arXiv subjects

Mohammad Javaheri

Publications and source records attributed to Mohammad Javaheri.

At least 19 recordsLinked to original sources

arXiv2026

O-segments on topological measure spaces

Let $μ$ be a finite non-atomic measure on a set $X$. A \emph{segment} for $μ$ is an increasing family $u(t)$, $t\in[0,1]$, of measurable sets with $u(0)=\emptyset$, $u(1)=X$ and $μ(u(t))=t\,μ(X)$; by a theorem of Fryszkowski, finitely many measures $f\,dμ$ admit a common segment. We study \emph{open segments}, i.e.\ segments consisting of open sets, when $X$ carries a topology. We show that weak outer regularity of $μ$ is equivalent to the approximability of measurable sets by open sets, and that it holds for every finite Borel measure on a metrizable space. Our main result states that if $μ$ is finite, non-atomic and weakly outer regular, then for all $f,g\in L^1(μ)$ with $(\int f\,dμ,\int g\,dμ)\neq(0,0)$ there is a common open segment for $f\,dμ$ and $g\,dμ$; in particular, for every $f\in L^1(μ)$ there is an open segment for $μ$ which is simultaneously a segment for $f\,dμ$. Relative versions allow one to prescribe an initial set. We also characterize open segments as upper semicontinuous functions with uniform distribution and show that on separable metrizable spaces a single measure admits such a function which is continuous.

math.FA↗

arXiv2026

On a property of plane curves

Let $γ:[0,1]\to \mathbb{R}^2$ be a continuous curve with $γ(0)=(0,0)$ and $γ(1)=(1,1)$. We prove that if $γ(t)\in(0,1)^2$ for all $t\in(0,1)$, then for each $n\in \mathbb{N}$ there exist points $A_1,\dots,A_n$ on $γ$ such that, with $A_0=(0,0)$ and $A_{n+1}=(1,1)$, the horizontal increments $π_1(\overrightarrow{A_iA_{i+1}})$ and the vertical increments $π_2(\overrightarrow{A_iA_{i+1}})$, $0\le i\le n$, are positive and coincide up to order. More generally, the same conclusion holds as soon as $γ$ contains an arc that joins two points of the diagonal segment, lies on one side of the diagonal, and satisfies a mild one-sided boundary condition. This covers, for instance, the graph of every continuous function $f$ with $f(0)=0$ and $f(1)=1$, with no restriction on the range of $f$. In all cases the rearrangement can be taken to be a cyclic shift by one place of a block of consecutive increments. The main tool is a version of the mountain climbers' theorem in which one function is piecewise monotone and the other is an arbitrary continuous function. As an application, any two non-atomic probability measures on $[0,1]$ satisfying a mild condition admit, for each $n$, a partition of $[0,1]$ into $n+1$ consecutive intervals whose two lists of measures are rearrangements of each other.

math.CA↗

arXiv2026

A generalization of the Dirichlet approximation theorem for affine actions on the real line

We study quantitative density of orbits of the free semigroup $Γ$ generated by two maps of $(0,\infty)$. For the Möbius pair $x\mapsto 1/x$, $x\mapsto x+1$ we show that every orbit is $O(n^{-1})$-dense in $[0,\infty)$, that every irrational number $u$ is approached at a rate governed by the continued fraction of $u$ (at least $n^{-2}$, exponential when $u$ has bounded partial quotients, and $\exp(-cn/\log n)$ for almost every $u$), and that for rational $u$ and rational starting points the rate $n^{-1}$ is optimal. For the affine pair $T_0(x)=x/a$, $T_1(x)=bx+1$ with $a,b>1$, which was studied by Bergelson, Misiurewicz and Senti, we prove that for every $x>0$ and $u\ge 0$ there are infinitely many $γ\inΓ$ with \[ |γ(x)-u| 1, \] where $|γ|$ is the word length of $γ$ and $C$ is independent of $γ$. This exponential rate holds whether or not $\log_a b$ is rational, and the exponential order cannot be improved for almost every $u$. We also give a corrected version of the Kronecker-type estimate $|γ(x)-u|<(u+1)(t^{1/|γ|}-1)$. Finally, we discuss the approximation of real numbers by ratios of primes and by ratios of logarithms of primes, updated in light of recent results on gaps between primes.

math.NT↗

arXiv2026

Non-collapsing of fibers in warped-product Poincare--Einstein manifolds

Let $g=g_M+u^2h$ be a Poincaré--Einstein (asymptotically hyperbolic Einstein, AHE) metric of class $C^{3,α}$ on $N^{n+1}=M^q\times F^p$, where $(F,h)$ is compact with unit volume and $\Ric_h=λh$, $λ\le 0$. We show that if the conformal infinity $[γ]$ has positive Yamabe constant, then \[ u\;\ge\;Θ[γ]:=\sup_η\Big(\min_{\dN}ψ_η\Big)\Big(\frac{\min_{\dN} s_η}{n(n-1)}\Big)^{1/2}>0, \] the supremum being over representatives $η\in[γ]$ of positive scalar curvature and $ψ_η$ the boundary warping function of $η$. In particular $u\ge (\Y[γ]/n(n-1))^{1/2}\minψ_{η_Y}$ for the Yamabe metric $η_Y$. The estimate rests on a comparison principle between $\ln u$ and Lee's eigenfunction, which also applies to each factor of a multiply warped product, and pointwise yields $u\ge(\minψ_η)\,w_η$ when $\Y[γ]\ge 0$. We give an explicit bound in terms of the product representative of $[γ]$, show that the hypotheses force $q\ge 3$ and $\partial M$ connected, prove lower semicontinuity of $Θ$ in the boundary data, and discuss examples (hyperbolic Fermi coordinates, AdS--Schwarzschild) showing that the hypothesis $λ\le0$ cannot be removed.

math.DG↗

arXiv2026

Chord theorems on graphs (revised and extended)

The Horizontal Chord Theorem states that if a continuous curve connects points $A$ and $B$ in the plane, then for every positive integer $k$ there are points $C$ and $D$ on the curve with $\overrightarrow{AB}=k\,\overrightarrow{CD}$. We reformulate the one-dimensional chord problem on finite metric graphs: the \emph{chord set} $\cH(G)$ consists of those $r$ such that every function of mean zero on $G$ integrates to zero over some closed connected subset of $G$ of measure $r$. We prove that the space of connected subsets of fixed measure is compact and path-connected, that $\cH(G)$ always contains the closure of a combinatorially defined \emph{partition set} $\cP(G)$, and that the two sets agree at every rational number; as a consequence, membership of a rational number in $\cH(G)$ is decided by a finite linear program, and the conjecture $\cH(G)=\overline{\cP(G)}$ reduces to a statement about irrational points. We give a covering lemma that yields $\cH(G)=[0,|E|]$ for Eulerian graphs, $[0,1]\subseteq\cH(G)$ when $G$ has no vertex of degree one, and $[0,g(G)]\subseteq\cH(G)$ when $G$ is bridgeless with girth $g(G)$. Conversely, a bridge forces a gap in $\cH(G)$ below $|E|$. Computations on small graphs support the conjecture that $\cH(G)=[0,|E|]$ exactly when $G$ is bridgeless.

math.GR↗

arXiv2026

A new arrangement inequality (revised and extended version)

We study the cyclic inequality \[ \sum_{i=1}^n x_i\,\sum_{i=1}^n x_i^a x_{i+1}^b\;\le\;\Big(\sum_{i=1}^n x_i^{(1+a+b)/2}\Big)^2, \] where $x_1,\dots,x_n\ge 0$ and indices are taken modulo $n$. Let $N(a,b)$ be the largest $n$ for which the inequality holds for all nonnegative $x_i$. We show that $N(a,b)$ is bounded above by an explicit quantity $N^*(a,b)$ obtained from the second variation at the constant vector, and we conjecture that $N(a,b)=N^*(a,b)$. For $a=b=1$ we prove that the inequality holds if and only if $n\le 8$, which settles the case left open in the original version of this paper. For $n=4$ the inequality holds whenever $1,a,b$ are the sides of a triangle, with an explicit remainder term. We show that $N$ is monotone along rays emanating from the line $a+b=1$; this corrects and extends an earlier monotonicity statement. Finally, any $n$ nonnegative numbers can be arranged so that the inequality holds; for $n\neq 6$ this is achieved by one of $n-1$ prescribed arrangements, which yields a volume bound for every such $n$.

math.GM↗

arXiv2026

Dirichlet problem on locally finite graphs (revised and extended version)

We study existence and uniqueness of solutions to the vertex-weighted Dirichlet problem on locally finite graphs. Let $B$ be a subset of the vertices of a connected graph $G$. The Dirichlet problem asks for a function whose discrete Laplacian is prescribed on $V\setminus B$ and whose values are prescribed on $B$. Each infinite connected component of $G\setminus B$ is called an end of $G$ relative to $B$. On finite graphs we show that the solution is the limit of an averaging process, with an explicit geometric rate given by the spectrum of the random walk, and is also the limit of the heat flow and the minimizer of an energy functional. On infinite graphs we prove two existence theorems: one with no assumption on vertex degrees, under a finite-width condition and summable data, and one for bounded data on graphs of bounded degree, where the solution is also unique among bounded functions. If there are no ends, the Dirichlet problem has exactly one solution. If there are $l$ ends, the solution set, when nonempty, has dimension at least $l$; each end carries a positive harmonic function vanishing on $B$. Calling an end parabolic when the random walk started in it hits $B$ almost surely, we show that this function can be taken bounded exactly on non-parabolic ends and is necessarily unbounded on parabolic ones, a discrete analogue of a theorem of Li and Tam. Consequently, bounded solutions are unique if and only if every end is parabolic.

math.AP↗

arXiv2025

R-harmonious groups

A group is R-harmonious if there exists a permutation $g_1,g_2,\ldots, g_{n-1}$ of the non-identity elements of $G$ such that the consecutive products $g_1g_2$, $g_2g_3$, $\ldots, g_{n-1}g_1$ also form a permutation of the non-identity elements, where $n=|G|$. We investigate R-harmonious groups via cyclic and split extensions. Among our results, we prove that every group of odd-order not divisible by 3 is R-harmonious.

math.GR↗

arXiv2025

Symmetric sequencings and other combinatorial properties of large groups

We prove that Anderson's conjecture on symmetric sequencings and Bailey's conjecture on 2-sequencings hold for sufficiently large groups. In addition, we discuss extensions of partial harmonious sequences and partial R-sequencings. Several further results on double sequencings are presented, both in the context of abelian groups and for sufficiently large non-abelian groups.

math.CO↗

arXiv2025

Symmetric harmoniousness of odd-order groups

We prove that every odd-order group is symmetric harmonious: there exists a permutation $g_0,g_1,\ldots, g_{\ell-1}$ of elements of $G$ such that the consecutive products $g_0g_1,g_1g_2,\ldots, g_{\ell-1}g_0$ also form a permutation of elements of $G$ and $g_{\ell-i}=g_i^{-1}$ for all $1\leq i \leq \ell-1$. We apply this result to obtain new examples of R*-sequenceable groups.

math.GR↗

arXiv2024

Harmonious sequences in groups with a unique involution

We study several combinatorial properties of finite groups that are related to the notions of sequenceability, R-sequenceability, and harmonious sequences. In particular, we show that in every abelian group $G$ with a unique involution $\imath_G$ there exists a permutation $g_0,\ldots, g_{m}$ of elements of $G \backslash \{\imath_G\}$ such that the consecutive sums $g_0+g_1, g_1+g_2,\ldots, g_{m}+g_0$ also form a permutation of elements of $G\backslash \{\imath_G\}$. We also show that in every abelian group of order at least 4 there exists a sequence containing each non-identity element of $G$ exactly twice such that the consecutive sums also contain each non-identity element of $G$ twice. We apply several results to the existence of transversals in Latin squares.

math.GR↗

arXiv2024

Doubly Sequenceable Groups

Given a sequence ${\bf g}: g_0,\ldots, g_{m}$, in a finite group $G$ with $g_0=1_G$, let ${\bf \bar g}: \bar g_0,\ldots, \bar g_{m}$, be the sequence defined by $\bar g_0=1_G$ and $\bar g_i=g_{i-1}^{-1}g_i$ for $1\leq i \leq m$. We say that $G$ is doubly sequenceable if there exists a sequence ${\bf g}$ in $G$ such that every element of $G$ appears exactly twice in each of ${\bf g}$ and ${\bf \bar g}$. If a group $G$ is abelian, odd, sequenceable, R-sequenceable, or terraceable, then $G$ is doubly sequenceable. In this paper, we show that if $N$ is an odd or sequenceable group and $H$ is an abelian group, then $N \times H$ is doubly sequenceable.

math.GR↗

arXiv2022

On a Cyclic Inequality Related to Chebyshev Polynomials

We show that any weighted geometric mean of Chebyshev polynomials is bounded from above by another Chebyshev polynomial. We also study a related homogeneous cyclic inequality $$ \left (\sum_{i=1}^n x_i^{(a+b+1)/2} \right )^2 \geq \sum_{i=1}^n x_i \sum_{i=1}^n x_i^a x_{i+1}^b,$$ where $a,b,x_1,\ldots, x_n$ (with $x_{n+1}=x_1$) are nonnegative. In particular, we prove that the inequality holds when $a=b=1$ and $n\leq 8$ for all nonnegative numbers $x_1,\ldots, x_n$.

math.CA↗

arXiv2019

Permutations with a distinct divisor property

A finite group of order $n$ is said to have the distinct divisor property (DDP) if there exists a permutation $g_1,\ldots, g_n$ of its elements such that $g_i^{-1}g_{i+1} \neq g_j^{-1}g_{j+1}$ for all $1\leq i<j<n$. We show that an abelian group is DDP if and only if it has a unique element of order 2. We also describe a construction of DDP groups via group extensions by abelian groups and show that there exist infinitely many non abelian DDP groups.

math.GR↗

arXiv2018

A note on polynomial sequences modulo integers

We study the uniform distribution of the polynomial sequence $λ(P)=(\lfloor P(k) \rfloor )_{k\geq 1}$ modulo integers, where $P(x)$ is a polynomial with real coefficients. In the nonlinear case, we show that $λ(P)$ is uniformly distributed in $\mathbb{Z}$ if and only if $P(x)$ has at least one irrational coefficient other than the constant term. In the case of even degree, we prove a stronger result: $λ(P)$ intersects every congruence class modulo every integer if and only if $P(x)$ has at least one irrational coefficient other than the constant term.

math.NT↗

arXiv2018

Complete Generalized Fibonacci Sequences Modulo Primes

We study generalized Fibonacci sequences $F_{n+1}=PF_n-QF_{n-1}$ with initial values $F_0=0$ and $F_1=1$. Let $P,Q$ be nonzero integers such that $P^2-4Q$ is not a perfect square. We show that if $Q=\pm 1$ then the sequence $\{F_n\}_{n=0}^\infty$ misses a congruence class modulo every prime large enough. On the other hand, if $Q \neq \pm 1$, we prove that (under GRH) the sequence $\{F_n\}_{n=0}^\infty$ hits every congruence class modulo infinitely many primes.

math.NT↗

arXiv2017

On the subgroup generated by solutions of Pell's equation

Equivalence classes of solutions of the Diophantine equation $a^2+mb^2=c^2$ form an infinitely generated abelian group $G_m$, where $m$ is a fixed square-free positive integer. Solutions of Pell's equation $x^2-my^2=1$ generate a subgroup $P_m$ of $G_m$. We prove that $P_m$ and $G_m/P_m$ have infinite rank for all $m>1$. We also give several examples of $m$ for which $G_m/P_m$ has nontrivial torsion.

math.NT↗

arXiv2017

Dense semigroups of triangular matrices

Let $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$, and $T_n(\mathbb{K})$ be the set of $n\times n$ lower triangular matrices with entries in $\mathbb{K}$. We show that $T_n(\mathbb{K})$ has dense subsemigroups that are generated by $n+1$ matrices.

math.DS↗