For a composite number 𝒏, we explore a topological space on the ring of integers modulo 𝒏, we use basis sets formed by solutions to quadratic residue equations, where elements are units modulo 𝒏. This definition allows us to investigate the algebraic relationships among solutions, focusing on properties like clopen sets, closure, and interior operations. Additionally, we examine the continuity of mappings between the group of units in ℤ/𝒏ℤ and quadratic residues, alongside the quotient topology induced on this group of units. A comparison of the separation properties of these topological spaces is also presented

Abstract

Polycystic Ovarian Syndrome (PCOS) is one of the most common endocrine disorders affecting women of reproductive age. This study investigates the prevalence of PCOS, focusing on patient-related and disease-related factors. A cross-sectional survey was conducted to explore obstetricians' and gynecologists' perspectives on the diagnosis, prevalence, and management of PCOS. A total of 35 specialists, each with at least three years of clinical experience, participated. The survey included open-ended questions covering patient-related factors (e.g., symptoms, lifestyle) and disease-related aspects (e.g., prevalence, diagnostic methods, treatment). Data were collected through face-to-face interviews from January to April 2022, ensuring confidentiality and standardized procedures to minimize bias. This study analyzed PCOS's prevalence, etiology, symptoms, diagnosis, and treatment. The highest prevalence (62.9%) was observed in the 20–29 age group, declining with age. Unknown causes accounted for 51.42% of cases, followed by hormonal and genetic factors (41.42% each). Menstrual irregularity (80%) was the most common symptom. Ultrasound was the primary diagnostic tool (82.85%), while metformin was the most prescribed treatment (71.42%). PCOS was the most prevalent condition among gynecological disorders (57.14%). These findings highlight the significant burden of PCOS and the need for improved management strategies. Our study concluded that the PCOS remains a significant health concern both globally and within Libya. Highlights the necessity for heightened awareness, early screening, and tailored management strategies. Further research is warranted to explore the underlying causes of regional prevalence variations and to develop culturally appropriate interventions that address the unique needs of the Libyan female population.

Keywords: PCOS, Hyperandrogenism, Endocrine Disorders, Infertility.

Abstract:

The construction of residential buildings in the state of Libya has become increasingly expensive, necessitating cost-effective design solutions. Structural engineers play a critical role in reducing construction costs while ensuring safety and efficiency. One approach to achieving an optimal design is minimizing the dimensions of structural elements ,a critical factor influencing structural performance and economy is the rigidity of beam-to-column connections, which significantly affects deformations and then internal forces. According to beam bending theory, bending moments and shear forces are directly proportional to deformation. Therefore, reducing primary curvature leads to a decrease in design moments, allowing for more economical structural sections. This study investigates the impact of considering partial rigidity in beam-to-column connections within reinforced concrete (RC) frame, particularly for single-story buildings. In conventional structural design, connections are often assumed to be either fully rigid or fully pinned, neglecting partial rigidity effects. This oversimplified modeling approach results in overdesign and increased material consumption, deviating from sustainability principles. The research use SAP2000 structural analysis software to assess various degrees of connection rigidity and their influence on member deformation. The findings indicate that incorporating realistic connection rigidity can reduce beam deformation by up to 20% (at 0.7 rigidity) , leading to smaller and more cost-effective frame sections. Furthermore, common construction methods in the state of Libya inherently provide a certain degree of rigidity at beam-to-column interfaces, yet current design practices often overlook this advantage. This study underscores the importance of optimizing beam-to-column connection rigidety to enhance structural performance, reduce material usage, and align with sustainable design principles. The findings contribute to improving cost efficiency in RC frame construction, providing valuable insights for engineers seeking to optimize structural design in residential buildings

Keywords:Beam-to-Column,Connection, Sustainability, Optimum Design , Rigidety

The electric power system network has become more self-sufficient and less dependent on fossil fuel-based units due to the increasing integration of renewable energy resources. It is crucial to have an efficient method or technology for managing the system’s economics, security, reliability,  environmental damage, and the un- certainties that come with fluctuating loads. In this context, this paper utilizes a framework based on probabi- listic simulation of a demand-side management approach and computational intelligence to calculate the optimal value of saving utility cost (SUC). Unlike traditional methods that dispatch peak-clipped resource blocks sequentially, a modified artificial bee colony (MABC) algorithm is employed. The SUC is then reported through a sequential valley-filling procedure. Consequently, the SUC is derived from the overall profitability of the gen- eration system and includes savings in energy costs, capacity costs, and expected cycle costs. Further investi- gation to obtain the optimal value of SUC was conducted by comparing the SUC determined directly and indirectly, explicitly referring to the peak clipping energy of thermal units (PCETU). The comparisons utilized the MABC algorithm and a standard artificial bee colony, and the results were verified using the modified IEEE RTS- 79 with varying peak load demands. The findings illustrate that the proposed method demonstrated robustness in determining the global optimal values of SUC increments, achieving increases of 7.26 % for 2850 MW and 5 % for 3000 MW, compared to indirect estimation based on PCETU. Moreover, SUC increments of 18.13 % and 25.47 % were also achieved over the conventional method.