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Oleg R. Musin

Publications and source records attributed to Oleg R. Musin.

At least 19 recordsLinked to original sources
arXiv2026

Borsuk-Ulam type theorem for the orthogonal group and orthogonal four-partitions

Makeev [2] stated that every finite Borel measure in $\mathbb R^d$ assigning zero mass to hyperplanes can be cut by $d$ mutually orthogonal hyperplanes so that every pair divides the measure into four equal parts, and outlined a proof strategy, but the key steps were left incomplete. We give a direct and elementary proof. The main ingredient is a Borsuk--Ulam theorem for $O(k)$: every $B_k$-equivariant map from $O(k)$ to a natural representation of the hyperoctahedral group $B_k$ has a zero. An explicit model map has one free orbit of zeros, consisting of the signed eigenbases of a diagonal operator with simple spectrum. A derivative computation and mod-$2$ equivariant degree complete the proof. This is a self-contained proof of a special case of the general Stiefel-manifold theorem in [5].

math.CO↗
arXiv2026

Borsuk-Ulam type theorem for Stiefel manifolds and orthogonal mass partitions

We prove a Borsuk--Ulam-type zero theorem for the Stiefel manifold $V_{n,k}$ equipped with a free action of the hyperoctahedral group $B_k=(\mathbb{Z}/2)^k\rtimes S_k$. The existence of a zero reduces to a nonvanishing condition for explicit polynomials in the truncated ring ${R}_{n,k}=\mathbb{F}_2[a_1,\ldots,a_k]/(a_1^n,a_2^{n-1},\ldots,a_k^{n-k+1})$, converting a topological problem into finite algebra over~$\mathbb{F}_2$. As an application we study equipartitions by mutually orthogonal hyperplanes. We prove that if $(P_{k,n})^m\ne0$ in the partition ring ${R}_{d,k}=\mathbb{F}_2[a_1,\ldots,a_k]/(a_1^{d+1},a_2^{d},\ldots,a_k^{d-k+2})$, then for any $m$ finite Borel measures in $\mathbb{R}^d$ there exist $k$ mutually orthogonal hyperplanes such that every $n$-element subfamily partitions each measure into $2^n$ equal parts. Let $Δ^*(m,k,n)$ denote the smallest such dimension~$d$. We prove lower bounds on $Δ^*(m,k,n)$ for all $m$, $k$, $n$ via a Sard-theoretic dimension argument, and the nonvanishing condition above provides algebraic upper bounds. We establish these bounds in several cases and derive exact values, including $Δ^*(2^j-1,k,2)=2^{j-1}(k+1)-1$ for all $j\ge1$, $k\ge2$. In particular, the MVZ upper bound on $Δ(m,k)$ \cite{MSZ} is achieved by mutually orthogonal hyperplanes: orthogonality comes for free.

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arXiv2026

The Five Distance Theorem For An Arbitrary Norm

The three gap theorem states that the points of the Kronecker sequence $α,2α,\ldots,Nα$, considered modulo one, divide the circle into intervals of at most three distinct lengths. In a two-dimensional nearest-neighbour analogue, Haynes and Marklof proved that the Kronecker sequence modulo an arbitrary unimodular lattice determines at most five distinct nearest-neighbour distances in the Euclidean norm, and that this bound is sharp. Dettmann subsequently constructed examples attaining five distinct distances for every $\ell_p$-norm, $1\leq p\leq\infty$. We prove the corresponding upper bound for every norm on $\mathbb R^2$: for every full-rank lattice $L$, every $\boldsymbolα\in\mathbb R^2$, and every $N\in\mathbb N$, the number of distinct nearest-neighbour distances is at most five. For strictly convex norms, the proof extends the lattice-theoretic argument of Haynes and Marklof by replacing the Euclidean angular estimates with a cone lemma based on a proper Brass angular measure. The result for arbitrary norms is then obtained by a strictly convex perturbation and a limiting argument.

math.MG↗
arXiv2026

On Scarf's theorem for Generalized Cooperative Games

In this paper, we study a generalization of cooperative games with non-transferable utility. In our model, coalitions are replaced by firms: each firm is assigned a resource vector, while the set of utility vectors of a coalition is replaced by a general comprehensive set of feasible payoff vectors of this firm in a common payoff space. We introduce the notions of core and fractional core for such games and relate their existence to homotopy invariants of covers associated with the game. The main result shows that the fractional core is nonempty precisely when the associated cover is homotopicaly nontrivial. As a consequence, we obtain a Scarf-type theorem for generalized cooperative games.

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arXiv2026

Algorithms for orthogonal partitioning into four parts

The famous pancake theorem states that for every finite set $X$ in the plane, there exist two orthogonal lines that divide $X$ into four equal parts. We propose an algorithm whose running time is linear in the number of points in $X$ and prove that this complexity is optimal. We also consider generalizations of the pancake theorem and show that orthogonal hyperplanes can be found in polynomial time.

math.CO↗
arXiv2026

Minimal simplicial spherical mappings with a given degree

This paper studies the minimal number of vertices $λ(n,d)$ required in a triangulation of the $n$-sphere to admit a simplicial map to the boundary of a $(n+1)$-simplex with a given degree $d$. We establish upper bounds for $λ(n,d)$ in dimensions $n \geq 3$. Furthermore, we provide exact formulas for small values of $d$, showing that $λ(n,d)=n+d+3$ for $n \geq 3$ and $d=2,3,4$. A key technical result is the identity $λ(n,d) = λ(d-1,d) + n - d + 1$ for $n \geq d$, which allows us to reduce higher-dimensional cases to lower-dimensional ones. The proofs involve constructive methods based on local modifications of triangulations and combinatorial arguments.

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arXiv2025

On essential simplicial maps $S^3 \rightarrow S^2$

A fiber-uniform bound on the complexity of an essential simplicial map $S^3\rightarrow S^2$ is proven, and the tightness of the bound is investigated. It follows that the triangulation of the Hopf map constructed by Madahar and Sarkaria is minimal in its homotopy class in terms of the number of 3-simplices in the triangulation of $S^3$.

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arXiv2025

Computing the Hopf invariant

We consider Whitehead's integral formula and propose an algorithm for computing the Hopf invariant for simplicial mappings.

math.CO↗
arXiv2025

Borsuk's conjecture for two-distance sets and its equivalent formulation for graphs

Every graph G can be embedded in a Euclidean space as a two-distance set. This allows us to reformulate the analogue of Borsuk's conjecture for two-distance sets in terms of graphs. This conjecture remains open for dimensions from 4 to 63. This short note also discusses an approach for finding counterexamples using graphs, as well as its generalization for s-distance sets.

math.CO↗
arXiv2025

Topology-Aware Activation Functions in Neural Networks

This study explores novel activation functions that enhance the ability of neural networks to manipulate data topology during training. Building on the limitations of traditional activation functions like $\mathrm{ReLU}$, we propose $\mathrm{SmoothSplit}$ and $\mathrm{ParametricSplit}$, which introduce topology "cutting" capabilities. These functions enable networks to transform complex data manifolds effectively, improving performance in scenarios with low-dimensional layers. Through experiments on synthetic and real-world datasets, we demonstrate that $\mathrm{ParametricSplit}$ outperforms traditional activations in low-dimensional settings while maintaining competitive performance in higher-dimensional ones. Our findings highlight the potential of topology-aware activation functions in advancing neural network architectures. The code is available via https://github.com/Snopoff/Topology-Aware-Activations.

cs.LG↗
arXiv2025

Homotopy groups and quantitative Sperner-type lemma

We consider a generalization of Sperner's lemma for a triangulation $T$ of $(m+1)$-discs $D$ whose vertices are colored in $n+2$ colors. A proper coloring of $T$ on the boundary of $D$ determines a simplicial mapping $f:S^m \to S^n$ and the element $x=[f]$ in $π_m(S^n)$. For any $x$ in this homotopy group we define a non-negative integer $μ(x)$. For some cases this invariant can be found explicitly. Namely, if $m=n$ then this number is the Brouwer degree of the mapping $f$. For the case $m=3, n=2$ we found a lower bound for $μ(x)$, where $x$ is the Hopf invariant, and proved that $μ(1)=μ(2)=9$. The main result of this paper is the theorem that the number of fully colored $n$-simplexes in $T$ is not less than $μ([f])$. To prove this theorem we use a generalization of Pontryagin's theorem for manifolds with respect to their boundaries.

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arXiv2023

Logarithmic algorithms for fair division problems

We study the algorithmic complexity of fair division problems with a focus on minimizing the number of queries needed to find an approximate solution with desired accuracy. We show for several classes of fair division problems that under certain natural conditions on sets of preferences, a logarithmic number of queries with respect to accuracy is sufficient.

math.CO↗
arXiv2023

Semidefinite programming bounds for distance distribution of spherical codes

We present an extension of known semidefinite and linear programming upper bounds for spherical codes. We apply the main result for the distance distribution of a spherical code and show that this method can work effectively In particular, we get a shorter solution to the kissing number problem in dimension 4.

math.OC↗
arXiv2023

Circle actions on oriented 4-manifolds

In the present paper, we consider an action of the circle group on a compact oriented 4-manifold. We derive the Atiyah-Hirzebruch formula for the manifold, and associate a graph in terms of data on the fixed point set. We show in the case of isolated fixed points that if an abstract graph satisfies the Atiyah-Hirzebruch formula, then there exists a corresponding 4-dimensional oriented $S^1$-manifold.

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arXiv2023

Balanced 2-subsets

Balanced sets appeared in the 1960s in cooperative game theory as a part of nonempty core conditions. In this paper we present a classification of balanced families containing only 2-element subsets. We also discuss generalizations of the classical Sperner and Tucker lemmas using balanced sets.

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arXiv2022

Log-optimal (d+2)-configurations in d-dimensions

We enumerate and classify all stationary logarithmic configurations of d+2 points on the unit (d-1)-sphere in d-dimensions. In particular, we show that the logarithmic energy attains its relative minima at configurations that consist of two orthogonal to each other regular simplexes of cardinality m and n. The global minimum occurs when m=n if d is even and m=n+1 otherwise. This characterizes a new class of configurations that minimize the logarithmic energy on the (d-1)-sphere for all d. The other two classes known in the literature, the regular simplex and the cross polytope, are both universally optimal configurations.

math.MG↗
arXiv2022

Neighboring mapping points theorem

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 goes to a subset that is small in some sense'. This approach covers the case of mappings m-sphere to n-space with m<n and extends to wider classes of spaces.

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