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In this paper, we obtain some sufficient conditions to guarantee the existence of multiple points of maps from $S^m$ to $\mathbb{R}^d$. Our main tool is the ideal-valued index of $G$-space defined by E. Fadell and S. Husseini. We obtain…

We give a sharp lower bound to the largest possible Euclidean norm of signed sums of $n$ vectors in the plane. This is achieved by connecting the signed vector sum problem to the isoperimetric problem for the circumradius of polygons. In…

Let $X$ be a finite set in the Euclidean space $\mathbb{R}^d$. If the squared distance between any two distinct points in $X$ is an odd integer, then the cardinality of $X$ is bounded above by $d+2$, as shown by Rosenfeld (1997) or Smith…

We introduce and discuss the concept of \emph{$n$-distance}, a generalization to $n$ elements of the classical notion of distance obtained by replacing the triangle inequality with the so-called simplex inequality $$ d(x_1, \ldots,…

Following Ghomi and Tabachnikov we study topological obstructions to totally skew embeddings of a smooth manifold M in Euclidean spaces. This problem is naturally related to the question of estimating the geometric dimension of the stable…

A finite set $X$ in the $d$-dimensional Euclidean space is called an $s$-distance set if the set of distances between any two distinct points of $X$ has size $s$. In 1977, Larman-Rogers-Seidel proved that if the cardinality of an…

We investigate the average minimum cost of a bipartite matching between two samples of n independent random points uniformly distributed on a unit cube in d $\ge$ 3 dimensions, where the matching cost between two points is given by any…

This paper is devoted to the investigation of generalized Abel equation $\dot{x}=S(x,t)=\sum^m_{i=0}a_i(t)x^i$, where $a_i\in\mathrm C^{\infty}([0,1])$. A solution $x(t)$ is called a {\em periodic solution} if $x(0)=x(1)$. In order to…

Let P be a set of n points in $\mathbb{R}^d$. A point x is said to be a centerpoint of P if x is contained in every convex object that contains more than $dn\over d+1$ points of P. We call a point x a strong centerpoint for a family of…

Given a finite set of points $S\subset\mathbb{R}^d$, a $k$-set of $S$ is a subset $A \subset S$ of size $k$ which can be strictly separated from $S \setminus A $ by a hyperplane. Similarly, a $k$-facet of a point set $S$ in general position…

Let $F$ be an $n$-point set in $\mathbb{K}^d$ with $\mathbb{K}\in\{\mathbb{R},\mathbb{Z}\}$ and $d\geq 2$. A (discrete) X-ray of $F$ in direction $s$ gives the number of points of $F$ on each line parallel to $s$. We define…

A design is a finite set of points in a space on which every "simple" functions averages to its global mean. Illustrative examples of simple functions are low-degree polynomials on the Euclidean sphere or on the Hamming cube. We prove lower…

We derive analytic solutions for the potential and field in a one-dimensional system of masses or charges with periodic boundary conditions, in other words Ewald sums for one dimension. We also provide a set of tools for exploring the…

Let $\mathbb{F}_q^d$ be a $d$-dimensional vector space over a finite field $\mathbb{F}_q$ with $q$ elements. For $x\in \mathbb{F}_q^d$, let $\|x\| = x_1^2+\dots+x_d^2$. By abuse of terminology, we shall call $\|\cdot\|$ a norm on…

The proof of the theorem, which states that the Euclidean metric on the set of random points in an $n$-dimensional Euclidean space with the distribution of a special class, converges in probability in the limit $n\rightarrow\infty$ to the…

For the case of the single-O($N$)-vector linear sigma models the critical behaviour following from any $A_k$ singularity in the action is worked out in the double scaling limit $N \rightarrow \infty$, $f_r \rightarrow f_r^c$, $2 \leq r \leq…

For a finite point set $P \subset \mathbb{R}^d$, denote by $\text{diam}(P)$ the ratio of the largest to the smallest distances between pairs of points in $P$. Let $c_{d, \alpha}(n)$ be the largest integer $c$ such that any $n$-point set $P…

The center of distances of a metric space $(X,d)$ is the set $C(X)$ of all $t\in \mathbb R^+$ for which the equation $d(x,p)=t$ has a solution for each $p\in X$. We prove the inequality $|C(X)| \le 1 + \lfloor \log_2 n \rfloor$ for all…

We consider, for $a,l\geq1,$ $b,s,\alpha>0,$ and $p>q\geq1,$ the homogeneous Dirichlet problem for the equation $-\Delta_{p}u=\lambda u^{q-1}+\beta u^{a-1}\left\vert \nabla u\right\vert ^{b}+mu^{l-1}e^{\alpha u^{s}}$ in a smooth bounded…

We propose a new $(1+O(\varepsilon))$-approximation algorithm with $O(n+ 1/\varepsilon^{\frac{(d-1)}{2}})$ running time for computing the diameter of a set of $n$ points in the $d$-dimensional Euclidean space for a fixed dimension $d$,…