Related papers: A remark on sums of squares of complex vector fiel…
We refine and extend quantitative bounds, on the fraction of nonnegative polynomials that are sums of squares, to the multihomogenous case.
We consider second order elliptic operators with real, nonsymmetric coefficient functions which are subject to mixed boundary conditions. The aim of this paper is to provide uniform resolvent estimates for the realizations of these…
We consider a class of non-doubling manifolds $\mathcal{M}$ that are the connected sum of a finite number of $N$-dimensional manifolds of the form $\mathbb{R}^{n_{i}} \times \mathcal{M}_{i}$. Following on from the work of Hassell and the…
Extending the corresponding notion for matrices or bounded linear operators on a Hilbert space we define a generalized Schur complement for a non-negative linear operator mapping a linear space into its dual and derive some of its…
Counting integral binary quadratic forms with certain restrictions is a classical problem. In this paper, we count binary quadratic forms of fixed discriminant given restrictions on the size of their coefficients. We accomplish this by…
In this chapter we present the sums of Hermitian squares approach to noncommutative polynomial optimization problems. This is an extension of the sums of squares approach for polynomial optimization arising from real algebraic geometry. We…
In this paper, we establish a condition on the coefficients of differential operators generated in the space of square-integrable functions on the entire real line by an ordinary differential expression with periodic, complex-valued…
We use a variation of the Circle Method, along with the Saddle Point Method, to obtain an asymptotic formula for the number of partitions of a number n into integers which are sums of two squares. Unlike previous work on partitions into…
These notes present Sobolev-Gagliardo-Nirenberg endpoint estimates for classes of homogeneous vector differential operators. Away of the endpoint cases, the classical Calder\'on-Zygmund estimates show that the ellipticity is necessary and…
The purpose of this note is to extend the results of V. Guillemin on elliptic self-adjoint pseudodifferential operators of order one, from operators defined on smooth functions on a closed manifold to operators defined on smooth sections in…
The summatory function of $t_j(n^2)$ is estimated, where $H_j(s) = \sum_{n=1}^\infty t_j(n)n^{-s}$ is the Hecke series of a non-holomorphic cusp form. The analogous problem of holomorphic cusp forms is also treated.
The spectral decomposition for an explicit second-order differential operator $T$ is determined. The spectrum consists of a continuous part with multiplicity two, a continuous part with multiplicity one, and a finite discrete part with…
An orbitope is the convex hull of an orbit of a point under the action of a compact group. We derive bounds on volumes of sections of polar bodies of orbitopes, extending our previously developed methods. As an application we realize the…
Computers are good at evaluating finite sums in closed form, but there are finite sums which do not have closed forms. Summands which do not produce a closed form can often be ``fixed'' by multiplying them by a suitable polynomial. We…
We consider an arbitrary metric graph, to which we glue another graph with edges of lengths proportional to $\varepsilon$, where $\varepsilon$ is a small positive parameter. On such graph, we consider a general self-adjoint second order…
We analyze geometry of the second order differential operators, having in mind applications to Batalin--Vilkovisky formalism in quantum field theory. As we show, an exhaustive picture can be obtained by considering pencils of differential…
In this paper, we revise the concept of noncommutative vector fields introduced previously in Ref. [1,2], extending the framework, adding new results and clarifying the old ones. Using appropriate algebraic tools certain shortcomings in the…
The paper is devoted to a comprehensive second-order study of a remarkable class of convex extended-real-valued functions that is highly important in many aspects of nonlinear and variational analysis, specifically those related to…
We connect the existence of a ternary classical universal quadratic form over a totally real number field $K$ with the property that all totally positive multiples of 2 are sums of squares (if $K$ does not contain $\sqrt 2$ or contains a…
We apply Kr\"{o}necker's approximation theorem to measure (in a topological sense) a set of constants which turn a vector field into a non-globally hypoelliptic operator. We present situations in which this set is a discrete enumerable…