Related papers: Sum of squares I: scalar functions

This is the second in a series of three papers dealing with sums of squares and hypoellipticity in the infinitely degenerate regime. We give sharp conditions on the entries of a positive semidefinite NxN matrix function F on n-dimensional…

This is the third in a series of papers dealing with sums of squares and hypoellipticity in the infinitely degenerate regime. We establish a C^2,delta generalization of M. Christ's sum of squares theorem, and use a bootstrap argument with…

In the process of proving a sharpened form of G\r{a}rding's inequality, Fefferman & Phong demonstrated that every non-negative function $f\in C^{3,1}(\mathbb{R}^n)$ can be written as a finite sum of squares of functions in…

In this article we present certain formulas involving arithmetical functions. In the first part we study properties of sums and product formulas for general type of arithmetic functions. In the second part we apply these formulas to the…

In this paper certain classes of infinite sums involving special functions are evaluated analytically by application of basic quantum mechanical principles to simple models of half harmonic oscillator and a particle trapped inside an…

We investigate the number of half-regular squares required to decompose a non-negative $C^{k,\alpha}(\mathbb{R}^n)$ function into a sum of squares. Each non-negative $C^{3,1}(\mathbb{R}^n)$ function is known to be a finite SOS in…

Let X be a finite set of points in R^n. A polynomial p nonnegative on X can be written as a sum of squares of rational functions modulo the vanishing ideal I(X). From the point of view of applications, such as polynomial optimization, we…

Let $f(n)$ be an arithmetic function with $f(n) \ll n^\alpha$ for some $\alpha\in[0,1)$ and let $\lfloor .\rfloor $ denote the integer part function. In this paper, we evaluate asymptotically the sums $$\sum_{n_{1}n_{2}\leq x}f \left(…

It is shown how sums of squares of real valued functions can be used to give new proofs of the reality of the zeros of the Bessel functions $J_\alpha (z)$ when $\alpha \ge -1,$ confluent hypergeometric functions ${}_0F_1(c\/; z)$ when $c>0$…

Inspired by the fact that the sum of the cubes of the first $n$ naturals is equal to the square of their sum, we explore, for each $n$, the Diophantine equation representing all non-trivial sets of $n$ integers with this property. We find…

We ask whether every polynomial function that is non-negative on a real algebraic curve can be expressed as a sum of squares in the coordinate ring. Scheiderer has classified all irreducible curves for which this is the case. For reducible…

Let $p$ be an odd prime and let $f(x)=\sum_{i=1}^ka_ix^{p^{\alpha_i}+1}\in\Bbb F_{p^n}[x]$, where $0\le \alpha_1<...<\alpha_k$. We consider the exponential sum $S(f,n)=\sum_{x\in\Bbb F_{p^n}}e_n(f(x))$, where $e_n(y)=e^{2\pi…

Let $n\in\mathbb{N}$ and let $K$ be a field with a henselian discrete valuation of rank $n$ with hereditarily euclidean residue field. Let $F/K$ be an algebraic function field in one variable. We show that the Pythagoras number of $F$ is…

We begin by studying the surface area of an ellipsoid in n-dimensional Euclidean space as the function of the lengths of the semi-axes. We write down an explicit formula as an integral over the unit sphere in n-dimensions and use this…

Finite trigonometric sums appear in various branches of Physics, Mathematics and their applications. For p; q to coprime positive integers and r we consider the finite trigonometric sums involving the product of three trigonometric…

We simplify and give an alternative proof of hypoellipticity for generalizations of the singular sum of squares of complex vector fields studied by Kohn, with an appendix by Derridj and Tartakoff, in the Annals of Mathematics, vol. 162 no.…

We establish a lower bound for the frequency with which an irreducible monic cubic polynomial with negative discriminant can be expressed as a sum of two squares ($\square_{2}$). This provides a quantitative answer to a question posed by…

We calculate some infinite sums containing the digamma function in closed-form. These sums are related either to the incomplete beta function or to the Bessel functions. The calculations yield interesting new results as by-products, such as…

We study the surface area of an ellipsoid in n-dimensional Euclidean space as the function of the lengths of their major semi-axes. We write down an explicit formula as an integral over the unit sphere, use the formula to derive convexity…

We argue that a necessary condition for hypoellipticity is that the polar is not a spiral domain.