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Ant-inspired metaheuristic algorithms known as ant colony optimization (ACO) offer an approach that has the ability to solve complex problems in both discrete and continuous domains. ACOs have gained significant attention in the field of water resources management, since many problems in this domain are non-linear, complex, challenging and also demand reliable solutions. The aim of this study is to critically review the applications of ACO algorithms specifically in the field of hydrology and hydrogeology, which include areas such as reservoir operations, water distribution systems, coastal aquifer management, long-term groundwater monitoring, hydraulic parameter estimation, and urban drainage and storm network design. Research articles, peer-reviewed journal papers and conference papers on ACO were critically analyzed to identify the arguments and research findings to delineate the scope for future research and to identify the drawbacks of ACO. Implementation of ACO variants is also discussed, as hybrid and modified ACO techniques prove to be more efficient over traditional ACO algorithms. These algorithms facilitate formulation of near-optimal solutions, and they also help improve cost efficiency. Although many studies are attempting to overcome the difficulties faced in the application of ACO, some parts of the mathematical analysis remain unsolved. It is also observed that despite its popularity, studies have not been successful in incorporating the uncertainty in ACOs and the problems of dimensionality, convergence and stability are yet to be resolved. Nevertheless, ACO is a potential area for further research as the studies on the applications of these techniques are few. [ABSTRACT FROM AUTHOR]

The Internet of Things (IoT) is the underlying technology that has enabled connecting daily apparatus to the Internet and enjoying the facilities of smart services. IoT marketing is experiencing an impressive 16.7% growth rate and is a nearly USD 300.3 billion market. These eye-catching figures have made it an attractive playground for cybercriminals. IoT devices are built using resource-constrained architecture to offer compact sizes and competitive prices. As a result, integrating sophisticated cybersecurity features is beyond the scope of the computational capabilities of IoT. All of these have contributed to a surge in IoT intrusion. This paper presents an LSTM-based Intrusion Detection System (IDS) with a Dynamic Access Control (DAC) algorithm that not only detects but also defends against intrusion. This novel approach has achieved an impressive 97.16% validation accuracy. Unlike most of the IDSs, the model of the proposed IDS has been selected and optimized through mathematical analysis. Additionally, it boasts the ability to identify a wider range of threats (14 to be exact) compared to other IDS solutions, translating to enhanced security. Furthermore, it has been fine-tuned to strike a balance between accurately flagging threats and minimizing false alarms. Its impressive performance metrics (precision, recall, and F1 score all hovering around 97%) showcase the potential of this innovative IDS to elevate IoT security. The proposed IDS boasts an impressive detection rate, exceeding 98%. This high accuracy instills confidence in its reliability. Furthermore, its lightning-fast response time, averaging under 1.2 s, positions it among the fastest intrusion detection systems available. [ABSTRACT FROM AUTHOR]

Fuzzy logic is becoming one of the most-influential fields of modern mathematics with applications that impact not only other sciences, but society in general. This newly found interest in fuzzy logic is in part due to the crucial role it plays in the development of artificial intelligence. As a result, new tools and practices for the development of the above-mentioned field are in high demand. This is one of the issues this paper was composed to address. To be more specific, a sizable part of fuzzy logic is the study of fuzzy connectives. However, the current method used to generalize them is restricted to the use of basic automorphisms, which hinders the creation of new fuzzy connectives. For this reason, in this paper, a new method of generalization is conceived of that aims to generalize the fuzzy connectives using polynomial automorphism functions instead. The creation of these automorphisms is achieved through numerical analysis, an endeavor that is supported with programming applications that, using mathematical modeling, validate and visualize the research. Furthermore, the automorphisms satisfy all the necessary criteria that have been established for use in the generalization process and, consequently, are used to successfully generalize fuzzy connectives. The result of the new generalization method is the creation of new usable and flexible fuzzy connectives, which is very promising for the future development of the field. [ABSTRACT FROM AUTHOR]

In our paper, we propose a generalized version of the Alternating Projections Digital Hard Successive Interference Cancellation (AP-HSIC) algorithm that is capable of decoding any order of constellation M in an M-Quadrature Amplitude Modulation (QAM) system. Our approach applies to Rayleigh deep-fading Multiple-Input Multiple-Output (MIMO) channels with high-level Additive White Gaussian Noise (AWGN). It can handle various destructive phenomena without restricting the number of antenna arrays in the transmitter/receiver. Importantly, it does not rely on closed-loop MIMO feedback or the need for Channel-State Information Transmission (CSIT). We have demonstrated the effectiveness of our approach and provided a Bit Error Rate (BER) analysis for 16-, 32-, and 64-QAM modulation systems. Real-time simulations showcase the differences and advantages of our proposed algorithm compared to the Multi-Group Space-Time Coding (MGSTC) decoding algorithm and the Lagrange Multipliers Hard Successive Interference Cancellation (LM-HSIC) algorithm, which we have also developed here. Additionally, our paper includes a mathematical analysis of the LM-HSIC algorithm. The AP-HSIC algorithm is not only effective and fast in decoding, including interference cancellation computational feedback, but it can also be integrated with any Linear Processing Complex Orthogonal Design (LPCOD) technique, including Complex Orthogonal Design (COD) schemes such as high-order Orthogonal Space–Time Block Code (OSTBC) with high-order QAM symbols. [ABSTRACT FROM AUTHOR]

The history of variational calculus dates back to the late 17th century when Johann Bernoulli presented his famous problem concerning the brachistochrone curve. Since then, variational calculus has developed intensively as many problems in physics and engineering are described by equations from this branch of mathematical analysis. This paper presents two non-classical, distinct methods for solving such problems. The first method is based on the differential transform method (DTM), which seeks an analytical solution in the form of a certain functional series. The second method, on the other hand, is based on the physics-informed neural network (PINN), where artificial intelligence in the form of a neural network is used to solve the differential equation. In addition to describing both methods, this paper also presents numerical examples along with a comparison of the obtained results.Comparingthe two methods, DTM produced marginally more accurate results than PINNs. While PINNs exhibited slightly higher errors, their performance remained commendable. The key strengths of neural networks are their adaptability and ease of implementation. Both approaches discussed in the article are effective for addressing the examined problems. [ABSTRACT FROM AUTHOR]

The main contribution of this paper is to present a simple algorithm that theoretically and numerically assesses the switching angles of an inverter operated with the SPWM technique. This technique is the most widely used for eliminating harmonics in DC-AC converters for powering motors, renewable energy applications, household appliances, etc. Unlike conventional implementations of the SPWM technique based on the analog or digital comparison of a sinusoidal signal with a triangular signal, this paper mathematically performs this comparison. It proposes a simple solution to solve the transcendental equations arising from the mathematical analysis numerically. The technique is validated by calculating the total harmonic distortion (THD) of the generated signal theoretically and numerically, and the results indicate that the calculated angles produce the same distribution of harmonics calculated analytically and numerically. The algorithm is limited to single-phase inverters with unipolar SPWM. [ABSTRACT FROM AUTHOR]

This paper explores the Apéry-like series and demonstrates the derivation of closed-form expressions using fractional calculus. We consider a variety of Apéry-like functions, which were categorized by their functional forms and coefficients by applying the Riemann–Liouville fractional integral and derivative to examine their properties across various domains. The study focuses on establishing rigorous mathematical frameworks that unveil new insights into the behaviors of these series, contributing to a deeper understanding of number theory and mathematical analysis. Key results include proofs of convergence and divergence within specified intervals and the derivation of closed-form solutions through fractional integration and differentiation. This paper also introduces a method aimed at conjecturing mathematical constants through continued fractions as an application of our results. Finally, we provide the proof of validation for three unproven conjectures of continued fractions obtained from the Ramanujan Machine. [ABSTRACT FROM AUTHOR]

Calculus of Variations In the paper I On a Non-Newtonian Calculus of Variations i , Delfim F. M. Torres presents, for the first time in the literature, a non-Newtonian calculus of variations that involves the minimization of a function defined by a non-Newtonian integral with a Lagrangian depending on the non-Newtonian derivative. This publication is an editorial for the Special Issue of Axioms "Calculus of Variations, Optimal Control and Mathematical Biology: A Themed Issue Dedicated to Professor Delfim F. M. Torres on the Occasion of His 50th birthday". This Special Issue is dedicated to Professor Delfim F. M. Torres on the occasion of his 50th birthday, as recognition of his significant contributions to Mathematics, in particular in the calculus of variations, optimal control, and mathematical biology. [Extracted from the article]

Variable (feature) selection plays an important role in data analysis and mathematical modeling. This paper aims to address the significant lack of formal evaluation benchmarks for feature selection algorithms (FSAs). To evaluate FSAs effectively, controlled environments are required, and the use of synthetic datasets offers significant advantages. We introduce a set of ten synthetically generated datasets with known relevance, redundancy, and irrelevance of features, derived from various mathematical, logical, and geometric sources. Additionally, eight FSAs are evaluated on these datasets based on their relevance and novelty. The paper first introduces the datasets and then provides a comprehensive experimental analysis of the performance of the selected FSAs on these datasets including testing the FSAs' resilience on two types of induced data noise. The analysis has guided the grouping of the generated datasets into four groups of data complexity. Lastly, we provide public access to the generated datasets to facilitate bench-marking of new feature selection algorithms in the field via our Github repository. The contributions of this paper aim to foster the development of novel feature selection algorithms and advance their study. [ABSTRACT FROM AUTHOR]

In this paper, we investigate mathematical models of meteorological forecasting based on the work of neural networks, which allow us to calculate presumptive meteorological parameters of the desired location on the basis of previous meteorological data. A new method of grouping neural networks to obtain a more accurate output result is proposed. An algorithm is presented, based on which the most accurate meteorological forecast was obtained based on the results of the study. This algorithm can be used in a wide range of situations, such as obtaining data for the operation of equipment in a given location and studying meteorological parameters of the location. To build this model, we used data obtained from personal weather stations of the Weather Underground company and the US National Digital Forecast Database (NDFD). Also, a Google remote learning machine was used to compare the results with existing products on the market. The algorithm for building the forecast model covered several locations across the US in order to compare its performance in different weather zones. Different methods of training the machine to produce the most effective weather forecast result were also considered. [ABSTRACT FROM AUTHOR]

This paper presents the mathematical and physical analysis, as well as distinct types of exact wave solutions, of an important fluid flow dynamics model called the truncated M-fractional (1+1)-dimensional nonlinear Estevez–Mansfield–Clarkson (EMC) equation. This model is used to explain waves in shallow water, fluid dynamics, and other areas. We obtain kink, bright, singular, and other types of exact wave solutions using the modified extended direct algebraic method and the improved (G ′ / G) -expansion method. Some solutions do not exist. These solutions may be useful in different areas of science and engineering. The results are represented as three-dimensional, contour, and two-dimensional graphs. Stability analysis is also performed to check the stability of the corresponding model. Furthermore, modulation instability analysis is performed to study the stationary solutions of the corresponding model. The results will be helpful for future studies of the corresponding system. The methods used are easy and useful. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel system integrating reconfigurable intelligent surfaces (RISs) with cognitive radio (CR) technology, presenting a forward-looking solution aligned with the evolving standards of 6G and beyond networks. The proposed RIS-assisted CR networks operate with a base station (BS) transmitting signals to two users, the primary user (PU) and secondary user (SU), through direct and reflected signal paths, respectively. Our mathematical analysis focuses on deriving expressions for SU in the RIS-assisted CR system, validated through Monte Carlo simulations. The investigation covers diverse aspects, including the impact of the signal-to-noise ratio (SNR), power allocations, the number of reflected surfaces, and blocklength variations. The results provide nuanced insights into RIS-assisted CR system performance, highlighting its sensitivity to factors like the number of reflectors, fading severity, and correlation coefficient. Careful parameter selection, such as optimizing the configuration of reflectors, is shown to prevent a complete outage, showcasing the system's robustness. Additionally, the work suggests that the optimization of reflectors configuration can significantly enhance overall system performance, and RIS-assisted CR systems outperform reference schemes. This work contributes a thorough analysis of the proposed system, offering valuable insights for efficient performance evaluation and parameter optimization in RIS-assisted CR networks. [ABSTRACT FROM AUTHOR]

In theory, two extreme forms of substances exist: the pure form and the single-molecule mixture form. The latter contains a mixture of molecules with molecularly different structures. Inspired by the "chemical space" concept, in this paper, I report a study of the single-molecule mixture state that combines model construction and mathematical analysis, obtaining some interesting results. These results provide theoretical evidence that the single-molecule mixture state may indeed exist in realistic synthetic or natural polymer systems. [ABSTRACT FROM AUTHOR]

This article presents a comparative study of two different kinds of processes that produce oscillatory motion on a work surface during the mechanical separation process. The investigation began with determining the trajectory produced by the oscillating separator's active component of the classical drive mechanism. Based on this, a second mechanism—the six-bar mechanism—was created using the WATT program, and a mathematical analysis was conducted. The comparative examination of the two mechanisms was carried out using OriginPro, Mathcad, and Roberts software. This study's findings all point to the same conclusion: the newly developed mechanism produces the same trajectory as the classical mechanism when viewed through the lens of the reference element, or the element that causes the oscillatory movement. However, when looking at the operating parameters, there was a noticeable difference in the movement's speed and the angle of the crank when producing its maximum speed. Theoretically, this new mechanism increases the speed at which solid particles move across a work surface. However, this difference cannot be characterized as positive or negative because further research is required to determine how the nature of solid particles and the work surface's inclination affect this process, in addition to this mechanism. The identification of the mathematical equations of motion for the constituent parts of the mechanism under study is the novelty produced of this paper. [ABSTRACT FROM AUTHOR]

In this paper, a signal model and a mathematical analysis of an efficient approach are derived to acquire the approximate magnitude of a complex signal by inducing a pre-biased or a pre-scaled factor in the design criteria. According to the deductive results, the pre-biased or pre-scaled factor can be 2∼3 dB, which is determined through its application. The numerical evaluations show that the mean square error (MSE) of the proposed efficient approach for the random signal is around −33 dB. Based on the design templates for the considered approaches, the occupied areas of the proposed type-1 and -2 approaches are merely 0.13 and 0.09 times the area of the direct-method, respectively. As a result, the proposed efficient approach is certainly a cost-efficient method for obtaining the approximate magnitude of a complex signal. [ABSTRACT FROM AUTHOR]

This paper investigates a cellular network assisted by unmanned aerial vehicles (UAVs) in the presence of a fluctuating 3-dimensional (3D) antenna beamwidth. The primary objective is to perform an analysis of typical user equipment (T-UE) performance with a specific focus on coverage probability and spectral efficiency (SE) in the presence of fluctuations of 3D antenna beamwidth. Within this analytical framework, the macro base stations (MBSs) are meticulously characterized through the application of an independent 2D homogeneous Poisson point process (PPP), while the low-altitude platforms (LAPs) are described using an independent 3D PPP. The study entails the derivation of association probabilities, determining the likelihood of the T-UE associating with MBSs, line-of-sight LAPs, and non-line-of-sight LAPs. Through rigorous mathematical analysis, the paper formulates precise analytical expressions that encapsulate the association and coverage probabilities, taking into account the inherent variability in the UAV antenna beamwidth. This research focuses on a thorough performance evaluation of the T-UE across diverse network configurations, encompassing LAP density, the transmission power of LAPs, and the critical signal-to-interference ratio threshold. The outcomes of this study distinctly underscore the substantial disruptive impact resulting from fluctuating beamwidth on the performance of the T-UE within the UAV-assisted cellular network. Additionally, this performance is further impacted by larger densities and transmission power of the LAP. Hence, it is imperative to take into account the influence of these fluctuations on network association, coverage, and SE whenever contemplating a UAV-assisted cellular network. [ABSTRACT FROM AUTHOR]

This paper is devoted to the analysis of mathematical models of chemotherapy for malignant tumors growing according to the Gompertz law or the generalized logistic law. The influence of the therapeutic agent on the tumor dynamics is determined by a therapy function depending on the time-varying concentration of the drug in the patient's body. The case of a non-monotonic therapy function with two maxima is studied. It reflects the use of two different therapeutic agents. The state variables of the dynamics are the tumor volume and the amount of the therapeutic agent able to suppress malignant cells (concentration of the drug in the body). The treatment protocol (the rate of administration of the therapeutic agent) is the control in the dynamics. The optimal control problem for this models is considered. It is the problem of the construction of treatment protocols that provide the minimal tumor volume at the end of the treatment. The solution of this problem was obtained by the authors in previous works via the optimal control theory. The form of the considered therapy functions provides a specific structure for the optimal controls. The managerial insights of this structure are discussed. In this paper, the structure of the viability set is described for the model according to the generalized logistic law. It is the set of the initial states of the model for which one can find a treatment protocol that guarantees that the tumor volume remains within the prescribed limits throughout the treatment. The description of the viability set's structure is based on the optimal control theory and the theory of Hamilton–Jacobi equations. An inverse problem of therapy is also considered, namely the problem of reconstruction of the treatment protocol and identification of the unknown parameter of the intensity of the tumor growth. Reconstruction is carried out by processing information about the observations of the tumor volume dynamics and the measurements of the drug concentration in the body. A solution to this problem is obtained through the use of a method based on the calculus of variations. The results of the numerical simulations are presented herein. [ABSTRACT FROM AUTHOR]

The Ebola virus continues to be the world's biggest cause of mortality, especially in developing countries, despite the availability of safe and effective immunization. In this paper, we construct a fractional-order Ebola virus model to check the dynamical transmission of the disease as it is impacted by immunization, learning, prompt identification, sanitation regulations, isolation, and mobility limitations with a constant proportional Caputo (CPC) operator. The existence and uniqueness of the proposed model's solutions are discussed using the results of fixed-point theory. The stability results for the fractional model are presented using Ulam–Hyers stability principles. This paper assesses the hybrid fractional operator by applying methods to invert proportional Caputo operators, calculate CPC eigenfunctions, and simulate fractional differential equations computationally. The Laplace–Adomian decomposition method is used to simulate a set of fractional differential equations. A sustainable and unique approach is applied to build numerical and analytic solutions to the model that closely satisfy the theoretical approach to the problem. The tools in this model appear to be fairly powerful, capable of generating the theoretical conditions predicted by the Ebola virus model. The analysis-based research given here will aid future analysis and the development of a control strategy to counteract the impact of the Ebola virus in a community. [ABSTRACT FROM AUTHOR]

In this paper, we have provided a mathematical analysis of an empirical result, namely, the Fahraeus–Lindqvist effect, a phenomenon that occurs in capillary tubes with a diameter lower than 0.3 mm. According to this effect, in capillary tubes under 0.3 mm in diameter, the apparent viscosity of blood decreases as the diameter of the tube decreases, making flow possible in these vessels. A two-phase blood flow mathematical model for human capillaries has been presented here. According to Haynes' theory, blood is separated into two layers when it flows from the capillary. It is assumed that the first layer is plasma, and the second layer is the core layer. The plasma layer flows near the wall of the capillary, and the core layer flows along the axis of the capillary. Further, the core layer is assumed to be a mixture of two phases: one is the plasma, and the other is that of RBCs. For mathematical modeling purposes, a curvilinear coordinate system has been adopted, with physical quantities used in tensorial form. Derived equations are solved to find the effective viscosity, which depends upon the radius of the capillary; that is, it reduces viscosity to make blood flow possible. A comparative study was conducted with the experimental result of this effect, and it was observed that the proposed two-phase blood flow model is much closer to the experimental data than the single-phase blood flow model, and both have the same trends. After validation of the model with the experimental result, this model was applied to human capillaries (diameter lower than 10 μ m ) to show the F-L effect, and the impact of various physiological quantities that are relevant to the flow of blood into human capillaries are also discussed here. The impact of hematocrit on various parameters has been demonstrated explicitly. [ABSTRACT FROM AUTHOR]

Experimental evidence supports the counterintuitive notion that rapid eradication of pathogens within a host, infected with both drug-sensitive and -resistant malaria parasites, can actually accelerate the evolution of drug-resistant pathogens. This study aims to analyze the competitive dynamics between these two strains through a mathematical model and evaluate the impact of aggressive treatment on the spread of drug resistance. We conducted equilibrium, uncertainty, and sensitivity analyses to assess the model, identifying and measuring the influence of key factors on the outcome variable (the population of drug-resistant parasites). Both equilibrium and local sensitivity analyses concurred that the density of drug-resistant parasites is notably affected by genetic instability, the production rate of red blood cells, the number of merozoites, and competition factors. Conversely, there is a negative relationship between genetic instability and one of the competition coefficients. Global sensitivity analysis offers a comprehensive examination of the impact of each input parameter on the temporal propagation of drug resistance, effectively accounting for the interplay among parameters. Both local and global sensitivity analyses underscore the continuous impact of drug treatment on the progression of drug resistance over time. This paper anticipates exploring the underlying mechanisms of drug resistance and providing theoretical support for developing more effective drug treatment strategies. [ABSTRACT FROM AUTHOR]

A combination of experimental measurement preparations using pressure and temperature sensors in conjunction with the theory of one-dimensional isentropic flow and mathematical physics analyses is presented as a tool for analysis in this paper. Furthermore, the subsequent development of a nozzle for use in environmental electron microscopy between the specimen chamber and the differentially pumped chamber is described. Based on experimental measurements, an analysis of the impact of the nozzle shaping located behind the aperture on the character of the supersonic flow and the resulting dispersion of the electron beam passing through the differential pumped chamber is carried out on the determined pressure ratio using a combination of theory and mathematical physics analyses. The results show that nozzle shapes causing under-expanded gas outflow from the aperture to the nozzle have a worse impact on the dispersion of the primary electron beam. This is due to the flow velocity control. The controlled reduction in the static pressure curve on the primary electron beam path thus causes a significantly higher course of electron dispersion values than variants with shapes causing over-expanded gas outflow. [ABSTRACT FROM AUTHOR]

In this paper, we explore the properties of the generalized Lambert transform, the L-transform, the generalized Stieltjes transform, and the Stieltjes–Poisson transform within the framework of Lebesgue spaces. We establish Parseval-type relations for each transform, providing a comprehensive analysis of their behaviour and mathematical characteristics. [ABSTRACT FROM AUTHOR]

Load-capacity has always been one of the performances that is paid much attention to in the development of bevel gear transmission applications. Consequently, the mathematical model of novel bevel gear with high load-capacity based on geometric elements is proposed in this paper, which could be applied to the aviation, aerospace and other fields. In parallel, the design principle and design method of the novel bevel gear are introduced in detail. Subsequently, the conditions for tooth surface continuity and non-interference are derived. Furthermore, the model of novel bevel gear is established. Finally, the load-bearing characteristics are analyzed, revealing that an increase in the number of contact points could significantly enhance the load capacity of the bevel gear pairs. When the load torque applied to bevel gear II is 100 Nm, the contact pressure endured by the bevel gear pair with five-point contact is decreased by 41.37% compared to the bevel gear pair with single-point contact. When the number of contact points is the same, increasing the distance between the contact points could also reduce the contact stress. This provides strong theoretical support for the application of the bevel gear based on the geometric elements. [ABSTRACT FROM AUTHOR]

The Sturm–Liouville boundary value problem (SLBVP) stands as a fundamental cornerstone in the realm of mathematical analysis and physical modeling. Also known as the Sturm–Liouville problem (SLP), this paper explores the intricacies of this classical problem, particularly the relationship between its canonical and Liouville normal (Schrödinger) forms. While the conversion from the canonical to Schrödinger form using Liouville's transformation is well known in the literature, the inverse transformation seems to be neglected. Our study attempts to fill this gap by investigating the inverse of Liouville's transformation, that is, given any SLP in the Schrödinger form with some invariant function, we seek the SLP in its canonical form. By closely examining the second Paine–de Hoog–Anderson (PdHA) problem, we argue that retrieving the SLP in its canonical form can be notoriously difficult and can even be impossible to achieve in its exact form. Finding the inverse relationship between the two independent variables seems to be the main obstacle. We confirm this claim by considering four different scenarios, depending on the potential and density functions that appear in the corresponding invariant function. In the second PdHA problem, this invariant function takes a reciprocal quadratic binomial form. In some cases, the inverse Liouville transformation produces an exact expression for the SLP in its canonical form. In other situations, however, while an exact canonical form is not possible to obtain, we successfully derived the SLP in its canonical form asymptotically. [ABSTRACT FROM AUTHOR]

This paper presents an analysis of the effectiveness of the element interaction model in a vehicular ad hoc network (VANET). An analysis of the mathematical model and its numerical solution for the system of boundary device interactions in the traditional configuration of roadside unit (RSU) placement using single- and dual-channel connection between on-board units (OBU) and RSU is given. In addition, the model efficiency is improved using a clustering approach. The efficiency evaluation is based on calculating the percentage of unprocessed requests generated by OBUs during their mobility, the average power consumption and the magnitude of the delay in transmitting and processing the generated requests in the OBU–RSU system. The traditional and cluster models are compared. The results obtained in this paper show that each of the proposed models can be effectively implemented in mobile nodes and will significantly reduce the overall expected query processing time to improve the organization and algorithmic support of VANET. Along with this, it is shown that the developed approach allows for efficient power consumption when combining RSUs into clusters with a shared cache. The novelty of solving the problems is due to the lack of a comprehensive model that allows the distribution and prediction of the parameters and resources of the system for different computational tasks, which is essential when implementing and using V2X technology to solve the problems of complex management of VANET elements. [ABSTRACT FROM AUTHOR]

Featured Application: The methodology proposed in this paper studies the mobility of a multi-loop spherical mechanism with a spherical slider containing a spherical prismatic pair. By analyzing the motion of each closed kinematic chain that the mechanism consists of, not only can the proposed methodology thoroughly and accurately identify the mechanism's singularity configurations, but the method can also demonstrate the mechanism's range of motion. Therefore, the proposed method ultimately identifies the mechanism's motion continuity and smoothness, which are two of the most essential characteristics a multi-loop spherical mechanism possesses. Hence, the proposed methodology can help mechanism designers create multi-loop mechanisms with spherical sliders without motion deficiencies by avoiding the mechanisms' singularity configurations. Multi-loop spherical mechanisms are extremely beneficial for creating versatile mechanical devices, including robotic joints and surgical tools, since multi-loop spherical mechanisms possess unique capabilities to operate in spatial situations with relatively simple movement. Nevertheless, the research on multi-loop spherical mechanisms with spherical sliders containing spherical prismatic pairs is lacking. Therefore, the main innovation of this paper is to propose the Stephenson-III two-loop spherical mechanism that possesses a spherical slider containing a spherical prismatic pair and to analyze the proposed spherical mechanism's motion characteristics. An algebraic approach was employed to obtain the branch graphs of the proposed spherical mechanism with a spherical slider. The branch graphs were categorized into two types, according to whether branch points existed. With the algebraic approach, loop equations of the two spherical kinematic chains inside the proposed spherical mechanism were established to identify the input–output curves and singularity curves, with which the branch graphs were obtained. With the branch graphs, the joint rotation spaces (JRSs) of the proposed mechanism were recognized and so were the dead center positions, branches, sub-branches, and branch points. The results from the mathematical analysis were simulated and verified by three-dimensional (3D) models of the proposed spherical mechanism. The analytical results demonstrate that the spherical prismatic pair diversifies the motion of the proposed spherical mechanism by producing rotational sliding movement, which can cover the entire circumference of a specific greater circle on the proposed mechanism's sphere. [ABSTRACT FROM AUTHOR]

In this study, we explore a downlink cellular network where each base station (BS) engages in simultaneous communication with multiple users through spatial division multiple access (SDMA). The positions of both BSs and users are established through independent random point processes, effectively representing the cellular network. SDMA utilizes block diagonalization (BD) at each BS, employing multiple receive antennas for each user. To implement BD, users quantize and provide feedback on their downlink channels to their respective BSs. The net spectral efficiency, measuring the effective rate accounting for both downlink and uplink resource usage, serves as a performance metric. In prior research, the optimal feedback rate in terms of maximizing net spectral efficiency has been approximated in this scenario. The corresponding approximations effectively illustrate the asymptotic behavior of the optimal number as a function of the length of the coherent channel block. However, the accuracy of the approximation diminishes when the length of the coherent channel block is relatively small. Given that the length of the coherent channel block can assume relatively small values depending on wireless environments, achieving a precise estimate across the entire range of the coherent block length holds significant importance. Consequently, this paper focuses primarily on enhancing the accuracy of the approximation for the optimal feedback rate. In order to achieve a more precise estimation, we analyze the derivative of the net spectral efficiency, which encompasses two functions that demonstrate distinct growth rates. In contrast to prior studies, both functions are rigorously approximated through mathematical analysis. As a result, the proposed approximation significantly improves the accuracy compared to previous studies, particularly when dealing with short coherent channel block lengths. Moreover, this approximation generally achieves near-optimal performance, regardless of system parameters. [ABSTRACT FROM AUTHOR]

In the containment control problem of multi-agent systems (MASs), the convergence of followers is always a potential threat to the security of system operations. From the perspective of system topology, the inherently non-linear properties of the algebraic connectivity of the follower2follower (F2F) network, combined with the influence of the leader2follower (L2F) topology on the system, make it difficult to design the convergence positions of the followers through mere mathematical analysis. Therefore, in the background of temporary networking tasks for large-scale systems, to achieve the goal of forecasting the performance of the whole system when networking is only completed with local information, this paper investigates the application and effectiveness of recurrent neural networks (RNNs) in the containment control system performance identification, thus improving the efficiency of system networking while ensuring system security. Two types of identification models based on two types of neural networks (NNs), MLP and standard RNN are developed, according to the range of information required for performance identification. Evaluation of the models is carried out by means of the coefficient of determination ( R 2 ) as well as the root-mean-square error (RMSE). The results show that each model may produce a better forecasting accuracy than the other models in specific cases, with models based on the standard RNN possessing smaller errors. With the proposed method, model identification can be achieved, but in-depth development of the model in further studies is still necessary to the extent the accuracy of the model. [ABSTRACT FROM AUTHOR]

This paper introduces a mathematical design and analysis of three-phase inverters used in electric drive applications such as aerospace, electric vehicles, and pumping applications. Different wide bandgap (WBG) semiconductor technologies are considered in this analysis. Using SiC MOSFETs and Si IGBTs, two drive systems are developed in order to show the improvement in the efficiency of the inverter. The efficiency, total losses of the drive systems and the power losses of two inverters are computed and compared for both drive systems at the same operating condition. The drive system with SiC MOSFET shows much better performance compared to the drive system with Si IGBT. The SiC MOSFET system provides a 59.39%, 86.13%, and 29.76% lower conduction losses, switching losses and drive's total losses, respectively, compared to the Si IGBT system. The efficiency of the SiC MOSFET system is 2.46%pu higher than the efficiency of the Si IGBT drive system. Moreover, this paper introduces a detailed analysis for the dc-link voltage and current ripples in three-phase inverters. Furthermore, the minimal dc-link capacitor needed to deal with the ripple current and voltage is investigated. Finally, the performance of the drive with Si IGBT is experimentally tested under different operating speeds and loads. [ABSTRACT FROM AUTHOR]

This paper presents the common methods and corresponding drawbacks concerning nonlinear analysis of fluxgate excitation circuits and emphasizes the importance of nonlinear analysis for these circuits. With regard to the nonlinearity of the excitation circuit, this paper proposes the use of the core-measured hysteresis curve for mathematical analysis and the use of a nonlinear model that considers the coupling effect of the core and winding and influence of the historical magnetic field on the core for simulation analysis. The feasibility of mathematical calculations and simulation for the nonlinear study of fluxgate excitation circuit is verified via experiments. The results demonstrate that, in this regard, the simulation is four times better than a mathematical calculation. The simulation and experimental results of the excitation current and voltage waveforms under different excitation circuit parameters and structures are essentially consistent, with a difference in current of no more than 1 mA, thereby verifying the effectiveness of the nonlinear excitation analysis method. [ABSTRACT FROM AUTHOR]

At present, diagnostics of the current technical condition of high-voltage power equipment in power systems have become more important. This allows the estimation of the real technical condition of power equipment more accurately with its removal into repair based on the results of the diagnostics. This paper presents the comparative analysis of expert evaluations with the use of the arithmetical mean and median values of expert evaluations. In this case, individual expert opinions, influenced by a level of competence, correspond to each other in a different manner, depending on the applied approach. As the comparison of the consistency of expert opinions is the basis for decision-making, it is recommended to make a decision on the technical condition using median estimations because these estimations are less subjected to distortions from single outliers of judgments. This provides more reliable information for making key decisions. Three approaches are considered in this paper: the method of arithmetical mean estimations, the method of median estimations based on the Kemeny median method, and the analytic hierarchy process of Saaty. The considered methods allow decisions on power equipment operation to be made very quickly; namely, if the power equipment is in an operable state and may remain operated, or it has considerable defects and should be removed from operation for routine maintenance, or it has reached the final technical state and needs to be removed from operation. [ABSTRACT FROM AUTHOR]

Optimization problems relating to heat source operation are solved via linear (LP) or nonlinear (NLP) programming. The optimization methods used can be compared based on the complexity of the computational program setup, the time required to input and update data, and the computational time required for the optimal shifting and loading of installed power machinery. LP methods are preferred (e.g., the simplex method, the method of characteristics), but the Lagrange multiplier NPL method is sometimes applied. In this paper, the method of proportional increments (LP and NLP) is applied to compare the optimal loadings of power machinery installed in a thermal power plant using linear and quadratic economic characteristics. [ABSTRACT FROM AUTHOR]

In mathematical analysis, the q-analogue of a function refers to a modified version of the function that is derived from q-series expansions. This paper is focused on the q-analogue of the exponential function and investigates a class of convex functions associated with it. The main objective is to derive precise inequalities that bound the coefficients of these convex functions. In this research, the initial coefficient bounds, Fekete–Szegő problem, second and third Hankel determinant have been determined. These coefficient bounds provide valuable information about the behavior and properties of the functions within the considered class. [ABSTRACT FROM AUTHOR]

In a series of papers, Dimitrov suggested the classical D-scoring rule for scoring items that give difficult items a higher weight while easier items receive a lower weight. The latent D-scoring model has been proposed to serve as a latent mirror of the classical D-scoring model. However, the item weights implied by this latent D-scoring model are typically only weakly related to the weights in the classical D-scoring model. To this end, this article proposes an alternative item response model, the modified Ramsay quotient model, that is better-suited as a latent mirror of the classical D-scoring model. The reasoning is based on analytical arguments and numerical illustrations. [ABSTRACT FROM AUTHOR]

The internet of things (IoT) revolutionized human life, whereby a large number of interrelated devices are connected to exchange data in order to accomplish many tasks, leading to the rapid growth of connected devices, reaching the tens of billions. The Low Power Wide Area (LPWA) protocols paradigm has emerged to satisfy the IoT application requirements, especially in terms of long-range communication and low power consumption. However, LPWA technologies still do not completely meet the scalability requirement of IoT applications. The main critical issues are the restrictive duty cycle regulations of the sub-GHz band in which most LPWA technologies operate, as well as the random access to the medium. Ingenu Random Phase Multiple Access (RPMA) is an LPWA technology that uses the 2.4 GHz band that is not subject to the duty cycle constraint. Furthermore, RPMA uses Direct-Sequence Spread Spectrum (DSSS) as a modulation technique; hence, it is an excellent candidate technology for handling scalable LPWA networks. In this paper, we perform mathematical and simulation analysis to assess RPMA scalability and the factors that affect it, especially when all the available channels are used. The results indicate that RPMA has impressive scalability. Indeed, by taking advantage of the multichannel feature in RPMA, the network capacity can be increased by up to 38 times. Aditionally, randomly selecting the Spreading Factors (SF) degrades the network scalability, as working on higher SFs will increase the probability of collision. Thus, we proposed an SF distribution algorithm that ensures effective packet delivery with minimum collision. [ABSTRACT FROM AUTHOR]

This paper analyses the possibility of damaging and destroying an identification chip of the Mifare type in a frequently used contactless identification card of size ID-1, following the standard ISO/IEC 7810 (i.e., with dimensions 85.60 × 53.98 × 0.76 mm), using the magnetic field of an adjacent conductor in which a current pulse of a defined shape and amplitude is flowing. For analysis purposes, the nonlinear current–voltage characteristic of the Mifare chip voltage limiter was measured and approximated, and the mutual inductance of the straight conductor and the rectangle coil antenna in the card was calculated. Next, a mathematical analysis was conducted based on the description of the equivalent electrical circuit by the differential equations. The results of the mathematical analysis were verified by a simulation in the free simulation software Micro-Cap 12. The peak value of the current pulse that can damage the Mifare chip was measured by a combination wave generator. Based on these measurements and the chip characteristics, the energy capable of destroying the chip was calculated. The characteristics of chip damage were determined using a comparison of the resonant characteristics of undamaged and damaged RFID cards with Mifare chips. [ABSTRACT FROM AUTHOR]

In this paper, we employ the new iterative method to investigate two prominent nonlinear partial differential equations, namely the modified Camassa–Holm (mCH) equation and the modified Degasperis–Procesi (mDP) equation, both within the framework of the Caputo operator. The mCH and mDP equations are fundamental in studying wave propagation and soliton dynamics, exhibiting complex behavior and intriguing mathematical structures. The new iterative method (NIM), a powerful numerical technique, is utilized to obtain analytical and numerical solutions for these equations, offering insights into their dynamic properties and behavior. Through systematic analysis and computation, we unveil the unique features of the mCH and the mDP equations, shedding light on their applicability in various scientific and engineering domains. This research contributes to the ongoing exploration of nonlinear wave equations and their solutions, emphasizing the versatility of the new iterative method in tackling complex mathematical problems. Numerical results and comparative analyses are presented to validate the effectiveness of the new iterative method in solving these equations, highlighting its potential for broader applications in mathematical modeling and analysis. [ABSTRACT FROM AUTHOR]

DC–DC converters are gaining attention due to their importance in key applications like renewable energy generation. A desirable feature in new DC–DC converters is a reduction in the size, which can be achieved with a reduction in the energy stored in the inductors. This article introduces a new step-up DC–DC converter topology with the following advantages: (i) a larger relation of duty cycle vs. voltage gain compared with the classical boost topology and (ii) an inductor with smaller current and smaller inductance (for the same power conversion characteristics) compared to the traditional boost converter. The smaller inductor current is an advantage against many step-up topologies with the inductor in series with the input (and then the input and the inductor currents are equal). The necessary inductance to achieve a certain current ripple is also reduced compared to the classical boost topology. This results in an inductor with a smaller amount of stored energy, lower inductance, and lower current. The proposed topology can be scaled to have a full family of large-voltage-gain converters. This paper presents the mathematical analysis, simulations, and experiments to assess the benefits of the proposition. [ABSTRACT FROM AUTHOR]

In this paper, a mathematical analysis is presented to show the effect of grid-connected inverter (GCI) parameters on its emissions in the supraharmonic range. This analysis is extended to explain the effect of asymmetry on the emissions of parallel-connected GCIs on distributed power generation systems. The switching harmonics of a GCI appear as bands around the switching frequency and its multiples. A MATLAB/Simulink model is built to perform two studies. In the first study, we use one GCI to examine the effect of the parameters on the emissions, while in the second study, we examine the effect of the asymmetry of two parallel-connected GCIs on the total emission toward the grid. An actual setup is built to verify the results of the mathematical analysis and the simulation study. It is found that the SHs of single-phase GCI amplitude are strongly dependent on the DC-link voltage and the coupling inductor, while the phases of the sideband harmonics only change with changing the injected power. The variation of the injected power does not have any tangible effect on the carrier harmonics. [ABSTRACT FROM AUTHOR]

Long COVID, a long-lasting disorder following an acute infection of COVID-19, represents a significant public health burden at present. In this paper, we propose a new mechanistic model based on differential equations to investigate the population dynamics of long COVID. By connecting long COVID with acute infection at the population level, our modeling framework emphasizes the interplay between COVID-19 transmission, vaccination, and long COVID dynamics. We conducted a detailed mathematical analysis of the model. We also validated the model using numerical simulation with real data from the US state of Tennessee and the UK. [ABSTRACT FROM AUTHOR]

In this paper, we investigate two within-host mathematical models that are based on differential equations. These mathematical models include healthy cells, tachyzoites, and bradyzoites. The first model is based on ordinary differential equations and the second one includes a discrete time delay. We found the models' steady states and computed the basic reproduction number R 0 . Two equilibrium points exist in both models: the first is the disease-free equilibrium point and the second one is the endemic equilibrium point. We found that the initial quantity of uninfected cells has an impact on the basic reproduction number R 0 . This threshold parameter also depends on the contact rate between tachyzoites and uninfected cells, the contact rate between encysted bradyzoite and the uninfected cells, the conversion rate from tachyzoites to bradyzoites, and the death rate of the bradyzoites- and tachyzoites-infected cells. We investigated the local and global stability of the two equilibrium points for the within-host models that are based on differential equations. We perform numerical simulations to validate our analytical findings. We also demonstrated that the disease-free equilibrium point cannot lose stability regardless of the value of the time delay. The numerical simulations corroborated our analytical results. [ABSTRACT FROM AUTHOR]

In the present study, we introduce a four-dimensional deterministic mathematical pattern in order to study the interactions between the human immune system and a virus. The model is based on a system with four first-order ordinary differential equations, and the main aim of the paper is to perform a mathematical analysis of the local behavior of the associated dynamical system using the tools of the qualitative theory of dynamical systems. Moreover, two types of patterns with controls were introduced; consequently, some very interesting theoretical conclusions with medical relevance were obtained. [ABSTRACT FROM AUTHOR]

A 3 × 3 Mueller matrix polarimeter, as one of the primary polarization measurement tools, has attracted intensive interest in various fields of polarization optics. In this paper, from a novel viewpoint with more mathematical analysis of the basic configuration and the working performance of a 3 × 3 Mueller matrix polarimeter, we systematically study the relationship between Mueller matrix measurement errors with the configuration and characteristics of polarizers in the polarimeter. We figure out that, mathematically, there is an optimal orientation angle configuration (0°, 60°, and 120°) of polarizers, which inherently ensures a minimal relative measurement error in general. Moreover, we interpret the internal source of such a robust error resistance under this optimized configuration mathematically for the first time. Simulation and practical experiments show the effectiveness of our mathematical analyses, and compared to the frequently used orientation angle configuration, the average relative error is reduced by up to 32% under such an optimized configuration. [ABSTRACT FROM AUTHOR]

Research into the recent developments for solving fractional mathematical equations requires accurate and efficient numerical methods. Although many numerical methods based on Caputo's fractional derivative have been proposed to solve fractional mathematical equations, the efficiency of obtaining solutions using these methods when dealing with a large matrix requires further study. The matrix size influences the accuracy of the solution. Therefore, this paper proposes a quarter-sweep finite difference scheme with a preconditioned relaxation-based approximation to efficiently solve a large matrix, which is based on the establishment of a linear system for a fractional mathematical equation. The paper presents the formulation of the quarter-sweep finite difference scheme that is used to approximate the selected fractional mathematical equation. Then, the derivation of a preconditioned relaxation method based on a quarter-sweep scheme is discussed. The design of a C++ algorithm of the proposed quarter-sweep preconditioned relaxation method is shown and, finally, efficiency analysis comparing the proposed method with several tested methods is presented. The contributions of this paper are the presentation of a new preconditioned matrix to restructure the developed linear system, and the derivation of an efficient preconditioned relaxation iterative method for solving a fractional mathematical equation. By simulating the solutions of timefractional diffusion problems with the proposed numerical method, the study found that computing solutions using the quarter-sweep preconditioned relaxation method is more efficient than using the tested methods. The proposed numerical method is able to solve the selected problems with fewer iterations and a faster execution time than the tested existing methods. The efficiency of the methods was evaluated using different matrix sizes. Thus, the combination of a quarter-sweep finite difference method, Caputo's time-fractional derivative, and the preconditioned successive over-relaxation method showed good potential for solving different types of fractional mathematical equations, and provides a future direction for this field of research. [ABSTRACT FROM AUTHOR]

Transformer-less connection schemes can provide a feasible solution for lowering the economic cost, occupied space, and device weight of modular multilevel converter (MMC) systems. However, due to the reduction in the converter transformer, the current flow loop is changed; as a result, the existing MMC model is not suitable. In this paper, the ac- and dc-side equivalent circuit models of the MMC system using a transformer-less connection scheme are established in both a–b–c stationary and d–q rotating coordinate systems. Then, a highly accurate mathematical analysis model is proposed, in which the interactions among the electrical quantities can be fully seen. The mathematical model is established in the time domain, and hence the amplitude and phase angle of every harmonic component in each quantity can be directly obtained. The proposed model is verified under various typical situations by comparing the calculated values with the actual waveforms. The comparison results prove that the calculation error is small enough to be negligible. The mathematical model in this paper can provide a powerful tool in terms of the performance analysis and the main circuit parameter design for MMCs in transformer-less applications. [ABSTRACT FROM AUTHOR]

Complex dynamical networks with time-varying connections have characteristics that allow a better representation of real-world complex systems, especially interest in their not static behavior and topology. Their applications reach areas such as communication systems, electrical systems, medicine, robotic, and more. Both continuous and discrete-time complex dynamical networks and the pinning control technique have been studied. However, even with interest in the research on complex networks combining characteristics of discrete-time, time-varying connections, pinning control, and impulsive control, there are few studies reported in the literature. There are some previous studies dealing with impulsively pin-controlling a discrete-time complex network. Nevertheless, they neglect to deal with time-varying connections; they deal with these systems by experimentally using continuous-time methods or linearizing the node dynamics. In this manner, this paper presents a control scheme that not only deals with pin control on discrete-time complex networks but also includes time-varying connections. This paper proposes an impulsive pin control to a zero state using passivity degrees considering a discrete-time complex network with undirected, linear, and diffusive couplings. Additionally, a corresponding mathematical analysis, which allows the representation of the dynamics as a set of symmetric matrices, is presented. With this, certain kinds of time-varying connections can be integrated into the analysis. Moreover, a particular criterion for selecting nodes to pin is also presented. The behavior of the controller for the non-varying and time-varying coupling cases is shown via numeric simulations. [ABSTRACT FROM AUTHOR]

The results presented in this paper are obtained as part of the continued development and research of clustering algorithms based on the discrete mathematical analysis. The article briefly describes the theory of Discrete Perfect Sets (DPS-sets) that is the basis for the construction of DPS-clustering algorithms. The main task of the previously constructed DPS-algorithms is to search for clusters in multidimensional arrays with noise. DPS-algorithms have two stages: the first stage is the recognition of the maximum perfect set of a given density level from the initial array, the second stage is the partitioning of the result of the first stage into connected components, which are considered to be clusters. Study of qualities of DPS-algorithms showed that, in a number of situations in the first stage, the result does not include all clusters which have practical sense. In the second stage, partitioning into connected components can lead to unnecessarily small clusters. Simple variation of parameters in DPS-algorithms does not allow for eliminating these drawbacks. The present paper is devoted to the construction on the basis of DPS-algorithms of their new versions, more free from these drawbacks. [ABSTRACT FROM AUTHOR]

This paper presents a photovoltaic (PV) system designed to reduce the DC-link capacitance present in double-stage PV microinverters without increasing the capacitor interfacing the PV source. This solution is based on a modified boost topology, which exhibits continuous current in both input and output ports. Such a characteristic enables the implementation of PV microinverters without electrolytic capacitors, which improves the reliability in comparison with solutions based on classical converters with discontinuous output current and electrolytic capacitors. However, the modified boost converter exhibits different dynamic behavior in comparison with the classical boost converter; thus, design processes and controllers developed for the classical boost converter are not applicable. This paper also introduces a sliding-mode controller designed to ensure the stable operation of the PV microinverter around the maximum power point. Moreover, this solution also rejects the voltage oscillations at double the grid frequency generated by the grid-connection. The global stability of the complete PV system is formally demonstrated using mathematical analyses, and a step-by-step design process for both the power stage and control system is proposed. Finally, the design process is illustrated using a representative application example, and the correct operation of the PV system is validated using realistic circuital simulations. The results validate the accuracy of the theoretical equations proposed for both the design and control of the novel PV system, where errors below 4.5% were obtained for the ripple prediction, and below 1% for the prediction of the dynamic behavior. [ABSTRACT FROM AUTHOR]

A tensor representation structure based on the multilayer tensor spectrum pyramid (MLTSP) is introduced in this work. The structure is "truncated", i.e., part of the high-frequency spectrum coefficients is cut-off, and on the retained low-frequency coefficients, obtained at the output of each pyramid layer, a hierarchical tensor SVD (HTSVD) is applied. This ensures a high concentration of the input tensor energy into a small number of decomposition components of the tensors obtained at the coder output. The implementation of this idea is based on a symmetrical coder/decoder. An example structure for a cubical tensor of size 8 × 8 × 8, which is represented as a two-layer tensor spectrum pyramid, where 3D frequency-ordered fast Walsh–Hadamard transform and HTSVD are used, is given in this paper. The analysis of the needed mathematical operations proved the low computational complexity of the new approach, due to a lack of iterative calculations. The high flexibility of the structure in respect to the number of pyramid layers, the kind of used orthogonal transforms, the number of retained spectrum coefficients, and HTSVD components, permits us to achieve the desired accuracy of the restored output tensor, imposed by the application. Furthermore, this paper presents one possible application for 3D object searches in a tensor database. In this case, to obtain the invariant representation of the 3D objects, in the spectrum pyramid, the 3D modified Mellin–Fourier transform is embedded, and the corresponding algorithm is shown. [ABSTRACT FROM AUTHOR]

In addition to other things, a robot's design also determines its singularity configurations and points in the workspace. In designing the robot's working trajectory, one of the main issues of robot steering is avoiding singularities. The article proposes a different approach to calculating the inverse task, which lies in the random mapping of the robot mechanism's workspace through searching for points closest in proximity to the trajectory in question. The new methodology of mapping and detecting the states of singularity in the workspace is actually based on Monte Carlo analysis, since we were also interested in the number of occurrences. In terms of mathematical analysis, this method is less demanding, because in searching for joint variables suitable for the given trajectory, it does not use inverse calculation. It is important that the method is chosen appropriately. The method is sufficiently illustrative in the form of a graph, making, e.g., programming optimization simpler. The ultimate effect is the reduced time needed for computing joint variables and the availability of an option to select a robot configuration suitable for carrying out the required work. The paper offers an example of an analysis concerning three different robots. [ABSTRACT FROM AUTHOR]