相关论文: The Bramble-Hilbert Lemma
The multiplication theorem for univariate Hermite polynomials $H_k(\lambda x)$ is well-known. In this paper we generalize this result to multivariate Hermite polynomials ${\rm H}_{\bf k}({\mathbf{\Lambda}}{\bf x};{\mathbf{\Sigma}})$, and…
We first propose two conjectural estimates on Diophantine approximation of logarithms of algebraic numbers. Next we discuss the state of the art and we give further partial results on this topic.
In the study of Hilbert schemes, the integer partition $\lambda$ helps researchers identify some geometric and combinatorial properties of the scheme in question. To aid researchers in extracting such information from a Hilbert polynomial,…
In this expositional paper, we discuss commutative algebra -- a study inspired by the properties of integers, rational numbers, and real numbers. In particular, we investigate rings and ideals, and their various properties. After, we…
We establish a new perturbation theory for orthogonal polynomials using a Riemann--Hilbert approach and consider applications in numerical linear algebra and random matrix theory. This new approach shows that the orthogonal polynomials with…
The Nyman-Beurling criterion, equivalent to the Riemann hypothesis (RH), is an approximation problem in the space of square integrable functions on $(0,\infty)$, involving dilations of the fractional part function by factors…
We give a direct harmonic approximation lemma for local minima of quasiconvex multiple integrals that entails their $\mathrm{C}^{1,\alpha}$ or $\mathrm{C}^{\infty}$-partial regularity. Different from previous contributions, the method is…
A fundamental problem in numerical analysis and approximation theory is approximating smooth functions by polynomials. A much harder version under recent consideration is to enforce bounds constraints on the approximating polynomial. In…
For the bi-orthogonal polynomials with the third degree polynomial potential functions, the 3 x 3 matrix Riemann-Hilbert problem is explicitly constructed. The developed approach admits an extension to the bi-orthogonal polynomials with…
The paper is mostly a survey on recent results in Diophantine approximation, with emphasis on properties of exponents measuring various notions of Diophantine <approximation.
We give a bracket polynomial expression for intermediate terms between discriminant and resultant for pair of binary forms. As an application of the bracket polynomial expression, we give an algebraic proof of the algebraic independence of…
The Bohnenblust-Hille inequality and its variants have found applications in several areas of Mathematics and related fields. The control of the constants for the variant for complex $m$-homogeneous polynomials is of special interest for…
The inverse of a large matrix can often be accurately approximated by a polynomial of degree significantly lower than the order of the matrix. The iteration polynomial generated by a run of the GMRES algorithm is a good candidate, and its…
This paper describes infinite sets of polynomial equations in infinitely many variables with the property that the existence of a solution or even an approximate solution for every finite subset of the equations implies the existence of a…
This work provides closed-form solutions and minimum achievable errors for a large class of low-rank approximation problems in Hilbert spaces. The proposed theorem generalizes to the case of bounded linear operators the previous results…
Polynomials are common algebraic structures, which are often used to approximate functions including probability distributions. This paper proposes to directly define polynomial distributions in order to describe stochastic properties of…
The aim of this paper is to introduce Bell polynomials and numbers of the second kind and poly-Bell polynomials and numbers of the second kind, and to derive their explicit expressions, recurrence relations and some identities involving…
This note gives a simple approach to q-analogues of some results associated with Abel polynomials.
We give a formula and an estimation for the number of irreducible polynomials in two (or more) variables over a finite field.
We study the Rokhlin lemma in the context of infinite measure-preserving bijections, and completely classify such bijections up to $\lambda$-approximate conjugacy, where $\lambda$ is the infinite measure which is preserved. This sharpens…