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…

Study exact partition recovery with same-cluster oracle, bounded error.

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.

The paper studies self-Bäcklund curves in centroaffine geometry using elliptic functions.

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.

Let $M$ be a manifold, $V$ be a vector field on $M$, and $B$ be a Banach space. For any fixed function $f:M\rightarrow B$ and any fixed complex number $λ$, we study Hyers-Ulam stability of the global differential equation $Vy=λy+f$.

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.

Proves existence of sentences to identify homeomorphic manifolds.

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…

The abstract discusses transversality for infinite dimensional manifolds.

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…

This note explores norms beyond ultrametric inequalities in non-Archimedean analysis.

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…

Stability result for nearly isometric subspaces and Finsler surfaces.

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…

This paper is an expository account of the development of soliton mathematics, from its inception in famous numerical experiments of Fermi-Pasta-Ulam and Zabusky-Kruskal to the recent synthesis of Terng-Uhlenbeck (dg-ga/9707004) that explains hidden symmetries of soliton equations in terms of loop-groups acting by dres…

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}…

New family of measurable pseudo-Anosov maps on spheres.

New quasimorphisms constructed for manifolds with negative curvature.

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

The study applies wealth thermalization hypothesis to social networks and explains inequality.

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.