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.

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.

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. 

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

Thin films deposited on glass surfaces can be produced by various

methods, including physical methods (thermal evaporation in

vacuum, sputtering, etc.) and chemical methods (chemical

deposition, chemical vapor deposition, thermal chemical analysis,

etc.). In this paper, thin films were deposited on glass slides using

the plasma-enhanced chemical vapor deposition (PECVD)

technique at pressures (10,30 and 50 mTorr) for varying durations

of (5,10 and 15 minutes). In addition, the results indicates that

increasing the deposition time significantly influenced the film

thickness while the adhesion does not directly depend on time.

In this study, the morphological and thermal properties of a blend consisting of 75% polyvinyl chloride (PVC) and

25% low-density polyethylene (LDPE), reinforced with nano-refractory bricks (NRB) at varying ratios (1%, 3%, 5%, and 7%),

were prepared and investigated. The objective was to enhance the blend’s thermal stability and surface structure homogeneity.

The morphological structure of the prepared samples was analyzed using scanning electron microscopy (SEM) and X-ray

diffraction (XRD). The obtained images revealed that the nano-refractory brick particles remained predominantly within the

PVC phase, resulting in an increased composite density without significantly affecting the LDPE phase. This suggests that the

overall morphology of the composite is largely unaffected by the presence of the nano-refractory bricks. From a thermal

perspective, thermogravimetric analysis (TGA/DTG) was conducted on all samples from room temperature up to 600 °C at a

heating rate of 10 °C/min. FTIR and TGA results indicated that the decomposition onset temperature shifted to higher values

due to the presence of nano-refractory bricks, with the most pronounced shift observed in the sample containing 3%

reinforcement. Additionally, the rate of mass loss was reduced. These findings demonstrate that reinforcing the PVC/LDPE

blend with nano-refractory bricks improves its performance, making it more suitable for applications that demand enhanced

thermal and mechanical properties, such as those in the construction and electronics industries.

This study aimed to assess surface contamination levels of children’s playground equipment in two public parks in AL-Zawiya City, Libya (Jdayem Park and AL-Zawiya Park), using adenosine triphosphate (ATP) bioluminescence monitoring. The equipment surveyed included plastic slides, metal slides, iron swings, plastic swings, and rope swings. A total of 24 samples were collected from these surfaces using standardized ATP swabs, and results were classified based on established RLU thresholds (<100 RLU: clean, 100–300 RLU: marginally contaminated, >300 RLU: contaminated). The results revealed significant variability across equipment and surface types. The highest contamination level was observed on the iron swing at Jdayem Park (21–416 RLU), while the lowest levels were recorded on rope swings in AL-Zawiya Park (11–25 RLU). Plastic surfaces and rope swings generally exhibited lower contamination compared to metal surfaces, suggesting that surface material and design, combined with usage frequency, play a pivotal role in contamination accumulation. These findings align with previous studies that highlight the role of surface characteristics and environmental exposure in influencing microbial load. The results underscore the urgent need for targeted cleaning, disinfection, and regular quality

.monitoring of public playground equipment to maintain a safe, hygienic, and child-friendly recreational environment

This paper investigates human-induced signal attenuation in indoor mm-wave communications at 32.5 GHz, a critical concern for 5G systems. Two distinct diffraction-based models are applied to the same indoor scenario to assess human blockage effects: one employs the Double Knife-Edge Diffraction (DKED) approach, and the other uses Fresnel diffraction principles with complex Fresnel integrals. Controlled experiments with a human subject moving between a transmitter (TX) and a receiver (RX) reveal that the DKED model consistently underestimates the received power by 2 6 dB, while the Fresnel diffraction approach underestimates it by 2–5 dB Based on the comparative results, the DKED model demonstrates higher accuracy in predicting signal attenuation, offering valuable insights for improving indoor 5G network performance