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This comment is devoted to the paper “Lie Symmetry Analysis, Explicit Solutions, and Conservation Laws of a Spatially Two-Dimensional Burgers–Huxley Equation” (Symmery, 2020, vol.12, 170), in which several results are either incorrect, or incomplete, or misleading.

The class of log-elliptical distributions is well used and studied in risk measurement and actuarial science. The reason is that risks are often skewed and positive when they describe pure risks, i.e., risks in which there is no possibility of profit. In practice, risk managers confront a system of mutually dependent risks, not only one risk. Thus, it is important to measure risks while capturing their dependence structure. In this short paper, we compute the multivariate risk measures, multivariate tail conditional expectation, and multivariate tail covariance measure for the family of log-elliptical distributions, which captures the dependence structure of the risks while focusing on the tail of their distributions, i.e., on extreme loss events. We then study our result and examine special cases, as well as the optimal portfolio selection using such measures. Finally, we show how the given multivariate tail moments can also be computed for log-skew elliptical models based on similar approaches given for the log-elliptical case.

In order to solve the Sylvester equations more efficiently, a new four parameters positive and skew-Hermitian splitting (FPPSS) iterative method is proposed in this paper based on the previous research of the positive and skew-Hermitian splitting (PSS) iterative method. We prove that when coefficient matrix A and B satisfy certain conditions, the FPPSS iterative method is convergent in the parameter&rsquo, s value region. The numerical experiment results show that compared with previous iterative method, the FPPSS iterative method is more effective in terms of iteration number IT and runtime.

This paper discusses and provides some analytical studies for a modified fractional-order SIRD mathematical model of the COVID-19 epidemic in the sense of the Caputo–Katugampola fractional derivative that allows treating of the biological models of infectious diseases and unifies the Hadamard and Caputo fractional derivatives into a single form. By considering the vaccine parameter of the suspected population, we compute and derive several stability results based on some symmetrical parameters that satisfy some conditions that prevent the pandemic. The paper also investigates the problem of the existence and uniqueness of solutions for the modified SIRD model. It does so by applying the properties of Schauder’s and Banach’s fixed point theorems.

In this paper, we consider two single server queueing systems to which customers of two distinct priorities (P1 and P2) arrive according to a Marked Markovian arrival process (MMAP). They are served according to two distinct phase type distributions. The probability of a P1 customer to feedback is θ on completion of his service. The feedback (P1) customers, as well as P2 customers, join the low priority queue. Low priority (P2) customers are taken for service from the head of the line whenever the P1 queue is found to be empty at the service completion epoch. We assume a finite waiting space for P1 customers and infinite waiting space for P2 customers. Two models are discussed in this paper. In model I, we assume that the service of P2 customers is according to a non-preemptive service discipline and in model II, the P2 customers service follow a preemptive policy. No feedback is permitted to customers in the P2 line. In the steady state these two models are compared through numerical experiments which reveal their respective performance characteristics.

In some previous works, two of the authors introduced a technique to design high-order numerical methods for one-dimensional balance laws that preserve all their stationary solutions. The basis of these methods is a well-balanced reconstruction operator. Moreover, they introduced a procedure to modify any standard reconstruction operator, like MUSCL, ENO, CWENO, etc., in order to be well-balanced. This strategy involves a non-linear problem at every cell at every time step that consists in finding the stationary solution whose average is the given cell value. In a recent paper, a fully well-balanced method is presented where the non-linear problems to be solved in the reconstruction procedure are interpreted as control problems. The goal of this paper is to introduce a new technique to solve these local non-linear problems based on the application of the collocation RK methods. Special care is put to analyze the effects of computing the averages and the source terms using quadrature formulas. A general technique which allows us to deal with resonant problems is also introduced. To check the efficiency of the methods and their well-balance property, they have been applied to a number of tests, ranging from easy academic systems of balance laws consisting of Burgers equation with some non-linear source terms to the shallow water equations—without and with Manning friction—or Euler equations of gas dynamics with gravity effects.

The purpose of image registration is to find the symmetry between the reference image and the image to be registered. In order to improve the registration effect of unmanned aerial vehicle (UAV) remote sensing imagery with a special texture background, this paper proposes an improved scale-invariant feature transform (SIFT) algorithm by combining image color and exposure information based on adaptive quantization strategy (AQCE-SIFT). By using the color and exposure information of the image, this method can enhance the contrast between the textures of the image with a special texture background, which allows easier feature extraction. The algorithm descriptor was constructed through an adaptive quantization strategy, so that remote sensing images with large geometric distortion or affine changes have a higher correct matching rate during registration. The experimental results showed that the AQCE-SIFT algorithm proposed in this paper was more reasonable in the distribution of the extracted feature points compared with the traditional SIFT algorithm. In the case of 0 degree, 30 degree, and 60 degree image geometric distortion, when the remote sensing image had a texture scarcity region, the number of matching points increased by 21.3%, 45.5%, and 28.6%, respectively and the correct matching rate increased by 0%, 6.0%, and 52.4%, respectively. When the remote sensing image had a large number of similar repetitive regions of texture, the number of matching points increased by 30.4%, 30.9%, and −11.1%, respectively and the correct matching rate increased by 1.2%, 0.8%, and 20.8% respectively. When processing remote sensing images with special texture backgrounds, the AQCE-SIFT algorithm also has more advantages than the existing common algorithms such as color SIFT (CSIFT), gradient location and orientation histogram (GLOH), and speeded-up robust features (SURF) in searching for the symmetry of features between images.

This paper explains a method of constructing algebras, starting with the properties of discrimination in elementary discrete systems. We show how to use points of view about these systems to construct what we call iterant algebras and how these algebras naturally give rise to the complex numbers, Clifford algebras and matrix algebras. The paper discusses the structure of the Schrödinger equation, the Dirac equation and the Majorana Dirac equations, finding solutions via the nilpotent method initiated by Peter Rowlands.

In recent years, crowded stampede incidents have occurred frequently, resulting in more and more serious losses. The common cause of such incidents is that when large-scale populations gather in a limited area, the population is highly unstable. In emergency situations, only when the crowd reaches the safe exit as soon as possible within a limited evacuation time to complete evacuation can the loss and casualties be effectively reduced. Therefore, the safety evacuation management of people in public places in emergencies has become a hot topic in the field of public security. Based on the analysis of the factors affecting the crowd path selection, this paper proposes an improved path-planning algorithm based on BEME (Balanced Evacuation for Multiple Exits). And pedestrian evacuation simulation is carried out in multi-exit symmetrical facilities. First, this paper optimizes the update method of the GSDL list in the BEME algorithm as the basis for evacuating pedestrians to choose an exit. Second, the collision between pedestrians is solved by defining the movement rule and collision avoidance strategy. Finally, the algorithm is compared with BEME and traditional path-planning algorithms. The results show that the algorithm can further shorten the global evacuation distance of the symmetrical evacuation scene, effectively balance the number of pedestrians at each exit and reduce the evacuation time. In addition, this improved algorithm uses a collision avoidance strategy to solve the collision and congestion problems in path planning, which helps to maximize evacuation efficiency. Whether the setting of the scene or the setting of the exit, all studies are based on symmetric implementation. This is more in line with the crowd evacuation in the real scene, making the experimental results more meaningful.

This paper is devoted to the study of the quadruple Roman domination in trees, and it is a contribution to the Special Issue “Theoretical computer science and discrete mathematics” of Symmetry. For any positive integer k, a [k]-Roman dominating function ([k]-RDF) of a simple graph G is a function from the vertex set V of G to the set {0,1,2,…,k+1} if for any vertex u∈V with f(u)&lt, k, ∑x∈N(u)∪{u}f(x)≥|{x∈N(u):f(x)≥1}|+k, where N(u) is the open neighborhood of u. The weight of a [k]-RDF is the value Σv∈Vf(v). The minimum weight of a [k]-RDF is called the [k]-Roman domination number γ[kR](G) of G. In this paper, we establish sharp upper and lower bounds on γ[4R](T) for nontrivial trees T and characterize extremal trees.

This paper deals with symmetrical data that can be modelled based on Gaussian distribution, such as linear mixed models for longitudinal data. The latent factor linear mixed model (LFLMM) is a method generally used for analysing changes in high-dimensional longitudinal data. It is usual that the model estimates are based on the expectation-maximization (EM) algorithm, but unfortunately, the algorithm does not produce the standard errors of the regression coefficients, which then hampers testing procedures. To fill in the gap, the Supplemented EM (SEM) algorithm for the case of fixed variables is proposed in this paper. The computational aspects of the SEM algorithm have been investigated by means of simulation. We also calculate the variance matrix of beta using the second moment as a benchmark to compare with the asymptotic variance matrix of beta of SEM. Both the second moment and SEM produce symmetrical results, the variance estimates of beta are getting smaller when number of subjects in the simulation increases. In addition, the practical usefulness of this work was illustrated using real data on political attitudes and behaviour in Flanders-Belgium.

This is a continuation of our previous works on entanglement entropy (EE) in interacting field theories. In arXiv:2103.05303, we have proposed the notion of $\mathbb{Z}_M$ gauge theory on Feynman diagrams to calculate EE in quantum field theories and shown that EE consists of two particular contributions from propagators and vertices. As shown in the next paper arXiv:2105.02598, the purely non-Gaussian contributions from interaction vertices can be interpreted as renormalized correlation functions of composite operators. In this paper, we will first provide a unified matrix form of EE containing both contributions from propagators and (classical) vertices, and then extract further non-Gaussian contributions based on the framework of the Wilsonian renormalization group. It is conjectured that the EE in the infrared is given by a sum of all the vertex contributions in the Wilsonian effective action., Comment: 29 pages, 10 figures; typos corrected, published version in Symmetry (v2)

The concept of an endpoint is a relatively new concept compared to the concept of a fixed point. The aim of this paper is to perform a convergence analysis of M—iteration involving α—Reich–Suzuki nonexpansive mappings. In this paper, we prove strong and Δ—convergence theorems in a hyperbolic metric space. Thus, our results generalize and improve many existing results.

In this paper, the existence of the solution and its stability to the fractional boundary value problem (FBVP) were investigated for an implicit nonlinear fractional differential equation (VOFDE) of variable order. All existence criteria of the solutions in our establishments were derived via Krasnoselskii’s fixed point theorem and in the sequel, and its Ulam–Hyers–Rassias (U-H-R) stability is checked. An illustrative example is presented at the end of this paper to validate our findings.

With the gradual popularity of online sales and the enhancement of consumers’ low-carbon awareness, the low-carbon e-commerce supply chain (LCECSC) has developed rapidly. However, most of the current research on LCECSC assumes that the decision-making body is rational, and there is less research on the irrational behavior of the e-platform altruistic preference. Therefore, aiming at the LCECSC composed of a single e-platform and a single manufacturer, this paper establishes two basic models with or without altruistic preference. Additionally, this paper combines the characteristics of online sales and assumes that altruistic preference is a proportional function of commission, then establishes a commission-based extended model with altruistic preference to further explore the influence of commission on its altruistic preference. The current literature does not consider this point, nor does it analyze the influence of other parameters on the degree of altruism preference. By comparing the optimal decisions and numerical analysis among the models, the following conclusions can be drawn that: (1) different from the traditional offline supply chain, the profit of the dominator e-platform is lower than the profit of the follower manufacturer, (2) when the consumers’ carbon emission reduction elasticity coefficient increases, service level, sales price, carbon emission reduction, sales, supply chain members profits, and system profit increase, ultimately improving economic and environmental performances, (3) the altruistic preference behavior of the e-platform is a behavior of ‘profit transferring’. The moderate altruistic preference is conducive to the stable operation and long-term development of LCECSC.

This paper presents a computational tool for estimating energy generated by low-power photovoltaic systems based on the specific conditions of the study region since the characteristic energy equation can be obtained considering the main climatological factors affecting these systems in terms of the symmetry or skewness of the random distribution of the generated energy. Furthermore, this paper is aimed at determining any correlation that exists between meteorological variables with respect to the energy generated by 5-kW solar systems in the specific climatic conditions of the Republic of Cuba. The paper also presents the results of the influence of each climate factor on the distribution symmetry of the generated energy of the solar system. Studying symmetry in statistical models is important because they allow us to establish the degree of symmetry (or skewness), which is the probability distribution of a random variable, without having to make a graphical representation of it. Statistical skewness reports the degree to which observations are distributed evenly and proportionally above and below the center (highest) point of the distribution. In the case when the mentioned distribution is balanced, it is called symmetric.

In this paper, we construct and study a new family of multi-point Ehrlich-type iterative methods for approximating all the zeros of a uni-variate polynomial simultaneously. The first member of this family is the two-point Ehrlich-type iterative method introduced and studied by Trićković and Petković in 1999. The main purpose of the paper is to provide local and semilocal convergence analysis of the multi-point Ehrlich-type methods. Our local convergence theorem is obtained by an approach that was introduced by the authors in 2020. Two numerical examples are presented to show the applicability of our semilocal convergence theorem.

Canonical correlation analysis (CCA) is the default method for investigating the linear dependence structure between two random vectors, but it might not detect nonlinear dependencies. This paper models the nonlinear dependencies between two random vectors by the perturbed independence distribution, a multivariate semiparametric model where CCA provides an insight into their nonlinear dependence structure. The paper also investigates some of its probabilistic and inferential properties, including marginal and conditional distributions, nonlinear transformations, maximum likelihood estimation and independence testing. Perturbed independence distributions are closely related to skew-symmetric ones.

The objective of this paper is to define one class of plane curves with arc-length parametrization. To accomplish this, we constructed a novel class of special polynomials and special functions. These functions form a basis of L2(R) space and some of their interesting properties are discussed. The developed curves are used for the linear static analysis of curved Bernoulli–Euler beam. Due to the parametrization with arc length, the exact analytical solution can be obtained. These closed-form solutions serve as the benchmark results for the development of numerical procedures. One such example is provided in this paper.

In this paper, the notions of post-quantum integrals for two-variable interval-valued functions are presented. The newly described integrals are then used to prove some new Hermite-Hadamard inclusions for co-ordinated convex interval-valued functions. Many of the findings in this paper are important extensions of previous findings in the literature. Finally, we present a few examples of our new findings. Analytic inequalities of this nature and especially the techniques involved have applications in various areas in which symmetry plays a prominent role. WOS:000677046700001 2-s2.0-85110868353

In this paper, some coefficient problems for starlike analytic functions with respect to symmetric points are considered. Bounds of several coefficient functionals for functions belonging to this class are provided. The main aim of this paper is to find estimates for the following: coefficients, logarithmic coefficients, some cases of the generalized Zalcman coefficient functional, and some cases of the Hankel determinant.

In this paper, we consider fractional ordinary differential equations and the fractional Euler–Poisson–Darboux equation with fractional derivatives in the form of a power of the Bessel differential operator. Using the technique of the Meijer integral transform and its modification, fundamental solutions to these equations are derived in terms of the Fox–Wright function, the Fox H-function, and their particular cases. We also provide some explicit formulas for the solutions to the corresponding initial-value problems in terms of the generalized convolutions introduced in this paper.

This paper provides three classes of q-summation formulas in the form of general contiguous extensions of the first q-Kummer summation theorem. Their derivations are presented by using three methods, which are along the lines of the three types of well-known proofs of the q-Kummer summation theorem with a key role of the q-binomial theorem. In addition to the q-binomial theorem, the first proof makes use of Thomae’s q-integral representation and the second proof needs Heine’s transformation. Whereas the third proof utilizes only the q-binomial theorem. Subsequently, the applications of these summation formulas in obtaining the general contiguous extensions of the second and the third q-Kummer summation theorems are also presented. Furthermore, the investigated results are specialized to give many of the known as well as presumably new q-summation theorems, which are contiguous to the three q-Kummer summation theorems. This work is motivated by the observation that the basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) gamma and q-hypergeometric functions and basic (or q-) hypergeometric polynomials, are applicable particularly in several diverse areas including Number Theory, Theory of Partitions and Combinatorial Analysis as well as in the study of Combinatorial Generating Functions. Just as it is known in the theory of the Gauss, Kummer (or confluent), Clausen and the generalized hypergeometric functions, the parameters in the corresponding basic or quantum (or q-) hypergeometric functions are symmetric in the sense that they remain invariant when the order of the p numerator parameters or when the order of the q denominator parameters is arbitrarily changed. A case has therefore been made for the symmetry possessed not only by hypergeometric functions and basic or quantum (or q-) hypergeometric functions, which are studied in this paper, but also by the symmetric quantum calculus itself.

The purpose of this paper is to analyze solutions of a non-local nonlinear partial integro-differential equation (PIDE) in multidimensional spaces. Such class of PIDE often arises in financial modeling. We employ the theory of abstract semilinear parabolic equations in order to prove existence and uniqueness of solutions in the scale of Bessel potential spaces. We consider a wide class of Lévy measures satisfying suitable growth conditions near the origin and infinity. The novelty of the paper is the generalization of already known results in the one space dimension to the multidimensional case. We consider Black–Scholes models for option pricing on underlying assets following a Lévy stochastic process with jumps. As an application to option pricing in the one-dimensional space, we consider a general shift function arising from a nonlinear option pricing model taking into account a large trader stock-trading strategy. We prove existence and uniqueness of a solution to the nonlinear PIDE in which the shift function may depend on a prescribed large investor stock-trading strategy function.

The paper presents k-version of the finite element method for boundary value problems (BVPs) and initial value problems (IVPs) in which global differentiability of approximations is always the result of the union of local approximations. The higher order global differentiability approximations (HGDA/DG) are always p-version hierarchical that permit use of any desired p-level without effecting global differentiability. HGDA/DG are true Ci, Cij, Cijk, hence the dofs at the nonhierarchical nodes of the elements are transformable between natural and physical coordinate spaces using calculus. This is not the case with tensor product higher order continuity elements discussed in this paper, thus confirming that the tensor product approximations are not true Ci, Cijk, Cijk approximations. It is shown that isogeometric analysis for a domain with more than one patch can only yield solutions of class C0. This method has no concept of finite elements and local approximations, just patches. It is shown that compariso of this method with k-version of the finite element method is meaningless. Model problem studies in R2 establish accuracy and superior convergence characteristics of true Cijp-version hierarchical local approximations presented in this paper over tensor product approximations. Convergence characteristics of p-convergence, k-convergence and pk-convergence are illustrated for self adjoint, non-self adjoint and non-linear differential operators in BVPs. It is demonstrated that h, p and k are three independent parameters in all finite element computations. Tensor product local approximations and other published works on k-version and their limitations are discussed in the paper and are compared with present work.

In this paper, we propose the fractal (2 + 1)-dimensional Zakharov–Kuznetsov equation based on He’s fractal derivative for the first time. The fractal generalized variational formulation is established by using the semi-inverse method and two-scale fractal theory. The obtained fractal variational principle is important since it not only reveals the structure of the traveling wave solutions but also helps us study the symmetric theory. The finding of this paper will contribute to the study of symmetry in the fractal space.

The object of investigation in this paper is a scalar linear fractional differential equation with generalized proportional derivative of Riemann–Liouville type (LFDEGD). The main goal is the obtaining an explicit solution of the initial value problem of the studied equation. Note that the locally solvability, being the same as the existence of solutions to the initial value problem, is connected with the symmetry of a transformation of a system of differential equations. At the same time, several criteria for existence of the initial value problem for nonlinear fractional differential equations with generalized proportional derivative are connected with the linear ones. It leads to the necessity of obtaining an explicit solution of LFDEGD. In this paper two cases are studied: the case of no impulses in the differential equation are presented and the case when instantaneous impulses at initially given points are involved. All obtained formulas are based on the application of Mittag–Leffler function with two parameters. In the case of impulses, initially the appropriate impulsive conditions are set up and later the explicit solutions are obtained.

The present paper deals with a class of second-order PDE constrained controlled optimization problems with application in Lagrange–Hamilton dynamics. Concretely, we formulate and prove necessary conditions of optimality for the considered class of control problems driven by multiple integral cost functionals involving second-order partial derivatives. Moreover, an illustrative example is provided to highlight the effectiveness of the results derived in the paper. In the final part of the paper, we present an algorithm to summarize the steps for solving a control problem such as the one investigated here.

This Special Issue covers symmetric and asymmetric data that occur in real-life problems. We invited authors to submit their theoretical or experimental research to present engineering and economic problem solution models that deal with symmetry or asymmetry of different data types. The Special Issue gained interest in the research community and received many submissions. After rigorous scientific evaluation by editors and reviewers, seventeen papers were accepted and published. The authors proposed different solution models, mainly covering uncertain data in multi-criteria decision-making problems as complex tools to balance the symmetry between goals, risks, and constraints to cope with the complicated problems in engineering or management. Therefore, we invite researchers interested in the topics to read the papers provided in the Special Issue.

Since Gottlieb introduced and investigated the so-called Gottlieb polynomials in 1938, which are discrete orthogonal polynomials, many researchers have investigated these polynomials from diverse angles. In this paper, we aimed to investigate the q-extensions of these polynomials to provide certain q-generating functions for three sequences associated with a finite power series whose coefficients are products of the known q-extended multivariable and multiparameter Gottlieb polynomials and another non-vanishing multivariable function. Furthermore, numerous possible particular cases of our main identities are considered. Finally, we return to Khan and Asif’s q-Gottlieb polynomials to highlight certain connections with several other known q-polynomials, and provide its q-integral representation. Furthermore, we conclude this paper by disclosing our future investigation plan.

This paper summarizes a mathematical model for the industrial heating and cooling processes of a steel workpiece corresponding to the steering rack of an automobile. The general purpose of the heat treatment process is to create the necessary hardness on critical parts of the workpiece. Hardening consists of heating the workpiece up to a threshold temperature followed by a rapid cooling such as aquaquenching. The high hardness is due to the steel phase transformation accompanying the rapid cooling resulting in non-equilibrium phases, one of which is the hard microconstituent of steel, namely martensite. The mathematical model describes both processes, heating and cooling. During the first one, heat is produced by Joule's effect from a very high alternating current passing through the rack. This situation is governed by a set of coupled PDEs/ODEs involving the electric potential, the magnetic vector potential, the temperature, the austenite transformation, the stresses and the displacement field. Once the workpiece has reached the desired temperature, the current is switched off an the cooling stage starts by aquaquenching. In this case, the governing equations involve the temperature, the austenite and martensite phase fractions, the stresses and the displacement field. This mathematical model has been solved by the FEM and 2D numerical simulations are discussed along the paper., This research was partially supported by Ministerio de Educacion y Ciencia under grants MTM2010-16401 and TEC2017-86347-C2-1-R with the participation of FEDER, and Consejeria de Educacion y Ciencia de la Junta de Andalucia, research group FQM-315. Giuseppe Viglialoro is a member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilita e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and he is partially supported by the research projects Evolutive and stationary Partial Differential Equations with a focus on biomathematics, funded by Fondazione di Sardegna (2019), and by MIUR (Italian Ministry of Education, University and Research) Prin 2017 Nonlinear Differential Problems via Variational, Topological and Set-valued Methods (Grant Number: 2017AYM8XW).

This paper provides several error estimations for total variation (TV) type regularization, which arises in a series of areas, for instance, signal and imaging processing, machine learning, etc. In this paper, some basic properties of the minimizer for the TV regularization problem such as stability, consistency and convergence rate are fully investigated. Both a priori and a posteriori rules are considered in this paper. Furthermore, an improved convergence rate is given based on the sparsity assumption. The problem under the condition of non-sparsity, which is common in practice, is also discussed, the results of the corresponding convergence rate are also presented under certain mild conditions.

What are the biomechanical implications in the dynamics and evolution of a growing solid tumor? Although the analysis of some of the biochemical aspects related to the signaling pathways involved in the spread of tumors has advanced notably in recent times, their feedback with the mechanical aspects is a crucial challenge for a global understanding of the problem. The aim of this paper is to try to illustrate the role and the interaction between some evolutionary processes (growth, pressure, homeostasis, elasticity, or dispersion by flux-saturated and porous media) that lead to collective cell dynamics and defines a propagation front that is in agreement with the experimental data. The treatment of these topics is approached mainly from the point of view of the modeling and the numerical approach of the resulting system of partial differential equations, which can be placed in the context of the Hele-Shaw-type models. This study proves that local growth terms related to homeostatic pressure give rise to retrograde diffusion phenomena, which compete against migration through flux-saturated dispersion terms., MINECO-Feder (Spain) research grant numbers RTI2018-098850-B-I00 (J.C., J.S.) & EQC2018-004508-P (B.B., J.M.), Junta de Andalucía (Spain) Projects PY18-RT-2422 (J.C., J.S.), A-FQM-311-UGR18 (J.C., J.S.) & IE2017-5537 (B.B., J.M.), Instituto de Salud Carlos III, project number DTS17/00087 (J.M., J.S.), Ministry of Science, Innovation and Universities of Spain, project numbers DPI2017-85359-R (B.B., J.M.) & PID2019-106947RA-C22 (B.B., J.M.), Consejería de Economía, Conocimiento, Empresas y Universidad and European Regional Development Fund (ERDF), ref. SOMM17/6109/UGR (J.C., J.M., J.S.), Ministry of Science, Innovation and Universities of Spain, FPU2017/01415

In this paper, we analyze involutes of pseudo-null curves in Lorentz–Minkowski 3-space. Pseudo-null curves are spacelike curves with null principal normals, and their involutes can be defined analogously as for the Euclidean curves, but they exhibit properties that cannot occur in Euclidean space. The first result of the paper is that the involutes of pseudo-null curves are null curves, more precisely, null straight lines. Furthermore, a method of reconstruction of a pseudo-null curve from a given null straight line as its involute is provided. Such a reconstruction process in Euclidean plane generates an evolute of a curve, however it cannot be applied to a straight line. In the case presented, the process is additionally affected by a choice of different null frames that every null curve allows (in this case, a null straight line). Nevertheless, we proved that for different null frames, the obtained pseudo-null curves are congruent. Examples that verify presented results are also given., MTM2017-89677-P, MINECO/ AEI/FEDER, UE.

After the death of Peter Wynn in December 2017, manuscript documents he left came to our knowledge. They concern continued fractions, rational (Padé) approximation, Thiele interpolation, orthogonal polynomials, moment problems, series, and abstract algebra. The purpose of this paper is to analyze them and to make them available to the mathematical community. Some of them are in quite good shape, almost finished, and ready to be published by anyone willing to check and complete them. Others are rough notes, and need to be reworked. Anyway, we think that these works are valuable additions to the literature on these topics and that they cannot be left unknown since they contain ideas that were never exploited. They can lead to new research and results. Two unpublished papers are also mentioned here for the first time.

In this paper, we introduce a general family of $q$-hypergeometric polynomials and investigate several $q$-series identities such as an extended generating function and a Srivastava-Agarwal type bilinear generating function for this family of $q$-hypergeometric polynomials. We give a transformational identity involving generating functions for the generalized $q$-hypergeometric polynomials which we have introduced here. We also point out relevant connections of the various $q$-results, which we investigate here, with those in several related earlier works on this subject. We conclude this paper by remarking that it will be a rather trivial and inconsequential exercise to give the so-called $(p,q)$-variations of the $q$-results, which we have investigated here, because the additional parameter $p$ is obviously redundant., 14 pages

Power electronic converters are mathematically represented by a system of ordinary differential equations discontinuous right-hand side that does not verify the conditions of the Cauchy-Lipschitz Theorem. More generally, for the properties that characterize their discontinuous behavior, they represent a particular class of systems on which little has been investigated over the years. The purpose of the paper is to prove the existence of at least one global solution in Filippov’s sense to the Cauchy problem related to the mathematical model of a power converter and also to calculate the error in norm between this solution and the integral of its averaged approximation. The main results are the proof of this theorem and the analytical formulation that provides to calculate the cited error. The demonstration starts by a proof of local existence provided by Filippov himself and already present in the literature for a particular class of systems and this demonstration is generalized to the class of electronic power converters, exploiting the non-chattering property of this class of systems. The obtained results are extremely useful for estimating the accuracy of the averaged model used for analysis or control of the effective system. In the paper, the goodness of the analytical proof is supported by experimental tests carried out on a converter prototype representing the class of power electronics converter.

The aim of this paper is to show analytically and empirically how ant-based algorithms for medical image edge detection can be enhanced by using an admissible perturbation of demicontractive operators. We thus complement the results reported in a recent paper by the second author and her collaborators, where they used admissible perturbations of demicontractive mappings as test functions. To illustrate this fact, we first consider some typical properties of demicontractive mappings and of their admissible perturbations and then present some appropriate numerical tests to illustrate the improvement brought by the admissible perturbations of demicontractive mappings when they are taken as test functions in ant-based algorithms for medical image edge detection. The edge detection process reported in our study considers both symmetric (Head CT and Brain CT) and asymmetric (Hand X-ray) medical images. The performance of the algorithm was tested visually with various images and empirically with evaluation of parameters.

In this paper, we introduce the concept of n-dimensional Diamond-Alpha integral on time scales. In particular, it transforms into multiple Delta, Nabla and mixed integrals by taking different values of alpha. Some of its properties are explored, and the relationship between it and the multiple mixed integral is provided. As an application, we establish some weighted Ostrowski type inequalities through the new integral. These new inequalities expand some known inequalities in the monographs and papers, and in addition, furnish some other interesting inequalities. Examples of Ostrowski type inequalities are posed in detail at the end of the paper.

The present paper introduces new classes of Stancu–Kantorovich operators constructed in the King sense. For these classes of operators, we establish some convergence results, error estimations theorems and graphical properties of approximation for the classes considered, namely, operators that preserve the test functions e0(x)=1 and e1(x)=x, e0(x)=1 and e2(x)=x2, as well as e1(x)=x and e2(x)=x2. The class of operators that preserve the test functions e1(x)=x and e2(x)=x2 is a genuine generalization of the class introduced by Indrea et al. in their paper “A New Class of Kantorovich-Type Operators”, published in Constr. Math. Anal.

In tokamak nuclear fusion reactors, one of the main issues is to know the total emission of radiation, which is mandatory to understand the plasma physics and is very useful to monitor and control the plasma evolution. This radiation can be measured by means of a bolometer system that consists in a certain number of elements sensitive to the integral of the radiation along straight lines crossing the plasma. By placing the sensors in such a way to have families of crossing lines, sophisticated tomographic inversion algorithms allow to reconstruct the radiation tomography in the 2D poloidal cross-section of the plasma. In tokamaks, the number of projection cameras is often quite limited resulting in an inversion mathematic problem very ill conditioned so that, usually, it is solved by means of a grid-based, iterative constrained optimization procedure, whose convergence time is not suitable for the real time requirements. In this paper, to illustrate the method, an assumption not valid in general is made on the correlation among the grid elements, based on the statistical distribution of the radiation emissivity over a set of tomographic reconstructions, performed off-line. Then, a regularization procedure is carried out, which merge highly correlated grid elements providing a squared coefficients matrix with an enough low condition number. This matrix, which is inverted offline once for all, can be multiplied by the actual bolometer measures returning the tomographic reconstruction, with calculations suitable for real time application. The proposed algorithm is applied, in this paper, to a synthetic case study.

The paper is devoted to the guaranteeing estimation of parameters in the uncertain stochastic nonlinear regression. The loss function is the conditional mean square of the estimation error given the available observations. The distribution of regression parameters is partially unknown, and the uncertainty is described by a subset of probability distributions with a known compact domain. The essential feature is the usage of some additional constraints describing the conformity of the uncertain distribution to the realized observation sample. The paper contains various examples of the conformity indices. The estimation task is formulated as the minimax optimization problem, which, in turn, is solved in terms of saddle points. The paper presents the characterization of both the optimal estimator and the set of least favorable distributions. The saddle points are found via the solution to a dual finite-dimensional optimization problem, which is simpler than the initial minimax problem. The paper proposes a numerical mesh procedure of the solution to the dual optimization problem. The interconnection between the least favorable distributions under the conformity constraint, and their Pareto efficiency in the sense of a vector criterion is also indicated. The influence of various conformity constraints on the estimation performance is demonstrated by the illustrative numerical examples.

In this paper, we use integral equations of non-integer orders to derive discrete maps with memory. Note that discrete maps with memory were not previously derived from fractional integral equations of non-integer orders. Such a derivation of discrete maps with memory is proposed for the first time in this work. In this paper, we derived discrete maps with nonlocality in time and memory from exact solutions of fractional integral equations with the Riemann–Liouville and Hadamard type fractional integrals of non-integer orders and periodic sequence of kicks that are described by Dirac delta-functions. The suggested discrete maps with nonlocality in time are derived from these fractional integral equations without any approximation and can be considered as exact discrete analogs of these equations. The discrete maps with memory, which are derived from integral equations with the Hadamard type fractional integrals, do not depend on the period of kicks.

In this paper we study a certain differential subordination related to the harmonic mean and its symmetry properties, in the case where a dominant is a linear function. In addition to the known general results for the differential subordinations of the harmonic mean in which the dominant was any convex function, one can study such differential subordinations for the selected convex function. In this case, a reasonable and difficult issue is to look for the best dominant or one that is close to it. This paper is devoted to this issue, in which the dominant is a linear function, and the differential subordination of the harmonic mean is a generalization of the Briot–Bouquet differential subordination.

Previous works were dedicated to the functional k-Nearest Neighbors (kNN) and the local linearity method estimations of a regression operator. In this paper, a sequence pair of (Xi,Yi)i=1,…,n of functional mixing observations are considered. We treat the local linear estimation of the cumulative function of Yi given functional input variable Xi. Precisely, we combine the kNN method with the local linear algorithm to construct a new and fast efficiency estimator of the conditional distribution function. The main purpose of this paper is to prove the strong convergence of the constructed estimator under mixing conditions. An application to the functional times series prediction is used to compare our proposed estimator with the existing competitive estimators, and show its efficiency and superiority.

In previous papers, a technique has been suggested to avoid order reduction when integrating initial boundary value problems with several kinds of exponential methods. The technique implies in principle to calculate additional terms at each step from those already necessary without avoiding order reduction. The aim of the present paper is to explain the surprising result that, many times, in spite of having to calculate more terms at each step, the computational cost of doing it through Krylov methods decreases instead of increases. This is very interesting since, in that way, the methods improve not only in terms of accuracy, but also in terms of computational cost.

In this paper, we introduce a new representation of semiopenness of L-fuzzy sets in RL-fuzzy bitopological spaces based on the concept of pseudo-complement. The concepts of pairwise RL-fuzzy semicontinuous and pairwise RL-fuzzy irresolute functions are extended and discussed based on the (i,j)-RL-semiopen gradation. Further, pairwise RL-fuzzy semi-compactness of an L-fuzzy set in RL-fuzzy bitopological spaces are given and characterized. As RL-fuzzy bitopology is a generalization of L-bitopology, RL-bitopology, L-fuzzy bitopology, and RL-fuzzy topology, the results of our paper are more general.

This article belongs to the Special Issue Discrete and Fractional Mathematics: Symmetry and Applications. In this paper we introduce a generalized Laplace transform in order to work with a very general fractional derivative, and we obtain the properties of this new transform. We also include the corresponding convolution and inverse formula. In particular, the definition of convolution for this generalized Laplace transform improves previous results. Additionally, we deal with the generalized harmonic oscillator equation, showing that this transform and its properties allow one to solve fractional differential equations. We would like to thank the referees for their comments, which have improved the paper. The research of José M. Rodríguez and José M. Sigarreta was supported by a grant from Agencia Estatal de Investigación (PID2019-106433GB-I00/AEI/10.13039/501100011033), Spain. The research of José M. Rodríguez is supported by the Madrid Government (Comunidad de Madrid-Spain) under the Multiannual Agreement with UC3M in the line of Excellence of University Professors (EPUC3M23), and in the context of the V PRICIT (Regional Programme of Research and Technological Innovation).

Here, we review some results of fractional volatility models, where the volatility is driven by fractional Brownian motion (fBm). In these models, the future average volatility is not a process adapted to the underlying filtration, and fBm is not a semimartingale in general. So, we cannot use the classical Itô’s calculus to explain how the memory properties of fBm allow us to describe some empirical findings of the implied volatility surface through Hull and White type formulas. Thus, Malliavin calculus provides a natural approach to deal with the implied volatility without assuming any particular structure of the volatility. The aim of this paper is to provides the basic tools of Malliavin calculus for the study of fractional volatility models. That is, we explain how the long and short memory of fBm improves the description of the implied volatility. In particular, we consider in detail a model that combines the long and short memory properties of fBm as an example of the approach introduced in this paper. The theoretical results are tested with numerical experiments.

In this paper, we introduce the general fractional integrals and derivatives of arbitrary order and study some of their basic properties and particular cases. First, a suitable generalization of the Sonine condition is presented and some important classes of the kernels that satisfy this condition are introduced. Whereas the kernels of the general fractional derivatives with these kernels possess the integrable singularities at the point zero, the kernels of the general fractional integrals can be - depending on their order - both singular and continuous at the origin. For the general fractional integrals and derivatives of arbitrary order with the kernels introduced in this paper, two fundamental theorems of fractional calculus are formulated and proved., 15 pages