arXiv subjects

Jasem Hamoud

Publications and source records attributed to Jasem Hamoud.

34 records · Page 2Linked to original sources

arXiv2025

The Effect of Using Popular Mathematical Puzzles on The Mathematical Thinking of Syrian Schoolchildren

In this paper we provide a good overview of the problems and the background of mathematics education in Syrian schools. We aimed to study the effect of using popular mathematical puzzles on the mathematical thinking of schoolchildren, by conducting a paired experimental study (pre-test and post-test control group design) of the data we obtained through a sample taken from students of sixth-grade primary school students in Syria the Lady Mary School in Syria, in order to evaluate the extent of the impact of popular mathematical puzzles on students' ability to solve problems and mathematical skills, and then the skills were measured and the results were analyzed using a t-test as a tool for statistical analysis.

econ.TH↗

arXiv2025

A Study of NP-Completeness and Undecidable Word Problems in Semigroups

In this paper we explore fundamental concepts in computational complexity theory and the boundaries of algorithmic decidability. We examine the relationship between complexity classes \textbf{P} and \textbf{NP}, where $L \in \textbf{P}$ implies the existence of a deterministic Turing machine solving $L$ in polynomial time $O(n^k)$. Central to our investigation is polynomial reducibility. Also, we demonstrate the existence of an associative calculus $A(\mathfrak{T})$ with an algorithmically undecidable word problem, where for a Turing machine $\mathfrak{T}$ computing a non-recursive function $E(x)$, we establish that $q_1 01^x v \equiv q_0 01^i v \Leftrightarrow x \in M_i$ for $i \in \{0,1\}$, where $M_i = \{x \mid E(x) = i\}$. This connection between computational complexity and algebraic undecidability illuminates the fundamental limits of algorithmic solutions in mathematics.

cs.CC↗

arXiv2025

Density Characterization with The Upper Bound of Density of Fibonacci Word

This paper investigates the natural density and structural relationships within Fibonacci words, the density of a Fibonacci word is $\operatorname{DF}(F_k)=n/(n+m),$ where $m$ denote the number of zeros in a Fibonacci word and $n$ denote the units digit. Through analysis of these ratios and their convergence to powers of $φ$, we illustrate the intrinsic exponential growth rates characteristic of Fibonacci words. By considering the natural density concept for sets of positive integers, it is demonstrated that the density of Fibonacci words approaches unity, correlating with classical results on Fibonacci number distributions as \[ \operatorname{DF}(F_k) <\frac{m(m+1)}{n(2m-n+1)}. \] Furthermore, generating functions and combinatorial formulas for general terms of Fibonacci words are derived, linking polynomial expressions and limit behaviors integral to their combinatorial structure. The study is supplemented by numerical data and graphical visualization, confirming theoretical findings and providing insights into the early transient and asymptotic behavior of Fibonacci word densities.

math.CO↗

arXiv2025

The Role of Mathematical Folk Puzzles in Developing mathematical Thinking and Problem-Solving Skills

This paper covers a variety of mathematical folk puzzles, including geometric (Tangrams, dissection puzzles), logic, algebraic, probability (Monty Hall Problem, Birthday Paradox), and combinatorial challenges (Eight Queens Puzzle, Tower of Hanoi). It also explores modern modifications, such as digital and gamified approaches, to improve student involvement and comprehension. Furthermore, a novel concept, the "Minimal Dissection Path Problem for Polyominoes," is introduced and proven, demonstrating that the minimum number of straight-line cuts required to dissect a polyomino of N squares into its constituent units is $\mathrm{N}-1$. This problem, along with other puzzles, offers practical classroom applications that reinforce core mathematical concepts like area, spatial reasoning, and optimization, making learning both enjoyable and effective.

econ.TH↗

arXiv2025

The Exponential Hyper-Zagreb Indices and Structural Properties of Trees and Bipartite Graphs

In this paper, we investigate the structural properties of trees and bipartite graphs through the lens of topological indices and combinatorial graph theory. We focus on the First and Second Hyper-Zagreb indices, $HM_1(G)$ and $HM_2(G)$, for trees $T \in T(n, Δ)$ with $n$ vertices and maximum degree $Δ$. Key propositions demonstrate that the presence of end-support or support vertices of degree at least three, distinct from a vertex of maximum degree, implies the existence of another tree $T' \in T(n, Δ)$ with strictly smaller Hyper-Zagreb indices. These results are extended to exponential forms, highlighting the influence of high-degree vertices. Additionally, we explore structural characterizations of trees via degree sequence majorization and $S$-order, establishing conditions for the first and last trees in specific classes. For bipartite graphs, we examine equitable coloring, cycle lengths, and $k$-redundant tree embeddings, supported by theorems on connectivity and minimum degree constraints. The paper also addresses the independence number of bipartite graphs, exterior covers, and competition numbers of complete $r$-partite graphs, providing bounds and structural insights. Finally, we discuss Markov-chain algorithms for generating bipartite graphs and tournaments with prescribed degree sequences, analyzing their mixing times and convergence properties. These results contribute to the understanding of extremal properties and combinatorial structures in graph theory, with applications in chemical graph theory and network analysis.

math.CO↗

arXiv2025

Bounds of Trees with Degree Sequence-Based Topological Indices on Specialized Graph Classes

In this paper, the investigates Adriatic indices, specifically the sum lordeg index where it defined as $SL(G) = \sum_{u \in V(G)} °_G(u) \sqrt{\ln °_G(u)}$ and the variable sum exdeg index $SEI_a(G)$ for $a>0$, $a\neq 1$. We present several sharp bounds and characterizations of these and related topological indices on specialized graph classes, including regular graphs, thorny graphs, and chemical trees. Using the strict convexity of function $f$, inequalities for degree-based graph invariants $H_f(T)$ are derived under structural constraints on trees such as branching vertices and maximum degree. Examples on caterpillar trees illustrate the computation of indices like $^{m}M_2(G)$, $F(G)$, $M_2(G)$, and others, revealing the interplay between degree sequences and index values. Additionally, upper and lower bounds on the Sombor index $SO(G^*)$ of thorny graphs $G^*$ are established as \[ \operatorname{SO} \leqslant \sum_{uv\in E(G)}\sqrt{\frac{1}{°_{G}(u)^2+°_{G}(v)^2}+°_{G}(u)+°_{G}(v)}, \] including criteria for equality, with implications for regular and thorn-regular graphs. The treatment includes detailed formulas, constructive examples, and inequalities critical for understanding the relationship between graph topology and vertex-degree-based descriptors.

math.CO↗

arXiv2025

The Density of Zero and One in Fibonacci Word for Subwords and Their Palindromes

This paper studies the density of zero and one in subwords of the Fibonacci word with lengths less than thirty and compares them to the densities of their corresponding palindromes. We used computational methods to produce a sufficiently large piece of the Fibonacci word, extract all unique subwords up to a predetermined length, and calculate their palindrome. The density of each character (0 and 1) was then determined for both the original subwords and their palindromic counterparts. This study contributes to a deeper understanding of the combinatorial properties of the Fibonacci word and the behavior of its constituent elements under reversal.

math.CO↗

arXiv2025

On Topological Indices in Trees: Fibonacci Degree Sequences and Bounds

In this paper, we have studied bounds based on topological indicators, from which we selected Albertson index $\mathrm{irr}$ and the Sigma index $σ$. The Sigma index was defined through the following relationship: \[ σ(G)=\sum_{uv\in E(G)}\left( d_u(G)-d_v(G) \right)^2. \] We establish a precise formula for the Albertson index of a tree $T$ of order $n$ with a Fibonacci degree sequence $\mathscr{D} = (F_3, \dots, F_n)$. Additionally, we derive bounds for the minimum and maximum Albertson indices ($\irr_{\min}$ and $\irr_{\max}$) across various tree structures. Propositions and lemmas provide upper and lower bounds, incorporating parameters such as the maximum degree $ Δ$, minimum degree $δ$. We further relate the Albertson index to the second Zagreb index $M_2(T)$ and the forgotten index $F(T)$, establishing a new upper bound.

math.CO↗

arXiv2025

Bounds on Trees with Topological Indices Among Degree Sequence

In this paper, we investigate The relationship between the Albertson index and the first Zagreb index for trees. For a tree $T=(V,E)$ with $n=|V|$ vertices and $m=|E|$ edges, we provide several bounds and exact formulas for these two topological indices, and we show that the Albertson index $\irr(T)$ and the first Zagreb index $M_1(T)$ satisfy the association \[ \operatorname{irr}(T)=d_1^2+d_n^2+(n-2)\left(\frac{Δ+ δ}{2}\right)^2+\sum_{i=2}^{n-1} d_i+d_n - d_1-2n-2.\] Our goal of this paper is provide a topological indices, Albertson index, Sigma index among a degree sequence $\mathscr{D}=(d_1,\dots,d_n)$ where it is non-increasing and non-decreasing of tree $T$.

math.CO↗

arXiv2025

Novel Constructions of Words with Strong Avoidance Properties and their Combinatorial Analysis

This paper begins with a comprehensive overview of combinatorics on words and symbolic dynamics, covering their historical origins, fundamental concepts, and interconnections. Building upon this foundation, we introduce novel mathematical constructions related to pattern avoidance in infinite words. Specifically, we define Strongly $(k, δ)$-Free Words generated via cyclic shift morphisms and present a theorem establishing specific avoidance properties for these words, along with a detailed proof. Furthermore, we propose a conjecture regarding their factor complexity. These original results contribute to the theoretical understanding of word structures and their combinatorial properties, opening avenues for further research in discrete mathematics.

math.CO↗

arXiv2025

Topological Indices Among Strong Support Vertex

In this paper, we provide the irregularity properties of trees with strong support vertex by analyzing two prominent topological indices: the Albertson index and the Sigma index. We further establish extremal bounds for both indices across families of trees defined by given degree sequences. Let $T_1$ and $T_2$ be a star trees of order $n$, where $T_1 \cong T_2$, then, we provide Albertson index of $T_1 \cong T_2$. Let $\mathcal{T}_{n, Δ}$ be a class of trees with $n$ vertices, there are a tree $T^{\prime} \in \mathcal{T}_{n, Δ}$ such that $\irr(T^{\prime}) < \irr(T)$.

math.CO↗

arXiv2025

Irregularity and Topological Indices in Fibonacci Word Trees and Modified Fibonacci Word Index

This paper introduces the concept of the Fibonacci Word Index $\operatorname{FWI}$, a novel topological index derived from the Albertson index, applied to trees constructed from Fibonacci words. Building upon the classical Fibonacci sequence and its generalizations, we explore the structural properties of Fibonacci word trees and their degree-based irregularity measures. We define the $\operatorname{FWI}$ and its variants, including the total irregularity and modified Fibonacci Word Index where it defined as \[ \operatorname{FWI}^*(T)=\sum_{n,m\in E(\mathscr{T})}[deg F_n^2-deg F_m^2], \] and establish foundational inequalities relating these indices to the maximum degree of the underlying trees. Our results extend known graph invariants to the combinatorial setting of Fibonacci words, providing new insights into their algebraic and topological characteristics.

math.CO↗

arXiv2025

Generalized Natural Density $\DF(\mathfrak{F}_n)$ of Fibonacci Word

This paper explores profound generalizations of the Fibonacci sequence, delving into random Fibonacci sequences, $k$-Fibonacci words, and their combinatorial properties. We established that the $n$-th root of the absolute value of terms in a random Fibonacci sequence converges to $1.13198824\ldots$, a symmetry identity for sums involving Fibonacci words, $\sum_{n=1}^{b} \frac{(-1)^n F_a}{F_n F_{n+a}} = \sum_{n=1}^{a} \frac{(-1)^n F_b}{F_n F_{n+b}}$, and an infinite series identity linking Fibonacci terms to the golden ratio. These findings underscore the intricate interplay between number theory and combinatorics, illuminating the rich structure of Fibonacci-related sequences. We provide, according to this paper, new concepts of density of Fibonacci word.

math.CO↗

arXiv2025

Dynamic and Programmatic Analysis of Fibonacci Word Density

Fibonacci word is the archetype of the Sturmian word, and it is one of the most studied of combinatorics on words. We studied the properties of the Fibonacci word and found its density for limited value then by calculating the limit associated with the general relation of $\frac{F(n)}{F(n+1)}$. We also generated the density function through the integral relation of $\int_a^b f(x) dx $ where $D(n)=\int_a^b f(x) dx$ and $f(x) = e^{-x/τ} g(x)$. we presented the first study of the density of the Fibonacci word, in addition to some analysis through programming.

math.CO↗

arXiv2025

Improvement Ergodic Theory For The Infinite Word $\mathfrak{F}=\mathfrak{F}_{b}:=\left({ }_{b} f_{n}\right)_{n \geqslant 0}$ on Fibonacci Density

The paper explores combinatorial properties of Fibonacci words and their generalizations within the framework of combinatorics on words. These infinite sequences, measures the diversity of subwords in Fibonacci words, showing non-decreasing growth for infinite sequences. Extends factor analysis to arithmetic progressions of symbols, highlighting generalized pattern distributions. Recent results link Sturmian sequences (including Fibonacci words) to unbounded binomial complexity and gap inequivalence, with implications for formal language theory and automata. This work underscores the interplay between substitution rules, algebraic number theory, and combinatorial complexity in infinite words, providing tools for applications in fractal geometry and theoretical computer science.

math.CO↗

arXiv2024

Sigma index in Trees with Given Degree Sequences

The sigma index in graph theory refers to a measure of the degree differences between vertices in a graph. The goal is to determine the graphs that have the maximum sigma index within certain classes of graphs. Abdo, Dimitrov, and Gutman characterized the graphs with the greatest sigma index among all connected graphs of a fixed order.

math.GM↗