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.

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

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.

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.

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.

Argemone ochroleuca Sweet, Brit. Fl. Gard. Ser.1 3(2): pl. 242 (1828) of the family Papaveraceae is reported here for the first time from Libya. This species is native to Mexico. Flowering specimens of A. ochroleuca were collected from the region Ariggiba (26°35’11.8”N 13°29’24.8”E), 110 km southwest of the city of Sabha, which lies about 700 km south of Tripoli. It is presented with updated nomenclature, taxonomic description, geographical distribution, key to genera, location in Libya, and colour photographs taken in the field. The voucher specimens are deposited in the Herbarium of Botany Department (ULT), University of Tripoli, Libya.