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 ).

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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.

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.

This paper presents a comprehensive study of modelling human body blockage (the most critical challenges in fifth-generation (5G)) effects on indoor millimetre wave (mmWave) communication links at 32.5 GHz, a key frequency for 5G networks. Through controlled experiments in a laboratory environment, we analyse signal attenuation as a human subject obstructs the line-of-sight (LOS) path between transmitter and receiver, recording received power at incremental positions. To model the observed phenomena, we propose a hybrid framework integrating deterministic and statistical components: (1) a modified Double Knife-Edge Diffraction (DKED) model with Gaussian-shaped blockage attenuation (20.8 dB peak at full blockage) and reflection-induced signal enhancement (−15.0 dB peak from nearby objects), and (2) a log-normal shadowing component (σ = 11.8 dB) capturing environmental randomness. Our results reveal strong agreement between simulations and measurements, achieving a mean absolute error of 3.2 dB and a correlation coefficient R² = 0.89. The analysis demonstrates that human-induced diffraction dominates near the LOS centre, while multipath reflections significantly alter signal strength at peripheral positions. We further derive practical guidelines for 5G network design, recommending a 44.4 dB link budget safety margin to account for combined blockage and shadowing effects. This work advances indoor mmWaves channel modelling by unifying physics-based diffraction analysis with empirical reflection characterization, the framework achieves strong experimental validation and offers actionable insights for 5G network design

The advent of 5G networks has revolutionized wireless communications by unlocking unprecedented data rates through millimeter-wave (mmWave) frequencies. However, the short wavelengths of mmWave signals (e.g., 32.5 GHz) make them highly vulnerable to obstructions, particularly human blockage, posing significant challenges for reliable link prediction and network planning. Existing models often oversimplify human-induced attenuation, limiting their accuracy in real-world scenarios. This work addresses this gap by proposing a cylindrical diffraction model to quantify human blockage effects at 32.5 GHz—the first application of such a model at this frequency. Through controlled experiments, we measured signal degradation as a human subject progressively blocked a 2-meter mmWave link, revealing a sharp decline in received power from −41.2 dBm (no blockage) to −69.7 dBm (full blockage). The cylindrical model demonstrated strong alignment with empirical trends, accurately capturing the nonlinear increase in attenuation as the human approached the line-of sight path. Notably, the model matched baseline measurements within 1.4 dB and predicted full-blockage loss within 7 dB of observed values, despite inherent simplifications. This study underscores the efficacy of cylindrical modelling for mmWave blockage prediction while highlighting critical refinements needed for practical deployment, such as incorporating material properties and antenna radiation patterns. By bridging theoretical and empirical insights, our work provides a foundational framework for enhancing 5G/6G network resilience in human-dense environments, ensuring robust performance for high-data-rate applications. 

ZnO thin films were deposited on stainless steel substrates using the immersion method by immersing the

substrates in a sol-gel coating solution at temperatures of 70 °C and 80 °C for varying durations of (1, 2, 3, and 4minutes).

The results indicated that increasing the immersion time significantly influenced the film thickness. Optical measurements

showed that transmittance at a wavelength of 350nm increased with higher deposition temperatures. Additionally, the

Urbach tail energy increased with temperature, whereas the band gap (Eg) decreased markedly. Furthermore, the

electrical conductivity of the ZnO films improved with increased temperature and immersion time.