arXiv subjects

Mehdi Golafshan

Publications and source records attributed to Mehdi Golafshan.

6 recordsLinked to original sources

arXiv2026

Asymptotic counting of integers with prime factors $p_{r^a s^b}$

Let $p_n$ be the $n$th prime, and let $r,s\ge2$ be fixed multiplicatively independent integers. We count the integers up to $x$ whose prime factors all have the form $p_{r^a s^b}$ with integers $a,b\ge0$. Our asymptotic formula for this count has relative error $o(1)$ and is explicit down to the multiplicative constant. For $(r,s)=(2,3)$ these integers are the prime codes of the ordinals below $ω^{ω^2}$, so the formula settles that case of the counting problem of Vernaeve, Vindas and Weiermann.

math.NT↗

arXiv2026

Binary binomial equivalence via hyperplane arrangements

Rigo and Salimov (2015) proved that the number of binary $2$-binomial equivalence classes of words of length $n$ is the $n^{\text{th}}$ cake number. We give a geometric explanation of this identity by constructing an explicit arrangement of $n$ planes in three-dimensional space whose chambers are naturally indexed by these equivalence classes. This arrangement is the three-dimensional member of an infinite family of hyperplane arrangements. In dimensions $1$, $2$, and $3$, the corresponding quotients recover abelian equivalence, a natural intermediate equivalence between abelian and $2$-binomial equivalence, and binary $2$-binomial equivalence itself. In higher dimensions, the same family realises natural refinements of $2$-binomial equivalence. We also determine the sizes of the resulting classes. Each size is given by a coefficient of a suitable Gaussian binomial coefficient. This yields the full class-size distribution for binary $2$-binomial equivalence and stabilisation results for the number of classes of any fixed cardinality.

math.CO↗

arXiv2026

Enumeration of Factor Occurrences in $k$-Bonacci Words over an Infinite Alphabet

We study the $k$-Bonacci word over the infinite alphabet $\mathbb{N}$. Since the alphabet is infinite, the usual factor complexity is infinite and does not provide any information. We therefore investigate factor occurrence statistics in the finite iterates. For $k \ge 3$, we obtain closed forms for the generating functions (with respect to the iteration index) that count the number of occurrences of an arbitrary digit in the $n$th iterate. We then characterize the complete set of length-$2$ factors occurring in the infinite word and compute, for each such factor, a closed form for the generating function encoding its number of occurrences across all finite iterates. As a consequence, the associated counting sequences satisfy uniform $(k\!-\!1)$-step Fibonacci-type recurrences and admit a description in terms of $(k\!-\!1)$-Bonacci enumeration phenomena, including self-convolution structures.

math.CO↗

arXiv2024

Factor Complexity of the Most Significant Digits of~$a^{n^d}$

We investigate unipotent dynamics on a torus and apply these techniques to the following problem. Let $d$ be a positive integer, and let $a > 0$ be a real number. For an integer $b \geqslant 5$, such that $a$ and $b$ are multiplicatively independent, consider the sequence $(\mathbf{w}_n)$, where $\mathbf{w}_n$ is the most significant digit of $a^{n^d}$ when expressed in base $b$. We prove that the complexity function of the sequence $(\mathbf{w}_n)$ is, up to finitely many exceptions, a polynomial function.

math.DS↗

arXiv2023

An Indirect Method for Solving Combinatorial Problems

This article introduces a pedagogical method for {\it solving combinatorial problems} that frequently involve structures that are unfamiliar or less familiar. Indeed, an indirect method has been proposed in order to evade any possible questions that might obstruct our path to the correct answer and final solution. With this method, the desired configuration can be formulated. This method is effective for solving problems in Olympiads and mathematical competitions as well as for teaching combinations and counting science in the elementary school mathematics curriculum.

math.HO↗

arXiv2022

On the Distribution function of area and perimeter for planar poisson line process

The challenges of examining random partitions of space are a significant class of problems in the theory of geometric transformations. Richard Miles calculated moments of areas and perimeters of any order (including expectation) of the random division of space in 1972. In the paper we calculate whole distribution function of random divisions of plane by poisson line process. The idea is to interpret a random polygon as the evolution of a segment along a moving straight line. In the plane example, the issue connected with an infinite number of parameters is overcome by considering a secant line. We shall take into account the following tasks: {\textbf 1.} On the plane, a random set of straight lines is provided, all shifts are equally likely, and the distribution law is of the form $F(φ).$ What is the area distribution of the partition's components? {\textbf 2.} On the plane, a random set of points is marked. Each point $A$ has an associated area of attraction, which is the collection of points in the plane to which the point $A$ is the nearest of the designated ones. In the first problem, the density of moved sections adjacent to the line allows for the expression of the balancing ratio in kinetic form. Similarly, you can write the perimeters kinetic equations. We will demonstrate how to reduce these equations to the Riccati equation using the Laplace transformation in this paper.

math.PR↗