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 ).

Abstract— Crime prediction has gained increasing attention due to the growing availability of historical crime data and the need for data-driven decision-making in public safety. This study presents a comparative analysis of Long Short-Term Memory (LSTM) architectures for predicting the exact occurrence time of crimes based on temporal patterns. Three LSTM-based models are evaluated: Vanilla LSTM, Stacked LSTM, and Bidirectional LSTM.

The proposed approach integrates time-based features and lag features to capture temporal dependencies within crime data. Model performance is assessed using standard regression metrics, including Mean Absolute Error (MAE), Mean Squared Error (MSE), and Root Mean Squared Error (RMSE). Experimental results indicate that deeper LSTM architectures combined with temporal lag information improve prediction accuracy compared to the baseline model.

This study demonstrates the effectiveness of LSTM-based models for crime occurrence prediction and provides insights into selecting suitable deep learning architectures for time-series crime analysis, supporting the development of more reliable tools for proactive crime prevention.

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.

Abstract—

The verification of Arabic handwritten signatures is an important research area in the fields of computer vision, machine learning, and deep learning. It is essential to distinguish between genuine and forged signatures due to its significance in preventing fraud and forgery, as well as in various legal contexts. Handwritten signatures vary due to several influencing factors, such as mood, writing surface, and writing instrument. In this research paper, three different datasets were utilized: the first dataset (Dataset1) is an English signature dataset known as CEDAR; the second dataset (Dataset2) is an Arabic signature dataset specifically created for this research; and the third dataset (Dataset3) combines the CEDAR dataset with the Arabic signatures. These datasets were utilized to develop a transfer learning model for signature recognition called MobileNet-V2, which is a deep neural network (DNN) model developed by Google, aimed at achieving a balance between accuracy and performance speed. It is primarily designed for image classification tasks in resource-constrained environments. Standard training and k-fold cross-validation (CV) methods were applied to evaluate the model's performance. The model achieved an accuracy of 97.74% on the CEDAR dataset, 98.49% on the Arabic dataset, and 99.49% on the combined dataset.

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.

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A new mechanical framework is introduced to describe bodies with time-continuous mass

variation, incorporating a functional dependence on mass. In this approach, the dynamics

are governed by momentum balance equations formulated using Caputo fractional

derivatives, which adhere to a weak form of Galilean invariance. The formulation is

particularly focused on the Meshchersky kinetics, accounting for both mass and velocity

changes. As a practical example, this paper presents a novel model for the motion of a

material body with continuously varying mass in a constant gravitational field—leading to

a time-fractional version of the Tsiolkovsky rocket equation, augmented by a dissipative

term. Under time-based approximation, deviations from vertical projectile motion are

analyzed to assess the internal consistency of the proposed model.

A three-parameter generalization of the Tsallis entropy based on the properties of the power functions and Weyl fractional calculus like extension of quantum calculus, are introduced. The generalization of the Shannon-Khinchin axioms corresponding to the fractional Tsallis entropy is verified and proposed. These axioms uniquely characterize new entropy function. For a certain sets of parameter values satisfied the second and third law of thermodynamics, the Lesche and thermodynamic stability criteria.

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.