相关论文: Partition regularity and multiplicatively syndetic…

The first half of this paper is largely expository, wherein we present a systematic combinatorial approach to the theory of polynomial (semi)invariants and multilinear invariants of several vectors and covectors, for the classical groups.…

In this paper we consider the computation of approximate solutions for inverse problems in Hilbert spaces. In order to capture the special feature of solutions, non-smooth convex functions are introduced as penalty terms. By exploiting the…

We consider the inverse problem of determining the fragmentation rate from noisy measurements in the growth-fragmentation equation. We use Fourier transform theory on locally compact groups to treat this problem for general fragmentation…

Symmetry group methods are applied to obtain all explicit group-invariant radial solutions to a class of semilinear Schrodinger equations in dimensions $n\neq 1$. Both focusing and defocusing cases of a power nonlinearity are considered,…

We consider a quasilinear parabolic stochastic partial differential equation driven by a multiplicative noise and study regularity properties of its weak solution satisfying classical a priori estimates. In particular, we determine…

Invariant sets are a key ingredient for verifying safety and other properties of cyber-physical systems that mix discrete and continuous dynamics. We adapt the elimination-theoretic Rosenfeld-Gr\"{o}bner algorithm to systematically obtain…

We devise a simple but remarkably accurate iterative routine for calculating the roots of a polynomial of any degree. We demonstrate that our results have significant improvement in accuracy over those obtained by methods used in popular…

Given a subspace arrangement, there are several De Concini-Procesi models associated to it, depending on distinct sets of initial combinatorial data (building sets). The first goal of this paper is to describe, for the root arrangements of…

In this work, we develop an adaptive, multivariate partitioning algorithm for solving mixed-integer nonlinear programs (MINLP) with multi-linear terms to global optimality. This iterative algorithm primarily exploits the advantages of…

Through dualities on representations on tensor powers and symmetric powers respectively, the partition algebra and multiset partition algebra have been used to study long-standing questions in the representation theory of the symmetric…

We use an entropy based method to study two graph maximization problems. We upper bound the number of matchings of fixed size $\ell$ in a $d$-regular graph on $N$ vertices. For $\frac{2\ell}{N}$ bounded away from 0 and 1, the logarithm of…

This paper presents two new constructions related to singular solutions of polynomial systems. The first is a new deflation method for an isolated singular root. This construction uses a single linear differential form defined from the…

We study density and partition properties of polynomial equations in prime variables. We consider equations of the form $a_1h(x_1) + \cdots + a_sh(x_s)=b$, where the $a_i$ and $b$ are fixed coefficients, and $h$ is an arbitrary integer…

We develop a new algebraic setting for treating piecewise functions and distributions together with suitable differential and Rota-Baxter structures. Our treatment aims to provide the algebraic underpinning for symbolic computation systems…

We show that sets of integers lacking the configuration $x$, $x+y$, $x+y^2$ have at most polylogarithmic density.

The complexity of computing the solutions of a system of multivariate polynomial equations by means of Groebner bases computations is upper bounded by a function of the solving degree. In this paper, we discuss how to rigorously estimate…

When we consider the action of a finite group on a polynomial ring, a polynomial unchanged by the action is called an invariant polynomial. A famous result of Noether states that in characteristic zero the maximal degree of a minimal…

We show that a set is almost periodic if and only if the associated exponential sum is concentrated in the minor arcs. Hence binary additive problems involving almost periodic sets can be solved using the circle method.

This paper presents a program analysis method that generates program summaries involving polynomial arithmetic. Our approach builds on prior techniques that use solvable polynomial maps for summarizing loops. These techniques are able to…

We study exponential sums whose coefficients are completely multiplicative and belong to the complex unit disc. Our main result shows that such a sum has substantial cancellation unless the coefficient function is essentially a Dirichlet…