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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.

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.

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 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.

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.

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.

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.

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 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 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.

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).

The fractional integral is a function known for the elegant results obtained when introducing new operators, it has proved to have interesting applications. In the present paper, differential subordinations and superodinations for the fractional integral of the confluent hypergeometric function introduced in a previously published paper are presented. A sandwich-type theorem at the end of the original part of the paper connects the outcomes of the studies done using the dual theories.

The paper proposes composite materials for the manufacturing of parts of the car body structure, namely a door. This work aims to analyze the possibility of replacing the metal door of a vehicle with a door made of composite materials. Specific issues related to this replacement are analyzed in the paper. Test specimens were made of composite materials of different sizes, using several types of constituents to determine which material might be most suitable to replace metal in the manufacturing of the door. The choice of materials for the car door was made starting from the characteristics of the analyzed composite materials, but also taking into account the manufacturing possibilities and other engineering limitations. The behavior of the automotive structure as analyzed, using the finite element method for determining the stresses in the structure. Experimental verifications were performed on an experimental stand which has been specially designed for this purpose, to validate the proposed model.

Both the theory of differential subordination and its dual, the theory of differential superordination, introduced by Professors Miller and Mocanu are based on reinterpreting certain inequalities for real-valued functions for the case of complex-valued functions. Studying subordination and superordination properties using different types of operators is a technique that is still widely used, some studies resulting in sandwich-type theorems as is the case in the present paper. The fractional integral of confluent hypergeometric function is introduced in the paper and certain subordination and superordination results are stated in theorems and corollaries, the study being completed by the statement of a sandwich-type theorem connecting the results obtained by using the two theories.

The idea of inequality has been extended from the real plane to the complex plane through the notion of subordination introduced by Professors Miller and Mocanu in two papers published in 1978 and 1981. With this notion came a whole new theory called the theory of differential subordination or admissible functions theory. Later, in 2003, a particular form of inequality in the complex plane was also defined by them as dual notion for subordination, the notion of differential superordination and with it, the theory of differential superordination appeared. In this paper, the theory of differential superordination is applied to confluent hypergeometric function. Hypergeometric functions are intensely studied nowadays, the interest on the applications of those functions in complex analysis being renewed by their use in the proof of Bieberbach’s conjecture given by de Branges in 1985. Using the theory of differential superodination, best subordinants of certain differential superordinations involving confluent (Kummer) hypergeometric function are stated in the theorems and relation with previously obtained results are highlighted in corollaries using particular functions and in a sandwich-type theorem. An example is also enclosed in order to show how the theoretical findings can be applied.

Some generalizations of the Gronwall&ndash, Bellman (G&ndash, B) inequality are presented in this paper in continuous form and on time scales. After S. Hilger introduced the time scales theory in 1988, over the years many mathematicians have studied new versions of this inequality according to new results, the purpose of this paper is to present some of them. Therefore, in the Introduction, some generalizations of G&ndash, B inequality in continuous forms, linear and nonlinear are presented. In the second section, some important and interesting results on time scales theory are given. In the third and main part of our paper, G&ndash, B inequalities on time scales and their possible connection with G&ndash, B inequalities presented in the introduction are investigated. In particular, in the third section of this work, more attention is given to G&ndash, B type inequalities on time scales discussed in the last four years.

This paper is concerned with a type of time-symmetric stochastic system, namely the so-called forward&ndash, backward doubly stochastic differential equations (FBDSDEs), in which the forward equations are delayed doubly stochastic differential equations (SDEs) and the backward equations are anticipated backward doubly SDEs. Under some monotonicity assumptions, the existence and uniqueness of measurable solutions to FBDSDEs are obtained. The future development of many processes depends on both their current state and historical state, and these processes can usually be represented by stochastic differential systems with time delay. Therefore, a class of nonzero sum differential game for doubly stochastic systems with time delay is studied in this paper. A necessary condition for the open-loop Nash equilibrium point of the Pontriagin-type maximum principle are established, and a sufficient condition for the Nash equilibrium point is obtained. Furthermore, the above results are applied to the study of nonzero sum differential games for linear quadratic backward doubly stochastic systems with delay. Based on the solution of FBDSDEs, an explicit expression of Nash equilibrium points for such game problems is established.

Mathematical models are useful tools to describe the dynamics of infection and predict the role of possible drug combinations. In this paper, we present an analysis of a hepatitis B virus (HBV) model including cytotoxic T lymphocytes (CTL) and antibody responses, under distributed feedback control, expressed as an integral form to predict the effect of a combination treatment with interleukin-2 (IL-2). The method presented in this paper is based on the symmetry properties of Cauchy matrices C(t,s), which allow us to construct and analyze the stability of corresponding integro-differential systems.

In this paper, we give some classifications of the k-Yamabe solitons on the hypersurfaces of the Euclidean spaces from the vector field point of view. In several results on k-Yamabe solitons with a concurrent vector field on submanifolds in Riemannian manifolds, is proved that a k-Yamabe soliton (Mn,g,vT,λ) on a hypersurface in the Euclidean space Rn+1 is contained either in a hypersphere or a hyperplane. We provide an example to support this study and all of the results in this paper can be implemented to Yamabe solitons for k-curvature with k=1.

In this paper, sufficient conditions ensuring existence and multiplicity of positive solutions for a class of nonlinear singular fractional differential systems are derived with Riemann&ndash, Stieltjes coupled integral boundary value conditions in Banach Spaces. Nonlinear functions f(t,u,v) and g(t,u,v) in the considered systems are allowed to be singular at every variable. The boundary conditions here are coupled forms with Riemann&ndash, Stieltjes integrals. In order to overcome the difficulties arising from the singularity, a suitable cone is constructed through the properties of Green&rsquo, s functions associated with the systems. The main tool used in the present paper is the fixed point theorem on cone. Lastly, an example is offered to show the effectiveness of our obtained new results.

During the last few decades, domination theory has been one of the most active areas of research within graph theory. Currently, there are more than 4400 published papers on domination and related parameters. In the case of total domination, there are over 580 published papers, and 50 of them concern the case of product graphs. However, none of these papers discusses the case of rooted product graphs. Precisely, the present paper covers this gap in the theory. Our goal is to provide closed formulas for the total domination number of rooted product graphs. In particular, we show that there are four possible expressions for the total domination number of a rooted product graph, and we characterize the graphs reaching these expressions.

In this paper, we study some optimal inequalities of the Riccati type and of the Bihari type. We also consider nonoptimal inequalities of the Wendorf type. At the same time, we get a partial answer to Problems 5 and 9, formulated by I. A. Rus. This paper is also motivated by the fact that, in many inequalities, the upper bound is not an optimal one.

In this paper, we study Carlitz&rsquo, s type higher-order (p,q)-Genocchi polynomials. To be specific, we define Carlitz&rsquo, s type higher-order (p,q)-Genocchi polynomials and Carlitz&rsquo, s type higher-order (h,p,q)-Genocchi polynomials. This paper also explores properties including distribution relation and symmetric identities. In addition, we find alternating (p,q)-power sums. We identify symmetric identities using Carlitz&rsquo, s type higher-order (h,p,q)-Genocchi polynomials and alternating (p,q)-power sums.

In this paper, we consider the matrix expression of convolution, and its generalized continuous form. The matrix expression of convolution is effectively applied in convolutional neural networks, and in this study, we correlate the concept of convolution in mathematics to that in convolutional neural network. Of course, convolution is a main process of deep learning, the learning method of deep neural networks, as a core technology. In addition to this, the generalized continuous form of convolution has been expressed as a new variant of Laplace-type transform that, encompasses almost all existing integral transforms. Finally, we would, in this paper, like to describe the theoretical contents as detailed as possible so that the paper may be self-contained.

Let C0(h) be a subclass of analytic and close-to-convex functions defined in the open unit disk by the formula ()(1&minus, z2)f&prime, (z)}&gt, 0. In this paper, some coefficient problems for C0(h) are considered. Some properties and bounds of several coefficient functionals for functions belonging to this class are provided. The main aim of this paper is to find estimates of the difference and of sum of successive coefficients, bounds of the sum of the first n coefficients and bounds of the n-th coefficient. The obtained results are used to determine coefficient estimates for both functions convex in the imaginary-axis direction with real coefficients and typically real functions. Moreover, the sum of the first initial coefficients for functions with a positive real part and with a fixed second coefficient is estimated.

This paper presents a method of establishing the D-stability terms of the symmetric solution of scalar symmetric linear and nonlinear functional differential equations. We determine the general conditions of the unique solvability of the initial value problem for symmetric functional differential equations. Here, we show the conditions of the symmetric property of the unique solution of symmetric functional differential equations. Furthermore, in this paper, an illustration of a particular symmetric equation is presented. In this example, all theoretical investigations referred to earlier are demonstrated. In addition, we graphically demonstrate two possible linear functions with the required symmetry properties.

Tropical Algebra is used to model the dynamics of Timed Event Graphs (TEG), a particular class of Timed Discrete-Event System (TDES) in which we are interested only in synchronization and delay phenomena. Whenever this TEG has control inputs, we can use them to control the synchronization of the system to achieve some objective. Thus, this paper formulates a framework based on tropical algebra and lexicographic optimization to synchronize a TEG when dealing with many synchronization objectives that are ranked in previous priority order. We call this kind of problem the Tropical Lexicographic Synchronization Optimization (TLSO). This work develops a solution to this problem, based on Tropical Fractional Linear Programming (TFLP) and lexicographic optimization concepts. In this way, the basics of tropical algebra are determined, including essential terms to this paper, such as left and right residuations, and the following stages of the solution to the TLSO problem are explained. Therefore, this work presents a general framework based on structured algebraic models with application to TEG synchronization. By synchronization, we mean balancing and organizing events chronologically in order to achieve the desired goal. So, we are dealing with concepts closely related to symmetry ones. An illustrative numerical example is presented, which demonstrates the implementation of the proposed algorithms. The acquired results confirm the efficiency of the proposed methodology. Codes used for implementing the algorithms are listed in the appendix section of the article.

The natural transformation constitutes one of the most important entity of category theory and it introduces a piece of sophisticated dynamism to the categorial structures. Each natural transformation forms a unique mapping between the so-called functors, which live between categories. In the most simple contexts, natural transformations may be recognized by commutativity of diagrams, which determine them. In fact, the natural transformation does not form any single mapping, but a pair of two components, which&ndash, together with the commutativity condition itself&ndash, introduces a kind of a symmetry to the functor diagrams. Meanwhile, the general form of the natural transformation may be predicted by means the so-called Yoneda&rsquo, s lemma in each scenario based on two-valued logic. Meanwhile, the situation may be radically different if we deal with multi-diagrams (instead of the single ones) and if we exchange the two-valued scenario for a multi-valued or fuzzy one. Due to this background&ndash, the paper introduces a new concept of multi-fuzzy natural transformation. Its definition exploits the notion of fuzzy natural transformation. Moreover, a multi-fuzzy Yoneda&rsquo, s lemma is formulated and proved. Finally, some references of these constructions to coding theory are elucidated in last parts of the paper.

The paper aims to present advanced algorithms arising out of adding the inertial technical and shrinking projection terms to ordinary parallel and cyclic hybrid inertial sub-gradient extra-gradient algorithms (for short, PCHISE). Via these algorithms, common solutions of variational inequality problems (CSVIP) and strong convergence results are obtained in Hilbert spaces. The structure of this problem is to find a solution to a system of unrelated VI fronting for set-valued mappings. To clarify the acceleration, effectiveness, and performance of our parallel and cyclic algorithms, numerical contributions have been incorporated. In this direction, our results unify and generalize some related papers in the literature. This work was supported in part by the Basque Government under Grant IT1207-19.

The fundamental groups and homotopy decompositions of algebraic topology have applications in systems involving symmetry breaking with topological excitations. The main aim of this paper is to analyze the properties of homotopy decompositions in quotient topological spaces depending on the connectedness of the space and the fundamental groups. This paper presents constructions and analysis of two varieties of homotopy decompositions depending on the variations in topological connectedness of decomposed subspaces. The proposed homotopy decomposition considers connected fundamental groups, where the homotopy equivalences are relaxed and the homeomorphisms between the fundamental groups are maintained. It is considered that one fundamental group is strictly homotopy equivalent to a set of 1-spheres on a plane and as a result it is homotopy rigid. The other fundamental group is topologically homeomorphic to the first one within the connected space and it is not homotopy rigid. The homotopy decompositions are analyzed in quotient topological spaces, where the base space and the quotient space are separable topological spaces. In specific cases, the decomposed quotient space symmetrically extends Sierpinski space with respect to origin. The connectedness of fundamental groups in the topological space is maintained by open curve embeddings without enforcing the conditions of homotopy classes on it. The extended decomposed quotient topological space preserves the trivial group structure of Sierpinski space.

This paper evaluates the limit behavior for symmetry interactions networks of set points for nonlinear mathematical models. Nonlinear mathematical models are being increasingly applied to most software and engineering machines. That is because the nonlinear mathematical models have proven to be more efficient in processing and producing results. The greatest challenge facing researchers is to build a new nonlinear model that can be applied to different applications. Quadratic stochastic operators (QSO) constitute such a model that has become the focus of interest and is expected to be applicable in many biological and technical applications. In fact, several QSO classes have been investigated based on certain conditions that can also be applied in other applications such as the Extreme Doubly Stochastic Quadratic Operator (EDSQO). This paper studies the behavior limitations of the existing 222 EDSQ operators on two-dimensional simplex (2DS). The created simulation graph shows the limit behavior for each operator. This limit behavior on 2DS can be classified into convergent, periodic, and fixed.

The continuous and injective embeddings of closed curves in Hausdorff topological spaces maintain isometry in subspaces generating components. An embedding of a circle group within a topological space creates isometric subspace with rotational symmetry. This paper introduces the generalized algebraic construction of functional groups and its topological embeddings into normal spaces maintaining homeomorphism of functional groups. The proposed algebraic construction of functional groups maintains homeomorphism to rotationally symmetric circle groups. The embeddings of functional groups are constructed in a sequence in the normal topological spaces. First, the topological decomposition and associated embeddings of a generalized group algebraic structure in the lower dimensional space is presented. It is shown that the one-point compactification property of topological space containing the decomposed group embeddings can be identified. Second, the sequential topological embeddings of functional groups are formulated. The proposed sequential embeddings follow Schoenflies property within the normal topological space. The preservation of homeomorphism between disjoint functional group embeddings under Banach-type contraction is analyzed taking into consideration that the underlying topological space is Hausdorff and the embeddings are in a monotone class. It is shown that components in a monotone class of isometry are not separable, whereas the multiple disjoint monotone class of embeddings are separable. A comparative analysis of the proposed concepts and formulations with respect to the existing structures is included in the paper.

The aim of this paper is to give some fixed point results in generalized metric spaces in Perov&rsquo, s sense. The generalized metric considered here is the w-distance with a symmetry condition. The operators satisfy a contractive weakly condition of Hardy&ndash, Rogers type. The second part of the paper is devoted to the study of the data dependence, the well-posedness, and the Ulam&ndash, Hyers stability of the fixed point problem. An example is also given to sustain the presented results.

This paper is devoted to the derivation and mathematical analysis of new thermostatted kinetic theory frameworks for the modeling of nonequilibrium complex systems composed by particles whose microscopic state includes a vectorial state variable. The mathematical analysis refers to the global existence and uniqueness of the solution of the related Cauchy problem. Specifically, the paper is divided in two parts. In the first part the thermostatted framework with a continuous vectorial variable is proposed and analyzed. The framework consists of a system of partial integro-differential equations with quadratic type nonlinearities. In the second part the thermostatted framework with a discrete vectorial variable is investigated. Real world applications, such as social systems and crowd dynamics, and future research directions are outlined in the paper.

Skew-symmetric distributions are a popular family of flexible distributions that conveniently model non-normal features such as skewness, kurtosis and multimodality. Unfortunately, their frequentist inference poses several difficulties, which may be adequately addressed by means of a Bayesian approach. This paper reviews the main prior distributions proposed for the parameters of skew-symmetric distributions, with special emphasis on the skew-normal and the skew-t distributions which are the most prominent skew-symmetric models. The paper focuses on the univariate case in the absence of covariates, but more general models are also discussed.

In this paper, new sufficient conditions for oscillation of fourth-order neutral differential equations are established. One objective of our paper is to further improve and complement some well-known results which were published recently in the literature. Symmetry ideas are often invisible in these studies, but they help us decide the right way to study them, and to show us the correct direction for future developments. An example is given to illustrate the importance of our results.

The main purpose of this paper is to use the Hardy&ndash, Littlewood method to study the solvability of mixed powers of primes. To be specific, we consider the even integers represented as the sum of one prime, one square of prime, one cube of prime, and one biquadrate of prime. However, this representation can not be realized for all even integers. In this paper, we establish the exceptional set of this kind of representation and give an upper bound estimate.

This paper defines two new concepts: the concept of multivalued left-weighted mean contractions in the generalized sense of Nadler in a symmetric generalized metric space and the concept of multivalued right-weighted mean contractions in the generalized sense of Nadler in a symmetric generalized metric space, and demonstrates fixed-point theorems for them. For these, we demonstrated two fixed-point existence theorems and their corollaries, by using the properties of the regular-global-inf function and the properties of symmetric generalized metric spaces, respectively. Moreover, we demonstrated that the theorems can be applied in particular cases of inclusion systems. This article contains not only an example of application in science, but also an example of application in real life, in biology, in order to find an equilibrium solution to a prey&ndash, predator-type problem. The results of this paper extend theorems for multivalued left-weighted mean contractions in the generalized sense of Nadler, demonstrating that some of the results given by Rus (2008), Mureșan (2002), and Nadler (1969) in metric spaces can also be proved in symmetric generalized metric spaces.

This paper presents a new approach to the advanced dynamics of mechanical systems. It is known that in the movements corresponding to some mechanical systems (e.g., robots), accelerations of higher order are developed. Higher-order accelerations are an integral part of higher-order acceleration energies. Unlike other research papers devoted to these advanced notions, the main purpose of the paper is to present, in a matrix form, the defining expressions for the acceleration energies of a higher order. Following the differential principle in generalized form (a generalization of the Lagrange&ndash, D'Alembert principle), the equations of the dynamics of fast-moving systems include, instead of kinetic energies, the acceleration energies of higher-order. To establish the equations which characterize both the energies of accelerations and the advanced dynamics, the following input parameters are considered: matrix exponentials and higher-order differential matrices. An application of a 5 d.o.f robot structure is presented in the final part of the paper. This is used to illustrate the validity of the presented mathematical formulations.

In this paper, we investigate the co-amenability of compact quantum groups. Combining with some properties of regular C*-norms on algebraic compact quantum groups, we show that the quantum double of co-amenable compact quantum groups is unique. Based on this, this paper proves that co-amenability is preserved under formulation of the quantum double construction of compact quantum groups, which exhibits a type of nice symmetry between the co-amenability of quantum groups and the amenability of groups.

To rapidly detect the wheat moisture content (WMC) without harm to the wheat and before harvest, this paper measured wheat and panicle moisture content (PMC) and the corresponding spectral reflectance of panicle before harvest at the Beijing Tongzhou experimental station of China Agricultural University. Firstly, we used correlation analysis to determine the optimal regression model of WMC and PMC. Secondly, we derived the spectral sensitive band of PMC before filtering the redundant variables competitive adaptive reweighted sampling (CARS) to select the variable subset with the least error. Finally, partial least squares regression (PLSR) was used to build and analyze the prediction model of PMC. At the early stage of wheat harvest, a high correlation existed between WMC and PMC. Among all regression models such as exponential, univariate linear, polynomial models, and the power function regression model, the logarithm regression model was the best. The determination coefficients of the modeling sample were: R2 = 0.9284, the significance F = 362.957, the determination coefficient of calibration sample R2v = 0.987, the root mean square error RMSEv = 3.859, and the relative error REv = 7.532. Within the range of 350&ndash, 2500 nm, bands of 728&ndash, 907 nm, 1407&ndash, 1809 nm, and 1940&ndash, 2459 nm had a correlation coefficient of PMC and wavelength reflectivity higher than 0.6. This paper used the CARS algorithm to optimize the variables and obtained the best variable subset, which included 30 wavelength variables. The PLSR model was established based on 30 variables optimized by the CARS algorithm. Compared with the all-sensitive band, which had 1103 variables, the PLSR model not only reduced the number of variables by 1073, but also had a higher accuracy in terms of prediction. The results showed that: RMSEC = 0.9301, R2c = 0.995, RMSEP = 2.676, R2p = 0.945, and RPD = 3.362, indicating that the CARS algorithm could effectively remove the variables of spectral redundant information. The CARS algorithm provided a new way of thinking for the non-destructive and rapid detection of WMC before harvest.

This paper presents many new complex combined dark-bright soliton solutions obtained with the help of the accurate sine-Gordon expansion method to the B-type Kadomtsev-Petviashvili-Boussinesq equation with binary power order nonlinearity. With the use of some computational programs, we plot many new surfaces of the results obtained in this paper. In addition, we present the interactions between complex travelling wave patterns and their solitons.

For any odd positive integer x, define ( x n ) n ⩾ 0 and ( a n ) n ⩾ 1 by setting x 0 = x , x n = 3 x n - 1 + 1 2 a n such that all x n are odd. The 3 x + 1 problem asserts that there is an x n = 1 for all x. Usually, ( x n ) n ⩾ 0 is called the trajectory of x. In this paper, we concentrate on ( a n ) n ⩾ 1 and call it the E-sequence of x. The idea is that we generalize E-sequences to all infinite sequences ( a n ) n ⩾ 1 of positive integers and consider all these generalized E-sequences. We then define ( a n ) n ⩾ 1 to be &Omega, convergent to x if it is the E-sequence of x and to be &Omega, divergent if it is not the E-sequence of any odd positive integer. We prove a remarkable fact that the &Omega, divergence of all non-periodic E-sequences implies the periodicity of ( x n ) n ⩾ 0 for all x 0 . The principal results of this paper are to prove the &Omega, divergence of several classes of non-periodic E-sequences. Especially, we prove that all non-periodic E-sequences ( a n ) n ⩾ 1 with lim &macr, n &rarr, &infin, b n n &gt, log 2 3 are &Omega, divergent by using Wendel&rsquo, s inequality and the Matthews and Watts&rsquo, formula x n = 3 n x 0 2 b n &prod, k = 0 n - 1 ( 1 + 1 3 x k ) , where b n = &sum, k = 1 n a k . These results present a possible way to prove the periodicity of trajectories of all positive integers in the 3 x + 1 problem, and we call it the E-sequence approach.

In this paper, the class of the twice-iterated 2D q-Appell polynomials is introduced. The generating function, series definition and some relations including the recurrence relations and partial q-difference equations of this polynomial class are established. The determinant expression for the twice-iterated 2D q-Appell polynomials is also derived. Further, certain twice-iterated 2D q-Appell and mixed type special q-polynomials are considered as members of this polynomial class. The determinant expressions and some other properties of these associated members are also obtained. The graphs and surface plots of some twice-iterated 2D q-Appell and mixed type 2D q-Appell polynomials are presented for different values of indices by using Matlab. Moreover, some areas of potential applications of the subject matter of, and the results derived in, this paper are indicated.

Recently, two classes of univalent functions S e * and K e were introduced and studied. A function f is in S e * if it is analytic in the unit disk, f ( 0 ) = f &prime, ( 0 ) - 1 = 0 and z f &prime, ( z ) f ( z ) ≺ e z . On the other hand, g &isin, K e if and only if z g &prime, &isin, S e * . Both classes are symmetric, or invariant, under rotations. In this paper, we solve a few problems connected with the coefficients of functions in these classes. We find, among other things, the estimates of Hankel determinants: H 2 , 1 , H 2 , 2 , H 3 , 1 . All these estimates improve the known results. Moreover, almost all new bounds are sharp. The main idea used in the paper is based on expressing the discussed functionals depending on the fixed second coefficient of a function in a given class.

In this paper, we propose a class of simple numerical methods for approximating solutions of one-dimensional mixed Volterra&ndash, Fredholm integral equations of the second kind. These methods are based on fixed point results for the existence and uniqueness of the solution (results which also provide successive iterations of the solution) and suitable cubature formulas for the numerical approximations. We discuss in detail a method using Picard iteration and the two-dimensional composite trapezoidal rule, giving convergence conditions and error estimates. The paper concludes with numerical experiments and a discussion of the methods proposed.