mm-2929 === Subject: Re: differentiable structure on S^2 using spherical coordiates That map, and no other? It can't be done in any principled way. ...And in particular it can't be done the way you're trying to do it, which is (in principle) the right thing to try to do. In general, to define a differentiable structure on a manifold M means to find a collection of open sets in M that covers M, together with a homeomorphism from each of those open sets to some (open) set in (some) R^n, so that where two of the open sets in M have non-empty intersection, the resulting map between open sets in a couple of R^ns is a diffeomorphism (as defined for open subsets of R^n, and of whatever smoothness class you were looking to impose on M). Now, you don't *have* any such collection of open-sets-together-with- -homeomorphisms on S^2, and you aren't going to find one if you insist on constructing your collection using nothing but your one function F. As it happens, for small n (for instance, 2), there are (more or less difficult) theorems asserting that in your situation, where you have a differentiable structure on all but finitely many points of a topological n-manifold M, there exists a differentiable structure on M extending that structure. This theorem fails in sufficiently high dimensions (I think 8 is high enough). And even in low dimensions (other than 0...) there will be unequal structures of the required sort (though they will be diffeomorphic). For instance, if n=1, then the differentiable structure on R^1 given by the single coordinate system (R^1, identity) and the differentiable structure on R^1 given by the single coordinate structure on the complement of 0 (that is, on any open subset of R not containing 0, exactly the same continuous real-valued functions are smooth in one structure as are in the other structure), but they are different (albeit diffeomorphic) differentiable structures on R as a whole (because on any open subset of R containing 0, there exist functions that are differentiable in one structure but not in the other). So there's an example where one and the same differentiable structure on M-{point} extends to two distinct differentiable structures on M. The same, and much worse, happens for 2-manifolds, 3-manifolds, .... . Lee Rudolph == Subject: Re: A Simple Functional Equation Good grief. I could see calling it the phase, but the sign? In what circles have you seen that? Lee Rudolph (yeah, yeah, I know, the unit circle) == Subject: Re: A Simple Functional Equation I'm not sure whether I've seen it called that, but the terminology makes perfect sense to me. Calling it the phase, no, that can't be right - the phase is just the argument... David C. Ullrich == Subject: Re: Nonzero derivative at 05:40 PM, ullrich@math.okstate.edu (David C. Ullrich) said: My first thought was that he meant a function whose derivative nowhere vanishes, in which case your f would not be a counterexample. On second thought I can see a couple of other things that he might have meant, so I endorse your request that the OP clarify his intent. -- 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 reply to spamtrap@library.lspace.org == Subject: Re: Nonzero derivative On Sat, 24 Sep 2005 22:30:34 -0300, Shmuel (Seymour J.) Metz Certainly it is. The derivative is the matrix [[1,0], [0,0]], which is nowhere zero. You must be taking the derivative of f to mean something else. What, exactly? David C. Ullrich == Subject: Re: Nonzero derivative Where does [1,0;0,0] vanish? On Maybe 'invertible everywhere'? == Subject: Re: Is math a real science? Demers replied: The OP never said or implied that the calculus or any mathematics exists outside human culture. And it is a certain fact that the ability to do calculus contributes little to one's survival in the Darwinian sense. Your ability to state the trivial, however, does not address the point that the OP did make. Perhaps truth does not exist for you? That is poverty, no matter how many things or how much money one accumulates. Tom == Subject: Re: Is math a real science? Demers: No. It is a way of saying that value assigned by personal belief differs from value deduced from first principles or from empirical observations. This was said in response to your imlying that worth should be judged by opinion polls. Tom I see only small, easily repaired, faults: + between (1) and (2): In English we say differentate and not derivate + between (4) and (5) ... what if f(x)=0 for some x and not others? + (6) what if f(x)<0 for some x ? + (7) why do you suppose the solution of a differential equation is more elementary than the solution of the functional equation we are working on? y, so... == Subject: Re: anything to the power of 0 = 1, why???? Ah, well. I guessed I learned something, then. :-) Tom == Subject: Re: Area of X^2*n+Y^2*n=a^2*n If my second guess is correct (which I'm probably about 75% confident it is) then, normalising the equation to |x|^n + |y|^n = 1 area of the largest triangle equal to x^2/2 + c*x + c*(1-x^n)^(1/n) - 1/2*(1-x^n)^(2/n) where x is the root between 0 and 1 of x + c - c*x^(n-1)*(1-x^n)^(1/n-1) + x^(n-1)*(1-x^n)^(2/n-1) = 0 and c = (1/2)^(1/n) == Subject: Re: Area of X^2*n+Y^2*n=a^2*n Did you check this at n=2 Is the area = 3*3^(1/2)/4 == Subject: Re: operations between different numbers that one started with the naturals, then extended to the integers (and identified the naturals with a subset of that), then went to the field of quotients (and identified the integers with a subset of that) and then to the reals by completeness (and identified the rationals with a subset of that). In the context of the question I was responding to, it seemed sufficient. Well, since I was only talking about the reals, that paucity doesn't really bother me too much. [Lots of fun stuff deleted.] == Subject: Re: operations between different numbers Ok - but Sizor asked about different number sets. My point is that reals are only one number set, very convenient and well understood, but highly constrained. Confession time - I had confused this thread with that on Noether's theorem, and my comments were more relevant there (I hope). Anno domini! Sorry. I hope you meant it about fun stuff. Roger Beresford. ! have a theory! (John Cleese.) == Subject: Re: operations between different numbers Assuming that that was a reply to what I just posted, yes the reals are one set. But it is true that the naturals are one 'kind' of object, the integers another 'kind', the rationals yet another, and the reals still another. Nevertheless, we can naturally identify each kind with a subset of the next. Then, being grown-up, we forget all that, and just treat the reals as if we always had them. Sure. It looks related to some of what John Baez sometimes posts about Tony Smith's work (about which all I know is that it looks like fun). Life is short. == Subject: Re: Serge Lang dead == Subject: Re: Serge Lang dead September 25, 2005. Here is a link, which may get you in without registering (or maybe not). http://www.nytimes.com/2005/09/25/national/25lang.html?ex=1128312000&en=05db bd8f4cb4315a&ei=5070&emc=eta1 Jim Buddenhagen == Subject: Re: Theorems with short proofs in analysis and long proofs in PA Harvey Friedman at Ohio State University has probably done the most work on what you're looking for. http://www.math.ohio-state.edu/~friedman/publications.html See the following under his Lecture Notes. They appear to be about the same, but there might be some additional comments in one that is not in the others. [17] Enormous Integers in Real Life, June 1, 2000, 11 pages. [20] Does normal mathematics need new axioms?, October 26, 2001, 12 pages. [22] Lecture notes on enormous integers, November 22, 2001, 11 pages. In particular, you might find Friedman's story about Paul Sally's gifted High School class interesting (see [17] [22] 8. BLOCKS IN SEQUENCES FROM {1,...,k} and [20] 1. EXOTIC HIGH SCHOOL MATH), which I also posted an excerpt from in my 8 April 2002 post Big Numbers #3 (see below). It is very difficult to get a feel for how large these numbers are, but these old posts of mine (one day to be greatly expanded) might help: BIG NUMBERS #1, 2, 3 By the way, Harvey Friedman has quite an impressive background, being a former prodigy of at least the same order that Stephen Wolfram and Lenny Ng were: http://www.math.ohio-state.edu/~friedman/distinctions.html Dave L. Renfro == Subject: Re: Theorems with short proofs in analysis and long proofs in PA I'm sure Peter is aware of this work, but it isn't about ordinary theorems of number theory. == Subject: Re: Theorems with short proofs in analysis and long proofs in PA Ooops! The correct URL should have been: http://www.math.ohio-state.edu/~friedman/manuscripts.html Lecture Notes #17, 20, 22 may have the kinds of things that Peter is interested in -- in particular, where I mentioned that Friedman talks about Paul Sally's gifted high school class. Dave L. Renfro == Subject: Re: Theorems with short proofs in analysis and long proofs in PA Indeed not provable, so not an example of speed-up! I'm hoping for examples where (1) both PA |- S and PA + whatever |- S, but proof in the second case is wildly shorter than the first. And (2) S is a natural theorem (not a cleverly hoked up proposition constructed with an eye on Godelian considerations). == Subject: Re: Theorems with short proofs in analysis and long proofs in PA To provide some irrelevant arrant pedantry let me note that taking S itself as whatever for any natural number theoretic S with a non-trivial shortest proof in PA fits your requirement. -- Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus == Subject: Re: Theorems with short proofs in analysis and long proofs in PA So just what do you mean here by a proof in PA? == Subject: Re: Theorems with short proofs in analysis and long proofs in PA Do you mean PA, or extensions by definition of PA? == Subject: Re: Theorems with short proofs in analysis and long proofs in PA Hi Torkel. Definitions allowed (as they are for working mathematicians). Background. Without thinking about it too much, I've said in the past (in the category of throw-away-remarks in lectures) something like this: It's well known that there are number theoretic props which have snappy proofs if you help yourself to classical analysis and which it seems can only be proved in pure arithmetic with vastly more effort. Are we just being dim in not spotting shorter purely arithmetic proofs? Maybe not so. There are Godelian speed-up results which indicate that bumping up the axioms of PA (say) not only allows us to prove new things but will also always radically shorted the proof of some old theorems. And I would like to be able to point to some tolerably accessible examples which illustrate what I (foolishly? ignorantly??) said was well known. So I guess I'm thinking of examples where the pure arithmetic is mathematician's working arithmetic with whatever abbreviatory definitions you like rather the the logical purist's unaugmented PA. == Subject: Re: Theorems with short proofs in analysis and long proofs in PA Here are two books that might contain useful information: Elementary Methods in the Analytic Theory of Numbers Authors: A. O. Gel'fond and Yu. V. Linnik Elementary Methods in Number Theory Author: Melvyn B. Nathanson In addition to the already-mentioned Prime Number Theorem and Dirichlet's Theorem, these books contain elementary proofs/solutions of Waring's Problem, Liouville's method for representing integers as sums of squares, Erdos's asymptotic estimates for partition functions, and lots more. == Subject: Re: Theorems with short proofs in analysis and long proofs in PA In that case, I doubt that you can do better than Dirichlet's theorem or the prime number theorem. After all, you said with vastly more effort, which seems a reasonable description of such cases. == Subject: Re: Theorems with short proofs in analysis and long proofs in PA What do you take to be ultimate about it? Anyway, it's not an 'ordinary' theorem that you might encounter in a first course on number theory. == Subject: Re: Continuous Injections dictionary.com refers us to an acronym finder, which tells us WLOG = without loss of generality It is a common math abbreviation. == Subject: Re: Linguistic ambiguity vs. mathematical precision My sig. is from a song that a character in a comedy called the Goons used to sing. It was not addressed to you personally. -- I don't know who you are Sir, or where you come from, but you've done me a power of good. == Subject: Re: How to evaluate this integral using pencil and paper? A strange suggestion from someone whose nickname is Mathematics Lover. Jose Carlos Santos == Subject: Re: Speed and acceleration LOL :)))) == Subject: Re: Complex conjugate of an integral Good point, but I still think that I'm right. Do you have another conjecture about what the OP meant by that notation? Jose Carlos Santos == Subject: Re: Complex conjugate of an integral The a and b are not the limits, they are arbitrary constant terms which may or may not be complex. The variable x was also potentially complex. I wanted to find out the generality of the equation i.e. whether it held for complex constant factors or indeed complex variables. Hope this clarifies. Zinc -- zincnews123 at tiscali.c123o.u123k To reply to address don't click. Cut and paste, change at to symbol then delete all 123's == Subject: Re: Complex conjugate of an integral Then I do not understand your question (and, of course, it follows from this fact that I now withdraw my original reply) Not really, since now I do to know what's the meaning of int[a b f(x)] dx. Perhaps you might clarify this. Jose Carlos Santos == Subject: Re: Celestial Mechanics On Sun, 25 Sep 2005 06:09:34 EDT, Maury Barbato Do you know for ENTER key? == Subject: Re: Celestial Mechanics Sorry, but I can't see this layout error, probably because of the way my reader is set up. However, is Do you know for ENTER key? a correct sentence? Is to know an intransitive verb? But, I'd like to talk about mathematics: do you have some idea to solve the problem? Maury == Subject: Re: Celestial Mechanics You can see it here: Jose Carlos Santos == Subject: Re: Where is my error? So there is my mistake. Can you please tell me how to solve this correctly? == Subject: Re: Where is my error? I don't know what you mean with this question. You've shown us an argument that seemed to prove that 2*e = 2 and then I showed you which step of the argument is wrong. What's left to solve? Jose Carlos Santos == Subject: Re: Where is my error? are not the same. Those unfamiliar with doing arithmetic with sets of values might find that fact to be rather curious. with sets. == Subject: Re: Where is my error? Earlier, I had written: It's trivial, really. Suppose that A and B are sets of numbers of some type, and that ~ and # are, resp., unary and binary operations applicable to numbers of that type. Then we say that ~A = {~a | a in A} and A#B = {a#b | a in A, b in B}. [BTW, you might, justifiably, be confused by 2*ln(-1), since the second operand is a set of values while the first is not. There are two reasonable ways to think of 2*ln(-1) in this context, and happily, they both give the same result. One could choose to think of it as actually {2}*ln(-1), and proceed to apply the binary operation of multiplication. Alternatively, one could choose to think of 2* as being the unary operation doubling.] David == Subject: Re: easy question about supremum That's one use of the term - in another perfectly reasonable and standard use of the term an unbounded sequence has supremum equal to +infinity. Honest. David C. Ullrich == Subject: Re: new easy question about supremmum I doubt that. Well, I suppose you _may_ have seen a_n < oo used as shorthand for I am the king of France. But it does not actually mean either one. In a context where the variables are allowed to take infinite values the notation a_n < oo means exactly what it appears to mean: a_n is finite. David C. Ullrich == Subject: Re: Topology Question On Sun, 25 Sep 2005 06:28:29 EDT, Narcoleptic Insomniac The problem is much easier than all this. Hint: What is the definition of topology? The solution to the problem is immediate from that. David C. Ullrich == Subject: Re: Topology Question Yes. Do you see the two lines above which contain two questions? Well, at first I saw them in a single line. As you may imagine, they actually did not fit in a single line and I had to choose between breaking them manually (that was my option) or to move to the right the bar at the bottom of my screen in order to read it all. I surely believe you, but it would be in your best interest from to on to break the lines manually. Then my question becomes: what is x? You are making a typical newbie error, which consists in introducing in a proof mathematical objects which were not defined previously. Jose Carlos Santos == Subject: Re: Topology Question In other words, I got something like this: Jose Carlos Santos == Subject: Re: Topology Question Please don't use special characters, those that require other than the characters on the top of the keys and shift key. As you did, I don't base for the topology, that is trivial observation. The real definition, the second one you gave. There is no definition for open for a base generated topology. A topology generated by a base B, or any subset A of P(space), is the smallest topology for space containing A. Then it is theorem to prove that when B is a base for the space S { /C | C subset B } is a topology for S. Thus is proven that when B is a base for the topology T of the space S, U is open subset of S, ie member of T iff U is a union of base sets from B. Is the confusion about definition or theorem of open for a base generated topology your confusion or the text's? Topology class or analysis class? at.yorku.ca/topology == Subject: Re: Topology Question Oppps, my mistake. I've been just terrible lately at posting ediqutte! Anyhow, the character that didn't show up was the empty set. Well ugly definition I gave came from the chapter Basis for a Topology, which is where the exercise was too. After the definition of a basis it says, A subset U of X is said to be open in X (that is, to be an element of T) if for each x in U, there is a basis element B in C such that x is in B and B is contained in U. Here T is the topology on X and C is the collection of basis elements. The 'real' definition was given in the previous chapter. Either way I was making it harder than it needed to be by trying to use a basis. Kyle == Subject: Re: Sphere inside a pyramid I found a good refernce,here it is: Bassam Karzeddin == Subject: Re: Sphere inside a pyramid On the other hand, in general there isn't a sphere tangent to all 6 edges - the best you can do is a sphere tangent to 4 edges. My question If you make a perpendicular planes on the lines bisecting the angels of a tetrahedron you will get such a sphere I hoop Iam wrong Bassam Karzeddin AL-Hussein Bin Talal University JORDAN == Subject: Re: math is more subjective than art appreciation I have this nagging suspicion that I'm being mocked. But what are the odds? -- Jesse F. Hughes Our enemies are innovative and resourceful, and so are we. They never stop thinking about new ways to harm our country and our people, and neither do we.-- George W. Bush == Subject: Re: Revisiting my (alleged) proof of the Goldbach Conjecture Arggghhh. How dumb of me. == Subject: Problem solving a linear diophantine rquation with three variables I am trying to find integer solutions to the equation 35x + 55y + 77z = 1 using modular arithmetic and linear congruence. I see that gcd(15, 55, 77) = 1 and 1 divides 1, so the equation should have integer solution (this is a theorem I vaguely remember from an intro to number theory class. Kindly correct me if I am wrong.) I know that the equation is equivalent to the linear congruence: 35x + 55y == 1 mod 77 But when I try to solve this linear congruence, I am stuck. The only approach I know to solve such an equation is the extended Euclidian Algorithm. I am however confused because I notice that the coefficients of x, y, and z share a common factor when paired. So that gcd(35,77) = 7, gcd(35,55) = 5, and gcd(55,77) = 11. I do not therefore get the remainder 1 typical to the Euclidian Algorithm. Any help or hint is greatly appreciated. == Subject: Problem solving a linear diophantine rquation with three variables I am trying to find integer solutions to the equation 35x + 55y + 77z = 1 using modular arithmetic and linear congruence. I see that gcd(15, 55, 77) = 1 and 1 divides 1, so the equation should have integer solution (this is a theorem I vaguely remember from an intro to number theory class. Kindly correct me if I am wrong.) I know that the equation is equivalent to the linear congruence: 35x + 55y == 1 mod 77 But when I try to solve this linear congruence, I am stuck. The only approach I know to solve such an equation is the extended Euclidian Algorithm. I am however confused because I notice that the coefficients of x, y, and z share a common factor when paired. So that gcd(35,77) = 7, gcd(35,55) = 5, and gcd(55,77) = 11. I do not therefore get the remainder 1 typical to the Euclidian Algorithm. Any help or hint is greatly appreciated. == Subject: Re: Problem solving a linear diophantine rquation with three variables I think you have to do two linear congruences, one for x and again for y. In terms of x, you have 35x == (1-55y) mod 77 So GCD(35,77)=7 must divide 1-55y. But must divide means 1-55y == 0 mod 7 or 55y == 1 mod 7 which results in y=6. In terms of y, you have 55y == (1-35x) mod 77 So GCD(55,77)=11 must divide 1-35x. But must divide means 1-35x == 0 mod 11 or 35x == 1 mod 7 which also results in x=6. Thus, the first solution occurs at x=6,y=6 with solutions repeating every 11x and 7y. So solutions occur at (x,y) (6,6) (6,13) (6,20) (6,27) . . . (17,6) (17,13) (17,20) (17,27). . . (28,6) (28,13) (28,20) (28,27). . . (39,6) (39,13) (39,20) (39,27). . . . . . . . . . . . . . . Z is found by plugging the (x,y) values into your original equation. == Subject: Re: Problem solving a linear diophantine rquation with three variables ^^^ typo, should be 11 == Subject: Re: Proof that 1 is the only Odd Perfect Number and thus the oldest math conjecture is conquered That _is_ a question of taste. A definition is arbitrary as long as it is well-defined. To say that a definition is well-defined doesn't mean that the definition is aesthetic, or useful, or strategically the right choice. It means only that when the definition is applied, it gives a definite answer as to whether or not an object matches the specifications of the definition. Thus it can be applied by anyone, anywhere and it will give the _same_ results. So what? What you're saying is they made a poor choice by insisting on distinct factors. That does not invalidate the definition. You can argue that it would have been a more sensible choice to have defined the term differently, but the definition as it stands is still well-defined. To see this, apply the standard definition to some number and see what answer you get as to whether or not it's perfect. If someone else applies the same definition to the same number you chose, then you will both get the same answer (assuming neither of you made a numerical error). That's because it's well-defined. On the other hand, if a definition could be applied to give different answers, then such a definition is truly flawed and in that case must be revised. That's taste. You are arguing that the definition is not aesthetically satisfying. That doesn't invalidate the definition. The validation is based only on whether or not the definition is well defined. Now you're arguing that the definition is not useful. All you're saying is they made a poor choice, but once again, that doesn't invalidate the definition. To call it random is misleading. For a given number n, you always get the same answer as to whether n is or is not perfect. You get the same answer every time you apply the definition to the number n. Anyone else will also get the same answer. That's what it means to say a term is well-defined. To call the definition ill-defined is expressing your dissatisfaction with the definition. To show the definition is invalid, you have to show it's not well-defined -- that it's not precise enough to force a unique answer. As an example, call a set of positive mostly odd if it has more odd elements than even ones. Is the term mostly odd well defined? For finite sets, yes. For infinite sets, more has to be said, otherwise different answers are possible. So as it stands, the term mostly odd is not well-defined since the phrase more odd elements than even ones has not been fully specified so as to be unambiguous for the case of infinite sets. In other words, the definition of mostly odd given above is truly flawed and needs to be revised. One possible revision is to only define it for finite sets. It is possible to make a revised definition that also deals with infinite sets, and there are various ways to do this, but as long as the definition when applied to a given set yields a definite answer, then the term mostly odd becomes well-defined. Sure you can, but if you do that, no one will understand what you are talking about. Sure, definitions can be changed -- typically by consensus, if it's felt that there is a better choice as far as aesthetics, usefulness, etc. But such a change is not mandatory unless it can be shown to be not well-defined (i.e. is not precise enough to force a definite answer in all cases). And I am trying to tell you that what you see as a flaw is really an objection based on aesthetics, usefulness, etc. So let's try this example. Suppose a student argues that 1 should be a prime since it can't be factored -- its only positive integer factors are 1 and itself. The student therefore claims that the definition is flawed and therefore should be changed. So should we define prime so as to allow 1 to be prime? It's an arbitrary choice -- it's ok either way, but we need to agree on a standard. Of course, if we were to change the definition of prime to allow 1 as a prime, then the statements of many theorems and conjectures would have to be changed to be consistent with the new definition. But the point is that the objection raised by the student to the standard definition of prime does not invalidate the standard definition. Bravo. If you change a definition without using a new name for the term, then you are a redefiner. You can choose a new name for your concept -- that's ok, but you can't use the name perfect since it's already in use. You can't steal it. You can't insist that existing term must be changed by claimng that it is flawed, unless you can show it is not well-defined. But the standard definition of the term perfect number is well-defined in the sense of yielding a definite answer for an given positive integer n is or is not perfect, so a revision is not mandatory. Just choose a new name for your concept. Then you can freely use either one. On the other hand, if you insist on confusing things by using your own definitions instead of the standard ones, then you are forcing the reader to constantly translate back and forth between the standard definition and yours. If such a confusion was inadvertent, then we could simply point it out and you could change to match the standard definitions. But for you to insist that everyone else change their use of the language is arrogant, and unacceptable. It's also counter-productive, since it would make it almost impossible to have a meaningful dialog with anyone except yourself. quasi == Subject: Re: Proof that 1 is the only Odd Perfect Number and thus the oldest math conjecture is conquered (snipped) Quasi, I disagree. A definition, like a tape-measure to measure distance should have the same uniform standards when measuring. When a tape measure is not applied to one end of the object but crassly laid on the ground and thus giving an approximate distance in some cases while in other cases two people are involved holding the tape measure carefully on one end and getting a precise measure of distance. The definition of Perfect Number applies a different measure to numbers such as 1,4,9,16,25,etc etc than it applies a measure to 2,3,5,6,7,8,10,etc etc. This is not a well defined definition. An example in common life is that we want to treat our children all fairly and not play favorites with one and bias towards the other. So that at dinnertime, we do not give one child the best food and the other child the lesser food. Nor do we want to praise one child and always scold the other. So when you have a definition in mathematics that does not treat every number to the same standard, you have a ill-defined definition. It is important to trashcan the old definition of Perfect Number because the theory of abundance and deficit from being perfect is nothing but junk under the old definition and any conjecture concerning defiict or abundance cannot be proven because it is based on a ill-defined definition. Archimedes Plutonium www.iw.net/~a_plutonium whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies == Subject: Re: Proof that 1 is the only Odd Perfect Number and thus the oldest math conjecture is conquered What's so HOT about Number ONE (1).?!! ONE's NOT EVEN a PRiME MUMBER ..like it WAS.!! ```Brian == Subject: Re: Analogue of Noether's Theorem for finite groups? I was thinking more of a group like the Rubik's cube. If one takes apart a Rubik's cube and puts it back together blindfolded, there is a one out of twelve chance that the cube will be solvable. I remember reading somewhere that Golomb observed in the early 1980's, when the cube was very popular, that as long as the cube is in a solvable state, a certain quantity is conserved, but once someone, for instance, twists one of the corners of the cube or flips one of the middle pieces on the edge of the cube, that quantity is no longer conserved. Does anyone have any more details about this? Can this idea be generalized to all finite groups or at least to a class of finite groups? Craig == Subject: curvature of curve given by smooth function Let a curve be given by x(t) = (t, f(t)) for some smooth real function f. Let x_0 be a point such that f'(x_0)=0 and f''(x_0) =/= 0. I want to prove that |k| = |f''(x_0)| where k is the curvature of x(t) at x_0. I am given the hint to use taylor's theorem, but that doesn't help me much. Taylor tells me that f(x) = f(x_0) + f''(x_0)/2 (x-x_0)^2 + r(x, x_0) I should note that I am trying to derive this using the definition of curvature that says |k| = 1/R, where R is the radius of the osculating circle. I know there is an equivalent definition, something like k = f'' / (1 + (f')^2)^{3/2} but I don't want to use it. == Subject: Re: curvature of curve given by smooth function and In what sense are these two equivalent? The second clearly gives a different answer to the first whenever f'' and f' are nonzero. == Subject: Re: curvature of curve given by smooth function Sorry, I meant that k = f'' / (1 + (f')^2)^{3/2} implies the previous. == Subject: Re: curvature of curve given by smooth function Please include enough context that somebody has some idea what you're talking about. But if you're talking about |k|=|f''|, I don't see how k=f''/(1+(f')^2)^(3/2) implies that, because (as I said earlier), they are different when f'' and f' are nonzero. == Subject: Re: curvature of curve given by smooth function Well, if we had already established k=f''/(1+(f')^2)^(3/2) , then using the hypotheses I stated (f'(x_0)=0 and f''(x_0) =/= 0) we would have k(x_0) = f''(x_0) / (1+f'(x_0)^2)^{3/2} = f''(x_0) / (1+0)^{3/2} = f''(x_0) so it follows easily from k=f''/(1+(f')^2)^(3/2) . However, I am wanting to use a more basic definition of k. That is, it is 1/R where R is the radius of the osculating circle at x_0. == Subject: Re: curvature of curve given by smooth function Oops, sorry: I didn't read carefully. Of course, that is something of a limitation... OK, so you're looking for the osculating circle to a parabola, y=(k/2)x^2 at x=0. Then fix h, and find the circle through (-h,(k/2)h^2), (0,0) and (h,(k/2)h^2), then let h tend to zero in that. Fortunately, this is an easy problem. If you take the line segment connecting (-h,(k/2)h^2) to (0,0) and the one connecting (h,(k/2)h^2) to (0,0), then find the point of intersection of the perpendicular bisectors of those two, that will be the centre of the osculating circle. And finally, you don't even need to do that. Just find where one of the perpendicular bisectors cuts the y-axis. By symmetry, that will be the centre of the osculating circle, and so the y coordinate of that point will give the radius of the osculating circle. == Subject: pointwise convergence question Suppose we have a sequence of bounded continuous functions f^n, each f^n mapping R^n to R, that converges POINTWISE to a bounded continuous function f. Does the sequence f^n converge UNIFORMLY to f? If not, what is the counterexample? == Subject: Re: pointwise convergence question f^n is usually used to mean f to the nth power. For subscripts you might write f_n. Also I doubt you mean what you have asked i.e., that f_n maps R^n to R, giving each function a different domain: etc. ??? --Lynn == Subject: Re: pointwise convergence question is such a counterexample. Can you see what the graphic of f_n looks like? Jose Carlos Santos == Subject: Re: 0.999... = 1? (I know, a beaten dead horse) My answer is: 0,99999.... = 1 - 10^-N, where N is number which counts used digits. In nowadays Cantorian math it is impossible to find these numbers - like 10^N or 10^-N, but it is quite simple to show these numbers with nonstandard theory, for example. In standard theory of real numbers all numbers, which has difference smaller then 1/n for every small n in N - are the same by definition - and 10^-N is smaller then 1/n for all such small numbers. So the answer is: number 0.99999 a and 1.0 are very similar to each other and from the point of view of classical thery which can do only with potential infinity - are these two numbers the same number. In other Math - which are able to do with true actual infinity - one can show how big this difference of these two numbers are. MS == Subject: Re: Rational vs irrational Robert Israel has given a convincing proof that any expressible number can be expressed as a converging sequence of rationals, but I was under the impression that expressible number meant exactly what it said - namely that it can be expressed precisely in English (or any other language). Why should there be requirement to be able to compare it algorithmically with a given number (and in any case, what do you mean by a given number?). and prove that f has a unique fixed point, then the unique fixed point of f certainly defines an expressible number. It is not at all clear that the set of expressible numbers is well-defined, but specific numbers can be shown to be expressible simply by expressing them. Derek Holt. == Subject: Re: Rational vs irrational depending on what is meant by expressible. It says nothing exactly. The word expressible is far too vague to have any automatic interpretation. In a given context, you need to provide a meaning by giving a definition. So you are saying is that a number is expressible if it can be expressed precisely in some language. In other words, a number is expressible if it can be expressed. Doesn't that strike you as circular? Because that's my definition of expressible. At least it's precise. I'll settle for any given rational number, where by given is meant that the numerator and denominator are specified. Ok, but make sure your function f is alsoexpressible. This is way too hypothetical. If you can give a precise definition of a function f that relates to this discussion, then go ahead and give one, don't just talk about it. Certainly defines an expressible number? I don't see how you can be so certain if you can't provide a precise definition of expressible. But show us an example so we have an actual number to talk about. Exactly. It's not well-defined unless you define it. That's called hand waving. You can't show anything rigorously without a definition. quasi == Subject: Re: good finanical math/engineering forums? Kiki, please ignore this fictional character. He has directed you to his own fictional site. He is, sadly, an idiot in real life. Sorry I can't suggest good financial math/engineering forums. But better than directing you to a hoax group. (Mark: Your investment firm seems to be languishing. No new posts since you flirted online with Carol! Your peons need intellectual direction!) -- Jesse F. Hughes Casting [Demi] Moore as a woman who has come to the New World so that she can 'worship without fear or persecution' in _The_Scarlet_Letter_ is like casting Bruce Willis as Young Rene Descartes. -Joe Queenan == Subject: Re: fractal used in tv show After some experimentation I found that a Julia of power four gives a three cycle spiral like the TV one. My cyclotomic cycles julias give an approximation at m=9, n=3: r(z)=((a+1)*z-a*z^(n+1))/sd a=Exp{I*2*Pi/m] n=3 m=9 sd=1.5 close up at 0.5 to 0.25 == Subject: Re: infinity Then what about 1/(.999...) ? == Subject: Re: infinity Surely you're not asking me to defend a .sig. -- Now I'm informing all of you that the people arguing against me are EVIL, yes they are real, live EVIL people as mathematics is that important, so it's important enough for Evil itself to send minions like them. -- James Harris on Evil's interest in Algebraic Number Theory == Subject: Re: infinity Just curious. I enjoy your Harris .sigs. == Subject: Re: What does Infinity really mean? It's too mutch, so I pick just some of the aspects to respond. A set is not finite if there exists a mapping like this. Yes. Shure. But is this enough? What is infinity? Your (and Bolzano's or Dedekind's) definition: Infinity is that which posseses parts which are as big as the whole. Maybe it is an aspect of infinity. But the only important one? And what is cardinality? A measure of the size of a set. Size normaly means number (for the discret case usually). Now your definition leads to the consequence: In the infinite case there is no size. Consider: Something which containes in parts as much as it has itself in the whole could not have a size in the usual understanding. You say, the cardinality of a set like this is infinity. This implies the idea of a sequence of cardinalities, starting from 0, going through the whole field of natural numbers and containes as an element, greater than all natural numbers, the number infinity. At first: there is no sequence starting from finite elements which contains an infinite element. You are not able to build a sequence like this. So you had to take the set IN_0 and add the element infinity by an artificial way. You may do this. But this element has no connection with the other elements of the set. If you go through the set, you never will reach this last element. Consider we had a set with just IN_0 plus infinity. So you have a set, which contains three totally different kinds of elements: 0 (a number without predecessor), finite numbers, infinity (also without predecessor and shurely no number). This is exact the point which uncovers the misleading idea of Cantor: there is no bijection between this infinite set of cardinalities and the infinite set of the natural numbers. This two sets are the stamp forms for the two kinds of infinity: the approximated infinity and the perfect infinity. This two kinds of infinity are mixed up in Cantor's second diagonal proof and in all proofs which seemingly show that there are transfinite numbers. Albrecht Storz == Subject: Differentiating using chain rule Am I properly showing work for each step on this problem using the chain rule? Does the line f'(x)=dy/du[sin(u)] * du/dx[2x] correctly show that I am trying to apply the chain rule in that step? Find f'(x): f(x)=sin(2x) Solution ------- let u = 2x f'(x)=dy/du[sin(u)] * du/dx[2x] f'(x)=cos(u)*2 f'(x)=2cos(2x) == Subject: Re: question about subspaces hi i finally solved the problem. lets consider basis of {W1 intersection W2}. we can extend this basis for the basis of W1. lets call this basis b1. we can as well extend the basis for basis of W2. lets call that basis b2. now consider {b1 U b2}. i could prove that this generates W1+W2, and its linearly independent. so it must be basis for W1+W2. so if dim{W1 intersetion W2} = r, then by our extention of bases dim{W1} = r+p (say) dim{W2}= r+q (say) since i just proved that {b1 U b2} is basis of W1+W2. dim{W1+W2} = r+p+q = (r+p)+(q+r) - r dim{W1+W2} = dim{W1} + dim{W2}-dim{W1 intersection W2} so i actually proved much more general result. and from this it easily follows what i wanted to prove. wood == Subject: Re: revisiting my old alleged proof of Infinitude of Perfect Numbers Okay, I think this proof method will work and I think it will be easier than what I expect. I believe the factorial can be combined with a Mathematical-Induction. So that I find a prime of form (2^n)-1 and its consecutive composites that precede the prime. I find a prime of this form for which the consecutive composites obeys the Chebyshev Theorem between M and 2M exists a prime. Then I assume true for N (math induction) and show it to be true for N+1. I think the factorial will do that. So I find the (2^n)-1 prime with its preceding long string of consecutive composites and use it for the case of 1 in Math Induction. I then assume true for case N in Math Induction and must show it true for N+1. Archimedes Plutonium www.iw.net/~a_plutonium whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies == Subject: Re: revisiting my old alleged proof of Infinitude of Perfect Numbers According to Wikipedia, these are the first six perfect numbers for n=2; 2^1(2^2 - 1) = 6 is perfect because of prime 3 in (2^n)-1 for n = 3; 2^2(2^3 - 1) = 28 is perfect because of prime 7 in (2^n) -1 for n = 5; 2^4(2^5 - 1) = 496 is perfect because of prime 31 in (2^n)-1 for n = 7; 2^6(2^7 - 1) = 8128 is perfect because of prime 127 in (2^n)-1 for n = ; 2^12(2^13 - 1) = 33 550 336 is perfect because of prime 8191 in (2^n)-1 8 589 869 056 I wonder if someone can do me the favor of telling me the 7th and 8th and 9th and 10th perfect numbers are along with their primes of form (2^n)-1. And if someone can please tell me what the longest string of consecutive composite numbers are that precedes the prime of form (2^n)-1 For 31; longest string of composites 28,27,26,25,24 For 127, longest string of composites 126,125,124,123,122,121,120,119,118 Can someone please post what the longest strings of composites are for the fifth up to the tenth perfect numbers are. Archimedes Plutonium www.iw.net/~a_plutonium whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies