Related papers: [0,1] is not a Minimality Detector for [0,1]^2

We consider the modulus of noncompact convexity $\Delta_{X,\phi}(\varepsilon)$ associated with the minimalizable measure of noncompactness $\phi$. We present some properties of this modulus, while the main result of this paper is showing…

Consider a semi-algebraic function $f\colon\mathbb{R}^n \to {\mathbb{R}},$ which is continuous around a point $\bar{x} \in \mathbb{R}^n.$ Using the so--called {\em tangency variety} of $f$ at $\bar{x},$ we first provide necessary and…

Given a multiplicative function $f$, we let $S(x,f)=\sum_{n\leq x}f(n)$ be the associated partial sum. In this note, we show that lower bounds on partial sums of divisor-bounded functions result in lower bounds on the partial sums…

Let $ (\bx(n))_{n \geq 1} $ be an $s-$dimensional Niederreiter-Xing sequence in base $b$. Let $D((\bx(n))_{n = 1}^{N})$ be the discrepancy of the sequence $ (\bx(n))_{n = 1}^{N} $. It is known that $N D((\bx(n))_{n = 1}^{N}) =O(\ln^s N)$ as…

We prove that for any $\alpha\in[0,\omega_1)$ there exists a strongly separately continuous function $f:\ell_\infty\to [0,1]$ such that $f$ belongs to the $(\alpha+1)$'th /$(\alpha+2)$'th/ Baire class and does not belong to the $\alpha$'th…

We prove that if $X$ is a quasi-normed space which possesses an infinite countable dimensional subspace with a separating dual, then it admits a strictly weaker Hausdorff vector topology. Such a topology is constructed explicitly. As an…

Estimates based on 2x2 tables of frequencies are widely used in statistical applications. However, in many cases these tables are incomplete in the sense that the data required to compute the frequencies for a subset of the cells defining…

Given a continuous function $f(x)$, suppose that the sign of $f$ only has finitely many discontinuous points in the interval $[0,1]$. We show how to use a sequence of one dimensional deterministic binary cellular automata to determine the…

We prove a weak-type (1,1) inequality for square functions of non-commutative martingales that are simultaneously bounded in $L^2$ and $L^1$. More precisely, the following non-commutative analogue of a classical result of Burkholder holds:…

This article shows a very elementary and straightforward proof of the Implicit Function Theorem for differentiable maps $F(x,y)$ defined on a finite-dimensional Euclidean space. There are no hypothesis on the continuity of the partial…

The continued fraction mapping maps a number in the interval $[0,1)$ to the sequence of its partial quotients. When restricted to the set of irrationals, which is a subspace of the Euclidean space $\mathbb{R}$, the continued fraction…

We prove that the invariant subspaces of the Hardy operator on $L^2[0,1]$ are the spaces that are limits of sequences of finite dimensional spaces spanned by monomial functions.

In this paper we prove that if a pair of graphs G,H have senders, then they necessarily have connected minimal senders; we also prove that given two fixed graphs that are either 3-connected or triangles there are minimal (G,H)-senders with…

It is shown that if $p$ is a complete type of Lascar rank at least 2 over $A$, in the theory of differentially closed fields of characteristic zero, then there exists a pair of realisations, $a_1$ and $a_2$, such that $p$ has a nonalgebraic…

Let $\mathcal{M}$ be a compact $d$-dimensional submanifold of $\mathbb{R}^N$ with reach $\tau$ and volume $V_{\mathcal M}$. Fix $\epsilon \in (0,1)$. In this paper we prove that a nonlinear function $f: \mathbb{R}^N \rightarrow…

One of the classical results concerning differentiability of continuous functions states that the set $\mathcal{SD}$ of somewhere differentiable functions (i.e., functions which are differentiable at some point) is Haar-null in the space…

The recent literature has intensively studied two classes of nonlocal variational problems, namely the ones related to the minimisation of energy functionals that act on functions in suitable Sobolev-Gagliardo spaces, and the ones related…

We consider the problem of computing the minimum length of functional batch and PIR codes of fixed dimension and for a fixed list size, over an arbitrary finite field. We recover, generalize, and refine several results that were previously…

We introduce adequate concepts of expansion of a digraph to obtain a sequential construction of minimal strong digraphs. We characterize the class of minimal strong digraphs whose expansion preserves the property of minimality. We prove…

Let $S$ be a finite set, $s=|S|\ge6$. Given a non-negative integer $t$, there exists an inclusion-minimal non-Bondy system $\mathscr{A}$ of size $t$ on $S$ if and only if $s+1\le t\le2s$.