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.

The healthcare system in Libya faces significant challenges due to political instability, fragmented infrastructure, and inconsistent medical practices. Clinical Practice Guidelines (CPGs) serve as essential tools for standardising care, ensuring evidence- based treatment, and optimising healthcare resources. In Libya, the lack of structured guidelines has contributed to disparities in disease management, affecting patient outcomes and overall healthcare efficiency. This commentary explores the critical need for CPGs in Libya, highlighting their potential to improve healthcare delivery, minimise variability in treatment, and enhance patient safety. While implementation poses challenges, including centralisation, limited research capacity, and resource constraints, integrating CPGs through a phased implementation framework could be a transformative step toward a more resilient and equitable healthcare system. By fostering collaboration among policymakers, healthcare professionals, and international organisations, Libya can lay the foundation for a systematic approach to disease management, ultimately improving the quality of care for its population. Healthcare reform in Libya is urgently needed, and strategic investments in CPG development and dissemination could drive the necessary transformation in Libyan healthcare.

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 Micro-hardness is a fundamental property of prosthodontic restorative materials, as it affects their resistance to surface deformation, wear, and long-term clinical performance. This study aimed to compare the Vickers micro-hardness of three widely used CAD/CAM materials: lithium disilicate, monolithic zirconia, and multilayered zirconia. A total of 30 specimens (N = 30) were fabricated, with 10 samples allocated to each material group. Lithium disilicate specimens were prepared as rectangular plates (18 × 15 × 1 mm), whereas zirconia specimens—both monolithic and multilayered—were fabricated as discs (10 mm in diameter and 1.5 mm in thickness) following standardized CAD/CAM milling and sintering protocols. Vickers micro-hardness testing was conducted using a digital micro-hardness tester under material-specific conditions: a load of 1 kg and a dwell time of 10 s for lithium disilicate, and a load of 500 g with a 20 s dwell time for zirconia. Statistical analysis was performed using one-way ANOVA followed by Tukey’s post-hoc test, with the significance level set at p < 0.05. Significant differences in micro-hardness were identified among the three materials (p < 0.001). Monolithic zirconia demonstrated the highest mean hardness (680 ± 19 HV), followed by multilayered zirconia (623 ± 47 HV), while lithium disilicate exhibited the lowest values (553 ± 32 HV). Post-hoc analysis confirmed that all pairwise comparisons were statistically significant. The findings indicate that monolithic zirconia possesses superior micro-hardness compared to multilayered zirconia and lithium disilicate, supporting its suitability for high-stress clinical applications. Multilayered zirconia offers a balance between mechanical performance and esthetics, whereas lithium disilicate remains optimal for highly esthetic anterior restorations. These results provide clinicians with evidence-based guidance for selecting CAD/CAM materials in fixed prosthodontic rehabilitation. 

The surface integrity of CAD/CAM restorative materials is critical for their esthetic and mechanical performance. Acidic challenges in the oral environment can degrade ceramic surfaces, increasing roughness and potentially compromising restoration longevity. This study aimed to evaluate and compare the effect of 4% acetic acid immersion on the surface roughness (Ra) of lithium disilicate glass-ceramic and hybrid ceramic (Vita Enamic) CAD/CAM restorative materials. A total of 40 rectangular specimens (18×15×1 mm) were fabricated, 10 per material per condition. Specimens were polished according to manufacturer protocols and immersed in 4% acetic acid (v/v) at 80∘C for 16 hours, following ISO 6872 standards. Surface roughness was measured using a Roughness Tester (PCE-RT 2000). Data were analyzed using independent t-tests, and effect sizes were calculated (Cohen’s d). Both materials exhibited significant increases in surface roughness after acid exposure. Lithium disilicate showed Ra values increasing from 0.0676±0.0187 µm to 0.5129±0.0912 µm (p<0.001, d=4.79). The hybrid ceramic exhibited a larger increase, from 0.2063±0.0568 µm to 1.4409±0.4951 µm (p<0.001, d=2.48). The hybrid material was therefore more susceptible to acid-induced degradation than lithium disilicate. Immersion in 4% acetic acid significantly deteriorates the surface of both lithium disilicate and hybrid ceramic, with hybrid ceramic showing greater roughness changes. Lithium disilicate may be preferred for patients at high risk of acidic challenges. These findings emphasize the importance of material selection and surface management to optimize restoration longevity.

Fracture toughness is a critical mechanical property influencing the clinical durability of prosthodontic materials, especially those used in high-stress posterior regions. Differences in composition, microstructure, and fabrication technologies can significantly affect resistance to crack propagation. This experimental study evaluated the fracture toughness of three prosthodontic material groups (n = 10 each). Standardized samples were prepared and tested using the Vickers indentation technique under controlled laboratory conditions. Descriptive statistics were calculated for each group, and a one-way analysis of variance (ANOVA) was performed to determine significant differences among the groups. Post-hoc pairwise comparisons were conducted using Tukey’s Honest Significant Difference (HSD) test. Statistically significant differences in fracture toughness were observed among the three groups (p < 0.001). Group II demonstrated the highest mean fracture toughness (5.39 ± 0.34 MPa·m^1/2), followed by Group III (4.34 ± 0.36 MPa·m1/2). Group I recorded the lowest mean value (2.82 ± 0.56 MPa·m1/2). Tukey’s post-hoc analysis confirmed that all pairwise comparisons were significant (p < 0.001), indicating that each material group exhibited distinct mechanical performance profiles. The findings show that the fracture toughness of prosthodontic materials vary significantly depending on their composition and manufacturing technology. Materials in Group II outperformed the other groups, suggesting greater suitability for clinical situations where high resistance to fracture is required. Further studies incorporating additional mechanical tests and long-term clinical evaluation are recommended to validate these results under functional oral conditions. 

Abstract— This study examines the current state of conformity assessment (CA) in Libya, with a specific focus on the cement manufacturing sector as a case study. Conformity assessment, encompassing testing, inspection, certification, and accreditation, plays a critical role in ensuring product quality, safety, and market access. Libya faces challenges in implementing effective CA practices, hindering its economic diversification. This research evaluates stakeholder awareness, infrastructure adequacy, legal/institutional frameworks, and alignment with international standards using a mixed-methods approach. A stakeholder survey (N=54) and qualitative analysis of Libyan legal and regulatory documents were employed. Key findings reveal nominal CA awareness (82.69%), yet practical implementation gaps exist due to inadequate infrastructure (18.8% of respondents citing this as an obstacle), weak enforcement (18.8%), and limited technical expertise. The cement sector showed low Quality Management Systems (QMS) adoption (48%) and inconsistent adherence to the Libyan Portland Cement Standard LNS 340:2009. While support for aligning with international standards is strong (average rating 4.04/5), obstacles like lack of awareness (31.1%) and technical expertise (30.2%) impede progress. The study proposes actionable recommendations to strengthen Libya’s CA system, including developing a unified national framework, investing in accredited laboratories, and promoting collaboration.

Keywords— Conformity Assessment, Cement Industry, Libya, Quality Standards, Economic Diversification, Stakeholder Awareness.