arXiv subjects

Dmitri Fomin

Publications and source records attributed to Dmitri Fomin.

6 recordsLinked to original sources

arXiv2026

Closed polylines with fixed self-intersection index

We investigate the existence of closed polylines (also known as closed polygonal chains or self-crossing polygons) that intersect each of their edges the same number of times. The most general question in this corner of combinatorial geometry asks for all pairs $(n, k)$ such that there exists a closed polyline with $n$ edges, each intersecting the same polyline exactly $k$ times. For $k = 1$ and $k = 2$, this is a very simple question answered several decades ago. In this article, we present a complete solution for $k = 3, 4, 6$, as well as the proof of some non-existence theorems. In conclusion, we show that, for an arbitrary positive integer $k$, a polyline of the required type exists for any sufficiently large integer $n$ such that $nk$ is even.

math.MG↗

arXiv2025

Math Matters of the Past (The Very First Mathematical Olympiads)

The history of the very first mathematical contests for high school students is discussed. The main body of the article is dedicated to the mathematical and scientific contests held in imperial Russia in the XIX century. More specifically, we discuss and analyze the recently discovered evidence of the city-level official "olympiads" in several school subjects (mathematics, history, and languages), which were organized in the Russian capital city of Saint Petersburg in 1840-1842. Educational and social aspects of the high school educational system of that time are discussed as well.

math.HO↗

arXiv2023

Is the Multiset of $n$ Integers Uniquely Determined by the Multiset of Its $s$-sums?

In 1957 Leo Moser published a problem in American Mathematical Monthly asking whether knowing the set of all pairwise sums of five numbers one could determine the original numbers. Problem was quickly generalized as "Is it always possible to restore a collection of $n$ numbers from the collection of its $s$-sums?"; it turned out to be a very interesting and nontrivial question in additive number theory and combinatorics. On its sixtieth anniversary we present here a survey of all the known research, results and techniques used in various attempts to solve this problem. Some new findings and open questions are presented as well.

math.NT↗

arXiv2021

Elementary proof for the bounds of the complexity of a planar multigraph and the size of a prime rectangular squaring

Two results (together with their relatively elementary proofs) are presented. The first one presents the upper boundary on the number of spanning trees in a finite planar multigraph, proving that the complexity (the number of spanning trees) of a planar multigraph with $n$ edges does not exceed $τ^n$, where $τ\approx 1.8637$. This result is, quite possibly, already known and/or published -- my quick web search did not turn up anything but that does not really prove much. It also seems plausible that this inequality is actually true for the "best possible" value of $τ^* \approx 1.7916$. The second result uses the above theorem to improve on the well-known Conway's inequality for the number of tiles in a prime rectangular squaring.

math.CO↗

arXiv2019

Moser Polynomials and Eulerian Numbers

Article presents a short investigation into some properties of the Moser polynomials which appear in various problems from algebraic combinatorics. For instance, these polynomials can be used to solve the Generalized Moser's Problem on multiset recovery: Can a collection (multiset) of $n$ numbers can be uniquely restored given the collection of its $s$-sums? We prove some explicit formulas showing relationships between Moser polynomials and such popular algebraic combinatorial sequences as Eulerian and Stirling numbers.

math.CO↗

arXiv2017

Upper Bounds For Hitting Times Of Random Walks On Sparse Graphs

We obtain upper bounds (in most cases, sharp) for the hitting times of random walks on finite undirected graphs expressed as functions of the graph's number of edges. In particular, we show that the maximum hitting time for a simple random walk on a connected graph with $m$ edges is at most $m^2$. Similar bounds are given for the settings involving arbitrary edge-weight and edge-cost functions. Upper bounds of this type are especially useful for sparse graphs.

math.CO↗