In this paper, a new continuous probability distribution has been proposed. The proposed distribution is considered as one of the extended distributions of Rayleigh distribution, which is well known in probability Theory and its applications. The distribution under consideration is titled "Exponential Log-compound Rayleigh distribution". proposed. The statistical properties of distribution parameters estimating using Maximum likelihood technique and Fisher information matrix are investigated theoretically. The evaluating of distribution performance is demonstrated on both simulated data and real data. Based on the values of bias, MSE and Fisher information matrix of the parameters estimates, the proposed distribution is considered to be working well.

In this paper, we consider the Bayesian estimation of the parameters and reliability function for a Cosine inverse log compound Rayleigh distribution under squared error and squared logarithmic loss functions. We use Lindley’s approximation to compute the Bayesian estimates. This method is evaluated using mean square error through simulation study with varying sample size.

The aim of this paper is to represent a bitopological representation (𝑉,𝜏𝑆1,𝜏𝑆2) of divisor graph 𝐺=(𝑉,𝐸) defining in a finite commutative rings in which every vertex 𝑣 is adjacent with a vertex 𝑢 if and only if 𝑔.𝑐.𝑑(𝑢,𝑣)=1. Then some properties of this bitopological space were investigated.

This paper introduces a finite topological space 𝜏𝑝 on the group of units modulo a prime 𝑝, defined by its basis of conjugate residue pairs {𝛼, 𝑝 − 𝛼} for all units 𝛼 ∈ 𝑈𝑝. We investigate the fundamental topological concepts such as point-set topology, separation axioms, and characterise the structure and behaviour of this topology. Additionally, we examine a function 𝑓 from 𝜏𝑝 to the topology of quadratic residues 𝜏𝑄, mapping each unit to its square modulo 𝑝. We analyse the continuity, openness of 𝑓, and explore its implications for separation properties. Furthermore, we define a quotient topology on 𝑈𝑝 based on the equivalence relation 𝑥 ∼ 𝑦 if and only if 𝑥2 ≡ 𝑦2 𝑚𝑜𝑑 𝑝, showing that the resulting quotient space is homeomorphic to (𝑄𝑝 , 𝜏𝑄 ).

This study presents a descriptive statistical analysis of a retrospective study on Polycystic Ovary Syndrome (PCOS), a common endocrine disorder among women of reproductive age that can lead to various metabolic, hormonal, and reproductive complications. Data were collected from the Misurata Infertility Center between 2023 and 2024 from 250 women undergoing infertility evaluation. Demographic data, including age and body mass index (BMI), along with medical records, were reviewed. Hormonal parameters assessed were Luteinizing Hormone (LH), Follicle-Stimulating Hormone (FSH), Estradiol, Prolactin, Anti-Müllerian Hormone (AMH), and Thyroid-Stimulating Hormone (TSH). Biochemical markers included blood sugar (BS), HbA1c, and Vitamin D levels. Results indicated that the average BMI was 28.72, which is above the World Health Organization's ideal range (18.5–24.9), suggesting that most patients were overweight. The mean Estradiol level was 47.95 pg/mL (within the ideal range of 0–80), and the mean FSH was 7.704 mIU/mL (within the ideal range of 1.37–9.9). In contrast, the mean Prolactin level was 23.148 ng/mL, exceeding the ideal range (4.5–21.5), indicating hyperprolactinemia in most patients. The mean Vitamin D level was 21.22 ng/mL, below the ideal range (30–50), indicating a deficiency. The mean BS was 105.92 mg/dL (within normal range: 70–110), and the average HbA1c was 5.622%, suggesting most patients were not diabetic. Keywords: Polycystic Ovary Syndrome; Demographic Data; Medical Records; Hormonal Levels; Misurata Infertility Center.

We investigate the structural properties of a graph defined on the ring. The Adjacency between two different vertices and is determined by the bilinear congruence. We analyze three fundamental cases, and for distinct odd primes. We describe the graph's breakdown into unit and non-unit vertex subsets. The unit subgraph forms disjoint cliques, with sizes depending on Euler's totient function. In contrast, the zero-divisor subgraph shows more complex behaviour governed by annihilation ideals. We establish general properties, including degree formulas, determination of maximum clique sizes in each component, determining the diameter, computing the girth, locating the graph centers, and finding the measures of vertex and edge connectivity. Additionally, we characterize independent sets and prove the existence of Hamiltonian cycles and supereulerian properties under certain connectivity conditions. Our results show how the prime factorization of influences these properties.

The aim of this paper is to prove that the Diophantine exponent 𝐷𝑖𝑜( 𝑑−𝜷(𝑙𝛽)) is bounded by 𝑙𝑜𝑔𝑀(𝛽)𝑙𝑜𝑔𝛽+(𝑑−1)𝑙𝑜𝑔|𝑎𝑑|+1𝑙𝑜𝑔𝛽, where 𝛽 is an algebraic number, and 𝑀(𝛽) is the Mahler measure of 𝛽. In addition, a new result will be founded about transcendental number 𝑎 when 𝐷𝑖𝑜(𝑎) is infinite.

For a composite number 𝒏, we explore a topological space on the ring of integers modulo 𝒏, we use basis sets formed by solutions to quadratic residue equations, where elements are units modulo 𝒏. This definition allows us to investigate the algebraic relationships among solutions, focusing on properties like clopen sets, closure, and interior operations. Additionally, we examine the continuity of mappings between the group of units in ℤ/𝒏ℤ and quadratic residues, alongside the quotient topology induced on this group of units. A comparison of the separation properties of these topological spaces is also presented

Abstract: The exact solution of a nonlinear fractional Volterra-Fredholm integro-differential equation is found in this paper through the successful application of the Abaoub Shkheam decomposition method. These techniques have a wider range of applications due to their dependability and decreased computational effort. 

Abstract: In this paper, a nonlinear fractional diffusion and wave equations are solved using the Abaoub-Shkheam Decomposition Method (QDM). The Caputo sense is used to characterise the fractional derivative. An example is provided to demonstrate the method's efficiency and practicality.