The exponential growth of digital media content has introduced new challenges in managing and classifying internet traffic. Digital media traffic is composed of various applications such as video, audio, social media, and search, and its data structure is complex, incorporating a vast array of features. The classification of traffic data is a crucial aspect of internet traffic management and network security, and it forms the basis for several scenarios, including content distribution, advertising recommendations, and data analysis. Traditional classification methods rely mainly on deep packet inspection and port-based techniques, which have become increasingly ineffective due to the rapid evolution of network traffic. To address this issue, this study proposes a machine learning-based traffic classification method aimed at enhancing the accuracy and efficiency of digital media traffic classification to meet the current needs of traffic management and network security. The paper also analyzes and evaluates the classification effect and prediction capability of various algorithms under different training set sizes to validate the feasibility and effectiveness of the proposed method. The result demonstrates that the neural network algorithm has superior classification and prediction capabilities compared to the decision tree and support vector machine algorithms. Furthermore, our proposed method achieves the highest accuracy of 96.88% with a large training sample of 40,000 data streams, proving its superiority in handling high-dimensional data and complex datasets. The research results are significant for the development of digital media traffic classification and prediction methods and are expected to be applied in practical scenarios.

The Physics-Informed Neural Networks (PINNs) improve the efficiency of data utilization by combining physical principles with neural network algorithms and ensure that the predictions are consistent and stable with the physical laws. PINNs opens up a new approach to address inverse problems in fluid mechanics. Based on the single-relaxation-time lattice Boltzmann method (SRT-LBM) with the Bhatnagar-Gross-Krook (BGK) collision operator, the PINN-SRT-LBM model is proposed in this paper for solving the inverse problem in fluid mechanics. The PINN-SRT-LBM model consists of three components. The first component involves a deep neural network that predicts the equilibrium control equations in different discrete velocity directions within SRT-LBM. The second component employs another deep neural network to predict non-equilibrium control equations, enabling inference of the fluid's non-equilibrium characteristics. The third component, a physics informed function translates the outputs of the first two networks into physical infor-mation. By minimizing the residuals of the physical partial differential equations (PDEs), the physics informed function infers relevant macroscopic quantities of the flow. The model evolves two sub-models applicable to different dimensions, named PINN-SRT-LBM-I and PINN-SRT-LBM-II models according to the construction of the physical informed function. The innovation of this work is the introduction of SRT-LBM and discrete velocity models as physical drivers into the neural network through the interpretation function. Therefore, PINN-SRT-LBM allows the neural network to handle inverse problems of various dimensions and focus on problem-specific solving. Experimental results confirm the accurate prediction of flow infor-mation at different Reynolds numbers within the computational domain. Relying on the PINN-SRT-LBM models, inverse problems in fluid mechanics can be solved efficiently.

Inverse problems of recovering surface fluxes on the boundary of a domain from pointwise observations are considered. Sharp conditions on the data ensuring existence and uniqueness of solutions in Sobolev classes are exposed. They are smoothness conditions on the data, geometric conditions on the location of measurement points, and the boundary of a domain. The proof relies on asymptotics of fundamental solutions to the corresponding elliptic problems and the Laplace transform. The problem is reduced to a linear algebraic system with a nondegerate matrix.

The objective of this study is to examine a nonparametric estimate , using the kernel approach, of the conditional distribution function of a scalar response variable that is given a random variable whose values take place in a separable real Hilbert space. The observations will be dependent on one another in a quasi-associated fashion. The pointwise practically perfect consistencies with rates of this estimator are established by us under some broad conditions. The study’s major objective is to investigate the convergence rate of the proposed estimator and its application in the convergence rate and asymptotic normality of the hazard function. The asymptotic normality of the developed estimator is established precisely. Simulation studies were conducted to investigate the behavior of the asymptotic property in the context of finite sample data.

The key concepts in symmetry detection and similarity, identity measures are automorphism and isomorphism respectively. Therefore, methods for symmetry detection and similarity, identity measures should be functionally bijective, inverse, and invariance under a set of mathematical operations. Nevertheless, few or no existing method is functional for these properties. In this paper, a new methodological paradigm, called optinalysis, is presented for symmetry detections, similarity, and identity measures between isoreflective or autoreflective pair of mathematical structures. The paradigm of optinalysis is the re-mapping of isoreflective or autoreflective pairs with an optical scale. Optinalysis is characterized as invariant under a set of transformations and its isoreflective polymorphism behaves on polynomial and non-polynomial models.

A new set of basis functions is presented. The foundation in mathematics is established. To reconstruct a signal, it is compared to the Fourier basis functions. The M-basis functions' potential applications are also presented.

Voronoi mosaics inspired by the seed points placed on the Archimedes Spirals are reported. Voronoi entropy was calculated for these patterns. Equidistant and non-equidistant patterns are treated. Voronoi mosaics built from a sells of equal size which are of a primary importance for decorative arts are reported. The pronounced prevalence of hexagons is inherent for the patterns with an equidistant and non-equidistant distribution of points, when the distance between the seed points is of the same order of magnitude as the distance between the turns of the spiral. Penta- and heptagonal “defected” cells appeared in the Voronoi diagrams due to the finite nature of the pattern. The ordered Voronoi tessellations demonstrating the Voronoi entropy larger than 1.71, reported for the random 2D distribution of points, were revealed. The dependence of the Voronoi entropy on the total number of the seed points located on the Archimedes Spirals is reported. Voronoi tessellations generated by the phyllotaxis-inspired patterns are addressed. The aesthetic attraction of the Voronoi mosaics arising from seed points placed on the Archimedes Spirals is discussed.

This study presents the solution of the second type of a two-dimensional nonlinear integral equation in Banach space. Also, the existence and uniqueness of this equation’s solution are discussed. We utilize a numerical approach involving hybrid and block-pulse functions to obtain the approximate solution of a two-dimensional nonlinear integral equation. Nonlinear integral equation in two dimensions is reduced numerically to a system of nonlinear algebraic equations that can be solved using numerical methods. This study focuses on showing the convergence analysis for the numerical approach and obtaining an error estimate. Some numerical examples have been provided to demonstrate the approach’s viability and efficacy

In this paper a new monolithic Eulerian formulation in the framework of non-classical continuum is presented for the analysis of fluid-strucutre interaction problems. In this respect, Cosserat continuum theory taking into account the micro-rotational degrees of freedom of the particles is considered. Continuum description of the model and variational formulation of the governing flow dynamics for non-classical -fluid-structure interaction nCFSI is presented. The model is analyzed by computing a well known benchmark problem by Hecht and Pironneau [16]. The algorithmic description is presented and implemented using FreeFEM++. Code is validated with the benchmark solution of Turek and Hron [38] in case of flow around a flag attached with cylinder. New microstructral behavior of the solution is studied and numerical simulations and results are shown in the form of figures. Some interesting feature of the flow is observed and microstructural characteristics are discussed.

University course scheduling (UCS) is one of the most important and time-consuming issues that all educational institutions face yearly. Most of the existing techniques to model and solve UCS problems have applied approximate methods, which are different in terms of efficiency, performance, and optimization speed. Accordingly, this research aims to apply an exact optimization method to provide an optimal solution to the course scheduling problem. In other words, in this research, an integer programming model is presented to solve the USC problem. In this model, hard and soft constraints include the facilities of classrooms, courses of different levels and compression of students' curriculum, courses outside the faculty and planning for them, and the limited time allocated to the professors. The objective is to maximize the weighted sum of allocating available times to professors based on their preferences in all periods. To evaluate the presented model's feasibility, it is implemented using the GAMS software. Finally, the presented model is solved in a larger dimension using a real data set from a college in China and compared with the current program in the same college. The obtained results show that considering the mathematical model's constraints and objective function, the faculty courses' timetable is reduced from 4 days a week to 3 working days. Moreover, master courses are planned in two days, and the courses in the educational groups do not interfere with each other. Furthermore, by implementing the proposed model for the real case study, the maximum teaching hours of the professors are significantly reduced. The results demonstrate the efficiency of the proposed model and solution method in terms of optimization speed and solution accuracy.

This short report contains a result that characterizes almost global pullback attractor of discrete-time non-autonomous systems. Analogously to the multiple Lyapunov functions approach for switched systems, we show here that existence of multiple Lyapunov densities implies pullback convergence of almost all initial states to the origin for a discrete-time non-autonomous system.

Incremental stability analysis for time-delay systems has attracted more and more attentions for its contemporary applications in transportation processes, population dynamics, economics, satellite positions, etc. This paper researches criteria on v(t)-incremental stability, which is defined to demonstrate the convergence rate for incremental stability, for time-delay systems. Firstly, the sufficient conditions for v(t)-incremental stability for time-delay systems with continuous right-hands are studied, and several corollaries for specific cases are provided. As for time-delay systems with discontinuous right-hands, after expounding the relevant conditions for the existence and uniqueness of the Filippov solution, by using approximation methods, sufficient conditions for v(t)-incremental stability are obtained. The conclusions are applied to linear switched time-delay systems and Hopfield neural network system with composite right-hand afterwards.

Global optimization is an imperative development of local optimization because there are many problems in artificial intelligence and machine learning requires highly acute solutions over entire domain. There are many methods to resolve the global optimization, which can be classified into three groups such as analytic methods (purely mathematical methods), probabilistic methods, and heuristic methods. Especially, heuristic methods like particle swarm optimization and ant bee colony attract researchers because their effective and practical techniques which are easy to be implemented by computer programming languages. However, these heuristic methods are lacking in theoretical mathematical fundamental. Fortunately, minima distribution establishes a strict mathematical relationship between optimized target function and its global minima. In this research, I try to study minima distribution and apply it into explaining convergence and convergence speed of optimization algorithms. Especially, weak conditions of convergence and monotonicity within minima distribution are drawn so as to be appropriate to practical optimization methods.

Wireless cellular communication technology has developed into a very resourceful commodity worldwide. Today, people of all races can hardly live without means of voice and data cellular communication technology. Imprecise propagation loss estimation leads to high power waste, high co-channel interference and poor service quality in cellular communication system networks. This paper proposes a realistic adaptive fine-tuning method for distinctive propagation loss estimation over a microcellular communication radio links based on signal power measurements from Long Term Evolution radio broadband networks, taking non-line of sight (NLOS) and line of sight (LOS) environments into consideration. The methodology is verified by measurements taken in non-line of sight and line of sight signal propagation scenarios. The results showed that the estimated propagation losses using the proposed realistic adaptive tuning models were more accurate than the existing Cost -231 modelling estimation approach

The fast and reliable processing of medical images is of paramount importance to adequately generate data to feed machine learning algorithms that can prevent and diagnose health issues. Here we benchmark different compressed sensing techniques applied to magnetic resonance imaging as a means to reduce the acquisition time spent in the collection of data and signals that form the image. We show that by using these techniques, it is possible to reduce the number of signals needed and, therefore, substantially decrease the time to acquire the measurements. To this end, we have considered and compared different algorithms: the Iterative Re-Weighted Least Squares, the Iterative Soft Thresholding Algorithm, the Iterative Hard Thresholding Algorithm, the Primal Dual Algorithm and the Log-Barrier Algorithm. We have implemented such algorithms in different analysis programs that have been used to perform the reconstruction of the images and found that the Iterative Soft Thresholding Algorithm gives the optimal results. We found that the images obtained with this algorithm have less quality than the original ones, but the quality is good enough to distinguish each body structure and detect any health problems.

The present paper introduces a method of basis transformation of vector fields that is specifically applicable to polynomials space and differential equations with certain polynomials solutions such as Hermite, Laguerre and Legendre polynomials. The method based on separated transformation of vector space basis by a set of operators that are equivalent to the formal basis transformation and connected to it by linear combination with projection operators. Applying the Forbenius covariants yields a general method that incorporates the Rodrigues formula as a special case in polynomial space. Using the Lie algebra modules, specifically , on polynomial space results in isomorphic algebras whose Cartan sub-algebras are Hermite, Laguerre and Legendre differential operators. Commutation relations of these algebras and Baker-Campbell-Hausdorff formula gives new formulas for special polynomials

An embryonic version of membrane theory can be date back to the Bernstein's work reported more than a hundred years ago. Such an originally old work has evolved conceptually and mathematically up until today, and it plays a central role in current membrane theory. Goldman-Hodgkin-Katz equation (GHK eq.) is one of the math-based monumental works, which constitutes the present membrane theory. Goldman theoretically derived GHK eq., but its physiological meaning was provided by the two renowned scientists, Hodgkin and Katz. These two employed an assumption that the electric field (EF) across the plasma membrane is constant to validate the GHK eq. physiologically. Proposal of Hodgkin-Huxley model (HH model) is another math-based monumental works developed from the membrane theory and now forms a fundamental part of the current membrane theory. GHK eq. and HH model are quite fundamental central concepts in the current physiology. Despite the broad acceptance of GHK eq. at present time, its prerequisite that the EF within the plasma membrane is constant is hardly believable. Especially when the action potential is generated, it sounds totally nonsense. Furthermore, the existence of constant EF within the plasma membrane is conceptually almost in conflict with the HH model. The authors will discuss those problematic issues the membrane theory inherits.

Many authors proposed analytical methods for solving fully fuzzy Sylvester matrix equation (FFSME) based on Vec-operator and Kronecker product. However, these methods are restricted to nonnegative fuzzy numbers and cannot be extended to FFSME with near-zero fuzzy numbers. The main intention of this paper is to develop a new numerical method for solving FFSME with near-zero trapezoidal fuzzy numbers that provides a wider scope of trapezoidal fully fuzzy Sylvester matrix equation (TrFFSME) in scientific applications. This numerical method can solve the trapezoidal fully fuzzy Sylvester matrix equation with arbitrary coefficients and find all possible finite arbitrary solutions for the system. In order to obtain all possible fuzzy solutions, the TrFFSME is transferred to a system of non-linear equations based on newly developed arithmetic fuzzy multiplication between trapezoidal fuzzy numbers. The fuzzy solutions to the TrFFSME are obtained by developing a new two-stage algorithm. To illustrate the proposed method numerical example is solved.

This paper introduces extensions H4,p and X8,p of Horn’s double hypergeometric function H4 and Exton’s triple hypergeometric function X8, taking into account recent extensions of Euler’s beta function, hypergeometric function, and confluent hypergeometric function. Among the numerous extended hypergeometric functions, the primary rationale for choosing H4 and X8 is their comparable extension type. Next, we present various integral representations of Euler and Laplace type, Mellin and inverse Mellin transforms, Laguerre polynomial representation, transformation formulae and a recurrence relation for the extended functions. In particular, we provide a generating function for the X8,p and several bounding inequalities for the H4,p and X8,p.

The purpose of this paper is to build a new bridge between category theory and a generalized probability theory known as noncommutative probability or quantum probability, which was originated as a mathematical framework for quantum theory, in terms of states as linear functional defined on category algebras. We clarify that category algebras can be considered as generalized matrix algebras and that the notions of state on category as linear functional defined on category algebra turns out to be a conceptual generalization of probability measures on sets as discrete categories. Moreover, by establishing a generalization of famous GNS (Gelfand-Naimark-Segal) construction, we obtain a representation of category algebras of †-categories on certain generalized Hilbert spaces which we call semi-Hilbert modules over rigs.

This paper presents a new approach to a different proof of "Fermat's Last Theorem.” For the proof of the theorem, it is proposed to use a more straightforward geometrical approach. A special family of curves, the Elba curves, is introduced to facilitate the proof. This approach makes the proof easier to comprehend by mathematicians and those interested in the subject.

The basic challenges of this work are twofold: demonstrating the dependence between the functional and topological qualities of partition networks and finding the simplest—with respect to algorithmic complexity—network elements. The study of these problems is based on finding the solution to an appropriate antagonistic vertex game. The results of the numerical simulations of antagonistic partition games demonstrate that the winner’s graphs are “almost always” dense and hyperenergetic compared to the loser’s graphs. These observations reveal that successful evolutionary mechanisms can be realized, in principle, by the simplest objects (such as viruses).

Abstract:Shown is the derivation of Lorentz-Einstein k-factor in SRT as an amplitude-term of oscillation-differential equations of second order.This case is shown for classical Lorentz-factor as solution of an equation for undamped oscillation as well as the developed theorem as a second solution for advanced SRT of fourth order with an equation for damped oscillation-states.This advanced term allows a calculation for any velocities by real rest mass

With Unmanned Aerial Vehicles (UAV), a swift response to urgent needs like search \& rescue missions or medical deliveries can be realized. Simultaneously, the legislator is establishing so-called geographical zones, which restrict UAV operations to mitigate the air and ground risk to third parties. These geographical zones serve a particular safety interest, but they may also hinder the efficient usage of UAVs on time-critical missions with a range-limiting battery capacity. In this study, we address a facility location problem for up to two UAV hangars with a robust optimization model considering demand hotspots, geographical zones as restricted areas, a standard mission to satisfy battery capacity constraints, and the impact of wind scenarios. To this end, water rescue missions are used exemplary, for which positive and negative location factors for UAV hangars and areas of increased drowning risk as demand points are derived from open-source georeferenced data. Optimal UAV mission trajectories are computed with an A* algorithm considering five different restriction scenarios. As this pathfinding is very time-consuming, binary occupancy grids and image processing algorithms accelerate the computation by identifying either entirely inaccessible or restriction-free connections beforehand. For the optimal UAV hangar locations, we maximize accessibility while minimizing the service time to the hotspots, resulting in a decrease from the average service time of 570.4 s for all facility candidates to 351.1 s for one and 287.2 s for two optimal UAV hangar locations.

Background: With the trend of intelligent display, the interface design of in-vehicle HUD is an expanding research field; Methods: To solve the subjectivity and uncertainty in the optimization of HUD interface design schemes, this paper proposed a hybrid scheme evaluation and optimization method based on entropy weight and VIKOR. The entropy weight method was used to reduce the subjectivity the decision-maker's weighting and obtain the objective weight of each indicator; The VIKOR method was used to obtain the best ranking of alternative schemes, and then the optimal interface design scheme was selected; Results: The evaluation of in-vehicle HUD interface design schemes were taken as an example for verification and calculation. The results showed that this method considers the subjectivity and uncertainty of the decision-making process in the optimization of design scheme, which can effectively improve the objectivity and accuracy of the evaluation results, and provide a reference for designers to optimize interface design schemes.

The fractional diffusion equations with integrals play a significant role in describing anomalous diffusion phenomena. We first construct an appropriate resolvent family, and use the convolution theorem of Laplace transform, the probability density function, Cauchy integral formula and Fubini theorem to study its related equicontinuity, strongly continuity, compactness, etc. Then, a reasonable mild solution is defined. Finally, by combining some fixed point theorems, we obtain some conclusions on the existence and uniqueness of mild solutions.

In this paper we totally discard the traditional trial-and-error algorithms of choosing acceptable shape parameter c in the multiquadrics $-\sqrt{c^{2}+\|x\|^{2}}$ when dealing with differential equations. Instead, we choose c directly by the MN-curve theory and hence avoid the time-consuming steps of solving a linear system required by each trial of the c value in the traditional methods. The quality of the c value thus obtained is supported by the newly born choice theory of the shape parameter. Experiments show that the approximation error of the approximate solution to the differential equation is very close to the best approximation error among all possible choices of c.

We apply the Ramsey theory to the analysis of geometrical properties of closed contours. Consider a set of six points placed on a closed contour. The straight lines connecting these points are yikx=αikx+βik i,k=1...6, αik≠0. We color the edges connecting the points for which αik>0 holds with red, and the edges for which αik<0 with green with red. At least one monochromic triangle should necessarily appear within the curve (according to the Ramsey number R3,3=6). This result is immediately applicable for the analysis of dynamical billiards. The second theorem emerges from the combination of the Jordan curve and Ramsey theorem. The closed curve is considered. We connect the points located within the same region with green links and the points placed within the different regions with red links. In this case, transitivity/intransitivity of the relations between the points should be considered. Ramsey constructions arising from the differential geometry of closed contours are discussed.

Classical numerical methods for solving ordinary differential equations often produce less accurate results when applied to problems with oscillatory or periodic behaviour. To adapt them for such problems, they are usually modified using the exponential fitting technique. This adaptation allows for the construction of new methods from their classical counterparts. The new methods are usually more accurate, efficient and suitable for handling the oscillatory or periodic behaviour of the problem. In this work, we construct a two-parameter exponentially-fitted Taylor method suitable for solving oscillatory or periodic problems that possess two frequencies. The construction algorithm is based on a proposed six-step flowchart discussed by authors in related literature. Two standard test problems were used to illustrate the accuracy and performance of the proposed method.

Offshore grids can have a key role in the transition of the energy system to sustainability. Although they require extensive infrastructure investments, they open up for the exploitation of additional resources and may be important to provide for part of the increasing electricity demand driven by sector coupling. This paper quantifies the socio-economic value of offshore grids and identifies their major drivers, performing energy system optimization in a model application of the Northern-central European energy system and the North Sea offshore grid towards 2050. The increasing wake loss with the size of hub-connected wind farms is integrated in the modelling. We find that without sector coupling no offshore grid may develop, and that the higher the level of sector coupling, the higher the value of offshore grids. Therefore, it can be strongly stated that offshore grid infrastructure development should not be discussed as a separate political topic, but seen in connection to sector coupling.

Endoscopic medical images can suffer from uneven illumination, low contrast, and lack of texture information due to the use of point directional light sources and the presence of narrow tissue structures, posing diagnostic difficulties for physicians. In this paper, a deep learning-based su-pervised illumination enhancement network is designed for low-light endoscopic images, aiming to improve both global illumination and local details. Initially, a global illumination enhancement module is formulated utilizing a higher-order curve function to improve global illumination. Sec-ondly, a local feature extraction module incorporating dual attention is designed to capture local detailed features. Considering the significance of color fidelity in biomedical scenarios, the designed loss function prioritizes introducing color loss to alleviate image color distortion. Compared with seven state-of-the-art enhancement algorithms on Endo4IE endoscopic datasets, experimental re-sults show that the proposed method can better enhance low-light endoscopic images and avoid image color distortion. It provides an efficient method to enhance images captured by endoscopes which can effectively assist clinical diagnosis and treatment.

In applied mathematics, statistics, operation research, physics, and engineering mathematics, confluent representations of hypergeometric functions in one and two variables are known to exist, and their occurrence in a variety of applications is also well recognised. In this article, we intend to present a new derivation of the extended Watson summation theorem for the Kim et al. given series 4F3. We assessed four attractive integrals involving generalized hypergeometric function as an application. With a few particular cases, this note will come to an end. In the results given above, symmetry appears on its own.

The splitting method developed in a previous study for linear, multi – component, convection-diffusion mass transfer equations is extended to non-linear, multi-component, convection-diffusion mass transfer equations. The diffusion term is non-linear (the diffusion coefficients depend on chemical species concentrations). Theoretical results about the splitting error are presented, as well as numerical simulations for mixtures with n ≤ 5. The numerical experiments show second-order accuracy and stability for the present splitting algorithm.

This note addresses a fundamental problem in matrix theory on establishing and characterizing range equalities for matrix expressions that involve generalized inverses. We first establish a group of necessary and sufficient conditions for the matrix range equality ${\rm range}(D_1 - C_1A_1^{\dag}B_1) = {\rm range}(D_2 - C_2A_2^{\dag}B_2)$ to hold, where $(\cdot)^{\dag}$ denotes the Moore--Penrose inverse of matrix. We then give several groups of range equalities with extrusion properties for multiple matrix products associated with two matrices and their conjugate transposes and Moore--Penrose inverses.

The work proposes a new approach for evolutionary global optimization which, in simple terms, may be described as a fusion of quantum and simulated annealing. This becomes possible by using basic concepts from Homotopy Theory, conjugated to the already existing Fuzzy Adaptive Simulated Annealing paradigm and other components. In this fashion, it occurs a temporal and nonlinear superposition of the two types of annealing, provoking interesting effects in runtime, including the so-called quantum tunneling, when there is a sudden transition between attraction basins of different minima without "climbing" potential barriers. In addition, the proposal includes a generalization of the "deformation" of Hamiltonians used in quantum annealing by suggesting the use of general homotopies between the surface corresponding to the cost function under processing and another specific landscape, being this possible because of the ability of Fuzzy ASA of handling time-varying objective functions. The proposed paradigm shows that it is possible and beneficial to superpose the two types of annealing, displaying amazing numerical results and suggesting a kind of interlacing or synergic behavior - perhaps we could call it an "annealing entanglement". Finally, given the fast advances in commercial quantum annealing processors, it seems sensible to expect that an implementation of the proposed approach using such devices may occur very soon. In order to demonstrate the efficacy of the proposed algorithm, some significant and detailed examples are included in the text, so as to illustrate and clarify the presented ideas.

This paper presents the symmetries of differential equations associated with one-variable and Bivariate Hermite polynomials by proposing a representation of Lie algebra for these differential operators. Applying the Baker-Campbell-Hausdorff formula to these algebras, results in new relations and generating functions in one-variable and Bivariate Hermite polynomials. A general form of representation for other orthogonal polynomials such as Laguerre polynomials is introduced.

Non-normal mode stability analysis has long been employed to study linear transient characteristics of wall bounded shear flows. However, the application of this approach to investigate shear flows with stratified thermal conditions (STCF) has received limited attention. This paper examines thermal effects on Taylor-Couette flow. The results in this study indicate the possibility of a subcritical bifurcation in stratified Taylor-Couette flow.

Work done by us shows that rough topology is a good tool for dealing with things that happen in the real world. The concept of ”basis” has been used to show out the deciding elements of the recent outbreak of COVID-19 that was reported all over the world.

The present work concerns with presenting an explicit formula for new shifted wavelet (NSW) functions. Wavelet functions have many applications and advantages in both applied and theoretical fields. They are formulated with different orthogonal polynomials to construct new techniques for treating some problems in sciences, and engineering. A new important differentiation property of NSW in terms of NSW themselves is obtained and proved in this paper. Then it is utilized together with the state parameterization technique to find solution of optimal control problem (OCP) approximately. The suggested method converts the OCP into a quadratic programming problem, which can be easily determined on computer. As a result, the approximate solution closes with the exact solution even with a small number of NSW utilized in estimation. The error bound estimation for the proposed method is also discussed. Some test numerical examples are solved to demonstrate the applicability of the suggested method. For comparison, the exact known solutions against the obtained approximate results are listed in Tables.

We examine a family of linear partial differential equations both singularly perturbed in a complex parameter and singular in complex time at the origin. These equations entail forcing terms which combine polynomial and logarithmic type functions in time and that are bounded holomorphic on horizontal strips in one complex space variable. A set of sectorial holomorphic solutions are built up by means of complete and truncated Laplace transforms w.r.t time and parameter and Fourier inverse integral in space. Asymptotic expansions of these solutions relatively to time and parameter are investigated and two distinguished Gevrey type expansions in monomial and logarithmic scales are exhibited.

In the present paper, we introduced a quadratically convergent Newton’s like normal S2 iteration method free from the second derivative for the solution of nonlinear equations permitting 3 f'(x) = 0 at some points in the neighborhood of the root. Our proposed method works well 4 when the Newton method fails. Numerically it has been verified that the Newton’s like normal 5 S-iteration method converges faster than Fang et al. method [A cubically convergent Newton-type 6 method under weak conditions, J. Compute. and Appl. Math., 220 (2008), 409-412]. We studied 7 different aspects of normal S-iteration method. Lastly, fractal patterns support the numerical 8 results and explain the convergence, divergence, and stability of method.

We point out an inconsistency in Newton's equations of celestial mechanics. A set of differential equations implied by Newton's equations are shown to be free of this inconsistency. We then investigate the integrals of motion associated with this relative difference system.

Let G be a connected Lie group with Lie algebra g. This review is devoted to studying the fundamental dynamic properties of elements in the normalizer NG of G. Through an algebraic characterization of NG we analyze the different dynamics inside the normalizer. NG contains the well-known left-invariant vector fields and the linear and affine vector fields on G. In any case, we show the shape of the solutions of these ordinary differential equations on G. We give examples in low-dimensional Lie groups. It is worth saying that these dynamics generate the linear and bilinear control systems on Euclidean spaces and the invariant and linear control systems on Lie groups. Moreover, the Jouan Equivalence Theorem shows how to extend this theory to control systems on manifolds.

In this study, we consider a Lambert series whose coefficients are the sum of divisors function. Utilizing the Lambert series in the sequel we introduce a normalized linear operator JR_(α,β) (z) by applying the convolution with Rabotnov function. We then, acquire sufficient conditions for JR_(α,β) (z) to be Univalent, Starlike and Convex respectively. In each component of this study, we expand the derived results by applying two Robin's inequalities, one of which is equivalent to the Riemann hypothesis.

The purpose of the research is to investigate the unique solvability of a nonlocal boundary value problem for a loaded equation of parabolic-hyperbolic type in a special domain. Methods: Using representations of a general regular solution, the existence and uniqueness of the problem posed is proved. Results: A new method for proving the unique solvability of nonlocal boundary value problems for one loaded equation of mixed type in a special domain has been formed and developed. Conclusions: The obtained results make it possible to solve some problems of genetics, immunology, transonic gas dynamics and thermal physics, which are associated with displacement.

Stabilization of planar premixed flame front on a lattice (porous) burner is considered. It is rigorously mathematically proved that there exist two different stabilization regimes: one with flame front located nearby the surface of the burner, and another with flame front located inside the lattice. These two regimes result in qualitatively different gas temperature profiles along the flow, that is monotonic and non-monotonic, respectively. Boundary between the two regimes is described in terms of dependence of the lattice solid material temperature on flow Peclet number. Likewise temperature profiles, such dependencies may be both monotonic and non-monotonic. Transition between the two types of dependencies is controlled by Arrhenius number. Conclusions of the study are supported by numerical analysis. They also compare favourably with available experimental data. Novelty of the present approach is fundamentally rigorous analytical analysis of the problem.

In this study, we have considered for numerical solution of a Poisson equation in a domain. we have presented a first and second order finite difference method for solving the system of the intial boundary value. We have presented the numerical solution of the Poisson equations in a two-dimensional finite region. Numerical Solution and exact solution for the presented method has been compared and the figures presented. The present error analysis tables demonstrate the efficiency of the method.

This paper presents a new approach to a different proof of "Fermat's Last Theorem.” For the proof of the theorem, it is proposed to use a more straightforward geometrical approach. A special family of curves, the Elba curves, is introduced to facilitate the proof. This approach makes the proof easier to comprehend by mathematicians and those interested in the subject.

Teaching kinematic rotations is a daunting task for even some of the most advanced mathematical minds. However, envisioning and explaining the three-dimensional rotations can be highly simplified by changing the paradigm. This paradigm change allows for a high school student with a understanding of geometry to be able to not only develop the matrix but also explain the rotations at a collegiate level. The proposed method includes the assumption of a point (P) within the initial three-dimensional frame with axes (x ̂_i, y ̂_i, z ̂_i). It then utilizes a two-dimensional rotation view (2DRV) to measure how the coordinates of point P translate after the initial axis is rotated instead of using the established Euler's formula. The equations are used in matrix notation to develop a direction cosine matrix (DCM) for future equations. This method provides a high school student with an elementary and repeatable process to compose and explain kinematic rotations.

Here, we describe the bicomplex (p,q)- Fibonacci numbers and the bicomplex (p,q)- Fibonacci quaternions that are based on these numbers and give some of their equations, including the Binet formula, generating function, Catalan, Cassini, d’Ocagne’s identities, and some summation formulas for both of them. Finally, we create a matrix for bicomplex (p,q)- Fibonacci quaternions, and we obtain a determinant of a special matrix that gives the terms of that quaternion.

Missing observation is a common problem in scientific and industrial experiments, particularly in a small-scale experiment. They often present significant challenges when experiment repetition is infeasible. In this research, we propose a multi-objective genetic algorithm as a practical alternative for generating optimal mixture designs that remain robust in the face of missing observation. Our algorithm prioritizes designs that exhibit superior D-efficiency while maintaining a high minimum D-efficiency due to missing observation. The focus on D-efficiency stems from its ability to minimize the impact of missing observations on parameter estimates, ensure reliability across the experimental space, and maximize the utility of available data. We study problems with three mixture components where the experimental region is an irregularly shaped polyhedral within the simplex. Our designs have proven to be D-optimal designs, demonstrating exceptional performance in terms of D-efficiency and robustness against missing observations. We provide a well-distributed set of optimal designs derived from the Pareto front, enabling experimenters to select the most suitable design based on their priorities using the desirability function.

While the language model using the stop sign as an independent token has been widely used to decide when the model should stop, it may lead to the growth of vocabulary dimensions and further problems. Similarly, present research on game algorithms usually estimate stopping point related problems based on the evaluation of the winning rate. However, information redundancy may also exist in such models, thus increasing the training difficulty. Above two types of tasks (and similar autoregressive tasks) show a common problem of stopping point prediction. In this paper, we describe a design of separated model, trying to separate the complexity of stopping point prediction from the main task model, so that the information used for estimating stopping point can be reduced. On this basis, in order to verify the rationality of using separated model, we propose a model-free test method. It judges the separability of transformed data based on point difference and sequence difference metrics. In this way, it can predict the credibility of the separated model inference.

Calculus, as a starting point to everyone's study of mathematics, was developed along with classical mechanics. Though not rigorously formulated, the intuition for differential form and infinitesimal motion makes calculus not hard to understand. And hence it does not require great effort to grasp the principles of classical mechanics. However, the situation becomes quite different when it comes to the quantum world. Given a particle system, what of interest is always the physical quantity of this system, such as energy, position, momentum, entropy, etc. But it is no longer straightforward.

Since its original publication 1 in 1978, Lozi’s chaotic map has been thoroughly explored and continues to be. Hundreds of publications analyze its particular structure or apply its properties in many fields (electronic devices like memristor, A.I. with swarm intelligence, etc.). Several generalizations have been proposed, transforming the initial two-dimensional map into multidimensional one. However, they do not respect the original constraint that allows this map to be one of the few strictly hyperbolic: a constant Jacobian. In this paper we introduce a three-dimensional piece-wise linear extension respecting this constraint and we explore a special property never highlighted for chaotic mappings: the coexistence of thread-chaotic attractors (i.e., attractors which are formed by collection of lines) and sheet-chaotic attractors (i.e., attractors which are formed by collection of planes). This new 3-dimensional mapping can generate a large variety of chaotic and hyperchaotic attractors. We give five examples of such behavior in this article. In the first three examples, there is coexistence of thread and sheet-chaotic attractors. However, their shape are different and they are constituted by a different number of pieces. In the two last examples, the blow up of the attractors with respect to parameter a and b is highlighted.

In real-life scenario, there are many mathematical tools to handle the incomplete and imprecise data. One of them is the fuzzy approach. This article aims to contribute to the literature of fuzzy multi-objective dynamic programming issues involving the fuzzy objective functions. The piecewise quadratic fuzzy numbers characterize these fuzzy parameters. Some basic notions in the problem under the α-pareto optimal solution concept is redefined and analyzed to study the stability of the problem. Furthermore, a technique, named as enhancing decomposition approach, is presented for achieving a subset for the parametric space that contains the sameα-pareto optimal solution. For a better understanding and comprehension of the suggested concept, a numerical example is provided.

Based on the definition of double fuzzy relations, two types of new rough set models are constructed, which are multi-granulation double fuzzy rough sets called optimistic and pessimistic multi-granulation double fuzzy rough sets, and some of their properties were studied. In addition, we study the relationship between optimistic and pessimistic multi-granulation double fuzzy rough sets and double fuzzy rough sets.

Currently, Pythagorean fuzzy sets (PFSs) have been widely applied in various fields due to their substantial advantages in expressing and dealing with uncertainty. However, measuring the similarity and difference between PFSs effectively remains an unresolved issue. Inspired by Tanimoto similarity, we propose a novel set of similarity and distance measures for PFSs. We delve into the theoretical properties of the proposed measures and compare it with existing PFSs measures. Numerous numerical examples validate their rationality and effectiveness. Furthermore, our experimental findings suggest that in contrast to existing measures, the introduced measures successfully circumvent various counter-intuitive issues encountered by current measures, and yield more pronounced outcomes in the discrimination of different fuzzy sets. This enhances the uniqueness and superiority of our measures. Finally, we developed two decision models based on the proposed measures and validated their applicability in three applications.

In this compact and focused paper, we revisit the interplay between Bramble and Filter through the lens of a submodular partition function.

The paper describes the use of topological indices in conjunction with high cholesterol drugs, specifically Fibrates, to predict their physicochemical properties and biological activities. Fibrates are known to lower high triglycerides, increase HDL cholesterol, and reduce the small dense fraction of LDL cholesterol. The study uses a quantitative structural-property relationships (QSPR) approach, which involves analyzing the relationships between physicochemical properties and topological indices using curvilinear regression. The QSPR model predicts the physicochemical properties of the drugs based on degrees and distances determined from topological indices. The study also conducted (DFT) calculations at the B3LYP/6-31G(d,p) level on the four investigated derivatives to gain insights into their optimized geometries, DOS plots, HOMO and LUMO orbital energies, and distribution. The theoretical results presented in the study suggest that the use of topological indices in QSPR models could provide a powerful tool for predicting the physicochemical properties and biological activities of molecules, including drugs. These findings could lead to the development of new cholesterol-lowering drugs with desirable properties.

In this article, an iterative method of two step is proposed for finding all roots simultaneously of polynomial equations. The order of convergence of the proposed algorithm is 2m, by using any iterative scheme of order m. Numerical tests are performed to confirm the theoretical results and to compare the proposed scheme with existing methods for finding all roots simultaneously.

Proxy re-encryption (PRE) is a momentous and widely used cryptographic technique. It enables a proxy to forward ciphertext without the need of decryption. PRE has received significant interest in applications like cloud computing, blockchain, and the Internet of Things. Despite its wide range of uses, PRE has also been subject to new security and privacy regulations. In PKC’19, Cohen $et~al.$~first drew attention to the weakness in PRE's security against chosen-plaintext attacks (CPA) and put up a more stringent security concept known as security against honest re-encryption attacks (HRA). Notably, Cohen provides a beneficial conclusion as well. It is proveded that PRE schemes with re-encryption simulatability property can be elevated from CPA to HRA security. It is also proved that CPA-secure PRE schemes with re-encryption simulatability property can be directly elevated to those satisfying HRA security. However, those PRE schemes with re-encryption simulatability are almost always based on pairings. In this study, to the best of our knowledge, we directly construct HRA-secure PRE with re-encryption simulatability for the first time based on the learning with errors (LWE) assumption, which was widely believed to be quantum-resistant. Based on the re-encryption key generation algorithm and the re-encryption algorithm construction method of the above scheme, we can modify attribute-based conditional proxy re-encryption (AB-CPRE) as well as the corresponding attribute-based proxy re-encryption (AB-PRE) algorithm to make them have re-encryption simulatability properties. Finally, by using this property, We boost the security of AB-CPRE scheme of ESORICS'21 from CPA to HRA and simplify the HRA-security proof for the AB-PRE of ESORICS'21.

We discuss two queueing-inventory systems with catastrophes in the warehouse. Catastrophes occur according to Poisson process and upon arrival of a catastrophe all inventory in the system is instantly destroyed. But consumer customers in the system (in the server or in the buffer) continue still waiting for the replenishment of the stock. The arrivals of the consumer customers follow a Markovian Arrival Process (MAP) and they can be queued in an infinite buffer. Service time of a consumer customer follows a phase-type distribution. The system receives negative customers whose have Poisson flows to service facility and upon arrival of a negative customer one consumer customer is pushed out from the system, if any. One of two replenishment policies can be used in the system: either (s,S) or (s,Q). If upon arrival of the consumer customer, the inventory level is zero, then according to the Bernoulli scheme, this customer is either lost (lost sale scheme) or join the queue (backorder sale scheme). The system is formulated by a four-dimensional continuous-time Markov chain. Steady state distribution is obtained using the matrix-geometric method. A comprehensive numerical study is performed on the performance measures under various replenishment policies. Finally, an optimization study is presented.

In an $m$-person symmetric game, all players are identical and indistinguishable. In this paper, we find that the payoff tensor of the player $k$ in an $m$-person symmetric game is $k$-mode symmetric, and the payoff tensors of two different individuals are the transpose of each other. Furthermore, we reformulate the $m$-person symmetric game as a tensor complementary problem and demonstrate that locating a symmetric Nash equilibrium is equivalent to finding a solution to the resulting tensor complementary problem. Finally, we use the hyperplane projection algorithm to solve the resulting tensor complementary problem, and we present some numerical results to find the symmetric Nash equilibrium.

We formulate a more straightforward, symmetry-based techniques in manually computing the determinant of any n×n matrix by revisiting Dodgson’s condensation method and strategically applying elementary row operations and the definition and properties of determinants. The result yields a more simplified mnemonical algorithm than the usual reduction by elementary row operations and co-factor expansion and with minimal terminal divisions performed compared to Dodgson’s internal divisions.

In this work, we consider two SIR patterns with demography: the classical pattern and a modified pattern with a linear transmission coefficient of the infection. By reformulating of each first order differential systems as a system with two second-order differential equations, we investigate the nonlinear dynamics of the system from the Jacobi stability point of view by using the KCC geometric theory. We will study the intrinsic geometric properties of the systems by determining the geometric associated objects: the zero-connection curvature tensor, the nonlinear connection, the Berwald connection, and the five KCC invariants: the first invariant - the external force εi, the second invariant - the deviation curvature tensor Pji, the third invariant - the torsion tensor Pjki, the fourth invariant - the Riemann-Christoffel curvature tensor Pjkli, and the fifth invariant - the Douglas tensor Djkli. In order to obtain necessary and sufficient conditions for the Jacobi stability near the equilibrium points, the deviation curvature tensor will be determined at each equilibrium points. Furthermore, we will compare the Jacobi stability with the classical linear stability, inclusive by diagrams related to the values of parameters of the system.

In this paper, we will extend the expected value of the function w.r.t the uniform probability measure on sets measurable in the Caratheodory sense to be finite for a larger class of functions, since the set of all measurable functions with infinite or undefined expected values may form a prevalent subset of the set of all measurable functions. This means "almost all" measurable functions have infinite or undefined expected values. Before we define the specific problem in section 2, with a unique solution that allows "more" functions to have finite expected values, we'll outline some preliminary definitions. We'll then define the specific problem in section 2 (with a partial solution in section 3) to visualize the complete solution to the problem. Along the way, we will ask a series of questions to clarify our understanding of the paper.

The paper considers coherent states for the representation of Weyl commutation relations over a field of $p$-adic numbers. A geometric object, a lattice in vector space over a field of p-adic numbers, corresponds to the family of coherent states. It is proved that the bases of coherent states corresponding to different lattices are mutually unbiased, and the operators defining the quantization of symplectic dynamics are Hadamard operators.

This paper focuses on robust multi-criteria traffic network equilibrium problems with path capacity constraints. The concepts of robust vector equilibrium and robust vector equilibrium with respect to worst case are introduced, respectively. To generate a subset of robust vector equilibrium, an equivalent min-max optimization problem is constructed. In addition, a direct search algorithm, in which the step size without derivatives and redundant parameters, for solving this min-max problem is proposed. To find a subset of robust vector equilibrium with respect to worst case, an equivalent smoothing optimization problem based on a variant version of ReLU activation function is established. Numerical experiments are also presented to expound the advantages of the algorithms.

Nonnegative matrix factorization (NMF) has been shown to be a strong data representation technique, with applications in text mining, pattern recognition, image processing, clustering and other fields. In this paper, we propose a hypergraph regularized Lp smooth nonnegative matrix factorization (HGSNMF), by incorporating hypergraph regularization and Lp smoothing constraint terms into the standard NMF. The hypergraph regularization term can capture the intrinsic geometry structure of the high dimension space data more comprehensively than simple graph, the Lp smoothing constraint may yield a smooth and more accurate solution to the optimization problem. The updating rules are given using multiplicative update techniques, and the convergence of HGSNMF is theoretically investigated. The experimental results on four different data sets show that the proposed method has better clustering effect than the related state-of-the-art methods in the vast majority of cases.

The notion of oriented set is the basic elementary concept of the theory of changeable sets. The main motivation for the introduction of changeable sets was the sixth Hilbert problem, that is, the problem of mathematically rigorous formulation of the fundamentals of theoretical physics. In the present paper the necessary and sufficient condition of the existence of one-point time on an oriented set is established. From the intuitive point of view, one-point time is the time associated with the evolution of a system consisting of only one object (for example, from one material point). Namely, it is proven that the one-point time exists on the oriented set if and only if this oriented set is a quasi-chain. Also, using the obtained result, the problem of describing all possible images of linearly ordered sets is solved. This problem naturally arises in the theory of ordered sets.

This work aims to introduce a novel concept: the Riesz-Dunkl fractional derivatives, within the context of Dunkl type operators. A particularly noteworthy revelation is that when a specific parameter $\kappa$ equals zero, the Riesz-Dunkl fractional derivative smoothly reduces to both the well-known Riesz fractional derivative and the fractional second-order derivative. Furthermore, we introduce a new concept: the fractional Sobolev space. This space is defined and characterized using the versatile framework of the Dunkl transform.

The article outlines a creative approach by fusing ideas from fuzzy contractions, graph mappings, and Kannan mappings to create a brand-new notion known as "Kannan-graph-fuzzy contraction." This combination of concepts makes it possible to create a unique framework for resolving real-world issues controlled by nonlinear models. It is anticipated that using Kannan-graph-fuzzy contractions will offer a fresh approach to discussing solution existence in situations where these ideas overlap. This strategy might have a significant impact on a lot of real-world applications, proving how practical multidisciplinary techniques are for tackling issues in the real world originating from technology and engineering. The article includes several illustrated examples based on computer simulations, making the results more explicit.

In this paper, we consider a class of bilevel programming problems (abbreviated as, BLPP). Exploiting the generalized approximate convexity assumptions, we investigate the relations among the solutions of approximate Minty (respectively, Stampacchia) type variational inequalities (abbreviated as, AMTVI (respectively, ASTVI)), and the local ϵ-quasi solutions of the BLPP. Moreover, by employing the generalized Knaster–Kuratowski–Mazurkiewicz (abbreviated as, KKM)-Fan’s lemma, we derive some existence results for the solutions of approximate variational inequalities (abbreviated as, AVI), namely, AMTVI and ASTVI. A non-trivial example is given to highlight the importance of the established results. To the best of our knowledge, there is no research paper available in the literature that establishes relationships between the AVI and the BLPP under the assumptions of generalized approximate convexity in terms of limiting subdifferentials.

This paper proposed an optimal tracking control scheme through adaptive dynamic programming(ADP) for a class of partially unknown discrete-time nonlinear systems based on radial basis function neural network(RBF-NN). In order to acquire the unknown system dynamics, we use two RBF-NNs, the one is used to construct the identifier, and the another is used to directly approximate the steady-state control input, where a novel adaptive law is proposed to update neural network weights. While the optimal feedback control and the cost function are derived via feedforward neural networks approximating, it is proposed to regulate the tracking error, the critic network and the actor network are then trained online to obtain the solution of the associated Hamilton–Jacobi–Bellman (HJB) equation being built under the ADP framework. Simulations verify the effectiveness of the optimal tracking control technique using the neural networks.

In different fields in our life, like physics, coding theory and computer science, graph labeling dramas an vital role and appears in many applications. A labeling of a graph is a map which assign each element in with a positive integer number. An edge irregular total -labeling is a function such that where and are weights for any two distinct edges. In this case, has total edge irregularity strength (TEIS) if is minimum. In our paper, we defined a new type of graphs called a triple star snake graph and star snake graph . Also, we investigated TEIS for a triple star snake graph . After that, we generalized the results for star snake graph .

We obtain some generalised Minkowski-type integral formulas for compact Riemannian (resp. spacelike) hypersurfaces in Riemannian (resp. Lorentzian) manifolds admitting an arbitrary vector field that we assume to be timelike in the case where the ambient space is Lorentzian. Some of these formulas generalize existing formulas in the case of conformal and Killing vector fields. We apply these integral formulas to obtain interesting results concerning the characterization of such hypersurfaces in some particular cases such as when the ambient space is Einstein admitting an arbitrary (in particular, conformal or Killing) vector field, and when the hypersurface has constant mean curvature.

This work uses a new approach to investigate the Riemann hypothesis, drawing conclusions about its trueness. It is based on the more general method presented in a very recent publication, by the same author, showing in detail how to approach that very well-known and interesting problem. This is achieved by means of the theory of dynamical systems (Poincaré index associated to equilibria of 2-dimensional systems) and the study of the zeros of the Dirichlet eta function, defined by a Dirichlet series. By using the well-known fact that the zeros of the eta function include all zeros of the Riemann zeta function in the (open) critical strip, excluded the critical line, ((0,1/2) ∪ (1/2,1)) × (−∞,∞), the development proceeds using only the eta function. In addition, the open and simply connected region (1/2,1) × (0,∞) is used along the text, taking into account the symmetries of zeros of the functions under analysis in the critical strip. The basic line of proof is to find the mathematical expression for the Poincaré index of the vector field associated to the eta function, assuming the existence of a zero of the eta function outside the critical line (in (1/2,1) × (0,∞)), and investigating the resulting unfoldings. Eventually, an inconsistency occurs and the proof ends by contradiction.

The aim of this paper is to deep into the dynamic study of the well known Chebyshev-Halley family of iterative methods for solving nonlinear equations. Our objetives are twofold: one the one hand, we are interested in characterize the existence of extraneous attracting fixed points when the methods in the family are applied to polynomial equations. On the other hand, we are also interested in study the free critical points of the methods in the family, as a previous step to determine the existence of attracting cycles. In both cases, we want to identify situations where the methods in the family have a bad behavior from the root-finding point of view. Finally, and joining these two studies, we look for polynomials for which there are methods in the family where this two situations happen simultaneously. That is, the rational map obtained by applying a method in the Chebyshev-Halley family to a polynomial has both super-attracting extraneous fixed point and super-attracting cycles different to the roots of the polynomial.

This article introduces a new fractional approach to the concept of information dimension of complex networks, based on a (q,q′)-entropy proposed in the literature. The q parameter measures how far is the number of subsystems (for a given size ε) from the mean number of overall sizes. The q′ (interaction index) measures when the interactions between subsystems are greater (q′>1), lesser (q′<1) or equal to the interactions into these subsystems. The computations of the proposed information dimension are carried out on several real-world and synthetic complex networks. The results from the proposed information dimension are compared with those from the information dimension based on the Shannon entropy.

We firstly present two popular space discretization models of the Nonlinear Schrödinger Equation in dark solitons motion : the Direct-Discrete model and the Ablowitz-Ladik model. Applying the midpoint scheme to the space discretization models, we get two time-space discretization models: the Crank-Nicolson method and the New-Difference method. Secondly, we demonstrate that the solutions of the two space discretization models converge to the solution of the Nonlinear Schrödinger Equation. Also, we prove that the convergence order of the two time-space discretization models are O(h2+τ2) in discrete L2-norm error estimates. Finally, the numerical experiments agree well with the proven theoretical results.

In this paper, we will use the maximum likelihood estimation (MLE) and the Bayes methods to perform estimation procedures for the reliability of stress-strength ℜ=P[X>Y] based on independent adaptive progressive censored samples that were taken from the Chen distributions. An approximate confidence interval of ℜ is constructed using a variety of classical techniques, such as the normal approximation of the MLE, the normal approximation of the log-transformed MLE, and the percentile bootstrap (Boot-p) procedure. The Bayesian estimation of ℜ is obtained based on the balanced loss function, which comes in two versions here, the symmetric balanced squared error (BSE) loss function and the asymmetric balanced linear exponential (BLINEX) loss function. When estimating ℜ using the Bayesian approach, all the unknown parameters of the Chen distributions are assumed to be independently distributed and to have informative gamma priors. Further, a mixture of Gibbs sampling algorithm and Metropolis-Hastings algorithm is used to compute the Bayes estimate of ℜ and the associated highest posterior density credible interval. In the end, simulation research are used to assess the general overall performance of the proposed estimators and a real data set is provided to exemplify the theoretical results.

Matrix-variate Gaussian type or Wishart type distributions in the real domain are widely used in the literature. When the exponential trace has an arbitrary power and when a factor involving a determinant enters into the model or a matrix-variate gamma type or Wishart type model with exponential trace having an arbitrary power, is extremely difficult to handle. Evaluation of the normalizing constant in such a model is the most important part because when studying the properties of such a model, the method used in the evaluation of the normalizing constant will be the relevant steps in all the computations involved. One such model with a factor involving a trace and the exponential trace having an arbitrary power, in the real domain, is known in the literature as Kotz' model. No explicit evaluation of the normalizing constant in the model involving trace with an exponent and determinant with an exponent entering into the model and at the same time the exponential trace having an arbitrary exponent seems to be available in the literature. The normalizing constant widely used in the literature and interpreted as the normalizing constant in the general model and refers to as a Kotz' model does not seem to be correct. Corresponding model in the complex domain, with the correct normalizing constant, does not seem to be available in the literature. One of the main contributions in this paper is the matrix-variate distributions in the complex domain belonging to Gaussian type, gamma type, type-1 and type-2 beta type when the exponential trace has an arbitrary power. All these models are believed to be new. A second main contribution is the explicit evaluation of the the normalizing constants, in the real and complex domains especially in the complex domain, in a matrix-variate model involving a determinant and a trace as multiplicative factors and at the same time the exponential trace having an arbitrary power. Another main contribution is the introduction of matrix-variate models with exponential trace having an arbitrary exponent, in the categories of type-1 beta, type-2 beta and gamma distributions or in the family of Mathai's pathway models [1], both in the real and complex domains. Another new contribution is the logistic-based extensions of models in the real and complex domains with exponential trace having an arbitrary exponent and connecting to extended zeta functions introduced by this author recently. Some properties of such models are indicated but not derived in detail in order to limit the size of the paper. The techniques and steps used at various stages in this paper will be highly useful for people working in multivariate statistical analysis as well as people applying such models in engineering problems, communication theory, quantum physics and related areas, apart from statistical applications.

The paper discusses the classification of fuzzy metrics based on their continuity conditions, dividing them into Erceg, Deng, Yang-Shi, and Chen metrics. It explores the relationships between these types of fuzzy metrics, concluding that a Deng metric in [0,1]-topology must also be Erceg, Chen, and Yang-Shi metrics. The paper also proves that the product of countably many Deng pseudo-metric spaces remains a Deng pseudo-metric space, and demonstrates some σ-locally finite properties of Deng metric space. Additionally, the paper constructs two interrelated mappings based on normal space and concludes that if a [0,1]-topological space is T1 and regular, and its topology has a σ-locally finite base, then it is Deng metrizable, and thus Erceg, Yang-Shi, and Chen metrizable as well.

In this study, we first discover some new concept coordinated UD-convex mappings with fuzzy-number values. After that, we look into Hermite-Hadamard type inequalities via fuzzy-number-valued coordinated UD-convex fuzzy-number-valued mapping (coordinated UD-convex FNVM). In the case of coordinated UD-convex FNVM, novel conclusions are derived by making particular decisions in recently proven inequalities. Additionally, it is demonstrated that the recently discovered inequalities are expansions of comparable findings in the literature. It is important to note that the main outcomes are validated by nontrivial examples.

Recently, Gohar et al. introduced a novel, local, and well-behaved fractional calculus. It possesses all the classical properties, and Its locality imposes simplicity and accuracy in modeling fractional order systems. In this article, we further develop the definitions and extend the classical properties of Gohar fractional calculus to address some of the open problems in Calculus. The fractional Gronwall's integral inequality, Taylor power series expansion, and Laplace transform are defined and applied to overcome some of the limitations in the classical integer-order calculus. The fractional Laplace transform is applied to solve Bernoulli-type logistic and Bertalanffy nonlinear fractional differential equations, and the criteria under which it can be applied to solve linear differential equations are investigated.

In the paper, we develop some numerical schemes including finite difference scheme and finite volume scheme for the fractional Laplacian operator, and apply the resulted numerical schemes to solve some fractional diffusion equations. First, the fractional Laplacian operator can be characterized as the weak singular integral by a integral operator with zero boundary condition. Second, because of the solution of fractional diffusion equation is usually singular near the boundary, we apply finite difference scheme to discrete fractional Laplacian operator and fractional diffusion equation by fractional interpolation function. Moreover, it is find that the differential matrix of the above scheme is symmetric matrix, and strictly row-wise diagonally dominant in special fractional interpolation function. Third, we show finite volume scheme to discrete fractional diffusion equation by fractional interpolation function, and analyze the properties of differential matrix. Finally, the numerical experiments are given, and verify the correctness of theoretical results and the efficiency of the schemes.

Fractional differential systems have received much attention in recent decades likely due to its powerful ability in modeling memory processes, which are mostly observed in real world. Fractional models have been widely investigated and applied in many fields such as physics, biochemistry, electrical engineering, continuum and statistical mechanics. It is shown by many researchers that fractional derivatives can provide more accurate models than integer order derivatives do. Obviously, bang-bang property is of a significant importance in optimal control theory. In this paper a Bang-Bang property and time-optimal control problem for time-Fractional Differential System (FDS) is under consideration. First we formulate our problem and we prove the existence theorem. We state and prove the bang-bang theorem. Finally we give the optimality conditions that characterized the optimal control. Some illustrate applications are given to clarify our results.

In this paper, we establish the necessary and sufficient conditions for solving a dual split quaternion matrix equation AXB=C, and present the general solution expression when solvability is achieved. As an application, we delve into the necessary and sufficient condition for the existence of Hermitian solution to this equation by using a newly defined real representation method. Furthermore, we obtain the solutions for the dual split quaternion matrix equations AX=C and XB=C. Finally, we provide a numerical example to demonstrate the findings of this paper.

The study aimed to investigate the impact of incorporating deep learning strategies in mathematics instruction on students' problem-solving abilities, critical thinking skills, academic achievement, and mathematical skills development. The study population consisted of 280 students, with a sample size of 162 participants. Employing a purely quantitative approach, data collection utilized questionnaires and achievement tests as instruments. Data analysis involved simple linear regression, descriptive analysis, and Pearson correlation. The results of the study revealed that there was no significant enhancement observed in students' problem-solving abilities and critical thinking skills when deep learning strategies were incorporated into mathematics instruction. Additionally, students' attitudes towards deep learning activities in mathematics showed no effect on their academic achievement. Furthermore, students' engagement levels in deep learning activities did not impact their development of mathematical skills. Lastly, the frequency of deep learning activities in the mathematics classroom did not show a significant relationship with students' academic performance. These findings suggest that the implementation of deep learning strategies in mathematics instruction did not yield notable improvements in the specified areas based on the study outcomes.

We consider two abstract systems of parabolic-hyperbolic type that model thermoelastic problems. We study the influence of the physical constants and the initial data on the nonexistence of global solutions that in our framework are produced by the blow-up in finite time of the norm of the solution in the phase space. We employ a differential inequality to find sufficient conditions that produce the blow-up. To that end, we construct a set that is positive invariant for any positive value of the initial energy. As a result we found that the coupling with the parabolic equation stabilizes the system, as well as the damping term in the hyperbolic equation. Moreover, for any pair of positive values $(\xi,\epsilon)$, there exist initial data such that the corresponding solution with initial energy $\xi$ blows-up at a finite time less than $\epsilon$. Our purpose is to improve results previously published in the literature.

In this paper, we give the Galois adjunction form of the regular relation, and by using residuated mapping, we find the connection between it and order-generating set. Furthermore, we obtain the equivalent conclusion of "regular". In addition, by using the specialization order of topological space and the lower approximate operator in the rough set, we find a continuous homeomorphic mapping from a regular space to a Alexandrov topological space

Fermatean fuzzy sets (FFSs) are a versatile tool for handling uncertain problems and have shown great effectiveness in practical applications. However, many existing Fermatean fuzzy similarity measures exhibit counter-intuitive situations, making it challenging to accurately measure the similarity or difference between FFSs. To address this issue, several similarity and distance measures for FFSs are proposed, inspired by the Tanimoto similarity measure. The properties of the proposed measures are also explored, along with several comparative examples with existing measures for FFSs, which illustrate their superior performance in processing fuzzy information from FFSs. The proposed measures have effectively overcome the counter-intuitive challenges posed by many existing measures and significantly outperforms existing measures in differentiating different FFSs. Furthermore, we demonstrate the practical applicability of the proposed measures in solving pattern recognition, medical diagnosis and multiple-attribute decision-making problems.

It has been more than a century since Erlang published his paper in 1909. The classical queueing system for the M/D/c queue has been investigated by many authors. The key is to find the roots of an equation using explicit or numerical methods. We present an alternate form of explicit solution(s) of this/these root(s) using the Lambert-W function in this paper. In addition, the unsaturated probabilities of the number of customers in the system is derived explicitly in terms of this/these root(s). Finally, the mean number of customers in the system and the mean queueing delay are obtained from these probabilities. The exact queueing delay formula for the M/D/c queue is expressed in closed-form in terms of the root(s). Numerical results show that they are in excellent agreement with the numerical results published by Seelen, Tijms and Van Hoorn.

Recently, Li et al [1] investigated the nonlinear Langevin equations existence including two fractional orders with infinite-point boundary conditions. It has shown that the outcomes given it does count on the solution form for which it has boundary values. It means that their solution needs more to be in the final step. In current work, we will have discussion about the existence and uniqueness for the same boundary conditions by investigating the closely explicit solution form that has no boundary values. Numerical example is supported to have more vision to compare the between new and previous results. It turns out that our results are superior.

The duality is one of the most interesting properties of the Laplace and Fourier transforms associated to the integer order derivative. Here, we will generalize it for fractional derivatives and extend the results to the Mellin, Z and discrete-time Fourier transforms. The scale and nabla derivatives are used.

. Interconnected control systems (ICS) widely apply to various technical systems. ICS has a complex description. Therefore, simplified models use, which the specifics of an object do not always reflect. So, the problem of synthesizing mathematical models is topical. The work purpose is to develop an approach to obtaining models in conditions of incomplete a priori information. We use an adaptive approach. As examples, two-channel systems (TCS) with cross-links and identical channels consider, and a method for mathematical model design. We consider the system with asymmetric cross-links and evaluate their impact on properties of an adaptive identification system. Approach proposed to assessing the TCS identifiability with cross-links. The excitation constancy influence on TСS parameter estimates is studied. A generalization of the proposed approach is given for an interconnected system.

Quantum logic is a well-structured theory, which has recently received some attention because of its fundamental relation to quantum computing. However, the complex foundation of quantum logic borrowing concepts from different branches of mathematics as well as its peculiar settings have made it a non-trivial task to device suitable applications. This article aims to propose for the first time an approach to use quantum logic in image processing at hand of a process for shape classification. We show how to make use of the principal component analysis to realize quantum logical propositions. In this way we are able to assign a concrete meaning to the rather abstract quantum logical concepts, and we are able to compute a probability measure from the principal components. For shape classification we consider encrypting given point clouds of different objects by making use of specific distance histograms. This enables to initiate the principal component analysis. At hand of experiments, we explore the possibility to distinguish between different geometrical objects and discuss the results in terms of quantum logical interpretation.

Real and complex analytic functions are largely studied in the field of complex analysis and are seen as very useful tools in solving problems in mathematics, physics and engineering. Real and complex numbers are a subset of semi-structured complex numbers (a new number set created to algebraically solve division by zero). Nevertheless, the properties of analytic functions made up of semi-structured complex variables (called semi-structured complex analytic functions) is yet to be explored. This limits the range of possible problems that can be resolved using analytic functions). In this regard, the aim of this paper was to expound upon the properties of semi-structured complex analytic functions and show their application in solving engineering problems. The results of this paper included (1) developing a full set of Cauchy–Riemann Equations for the semi-structured complex -space; (2) define sufficient and necessary conditions for a semi-structured complex function to be analytic; (3) determine the relationship between semi-structured complex analytic functions, Laplace’s Equations and Poisson’s Equations; and (5) provide an example of the use of these functions.

Extension of a derivation on a universally reversible JC-algebra A⊆B(H)_{sa} to the C*-algebra [A] generated by A in B(H) was studied by Upmier [20, Theorem 2.5]. In this article we study the extension of a Jordan derivation on a universally JC-algebra A to its universal enveloping real and complex C*-algebras R and U, respectively. Also, we establish the relationship between local derivations (resp., 2-local derivations, weak local derivations, weak-2-local derivations) of a universally JC-algebra A and the corresponding maps on its universal enveloping real and complex C*-algebras R and A, respectively.

A novel theoem is presented about a circle with four antipodal points, arbitrarily ordered counterclockwise (or clockwise), in a Euclidean plane.