arXiv research

Study determines Borsuk-Ulam property for maps between torus and Klein bottle.

The paper connects sectional category and parametrized Borsuk-Ulam property for fibrations.

In this paper we present a Kakutani type theorem that is equivalent to the Borsuk--Ulam theorem for manifolds.

Extends Borsuk-Ulam theorem with applications in sphere coverings and colorings.

New example solves topological dynamics problem.

The paper generalizes the Borsuk-Ulam theorem to surfaces and cyclic actions.

Let M be a closed, connected 3-manifold which admits Nil geometry, we determine all free involutions $τ$ on M and the Borsuk-Ulam index of $(M,τ)$.

We consider spaces with free involutions that satisfy the Borsuk - Ulam theorems (BUT-spaces). There are several equivalent definitions for BUT-spaces that can be considered as their properties. Our main technical tool is Yang's cohomological index.

Let M be a Seifert manifold which belongs to the geometry Flat. In this work we determine all the free involutions τ on M, and the Borsuk-Ulam indice of (M,τ).

Let M and N be topological spaces such that M admits a free involution $\τ$. A homotopy class $β$ $\in$ [M, N ] is said to have the Borsuk-Ulam property with respect to $\τ$ if for every representative map f : M $\rightarrow$ N of $β$, there exists a point x $\in$ M such that f ($\τ$ (x)) = f (x). In the case where M i…

The paper proves no multiple equichordal points exist in convex bodies.

Let $M$ be a topological space that admits a free involution $τ$, and let $N$ be a topological space. A homotopy class $β\in [ M,N ]$ is said to have {\it the Borsuk-Ulam property with respect to $τ$} if for every representative map $f: M \to N$ of $β$, there exists a point $x \in M$ such that $f(τ(x))= f(x)$. In this …

Let (X, t, S) be a triple, where S is a compact, connected surface without boundary, and t is a free cellular involution on a CW-complex X. The triple (X, t, S) is said to satisfy the Borsuk-Ulam property if for every continuous map f:X-->S, there exists a point x belonging to X satisfying f(t(x))=f(x). In this paper, …

New theorem connects distant points and identical points on manifolds.

Thickenings of a metric space capture local geometric properties of the space. Here we exhibit applications of lower bounding the topology of thickenings of the circle and more generally the sphere. We explain interconnections with the geometry of circle actions on Euclidean space, the structure of zeros of trigonometr…

The paper studies which branched covers can be lifted to braided embeddings.

The paper connects geometric and topological concepts to bound distances between metric spaces.

We compute the rings $H^*(N;\mathbb{F}_2)$ for $N$ a closed $\mathbb{S}ol^3$-manifold and then determine the Borsuk-Ulam indices $BU(N,φ)$ with $φ\not=0$ in $H^1(N;\mathbb{F}_2)$.

Bounds and constructions for Gromov-Hausdorff distance between spheres.

We prove several new results around Gromov's waist theorem. We give a simple proof of Vaaler's theorem on sections of the unit cube using the Borsuk--Ulam--Crofton technique. We consider waists of real and complex projective spaces, flat tori, convex bodies in Euclidean space. We establish waist-type results in terms o…

The paper explores how to reduce classification tasks to optimization problems in Euclidean space.

The study connects projective codes to the distribution of zeros of odd maps.

Introduces injective category number for continuous maps, linking classical and contemporary research.

Researchers create black holes isometric to extremal Reissner-Nordström.

Tucker and Ky Fan's lemma are combinatorial analogs of the Borsuk-Ulam theorem (BUT). In 1996, Yu. A. Shashkin proved a version of Fan's lemma, which is a combinatorial analog of the odd mapping theorem (OMT). We consider generalizations of these lemmas for BUT-manifolds, i.e. for manifolds that satisfy BUT. Proofs rel…

We introduce and study a new family of extensions for the Borsuk-Ulam and topological Radon type theorems. The defining idea for this new family is to replace requirements of the form `a subset that is large in some sense goes to a singleton' with requirements of the milder form `a subset that is large in some sense go…

Let $V$ and $W$ be orthogonal representations of $G$ with $V^G= W^G=\{0\}$. Let $S(V )$ be the sphere of $V$ and $f : S(V ) \to W$ be a $G$-equivariant mapping. We give an estimate for the dimension of the set $Z_f=f^{-1}\{0\}$ in terms of $\dim V$ and $\dim W$, if $G$ is the torus $\mathbb T^k$, or the $p$-torus $\ma…

The width $w$ of a curve $γ$ in Euclidean space $R^n$ is the infimum of the distances between all pairs of parallel hyperplanes which bound $γ$, while its inradius $r$ is the supremum of the radii of all spheres which are contained in the convex hull of $γ$ and are disjoint from $γ$. We use a mixture of topological and…

It is proved that for a product action of $(\mathbb Z_p)^k$ on a product of (mod p) homology spheres $N^{n_1}\times...\times N^{n_k}$, where all $n_i$'s are assumed to be odd if $p$ is odd, and any continuous map $f\colon N^{n_1}\times...\times N^{n_k}\to \mathbb R^m$ the set $A(f)=\{x\in N^{n_1}\times...\times N^{n_k}…

Study of $f$-neighbors in Riemannian manifolds, proving infinite set of distances.

We survey the status of some decision problems for 3-manifolds and their fundamental groups. This includes the classical decision problems for finitely presented groups (Word Problem, Conjugacy Problem, Isomorphism Problem), and also the Homeomorphism Problem for 3-manifolds and the Membership Problem for 3-manifold gr…

Optimal transport reformulates multiple quantile hedging problem.

Solves four problems related to circle families in the plane.

Solves four problems related to sphere families in 3D space.

The paper solves optimal control problems for various convex sets using convex trigonometry.

This paper is a tutorial for eigenvalue and generalized eigenvalue problems. We first introduce eigenvalue problem, eigen-decomposition (spectral decomposition), and generalized eigenvalue problem. Then, we mention the optimization problems which yield to the eigenvalue and generalized eigenvalue problems. We also prov…

This paper solves the Christoffel problem in hyperbolic space and its equivalent on spheres.

In the present paper, the primal-dual problem consisting of the investment risk minimization problem and the expected return maximization problem in the mean-variance model is discussed using replica analysis. As a natural extension of the investment risk minimization problem under only a budget constraint that we anal…

Study proves only origin-centered spheres solve certain curvature problems.

MathChat uses LLM agents to solve challenging math problems through conversational problem-solving.

The paper solves a generalized Christoffel-Minkowski problem using a curvature flow.

Paper solves Gromov-Wasserstein for point clouds efficiently.

The paper explains how microlocal analysis solves geometric inverse problems.

Proves NP and co-NP status for knot core recognition in solid torus.

The min-max problem, also known as the saddle point problem, is a class of optimization problems which minimizes and maximizes two subsets of variables simultaneously. This class of problems can be used to formulate a wide range of signal processing and communication (SPCOM) problems. Despite its popularity, most exist…

A new method solves complex control problems with random coefficients.

This is a survey of some problems in geometric group theory which I find interesting. The problems are from different areas of group theory. Each section is devoted to problems in one area. It contains an introduction where I give some necessary definitions and motivations, problems and some discussions of them. For ea…

We present updates to the problems on Hirzebruch's 1954 problem list focussing on open problems, and on those where substantial progress has been made in recent years. We discuss some purely topological problems, as well as geometric problems about (almost) complex structures, both algebraic and non-algebraic, about co…