Related papers: A discrete form of the Beckman-Quarles theorem for…

A homemorphism between domains in $\mathbb R^n$, $n\ge 2$ is quasiconformal, with its intricate analytic and geometric consequences, if the (pointwise) linear dilatation -- a purely metric quantity -- is uniformly bounded. Gehring proved…

Let $(X,d)$ be a finite metric space with $|X|=n$. For a positive integer $k$ we define $A_k(X)$ to be the quotient set of all $k$-subsets of $X$ by isometry, and we denote $|A_k(X)|$ by $a_k$. The sequence $(a_1,a_2,\ldots,a_{n})$ is…

In this paper we study the problems of the following kind: For a pair of topological spaces $X$ and $Y$ find sufficient conditions that under every continuous map $f : X\to Y$ a pair of sufficiently distant points is mapped to a single…

We present a new approach to the theory of asymptotic properties of solutions of difference equations. Usually, two sequences $x,y$ are called asymptotically equivalent if the sequence $x-y$ is convergent to zero i.e., $x-y\in c_0$, where…

A (positive definite and non-classic integral) quadratic form is called strongly $s$-regular if it satisfies a strong regularity property on the number of representations of squares of integers. In this article, we prove that for any…

A parametric version of Brouwer's Fixed Point Theorem, which is proven using the fixed-point index, states that for every continuous mapping $f : (X \times Y) \to Y$, where $X$ is nonempty, compact, and connected subset of a Hausdorff…

A finite set of unlabelled points in Euclidean space is the simplest representation of many real objects from mineral rocks to sculptures. Since most solid objects are rigid, their natural equivalence is rigid motion or isometry maintaining…

We show quantitative versions of classic results in discrete geometry, where the size of a convex set is determined by some non-negative function. We give versions of this kind for the selection theorem of B\'ar\'any, the existence of weak…

Building on the work of Avraham, Rubin, and Shelah, we aim to build a variant of the Fra\"iss\'e theory for uncountable models built from finite submodels. With this aim, we generalize the notion of an increasing set of reals to other…

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 Erdos-Szekeres theorem states that for any natural k there is a natural number g(k) such that any set of at least g(k) points on a plane in general position contains a set of k points that are the extreme points of a convex polytope. We…

We construct a consistent example of a topological space $Y=X \cup \{\infty\}$ such that: 1) $Y$ is regular. 2) Every $G_\delta$ subset of $Y$ is open. 3) The point $\infty$ is not isolated, but it is not in the closure of any discrete…

Let X be a smooth projective variety over the complex numbers, and let D be an ample divisor in X. For which spaces Y is the restriction map r: Hom(X, Y) -> Hom(D, Y) an isomorphism? Using positive characteristic methods, we give a fairly…

This paper concerns the characterisation of second order marginals for random sets in a discrete setting. Under the instance of unit covariances, this problem possesses a combinatorial symmetry, exploited jointly in the companion paper to…

The classical Steinhaus theorem (\cite{Steinhaus1920}) says that if $A \subset {\Bbb R}^d$ has positive Lebesgue measure than $A-A=\{x-y: x,y \in A\}$ contains an open ball. We obtain some quantitative lower bounds on the size of this ball…

Following a recent idea by Ball, we introduce the notion of strongly truncated Riesz space with a suitable spectrum. We prove that, under an extra Archimedean type condition, any strongly truncated Riesz space is isomorphic to a uniformly…

A linear mapping upon real n-dimensional space, where the dimension n is odd, has a real eigenvalue-eigenvector pair. The corresponding statement for complex vector spaces holds true for any dimension n, but should be easy to demonstrate…

Let $S$ be a smooth, totally real, compact immersion in $\mathbb{C}^n$ of real dimension $m \leq n$, which is locally polynomially convex and it has finitely many points where it self-intersects finitely many times, transversely or…

It is shown that the existence of a biseparating map between a large class of spaces of vector-valued continuous functions A(X,E) and A(Y,F) implies that some compactifications of X and Y are homeomorphic. In some cases, conditions are…

The visibility graph Vis(X) of a discrete point set X in the plane has vertex set X and an edge xy for every two points x,y\in X whenever there is no other point in X on the line segment between x and y. We show that for every graph G,…