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This study delves into special class of generalized p-trigonometric functions, examining their connection to the established counterparts like p-cosine and p-sine functions. We here explore the new class of functions called the p-versine, p-coversine, p-haversine, and p-hacovercosine, by providing comprehensive definitions and properties. Grounded in the characteristics of p-cosine and p-sine functions, the newly proposed functions offer unique mathematical insights. Our work contributes towards a thorough understanding of these new special functions, showcasing their potential applications in diverse scientific domains, from mathematical analysis to physics and engineering. This paper contributes as a valuable resource for future applied mathematics researchers, engaging with these new mathematical functions, enhancing the ability to model complex patterns from diverse real-world applications. [ABSTRACT FROM AUTHOR]

In this paper, a new microstrip triplexer is designed to work at 2.5 GHz, 4.4 GHz and 6 GHz for mid-band 5G applications. All channels are flat with three low group delays (GDs) of 0.84 ns, 0.75 ns and 0.49 ns, respectively. Compared to the previously reported works, the proposed triplexer has the minimum group delay. The designed triplexer has 18.2%, 13.7%, 23.6% fractional bandwidths (FBW%) at 2.5 GHz, 4.4 GHz and 6 GHz, respectively. The obtained insertion losses (ILs) are low at all channels. These features are obtained without a noticeable increase in the overall size. A novel and simple resonator is used to design the proposed triplexer, which includes two pairs of coupled lines combined with a shunt stub. A perfect mathematical analysis is performed to find the resonator behavior and the layout optimization. The type of shunt stub is determined mathematically. Also, the smallness or largeness of some important physical dimensions is determined using the proposed mathematical analysis. Finally, the designed triplexer is fabricated and measured, where the measurement results verify the simulations. [ABSTRACT FROM AUTHOR]

From the facts that numerous regions with initially similar economic conditions end up with different levels of entrepreneurial agglomeration, this paper constructs a model assuming that the sequential entrants make their career choices based on existing entrepreneurial ratio and describing the dynamics of entrepreneurial agglomeration formation. After mathematical analysis and numerical simulation using NetLogo, it is found that under social selection, a nonlinear Polya process with self-reinforcing and path-dependency characters will emerge, and the repeated entrants' career choices will lead to the agglomeration of entrepreneurship; the agent's risk compensation value, the initial population of agents, the number of role models in the early stage and the initial entrepreneurial ratio are determinants to the formation of entrepreneurial agglomeration. The findings confirm that entrepreneurship has "memory" and the entrepreneurial history could have influence on the future. In order to forge the entrepreneurial agglomeration, our suggestions include exerting influence on the determinants from an early age, and improving the individual's risk-taking abilities. [ABSTRACT FROM AUTHOR]

Support vector machine (SVM) is a new machine learning method developed from statistical learning theory. Since the objective function of the unconstrained SVM model is a non-smooth function, a lot of fast optimization algorithms can't be used to find the solution. Firstly, to overcome the non-smooth property of this model, a new padé33 approximation smooth function is constructed by rational approximation method, and a new smooth support vector machine model (SSVM) is established based on the smooth function. Then, by analyzing the performance of the smooth function, we find that the smooth precision is significantly higher than existing smooth functions. Moreover, theoretical and rigorous mathematical analyses are given to prove the convergence of the new model. Finally, it is applied to the heart disease diagnosis. The results show that the Padé33-SSVM model has better classification capability than existing SSVMs. [ABSTRACT FROM AUTHOR]

We construct a model to investigate HIV/AIDS dynamics in real cases and study its mathematical analysis. The study examines the qualitative outcomes and confirms the local and global asymptotic stability of both the endemic equilibrium and the disease-free equilibrium. The model's criteria for exhibiting both local and global asymptotically stable behavior are examined. We compute the endemic equilibria and obtain the existence of a unique positive endemic equilibrium. The data is fitted to the model using the idea of nonlinear least-squares fitting. Accurate parameter values are achieved by fitting the data to the model using a 95% confidence interval. The basic reproduction number is computed using parameters that have been fitted or estimated. Sensitivity analysis is performed to discover the influential parameters that impact the reproduction number and the eradication of the disease. The results show that implementing preventive measures can reduce HIV/AIDS cases. [ABSTRACT FROM AUTHOR]

Fractional order algorithms demonstrate superior efficacy in signal processing while retaining the same level of implementation simplicity as traditional algorithms. The self-adjusting dual-stage fractional order least mean square algorithm, denoted as LFLMS, is developed to expedite convergence, improve precision, and incurring only a slight increase in computational complexity. The initial segment employs the least mean square (LMS), succeeded by the fractional LMS (FLMS) approach in the subsequent stage. The latter multiplies the LMS output, with a replica of the steering vector (Ŕ) of the intended signal. Mathematical convergence analysis and the mathematical derivation of the proposed approach are provided. Its weight adjustment integrates the conventional integer ordered gradient with a fractional-ordered. Its effectiveness is gauged through the minimization of mean square error (MSE), and thorough comparisons with alternative methods are conducted across various parameters in simulations. Simulation results underscore the superior performance of LFLMS. Notably, the convergence rate of LFLMS surpasses that of LMS by 59%, accompanied by a 49% improvement in MSE relative to LMS. So it is concluded that the LFLMS approach is a suitable choice for next generation wireless networks, including Internet of Things, 6G, radars and satellite communication. [ABSTRACT FROM AUTHOR]

This research establishes an emergency evacuation time model specifically designed for subway stations with complex structures. The model takes into account multiple factors, including passenger flow rate, subway facility parameters, and crowd density, to accurately assess evacuation times. It considers the impact of horizontal walking distance, flow rate, subway train size, and stair parameters on the overall evacuation process. By identifying bottleneck points such as gates, car doors, and stairs, the model facilitates the evaluation of evacuation capacity and the formulation of effective evacuation plans, particularly in multiline subway transfer stations. The good consistency is achieved between the calculated evacuation time and simulated results using the Pathfinder software (with the relative error of 5.4%). To address urban traffic congestion and enhance subway station safety, the study recommends implemented measures for emergency diversion and passenger flow control. Additionally, the research presents characteristic mathematical models for various evacuation routes by considering the structural and temporal characteristics of metro systems. These models provide valuable guidance for conducting large-scale passenger evacuation simulations in complex environments. Future research can further enhance the model by incorporating psychological factors, evacuation signage, and strategies for vulnerable populations. Overall, this study contributes to a better understanding of evacuation dynamics and provides practical insights to improve safety and efficiency in subway systems. [ABSTRACT FROM AUTHOR]

Alfvén's theorem concerning "frozen-in magnetic field" states that the magnetic field lines, passing through an ideal conductive fluid, are frozen with the fluid, and the fluid can move along the magnetic field lines but cannot cross them. Here, we present two negative experimental results and an analysis of the theorem. The experimental results prove that Alfvén's theorem cannot correctly explain the evolution of the magnetic field and magnetic resistance; therefore, it is incorrect. Mathematical analysis explains why this theorem is incorrect. [ABSTRACT FROM AUTHOR]

The purpose of a rock bolt is to improve the strength capacity of a jointed rock mass. The strengthened arch controlling area can be formed based on the superposition of the controlling influence range of the bolt with the controlling influence angle of rock bolt playing an important role. However, quantitative research on the influence angle is still rare. In this study, numerical simulations and mathematical analysis are used to study the law of stress field distribution and the controlling influence angle through a single bolt, and the following conclusions can be obtained. (1) The compressive stress field is roughly distributed in an "Apple shape" and in a "conical" spatial distribution. (2) The bolt controlling angle is not a constant 45°, and it is influenced by the rock mass strength and bolt parameters. It decreases with the increasing elastic modulus of the bolt, bolt diameter and bolt length. It also increases with the increasing pretension and rock mass strength. The length has less influence on the supporting range. (3) Based on the experimental results, an optimal analytical model to predict the bolt's controlling influence angle was developed. The analytical model includes the influences of the rock mass strength and bolt parameters. (4) A comparison between the model predictions with the results from the Dabei Mining 103 face transportation tunnel and the existing results shows the rationale behind the original support design scheme and an improvement over the existing results. [ABSTRACT FROM AUTHOR]

Machine learning is widely used for personalisation, that is, to tune systems with the aim of adapting their behaviour to the responses of humans. This tuning relies on quantified features that capture the human actions, and also on objective functions—that is, proxies – that are intended to represent desirable outcomes. However, a learning system's representation of the world can be incomplete or insufficiently rich, for example if users' decisions are based on properties of which the system is unaware. Moreover, the incompleteness of proxies can be argued to be an intrinsic property of computational systems, as they are based on literal representations of human actions rather than on the human actions themselves; this problem is distinct from the usual aspects of bias that are examined in machine learning literature. We use mathematical analysis and simulations of a reinforcement-learning case study to demonstrate that incompleteness of representation can, first, lead to learning that is no better than random; and second, means that the learning system can be inherently unaware that it is failing. This result has implications for the limits and applications of machine learning systems in human domains. [ABSTRACT FROM AUTHOR]

Capacitation is a complex maturation process mammalian sperm must undergo in the female genital tract to be able to fertilize an egg. This process involves, amongst others, physiological changes in flagellar beating pattern, membrane potential, intracellular ion concentrations and protein phosphorylation. Typically, in a capacitation medium, only a fraction of sperm achieve this state. The cause for this heterogeneous response is still not well understood and remains an open question. Here, one of our principal results is to develop a discrete regulatory network, with mostly deterministic dynamics in conjunction with some stochastic elements, for the main biochemical and biophysical processes involved in the early events of capacitation. The model criterion for capacitation requires the convergence of specific levels of a select set of nodes. Besides reproducing several experimental results and providing some insight on the network interrelations, the main contribution of the model is the suggestion that the degree of variability in the total amount and individual number of ion transporters among spermatozoa regulates the fraction of capacitated spermatozoa. This conclusion is consistent with recently reported experimental results. Based on this mathematical analysis, experimental clues are proposed for the control of capacitation levels. Furthermore, cooperative and interference traits that become apparent in the modelling among some components also call for future theoretical and experimental studies. [ABSTRACT FROM AUTHOR]