Related papers: Integer Valued Definable Functions in $\mathbb{R}_…

Fix an odd prime $p$. If $r$ is a positive integer and $f$ a polynomial with coefficients in $\mathbb{F}_{p^r}$, let $P_{p,r}(f)$ be the proportion of $\mathbb{P}^1(\mathbb{F}_{p^r})$ that is periodic with respect to $f$. We show that as…

This paper deals with approximation of smooth convex functions $f$ on an interval by convex algebraic polynomials which interpolate $f$ at the endpoints of this interval. We call such estimates "interpolatory". One important corollary of…

Interpolation and approximation of functionals with conditionally positive definite kernels is considered on sets of centers that are not determining for polynomials. It is shown that polynomial consistency is sufficient in order to define…

Response functions of quantum systems, such as electron Green's functions, magnetic, or charge susceptibilities, describe the response of a system to an external perturbation. They are the central objects of interest in field theories and…

Let $A \cong k\langle X \rangle / I$ be an associative algebra. A finite word over alphabet $X$ is $I${\it-reducible} if its image in $A$ is a $k$-linear combination of length-lexicographically lesser words. An {\it obstruction} in a…

The following result, a consequence of Dumas criterion for irreducibility of polynomials over integers, is generally proved using the notion of Newton diagram: Let $f(x)$ be a polynomial with integer coefficients and $k$ be a positive…

We prove a strengthened form of convexity for operator monotone decreasing positive functions defined on the positive real numbers. This extends Ando and Hiai's work to allow arbitrary positive maps instead of states (or the identity map),…

Let $\mathcal{A}$ denote the class of normalized analytic functions $f$ in the open unit disk defined as $ \mathbb{D}:=\{z\in\mathbb{C}:|z|<1\} $ with $f(0)=0$ and $f'(0)=1$. A function $f\in\mathcal{A}$ is said to be starlike if…

The stretched Gau{\ss}ian function $f(\mathbf{x})=\exp \left(-\|\mathbf{x}\|^s\right)$, as a real function defined on $\mathbb{R}^d$, has found numerous applications in mathematics and physics. For instance, to describe results from…

The arbitrary functions principle says that the fractional part of $nX$ converges stably to an independent random variable uniformly distributed on the unit interval, as soon as the random variable $X$ possesses a density or a…

An almost perfect nonlinear (APN) function (necessarily a polynomial function) on a finite field $\mathbb{F}$ is called exceptional APN, if it is also APN on infinitely many extensions of $\mathbb{F}$. In this article we consider the most…

Let $R^h$ denote the polynomial ring in variables $x_1,\,\ldots,\, x_h$ over a specified field $K$. We consider all of these rings simultaneously, and in each use lexicographic (lex) monomial order with $x_1 > \cdots > x_h$. Given a fixed…

Let A be an arbitrary countable set of reals, for example A=Q. Let g be an arbitrary mapping from A into the positive reals, for example g(a)=2^a. We show how a strictly increasing real function f can be constructed such that f'(x)=g(x) for…

We study holomorphic functions attaining weighted norms and its connections with the classical theory of norm attaining holomorphic functions. We prove that there are polynomials on $\ell_p$ which attain their weighted but not their…

For an infinite iterated function system $\mathbf{f}$ on $[0,1]$ with an attractor $\Lambda(\mathbf{f})$ and for an infinite subset $D\subseteq \mathbb{N}$, consider the set \[ \mathbb E(\mathbf{f},D)= \{ x \in \Lambda(\mathbf{f}):…

We obtain the best approximation in $L^1(\R)$, by entire functions of exponential type, for a class of even functions that includes $e^{-\lambda|x|}$, where $\lambda >0$, $\log |x|$ and $|x|^{\alpha}$, where $-1 < \alpha < 1$. We also give…

In this paper, we obtain several new factorization results for certain classes of polynomials having integer coefficients. In doing so, we use the information about prime factorization of the value taken up by such polynomials and their…

In this paper, we study the properties of integral functionals induced on $L^1_E (S,\mu)$ by closed convex functions on a Euclidean space $E$. We give sufficient conditions for such integral functions to be strongly rotund (well-posed). We…

Extension problems for polynomial valuations on different cones of convex functions are investigated. It is shown that for the classes of functions under consideration, the extension problem reduces to a simple geometric obstruction on the…

We discuss conditions under which a convex cone $\K\subset \R^{\Omega}$ admits a probability $m$ such that $\sup_{k\in \K} m(k)\leq0$. Based on these, we also characterize linear functionals that admit the representation as finitely…