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1.
In a recent work, S. Cooper (J. Number Theory 103:135–162, [1988]) conjectured a formula for r
2k+1(p
2), the number of ways p
2 can be expressed as a sum of 2k+1 squares. Inspired by this conjecture, we obtain an explicit formula for r
2k+1(n
2),n≥1.
Dedicated to Srinivasa Ramanujan. 相似文献
2.
Shaun Cooper 《Journal of Number Theory》2003,103(2):135-162
Let rk(n) denote the number of representations of an integer n as a sum of k squares. We prove that for odd primes p,
3.
In 2005, Boman et al. introduced the concept of factor width for a real symmetric positive semidefinite matrix. This is the smallest positive integer k for which the matrix A can be written as with each column of V containing at most k non-zeros. The cones of matrices of bounded factor width give a hierarchy of inner approximations to the PSD cone. In the polynomial optimization context, a Gram matrix of a polynomial having factor width k corresponds to the polynomial being a sum of squares of polynomials of support at most k. Recently, Ahmadi and Majumdar [1], explored this connection for case and proposed to relax the reliance on polynomials that are sums of squares in semidefinite programming to polynomials that are sums of binomial squares In this paper, we prove some results on the geometry of the cones of matrices with bounded factor widths and their duals, and use them to derive new results on the limitations of certificates of nonnegativity of quadratic forms by sums of k-nomial squares using standard multipliers. In particular we show that they never help for symmetric quadratics, for any quadratic if , and any quaternary quadratic if . Furthermore we give some evidence that those are a complete list of such cases. 相似文献
4.
Lovász and Schrijver (SIAM J. Optim. 1:166–190, 1991) have constructed semidefinite relaxations for the stable set polytope
of a graph G = (V,E) by a sequence of lift-and-project operations; their procedure finds the stable set polytope in at most α(G) steps, where α(G) is the stability number of G. Two other hierarchies of semidefinite bounds for the stability number have been proposed by Lasserre (SIAM J. Optim. 11:796–817,
2001; Lecture Notes in Computer Science, Springer, Berlin Heidelberg New York, pp 293–303, 2001) and by de Klerk and Pasechnik
(SIAM J. Optim. 12:875–892), which are based on relaxing nonnegativity of a polynomial by requiring the existence of a sum
of squares decomposition. The hierarchy of Lasserre is known to converge in α(G) steps as it refines the hierarchy of Lovász and Schrijver, and de Klerk and Pasechnik conjecture that their hierarchy also
finds the stability number after α(G) steps. We prove this conjecture for graphs with stability number at most 8 and we show that the hierarchy of Lasserre refines
the hierarchy of de Klerk and Pasechnik.
相似文献
6.
Koji Tasaka 《The Ramanujan Journal》2014,33(1):1-21
In this paper, we prove a conjecture of Chan and Chua for the number of representations of integers as sums of $8s$ integral squares. The proof uses a theorem of Imamo?lu and Kohnen, and the double shuffle relations satisfied by the double Eisenstein series of level 2. 相似文献
7.
Jean-Michel Bony 《Journal of Functional Analysis》2006,232(1):137-147
We prove that, for n?4, there are C∞ nonnegative functions f of n variables (and even flat ones for n?5) which are not a finite sum of squares of C2 functions. For n=1, where a decomposition in a sum of two squares is always possible, we investigate the possibility of writing f=g2. We prove that, in general, one cannot require a better regularity than g∈C1. Assuming that f vanishes at all its local minima, we prove that it is possible to get g∈C2 but that one cannot require any additional regularity. 相似文献
8.
Starting with a new formula for the regression of sum of squares of spacings (SSS) with respect to the maximum we present
a characterization of a family of beta type mixtures in terms of the constancy of regression of normalized SSS of order statistics.
Related characterization for records describes a family of minima of independent Weibull distributions. 相似文献
9.
Given a fixed family of polynomials , we study the problem of representing polynomials in the form(*)
f=s0+s1h1++srhr