On sums of squares of k-nomials

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 $A=V{V}^T$ 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 $k=2$ 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 $k=2$, and any quaternary quadratic if $k=3$. Furthermore we give some evidence that those are a complete list of such cases.     

Stability of sums of weighted nonnegative random variables

A stability result for sums of weighted nonnegative random variables is established and then it is utilized to obtain, among other things, a slight generalization of the Borel-Cantelli lemma and to show that the work of Jamison, Orey, and Pruitt (Z. Wahrsch. Verw. Gebiete4 (1965), 40–44) on almost sure convergence of weighted averages of independent random variables remains valid if the assumption of independence on the random variables is replaced by pairwise independence.     

Constructing all magic squares of order three

We find by applying MacMahon's partition analysis that all magic squares of order three, up to rotations and reflections, are of two types, each generated by three basis elements. A combinatorial proof of this fact is given.     

Sparsity in sums of squares of polynomials

Representation of a given nonnegative multivariate polynomial in terms of a sum of squares of polynomials has become an essential subject in recent developments of sums of squares optimization and semidefinite programming (SDP) relaxation of polynomial optimization problems. We discuss effective methods to obtain a simpler representation of a sparse polynomial as a sum of squares of sparse polynomials by eliminating redundancy.A considerable part of this work was conducted while this author was visiting Tokyo Institute of Technology. Research supported by Kosef R004-000-2001-00200Mathematics Subject Classification (1991): 90C22, 90C26, 90C30     

The limit of the partial sums process of spatial least squares residuals

We establish a functional central limit theorem for a sequence of least squares residuals of spatial data from a linear regression model. Under mild assumptions on the model we explicitly determine the limit process in the case where the assumed linear model is true. Moreover, in the case where the assumed linear model is not true we explicitly establish the limit process for the localized true regression function under mild conditions. These results can be used to develop non-parametric model checks for linear regression. Our proofs generalize ideas of a univariate geometrical approach due to Bischoff [W. Bischoff, The structure of residual partial sums limit processes of linear regression models, Theory Stoch. Process. 8 (24) (2002) 23–28] which is different to that proposed by MacNeill and Jandhyala [I.B. MacNeill, V.K. Jandhyala, Change-point methods for spatial data, in: G.P. Patil, et al. (Eds.), Multivariate Environmental Statistics. Papers Presented at the 7th International Conference on Multivariate Analysis held at Pennsylvania State University, University Park, PA, USA, May 5–9 1992, in: Ser. Stat. Probab., vol. 6, North-Holland, Amsterdam, 1993, pp. 289–306 (in English)]. Moreover, Xie and MacNeill [L. Xie, I.B. MacNeill, Spatial residual processes and boundary detection, South African Statist. J. 40 (1) (2006) 33–53] established the limit process of set indexed partial sums of regression residuals. In our framework we get that result as an immediate consequence of a result of Alexander and Pyke [K.S. Alexander, R. Pyke, A uniform central limit theorem for set-indexed partial-sum processes with finite variance, Ann. Probab. 14 (1986) 582–597]. The reason for that is that by our geometrical approach we recognize the structure of the limit process: it is a projection of the Brownian sheet onto a certain subspace of the reproducing kernel Hilbert space of the Brownian sheet. Several examples are discussed.     

Packing equal squares into a large square

Let s(x) denote the maximum number of non-overlapping unit squares which can be packed into a large square of side length x. Let W(x)=x²−s(x) denote the “wasted” area, i.e., the area not covered by the unit squares. In this note we prove that

     

Error estimation for a weighted minimum-norm least squares solution with positive definite weights

The weighted least squares problem is considered. Given a generally inconsistent system of linear algebraic equations, error estimates are obtained for its weighted minimum-norm least squares solution under perturbations of the matrix and the right-hand side, including the case of rank modifications of the perturbed matrix.     

On the number of primitive representations of integers as sums of squares

Formulas for the number of primitive representations of any integer n as a sum of k squares are given, for 2 ≤ k ≤ 8, and for certain values of n, for 9 ≤ k ≤ 12. The formulas have a similar structure and are striking for their simplicity. Dedicated to Richard Askey on the occasion of his 70th birthday. 2000 Mathematics Subject Classification Primary—11E25; Secondary—05A15, 33E05.     

On sums of a prime and four prime squares in short intervals

In this paper, we prove that each sufficiently large integer N ≠1（mod 3） can be written as N=p＋p1^2＋p2^2＋p3^2＋p4^2, with
｜p-N/5｜≤U,｜pj-√N/5｜≤U,j=1,2,3,4,
where U=N^2/20＋c and p,pj are primes.     

Complete convergence of weighted sums of martingale differences

LetF _oF ₁ ... be an increasing family of -algebras. For eachn1,X _n isF _n-measurable, andE(X _n|F_n–1) is zero almost surely, andE(|E_n|^p|F_n–1) is bounded by a finite constant almost surely for somep2. Leta _n1,...,a _nn be constants. Conditions are given to establish the complete convergence of (a _n1 X ₁+...+a _nnX_n)/n ^1/p , thereby obtaining an extension of Chow's (1966) result for the case of independent and identically distributed random variables. Whenp>2, the conditions are an improvement on existing results for the case of independence and identical distribution.     

Sums of squares in octonion algebras

Sums of squares in composition algebras are investigated using methods from the theory of quadratic forms. For any integer octonion algebras of level and of level are constructed.

     

Regressions for sums of squares of spacings

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.     

A weighted least squares method for scattered data fitting

In this paper, we present a weighted least squares method to fit scattered data with noise. Existence and uniqueness of a solution are proved and an error bound is derived. The numerical experiments illustrate that our weighted least squares method has better performance than the traditional least squares method in case of noisy data.     

The Nicolas and Robin inequalities with sums of two squares

In 1984, G. Robin proved that the Riemann hypothesis is true if and only if the Robin inequality σ(n) < e ^γ n log log n holds for every integer n > 5040, where σ(n) is the sum of divisors function, and γ is the Euler–Mascheroni constant. We exhibit a broad class of subsets of the natural numbers such that the Robin inequality holds for all but finitely many . As a special case, we determine the finitely many numbers of the form n = a ² + b ² that do not satisfy the Robin inequality. In fact, we prove our assertions with the Nicolas inequality n/φ(n) < e ^γ log log n; since σ(n)/n < n/φ(n) for n > 1 our results for the Robin inequality follow at once.      

On the representation of integers as sums of an odd number of squares

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.     

Generation of symmetric exponential sums

In this paper, a new method for generation of infinite series of symmetric identities written for exponential sums in real numbers is proposed. Such systems have numerous applications in theory of numbers, chaos theory, algorithmic complexity, dynamic systems, etc. Properties of generated identities are studied. Relations of the introduced method for generation of symmetric exponential sums to the Morse-Hedlund sequence and to the theory of magic squares are established.     

Nonnegative functions as squares or sums of squares

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.     

Equal sums of four seventh powers

In this paper, the method used to find the smallest, nontrivial, positive integer solution of is discussed. The solution is

Factors enabling this discovery are advances in computing power, available workstation memory, and the appropriate choice of optimized algorithms.

     

Non-existence of degree bounds for weighted sums of squares representations

Given a fixed family of polynomials , we study the problem of representing polynomials in the form

(*)

f=s₀+s₁h₁++s_rh_r