. Seriously though, Bubba, ever thought of seeing a doctor? Maybe if you take JSH along you will get a discount. Heck, the doctor might even say that you are right. Good luck. === Subject: Re: Possible proof of Gabriel's Theorem? >>Truth is you may end up convincing me >>that gabriel is indeed wrong because I still have some questions. What >>really keeps me going is the fact neither you nor anyone else has been >>able to disprove or prove *anything* aside from rhetoric, ignorant >>posts and irrelevant troll-like comments. >I don't believe that anyone has _said_ that his theorem is false >as stated. But nobody has given a proof of it, and people >_suspect_ it's false, for example if there does indeed exist >a non-constant differentiable f such that f'(r) = 0 for all >rational r, as I suspect, then the theorem's certainly false. Yikes! I wasn't expecting that. I killfiled Jason in exasperation some time ago, and I haven't been following these threads regularly, but this made me sit up and take notice. You're saying that, in spite of the strange presentation on Gabriel's website, and in spite of Jason's many hilarious misconceptions (about constant functions not being differentiable, and so on), there is a coherent conjecture here that hasn't been settled? I don't suppose you could save me the effort of deciphering Gabriel's been competently formulated, could you? Sorry I haven't been paying attention. I wasn't expecting actually to have to look at the maths, and I only hope my rusty analysis is up to the job. I'd better get that hat and humble pie ready! -- Angus Rodgers Contains mild peril === Subject: Re: Possible proof of Gabriel's Theorem? >[...] >You're saying that, in spite of the strange presentation on Gabriel's >website, and in spite of Jason's many hilarious misconceptions (about >constant functions not being differentiable, and so on), there is a >coherent conjecture here that hasn't been settled? >I don't suppose you could save me the effort of deciphering Gabriel's >been competently formulated, could you? Suppose that f is differentiable on [0,1] (or differentiable on (0,1) and continuous on [0,1]). Does it follow that f(1) - f(0) = lim_N sum_1^N f'(j/N)/N ? Of course this is clear from the fundamental theorem of calculus if f' is continuous. It seems unlikely that it's true under the stated hypotheses, but seems unlikely is not quite a counterexample. (Also it's not entirely clear whether this is exactly Gabriel's claim, because of confusion over the definition of derivative; we've been given several mutually inconsistent definitions, together with the assertion that we mean the same thing as usual by f'... The question I was talking about takes the standard definition of the derivative.) >Sorry I haven't been paying attention. I wasn't expecting actually >to have to look at the maths, and I only hope my rusty analysis is >up to the job. I'd better get that hat and humble pie ready! ************************ David C. Ullrich === Subject: Re: Possible proof of Gabriel's Theorem? <722m21tqviiidui5s2jhauha932hn2199l@4ax.com> >[...] > Suppose that f is differentiable on [0,1] (or differentiable > on (0,1) and continuous on [0,1]). Does it follow that > f(1) - f(0) = lim_N sum_1^N f'(j/N)/N ? > Of course this is clear from the fundamental theorem of calculus > if f' is continuous. It seems unlikely that it's true under the > stated hypotheses, but seems unlikely is not quite a counterexample. This is just the first part of Gabriel's theorem. Part ii) states the Integral (x to x+w) f'(t)dt = some limit on the right. We have many counterexamples which show that if f is differentiable, then f' need not be Riemann (or even Lebesgue) integrable. So the theorem as stated is wrong. Part i) might be true, but I am confident there will be counterexamples to this. I read somewhere that there are strictly increasing differentiable functions g such that g'(x) = 0 almost everywhere, (supposedly this is given as Example 2.1.2.1 in B.R. Gelbaum and J M H Olmstead, Theorems and counterexamples in mathematics, Springer 1990) which leads me to believe there will be functions such that f(0) < f(1) and f'(r) = 0 for rational r. === Subject: Re: Possible proof of Gabriel's Theorem? <722m21tqviiidui5s2jhauha932hn2199l@4ax.com> No fool! This is not the first or any part of Gabriel's theorem. Can't you read anything correctly? Go back and read Gabriel's theorem. It is usually a good idea to look at something carefully before you open your ass and utter so much ! Just compare the above formula to Gabriel's theorem and you will see they are not the same! Not even remotely!!!! Fool!!!! === Subject: Re: Possible proof of Gabriel's Theorem? <722m21tqviiidui5s2jhauha932hn2199l@4ax.com> > No fool! This is not the first or any part of Gabriel's theorem. Can't > you read anything correctly? > Go back and read Gabriel's theorem. It is usually a good idea to look > at something carefully before you open > your ass and utter so much ! Just compare the above formula to > Gabriel's theorem and you will see they are not the same! > Not even remotely!!!! > Fool!!!! Speak when spoken to. Idiot. I was talking to Prof Ullrich, not to you. Kids these days... === Subject: Ullrich the Mathematician! <722m21tqviiidui5s2jhauha932hn2199l@4ax.com> > Suppose that f is differentiable on [0,1] (or differentiable > on (0,1) and continuous on [0,1]). Does it follow that > f(1) - f(0) = lim_N sum_1^N f'(j/N)/N ? > Of course this is clear from the fundamental theorem of calculus > if f' is continuous. It seems unlikely that it's true under the > stated hypotheses, but seems unlikely is not quite a counterexample. Now you talk about my posts being funny! Let's see how funny yours is: First you say suppose f is differentiable on [0,1] and in parentheses you write or differentiable on (0,1) and continuous on [0,1]. May I remind you the two are not the *equivalent*. Your statement seems to imply they are. In mathematics you need to be precise - either/or makes a big difference. Then you casually proceed to ask a question which follows on a shaky statement. Okay, let's see: Does your question pertain to the first part or second part of your statement or both? Wow! And you call yourself a mathematician. But wait, I am not done with you yet! > f(1) - f(0) = lim_N sum_1^N f'(j/N)/N ? What are you writing here? Let me see. What does *j* represent ?? Is it a variable of summation maybe? Giggle. Are you summing from 1 - N ? Cause if you are, it is incorrect. What does j/N mean? Pheeeww. I think I'll stop here. Talk about clarity!! Just out of curiousity, what percentage of your students pass? Or maybe I should ask: How many of them have learned anything from your classes? Jason Wells === Subject: Re: Ullrich the Mathematician! >> Suppose that f is differentiable on [0,1] (or differentiable >> on (0,1) and continuous on [0,1]). Does it follow that >> f(1) - f(0) = lim_N sum_1^N f'(j/N)/N ? >> Of course this is clear from the fundamental theorem of calculus >> if f' is continuous. It seems unlikely that it's true under the >> stated hypotheses, but seems unlikely is not quite a > counterexample. > Now you talk about my posts being funny! Let's see how funny yours is: > First you say suppose f is differentiable on [0,1] and in parentheses > you write or differentiable on (0,1) and continuous on [0,1]. > May I remind you the two are not the *equivalent*. I think they are equivalent. To be differentiable at the endpoints means one-sided differentiability and continuity there is enough. === Subject: Re: Ullrich the Mathematician! <722m21tqviiidui5s2jhauha932hn2199l@4ax.com> Hee, hee. In your tiny brain anything is possible you nasty toad! I rest my case with this psychopath. === Subject: Re: Ullrich the Mathematician! >>First you say suppose f is differentiable on [0,1] and in parentheses >>you write or differentiable on (0,1) and continuous on [0,1]. >>May I remind you the two are not the *equivalent*. > I think they are equivalent. To be differentiable at the endpoints means > one-sided differentiability and continuity there is enough. Continuous differentiability on an open interval, together with continuity at the endpoints, does not buy you even one-sided differentiability at the endpoints. You can see this from a minor variation of the last example I posted. Let | x sin(1/x) , 0 < x <= 1 f(x) = | | 0, , x = 0 f is continuous at 0 and (continuously) differentiable on (0,1), but not differentiable at 0. === Subject: Re: Ullrich the Mathematician! >First you say suppose f is differentiable on [0,1] and in parentheses >you write or differentiable on (0,1) and continuous on [0,1]. >May I remind you the two are not the *equivalent*. >> I think they are equivalent. To be differentiable at the endpoints means >> one-sided differentiability and continuity there is enough. > Continuous differentiability on an open interval, together with > continuity at the endpoints, does not buy you even one-sided > differentiability at the endpoints. You can see this from > a minor variation of the last example I posted. Let > | x sin(1/x) , 0 < x <= 1 > f(x) = | > | 0, , x = 0 > f is continuous at 0 and (continuously) differentiable on (0,1), but not > differentiable at 0. I'm glad somebody has an analysis book or a counterexamples in analysis book. === Subject: Re: Ullrich the Mathematician! <722m21tqviiidui5s2jhauha932hn2199l@4ax.com> <396nh1F5t6nj0U1@individual.net> > I'm glad somebody has an analysis book > or a counterexamples in analysis book. There was no correction needed you stupid tit! He didn't read the theorem correctly - just like you and all the others like you!! Bozo!! === Subject: Re: Ullrich the Mathematician! >> I'm glad somebody has an analysis book >> or a counterexamples in analysis book. > There was no correction needed you stupid tit! > He didn't read the theorem correctly - just like you and all the others > like you!! Bozo!! Whatever case you are trying to make, this isn't the way to make it. === Subject: Re: Ullrich the Mathematician! <722m21tqviiidui5s2jhauha932hn2199l@4ax.com> <396nh1F5t6nj0U1@individual.net> <0qqXd.687402$Zm5.98537@news.easynews.com> You are quite right! The trolls have accused me of being a troll. Now that I am retaliating, they are all offended. Oh shame, oh shame. They can dish it out just fine but when it gets dished right back at them - then they can't take it! I am not angry. Really I am not. But I take pleasure in making these trolls angry. Every response I get from a troll pleases me extremely for they show how stupid they are and I know they are angry which is bad for their body chemistry. With some luck, their blood pressure will rise, a heart-attack will follow and they can become daisy manure which is far more profitable for those who are really trying to accomplish something. lol A bit extreme. Yeah, but who cares. Most people here will not accept responsibility for a single bad word they say. === Subject: Re: Ullrich the Mathematician! >> I'm glad somebody has an analysis book >> or a counterexamples in analysis book. > There was no correction needed you stupid tit! > He didn't read the theorem correctly - just like > you and all the others like you!! Bozo!! Pipe down, sonnyboy. It's unnecessary to bring this verbal abuse here. Did you use your parents as sparring partners? === Subject: Mike the turd! (Was Ullrich the Mathematician!) <722m21tqviiidui5s2jhauha932hn2199l@4ax.com> <396nh1F5t6nj0U1@individual.net> > Continuous differentiability on an open interval, together with > continuity at the endpoints, does not buy you even one-sided > differentiability at the endpoints. You can see this from > a minor variation of the last example I posted. Let > | x sin(1/x) , 0 < x <= 1 > f(x) = | > | 0, , x = 0 > f is continuous at 0 and (continuously) differentiable on (0,1), but not > differentiable at 0. Of course you are correct - you turd! But if you bothered reading Gabriel's theorem, you would see that it requires the function to be differentiable over [x;x+w) and continuous over [x;x+w]. Are there only fools on this forum?! Jason === Subject: Jason The Dolt! (Was: Ullrich the Mathematician!) >> Suppose that f is differentiable on [0,1] (or differentiable >> on (0,1) and continuous on [0,1]). Does it follow that >> f(1) - f(0) = lim_N sum_1^N f'(j/N)/N ? >> Of course this is clear from the fundamental theorem of calculus >> if f' is continuous. It seems unlikely that it's true under the >> stated hypotheses, but seems unlikely is not quite a > counterexample. > Now you talk about my posts being funny! Let's see how funny yours is: > First you say suppose f is differentiable on [0,1] and in parentheses > you write or differentiable on (0,1) and continuous on [0,1]. > May I remind you the two are not the *equivalent*. > I think they are equivalent. To be differentiable at the endpoints means > one-sided differentiability and continuity there is enough. Take something like f (x) = x * sin (1 / x^2) if x > 0 f (x) = 0 if x = 0 g (x) = f (x) + f (1 - x). === Subject: Re: Ullrich the Mathematician! days. My association with the Department is that of an alumnus. >> Suppose that f is differentiable on [0,1] (or differentiable >> on (0,1) and continuous on [0,1]). Does it follow that >> f(1) - f(0) = lim_N sum_1^N f'(j/N)/N ? >> Of course this is clear from the fundamental theorem of calculus >> if f' is continuous. It seems unlikely that it's true under the >> stated hypotheses, but seems unlikely is not quite a >counterexample. >Now you talk about my posts being funny! Let's see how funny yours is: >First you say suppose f is differentiable on [0,1] and in parentheses >you write or differentiable on (0,1) and continuous on [0,1]. >May I remind you the two are not the *equivalent*. Your statement seems >to imply they are. In mathematics you need to be precise - either/or >makes a big difference. Then you casually proceed to ask a question >which follows on a shaky statement. Sigh. He gives you two conditions: one is clearly stronger than the second. The statement CLEARLY is saying that while the second (weaker) condition is enough for what follows, you are welcome to proceed with the first (stronger) assumption. > Okay, let's see: Does your question >pertain to the first part or second part of your statement or both? Either. The parenthetical comment is weaker than the original. If you can answer the question assuming only the weaker statement, then the same answer holds if you assume the stronger statement. But if you prefer to work with stronger hypothesis, you can use the stronger statement to see the same answer. >Wow! And you call yourself a mathematician. But wait, I am not done >with you yet! Wow. And you think this sort of attitude will engender anything but contempt and will buy you anything other than very little help in the future? >> f(1) - f(0) = lim_N sum_1^N f'(j/N)/N ? >What are you writing here? Let me see. What does *j* represent ?? Is it >a variable of summation maybe? Giggle. Perhaps if instead of laughing like a simpleton you had simply asked a question or two because you didn't get it, you would have gotten very nice and helpful responses... >Are you summing from 1 - N ? He is summing from one to N; the limit is being taken as N goes to infinity; j is the index of summation. This is all apparent, and j is ommitted to avoid overly cluttering the ASCII line. Would you have understoof better something like f(1) - f(0) = lim_{N->infinity}( sum_{from j = 1 to N} (f'(j/N)/N) ) ? >Cause if you are, it is incorrect. What does j/N mean? It seems clear that we are partitioning the interval [0,1] into N parts, and j/N is the right end point of the j-th subinterval. Then we are evaluating the defivative at this point. Does the equation hold, if f'(x) is continuous, by the fundamental theorem of calculus? The Fundamental Theorem of Calculus, applied to a continuous function f'(x), would tells us that integral_0^1 f'(x) dx = f(1) - f(0). So the left hand side of the equation would be equal to the integral. By the definition of integrable (and the theorem that continuous functions are integrable), we can evaluate the integral by taking the limit over any sequence of partitions, as the norm of the partition goes to 0, of any Riemann sum over that partition. Choosing the sequence of regular partitions into N intervals as N->infinity, and choosing the right point Riemman sum for each partition, yiels that the Riemann sum associated to the N-th partition is exactly equal to sum_{from j = 1 to N} f'(j/N)*(1/N) since the j-th subinterval goes from (j-1)/N to j/N, and we are taking the right endpoint; and the subinterval has length 1/N. So the right side is the limit as the norm of the partitions go to 0 of certain Riemann sums, and therefore, IF f'(x) is continuous, the right hand side is equal to the integral from 0 to 1 of f'(x). It follows from the Fundamental Theorem of Calculus that the left hand side is then equal to the right hand side. The question asked was: does this also follow if, rather than the hypothesis f'(x) is continuous, we only have the hypothesis f is differentiable on (a,b) and continuous on [a,b], or even the stronger hypothesis f is differentiable on [a,b] And his answer was, it seems unlikely, but 'seems unlikely' is not a counterexample. All perfectly clear. >Pheeeww. I think I'll stop here. Talk about clarity!! Just out of >curiousity, what percentage of your students pass? Or maybe I should >ask: How many of them have learned anything from your classes? Lovely, lovely man you are. Is this how you interact with your professors and peers? Just out of curiosity, what percentage of your professors hold you in personal contempt? Or maybe I should ask: have you managed to learn anything from your classes? I might add: I have not read any other posts on these threads; I chanced on this by mistake (selected the wrong thread to read in trn). I shan't be reading the rest, given your attitude. -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: Ullrich the Mathematician! See, exactly what I predicted. This Arturo fellow must have been an ex-student of Ullrich's - why else would he be so compassionate? :-) He also uses phrases like: non-sequitor, mature enough, etc. Oh it all smacks of one club: The Usenet Trolls and Thugs. I thought you were not going to respond? Giggle. (Oh, this one rubbed off onto me from Ullrich! I mean the 'giggle' word. Just hope nothing else rubs off!) pay attention to the subject at hand. Oh and I cannot resist this one: > Lovely, lovely man you are. Is this how you interact with your > professors and peers? Just out of curiosity, what percentage of your > professors hold you in personal contempt? Or maybe I should ask: have > you managed to learn anything from your classes? I hold most professors in contempt: They are nothing but closet paedophiles and perverts. Their arrogance far exceeds their knowledge and they don't know their ear from their ass! Unlike most of my peers, I did not emulate any professor - most of them are dispicable, dirty, worthless and failures as human beings. That's why they end up in colleges because they have failed everywhere else. Arriverderci! (Don't know if I spelled this correctlt and I don't care - Italian is not one of my languages) === Subject: Re: Ullrich the Mathematician! Really? What does this have to do with Gabriel's formula? Absolutely nothing! And you are a pathetic liar!! First of all you say you *chanced* on this thread. Giggle, giggle. For someone who chances on a thread and starts to cast your harsh judgement, I would say you are a liar! You first read much of the thread, decided you did not like me because of how I reacted to others and then you answered this post. For if you did not read the other messages, where would you come up with statements such as personal fountain etc? And you call me a simpleton? Sheeesh, check your logic first guy!!! It seems as if you are the simpleton now! Your exit paragraph is such a joke: How old do you think I am? :-) Oh, this is a rhetorical question in case you didn't know! Ha, ha === Subject: Re: Ullrich the Mathematician! >How old do you think I am? :-) Oh, >this is a rhetorical question in case you didn't know! Ha, ha That question is more difficult that it appears at first glance. It would appear that you have had exposure to college level mathematics which would put the expected age in the 17 - 22 years or older yet your rhetoric would lead me to guess 12 or 13. --Lynn === Subject: Re: Ullrich the Mathematician! Oh really madam? Where do you hail from? Aside from your keen observational skills, do you have anything worthwhile to contribute? === Subject: Re: Ullrich the Mathematician! [Jason, to Arturo Magidin] > Really? What does this have to do with Gabriel's formula? Absolutely > nothing! What do you think Gabriel's formula becomes if you fix x=0 and w=1? Exactly what David posted, except David used the right side of the partitions instead of the left side. Maybe David's use of j as the summation index instead of Gabriel's s threw you too, but that's so trivial a substitution it shouldn't need explanation. > And you are a pathetic liar!! > ... [and so on] ... Personal attacks are what you do most here, Jason -- you've amply earned the poor opinions of your behavior here. === Subject: Re: Ullrich the Mathematician! <1OKdnam7kvDbYLDfRVn-1w@comcast.com> No Tim. Ullrich's formula does not become gabriel's under any circumstances. It is a load of crap Ullrich posted which only makes sense to Ullrich! Have you bothered to read any of Gabriel's stuff? Nothing threw me out about Ullrich's babbles - there is no trivial substitution or anything that is self-evident. write out an attempted proof of gabriel's theorem and what squiggles Ullrich typed up in his response. His response tells me a lot about the state of his office and maybe even his home: It is probably untidy with loads of paper strewn everywhere and no particular order. Finally, you don't seem to know what you are talking about. Go and read some more. Personal attacks are justified when one is attacked. Truth is that words hurt and make people angry. You need to watch your words. If you or any other contributor wants my respect, then you better learn to give me respect. Otherwise be prepared - I will rip you to shreads! === Subject: Re: Ullrich the Mathematician! [Jason] > No Tim. Ullrich's formula does not become gabriel's under any > circumstances. I didn't say that it does. I said that Gabriel's reduces to Ullrich's, under conditions I already explained. I'll spell it out below. > It is a load of crap Ullrich posted which only makes sense to Ullrich! > Have you bothered to read any of Gabriel's stuff? You confirm David's assessment of you most times you post: you post nonsense, and then resort to personal attack when called on it. Here's equation 1 from : f(x+w) - f(x) 1 ------------- = lim - sum f'(x + ws/n) w n->oo n 0 <= s <= n-1 As I said, substitute x=0 and w=1 (the starting point and range are inessential to anything _interesting_ about this, and normalizing them reduces formal clutter). Then it becomes: f(1+0) - f(0) 1 ------------- = lim - sum f'(0 + 1s/n) 1 n 0 <= s <= n-1 or (simplifying expressions): f(1) - f(0) = lim 1/n sum f'(s/n) = by the distributive law lim sum f'(s/n)/n = renaming the formal summation index lim sum f'(j/n)/n = making n a capital letter lim sum f'(j/N)/N As I also said, you need to switch to using the right side of the partitions to get exactly what David posted (he summed from 1 to N instead of from 0 to N-1 -- which makes no real difference either): f(1) - f(0) = lim_N sum_1^N f'(j/N)/N Are you going to say that was too hard for you to see from the explanation I gave? You need every little step spelled out like this? That's fine if so, but then stop assuming you understand what posters have been telling you, and sanely assume that you're missing their meaning instead of that they're pathetic liars, clowns, etc. More thought, less attack. > ... [rant deleted] ... === Subject: Re: Ullrich the Mathematician! [Tim Peters] [...] >> Here's equation 1 from : >> f(x+w) - f(x) 1 >> ------------- = lim - sum f'(x + ws/n) >> w n->oo n 0 <= s <= n-1 >> As I said, substitute x=0 and w=1 (the starting point and range are >> inessential to anything _interesting_ about this, and normalizing them >> reduces formal clutter). Then it becomes: >> f(1+0) - f(0) 1 >> ------------- = lim - sum f'(0 + 1s/n) >> 1 n 0 <= s <= n-1 >> or (simplifying expressions): >> f(1) - f(0) = lim 1/n sum f'(s/n) = by the distributive law >> lim sum f'(s/n)/n = renaming the formal summation index [Jason] > So far I agree Since that's near the end, I take it that means you no longer wish to claim If this is what he means, it makes no sense at all! or Really? What does this have to do with Gabriel's formula? Absolutely nothing! or It is a load of crap Ullrich posted which only makes sense to Ullrich! Have you bothered to read any of Gabriel's stuff? Good. > However, I do not agree you can rename the formal summation index Sorry, you can't sanely disagree with the renaming step. I'll do you the small favor of moving your disagreement down to the step where it belongs. >> lim sum f'(j/n)/n = making n a capital letter >> lim sum f'(j/N)/N >> As I also said, you need to switch to using the right side of the >> partitions to get exactly what David posted (he summed from 1 to N >> intead of from 0 to N-1 -- which makes no real difference either): _That's_ the step you disagree with: changing the summation limits, not renaming the formal index. >> f(1) - f(0) = lim_N sum_1^N f'(j/N)/N > because if you do, then you lose f'(0) which is what gabriel says is > needed in his theorem. This also has to do with the function f being > differentiable at x=0 for if it is not, gabriel's theorem does not > apply. It _should_ go without saying that if you switch from using the left sides of the partitions to the right sides, then the precise conditions that need to obtain at the endpoints need to switch too. If you don't see that, then sure, throwing a fit over it makes a lot of sense . In David's original post, he said Suppose that f is differentiable on [0,1], so that he didn't have to pick nits at either endpoint when evaluating f'. He's not Matthew, Mark, Luke or John to Gabriel's Jesus here, he was responding to Angus's request for a competent formulation of the issue in question. OTOH, if you believe the original formula holds when f is continuous on [x, x+w] and differentiable on [x, x+w), could you possibly believe that it would _not_ hold if: 1. The summation limits changed, from 0 <= s < N to 0 < s <= N; and, 2. The prerequisite changed to match, from differentiability on [x, x+w) to on (x, x+w]? IOW, are you thinking here -- or just arguing for the sake of being disagreeable? > Now what's so hard about that? Nothing I can see. Reread the posts about Riemann sums, with your brain engaged, and your confusions may vanish too. > I guess now it's a good time for you to apologize, is it not? It's always an excellent time for an apology -- show a little class and start by apologizing to David. I apologize to you for ... oh, I don't know. Make something up that you think fits; I drew a blank. === Subject: Re: Ullrich the Mathematician! <1OKdnam7kvDbYLDfRVn-1w@comcast.com> > OTOH, if you believe the original formula holds when f is continuous on [x, > x+w] and differentiable on [x, x+w), could you possibly believe that it > would _not_ hold if: > 1. The summation limits changed, from 0 <= s < N to 0 < s <= N; > and, > 2. The prerequisite changed to match, from differentiability on [x, x+w) > to on (x, x+w]? Okay, I rushed that last post but you interpreted it correctly. What I wanted to say is that you cannot change the summation limits. Yes, it makes a big difference if you begin to sum from 1 rather than 0. As I pointed out, if you start at 1, then you lose f'(0) which is required for gabriel's theorem. Also, the upper limit has to be n-1 and not n for the same reason, i.e. f'(n) may not exist! And furthermore, Gabriel's theorem does not require f'(n) to exist. Does this make sense to you now? > IOW, are you thinking here -- or just arguing for the sake of being > disagreeable? What do you think eh? Do you believe I am enjoying having to throw back at fools what they continue to throw at me? It's a no-brainer. I am surprised you asked this question or maybe I am not. > Nothing I can see. Reread the posts about Riemann sums, with your brain > engaged, and your confusions may vanish too. Riemann sums have *nothing* to do with gabriel's theorem. Sooner you realize this, easier it might be to find out how it works. Riemann sums are not even precise - they are approximations. Gabriel's Average derivative is not an approximation. In one of his web pages he shows how the different limits in his formula would produce non-sense results, i.e. the only correct summation limits are from 0 to n-1. > It's always an excellent time for an apology -- show a little class and > start by apologizing to David. I apologize to you for ... oh, I don't know. > Make something up that you think fits; I drew a blank. Man, I am the only one with any class on this forum. Pull up your pants and give your mouth a chance! Jason === Subject: Re: Ullrich the Mathematician! <1OKdnam7kvDbYLDfRVn-1w@comcast.com> > f(1) - f(0) = lim 1/n sum f'(s/n) = by the distributive law So far I agree. However, I do not agree you can rename the formal summation index because if you do, then you lose f'(0) which is what gabriel says is needed in his theorem. This also has to do with the function f being differentiable at x=0 for if it is not, gabriel's theorem does not apply. Now what's so hard about that? I guess now it's a good time for you to apologize, is it not? === Subject: Re: Ullrich the Mathematician! > [Jason] > No Tim. Ullrich's formula does not become gabriel's under any > circumstances. > I didn't say that it does. I said that Gabriel's reduces to Ullrich's, > under conditions I already explained. I'll spell it out below. > It is a load of crap Ullrich posted which only makes sense to Ullrich! > Have you bothered to read any of Gabriel's stuff? > You confirm David's assessment of you most times you post: you post > nonsense, and then resort to personal attack when called on it. Here's > equation 1 from : > f(x+w) - f(x) 1 > ------------- = lim - sum f'(x + ws/n) > w n->oo n 0 <= s <= n-1 > As I said, substitute x=0 and w=1 (the starting point and range are > inessential to anything _interesting_ about this, and normalizing them > reduces formal clutter). Then it becomes: > f(1+0) - f(0) 1 > ------------- = lim - sum f'(0 + 1s/n) > 1 n 0 <= s <= n-1 Recently, in another thread, mention was made of an example [appearing in Stromberg's Intro. to Classical Real Analysis] of a non-constant differentiable function f with f'(x)=0 for x in a dense set. Doubtless the dense set can be chosen to contain the rationals. For such a function (chosen so that f(1)=1 and f(0)=0), the above corollary of Gabriel's ATT would seem to fail. -- A. === Subject: Re: Ullrich the Mathematician! days. My association with the Department is that of an alumnus. >Really? What does this have to do with Gabriel's formula? Absolutely >nothing! The thread was titled Ullrich the Mathematician! The excerpt that appeared in the post made no mention of any Gabriel. >And you are a pathetic liar!! Whatever. >First of all you say you *chanced* on this thread. Giggle, giggle. For >someone who chances on a thread and starts to cast your harsh >judgement, I would say you are a liar! Non sequitur. > You first read much of the >thread, decided you did not like me because of how I reacted to others >and then you answered this post. If that makes you feel better, by all means continue to believe that this is the case. >For if you did not read the other messages, where would you come up >with statements such as personal fountain etc? little evidence from time to time. While I sometimes find to my chagrin I am incorrect, I am happy you have decided to take it upon yourself to give such strong evidence in my favor here. >And you call me a simpleton? Sheeesh, check your logic first guy!!! Non sequitur. > It >seems as if you are the simpleton now! Whatever. >Your exit paragraph is such a joke: How old do you think I am? :-) > Oh, this is a rhetorical question in case you didn't know! Ha, ha At a guess: not mature enough for adult discussion. Bye. -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: Ullrich the Mathematician! <722m21tqviiidui5s2jhauha932hn2199l@4ax.com> Discussion, linux) >> Suppose that f is differentiable on [0,1] (or differentiable >> on (0,1) and continuous on [0,1]). Does it follow that >> f(1) - f(0) = lim_N sum_1^N f'(j/N)/N ? >> Of course this is clear from the fundamental theorem of calculus >> if f' is continuous. It seems unlikely that it's true under the >> stated hypotheses, but seems unlikely is not quite a > counterexample. > Now you talk about my posts being funny! Let's see how funny yours is: > First you say suppose f is differentiable on [0,1] and in parentheses > you write or differentiable on (0,1) and continuous on [0,1]. > May I remind you the two are not the *equivalent*. Your statement seems > to imply they are. His statement implies no such thing. If they were equivalent, he might right i.e., differentiable.... or in other words, differentiable... or *equivalently*, differentiable... He didn't write any of those things. His phrase strongly implied that, in what follows, the parenthetical condition was sufficient for his argument. It perhaps implied that the non-parenthetical condition implied the parenthetical condition. But it didn't imply equivalence. > In mathematics you need to be precise - either/or makes a big > difference. He was precise. And you're a boob. -- I don't know why I live in a world with so many supposed mathematicians who are all so dumb AND rude. Why oh why couldn't someone like Gauss or Dedekind still be around? Shoot, I'd even take someone like Hardy at this point. -- James S Harris compromises === Subject: Re: Ullrich the Mathematician! <722m21tqviiidui5s2jhauha932hn2199l@4ax.com> <87sm36xamq.fsf@phiwumbda.org> > His phrase strongly implied that, in what follows, the parenthetical > condition was sufficient for his argument. It perhaps implied that > the non-parenthetical condition implied the parenthetical condition. > But it didn't imply equivalence. What a load of bull-!! It can be interpreted in a number of different ways. Ullrich complains about others being precise but does not follow his own advice! Do you have any other talents aside from those of a clown? Don't know what a boob is and quite frankly, don't care. Unless you have anything pertaining to the subject, I suggest you find yourself another sandbox to play in. === Subject: Re: Ullrich the Mathematician! <722m21tqviiidui5s2jhauha932hn2199l@4ax.com> <87sm36xamq.fsf@phiwumbda.org> Discussion, linux) >> His phrase strongly implied that, in what follows, the parenthetical >> condition was sufficient for his argument. It perhaps implied that >> the non-parenthetical condition implied the parenthetical condition. >> But it didn't imply equivalence. > What a load of bull-!! It can be interpreted in a number of > different ways. Ullrich complains about others being precise but does > not follow his own advice! > Do you have any other talents aside from those of a clown? I learned enough logic to understand the standard meanings of or. Does that count? > Don't know what a boob is and quite frankly, don't care. Unless you > have anything pertaining to the subject, I suggest you find yourself > another sandbox to play in. Yes sir, thank you sir. Boob. -- But remember, as long as one human being follows the rules of mathematics, then mathematics as a human discipline survives. Right now I'm that one human being, so mathematics survives. -- James S. Harris === Subject: Re: Ullrich the Mathematician! <722m21tqviiidui5s2jhauha932hn2199l@4ax.com> <87sm36xamq.fsf@phiwumbda.org> <87psyax7op.fsf@phiwumbda.org> > Yes sir, thank you sir. > Boob. Whatever boob means, I think I'd rather be a boob than a dickhead like you! LOL Every piss-willie on this forum has a little quote at the end of their post which is supposed to say something really profound or humourous. :-( That it's out of place or irrelevant does not really cross their minds, neither does it matter at the end of the day. In my next book which will be called: I moved your cheese - Dickhead!, I will be sure to announce a new rule: No more silly quotes at the end of pathetic and useless posts in usenet groups! Look out for it! - will be on the shelf in a few months time with ultra large print and only 30 pages cause most of you can't read fast enough or even concentrate longer than 10 minutes. No doubt it will be a best seller like Who moved my cheese. Wow, what a great country the United States is - filled with blundering idiots and sheep. No doubt they will all be lining up to buy the latest and greatest self-help book. Ha, ha. === Subject: Re: Ullrich the Mathematician! <722m21tqviiidui5s2jhauha932hn2199l@4ax.com> <87sm36xamq.fsf@phiwumbda.org> <87psyax7op.fsf@phiwumbda.org> Discussion, linux) > In my next book which will be called: I moved your cheese - > Dickhead!, I will be sure to announce a new rule: No more silly > quotes at the end of pathetic and useless posts in usenet groups! This is too obvious. No self-respecting .sig man would touch this one. You have to act like being quoted is the farthest thing from your mind, don't you know? -- Jesse F. Hughes [Iota]'s the smallest infinitesimal, Russell, there are smaller infinitesimals. -- Ross Finlayson === Subject: Re: Ullrich the Mathematician! >> Suppose that f is differentiable on [0,1] (or differentiable >> on (0,1) and continuous on [0,1]). Does it follow that >> f(1) - f(0) = lim_N sum_1^N f'(j/N)/N ? >> Of course this is clear from the fundamental theorem of calculus >> if f' is continuous. It seems unlikely that it's true under the >> stated hypotheses, but seems unlikely is not quite a > counterexample. > Now you talk about my posts being funny! Let's see how funny yours is: > First you say suppose f is differentiable on [0,1] and in parentheses > you write or differentiable on (0,1) and continuous on [0,1]. > May I remind you the two are not the *equivalent*. Your statement seems > to imply they are. In mathematics you need to be precise - either/or His statement does not imply that they are equivalent, just that either is sufficient for what follows. > makes a big difference. Then you casually proceed to ask a question > which follows on a shaky statement. Okay, let's see: Does your question > pertain to the first part or second part of your statement or both? > Wow! And you call yourself a mathematician. But wait, I am not done > with you yet! >> f(1) - f(0) = lim_N sum_1^N f'(j/N)/N ? > What are you writing here? Let me see. What does *j* represent ?? Is it > a variable of summation maybe? Giggle. Are you summing from 1 - N ? > Cause if you are, it is incorrect. What does j/N mean? I would read that as: The limit as N tends to infinity of: The sum from j=1 to j=N of: the value of the first differential of f at (j divided by N) divided by N That is why it does not matter whether f is differentiable at 0 or 1, as f' is never evaluated there. > Pheeeww. I think I'll stop here. Talk about clarity!! Just out of > curiousity, what percentage of your students pass? Or maybe I should > ask: How many of them have learned anything from your classes? Seems like the problem is on your end. === Subject: Re: Ullrich the Mathematician! <722m21tqviiidui5s2jhauha932hn2199l@4ax.com> > I would read that as: > The limit as N tends to infinity of: > The sum from j=1 to j=N of: > the value of the first differential of f at (j divided by N) divided by N If this is what he means, it makes no sense at all! And of course the problem is on his end, not mine. My question was: what is he trying to say with the above? What function is he using? Whose formula is this - anyway? === Subject: Re: Ullrich the Mathematician! days. My association with the Department is that of an alumnus. >> Suppose that f is differentiable on [0,1] (or differentiable >> on (0,1) and continuous on [0,1]). Does it follow that >> f(1) - f(0) = lim_N sum_1^N f'(j/N)/N ? >> Of course this is clear from the fundamental theorem of calculus >> if f' is continuous. It seems unlikely that it's true under the >> stated hypotheses, but seems unlikely is not quite a >counterexample. >Now you talk about my posts being funny! Let's see how funny yours is: >First you say suppose f is differentiable on [0,1] and in parentheses >you write or differentiable on (0,1) and continuous on [0,1]. >May I remind you the two are not the *equivalent*. Your statement seems >to imply they are. Ehr, no, it does not. If instead of or he had written that is, then the statement would indeed imply the two conditions are equivalent. But the statement, as written, implies no such thing: it says suppose A (or, if you prefer, suppose B instead). You seem to be reading the statement as if he had written Suppose A (i.e., B). But he did not. -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: Ullrich the Mathematician! He is implying that I can assume either one of the statements: f is differentiable on [0,1] A OR f differentiable on (0,1) and continuous on [0,1] B These two statements are not equivalent. Now when I read his question, I ask myself to which of these statements is it a follow up on A or B or both. There is confusion and uncertainty. Did you understand his entire post? If yes, could you answer the rest of my questions? === Subject: Re: Ullrich the Mathematician! days. My association with the Department is that of an alumnus. >He is implying that I can assume either one of the statements: > f is differentiable on [0,1] > f differentiable on (0,1) and continuous on [0,1] B >These two statements are not equivalent. Yes. > Now when I read his question, >I ask myself to which of these statements is it a follow up on A or B >or both. There is confusion and uncertainty. Nonsense. You can assume either, whichever you are most comfortable with. Perhaps the doubt and uncertainty spring from somewhere other than the statement? The same fountain as your personal attacks, perhaps? > Did you understand his >entire post? If yes, could you answer the rest of my questions? I'll go back and try... -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: Ullrich the Mathematician! > Nonsense. You can assume either, whichever you are most comfortable > with. Perhaps the doubt and uncertainty spring from somewhere other > than the statement? The same fountain as your personal attacks, > > perhaps? Excuse me?! Have we met before? Do you have a problem with me? Look, I can see you have a personal problem. I don't know from where your resentment of me stems from, but it is not justified and it's unfair!! Who are you to pass such judgement on me? If this is going to be your attitude, then don't bother because you are probably going to post a lot of crap just like everyone else on this What a friendly guy you are - do you always introduce yourself this way? Have I said anything nasty to you before that you should say what you did? Stuff you man!!!!! === Subject: Re: Ullrich the Mathematician! > Nonsense. You can assume either, whichever you are most comfortable > with. Perhaps the doubt and uncertainty spring from somewhere other > than the statement? The same fountain as your personal attacks, > > perhaps? > Excuse me?! Have we met before? Do you have a problem with me? I read three posts of yours, and it is enough to convince me that you have serious personality problems. > Look, I can see you have a personal problem. I don't know from where > your resentment of me stems from, but it is not justified and it's > unfair!! > Who are you to pass such judgement on me? > If this is going to be your attitude, then don't bother because you are > probably going to post a lot of crap just like everyone else on this > What a friendly guy you are - do you always introduce yourself this > way? > Have I said anything nasty to you before that you should say what you > did? > Stuff you man!!!!! That confirms it. === Subject: Re: Ullrich the Mathematician! days. My association with the Department is that of an alumnus. >> Nonsense. You can assume either, whichever you are most comfortable >> with. Perhaps the doubt and uncertainty spring from somewhere other >> than the statement? The same fountain as your personal attacks, > >> perhaps? >Excuse me?! Have we met before? Do you have a problem with me? I find the attitude displayed in the message I first replied to to be contemptible. That's what I know so far. >Look, I can see you have a personal problem. I don't know from where >your resentment of me stems from, but it is not justified and it's >unfair!! Your post gave me ample justification for my preliminary judgement. If it is unfair, the only person at fault is you. >Who are you to pass such judgement on me? I'm not passing judgement on you. I am expressing my impression of your attitude and my particular distaste for it. >If this is going to be your attitude, then don't bother because you are >probably going to post a lot of crap just like everyone else on this >What a friendly guy you are - do you always introduce yourself this >way? No. Only when I encounter the sort of attitude you have so amply displayed. >Have I said anything nasty to you before that you should say what you >did? Oh, I see. I can only find your attitude offensive if you happen to be directing insults towards me? Whatever. >Stuff you man!!!!! I guess I can now take official notice of your attitude, then? -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: FLTMA: How to write (a,b,c) mod (a,b,c): six combinations How do I use the verb mod (modulus) to write a vector or matrix of three numbers a, b, and c, modulo the same three numbers? Sure, I could write out a mod a a mod b a mod c b mod a b mod b b mod c c mod a c mod b c mod c as a matrix, but isn't there an operator that does this, that is, combine three numbers in a binary operation (modulo) in all nine possible ways? Of course, three of the nine are zero: the three on the diagonal. I guess that's all I have to do. If W is a vector (a,b,c) then W (some symbol) W = the above matrix. Can you suggest a symbol? Should it be a binary operator, or prefix, or postfix? I can make an analogy to multiplication. The column vector (a,b,c) times the row vector (a,b,c) gives the pattern above. Modulus is somewhat analogous to division. Multiplication and division should distribute over addition. Is there a distributive property related to the modulus? Let's see (a+b) mod c = (( a mod c ) + ( b mod c) ) mod c. If for convenience, all vector are row vectors then: (transpose of (a,b,c)) * (a,b,c) gives the same pattern. Yours, Doug Goncz Replikon Research Falls Church, VA 22044-0394 === Subject: Solving an integral... How can I show that Integral from -inf to +inf over x of exp[-(ax+b)^2] erf(cx+b) is equal to sqrt(pi)/a*erf[(ad-bc)/sqrt(a^2+c^2)] Didier -- Didier A Depireux ddepi001@umaryland.edu didier@isr.umd.edu 20 Penn Str - S218E http://neurobiology.umaryland.edu/depireux.htm Anatomy and Neurobiology Phone: 410-706-1272 (lab) University of Maryland -1273 (off) Baltimore MD 21201 USA Fax: 1-410-706-2512 === Subject: Re: Solving an integral... >How can I show that >Integral from -inf to +inf over x of >exp[-(ax+b)^2] erf(cx+b) >is equal to >sqrt(pi)/a*erf[(ad-bc)/sqrt(a^2+c^2)] Where did that d come from? I guess you meant to say cx+d. For simplicity suppose a>0 (note that both your answer and the integral are affected in the same way by changing signs of all the parameters). Write erf(cx+d) = 2a/sqrt(pi) int_0^((cx+d)/a) dy exp(-a^2 y^2) so your integral is int_{-infty}^infty dx int_0^{(cx+d)/a} dy f(x,y) where f(x,y) = 2 a /sqrt(pi) exp(-(ax+b)^2- a^2 y^2) = 2 a /sqrt(pi) exp(-a^2 r^2) where r is distance in the plane from the point (-b/a, 0). Note that the double integral can be written as int_{H1} dx dy f(x,y) - int_{H2} dx dy f(x,y) where H1 is the halfplane y < (cx+d)/a and H2 is the halfplane y < 0. The integral over H2 can be done easily; that over H1 is helped by a linear change of variables. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: constructing a 3d fibonacci spiral Hey folks - I'm trying to sculpt a flower in the shape of a fibonacci spiral like that found in these plants: http://wussu.com/fractals/romanesco.htm http://www.discover.com/images/webexclusive/fibonacci.jpg I'm able to draw a 2d spiral like this: http://www.elliottrader.com/webpagina%20afbeeldingen/fibonacci%20zonnebloem. gif but i don't know how to convert it to 3d. Can anyone help me? === Subject: Help or Hints on semiprime problem Let R be semiprime and let 0 not equal to a belong to R. 1) Show that there exists an infinite sequence a=a1,a2,a3,.... of nonzero elements of R such taht a_j belongs to a_iRa_i whenever 1 less than equal i less than j. 2) Let S={a_i|1 less than equal i less than infinity} as in part 1. IF I,J are ideals of R with I intersect S not equal to the empty set not equal to J intersect S, show that IJ intersect S is not the empty set. 3) Let P be an ideal maximal in R with the property that P intersect S equals the empty set where S is as in part 2. Show that P is prime. === Subject: Gabriel's AFD - Thread 8th March 2005 Here is my attempt to prove Gabriel's Theorem: Have changed some of his wording and am calling it the AFD rather than ATG as he calls it. Given an interval [x;x+w] subdivided into n equal parts and a function f which is continuous on [x;x+w] and differentiable on [x;x+w), the secant gradient or mean value of f is equal to the average first derivative of f [or AFD(f)] defined as follows: w is the width of the interval n is the number of partitions f(x + w/n) - f(x) AFD(f) = ------------------ --- L.H.S w/n 1 n-1 ws AFD(f) = Lim - SIGMA f'(x+ --- ) --- R.H.S n->oo n s=0 n Proof: Start with RHS. 1 AFD(f) = Lim - [ f'(x) + f'(x+w/n) + f'(x+2w/n) + ... n->oo n f'(x+(n-2)w/n) + f'(x+(n-1)w/n)] Now we simplify Gabriel's proof by keeping the ens fixed and using t in the definition: i.e. we let n be fixed for each of f'(x), f'(x+w/n), f'(x+2w/n), etc. In this case, the following are true: f(x+w/t)-f(x) f'(x) = Lim ------------- t->oo w/t f'(x+w/n) = f(x+w/n+w/t)-f(x+w/n) Lim --------------------- t->oo w/t f'(x+2w/n) = f(x+2w/n+w/t)-f(x+2w/n) Lim ----------------------- t->oo w/t f'(x+ 3w/n) = f(x+3w/n+w/t)-f(x+3w/n) Lim ----------------------- t->oo w/t and so on: Now, 1 f(x+w/t)-f(x) AFD(f) = Lim - [ Lim ------------- + n->oo n t->oo w/t f(x+w/n+w/t)-f(x+w/n) Lim --------------------- + t->oo w/t f(x+2w/n+w/t)-f(x+2w/n) Lim ----------------------- + t->oo w/t f(x+3w/n+w/t)-f(x+3w/n) Lim ----------------------- + t->oo w/t + ... f(x+(n-2)w/n+w/t)-f(x+(n-2)w/n) Lim ------------------------------- + t->oo w/t f(x+(n-1)w/n+w/t)-f(x+(n-1)w/n) Lim ------------------------------- ] t->oo w/t Looking at the above we have two limits to infinity. i.e. n and t. However, since both n and t tend to infinity, the outer limit is taken once over the entire sum. With n and t coinciding, we have: 1 f(x+w/n)-f(x) AFD(f) = Lim - [ Lim ------------- + n->oo n n->oo w/n f(x+2w/n)-f(x+w/n) Lim ------------------ + n->oo w/n f(x+3w/n)-f(x+2w/n) Lim ------------------- + n->oo w/n f(x+4w/n)-f(x+3w/n) Lim ------------------- + n->oo w/n + ... f(x+(n-1)w/n)-f(x+(n-2)w/n) Lim --------------------------- + n->oo w/n f(x+w)-f(x+(n-1)w/n) Lim -------------------- ] n->oo w/n Which leads to (**) : 1 f(x+w/n)-f(x) AFD(f) = Lim - [ ------------- + n->oo n w/n f(x+2w/n)-f(x+w/n) ------------------ + w/n f(x+3w/n)-f(x+2w/n) ------------------- + w/n f(x+4w/n)-f(x+3w/n) ------------------- + w/n + ... f(x+(n-1)w/n)-f(x+(n-2)w/n) --------------------------- w/n f(x+w)-f(x+(n-1)w/n) -------------------- ] w/n Some more simplication: 1 f(x+w)-f(x) AFD(f) = Lim - [ ----------- ] n->oo n w/n Then, 1 n f(x+w)-f(x) AFD(f) = Lim -. [ ----------- ] n->oo n w So, f(x+w)-f(x) AFD(f) = Lim [ ----------- ] n->oo w And finally, f(x+w)-f(x) AFD(f) = ----------- w which is the desired result. (**) Dropping the inner limit is what I am not sure about. Jason Wells NOTE: **PLEASE** DO NOT POST TO THIS THREAD unless it's related to the mathematics in this proof. If you have a question, I will attempt to answer it. Please note this is not my theorem but I am trying to prove it. Why? Well, it works and I don't believe this is mere coincidence. Does Gabriel use *infinitesimals* ? I don't know. Does the mean value theorem use infinitesimals? Most will say no. I think gabriel's theorem may use infinitesimals. Gabriel's average sum theorem works. It is not easy to understand. Thus far, most reponses to this subject have been extremely ignorant, hostile and prejudiced. Is the majority wrong? I hate to say this but I think the answer is yes. So far I stand alone save for a very few individuals who think the theorem has value. But even they have not proved it. Why I want to prove it: I believe it's a great way to teach all the fundamental calculus - free of real analysis rubbish. Real analysis is riddled with discrepancies and unresolved problems. In fact, real analysis is a *big lie* for much of the terminology it uses is in fact unsound. Its origins are traced back to drunks, nasty men whose morality and virtues were highly in doubt. They took the mathematics world hostage. Today every Tom, Dick and Harry has some Phd in Group Algebra or Real Analysis. Many deep problems remain unsolved yet paper after paper is published and Phd begets Phd. There is a serious crisis in education especially in the field of mathematics where most students do badly. === Subject: Fuller story, brainstorming and math freaks I have this HUGE reputation as a crank or crackpot on math newsgroups when I state very clearly what I do, and why I do it. I brainstorm. Some of you may know about brainstorming and you may also know that part of the creative process is a period where you don't carefully critique ideas as you generate them--LOTS of them. Now I know most of you are probably people who would prefer not to toss out ideas, many of which you know will probably be crappy, to a hostile audience where people may make fun of you or call you names. I'm just not that sensitive. I figured that I could use Usenet to jumpstart the brainstorming process by generating a lot of ideas and letting OTHER people critique them at first, later going through myself in the second half of the process to carefully critique and see if anything was valid. It actually works to do it that way, but there's one problem as there are posters who basically live on Usenet, who post just about every day, and they see it as their territory. These people try to control posting in what they see as their space, and over the years I've ended up with a dedicated crew of them that I see as being like barnacles on a ship. They attach themselves to you and slow down progress, often doing so deliberately. In my case, posters will deliberately make a lot of postings in threads I create to obscure whatever mathematics I'm currently brainstorming. They will make repeated nonsensical requests, or replies that bounce all over the map, never settling on a particular issue. Years ago, while I was till perfecting using Usenet, there was even a period when posters on the sci.math newsgroup, coordinated their efforts enough to not make replies that critiqued the mathematical ideas I had at all. They figured out what I was doing, and worked very hard for a while to counteract it by refusing to critique. I ended up making a bet, which enticed some math people to actually critique, and even shelled out over $300 US, when it turned out that the ideas were faulty. I figured out better ways since then to counteract these people. Some of you may think it's wrong for me to use Usenet in this way. However, the gist of it is that I post mathematical ideas primarily on math newsgroups. So what's really wrong with that? And the method HAS born fruit, not matter what the barnacles say to the contrary, as you can't be a rational adult and just believe that anyone actually gets published, even if it is a small electronic math journal, which backs down under pressure, or that they can then just write up another paper and get it to an even more prestigious journal, when they're not at all rational, or have nothing. The Usenet posters who try to control my posts bounce all over the map there as well, routinely calling me crazy, when I explain what I do, and the results are not what you get if you are not rational. Irrational people don't write up math papers that get taken seriously by math journals. So why talk about it? Why continue when I face so much opposition? Well, I am a strong-willed person, and like I said above, I'm not very sensitive. I think many of you are, and some of you are hurt emotionally far more than I am. I'm sorry about that, but it's not me doing it to you. I use Usenet to brainstorm and critique math ideas. I am good at it and I've done it for years. There are some weird people in this space who spend a lot of time and effort trying to manage particular newsgroups who help to make things nasty. At times I may push things because I've learned that controversy, and flame posting here and there can bring more attention meaning I get more critiques. I use to talk about expediency, when I was working through the ethics of my techniques, and now I've settled on practicality, and the fact that I really enjoy what I'm doing and I get results. I like to brainstorm, and I like to put my ideas out there to see who is willing to try and shoot them down. If no one is at first, then I have my ways... And the world keeps going round and round.... James Harri === Subject: Re: Fuller story, brainstorming and math freaks > I have this HUGE reputation as a crank or crackpot on math newsgroups > when I state very clearly what I do, and why I do it. Mr. Harris, I think it would be more correct to label you as a Super Crank,Super Crackpot and Super Troll. You actually have no desire to see work that is correct. Your only indulgence is in perpetrating your lies, cheating and crying about how unfair the math world is to you. Some of us would term you an ignoramus! However, I understand your medical condition of NPD! You are sinking further and further into a hole which you will not be able to get out of. Someday, we will see wearing a white jacket in a padded cell screaming what about all of my research. Truth is, you have nothing! Deep in your heart (as dark with hatred as that place may be), you know you have nothing and still you scream and defend. I am amazed at the amount of time and energy which you have been given by very capable sci.math posters. They have even provided refutations at a level you can understand (and that is pretty hard) and still you ignore them. You have become so translucent as you have finally realized (after ten years) that you don't have a clue or any accomplishments of any merit! You wouldn't understand a proof if it bit you in the ass, although you have proven more than capable of generating garbage! Are you not disgusted with yourself and with this charade? Go see a doctor before it is too late! Take up needle point or abstract painting (something your mind is more suited for).. === Subject: Re: Fuller story, brainstorming and math freaks <1110338630.391848@news-1.nethere.net> Message: Narcissistic Personality Disorder: Reason for posting? === Subject: Narcissistic Personality Disorder: Reason for posting? Author: James Harris I've become really bothered by my posting, but I can't seem to stop. However, I realized that part of the sick cycle are some of those people who reply to me so negatively, so in the hope that these people might show mercy I thought I'd post something I found on the web, with limited hope, I admit, that anything will change. Alternatively, the narcissist feels victimized by mediocre bureaucrats and intellectual dwarves who consistently fail to appreciate his outstanding - really, unparalleled - talents, skills, and accomplishments. Being haunted by his challenged inferiors substantiates the narcissist's comparative superiority. Driven by pathological envy, these pygmies collude to defraud him, badger him, deny him his due, denigrate, isolate, and ignore him. The narcissist projects onto this second class of lesser persecutors his own deleterious emotions and transformed aggression: hatred, rage, and seething jealousy. But here are the scary quotes: Paranoid ideation - the narcissist's deep-rooted conviction that he is being persecuted by his inferiors, detractors, or powerful ill-wishers - serves two psychodynamic purposes. It upholds the narcissist's grandiosity and it fends off intimacy. The paranoid delusions of the narcissist are always grandiose, cosmic, or historical. His pursuers are influential and formidable. They are after his unique possessions, out to exploit his expertise and special traits, or to force him to abstain and refrain from certain actions. The narcissist feels that he is at the center of intrigues and conspiracies of colossal magnitudes. The paranoid narcissist ends life as an oddball recluse - derided, feared, and loathed in equal measures. His paranoia - exacerbated by repeated rejections and ageing - pervades his entire life and diminishes his creativity, adaptability, and functioning. The narcisstis personality, buffetted by paranoia, turns ossified and brittle. Finally, atomized and useless, it succumbs and gives way to a great void. The narcissist is consumed. It looks like a person in the grips of this narcissism thing loses connection with reality but somehow *feeds* on ANY attention, including negative attention. They called it narcissistic supply. It really is a twisted illness that apparently can lead to even nastier illnesses. I am self-diagnosed (typical narcissistic behavior) but it looks like it fits to me. However, I've decided that I don't want to succumb to mental woes, so I am fighting the dark narcissist within by making this informative post, though it's probably a sign of the illness. In any event, maybe it will help in the long run as information is power. I will do my best to stop posting, but it will help if some of you try to help out by not replying to me, or if you reply, by refraining from the attacks that feed my narcissistic side, and its irrational beliefs that everyone is out to get me. If you have facts, sure, no problem. My inability to face facts presented without emotion will just be a sign of the dark side. My only fear now is that some of you are sick as well, and act from your own illness, so that you can't stop attacking in your posting. But on this Christmas Day, I will act from hope, and hopefully this New Year will bring a change, and an escape from the madness. James Harris === Subject: Re: Fuller story, brainstorming and math freaks >I have this HUGE reputation as a crank or crackpot on math newsgroups For good reason.... you are. === Subject: Re: Please help with this Excel calculation <6TcqQuV3WaTuBYIjYLoaFolM=TUr@4ax.com> >Just for the heck of it I put it in a spreadsheet for you to try: >http://math.asu.edu/~kurtz/CityFunding.xls That's a neat-looking spreadsheet. It's interesting how your grants differ slightly from r.e.s.'s. With a total available of 1,500,000, the amounts requested were: $7,244 $11,234 $18,356 $112,562 $113,374 $115,134 $146,251 $293,561 $369,471 $583,351 Your grants were: $7,244.00 $10,991.89 $17,564.81 $105,284.40 $103,600.53 $102,727.51 $127,339.53 $249,274.47 $305,770.09 $470,202.78 And r.e.s.'s were: 7,244.00 11,012.77 17,598.44 99,399.93 100,084.36 101,550.27 127,625.92 248,211.48 309,155.03 478,117.80 Would either of you care to state why one or the other is better in some way? Ron M. === Subject: Re: Please help with this Excel calculation r.e.s., this is excellent. I'm wondering if there's one more modification that can be done. That is to have it be able to automatically calculate the value of p that would produce the maximal variation of f(d), as in your last example. That is, which would produce an allocation equal to the smallest demand, so you wouldn't have to sit there tweaking p $.0000001 at a time trying to find it. Ron M. > If you come up with any other ideas, feel free, and in the meantime I > will try to use these suggestions and see what I come up with. > Suppose the total T is to be allocated to demands (d_1,...,d_n) > according to an allocation function f, such that > (1) sum f(d_i) d_i = T; > (2) 0 <= f(d) <= 1 for all d >= 0; > (3) f(d) is a non-increasing function of d. > (The focus is on the function f(d) because of your expressed > interest in controlling to some extent the values f(d_i).) > As other posters have pointed out, one obvious choice is > f(d) = (T/d)* d^p / sum (d_i)^p > = T / ( d^(1-p) * sum (d_i)^p ), > but, as you've discovered, there must be some constraints > on the values of p. The constraints are that p must be in an > interval [p_min, 1), as follows: > (3) ==> p <= 1 > (2) ==> p >= p_min > where p_min is the least p for which f(d_i) <= 1 for all i. > (Letting d_min be the least demand, p_min is the solution of > T / ( d_min^(1-p) * sum (d_i)^p ) = 1, but because of the sum, > there is no nice formula for p_min.) > In Excel you can simply reserve a cell for the value of the > parameter p and adjust p downward from p=1 to p=p_min. Here > are three illustrations using your example (I used Excel's > Solver tool to find p_min): > Case p = 1 (produces constant f(d)): > demand weight alloc.fn. allocation shortage > i d_i (d_i)^p f(d_i) f(d_i)*d_i (demand - alloc.) > -- ---------- ------------ ------ ---------- ------------- > 1 6,136.00 6,136.00 0.57 3,467.78 2,668.22 > 2 11,234.00 11,234.00 0.57 6,348.94 4,885.06 > 3 18,356.00 18,356.00 0.57 10,373.96 7,982.04 > 4 112,562.00 112,562.00 0.57 63,614.84 48,947.16 > 5 113,374.00 113,374.00 0.57 64,073.74 49,300.26 > 6 115,134.00 115,134.00 0.57 65,068.41 50,065.59 > 7 146,251.00 146,251.00 0.57 82,654.30 63,596.70 > 8 293,561.00 293,561.00 0.57 165,907.10 127,653.90 > 9 369,471.00 369,471.00 0.57 208,807.92 160,663.08 > 10 583,351.00 583,351.00 0.57 329,683.01 253,667.99 > 1,769,430.00 1,000,000.00 > Case p = .9 (produces less-than-maximal variation of f(d)) > demand weight alloc.fn. allocation shortage > i d_i (d_i)^p f(d_i) f(d_i)*d_i (demand - alloc.) > -- ---------- ------------ ------ ---------- ------------- > 1 6,136.00 2,565.06 0.83 5,075.40 1,060.60 > 2 11,234.00 4,420.60 0.78 8,746.91 2,487.09 > 3 18,356.00 6,877.02 0.74 13,607.36 4,748.64 > 4 112,562.00 35,176.50 0.62 69,602.75 42,959.25 > 5 113,374.00 35,404.80 0.62 70,054.48 43,319.52 > 6 115,134.00 35,899.07 0.62 71,032.48 44,101.52 > 7 146,251.00 44,523.47 0.60 88,097.32 58,153.68 > 8 293,561.00 83,354.40 0.56 164,931.00 128,630.00 > 9 369,471.00 102,523.24 0.55 202,859.84 166,611.16 > 10 583,351.00 154,645.38 0.52 305,992.45 277,358.55 > 1,769,430.00 1,000,000.00 > Case p = p_min = 0.849539888 (produces maximal variation of f(d)) > demand weight alloc.fn. allocation shortage > i d_i (d_i)^p f(d_i) f(d_i)*d_i (demand - alloc.) > -- ---------- ------------ ------ ---------- ------------- > 1 6,136.00 1,651.81 1.00 6,136.00 0.00 > 2 11,234.00 2,761.15 0.91 10,256.90 977.10 > 3 18,356.00 4,190.34 0.85 15,565.93 2,790.07 > 4 112,562.00 19,559.55 0.65 72,658.17 39,903.83 > 5 113,374.00 19,679.35 0.64 73,103.21 40,270.79 > 6 115,134.00 19,938.58 0.64 74,066.19 41,067.81 > 7 146,251.00 24,431.91 0.62 90,757.63 55,493.37 > 8 293,561.00 44,159.88 0.56 164,041.41 129,519.59 > 9 369,471.00 53,688.54 0.54 199,437.69 170,033.31 > 10 583,351.00 79,138.44 0.50 293,976.85 289,374.15 > 1,769,430.00 1,000,000.00 > f(d) is made to decrease over a greater-and-greater range as p is > decreased from 1 to p_min. The idea is that the immediate feedback > of a spreadsheet may allow you to tweak p to find a suitable set of > allocations. (BTW, min f(d) = 0.50 appears to be a coincidence -- > it's not generally 1/2 when p=p_min.) > Also, instead of the specific allocation function used above, f(d) > can be of form f(d) = (T/d)*w(d), where w(d) is any non-negative > function such that sum w(d_i) = 1 -- so it might be interesting > to consider other parametric families as well. > The spreadsheet I used for this is at > http://r.s.home.mindspring.com/Misc/Allocation.xls > --r.e.s. === Subject: Re: Please help with this Excel calculation > I'm wondering if there's one more > modification that can be done. That is to have it be able to > automatically calculate the value of p that would produce the maximal > variation of f(d), as in your last example. That is, which would > produce an allocation equal to the smallest demand, so you wouldn't > have to sit there tweaking p $.0000001 at a time trying to find it. That's what I was referring to when I mentioned using the Solver tool, which is an add-in that comes as part of Excel's Analysis Toolpack. Once you've installed it, go to Tools/Solver, and in the resulting window try the following ... Set the target cell to Equal to By changing cells Subject to constraints All this can be done by writing a small macro and attaching it to a command button in the spreadsheet, so a click of the button does the constrained minimisation of the parameter p. I've uploaded a revised worksheet to do this (remember, you'll need to have Solver installed before the macro can function) -- BTW, I recommend *not* running it from within your browser, but rather save it to disk, and run it from there. (Details on runnning Solver by macros can be found in Tools/Macro/Visual Basic Editor/Help.) --r.e.s. === Subject: Re: New integer multiplication algorithm > ) You're comparing apples with oranges here. NP problems are problems > ) where the answer is YES or NO (only one bit long). > I'm pretty sure you're wrong there. Not according to Papadimitriou's _Computational Complexity_. Problems in NP are decision (YES/NO) problems. > For example, the Traveling Salesman problem definately isn't a > problem where the answer is yes or no. But the threshold problem is: Input: a graph, the cost function, and a number B. Output: YES if there is a TSP tour with cost <= B. Now, if you have a TSP problem with integer weights, you can use the decision problem to find the minimum tour, using a binary-search method. > You may be confused in that NP problems are such that a solution can > be checked to be correct in polynomial time. No, I'm not, and I don't see how you could assume such a thing. It would be as if I said, The sun is red, and you said, No, it burns using nuclear fusion. I never said anything about the definition of NP, only a property. > (As you may have guessed, those don't actually exist.) No duh. (Well, maybe using a quantum computing model, they might.) > Disclaimer: I am in no way responsible for any of the statements > made in the above text. For all I know I might be > drugged or something.. > No I'm not paranoid. You all think I'm paranoid, don't > you ! I could make a comment about this, but I won't. --- Christopher Heckman === Subject: Re: New integer multiplication algorithm >> >>You need NP-completeness to push P=NP home, but why do you need it >for >>P!=NP? _Anything_ in NPP proves NP!=P. >> Integer factorization isn't known to be in NP, either. >You're comparing apples with oranges here. NP problems are problems >where the answer is YES or NO (only one bit long). The relevant NP >problem is whether a number N is prime. The relvant NP problem is whether a number N has a factor less than some other number. That's in NP, but not known to be NP-hard. -- There's no such thing as a free lunch, but certain accounting practices can result in a fully-depreciated one. === Subject: SF: Sign change I think that surrogate factoring preferentially factors the smaller number, so I need to use a j that is bigger than M, not less. That changes signs in key equations, and with THAT minor change (or major depending on how you look at it), my basic algorithm just might work. Here it is again. Given a target natural M that you wish to factor, you must first select a non-zero integer j, such that j is coprime to M, and you need to pick one that is larger than M, like j = M + 1 and most importantly you want to pick j such that T that is easy to factor. Next you calculate T, where T = M^2 - j^2 and you factor T. Notice that T is negative. Now you consider integers f_1 and f_2, such that f_1 f_2 = T^2 and you need to consider all such solutions, including seemingly trivial ones like where f_1 = T^2, or f_1 = 1, or -1. Then you calulate a number I call y/A^2, where the numerator n is given by n = +/-(f_1 + f_2) - 2(2j^2 + T) and you divide off multiples of M, if any, and then take the gcd with M, and if it's not 1, then you've succeeded. Not surprisingly I've focused on j less than M, because that gives smaller numbers but I think that then j is preferentially factored. There IS a solid mathematical reason for having j larger than M, and if it's solid enough that could be the final answer. James Harris === Subject: Re: SF: Sign change > I think that surrogate factoring preferentially factors the smaller > number, so I need to use a j that is bigger than M, not less. > That changes signs in key equations, and with THAT minor change (or > major depending on how you look at it), my basic algorithm just might > work. It should but I'd like to use j smaller than M, and I think the way is simple enough as I need yx^2 - Ax + M^2 = 0 and yz^2 - Az + j^2 = 0 and it might seem strange that a simple shift in signs could mean all the difference, but I think it does. It comes down to getting multiples of two squares subtracting from each other versus adding which is what they've been doing with my other equations. I think that it is actually that easy. Redo all the equations with that shift in signs and if my guess is right the algorithms will now ALWAYS factor. James Harris === Subject: Re: SF: Sign change > I think that surrogate factoring preferentially factors the smaller > number, so I need to use a j that is bigger than M, not less. > That changes signs in key equations, and with THAT minor change (or > major depending on how you look at it), my basic algorithm just might > work. > It should but I'd like to use j smaller than M, and I think the way is > simple enough as I need > yx^2 - Ax + M^2 = 0 > and > yz^2 - Az + j^2 = 0 > and it might seem strange that a simple shift in signs could mean all > the difference, but I think it does. And so help me, if all I had to do was pick the right signs in the first place... Oh well, that's how basic research is, and that's why the discovery process can be such an amazing thing. How do people figure anything out at all? Well in my case, a lot of experimenting and brainstorming. Here for simple reasons I focused on one set of signs and spent a lot of time exploring a rather rich problem space, and now I finally start to say to myself that hey, maybe I picked the wrong signs!!! > It comes down to getting multiples of two squares subtracting from each > other versus adding which is what they've been doing with my other > equations. Deep within the methods there is a key area where these squares are either subtracting from multiples of each other or adding, and with the signs the way I had them, for factors of M, they are freaking adding, but for the factors of j, they are subtracting. That little difference of multiples of squares adding versus subtracting probably explains a LOT. > I think that it is actually that easy. Redo all the equations with > that shift in signs and if my guess is right the algorithms will now > ALWAYS factor. And I'm too sleepy to do it now, so, later... Change the signs and that's it? If this works then it will be one of the oddest hold-ups in my research history. Whether you realize it or not, I tend to zoom in on problems rather quickly, even if it takes years, as some problems just take years. Here I've been held up for over two months with what I now think is a sign problem. Well, I guess a little over two months isn't that big of a deal. Now if it had been two years, then that would have been something, or five years, and then I'd have reason to really be upset. A few months? Not a big deal. James Harris === Subject: Re: SF: Sign change Message: Narcissistic Personality Disorder: Reason for posting? === Subject: Narcissistic Personality Disorder: Reason for posting? Author: James Harris I've become really bothered by my posting, but I can't seem to stop. However, I realized that part of the sick cycle are some of those people who reply to me so negatively, so in the hope that these people might show mercy I thought I'd post something I found on the web, with limited hope, I admit, that anything will change. Alternatively, the narcissist feels victimized by mediocre bureaucrats and intellectual dwarves who consistently fail to appreciate his outstanding - really, unparalleled - talents, skills, and accomplishments. Being haunted by his challenged inferiors substantiates the narcissist's comparative superiority. Driven by pathological envy, these pygmies collude to defraud him, badger him, deny him his due, denigrate, isolate, and ignore him. The narcissist projects onto this second class of lesser persecutors his own deleterious emotions and transformed aggression: hatred, rage, and seething jealousy. But here are the scary quotes: Paranoid ideation - the narcissist's deep-rooted conviction that he is being persecuted by his inferiors, detractors, or powerful ill-wishers - serves two psychodynamic purposes. It upholds the narcissist's grandiosity and it fends off intimacy. The paranoid delusions of the narcissist are always grandiose, cosmic, or historical. His pursuers are influential and formidable. They are after his unique possessions, out to exploit his expertise and special traits, or to force him to abstain and refrain from certain actions. The narcissist feels that he is at the center of intrigues and conspiracies of colossal magnitudes. The paranoid narcissist ends life as an oddball recluse - derided, feared, and loathed in equal measures. His paranoia - exacerbated by repeated rejections and ageing - pervades his entire life and diminishes his creativity, adaptability, and functioning. The narcisstis personality, buffetted by paranoia, turns ossified and brittle. Finally, atomized and useless, it succumbs and gives way to a great void. The narcissist is consumed. It looks like a person in the grips of this narcissism thing loses connection with reality but somehow *feeds* on ANY attention, including negative attention. They called it narcissistic supply. It really is a twisted illness that apparently can lead to even nastier illnesses. I am self-diagnosed (typical narcissistic behavior) but it looks like it fits to me. However, I've decided that I don't want to succumb to mental woes, so I am fighting the dark narcissist within by making this informative post, though it's probably a sign of the illness. In any event, maybe it will help in the long run as information is power. I will do my best to stop posting, but it will help if some of you try to help out by not replying to me, or if you reply, by refraining from the attacks that feed my narcissistic side, and its irrational beliefs that everyone is out to get me. If you have facts, sure, no problem. My inability to face facts presented without emotion will just be a sign of the dark side. My only fear now is that some of you are sick as well, and act from your own illness, so that you can't stop attacking in your posting. But on this Christmas Day, I will act from hope, and hopefully this New Year will bring a change, and an escape from the madness. James Harris === Subject: recipe for the wavelet Please help the following: ___ If phi(x) = /__ Alpha_n Phi (2x-n), show that a recipe for the n wavelet Psi is given by ___ Psi(x) =/__ (-1)^n Alpha_(-n+1) Phi(2x-n) . n i understand that i need to show that = 0, but i need to know the starting point, how do i get start it??? === Subject: Publication not so important in mathematics The real point whether you pick it up or not is that in the mathematics field, having a published result is not given the weight that people outside of the math field tend to think it does. The reality is that mathematical results go through a highly democratic process, which is basically a checking by committee. It'd be much nicer, if say, a computer did proof-checking but that's not how it's done. So notice that I got a paper published, and the math people didn't pause before berating the journal, and some of them sent a bunch of emails and the editor caved. More results than are admitted are not counted as worth anything at all in the math field despite publication. Now I went to the trump card which was to go to a heavily respected journal, which now has that paper, and if they accept it for publication, it won't matter how many sci.math'ers email in protest! But just remember, in the math world, it's a lot more democratic than you may have realized as publication has to do with a committee decision, and people do make mistakes, and math people know it. They just don't go around admitting it readily. Here in their behavior with my paper you can see the reality. So it's not publication that's really important in mathematics. If you listen a lot you'll hear that the weight of the journal, and how But if you listen *really* carefully, you'll realize that mathematicians don't really have a well established system after all, and a lot of what they do is go with gut feelings. James Harris === Subject: Re: Publication not so important in mathematics Message: Narcissistic Personality Disorder: Reason for posting? === Subject: Narcissistic Personality Disorder: Reason for posting? Author: James Harris I've become really bothered by my posting, but I can't seem to stop. However, I realized that part of the sick cycle are some of those people who reply to me so negatively, so in the hope that these people might show mercy I thought I'd post something I found on the web, with limited hope, I admit, that anything will change. Alternatively, the narcissist feels victimized by mediocre bureaucrats and intellectual dwarves who consistently fail to appreciate his outstanding - really, unparalleled - talents, skills, and accomplishments. Being haunted by his challenged inferiors substantiates the narcissist's comparative superiority. Driven by pathological envy, these pygmies collude to defraud him, badger him, deny him his due, denigrate, isolate, and ignore him. The narcissist projects onto this second class of lesser persecutors his own deleterious emotions and transformed aggression: hatred, rage, and seething jealousy. But here are the scary quotes: Paranoid ideation - the narcissist's deep-rooted conviction that he is being persecuted by his inferiors, detractors, or powerful ill-wishers - serves two psychodynamic purposes. It upholds the narcissist's grandiosity and it fends off intimacy. The paranoid delusions of the narcissist are always grandiose, cosmic, or historical. His pursuers are influential and formidable. They are after his unique possessions, out to exploit his expertise and special traits, or to force him to abstain and refrain from certain actions. The narcissist feels that he is at the center of intrigues and conspiracies of colossal magnitudes. The paranoid narcissist ends life as an oddball recluse - derided, feared, and loathed in equal measures. His paranoia - exacerbated by repeated rejections and ageing - pervades his entire life and diminishes his creativity, adaptability, and functioning. The narcisstis personality, buffetted by paranoia, turns ossified and brittle. Finally, atomized and useless, it succumbs and gives way to a great void. The narcissist is consumed. It looks like a person in the grips of this narcissism thing loses connection with reality but somehow *feeds* on ANY attention, including negative attention. They called it narcissistic supply. It really is a twisted illness that apparently can lead to even nastier illnesses. I am self-diagnosed (typical narcissistic behavior) but it looks like it fits to me. However, I've decided that I don't want to succumb to mental woes, so I am fighting the dark narcissist within by making this informative post, though it's probably a sign of the illness. In any event, maybe it will help in the long run as information is power. I will do my best to stop posting, but it will help if some of you try to help out by not replying to me, or if you reply, by refraining from the attacks that feed my narcissistic side, and its irrational beliefs that everyone is out to get me. If you have facts, sure, no problem. My inability to face facts presented without emotion will just be a sign of the dark side. My only fear now is that some of you are sick as well, and act from your own illness, so that you can't stop attacking in your posting. But on this Christmas Day, I will act from hope, and hopefully this New Year will bring a change, and an escape from the madness. James Harris === Subject: As Seen on TV by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) with ESMTP id j2945Nm17508 HOME | ABOUT US | SCIENTIFI<S>C</S> APPROACH | FAQ's | HOLLYWOOD SECRETS | Privacy Policy and Terms of Use

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=== Subject: Rational expressions, Draining pool I've been trying to set up this problem, and haven't been making very good headway: Drains A and B are used to empty a swimming pool. Drain A alone can empty the pool in 4.75 hours. Let t be the time it takes for drain B alone to empty the pool. (Assume the pool drains at a constant rate.) Find an algebraic representation that gives the part of the drainage that can be done in 1 hour with both drains open at the same time as a function of t. In a similiar vein, I then have to find out the time it takes for drain B alone to empty the pool if both drains, when open at the same time, can empty the pool in 2.6 hours. I should be able to do this if I can figure out how to set up the above part.