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Fourier extension is an approximation method that alleviates the periodicity requirements of Fourier series and avoids the Gibbs phenomenon when approximating functions. We describe a similar extension approach using regular wavelet bases…

An old conjecture of Sierpinski asserts that for every integer k \ge 2, there is a number m for which the equation \phi(x)=m has exactly k solutions. Here \phi is Euler's totient function. In 1961, Schinzel deduced this conjecture from his…

We introduce and study the minimum distance function of a graded ideal in a polynomial ring with coefficients in a field, and show that it generalizes the minimum distance of projective Reed-Muller-type codes over finite fields. This gives…

We construct a vector field E from the real and imaginary parts of an entire function xi (z) which arises in the quantum statistical mechanics of relativistic gases when the spatial dimension d is analytically continued into the complex z…

Let F be either R or C. Let $(\pi,V)$ be an irreducible admissible smooth \Fre representation of GL(2n,F). A Shalika functional $\phi:V \to \C$ is a continuous linear functional such that for any $g\in GL_n(F), A \in \Mat_{n \times n}(F)$…

In this paper we consider a wide class of generalized Lipschitz extension problems and the corresponding problem of finding absolutely minimal Lipschitz extensions. We prove that if a minimal Lipschitz extension exists, then under certain…

In this paper we study number fields which are Euclidean with respect to a function different from the absolute value of the norm. We also show that the Euclidean minimum with respect to weighted norms may be irrational and not isolated.

A celebrated result of Dol'nikov, and of \v{Z}ivaljevi\'c and Vre\'cica, asserts that for every collection of $m$ measures $\mu_1,\dots,\mu_m$ on the Euclidean space $\mathbb R^{n + m - 1}$ there exists a projection onto an $n$-dimensional…

For a finite abelian group $G$, the Erd\H{o}s-Ginzburg-Ziv constant $\mathfrak{s}(G)$ is the smallest $s$ such that every sequence of $s$ (not necessarily distinct) elements of $G$ has a zero-sum subsequence of length…

Every o-minimal expansion R-tilde of the real field has an o-minimal expansion P(R-tilde) in which the solutions to Pfaffian equations with definable C^1 coefficients are definable.

Recently it is shown that the non-relativistic quantum formulations can be derived from a least observability principle [36]. In this paper, we apply the principle to massive scalar fields, and derive the Schr\"{o}dinger equation of the…

Several important conjectures in Fractal Geometry can be summarised as follows: If the dimension of a self-similar measure in $\mathbb{R}$ does not equal its expected value, then the underlying iterated function system contains an exact…

For a matrix $W \in \mathbb{Z}^{m \times n}$, $m \leq n$, and a convex function $g: \mathbb{R}^m \rightarrow \mathbb{R}$, we are interested in minimizing $f(x) = g(Wx)$ over the set $\{0,1\}^n$. We will study separable convex functions and…

This paper deals with extensions of vector-valued functions on finite graphs fulfilling distinguished minimality properties. We show that so-called lex and L-lex minimal extensions are actually the same and call them minimal Lipschitz…

In this paper, we investigate the minimality of the map $\frac{x}{\|x\|}$ from the euclidean unit ball $\mathbf{B}^n$ to its boundary $\mathbb{S}^{n-1}$ for weighted energy functionals of the type $E\_{p,f}= \int\_{\mathbf{B}^n}f(r)\|\nabla…

We consider the fundamental task of optimising a real-valued function defined in a potentially high-dimensional Euclidean space, such as the loss function in many machine-learning tasks or the logarithm of the probability distribution in…

Let $Z_n(s; a_1,..., a_n)$ be the Epstein zeta function defined as the meromorphic continuation of the function \sum_{k\in\Z^n\setminus\{0\}}(\sum_{i=1}^n [a_i k_i]^2)^{-s}, \text{Re} s>\frac{n}{2} to the complex plane. We show that for…

If alpha is an irrational number, we define Yoccoz's Brjuno function Phi by Phi(alpha)=sum_{n geq 0} alpha_0*alpha_1*...*alpha_{n-1}*log(1/alpha_n), where alpha_0 is the fractional part of alpha and alpha_{n+1} is the fractional part of…

This paper addresses to Nichtnegativstellens\"atze for definable functions in o-minimal structures on $(\mathbb{R}, +, \cdot).$ Namely, let $f, g_1, \ldots, g_l \colon \mathbb{R}^n \to \mathbb{R}$ be definable $C^p$-functions ($p \ge 2$)…

Given functions $f,g: [n] \rightarrow [n]$ do there exist $n$ points $A_1,A_2\ldots A_n$ in some metric space such that $A_{f(i)},A_{g(i)}$ are the points closest and farthest from point $A_i$? In this paper we characterize precisely which…