arXiv subjects

Eugene Stepanov

Publications and source records attributed to Eugene Stepanov.

At least 19 recordsLinked to original sources

arXiv2026

On a discrete max-plus transportation problem

We provide an explicit algorithm to solve the idempotent analogue of the discrete Monge-Kantorovich optimal mass transportation problem with the usual real number field replaced by the tropical (max-plus) semiring, in which addition is defined as the maximum and product is defined as usual addition, with minus infinity and zero playing the roles of additive and multiplicative identities. Such a problem may be naturally called tropical or "max-plus" optimal transportation problem. We show that the solutions to the latter, called the optimal tropical plans, may not correspond to perfect matchings even if the data (max-plus probability measures) have all weights equal to zero, in contrast with the classical discrete optimal transportation analogue, where perfect matching optimal plans in similar situations always exist. Nevertheless, in some randomized situation the existence of perfect matching optimal tropical plans may occur rather frequently. At last, we prove that the uniqueness of solutions of the optimal tropical transportation problem is quite rare.

math.OC↗

arXiv2025

Minimal points and non-holonomic controllability on compact manifolds

We study the problem of non-holonomic point-to-point controllability for ODEs with drift possessing some recursion property of the flow (nonwandering or chain recurrence) and satisfying various versions of H\"ormander condition (also known as Lie bracket generating condition). We show that for the flows on compact manifolds, it suffices to assume the validity of the H\"ormander condition on the closure of the set of their minimal points only. Also, we construct a 2-dimensional example of a drift defining a chain recurrent flow and the vector fields defining the non-holonomic constraint, which together satisfy the H\"ormander condition, but the flow is not controllable in the direction of the given vector fields.

math.DS↗

arXiv2025

Chow and Rashevskii meet Sobolev

We prove a weak version of the Chow-Rashevskii theorem for vector fields having only Sobolev regularity and generating suitable flows as selections of solutions to the respective ODEs, for a.e.\ initial datum.

math.DS↗

arXiv2025

On the signature of squared distance matrices of metric measure spaces

We consider the numbers of positive and negative eigenvalues of matrices of squared distances between randomly sampled i.i.d. points in a given metric measure space. These numbers and their limits, as the number of points grows, in fact contain some important information about the whole space. In particular, by knowing them, we can determine whether this space can be isometrically embedded in the Hilbert space. We show that the limits of these numbers exist almost surely, are nonrandom and the same for all Borel probability measures of full support, and, moreover, are naturally related to the operators defining the multidimensional scaling (MDS) method. We also relate them to the signature of the pseudo-Euclidean space in which the given metric space can be isometrically embedded. In addition, we provide several examples of explicit calculations or just estimates of those limits for sample metric spaces. In particular, for a large class of countable spaces (for instance, containing all graphs with bounded intrinsic metrics), we get that the number of negative eigenvalues increases to infinity as the size of samples grows. However, we are able to provide examples when the number of samples grows to infinity and the numbers of both negative and positive eigenvalues increases to infinity, or the number of positive eigenvalues is bounded (but as large as desired), and the number of positive ones is fixed. Finally, we consider the example of the universal countable Rado graph.

math.MG↗

arXiv2024

On the total surface area of potato packings

We prove that if we fill without gaps a bag with infinitely many potatoes, in such a way that they touch each other in few points, then the total surface area of the potatoes must be infinite. In this context potatoes are measurable subsets of the Euclidean space, the bag is any open set of the same space. As we show, this result also holds in the general context of doubling (even locally) metric measure spaces satisfying Poincar\'e inequality, in particular in smooth Riemannian manifolds and even in some sub-Riemannian spaces.

math.MG↗

arXiv2024

Reconstruction of manifold embeddings into Euclidean spaces via intrinsic distances

We consider the problem of reconstructing an embedding of a compact connected Riemannian manifold in a Euclidean space up to an almost isometry, given the information on intrinsic distances between points from its ``sufficiently large'' subset. This is one of the classical manifold learning problems. It happens that the most popular methods to deal with such a problem, with a long history in data science, namely, the classical Multidimensional scaling (MDS) and the Maximum variance unfolding (MVU), actually miss the point and may provide results very far from an isometry; moreover, they may even give no bi-Lipshitz embedding. We will provide an easy variational formulation of this problem, which leads to an algorithm always providing an almost isometric embedding with the distortion of original distances as small as desired (the parameter regulating the upper bound for the desired distortion is an input parameter of this algorithm).

math.OC↗

arXiv2024

On eigenvalues and eigenfunctions of the operators defining multidimensional scaling on some symmetric spaces

We study asymptotics of the eigenvalues and eigenfunctions of the operators used for constructing multidimensional scaling (MDS) on compact connected Riemannian manifolds, in particular on closed connected symmetric spaces. They are the limits of eigenvalues and eigenvectors of squared distance matrices of an increasing sequence of finite subsets covering the space densely in the limit. We show that for products of spheres and real projective spaces, the numbers of positive and negative eigenvalues of these operators are both infinite. We also find a class of spaces (namely $\mathbb{RP}^n$ with odd $n>1$) whose MDS defining operators are not trace class, and original distances cannot be reconstructed from the eigenvalues and eigenfunctions of these operators.

math.MG↗

arXiv2023

On the Steiner tree connecting a fractal set

We construct an example of an infinite planar embedded self-similar binary tree $\Sigma$ which is the essentially unique solution to the Steiner problem of finding the shortest connection of a given planar self-similar fractal set $C$ of positive Hausdorff dimension. The set $C$ can be considered the set of leaves, or the ``boundary``, of the tree $\Sigma$, so that $\Sigma$ is an irreducible solution to the Steiner problem with datum $C$ (i.e. $\Sigma\setminus C$ is connected).

math.MG↗

arXiv2022

On the geometric structure of currents tangent to smooth distributions

It is well known that a k-dimensional smooth surface in a Euclidean space cannot be tangent to a non-involutive distribution of k-dimensional planes. In this paper we discuss the extension of this statement to weaker notions of surfaces, namely integral and normal currents. We find out that integral currents behave to this regard exactly as smooth surfaces, while the behaviour of normal currents is rather multifaceted. This issue is strictly related to a geometric property of the boundary of currents, which is also discussed in details.

math.DG↗

arXiv2022

Isoperimetric planar clusters with infinitely many regions

An infinite cluster $\mathbf E$ in $\mathbb R^d$ is a sequence of disjoint measurable sets $E_k\subset \mathbb R^d$, $k\in \mathbb N$, called regions of the cluster. Given the volumes $a_k\ge 0$ of the regions $E_k$, a natural question is the existence of a cluster $\mathbf E$ which has finite and minimal perimeter $P(\mathbf E)$ among all clusters with regions having such volumes. We prove that such a cluster exists in the planar case $d=2$, for any choice of the areas $a_k$ with $\sum \sqrt a_k < \infty$. We also show the existence of a bounded minimizer with the property $P(\mathbf E)=\mathcal H^1(\partial \mathbf E)$, where $\partial mathbf E$ denotes the measure theoretic boundary of the cluster. We also provide several examples of infinite isoperimetric clusters for anisotropic and fractional perimeters.

math.AP↗

arXiv2022

Infinite multidimensional scaling for metric measure spaces

For a given metric measure space $(X,d,μ)$ we consider finite samples of points, calculate the matrix of distances between them and then reconstruct the points in some finite-dimensional space using the multidimensional scaling (MDS) algorithm with this distance matrix as an input. We show that this procedure gives a natural limit as the number of points in the samples grows to infinity and the density of points approaches the measure $μ$. This limit can be viewed as "infinite MDS" embedding of the original space, now not anymore into a finite-dimensional space but rather into an infinitedimensional Hilbert space. We further show that this embedding is stable with respect to the natural convergence of metric measure spaces. However, contrary to what is usually believed in applications, we show that in many cases it does not preserve distances, nor is even bi-Lipschitz, but may provide snowflake (Assouad-type) embeddings of the original space to a Hilbert space (this is, for instance, the case of a sphere and a flat torus equipped with their geodesic distances).

math.MG↗

arXiv2022

Constructive controllability for incompressible vector fields

We give a constructive proof of a global controllability result for an autonomous system of ODEs guided by bounded locally Lipschitz and divergence free (i.e.\ incompressible) vector field, when the phase space is the whole Euclidean space and the vector field satisfies so-called vanishing mean drift condition. For the case when the ODE is defined over some smooth compact connected Riemannian manifold, we significantly strengthen the assertion of the known controllability theorem in absence of nonholonomic constraints by proving that one can find a control steering the state vector from one given point to another by using the observations of only the state vector, i.e., in other words, by changing slightly the vector field, and such a change can be made small not only in uniform, but also in Lipschitz (i.e. $C^1$) topology.

math.DS↗

arXiv2021

Isoperimetric clusters in homogeneous spaces via concentration compactness

We show the existence of generalized clusters of a finite or even infinite number of sets, with minimal total perimeter and given total masses, in metric measure spaces homogeneous with respect to a group acting by measure preserving homeomorphisms, for a quite wide range of perimeter functionals. Such generalized clusters are a natural "relaxed'' version of a cluster and can be thought of as ``albums'' with possibly infinite pages, having a minimal cluster drawn on each page, the total perimeter and the vector of masses being calculated by summation over all pages, the total perimeter being minimal among all generalized clusters with the same masses. The examples include any anisotropic perimeter in a Euclidean space, as well as a hyperbolic plane with the Riemannian perimeter and Heisenberg groups with a canonical left invariant perimeter or its equivalent versions.

math.AP↗

arXiv2021

On exterior differential systems involving differentials of Hölder functions

We study the validity of an extension of Frobenius theorem on integral manifolds for some classes of Pfaff-type systems of partial differential equations involving multidimensional "rough" signals, i.e. "differentials" of given Hölder continuous functions interpreted in a suitable way, similarly to Young Differential Equations in Rough Paths theory. This can be seen as a tool to study solvability of exterior differential systems involving rough differential forms, i.e. the forms involving weak (distributional) derivatives of highly irregular (e.g. Hölder continuous) functions; the solutions (integral manifolds) being also some very weakly regular geometric structures.

math.AP↗

arXiv2020

Lie brackets of nonsmooth vector fields and commutation of their flows

It is well-known that the flows generated by two smooth vector fields commute, if the Lie bracket of these vector fields vanishes. This assertion is known to extend to Lipschitz continuous vector fields, up to interpreting the vanishing of their Lie bracket in the sense of almost everywhere equality. We show that this cannot be extended to general a.e. differentiable vector fields admitting a.e. unique flows. We show however that the extension holds when one field is Lipschitz continuous and the other one is merely Sobolev regular (but admitting a regular Lagrangian flow).

math.FA↗

arXiv2020

Decomposition of acyclic normal currents in a metric space

We prove that every acyclic normal one-dimensional real Ambrosio-Kirchheim current in a Polish (i.e. complete separable metric) space can be decomposed in curves, thus generalizing the analogous classical result proven by S. Smirnov in Euclidean space setting. The same assertion is true for every complete metric space under a suitable set-theoretic assumption.

math.DG↗

arXiv2020

Towards geometric integration of rough differential forms

We provide a draft of a theory of geometric integration of rough differential forms which are generalizations of classical (smooth) differential forms to similar objects with very low regularity, for instance, involving Hölder continuous functions that may be nowhere differentiable. Borrowing ideas from the theory of rough paths, we show that such a geometric integration can be constructed substituting appropriately differentials with more general asymptotic expansions. This can be seen as the basis of geometric integration similar to that used in geometric measure theory, but without any underlying differentiable structure, thus allowing Lipschitz functions and rectifiable sets to be substituted by far less regular objects (e.g. Hölder functions and their images which may be purely unrectifiable). Our construction includes both the one-dimensional Young integral and multidimensional integrals introduced recently by R. Züst. To simplify the exposition, we limit ourselves to integration of forms of dimensions not exceeding two.

math.DG↗

arXiv2019

Integration of nonsmooth $\boldsymbol{2}$-forms: from Young to Itô and Stratonovich

We show that geometric integrals of the type $\int_Ωf\, d g^1\wedge \, d g^2$ can be defined over a two-dimensional domain $Ω$ when the functions $f$, $g^1$, $g^2\colon \mathbb{R}^2\to \mathbb{R}$ are just Hölder continuous with sufficiently large Hölder exponents and the boundary of $Ω$ has sufficiently small dimension, by summing over a refining sequence of partitions the discrete Stratonovich or Itô type terms. This leads to a two-dimensional extension of the classical Young integral that coincides with the integral introduced recently by R.~Züst. We further show that the Stratonovich-type summation allows to weaken the requirements on Hölder exponents of the map $g=(g^1,g^2)$ when $f(x)=F(x,g(x))$ with $F$ sufficiently regular. The technique relies upon an extension of the sewing lemma from Rough paths theory to alternating functions of two-dimensional oriented simplices, also proven in the paper.

math.FA↗