mm-2999 === Subject: Compact Metric Space Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Im reading a particular paper, Equivalent Martingale Measures and No-Arbitrage by Rogers. In it he makes the claim that, for a fixed n, the set of all nxn perpendicular projection matrices is a compact set. A perpendicular projection matrix can be characterized as symmetric and idempotent which means, of course, that its eigenvalues are 0s or 1s(As many 1s as its rank). Any ideas on how you would prove this space is compact? Additionally he follows up with the claim that Li, where Li is the set of all rank i nxn perpendicular projection matrices, is a closed set. If nobody has any suggestions could you possibly suggest a more appropriate newsgroup? --Scott === Subject: Re: Compact Metric Space Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > Im reading a particular paper, Equivalent Martingale Measures and > No-Arbitrage by Rogers. In it he makes the claim that, for a fixed n, the > set > of all nxn perpendicular projection matrices is a compact set. A > perpendicular projection matrix can be characterized as symmetric and > idempotent which means, of course, that its eigenvalues are 0s or 1s(As many > 1s as its rank). You could use the criterion that this set of endomorphism is closed (as defined by a continuous equation) and bounded (since any projection is norm 1). > Any ideas on how you would prove this space is compact? > Additionally he follows up with the claim that Li, where Li is the set of > all rank i nxn perpendicular projection matrices, is a closed > set. Use the dimension formula for linear maps and the semicontinuity of the map A -> dim ker(A). > If nobody has any suggestions could you possibly suggest a more appropriate > newsgroup? > --Scott J. === Subject: Expressing a polynomial as a sum of squares Originator: bergv@math.uiuc.edu (Maarten Bergvelt) With u and v being real parameters, the following polynomial is never negative: 27-10u^2+27u^4-128uv-10v^2+108u^2v^2+54u^4v^2+27v^4 +54u^2v^4+27u^4v^4 (a) Can this polynomial be written as the sum of squares of polynomials? (b) If so, can you find an explicit representation? (c) Is there a general algorithm that can be used to express a polynomial as the sum of squares of polynomials when such a representation exists? (d) Is the problem any easier for univariate polynomials? === Subject: Re: Expressing a polynomial as a sum of squares Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > With u and v being real parameters, the following polynomial is never > negative: > 27-10u^2+27u^4-128uv-10v^2+108u^2v^2+54u^4v^2+27v^4 +54u^2v^4+27u^4v^4 > (a) Can this polynomial be written as the sum of squares of > polynomials? > (b) If so, can you find an explicit representation? > (c) Is there a general algorithm that can be used to express a > polynomial as the > sum of squares of polynomials when such a representation exists? > (d) Is the problem any easier for univariate polynomials? At least I can answer yes to (d): if f in R[x] has only nonnegative values on R, then it is the sum of two squares. First consider the subset W := { g^2 + h^2 | g,h in R[x] } and observe that W is closed under multiplication. Then assume f =/= 0 with leading coefficient c, and write f/c as a product of monic irreducible factors. (1) each monic irreducible quadratic polynomial is in W (2) each monic irreducible linear factor of f/c occurs with even order in f/c Therefore f/c is in W. Since there is some point a in R where f(a) > 0 one has c > 0 and hence f = c(f/c) is in W. Marc === Subject: Re: Expressing a polynomial as a sum of squares Originator: bergv@math.uiuc.edu (Maarten Bergvelt) >> With u and v being real parameters, the following polynomial is never >> negative: >> 27-10u^2+27u^4-128uv-10v^2+108u^2v^2+54u^4v^2+27v^4 +54u^2v^4+27u^4v^4 >> (a) Can this polynomial be written as the sum of squares of >> polynomials? >> (b) If so, can you find an explicit representation? >> (c) Is there a general algorithm that can be used to express a >> polynomial as the >> sum of squares of polynomials when such a representation exists? Not all non-negative polynomials in two variables can be written as sums of squares of polynomials; Hilbert was, I believe, the first to give a counterexample. It is the case that all non-negative rational functions in k variables can be written as a sum of squares of rational functions. >> (d) Is the problem any easier for univariate polynomials? >At least I can answer yes to (d): >if f in R[x] has only nonnegative values on R, then it is the sum of >two squares. >First consider the subset W := { g^2 + h^2 | g,h in R[x] } and observe >that W is closed under multiplication. >Then assume f =/= 0 with leading coefficient c, and write f/c >as a product of monic irreducible factors. >(1) each monic irreducible quadratic polynomial is in W >(2) each monic irreducible linear factor of f/c occurs with even order in f/c >Therefore f/c is in W. Since there is some point a in R where f(a) > 0 >one has c > 0 and hence f = c(f/c) is in W. >Marc Here is an easier proof. Since a polynomial can be factored over the complex numbers, consider a factorization a*prod(x-b_j) = a*prod(x-c_j-i*d_j). As this is a real polynomial, each b occurs with its conjugate, and as the polynomial is non-negative, each real c occurs to an even power. Also, a > 0. The right-hand product can be written as the product of a polynomial times its conjugate; i.e. C^2 + D^2. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: Expressing a polynomial as a sum of squares Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Yes, there's a simple explicit representation. Define the symmetric matrix Q as [ 81, -14, -53, -14, 0, 0, 0, 0, 0] [ -14, 9, 7/2, 5/2, -1, 0, 0, 0, 0] [ -53, 7/2, 45, 7/2, 1, 0, 0, 0, 0] [ -14, 5/2, 7/2, 9, -1, 0, 0, 0, 0] [ 0, -1, 1, -1, 1, 0, 0, 0, 0] [ 0, 0, 0, 0, 0, 18, -11, -7/2, -7/2] [ 0, 0, 0, 0, 0, -11, 18, -7/2, -7/2] [ 0, 0, 0, 0, 0, -7/2, -7/2, 8, -1] [ 0, 0, 0, 0, 0, -7/2, -7/2, -1, 8] This matrix is positive semidefinite (all eigenvalues are nonnegative). Define the polynomial R(u,v) as: R(u,v) := z' * Q * z where z = [1 , v^2, u*v, u^2, u^2*v^2, v, u, u*v^2, u^2*v]'. This polynomial R(u,v) is a sum of squares (just factorize the PSD matrix Q as Q = R'*R). Then, the given polynomial P(u,v) is also a sum of squares since: P(u,v) = (1/3) * R( sqrt(3)*u , sqrt(3)*v ) How did I obtain this matrix? For the construction, I used the parser/solver SOSTOOLS (check it out at www.mit.edu/~parrilo/sostools), which solves this kind of problems (and much more difficult ones). You'll find more details there, including references to the relevant papers. The methods are based on convex optimization, and in particular, semidefinite programming. Let me know if this is not clear, or if I screwed up the equations... Best, -p > With u and v being real parameters, the following polynomial is never > negative: > 27-10u^2+27u^4-128uv-10v^2+108u^2v^2+54u^4v^2+27v^4 +54u^2v^4+27u^4v^4 > (a) Can this polynomial be written as the sum of squares of > polynomials? > (b) If so, can you find an explicit representation? > (c) Is there a general algorithm that can be used to express a > polynomial as the > sum of squares of polynomials when such a representation exists? === Subject: Re: Expressing a polynomial as a sum of squares Originator: bergv@math.uiuc.edu (Maarten Bergvelt) >With u and v being real parameters, the following polynomial is never >negative: >27-10u^2+27u^4-128uv-10v^2+108u^2v^2+54u^4v^2+27v^4 +54u^2v^4+27u^4v^4 >(a) Can this polynomial be written as the sum of squares of >polynomials? >(b) If so, can you find an explicit representation? >(c) Is there a general algorithm that can be used to express a >polynomial as the > sum of squares of polynomials when such a representation exists? > Not all non-negative polynomials in two variables can > be written as sums of squares of polynomials; Hilbert > was, I believe, the first to give a counterexample. Indeed: .86ber die Darstellung definiter Formen als Summe von Formenquadraten, Math. Ann. 32 (1888), 342.9a350 Reproduced at: Ges. Abh. vol. 2, 154.9a161. Chelsea Publishing Company, 1965 Jose Carlos Santos === Subject: Re: Expressing a polynomial as a sum of squares Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > With u and v being real parameters, the following polynomial is never > negative: > 27-10u^2+27u^4-128uv-10v^2+108u^2v^2+54u^4v^2+27v^4 +54u^2v^4+27u^4v^4 > (a) Can this polynomial be written as the sum of squares of > polynomials? > (b) If so, can you find an explicit representation? > (c) Is there a general algorithm that can be used to express a > polynomial as the > sum of squares of polynomials when such a representation exists? > (d) Is the problem any easier for univariate polynomials? Walter Rudin, Sums of squares of polynomials, American Mathematical Monthly 107, No.9, 813-821 (2000) M. D. Choi, T. Y. Lam, and B. Reznick, Sums of squares of real polynomials, in K-theory and algebraic geometry: connections with quadratic forms and division algebras, Proceedings of Symposia in Pure Mathematics 58, Part 2, 103-126 (1995), American Mathematical Society Jose Carlos Santos === Subject: Re: Expressing a polynomial as a sum of squares Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > 27-10u^2+27u^4-128uv-10v^2+108u^2v^2+54u^4v^2+27v^4 +54u^2v^4+27u^4v^4 > (a) Can this polynomial be written as the sum of squares of > polynomials? > (b) If so, can you find an explicit representation? > (c) Is there a general algorithm that can be used to express a > polynomial as the > sum of squares of polynomials when such a representation exists? > (d) Is the problem any easier for univariate polynomials? There has been interesting work on using conic optimization to determine whether or not a multivariate polynomial can be written as a sum of squares. The software package SOSTOOLS available from http://www.cds.caltech.edu/sostools/ can be used to do the calculations. There are several relevant papers at this web site. In solving these problems there are some difficult numerical issues: It may or may not be possible to round off the solution to the conic optimization problem to an exact rational solution to the sum of squares problem. > (a) Can this polynomial be written as the sum of squares of > polynomials? > (b) If so, can you find an explicit representation? Here's some output from SOSTOOLS. First, we use floating point arithmetic. >>Size: 81 25 >>Alg = 2: xz-corrector, theta = 0.250, beta = 0.500 >>eqs m = 25, order n = 10, dim = 82, blocks = 2 >>nnz(A) = 45 + 0, nnz(ADA) = 625, nnz(L) = 325 >> it : b*y gap delta rate t/tP* t/tD* feas cg cg >> 0 : 1.37E+01 0.000 >> 1 : -1.51E+01 3.77E+00 0.000 0.2745 0.9000 0.9000 1.66 1 1 >> 2 : -7.48E-01 3.07E-01 0.000 0.0814 0.9900 0.9900 1.58 1 1 >> 3 : -1.69E-02 7.37E-03 0.000 0.0240 0.9900 0.9900 1.12 1 1 >> 4 : -1.78E-03 8.17E-04 0.000 0.1110 0.9450 0.9450 0.95 1 1 >> 5 : -7.97E-05 3.43E-05 0.000 0.0419 0.9900 0.9900 0.96 1 1 >> 6 : -1.47E-05 6.19E-06 0.000 0.1806 0.9000 0.9000 0.95 1 2 >> 7 : -7.40E-07 2.30E-07 0.424 0.0372 0.9900 0.9900 1.02 2 2 >> 8 : -8.76E-08 2.78E-08 0.007 0.1209 0.9450 0.9450 1.16 4 4 >> 9 : -1.55E-08 5.25E-09 0.000 0.1885 0.9000 0.9000 1.13 4 4 >> 10 : -1.65E-09 5.84E-10 0.188 0.1112 0.9450 0.9450 1.06 6 6 >> 11 : -1.97E-10 7.09E-11 0.248 0.1215 0.9450 0.9450 1.03 7 9 >>Run into numerical problems. >>iter seconds digits c*x b*y >> 11 0.1 11.1 0.0000000000e+00 -1.9717003602e-10 >>|Ax-b| = 6.7e-10, [Ay-c]_+ = 1.3E-12, |x|= 1.7e+02, |y|= 1.8e-01 >>Max-norms: ||b||=128, ||c|| = 0, >>Cholesky |add|=0, |skip| = 2, ||L.L|| = 4.686. >>Residual norm: 6.6677e-10 >> cpusec: 0.1400 >> iter: 11 >> feasratio: 1.0296 >> pinf: 0 >> dinf: 0 >> numerr: 0 >>Q = >> Columns 1 through 7 >> 27.0000 -0.0000 -16.8611 0.0000 -41.6518 0.0000 -16.8611 >> -0.0000 23.7221 -0.0000 -22.3482 -0.0000 -14.1962 -0.0000 >> -16.8611 -0.0000 27.0000 -0.0000 14.1962 0.0000 8.2380 >> 0.0000 -22.3482 -0.0000 23.7223 0.0000 10.0744 -0.0000 >> -41.6518 -0.0000 14.1962 0.0000 84.9823 -0.0000 14.1964 >> 0.0000 -14.1962 0.0000 10.0744 -0.0000 47.1070 0.0000 >> -16.8611 -0.0000 8.2380 -0.0000 14.1964 0.0000 27.0000 >> -0.0000 10.0746 0.0000 -14.1964 0.0000 -34.7415 0.0000 >> -16.8781 0.0000 3.4465 -0.0000 34.7415 -0.0000 3.4465 >> Columns 8 through 9 >> -0.0000 -16.8781 >> 10.0746 0.0000 >> 0.0000 3.4465 >> -14.1964 -0.0000 >> 0.0000 34.7415 >> -34.7415 -0.0000 >> 0.0000 3.4465 >> 47.1070 0.0000 >> 0.0000 27.0000 >>Z = >>[ 1] >>[ v] >>[ v^2] >>[ u] >>[ u*v] >>[ u*v^2] >>[ u^2] >>[ u^2*v] >>[ u^2*v^2] This says that papprox(u,v)=Z'*Q*Z is a close numerical approximation to the original polynomial. However, when we attempt to get an exact solution with rational coefficients, we get: >>Could not compute a rational SOS! >>Q = >>[ empty sym ] >>Z = >> [] This suggests that this problem is numerically challenging. I'd encourage you to contact the authors of SOSTOOLS- they might be able to solve your problem by finding a solution in rational arithmetic, or show that no such solution exists. > (c) Is there a general algorithm that can be used to express a > polynomial as the sum of squares of polynomials when such a > representation exists? The conic programming approach can in theory do this, although in practice you can run into numerical problems as we did here. Note that being able to find a sum of squares representation does not solve the problem of Is this polynomial always nonnegative?, since a polynomial might be nonnegative but not have a sum of squares representation. However, For univariate polynomials, p(x)>=0 for all x if and only if p(x) can be written as a sum of squares. For bivariate quartic polynomials (like the one given here), it can also be shown that p(u,v)>=0 for all u and v if and only if p(u,v) can be written as a sum of squares. In these particular cases, nonnegativity of the polynomial is equivalent to the polynomial having a sum of squares representation. > (d) Is the problem any easier for univariate polynomials? What do you mean by easier? In theory, for either the univariate or multivariate cases, the conic optimization problems can be solved to any desired accuracy in polynomial time, thus providing a solution to the problem. -- Brian Borchers borchers@nmt.edu Department of Mathematics http://www.nmt.edu/~borchers/ New Mexico Tech Phone: 505-835-5813 Socorro, NM 87801 FAX: 505-835-5366 === Subject: Re: IS quantum mechanics a limit cycle theory?? Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > V(x,y,z) = (1 - x^2 - y^2)^2 z + z^3 + x^2 + y^2 > Note that dV/dz > 0 except on the circle x^2 + y^2 = 1, z = 0, so > the only possible closed orbits are on that circle. On that circle > there are closed orbits in both directions. IMHO this cannot have any physical relevance, because these orbits are highly _unstable_. Am I wrong? Han de Bruijn . === Subject: Adjoint Action of Lie-type Groups Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Does anybody know if the number of orbits on vectors (or 1-spaces) of the finite groups of Lie-type in their adjoint action on the Lie algebra are known? Ideally I'd like to know the conjugacy classes of stabilizers and in particular would like this information for the exceptional groups. J.B. === Subject: Re: IS quantum mechanics a limit cycle theory?? Originator: bergv@math.uiuc.edu (Maarten Bergvelt) the domain of hamiltonian is even dimensional space! But can Morse theory help to my question, namely is dynamic fix if we do not pass a critical value (However I prefer to not change the main subject of my question, a possible relation between limit cycle theory and quantization) > potential > V(x,y,z) = (1 - x^2 - y^2)^2 z + z^3 + x^2 + y^2 > Note that dV/dz > 0 except on the circle x^2 + y^2 = > 1, z = 0, so > the only possible closed orbits are on that circle. > On that circle > there are closed orbits in both directions. > Robert Israel > israel@math.ubc.ca > Department of Mathematics > http://www.math.ubc.ca/~israel > University of British Columbia Vancouver, > BC, Canada === Subject: Re: IS quantum mechanics a limit cycle theory?? Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > the domain of hamiltonian is even dimensional space! > potential > V(x,y,z) = (1 - x^2 - y^2)^2 z + z^3 + x^2 + y^2 > Note that dV/dz > 0 except on the circle x^2 + y^2 = > 1, z = 0, so > the only possible closed orbits are on that circle. > On that circle > there are closed orbits in both directions. I think you may have misunderstood my example. The Hamiltonian system is 6-dimensional (3 components for velocity and 3 for position), Hamiltonian is u^2/2 + v^2/2 + w^2/2 + V(x,y,z) where u, v, w are conjugate to x, y, z respectively. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: This week in the mathematics arXiv (30 Sep - 6 Oct) Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Here are this week's titles in the mathematics arXiv, available at: http://front.math.ucdavis.edu/ http://front.math.ucdavis.edu/submissions This week in the mathematics arXiv may be freely redistributed with attribution and without modification. Titles in the mathematics arXiv (30 Sep - 6 Oct) ------------------------------------------------ AC: Commutative Algebra ----------------------- math.AC/0510030 Margherita Barile: On ideals generated by monomials and one binomial math.AC/0510025 George Kirkup: Minimal Primes Over Permanental Ideals AG: Algebraic Geometry ---------------------- math.AG/0510128 Alastair Craw, Diane Maclagan: Fiber fans and toric quotients math.AG/0510126 Alicia Dickenstein, Eva Maria Feichtner, Bernd Sturmfels: Tropical Discriminants math.AG/0510125 Patrick M. Gilmer: Arf invariants of real algebraic curves math.AG/0510123 St'ephane Druel: D'eformations des courbes rationnelles math.AG/0510110 Sergey Lysenko: Geometric Waldspurger periods hep-th/0408033 Andrey Todorov: Weil-Petersson Volumes of the Moduli Spaces of CY Manifolds math.AG/0510093 Alex Kasman, Kathryn Pedings, Amy Reiszl, Takahiro Shiota: Universality of Rank 6 Plucker Relations and Grassmann Cone Preserving Maps math.AG/0510088 Xuhua He, Jesper Funch Thomsen: Geometry of $B times B$-orbit closures in equivariant embeddings math.AG/0510085 Haibao Duan; Xuezhi Zhao: Appendix to The Chow rings of generalized Grassmannians math.AG/0510063 Matthias Schuett, Jaap Top: Arithmetic of the [19,1,1,1,1,1] fibration math.AG/0510059 Yoshinori Namikawa: Flops and Poisson deformations of symplectic varieties math.AG/0510055 Nicholas J. Proudfoot: All the GIT quotients at once math.AG/0510049 A.Libgober: Lectures on topology of complements and fundamental groups math.AG/0510011 Spencer Bloch, H'el`ene Esnault, Dirk Kreimer: On Motives Associated to Graph Polynomials math.AG/0510006 Amin Gholampour, Yinan Song: Evidence for the Gromov-Witten/Donaldson-Thomas Correspondence math.AG/0509727 Alexey Glutsyuk: Upper bounds of topology of complex polynomials in two variables math.AG/0509725 Fabrizio Catanese: Q.E.D. for algebraic varieties math.AG/0509722 Dominic Joyce: Motivic invariants of Artin stacks math.AG/0509710 Euisung Park: Some effects of veronese map on syzygies of projective varieties AP: Analysis of PDEs -------------------- math.AP/0510107 Pascal Azerad, Mohamed Mellouk: On a stochastic partial differential equation with non-local diffusion math.AP/0510096 Malte Henkel, Rene Schott, Stoimen Stoimenov, Jeremie Unterberger: On the dynamical symmetric algebra of ageing: Lie structure, representations and Appell systems math.AP/0510080 S. Gustafson, K. Nakanishi, T.-P. Tsai: Scattering for the Gross-Pitaevskii equation math.AP/0510076 A.G.Ramm: Inverse problems for parabolic equations 2 math.AP/0510053 Zhongwei Shen: On Estimates of Biharmonic Functions on Lipschitz and Convex Domains math.AP/0510041 Gerd Grubb: Remarks on nonlocal trace expansion coefficients math.AP/0509704 Piero D'Ancona, Vittoria Pierfelice, Alessandro Teta: Dispersive estimate for the Schroedinger equation with point interactions AT: Algebraic Topology ---------------------- math.AT/0510026 Behrens: Buildings, elliptic curves, and the K(2)-local sphere CA: Classical Analysis and ODEs ------------------------------- math.CA/0510130 Gady Kozma, Alexander Olevskii: Is PLA large? math.CA/0510084 Feng Dai: Characterizations of function spaces on the sphere using frames math.CA/0510050 Dominici: Some observations on a Kapteyn series math.CA/0510007 Gavin Brown, Feng Dai: Approximation of smooth functions on compact two-point homogeneous spaces math.CA/0509726 Enrico De Micheli, Nicodemo Magnoli, Giovanni Alberto Viano: On the regularization of Fredholm integral equations of the first kind CO: Combinatorics ----------------- math.CO/0510121 William Y. C. Chen, Kathy Q. Ji: Weighted Forms of Euler's Theorem math.CO/0510102 V. Farmaki, S. Negrepontis: Schreier Sets in Ramsey Theory math.CO/0510098 Luigi Santocanale: Congruences of Multinomial Lattices math.CO/0510094 Anant Godbole, Debra Knisley, Rick Norwood: Some Properties of Alphabet Overlap Graphs math.CO/0510092 Le Anh Vinh: On chromatic number of unit-quadrance graphs (finite Euclidean graphs) math.CO/0510079 Nicholas A. Loehr, Bruce E. Sagan, Gregory S. Warrington: A human proof for a generalization of Shalosh B. Ekhad's 10^n Lattice Paths Theorem math.CO/0510051 Vida Dujmovi'c, David R. Wood: Upward Three-Dimensional Grid Drawings of Graphs math.CO/0510045 Melody Chan, Anant P. Godbole: Improved Pebbling Bounds math.CO/0510044 Vincent Vatter: Enumeration schemes for restricted permutations math.CO/0510027 A. K. Kwasniewski: Prefab posets` Whitney numbers math.CO/0509716 Frank Vallentin: Optimal Embeddings of Distance Transitive Graphs into Euclidean Spaces math.CO/0509715 William Y.C. Chen, Sherry H.F. Yan: Noncrossing Trees and Noncrossing Graphs CT: Category Theory ------------------- quant-ph/0510032 Bob Coecke: Kindergarten Quantum Mechanics math.CT/0510072 Yaroslav Kopylov: On the Lambek Invariants of Commutative Squares in a Quasi-Abelian Category math.CT/0510057 Kostas Zotos, Andreas Litke: Cryptography and Encryption math.CT/0510039 K. Dosen, Z. Petric: Symmetric Self-Adjunctions and Matrices CV: Complex Variables --------------------- math.CV/0510127 Alexander Yu. Solynin, Victor A Zalgaller: An isoperimetric inequality for logarithmic capacity of polygons math.CV/0510071 Alexey Glutsyuk: Simple proofs of uniformization theorems math.CV/0509708 G'abor Francsics, Peter D. Lax: An explicit fundamental domain for the Picard modular group in two complex dimensions math.CV/0509703 David H Hamilton: Mapping Theorems DG: Differential Geometry ------------------------- math.DG/0510131 Claus Jeschek, Frederik Witt: Generalised geometries, constrained critical points and Ramond-Ramond fields math.DG/0510099 Jos'e M. M. Senovilla: Second-order symmetric Lorentzian manifolds math.DG/0510097 Andrew Stacey: The differential topology of loop spaces math.DG/0510087 Simon G. Chiossi, Anna Fino: Special metrics in $G_2$ geometry math.DG/0510083 Xianzhe Dai, Li Ma: Mass under the Ricci flow math.DG/0510078 Branislav Jurco: Crossed Module Bundle Gerbes; Classification, String Group and Differential Geometry math.DG/0510075 Joel Fine: Fibrations with constant scalar curvature Kahler metrics and the CM-line bundle math.DG/0510069 K. Grabowska, J. Grabowski, P. Urbanski: Frame-independent mechanics:geometry on affine bundles math.DG/0510062 Max Karoubi: Fibre bundles, connections and cyclic homology math.DG/0510061 Raphael Ponge: New invariants for CR and contact manifolds. I math.DG/0510031 Janusz Grabowski, Norbert Poncin: On quantum and classical Poisson algebras math.DG/0510021 Zhiqin Lu, Xiaofeng Sun: On the Weil-Petersson volume and the first Chern Class of the moduli space of Calabi-Yau manifolds math.DG/0510020 Zhiqin Lu, Xiaofeng Sun: Weil-Petersson geometry on moduli space of polarized Calabi-Yau manifolds math.DG/0510016 Julie Clutterbuck: Interior Gradient Estimates for Anisotropic Mean Curvature Flow math.DG/0510010 Yi Lin, Susan Tolman: Reduction of twisted generalized Kahler structure math.DG/0510002 Alexander Yampolsky: On special types of minimal and totally geodesic unit vector fields hep-th/0509224 Michael R. Douglas, Zhiqin Lu: Finiteness of volume of moduli spaces gr-qc/0509117 R. Casana, C. A. M. de Melo, B. M. Pimentel: Massless DKP field in Lyra manifold DS: Dynamical Systems --------------------- math.DS/0510082 Laurent Bartholdi, Volodymyr Nekrashevych: Thurston equivalence of topological polynomials hep-th/0510012 Andreas Fring, Nenad Manojlovic: G(2)-Calogero-Moser Lax operators from reduction math.DS/0510032 Konstantin Medynets: On approximation of homeomorphisms of a Cantor set math.DS/0510022 Nandor Simanyi: The Boltzmann-Sinai Ergodic Hypothesis in Full Generality (Without Exceptional Models) math.DS/0510018 Hailun Zhou: The Cost of Two-dimensional Rearrangement math.DS/0510014 Boris Solomyak: Pseudo-self-affine tilings in R^d math.DS/0510012 Tanya Schmah, Cristina Stoica: Saari's Conjecture is True for Generic Vector Fields math.DS/0509719 Hiroki Sumi: Semi-hyperbolic fibered rational maps and rational semigroups math.DS/0509717 Gheorghe Tigan: Analysis of the reconnection process in nontwist cubic maps FA: Functional Analysis ----------------------- math.FA/0510101 Javier H. Guachalla: A Direct Sum decomposition for Dual Spaces math.FA/0510091 Peter Balazs: Bessel, Frame and Riesz Multipliers math.FA/0510067 Jean-christophe Bourin: A concavity inequality for symmetric norms math.FA/0510064 Gabriel Maresch, Reinhard Winkler: Hartman functions and (weak) almost periodicity math.FA/0510024 Peter G. Casazza, Matt Fickus, Janet C. Tremain, Eric Weber: The Kadison-Singer Problem in Mathematics and Engineering math.FA/0510005 Wladyslaw A. Majewski, Marcin Marciniak: Decomposability of extremal positive maps on $M_2$ GM: General Mathematics ----------------------- math.GM/0510081 Jamel Ghanouchi: Proof of Fermat theorem math.GM/0510056 Kostas Zotos, Andreas Litke: An Introduction to Zoli Numbers GN: General Topology -------------------- math.GN/0510120 Vladimir P. Fonf, Matatyahu Rubin: Reconstruction theorem for homeomorphism groups without small sets and non-shrinking functions of a normed space math.GN/0510118 Matatyahu Rubin, Yosef Yomdin: Reconstruction of manifolds and subsets of normed spaces from subgroups of their homeomorphism groups GR: Group Theory ---------------- math.GR/0510116 Ursula Hamenstaedt: Geometry of the mapping class groups I: Boundary amenability math.GR/0510112 Manoj K. Yadav: Class Preserving Automorphisms of Finite p-groups math.GR/0510015 Xingzhong You, Guohua Qian, Wujie Shi: A new graph related to conjugacy classes of finite groups GT: Geometric Topology ---------------------- math.GT/0510129 Nathan M. Dunfield, Dylan P. Thurston: A random tunnel number one 3-manifold does not fiber over the circle math.GT/0510105 Cormac Walsh: The horofunction boundary of finite-dimensional normed spaces math.GT/0510095 I. G. Korepanov: Pachner move 3->3 and affine volume-preserving geometry in R^3 math.GT/0510086 Yuri A. Turygin: A Borsuk-Ulam theorem for $(mathbb Z_p)^k$-actions on products of (mod $p$) homology spheres math.GT/0510065 Thierry Barbot: Causal properties of AdS-isometry groups II: BTZ multi black-holes math.GT/0510048 Francesco Costantino: Colored Jones invariants of links in S^3 # k S^2 X S^1 and the Volume Conjecture math.GT/0510008 John B. Etnyre, Terry Fuller: Realizing 4-manifolds as achiral Lefschetz fibrations HO: History and Overview ------------------------ math.HO/0510054 Jordan Bell: Euler and the pentagonal number theorem LO: Logic --------- math.LO/0510122 Stephen McCleary, Matatyahu Rubin: Locally Moving Groups and the Reconstruction Problem for Chains and Circles math.LO/0510004 Rami Grossberg, Monica VanDieren: Categoricity from one successor cardinal in Tame Abstract Elementary Classes math.LO/0509707 Rami Grossberg, Olivier Lessmann: Abstract decomposition theorem and applications MP: Mathematical Physics ------------------------ math-ph/0510026 Roldao da Rocha, Waldyr A. Rodrigues, Jr: Diffeomorphism Invariance and Local Lorentz Invariance math-ph/0510025 Farrukh Mukhamedov, Utkir Rozikov: On Gibbs Measures of $P$-Adic Potts Model on the Cayley Tree math-ph/0510024 Farrukh Mukhamedov, Utkir Rozikov: On Inhomogeneous $p$-Adic Potts Model on a Cayley Tree math-ph/0510023 Evangelos Chaliasos: The modified Magnetohydrodynamic equations math-ph/0510022 Farrukh Mukhamedov, Utkir Rozikov: On some remarks on the Ising model with competing interactions on a Cayley tree math-ph/0510021 Murod Khamraev, Farrukh Mukhamedov, Utkir Rozikov: On Uniqueness of Gibbs Measures for $P$-Adic Nonhomogeneous $l$-Model on the Cayley Tree math-ph/0510020 Farrukh Mukhamedov, Utkir Rozikov: On Gibbs Measures of Models with Competing Ternary and Binary Interactions and Corresponding Von Neumann Algebras II math-ph/0510019 F. Peherstorfer, A. Volberg, P. Yuditskii: Limit periodic Jacobi matrices with a singular continuous spectrum and the renormalization of periodic matrices hep-th/0510034 Bogdan Morariu, Alexios P. Polychronakos: Fractional quantum Hall effect on the two-sphere: a matrix model proposal math-ph/0510018 D. Arnaudon, N. Crampe, A. Doikou, L. Frappat, Eric Ragoucy: Analytical Bethe ansatz in gl(N) spin chains math-ph/0510017 M. A. Agrotis, P. A. Damianou: Volterra's realization of the KM-system hep-th/0510028 Hanno Hammer: Compactification along Lightlike Lattices gr-qc/0510014 Muxin Han, Yongge Ma: Master Constraint Operator in Loop Quantum Gravity cond-mat/0510092 Petr Jizba, Toshihico Arimitsu: Towards information theory for q-nonextensive statistics without q-deformed distributions quant-ph/0509205 V. P. Belavkin: Continuous Non-Demolition Observation, Quantum Filtering and Optimal Estimation physics/0509242 O. Cornejo-Perez: Mathematical methods of factorization and a feedback approach for biological systems math-ph/0510016 Evangelos Chaliasos: The modified Vlasov equations math-ph/0510015 Maryna Nesterenko, Roman Popovych: Realizations of real semisimple low-dimensional Lie algebras math-ph/0510014 Giovanni Gallavotti: Constructive Quantum Field Theory math-ph/0509075 Hans F. de Groote: Observables II : Quantum Observables gr-qc/0510001 WKB approximation quant-ph/0509055 C. B. Compean, M. Kirchbach: The Trigonometric Rosen-Morse Potential in the Supersymmetric Quantum Mechanics and its Exact Solutions hep-th/0508091 Stoil Donev, Maria Tashkova: Integrability-Nonintegrability Structures and Individual Photons' Description as Finite Field Objects gr-qc/0509113 Alan Coley, Sigbjorn Hervik, Nicos Pelavas: On Spacetimes with Constant Scalar Invariants NA: Numerical Analysis ---------------------- math.NA/0510070 Steffen Hein: DSC Approach to Computational Fluid Dynamics math.NA/0510066 Bruno Lombard, Jo{e}l Piraux: Modeling 1-D elastic P-waves in a fractured rock with hyperbolic jump conditions math.NA/0509724 Esteban Moro, Henri Schurz: Non-negativity preserving numerical algorithms for stochastic differential equations NT: Number Theory ----------------- math.NT/0510114 Aleksandar Ivic: On the integral of the error term in the Dirichlet divisor problem math.NT/0510113 Dinakar Ramakrishnan, Jonathan Rogawski: Average values of modular L-series via the relative trace formula, and a curious measure math.NT/0510100 Sandro Mattarei: Modular periodicity of binomial coefficients math.NT/0510090 Laurent Berger: Repr'esentations modulaires de $mathrm{GL}_2(mathbf{Q}_p)$ et repr'esentations galoisiennes de dimension 2 math.NT/0510089 Farrell Brumley: Second order average estimates on local data of cusp forms cs.LO/0510011 David Delahaye, Micaela Mayero: Diophantus' 20th Problem and Fermat's Last Theorem for n=4: Formalization of Fermat's Proofs in the Coq Proof Assistant math.NT/0510052 N. A. Carella: On the Coefficients of Primitive Normal polynomials math.NT/0510023 Matthew Papanikolas, Christopher Rasmussen: On the torsion of Jacobians of principal modular curves of level 3^n math.NT/0510003 Pieter Moree: Counting carefree couples math.NT/0510001 Sungkon Chang: Note on the rank of quadratic twists of Mordell equations math.NT/0509723 R. Cluckers, F. Loeser: Exponential constructible functions, motivic Fourier transformation and transfer principle OA: Operator Algebras --------------------- math.OA/0510111 Benjam'in Itz'{a}-Ortiz: Continuous and discrete flows on operator algebras math.OA/0510103 D. Shlyakhtenko: A Free Analogue of Shannon's Problem on Monotonicity of Entropy math.OA/0510073 Maria Joita: A Radon-Nikodym theorem for completely multi-positive linear maps and applications math.OA/0509706 Stefaan Vaes, Roland Vergnioux: The boundary of universal discrete quantum groups, exactness and factoriality OC: Optimization and Control ---------------------------- math.OC/0510115 K. Eisenack, J. Scheffran, J.P. Kropp: Viability Analysis of Management Frameworks for Fisheries cs.SC/0510014 Cl'{e}ment Pernet: Computing the Kalman form math.OC/0510009 Islam I. Hussein, Anthony M. Bloch: Optimal Control of Underactuated Nonholonomic Mechanical Systems math.OC/0509718 Alexander L. Fradkov: Conic S-Procedure And Constrained Dissipativity math.OC/0509705 D. Cheng, L. Guo, Y. Lin, Y. Wang: A Note on Overshoot Estimation in Pole Placements PR: Probability --------------- math.PR/0510117 Marc Lelarge: Tail asymptotics for the supremum of an independent subadditive process, with applications to monotone-separable networks math.PR/0510077 Vitalii A. Gasanenko: On invariance of domains with smooth boundaries with respect to stochastic differential equations math.PR/0510047 G. Giacomin, F. L. Toninelli: The localized phase of disordered copolymers with adsorption math.PR/0510046 Alexandre Rybko, Senya Shlosman, Alexandre Vladimirov: Self-averaging property of queuing systems math.PR/0510043 Didier Piau: Maximal generalization of Baum-Katz theorem and optimality of sequential tests math.PR/0510042 Alexander V. Gnedin: Counting the Chain Records: The Product Case math.PR/0510038 Didier Piau: On two duality properties of random walks in random environment on the integer line math.PR/0510037 Didier Piau: Asymptotics of iterated branching processes math.PR/0510036 Didier Piau: Invariance principle for the coverage rate of genomic physical mappings math.PR/0510035 Didier Piau: Harmonic moments of non homogeneous branching processes math.PR/0510034 Jean B'erard, Jean-Baptiste Gou'er'e, Didier Piau: Solvable models of neighbor-dependent nucleotide substitution processes math.PR/0510029 R. Liptser: Large deviations for two scaled diffusions math.PR/0510028 Robert Sh. Liptser, Anatolii A. Pukhalskii: Limit theorems on large deviations for semimartingales math.PR/0509721 Amine Asselah Fabienne Castell: Self-Intersection Times for Random Walk, and Random Walk in Random Scenery in dimensions d>4 math.PR/0509720 Jon Warren: Dyson's Brownian motions, intertwining and interlacing math.PR/0509713 Jacky Cresson, S'{e}bastien Darses: Stochastic embedding of dynamical systems math.PR/0509712 Fabien Panloup: Recursive computation of the invariant measure of a stochastic differential equation driven by a L'{e}vy process math.PR/0509711 Huyen Pham: On some recent aspects of stochastic control theory and their applications QA: Quantum Algebra ------------------- math.QA/0510124 Nantel Bergeron, Yun Gao, Naihong Hu: Representations of Two-parameter Quantum Orthogonal and Symplectic Groups math.QA/0510119 Teodor Banica: Spectral measures of free quantum groups math.QA/0510109 Rita Fioresi, Fabio Gavarini: Quantum Duality Principle for Quantum Grassmannians math.QA/0510106 Shun-Jen Cheng, Weiqiang Wang, R.B. Zhang: A Fock space approach to representation theory of osp(2|2n) math.QA/0510040 Florin Panaite, Mihai D. Staic, Freddy Van Oystaeyen: Pure lazy cocycles and entwined monoidal categories math.QA/0510017 Satoshi Naito, Daisuke Sagaki: Crystal structure of the set of Lakshmibai-Seshadri paths of a level-zero shape for an affine Lie algebra RA: Rings and Algebras ---------------------- math.RA/0510104 Alberto Facchini, Dolors Herbera: Local morphisms and modules with a semilocal endomorphism ring math.RA/0510074 Francois Couchot: Localization of injective modules over valuation rings math.RA/0510068 Francois Couchot: Indecomposable modules and Gelfand rings math.RA/0509709 Edward S. Letzter: Noncommutative Images of Commutative Spectra RT: Representation Theory ------------------------- math.RT/0510060 Katsuhiro Uno, Hiro-Fumi Yamada: Elementary divisors of Cartan matrices for symmetric groups math.RT/0510058 Dimitar Grantcharov, Vera Serganova: Category of sp(2n)-modules with bounded weight multiplicities math.RT/0510033 Soo Teck Lee, Kyo Nishiyama, Akihito Wachi: Intersection of harmonics and Capelli identities for symmetric pairs math.RT/0510019 Dimitry Leites, Elena Poletaeva: Defining relations for classical Lie algebras of polynomial vector fields hep-th/9702120 Pavel Grozman, Dimitry Leites, Irina Shchepochkina: Lie superalgebras of string theories hep-th/9702073 Pavel Grozman, Dimitry Leites: Defining Relations for Lie Superalgebras with Cartan matrix SG: Symplectic Geometry ----------------------- math.SG/0510108 R. Hind, A. Ivrii: Ruled 4-manifolds and isotopies of symplectic surfaces math.SG/0509714 Paolo Ghiggini, Paolo Lisca, Andras I. Stipsicz: Tight contact structures on some small Seifert fibered 3--manifolds ST: Statistics -------------- cs.NI/0510007 Fabien Viger, Alain Barrat, Luca Dall'Asta, Cun-Hui Zhang, Eric D. Kolaczyk: Network Inference from TraceRoute Measurements: Internet Topology `Species' math.ST/0510013 David B. Chua, Eric D. Kolaczyk, Crovella: Network Kriging -- / Greg Kuperberg (UC Davis) / Home page: http://www.math.ucdavis.edu/~greg/ / Visit the Math ArXiv Front at http://front.math.ucdavis.edu/ / * All the math that's fit to e-print * === Subject: NSERC University Faculty Award Originator: bergv@math.uiuc.edu (Maarten Bergvelt) The Department of Mathematical and Statistical Sciences is actively seeking to nominate a candidate for a NSERC University Faculty Award in the Fall 2006 competition. The University Faculty Award was created by NSERC to encourage Canadian universities to appoint outstanding women and aboriginal researchers to tenure-track positions in science and engineering. Further information on the program can be found at the following webpage: http://www.nserc.gc.ca/professors_e.asp?nav=profnav&lbi=c7 The nominee will have an excellent record of research and publication. We are particularly interested in candidates who work in a field related to an area of existing or emerging strength in the Department, although other areas will be considered. Some areas of research excellence, recently highlighted by the University of Alberta Faculty of Science, include algebra, functional analysis, fluid dynamics, statistics, mathematical biology, and scientific computing. The candidate will also have a strong commitment to and aptitude for teaching undergraduate students, and will be expected to supervise graduate theses. This tenure-track appointment is scheduled to begin on or near July 1, 2007. Applicants should submit a curriculum vitea, research and teaching profiles outlining experience and/or interests, and at least three confidential letters of reference to: Anthony To-Ming Lau, Chair Department of Mathematical and Statistical Sciences University of Alberta Edmonton, Alberta, Canada T6G 2G1 The closing date for applications is March 1, 2006. Early applications are encouraged. According to NSERC regulations, applicants must be Canadian citizens or permanent residents of Canada. Please Note: Applicants being considered will generally be contacted within 3-4 weeks of the deadline date. Those not contacted are thanked for their interest and encouraged to apply for future positions advertised by the University. The University of Alberta hires on the basis of merit. We are committed to the principle of equity in employment. We welcome diversity and encourage applications from all qualified women and men, including persons with disabilities, members of visible minorities, and Aboriginal persons. === Subject: Re: IS quantum mechanics a limit cycle theory?? Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Let's try my question about the following example in R^4: H=(y-(x^3-x))z-xw consider the corresponding hamiltonian vector field X_H in R^4 with respect to standard symplectic structure of R^4 (how many periodic solutions do exist)?? Now try to construct a Hermitian operator with H(as usual methods), my question is that what is the operator theoretic interpretation for the number of closed orbits of X_H... > the domain of hamiltonian is even dimensional space! > But can Morse > theory help to my question, namely is dynamic fix if > we do not pass a > critical value (However I prefer to not change the > main subject of my > question, a possible relation between limit cycle > theory and > quantization) > potential > V(x,y,z) = (1 - x^2 - y^2)^2 z + z^3 + x^2 + y^2 Note that dV/dz > 0 except on the circle x^2 + y^2 > 1, z = 0, so > the only possible closed orbits are on that circle. > On that circle > there are closed orbits in both directions. Robert Israel > israel@math.ubc.ca > Department of Mathematics > http://www.math.ubc.ca/~israel > University of British Columbia > Vancouver, > BC, Canada === Subject: Radical Expression For cos(2*pi/31) Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Introduction ------------ Trigonometric functions of for prime have an especially complicated Galois-minimal representation. In particular, the case Cos[Pi/23] requires approximately 500 MB of space using the Mathematica command FunctionExpand[Cos[Pi/23]]. However, they can be expressed concisely as algebraic numbers. For example, letting p(x)n denote the nth root of the polynomial p(x) using the ordering of Mathematica's Root function, sin(pi/23) is given by see http://mathworld.wolfram.com/TrigonometryAnglesPi23.html Also, there is a several researches to get minimal polynomial of cos(2*pi/p) where p is prime see http://www.math.ksu.edu/~dbski/cosine-mo-jo.pdf Moreover, there were several discussions about same subject see Table[Timing[FullSimplify[{i, Cos[Pi/i], Sin[Pi/i]}]], {i, 1, 22}] The first run of this command gives very varying times, from 0. Second for i=2 to 2.3 Second for i=19. If we change the limits of table, Mathematica get completely stuck at i=23 (?!?!). For i=29, it takes 119.73 Seconds, while i=36 requires 0.06 Second. So, I feel this subject is quiet interesting see http://mathforum.org/kb/thread.jspa?threadID=1267682&tstart=15 http://mathforum.org/kb/thread.jspa?threadID=1273051&tstart=15 Radical Expression For cos(2*pi/31) ----------------------------------- cos(2*pi/31)=1/6*(8*n1^3-18*n1*n2-27*n1+9*n3+3*(8*n1^6-36*n1^4*n2-60*n1^4+42 *n1^2*n2^2+144*n1^2*n2+135*n1^2-3*n2^3-27*n2^2-81*n2-81+16*n1^3*n3-36*n1*n2* n 3-54*n1*n3+9*n3^2)^(1/2))^(1/3)-6*(1/18*n1^2-1/12*n2-1/4)/(8*n1^3-18*n1*n2-2 7 *n1+9*n3+3*(8*n1^6-36*n1^4*n2-60*n1^4+42*n1^2*n2^2+144*n1^2*n2+135*n1^2-3*n2 ^ 3-27*n2^2-81*n2-81+16*n1^3*n3-36*n1*n2*n3-54*n1*n3+9*n3^2)^(1/2))^(1/3)+1/3* n 1 n1=(g1+g2+g3+g4-1/2)/5 n2=(g1*w^4+g2*w^3+g3*w^2+g4*w-1/2)/5 n3=(g1*w+g2*w^2+g3*w^3+g4*w^4-1/2)/5 w=exp(2*pi*i/5)= 1/4*(5^(1/2)-1+i*(10+2*5^(1/2))^(1/2)) g1=(-12679/128-3875/128*5^(1/2)+155/128*(-9250+4090*5^(1/2))^(1/2))^(1/5) g2=(-12679/128+3875/128*5^(1/2)-155/128*(-9250-4090*5^(1/2))^(1/2))^(1/5) g3=(-12679/128+3875/128*5^(1/2)+155/128*(-9250-4090*5^(1/2))^(1/2))^(1/5) g4=(-12679/128-3875/128*5^(1/2)-155/128*(-9250+4090*5^(1/2))^(1/2))^(1/5) === Subject: Density in a Lie group Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Do you know something on the following questions? When do 2 matrices A,B in SU(n) generate a dense (not Zariski dense) subgroup? To have no common invariant C^n subspace is obviously a required condition. Is it also sufficient? Is it sufficient at least if A,B have infinite order? If their cyclic groups are dense in a maximal torus? Is a conjugacy class in SU(n) (of non-central matrices) always Zariski dense? (It is sufficient for me if the matrix elements of the conjugacy class satisfy no non-trivial linear relation.) What is known on the questions for SL(n)? Alexander Stoimenow === Subject: Re: IS quantum mechanics a limit cycle theory?? Originator: bergv@math.uiuc.edu (Maarten Bergvelt) May you more Explain on Physical relevance? Further in this example the hamiltonian vector field in R^6 has an invariant torus with infinite number of closed orbits. I search for an example of an analytic Hamiltonian in R^2n with a finit number of closed orbits.... Is there is any example of such Hamiltonian, with k number of closed orbits, what is the quantum interpretation for this k,after an appropriate quantization of H! Let's try this question for the following this question for the following hamiltonian: H=(y-(x^3-x))z-zw (in R^4) > potential > V(x,y,z) = (1 - x^2 - y^2)^2 z + z^3 + x^2 + y^2 Note that dV/dz > 0 except on the circle x^2 + y^2 > = 1, z = 0, so > the only possible closed orbits are on that circle. > On that circle > there are closed orbits in both directions. > IMHO this cannot have any physical relevance, because > these orbits are > highly _unstable_. Am I wrong? > Han de Bruijn === Subject: Extended Beta function? Originator: bergv@math.uiuc.edu (Maarten Bergvelt) For Numerical Analysis applications, I am interested in the following integral: int_0^1 (p + x)^m (q - x)^n dx where p, q, m, n are integers. For p = 0 and q = 1, this is B(m, n), the Beta function. AB (Bossavit at lgep dot supelec dot fr) === Subject: Re: Extended Beta function? Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > For Numerical Analysis applications, I am interested in the following integral: > int_0^1 (p + x)^m (q - x)^n dx > where p, q, m, n are integers. For p = 0 and q = 1, this is B(m, > n), the Beta function. > AB > (Bossavit at lgep dot supelec dot fr) int_0^1 (p + x)^m (q - x)^n dx = int_p^{1+p} y^m (p+q - y)^n dy (using y= p+x) = (p+q)^(m+n+1) int_{p/(p+q)}^{(1+p)/(p+q)} t^m (1-t)^n dt (I think using t= y/(p+q)) so that the problem is to evaluate : int_z1^z2 t^m (1-t)^n dt this is sometimes called 'generalized incomplete beta function' and handled in details here (recurrence relations and so on...) : http://functions.wolfram.com/GammaBetaErf/Beta4/ Hoping it helped, Raymond === Subject: Re: Extended Beta function? Originator: bergv@math.uiuc.edu (Maarten Bergvelt) To my query, I am interested in the following integral: > (1) int_0^1 (p + x)^m (q - x)^n dx Raymond Manzoni answered >the problem is to evaluate : > (2) int_z1^z2 t^m (1-t)^n dt >sometimes called 'generalized incomplete >beta function'[cf.] http://functions.wolfram.com/GammaBetaErf/Beta4/ Indeed, I had transformed (2) into (1), so this link is exactly what I A.B. === Subject: Re: local classification of riemannian manifolds Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > Is this more than knowing the curvature tensor and all its covariant > derivatives? I am not certain, but the parallel translation can > certainly be expressed in terms of covariant derivatives. >> But in particular cases, even the parallel transport >> is not enough to reconstruct the metric: >> If you are also given the metric tensor at a single point, >> then you can reconstruct the whole metric on the manifold by >> transporting it around (provided the manifold is connected, and the >> connection complete). >> In general, this metric at a single point can not be chosen arbitrarily: >> It has to be invariant under action of the holonomy group of the >> parallel transport. > I don't understand this at all. At a single point, all metrics are > equivalent. What are you choosing? Yes, at a given point in a manifold without further structure, all Riemannian metrics are equivalent. But in a manifold with a given additional covariant derivative on its tangent bundle, this is usually not true any longer. What I wanted to say is: When we say, we want to reconstruct a Riemannian metric (say, on a closed manifold M) from its Levi-Civita covariant derivative, then we have forget about this metric and have to start from the manifold M with this given torsion-free covariant derivative D on its tangent bundle TM. This covariant derivative allows us to define parallel transport, in particular parallel transport along closed curves, and thus the holonomy group of these parallel transports. Now the task is to find a metric tensor g on M such that g is parallel with respect to D; and the question is whether such a metric tensor is uniquely determined by D. And the answer is: *Not always!* - We are free to pick at a (single) point p of M *any* metric tensor g that is invariant under the action of the holonomy group of D at p, and this choice then determines the metric uniquely. - So, if this holonomy group is small - e.g. trivial - we may have more than one choice of metric; hence the Levi-Civita derivative will not determine its metric. But in the *generic* case, the holonomy group of D will be SO(dim M) (or even larger), and in this case the metric will be uniquely determined by its Levi-Civita derivative. Maybe one can apply the Ambrose-Singer theorem to deduce that for *generic* metrics even the curvature tensor determines the metric uniquely. === Subject: Re: local classification of riemannian manifolds Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > Is this more than knowing the curvature tensor and all its covariant > derivatives? I am not certain, but the parallel translation can > certainly be expressed in terms of covariant derivatives. >> But in particular cases, even the parallel transport >> is not enough to reconstruct the metric: >> If you are also given the metric tensor at a single point, >> then you can reconstruct the whole metric on the manifold by >> transporting it around (provided the manifold is connected, and the >> connection complete). >> In general, this metric at a single point can not be chosen arbitrarily: >> It has to be invariant under action of the holonomy group of the >> parallel transport. > I don't understand this at all. At a single point, all metrics are > equivalent. What are you choosing? Coefficients with respect to > a specified coordinate chart? Why? Parallel transport then defines the > metric at each point (assuming completeness) -- but there is a lot encoded > in the parallel transport. I believe someone provided the answer: just write the metric locally with respect to exponential (geodesic normal) co-ordinates. This determines the metric uniquely up to rotations (of the tangent space at the origin). Moreover, the metric is uniquely determined by the curvature tensor in the following sense: The metric is given locally by g = J^2 where the n-by-n matrix J is the unique solution to the system of ODE's J'' + KJ = 0, J(0) = I, J'(0) = 0, where ' denotes differentiation in the radial direction (on R^n) and K_{ij} = R_{ikjl}x^k x^l/|x|^2 This, I believe, reduces the question of parameterizing the space of local Riemannian metrics to characterizing what tensors can arise as the curvature tenor of a metric written in exponential coordinates. I haven't a clue, so someone else needs to take over from here. === Subject: Re: local classification of riemannian manifolds Originator: ilya@powdermilk Originator: bergv@math.uiuc.edu (Maarten Bergvelt) [A complimentary Cc of this posting was sent to Deane Yang > I believe someone provided the answer: just write the metric locally > with respect to exponential (geodesic normal) co-ordinates. This > determines the metric uniquely up to rotations (of the tangent space at > the origin). Moreover, the metric is uniquely determined by the > curvature tensor in the following sense: The metric is given locally by > g = J^2 > where the n-by-n matrix J is the unique solution to the system of ODE's > J'' + KJ = 0, J(0) = I, J'(0) = 0, > where ' denotes differentiation in the radial direction (on R^n) > and K_{ij} = R_{ikjl}x^k x^l/|x|^2 Now you can calculate the R-tensor R' of the obtained metric g. Since one should have R'=R, this gives one set of equations on the tensor R. If they are satisfied, the tensor R is (obviously) a Riemannian tensor of some metric. > This, I believe, reduces the question of parameterizing the space of > local Riemannian metrics to characterizing what tensors can arise as the > curvature tenor of a metric written in exponential coordinates. The only thing to consider is that straight lines passing through center are geodesics (in natural parametrization). Since the coefficients for ODE of geodesics are essentially the components of Riemannian tensor, this gives linear equations on values of R at non-0 points. (Hmm, maybe one needs to raise some indices; then it is a joint restrictions on R and g...) Now if these two sets of equations are satisfied, then R *is* as required. Unfortulately, IIRC, the first set of equations is not local. I do not remember whether one can rewrite these conditions as a finite set of differential equations on the tensor R. I remember there was some spike in research on this theme in '90s, but do not remember what it was about. :-( Should not be hard to find, though... Hope this helps, Ilya === Subject: Paper published by Geometry and Topology Originator: bergv@math.uiuc.edu (Maarten Bergvelt) The following paper has been published: Geometry and Topology, Volume 9 (2005) Paper no. 46, pages 2013--2078 URL: http://www.maths.warwick.ac.uk/gt/GTVol9/paper46.abs.html DOI: 10.2140/gt.2005.9.2013 Title: Contact homology and one parameter families of Legendrian knots Author(s): Tamas Kalman Abstract: We consider S^1-families of Legendrian knots in the standard contact R^3. We define the monodromy of such a loop, which is an automorphism of the Chekanov-Eliashberg contact homology of the starting (and ending) point. We prove this monodromy is a homotopy inhvariant of the loop. We also establish techniques to address the issue of Reidemeister moves of Lagrangian projections of Legendrian links. As an application, we exhibit a loop of right-handed Legendrian torus knots which is non-contractible in the space Leg(S^1,R^3) of Legendrian knots, although it is contractible in the space Emb(S^1,R^3) of smooth knots. For this result, we also compute the contact homology of what we call the Legendrian closure of a positive braid and construct an augmentation for each such link diagram. Secondary: 57M25 Keywords: Legendrian contact homology, monodromy, Reidemeister moves, braid positive knots, torus knots Revised: 24 July 2005 Accepted: 17 September 2005 Published: 26 October 2005 Proposed: Yasha Eliashberg Seconded: Peter Ozsvath, Tomasz Mrowka Author(s) address(es): Department of Mathematics, University of Southern California Los Angeles, CA 90089, USA Email: tkalman@usc.edu === Subject: Research Opportunity: Algs for Water Quality Prediction & Control in Pipeline Networks Originator: bergv@math.uiuc.edu (Maarten Bergvelt) I am looking for students with an interest or background in Applied Math / Physics / Computer Science / Engineering and who wish to branch out and apply their knowledge to important drinking water quality issues. Graduate student research assistantships are currently available at the University of Cincinnati to study mathematical models and numerical algorithms for predicting and controlling water quality changes in water distribution systems (see below for further details). Students would be working toward a Ph.D. degree in Environmental Engineering at the University of Cincinnati. This program is flexible and interdisciplinary. Stipends range from $23,000 - $26,000/year plus full tuition scholarship. The U.C. program is ranked in the top 20 among U.S. Environmental Engineering programs. Drinking water distribution systems are interesting looped networks of pipes that constitute a critical infrastructure blanketing large urban areas, yet are similar mathematically to electrical circuits. Water quality dynamics are complicated by the presence of storage, flow reversals, discrete operational changes, and time-varying spatially distributed water usage. Uncertainty and variability is large. There is much interest in understanding the behavior of these systems, for water quality management and fault-detection/diagnosis. These system have been literally and figuratively buried, and we have a long way to go in achieving this understanding. In short, there is significant opportunity to make an impact. Research opportunities exist in the following general areas: 1) predicting and controlling microbial health risks associated with drinking water; 2)developing real-time algorithms for detecting water quality anomalies and identifying their cause/source; 3) developing advanced modeling tools for Monte Carlo simulation of fate and transport in water distribution systems. These research areas are motivated by general concerns about microbial and chemical health risks of drinking water (and associated regulations), and more specifically by drinking water security. In each of these areas there are interesting applied math problems, some of which we have anticipated and certainly many that we have not. As one example, under certain assumptions the input-output water quality relationship in pipe networks is linear with time varying time delays; a problem of inverting real-time sensor data for fault detection and diagnosis can then be posed as a large scale underdetermined linear least squares problem. Robust techniques for solving such problems in real time (and even what we mean by solving them) are of interest, and will be developed and field deployed. Yes, you can get jobs doing this sort of thing, and the trend is upward, in research, government, and consulting. I am looking for creative people who are motivated to dig in, learn about these problems, and make a unique contribution. Our research environment is loosely structured and team oriented. If you are interested please contact me directly by phone or email (see below). Dr. James G. Uber Associate Professor University of Cincinnati Dept. Civil & Envr. Engr. 765 Baldwin Hall; PO Box 210071 Cincinnati, OH 45221-0071 (E) jim.uber@uc.edu (P) 513-556-3643 (F) 513-556-2599 === Subject: Radical Expression For cos(2*pi/53) Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Introduction ------------ Trigonometric functions of pi/p for p prime have an especially complicated Galois-minimal representation. In particular, the case Cos[Pi/23] requires approximately 500 MB of space using the Mathematica command FunctionExpand[Cos[Pi/23]]. However, they can be expressed concisely as algebraic numbers. For example, letting p(x)n denote the nth root of the polynomial p(x) using the ordering of Mathematica's Root function, sin(pi/23) is given by see http://mathworld.wolfram.com/TrigonometryAnglesPi23.html Also, there is a several researches to get minimal polynomial of cos(2*pi/p) where p is prime see http://www.math.ksu.edu/~dbski/cosine-mo-jo.pdf Moreover, there were several discussions about same subject see Table[Timing[FullSimplify[{i, Cos[Pi/i], Sin[Pi/i]}]], {i, 1, 22}] The first run of this command gives very varying times, from 0. Second for i=2 to 2.3 Second for i=19. If we change the limits of table, Mathematica get completely stuck at i=23 (?!?!). For i=29, it takes 119.73 Seconds, while i=36 requires 0.06 Second. So, I feel this subject is quiet interesting see http://mathforum.org/kb/thread.jspa?threadID=1267682&tstart=15 http://mathforum.org/kb/thread.jspa?threadID=1273051&tstart=15 http://mathforum.org/kb/thread.jspa?threadID=1278852&tstart=75 Minimal polynomial for 2*cos(2*pi/53) ------------------------------------- x^26+x^25-25*x^24-24*x^23+276*x^22+253*x^21-1771*x^20-1540*x^19+7315*x^18+59 85*x^17-20349*x^16-15504*x^15+38760*x^14+27132*x^13-50388*x^12-31824*x^11+43 7 58*x^10+24310*x^9-24310*x^8-11440*x^7+8008*x^6+3003*x^5-1365*x^4-364*x^3+91* x ^2+13*x-1=0 see http://www.math.ksu.edu/~dbski/cosine-mo-jo.pdf Radical Expression For cos(2*pi/53) ----------------------------------- cos(2*pi/53)=1/4*(k1+(-k1^2+2*k2+8)^(1/2)) k1=(g1+g2+g3+g4+g5+g6+g7+g8+g9+g10+g11+g12-1)/13 k2=(g1*w^12+g2*w^11+g3*w^10+g4*w^9+g5*w^8+g6*w^7+g7*w^6+g8*w^5+g9*w^4+g10*w^ 3+g11*w^2+g12*w-1)/13 g1=(m1/2-(m1^2/4-53^13)^(1/2))^(1/13)*w^3 g2=(m2/2-(m2^2/4-53^13)^(1/2))^(1/13)*w^3 g3=(m3/2+(m3^2/4-53^13)^(1/2))^(1/13)*w^2 g4=(m4/2-(m4^2/4-53^13)^(1/2))^(1/13)*w^6 g5=(m5/2-(m5^2/4-53^13)^(1/2))^(1/13)*w^3 g6=(m6/2+(m6^2/4-53^13)^(1/2))^(1/13)*w^10 g7=(m6/2-(m6^2/4-53^13)^(1/2))^(1/13)*w^3 g8=(m5/2+(m5^2/4-53^13)^(1/2))^(1/13)*w^10 g9=(m4/2+(m4^2/4-53^13)^(1/2))^(1/13)*w^7 g10=(m3/2-(m3^2/4-53^13)^(1/2))^(1/13)*w^11 g11=(m2/2+(m2^2/4-53^13)^(1/2))^(1/13)*w^10 g12=(m1/2+(m1^2/4-53^13)^(1/2))^(1/13)*w^10 w=exp(2*pi*i/13) cos(2*pi/13)=1/12*((64+18*(-260+156*i*3^(1/2))^(1/3)+18*(-260-156*i*3^(1/2)) ^(1/3)+6*(-1664+64*(-260+156*i*3^(1/2))^(1/3)+64*(-260-156*i*3^(1/2))^(1/3)+ 9 *(-260+156*i*3^(1/2))^(2/3)+18*(-260+156*i*3^(1/2))^(1/3)*(-260-156*i*3^(1/2 ) )^(1/3)+9*(-260-156*i*3^(1/2))^(2/3))^(1/2))^(2/3)+40-2*(64+18*(-260+156*i*3 ^ (1/2))^(1/3)+18*(-260-156*i*3^(1/2))^(1/3)+6*(-1664+64*(-260+156*i*3^(1/2))^ ( 1/3)+64*(-260-156*i*3^(1/2))^(1/3)+9*(-260+156*i*3^(1/2))^(2/3)+18*(-260+156 * i*3^(1/2))^(1/3)*(-260-156*i*3^(1/2))^(1/3)+9*(-260-156*i*3^(1/2))^(2/3))^(1 / 2))^(1/3))/(64+18*(-260+156*i*3^(1/2))^(1/3)+18*(-260-156*i*3^(1/2))^(1/3)+6 * (-1664+64*(-260+156*i*3^(1/2))^(1/3)+64*(-260-156*i*3^(1/2))^(1/3)+9*(-260+1 5 6*i*3^(1/2))^(2/3)+18*(-260+156*i*3^(1/2))^(1/3)*(-260-156*i*3^(1/2))^(1/3)+ 9 *(-260-156*i*3^(1/2))^(2/3))^(1/2))^(1/3) m1= 1/6*(36*c1*b1-108*d1-8*b1^3+12*(12*c1^3-3*c1^2*b1^2-54*c1*b1*d1+81*d1^2+12*d 1 *b1^3)^(1/2))^(1/3)-6*(1/3*c1-1/9*b1^2)/(36*c1*b1-108*d1-8*b1^3+12*(12*c1^3- 3 *c1^2*b1^2-54*c1*b1*d1+81*d1^2+12*d1*b1^3)^(1/2))^(1/3)-1/3*b1 m2=1/6*(36*c2*b2-108*d2-8*b2^3+12*(12*c2^3-3*c2^2*b2^2-54*c2*b2*d2+81*d2^2+1 2*d2*b2^3)^(1/2))^(1/3)-6*(1/3*c2-1/9*b2^2)/(36*c2*b2-108*d2-8*b2^3+12*(12*c 2 ^3-3*c2^2*b2^2-54*c2*b2*d2+81*d2^2+12*d2*b2^3)^(1/2))^(1/3)-1/3*b2 m3=-1/12*(36*c1*b1-108*d1-8*b1^3+12*(12*c1^3-3*c1^2*b1^2-54*c1*b1*d1+81*d1^2 +12*d1*b1^3)^(1/2))^(1/3)+3*(1/3*c1-1/9*b1^2)/(36*c1*b1-108*d1-8*b1^3+12*(12 * c1^3-3*c1^2*b1^2-54*c1*b1*d1+81*d1^2+12*d1*b1^3)^(1/2))^(1/3)-1/3*b1+1/2*i*3 ^ (1/2)*(1/6*(36*c1*b1-108*d1-8*b1^3+12*(12*c1^3-3*c1^2*b1^2-54*c1*b1*d1+81*d1 ^ 2+12*d1*b1^3)^(1/2))^(1/3)+6*(1/3*c1-1/9*b1^2)/(36*c1*b1-108*d1-8*b1^3+12*(1 2 *c1^3-3*c1^2*b1^2-54*c1*b1*d1+81*d1^2+12*d1*b1^3)^(1/2))^(1/3)) m4=-1/12*(36*c1*b1-108*d1-8*b1^3+12*(12*c1^3-3*c1^2*b1^2-54*c1*b1*d1+81*d1^2 +12*d1*b1^3)^(1/2))^(1/3)+3*(1/3*c1-1/9*b1^2)/(36*c1*b1-108*d1-8*b1^3+12*(12 * c1^3-3*c1^2*b1^2-54*c1*b1*d1+81*d1^2+12*d1*b1^3)^(1/2))^(1/3)-1/3*b1-1/2*i*3 ^ (1/2)*(1/6*(36*c1*b1-108*d1-8*b1^3+12*(12*c1^3-3*c1^2*b1^2-54*c1*b1*d1+81*d1 ^ 2+12*d1*b1^3)^(1/2))^(1/3)+6*(1/3*c1-1/9*b1^2)/(36*c1*b1-108*d1-8*b1^3+12*(1 2 *c1^3-3*c1^2*b1^2-54*c1*b1*d1+81*d1^2+12*d1*b1^3)^(1/2))^(1/3)) m5=-1/12*(36*c2*b2-108*d2-8*b2^3+12*(12*c2^3-3*c2^2*b2^2-54*c2*b2*d2+81*d2^2 +12*d2*b2^3)^(1/2))^(1/3)+3*(1/3*c2-1/9*b2^2)/(36*c2*b2-108*d2-8*b2^3+12*(12 * c2^3-3*c2^2*b2^2-54*c2*b2*d2+81*d2^2+12*d2*b2^3)^(1/2))^(1/3)-1/3*b2+1/2*i*3 ^ (1/2)*(1/6*(36*c2*b2-108*d2-8*b2^3+12*(12*c2^3-3*c2^2*b2^2-54*c2*b2*d2+81*d2 ^ 2+12*d2*b2^3)^(1/2))^(1/3)+6*(1/3*c2-1/9*b2^2)/(36*c2*b2-108*d2-8*b2^3+12*(1 2 *c2^3-3*c2^2*b2^2-54*c2*b2*d2+81*d2^2+12*d2*b2^3)^(1/2))^(1/3)) m6=-1/12*(36*c2*b2-108*d2-8*b2^3+12*(12*c2^3-3*c2^2*b2^2-54*c2*b2*d2+81*d2^2 +12*d2*b2^3)^(1/2))^(1/3)+3*(1/3*c2-1/9*b2^2)/(36*c2*b2-108*d2-8*b2^3+12*(12 * c2^3-3*c2^2*b2^2-54*c2*b2*d2+81*d2^2+12*d2*b2^3)^(1/2))^(1/3)-1/3*b2-1/2*i*3 ^ (1/2)*(1/6*(36*c2*b2-108*d2-8*b2^3+12*(12*c2^3-3*c2^2*b2^2-54*c2*b2*d2+81*d2 ^ 2+12*d2*b2^3)^(1/2))^(1/3)+6*(1/3*c2-1/9*b2^2)/(36*c2*b2-108*d2-8*b2^3+12*(1 2 *c2^3-3*c2^2*b2^2-54*c2*b2*d2+81*d2^2+12*d2*b2^3)^(1/2))^(1/3)) b1=-391237897837+14832792*265242757^(1/2) c1=31554631892503606510713/2-233360975575/2*807601609731642430810573^(1/2) d1=9999288597868043977278426171849201/2+1146885757067680981/2*38195184393986 1794970897567756013^(1/2) b2=-391237897837-14832792*265242757^(1/2) c2=31554631892503606510713/2+233360975575/2*807601609731642430810573^(1/2) d2=9999288597868043977278426171849201/2-1146885757067680981/2*38195184393986 1794970897567756013^(1/2) The numeric values of the used symbols as per MATLAP shall be as follows ------------------------------------------------------------------------- b1=-391237897837+14832792*265242757^(1/2) c1=31554631892503606510713/2-233360975575/2*807601609731642430810573^(1/2) d1=9999288597868043977278426171849201/2+1146885757067680981/2*38195184393986 1794970897567756013^(1/2) b2=-391237897837-14832792*265242757^(1/2) c2=31554631892503606510713/2+233360975575/2*807601609731642430810573^(1/2) d2=9999288597868043977278426171849201/2-1146885757067680981/2*38195184393986 1794970897567756013^(1/2) b1 = -149666966656.743 c1 = -8.90795381694495e+022 d1 = 1.62067758349671e+034 b2 = -632808829017.257 c2 = 1.20634170061953e+023 d2 = -6.20748723709901e+033 m1= 1/6*(36*c1*b1-108*d1-8*b1^3+12*(12*c1^3-3*c1^2*b1^2-54*c1*b1*d1+81*d1^2+12*d 1 *b1^3)^(1/2))^(1/3)-6*(1/3*c1-1/9*b1^2)/(36*c1*b1-108*d1-8*b1^3+12*(12*c1^3- 3 *c1^2*b1^2-54*c1*b1*d1+81*d1^2+12*d1*b1^3)^(1/2))^(1/3)-1/3*b1 m2=1/6*(36*c2*b2-108*d2-8*b2^3+12*(12*c2^3-3*c2^2*b2^2-54*c2*b2*d2+81*d2^2+1 2*d2*b2^3)^(1/2))^(1/3)-6*(1/3*c2-1/9*b2^2)/(36*c2*b2-108*d2-8*b2^3+12*(12*c 2 ^3-3*c2^2*b2^2-54*c2*b2*d2+81*d2^2+12*d2*b2^3)^(1/2))^(1/3)-1/3*b2 m3=-1/12*(36*c1*b1-108*d1-8*b1^3+12*(12*c1^3-3*c1^2*b1^2-54*c1*b1*d1+81*d1^2 +12*d1*b1^3)^(1/2))^(1/3)+3*(1/3*c1-1/9*b1^2)/(36*c1*b1-108*d1-8*b1^3+12*(12 * c1^3-3*c1^2*b1^2-54*c1*b1*d1+81*d1^2+12*d1*b1^3)^(1/2))^(1/3)-1/3*b1+1/2*i*3 ^ (1/2)*(1/6*(36*c1*b1-108*d1-8*b1^3+12*(12*c1^3-3*c1^2*b1^2-54*c1*b1*d1+81*d1 ^ 2+12*d1*b1^3)^(1/2))^(1/3)+6*(1/3*c1-1/9*b1^2)/(36*c1*b1-108*d1-8*b1^3+12*(1 2 *c1^3-3*c1^2*b1^2-54*c1*b1*d1+81*d1^2+12*d1*b1^3)^(1/2))^(1/3)) m4=-1/12*(36*c1*b1-108*d1-8*b1^3+12*(12*c1^3-3*c1^2*b1^2-54*c1*b1*d1+81*d1^2 +12*d1*b1^3)^(1/2))^(1/3)+3*(1/3*c1-1/9*b1^2)/(36*c1*b1-108*d1-8*b1^3+12*(12 * c1^3-3*c1^2*b1^2-54*c1*b1*d1+81*d1^2+12*d1*b1^3)^(1/2))^(1/3)-1/3*b1-1/2*i*3 ^ (1/2)*(1/6*(36*c1*b1-108*d1-8*b1^3+12*(12*c1^3-3*c1^2*b1^2-54*c1*b1*d1+81*d1 ^ 2+12*d1*b1^3)^(1/2))^(1/3)+6*(1/3*c1-1/9*b1^2)/(36*c1*b1-108*d1-8*b1^3+12*(1 2 *c1^3-3*c1^2*b1^2-54*c1*b1*d1+81*d1^2+12*d1*b1^3)^(1/2))^(1/3)) m5=-1/12*(36*c2*b2-108*d2-8*b2^3+12*(12*c2^3-3*c2^2*b2^2-54*c2*b2*d2+81*d2^2 +12*d2*b2^3)^(1/2))^(1/3)+3*(1/3*c2-1/9*b2^2)/(36*c2*b2-108*d2-8*b2^3+12*(12 * c2^3-3*c2^2*b2^2-54*c2*b2*d2+81*d2^2+12*d2*b2^3)^(1/2))^(1/3)-1/3*b2+1/2*i*3 ^ (1/2)*(1/6*(36*c2*b2-108*d2-8*b2^3+12*(12*c2^3-3*c2^2*b2^2-54*c2*b2*d2+81*d2 ^ 2+12*d2*b2^3)^(1/2))^(1/3)+6*(1/3*c2-1/9*b2^2)/(36*c2*b2-108*d2-8*b2^3+12*(1 2 *c2^3-3*c2^2*b2^2-54*c2*b2*d2+81*d2^2+12*d2*b2^3)^(1/2))^(1/3)) m6=-1/12*(36*c2*b2-108*d2-8*b2^3+12*(12*c2^3-3*c2^2*b2^2-54*c2*b2*d2+81*d2^2 +12*d2*b2^3)^(1/2))^(1/3)+3*(1/3*c2-1/9*b2^2)/(36*c2*b2-108*d2-8*b2^3+12*(12 * c2^3-3*c2^2*b2^2-54*c2*b2*d2+81*d2^2+12*d2*b2^3)^(1/2))^(1/3)-1/3*b2-1/2*i*3 ^ (1/2)*(1/6*(36*c2*b2-108*d2-8*b2^3+12*(12*c2^3-3*c2^2*b2^2-54*c2*b2*d2+81*d2 ^ 2+12*d2*b2^3)^(1/2))^(1/3)+6*(1/3*c2-1/9*b2^2)/(36*c2*b2-108*d2-8*b2^3+12*(1 2 *c2^3-3*c2^2*b2^2-54*c2*b2*d2+81*d2^2+12*d2*b2^3)^(1/2))^(1/3)) m1 = 236960482905.095 + 0.000335693359375i m2 = 297053239103.921 - 0.0035858154296875i m3 = -308787218822.853 + 3.94180678097079e-006i m4 = 221493702574.501 - 0.000339635166155971i m5 = 82519318467.4274 - 0.000916064571774443i m6 = 253236271445.909 + 0.00450188000146194i 1/12*((64+18*(-260+156*i*3^(1/2))^(1/3)+18*(-260-156*i*3^(1/2))^(1/3)+6*(-16 64+64*(-260+156*i*3^(1/2))^(1/3)+64*(-260-156*i*3^(1/2))^(1/3)+9*(-260+156*i * 3^(1/2))^(2/3)+18*(-260+156*i*3^(1/2))^(1/3)*(-260-156*i*3^(1/2))^(1/3)+9*(- 2 60-156*i*3^(1/2))^(2/3))^(1/2))^(2/3)+40-2*(64+18*(-260+156*i*3^(1/2))^(1/3) + 18*(-260-156*i*3^(1/2))^(1/3)+6*(-1664+64*(-260+156*i*3^(1/2))^(1/3)+64*(-26 0 -156*i*3^(1/2))^(1/3)+9*(-260+156*i*3^(1/2))^(2/3)+18*(-260+156*i*3^(1/2))^( 1 /3)*(-260-156*i*3^(1/2))^(1/3)+9*(-260-156*i*3^(1/2))^(2/3))^(1/2))^(1/3))/( 6 4+18*(-260+156*i*3^(1/2))^(1/3)+18*(-260-156*i*3^(1/2))^(1/3)+6*(-1664+64*(- 2 60+156*i*3^(1/2))^(1/3)+64*(-260-156*i*3^(1/2))^(1/3)+9*(-260+156*i*3^(1/2)) ^ (2/3)+18*(-260+156*i*3^(1/2))^(1/3)*(-260-156*i*3^(1/2))^(1/3)+9*(-260-156*i * 3^(1/2))^(2/3))^(1/2))^(1/3) ans = 0.88545602565321 cos(2*pi/13) ans = 0.88545602565321 (1-(1/12*((64+18*(-260+156*i*3^(1/2))^(1/3)+18*(-260-156*i*3^(1/2))^(1/3)+6* (-1664+64*(-260+156*i*3^(1/2))^(1/3)+64*(-260-156*i*3^(1/2))^(1/3)+9*(-260+1 5 6*i*3^(1/2))^(2/3)+18*(-260+156*i*3^(1/2))^(1/3)*(-260-156*i*3^(1/2))^(1/3)+ 9 *(-260-156*i*3^(1/2))^(2/3))^(1/2))^(2/3)+40-2*(64+18*(-260+156*i*3^(1/2))^( 1 /3)+18*(-260-156*i*3^(1/2))^(1/3)+6*(-1664+64*(-260+156*i*3^(1/2))^(1/3)+64* ( -260-156*i*3^(1/2))^(1/3)+9*(-260+156*i*3^(1/2))^(2/3)+18*(-260+156*i*3^(1/2 ) )^(1/3)*(-260-156*i*3^(1/2))^(1/3)+9*(-260-156*i*3^(1/2))^(2/3))^(1/2))^(1/3 ) )/(64+18*(-260+156*i*3^(1/2))^(1/3)+18*(-260-156*i*3^(1/2))^(1/3)+6*(-1664+6 4 *(-260+156*i*3^(1/2))^(1/3)+64*(-260-156*i*3^(1/2))^(1/3)+9*(-260+156*i*3^(1 / 2))^(2/3)+18*(-260+156*i*3^(1/2))^(1/3)*(-260-156*i*3^(1/2))^(1/3)+9*(-260-1 5 6*i*3^(1/2))^(2/3))^(1/2))^(1/3))^2)^(1/2) ans = 0.464723172043769 sin(2*pi/13) ans = 0.464723172043769 w=exp(2*pi*i/13) w = 0.88545602565321 + 0.464723172043769i g1=(m1/2-(m1^2/4-53^13)^(1/2))^(1/13)*w^3 g2=(m2/2-(m2^2/4-53^13)^(1/2))^(1/13)*w^3 g3=(m3/2+(m3^2/4-53^13)^(1/2))^(1/13)*w^2 g4=(m4/2-(m4^2/4-53^13)^(1/2))^(1/13)*w^6 g5=(m5/2-(m5^2/4-53^13)^(1/2))^(1/13)*w^3 g6=(m6/2+(m6^2/4-53^13)^(1/2))^(1/13)*w^10 g7=(m6/2-(m6^2/4-53^13)^(1/2))^(1/13)*w^3 g8=(m5/2+(m5^2/4-53^13)^(1/2))^(1/13)*w^10 g9=(m4/2+(m4^2/4-53^13)^(1/2))^(1/13)*w^7 g10=(m3/2-(m3^2/4-53^13)^(1/2))^(1/13)*w^11 g11=(m2/2+(m2^2/4-53^13)^(1/2))^(1/13)*w^10 g12=(m1/2+(m1^2/4-53^13)^(1/2))^(1/13)*w^10 g1 = 1.29068602985161 + 7.16478398643992i g2 = 0.653929479296265 + 7.25068108773976i g3 = 5.3383379122479 + 4.94996447812071i g4 = -7.16376618252864 + 1.29632329380409i g5 = 0.144793391783627 + 7.27866985607232i g6 = 1.24788059336569 - 7.17236321059534i g7 = 1.24788059336569 + 7.17236321059531i g8 = 0.144793391783623 - 7.27866985607232i g9 = -7.16376618252864 - 1.29632329380409i g10 = 5.33833791224789 - 4.94996447812071i g11 = 0.653929479296263 - 7.2506810877398i g12 = 1.2906860298516 - 7.16478398643992i k1=(g1+g2+g3+g4+g5+g6+g7+g8+g9+g10+g11+g12-1)/13 k2=(g1*w^12+g2*w^11+g3*w^10+g4*w^9+g5*w^8+g6*w^7+g7*w^6+g8*w^5+g9*w^4+g10*w^ 3+g11*w^2+g12*w-1)/13 k1 = 0.15567095754099 - 5.46571335200077e-015i k2 = 3.29401183119725 - 1.77635683940025e-015i 1/4*(k1+(-k1^2+2*k2+8)^(1/2)) ans = 0.992981096013516 - 1.42705743575742e-015i cos(2*pi/53) ans = 0.992981096013517 === Subject: Re: Expressing a polynomial as a sum of squares Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Hi again, Here's a very simple expression, obtained by simplifying the other one. The original polynomial is: P := 27-10*u^2+27*u^4-128*u*v-10*v^2+108*u^2*v^2+54*u^4*v^2+27*v^4+54*u^2*v^4+27* u^4*v^4; Define R1 and R2, which are sums of squares R1 := (3*u*v-1)^2 + 3*(u-v)^2 + (2/3)*(3*u+3*v-2*sqrt(3))^2 ; R2 := (3*u*v-1)^2 + 3*(u-v)^2 + (2/3)*(3*u+3*v+2*sqrt(3))^2 ; Then, P = (1/3) * R1 * R2, and therefore it's also a sum of squares. Best, -p >> With u and v being real parameters, the following polynomial is never >> negative: >> 27-10u^2+27u^4-128uv-10v^2+108u^2v^2+54u^4v^2+27v^4 +54u^2v^4+27u^4v^4 >> (a) Can this polynomial be written as the sum of squares of >> polynomials? >> (b) If so, can you find an explicit representation? >> (c) Is there a general algorithm that can be used to express a >> polynomial as the >> sum of squares of polynomials when such a representation exists? === Subject: Definition of little n-cubes action: disjointness Originator: bergv@math.uiuc.edu (Maarten Bergvelt) My question is about the requirement in the definition of the little n-cubes operad that in the space of j-cubes the j little cubes are disjoint. This does seem to be important in the action of the little cubes on an iterated loop space, since intuitively allowing overlapping little cubes would preclude in general the continuity of the action. However, allowing overlapping cubes doesn't appear to harm the associativity of the operad composition. My question then is about a space that allows a traditional little cube action but also seems to allow a continuous overlapping little cube action. Has such a thing been studied? Does such an action imply more or less than recognition of the loop degree of said space? If anyone is interested in more details about the space I happen to be looking at--continuous valued cellular automata--preliminary drafts are available at www.math.vt.edu/people/sforcey/automads.pdf === Subject: Paper published by Geometry and Topology Originator: bergv@math.uiuc.edu (Maarten Bergvelt) The following paper has been published: Geometry and Topology, Volume 9 (2005) Paper no. 47, pages 2079--2127 URL: http://www.maths.warwick.ac.uk/gt/GTVol9/paper47.abs.html DOI: 10.2140/gt.2005.9.2079 Title: Rohlin's invariant and gauge theory III. Homology $4$--tori Author(s): Daniel Ruberman, Nikolai Saveliev Abstract: This is the third in our series of papers relating gauge theoretic invariants of certain 4-manifolds with invariants of 3-manifolds derived from Rohlin's theorem. Such relations are well-known in dimension three, starting with Casson's integral lift of the Rohlin invariant of a homology sphere. We consider two invariants of a spin 4-manifold that has the integral homology of a 4-torus. The first is a degree zero Donaldson invariant, counting flat connections on a certain SO(3)-bundle. The second, which depends on the choice of a 1-dimensional cohomology class, is a combination of Rohlin invariants of a 3-manifold carrying the dual homology class. We prove that these invariants, suitably normalized, agree modulo 2, by showing that they coincide with the quadruple cup product of 1-dimensional cohomology classes. Secondary: 57R58 Keywords: Rohlin invariant, Donaldson invariant, equivariant perturbation, homology torus Received: 2 August 2005 Accepted: 25 October 2005 Published: 27 October 2005 Proposed: Ronald Stern Seconded: Ronald Fintushel, Simon Donaldson Author(s) address(es): Department of Mathematics, MS 050, Brandeis University Waltham, MA 02454, USA and Department of Mathematics, University of Miami PO Box 249085, Coral Gables, FL 33124, USA Email: ruberman@brandeis.edu, saveliev@math.miami.edu