arXiv subjects

Vladimir Gurvich

Publications and source records attributed to Vladimir Gurvich.

At least 19 recordsLinked to original sources

arXiv2026

On pairs of triangular numbers whose product is a perfect square and pairs of intervals of successive integers with equal sums of squares

In 1778 Leonhard Euler characterized triangular numbers that are perfect squares. Obviously, the product of any two such numbers is a perfect square too. Yet, there are many other solutions, that is, pairs $(k,k')$ such that $k(k+1)k'(k'+1)$ is a perfect square. We give explicit formulas characterizing all these square triangular pairs by means of some integer positive polynomials, which is the primary novelty of our work. This result allows us to find all pairs of intervals of successive integers with equal sums of squares in case when the lengths of two intervals in a pair differ by 1. It is known that there is a one-to-one correspondence between the square triangular numbers and nearly isosceles Pythagorean triples: $n^2 + (n+1)^2 = N^2$. Both are generated by the same Fermat-Pell recursion. This is a special case of our result, when the lengths of the two intervals are 2 and 1.

math.NT↗

arXiv2026

Subgroup Inconsistency and the Dilution Effect in Levene's Test for Homoscedasticity

Levene's test for homoscedasticity is a standard procedure used to evaluate whether multiple groups of independent observations share a common variance. Because Levene's test relies on an Analysis of Variance (ANOVA) applied to transformed absolute or squared deviations, it structurally mirrors the statistical properties of ANOVA itself. Let $μ_j$ and $σ^2_j$ denote the expectation and variance of the observations in group $j$. It was previously established that for a given significance level $α$, ANOVA can result in a logical contradiction: failing to reject the global null hypothesis $H_0: μ_1 = μ_2 = μ_3$ while simultaneously rejecting the localized hypothesis $H_0': μ_1 = μ_2$ with the same or higher confidence. In this paper, we show that Levene's test directly inherits this same ``paradox'' regarding group variances $σ^2_j$. We provide theoretical reasoning and a numerical illustration of this inconsistency, demonstrating how the addition of a well-behaved third group can dilute the test statistic and mask a significant localized variance discrepancy.

stat.ME↗

arXiv2026

Partitioning set $[n] = \{1, \dots, n\}$ into subsets of size at most $m$ such that all sums are powers of $m$

Given integers $m > 1$ and $n > 0$, we say that a partition of the set $[n] = \{1, \dots, n\}$ is {\em $m$-good} if the number of elements in each part is at most $m$ and their sum is a power of $m$. It is easily seen that for every $n$ there is a unique 2-good partition of $[n]$ and for each $m > 3$ there is no $m$-good partition for infinitely many $n$. Less is known for $m=3$. We conjecture that a 3-good partition of $[n]$ exists for each $n$ and prove that a minimal counter-example, if any, must be of the form: (i) $n = 3^t + 3k +2$, where $t > 0$ and (ii) $0 \leq k < \frac{3^{t-1}-1}{2}$; moreover, (iii) $k \neq \frac{3^\ell - 1}{2}$ for all nonnegative integers $\ell < t$. Obviously, these conditions can be equivalently rewritten as: (i$'$) $n \equiv 2 \; \pmod 3$, (ii$'$) $3k + 2 < \frac{3^{t+1} + 1}{2}$, and (iii$'$) $3k + 2 \neq \frac{3^{\ell + 1} + 1}{2}$ for $0 \leq \ell < t$. By computations, the above conjecture was verified for $n \leq 844$. We also modify the statement slightly and prove it for the 3-good quasi-partitions, which cover all numbers of $[n] = \{1, \dots, 3^t+3k+2\}$ once, except $3^t$, which is covered twice. Finally, we prove that a 3-good partition of $[n]$ is unique if {\centering $n \in \{1,2,3,4, 3^t-4, 3^t-2, 3^t-1, 3^t, 3^t+1, 3^t+2, 3^t+3, 3^t+5 \;\; \text{for} \;\; % \mid t \geq 2\}$}, and there are exactly two 3-good partitions of $[n]$ for $n = 3^t-3$. We conjecture that the number of 3-good partitions is greater than 2 for any other $n$, except 13.

math.CO↗

arXiv2026

Bridge distances for networks

Let $G = (V,E)$ be a finite directed graph with a non-negative real length $μ_e$ assigned to every directed edge $e \in E$. We assume that $μ_e = +\infty$ for every non-edge $e \not\in E$. Fix any two distinct vertices $a, b \in V$. A directed path from $a$ to $b$ is called an $(a,b)$-path. An edge $e$ is called an $(a,b)$-bridge if it belongs to all $(a,b)$-paths. Furthermore, it is not difficult to show that all $(a,b)$-paths pass all $(a,b)$-bridges in the same order. Define the distance $μ(a,b)$ from $a$ to $b$ as the sum of lengths of all $(a,b)$-bridges. Furthermore, $μ(a,b) = \infty$ if there are no $(a,b)$-paths and $μ(a,b) = 0$ if $(a,b)$-paths exist but there are no $(a,b)$-bridges. It is easily seen that $μ(a,b)$ can be computed in polynomial time and the metric inequality $μ(a,b) \leq μ(a,c) + μ(c,b)$ holds for every $a,b,c \in V$. Furthermore, equality holds if and only if each $(a,b)$-bridge is either an $(a,c)$- or a $(c,b)$-bridge. \newline We will show that this is a special limit case $r=s \rightarrow 0$ of the inequality $μ(a,b)^{s/r} \leq μ(a,c)^{s/r} + μ(c,b)^{s/r}$ obtained for all positive real parameters $r$ and $s$ in the paper ``Metric and ultrametric inequalities for directed graphs'', Discrete Appl. Math. 314 (2022) 93--104, along with 3 other limit cases $r=s \rightarrow \infty$, $r=1, s \rightarrow \infty$, and $s = 1, r \rightarrow 0$, considered in that paper.

math.CO↗

arXiv2025

Conformality of Minimal Transversals of Maximal Cliques

Given a hypergraph $H$, the dual hypergraph of $H$ is the hypergraph of all minimal transversals of $H$. A hypergraph is conformal if it is the family of maximal cliques of a graph. In a recent work, Boros, Gurvich, Milanič, and Uno (Journal of Graph Theory, 2025) studied conformality of dual hypergraphs and proved several results related to this property, leading in particular to a polynomial-time algorithm for recognizing graphs in which all minimal transversals of maximal cliques have size at most $k$, for any fixed $k$. In this follow-up work, we provide a novel aspect to the study of graph clique transversals, by considering the dual conformality property from the perspective of graphs. More precisely, we study graphs for which the family of minimal transversals of maximal cliques is conformal. Such graphs are called clique dually conformal (CDC for short). It turns out that the class of CDC graphs is a rich generalization of the class of $P_4$-free graphs. As our main results, we completely characterize CDC graphs within the families of triangle-free graphs and split graphs. Both characterizations lead to polynomial-time recognition algorithms. Generalizing the fact that every $P_4$-free graph is CDC, we also show that the class of CDC graphs is closed under substitution, in the strong sense that substituting a graph $H$ for a vertex of a graph $G$ results in a CDC graph if and only if both $G$ and $H$ are CDC.

math.CO↗

arXiv2025

Two-person Positive Shortest Path Games Have Nash Equilibria in Pure Stationary Strategies

We prove that every finite two-person shortest path game, where the local cost of every move is positive for each player, has a Nash equilibrium (NE) in pure stationary strategies, which can be computed in polynomial time. We also extend the existence result to infinite graphs with finite out-degrees. Moreover, our proof gives that a terminal NE (in which the play is a path from the initial position to a terminal) exists provided at least one of the two players can guarantee reaching a terminal. If none of the players can do it, in other words, if each of the two players has a strategy that separates all terminals from the initial position $s$, then, obviously, a cyclic NE exists, although its cost is infinite for both players, since we restrict ourselves to positive games. We conjecture that a terminal NE exists too, provided there exists a directed path from $s$ to a terminal. However, this is open. We extend our result to short paths interdiction games, where at each vertex, we allow one player to block some of the arcs and the other player to choose one of the non-blocked arcs. Assuming that blocking sets are chosen from an independence system given by an oracle, we give an algorithm for computing a NE in time $O(|E|(\log|V|+τ))$, where $V$ is the set of vertices, $E$ is the set of arcs, and $τ$ is the maximum time taken by the oracle on any input.

cs.DM↗

arXiv2025

Critical issues with the Pearson's chi-square test

Pearson's chi-square tests are among the most commonly applied statistical tools across a wide range of scientific disciplines, including medicine, engineering, biology, sociology, marketing and business. However, its usage in some areas is not correct. For example, the chi-square test for homogeneity of proportions (that is, comparing proportions across groups in a contingency table) is frequently used to verify if the rows of a given nonnegative $m \times n$ (contingency) matrix $A$ are proportional. The null-hypothesis $H_0$: ``$m$ rows are proportional'' (for the whole population) is rejected with confidence level $1 - α$ if and only if $χ^2_{stat} > χ^2_{crit}$, where the first term is given by Pearson's formula, while the second one depends only on $m, n$, and $α$, but not on the entries of $A$. It is immediate to notice that the Pearson's formula is not invariant. More precisely, whenever we multiply all entries of $A$ by a constant $c$, the value $χ^2_{stat}(A)$ is multiplied by $c$, too, $χ^2_{stat}(cA) = c χ^2_{stat} (A)$. Thus, if all rows of $A$ are exactly proportional then $χ^2_{stat}(cA) = c χ^2_{stat}(A) = 0$ for any $c$ and any $α$. Otherwise, $χ^2_{stat} (cA)$ becomes arbitrary large or small, as positive $c$ is increasing or decreasing. Hence, at any fixed significance level $α$, the null hypothesis $H_0$ will be rejected with confidence $1 - α$, when $c$ is sufficiently large and not rejected when $c$ is sufficiently small, Yet, obviously, the rows of $cA$ should be proportional or not for all $c$ simultaneously. Thus, any reasonable formula for the test statistic must be invariant, that is, take the same value for matrices $cA$ for all real positive $c$. KEY WORDS: Pearson chi-square test, difference between two proportions, goodness of fit, contingency tables.

stat.ME↗

arXiv2025

Avoidability beyond paths

The concept of avoidable paths in graphs was introduced by Beisegel, Chudnovsky, Gurvich, Milanič, and Servatius in 2019 as a common generalization of avoidable vertices and simplicial paths. In 2020, Bonamy, Defrain, Hatzel, and Thiebaut proved that every graph containing an induced path of order $k$ also contains an avoidable induced path of the same order. They also asked whether one could generalize this result to other avoidable structures, leaving the notion of avoidability up to interpretation. In this paper we address this question: we specify the concept of avoidability for arbitrary graphs equipped with two terminal vertices. We provide both positive and negative results, some of which appear to be related to the recent work by Chudnovsky, Norin, Seymour, and Turcotte [arXiv:2301.13175].

math.CO↗

arXiv2025

Growing Trees and Amoebas' Replications

An amoeba is a tree together with instructions how to iteratively grow trees by adding paths of a fixed length $\ell$. This paper analyses such a growth process. An amoeba is mortal if all versions of the process are finite, and it is immortal if they are all infinite. We obtain some necessary and some sufficient conditions for mortality. In particular, for growing caterpillars in the case $\ell=1$ we characterize mortal amoebas. We discuss variations of the mortality concept, conjecture that some of them are equivalent, and support this conjecture for $\ell\in\{1,2\}$.

math.CO↗

arXiv2025

On Nash Equilibria in Play-Once and Terminal Deterministic Graphical Games

We consider finite $n$-person deterministic graphical games and study the existence of pure stationary Nash-equilibrium in such games. We assume that all infinite plays are equivalent and form a unique outcome, while each terminal position is a separate outcome. It is known that for $n=2$ such a game always has a Nash equilibrium, while that may not be true for $n > 2$. A game is called {\em play-once} if each player controls a unique position and {\em terminal} if any terminal outcome is better than the infinite one for each player. We prove in this paper that play-once games have Nash equilibria. We also show that terminal games have Nash equilibria if they have at most three terminals.

econ.TH↗

arXiv2024

A counterexample to conjecture "Catch 22"

We construct a finite deterministic graphical (DG) game without Nash equilibria in pure stationary strategies. This game has 3 players $I=\{1,2,3\}$ and 5 outcomes: 2 terminal $a_1$ and $a_2$ and 3 cyclic. Furthermore, for 2 players a terminal outcome is the best: $a_1$ for player 3 and $a_2$ for player 1. Hence, the rank vector $r$ is at most $(1,2,1)$. Here $r_i$ is the number of terminal outcomes that are worse than some cyclic outcome for the player $i \in I$. This is a counterexample to conjecture ``Catch 22" from the paper ``On Nash-solvability of finite $n$-person DG games, Catch 22" (2021) arXiv:2111.06278, according to which, at least 2 entries of $r$ are at least 2 for any NE-free game. However, Catch 22 remains still open for the games with a unique cyclic outcome, not to mention a weaker (and more important) conjecture claiming that an $n$-person finite DG game has a Nash equilibrium (in pure stationary strategies) when $r = (0^n)$, that is, all $n$ entries of $r$ are 0; in other words, when the following condition holds: $\qquad\bullet$ ($C_0$) any terminal outcome is better than every cyclic one for each player. A game is play-once if each player controls a unique position. It is known that any play-once game satisfying ($C_0$) has a Nash equilibrium. We give a new and very short proof of this statement. Yet, not only conjunction but already disjunction of the above two conditions may be sufficient for Nash-solvability. This is still open.

math.CO↗

arXiv2024

Conformal Hypergraphs: Duality and Implications for the Upper Clique Transversal Problem

Given a hypergraph $\mathcal{H}$, the dual hypergraph of $\mathcal{H}$ is the hypergraph of all minimal transversals of $\mathcal{H}$. The dual hypergraph is always Sperner, that is, no hyperedge contains another. A special case of Sperner hypergraphs are the conformal Sperner hypergraphs, which correspond to the families of maximal cliques of graphs. All these notions play an important role in many fields of mathematics and computer science, including combinatorics, algebra, database theory, etc. In this paper we study conformality of dual hypergraphs and prove several results related to the problem of recognizing this property. In particular, we show that the problem is in co-NP and can be solved in polynomial time for hypergraphs of bounded dimension. In the special case of dimension $3$, we reduce the problem to $2$-Satisfiability. Our approach has an implication in algorithmic graph theory: we obtain a polynomial-time algorithm for recognizing graphs in which all minimal transversals of maximal cliques have size at most $k$, for any fixed $k$.

math.CO↗

arXiv2024

More on discrete convexity

In several recent papers some concepts of convex analysis were extended to discrete sets. This paper is one more step in this direction. It is well known that a local minimum of a convex function is always its global minimum. We study some discrete objects that share this property and provide several examples of convex families related to graphs and to two-person games in normal form.

math.CO↗

arXiv2023

Screw discrete dynamical systems and their applications to exact slow NIM

Given integers $n,k,\ell$ such that $0 m(x)$, take all $m$ such entries, if any, and add remaining $n-k-m$ entries arbitrarily, for example, take the largest ones. In one step, the chosen $n-k$ entries (bears) keep their values, while the remaining $k$ (bulls) are reduced by 1. Repeat such steps getting the sequence $S = S(n,k,\ell,x^0) = (x^0 \to x^1 \to \ldots \to x^j \to \ldots)$. It is ``quasi-periodic". More precisely, there is a function $N = N(n,k,\ell,x^0)$ such that for all $j \geq N$ we have $m(x^j) \geq n-k$ and $range(x^j) \leq \ell$, where $range(x) = (\max(x_i \mid i \in [n]) - \min(x_i \mid i \in [n])$. Furthermore, $N$ is a polynomial in $n,k,\ell,$ and $range(x^0)$ and can be computed in time linear in $n,k,\ell$, and $\log(1 + range(x^0))$. After $N$ steps, the system moves ``like a screw". Assuming that $x_1 \leq \dots \leq x_n$, introduce the cyclical order on $[n] = \{1, \ldots, n\}$ considering 1 and $n$ as neighbors. Then, bears and bulls partition $[n]$ into two intervals, rotating by the angle $2 πk /n$ with every $\ell$ steps. Furthermore, after every $p = \ell n / GCD(n,k) = \ell LCM(n,k) / k$ steps all entries of $x$ are reduced by the same value $δ= pk/n$, that is, $x_i^{j+p} - x_i^j = δ$ for all $i \in [n]$ and $j \geq N$. We provide an algorithm computing $N$ (and $x^j$) in time linear in $n,k,\ell, \log(1 + range(x^0))$ (and $\log (1+j)$). In case $k=n-1$ and $\ell = 2$ such screw dynamical system are applicable to impartial games.

math.CO↗

arXiv2023

Experimental Study of the Game Exact Nim(5, 2)

We compare to different extensions of the ancient game of nim: Moore's nim$(n, \leq k)$ and exact nim$(n, = k)$. Given integers $n$ and $k$ such that $0 < k \leq n$, we consider $n$ piles of stones. Two players alternate turns. By one move it is allowed to choose and reduce any (i) at most $k$ or (ii) exactly $k$ piles of stones in games nim$(n, \leq k)$ and nim$(n, = k)$, respectively. The player who has to move but cannot is the loser. Both games coincide with nim when $k=1$. Game nim$(n, \leq k)$ was introduced by Moore (1910) who characterized its Sprague-Grundy (SG) values 0 (that is, P-positions) and 1. The first open case is SG values 2 for nim$(4, \leq 2)$. Game nim$(n, = k)$, was introduced in 2018. An explicit formula for its SG function was computed for $2k \geq n$. In contrast, case $2k < n$ seems difficult: even the P-positions are not known already for nim$(5,=2)$. Yet, it seems that the P-position of games nim$(n+1,=2)$ and nim$(n+1,\leq 2)$ are closely related. (Note that P-positions of the latter are known.) Here we provide some theoretical and computational evidence of such a relation for $n=5$.

math.CO↗

arXiv2023

On remoteness functions of k-NIM with k+1 piles in normal and in misère versions

Given integer $n$ and $k$ such that $0 < k \leq n$ and $n$ piles of stones, two players alternate turns. By one move it is allowed to choose any $k$ piles and remove exactly one stone from each. The player who has to move but cannot is the loser. in the normal version of the game and (s)he is the winner in the misère version. Cases $k=1$ and $k = n$ are trivial. For $k=2$ the game was solved for $n \leq 6$. For $n \leq 4$ the Sprague-Grundy function was efficiently computed (for both versions). For $n = 5,6$ a polynomial algorithm computing P-positions was obtained for the normal version. \newline Then, for the case $k = n-1$, a very simple explicit rule that determines the Smith remoteness function was found for the normal version of the game: the player who has to move keeps a pile with the minimum even number of stones; if all piles have odd number of stones then (s)he keeps a maximum one, while the $n-1$ remaining piles are reduced by one stone each in accordance with the rules of the game. \newline Computations show that the same rule works efficiently for the misère version too. The exceptions are sparse and are listed in Section 2. Denote a position by $x = (x_1, \dots, x_n)$. Due to symmetry, we can assume wlog that $x_1 \leq \ldots \leq x_n$. Our computations partition all exceptions into the following three families: $x_1$ is even, $x_1 = 1$, and $x_1 \geq 3$ is odd. In all three cases we suggest explicit formulas that cover all found exceptions, but this is not proven.

math.CO↗

arXiv2023

GM-rule and its applications to impartial games

Given integer $n \geq 1, \ell \geq 2$, and vector $x = (x_1, \ldots, x_n)$ that has an entry which is a multiple of $\ell$ and such that $x_1 \leq \ldots \leq x_n$, the GM-rule is defined as follows: Keep the rightmost minimal entry $x_i$ of $x$, which is a multiple of $\ell$ and reduce the remaining $n-1$ entries of $x$ by~1. We will call such $i$ the {\em pivot} and $x_i$ the {\em pivotal entry}. The GM-rule respects monotonicity of the entries. It uniquely determines a GM-move $x^0 \to x^1$ and an infinite GM-sequence $S$ that consists of successive GM-moves $x = x^0 \to x^1 \to \ldots \to x^j \to \ldots$ . If $range(x) = x_n - x_1 \leq \ell$ then for all $j \geq 0$: (i) $range(x^j) \leq \ell$; (ii) the pivot of $x^{j + \ell}$ is one less than the pivot of $x^j$, assuming that $1 - 1 = 0 = n$. (iii) $x_i^j - x_i^{j + n \ell} = (n-1) \ell$ for all $i = 1,\ldots,n$. Due to (iii), we compute $x^j$ in time linear in $n, \ell, \log(j)$, and $\sum^n_{i=1}\log(|x_i|+1)$. For $\ell = 2$ a slighty modified version of the GM-rule was recently introduced by Gurvich, Martynov, Maximchuk, and Vyalyi, "On Remoteness Functions of Exact Slow $k$-NIM with $k+1$ Piles", arXiv:2304.06498 (2023), where applications to impartial games were considered.

math.CO↗

arXiv2023

Computing Remoteness Functions of Moore, Wythoff, and Euclid's games

We study remoteness function $\mathcal R$ of impartial games introduced by Smith in 1966. The player who moves from a position $x$ can win if and only if $\mathcal R(x)$ is odd. The odd values of $\mathcal R(x)$ show how soon the winner can win, while even values show how long the loser can resist, provided both players play optimally. This function can be applied to the conjunctive compounds of impartial games, in the same way as the Sprague-Grundy function is applicable to their disjunctive compounds. We provide polynomial algorithms computing $\mathcal R(x)$ for games Euclid and generalized Wythoff. For Moore's NIM we give a simple explicit formula for $\mathcal R(x)$ if it is even and show that computing it becomes an NP-hard problem for the odd values.

math.CO↗