mm-204 === Subject: Re: factoring to satisfiability this is an interesting & thoughtful observation but RE glosses over one presumably obvious consideration. let f(x) be the clause size of a formula with fewest variables to factor a x-bit number. as RE points out, the minimum-variable SAT formula to factor a x-bit number has approx x bits. (two factors of approx x/2 bits each. note sqrt(2^x) has approx x/2 bits) HOWEVER it is likely assured that f(x), the clause-to-bits relationship, grows exponentially. in fact proving this is likely equivalent to proving the difficulty of factoring. (I think it can be shown without a lot of trouble, something like, factoring is NP complete <=> f(x) grows exponentially on the other hand it would be very interesting to construct these minimal SAT factoring clauses recursively, & look at other structural properties. afaik nobody has ever done this. > I wonder if SAT solvers might be of use for factorization. > I have noticed that the factorization to SAT conversions > are not optimized for SAT solvers. > For example, using Purdom and Sabrys generator to factor 9 > produces a SAT instance with 22 variables and 73 clauses. > Factoring 9 can be converted to a Boolean formula with as few as 4 input > variables (two 2-bit multiplicands). > Most of the extra variables are intermediate results that are defined > from the multiplicands or carry bits. > All the factors of a 2N bit number can be found using an N-bit x (N+1) bit > multiplication circuit.There exists a Boolean formula with N+1 inputs that > will find all > factors of an N-bit number. Just wanted to mention that Springer-Verlags annual mathematics book sale, the Yellow Sale, is finally back. Heres the URL: http://www.springer-ny.com/yellowsale/ The book sale is available in bookstores across North America, as well as online at the URL above to customers in the Americas. Springer is the worlds leading publisher in the field of mathematics. Jason Roth Springer-Verlag NY === Subject: Re: Chess/Go/etc: Continuous Game-Boards? (1.) Two pieces of the same colour only form a connected group if they >are in physical contact. > In standard go, this is not true, unless you count diagonally adjacent pieces as being in physical contact. For example xx xxxxxx x xx x x xx x x x xx xxxxxx Even then, it depends on what one means by connected. xxx xxx x x x x xxx xxx -- Stewart Robert nsley === Subject: Re: Deep Thoughts # 1: A new limitation to the human mind > >> 1. Mathematics is the science in which we make something out of >> nothing. Wrong. Mathematics is built on the 13 Axioms, which are not nothing. Which 13? I am sorry, I let myself get carried away by indignation. I would have to look them up. My math has become extremely rusty since I studied, but I can recall that our first semester analysis class consisted in laying out the foundation of mathematics by learning 23 axioms and how they sufficed to build up the rest. I believe we started with the peano Axioms. === Subject: Re: Numeric one-way hash function > , > I need to find an algorithm that can produce a unique non-predictable 12 > digit (0-9) number for any given 12 digit number. This is to be used to > create a unique barcode on a ticket that cannot be predicted. It is not > required that the original seed number be computed from the resulting > barcode, so some form of one-way hashing function would be acceptable. > Any help in this problem would be appreciated. Ive seen wiser heads than mine recommend a Ruby-Lackov cipher for this kind of thing. First, you need a random function f(i) that takes a 6-digit decimal as input and produces a 6-digit decimal output. Start by splitting your original 12-digit decimal number into two 6-digit parts, A and B. Then perform four steps: A=A+f(B) mod 1000000 B=B+f(A) mod 1000000 A=A+f(B) mod 1000000 B=B+F(A) mod 1000000 Concatenate the final A and B to form the 12-digit output. The process is reversible, so there wont be any duplicates among the output values. f(i) should be a good randomizing function, such as the cryptographic hash of the concatenation of a secret key k with the operand i. While you could convert to binary and back, if thats convenient, its not necessary. If youre going to do this in decimal, you could use the ASCII decimal digits of A or B as the input to the hash, and take, say, the first 32 bits of the hashs output, convert it to decimal, and use the low-order 6 digits as the value of f. [I think Ruby-Lackov can tolerate a small amount of bias in f. If not, Im sure someone will post another suggestion.] --Mike Amling === Subject: Re: factoring to satisfiability by the way, the following observation by RE is also the roughly inspiration for an interesting recursive factoring algorithm, called the lankinen algorithm.. it is extremely unstudied/unknown/obscure or nonexistent in the factoring literature. see 40ccncsu.ColoState.EDU > The Mth output bit of a multiplication circuit is determined by the M low > order > bits in the multiplicands. For example, the formula for A is the always > determined > by the lowest order bit of the two multiplicands. If we interleave the input bits of the multiplicands then the formula for > the > lower order output bits will not change as we create larger multiplication > circuits. === Subject: irreducible polynomial ? Is the polynomial 2*cos(2^k*arccos(x/2)) irreducible for each positive integer k? If so, how is it proved? === Subject: Re: Fixed points > Conversation: Fixed points === > Subject: Fixed points > Suppose f : (0,1) --> (0,1) is continuous. Does f have to > have a fixed > point? If it was f : [0,1] ---> [0,1] or f : [0,1] ---(0,1), then yes. Since you have already been given the (an) answer..... What happens to (0,1) if you pick it up , §ip it over then set it back down? Cant you come up with an algebraic formula that describes this operation? === Subject: Re: MTL and MatLab I am trying to make Matrix Template Library > (http://www.osl.iu.edu/research/mtl/) interact with MatLab 6.5. If > anybody has done it, please give me your comments on what you think > about it. I would greatly appreciate it! > Iryna I have used both. What do you mean by having MTL ïinteract with Matlab? Do you want to call matlab functions via the mex interface? Or do want to generate a C++ program that is callable from matlab? I recently did a project where results had to be read in Matlab, but the simulation which generates them is far too complex to run efficiently as a Matlab program. I created a class for results that allowed easy measurments from a simulation, and automates the writing of a corresponding mat file. If this is what you are trying to do, let me know and I can send you the code (I have a version on the web, but it needs to be updated). Regarding MTL... Its been a long time since any updates came from MTL? In fact he object oriented numerics websit (oon.org) is looking a bit out of date. Is there a better reference? Has anything to replace blitz or MTL come out ? G.S. [ See http://www.gotw.ca/resources/clcm.htm for info about ] [ comp.lang.c++.moderated. First time posters: Do this! ] === Subject: Re: irreducible polynomial ? Visiting Assistant Professor at the University of Montana. >Is the polynomial 2*cos(2^k*arccos(x/2)) irreducible for each >positive integer k? If so, how is it proved? Is it a polynomial? In what? Irreducible over what? === Subject: Re: Is this newsgroup useless? The real > problem is that this is an *unmoderated* newsgroup. All posters are > permitted. I have two other main newsgroup interests. One of them is > christianity. In that case, I avoid the unmoderated groups > at like a Pharisee avoids a leper. They make sci.math look like > a candle next to the space shuttle taking off. It takes a hardworking, > patient and wise moderator to keeps christians behaving like...uh.. > christians. So in that sense, your half right. But my other interest is homebrewing, Whats your favorite style? I like to brew IPAs. But, so far, my best brew is a lagered Bohemian Pilsner. Smooooth. Im having a little problem with mashing ... poor yield. Math + Homebrew = FUN. > and that group is UNmoderated, > but the most pleasant and helpful and un§aming group on all of USENET. > Every newbie who comes along asking the same dumb question thats > been asked 17 trillion times is welcomed with open arms, given > kind advice and we cheer that another lost sheep has been converted > from the evils of Budmilloors and demon megaswill. So your half > wrong. Half the problem here is that its unmoderated. The other half is > that its populated by overly-anal-retentive jerks who think its > going to be important to point out that I spelled youre your > twice in the preceding paragraphs. Theres something the same about fanatic legalists in the religious > groups and anal mathematicians, in that they think theyve scored > something if they catch you in an error. AHA! Youve typed > slander when you should have typed libel! So what? Therefore > Im an idiot and hes a savant and now I have to erase his > chalkboards for him? Hardly. Im going to have a homebrew and > hang out with the cute chicks while he re-catalogues his PowerRanger > Collectors Cards. I _thought_ we had beaten these guys up sufficiently in the high > school locker room that theyd be quiet by now. All I did was > explain that if wed all place our duffel bags perpendicularly > on the benches, there would be room for all of us to get dressed > at once. I really dont think that was sufficient reason for > them to put Nair in my bottle of Prell. Maybe the real problem is that a guy cant e-mail a matburn. Bart === Subject: Re: irreducible polynomial ? > >Is the polynomial 2*cos(2^k*arccos(x/2)) irreducible for each >positive integer k? If so, how is it proved? Is it a polynomial? In what? Irreducible over what? Yes. In x. Over Z or Q. --JB === Subject: Re: irreducible polynomial ? >Is the polynomial 2*cos(2^k*arccos(x/2)) irreducible for each >positive integer k? If so, how is it proved? Is it a polynomial? In what? Irreducible over what? > in x, irreducible over Q For example, 2*cos(2^4*arccos(x/2)) = 16 14 12 10 8 6 4 2 x - 16 x + 104 x - 352 x + 660 x - 672 x + 336 x - 64 x + 2 is irredicible. === Subject: Re: Is this newsgroup useless? >so I dont see what the problem is. Well, you do need to be able to find the N key and the >I key. That can be tricky at first, but you only have to >do it once per session; just leave your fingers there. Actually that could be a hard task, to locate those two keys. Rght ïow, for ïstace, ï cat seem to do ït. dave (short for davd rus) === Subject: Re: function >can anyone show me an example of a function (R=real numbers) f:R-->R >such that for any a,b,c in R, there exists an x in R such that aand f(x)=c? Let f(x) = 5. === Subject: Re: irreducible polynomial ? Visiting Assistant Professor at the University of Montana. >Is the polynomial 2*cos(2^k*arccos(x/2)) irreducible for each >>positive integer k? If so, how is it proved? >> >> Is it a polynomial? In what? Irreducible over what? >> in x, irreducible over Q For example, 2*cos(2^4*arccos(x/2)) = 16 14 12 10 8 6 4 2 >x - 16 x + 104 x - 352 x + 660 x - 672 x + 336 x - 64 x + 2 is irredicible. Never seen that before; but would it not be possible to prove it irreducible by using Eisensteins Criterion? Looks like the leading coefficient is 1, the constant coefficient is 2, and all the other terms have even coefficient. If the pattern holds for arbitrary k, then there you are. === Subject: Re: Antidiagonal, Infinity > at 07:58 PM, raf@tiki-lounge.com (Ross A. Finlayson) said: > Im still thinking that f(x)=1 for irrational x and f(x)=0 for >rational x that f is everywhere discontinuous. That much is correct. What that describes is >that between each pair of any two rationals is at least one >irrational and between any two irrationals is at least one rational. That part is wrong; it describes no such thing. ?? Perhaps im missing something. Suppose that A and B are two subsets of R, mutually disjoint and their union is R. Define f to be 0 on A and 1 on B. If f is everywhere discontinuous, doesnt that imply that between any two elements of A there is an element of B and vice versa? >What I propose is that given any >rational that the value greater than it and less than any other >greater is irrational, There is no such number, as several different people have shown you. In non-standard analysis, there might be, however. See Alain Roberts book about NSA. Rather than being irrational, it would be non-standard, though. >Obviously enough then under this axiom Adding such an axiom to the standard axioms would yield an > inconsistent axiom system, and all statements would be provable. It > would not be an axiom system for the real numbers. Actually, there is an axiomatic approach of NSA in which a few axioms are added to ZF(C), and in which the above suggestion makes sense. The extra axioms are relatively consistent w.r.t. ZF(C). Again, see Alain Roberts book on NSA. The other poster might be interested in this approach towards NSA. Just for what its worth, Herman Jurjus === Subject: Re: Length of Stock > What is the formula for length of stock required to go around a circumference. >> Eg >> If I wanted to go around a piece of pipe 3 1/2 in diameter, with 1/4 §atbar, how long should I cut the bar. This is considering the fact that I can form it completely round >C = 2 * pi * r = pi * d, where C = circumference of the circle, >pi = 3.14159..., r = radius, d = diameter. The diameter of the neutral axis is to be used in this formula: http://archive.metalformingmagazine.com/1999/12/DieD.pdf And note that the outside diameter of a nominal 3.5-inch pipe is not 3.5 inches: http://mdmetric.com/tech/pipe0010.htm HTH Joe Geluso === Subject: Re: factoring to satisfiability > this is an interesting & thoughtful observation but RE > glosses over one presumably obvious consideration. let f(x) be the clause size of a formula > with fewest variables to factor a x-bit > number. as RE points out, the minimum-variable SAT formula to > factor a x-bit number has approx x bits. > (two factors of approx x/2 bits each. note > sqrt(2^x) has approx x/2 bits) HOWEVER it is likely assured that f(x), the > clause-to-bits relationship, grows exponentially. I wondered about this. Generating the clauses probably requires exponential effort, but the clause to variable ratio cant be exponential. Given that we have 2N literals, the maximum number of 3-clauses is O(2N^3). For a 100 variable problem there are C(200,3) possible 3-clauses = 1313400 ~ 100^3. Russell - 2 many 2 count === Subject: Re: irreducible polynomial ? > >Is the polynomial 2*cos(2^k*arccos(x/2)) irreducible for each >positive integer k? If so, how is it proved? this is up to normalization the n-th (=2^k) chebyshev polynomial (for |x|<=2) http://mathworld.wolfram.com/ ChebyshevPolynomialoftheFirstKind.html its roots are related to the n-th roots of unity, and the irreducibility to the special value of phi(n) hth (and hope it is not nonsense) klaus Is it a polynomial? In what? Irreducible over what? Arturo Magidin > magidin@math.berkeley.edu === Subject: Re: irreducible polynomial ? http://icm.mcs.kent.edu/reports/1998/ICM-199802-0001.pdf === Subject: Re: Brownian motion approximation Can you spell out how to do it? I can see that the general theorem follows easily if you can prove it where f(t) is a linear function(by scaling, Markov property, etc.) but how do you show easily that f(t) is approximated when f is linear? As I mentioned, I can handle the case f(t)=0 for all t, because then you can use the re§ection principal. Is B_t - f(t) with f linear a BM with drift(I have heard the term before but dont really know what it is)? > >> A while back I posted a question about whether or not > >> P[sup_{0<=t<=1}|B_t - f(t)| < d] > 0 for all d > 0 and f(t) continuous >> on [0,1] with f(0) = 0. > >> In other words, does Brownian motion uniformly approximate any >> continuous function(with f(0)=0) with positive probability? Someone >> replied that it does, and this follows from first proving it for f(t) >> = 0 for all t and then applying the Cameron-Martin Theorem. I can do >> it for f(t)=0, but I dont seem to be able to find a reference for the >> Cameron-Martin Theorem, though it seems to be related to Girsanovs >> Theorem, and maybe even follows from it. Can someone give me some help >> or lead me to a reference? > >I dont believe there is any significant difference between the >Cameron-Martin-Girsanov theorem, the Girsanov theorem and the Cameron-Martin >theorem. As I understand it the same theorem was discovered independently >and is now attributed to all three. > >Rather like the Green-Gauss-Ostrogradsky theorem. > This does not need quoting any complicated theorems. Construct > a polygonal function h such that |f - h| < d/3 on the interval, > bound (from below) the probability that |B_t - h(t)| < d/3 at > all vertices, and bound the probability that B differs by d/3 > from its polygon at those vertices. The Markov nature of B and > the boundedness and continuity of f enable all of this to be > carried out easily. === Subject: Vedctor Calculus Question Could someone help me to understand how to find the minimum distance between a surface (say f1(x,y,z)=c1) and a line (f2(x,y,z)=c2. i believe i should be using gradients. thank you very much! === Subject: Re: function >can anyone show me an example of a function (R=real numbers) f:R-->R >such that for any a,b,c in R, there exists an x in R such that aand f(x)=c? Not so that its continuous for the whole R, but it can be continuous within the open interval ]a, b[: f[x_]:=Tan[Pi/(b-a)*X-Pi*(a+b)/(2*(b-a))] -oo when x=a, oo when x=b and gets all values when between them. === Subject: Re: polysigned numbers Roger. Which form of terplex matches my construction? I do believe we have a subtle difference in our maths. Certainly yours is far more generalized. But I would differ with (a,b,-a-b) being zero-sized (or just plain zero in my terms). I cant really handle something like (a,b,-a-b) in my representation due to the negatives, but I would compensate for the negative signs by adding the magnitude of the sum of a and b to each ordinate to yield ( Sum(2a,b), sum(a,2b), 0 ). This would yield (in my three-signed representation) * 2a * b - a - 2b, not zero. I assume by zero size you mean equivalent to zero. > In Terplex, {a,a,a} and {a,b,-a-b} are zero-sized. Multiplying or > dividing by triples with these characteristics constrains the result > to a sub-algebra in which that size is always zero. What is product( (a,a,a), (1,2,3))? I suppose it is: a + 2aJ + 3aJJ + aJ + 2aJJ + 3a + aJJ +2a +3aJ. = 6a + 6aJ +6aJJ. = 6a(1,1,1) > The triple {a,b,c} can be written a+ bJ +cJJ, where ïJ^3=1. a,b,c > are in a field, and so can be real or complex, etc.; they have the > signs from the field, such as the complex signs {+,i,-,-i} as well as > the signs ïJ & ïJJ. Terplex can also be written in polar form, > {a+b+c,((a-b)^2+(b-c)^2+(c-a)^2)/2, ArcTan[2a-b-c,Sqrt(3)(c-b)]}, > where the second term is a squared radius and the third a polar angle. > On multiplication, the first two terms multiply and the angles add. Also you have stated that these terplex values fit in between the > reals and the complex numbers. The construction I am using produces > the complex numbers on the three signed stage. > snip You are re-developing a 3-phase description of the complex plane - > something that I taught to electrical engineering students in 1948! In > Terplex, 1, ïJ, and ïJJ are orthogonal directions that can be > projected onto the complex plane. Projecting them destroys their > interesting properties. This seems to be another important difference. But here is a con§ict. If 1, ïJ, and ïJJ are indeed orthogonal then how can (a,a,a) be zero? If each direction is unique then (a,a,a) is different than (b,b,b) and each have their own unique position. If you agree with this then we certainly are in two different spaces. And it would seem that this is true especially since the superposition with the complex plane destroys none of the properties of my system. > Algebras work over fields (e.g. real, complex, quaternion, octonion) > that have their own signs. Terplex is just one of the many > conservative algebras that introduce another set of signs such as ïJ, > which are probably better thought of as directions. Real & complex > numbers work well at human-scale problems, but other systems are > more appropriate to the quantum and cosmological scales. I fear that > your approach does not break out of the complex strait-jacket. Three-signed numbers are quite equivalent to complex numbers. But there are higher signs which lead to higher dimensions. For example four-signed math yields a three dimensional space. This four signed math has a product unlike anything that I know of for traditional RxRxR. Could you comment on this? > I dont understand the symbolic system above. They are very long. the > ï=1 part seems redundant. Perhaps something got lost in the font > translation to my machine. Im not seeing anything double struck. Are > the letters s, d, n, o, g, h, p, I, J, Y, k, l, m important? The ïd (etc) terms are written that way to satisfy the Groups.Google > convention that material should be printable on simple equipment, > which precludes double-struck letters. My choice of letters is:- > ïd 12 dozen; ïn 9 nine; ïo 8 octal; g7 seventh letter > ïh 6 hex; ïp 5 penta; ïi 4 standard nomenclature; > ïj & ïk 3 & 2 following ïi; ïm &,n 2 minus and negative; > ïY 3 symmetry. The -1 terms completed the definitions. I tried to > make the symbols easily remembered; the result is messy but mnemonic. Roger Beresford. Now I get your mnemonics. I probably wont get any higher than # (fourth sign)for some time to come. I believe that the closest we can get to an overlap of our constructions is your primals as the field and your terplex algebra as the operators to approximate three-signed arithmetic. Why does (a,a,a) become zero in your system? === Subject: Vigier V Conference Topics for Debate Comments by Jack Sarfatti on excerpts from: attention.) ABSTRACT Each approach to the quantum-gravity problem originates from expertise in one or another area Jack, I can feel your intense interest to find the mechanism of gravity and objects but are instead wave structures in a quantum space. Our perception of their properties was ïschaumkommen of the wave structures. (appearances.) I disagree. I agree with the deBroglie-Bohm-Vigier pilot theory that from information waves. IT FROM BIT matter cores /zpf < 0 that balance the centrifugal repulsion from quantized rotation about their centers of mass and from the repulsive self electric charge. the pilot wave information BIT landscape it is rolling on in a generalized gradient §ow including the fiber space connections or gauge potentials as in the Bohm-Aharonov effect that is the Josephson effect in the macro-quantum case. That is action without reaction for the micro-quantum approximation with signal locality that applies to not apply to complex macro-quantum systems. Einstein agreed, but nobody worked it out. False. Its all worked out in Bohm and leys The Undivided Universe. Now, it has been worked out. see QuantumMatter.com and SpaceandMotion.com The results are amazing. 1) All the natural laws are found as properties of the wave structure of the electron. 2) Everything grows out of only two principles which are properties of one thing - space. Awesome. Gravity is the simplest piece of cake. Take a look. I would love to have your thoughts. I do not know what you mean. Have you derived the equations for general relativity from the information wave? That is precisely what I have done for the giant vacuum pilot wave along with the unified dark energy/matter local field. Any new proposal must be couched in mathematical language and must in suitable limiting cases yield the battle tested equation of theoretical physics such as Guv = (8piG/c^4)Tuv Maxwells equations etc. Otherwise it is not legitimate physics IMHO. Also there must be contact with experimental observations both in terms of prediction and explanation as nicely presented in David Deutschs book The Fabric of Reality for example in the chapters on proper methodology in theoretical physics. Ditto for the excess verbal baggage of less by David C. Williams below on the nature of c in E = mc^2. The hard core of what is behind this can be found in the book by Wheeler and Taylors introductory text on Einsteins relativity. The basic idea of geometrodynamics is the block universe of 4 dimensions with time and space mixing together in changes of perspective of uniformly moving observers in the globally §at case without gravity to begin with. The issue is the invariance of the 4D line element ds under the relevant groups of local frame transformations at a fixed spacetime event P. Given a local frame of reference with coordinates x,y,z, t ds^2 = dx^2 + dy^2 + dz^2 - c^2dt^2 = dx^2 + dy^2 + dz^2 - c^2dt^2 Here c must be invariant under the group as in the Lorentz transformations dx = (1 - (v/c)^2)^-1/2[dx - vdt] dt = (1 - (v/c)^2)^-1/2[t - vx/c^2] dy = dy dz = dz for a nonaccelerating frame shift at constant velocity v along the common direction x parallel to x of the two global inertial frames S and S. One cannot describe gravity this way if one insists on retarded causality of no teleological future causes of past effects associated with the ideas of destiny, fate and purpose. could use Newtons gravity with global special relativity to produce the three classic tests of GR provided one introduced the Wheeler-Feynman-Dirac-Hoyle-Narklikar trick of advanced potentials in addition to retarded potentials. The fact that Puthoff gets those tests as well with his variable dielectric vacuum model is no great achievement either because Einsteins classical geometrodynamics goes beyond those tests, e.g. gravimagnetism and gravity waves and black holes. With gravity one must use LOCALLY curved spacetime in which at spacetime point event P ds^2(P) = guv(P)dx^udx^v with summation convention u,v, = 0,1,2,3 There are two LOCAL symmetry groups here. The Poincare group is no good anymore. The translation subgroup symmetry of the Poincare group is broken by the locally variable Diff(4) curvature tensor that is the essence of gravity. The local Lorentz group of invariant tipped light cone structure is obeyed in the tangent fiber space attached to P. The base space of the tangent bundle obeys the Diff(4) group of LOCAL general coordinate tensor transformations xu = x^u(x^u) that replaces the translation subgroup of the globally §at Poincare group of special relativity. This is all for zero torsion of course. All LOCAL observables must be tensors or spinors under both groups. A spinor is a square root of a tensor. Einsteins equivalence principle is mathematically represented by the tetrad map eu^a(P) from locally §at tangent space inertial coordinates a to locally curved base space non-inertial coordinates u. The a -> a transformation is via the 6-parameter Lorentz group of special relativity. The u -> u transformation is via the 4-parameter Diff(4) group of general relativity. dx^u = eu^a(P)dx^a nab = ea^u(P)eb^v(P)guv(P) Local elimination of the non-tensor connection field for parallel transport of tensors along world line paths normally associated with Newtonian gravity acceleration g for example. These are g-forces or inertial forces from accelerating non-inertial frames like the surface of the rotating Earth for example. They are not inhomogeneous tidal variations in the g-force from the curvature tensor, which are never locally eliminated although they are here on Earth very small of order (scale of measurement) 10^-13 in centimeters. Where nab is the §at space-time constant metric tensor of special relativity and guv(P) is the locally variable metric which represents a real gravity field only when the 4th rank curvature tensor of tidal forces does not vanish. You can have a variable guv(P) without a real gravity field from using a non-inertial local frame that is accelerating without tidal forces. This is not physically of great interest however. A LOCAL tensor that vanishes in one LOCAL frame vanishes in ALL LOCAL frames at fixed point event P for ALL relevant symmetry groups. If tuv = 0 then tab = 0 and vice versa. There is no such thing as a gravity force anymore in this non-Newtonian paradigm of geometrodynamics. Newtons gravity equations are regained in the limit of weak curvature and slow speeds compared to c. If there that minimize their dynamical action in the given metric field guv(P) Light rays move on null geodesics ds^2 = 0. The invariance of the speed of light c in global special relativity is replaced by the above remark! In general there are gravimagnetic cross terms in the case of nonstationary metrics and this complicates what is meant by the speed of light. For example, if there are no gravimagnetic cross terms in a simple case with spherical polar coordinates for a non-geodesic observer subject to spin 1 gauge forces like ds^2 = grr(P)dr^2 - gthetatheta(P)dtheta^2 + gphiphi(P)dphi^2 - c^2gtt(P)dt^2 For a light ray we have 0 = grr(P)dr^2 - gthetatheta(P)r^2dtheta^2 + gphiphi(P)(rsin(theta))^2dphi^2 - c^2gtt(P)dt^2 with the usual convention that the LOCAL metric field guv(P) is a pure dimensionless number and r(P) is the Schwarzschild curvature radial coordinate defined such that the surface area surrounding the center in the static spherically symmetric spacetime geometry has the Euclidean area 4pir(P)^2 . Consider the null radial geodesic, dtheta = dphi = 0 0 = grr(P)dr^2 - c^2gtt(P)dt^2 0 = dR^2 - c^2dT^2 where dR = grr(P)^1/2dr dT = gtt(P)^1/2dt dR and dT are physically measured space and time intervals for the light ray using meter sticks and clocks or radars by the non-geodesic observer for small separations between two lightlike separated events P and P. Small means compared to the local radii of spacetime curvature. For example in the static spherically symmetric Schwarzschild vacuum metric solution of Ruv = 0 for r > 2Gm/c^2 gthetatheta = 1 gphiphi = 1 grr(P) = (1 - 2Gm/c^2r)^-1 gtt(P) = (1 - 2Gm/c^2r) The black hole event horizon is at gtt(P) = 0 dR = (1 - 2Gm/c^2r)^-1/2dr But the circumference C = 2pir The change in C for dr is dC = 2pidr Therefore dC/dR = 2pi(1 - 2Gm/c^2r)^1/2 --> 0 at the event horizon. The physical radius R = integral dR is much larger compared to physical circumference C as it would be in §at 3D space. If one keeps R fixed ~ h/mc ~ G*m/c^2, the micro-geon of Wheelers Mass without mass Charge without charge Spin without spin shrinks to a point in high resolution Heisenberg microscope scattering probes with r ~ h/p for momentum transfer p like SLAC deep inelastic electron scattering off protons. This is a semi-classical theory without quantum electrodynamic vacuum polarization zero point energy density effects, however the latter were shown by Feynman and Schwinger to obey the Poincare group in the absence of gravity. The price for this is the obscure renormalization, which may not be internally mathematically consistent although its predictions are in remarkable agreement with experiments. Indeed, the requirement of renormalization with a finite number of fudge factors or epicycles :-) has been a very useful rule of thumb. Directly micro-quantizing Einsteins general relativity is not renormalizable, i.e. one needs an infinity of epicycle fudge factors. That tells us we have asked the wrong question. As John A. Wheeler says: The Question is: What is The Question? Resemblances of Wheelers remark to Cantors diagonal argument and Godels incompleteness theorem of self-referential spontaneous self-organization are not accidental and random. :-) Wheeler thinks of the universe as a self-excited circuit of observer-participators. The velocity of light c in ordinary non-gravitating vacuum with /zpf ~ 0 is directly measured by a variety of techniques. It is an observable measurable property. The velocity of light in a medium is c/n where n is the index of refraction. The physical vacuum also has an index of refraction n(vac) that is very close to 1 in most situations. This small variation comes primarily from vacuum polarization zero point §uctuations of the off mass shell or virtual electron-positron plasma electrically neutral ionized plasma inside the vacuum. This is vacuum. One must be careful in how to use both special and general relativity when polarization effects in real on mass shell media are considered. In the case of special relativity one should, for example, consult the text books by Arnold Sommerfeld, Panofsky and Phillips, Landau and Lifz. A real on mass shell medium in a sense spontaneously breaks translational symmetry on scales larger than the unit cell of the lattice as described in more detail by P. W. Anderson in his More is different series of papers collected in A Career in Theoretical Physics (World Scientific) When one includes all dynamical degrees of freedom including those inside the unit cell of the lattice, global special relativity is restored to the propagation of light in a real medium in the usual situation where gravity tidal forces are negligible. appropriate parties (bccs to you and Toby) to stimulate wider discussion and understanding of the important points you have raised towards re-evaluation and correction of fundamental errors in scientific conceptualizations since the key departure of Newtons work neutering science by his using his mathematicalizations which did not incorporate properly his own determined religious faith in the absolute nature of truth as per his belief in God. I edited you posts lightly for clearer reading per common American English punctuation etc., and changed one word from proof to idea relating to the Theosophical science quote so that your point was not mistaken as that quote you cited being some kind of empirical experimental proof of the your choice of proof over idea please expand on this in your next post to correct my misunderstanding and I will forward appropriately with apologies. Your discussion properly considered, by those to whom it was sent, should also go a long way towards restoring, or integrating, ethics into the scientific discipline of thought since the only way out of present conundrums in science is to replace uncertainty about uncertainty with certainty about the absolute nature of truth itself and let the chips fall where they may in terms of how this change in principle modifies scientific perception of the nature of reality, the various theory equations etc. I noticed in your writeups that you reference C as the constant value of the speed of light in several places and then in one place you seem to reference it as the velocity of light. I understand that in his earlier works, especially in German, Einstein used the term velocity and then over years he too began routinely using speed apparently due to the mathematical convention dictating that by definition the value of the square of the direction component of the square of a vector property is defined identically equal to one. Thus, in trying to understand, for example, what is the true value of, say, (10mph-North)^2, the square of the velocity of ten miles per hour in a northerly direction, the simple (2+2=4) logic of ordinary math is thrown out the window by this mathematical convention per this dictum and the normal logical value of the square of this vector quantity (100m^2/h^2-North^2) is defined for all intents and purposes as equal to (100m^2/h^2), ie, like saying because we cannot mentally grasp or understand what it means (North)^2, we define it to be identical to one, ie, no meaning at all. While this may seem a trivial point for all purposes within the realm of considering physical systems dynamics from the point of view of current science since apparently Galileos time, ie, the exclusively objective nature of reality, ie, that observer and observed are separate and all physical systems that are real exist independenttly of and identically observable by all observers, or they are not considered real by science when they are not reproducible by all observers at all times, in the case of C^2 as the proportionality constant between the value of the Energy of any given system of reality under observation and the value of the mass of that system, and since C itself appears in so many equations of electrodynamics etc., herein apparently lies another overlooked fundamental misconception of scientific thought compromising the absolute nature of truth since C itself is not pegged to the notion of an external objective reality reference frame but is pegged to the identity of the observer. That is to say, since C is non-additive and its square (C^2) is the proportionality constant between the values of energy and mass of real physical systems then this mathematical convention seems evidently the main limit of current mathematics to overcome for a full understanding of the relative nature of reality, ie, the relationship depicted in Buddhism (and infinduism I believe) that each persons mind is the creator of the universe relative to the identity of that person, ie, the ego-centric nature of the universe. While this view seems rightly to folks like Dr. Sarfatti as psycho-babble it nevertheless is the view that offers a mathematical handle for a starting point to conceptualize and integrate into modern physical theories this precept that there is a consciousness factor at work between the nature of the observer and the nature of the system of reality under observation, ie, a relationship of mind between observer and observed as well acknowledged by quantum physics experiments over last decades where he describes this principle as observership. Many scientists are wrestling with this notion of the relationship of mind between observer and observed, eg, denoted in OLearys books as the consciousness factor and in Dr. Sarfattis post quantum physics of consciousness theory equations by their complexities that include the Uncertainty Principle mathematics and depict the operation of intent on the mental field of matter causing a ïback-action in time re§ecting back via that mental field of matter in a ïcybernetic feedback loop of consciousness of predictible ïmoment of consciousness duration which corellates well with latest experimental results in neuroscience etc even though in Sarfattis view, in my words, the observer is transparent ie, there is no accounting for variations between observer identities and corresponding variations in observable systems dynamics from observer to observed (non-reproducibility by all observers at all times of all physical systems dynamics, eg, psi-phenomena), ie, the differences in each observers ïmind impacting the physical systems dynamics of the ïreality which each observer observes. Back-action is not temporal. It is the direct reaction if IT back on its BIT field. ITs BIT field quantum potential Q now has sources and is not fragile, but has macro-quantum phase rigidity of which Andrei Sakharovs metric elasticity is an example. IT is no longer a test from in spontaneous self-organization - the participatory universe as a self-excited circuit. It is not to be confused with the Wheeler-Feynman advanced potential from future to past. It is true that when the limits of micro-quantum theory without back-action hence with signal locality are transcended, then one gets presponse signal nonlocality which mimics an advanced potential effect. Back-action in my sense as quantum action + post-quantum reaction, and advanced potentials are, of course, not incompatible in my theory in which EPR micro-quantum nonlocality with signal locality are as in John Cramers transactional generalization of the Wheeler-Feynman classical advanced potential as first noted by Costa de Beauregard back in the 1970s. Antony Valentini shows that post-quantum back-action that pushes the system away from sub-quantal heat death causes signal nonlocality like what is seen by Dick Bierman in his mind-matter presponse experiments. Macro-quantum BEC systems in particular have post-quantum back-action IMHO. The above is my latest attempt to explain in words what I have rendered since 1974 into low level mathematical equations of a unified field theory (referencing field of awareness and the human minds consciousness orientation function of light as correction factor to add back this omitted true extra value of the square of the direction component of C^2 which is more than (north)^2 because of this fact that C is non-additive and therefore as such is fundamental, being the only physical not ïpegged to the notion of the exclusively objective nature of reality but rather ïpegged to the conceptualization of the ïrelative (or ïego-centric) nature of reality a conceptualization which is obvious in human daily life via common sense in all other realms and is throughout the history of human thought a fundamental principle as discussed in new age psycho-babble in such terms as we are the center of our own universe and by application of the human mind we can in§uence and change the nature of our universe. The above paragraph by David Williams is New Age cargo cult pseudoscience IMHO on a par with crystal power, orgone, spiral dynamics, monoatomic highly deformed nuclei superconducting Ormus powder et-al. It is not even wrong in Wolfgang Paulis sense. Bruce DePalma is the only person who has looked at the four equations of my Tetron Natural Unified Field Theory and the minimal discussion of their meaning presented to him soon after I met him in May 1979, and his reaction in subsequent soon talk with me was, David, you know, I understand your theory. ïSeeing is believing, right? was his response with a glimmer of humor behind his eyes that made me know that he had hit the nail on the head with this response which zeros in on the paradoxical relationship of mind between seeing and believing as it relates to all levels of human thought including science, religion, spirituality, politics, etc. The above paragraph by David Williams is New Age cargo cult pseudoscience IMHO on a par with crystal power, orgone, spiral dynamics, monoatomic highly deformed nuclei superconducting Ormus powder et-al. It is not even wrong in Wolfgang Paulis sense. Exclusive of Dr. Sarfattis vigorous ... criticisms, everyone else since has either said so what? in response, or like OLeary have just ignored and refused to respond to this theory of mind, with the single exception of Dr. Fred Wood who listened to my first-ever public lecture on my theory, on September 10, 2001, at my alma mater California State University at Northridge, and after made a point of telling me that he would think seriously about what I had said (a prepared written formal read aloud lecture archived at http://groups.yahoo.com/group/gcsc-csun/message/6 ) Subsequently, I emailed Dr. Wood and suggested that the next step in the application of my tetron thesis is beyond the mathematics of my education as a Bachelor of Science in Chemistry, ie, how to understand the (tensor?) mathematical relationship between this correction factor Tetron -- the mathematical function applied to the square of the speed of light to correct it to its true value of the square of the velocity of light oriented relative to the identity of the observer, ie, adding back the square of the vector component of C to overcome above described mathematical convention, ie, compromise about the nature of light and the absolute nature of truth itself in all places where C^2 appears in physics equations -- as Tetron applies to C^2, and since many equations particularly in electromagnetism contain C, there must be a similar correction factor to apply in such equations which is also the missing link in understanding the consciousness factor as it relates to these new space energy technologies and the expected reconciliation of their presently con§icting theories of operation. The above paragraph by David Williams is New Age cargo cult pseudoscience IMHO on a par with crystal power, orgone, spiral dynamics, monoatomic highly deformed nuclei superconducting Ormus powder et-al. It is not even wrong in Wolfgang Paulis sense. Since your theoretical approach well includes deep background understanding on this Eastern philosophy concept of the mind as the creator of the universe per Vedic knowledge you write and feedback on my above views will help stimulate productive thinking towards a deeper understanding of this consciousness factor as it also needs to be formalized in order to correct the mind of science and fully understand what is going on with all these various new space energy technology experiments and the theories posited to try and understand these results as well as how some of the theories (like yours and Joseph Newmans) have predicted and perhaps even empowered the successful experimental results that you have each obtained by different configurations of rotating magnetic systems with entirely different theoretical underpinnings yet with similar overunity results. Consciousness IMHO is when one has post-quantum back-action which excites the MACRO-QUANTUM BIT BEC field into self-awareness. Our minds are macroscopic non-classical information fields of both active and inactive information in the Bohm-ley sense. Flashes of memory are the activation of unoccupied basins of attraction of the MACRO-QUANTUM BIT LANDSCAPE when the IT system point of sub-microtubular hydrophobically caged electric dipole Frohlich collective modes roll into that basin. In the torsion field theory developed in recent decades from the Russian language-mind views expressed in mathematical language and with deeper consideration of many documented psi-phenomena experiments in former Soviet Union, going back to the 1950s-60s apparently, there may also be this idea of a relationship of mind between observer and observed but the details of how this is represented in this theory are unclear to me at my level of math education and literature availability. But it is clear from my personal conversation with Dr. Shipov courtesy of Dr. Sarfattis kind invitation for my informal participation with his group in San Francisco one day a few years ago, that torsion field theory also is perplexed by the results of DePalma showing variation in gravitational behavior between spinning and non-spinning ball bearing drop results discussed in earlier post. I do not believe DePalmas claims. Gennadys beliefs in psi are not directly connected with his torsion math. Bill Page talked about the latter at Vigier IV. Creon Levit investigated such claims at ISSO 1999-2000 that Williams refers to here. Creon was not impressed with any of the New Age Free Energy claims promoted by David. But Creon can speak for himself. My hunch here on this is that rotating objects too have a vector property which analogous to C discussed above, is being overlooked in its importance as it relates to the observer because every rotating system also may, similar to C, be seen in the view of its orientation and rotational properties as relates to the relative identity of the observer. I think that this other application of what I talk about above will not come here until this issue about C is resolved, but that there is an important connection between the corrected true value of C^2 as per above and a deeper understanding and correction also with torsion field theory, although the math involved is way beyond my education level to deal with as a language to express the principles I have tried my best above to explain in my style of California Chemical English :-) The above paragraph by David Williams is New Age cargo cult pseudoscience IMHO on a par with crystal power, orgone, spiral dynamics, monoatomic highly deformed nuclei superconducting Ormus powder et-al. It is not even wrong in Wolfgang Paulis sense. David Crockett Williams 661-822-3309 Chartered Life Underwriter Bachelor Science Chemistry www.angelfire.com/on/GEAR2000/vision.html http://groups.yahoo.com/group/drums-of-peace http://groups.yahoo.com/group/new-energy-solutions http://www.josephnewman.com Red Silk Road Peace March Project USA, California, Japan, Korea, China, Nepal, India, Pakistan, Afghanistan, Iran, Iraq, Jerusalem, http://groups.yahoo.com/group/silk-road-to-peace http://groups.yahoo.com/group/dharma-walks One Person Can Make a Difference www.kucinich.us Rep. Dennis Kucinich http://groups.yahoo.com/group/Kucinich-for-President Spiritualism, The ghest Form of Politics For a Culture of Peace & Community Our Spiritual Unity Re-Awakening www.leonardpeltier.org www.horseforgovernor.com www.prophecykeepers.com http://web.mahatma.org.in www.peaceinspace.com www.cesarechavez.org www.brianwillson.com www.dharmawalk.org www.sathyasai.org www.tewari.org www.prop1.org === Subject: Re: What to tell students in a 10-15 minute talk > Well, for (1), all you need is a series. §ys speed and that time, work out the distance. I expect most people here have heard it, but Ill pass on the following > anecdote: > Someone presented Feynman with that problem [bug §ying between two > approaching vehicles], and he of course solved it very quickly. The other > guy said, would you believe some people solve it with series! To which > Feynman responded, whats the other way? B Oh Feynman not Neumann ;-) My apologies to both of them. -- G.C. === Subject: Re: What to tell students in a 10-15 minute talk > > Well, for (1), all you need is a series. §ys speed and that time, work out the distance. I expect most people here have heard it, but Ill pass on the following > anecdote: > Someone presented Feynman with that problem [bug §ying between two > approaching vehicles], and he of course solved it very quickly. The other > guy said, would you believe some people solve it with series! To which > Feynman responded, whats the other way? :-) Here (USA), its almost always (i.e., every time Ive ever heard it) told > about Von Neumann himself: he solves the problem very quickly, exclaims, > Ah! Yes, it is 150 miles! or whatever, and, when the curious onlookers > ask him how he did it so quickly, he gives a blank look and replies, > I summed the series. I never heard the algebraic approach referred to as the Von Neumann > approach before. -Arthur I meant the sum the series approach is the von Neumann approach, only to be told that the clever fellow was Feynman not Neumann. Now Im utterly confused. Of course its possible that von Feynmann _didnt_ sum the series and was pulling his interlocutors leg. === Subject: Re: function >can anyone show me an example of a function (R=real numbers) f:R-->R >>such that for any a,b,c in R, there exists an x in R such that a>and f(x)=c? Not so that its continuous for the whole R, but it can be continuous >within the open interval ]a, b[: f[x_]:=Tan[Pi/(b-a)*X-Pi*(a+b)/(2*(b-a))] -oo when x=a, oo when x=b and gets all values when between them. Another example: g[x_]:=Pi/(b-a)*x-Pi*(a+b)/(2*(b-a)) f[x_]:=Cos[g[x]]/g[x] This one is continous everywhere except at x=(a+b)/2, and gets all values between a and b. === Subject: Re: Is this newsgroup useless? > Just use a decent news-reader and ignore certain Subject-lines and > authors. He does use a decent newsreader, and he could ignore certain Subject-lines and authors with it. === Subject: Re: Is this newsgroup useless? Moderating this group is not an option. The USENET Powers that Be > (otherwise knows as the moderators of news.announce.newgroups) have > decreed that proposals for changing the moderation status of existing > groups will be not be accepted. If you want a moderated version of > sci.math then youll have to propose a new newsgroup. But doesnt sci.math.research exist? And isnt it moderated? So what is all the fuss about? Joachim -- Trau niemals einem Stollentroll! === Subject: hw help -- continuity Folks, I have a couple questions. This is homework, so please post a nudge, not a solution. 1)prove that if f,g continuous, then so are max(f,g) and min(f,g) After drawing some graphs, this seems pretty obvious for the single point a0 -- max(f,g) has 2 cases: it equals to f or g. Either is continuous. However, this question implies continuous on R, not just at a single point. Any ideas how to approach this? 2)Let f be a function with the property that every point of discontinuity (ie the lim (x->a) f(x) exists, but is not equal to f(x)) is a removeable discontinuity. This implies lim (y->x)f(y) exists for all x, but f may be discontinuous at some (even infinitely many) numbers x. Define g(x) = lim (y->x) f(x). Prove g is continuous. --I dont even know where to start with this one. -earl- === Subject: Re: After Coxeter At one point or another I read parts of these books: * Manfredo P. do Carmo: Riemannian Geometry > a very nice reading. good exercises > * R.W. Sharpe : Differential Geometry. Cartans Generalization of > Kleins Erlangen Program. > very broad minded. not allways accurate. > * Kobayashi and Numizu > enciclopedic. > * J. Jost > the analytic view. very dense. > * Spivaks pentalogy > surpsingly, some parts of it are actually nice. Yes, now its manufactured as a proper book, Im saving up my pennies to buy it. === Subject: Exponentative closure Can the reals be defined using repeated exponentative closure? By the exponentative closure F, I define F/x as the set of all the zeroes of all the polynomial functions with coeffeicients AND exponents in F. For example, the the algebraics are the exponentative closure of the integers. Thus, it can be written A=Z/x. Does C=A/x? If not what does A/x equal? Can C be generated by repeatedly exponentatively closing the integers a finite number of times? If so, how many? A countable number of times? An uncountable number of times? === Subject: Re: Yet another question about circular sectors :) > When I run your numbers backwards, I get L => 1.4142. > Using your first answer where theta = 90 deg (or Pi[/2 you meant] radians) and radius = 1: L = 2 (r * sin(theta/2)) = 1.4142 > And, your second where theta = 180 deg (Pi radians) and radius = 1/sqrt(2) = > 0.7071: L = 2 (r * sin(theta/2)) = 1.4142 Please see my above post again ; L is the SEMI- chord, not full chord length. === Subject: Given the radius of convergence for one series, Im trying to find the radius for 2 simliar series Let f(z) = sum_k a_k*z^k be a formal power series with radius of convergence 1. Put s_n = a_0 + ... + a_n and t_n = (s_0 + ... + s_n)/(n+1). Let g(z) = sum_k s_k*z^k and h(z) = sum_k t_k*z^k. How would I show that the radius of convergence of h and g are both 1 as well? (The lim sup method doesnt seem to work). === Subject: Re: Rudin question >Why would you think that? Rudins texts are known for their concinnity >and concision, not for detailed proofs. > concinnity. What a wonderful word! I had never seen that before. (To save others the trip, here is the first definition from http://dictionary.reference.com/search?q=concinnity: 1. Harmony in the arrangement or interarrangement of parts with respect to a whole. 2. Studied elegance and facility in style of expression: .96He has what one character calls .95the gifts of concinnity and concision,ê that deft swipe with a phrase that can be so devastating in children.94 (Elizabeth Ward). 3. An instance of harmonious arrangement or studied elegance and facility.) === Subject: Re: a baseball odds calculation problem > a few weeks ago, i made an observation in alt.sports.baseball.mn-twins about > what seemed to be a rarity: all three teams in baseballs american league > clinched their respective divisions on the same date. my exact remark was > i bet this has never happened before. several others agreed and then tried to pin down the probability of this > happening in any given season. at this point the discussion turned a heated > argument over the probability issue, with widely disparate answers--were > talking answers not even in the same universe. worlds apart. anyway, im posting this problem here hoping that a few knowledgeable folks > will be willing to take a crack at it. while it helps to have a knowledge > of how competition among baseball clubs works mathematically, i dont > believe thorough understanding of the sport is required. it seems to be > more of a math issue than a baseball issue. heres the problem, specifically: there are three divisions in the american league. every season, each one of > these divisions will be clinched by one team on a certain date. this is a > mathematical certainty. past baseball history tells us that dates before september are extremely > rare for teams clinching. in fact, the later days of september, roughly the > 15th through the 28th, are where the likelihood of teams clinching is the > greatest. > An important fact that you left out is how the divisions are clinched. There are 5 teams in each division and 180 games per season spread over approximately 200 days. Typically all teams will play or very few will play in a given day. A division is clinched when one team has more wins the (180-losses) of any other team in the division. It should also be considered that certain teams have better chances of winning than others and different teams are better against other teams. If you ignore that consideration it can be determined by investigating the rows 90-180 of pascals triangle. I would say that the probability is about 1%. > i hope ive done an adequate job of spelling out the problem. what are the > approximate odds that all three divisions will be clinched on the same date? === Subject: What is C[[x]] (generating functions) : algebra structure I have a problem in combinatorics (chapter is generating functions) where f(x) = sum_{x>=0} a_n x^n is in C[[x]]. What is C[[x]]? I have to show that composition of two f,g in C[[x]] is again in C[[x]] if the constant term of g is 0. Our professor defined this structure in class but I cant find the definition. Any help on the definition of this structure is greatly appreciated. Also, what are WEIGHTED generating functions? === Subject: Re: Best differential equation solving software ? I like Mathematica. Never had my computer crash. Lurch > What is the best software for solving differential > equations? Is there anything better than Matlab -- > that at least wont crash and necessitate a reboot > of my computer when I give it a differential equation > it cant solve? --Bob Day > http://www.bob.day.name === Subject: Re: irreducible polynomial ? > Is the polynomial 2*cos(2^k*arccos(x/2)) irreducible for each > positive integer k? If so, how is it proved? and the proof is essentially trivial: Each polynomial is the square of its predecessor less 2. This follows immediately from the identity (cos z)^2 = (1 + cos(2*z))/2. Now for k=0,1 the polynomials are x and x^2-2 so it is clear that for each k the leading coeff is 1, the constant is 2 and all other coeffs are even, so Eisenstein applies as Arturo suggests. --Jim Buddenhagen === Subject: Re: irreducible polynomial ? > >Is the polynomial 2*cos(2^k*arccos(x/2)) irreducible for each >>positive integer k? If so, how is it proved? >Is it a polynomial? In what? Irreducible over what? >>in x, irreducible over Q >>For example, >>2*cos(2^4*arccos(x/2)) = >>16 14 12 10 8 6 4 2 >>x - 16 x + 104 x - 352 x + 660 x - 672 x + 336 x - 64 x + 2 >>is irredicible. Never seen that before; but would it not be possible to prove it > irreducible by using Eisensteins Criterion? Looks like the leading > coefficient is 1, the constant coefficient is 2, and all the other > terms have even coefficient. If the pattern holds for arbitrary k, > then there you are. > Right you are. Let p_k(x) = cos(2^{k} * arccos(x/2)), then p_1(x) = x^{2} / 2 - 1 and p_{k+1}(x) = 2 * (p_k(x))^2 - 1 in x^2, has constant term equal to 2 or -2 and that 2p_k(1) = -1 and 2p_k(2) = 2. As you say, there you are. You dont even need to invoke Eisenstein. Rick === Subject: Re: factoring to satisfiability > this is an interesting & thoughtful observation but RE >> glosses over one presumably obvious consideration. >> let f(x) be the clause size of a formula >> with fewest variables to factor a x-bit >> number. as RE points out, the minimum-variable SAT formula to >> factor a x-bit number has approx x bits. >> (two factors of approx x/2 bits each. note >> sqrt(2^x) has approx x/2 bits) >> HOWEVER it is likely assured that f(x), the >> clause-to-bits relationship, grows exponentially. I wondered about this. >Generating the clauses probably requires exponential effort, >but the clause to variable ratio cant be exponential. >Given that we have 2N literals, the maximum number of 3-clauses >is O(2N^3). For a 100 variable problem there are C(200,3) >possible 3-clauses = 1313400 ~ 100^3. Why should all the clauses in this formula have at most three literals? You cant necessarily represent a function in N variables using a 3-CNF formula. The number of functions that can be represented that way is very small by a counting argument: there are 2^(2^N) functions in N variables, but only 2^(O(N^3)) possible 3-CNF formulas. What are the odds that factoring is one of those lucky functions for interesting values of N? -- Daniel A. Jim.8enez djimenez@cs.utexas.edu Assistant Professor djimenez@cs.rutgers.edu Department of Computer Science Rutgers University === Subject: Re: Measure extension proof >In the material I have read, the terms Borel >Field, Sigma Field, and Sigma Algebra appear to mean the same, >that is a collection of sets closed under complementation and >countable unions. Is this not correct? > No, as I understand the terms Sigma fields and sigma algebras are indeed the same thing. The Borel field is the smallest sigma field containing the topology (i.e., all the open sets) of a topological space. When refering to R^n, the usually unstated topology is the usual one. -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu === Subject: looking for smooth function Hello sci.math, Ive got an array size 100 of integer values (range: 0-5). Each plot in the array represents a 1ms time window. I currently have these graphed as stairs in MatLab (which looks like a bar graph). I want to make this function smooooooth, but im not sure of a method to apply to it. What steps should I take to complete this? Dustin === Subject: Re: Fixed points >Since you have already been given the (an) answer..... What happens to (0,1) if you pick it up , §ip it over then set it back >down? > Er, doesnt that leave 1/2 fixed? -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu === Subject: Re: What to tell students in a 10-15 minute talk >>Also, when are the hour, minute and second hands positioned so that >>they divide the clockface in three equal sectors? >>The short answer is never. >>I was aware of the result. The interesting bit is thinking of >>different mathematical solutions to solving it, which leads us to the >>topic of having multiple ways to reach a given mathematical result. > So if you think of the three hands as rotating unit vectors, > their sum will never be zero. So heres another problem. Assign > angular velocities to the three unit vectors so that the mimimum > length of their sum is as large as possible. There are various answers depending on what additional restrictions you apply. For j = 1,2,3, let Case j allow up to j hands to have the same angular velocity. Define subcase (a) by requiring all hands to meet at some time, and subcase (b) by not requiring this. (Applying no additional restrictions yields the trivial case 3b.) Solutions: Cases 3a and 3b: maxmin length is 3. All hands have equal angular velocities and initial positions. Cases 2a, 2b, and 1b: maxmin length is 1. 2a: two hands have the same angular velocity, and one hand has an angular velocity different from the other two. 2b: The (2a) solution + another. The second solution is the same as (2a) except that the first two hands have opposite initial positions. 1b: The hour and minute hand start at the same initial position; the second hand, at the opposite position. The second hand moves twice as fast as the minute hand in the frame of reference of the hour hand. Case 1a: maxmin length is sqrt((47 - 14 sqrt(7))/27) ~= 0.607346. The second hand moves three times as fast as the minute hand in the frame of reference of the hour hand. I cant swear by these results: its easy to slip up when cases proliferate. Nor have I proven them. I found it interesting to compute the minimal length for our standard clocks. It turns out to be 0.0025408119679, and occurs at 2:54:34.56169 and 9:05:25.43831. Note: none of the above results apply to digital clocks. -- | Jim Ferry | Center for Simulation | +------------------------------------+ of Advanced Rockets | | http://www.uiuc.edu/ph/www/jferry/ +------------------------+ | jferry@[delete_this]uiuc.edu | University of Illinois | === Subject: Re: What is C[[x]] (generating functions) : algebra structure I have a problem in combinatorics (chapter is generating functions) > where f(x) = sum_{x>=0} a_n x^n is in C[[x]]. What is C[[x]]? I have to show that composition of two f,g in C[[x]] is again in C[[x]] > if the constant term of g is 0. Our professor defined this structure in class but I cant find the > definition. Any help on the definition of this structure is greatly appreciated. Also, what are WEIGHTED generating functions? C[[x]] are simply formal power series, i e you do not care for convergence. Weighted? Just ask him (as you could do for the first Q), may be he says: ïput some factors at the coefficients. Hm ... there is no shame to ask questions in lectures ... === Subject: Re: Best differential equation solving software ? > I like Mathematica. Never had my computer crash. -- Bob Day What is the best software for solving differential > equations? Is there anything better than Matlab -- > that at least wont crash and necessitate a reboot > of my computer when I give it a differential equation > it cant solve? --Bob Day > http://www.bob.day.name === Subject: Re: a baseball odds calculation problem >heres the problem, specifically: there are three divisions in the american league. every season, each one of >these divisions will be clinched by one team on a certain date. this is a >mathematical certainty. past baseball history tells us that dates before september are extremely >rare for teams clinching. in fact, the later days of september, roughly the >15th through the 28th, are where the likelihood of teams clinching is the >greatest. > You need to specify the probability distribution of the date of clinching for each division. For example, assuming the the three divisions are independent and have the same distribution, completely supported (a model of your approximation) by {15, 16,..., 28}, then the probability is sum(k=15..28, p(k)^3), where p(k) is the probability of clinching on the date k. So if you are saying that each division is equally likely to be clinched on each day and is independ of the others, the probability is 14*(1/14)^3 = 1/196 = .0051 (0.51%), approximately. Or if you assume identical triangular distributions, p(k) = min(k-14,29-k)/56, the probability (still assuming independence) is 2/56^3 sum(k=1..7, k^3) = 1/112 = .0089 (0.89%), approximately. Another model would be a discretized normal over the whole season. Perhaps you want to make an empirical estimate of the distribution; you may then do the calculation yourself. -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu === Subject: Re: What is C[[x]] (generating functions) : algebra structure Visiting Assistant Professor at the University of Montana. >I have a problem in combinatorics (chapter is generating functions) >where f(x) = sum_{x>=0} a_n x^n is in C[[x]]. What is C[[x]]? Usually, power series on nonnegative powers of x. Like polynomials with coefficients in C, except they are allowed to have infinitely many nonzero monomials. === Subject: re: Vigier V Topics Some more corrections: The physical radius R = integral dR is much larger compared to physical circumference C as it would be in §at 3D space. If one keeps R fixed ~ h/mc ~ G*m/c^2, the micro-geon of Wheelers Mass without mass Charge without charge Spin without spin shrinks to a point in high resolution Heisenberg microscope scattering probes with r ~ h/p for momentum transfer p like SLAC deep inelastic electron scattering off protons. Should be: The physical radius R = integral dR is much larger compared to physical circumference C as it would be in §at 3D space. If one keeps R fixed ~ e^2/mc^2 ~ G*m/c^2 (Blackett effect), the micro-geon of Wheelers Mass without mass Charge without charge Spin without spin shrinks horizon. We will see later that this is why lepto-quarks look like probes with r ~ h/p for momentum transfer p like SLAC deep inelastic electron scattering off protons. Note that G(rho + 3p/c^2) = Gphro(1 + 3w) is replaced by c^2/zpf in the w = -1 exotic vacuum case that included both dark energy of negative pressure and dark matter of positive pressure controlled by the vacuum polarization macro-quantum coherent local order parameter (0|e+(x)e-(x)|0) completely missed in the Puthoff-Haisch-Rueda approach to these problems for the origin of inertia and the origin of gravity from zero point vacuum quantum §uctuations in the sense proposed by Andrei Sakharov in 1967. Sakharovs metric elasticity is a special case of P.W. Andersons More is different generalized phase rigidity of the macro-quantum ground state coherence from spontaneous symmetry breaking. Imagine a dark matter exotic vacuum core of radius R = e^2/mc^2 c^2/zpfR^3/G* ~ m The Kerr-Newmann micro-geon picture for spatially extended IT hidden variable is that the inner event horizon is of order classical electron radius of ~ 1 fermi with outer event horizon out to ~ 137 fermi = h/mc. The in-between ergosphere is ionized vacuum polarization plasma of virtual electron-positron pairs. /zpf ~ 1/Lp*^2 Lp* ~ Lp^2/3(c/Ho)^1/3 ~ 1 fermi (tHooft-Susskind world hologram) m ~ (e^2/c^2)/zpf^1/2 ~ 1 Mev generic lepto-quark rest ggs field mass from zero point exotic vacuum energy, hadronic mass ~ 1 GEV from Heisenberg kinetic energy of confined lepto-quarks as in QCD lite bag model (Frank Wilczek). G*m^2 ~ e^2 (Blackett effect) G*m^2/hc ~ fine structure constant One must be careful in how to use both special and general relativity when polarization effects in real on mass shell media are considered. In the case of special relativity one should, for example, consult the text books by Arnold Sommerfeld, Panofsky and Phillips, Landau and Lifz. A real on mass shell medium in a sense spontaneously breaks translational symmetry on scales larger than the unit cell of the lattice as described in more detail by P. W. Anderson in his More is different series of papers collected in A Career in Theoretical Physics (World Scientific) When one includes all dynamical degrees of freedom including those inside the unit cell of the lattice, global special relativity is restored to the propagation of light in a real medium in the usual situation where gravity tidal forces are negligible. should be One must be careful in how to use both special and general relativity when polarization effects in real on mass shell media are considered. In the case of special relativity one should, for example, consult the text books by Arnold Sommerfeld, Panofsky and Phillips, Landau and Lifz. A real on mass shell medium in a sense spontaneously breaks translational symmetry on scales smaller than the unit cell of the lattice as described in more detail by P. W. Anderson in his More is different series of papers collected in A Career in Theoretical Physics (World Scientific) When one includes all dynamical degrees of freedom including those inside the unit cell of the lattice, global special relativity is restored to the propagation of light in a real medium in the usual situation where gravity tidal forces are negligible. Some corrections: Ditto for the excess verbal baggage of less by David C. Williams below on the nature of c in E = mc^2. should have been Ditto for the trite excess verbal baggage of less with more by David C. Williams below on the nature of c in E = mc^2. could use Newtons gravity with global special relativity to produce the three classic tests of GR provided one introduced the Wheeler-Feynman-Dirac-Hoyle-Narklikar trick of advanced potentials in addition to retarded potentials. The fact that Puthoff gets those tests as well with his variable dielectric vacuum model is no great achievement either because Einsteins classical geometrodynamics goes beyond those tests, e.g. gravimagnetism and gravity waves and black holes. The professor also got the basic black hole effective potential of the Schwarzschild /zpf = 0 vacuum metric with his advanced potential method. Interesting, but like the stochastic electrodynamics approach to EM zero point energy and the semi-classical attempt by Ed Jaynes not to quantize the EM field, like Puthoffs PV approach et-al these attempts at the very best are fragmentary incomplete and fail to ask and answer many of the really important fundamental questions e.g. What is consciousness physically? What is the universe made from? We are such dreams (BIT) as stuff (IT) is made from. === Subject: Express As Single Fraction How do I do this? Express the following as a single fraction: 4/3ab - 5/6bc and (m^2 + 2)/(m^2 + m) - (m - 2)/m === Subject: Re: mandelbrot iterations surely, and you might be the first. after all, I said that it was quite trivial. seriously, when I was passing though Santa Cruz, I stopped at Otto-Pagans office to pursue this, but he was on his European half of academia. when I suggested that, changing the hardware setting on the machine from double-precision to single, he pooh-poohed it -- the grad student that I found. anyway, as I said, monsieur M. had already done it, or at least he made that inference at Young Hall. correction to what I typed: the mini-Ms do not appear at *every* magnification, since the rounding-errors are tied to the lengths of the registers, which is enough for a few iterations. as your statement implies e.g. the specification is inherently chaotic, as the term of art goes, no matter how variously implimented. > that is the recurrence of mini-bugs or cardioids, > at every level of magnification, is just an artifact > of the §oating-point ops (IEEE-755, -855, I think). > this was (really/partially) confirmed by monsieur M, > when he glroriously begged my (only) technical question > at a talk for a general audience. > Its quite simple to disprove your claim by setting the rounding > method, which you can do in hardware on the Pentium (and many > other processors as well) to all its values and seeing what changes, > or doesnt change, in calculations. FP doubles are good for several > tens of thousand of iterations, down to an area of 10E-10 or so, > before the precision gives out. The cartoids are visible several > orders of magnitude above this. --les ducs de Buffet; vote NONE OF THE BELOW on Trickier Dick Cheneys California Recall/e-Dereg! http://larouchepub.com === Subject: consecutive composite integers Examining a table of factors and primes, I found that for any sequence of consecutive composite numbers there is always one integer that has a prime factor larger than any other prime factor of any of the other integers. Further, this prime is not raised to any power. My question is: Is this true for all consecutive composite sequences and if so, is there a proof? (Heres an example: 1500, 1501, ... 1509. The last integer has the prime factor 503.) === Subject: Re: Did mathematicians know? well, isnt this teensy duality the same, for any proof that one hasnt actually read, all the way through (or to the point where one can see it through, already) ?? > youre saying that the L-wing thing is just an argument > *about* an actual (R-wing) proof? --les ducs de Buffet et Schulz (R&D Chair Assoc.Intl.); vote NONE OF THE BELOW on Trickier Dick Cheneys California Recall/e-Dereg! http://larouchepub.com http://members.tripod.com/~american_almanac/ === Subject: Re: What is C[[x]] (generating functions) : algebra structure NNTP-Posting-User: [DNl+kfRPf6mkEI2haabWc7pkPCvSlQMj] > I have a problem in combinatorics (chapter is generating functions) > where f(x) = sum_{x>=0} a_n x^n is in C[[x]]. What is C[[x]]? > Ring of formal power series. Elements are power series in x with coefficients in C, not necessarily convergent. > I have to show that composition of two f,g in C[[x]] is again in C[[x]] > if the constant term of g is 0. Our professor defined this structure in class but I cant find the > definition. Any help on the definition of this structure is greatly appreciated. > Also, what are WEIGHTED generating functions? Michael A. Van Opstall Padelford C-113 opstall@math.washington.edu http://www.math.washington.edu/~opstall/ === Subject: Re: The Bible Code its not even wrong. its just simple numbertheory -- skipcodes are that, and they were apparently used by (some?) Torah writers/copyists to ensure accuracy, as with the old CRC in 8-bit communications programs. I read taht they summed the letters on every 70th, or skipped to every 70th, or some thing. the computer can be set to find any message in any ring of an alphabet, and Drosnin et al know this ... or maybe they cant learn it, not because theyre dumb. there was ambiguity in _The Bible Code_ taht he ignored, like with the variant translations and the fact that Old Hebrew has no vowels. repeat, _War and Peace_ or just the 26 letters in any order can be used with the infinite set of co-prime skips, with teh resulting hits being further massaged into some m by n array (or what ever). >FAILED. Is this accurate? And does this say something about the New Testament >and the belief in a Christ figure? --les ducs de Buffet et Schulz (R&D Chair Assoc.Intl.); vote NONE OF THE BELOW on Trickier Dick Cheneys California Recall/e-Dereg! http://larouchepub.com http://members.tripod.com/~american_almanac/ === Subject: Re: a baseball odds calculation problem >> heres the problem, specifically: >> there are three divisions in the american league. every season, each >> one of >> these divisions will be clinched by one team on a certain date. this >> is a >> mathematical certainty. >> past baseball history tells us that dates before september are extremely >> rare for teams clinching. in fact, the later days of september, >> roughly the >> 15th through the 28th, are where the likelihood of teams clinching is >> the >> greatest. > > You need to specify the probability distribution of the date of > clinching for each division. For example, assuming the the three > divisions are independent and have the same distribution, completely > supported (a model of your approximation) by {15, 16,..., 28}, then > the probability is sum(k=15..28, p(k)^3), where p(k) is the > probability of clinching on the date k. So if you are saying that > each division is equally likely to be clinched on each day and is > independ of the others, the probability is 14*(1/14)^3 = 1/196 = > .0051 (0.51%), approximately. Or if you assume identical triangular > distributions, p(k) = min(k-14,29-k)/56, the probability (still > assuming independence) is 2/56^3 sum(k=1..7, k^3) = 1/112 = .0089 > (0.89%), approximately. Another model would be a discretized normal over the whole season. > Perhaps you want to make an empirical estimate of the distribution; > you may then do the calculation yourself. > And as approximation of the discretized normal, suppose the date less 15 has binomial distribution with parameters 13 and 1/2. Then the probability is approximately the much larger 2.76%, by my calculations -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu === Subject: Re: Re fermat by Tomas what is found at your site is known as Sophistry. the fact that Fernats Last Theorem is negative is not problem, just as proofs (of neg or pos statements) by contradiction are good. of course, three per cent is rational by definition. Negative statements about numbers are unverifiable. Take the definition of an irrational as a number that is not rational, where being rational means having periodic decimal expansion. Suppose someone wants to verify that %3 is irrational. Since the indirect proof is out he computes its decimal expansion to the trillionth digit and, satisfied that no evidence of periodicity looms on the horizon, declares that %3 is irrational. A mischievous fellow quickly forms a periodic decimal whose first period is that first trillion > http://www.users.bigpond.com/pidro/home.htm > --les ducs de Buffet et Schulz (R&D Chair Assoc.Intl.); vote NONE OF THE BELOW on Trickier Dick Cheneys California Recall/e-Dereg! http://larouchepub.com http://members.tripod.com/~american_almanac/ === Subject: Re: function >>can anyone show me an example of a function (R=real numbers) f:R-->R >such that for any a,b,c in R, there exists an x in R such that aand f(x)=c? >>Not so that its continuous for the whole R, but it can be continuous >>within the open interval ]a, b[: >>f[x_]:=Tan[Pi/(b-a)*X-Pi*(a+b)/(2*(b-a))] >>-oo when x=a, oo when x=b and gets all values when between them. Another example: g[x_]:=Pi/(b-a)*x-Pi*(a+b)/(2*(b-a)) >f[x_]:=Cos[g[x]]/g[x] This one is continous everywhere except at x=(a+b)/2, and gets all >values between a and b. You are not reading the question! It says for all a,b,c, so a and b are not constants. And the question does not say anything about continuity. I think there was one correct solution posted. Here is another possibility. I will define f:R -> [0,1], which is good enough, because there are bijections from [0,1] to R. Let x in R and look at the decimal expansion of x. Choose m maximal such that x = n . a_1 a_2 ... a_m a_1 a_2 ... a_m ...; that is, the first m digits after the decimal point repeat. If there is no maximal m with this property, then define f(x) = 0. Otherwise, let x = n . a_1 a_2 ... a_m a_1 a_2 ... a_m b_1 b_2 b_3 ... and define f(x) = 0.b_1 b_3 b_5 ... (The point of taking only the odd b_i is to enable me to select the even b_i so as to prevent me accidentally getting a larger value of m in x.) Derek Holt. === Subject: Re: Express As Single Fraction > How do I do this? Express the following as a single fraction: 4/3ab - 5/6bc and (m^2 + 2)/(m^2 + m) - (m - 2)/m (1) find a common denominator. The least common denominator is a good one to use. (2) convert each fraction to an equivalent fraction, all having the same (common) denominator found in step 1. (3) Add (or subtract) numerators and put the result over the common denominator from step 1. (4) Reduce the result to lowest terms. Note that if you have used the least common denominator, the fraction may already be in lowest terms. === Subject: Re: a baseball odds calculation problem >heres the problem, specifically: there are three divisions in the american league. every season, each one of >these divisions will be clinched by one team on a certain date. this is a >mathematical certainty. past baseball history tells us that dates before september are extremely >rare for teams clinching. in fact, the later days of september, roughly the >15th through the 28th, are where the likelihood of teams clinching is the >greatest. > You need to specify the probability distribution of the date of > clinching for each division. For example, assuming the the three > divisions are independent and have the same distribution, completely > supported (a model of your approximation) by {15, 16,..., 28}, then the > probability is sum(k=15..28, p(k)^3), where p(k) is the probability of > clinching on the date k. So if you are saying that each division is > equally likely to be clinched on each day and is independ of the others, > the probability is 14*(1/14)^3 = 1/196 = .0051 (0.51%), approximately. > Or if you assume identical triangular distributions, p(k) = > min(k-14,29-k)/56, the probability (still assuming independence) is > 2/56^3 sum(k=1..7, k^3) = 1/112 = .0089 (0.89%), approximately. Another model would be a discretized normal over the whole season. > Perhaps you want to make an empirical estimate of the distribution; you > may then do the calculation yourself. last 27 full baseball seasons. in those seasons, there have been 126 different clinches (4 clinches per year from 1975-1993 excluding 1981 strike year and then 6 and i was able to find 96 of those dates (theyre not very easy to find). i did an estimate based on this heres the distribution of clinches i found: sept 7 1% 8 2% 9 3% 10-11 0% 12 1% 13 0% 14 1% 15 1% 16 0% 17 3% 18 2% 19 2% 20 4% 21 3% 22 2% 23 6% 24 4% 25 7% 26 5% 27 7% 28 7% 29 4% 30 6% oct 1 3% 2 5% 3 6% 4 3% 5 6% 6 1% 7 1% btw, the earliest clinch in baseball history, not just the 27 seasons i looked at, is apparently september 7th. > -- > Stephen J. Herschkorn herschko@rutcor.rutgers.edu > === Subject: Re: function > >can anyone show me an example of a function (R=real numbers) f:R-->R >such that for any a,b,c in R, there exists an x in R such that aand f(x)=c? Let f(x) = 5. > No, let c=6, there is no x such that 5=6. > Doug === Subject: Re: Mass is a quantity of matter > Cut< Im sure that people would stand in line for blocks to get a signed copy!! RJ P As a matter of fact RJ, I have already written a couple, and cant even give ïem away! As long as the gravy train keeps running, nobody wants to rock the gravy boat. === Subject: A polynomial problem all, Im trying to prove the following theorem: Let P be a polynomial with real coefficients such that P(x) >=0 for every real x. Then, there are polynomials R and S such that P(x) = R^2(x) + S^2(x) for every complex x. Its easy to see that the degree of P must be even. If r is a real root of P, then the restriction of P to the reals has an absolute minimum at r and, from the differentiability of P, it follows theres an even number k such that the k-1 first derivatives of P vanish at r and the k_th is positive. Therefore, the k-1 derivatives admits the root r with multiplicity 1, which implies P admits r as a rooth with multiplicity k. So, we see every real root of P, when they exist, must have an even multiplicity (I think we could come to this same conclusion a bit faster, considering only the continuity of polynomial functions). Corollary - If all of the roots of P are real, then, P = Q^2 for some polynomial Q. So, for this particular case the theorem has just been proved. To prove the theorem, for the general case, I tried to use mathematical induction. It didnt work, thats why Im asking for help. What I did is as folows: the previous paragraph, its also enough to cover the case of even-degree polynomials with real coefficients and no real roots. Its well known that every monic trinomial T of the 2nd degree that satisfies such conditions can be written as T(x) = (x-a)^2 + C^2, where a and C<>0 are real. Therefore, for such trinomials the theorem holds trivially. Now, suppose theres a natural k such that the theorem holds for i=1,...k-1 for every 2i-degree polynomial with real coefficients and no real roots. If P is a 2k-degree polynomial with these same properties, then at some (or several) real rs the restriction of P to the reals attains an absolute minimum m>0. This implies that, for every real x, the polynomial P-m is non-negative, has one (or several) real root(s) and (i) P(x) - m = (x-r_1)^p1 *...(x-r^_n)^p_n * Q(x), where r_1, ..r_n are the real roots and the numbers p_1,...p_n are even. In addition, Q is monic, has even degree < 2k, has no real root and is strictly positive on the real line. By the induction assumption, the theorem holds for Q and, in virtue of (i), also holds for P-m. But now, to complete the induction, it remains to prove the theorem is good for P, in other words, it remains to prove that if the theorem holds for some polynomial P then it holds for P+m for every m>0. Thats where I got stuck. Actually, I think I chose a very cumbersome way to prove the theorem, there certainly is a neater proof. Any suggestions are welcome. Amanda. === Subject: Re: Chess/Go/etc: Continuous Game-Boards? > I think, with Chess, what is considered a capture might be ambiguous. Each piece has a value and a circle of control. After a piece moves, it attacks everything in its circle. You must stop movement when you enter a hostile circle. Defenders sum their values. Pawns would get promoted after moving over a line. === Subject: Re: Measure extension proof %-nbwyHlYn=yh4r^* v|!,o}OFN$97k ?cjI1!x?l>5*VZ)c/:of{IPQt >In the material I have read, the terms Borel >Field, Sigma Field, and Sigma Algebra appear to mean the same, >that is a collection of sets closed under complementation and >countable unions. Is this not correct? > No, as I understand the terms Sigma fields and sigma algebras are > indeed the same thing. The Borel field is the smallest sigma field > containing the topology (i.e., all the open sets) of a topological > space. When refering to R^n, the usually unstated topology is the usual > one. If I recall correctly, Mr. Martinez (quoted above by Mr. Herschkorn) is studying out of Chungs _Course_. Chung uses Borel field, in a now old-fashioned way, as a synonym for sigma-field (aka sigma-algebra). The same usage can be found, for instance, in Doobs classic book on stochastic processes. penetrating analysis of increasing families of sigma-fields (aka filtrations). Needless to say they used the term Borel field.] -- A. === Subject: Re: Fixed points Thinking with my toes again. : ) > -----Original Message----- > Conversation: Fixed points === > Subject: Re: Fixed points > >Since you have already been given the (an) answer..... What happens to (0,1) if you pick it up , §ip it over then > set it back >down? Er, doesnt that leave 1/2 fixed? -- > Stephen J. Herschkorn > herschko@rutcor.rutgers.edu > === Subject: Re: hw help -- continuity > Folks, I have a couple questions. This is homework, so please post a nudge, > not a solution. 1)prove that if f,g continuous, then so are max(f,g) and min(f,g) > After drawing some graphs, this seems pretty obvious for the single > point a0 -- max(f,g) has 2 cases: it equals to f or g. Either is > continuous. However, this question implies continuous on R, not just > at a single point. Any ideas how to approach this? Proving a function continuous at any point proves it continuous at every point. > 2)Let f be a function with the property that every point of > discontinuity (ie the lim (x->a) f(x) exists, but is not equal to > f(x)) is a removeable discontinuity. This implies lim (y->x)f(y) > exists for all x, but f may be discontinuous at some (even infinitely > many) numbers x. Define g(x) = lim (y->x) f(x). Prove g is continuous. --I dont even know where to start with this one. Does you definition of g(x) actually read g(x) = lim (y->x) f(x) or should it be g(x) = lim (y->x) f(y)? In the first case, g(x) = f(x) at all x, so is not continuous at discontinuities of f. In the latter case, does g(x) = lim (y -> x) g(y), for all x? === Subject: Re: Hey, look! It will not as easy to lie in future! > Heres 5 peoples posts to verify that mind reading technology is already > here. When you think, you are not silent, a radar can pick up your thoughts JUST LIKE SPEACH and sound them out. Because Im the truman I get it constantly. 100,000 people in townsville australia know for certain that 100% clear > mind reading > is possible, all my neighbours listen to my every thought every day. its the most hideous torture possible to constantly have youre thoughts > played > back to you audibly and be FORCED to answer truthfully every passing > remark, > like I do every time I go out. Your voice box gets a trace stimulus of every thought you think, makes a > small > noise just like speaking, it can be picked up. They can play with the > timing, > they can hear compressed phonetics of whole sentences you are about to > think, and tell you your thought a second before you are aware of it. Herc > I CANNOT LIEEEE or another Jim Carrey, Majestic costarring Laurie Holden > Exhibit A: http://tinyurl.com/fuf8 she looks exactly like Laurie Holden > Exhibit B: http://tinyurl.com/fuf2 government has spied on me so long clearly > in between the release dates of The Truman Show : 1998 : Jim Carrey > Majestic : 2002 : Jim Carrey and Laurie Holden Im from Townsville and YOU ARE the Truman! > http://tinyurl.com/iky5 > I was in Townsville over the weekend, and I heard him. > Very spooky! > http://tinyurl.com/iky8 >phone someone in Townsville, half of you must know someone there, >every day I go out people say THERES THE TRUMAN > Im in Townsville. Were sick of you. > http://tinyurl.com/iky9 > http://tinyurl.com/iky4 > You rule Truman! >Do you know if the truman is living in Townsville? > Ive been hearing stuff, yeah > http://tinyurl.com/p0w3 > I can remember listening to Amazing Randis radio show in the ï60s . > He was aware of the phenomenon. > Its called Subvocalization and apparently, some people can detect the > acoustial or electro/acoustical energy from you saying what youre thinking > under your breath . Thats how I assume its done, but its done remotely with machines, a satelite I think, you cant hear it yourself. Popular Science had a May issue with I was shot by the armys pain beam on the cover, which was a clue to the ïweather satelite story, that use trains of satelites pumping quote 94 GHZ radar beam and pulsed laser array. Somehow perple claiming to to mind reading suffer a stunning loss of > accuracy when isolated acoustically from their victim ( oops..) client. ( > oops!), I mean subject. the Scientific study of this is slim, but Magick is not an acceptable > alternative at this time, when this is at least in the hypothesis stage. > Drifting off topic for sci.math?? > yes, but Im not writing about esp, Im submitting data that requires a simple statistical analysis for 6 months now as to whether it occurs naturally. I need sci.math to stamp my stat analysis so other groups wont dismiss it. Its not unlike a 5 mark question from second year stats. H0 the correlation is evident.... H1 Herc is rambling about nothing : H0 : ... Spend 10 seconds checking each tiny url I gave. Why would a man yell his heart out hes the truman, then get numerous responses like this : >Do you know if the truman is living in Townsville? > Ive been hearing stuff, yeah > http://tinyurl.com/p0w3 > Do you want Duggy to post in sci.math and tell you to check my empiricial data showing who I am? Hes quite conversant and hell tell you just what my inner thoughts sound like. I found his James Cook University email address and hes used it since before I was in Townsville. I put the truman verifying posts there to show that lots of people know the technology is already here is complete form. You can listen to my thoughts today if you come to Townsville. Honestly, media cover up + internet global communications = apathy. Herc Now if you dont mind, I have a dole form to submit and I have to stand in queue for half an hour with 50 people around me all interrogating my thoughts, then Ill thaw my last half loaf of frozen bread for lunch and then return into the mysteries of the internet where the whole morning of EVERYONE in sight knowing who I am never happened, like I have been for 2 years now. and dont set the group header on me === Subject: Re: Numeric one-way hash function >> , >> I need to find an algorithm that can produce a unique non-predictable 12 >> digit (0-9) number for any given 12 digit number. This is to be used to >> create a unique barcode on a ticket that cannot be predicted. It is not >> required that the original seed number be computed from the resulting >> barcode, so some form of one-way hashing function would be acceptable. >> Any help in this problem would be appreciated. Ive seen wiser heads than mine recommend a Ruby-Lackov cipher for > this kind of thing. > First, you need a random function f(i) that takes a 6-digit decimal > as input and produces a 6-digit decimal output. > Start by splitting your original 12-digit decimal number into two > 6-digit parts, A and B. Then perform four steps: > A=A+f(B) mod 1000000 > B=B+f(A) mod 1000000 > A=A+f(B) mod 1000000 > B=B+F(A) mod 1000000 > Concatenate the final A and B to form the 12-digit output. The process > is reversible, so there wont be any duplicates among the output values. > f(i) should be a good randomizing function, such as the cryptographic > hash of the concatenation of a secret key k with the operand i. While you could convert to binary and back, if thats convenient, > its not necessary. If youre going to do this in decimal, you could use > the ASCII decimal digits of A or B as the input to the hash, and take, > say, the first 32 bits of the hashs output, convert it to decimal, and > use the low-order 6 digits as the value of f. [I think Ruby-Lackov can > tolerate a small amount of bias in f. If not, Im sure someone will post > another suggestion.] You would have to be careful in the selection of your hash function. All standard hash functions have 2**n different outputs, and I dont know any hash function that produces 10**6 outputs. An example of a bad hash function would be to take the first 20 bits mod 1000000 of a standard cryptographical hash, because this hash function is extremely biased (some values occur twice as often as the others). When you use a larger number of bits the bias is reduced. Is it possible to quantify the maximum allowable bias in your Luby-Rackoff construction? greetings, Ernst Lippe === Subject: Re: JSH your ship has come in!!!! > message Maybe the only point is that I fear James being overwhelmed by evil. > Hmmm. > I have to ask myself, Why should I care? James may be the > reincarnation > of Gauss, but is it really any skin off my nose if he goes > unrecognized? > Ive been worrying about the guy for months (ever since I realized > that > he > was not, in fact, a crank, but a genius) and defending him on this > newsgroup. > My reward? Laughter and bile from the peanut gallery. And not a > word > from > James himself. A word of advice to Prof. Connes: Dont waste your > time > on > James Harris. If he loves being the solitary genius so much, let > him > fight > his own damn battles. Hes not worth losing sleep over. Well damn it, Im losing sleep at least partly because of your scary > dream!!! Good writing their Jim Ferry, and I have to give you credit for that, > but hey, how many years were you ridiculing me, and now you expect me > to just go, hey, pal? Show me youre serious and wade in and respond to some of these > ant-mathematicians in the current battle. Prove your sincerity, and um, keep posting any interesting dreams you > have, as I found it interesting puzzling over that one. Exactly! This Jim Ferry now thinks you are a genius, but has he > defended > you in your current battle? No! I suspect that he is not *fully* committed to your view of mathematics. > That > is just a feeling, and I suppose I could be wrong. The primary focus is the odd definition error in core. Once thats accepted for what it is, and most importantly FIXED, then > its not about committing to my view of mathematics, but about showing > your commitment to mathematics itself. There is ONE mathematics. I, however, have been trying to find out more about your Object Ring, > but I > feel you are pushing me away. I have spent some effort guiding you along at the Mega Foundation > discussion area. Yes, indeed. But, I expect you remember that I did ask you a question that I was unable > to answer. I was using the notation [a,b,c] to represent an ordered triple of complex > numbers. And I was wondering if the ordered triple [1,2,8] was an element of > the Object Ring. You replied: > Quit being lazy!!! You have the definition, figure it out for yourself!!! > What amazes me is how often people are willing to ask someone else to do > their work for them. > If youre smart enough, answer your own question. > Im curious to see if you can. > Ive given the definition for the object ring, so no excuses. Well, I am ashamed to say I still cannot figure it out. Sounds like a ploy. You have to understand that your record Clive Tooth is rather long and > involves some rather...sleazy behavior...like that attack webpage, and > quite a few negative postings over a period of YEARS. That was just... just... boyish high spirits, James. And you joined in the fun by threatening to sue me for libel!! Ah... happy days... > You dont get the benefit of the doubt from me but have to make the > extra effort yourself, so quit being lazy!!! Oh... James... You have to admit that I did help you out, on the Mega board, with sqrt(i) which you didnt realize was a complex number. Cant you help me out just a little with this one? And I noticed that you said that Jim Ferry is on the team! How can I get on the team if you wont help me when I am struggling? By the way, is anybody else on the team? I think that all the team-members should have well-defined roles for the up-coming battle. > Could you help me please? It sounds to me like you think you have some angle for even more > negativity and Ive given enough time explaining. > Remember mathematics is a continuing process. The object ring is fascinating in and of itself, so you cant expect > me to know all the answers just because Im a discoverer. After all, if it were that easy, then math research would have ended > long ago with the first mathematician explaining it all. Your friend. Clive Time will tell. Yes, as I said, time will tell. Very true. -- Clive Tooth http://www.clivetooth.dk === Subject: Re: Vedctor Calculus Question > Could someone help me to understand how to find the minimum distance > between a surface (say f1(x,y,z)=c1) and a line (f2(x,y,z)=c2. i > believe i should be using gradients. thank you very much! A single equation, such as f2(x,y,z)=c2, can describe in a 3d space a surface, possibly a plane, but not a line. The vector parametric form for a line is g(t) = (u1 + u2*t, v1+v2*t,w1+w2*t), where (ui,v1,w1) is a point on the line and (u2,v2,w2) is a vector parallel to the line. If the surface and line intersect, i.e., the distance between the surface and the line is zero, then f1(u1+u2*t,v1+v2*t.w1+w2*t) = c1 is true for some real value of t. If the surface and line do not intersect and the surface has continuous gradients and no boundary curves, then the gradient of f2 at any extremal point (closest to or furthest from the line) must be perpendicular to (u2,v2,w2). === Subject: Re: Fixed points , > -----Original Message----- > [mailto:madrian@pool-151-197-8-253.phil.east.verizon.net]On Behalf Of > Marc > Conversation: Fixed points === > Subject: Fixed points > Suppose f : (0,1) --> (0,1) is continuous. Does f have to > have a fixed > point? If it was f : [0,1] ---> [0,1] or f : [0,1] ---(0,1), then yes. Any thoughts? Marc Since you have already been given the (an) answer..... What happens to (0,1) if you pick it up , §ip it over then set it back > down? > Cant you come up with an algebraic formula that describes this > operation? > Do you mean something like f(x) = 1-x, on (0,1)? === Subject: Re: Measure extension proof >>Given: >>1.- F is a field (not necessarily Borel) >>2.- u is a measure on F >>3.- G is the minimal Borel Field containing F. >> I really dont see what sense the terminology >> minimal Borel field makes. Maybe you meant >> that G is the minimal sigma-field containing F? >Thats what I meant. In the material I have read, the terms Borel >Field, Sigma Field, and Sigma Algebra appear to mean the same, >that is a collection of sets closed under complementation and >countable unions. Is this not correct? Yes. Some authors, I think mostly probabilists rather than analysts, use Borel field this way. See e.g. K.L. Chung in A Course in Probability Theory. One trouble with this terminology is that it becomes clumsy to talk about the main example of a Borel field, namely the field of Borel sets. I prefer to use sigma-algebra or sigma-field. >>4.- v is a measure on G. >>5.- v and u agree on F. >>6.- v and u are sigma-finite on F. >>It can be proved that v is the unique extension of u from F to G. >>Apparently 6.- is a sufficient but not necessary condition for this >>uniqueness. Can someone please indicate the necessary condition and >> I doubt that there _is_ a simple necessary and sufficient condition. Me too. Here, by the way, is an example to show that without something like 6.- you dont have uniqueness in general. Consider the field F of finite unions of intervals in the reals, and the measure u on F such that u(A) = 0 if A is a finite set and u(A) = infinity otherwise. Of course it is not sigma-finite. The minimal sigma-field containing F is the sigma-field B of Borel sets. There are lots of extensions of u to B, e.g. Hausdorff measure of any dimension d with 0 < d < 1. this >measure extension subject in terms of further development of measure >and probability theory? In other words, is understanding of it >important in terms of understanding new material down the road? Your >help is always appreciated. My personal opinion is that its a rather specialized topic, and its not worth getting too worked up about, say, the most general form of the uniqueness theorem, unless you run into a case where you really need it. 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: non-Hausdorff homeomorphism to R^n is there a non-Hausdorff space locally homeomorphic to R^n? === Subject: Re: Is this newsgroup useless? > But doesnt sci.math.research exist? And isnt it moderated? So what is > all the fuss about? sci.math.research is oriented towards mathematical research at a fairly high level. sci.math isnt (although such topics arise here); its scope is much broader. === Subject: Re: Antidiagonal, Infinity >What I propose is that given any >rational that the value greater than it and less than any other >greater is irrational, There is no such number, as several different people have shown you. In non-standard analysis, there might be, however. > See Alain Roberts book about NSA. Rather than being > irrational, it would be non-standard, though. I have yet to see any standard or non-standard model of the reals in which there is a smallest positive number. In the various non-standard versions, there tend to be rather more numbers between any positive number x and zero, there are all those y such that y/x are ifinitesimal but positive, then all those z such that z/y is infinitesimal but positive, and so on ad infinitum. === Subject: Re: Vedctor Calculus Question >A single equation, such as f2(x,y,z)=c2, can describe in a 3d space >a surface, possibly a plane, but not a line. I could *swear* that {(x,y,z) in R^3 : x^2+y^2=0} was a line, last time I looked. Lee Rudolph === Subject: Re: Measure extension proof >> >>Given: >>1.- F is a field (not necessarily Borel) >>2.- u is a measure on F >>3.- G is the minimal Borel Field containing F. >> >> I really dont see what sense the terminology >> minimal Borel field makes. Maybe you meant >> that G is the minimal sigma-field containing F? Thats what I meant. In the material I have read, the terms Borel >Field, Sigma Field, and Sigma Algebra appear to mean the same, >that is a collection of sets closed under complementation and >countable unions. Is this not correct? If I hadnt read the other replies I would have simply said no, this is not correct - the last two are the same, but a Borel field is a special case of a sigma field (the sigma field generated by the open sets in a topological space). Thats the standard way the terminology is used these days, but I gather there are people who do use the term Borel field the way youve been doing - I didnt know that. >>4.- v is a measure on G. >>5.- v and u agree on F. >>6.- v and u are sigma-finite on F. >>It can be proved that v is the unique extension of u from F to G. >>Apparently 6.- is a sufficient but not necessary condition for this >>uniqueness. Can someone please indicate the necessary condition and >> >> I doubt that there _is_ a simple necessary and sufficient condition. >measure extension subject in terms of further development of measure >and probability theory? Well, it happens a lot that the _existence_ of the extension is used to define measures, by first defining them on a (non-sigma) field... when you do that you need the uniqueness to know that youve defined _a_ measure. >In other words, is understanding of it >important in terms of understanding new material down the road? Other people may have different opinions: If youre just learning measure theory my advice would be to skim through this part as quickly as possible and concentrate on the stuff coming up that gets used in applications of measures, as opposed to constructions of measures - if it turns out you get into something where this is important there will be plenty of time to go back to the details. >Your >help is always appreciated. >> ************************ >> >> David C. Ullrich ************************ David C. Ullrich === Subject: Re: non-Hausdorff homeomorphism to R^n >is there a non-Hausdorff space locally homeomorphic to R^n? Sure thing. Look for a current thread which mentions doubling points in one of the posts (I think by Simeon Stefanov). Not if n=0, though. Lee Rudolph === Subject: Re: Exponentative closure >Can the reals be defined using repeated exponentative closure? >By the exponentative closure F, I define F/x as the set of all the >zeroes of all the polynomial functions with coeffeicients AND >exponents in F. For example, the the algebraics are the exponentative >closure of the integers. Thus, it can be written A=Z/x. Does >C=A/x? If not what does A/x equal? Can C be generated by >repeatedly exponentatively closing the integers a finite number of >times? If so, how many? A countable number of times? An uncountable >number of times? Unless I misunderstand you, F/x is countable if F is countable. So no, a finite or even a countable number of exponentative closures wont do it: the union of countably many countable sets is countable. I dont know about an uncountable number of times. 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: Is this newsgroup useless? > But doesnt sci.math.research exist? And isnt it moderated? So what is > all the fuss about? sci.math.research doesnt allow discussions of elementary topics (homework problems), and other math related things like Latex, whether Maple is better than Mathematica, how easy it is to lose a job by going to the Joint Meetings, whether electronic journals are as good as paper ones, and so on. Bart === Subject: Re: Is this newsgroup useless? >[...] Several people have commented that the filtering systems should >work fine. Sure they do, if you continuously update them. > Youre certainly doing _something_ out loud - hard to say whether >> thinking is the right word, since the problems youre complaining >> about are so easy to fix. Luckily this is not a moderated group, >> so youre free to bounce ideas off us just like JSH is... I havent been complaining at all. Just chatting with the >original complainer. Everyone knows what the really, really, >absolutely true problem is, however, and it is the insufferable >rudeness of several posters. But like you say, these things >are easy to fix. See, that wasnt all that hard. ************************ David C. Ullrich === Subject: Re: Given the radius of convergence for one series, Im trying to find the radius for 2 simliar series >Let f(z) = sum_k a_k*z^k be a formal power series with radius of >convergence 1. Put s_n = a_0 + ... + a_n and t_n = (s_0 + ... + >s_n)/(n+1). >Let g(z) = sum_k s_k*z^k and h(z) = sum_k t_k*z^k. >How would I show that the radius of convergence of h and g are both 1 >as well? (The lim sup method doesnt seem to work). Some hints: Note that for any epsilon > 0, there is C such that |a_k| < C (1+epsilon)^k for all k. What does that say about |s_k|? For the other direction, note that a_n = s_n - s_{n-1}. 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: Open form for the integral of x^x dx by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h973a2W15776; Ive been trying to find a generalized (open) formula for the integral of x^x dx.....does anyone know if its been derived at all? Curious === Subject: real analysis: construct this set ... by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h974dnb19740; Can you construct a set E in [0, 1] s.t. for every open interval I in [0,1] m(I intersect E) > 0 & m(I intersect E^c) > 0 m is lebesgue measure E^c is the complement of E This is so tricky! I was thinking something with the generalized Cantor set but everything Im trying isnt working. Any suggestions? Ideas? === Subject: dissolving Russels Paradox by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h97D3uP19205; I studied an alternative set theory which dissolves Russels Paradox. In this theory, it is possible to get good theorems of ordinal number. More description about the set theory is available on my home page. http://boat.zero.ad.jp/~zbi74583/another02.htm I appreciate any comment about it. -------------------- È@ 1.Axiom of Free Class. È@ A. ma [A]A[A]B[A]x[A]y A ma x &È@B ma y & A=B -> x=y This axiom means that any Atsumari makes only one class. A. el x[A]A[A]B ([A]a a el A <-> a el B) -> A=B This axiom means that an Atsumari is decided by its members. A.F [E]x{E]B ([A}a a el B <-> F(a)) & B ma x This is the Schema of comprehention axioms. Where A,B,C,...are variables for Atsumari, which means collection or set or pile in Japanese. a,b,c,...are variables for Class, [A] means all, [E] means exist, [E]! means exist only one, ma means makes, el means element, -> means then, <-> means equivalent, V means or, & means and, Atsu is an abbreviation of Atsumari ~ means negation, {a,b,c} means the Atsu which has members a,b,c 2.What does Free Class mean? È@ [E]B a el B & B ma b means a el b in ZF (TR) For example if {a,c} ma b & {x,y} ma b are true, then a el {a,c} & {a,c} ma b so [E]B a el B & B ma b it means a el b in ZF. FC ZF {a,c} ma b corresponds a el b, c el b {x,y} ma b x el b, y el b 3. Russels class [A]a a el R <-> ~([E]B a el B & B ma a) The right side of formula means ~(a el a) in ZF. This Atsu R makes Russels Class r, so R ma r. Let a=r then, r el R <-> ~([E]B r el B & B ma r) If r el R is true, considering that R ma r is also true,then [E]B r el B & B ma r is true, then right side of formula is false. This is a contradiction. So, the following formulas are gotten. ~(r el R), [E]B r el B & B ma r. This conclusion means that the diagonal logic was dissolved. So, it is possible to think that 2^aleph(0)=aleph(0).È@ === Subject: a coloring problem by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h97D48o19230; I studied a generalized coloring problem. The ordinary coloring problem is defined as follows. È@ È@È@Place different colors on two vertices which are next each other in a plane graph. È@È@How many colors are necessary and enough? È@ È@È@And new coloring problem is the following. È@ È@È@A definition È@È@D.1. È@È@Place different colors on two vertices which are near each other in a plane graph. È@È@How many colors are necessary and enough? È@ È@È@The term near is defined as follows. È@È@For different two vertices a and b,È@either condition 1. or 2. is filled. È@È@c.1. a is next of b È@È@c.2. There are three paths of length 2 between a and b. È@ È@È@c.1 and c.2 is represented as (1,1) and (2,3) respectively. È@ È@È@If G is a plane graph and È@È@if edges are added between all pairs of vertices which satisfy (2,3)È@in G, È@È@then this graph is written as Near_2,3(G) È@ È@È@New problem is also defined as follows. È@ È@È@Another definition È@È@D.2. È@È@What is chr(Near_2,3(G))? È@È@È@È@ where chr(x) means the chromatic number of x. È@ I proved a theorem as follows. [ TheÈ@È@7 colors theorem. ] È@È@T.7C.È@È @7 colors are necessary and enough. More description about this problem is available on my HP. È@ http://boat.zero.ad.jp/~zbi74583/another02.htm === Subject: Re: The Octic x^8-x^7+29x^2+29 Revisited by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h97D4OF19281; Hello all, By a stroke of luck, I managed to find the missing piece to the puzzle of how to solve the resolvent septic. So here is the complete solution: Given: x^8-x^7+29x^2+29 = 0 Then, x1=(1+(a-b-c-d+e-f-g))/8 x2=(1-(a-b-c-d-e+f+g))/8 x3=(1-(a+b-c+d+e-f+g))/8 x4=(1+(a+b-c+d-e+f-g))/8 x5=(1-(a+b+c-d+e+f-g))/8 x6=(1+(a+b+c-d-e-f+g))/8 x7=(1-(a-b+c+d-e-f-g))/8 x8=(1+(a-b+c+d+e+f+g))/8 where the 7 variables a,b,c,d,e,f,g are the SQUARE ROOTS (positive case) of 4v+1 and the vs are the roots of the septic: 8903+47647v+39672v^2+7192v^3-522v^4-174v^5+v^7 = 0 (eq.3) with the solution: v=2(w^11+w^13+w^16+w^18)-2(w+w^12+w^17+w^28)-(w^2+w^5+w^24+w^ 27)+ (w^3+w^7+w^22+w^26)+(w^4+w^10+w^19+w^25)-(w^8+w^9+w^20+w^21) where w is ANY complex root of unity <>1 such that w^29=1. Note: Though there would be 28 such roots, the properties of these roots ensure that v will ONLY have 7 distinct values. I found the solution of v in Dave Rusins website, and its by P.Montgomery, though the solution wasnt used in the same way I used it. Eq.3 wasnt explicitly mentioned there but when I used the Integer Relations applet for a particular v, eq.3 popped out. It looked familiar and I realized i saw it before while trying to solve the resolvent (eq.2) of the prior post, namely: z^7-7z^6-2763z^5-19523z^4+1946979z^3+34928043z^2+119557031z- 3247^2=0 (eq.2) and by letting z=4v+1, we get the new septic: 8903+47647v+39672v^2+7192v^3-522v^4-174v^5+v^7 = 0 (eq.3) So there it is. Its so nice to have completion. :) By the way, do SOME solvable 12-deg polys need an 11-deg resolvent? --Tito === Subject: Re: Quaternion Extensions by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h97D4L919274; > I have recently extended the Quaternions to larger sets by requiring >> some (new) group elements to commute. In doing so, I found this process >> and its results to be very asthetic. For one, the law of association is >> regained. However, the algebra involved is no longer a division algebra, >> i.e. we may not always follow x = 0 or y = 0 from xy=0 (when x and y are >> certain elements taken from a linear combination of group vectors). >... >> Has this type of thing been done before and are its conclusions of >> interest? >>Its obvious that there exist extensions of the quaternions H, >eg H + H (direct sum), >or algebras of matrices with quaternion elements. >>Youd have to say what properties your extension has >before anyone could say if it is of interest. >>I am neither refering to the ring of quaternion matrices nor to the >group >product... have sent you a copy of this work. Apologies for not replying to your email ... lectures have just begun. >My point was that you seemed surprised to find that >there were algebras extending the quaternions, >and I noted that this wasnt too surprising. So the fact that you have constructed such an algebra >could not be considered interesting in itself -- >any interest would have to lie in the special properties of the algebra. >-- >Timothy Murphy >e-mail: tim /at/ birdsnest.maths.tcd.ie >(all email over 80k dispatched to /dev/null) >tel: +353-86-2336090, +353-1-2842366 >s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland Absolutely no need to apologize for not having replied yet; it is good to know you recieved it, as I now asume to be the case. The group diagram picture I made on page 4 or 5 (I forgot) and its explanation in the text should tell a lot about the groups properties very quickly. By the way, the same process of extension (I call it re§ection) can be used again and again to create more group elements, presumably proceeding to higher dimensions in the process- but Im not quite sure if this is somehow equivilant or to the procedure for extending Clifford Algebras (or, indeed, if my group is perhaps a Clifford algebra in disguise. Admitted, I dont know enough about Clifford algebras at the momement and am currently checking this possibility myself). Note also that many types of groups can be re§ected, but this does not always give rise to a new group. For example, the trivial group {1,-1} does not change after re§ection (the complex trivial group {1,-1, i, -i} does). Creighton Dement === Subject: Chessboard knight metric? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h97E9gZ23741; >Take a chessboard (with or without infinetely many squares) let the >distance d((x_1,x_2),(y_1,y_2)) between two squares x and y of the >chessboard be defined as the minimum number of moves a knight takes >to reach y from x. >Is d a metric? Not trying to suggest that this is some new >question that hasnt been asked/answered before. Is there a general >formula for calculating d? More generally, the same question may be >asked for the other pieces (queen, king, knight, biship)? Actually, I >asked myself this question a few years ago. If I remember back to the >notes I took, I had something like (x_1-y_1, x_2-y_2)= (even number, >even number), then d(x,y) = even number. If (x_1-y_1, x_2-y_2)= (even >number, odd number), then d(x,y) = odd number. Finally, if (x_1-y_1, >x_2-y_2)= (odd number, odd number), then d(x,y) = even number. In other words, the same rules for adding natural numbers... C.Dement === Subject: Re: Pointless by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h97E9jb23746; >mathedman scribbled the following >on comp.lang.c: >> Why is this being discussed in comp.lang.c??? >> >Obviously because IT IS FULL OF JEWS!!! Jews are taking over >comp.lang.c because their greedy grubby need to take over everything is >finally seeping into the C language. Next thing you know they will be >trying to rewrite the standard. The entire reason for my low IQ and >inability to succeed in life can be attributed to jews. If it wasnt >for all the damn JEWS in science I wouldnt have to study! They keep >taking all the women too, being all nice and treating them with >respect and making me look like a complete ass. They took all the >jobs, now there is no point even looking for one. All I can do is sit >around all day filled with self pity and loathing for the damn JEWS who >did this to me. I hate my life and its all the fault of the Jew! > YOU are a total idiot. I thought he was being sarcastic. -- >/-- Joona Palaste (palaste@cc.helsinki.fi) ---------------------------| Kingpriest of The Flying Lemon Tree G++ FR FW+ M- #108 D+ ADA N+++| >| http://www.helsinki.[Capi talThorn]/~ palaste W++ B OP+ | >----------------------------------------- Finland rules! ------------/ >It sure is cool having money and chicks. > - Beavis and Butt-head If he was, then he is very bad at it. === Subject: Re: Quaternion Extensions by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h97FaaV30213; > I have recently extended the Quaternions to larger sets by requiring > some (new) group elements to commute. In doing so, I found this process > and its results to be very asthetic. For one, the law of association is > regained. However, the algebra involved is no longer a division algebra, > i.e. we may not always follow x = 0 or y = 0 from xy=0 (when x and y are > certain elements taken from a linear combination of group vectors). >>... > Has this type of thing been done before and are its conclusions of > interest? >>Its obvious that there exist extensions of the quaternions H, >>eg H + H (direct sum), >>or algebras of matrices with quaternion elements. >>Youd have to say what properties your extension has >>before anyone could say if it is of interest. >>I am neither refering to the ring of quaternion matrices nor to the >group >>product... have sent you a copy of this work. >>Apologies for not replying to your email ... lectures have just begun. >>My point was that you seemed surprised to find that >>there were algebras extending the quaternions, >>and I noted that this wasnt too surprising. >>So the fact that you have constructed such an algebra >>could not be considered interesting in itself -- >>any interest would have to lie in the special properties of the algebra. >>-- >>Timothy Murphy >>e-mail: tim /at/ birdsnest.maths.tcd.ie >>(all email over 80k dispatched to /dev/null) >>tel: +353-86-2336090, +353-1-2842366 >>s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland Absolutely no need to apologize for not having replied yet; it is >good to know you recieved it, as I now asume to be the case. The >group diagram picture I made on page 4 or 5 (I forgot) and its >explanation in the text should tell a lot about the groups >properties very quickly. By the way, the same process of extension > (I call it re§ection) can be used again and again to create more > group elements, presumably proceeding to higher dimensions in the >process- but Im not quite sure if this is somehow equivilant or to >the procedure for extending Clifford Algebras (or, indeed, if my >group is perhaps a Clifford algebra in disguise. Admitted, I >dont know enough about Clifford algebras at the momement and am >currently checking this possibility myself). Note also that many >types of groups can be re§ected, but this does not always give rise >to a new group. For example, the trivial group {1,-1} does not change > after re§ection (the complex trivial group {1,-1, i, -i} does). >Creighton Dement by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) with ESMTP id h97G9do32618 id 1A6uP0-0003DA-6d include it with any abuse report 12]
 
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=== Subject: Re: The ... spacetime; answer to critic. by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h97JmJl15824; I know it is a bit stupid, but 1> how do you prove that a discrete topology is metrizable? 2> X is an set of all positive integers and T={{},{1,2,3,4...},{2,3,4...},{3,4...},{4...},...} Why is (X,T) not metrizable? As a topological space spacetime is metrizable, but one does not, > in general, look at any particular metric (in the topological sense) > on it. >> Why Severian thinks that this Hausdorf topology on Space-Time > is adequate to its physical sence and geometrical structure? >> I care nothing for physical sence (sic). All I was doing was pointing > out that the term metric was being used in two different ways > (like many words in mathematics). >> Do you mean as two ways::::::::::::::::::::::::::::::::::::::::::::::::::::::: > ### (pseudo-)Riemann metric -- quadratic form > and > ### metric (distance) > ? >> Yes, theyre different, but the first is applicable only to manifolds. >> Yes, I know that, but the original poster was confused by these two > distinct concepts having the same word. > Of course the two concepts are related this way. So? The > original poster was talking about one version of metric > in terms of the definition of the other version - thats going > to cause confusion, regardless of the fact that the two > are related, so pointing out that they are two different > things seems like a good idea. > >The original poster. In the original question. Where he said something >about metrics that applied to one version, then asked about what >he said in re an instance of the other version. >...And note how these concepts are related and how it can be applied to S.T. >>seems also a good idea... > More precisely: >> The standard topology of Euclidean space is induced by metric. >> You are failing to specify which usage of the term metric you are using > here. >>Directly, I used the metric (distance). >>But the Euclidean distance is a partial case of Riemann distance, >>so not important which I use distance or quadratic form. > The standard topology of Minkowski space isnt. >> Minkowski space is metrizable. >>mmel! ### Not by Minkowski metric ! ! ! He didnt say it _was_ metrizable by the Minkowski metric. >He said it was metrizable. It is. You said The standard topology of Euclidean space is induced >by metric. The standard topology of Minkowski space isnt. >We assumed that induced by metric meant induced by >some metric. What _did_ you mean by induced by metric??? If >induced by metric means induced by Minkowski >metric then the standard topology on euclidean >space is _not_ induced by metric... >You implicitly defined on it >>the standart (Hausdorf) topology of finite-dimension linear space >>and said that its metrizable, as all finite-dimension linear spaces! >>Original question was NOT ABOUT SUCH TOPOLOGY, >>but IS Minkowski A METRIC on the Space-Time? . >>The answer is: >>___________________________________________________________ ________ >>[ Minkowski metric on the Space Time ( dx0^2 -dx1^2 -dx2^2 -dx3^2 ) >>[ not leads to any metric (distance function) in classical sence. Thats correct. Several people have explained this already. Nobody has >disagreed with this. >~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ > The standard topology of Euclidean space is > the only (non-trivial) topology invariant to motions (rotation, mirror, shift). >> Is it? Proof? >>Sorry, in general it isnt. I mistaken. >>Its true that any non-one-point and non-discrete invariant topology >>is identical to standard IN ANY BOUNDED DOMAIN, Say _exactly_ what this means (and give a hint of the proof.) Seriously, I cant figure out what it means. Because I cant >see what it means for a topology on a BOUNDED DOMAIN >to be invariant under shifts, for one thing. Nor under rotations, >unless it happens that the domain is invariant under rotations. >but at infinity exist some different cases... >>Except standard topology, it may be some COMPACTIFIED topologies. >>One of them is like standard but accepts only bounded closed subsets >>(or, the same, only open subsets which are neighbourhoods of infinity). >>Is there another cases, I dont know yet. >>I would ignore this fact a long time... > The standard topology of Minkowski space is invariant but not unique. >> In Euclidean case, all smooth maps from Rê to the Space are CONTINIOUS > and may represent a point trajectory in Newtonian mechanics. > In the Space-Time, all smooth maps Rê->M are continious (in std. top.), > but not all are physically allowed. >> physically allowed! >>This mean that the 4-speed is not space-like: >>( d x / d tau )^2 >= 0 >>and that we have a correct time sign: >>( d x0 / d tau ) >= 0 > What bull. >>*** Anecdote: >>enter expression: cos ( pi / 2 ) >>Syntax error! >>enter expression: 1 * 0 >>Syntax error! >>enter expression: 2 + 2 >>Syntax error! >>enter expression: .8f.9b.90 .99.8c.89.8c >>.8f.9b.90 is not an argument! >>-- >>qq~~~~>/ / > /_/ / >> ____/ > === Subject: Re: Chessboard knight metric? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h97JmW515882; >>Take a chessboard (with or without infinetely many squares) let the >>distance d((x_1,x_2),(y_1,y_2)) between two squares x and y of the >>chessboard be defined as the minimum number of moves a knight takes >>to reach y from x. >>Is d a metric? Not trying to suggest that this is some new >>question that hasnt been asked/answered before. Is there a general >>formula for calculating d? More generally, the same question may be >>asked for the other pieces (queen, king, knight, biship)? Actually, I >>asked myself this question a few years ago. If I remember back to the >>notes I took, I had something like (x_1-y_1, x_2-y_2)= (even number, >>even number), then d(x,y) = even number. If (x_1-y_1, x_2-y_2)= (even >>number, odd number), then d(x,y) = odd number. Finally, if (x_1-y_1, >>x_2-y_2)= (odd number, odd number), then d(x,y) = even number. In other words, the same rules for adding natural numbers... C.Dement >Sorry for not quite completing the question above (even though, >perhaps, obvious): >Is d a metric over the product space of whole/natural numbers >(corresponding to two different cases of an infinite chess board) >or the restriction of the product space of natural numbers to 64 elements? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h97JmTA15867; A grammar would be S -> BaB B -> BB B -> aBb B -> bBa B -> lambda I doubt there is a way to prove it, or find a base case === Subject: Does Goldbach imply Reimann by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h97KV2u19406; It is possible prove the Ternary Goldbach Conjecture (TGC) and the Twin Prime Conjecture (TPC) are true, if the Generalized Riemann Hypothesis (GRH) is true. See http://www.ams.org/era/1997-03-15/S1079-6762-97-00031-0/S1079 -6762-97-00031- 0.pdf for the proof of GRH-->TGC. Is there a similar paper for the converse? If the TGC or TPC is true then, the GRH or (RH) is true. My question is can either of the following be proved? TGC-->GRH or TPC-->GRH John Washburn === Subject: Re: Class of computable functions >Suppose I want a ïlarge computably enumerable collection of functions >f_i : N --> N with the following properties: >1. Each primitive recursive function is included. >2. Each f_i is total by construction. >3. Given i and n in N, there is a computable function time: N x N --N which tells me that the value of f_i(n) will take at most time(i,n) >to compute by a turing machine or equivalent. >[ïTake as long as you want, but PLEASE tell me when you will be >done!] >4. The function time is computable in ïAckermann + constant time, >or at least its behavior is boundedly nasty in some sense. :) There was a paper 40 years ago somewhat along these lines. If you want to be able to predict how long a computation will take, then you have severely restricted the class of functions. Here is the reference: Robert W. Ritchie Classes of predictably computable functions Transactions of the American Mathematical Society, vol. 106 (1963), pp. 139-173 It might help. --Herb Enderton === Subject: Real world applications for Gauss-Jordan Gauss-Jordan elimination. I know text books are constantly using one-way traffic §ow analysis and such, and I really enjoyed this subject. However, I am taking a technical writing class and wanted to do a paper on using gauss-jordan elimination to solve real world problems....The problem is I cant find any resources for my research. Any help would be greatly appreciated. Keith === Subject: Re: division by zero...?? Im afraid that I dont understand what you mean You should only be afraid when you *do* understand what Tapio means. Exactly! :-) Tapio > -- > Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) > Waht is really scary is I am sure that hes saying Look guys, The question doesnt make sense! Zero is really the same thing as a decimal point. Since the decimal point is a marker, how do you divide by a marker? Why even post to math,sci with this kind of stuff? Its time for the bus to Plonk City to leave! Its called the Cerry bus because it crosses the River Schticks!! Bob Pease === Subject: Re: consecutive composite integers >Examining a table of factors and primes, I found that for any sequence >of consecutive composite numbers there is always one integer that has >a prime factor larger than any other prime factor of any of the other >integers. Further, this prime is not raised to any power. My question >is: Is this true for all consecutive composite sequences and if so, is >there a proof? >(Heres an example: 1500, 1501, ... 1509. The last integer has the >prime factor 503.) How does the Further,... apply to this example: 8, 9? 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: Chessboard knight metric? >Take a chessboard (with or without infinetely many squares) let the >distance d((x_1,x_2),(y_1,y_2)) between two squares x and y of the >chessboard be defined as the minimum number of moves a knight takes >to reach y from x. >Is d a metric? With this distance, the triangular equation is obviously true, and that makes it a metric. > Not trying to suggest that this is some new >question that hasnt been asked/answered before. Is there a general >formula for calculating d? More generally, the same question may be >asked for the other pieces (queen, king, knight, biship)? Actually, I >asked myself this question a few years ago. If I remember back to the >notes I took, I had something like (x_1-y_1, x_2-y_2)= (even number, >even number), then d(x,y) = even number. If (x_1-y_1, x_2-y_2)= (even >number, odd number), then d(x,y) = odd number. Finally, if (x_1-y_1, >x_2-y_2)= (odd number, odd number), then d(x,y) = even number. In other >words, the same rules for adding natural numbers... C.Dement > === Subject: Re: factoring to satisfiability > Why should all the clauses in this formula have at most three literals? > You cant necessarily represent a function in N variables using a 3-CNF > formula. The number of functions that can be represented that way is very > small by a counting argument: there are 2^(2^N) functions in N variables, > but only 2^(O(N^3)) possible 3-CNF formulas. What are the odds that > factoring is one of those lucky functions for interesting values of N? You are correct. I cant assume that the formula can be reduced to 3CNF unless I am willing to allow auxiliary variables. One of the references given by David Eppstein: The Propositional Formula Checker HeerHugo, J. F. Groote and J. P. Warners, http://ftp.cwi.nl/CWIreports/SEN/SEN-R9905.pdf states: using auxiliary variables, any formula can be put into (<=3)CNF form in linear time. Your counting argument shows why the number of auxiliary variables tends to grow rapidly. Let N be the minimum number of variables in a formula and M be the number of variables after adding the auxiliary variables. 2^(2^N) = 2^(O(M^3)) Russell - integers are an illusion === Subject: Re: Folk Psychology and Social Convention >The peephole you view your world through is very narrow, Longley. >There are many here who think the following matters are the crux of >building AIs: How do you recognize what you see? >How do you know how to move your arm? >How do you choose which words to say? >How do you understand what they mean? >How does commonsense reasoning work? Stop misquoting Lennon & McCartney! how do you recognize just what youve seen? > i cant tell you but i know its mine > how do you know how to move your own arm? > i get by with a little help from my friends > how do you choose which big words to say? > lend me your ears and ill sing you a song > how do you understand what they mean? > i try not to sing out of key > does commonsense reasoning work? > yes im certain that it happens all the time > oh building AIs with a little help from my friends > Brilliant, Lisa - I hope the originator of the quotes I gave can make use of your added insights. > * * * > weary with foils, i fall into my bed > and sought some rest for limbs with dust attired, > but there began a trial in my head > to obsess, when bodys overtired: > my thoughts then race, unlike their daily pace > when they escape and leave me little grace; > for verbal slings and arrows of bon mots > i am left gawking bereft of ripostes. > look in darkness on what might have been said > had i the wits to speak as those admired: > rejoinders, like galaxies, hang in space > and taunt me in my unrestful repose. > this sort of pesky followup is never very far > when you keep sending crosspostings to here, talk dot bizarre. > . > . thank you thank you ill be here at dogberrys fencing school all weekend Why this thread was x-posted to the other forums, I dont know - but theres clearly a lot more humor on t.dot.b than on c.a.p. - bon mots, rather than non bots [... had i the wits ...]. So, I shall assume Cyrano is the patron saint of t.dot.b. === Subject: Re: Numeric one-way hash function As usual, Im missing some of the intermediate postings, so this is a reply to all of the parties so far. >> , > I need to find an algorithm that can produce a unique non-predictable 12 > digit (0-9) number for any given 12 digit number. This is to be used to > create a unique barcode on a ticket that cannot be predicted. It is not > required that the original seed number be computed from the resulting > barcode, so some form of one-way hashing function would be acceptable. > Any help in this problem would be appreciated. that youve thought out your threat model. You say that the hash doesnt need to be reversible, so what happens if I just make up a 12-digit barcode and print my own ticket? I have a funny feeling that if you present the actual problem here, you might get other suggestions for solving it or have possible problems with your proposed solution identified. For example, your use of the word ticket makes me think authentication. For that you probably really want something like a ticket number and a MAC appended to it. Or something. >> Ive seen wiser heads than mine recommend a Ruby-Lackov cipher for >> this kind of thing. ... [snipped] ... >> ... [I think Ruby-Lackov can >> tolerate a small amount of bias in f. If not, Im sure someone will post >> another suggestion.] Depends what you mean by tolerate. The security proofs for Luby-Rackov certainly dont hold up if theres any knowledge at all of the f() functions. While on that subject, Ill also point out that the proofs require four *independent* f() functions, not one that is reused. You can simulate this with f_n = hash(key, n, data) for n = 1..4. That said, for such a small data input, youd probably be safe ignoring those two nits. Problems are more likely to surface somewhere else. >You would have to be careful in the selection of your hash function. >All standard hash functions have 2**n different outputs, and >I dont know any hash function that produces 10**6 outputs. An example of a bad hash function would be to take the first 20 bits >mod 1000000 of a standard cryptographical hash, because this hash >function is extremely biased (some values occur twice as often as the >others). When you use a larger number of bits the bias is reduced. Yes, theres a regular subject here about how to get unbiased uniform random in [0..N-1] given a pseudorandom bitstream (such as generated by a hash function), avoiding this bias that creeps in when you least expect it. Greg. -- Greg Rose 232B EC8F 44C6 C853 D68F E107 E6BF CD2F 1081 A37C Crypto Mini-FAQ: http://www.schla§y.net/crypto/faq.txt Qualcomm Australia: http://www.qualcomm.com.au === Subject: Re: Real world applications for Gauss-Jordan > Gauss-Jordan elimination. I know text books are constantly using > one-way traffic §ow analysis and such, and I really enjoyed this > subject. I think you are asking when might one use numerical methods to solve large systems of linear equations. One application is in economic planning using Leontief input-output planning. I actually read a science fiction novel last week joked about this application. I have Gauss-Jordan elimination built into my applet for Linear Programming mentioned in my sig. LPs have a wide variety of applications. -- Try http://csf.colorado.edu/pkt/pktauthors/Vienneau.Robert/ Bukharin.html To solve Linear Programs: .../LPSolver.html r c A game: .../Keynes.html v s a Whether strength of body or of mind, or wisdom, or i m p virtue, are found in proportion to the power or wealth e a e of a man is a question fit perhaps to be discussed by n e . slaves in the hearing of their masters, but highly @ r c m unbecoming to reasonable and free men in search of d o the truth. -- Rousseau === Subject: Re: A polynomial problem > all, Im trying to prove the following theorem: Let P be a polynomial with real coefficients such that P(x) >=0 for >every real x. Then, there are polynomials R and S such that P(x) = >R^2(x) + S^2(x) for every complex x. Its easy to see that the degree of P must be even. If r is a real >root of P, then the restriction of P to the reals has an absolute >minimum at r and, from the differentiability of P, it follows theres >an even number k such that the k-1 first derivatives of P vanish at r >and the k_th is positive. Therefore, the k-1 derivatives admits the >root r with multiplicity 1, which implies P admits r as a rooth with >multiplicity k. So, we see every real root of P, when they exist, must >have an even multiplicity (I think we could come to this same >conclusion a bit faster, considering only the continuity of polynomial >functions). Corollary - If all of the roots of P are real, then, P = Q^2 for some >polynomial Q. So, for this particular case the theorem has just been >proved. To prove the theorem, for the general case, I tried to use >mathematical induction. It didnt work, thats why Im asking for >help. What I did is as folows: the previous paragraph, its also enough to cover the case of >even-degree polynomials with real coefficients and no real roots. Its >well known that every monic trinomial T of the 2nd degree that >satisfies such conditions can be written as T(x) = (x-a)^2 + C^2, >where a and C<>0 are real. Therefore, for such trinomials the theorem >holds trivially. Now, suppose theres a natural k such that the >theorem holds for i=1,...k-1 for every 2i-degree polynomial with real >coefficients and no real roots. If P is a 2k-degree polynomial with >these same properties, then at some (or several) real rs the >restriction of P to the reals attains an absolute minimum m>0. This >implies that, for every real x, the polynomial P-m is non-negative, >has one (or several) real root(s) and (i) P(x) - m = (x-r_1)^p1 >*...(x-r^_n)^p_n * Q(x), where r_1, ..r_n are the real roots and the >numbers p_1,...p_n are even. In addition, Q is monic, has even degree >< 2k, has no real root and is strictly positive on the real line. By >the induction assumption, the theorem holds for Q and, in virtue of >(i), also holds for P-m. But now, to complete the induction, it >remains to prove the theorem is good for P, in other words, it remains >to prove that if the theorem holds for some polynomial P then it holds >for P+m for every m>0. Thats where I got stuck. Actually, I think I chose a very cumbersome way to prove the theorem, >there certainly is a neater proof. I believe there is. The real roots all have even order and the complex roots come in conjugate pairs - this means there exists a polynomial F (with complex coefficients) such that P is the product of F and the complex conjugate of F... >Any suggestions are welcome. >Amanda. ************************ David C. Ullrich === Subject: Re: Numeric one-way hash function > , >> I need to find an algorithm that can produce a unique non-predictable 12 >> digit (0-9) number for any given 12 digit number. This is to be used to >> create a unique barcode on a ticket that cannot be predicted. It is not >> required that the original seed number be computed from the resulting >> barcode, so some form of one-way hashing function would be acceptable. >> Any help in this problem would be appreciated. > The simplest way is to encrypt the first number using AES or > 3DES. You will have to convert the result from binary to decimal. > Any extra digits can be thrown away. Change the key variable > regularly and keep the old ones secret. But shouldnt the bar codes be unique? > Your procedure can generate duplicate bar codes. > With high grade ciphers like AES there will be very few duplicates. If you do not know the key variable then the conversion is unpredictable. The simplest way to ensure that the bar codes are unique is to add a prime number to the previous value. Lap round when you get to the top (or a prime number near the top). Start at a weird value. === Subject: Re: Minimal Graph, Four Color Theorem >> Certainly it is possible to formulate the idea of a minimal 5-chromatic >> graph in a way you might find more pleasing. For instance, we could say >> that its equivalent to a graph that is 5-chromatic in such a way that >> for every vertex v there exists a 5-colouring of the graph in which v >> is the sole vertex with the colour blue. >> >The only 5-chroma graph that I am likely to find pleasing is K5! Fair enough, but there are other minimal 5-chromatic graphs besides K5 > even if you arent pleased by them :). For example, glue two regular > pentagonal cones together at the base to get a polyhedron with 7 vertices, > and form the natural adjacency graph on those vertices (of course, we get > a planar graph). Then add one more edge joining the apex vertices. The resulting graph is 5-chromatic, but removing any vertex, no matter > which one, always gives a graph that is 4-chromatic (and also planar, > if Im not mistaken). Personally I find it just as pleasing as K5 :). > The 5-chroma graph is non-planar and therefore, cannot be an mc-e of the FCT. I too find it ïpleasing. Bill J. === Subject: Re: Minimal Graph, Four Color Theorem > No, you are confused because you read badly. Nobody has been arguing that > the four color theorem is false. Nobody! > I will concede that you are not arguing against the FCT if you will concede that I have no qualms about ïproof by contradiction. Bill J. === Subject: limit points of Q are the limit points of Q exactly R? well, I know this is true, and I can prove it. the reason i ask though is because i doubt myself on this. let (xn) be a sequence in the set of all rationals 0<=p<=1. Further, suppose xn is an enumeration of the rationals between 0 and 1. i think that all rationals between 0 and 1 are cluster points of xn, mainly because all are limit points. i could also work it from the definition since a rational lies between any two rationals, but i started doubting myself for some reason. can someone tell me if I am wrong about anything in the above? === Subject: matrix differential equation. hello group. im extremely confused as to how to change the system of odes into a matrix version. lets say you have the the lorenz system. { x[t] = - a*y[t] - a*z[t], y[t] = r*x[t] + y[t] - x[t]*z[t] , z[t] = x[t]*y[t]- b*z[t]} where a, r, b, are constants, how do i represent the above as a matrix differential equations system? and then find the eigenvalues and so on. any help is most appreciated. john === Subject: Re: Yet another question about circular sectors :) Oops! > When I run your numbers backwards, I get L => 1.4142. > Using your first answer where theta = 90 deg (or Pi[/2 you meant] radians) and radius = 1: L = 2 (r * sin(theta/2)) = 1.4142 > And, your second where theta = 180 deg (Pi radians) and radius = 1/sqrt(2) = > 0.7071: L = 2 (r * sin(theta/2)) = 1.4142 > Please see my above post again ; L is the SEMI- chord, not full chord length. === Subject: Re: Vedctor Calculus Question A single equation, such as f2(x,y,z)=c2, can describe in a 3d space >a surface, possibly a plane, but not a line. I could *swear* that {(x,y,z) in R^3 : x^2+y^2=0} was a line, > last time I looked. Lee Rudolph This actually happened. Many years ago a problem similar to this came up in a CalcIII class I was teaching. I mentioned that situations like this are called Degenerate case A girl raised her hand and said But thats what my Father says *I* am!! Bob Pease === Subject: Re: matrix differential equation. > hello group. im extremely confused as to how to change the system of odes into a > matrix version. lets say you have the the lorenz system. { x[t] = - a*y[t] - a*z[t], > y[t] = r*x[t] + y[t] - x[t]*z[t] , > z[t] = x[t]*y[t]- b*z[t]} where a, r, b, are constants, how do i represent the above as a matrix differential equations > system? and then find the eigenvalues and so on. You want to write [x y z]^T as a column vector: [ x ] X = [ y ] [ z ] and then express the system of DEs as follows: X = AX where A is some 3x3 matrix. Since the ith row of the product AX is the product of the ith row of the matrix A with the single column of X, and since the individual equations are sums of (something) times (x,y,z) in various combinations, what you try to do is to populate the matrix A with suitable coefficients that make that happen. For instance, x = -a y - a z can be written as follows: [ x ] x = [0 -a -a] [ y ] [ z ] Note how the coefficient of x (namely, 0) got put into the x position, the coefficient of y (that is, -a), and the coefficient of z (again -a), were placed into the y and z positions, respectively, of the row vector. The equations for y and z are not remarkably different, except for the fact that you cant use constant coefficients, but then this isnt a DE with constant coefficients. Once you have each DE written in this form: x_i = R_i*X with a row vector R_i, then put them together to build the matrix A, mentioned earlier, together. Dale. > any help is most appreciated. john === Subject: Re: The Bible Code > its not even wrong. its just simple numbertheory -- > skipcodes are that, and they were apparently used > by (some?) Torah writers/copyists to ensure accuracy, > as with the old CRC in 8-bit communications programs. > I read taht they summed the letters on every 70th, > or skipped to every 70th, or some thing. > the computer can be set to find any message > in any ring of an alphabet, and Drosnin et al know this ... or > maybe they cant learn it, not because theyre dumb. > there was ambiguity in _The Bible Code_ taht he ignored, > like with the variant translations and the fact that > Old Hebrew has no vowels. repeat, _War and Peace_ or just the 26 letters in any order > can be used with the infinite set of co-prime skips, > with teh resulting hits being further massaged > into some m by n array (or what ever). >FAILED. Is this accurate? And does this say something about the New Testament >and the belief in a Christ figure? --les ducs de Buffet et Schulz (R&D Chair Assoc.Intl.); > vote NONE OF THE BELOW > on Trickier Dick Cheneys California Recall/e-Dereg! > http://larouchepub.com > http://members.tripod.com/~american_almanac/ A defective theory when applied does neither prove or disprove a hypothesis. The Code stuff has been applied successfully to Moby Dick to predict moderns events with Astounding accuracy. (NOT!) rj Pease === Subject: The inertia of a slowly moving mass of matter The inertia of a slowly moving object; body or mass of material substance will increase, or decrease; in proportion to an increase, or decrease in its speed, will it not? Can this be plotted as a component of its motion? Maybe called its momentum, or impetus? If so, what would this component look like for a body moving tranversely across a slope? === Subject: Re: matrix differential equation. ... stuff deleted ... Some stuff, with a badly-spaced equation: > For instance, x = -a y - a z can be written as follows: [ x ] > x = [0 -a -a] [ y ] > [ z ] > I just have to fix the spacing here. Wont be a minute. [ x ] x = [0 -a -a] [ y ] [ z ] Lets see whether that works. === Subject: Number Theory Question I would greatly appreciate some comments upon the correctness of the assertion about the following equation (1) under the given conditions. y = (a^m + b^m)(a^m - b^m) (1) Conditions: (y, a, b ) = 1; m is a non-integer > 0; prime p > 3. Assertion: If y is a p-th power then both a^m + b^m and a^m - b^m separately be p-th power. === Subject: Re: Hey, look! It will not as easy to lie in future! and dont set the group header on me > my mistake, cutting a group is fine, I jumped to the conclusion you set the forward, I often find myself posting just to alt.kibble once day ill go there and itll be all posts from me. Herc === Subject: Re: Hey, look! It will not as easy to lie in future! and dont set the group header on me > my mistake, cutting a group is fine, I jumped to the conclusion you set the forward, > I often find myself posting just to alt.kibble once day ill go there and itll be > all posts from me. > dole office took nearly 2 hours btw, had to murmur a sermon to everyone to keep them quiet, then next month you can all tune into it word for word on everyone loves raymond or nic cages new prison scene. Herc aka the very poor star who gets abused for it === Subject: The Riemann Hypothesis Can the Riemann-Zeta Hypothesis possibly be explained to a math major who has only just begun to study Real Analysis? What is the significance of zero solutions lying on a critical line? And what substantial mathematical theorems are dependent on a hypothesis that has yet to be proven? I feel like a twelve year old leafing through Wiles proof of Fermats Last Theorem. Xevious === Subject: Re: deep holes of leech lattice Is there any algorithm to calculate the deep holes of leech lattice? Yes, its not hard to find one that calculates all holes of any lattice. Could you please tell me the method? I probably dont need all the deep holes Any software can do that? I tried MAGMA but MAGMA gave no result Im sure that on a lattice as complicated as Leech any known > algorithm would be totally intractible. I read SPLAG however I am not clear about it Pity. I wanna have the whole collection of leech lattice deep holes is the best account youre going to find. Finding > the Leech lattice holes was a major piece of research. Although > finding holes is a computable problem, just running some > general-purpose algorithm on Leech would take too much time. === Subject: Re: Factorial/Exponential Identity, Infinity lim n->oo ((sum n)^n - sum(n^n)) / n!^2 = 1 (sum n)^n - sum(n^n) = n! ^2 (sum n)^n = sum(n^n) + n! ^2 ((n^2+n)/2)^n = sum(n^n) + n! ^2 (n^2+n)^n = 2^n ( sum (n^n) + n! ^2 ) n^n (n+1)^n = 2^n ( sum (n^n) + n! ^2 ) The sum of the integers from {1, 2, ..., n ,...} to the nth power, (1+2+...+n+...)^n is greater than the sum of the numbers of the nth power {1^n, 2^n, 3^n, ..., n^n, ...}, (1^n + 2^n + ... + n^n + ... ). Their difference is the square of the factorial of n: 1^2 2^2 ... (n-1)^2 n^2 .... That is to say, lim n->oo ((sum n)^n - sum(n^n)) / n!^2 = 1. Stirlings approximation for n! is as so: lim n->oo ( n! e^n ) / ( n^n sqrt(2pi n) ) = 1 Thusly: lim n->oo ( (sum n)^n - sum(n^n) ) e^2n / ( n^2n 2 pi n) ) = 1 lim n->oo ( (n+1)^n - ( sum(n^n) / n^n ) ) e^2n / (2^n n^n 2pi n ) = 1 lim n->oo ( (n+1)^n - (sum(n^n)/n^n) ) e^2n / ( 2^(n+1) n^(n+1) pi) = 1 As described earlier in this thread, other equations describe n! in the limit. Euler lived in the 1700s. Euler did some amazing work. I wonder if there is a closed for sum(n^n). There are closed forms for sum(n^x), they are defined in terms of Bernoulli polynomials the coefficients of which are generated by recurrence relation. For small values of n, sum(n^n): n = 1, sum (n^n) = 1 n = 2, sum (n^n) = 5 n = 3, sum (n^n) = 36 n = 4, sum (n^n) = 354 n = 5, sum (n^n) = 4425 With (sum n)^n: n = 1, (sum n)^n = 1 n = 2, (sum n)^n = 9 n = 3, (sum n)^n = 216 n = 4, (sum n)^n = 10000 n = 5, (sum n)^n = 759375 With n! ^2: n = 1, n! ^2 = 1 n = 2, n! ^2 = 4 n = 3, n! ^2 = 36 n = 4, n! ^2 = 576 n = 5, n! ^2 = 14400 It appears that as n diverges that (sum n)^n - sum (n^n) ~= (sum n)^n, yet it is of course less than (sum n)^n, and it appears to be n! ^2. I guess I should start calculating ((sum n)^n - sum (n^n)) / n!^2 for small and increasing values of n and see if it holds true that it tends towards unity. I read Richmond and Merlinis paper, as mentioned earlier, about generalizations of Stirling cycle numbers [ n x ] to complex arguments where n-x is an integer, and dont understand it but it looks interesting. Im more interested currently in finding closed form solutions for [ n+1 n-x+1 ], or rather |s(n+1, n-x+1)|, to evaluate (n+1)(n+2)...(n+n), in the strange case of the reformulated Gamma function. Ross === Subject: Re: Grid: Integers = To Sum Of Some Divisors > Take an n-by-n-grid, n>= 3. Place the integers 2 to (n^2 +1) into the grid, > one DISTINCT integer per grid-square, so that: If s(k,j) = a grid-square (ie. an element of an > n-by-n matrix), then > (for all k and j where n >= k >= 3 and n >= j >= 1) > s(k,j) = (any divisor >= 2 of s(k-1,j)) + > (any divisor >= 2 of s(k-2,j)), > and> (for all k and j where n >= k >= 1 and n >= j >= 3) > s(k,j) =(any divisor >= 2 of s(k,j-1)) + > (any divisor >= 2 of s(k,j-2)) >[...] an n=3 example is: > 5 3 8 > 2 6 4 > 7 9 10 > > Is there an n=4 example?? There appear to be 2 basic solutions and of course their transposes: > 5 2 7 9 > 3 12 6 15 > 8 4 10 14 > 11 16 13 17 5 3 8 11 > 2 12 4 16 > 7 6 10 13 > 9 15 14 17 11 2 13 15 > 3 12 6 9 > 14 4 16 8 > 5 10 7 17 11 3 14 5 > 2 12 4 10 > 13 6 16 7 > 15 9 8 17 which a bit tediously could be extended to 5x5 or 6x6 > but not much further because it uses nested for loops, > one level per cell, rather than a recursive approach. > The program takes about 1 second to exhaust the 4x4 case. > -jiw Ahh...So there ARE solutions after all! I was neglecting the likelyhood of bigger integers, such as the 11 and 12, being in the upper-left, perhaps. Leroy Quet === Subject: Re: Minimal Graph, Four Color Theorem > >The only minimal counter-example to the FCT is K5! No, K5 is NOT a counterexample to the Four Color Theorem, because the > 4 color theorem states that any ->planar<- graph can be colored with > at most 4 colors in such a way that no two adjacent vertices share the > same color. > >The conjecture that there exists a 5-chroma graph may be recolored to >4-chroma is false. There is no such conjecture. Let H be any subgraph of G, where G has n vertices and H has n-1 >vertices. Then, the description of H seems to imply that the deletion >of ïany vertex from G will make chi(H)<=4. This is true if G is a minimal counterexample for the 4 Color Theorem. > >But this interpretation is generally false and is valid only for >n=5!!! The triple exclamation points make you look like a raving loon. So > start by removing them. Point taken, thank you. Could you explain why? > Then note that the original argument started by ->assuming<- that the > FCT is ->false<-, from which we deduce that if this is the case, then > among them, there is one with the least number of vertices. Call n the > number of vertices of this HYPOTHETICAL counterexample. Then, by the > definition of n, any graph with fewer than n vertices must be > 4-colorable. In particular, if you took this HYPOTHETICAL example G, > and removed one vertex, then the resulting graph would be 4-colorable. What exactly are you having trouble understanding about the above > argument? Try to answer without using a ->single<- exclamation point. > I understand the argument perfectly. I have given the problem some thought and I have concluded that HYPOTHETICAL G is impossible. No graph meets all three criteria; ie, G is 5-chroma, G is planar, H is 4-chroma. === Subject: Re: Antidiagonal, Infinity > >What I propose is that given any >rational that the value greater than it and less than any other >greater is irrational, There is no such number, as several different people have shown you. In non-standard analysis, there might be, however. > See Alain Roberts book about NSA. Rather than being > irrational, it would be non-standard, though. I have yet to see any standard or non-standard model of the reals in > which there is a smallest positive number. In the various non-standard versions, there tend to be rather more > numbers between any positive number x and zero, there are all those > y such that y/x are ifinitesimal but positive, then all those z such > that z/y is infinitesimal but positive, and so on ad infinitum. Between any two odd integers is an even integer, between any two even integers is an odd integer. The density in their union of either is one half. Here I equate density with measure in the unit interval. I dont care if you ignore gravity, it wont do you much good, Im here only concerned with considering a model where the rationals and irrationals alternate in the reals. If there are more irrationals than rationals and rationals and irrationals are disjoint and distinct, then, where they are each totally ordered, then there necessarily would be irrationals with no rationals between them. Yet, there are not. Im trying to think of a function between the unit intervals reals and irrationals. The claim is that one exists because the rationals map onto the integers and the integers dont map to the reals, thus that the irrationals map onto the reals else the reals would be a union of two sets that dont map onto the reals. Yet, a construction explicitly mapping each element of the irrationals to each element of the reals is not given. Im also still looking for a mapping between R[0,1)^N and R[0,1). I like to think that the rationals and irrationals alternate and that the function f(x)=x+iota maps Q[0,1) onto P(0,1), and f(x)=x-iota maps Q(0,1] to P(0,1). Then again I think the impulse function evaluates to half infinity at zero, and consider the Gamma function on negative integers to have values of various finite multiples of a scalar infinity. Now Im looking at the post about mapping R <-> P. I dont immediately grasp vector space over a field and linearly independent. You have the sequence b being a sequence of reals each linearly independent over Q, and a set C of reals of {b_0, b_1, ...} linearly independent over Q, with the initial sequence element b_0 being a rational. RQ=P, you claim that R injects into P by f(b_n)=b_{n+1} and f(c)=c. Why do you have braces around n+1 instead of parentheses? Then you have F(c)=c, for c in C. I think you mean that c in C is not an element of the sequence b. Then you say to extend that to all of R by linearity over Q. So you claim a function f(r)=p for r in R and p in P to be defined for all reals. Whats r for f(r)=pi? Whats p for f(2)? Why f and F, presumably a shift-key error? http://mathworld.wolfram.com/LinearlyIndependent.html http://mathworld.wolfram.com/VectorSpace.html http://www.wikipedia.org/wiki/Vector_space: A set V is a vector space over a field F, if given an operation vector addition defined in V, denoted v+w for all v, w in v, and an operation scalar multiplication in V, denoted a*v for all v in V and a in F, the following 10 properties hold for all a, b, in F and u, v, and w in V: 1. v+w E V 2. u+(v+w) = (u+v)+w 3. v+0 = v 4. v-v = 0 5. v+w = w+v 6. a*v E V 7. a*(b*v) = (a*b)*v 8. 1*v=v 9. a*(v+w) = a*v + a*w 10. (a+b)*v = a*v + b*v Those each hold for V = R and F = Q. Properties 1 through 5 indicate that V is an abelian group under vector addition. The intersection of all subspaces containing a given set of vectors is called their span; if no vector can be removed without diminishing the span, the set is called linearly independent. So you say each element of the sequence represents a set of vectors or a set of a vector, Im not sure which, and that it is linearly independent over Q because removing that vector from the set of vectors would diminish the span of the intersection of the subspaces of the vector space. Please neaten that up provide a more self-contained explanation. Also explain. While youre at it show a bijection between R^N and R. Some talk here is about the nosntandard treatment of the reals: the hyperreals. One thing to note is that *R, the hyperreals, as a set contains the same elements as R, the reals. Its just a different way to consider them. About the uniform probability distributions over intervals of reals, thats not about making some new definition of what a probability distribution is. Its about applying the characteristics of a probability distribution to an infinite population. We were talking about the probability of an infinite binary seqence having one element being on, the rest off. That probability is expressed as n/2^n, as n diverges to infinity. The probability of any possible sequence is equal to 2^n/2^n, in the limit: one. So anyways out of those n possible sequences with one on bit and the rest off bits, each is equally probable. The probability of each among all possible infinite binary sequences is being1/2^n, the probability of each among all infinite binary sequences with one on bit is 1/n. So a theoretical (read: thought experiment) method to generate an element of N is to once again §ip infinitely many coins. At this point its a crazy, or rather, unconventional thought experiment in that the first coin toss says whether it is oo/2 or greater or less than oo/2. Assume its a long sequence of zeros, then it would be saying about whether the result is greater than or equal to oo/4, oo/8, oo/16, etcetera. Without a method to generate a sample from a uniform distribution over the natural numbers, its still that the probability of selecting any is 1/|N|. Of course thats ludicrous but at the same time it allows us to consider the realm of thought in concern of this issue and to then talk about the probability of selecting a given element of the natural integers assuming a uniform probability distribution over the integers. At least we seem to have some agreement that a uniform probability distribution over an interval of the reals exists, and a simple method to sample an element of an interval of the reals exists. infinitesimals, it talks about 1-infinitesimals, 2-infinitesimals, etcetera, n-infinitesimals, with the oo-infinitesimal being zero. Ross === Subject: Divisibility Of A Derivative By...(Calculus /#-Theory) This is a slight generalization of the theorem in the Prime-Derivative Puzzle thread threadm=b4be2fdf from mid August. Let q and r be any positive integers. Let, for all x where -1 < x < 1, f(x) = (1-x)^((1-x)^(-q)) *(1+x)^((1+x)^(-r)) In ascii-art mode: f(x) = -q -r (1-x) (1+x) (1-x) *(1+x) Then: GCD(q+r ,m) always divides the (m+1)th derivative of f(x) at x = 0. (This derivative, and all derivatives, of f(x), at x =0, are integers.) Leroy Quet === Subject: Re: The problem statement is true, or false? I have been arguing the question, “Two coins were §ipped, and > at least one is a head. What are the chances for two heads?”, in > sci.math, for some time. I argue that the correct answer is 1/2. Unambiguously!!! Our question, as written, has correct answer 1/2. Dr. Holt should > concur. > To answer 1/3, you must assume a slightly different question. When we > are willing to assume a different questions, we can get different > answers. Eldon Moritz Apparently there is something missing from the exposition which was not > included in the (extensive) argument I snipped. This is a standard logic > problem, which yields to the following: 1) There are exactly four possible outcomes for §ipping two coins, HH, > HT, TH, TT. > 2) For ïfair coins, any of the four outcomes is equally likely. > 3) There are 3 outcomes with ïat least one head. > 4) If we choose the universe of discourse as the set of trials with ïat > least one head, the occurrence of two heads will happen 1 out of 3 times. In short, if you run the experiment 100 times, 75 results (should) have > ïat least one head. 25 (should) have two heads. 25/75 = 1/3. Are you discussing a different problem? > There are four equally likely outcomes for the toss. Prior to the statement at least one is a head. After the statement, there are three left, they are no longer equally likely. It is more likely to get the ïheads statement with HH, than with HT, or with TH. That is, assuming that the statement is true. As I showed earlier, assume it false and you assume a different question. Eldon > -- > There are two things you must never attempt to prove: the unprovable -- > and the obvious. > -- > Democracy: The triumph of popularity over principle. === Subject: Re: Exponentative closure >Can the reals be defined using repeated exponentative closure? >By the exponentative closure F, I define F/x as the set of all the >zeroes of all the polynomial functions with coeffeicients AND >exponents in F. For example, the the algebraics are the exponentative >closure of the integers. Thus, it can be written A=Z/x. Does >C=A/x? If not what does A/x equal? Can C be generated by >repeatedly exponentatively closing the integers a finite number of >times? If so, how many? A countable number of times? An uncountable >number of times? Unless I misunderstand you, F/x is countable if F is countable. So no, > a finite or even a countable number of exponentative closures wont > do it: the union of countably many countable sets is countable. I dont > know about an uncountable number of times. Robert Israel israel@math.ubc.ca > Department of Mathematics http://www.math.ubc.ca/~israel > University of British Columbia > Vancouver, BC, Canada V6T 1Z2 A further question along those lines is what is is the set, E, of numbers generated by repeated sums of products of rational powers of rationals? It is a subset of the algebraics, but does it form a field? What can be said about the set of sums and products of elements of E to the power of elements of E? === Subject: combination Suppose we take x numbers out of y numbers in a decreasing sequence. Say, take 2 numbers from {1,2,3} and arrange as {2,1}. What is the number of possible combinations ? === Subject: Re: The Riemann Hypothesis > Can the Riemann-Zeta Hypothesis possibly be explained to a math major > who has only just begun to study Real Analysis? What is the > significance of zero solutions lying on a critical line? And what > substantial mathematical theorems are dependent on a hypothesis that > has yet to be proven? I feel like a twelve year old leafing through Wiles proof of Fermats > Last Theorem. Interestingly, there are no fewer than four recent pop books on the Riemann Hypothesis. (Probably because of its appearance on the Clay Mathematics Institute list of problems.) One sees: Derbyshire, Prime Obsession du Sautoy, The Music of the Primes Sabbagh, The Riemann Hypothesis Edwards, Riemanns Zeta Function Ive read only Derbyshire and du Sautoy, and despite the fact that Derbyshire is a journalist and columnist, he illustrates (not proves) more math than du Sautoy, who teaches math at Oxford (though he is a journalist too). Both of these have a (fairly consistent) account of the history, but Debyshire spends more time in giving a taste of complex analysis, while du Sautoy is broader in his approach to the story parts, especially more recent stuff. Dennis === Subject: Re: Express As Single Fraction > How do I do this? > Express the following as a single fraction: 4/3ab - 5/6bc > (m^2 + 2)/(m^2 + m) - (m - 2)/m > You do it the same way you do it for fractions in arithematic. The general formula is derived thus a/b + r/s = as/bs + br/bs = (as + br)/bs === Subject: Re: combination > Suppose we take x numbers out of y numbers in a decreasing sequence. > Say, take 2 numbers from {1,2,3} and arrange as {2,1}. What is the > number of possible combinations ? y choose x (assuming the y numbers are distinct). -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Z transform, Integrator. , All: Could you please explain these integrators differece? 1. 1/(1-z^(-1)) 2. z^(-1)/(1-z^(-1)) 3. Tz^(-1)/(1-z^(-1)) Boki. === Subject: Re: Joe Uptaught (Was Re: David Ullrich on Identity) A propos, here are cites from Pierre Bourdieus > _Language & Symbolic Power_ (the titles are mine). > Enjoy! Censorship There Oughta Be A Law Cool-Hand Luke > The Social Conditions for the Effectiveness of Ritual Discourse Heretical Discourse > The *Skeptron* The skeptron is passed to the orator before he begins his speech so that he may speak with authority (......). It is an attribute of the person who brings a message, a sacred personage whose mission is to transmit the message of authority. E. Benveniste, in Indo-European Language and Society If, as Austin observes, there are utterances whose role is not only to ïdescribe a state of affairs or state some fact, but also to ïexecute an action, this is because the power of words resides in the fact that they are not pronounced on behalf of the person who is only the ïcarrier of these words: the authorized spokesperson is only able to use words to act on other agents and, through their action, on things themselves, because his speech concentrates within it the accumulated symbolic capital of the group which has delegated him and of which he is the *authorized representative*. The laws of social physics are only apparently independent of the laws of physics, and the power which certain *slogans* have to secure efforts from others without expending efforts themselves--which is the very aim of magical action--is rooted in the capital which the group has accumulated through its effort and whose effective use is subordinated to a whole set of conditions, those which define the *rituals of social magic*. Most of the conditions that have to be fulfilled in order for a performative utterance to succeed come down to the question of the appropriateness of the speaker--or better still, his social function--and of the discourse he utters. A performative utterance is destined to fail each time that it is not pronounced by a person who has the ïpower to pronounce it, or, more generally, each time that the ïparticular persons and circumstances in a given case are not ïappropriate for the invocation of the particular speaker invoked; in short, each time that the speaker does not have the authority to emit the words that he utters. But perhaps the most important thing to remember is that the success of these operations of social magic--comprised by *acts of authority*, or, what amounts to the same, *authorized acts*--is dependent on the combination of a systematic set of interdependent conditions which constitute social rituals. It is clear that all the efforts to find, in the specifically linguistic logic of different forms of argumentation, rhetoric and style, the source of their symbolic efficacy are destined to fail as long as they do not establish the relationship between the properties of discourses, the properties of the person who pro- nounces them and the properties of the institution which authorizes him to pronounce them. The limits (and the interest) of Austins attempt to define performative utterances lie in the fact that he does not exactly do what he thinks he is doing, and this prevents him from following it through to the end. Believing that he was contributing to the philosophy of language, he was in fact working out a theory of a particular class of symbolic expressions, of which the discourse of authority is only the paradigmatic form, and whose specific efficacy stems from the fact that they seem to possess *in themselves* the source of a power which in reality resides in the institutiional conditions of their production and reception. The specificity of the discourse of authority (e.g. a lecture, a sermon, etc.) consists in the fact that it is not enough for it to be *understood* (in certain cases it may even fail to be understood without losing its power), and that it exercises its specific effect only when it is *recognized* as such. This recognition, whether accompanied by understanding or not--is granted, in the manner of something taken for granted, only under certain conditions, namely, those which define legitimate usage; namely, it must be uttered by the person legitimately licensed to do so, the holder of the *skeptron*, known and recognized as being able and enabled to produce this particular class of discourse: a priest, a teacher, a poet, etc.; it must be uttered in a legitimate situation, that is, in front of legitimate receivers (one cannot read a piece of Dadaist poetry at a Cabinet meeting); finally, it must be enunciated according to the leg- itimate forms (syntactic, phonetic, etc.) What one might call the *liturgical* conditions, namely, the set of prescriptions which govern the *form* of the public manifestation of authority, like ceremonial etiquette, the code of gestures and officially prescribed rites, are clearly only an *element*, albeit the most visible one, in a system of conditions of which the most important and indispensable are those which produce the disposition towards recognition in the sense of misrecognition and belief, that is, the delegation of authority which confers its authority on authorized discourse. By focusing exclusively on the formal conditions for the effectiveness of ritual, one overlooks the fact that the ritual conditions that must be fulfilled in order for ritual to function and for the sacrament to be both *valid* and *effective* are never sufficient as long as the conditions which produce the recognition of this ritual are not met: the language of authority never governs without the collaboration of those that it governs, without the help of the social mechanisms capable of producing this complicity, based on misrecognition, which is the basis of all authority. In order to gauge the magnitude in Austins and other strictly formalist analyses of symbolic systems, it suffices to show that the language of authority is only the limiting case of the legitimate language, whose authority does not reside, as the racism of social class would have it, in the set of prosodic and articulatory variations which define distinguished pronunciation, or in the complexity of the syntax or the richness of the vocabulary, in other words in the intrinsic properties of discourse itself, but rather in the social conditions of production and reproduction of the distribution between the classes of knowledge and recognition of the legitimate language. (Pierre Bourdieu, _Language & Symbolic Power_, pp. 109-113) === Subject: Re: The Riemann Hypothesis > Xevious wonders in message > Can the Riemann-Zeta Hypothesis possibly be explained to a math major > who has only just begun to study Real Analysis? What is the > significance of zero solutions lying on a critical line? And what > substantial mathematical theorems are dependent on a hypothesis that > has yet to be proven? Interestingly, there are no fewer than four recent pop books > on the Riemann Hypothesis. (Probably because of its > appearance on the Clay Mathematics Institute list of problems.) One sees: > Derbyshire, Prime Obsession > du Sautoy, The Music of the Primes > Sabbagh, The Riemann Hypothesis > Edwards, Riemanns Zeta Function Another book that deserves mention here is Julian Havils book, Gamma. Its about Eulers constant, gamma, but it gets around to some useful material on the zeta function. I think all of these books are reviewed on the MAA website so you can see another opinion before you jump in. Executive summary: the zeta function can be thougnt of as a generating function for the primes. Anything we learn about the zeros of zeta has immediate implications for the distribution of the primes. Just about any quantitative question about prime numbers, or things that depend on prime numbers like divisors, can be answered more precisely the more we know about the zeros of zeta. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Number Theory Question > I would greatly appreciate some comments upon the correctness of the > assertion about the following equation (1) under the given conditions. y = (a^m + b^m)(a^m - b^m) (1) Conditions: (y, a, b ) = 1; m is a non-integer > 0; prime p > 3. Assertion: If y is a p-th power then both a^m + b^m and a^m - b^m > separately be p-th power. Huh? If m is a non-integer then it seems unlikely to me that a^m + b^m will be an integer. In what sense is sqrt 13 + sqrt 5 a 3rd power? -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: limit points of Q > are the limit points of Q exactly R? well, I know this is true, and I can prove it. the reason i ask though > is because i doubt myself on this. let (xn) be a sequence in the set of all rationals 0<=p<=1. Further, > suppose xn is an enumeration of the rationals between 0 and 1. i think > that all rationals between 0 and 1 are cluster points of xn, mainly > because all are limit points. i could also work it from the definition > since a rational lies between any two rationals, but i started > doubting myself for some reason. can someone tell me if I am wrong about anything in the above? Yes: you are wrong to doubt yourself. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Does Goldbach imply Reimann > It is possible prove the Ternary Goldbach Conjecture (TGC) and the Twin Prime > Conjecture (TPC) are true, if the Generalized Riemann Hypothesis (GRH) is > true. GRH implies twin primes? News to me. > Is there a similar paper for the converse? If the TGC or TPC is true then, > the GRH or (RH) is true. Not to my knowledge. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Fixed points > Suppose f : (0,1) --> (0,1) is continuous. Does f have to have a fixed > point? If it was f : [0,1] ---> [0,1] or f : [0,1] ---> (0,1), then yes. Any thoughts? Marc Doesnt Brouwer have something to say on this? I dont know. Lurch === Subject: Re: A polynomial problem > Im trying to prove the following theorem: Let P be a polynomial with real coefficients such that P(x) >=0 for > every real x. Then, there are polynomials R and S such that P(x) = > R^2(x) + S^2(x) for every complex x. Show P factors over the reals as a product of irreducible quadratics times a product of squares of linear polynomials. Show that an irreducible quadratic is a sum of 2 squares. Show that a product of two sums of two squares is a sum of two squares. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: real analysis: construct this set ... > Can you construct a set E in [0, 1] s.t. for every open interval I in [0,1] > m(I intersect E) > 0 & m(I intersect E^c) > 0 m is lebesgue measure > E^c is the complement of E take away fat disjoint Cantor sets K1 and J1. Now from B2 (K1 U J1) take away fat disjoint Cantor sets K2 and J2. Continue, and set E = ... (If this is a homework problem and you use this hint, be sure to give credit to sci.math.) === Subject: Re: JSH your ship has come in!!!! > Why do you take so much trouble to expose such a reasoner as > Mr. Smith? I answer as a deceased friend of mine used to answer > on like occasions - A mans capacity is no measure of his power > to do mischief. Mr. Smith has untiring energy, which does > something; self-evident honesty of conviction, which does more; > and a long purse, which does most of all. He has made at least > ten publications, full of figures few readers can critize. A great > many people are staggered to this extend, that they imagine there > must be the indefinite something in the mysterious all this. > They are brought to the point of suspicion that the mathematicians > ought not to treat all this with such undisguised contempt, > at least. > -- A Budget of Paradoxes, Vol. 2 p. 129 by Augustus de Morgan > A field is a structured social space, a field of forces, a force field. It contains people who dominate and others who are dominated. Constant, permanent relationships of inequality operate inside this space, which at the same time becomes a space in which the various actors struggle for the transformation or preservation of the field. All the individuals in this universe bring to the competition all the relative power at their disposal. It is this power that defines their position in the field and, as a result, their strategies. Economic competition between networks or newspapers for viewers, readers, or for marketshare, takes place concretely in the form of a contest between journalists. This contest has its own, specific stakes - the scoop, the ïexclusive, professional reputations, and so on. This kind of competition is neither experienced nor thought of as a struggle purely for economic gain, even though it remains subject to pressures deriving from the position the news medium itself occupies within a larger set of economic and symbolic power relations. Today, invisible but objective relations connect people and parties who may never meet - nevertheless, in everything these entities do, they are led, consciously or unconsciously, to take into account the same pressures and effects, because they belong to the same world! On Television, by Pierre Bourdieu === Subject: Re: billard mechanics I posted something like this in sci.physics but people seem to be mainly > with their heads in the stars there.. > I have programmed, ages ago, this billiard mechanics engine, using very > basic physical laws on momentum collision. It works in the sense that it > creates : > 1. good collisions between two balls that each have a specific velocity > vector, and > 2. it never draws balls over each other, > although Im not satisfied with the amount of calculation needed for those > two things (it involves solving a quadratic equation and then trigonometry). > But, as you may be aware, in snooker or pool one starts with the red balls > (and the pink) touching each other. > This kind of ruins this whole nice model since one can no longer use an > O(n^2) algorithm to scan the balls for possible collisions and then work > them, since..well it becomes a mess. When the white ball hits the pack of > red balls, I get a nice 4-dimensional representation of chaos, I think, > which isnt my intention. As you know, when balls touch each other then the > energy of the first ball gets transferred to the last. Most of it, anyway. > In fact I think it calls for an entirely new strategy. Does anybody have any > ideas how I could : > - devise a collision strategy that works on singular collisions as well > as on multiple simultaneous collisions > or > - treat these chained collisions separately (this would involve > sorting the touching balls with respect to the ball(s) that will hit them, > since a computer cant know in what direction the force gets transferred - meaning, this is probably impossible to accomplish. Correct. A triangle of balls is statically indeterminant, that is, they are just like a triangular pyramid of balls in static equilibrium. There are not enough constraints to specify the forces on all the balls. I would probably do best to treat the table of balls as a table of static > balls, with force fields on them, instead of each ball having its vx and vy, > but really, its not that easy.. > If anybody has any ideas or experience with this, Id be very grateful, I have toyed with this type of problem over the years, and I think the best thing is just to never let them touch. With double precision you can leave dither in the 8th place or so, and use your single collision algorithm.