Searcharxiv⌕ Search

arXiv subjects

Sergey Sadov

Publications and source records attributed to Sergey Sadov.

At least 19 recordsLinked to original sources
arXiv2026

The triplication method for constructing strong starters

The triplication method for constructing strong starters in $Z_{3m}$ from starters in $Z_{m}$ (say, a starter of order 21 from a starter of order 7) was proposed by the authors in 2025. The method reduced construction of the particular combinatorial design (a strong starter in a cyclic group) to solving a Sudoku-type problem -- an independent task with its own tools and techniques available. The Sudoku-type problem was formulated in terms of the so-called triplication table constructed from a starter of order $m$. The method was applicable for odd orders $m\ge 7$ not divisible by 3. In the present paper, our previous approach is developed in two directions: (1) the definition of the triplication table is generalized, which expands possibilities for its construction to include three base starters or even ``pseudostarters''; (2) the formulation of the Sudoku-type problem is broadened to embrace various scenarios of ``modular encoding'' and reconstruction of strong starters from its solution. A theoretical gain of these developments consists in the improved understanding of the general structure of the triplication approach. A practical outcome is elimination of the requirement that $m$ be not divisible by 3. This leads to a broader scope of strong starters obtainable by triplication: any latent strong starter of odd order $3m$ can emerge this way.

math.CO↗
arXiv2025

Around the Fej\'er-Jackson inequality: Tight bounds for certain oscillatory functions via Laplace transform representations

The error of approximation of the $2\pi$-periodic sawtooth function $(\pi-x)/2$, $0\leq x<2\pi$, by its $n$-th Fourier polynomial is shown to be bounded by arccot$((2n+1)\sin(x/2))$. Related asymptotically tight inequalities with explicit constants are given for the integral of the Dirichlet kernel interpolated to non-integer values of frequency parameter and for the Taylor series remainder of the logarithmic function $\log(1-z)$ in the unit circle. The proofs are based on the Laplace transform representation of the Lerch Zeta function with $s=1$.

math.CA↗
arXiv2025

Constructing strong starters of orders $3p$: triplication with SAT solver

A novel approach to building strong starters in cyclic groups of orders $n$ divisible by 3 from starters of smaller orders is presented. A strong starter in $Z_n$ ($n$ odd) is a partition of the set $\{1,2,\dots,n-1\}$ into pairs $\{a_i,b_i\}$ such that all pair sums $a_i+b_i$ are distinct and nonzero modulo $n$ and all differences $\pm(a_i-b_i)$ are distinct and nonzero modulo $n$. A special interest to strong starters of odd orders divisible by 3 is motivated by Horton's conjecture which claims that such starters exist (except when $n=3$ or $9$) but remains unproven since 1989. We begin with a strong starter of order $p$ coprime with 3 and describe an algorithm to obtain a Sudoku-type problem modulo 3 whose solution, if exists, yields a strong starter of order $3p$. The process leading from the original to the final starter is called {\em triplication}. Besides theoretical aspects of the construction, practicality of this approach is demonstrated. A general-purpose constraint-satisfaction (SAT) solver z3 is used to solve the Sudoku-type problem; various performance statistics are presented.

math.CO↗
arXiv2022

Precise asymptotics with log-periodic term in an elementary optimization problem

The function $\inf_n nx^{1/n}$ has the asymptotics $eu+e d^2(u)/(2u)+O(1/u^2)$ as $x\to\infty$, where $u=\log x$ and $d(u)$ is the distance from $u$ to the nearest integer. We generalize this observation. First, the curves $y=nx^{1/n}$ can be written parametrically as $\log x=nt$, $y=nt$. In general, let $(u_n(t),v_n(t))$ be a family of parametric curves with asymptotics $u_n=n p_1(t)+q_1(t)+r_1(t)/n+O(1/n^2)$ and $v_n=n p_2(t)+q_2(t)+r_2(t)/n+O(1/n^2)$. Suppose the function $p_1(t)/p_0(t)$ has a unique nondegenerate minimum in the parameter domain. It is shown that the asymptotics of their lower envelope $v(u)=\inf_{n,t} v_n(t)$, where $u=u_n(t)$, has the asymptotics of the form $v(u)=a_0 u+a_1+Φ(u)/u+O(1/u^2)$, where $Φ$ is an affinely transformed function $d^2(\cdot)$. Second, note that $nx^{1/n}$ is the minimum of the sum $t_1+t_2/t_1+\dots+t_{n}/t_{n-1}$ subject to the constraint $t_n=x$. We consider a similar asymptotic problem for the sums $t_1+t_2/(t_1+1)+\dots+t_n/(t_{n-1}+1)$. Let $F_n(x)$ is the minimum value of the $n$-term sum under the constraint $t_n=x$. Define $F(x)=\inf_n F_n(x)$. We show that $F(x)=eu-A+e d^2(u+b)/(2u)+O(1/u^2)$ with certain numerical constants $A$ and $b$. We present alternative forms of this optimization problem, in particular, a ``least action'' formulation. Also we find the asymptotics $F_n^{(p)}(x)=e\log n-A(p)+O(1/\log n)$ for the function arising from the sums with denominators of the form $t_j+p$ with arbitrary $p>0$ and establish some facts about the function $A(p)$.

math.CA↗
arXiv2022

Lower bound for cyclic sums with one-sided maximal averages in denominators

Let $\mathbf{x}=(x_1,\dots,x_n)$ be an $n$-tuple of positive real numbers and the sequence $(x_i)_{i\in\mathbb{Z}}$ be its $n$-periodic extension. Given an $n$-tuple $\mathbf{r}=(r_1,\dots,r_n)$ of positive integers, let $a_i$ be the arithmetic mean of $x_{i+1},\dots,x_{i+r_i}$. We form the cyclic sums $S_n(\mathbf{x},\mathbf{r})=\sum_{i=1}^n x_i/a_{i}$, following the pattern of the long studied Shapiro sums, which correspond to all $r_i=2$, and more general Diananda sums, where all $r_i$ are equal. We find the asymptotics of the $\mathbf{r}$-independent lower bounds $A_{n,*}=\inf_{\mathbf{r}}\inf_{\mathbf{x}} S_n(\mathbf{x},\mathbf{r})$ as $n\to\infty$: it is $A_{n,*}=e\log n - A+O(1/\log n)$.

math.CA↗
arXiv2022

Existence of convolution maximizers in $L_p(R^n)$ for kernels from Lorentz spaces

The paper extends an earlier result of G.V.~Kalachev and the author (Sb. Math. 2019 or arXiv:1712.08836) on the existence of a maximizer of convolution operator acting between two Lebesgue spaces on $R^n$ with kernel from some $L_q$, $1<q<\infty$. In view of Lieb's result of 1983 about the existence of an extremizer for the Hardy-Littlewood-Sobolev inequality it is natural to ask whether a convolution maximizer exists for any kernel from weak $L_q$. The answer in the negative was given by Lieb in the above citation. In this paper we prove the existence of maximizers for kernels from a slightly more narrow class than weak $L_q$, which contains all Lorentz spaces $L_{q,s}$ with $q\leq s<\infty$.

math.FA↗
arXiv2022

On the Cleaning Lemma of Quantum Coding Theory

The term "Cleaning Lemma" refers to a family of similar propositions that have been used in Quantum Coding Theory to estimate the minimum distance of a code in terms of its length and dimension. We show that the mathematical core is a simple fact of linear algebra of inner product spaces; moreover, it admits a further reduction to a combinatorial, lattice-theoretical level. Several concrete variants of the Cleaning Lemma and some additional propositions are derived as corollaries of the proposed approach.

quant-ph↗
arXiv2021

Three steps away from Shapiro's problem: lower bounds for graphic sums with functions `max' or `min' in denominators

Taking Shapiro's cyclic sums $\sum_{i=1}^n x_i/(x_{i+1}+x_{i+2})$ (assuming index addition mod $n$) as a starting point, we introduce a broader class of cyclic sums, called generalized Shapiro-Diananda sums, where the denominators are $p$-th order power means of the sets $\{x_{i+j_1},\dots,x_{i+j_k}\}$ with fixed distinct integers $j_1,\dots,j_k$ and $1\leq i\leq n$. Generalizing further, we replace the set of arguments of the power mean in the $i$-th denominator by an arbitrary nonempty subset of $\{1,\dots,n\}$ interpreted as the set of out-neighbors of the node number $i$ in a directed graph with $n$ nodes. We call such sums graphic power sums since their structure is controlled by directed graphs. The inquiry, as in the well-researched case of Shapiro's sums, concerns the greatest lower bound of the given ``sum'' as a function of positive variables $x_1,\dots,x_n$. We show that the cases of $p=+\infty$ (max-sums) and $p=-\infty$ (min-sums) are tractable. For the max-sum associated with a given graph the g.l.b. is always an integer; for a strongly connected graph it equals to graph's girth. For the similar min-sum, we could not relate the g.l.b. to a known combinatorial invariant; we only give some estimates and describe a method for finding the g.l.b., which has factorial complexity in $n$. A satisfactory analytical treatment is available for the secondary minimization -- when the g.l.b.'s of min-sums for individual graphs are mininized over the class of strongly connected graphs with $n$ nodes. The result (depending only on $n$) is found to be asymptotic to $e\ln n$.

math.CO↗
arXiv2020

Minimization of the sum under product constraints

We systematically explore a class of constrained optimization problems with linear objective function and constraints that are linear combinations of logarithms of the optimization variables. Such problems can be viewed as a generalization of the inequality between the arithmetic and geometric means. The existence and uniqueness of the minimizer is proved under natural assumptions in the general case. We study in detail special subclasses where the set of constraints is described in combinatorial terms (oriented graphs, rooted trees). In particular, given a directed, strongly connected graph, we seek to minimize the total of all arc values under cyclic product constraints. We obtain some estimates and asymptotics for the minimum in problems with given (large) number of variables. Also in this context we revisit an asymptotical result known as J. Shallit's minimization problem. The material is presented in the form of a problem book. Along with problems that constitute main theoretical threads, there are many exercises, some mini-paradoxes, and a touch of numerical methods.

math.CA↗
arXiv2018

On Shallit's minimization problem

We revisit J. Shallit's minimization problem from 1994 SIAM Review concerning a two-term asymptotics of the minimum of a certain rational sum involving variables and products of their reciprocals, the number of variables being the large parameter. Properties previously known numerically, most importantly, the existence of the constant in the asymptotics, are proved. We supply a sharp remainder estimate to the originally proposed asymptiotic formula. The proofs are based on the analysis of trajectories of a planar discrete dynamical system that determines the point of minimum.

math.CA↗
arXiv2017

On maximizers of convolution operators in $L_p$ spaces

A convolution operator in $\mathbb{R}^d$ with kernel in $L_q$ acts from $L_p$ to $L_s$, where $1/p+1/q=1+1/s$. The main theorem states that if $1<q,p,s<\infty$, then there exists an $L_p$ function of unit norm on which the $s$-norm of the convolution is attained. A number of questions, solved and open, related to the statement and proof of the main theorem, are discussed. The problem of computing best constants in the Hausdorff-Young inequality for the Laplace transform, which prompted this research, is considered.

math.CA↗
arXiv2015

Lower bound for cyclic sums of Diananda type

Let $C=\inf (k/n)\sum_{i=1}^n x_i(x_{i+1}+\dots+x_{i+k})^{-1}$, where the infimum is taken over all pairs of integers $n\geq k\geq 1$ and all positive $x_1,\dots,x_{n+k}$ subject to cyclicity assumption $x_{n+i}=x_i$, $i=1,\dots,k$. We prove that $\ln 2\leq C< 0.9305$. In the definition of the constant $C$ the operation $\inf_k\inf_n\inf_{\mathbf{x}}$ can be replaced by $\lim_{k\to\infty}\lim_{n\to\infty}\inf_{\mathbf{x}}$.

math.CA↗
arXiv2015

Asymptotics of alternating harmonic series with attenuation

We find the asymptotics of the series $\sum_{n=1}^\infty (-1)^n n^{-1} \exp(-t/n)$ as $t\to+\infty$. The answer is an oscillating function of $t$ dominated by $\exp(-(2πt)^{1/2})$. The intermediate step is to find the asymptotics of the two-dimensional Fourier transform $\hat F(ξ)$ of the function $F(x)=(1+\exp(\|x\|^2))^{-1}$ as $\|ξ\|\to\infty$.

math.CA↗
arXiv2014

On asymptotically free action of permutation groups on subsets and multisets

Let $G$ be a permutation group acting on a finite set $Ω$ of cardinality $n$. The number of orbits of the induced action of $G$ on the set $Ω_m$ of all size $m$ subsets of $Ω$ satisfies the trivial inequalities $|Ω_m|/|G|\leq |Ω_m/G|\leq |Ω_m|$. The paper offers improvements of the upper bound in terms of the minimal degree of $G$ or the minimal degree of some its subset with a small complement. Applications include asymptotic enumeration of point configurations in an affine space over a finite field, unlabeled graphs and hypergraphs. Finally, with references to known results of permutation groups theory it is shown that if $G$ is an arbitrary 2-transitive group except for $S_n$ and $A_n$, then $|Ω_m/G|\approx |Ω_m|/|G|$ for $m$ and $n$ large provided the ratio $m/n$ is bounded away from 0 and 1. Similar results hold for the induced action of $G$ on the set $Ω_{(m)}$ of all weight $m$ multisets on $Ω$ provided the ratio $m/n$ is not too small.

math.GR↗
arXiv2013

Counting fine gradings on matrix algebras and on classical simple Lie algebras

Known classification results allow us to find the number of (equivalence classes of) fine gradings on matrix algebras and on classical simple Lie algebras over an algebraically closed field $\mathbb{F}$ (assuming $\mathrm{char} \mathbb{F}\ne 2$ in the Lie case). The computation is easy for matrix algebras and especially for simple Lie algebras of type $B_r$ (the answer is just $r+1$), but involves counting orbits of certain finite groups in the case of Series $A$, $C$ and $D$. For $X\in\{A,C,D\}$, we determine the exact number of fine gradings, $N_X(r)$, on the simple Lie algebras of type $X_r$ with $r\le 100$ as well as the asymptotic behaviour of the average, $\hat N_X(r)$, for large $r$. In particular, we prove that there exist positive constants $b$ and $c$ such that $\exp(br^{2/3})\le\hat N_X(r)\le\exp(cr^{2/3})$. The analogous average for matrix algebras $M_n(\mathbb{F})$ is proved to be $a\ln n+O(1)$ where $a$ is an explicit constant depending on $\mathrm{char} \mathbb{F}$.

math.RA↗
arXiv2011

$L^p$ estimates for angular maximal functions associated with Stieltjes and Laplace transforms

Maximal angular operator sends a function defined in a sector of the complex plane to a Maximal angular operator sends a function defined in a sector of the complex plane with vertex at 0 to the function of modulus obtained by maximizing over argument. Compositions of the so defined maximal angular operator (in suitable sectors) with the Poisson, Stieltjes and Laplace transforms are shown to be bounded (nonlinear) operators from $L^p$ to $L^q$ for the same values of $p$ and $q$ as their standard counterparts.

math.CA↗