mm- Subject: Re: Riemannıs Hypothesis > Regarding an idea to solve a problem when I donıt know the subject: > Youıre right, I donıt know complex function analysis and am just > scratching the surface in real analysis, again. But, isnıt the > problem about prime numbers? The Riemann Hypothesis has *consequences* for the distribution of prime numbers, but it is most naturally stated and addressed as a problem about a function of a complex variable. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Seeking wealthy Paranormal Investigator we saw our hero |-|erc : >> I doubt very much if Herc is eligible for the challenge testing, >> unless Randi has had a change of heart. See March 15 2002 commentary >> (http://www.randi.org/jr/031502.html) for more details. > >So why didnıt they just test him? > >I mean, I can understand that they canıt simply drop everything and >run tests on anyone who happens to stroll through their door, but >couldnıt they arrange something? >> >> The Ŝrst test is to state clearly, in writing, exactly what it is >> that you claim to be able to do >> Herc failed. > nope. they weasled out. Nope. Iıve seen what you write. You canıt state *anything clearly. -- Mark K. Bilbo - a.a. #1423 EAC Department of Linguistic Subversion Alt-atheism website at: http://www.alt-atheism.org -------------------------------------------------- There is no system but GNU, and Linux is one of its kernels. === Subject: Re: Seeking wealthy Paranormal Investigator > we saw our hero |-|erc >: > I doubt very much if Herc is eligible for the challenge testing, > unless Randi has had a change of heart. See March 15 2002 commentary > (http://www.randi.org/jr/031502.html) for more details. >> >>So why didnıt they just test him? >> >>I mean, I can understand that they canıt simply drop everything and >>run tests on anyone who happens to stroll through their door, but >>couldnıt they arrange something? > > The Ŝrst test is to state clearly, in writing, exactly what it is > that you claim to be able to do > Herc failed. >> nope. they weasled out. >Nope. Iıve seen what you write. You canıt state *anything clearly. Now, now. Itıs not nice to tease the lunatic ;) -------------- Robyn Resident Witchypoo & EAC Spellcaster #1557 === Subject: Re: Seeking wealthy Paranormal Investigator <4haLc.243$L4.201@www.newsranger.com> we saw our hero Robibnikoff : > says... >> we saw our hero |-|erc >>: >> I doubt very much if Herc is eligible for the challenge testing, >> unless Randi has had a change of heart. See March 15 2002 >> commentary (http://www.randi.org/jr/031502.html) for more details. > >So why didnıt they just test him? > >I mean, I can understand that they canıt simply drop everything and >run tests on anyone who happens to stroll through their door, but >couldnıt they arrange something? >> >> The Ŝrst test is to state clearly, in writing, exactly what it is >> that you claim to be able to do >> >> Herc failed. > > nope. they weasled out. >>Nope. Iıve seen what you write. You canıt state *anything clearly. > Now, now. Itıs not nice to tease the lunatic ;) Party pooper! -- Mark K. Bilbo - a.a. #1423 EAC Department of Linguistic Subversion Alt-atheism website at: http://www.alt-atheism.org -------------------------------------------------- Come to think of it, there are already a million monkeys on a million typewriters, and the Usenet is NOTHING like Shakespeare! -- Blair Houghton === Subject: Re: Seeking wealthy Paranormal Investigator >> Herc failed. > nope. they weasled out. > Nope. Iıve seen what you write. You canıt state *anything clearly. >An interesting claim, which I have passed to our challenge committee for consideration >Barry Williams >Editor >the Skeptic YOUıRE A MORON UNDERSTAND THAT? Herc === Subject: Re: Seeking wealthy Paranormal Investigator we saw our hero |-|erc : > Herc failed. >> >> nope. they weasled out. >> Nope. Iıve seen what you write. You canıt state *anything clearly. >>An interesting claim, which I have passed to our challenge committee for >>consideration Barry Williams >>Editor >>the Skeptic > YOUıRE A MORON UNDERSTAND THAT? But at least Iım sane... -- Mark K. Bilbo - a.a. #1423 EAC Department of Linguistic Subversion Alt-atheism website at: http://www.alt-atheism.org -------------------------------------------------- Come to think of it, there are already a million monkeys on a million typewriters, and the Usenet is NOTHING like Shakespeare! -- Blair Houghton === Subject: Re: Seeking wealthy Paranormal Investigator Mark K. Bilbo > Herc failed. >> >> nope. they weasled out. >> >> Nope. Iıve seen what you write. You canıt state *anything clearly. >>An interesting claim, which I have passed to our challenge committee for >>consideration Barry Williams >>Editor >>the Skeptic > YOUıRE A MORON UNDERSTAND THAT? > But at least Iım sane... nope. a stitch in time saves Herc === Subject: Centrifugal force - real or Ŝctitious Newton demonstrated centrifugal - center ŝeeing - force with his spinning bucket experiment, and now farmers and scientists use centrifuges to separate milk and other substances. The sides of a spinning bucket exert a centripetal force on the ŝuid spinning with them; restraining the ŝuid to the conŜnes of the bucket. Spin drying washing machines have holes in the inner drum to allow the water to escape into an outer drum where it is pumped or otherwise drained away. Whatıs harder to understand is how centrifugal force causes the Earth to bulge at the equator, and the rate of free fall to be less there: To some, centrifugal force isnıt considered a real force as such, because if it was, masses would be thrown off into space due to Earthıs rotation. Such people canıt see that itıs a matter of _equilibrium_; that as long as the centripetal force is greater than the centrifugal force, mass will gravitate. === Subject: Re: Centrifugal force - real or Ŝctitious > Whatıs harder to understand is how centrifugal force causes the Earth > to bulge at the equator, and the rate of free fall to be less there: Earth bulge at equator is due to a REAL centrifugal force: http://observe.arc.nasa.gov/nasa/space/centrifugal/ centrifugal4.html http://observe.arc.nasa.gov/nasa/space/centrifugal/ centrifugal5.html The rate of free fall is less because the distance to Earth center increases and gravity decreases at the square of the distance. The tidal opposite Ocean bulge also is due to centrifugal force. (see NOAA explanation). > To some, centrifugal force isnıt considered a real force as such, > because if it was, masses would be thrown off into space due to > Earthıs rotation. > Such people canıt see that itıs a matter of _equilibrium_; that as > long as the centripetal force is greater than the centrifugal force, > mass will gravitate. NASA explains it all about centrifugal force starting here: http://observe.arc.nasa.gov/nasa/space/centrifugal/ centrifugal_entry.html The problem is that nobody can explain what inertia is. Because most of the people donıt like to show their ignorance they simply avoid the problem by means of the equivalence principle. Then they are like parrots after reading the book. Donıt ask intelligent questions to a parrot. === Subject: Re: Centrifugal force - real or Ŝctitious > Whatıs harder to understand is how centrifugal force causes the Earth > to bulge at the equator, and the rate of free fall to be less there: > Earth bulge at equator is due to a REAL centrifugal force: > http://observe.arc.nasa.gov/nasa/space/centrifugal/ centrifugal4.html > http://observe.arc.nasa.gov/nasa/space/centrifugal/ centrifugal5.html > The rate of free fall is less because the distance to Earth center increases > and gravity decreases at the square of the distance. > The tidal opposite Ocean bulge also is due to centrifugal force. > (see NOAA explanation). A Ŝctitious force is just as real as any other force. Thereıd be no confusion over this at all, if only people would understand the true deŜnition of a Ŝctitious force in physics. They are just forces that are dependent on the choice of coordinate system. Frankly, I prefer the alternative terminology inertial force, since it avoids such confusion. > To some, centrifugal force isnıt considered a real force as such, > because if it was, masses would be thrown off into space due to > Earthıs rotation. > Such people canıt see that itıs a matter of _equilibrium_; that as > long as the centripetal force is greater than the centrifugal force, > mass will gravitate. > NASA explains it all about centrifugal force starting here: > http://observe.arc.nasa.gov/nasa/space/centrifugal/ centrifugal_entry.html > The problem is that nobody can explain what inertia is. True, but totally irrelevant. > Because most of the people donıt like to show their ignorance they simply > avoid the problem by means of the equivalence principle. Ok, show me one inertial force that doesnıt obey the equivalence principle, at least locally. What do you think makes our experience of the so-called Ŝctitious forces just as real as any other forces? === Subject: Re: Centrifugal force - real or Ŝctitious > Newton demonstrated centrifugal - center ŝeeing - force with his > spinning bucket experiment, and now farmers and scientists use > centrifuges to separate milk and other substances. > The sides of a spinning bucket exert a centripetal force on the ŝuid > spinning with them; restraining the ŝuid to the conŜnes of the > bucket. Spin drying washing machines have holes in the inner drum to > allow the water to escape into an outer drum where it is pumped or > otherwise drained away. > Whatıs harder to understand is how centrifugal force causes the Earth > to bulge at the equator, and the rate of free fall to be less there: > To some, centrifugal force isnıt considered a real force as such, > because if it was, masses would be thrown off into space due to > Earthıs rotation. > Such people canıt see that itıs a matter of _equilibrium_; that as > long as the centripetal force is greater than the centrifugal force, > mass will gravitate. You need to look up noninertial frames in a physics or mechanical engineering textbook. Even Halliday & Resnick talks about them. === Subject: Re: Centrifugal force - real or Ŝctitious > You need to look up noninertial frames in a physics or mechanical > engineering textbook. Even Halliday & Resnick talks about them. Given sHeadıs age and understanding of basic physics, he probably believes Halliday & Resnick was a vaudeville team. Tom Davidson Richmond, VA === Subject: Re: Centrifugal force - real or Ŝctitious > Newton demonstrated centrifugal - center ŝeeing - force with his > spinning bucket experiment, and now farmers and scientists use > centrifuges to separate milk and other substances. > The sides of a spinning bucket exert a centripetal force on the ŝuid > spinning with them; restraining the ŝuid to the conŜnes of the > bucket. Spin drying washing machines have holes in the inner drum to > allow the water to escape into an outer drum where it is pumped or > otherwise drained away. > Whatıs harder to understand is how centrifugal force causes the Earth > to bulge at the equator, and the rate of free fall to be less there: > To some, centrifugal force isnıt considered a real force as such, > because if it was, masses would be thrown off into space due to > Earthıs rotation. > Such people canıt see that itıs a matter of _equilibrium_; that as > long as the centripetal force is greater than the centrifugal force, > mass will gravitate. If you actually knew any physics at all, you would understand the meanings of these various terms already and wouldnıt be perpetually trying to start all these silly debates over non-issues. Learn the true physical meaning of the term Ŝctictious force and there will no longer be any real problem. Same goes for all of your other nonsense. === Subject: Re: Centrifugal force - real or Ŝctitious I was taught that centripetal acceleration is real and that centrifugal force is Ŝctitious. However, the centrifugal force model could be used to solve certain problems as well as the centripetal acceleration model. === Subject: Re: Centrifugal force - real or Ŝctitious > I was taught that centripetal acceleration is real and that > centrifugal force is Ŝctitious. However, the centrifugal force model > could be used to solve certain problems as well as the centripetal > acceleration model. Pseudoforces are very real to those that experience them - e.g. stuff inside a centrifuge. === Subject: Re: Centrifugal force - real or Ŝctitious > Such people canıt see that itıs a matter of _equilibrium_; that as > long as the centripetal force is greater than the centrifugal force, > mass will gravitate. ing imbecile. Why are you trolling sci.math? Did a surd scare your momma when she went to have you aborted? http://www.pagetutor.com/idiot/idiot.html http://www.apa.org/journals/psp/psp7761121.html http://insti.physics.sunysb.edu/~siegel/quack.html -- Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) http://www.mazepath.com/uncleal/qz.pdf === Subject: Re: Centrifugal force - real or Ŝctitious Hi Donald, Please read this http://en.wikipedia.org/wiki/Centrifugal_force Maybe itıll help you understand. mate, you shouldıve paid more attention in your physics lessons! > Newton demonstrated centrifugal - center ŝeeing - force with his > spinning bucket experiment, and now farmers and scientists use > centrifuges to separate milk and other substances. > The sides of a spinning bucket exert a centripetal force on the ŝuid > spinning with them; restraining the ŝuid to the conŜnes of the > bucket. Spin drying washing machines have holes in the inner drum to > allow the water to escape into an outer drum where it is pumped or > otherwise drained away. > Whatıs harder to understand is how centrifugal force causes the Earth > to bulge at the equator, and the rate of free fall to be less there: > To some, centrifugal force isnıt considered a real force as such, > because if it was, masses would be thrown off into space due to > Earthıs rotation. > Such people canıt see that itıs a matter of _equilibrium_; that as > long as the centripetal force is greater than the centrifugal force, > mass will gravitate. === Subject: Re: JSHıs mistakes happen all the time >Now a poster has found a minor series of mistakes in my APF paper, and >I admit some chagrin. But mistakes happen. As an author you can >write something and for a lot of psychological reasons (and just plain >carelessness) miss mistakes. >I donıt know why I wasnıt notiŜed of that mistake by Ioannis Argyros, >but given what Iıve seen from editors at Southwest Journal of >Mathematics, Iım not surprised. >>So itıs *other* people again who are to blame? >OOPS! That should be Southwest Journal of Pure and Applied >Mathematics. Interesting that the poster didnıt point out that little >mistake. >>Should I have? >I think this post emphasizes my point that computers should check math >proofs, as notice below how *protective* this guy is about Andrew >Wiles. >>I am protective of: my lovely wife, my home, my family, etc. Why oh why >>should I be protective of Andrew Wiles? I like what he did, but he >>should protect himself and his; Iım sure he can. If I understood well, >>you couldnıt do it, the FLT thing. Neither could I have done it, ever. >>Difference is, I never pretended to. > Yet you act as if you know that his work is correct. I do not know. I *expect* it to be correct. I simply do not have the mathematical insight to understand one way or the other. Many mathematicians have checked his proof. If it turns out not to be correct, so be it. His contribution to mathematix in general will be *huge* nonetheless. Is yours? Mine is *none*, *zero*, *nothing*. But thatıs ok. I started out as a promising math student, a long time ago. But I took different paths. Now I am a Delphi programmer, and I can only do my best. I will never write a cipher that will shake cryptography at its foundations. I will never write a shorter proof for FLT. I will never factor a huge integer. Thatıs ok. I will just try to write an application thatıs as solid as it can be, and it will incorporate cryptography, and I will use standard algorithms like AES, even if I donıt understand exactly why AES is probably secure, for now. I simply donıt have the *time* to dive into Ŝnding out why Rijndael is secure. I will use standard algorithms like AES, even if thatıs more boring than using some algorithm of my own. It will be secure. Thatıs what my boss needs. Even if doesnıt know it... yet... > Actually you know what youıve been told. Yes. I have to believe what Iım told in some cases, like this particular one. As a programmer, I use techniques that a) Iıve learned, and b) Iıve developed myself. But even things I developed myself will oftentimes be founded on things Iıve learned. The greatest classical composer of all times (IMHO), J.S. Bach, learned most from *copying* work from others. It took his eyesight in the end, but his music turned out to be of a level never surpassed by anybody (IMHO, again). Iım not in any way comparing myself to JSB here (God forbid!), heıs just a *great* example of a genius learning the tricks of the trade. Where else can one learn, but from great teachers, they themselves having been students once? > Even if youıve looked over his paper, I doubt you have the ability to > check it thoroughly, and even if you do, Iım sure you didnıt. No I didnıt. I do *not* have the mathematical ability to even remotely understand what heıs talking about. But I play a mean guitar, can sing too, and know how to write a song. Iım not sure AW can. We all have our talents. > Why didnıt you? Because you believe what youıre told. Do I have to repeat myself? Even ifıd want to, I wouldnıt have the time... > You are a follower. Only when I need to. It gives me time to *not* be a follower in other areas. M. === etween. He has seen Pi and didnıt like it and didnıt think it resembled him at all. (Wrong, it Ŝts him to a tee) He doesnıt have total recall and has stated that he travels with a lap top to notate items. Also, he uses cut nı paste a lot if you read all the way through his ramblings. He is anti-social as shown by his angry statements towards those who, by his own admission, have been kind (but not kind enough) to him. Still, heıs intelligent and seems to be able to take a joke on occassion. Thatıs where I came in. ALOHA Reply to group (Unsolicited e-mail is deleted from the server unread if it comes from anyone not already in my addressbook. Iıll never even see it) === Subject: Force and counterforce Masses resting on the terra Ŝrma of planets like Earth are good examples of force and counterforce; where the weight-force exerted on the planet by those masses is equal in magnitude but opposite in direction to the counterweight exerted by the planet on them. A weight scale or toe between them will give a pretty good demonstration of the nature of force and weight. Since a planet is so much more massive than most masses resting on them it is common to attribute the acceleration of free fall (g) solely to that of the masses resting on the planet, because itıs acceleration is negligable in comparison. === Subject: Re: Force and counterforce > Masses resting on the terra Ŝrma of planets like Earth are good > examples of force and counterforce; where the weight-force exerted on > the planet by those masses is equal in magnitude but opposite in > direction to the counterweight exerted by the planet on them. > A weight scale or toe between them will give a pretty good > demonstration of the nature of force and weight. > Since a planet is so much more massive than most masses resting on > them it is common to attribute the acceleration of free fall (g) > solely to that of the masses resting on the planet, because itıs > acceleration is negligable in comparison. Put a block on the table, Shead. Measure all the forces on the block--primarily the force due to gravity, and the force from the table on the block. The net force is exactly zero, F = ma, there is no acceleration and the block doesnıt move. === Subject: Re: Force and counterforce > > Masses resting on the terra Ŝrma of planets like Earth are good > examples of force and counterforce; where the weight-force exerted on > the planet by those masses is equal in magnitude but opposite in > direction to the counterweight exerted by the planet on them. > > A weight scale or toe between them will give a pretty good > demonstration of the nature of force and weight. > Cut< > Put a block on the table, Shead. Measure all the forces on the > block--primarily the force due to gravity, and the force from > the table on the block. The net force is exactly zero, F = ma, > there is no acceleration and the block doesnıt move. Hereıs the point Iım trying to make Sam: Despite the fact that there is no motion as such between the table and the block, the table is _displacing_; forcibly decelerating the block from its usual free falling toward Earthıs center; which it would do if it was allowed to fall freely. === Subject: Re: Force and counterforce > Masses resting on the terra Ŝrma of planets like Earth are good > examples of force and counterforce; where the weight-force exerted on > the planet by those masses is equal in magnitude but opposite in > direction to the counterweight exerted by the planet on them. > A weight scale or toe between them will give a pretty good > demonstration of the nature of force and weight. > Since a planet is so much more massive than most masses resting on > them it is common to attribute the acceleration of free fall (g) > solely to that of the masses resting on the planet, because itıs > acceleration is negligable in comparison. If I have a ball of no weight, and another ball of no weight, with no earth, and nothing in between, Balls just touching, how much force and how much counterforce do they exert on each other? === Subject: Re: Force and counterforce >>Masses resting on the terra Ŝrma of planets like Earth are good >>examples of force and counterforce; where the weight-force exerted on >>the planet by those masses is equal in magnitude but opposite in >>direction to the counterweight exerted by the planet on them. >>A weight scale or toe between them will give a pretty good >>demonstration of the nature of force and weight. >>Since a planet is so much more massive than most masses resting on >>them it is common to attribute the acceleration of free fall (g) >>solely to that of the masses resting on the planet, because itıs >>acceleration is negligable in comparison. > If I have a ball of no weight, and another ball of no weight, with no earth, > and nothing in between, Balls just touching, how much force and how much > counterforce do they exert on each other? It depends whether the astronaut is trying not to pee in the suit. -- john === Subject: Re: Force and counterforce >>Masses resting on the terra Ŝrma of planets like Earth are good >>examples of force and counterforce; where the weight-force exerted on >>the planet by those masses is equal in magnitude but opposite in >>direction to the counterweight exerted by the planet on them. >> >>A weight scale or toe between them will give a pretty good >>demonstration of the nature of force and weight. >> >>Since a planet is so much more massive than most masses resting on >>them it is common to attribute the acceleration of free fall (g) >>solely to that of the masses resting on the planet, because itıs >>acceleration is negligable in comparison. > If I have a ball of no weight, and another ball of no weight, with no earth, > and nothing in between, Balls just touching, how much force and how much > counterforce do they exert on each other? > It depends whether the astronaut is trying not to pee in the suit. > -- > john Good one! That also means NASA has tried to solved it. === Subject: Re: Force and counterforce In the absence of a stronger gravitational Ŝeld wouldnıt they push away from each other due to the EM? Or would they transfer that force to orbiting each other? > Masses resting on the terra Ŝrma of planets like Earth are good > examples of force and counterforce; where the weight-force exerted on > the planet by those masses is equal in magnitude but opposite in > direction to the counterweight exerted by the planet on them. > A weight scale or toe between them will give a pretty good > demonstration of the nature of force and weight. > Since a planet is so much more massive than most masses resting on > them it is common to attribute the acceleration of free fall (g) > solely to that of the masses resting on the planet, because itıs > acceleration is negligable in comparison. > If I have a ball of no weight, and another ball of no weight, with no earth, > and nothing in between, Balls just touching, how much force and how much > counterforce do they exert on each other? === Subject: Re: Force and counterforce > Masses resting on the terra Ŝrma of planets like Earth are good > examples of force and counterforce; where the weight-force exerted on > the planet by those masses is equal in magnitude but opposite in > direction to the counterweight exerted by the planet on them. > A weight scale or toe between them will give a pretty good > demonstration of the nature of force and weight. > Since a planet is so much more massive than most masses resting on > them it is common to attribute the acceleration of free fall (g) > solely to that of the masses resting on the planet, because itıs > acceleration is negligable in comparison. Die howling in agony, with a brand-new hardcover college physics text rammed Ŝrmly up your back-side. === Subject: Re: Force and counterforce > Since a planet is so much more massive than most masses resting on > them Idiot. http://www.pagetutor.com/idiot/idiot.html http://www.apa.org/journals/psp/psp7761121.html http://insti.physics.sunysb.edu/~siegel/quack.html -- Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) http://www.mazepath.com/uncleal/qz.pdf === Subject: Best divisors approximation Hi all, Question for the math gurus. I have a clock circuit that divides a master clock to a requested frequency. There are two divisors, both integers. Divisor_1 can have the value 2-254, even values only. Divisor_2 can have the value 0-255. The problem is to select the two divisor values such that the resulting frequency is the closest possible to the requested frequency. The clock frequency is given by the equation: F_clk = F_master / (Divisor_1 * (Divisor_2 + 1)) Is there an algorithm that can do this fairly easily? Any input appreciated. Paul === Subject: Re: Best divisors approximation > Question for the math gurus. I have a clock circuit that divides a > master clock to a requested frequency. There are two divisors, both > integers. Divisor_1 can have the value 2-254, even values only. > Divisor_2 can have the value 0-255. The problem is to select the two > divisor values such that the resulting frequency is the closest > possible to the requested frequency. > The clock frequency is given by the equation: > F_clk = F_master / (Divisor_1 * (Divisor_2 + 1)) > Is there an algorithm that can do this fairly easily? Any input > appreciated. Christian Bauıs spreadsheet suggestion is easy, of course, or if you are going to do this frequently you could easily write a c or bc or perl program to try all the values in a few milliseconds. For bigger problems, the ofŜcial methods to solve in reasonable time are based on continued fractions and/or lattice reduction methods, unless my understanding is upside down. See, eg, the Re: How to Ŝnd near integer values of n*a and n*b? -jiw === Subject: Re: Best divisors approximation > Hi all, > Question for the math gurus. I have a clock circuit that divides a > master clock to a requested frequency. There are two divisors, both > integers. Divisor_1 can have the value 2-254, even values only. > Divisor_2 can have the value 0-255. The problem is to select the two > divisor values such that the resulting frequency is the closest > possible to the requested frequency. > The clock frequency is given by the equation: > F_clk = F_master / (Divisor_1 * (Divisor_2 + 1)) > Is there an algorithm that can do this fairly easily? Any input > appreciated. If F_clk, F_master and Divisor_1 are given, then Divisor_2 = F_master / (Divisor_1 * F_clk) - 1 except that Divisor_2 must be an integer in the range 0 to 255. There are 127 possible values for Divisor_1. For each of them, calculate Divisor_2 as above. Then let D1 = max (0, ŝoor (Divisor_2)) and D2 = min (255, ceil (Divisor_2)). Calculate F_clk substituting D1, D2 for Divisor_2. Find which pair of Divisor_1 and Divisor_2 produces the value closest to the F_clk that you want to achieve. Easiest way to do this is by using just about any available spreadsheet program. === Subject: Re: ŒUncountableı doesnıt exist > I realised some time ago that I would reject AC in favor of DC, which > allows only countable collections. The reason for this, I founc out, > was that I canıt accept any proof that requires an uncountable number > of steps - it is logically impossible. > I have never seen a proof that required even an inŜnite (let alone > uncountable) number of steps, and therefore the question has never > arisen. A proof by transŜnite induction, for example, has basically one > step, which often involves the consideration of three cases. Well, I guess we have different meanings here for Œnumber of stepsı. Regardless, I canıt see any other reason to reject AC. Andrew Usher === Subject: Re: ŒUncountableı doesnıt exist >> I realised some time ago that I would reject AC in favor of DC, which >> allows only countable collections. The reason for this, I founc out, >> was that I canıt accept any proof that requires an uncountable number >> of steps - it is logically impossible. >> I have never seen a proof that required even an inŜnite (let alone >> uncountable) number of steps, and therefore the question has never >> arisen. A proof by transŜnite induction, for example, has basically one >> step, which often involves the consideration of three cases. > Well, I guess we have different meanings here for Œnumber of stepsı. > Regardless, I canıt see any other reason to reject AC. The mere fact that a proof mentions inŜnite sets does not mean the proof requires inŜnitely many steps. How many steps does it take to demonstrate that the set of even numbers is countable? Answer: One. Let f: N -> 2N be given by f(n) = 2*n for each n. (Technically, if you Ŝll in all the gaps in that answer, there would be a few more steps than just one, but the number would still be small and Ŝnite.) By deŜnition, a proof is a *Ŝnite* sequence of formulas with the property that each formula is either an axiom, or a consequence of previous formulas. If you consider each formula to be a step in the proof, then every proof has a Ŝnite number of steps. Invoking the axiom of choice, even on an uncountable collection of sets, is just *one* step in a proof. -- Dave Seaman Judge Yohnıs mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: Algebra syntax > Iım wondering if anyone could explain some of the syntax in Algebra, or > direct me to a web source explaining it. > For instance what is the difference betwen the following. > xx (x)(x) > or the difference between 8(8)(8) and (8)(8)(8) > and is this true > y^2 = yy = y(y) = (y)(y) > I Ŝnd all the variations confusing. > tkns > Rob It should make clear when these types of expressions are equivalent or not. l8r, Mike N. Christoff === Subject: Re: PBZ: Short Proof of Fermatıs Last Theorem > Dude, youıre so smart. You have invented a completely new way of > proving things: Everything that ought to be true is true. Why did > nobody think of that before, I wonder? That is the nicest compliment that anyone has ever given me on usenet. I am touched. A tear just dripped down from my eye. My way of proving things comes from relativity theory. I believe that math is a subjective subject; it is relative to the person who is reading the proof. That is why I am able to prove things to myself that normal people have difŜculty understanding. Because I am a genius, relative to me, my proofs are correct, but relative to the typical average person, my proofs are wrong because the typical average person is just plain lazy and stupid! Prof. Ben Zona, PhD Doctor of Thinkology, University of San Moritz, 1968 === Subject: Re: Riemannıs Hypothesis by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i6K1BIK08520; Iım glad, in a way, for skepticism. At least, you think the problem wonıt be solved for some time, ... at least, not by me. I wanted to shortcut the process. While I intend to learn the math, because I had an idea that I thought would jumpstart the process, I didnıt want to hold it back. I thought the solution to the problem was a pressing issue with real applications. My idea doesnıt solve the complex analysis issue. Since I admitted to not knowing complex analysis, I thought that was obvious. Iım sorry I wasnıt clear on that. Actually, I donıt know what the problem is with Riemannıs hypothesis. That was one of my questions. My idea may lead someone knowledgeable in complex analysis to Ŝnd the solution. It isnıt a direct insight in complex function analysis. I think it would be in the category of number theory. And, Iım not versed in that subject, yet, either. >> Iım looking for collaborators to solve the Clay Mathematics Foundation >> Millennium Problem -- a proof of Riemannıs Hypothesis. >> I donıt have the competency, yet, to understand the problem from an analytic >> functions perspective. I need collaborator(s) to explain the problem and >> write the proof. >> I am re-studying Calculus and studying Topology and Number Theory, >> independently. It will be some time before I am competent to write the >> proof. >> What I have is an idea to solve the problem. >> It may have been tried; I donıt know and I havenıt researched that aspect, >> either, and I may not have the knowledge to do so. >> What do you think? >I think you have a better chance of success at building a perpetual >motion machine. And thereıs a lot more money in it than there is in >the Riemann Hypothesis. >-- >Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: HELP: INTEGER FUNCTION ?=? REAL FUNCTION by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i6K1BIJ08539; A standard example would be the Gamma Function and the Factorial Function. I want to know if there is a theorem that says... If you have an integer function, f(n) for n=0,1,2...., from the integers to the integers AND a real function, g(x) for x >= 0, from the reals to the reals AND if f(n) = g(x) for each x = n, THEN g(x) is unique. Is there such a theorem? Does it have a name? Where can I Ŝnd it? Are there conditions under which g(x) would not be unique? === Subject: Re: HELP: INTEGER FUNCTION ?=? REAL FUNCTION >A standard example would be the Gamma Function and the Factorial Function. >I want to know if there is a theorem that says... >If you have an integer function, f(n) for n=0,1,2...., from the integers to the integers >AND a real function, g(x) for x >= 0, from the reals to the reals AND > if f(n) = g(x) for each x = n, >THEN g(x) is unique. Of course not - youıre free to deŜne g(1/2) any way you like. >Is there such a theorem? >Does it have a name? For the gamma function in particular thereıs a thing called the Bohr-Mollerup Theorem that says g is unique if you add certain conditions. >Where can I Ŝnd it? Many books on complex analysis. Or >Are there conditions under which g(x) would not be unique? g is _not_ unique _without_ extra conditions... A general theorem (which doesnıt apply here!) is that if f : Z -> Z and sum |f(n)|^2 < inŜnity then f has exactly one extension to an entire function g : C -> C with the property that (1) |g(z)| <= c exp(pi|z|) and (2) int_R |g(x)|^2 < inŜnity. You could replace [2] by various similar conditions, but you canıt just leave it out (consider sin(pi z).) (Thatıs entire function as in complex analysis; int_R is the integral over the real line.) ************************ David C. Ullrich === Subject: Re: HELP: INTEGER FUNCTION ?=? REAL FUNCTION > If you have an integer function, f(n) for n=0,1,2...., from the > integers to the integers AND a real function, g(x) for x >= 0, from the > reals to the reals AND > if f(n) = g(x) for each x = n, THEN g(x) is unique. Is this what you mean? f:Z+ -> Z+ g:[0,oo) -> R for all n in Z+, g(n) = f(n) Then no, g isnıt unique. > Are there conditions under which g(x) would not be unique? The restriction of g to Z+ is uniquely determined by f. Other than that, g can be anything. === Subject: Re: HELP: INTEGER FUNCTION ?=? REAL FUNCTION > If you have an integer function, f(n) for n=0,1,2...., from the > integers to the integers AND a real function, g(x) for x >= 0, from the > reals to the reals AND > if f(n) = g(x) for each x = n, THEN g(x) is unique. > Is this what you mean? > f:Z+ -> Z+ > g:[0,oo) -> R > for all n in Z+, g(n) = f(n) > Then no, g isnıt unique. > Are there conditions under which g(x) would not be unique? > The restriction of g to Z+ is uniquely determined by f. > Other than that, g can be anything. So lets reŜne the OPs question. What additional restrictions can we place on g to make it unique. Particularly, if f(n) = n! then what makes g(z) = Gamma(z+1) so special? What properties does it have that singles this particular function from all the other candidates. Which of these properties can we generalize to the OP? I donıt have the other to these questions, but Iıd be very interested if someone can give me a clue. -Michael. === Subject: Re: HELP: INTEGER FUNCTION ?=? REAL FUNCTION >If you have an integer function, f(n) for n=0,1,2...., from the >integers to the integers AND a real function, g(x) for x >= 0, from the >reals to the reals AND > if f(n) = g(x) for each x = n, THEN g(x) is unique. >>Is this what you mean? >>f:Z+ -> Z+ >>g:[0,oo) -> R >>for all n in Z+, g(n) = f(n) >>Then no, g isnıt unique. >Are there conditions under which g(x) would not be unique? >>The restriction of g to Z+ is uniquely determined by f. >>Other than that, g can be anything. > So lets reŜne the OPs question. What additional restrictions can we place > on g to make it unique. Particularly, if f(n) = n! then what makes g(z) = > Gamma(z+1) so special? What properties does it have that singles this > particular function from all the other candidates. Which of these properties > can we generalize to the OP? > I donıt have the other to these questions, but Iıd be very interested if > someone can give me a clue. > -Michael. Artinıs theorem says that a function that agrees with n! on the integers and whose log is convex is gamma(x+1). Look in Bellmanıs book on the Gamma function. === Subject: Re: INTEGER FUNCTION ?=? REAL FUNCTION > I want to know if there is a theorem that says... > If you have an integer function, f(n) for n=0,1,2...., from the > integers to the integers AND a real function, g(x) for x >= 0, from > the reals to the reals AND if f(n) = g(x) for each x = n, > THEN g(x) is unique. No, there is no such theorem, because this conjecture is false. Counterexample: f(n) = 0 g1(x) = 0 g2(x) = sin(x * pi) Both g1 and g2 agree with f (that is, theyıre all zero) on every integer n, but g1 and g2 differ elsewhere. --Mark === Subject: Re: Polygrams and primes > Axel Harvey had written: > If n is the number of vertices of a regular polygon, then p_n , the > number of different ways that a regular polygram can be drawn inside > the polygon, is the number of non-divisors of n in [2, n/2). > Actually, that would have to be the number of integers in [2, n/2) which > are relatively prime to n. Consider 4 when n=10. You and Keith are right, of course. Sorry for the bother. === Subject: The real effect of centrifugal force Centrifugal force is mistakenly thought to cause a body to ŝy out of its circular path; along a tangent direction which represents the inertial path of Newtons inertial motion. Actually centrifugal force, when not restrained as by a string or sling causes a body to ŝee the center of an evolute radially; along involutes that remain directed radially away from the center: Like the sling used by David to slay Goliath; when it is released the missile travels radially away from the center to points equidistant along the tangent.; at potentially lethal speeds depending on the speed it has when it is released from the evolute. There is nothing Ŝctitious about centrifugal force! === Subject: Re: The real effect of centrifugal force > Actually centrifugal force, when not restrained as by a string or > sling causes a body to ŝee the center of an evolute radially; along > involutes that remain directed radially away from the center: Yes, its direction will be radial relative to the rotating reference frame center. But it will be tangential relative to any external inertial (stopped) reference frame. > Like the sling used by David to slay Goliath; when it is released the > missile travels radially away from the center to points equidistant > along the tangent.; at potentially lethal speeds depending on the > speed it has when it is released from the evolute. > There is nothing Ŝctitious about centrifugal force! The centrifugal force is very real for those that are in the rotating reference frame, not for those outside the rotating frame (in an inertial reference frame). NOAA explains Ocean tides Earth opposite bulge based on the centrifugal force due to Earth rotation. NASA also takes centrifugal forces very seriously. So far nobody has a good explanation for the circular motion. Nobody can explain this: http://physics.nad.ru/Physics/English/gyro_tmp.htm http://physics.nad.ru/Physics/English/gyro_txt.htm === Subject: Re: The real effect of centrifugal force > Centrifugal force is mistakenly thought to cause a body to ŝy out of > its circular path; along a tangent direction which represents the > inertial path of Newtons inertial motion. > Actually centrifugal force, when not restrained as by a string or > sling causes a body to ŝee the center of an evolute radially; along > involutes that remain directed radially away from the center: > Like the sling used by David to slay Goliath; when it is released the > missile travels radially away from the center to points equidistant > along the tangent.; at potentially lethal speeds depending on the > speed it has when it is released from the evolute. > There is nothing Ŝctitious about centrifugal force! not true. youıre mistaking the difference between letting go of the other end of the string and the string vanishing completely. if u let go of the string sure itıll ŝy off radially. but if the string itself disapeared. the rock would move in the direction it was.. tangentenial to the orbit of the rock. anyways have a nice day and please donıt forget to go yourself because youıre an illiterate dumb mother ing little wad. and btw. stop sending me emails about your gay black boyfriend and how you joined usenets to try to pick up little boys. everyone hates you you dumb old head. wit. useless old bag of . === Subject: Re: The real effect of centrifugal force > Centrifugal force is mistakenly thought to cause a body to ŝy out of > its circular path; along a tangent direction which represents the > inertial path of Newtons inertial motion. > > Actually centrifugal force, when not restrained as by a string or > sling causes a body to ŝee the center of an evolute radially; along > involutes that remain directed radially away from the center: > > Like the sling used by David to slay Goliath; when it is released the > missile travels radially away from the center to points equidistant > along the tangent.; at potentially lethal speeds depending on the > speed it has when it is released from the evolute. > > There is nothing Ŝctitious about centrifugal force! > not true. youıre mistaking the difference between letting go of the > other end of the string and the string vanishing completely. if u let > go of the string sure itıll ŝy off radially. but if the string itself > disapeared. the rock would move in the direction it was.. tangentenial > to the orbit of the rock. anyways have a nice day and please donıt > forget to go yourself because youıre an illiterate dumb mother > ing little wad. and btw. stop sending me emails about your gay > black boyfriend and how you joined usenets to try to pick up little > boys. everyone hates you you dumb old head. wit. useless old > bag of . Hey dirty mount, centripetal-centrifugal forces are action-reaction pair. When you whirl a stone attached to a rope (donıt break your empty head doing that) the centripetal force acts on the rock and the centrifugal force on the hand. If the rope is released, there is no longer a action-reaction, the centripetal force that caused the changes of momentum towards the center of forces (Iım sure you got no clue what Iım talking about) vanishes and the rock will move according to Newtonıs Ŝrst law in the direction of the right line of the initial velocity vecor at the time of release. If the centripetal force valished due to cutting the rope, the same will happen dummy because there is not any force acting on the rock any longer. Got it dirty mouth? If not, any Mechanics 101 text will do it if you can read. Mike === Subject: Re: The real effect of centrifugal force > Hey dirty mount, centripetal-centrifugal forces are action-reaction > pair. Yes, I think there are some textbooks that deŜne things this way -- that the centrifugal force is the force of the stone on the sling and that centripetal force is the force of the sling on the stone for instance. If I recall correctly, that is what I was taught. But that is not the usage that I prefer, that most of the people here prefer and that most textbooks prefer. In our preferred usage, centrifugal force is the imaginary force that you need to invoke in a rotating frame of reference so that Newtonıs second law (f=ma) works out. The terms Ŝctitious force or inertial force are often used to describe forces of this sort. Coriolis force is another example of a Ŝctitious force. Fictitious forces have no third law partners. John Briggs === Subject: Re: The real effect of centrifugal force > [snippo] > There is nothing Ŝctitious about centrifugal force! > not true. youıre mistaking the difference between letting go of the > other end of the string and the string vanishing completely. if u let > go of the string sure itıll ŝy off radially. Wrong. > but if the string itself > disapeared. the rock would move in the direction it was.. tangentenial > to the orbit of the rock. Yes. > everyone hates you you dumb old head. wit. useless old > bag of . Think about your answers before abusing Donny. ---DPM === Subject: Re: The real effect of centrifugal force >> Centrifugal force is mistakenly thought to cause a body to ŝy out of >> its circular path; along a tangent direction which represents the >> inertial path of Newtons inertial motion. >> Actually centrifugal force, when not restrained as by a string or >> sling causes a body to ŝee the center of an evolute radially; along >> involutes that remain directed radially away from the center: >> Like the sling used by David to slay Goliath; when it is released the >> missile travels radially away from the center to points equidistant >> along the tangent.; at potentially lethal speeds depending on the >> speed it has when it is released from the evolute. >> There is nothing Ŝctitious about centrifugal force! > not true. youıre mistaking the difference between letting go of the > other end of the string and the string vanishing completely. if u let > go of the string sure itıll ŝy off radially. No. Itıll ŝy off tangentially. Or, to be perfectly accurate, itıll ŝy off _both_ radially and tangentially. If you stay with the rotating frame, looking down your arm at the sling stone, itıll appear to ŝy off radially. At least until Coriolis rears its head and the stone veers off anti-spinward. (Ever played catch on a merry-go-round? Amazingly difŜcult.) If you work from an inertial frame, standing behind the slinger and watching the stone, itıll appear to ŝy off tangentially. Both descriptions work and both descriptions wind up with the stone impacting Goliathıs head at the same relative speed. > but if the string itself > disapeared. the rock would move in the direction it was.. tangentenial > to the orbit of the rock. For a string of negligible mass, it doesnıt matter whether you let go of it or disintegrate it or Ŝre the explosive bolts at the far end. If you let go and the string is of non-negligible mass, the string+stone assembly goes off radially (or tangentially depending on your point of view) and keeps on spinning. So the motion of the stone will be radial + circular orbit or tangential + circular orbit depending on your point of view. Note that in a traditional sling, you release the stone from the sling and keep hold of the sling. The sling doesnıt ŝy downrange. The stone does. > anyways have a nice day and please donıt > forget to go yourself because youıre an illiterate dumb mother > ing little wad. and btw. stop sending me emails about your gay > black boyfriend and how you joined usenets to try to pick up little > boys. everyone hates you you dumb old head. wit. useless old > bag of . Be careful not to make mistakes of your own when you go off the deep end like that. John Briggs === Subject: Re: The real effect of centrifugal force > Centrifugal force is mistakenly thought to cause a body to ŝy out of > its circular path; along a tangent direction which represents the > inertial path of Newtons inertial motion. > Actually centrifugal force, when not restrained as by a string or > sling causes a body to ŝee the center of an evolute radially; along > involutes that remain directed radially away from the center: > Like the sling used by David to slay Goliath; when it is released the > missile travels radially away from the center to points equidistant > along the tangent.; at potentially lethal speeds depending on the > speed it has when it is released from the evolute. > There is nothing Ŝctitious about centrifugal force! Consider a mass on a string one meter long spinning clockwise around a hub at one revolution per second. Letıs imagine the whole works out in space so that there is no stray gravity force to com0plicate the question. If the mass is released from the string as the string passes 12 oıclock, how long does it take for the involute path to pass the radius at 3 oıclock? At 6 oıclock? at 9 oıclock? === Subject: Re: The real effect of centrifugal force CUT< > Consider a mass on a string one meter long spinning clockwise around a hub > at one revolution per second. Letıs imagine the whole works out in space so > that there is no stray gravity force to com0plicate the question. If the > mass is released from the string as the string passes 12 oıclock, how long > does it take for the involute path to pass the radius at 3 oıclock? At 6 > oıclock? at 9 oıclock? As long as you are in no hurry, and weıre just imagining anyway; lets make the speed one twelfth [1/12] revolution per hour: In one hour [@1 oıclock], after releasing the string at noon [12 oıclock], the mass will have traveled 1/12 revolution, and be at the intersection of a tangent and a radial line at one oıclock: In two hours the mass will have traveled 2/12 revolution, and be at the intersection of a tangent and a radial line at two oıclock: In three hours the mass will have traveled 3/12 revolution, and be at the intersection of a tangent and a radial line at three oıclock: In four hours the mass will have traveled 4/12 revolution, and be at the intersection of a tangent and a radial line at four oıclock: With the whole works out in space, where there is no stray gravity force to complicate the situation, this will continue into a second revolution; even into a third, or more; but by 4 or 5 oıclock the conclusion is evident: A curved line drawn through the intersections of the tangent lines and the radial lines is the radial outward path followed by the mass as it centrifugally ŝees from the hub to which it was originally attached. This is the principle by which slings and similar weapons operate. === Subject: Re: The real effect of centrifugal force > Centrifugal force is mistakenly thought to cause a body to ŝy out of > its circular path; [snip crap] http://www.pagetutor.com/idiot/idiot.html http://www.apa.org/journals/psp/psp7761121.html http://insti.physics.sunysb.edu/~siegel/quack.html Hey Dumb Donny Head, why donıt you become obsessed with drinking birds? Lotta good possibilities with drinking birds, Dumb Donny Head. Google drinking birds 711 hits -- Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) http://www.mazepath.com/uncleal/qz.pdf === Subject: L-series of elliptic curves Could anyone lead me in the direction of a website which gives examples of an L-series of an elliptic curve (or even has a calculator of the series of a_n). I would like to see what they look like for some concrete examples without having to go through the trouble of calculating them myself. Craig === Subject: Re: L-series of elliptic curves > Could anyone lead me in the direction of a website which gives > examples of an L-series of an elliptic curve (or even has a calculator > of the series of a_n). I would like to see what they look like for > some concrete examples without having to go through the trouble of > calculating them myself. Craig Iıd suggest downloading GP-Pari on this and experimenting around with its elliptic functions, to see how they all Ŝt together. See http://www.skalatan.de/pariguide/doc/Functions_related_to_ elliptic_curves.ht ml and the GP-Pari site http://pari.math.u-bordeaux.fr/ Youıll need some understanding of modular forms to Ŝgure out how the L-series is related. === Subject: Re: Can you Ŝnd anything wrong with this solution to the Halting Problem? > int WillHalt(string SourceCode, string InputData) > { > if (TheProgramWillHalt(SourceCode, InputData)) > return TRUE; > else if (TheProgramWillNotHalt(SourceCode, InputData)) > return FALSE; > else > return UNKNOWN; > } > The whole purpose of representing the above code the way that I did > int WillHalt() functionıs point of view, since it can not determine > whether or not the LoopIfHalts() function will halt, and it also can > not > determine that it will not halt, this only leaves UNKNOWN as the > remaining choice, which is still a halting condition. > I donıt really have the time to continue this increasingly pointless > discussion. Youıre making so many invalid statements that it would take too > long to refute them all, and it doesnıt seem that youıre really > understanding my points anyway. If you are quitting because you have found there is too much truth in what I am saying to correctly refute, then at least be honest about it. I have pointed out several of your mistakes below. And no one has correctly pointed out any mistakes of mine. > At this point Iıll just point out that your > change to make WillHalt return a three-valued results rather than a boolean > indicates a fundamental misunderstanding. I donıt know what you mean by It might seem this way, but, then this would eliminate the possibility of (for example) an ill-formed statement that is neither true nor false. You do accept that it is possible for people to make mistakes donıt you, or are you simply assuming this away? If at least one element in the set of humans is not infallible, therefore NOT(TRUE) == FALSE is incorrect. > NOT(TRUE) != FALSE -- your own deŜnition of TRUE=1 and FALSE=0 means that > NOT(TRUE) does indeed equal FALSE. I still donıt know under what > circumstances you think that TheProgramWillHalt and TheProgramWillNotHalt > return different values. > Iıll also say that since it [WillHalt] can not determine whether or not the > LoopIfHalts() function will halt is meaningless. WillHalt doesnıt know > anything about LoopIfHalts. WillHalt is _called_ from LoopIfHalts (in the > proof), so WillHaltıs operation cannot depend on anything it knows or > doesnıt know about LoopIfHalts, the caller. You are forgetting the LoopIfHalts is a parameter to WillHalt(). Yes WillHalt() is called from LoopIfHalts, but LoopIfHalts is also passed as a parameter to WillHalt(). > Have you ever written any computer programs? You donıt seem to a very Ŝrm > grasp of either the principles or the terminology of programming. I have been a professional programmer for twenty years. I have a computer science degree, and graduated in the top of my class. I am the patent pending inventor of several different Ŝnite automata. >> 1. IF your function WillHalt works, then a new function WillHalt2 >> can be created, which also works, and happens to return its result >> rather than printing it to the screen. > 1. is a false assumption see this execution trace: > http://home.att.net/~olcott/halting/example.html > It is perfectly obvious to see that int WillHalt() returns UNKNOWN, > and void WillHalt() does not permit LoopIfHalts() to exist. The only > difference is that the former returns its value to the caller, and > the latter > does not. > Thatıs exactly right -- the ONLY difference is that one returns its value > and the other does not. Since WillHalt2 cannot exist, as I think you agree, > then WillHalt also cannot exist. The reason has nothing to do with using Sorry no this does not logically follow. http://home.att.net/~olcott/halting/example.html To see exactly how this does work, go see my updated execution trace. > WillHalt in a LoopIfHalts subroutine. Itıs liike saying, a perpetual > motion machine cannot exist, therefore a perpetual motion machine in a box > cannot exist. If object A cannot exist, and object B is almost identical > to A (such that A can easily be created from B), then object B cannot exist > either. Your function WillHalt cannot exist. If it did, I could create the > impossible WillHalt2 from it by making a trivial change. > --Mark === Subject: Re: Can you Ŝnd anything wrong with this solution to the Halting Problem? > And no one has correctly pointed out any mistakes > of mine. Well, that is incorrect Peter. It is also a deliberate lie. Lots of people have pointed out mistakes of yours, refuting your claims. You are lying above, because you _have_ recognized the validity of those refutations, and you _have_ made lots of attempts to patch your proof in response, right here in public, thus _indicating_ in front of all of us that you _do_ recognize those refutations to be valid. Your lie to the contrary avails you nothing. It just adds dishonest to your reputation next to ignorant. In any case, science works by peer review, and it is not you, but those peers, who get to decide whether your work has been refuted, whether the criticisms of your work are valid, not you. Science could not possibly work the way you want it to, or kooks just like you would fragment it into millions of incompatible parts, each championed by a single individual sticking his tongue out at all the others, as you are doing here. _You_ are the person so incompetent as to be making the same mistakes over and over, so _you_ are in no way competent to be judging whether the refutations are valid. You donıt have the skills to create a valid proof, you donıt have the skills to recognize a valid proof, much less do you have the skills to recognize an invalidation of that proof. Give it up, Peter. You are a kook, and an army of one. You are not going to rebut Goedel, you are not going to rebut Turing, you are not going to rebut Einstein. You are not smart enough for those tasks. You are not smart enough to study the material you need to learn to undertake those tasks. Had you studied it, youıd be well enough informed not to have begun what you cannot possibly complete. You are not smart enough to do science at all, or you wouldnıt be trying to treat science as a subset of politics, where the person shouting the loudest and longest wins. No one is giving up because you have _won_. People are giving up because you are a HOPELESS CASE, full stop. Look back to the URLs I gave you on invincible ignorance. Just because your ignorance is invincible doesnıt make it any less ignorance. xanthian. === Subject: Re: Can you Ŝnd anything wrong with this solution to the Halting Problem? >youıll have plenty of leisure [...] to >put towards your more theoretical research. Which is, I think, the root of this whole problem. -- BELANGER === Subject: Re: _Re: VOTE on whether 1/oo = 0 > Should the question be whether the limit of 1/x as x approaches > inŜnity equals 0? > And if not 0, what other value could it be. The other value could be undeŜned. The question is not even well-posed until the speciŜc number system is declared. The symbol oo is not a number in the reals, for example, so 1/oo is undeŜned in the reals. Itıs true that lim x->oo (1/x) is deŜned on the reals, but it is not the same as 1/oo, unless you stipulate a special meaning to the symbol 1/oo. Although there are exceptions, generally speaking, a limit and an evaluation are two different things. Patrick === Subject: what does Rudin mean in Section 11.17 in Real and Complex Analysis? In Rudinıs Real and Complex Analysis book (I have the 3rd edition), section 11.17 on Poisson Integrals of Measures (p.240 in my book) I donıt understand what Rudin means by u = P[d mu] and I donıt understand the notation d mu (e^(it)) in equation (1) - what does the (e^(it)) indicate? (On p. 233, Rudin deŜnes the notation P[f] for f in L^1, but I donıt see how to apply this to the P[d mu] case. Regarding my second question, I havenıt found where that notation is deŜned, but the Ŝrst instance of it Iım aware of is Theorem 10.7 (p.199).) Mike === Subject: Re: what does Rudin mean in Section 11.17 in Real and Complex Analysis? >In Rudinıs Real and Complex Analysis book (I have the 3rd edition), >section 11.17 on Poisson Integrals of Measures (p.240 in my book) I >donıt understand what Rudin means by u = P[d mu] and I donıt >understand the notation d mu (e^(it)) in equation (1) - what does the >(e^(it)) indicate? Itıs sort of unfortunate notation, deriving from the fact that if f is a function on the circle (the unit circle in the plane) it makes sense to talk about f(e^(it)), but such an f is the same as a 2pi-periodic function on the line, in which case one would write f(t). What it means is just that e^(it) is the variable: int_T f(e^(it)) d mu(e^(it)), int_T f d mu, int_0^{2pi) f(e^(it)) d mu(t) all mean the same thing. >(On p. 233, Rudin deŜnes the notation P[f] for f in L^1, but I donıt >see how to apply this to the P[d mu] case. Regarding my second >question, I havenıt found where that notation is deŜned, but the >Ŝrst instance of it Iım aware of is Theorem 10.7 (p.199).) I donıt have the third edition here at home. In the second edition the Poisson integral of a measure is deŜned in the paragraph immediately following the deŜnition of the Poisson integral of an integrable function. Hint: I found that page by looking for Poisson integral in the index. >Mike ************************ David C. Ullrich === Subject: connectedness and continuity Let X, Y be topological spaces and f: X -> Y a function. If f is continuous, then it maps any connected subspace of X to a connected subspace of Y. Conversely, suppose f maps every connected subspace of X to a connected subspace of Y. Is f then continuous? Iıd love to see David === Subject: Re: connectedness and continuity > Let X, Y be topological spaces and f: X -> Y a function. If f is > continuous, then it maps any connected subspace of X to a connected > subspace of Y. Conversely, suppose f maps every connected subspace of > X to a connected subspace of Y. Is f then continuous? Iıd love to see > David It is not hard to construct a function from R (the real line) into R such that the image of every proper interval is the whole R. Such a function maps every connected sets onto a connected set and it is not continuous at each point. J. Gerlits === Subject: Re: connectedness and continuity > Let X, Y be topological spaces and f: X -> Y a function. If f is > continuous, then it maps any connected subspace of X to a connected > subspace of Y. Conversely, suppose f maps every connected subspace of > X to a connected subspace of Y. Is f then continuous? Iıd love to see For a counterexample, take the function f:R --> R such that f(0) = 0 and that f(x) = sin(1/x) if x is different from 0. Then f is not continuous, but the image of every connected set (that is, each interval) is connected. Jose Carlos Santos === Subject: Re: connectedness and continuity >Let X, Y be topological spaces and f: X -> Y a function. If f is >continuous, then it maps any connected subspace of X to a connected >subspace of Y. Conversely, suppose f maps every connected subspace of >X to a connected subspace of Y. Is f then continuous? Iıd love to see For a counterexample, take any discontinuous function on the rationals Q. Since the only connected subspaces of Q are single points, the hypothesis f maps every connected subspace of X to a connected subspace of Y is trivially true. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: A problem in convexity > What if S has fewer than 2d points? S has >= 2d points. === Subject: Re: Prime Factorization and Digit Congruence coprime b and x and c > 2. It currently suffera from needing extended precision arithmetic to calculate b^c for relatively small values of b and c, as there is overŝow. There are some interesting things of note, though, that I had not calculated to see before. For example, in base 16, b=16, there are a variety of numbers that have a cycle length of 5, for example x = 25, 31, 33, 41, 55, 75, 93, 123, 155, 165, 205, .... A variety of numbers have cycles of length two, the plain alternating digit test would sufŜce for them: 3, 5, 15, 51, 85, and 255, where I charted from 2 to b^2. Thus if an algorithm summed the alternatingly negated digits, then it could test for divisibility of the original number in base 16 by 3, 5, 15, 51, 85, and 255. Fifteen is obviously b-1, 51 is b+1=17 *3, 85 is 17*5, and 255 is obviously enough 3*5*17. The value of b^2 is 256, b^2-1 is 255, and in base 256, 255 is tested for division. Yet, with the alternating sums of base 16: 3, 5, 15, 51, 85, and 255 can each granularly be tested. I generated for b from two to Ŝfty, itıs about a three megabyte Ŝle. I need to get some extended precision C libraries if I am going to compute some of the larger exponents, unless I can Ŝgure out how to calculate b^c mod x without having to calculate b^c, and otherwise keeping the intermediate result less than 2^32. Letıs see, I donıt recall, what is a way to test a*b mod x for congruence to a function of a mod x and b mod x. If I had a product congruence test for relatively small numbers that are exponentiated, then I could determine values of b^n mod x without exponentiating b. mod x is (b mod x)^c. While that is so, I implemented that and got spurious results. Ah, perhaps I know why, when x is greater than b, b mod x = b and b^c is some large number. I guess Iıll use a multiple precision big integer library. Iıve been using Java, this GNU MP BigNum Library, http://www.swox.com/gmp/, seems useful. First I change the type of b^c from unsigned long to unsigned long long, that appears to help, as it is deŜned in stdint.h as uintmax_t. Thatıs better, but still not genereal purpose. There might be some interesting phenomena that would not be visible until an integer base higher than 1000, decimal. Yet, for high values of b, the data set is unwieldy, for sampling each x coprime to b that is less than b^2. Of special interest might be the cases of b = 256, 2^8, and 2^16 and 2^32. These are of interest in the computational case because memory is addressable at each 8 bits and is often aligned on 16, 32, or power two bits, about a base thatıs a power of two of a power of two, a tetration. Anyways, in summing bytes instead of subsequences of the bytes, no logic is required to access the bits, and the divisibility test would still be accurate for any one that was good for a factor of 256. The program is generating a chart for 256. ... In reviewing the chart (a table) for b=64, there are a variety of values with a cycle of one or two, with the concept of using the plain sum an dalternating sum and not necessarily the linear coefŜcients on the summands that are not +-1. Ah, 256 is done. In reviewing the table for 256, these elements can be tested for divisibility into a big integer represented by an array of bytes by summing the bytes and testing for even divisibility by a zero remainder: 3, 5, 15, 17, 51, 85, 255 The only improvement over the base 16 (2^4) alternating test is the base 256 (2^8) test for 17. Thus presumably plainly summing 16 bit subsequences (base 2^16) would allow for constant time divisibility tests of the linear time summation for the same values as the alternating sum for 256, avoiding a sample division, which I donıt know is linear, polynomial, or exponential. Then the following elements can be tested for divisibility of the alternating sum of 256: 257, 771, 1285, 3855, 4369, 13107, 21845, and 65535 So, there is not some excellent advance so far in this post for better divisibility tests tractable to computers, those numbers are sparse within the integers. Ross F. ***** Table for base 256 of 256^c mod x ***** x= c= 1 2 3 4 5 6 7 8 ---------------------------------------------------------- 3 | 1 1 (1) 5 | 1 1 1 1 (1) 7 | 4 2 1 4 2 1 (3) 9 | 4 7 1 4 7 1 (3) 11 | 3 9 5 4 1 3 . . (5) 13 | 9 3 1 9 3 1 . . (3) 15 | 1 1 1 1 1 1 . . (1) 17 | 1 1 1 1 1 1 . . (1) 19 | 9 5 7 6 16 11 . . (.) 21 | 4 16 1 4 16 1 . . (3) 23 | 3 9 4 12 13 16 . . (.) 25 | 6 11 16 21 1 6 . . (5) 27 | 13 7 10 22 16 19 . . (.) 29 | 24 25 20 16 7 23 . . (.) 31 | 8 2 16 4 1 8 . . (5) 33 | 25 31 16 4 1 25 . . (5) 35 | 11 16 1 11 16 1 . . (3) 37 | 34 9 10 7 16 26 . . (.) 39 | 22 16 1 22 16 1 . . (3) 41 | 10 18 16 37 1 10 . . (5) 43 | 41 4 35 16 11 21 . . (.) 45 | 31 16 1 31 16 1 . . (3) 47 | 21 18 2 42 36 4 . . (.) 49 | 11 23 8 39 37 15 . . (.) 51 | 1 1 1 1 1 1 . . (1) 53 | 44 28 13 42 46 10 . . (.) 55 | 36 31 16 26 1 36 . . (5) 57 | 28 43 7 25 16 49 . . (.) 59 | 20 46 35 51 17 45 . . (.) 61 | 12 22 20 57 13 34 . . (.) 63 | 4 16 1 4 16 1 . . (3) 65 | 61 16 1 61 16 1 . . (3) 67 | 55 10 14 33 6 62 . . (.) 69 | 49 55 4 58 13 16 . . (.) 71 | 43 3 58 9 32 27 . . (.) 73 | 37 55 64 32 16 8 . . (.) 75 | 31 61 16 46 1 31 . . (5) 77 | 25 9 71 4 23 36 . . (.) 79 | 19 45 65 50 2 38 . . (.) 81 | 13 7 10 49 70 19 . . (.) 83 | 7 49 11 77 41 38 . . (.) 85 | 1 1 1 1 1 1 . . (1) 87 | 82 25 49 16 7 52 . . (.) 89 | 78 32 4 45 39 16 . . (.) 91 | 74 16 1 74 16 1 . . (3) 93 | 70 64 16 4 1 70 . . (5) 95 | 66 81 26 6 16 11 . . (.) 97 | 62 61 96 35 36 1 . . (6) 99 | 58 97 82 4 34 91 . . (.) 101 | 54 88 5 68 36 25 . . (.) 103 | 50 28 61 63 60 13 . . (.) 105 | 46 16 1 46 16 1 . . (3) 107 | 42 52 44 29 41 10 . . (.) 109 | 38 27 45 75 16 63 . . (.) 111 | 34 46 10 7 16 100 . . (.) 113 | 30 109 106 16 28 49 . . (.) 115 | 26 101 96 81 36 16 . . (.) 117 | 22 16 1 22 16 1 . . (3) 119 | 18 86 1 18 86 1 . . (3) 121 | 14 75 82 59 100 69 . . (.) 123 | 10 100 16 37 1 10 . . (5) 125 | 6 36 91 46 26 31 . . (.) 127 | 2 4 8 16 32 64 . . (.) 129 | 127 4 121 16 97 64 . . (.) 131 | 125 36 46 117 84 20 . . (.) 133 | 123 100 64 25 16 106 . . (.) 135 | 121 61 91 76 16 46 . . (.) 137 | 119 50 59 34 73 56 . . (.) 139 | 117 67 55 41 71 106 . . (.) 141 | 115 112 49 136 130 4 . . (.) 143 | 113 42 27 48 133 14 . . (.) 145 | 111 141 136 16 36 81 . . (.) 147 | 109 121 106 88 37 64 . . (.) 149 | 107 125 114 129 95 33 . . (.) 151 | 105 2 59 4 118 8 . . (.) 153 | 103 52 1 103 52 1 . . (3) 155 | 101 126 16 66 1 101 . . (5) 157 | 99 67 39 93 101 108 . . (.) 159 | 97 28 13 148 46 10 . . (.) 161 | 95 9 50 81 128 85 . . (.) 163 | 93 10 115 100 9 22 . . (.) 165 | 91 31 16 136 1 91 . . (5) 167 | 89 72 62 7 122 3 . . (.) 169 | 87 133 79 113 29 157 . . (.) 171 | 85 43 64 139 16 163 . . (.) 173 | 83 142 22 96 10 138 . . (.) 175 | 81 86 141 46 51 106 . . (.) 177 | 79 46 94 169 76 163 . . (.) 179 | 77 22 83 126 36 87 . . (.) 181 | 75 14 145 15 39 29 . . (.) 183 | 73 22 142 118 13 34 . . (.) 185 | 71 46 121 81 16 26 . . (.) 187 | 69 86 137 103 1 69 . . (5) 189 | 67 142 64 130 16 127 . . (.) 191 | 65 23 158 147 5 134 . . (.) 193 | 63 109 112 108 49 192 . . (.) 195 | 61 16 1 61 16 1 . . (3) 197 | 59 132 105 88 70 190 . . (.) 199 | 57 65 123 46 35 5 . . (.) 201 | 55 10 148 100 73 196 . . (.) 203 | 53 170 78 74 65 197 . . (.) 205 | 51 141 16 201 1 51 . . (5) 207 | 49 124 73 58 151 154 . . (.) 209 | 47 119 159 158 111 201 . . (.) 211 | 45 126 184 51 185 96 . . (.) 213 | 43 145 58 151 103 169 . . (.) 215 | 41 176 121 16 11 21 . . (.) 217 | 39 2 78 4 156 8 . . (.) 219 | 37 55 64 178 16 154 . . (.) 221 | 35 120 1 35 120 1 . . (3) 223 | 33 197 34 7 8 41 . . (.) 225 | 31 61 91 121 151 181 . . (.) 227 | 29 160 100 176 110 12 . . (.) 229 | 27 42 218 161 225 121 . . (.) 231 | 25 163 148 4 100 190 . . (.) 233 | 23 63 51 8 184 38 . . (.) 235 | 21 206 96 136 36 51 . . (.) 237 | 19 124 223 208 160 196 . . (.) 239 | 17 50 133 110 197 3 . . (.) 241 | 15 225 1 15 225 1 . . (3) 243 | 13 169 10 130 232 100 . . (.) 245 | 11 121 106 186 86 211 . . (.) 247 | 9 81 235 139 16 144 . . (.) 249 | 7 49 94 160 124 121 . . (.) 251 | 5 25 125 123 113 63 . . (.) 253 | 3 9 27 81 243 223 . . (.) 255 | 1 1 1 1 1 1 . . (1) 257 | 256 1 256 1 256 1 . . (2) 259 | 256 9 232 81 16 211 . . (.) 261 | 256 25 136 103 7 226 . . (.) 263 | 256 49 183 34 25 88 . . (.) 265 | 256 81 66 201 46 116 . . (.) 267 | 256 121 4 223 217 16 . . (.) 269 | 256 169 224 47 196 142 . . (.) 271 | 256 225 148 219 238 224 . . (.) 273 | 256 16 1 256 16 1 . . (3) 275 | 256 86 16 246 1 256 . . (5) 277 | 256 164 157 27 264 273 . . (.) 279 | 256 250 109 4 187 163 . . (.) 281 | 256 63 111 35 249 238 . . (.) 283 | 256 163 127 250 42 281 . . (.) 285 | 256 271 121 196 16 106 . . (.) 287 | 256 100 57 242 247 92 . . (.) 289 | 256 222 188 154 120 86 . . (.) 291 | 256 61 193 229 133 1 . . (6) 293 | 256 197 36 133 60 124 . . (.) 295 | 256 46 271 51 76 281 . . (.) 297 | 256 196 280 103 232 289 . . (.) 299 | 256 55 27 35 289 131 . . (.) 301 | 256 219 78 102 226 64 . . (.) 303 | 256 88 106 169 238 25 . . (.) 305 | 256 266 81 301 196 156 . . (.) 307 | 256 145 280 149 76 115 . . (.) 309 | 256 28 61 166 163 13 . . (.) 311 | 256 226 10 72 83 100 . . (.) 313 | 256 119 103 76 50 280 . . (.) 315 | 256 16 1 256 16 1 . . (3) 317 | 256 234 308 232 113 81 . . (.) 319 | 256 141 49 103 210 168 . . (.) 321 | 256 52 151 136 148 10 . . (.) 323 | 256 290 273 120 35 239 . . (.) 325 | 256 211 66 321 276 131 . . (.) 327 | 256 136 154 184 16 172 . . (.) 329 | 256 65 190 277 177 239 . . (.) 331 | 256 329 150 4 31 323 . . (.) 333 | 256 268 10 229 16 100 . . (.) 335 | 256 211 81 301 6 196 . . (.) 337 | 256 158 8 26 253 64 . . (.) 339 | 256 109 106 16 28 49 . . (.) 341 | 256 64 16 4 1 256 . . (5) 343 | 256 23 57 186 282 162 . . (.) 345 | 256 331 211 196 151 16 . . (.) 347 | 256 300 113 127 241 277 . . (.) 349 | 256 273 88 192 292 66 . . (.) 351 | 256 250 118 22 16 235 . . (.) 353 | 256 231 185 58 22 337 . . (.) 355 | 256 216 271 151 316 311 . . (.) 357 | 256 205 1 256 205 1 . . (3) 359 | 256 198 69 73 20 94 . . (.) 361 | 256 195 102 120 35 296 . . (.) 363 | 256 196 82 301 100 190 . . (.) 365 | 256 201 356 251 16 81 . . (.) 367 | 256 210 178 60 313 122 . . (.) 369 | 256 223 262 283 124 10 . . (.) 371 | 256 240 225 95 205 169 . . (.) 373 | 256 261 49 235 107 163 . . (.) 375 | 256 286 91 46 151 31 . . (.) 377 | 256 315 339 74 94 313 . . (.) 379 | 256 348 23 203 45 150 . . (.) 381 | 256 4 262 16 286 64 . . (.) 383 | 256 43 284 317 339 226 . . (.) 385 | 256 86 71 81 331 36 . . (.) 387 | 256 133 379 274 97 64 . . (.) 389 | 256 184 35 13 216 58 . . (.) 391 | 256 239 188 35 358 154 . . (.) 393 | 256 298 46 379 346 151 . . (.) 395 | 256 361 381 366 81 196 . . (.) 397 | 256 31 393 167 273 16 . . (.) 399 | 256 100 64 25 16 106 . . (.) 401 | 256 173 178 255 318 5 . . (.) 403 | 256 250 326 35 94 287 . . (.) 405 | 256 331 91 211 151 181 . . (.) 407 | 256 9 269 81 386 322 . . (.) 409 | 256 96 36 218 184 69 . . (.) 411 | 256 187 196 34 73 193 . . (.) 413 | 256 282 330 228 135 281 . . (.) 415 | 256 381 11 326 41 121 . . (.) 417 | 256 67 55 319 349 106 . . (.) 419 | 256 172 37 254 79 112 . . (.) 421 | 256 281 366 234 122 78 . . (.) 423 | 256 394 190 418 412 145 . . (.) 425 | 256 86 341 171 1 256 . . (5) 427 | 256 205 386 179 135 400 . . (.) 429 | 256 328 313 334 133 157 . . (.) 431 | 256 24 110 145 54 32 . . (.) 433 | 256 153 198 27 417 234 . . (.) 435 | 256 286 136 16 181 226 . . (.) 437 | 256 423 349 196 358 315 . . (.) 439 | 256 125 392 260 271 14 . . (.) 441 | 256 268 253 382 331 64 . . (.) 443 | 256 415 363 341 25 198 . . (.) 445 | 256 121 271 401 306 16 . . (.) 447 | 256 274 412 427 244 331 . . (.) 449 | 256 431 331 324 328 5 . . (.) 451 | 256 141 16 37 1 256 . . (5) 453 | 256 304 361 4 118 310 . . (.) 455 | 256 16 1 256 16 1 . . (3) 457 | 256 185 289 407 453 347 . . (.) 459 | 256 358 307 103 205 154 . . (.) 461 | 256 74 43 405 416 5 . . (.) 463 | 256 253 411 115 271 389 . . (.) 465 | 256 436 16 376 1 256 . . (5) 467 | 256 156 241 52 236 173 . . (.) 469 | 256 345 148 368 408 330 . . (.) 471 | 256 67 196 250 415 265 . . (.) 473 | 256 262 379 59 441 322 . . (.) 475 | 256 461 216 196 301 106 . . (.) 477 | 256 187 172 148 205 10 . . (.) 479 | 256 392 241 384 109 122 . . (.) 481 | 256 120 417 451 16 248 . . (.) 483 | 256 331 211 403 289 85 . . (.) 485 | 256 61 96 326 36 1 . . (6) 487 | 256 278 66 338 329 460 . . (.) 489 | 256 10 115 100 172 22 . . (.) 491 | 256 233 237 279 229 195 . . (.) 493 | 256 460 426 103 239 52 . . (.) 495 | 256 196 181 301 331 91 . . (.) 497 | 256 429 484 151 387 169 . . (.) 499 | 256 167 337 444 391 296 . . (.) 501 | 256 406 229 7 289 337 . . (.) 503 | 256 146 154 190 352 75 . . (.) 505 | 256 391 106 371 36 126 . . (.) 507 | 256 133 79 451 367 157 . . (.) 509 | 256 384 67 355 278 417 . . (.) 511 | 256 128 64 32 16 8 . . (.) 513 | 256 385 64 481 16 505 . . (.) === Subject: Re: ŒUncountableı doesnıt exist > Pick-a-proof; Cantor had *two*. His Ŝrst proof utilized properties > of the continuum to show its uncountability. > http://en.wikipedia.org/wiki/Cantor%27s_Ŝrst_uncountability_ proof > The second one, of course, is his diagonal argument, again proving > Rıs uncountability. > http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument > Neither one requires an inŜnite number of steps in the *proof*, > although both require an inŜnite number of steps to construct > a number. > ... The binary case is sufŜcient. The binary case is necessary. Any construction deŜned as a monotonic mapping from N onto any continuous segment of R avoids the nested interval dilemna. The set of all sets is its own powerset. In a model of ubiquitous ordinals, there may or may not be inŜnite ordinals, and dual and multiple representation of ordinals exists. In such a model the uncountable may or may not exist. Why deem the powersetıs existence axiomatically? Do you care for another that says the powerset is never the set? How many proper classes do you expect in your model? Newton and Leibniz sum the inŜnitesimals to get Ŝnite results, deriving exact, certain, empirical, geometrically provable results. The single point is more weighty than one on the line. What do you think about the Banach-Tarski ball cutting result? Around a point four spheres pack closest: four the for the spheres and one for the point equals Ŝve. The uncountable is deŜnitely useful for describing large cardinals, some of which shrink with time, ordered by their discovery. Ross F. === Subject: Re: ŒUncountableı doesnıt exist > The uncountable is deŜnitely useful for describing large cardinals, > some of which shrink with time... Depends on how you wash them. === Subject: Re: ŒUncountableı doesnıt exist by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i6K1BIX08528; >> I realised some time ago I would reject AC in favor of DC, which >> allows only countable collections. > >That seems to be a limit of your thinking rather than of logic. >Regardless, it has nothing to do with what follows. >-- >Will Twentyman >email: wtwentyman at copper dot net diagonal arguments about the rationals and the reals. Memories starting to stir now into half-clarity. Yes, I was muddling up several different things. Mark GrifŜth === Subject: Re: Length of sequence of consecutive primes starting at 2 >Hello >I would like to know how long the list of consecutive primes (i.e. not >skipping any primes in between) is and can I Ŝnd it somewhere. Also, >how far apart are the current largest prime (41st Mersenne prime?) and >the last prime in the above sequence? > This question is very poorly posed. What do you mean by THE list > of consecutive primes? > (1) By specifying ŒTHEı list, you assume such a list, if it exists, > is unique. What makes you think so? > (2) What do you mean by consecutive primes. Do you mean twin > primes? What does it mean for primes to be consecutive? If by > consecutive primes you mean all primes starting with 2, then the > answer to your question about the list of consecive primes is: > INFINITE > The list of all primes starting with 2 is inŜnite. If instead, you > mean the largest list of all primes starting with 2 that has ACTUALLY > been published, that is a DIFFERENT question. (1),(2) My apologies for not stating that I am looking for the list of known consecutive primes. What I mean by consecutive primes is the following. The sequence 2,3,5,7,11,13 is made up of consecutive primes while the sequence 2,3,5,11,13 is not (itıs missing the prime number 7). Understand what I mean? And yes, I mean the largest list that has ACTUALLY been published. > (3) If you mean that you are looking for a list of ALL primes below > a certain bound and you want the list with the largest bound that has > actually been calculated, I can tell you that such a list does not > exist per se. Or rather, that such a list changes constantly, so you > are asking for a moving target. It is also one thing to calculate > such a list, it is another to store and publish it. OK. Can I Ŝnd the largest upper bound published somewhere? Itıs OK if the bound in slightly outdated. > (4) If you want a list of primes up to some bound B, it would be > faster to generate it via a sieve than to read it over the internet. > And storage for any reasonable B would be problematic. You are > talking terabytes. Nope, Iım not interested in this. > (5) The answer to your last question is trivial. If B is the bound > on your list, then M41 - B is the number you are looking for. See (3) (I am looking for actual numbers, not theoretical results) > (6) At one time or another, lists of all primes up to at least 10^12 > have been generated. But I canıt imagine anyone putting such a list > into permanent or even semi-permanent storage. It would be pointless. > Further, if at anytime someone did store such a list, extending it > a little bit would be trivial. Really? How would you that (I mean extend the list)? Doesnıt this suggest that it would be trivial to Ŝnd the prime following M41? Note that I donıt consider methods that take months and many computers to Ŝnish trivial. > (7) If you tell me what you REALLY want, I can and will help. But > your questions are too imprecise for me to discern your real desire. > Please. If you are going to discuss mathematics, then you need to > learn how to pose your questions more precisely. What I am REALLY wondering about is the following. The largest known primes are Mersenne primes. The largest non-Mersenne prime is much smaller than the largest Mersenne prime [1]. It seems to me that there should be many primes between these two. I am wondering whether there exist any efŜcient methods of constructing primes that would fall into this range. Joe [1]: http://www.utm.edu/research/primes/largest.html#biggest === Subject: Re: Length of sequence of consecutive primes starting at 2 [...] > What I am REALLY wondering about is the following. The largest known > primes are Mersenne primes. The largest non-Mersenne prime is much > smaller than the largest Mersenne prime [1]. It seems to me that > there should be many primes between these two. I am wondering whether > there exist any efŜcient methods of constructing primes that would > fall into this range. To get an idea, which more general forms of primes have been investigated, have a look at H&R Lifchitzıs http://www.primenumbers.net/prptop/prptop.php A very efŜcient program that can test arbitrary numbers for primality is PrimeForm, see http://groups.yahoo.com/group/primeform/ If we assume for simplicity that you have tagged all atoms in the universe with the Ŝrst 10^80 prime numbers, then the largest number in this list might be ~= 1.9*10^82. To extend this list by 100 more consecutive primes you need to test the next 20000 numbers. I tried PrimeForm for 2*10^82+k and within a minute on a 550MHz PIII I had extended this Ŝctitious cosmic list by 100 additional terms (k=57,63,281,...). But where to store? You already used all atoms ;-) > Joe > [1]: http://www.utm.edu/research/primes/largest.html#biggest Hugo === Subject: Re: Length of sequence of consecutive primes starting at 2 |>> (6) At one time or another, lists of all primes up to at least 10^12 |>> have been generated. But I canıt imagine anyone putting such a list |>> into permanent or even semi-permanent storage. It would be pointless. |>> Further, if at anytime someone did store such a list, extending it |>> a little bit would be trivial. |>Really? How would you that (I mean extend the list)? Letıs pretend (since obviously I donıt have such a list in front of me) that the largest prime in the list is 10000000000037 (which happens to be the least prime > 10^13). As primes go, this is ridiculously small. Itıs very easy (with modern software) to check whether a number this size is prime, and the probability that a random number of this size is prime is not too small (on the order of .03), so we wonıt have very far to go to the next one. In fact, with Maple on an old and slow computer, it takes only .011 second to Ŝnd the next one (which happens to be 10000000000051). > Doesnıt this >suggest that it would be trivial to Ŝnd the prime following M41? Not at all. M41 is enormous, so itıs a huge task to check a number of that size for primality, and moreover there would probably be many candidates to check until you hit on one that is prime. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: Length of sequence of consecutive primes starting at 2 > What I am REALLY wondering about is the following. The largest known > primes are Mersenne primes. The largest non-Mersenne prime is much > smaller than the largest Mersenne prime [1]. It seems to me that > there should be many primes between these two. I am wondering whether > there exist any efŜcient methods of constructing primes that would > fall into this range. There are many primes between the two. Finding them - or rather proving youıve found one - is not so easy. Deciding whether a given 1000-digit number is prime is a piece of cake (if youıve got appropriate software & hardware). Deciding whether a given 1000000-digit number is prime is out of the question, unless the number is of a very special form, e.g., one less than a power of 2. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: foundations of mathematics The properties of equality, like all else in math, may have been abstracted from the physical world, but that happened so long ago that it has lost all signiŜcance and is now independent of it. All of math is self-contained, both philosophically and practically. It has nothing to do with observation any more. If something physical or real-worldy happens to agree with a math result, that is lovely OC, but is a matter for physics or sociology or whatever; not a matter for math. > People have never observed anything in the world where a=b,b=c,and a/=c, FALSE!! Such observations are ancient. example: a = Oedipusı mother b = Jocasta c = the person Oedipus wanted to It was long noted that a = b and b = c but deŜnitely a =/= c (!) ------------------------------------------------------------- --------------- -- Bill Taylor W.Taylor@math.canterbury.ac.nz ------------------------------------------------------------- --------------- -- Sex offenders arenıt completely evil. Even Oedipus loved his mother. ------------------------------------------------------------- --------------- -- === Subject: I am playing I am trying to learn about how to create pages. I am having a lot of trouble. It will not let me put a picture on a page. This is a test. http://hometown.aol.com/kurtstocklmeir/myhomepage/ personal.html === Subject: Re: I am playing > I am trying to learn about how to create pages. I am having a lot of trouble. > It will not let me put a picture on a page. This is a test. > http://hometown.aol.com/kurtstocklmeir/myhomepage/ personal.html get rid of most of the html code.. to insert an image just ust the === Subject: Re: I am playing >> I am trying to learn about how to create pages. I am having a lot of trouble. >> It will not let me put a picture on a page. This is a test. >> http://hometown.aol.com/kurtstocklmeir/myhomepage/ personal.html >get rid of most of the html code.. >to insert an image just ust the I tried that just now, didnıt work. Shows a funny little icon instead of the picture. (Whatıs picture.jpg a picture _of_, anyway?) ************************ David C. Ullrich === Subject: Re: I am playing >I am trying to learn about how to create pages. I am having a lot of trouble. >It will not let me put a picture on a page. This is a test. >http://hometown.aol.com/kurtstocklmeir/myhomepage/ personal.html ************************ David C. Ullrich === Subject: Re: I am playing > http://hometown.aol.com/kurtstocklmeir/myhomepage/ personal.html Itıs a mess. It looks like you pasted the source from one page into another page, without respecting section boundaries. There are non- matching tags; a closing without an opening