mm-3769 === Subject: Re: powering a conjugacy class in a simple group > Let S be a finite nonabelian simple group, and C an arbitrary > conjugacy class of elements in S, i.e. C={g*x*g^{-1} | g in S} for > some x in S. > Consider the set C^2 = { a*b | a,b in C }. > 1. How one can prove that C lies in C^2? > 2. What is known about the minimal number N such that C^N = S for all > classes C in S? Does it have any nice asymptotics for the series of > alternating groups A_n? The same question for other series of finite > simple nonabelian groups. > Ignat Soroko, > Institute of Mathematics, NASB, > Minsk, Belarus it could be hard to prove 1, seeing as it's false. or have i misread it? === Subject: Math. Models and Methods in Appl. Sci. - TOC alert Originator: israel@math.ubc.ca (Robert Israel) Mathematical Models and Methods in Applied Sciences (M3AS) Articles are available at http://www.worldscinet.com/m3as.html Table-of-Contents: A DISCRETE BOLTZMANN EQUATION BASED ON HEXAGONS BY LAEK S. ANDALLAH and HANS BABOVSKY A PHASE FIELD MODEL WITH THERMAL MEMORY GOVERNED BY THE ENTROPY BALANCE BY ELENA BONETTI, PIERLUIGI COLLI and MICHEL FREMOND ON RIGID AND INFINITESIMAL RIGID DISPLACEMENTS IN THREE-DIMENSIONAL ELASTICITY BY PHILIPPE G. CIARLET and CRISTINEL MARDARE A POSTERIORI ANALYSIS OF A PENALTY METHOD AND APPLICATION TO THE STOKES PROBLEM BY C. BERNARDI, V. GIRAULT and F. HECHT STABILITY OF SHEAR BANDS IN AN ELASTOPLASTIC MODEL FOR GRANULAR FLOW: THE ROLE OF DISCRETENESS BY MICHAEL SHEARER, DAVID G. SCHAEFFER and THOMAS P. WITELSKI DISCRETE COMPACTNESS FOR p AND hp 2D EDGE FINITE ELEMENTS BY DANIELE BOFFI, LESZEK DEMKOWICZ and MARTIN COSTABEL For more information, go to http://www.worldscinet.com/m3as.html === Subject: Re: rearranging a conditionally convergent integral Originator: israel@math.ubc.ca (Robert Israel) >The limit as b ---> infinity of the integral from 0 to b of >sin(x)/x dx is pi/2, and the integral over the half-line of >|sin(x)/x| is infinite. So there should be families >{ B_t : t > 0 } of bounded sets such that B_t is a subset of >B_s if t < s, and the union of the members of this parametrized >family is the half-line (0,infinity), and the limit as >t ---> infinity of the integral over B_t of sin(x)/x dx is >your favorite real number other than pi/2. >Is there any such family that is in some sense simple and >elegant and readily expressible in closed form? Ideally >the dependence of this family on the prescribe value of the >limit would be similarly simple and elegant. -- Mike Hardy Probably the best you're going to do is just imimitate the standard proof of the corresponding result for sequences: Let A_n be the integral of sin(x)/x from n pi to (n+1)pi. Given s, you find a permutation A_n_j with sum A_n_j = s by first throwing in enough positive A_n to make the partial sum > s, then adding enough negative A_n to make the sum < s, then adding just enough positive A_n to make the sum > s again, etc. Then to get your B_t, if k is the floor of t and f is the fractional part, let B_t be the union of the I_n_j for j < k, (where A_n is the integral over I_n) plus the subinterval of I_n_k with the same left endpoint as I_n_k and length f times the length of I_n_k. That's not closed form, but it's a fairly simple description - if someone comes up with something that's literally in closed form I'll be amazed. === Subject: Re: rearranging a conditionally convergent integral Originator: israel@math.ubc.ca (Robert Israel) > The limit as b ---> infinity of the integral from 0 to b of > sin(x)/x dx is pi/2, and the integral over the half-line of > |sin(x)/x| is infinite. So there should be families > { B_t : t > 0 } of bounded sets such that B_t is a subset of > B_s if t < s, and the union of the members of this parametrized > family is the half-line (0,infinity), and the limit as > t ---> infinity of the integral over B_t of sin(x)/x dx is > your favorite real number other than pi/2. > Is there any such family that is in some sense simple and > elegant and readily expressible in closed form? Ideally > the dependence of this family on the prescribe value of the > limit would be similarly simple and elegant. -- Mike Hardy The same as for any conditionally convergent series. Let a be the desired limit. Let B_{t+1} = B_t U {the support of the next positive arch} if the integral over B_t is < a, else let B_{t+1} = B_t U {the support of the next negative arch}. === Subject: Paper published by Geometry and Topology Originator: israel@math.ubc.ca (Robert Israel) The following paper has been published: URL: http://www.maths.warwick.ac.uk/gt/GTVol7/paper22.abs.html Title: Reidemeister-Turaev torsion modulo one of rational homology three-spheres Author(s): Florian Deloup Gwenael Massuyeau Abstract: Given an oriented rational homology 3-sphere M, it is known how to associate to any Spin^c-structure sigma on M two quadratic functions over the linking pairing. One quadratic function is derived from the reduction modulo 1 of the Reidemeister-Turaev torsion of (M,sigma ), while the other one can be defined using the intersection pairing of an appropriate compact oriented 4-manifold with boundary M. In this paper, using surgery presentations of the manifold M, we prove that those two quadratic functions coincide. Our proof relies on the comparison between two distinct combinatorial descriptions of Spin^c-structures on M Turaev's charges vs Chern vectors. Secondary: 57Q10, 57R15 Keywords: Rational homology 3-sphere, Reidemeister torsion, complex spin structure, quadratic function Proposed: Robion Kirby Seconded: Walter Neumann, Cameron Gordon Author(s) address(es): Laboratoire Emile Picard, UMR 5580 CNRS/Univ. Paul Sabatier 118 route de Narbonne, 31062 Toulouse Cedex 04, France and Laboratoire Jean Leray, UMR 6629 CNRS/Univ. de Nantes 2 rue de la Houssiniere, BP 92208, 44322 Nantes Cedex 03, France Email: deloup@picard.ups-tlse.fr, massuyea@math.univ-nantes.fr === Subject: Nilpotent matrices over GF(q) Epigone-thread: drungstronspol Originator: israel@math.ubc.ca (Robert Israel) Is there a simple proof that the number of nilpotent nXn matrices over the finite field GF(q) is q^(n^2-n) ? === Subject: restatement of problem on roots of polynomials Epigone-thread: taysherddex Originator: israel@math.ubc.ca (Robert Israel) In an earlier posting of a problem I had a mistake in one of the results. This is a reposting to correct the mistake. Let Z be a positive integer. Let M = Z/2 if Z is even and (Z+1)/2 if Z is odd. For each integer k, 1<= k <= M+1 let Z = kq(k) + r(k), where q(k) and r(k) are integers and 0<=r(k) < k. Let p be a real number , 0 < p < 1. Define V(k,p) = (k-r(k))q(k)p^q(k) + r(k)(q(k)+1)p^(q(k)+1). whenever 1<=k<= M+1. Define f_k(p) = V(k,p) - V(k-1,p), for 2<= k <= M+1 so f_k(p) is a polynomial in p. The polynomials f_k(p) have some interesting properties: 1. Whenever j > k , f'_j(1) >= f'_k(1), with equality iff f_j = f_k. 2. if f_i != f_j, then there is unique p(i,j) in (0,1) such that f_i(p(i,j)) = f_j(p(i,j)), and moreover p(i,j) is a simple root of f_i - f_j. 3. if j > i and 0 < p < 1 then f_j(p) - f_i(p) > 0 if and only if 0 < p < p(i,j). these 3 properties are not difficult to show. can one show the following? 4. if p(i,j) and p(i,k) are defined, and k > j, then p(i,k) <= p(i,j) ? Are there other families of polynomials that have this decreasing roots property? === Subject: Re: Clarify of Yang Mills and Mass Gap Hypothesis Problem Originator: israel@math.ubc.ca (Robert Israel) >Yang-Mills Field Equation and Mass Gap Hypothesis is one of the >problems which were included in Clay Millennium Problems. >A. Physicists had solved Yang-Mills Field Equations for past 40 years >using different numerical approximations and renormalization methods. >Why do we not consider these solutions as mathematically acceptable >solutions ? Are we looking for an exact and closed form solution for >Yang-Mills Field Equations ? Nobody's resolved a non-trivial quantum field theory in 3+1 dimensions. To get to the essence of the matter, you're looking to solve a system of non-linear PDE's over a non-commutative algebra in which the boundary conditions are singular: [G(x,0), G(y,0)] = 0 = [G_t(x,0), G_t(y,0)] [G(x,0), G_t(y,0)] ~~ delta(x-y). Since the differential equations are non-linear, the solutions are non-linear functionals of the boundary data. But that requires knowing how to form non-linear combinations of singular generalized functions. Classically, to get around that problem, the initial attempt was to force it back into the linear setting of Hilbert space representations via a perturbation theoretic approach. The problem is then framed as that of finding an appropriate Hilbert space operator U(t,s) such that = U(t,s)..U(s,t) where <...> stands for the Hilbert space representation of (...), and ().() composition; with U(t,t) = I; U(t,s).U(s,t) = I U(s,t).U(t,u) = U(s,u) U_s(s,t)|s=t = H(t)/(i h-bar) where H(t) is the Hamiltonian of the system. Perturbation theory compromises on this issue by only considering the limit S = U(+infinity,-infinity), and even then only as a formal series expansion in terms of H(t)/(i h-bar). In contrast to all of this, the axiomatic approach tried to bypass the whole issue of solving non-linear PDE's over singular boundary data, by just simply throwing out the whole starting point of equation-solving and instead focusing on the issue of what a solution ought to look like, clamping down on the range of possibilities by a suitable set of axioms. So, here (for instance), you may start only with the perturbation series for S = U(+infinity, -infinity) and try to (re)define what it means for S to be a semblance of the mythical solution to the system above. The requirement for the spectrum of H(t): spec(H(t)) subset of {0} union [m, infinity), for some m > 0. emerged as a key enabler of the asymptotic approach in the axiomatic setting. === Subject: Re: Clarify of Yang Mills and Mass Gap Hypothesis Problem Originator: israel@math.ubc.ca (Robert Israel) > > Yang-Mills Field Equation and Mass Gap Hypothesis is one of the > problems which were included in Clay Millennium Problems. > > > A. Physicists had solved Yang-Mills Field Equations for past 40 years > using different numerical approximations and renormalization methods. > Why do we not consider these solutions as mathematically acceptable > solutions ? > These are not solutions in the mathematical sense but uncontrolled > approximations. > Are we looking for an exact and closed form solution for > Yang-Mills Field Equations ? > No, but for a rous definition and proof of key properties of > the quantized version of Yang-Mills field theory. At present, it > is not even known how to formulate the problem precisely, lacking > a mathematical definition of a quantum field theory with a given > Lagrangian. > B. Physicists had solved zero mass issue by using Higgs mechanism and > Spontaneous Symmetry Broken, and produced mass that way. Again, why do > NOT we consider these method as mathematically acceptable solution or > proof ? > Again because the rous foundations are missing. All this stuff > uses mathematical language but is not mathematics since it is not > clearly defined. The millenium problems are problems in mathematics, > not in approximate reasoning. > But Higgs has nothing to do with the Millenium problem, which is > about pure YM without Higgs fields. Mass has to be be created by > quantum effects (zero point energy), while the Higgs mechanism > is basically a classical phenomenon. > Not even QED is a mathematical object, although it is the theory > that was able to reproduce experiments (Lamb shift) with an > accuracy of 1 in 10^12, and with less accuracy already in 1948. > But till today no one knows how to formulate the > theory in such a way that the relevant objects whose > approximations are calculated and compared with experiment > are logically well-defined. > Yang-Mills was chosen rather than QED since it is believed to > have properties (asymptotic freedom) that make it more > amenable to a rous treatment than QED (although this might > well be an illusion). Also, it is simpler in some sense > Arnold Neumaier Has any even partial attempt to solve this Clay problem been published? Any progress at all? === Subject: Re: Clarify of Yang Mills and Mass Gap Hypothesis Problem Originator: israel@math.ubc.ca (Robert Israel) > Not even QED is a mathematical object, although it is the theory > that was able to reproduce experiments (Lamb shift) with an > accuracy of 1 in 10^12, and with less accuracy already in 1948. > But till today no one knows how to formulate the > theory in such a way that the relevant objects whose > approximations are calculated and compared with experiment > are logically well-defined. I do not believe that this is correct. It is possible to have a rous QED, it is just that it is not clear (to me, at least, at this time) how one calculates the higher order effects (i.e. the Lamb shift in the one-electron atom & the anomalous magnetic moment of leptons). See http://www.cgoakley.demon.co.uk/qft/qedwip.pdf fields as sums of tensor products of free fields (Haag expansions) then one can obtain matrix elements by inspection knowing the free field (anti-)commutators. These can be compared with time-dependent perturbation theory & if the Haag expansions are based on the usual equations of motion of QED subject to normal ordering, scattering amplitudes can be reproduced that agree with Feynman graphs up to tree level. I might supplement Dr. Neumaier's comments with the observation that the fabled accuracy of renormalised QED is nothing of the sort. All the theory proves is that you can get any answer you like by subtracting infinity from infinity. === Subject: Re: Clarify of Yang Mills and Mass Gap Hypothesis Problem Originator: israel@math.ubc.ca (Robert Israel) > Not even QED is a mathematical object, although it is the theory > that was able to reproduce experiments (Lamb shift) with an > accuracy of 1 in 10^12, and with less accuracy already in 1948. > But till today no one knows how to formulate the > theory in such a way that the relevant objects whose > approximations are calculated and compared with experiment > are logically well-defined. > I do not believe that this is correct. You may wish to look at the discussion in sci.physics.research at > It is possible to have a rous > QED, it is just that it is not clear (to me, at least, at this time) how one > calculates the higher order effects (i.e. the Lamb shift in the one-electron > atom & the anomalous magnetic moment of leptons). But these are precisely the effects for which QED is famous. > See > http://www.cgoakley.demon.co.uk/qft/qedwip.pdf At the end of http://www.cgoakley.demon.co.uk/qft/corres.pdf which shows how many problems your work has, you give witness to the fact that your paper didn't generate any feedback and was never cited. If your paper were the answer to old unsolved questions, the response would have been quite different. > fields as sums of tensor products of free fields (Haag expansions) then one > can obtain matrix elements by inspection knowing the free field > (anti-)commutators. No, one only gets coefficients of an asymptotic expansion of matrix elements, which is likely to be divergent, and no one knows how to make rous sense of it. So what is rously defined are just approximations to something that should exist in a rous theory, and it happens that in QED these approximations are highly accurate. The state is comparable to the knowledge about pi at the time of Archimedes. He had good accuracy and a scheme to improve it, bu there was no theory in which real numbers like pi were well-defined. Your work does not go beyond this, since it only produces a power series expansion without a convergence proof, and is not even able to produce the Lamb shift. > I might supplement Dr. Neumaier's comments with the observation that the > fabled accuracy of renormalised QED is nothing of the sort. All the theory > proves is that you can get any answer you like by subtracting infinity from > infinity. No, one does not get anything one likes but a well-defined answer which agrees with experiment, and one does not need to subtract any infinities if the right approach is used. See the book on QED by Scharf. Arnold Neumaier === Subject: Re: Clarify of Yang Mills and Mass Gap Hypothesis Problem Originator: israel@math.ubc.ca (Robert Israel) >I do not believe that this is correct. It is possible to have a rous >QED, it is just that it is not clear (to me, at least, at this time) how one >calculates the higher order effects (i.e. the Lamb shift in the one-electron >atom & the anomalous magnetic moment of leptons). See >http://www.cgoakley.demon.co.uk/qft/qedwip.pdf >fields as sums of tensor products of free fields (Haag expansions) then one >can obtain matrix elements by inspection knowing the free field >(anti-)commutators. Haag's theorem is a no-go theorem for the Interacton picture (out of which perturbation theory arises). And, the perturbation theoretic expansion of the S matrix is only given on an order-by-order basis; there is no statement concerning the convergence of the expansion, itself. What's well-defined is not the S-matrix, itself, S = S(1,...,1) where S(g1,...,gn) = T[exp(i (L1(g1)+...+Ln(gn)))] Li(g) = integral Li(x) g(x) dx, i=1,..,n Li(x) = monomial corresponding T[] = time-ordering operator, but only its functional derivatives: T[y1,...,yn] = (1/n!) delta^n S(g)/delta g(y1)... delta g(yn), modulo an undetermined point-coincident distribution [simplifying matters by considering only the case of a single monomial]. The removal of infinities was already well-understood since the 1950's under the Bogoliubov/Epstein-Glaser/differential renormalizaiton approach, which (in fact) yields a resolution for T[] (modulo point-coincident distributions at each order) as an implicit solution to the cumulant expansion for the generating functional for T[]: TW[ ln(T[exp(A)]) - A ] = 0 TW[T[exp(A)]] = T[exp(A)] where TW[] is the Wick time-ordering prescription. Your reference didn't do anything more or different than any of this. You still have things defined by a series expansion, and until that's proven to be convergent in some sense, it's nothing but a formal expansion, too. Haag's theorem never got in the way of doing the above, either, on a term-by-term basis (only on the well-definedness of S itself). So, there's nothing new in that respect either when your reference cites Haag's Theorem as not being an impediment. One way to approach the issue to actually bypass the no-go, is to treat S(g) as the fundamental object; in particular S(chi_X g0), defining: for all sets X: chi_X(x) = 1 if x is in X; 0 otherwise; g0 a constant. In particular, if X is a compact spacetime region bounded by two Cauchy surfaces C+, C-: Boundary(X) = C+ - C- [with C+, C- therefore coinciding outside of X], then S(chi_X g0) is gives you a transition matrix between the state spaces corresponding to C- and C+, for an interaction with a coupling constant g0. This better captures the essence of the cut-off idea (arising via Heisenberg at the infrared scale from the fact that all processes and observations have finite space-time windows), in contrast to the more common asymptotic state formalism -- which the mass gap problem is actually pertaining to -- or Bogoliubov's adiabatic switching approach. My main contention has been that with the whole asymptotic state idea superseded by this approach, the relevance of the mass gap problem is no longer clear either.