Physical Mathematics

This paper examines the distribution of the prime numbers using epidemiological statistical methods. First, we will show more advanced conjectures of the prime number theorem in two forms. One is, like conventional conjecture, when drawn on coordinates, it becomes a curve. However, we aim for the curve to pass through the center of the dispersion of the prime counting function. And we obtain the results worthy of publication in this regard. The other one is what we call the corridors for the prime number theorem. As x increases, a larger proportion of π(x) gathers in what we call the main corridor. Next, we will clarify the process of acquiring this theory. Initially, we will clarify how the basic form of the conjecture is determined. Then, we will describe the process of obtaining the value for α used in this conjecture. Through this, we can share the results and origins of these conjectures. At the end, we declare our achievement, even though we acknowledge that this is only a prediction rather than a proof.

Microfluidic devices are essential tools in various scientific and industrial applications, including chemical analysis, medical diagnostics, and biosensing. These devices, which manipulate small volumes of fluids at the micron scale, offer significant advantages over traditional laboratory equipment, such as faster reaction times, lower reagent costs, and the ability to conduct experiments with minimal sample sizes. One of the critical factors in enhancing the performance of microfluidic devices is the optimization of the optical path length. The optical path length refers to the distance that light travels through a medium, and its length can significantly influence the accuracy and sensitivity of optical measurements. In many applications, such as spectrophotometry or fluorescence analysis, a longer optical path length increases the interaction time between the light and the sample, improving the sensitivity and detection limits of the device. A promising technique for increasing the optical path length in microfluidic devices is the use of a multi-pass cell.

Layered cuprates are a class of materials that exhibit high-temperature superconductivity and have been extensively studied due to their complex electronic properties. One of the fundamental phenomena in these materials is the behavior of longitudinal plasma waves, which are collective oscillations of charge carriers that play a crucial role in the understanding of their electronic properties, especially in relation to superconductivity and charge density waves. Investigating these longitudinal plasma waves in layered cuprates has proven to be a challenging task due to the intricate nature of the materials and the difficulty of directly measuring these waves. One promising method for studying longitudinal plasma waves is through the optical absorption in tilted geometries, a technique that allows for the indirect measurement of these waves. This method has shown significant potential in providing valuable insights into the dynamic behavior of the charge carriers in cuprates and their interactions with the underlying lattice. Optical absorption spectroscopy is a wellestablished technique that has been used to probe the electronic properties of various materials, including cuprates.

The study of multi-ply polymeric yarns is an essential area of research in the field of textile engineering, material science, and mechanics. These yarns are widely used in various industries, such as apparel manufacturing, automotive textiles, and industrial applications, due to their superior strength, durability, and flexibility. Multi-ply yarns, which are composed of multiple individual fibers twisted together, exhibit complex mechanical behavior that is influenced by their geometric structure. Understanding the geometry and mechanics of these yarns is crucial for optimizing their performance in practical applications. This report explores the geometrical and mechanical modeling of multi-ply polymeric yarns, focusing on their structure, mechanical properties, and the mathematical models used to predict their behavior. The geometry of multi-ply polymeric yarns plays a significant role in determining their mechanical properties.

The prediction of supersonic wave drag is a crucial aspect of the design and optimization of aerodynamic bodies, particularly in high-speed flight applications such as supersonic aircraft and spacecraft. Accurate wave drag prediction is essential for assessing the performance, fuel efficiency, and overall aerodynamic characteristics of vehicles that travel at speeds greater than the speed of sound. Among various approaches used to predict wave drag, multifidelity methods have emerged as an effective way to balance accuracy with computational efficiency. These methods combine models of varying fidelity to provide a more comprehensive and reliable assessment of wave drag, while minimizing computational cost. The application of multifidelity methods to the prediction of supersonic wave drag for axisymmetric bodies provides an interesting case study in balancing the trade-offs between model accuracy and computational resources. Supersonic wave drag, also known as shock drag, arises due to the formation of shock waves around an object moving through a compressible fluid, such as air at supersonic speeds. As a supersonic body moves through the air, it generates shock waves that cause a discontinuous change in pressure, temperature, and velocity.

Magnetostrictive materials exhibit unique properties that make them highly suitable for applications involving sensors, actuators, and energy harvesting devices. These materials can change their shape or dimensions in response to an applied magnetic field, making them attractive for use in composite structures, where the mechanical and magnetic responses can be precisely controlled. One area of interest in magnetostrictive materials is their application in cantilever resonators, which are widely used in various engineering fields, including vibration sensing, force measurement, and micro-electromechanical systems (MEMS). The vibration characteristics of magnetostrictive composite cantilever resonators have been extensively studied; however, many existing models fail to account for the nonlocal effects, which can significantly influence the resonator's behavior, especially at the micro- and nano-scales. In the traditional theory of elasticity, the material is assumed to have a local response to external forces, meaning that the stress at any point within the material is influenced only by the strain at that same point.

Bionic design has garnered significant attention in architecture and engineering due to its ability to mimic natural forms and functions, resulting in innovative and efficient solutions. Among these applications, bionic raster ceilings have emerged as a promising architectural element. These structures are inspired by natural patterns and aim to optimize performance in various aspects, including acoustics, airflow, structural integrity, and aesthetics. Understanding the dynamic properties of bionic raster ceilings is crucial for their effective design and application. This report presents a comprehensive numerical analysis of the dynamic performance of such ceilings, focusing on their vibrational behavior under various conditions. Bionic raster ceilings typically consist of repeating patterns that resemble natural structures such as honeycombs, leaves, or shells.

Cyclone separators are widely used in industrial applications for the separation of particles from air, liquids, or gases based on centrifugal forces. These devices are particularly efficient in separating solid particles, such as sand, dust, and even microplastics, from gas or liquid streams. The performance of cyclone separators depends on several factors, including the geometry of the cyclone shape, flow characteristics, and the physical properties of the particles to be separated. In recent years, there has been growing interest in understanding the efficiency of cyclone separators in separating different types of particles, such as sand and microplastics, especially considering the increasing environmental concerns related to microplastic pollution. One critical aspect of cyclone separator performance is the Reynolds number, which influences the flow behavior within the cyclone and, consequently, the efficiency of particle separation. This report investigates the relationship between cyclone shapes, separation efficiency, and Reynolds number in the context of sand and microplastic separation.

Wheel-rail impact dynamics is a critical area of study in railway engineering, especially concerning the interaction between the wheel and rail during train operations. The dynamic behavior of wheel-rail interactions can significantly affect the performance, safety, and longevity of rail systems. One of the most prominent issues in this area is the phenomenon of wheel flats. Wheel flats occur when a section of the wheel surface becomes flattened due to prolonged sliding or braking events. These flats lead to increased impact forces between the wheel and rail, causing undesirable effects such as noise, vibration, and accelerated wear. Understanding and modeling wheel-flat interactions are vital for improving the design and maintenance of rail systems. A parameter-driven methodology for wheel flat modeling offers a promising approach to accurately simulate and analyze the impact dynamics between wheels and rails. The presence of wheel flats creates localized contact points between the wheel and the rail, which disrupts the otherwise smooth rolling contact. This disturbance causes periodic impacts, and the magnitude of these impacts is influenced by various factors, such as the size and shape of the wheel flat, the speed of the train, and the material properties of the wheel and rail. The interactions between the wheel and the rail under impact conditions are highly dynamic, making them difficult to model accurately.

The increasing complexity of engineering problems, especially in structural design, has led to the development of advanced optimization methods aimed at improving the performance of multi-material structures under various loading conditions. Topology Optimization (TO) is a powerful technique that allows for the design of structures with optimal material distribution to achieve desired mechanical, thermal, and other performance criteria. In recent years, Isogeometric Analysis (IGA) has gained popularity as a method for integrating design and analysis, bridging the gap between Computer-Aided Design (CAD) and Finite Element Analysis (FEA). Combining isogeometric analysis with topology optimization presents a promising approach for designing multimaterial structures, especially when subject to complex thermal and mechanical loadings. The use of Neural Networks (NNs) further enhances this optimization process by improving efficiency and enabling real-time design adjustments. This report explores the integration of these advanced techniques and presents a numerical investigation into the isogeometric topology optimization of multi-material structures under thermal and mechanical loadings using neural networks. The fundamental goal of topology optimization is to find the optimal material distribution within a given design space, subjected to various constraints and loading conditions.