arXiv subjects

Mariya Naumova

Publications and source records attributed to Mariya Naumova.

16 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

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↗

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

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

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

On Nash-solvability of n-person graphical games under Markov's and a priori realizations

We consider graphical $n$-person games with perfect information that have no Nash equilibria in pure stationary strategies. Solving these games in mixed strategies, we introduce probabilistic distributions in all non-terminal positions. The corresponding plays can be analyzed under two different basic assumptions: Markov's and a priori realizations. The former one guarantees existence of a uniformly best response of each player in every situation. Nevertheless, Nash equilibrium may fail to exist even in mixed strategies. The classical Nash theorem is not applicable, since Markov's realizations may result in the limit distributions and effective payoff functions that are not continuous. The a priori realization does not share many nice properties of the Markov one (for example, existence of the uniformly best response) but in return, Nash's theorem is applicable. We illustrate both realizations in details by two examples with $2$ and $3$ players and also provide some general results.

math.CO↗

arXiv2023

Polynomial algorithms computing two lexicographically safe Nash equilibria in finite two-person games with tight game forms given by oracles

In 1975 the first author proved that every finite tight two-person game form $g$ is Nash-solvable, that is, for every payoffs $u$ and $w$ of two players the obtained game $(g;u,w)$, in normal form, has a Nash equilibrium (NE) in pure strategies. This result was extended in several directions; here we strengthen it further. We construct two special NE realized by a lexicographically safe (lexsafe) strategy of one player and a best response of the other. We obtain a polynomial algorithm computing these lexsafe NE. This is trivial when game form $g$ is given explicitly. Yet, in applications $g$ is frequently realized by an oracle $\cO$ such that size of $g$ is exponential in size $|\cO|$ of $\cO$. We assume that game form $g = g(\cO)$ generated by $\cO$ is tight and that an arbitrary {\em win-lose game} $(g;u,w)$ (in which payoffs $u$ and $w$ are zero-sum and take only values $\pm 1$) can be solved, in time polynomial in $|\cO|$. These assumptions allow us to construct an algorithm computing two (one for each player) lexsafe NE in time polynomial in $|\cO|$. We consider four types of oracles known in the literature and show that all four satisfy the above assumptions.

cs.GT↗

arXiv2022

Supercentenarian paradox

Consider the following statement: $B(t, Δt)$: a $t$ years old person NN will survive another $Δt$ years, where $t, Δt\in \mathbb{R}$ are nonnegative real numbers. We know only that NN is $t$ years old and nothing about the health conditions, gender, race, nationality, etc. We bet that $B(t, Δt)$ holds. It seems that our odds are very good, for any $t$ provided $Δt$ is small enough, say, $1 / 365$ (that is, one day). However, this is not that obvious and depends on the life-time probabilistic distribution. Let $F(t)$ denote the probability to live at most $t$ years and set $Φ(t) = 1 - F(t)$. Clearly, $Φ(t) \rightarrow 0$ as $t \rightarrow \infty$. It is not difficult to verify that $Pr(B(t, Δt)) \rightarrow 0$ as $t \rightarrow \infty$, for any fixed $Δt$, whenever the convergence of $Φ$ is fast enough (say, super-exponential). Statistics provide arguments (based on an extrapolation yet) that this is the case. Hence, for an arbitrarily small positive $Δt $ and $ε$ there exists a sufficiently large $t$ such that $Pr(B(t, Δt)) < ε$, which means that we should not bet... in theory. However, in practice we can bet safely, because for the inequality $Pr(B(t, Δt)) < 1/2$ a very large $t$ is required. For example, $Δt = 1/365$ may require $t > 125$ years for some typical distributions $F$ considered in the literature. Yet, on Earth there is no person of such age. Thus, our odds are good, either because the chosen testee NN is not old enough, or for technical (or, more precisely, statistical) reasons -- absence of a testee. This situation is similar to the famous St.Petersburg Paradox.

math.PR↗

arXiv2022

Lexicographically maximal edges of dual hypergraphs and Nash-solvability of tight game forms

Let $\mathcal{A} = \{A_1, \ldots, A_m\}$ and $\mathcal{B} = \{B_1, \ldots, B_n\}$ be a pair of dual multi-hypergraphs on the common ground set $O = \{o_1, \ldots, o_k\}$. Note that each of them may have embedded or equal edges. An edge is called containment minimal (or just minimal, for short) if it is not a strict superset of another edge. Yet, equal minimal edges may exist. By duality, (i) $A \cap B \neq \emptyset$ for every pair $A \in \mathcal{A}$ and $B \in \mathcal{B}$; (ii) if $A$ is minimal then for every $o \in A$ there exists a $B \in \mathcal{B}$ such that $A \cap B = \{o\}$. We will extend claim (ii) as follows. A linear order $\succ$ over $O$ defines a unique lexicographic order $\succ_L$ over the $2^O$. Let $A$ be a lexicographically maximal (lexmax) edge of $\mathcal{A}$. Then, (iii) $A$ is minimal and for every $o \in A$ there exists a minimal $B \in \mathcal{B}$ such that $A \cap B = \{o\}$ and $o \succeq o'$ for each $o' \in B$. This property has important applications in game theory implying Nash-solvability of tight game forms as shown in the old (1975 and 1989) work of the first author. Here we give a new, very short, proof of (iii). Edges $A$ and $B$ mentioned in (iii) can be found out in polynomial time. This is trivial if $\mathcal{A}$ and $\mathcal{B}$ are given explicitly. Yet, it is true even if only $\mathcal{A}$ is given, and not explicitly, but by a polynomial containment oracle, which for a subset $O_A \subseteq O$ answers in polynomial time whether $O_A$ contains an edge of $\mathcal{A}$.

math.CO↗

arXiv2022

On Nash-Solvability of Finite Two-Person Tight Vector Game Forms

We consider finite two-person normal form games. The following four properties of their game forms are equivalent: (i) Nash-solvability, (ii) zero-sum-solvability, (iii) win-lose-solvability, and (iv) tightness. For (ii, iii, iv) this was shown by Edmonds and Fulkerson in 1970. Then, in 1975, (i) was added to this list and it was also shown that these results cannot be generalized for $n$-person case with $n > 2$. In 1990, tightness was extended to vector game forms ($v$-forms) and it was shown that such $v$-tightness and zero-sum-solvability are still equivalent, yet, do not imply Nash-solvability. These results are applicable to several classes of stochastic games with perfect information. Here we suggest one more extension of tightness introducing $v^+$-tight vector game forms ($v^+$-forms). We show that such $v^+$-tightness and Nash-solvability are equivalent in case of weakly rectangular game forms and positive cost functions. This result allows us to reduce the so-called bi-shortest path conjecture to $v^+$-tightness of $v^+$-forms. However, both (equivalent) statements remain open.

cs.GT↗

arXiv2021

Logical contradictions in the One-way ANOVA and Tukey-Kramer multiple comparisons tests with more than two groups of observations

We show that the One-way ANOVA and Tukey-Kramer (TK) tests agree on any sample with two groups. This result is based on a simple identity connecting the Fisher-Snedecor and studentized probabilistic distributions and is proven without any additional assumptions; in particular, the standard ANOVA assumptions (independence, normality, and homoscedasticity (INAH)) are not needed. In contrast, it is known that for a sample with k > 2 groups of observations, even under the INAH assumptions, with the same significance level $α$, the above two tests may give opposite results: (i) ANOVA rejects its null hypothesis $H_0^{A}: μ_1 = \ldots = μ_k$, while the TK one, $H_0^{TK}(i,j): μ_i = μ_j$, is not rejected for any pair $i, j \in \{1, \ldots, k\}$; (ii) the TK test rejects $H_0^{TK}(i,j)$ for a pair $(i, j)$ (with $i \neq j$) while ANOVA does not reject $H_0^{A}$. We construct two large infinite pseudo-random families of samples of both types satisfying INAH: in case (i) for any $k \geq 3$ and in case (ii) for some larger $k$. Furthermore, in case (ii) ANOVA, being restricted to the pair of groups $(i,j)$, may reject equality $μ_i = μ_j$ with the same $α$. This is an obvious contradiction, since $μ_1 = \ldots = μ_k$ implies $μ_i = μ_j$ for all $i, j \in \{1, \ldots, k\}.$ Similar contradictory examples are constructed for the Multivariable Linear Regression (MLR). However, for these constructions it seems difficult to verify the Gauss-Markov assumptions, which are standardly required for MLR.

math.ST↗