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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.

Recently, various types of single valued neutrosophic (SVN) rough set models were presented based on the same inclusion relation. However, there is another SVN inclusion relation in SVN sets. In this paper, we propose a new type of SVN covering rough set model based on the new inclusion relation. Furthermore, the graph and matrix representations of the new SVN covering approximation operators are presented. Firstly, the notion of SVN β 2 -covering approximation space is proposed, which is decided by the new inclusion relation. Then, a type of SVN covering rough set model under the SVN β 2 -covering approximation space is presented. Moreover, there is a corresponding SVN relation rough set model based on a SVN relation induced by the SVN β 2 -covering, and two conditions under which the SVN β 2 -covering can induce a symmetric SVN relation are presented. Thirdly, the graph and matrix representations of the new SVN covering rough set model are investigated. Finally, we propose a novel method for decision making (DM) problems in paper defect diagnosis under the new SVN covering rough set model.

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.

We present three types of polyhedral surfaces, which are continuously flexible and have not only an initial pose, where all faces are coplanar, but pass during their self-motion through another pose with coplanar faces (“flat pose”). These surfaces are examples of so-called rigid origami, since we only admit exact flexions, i.e., each face remains rigid during the motion, only the dihedral angles vary. We analyze the geometry behind Miura-ori and address Kokotsakis’ example of a flexible tessellation with the particular case of a cyclic quadrangle. Finally, we recall Bricard’s octahedra of Type 3 and their relation to strophoids.

In this paper, it is explicitly determined the r-dynamic chromatic number of the direct product of any given path with either a complete graph or a wheel graph. Illustrative examples are shown for each one of the cases that are studied throughout the paper. Junta de Andalucía FQM-016

In this paper, variational discretization directed against the optimal control problem governed by nonlinear parabolic equations with control constraints is studied. It is known that the a priori error estimates is $|||u-u_h|||_{L^\infty(J; L^2(\Omega))} = O(h+k)$ using backward Euler method for standard finite element. In this paper, the better result $|||u-u_h|||_{L^\infty(J; L^2(\Omega))} = O(h^2+k)$ is gained. Beyond that, we get a posteriori error estimates of residual type.

This paper establishes a model of continuous-state branching processes with time inhomogeneous competition in Levy random environments. Some results on extinction are presented, including the distribution of the extinction time, the limiting distribution conditioned on large extinction times and the asymptotic behavior near extinction. This paper also provides a new time-space transformation which can be used for further exploration in similar models. The results are applied to an epidemic model to describe the dynamics of infectious population and a virus model to describe the dynamics of viral load.

The aim of this paper is to generalize the results regarding fractals and prove escape conditions for general complex polynomial. In this paper we state the orbit of a newly defined iterative scheme and establish the escape criteria in fractal generation for general complex polynomial. We use established escape criteria in algorithms to generate Mandelbrot and Multi-corns sets. In addition, we present some graphs of quadratic, cubic and higher Mandelbrot and Multi-corns sets and discuss how the alteration in parameters make changes in graphs.

In this paper, several oscillation criteria for a class of higher order dynamic equations with superlinear neutral term are established. The proposed results provide a unified platform that adequately covers both discrete and continuous equations and further sufficiently comments on oscillatory behavior of more general class of equations than the ones reported in the literature. We conclude the paper by demonstrating illustrative examples.

Covering rough set is a classical generalization of rough set. As covering rough set is a mathematical tool to deal with incomplete and incomplete data, it has been widely used in various fields. The aim of this paper is to extend the covering rough sets to interval neutrosophic sets, which can make multi-attribute decision making problem more tractable. Interval neutrosophic covering rough sets can be viewed as the bridge connecting Interval neutrosophic sets and covering rough sets. Firstly, the paper introduces the definition of interval neutrosophic sets and covering rough sets, where the covering rough set is defined by neighborhood. Secondly, Some basic properties and operation rules of interval neutrosophic sets and covering rough sets are discussed. Thirdly, the definition of interval neutrosophic covering rough sets are proposed. Then, some theorems are put forward and their proofs of interval neutrosophic covering rough sets also be gived. Lastly, this paper gives a numerical example to apply the interval neutrosophic covering rough sets.

In this paper a new approach to the use of kernel operators derived from fractional order differential equations is proposed. Three different types of kernels are used, power law, exponential decay and Mittag-Leffler kernels. The kernel's fractional order and fractal dimension are the key parameters for these operators. The main objective of this paper is to study the effect of the fractal-fractional derivative order and the order of the nonlinear term, 1

In the present paper, we consider an important problem from the point of view of application in sciences and mechanic, namely, a class of $ p(x) $-Laplacian type parabolic equation with weak-viscoelasticity. Here, we are concerned with global in time non-existence under suitable conditions on the exponents $ q(x) $ and $ p(x) $ with positive initial energy. We show that the weak-memory term is unable to stabilize problem (1.2) under conditions (1.5) and (1.7). Our main interest in this paper arose in the first place in consequence of a query to blow-up phenomenon.

In this paper, the invariance of separation in covering approximation spaces are discussed. This paper proves that some separations in covering approximation spaces are invariant to reducts of coverings, invariant to covering approximation subspaces and invariant under CAP -transformations of covering approximation spaces. These results deepen and enrich theory of separations in covering approximation spaces, which is helpful to give further researches and applications of Pawlak rough set theory in information sciences.

Spinors permeate all of modern physics and have also an important place in mathematics. Spinors are used intensively in modern theoretical physics and differential geometry. In this study, spinors are used for a different representation of differential geometry in $ \mathbb{E}^3 $. The goal of this study is also the spinor structure lying at the basis of differential geometry. In this paper, firstly, spinors are introduced algebraically. Then, the spinor construction of Bertrand curves is defined. Moreover, the angle notion for these spinors is given. In this way, a different geometric construction of spinors is established in this paper.

In this paper, several existence results of strongly essential set of the solution set for Ky Fan's section problems and vector Ky Fan's point problems are obtained. Firstly, two kinds of strongly essential sets of Ky Fan's section problems are defined, and some further results on existence of the strongly essential component of solutions set of Ky Fan's section problems are proved, which generalize the conclusion in [ 22 ], and further generalize the conclusions in [ 21 , 28 ]. Secondly, based on the above results, two classes of stronger perturbations of vector-valued inequality functions are proposed respectively, and several existence results of the strongly essential component of set of vector Ky Fan's points are obtained. By comparing several metrics, we give some strong and weak relationships among the various metrics involved in the text. The main results of this paper actually generalize the relevant conclusions in the current literature. Finally, as an application, we obtain an existence result of the strongly essential component of weakly Pareto-Nash equilibrium for multiobjective games.

This paper is concerned with the existence of periodic wave solutions for a type of non-Newtonian filtration equations with an indefinite singularity. A sufficient criterion for the existence of periodic wave solutions for non-Newtonian filtration equation is provided via an innovative method of combining a new continuation theorem with coincidence degree theory as well as mathematical analysis skills. The novelty of the present paper is that it is the first time to discuss the existence of periodic wave solutions for the indefinite singular non-Newtonian filtration equations. Finally, two numerical examples are presented to illustrate the effectiveness and feasibility of the proposed criterion in the present paper.

The new schemes of calculation of double integrals and triple integrals are proposed in this paper. The formulas in which the double integral is converted into a line integral with respect to the arc length, and the triple integral is converted into a surface integral with respect to the area or a line integral with respect to the arc length are given separately. The effectiveness of the proposed methods is verified by several examples. Under certain conditions, these methods become the normal iterated integrals in Cartesian coordinate system or polar coordinate system, and the commonly used triple iterated integrals in Cartesian coordinate system, Cylindrical coordinate system or Spherical coordinate system. The transformation calculation method promoted in this paper points out the intrinsic relationship among double integral, triple integral, line integral and surface integral, which further enriches the theories of multivariate integrals.

The paper deals with the existence of solutions for(p,Q)(p,Q)coupled elliptic systems in the Heisenberg group, with critical exponential growth at infinity and singular behavior at the origin. We derive existence of nonnegative solutions with both components nontrivial and different, that is solving an actual system, which does not reduce into an equation. The main features and novelties of the paper are the presence of a general coupled critical exponential term of the Trudinger-Moser type and the fact that the system is set inℍn{{\mathbb{H}}}^{n}.

Motivated by the stereological problem of volume estimation from parallel section profiles, the so-called Newton-Cotes integral estimators based on random sampling nodes are analyzed. These estimators generalize the classical Cavalieri estimator and its variant for non-equidistant sampling nodes, the generalized Cavalieri estimator, and have typically a substantially smaller variance than the latter. The present paper focuses on the following points in relation to Newton-Cotes estimators: the treatment of dropouts, the construction of variance estimators, and, finally, their application in volume estimation of convex bodies.Dropouts are eliminated points in the initial stationary point process of sampling nodes, modeled by independent thinning. Among other things, exact representations of the variance are given in terms of the thinning probability and increments of the initial points under two practically relevant sampling models.The paper presents a general estimation procedure for the variance of Newton-Cotes estimators based on the sampling nodes in a bounded interval. Finally, the findings are illustrated in an application of volume estimation for three-dimensional convex bodies with sufficiently smooth boundaries. Motivated by the stereological problem of volume estimation from parallel section profiles, the so-called Newton-Cotes integral estimators based on random sampling nodes are analyzed. These estimators generalize the classical Cavalieri estimator and its variant for non-equidistant sampling nodes, the generalized Cavalieri estimator, and have typically a substantially smaller variance than the latter. The present paper focuses on the following points in relation to Newton-Cotes estimators: the treatment of dropouts, the construction of variance estimators, and, finally, their application in volume estimation of convex bodies.Dropouts are eliminated points in the initial stationary point process of sampling nodes, modeled by independent thinning. Among other things, exact representations of the variance are given in terms of the thinning probability and increments of the initial points under two practically relevant sampling models.The paper presents a general estimation procedure for the variance of Newton-Cotes estimators based on the sampling nodes in a bounded interval. Finally, the findings are illustrated in an application of volume estimation for three-dimensional convex bodies with sufficiently smooth boundaries.

In this paper, we introduce a version of the Moebius function and other special functions on a particular class of intervals for groupoids, and study them to obtain results analogous to those obtained in the usual lattice, combinatorics and number theory setting, but of course much more general due to the viewpoint taken in this paper.

In this paper, first, we find the integrability theorems for the invariant and slant distributions which appeared in the concept of semi-slant submanifolds. Utilizing these theorems, we prove that a semi-slant submanifold reduces to be a warped product semi-slant submanifold, provided some necessary and sufficient conditions concerning the shape operators. Also, it is shown that a few earlier results are exceptional cases of this paper results.

The aim of the paper is to propose a new Reflected Power function distribution (RPFD). We provide the various properties of the new model in detail such as moments, vitality function and order statistics. We characterize the RPFD based on conditional moments (Right and Left Truncated mean) and doubly truncated mean. We also study the shape of the new distribution to be applicable in many real life situations. We estimate the parameters for the proposed RPFD by using different methods such as maximum likelihood method, modified maximum likelihood method, percentile estimator and modified percentile estimator. The aim of the study is to increase the application of the Power function distribution (PFD). Using two different data sets from real life, we conclude that the RPFD perform better as compare to different competitor models already exist in the literature. We hope that the findings of this paper will be useful for researchers in different field of applied sciences.

Crude oil being a significant source of energy, change of crude oil price can affect the global economy. In this paper, a new approach based on the intuitionistic fuzzy set theory has been implemented to predict the crude oil price. This paper presents the intuitionistic fuzzy time series forecasting algorithm to enhance the efficacy of time series forecasting which includes fuzzy c-means clustering to obtain the optimal cluster centers. Further, a computational technique is proposed for the construction of triangular fuzzy sets and these fuzzy sets are converted to intuitionistic fuzzy sets with the help of Sugeno type intuitionistic fuzzy generator. The popular benchmark dataset of West Texas Intermediate crude oil spot price is used for the validation process. The numerical results when compared with existing methods notify that the proposed method enhances the accuracy of the crude oil price forecasts.

In this study, at first we prove that the existence of best proximity points for cyclic nonexpansive mappings is equivalent to the existence of best proximity pairs for noncyclic nonexpansive mappings in the setting of strictly convex Banach spaces by using the projection operator. In this way, we conclude that the main result of the paper [Proximal normal structure and nonexpansive mappings, Studia Math. 171 (2005), 283–293] immediately follows. We then discuss the convergence of best proximity pairs for noncyclic contractions by applying the convergence of iterative sequences for cyclic contractions and show that the convergence method of a recent paper [Convergence of Picard's iteration using projection algorithm for noncyclic contractions, Indag. Math. 30 (2019), no. 1, 227–239] is obtained exactly from Picard’s iteration sequence.

In this paper, we discuss the existence of solutions to a boundary value problem of differential equations of variable order, which is a piecewise constant function. Our results are based on the Schauder fixed point theorem. Then, under some assumptions on the nonlinear term, we obtain a generalized Lyapunov-type inequality to the two-point boundary value problem considered. To the best of our knowledge, there is no paper dealing with Lyapunov-type inequalities for boundary value problems in term of variable order. In addition, some examples of the obtained inequalities are given.

In the late 1960s and early 1970s, a new class of problems appeared in the theory of optimal control. It was determined that the structure of a number of systems or processes is not homogeneous and can change over time. Therefore, new mathematical models of heterogeneous structure have been developed. Research methods for this type of system vary widely, reflecting various scientific schools and thought. One of the proposed options was to develop an approach that retains the traditional assumptions of optimal control theory. Its basis is Krotov's sufficient optimality conditions for discrete systems, formulated in terms of arbitrary sets and mappings. One of the classes of heterogeneous systems is considered in this paper: discretecontinuous systems (DCSs). DCSs are used for case where all the homogeneous subsystems of the lower level are not only connected by a common functional but also have their own goals. In this paper a generalization of Krotov's sufficient optimality conditions is applied. The foundational theory is the Krotov method of global improvement, which was originally proposed for discrete processes. The advantage of the proposed method is that its conjugate system of vector-matrix equations is linear; hence, its solution always exists, which allows us to find the desired solution in the optimal control problem for DCSs.

In this paper, we consider the solvability for boundary value problems of nonlinear fractional differential equations with mixed perturbations of the second type. The expression of the solution for the boundary value problem of nonlinear fractional differential equations with mixed perturbations of the second type is discussed based on the definition and the property of the Caputo differential operators. By the fixed point theorem in Banach algebra due to Dhage, an existence theorem for the boundary value problem of nonlinear fractional differential equations with mixed perturbations of the second type is given under mixed Lipschitz and Caratheodory conditions. As an application, an example is presented to illustrate the main results. Our results in this paper extend and improve some well-known results. To some extent, our work fills the gap on some basic theory for the boundary value problems of fractional differential equations with mixed perturbations of the second type involving Caputo differential operator.

Let w(x) be the Laguerre weight function, 1 ≤ p < ∞, and Lpw be the space of functions f, p-th power of which is integrable with the weight function w(x) on the non-negative axis. For a given positive integer r, let denote by WrLpw the Sobolev space, which consists of r−1 times continuously differentiable functions f, for which the (r−1)-st derivative is absolutely continuous on an arbitrary segment [a, b] of non-negative axis, and the r-th derivative belongs to the space Lpw. In the case when p = 2 we introduce in the space WrL2w an inner product of Sobolev-type, which makes it a Hilbert space. Further, by lαr,n(x), where n = r, r + 1, ..., we denote the polynomials generated by the classical Laguerre polynomials. These polynomials together with functions lαr,n(x) = xn / n! , where n = 0, 1, r − 1, form a complete and orthonormal system in the space WrL2w. In this paper, the problem of uniform convergence on any segment [0,A] of the Fourier series by this system of polynomials to functions from the Sobolev space WrLpw is considered. Earlier, uniform convergence was established for the case p = 2. In this paper, it is proved that uniform convergence of the Fourier series takes place for p > 2 and does not occur for 1 ≤ p < 2. The proof of convergence is based on the fact that WrLpw ⊂ WrL2w for p > 2. The divergence of the Fourier series by the example of the function ecx using the asymptotic behavior of the Laguerre polynomials is established.

The study of higher order energy functionals was first proposed by Eells and Sampson in 1965 and, later, by Eells and Lemaire in 1983. These functionals provide a natural generalization of the classical energy functional. More precisely, Eells and Sampson suggested the investigation of the so-called $ES-r$-energy functionals $ E_r^{ES}(\varphi)=(1/2)\int_{M}\,|(d^*+d)^r (\varphi)|^2\,dV$, where $r \geq 2 $ and $ \varphi:M \to N$ is a map between two Riemannian manifolds. The initial part of this paper is a short overview on basic definitions, properties, recent developments and open problems concerning the functionals $ E_r^{ES}(\varphi)$ and other, equally interesting, higher order energy functionals $E_r(\varphi)$ which were introduced and studied in various papers by Maeta and other authors. If a critical point $\varphi$ of $E_r^{ES}(\varphi)$ (respectively, $E_r(\varphi)$) is an \textit{isometric immersion}, then we say that its image is an $ES-r$-harmonic (respectively, $r$-harmonic) submanifold of $N$. We observe that \textit{minimal} submanifolds are trivially both $ES-r$-harmonic and $r$-harmonic. Therefore, it is natural to say that an $ES-r$-harmonic ($r$-harmonic) submanifold is \textit{proper} if it is not minimal. In the special case that the ambient space $N$ is the Euclidean space $\mathbb{R}^n$ the notions of $ES-r$-harmonic and $r$-harmonic submanifolds coincide. The Chen-Maeta conjecture is still open: it states that, for all $r \geq2$, any proper, $r$-harmonic submanifold of $\mathbb{R}^n$ is minimal. In the second part of this paper we shall focus on the study of $G={\rm SO}(p+1) \times {\rm SO}(q+1)$-invariant submanifolds of $\mathbb{R}^n$, $n=p+q+2$. In particular, we shall obtain an explicit description of the relevant Euler-Lagrange equations in the case that $r=3$ and we shall discuss difficulties and possible developments towards the proof of the Chen-Maeta conjecture for $3$-harmonic $G$-invariant hypersurfaces.

In this paper, we introduce the notion of an orthogonal (F, ψ)-contraction of Hardy-Rogers-type mapping and prove some fixed point theorem for such contraction mappings in orthogonally metric spaces. Our result extend and improve the main result of the paper by Sawangsup et al. [Fixed point theorems for orthogonal F-contraction mappings on O-complete metric spaces, J. Fixed Point Theory Appl. (2020) 22:10].

The focus of this paper is to present a new idea of fractional convex type contraction and establish some new results for such contraction under the improved approach of fractional convex type contractive condition in the context of $\mathcal{F}$ -complete $\mathcal{F}$ -metric space. The authors derive some results for Suzuki type contractions, orbitally T-complete and orbitally continuous mappings in $\mathcal{F}$ -metric spaces and obtain some consequences by using graphic contraction. The motivation of this paper is to observe the solution of fractional order differential equation with one of the boundary condition using fixed point technique in $\mathcal{F}$ -metric space.

In 1999, B. Y. Chen established a sharp inequality between the Ricci curvature and the squared mean curvature for an arbitrary Riemannian submanifold of a real space form. This inequality was extended in 2015 by M. E. Aydin et al. to the case of statistical submanifolds in a statistical manifold of constant curvature, obtaining a lower bound for the Ricci curvature of the dual connections. Also, the similar inequality for submanifolds in statistical manifolds of quasi-constant curvature studied by H. Aytimur and C. Ozgur in their recent article. In the present paper, we give a different proof of the same inequality but working with the statistical curvature tensor field, instead of the curvature tensor fields with respect to the dual connections. A geometric inequality can be treated as an optimization problem. The new proof is based on a simple technique, known as Oprea’s optimization method on submanifolds, namely analyzing a suitable constrained extremum problem. We also provide some examples. This paper finishes with some conclusions and remarks.

A new theory of analytic functions has been recently introduced in the sense of conformable fractional derivative. In addition, the concept of fractional contour integral has also been developed. In this paper, we propose and prove some new results on complex fractional integration. First, we establish necessary and sufficient conditions for a continuous function to have antiderivative in the conformable sense. Finally, some of the well-known Cauchy´s integral theorems will also be the subject of the extension that we do in this paper.

In this paper, we define and consider some new concepts of the higher order strongly general convex functions with respect to an arbitrary function. Some properties of the higher order strongly general convex functions are investigated under suitable conditions. It is shown that the optimality conditions of the higher order strongly general convex functions are characterized by a class of variational inequalities, which is called the higher order strongly variational inequality. Auxiliary principle technique is used to suggest an implicit method for solving strongly general variational inequalities. Convergence analysis of the proposed method is investigated using the pseudo-monotonicity of the operator. It is shown that the parallelogram laws for Banach spaces can be obtained as applications of higher order strongly affine convex functions. Some special cases also discussed. Results obtained in this paper can be viewed as refinement and improvement of previously known results.

This paper studies the mixed passive and $ H_{\infty} $ performance for uncertain neural networks with interval discrete and distributed time-varying delays via feedback control. The interval discrete and distributed time-varying delay functions are not assumed to be differentiable. The improved criteria of exponential stability with a mixed passive and $ H_{\infty} $ performance are obtained for the uncertain neural networks by constructing a Lyapunov-Krasovskii functional (LKF) comprising single, double, triple, and quadruple integral terms and using a feedback controller. Furthermore, integral inequalities and convex combination technique are applied to achieve the less conservative results for a special case of neural networks. By using the Matlab LMI toolbox, the derived new exponential stability with a mixed passive and $ H_{\infty} $ performance criteria is performed in terms of linear matrix inequalities (LMIs) that cover $ H_{\infty} $, and passive performance by setting parameters in the general performance index. Numerical examples are shown to demonstrate the benefits and effectiveness of the derived theoretical results. The method given in this paper is less conservative and more general than the others.

If $G$ is a connected graph, the $distance$ $d(u, v)$ between two vertices $u, v \in V(G)$ is the length of a shortest path between them. Let $W = \{w_1,w_2, \dots ,w_k\}$ be an ordered set of vertices of $G$ and let $v$ be a vertex of $G$. The $representation$ $r(v|W)$ of $v$ with respect to $W$ is the k-tuple $(d(v,w_1), d(v,w_2), \dots , d(v,w_k))$. $W$ is called a $resolving set$ or a $locating set$ if every vertex of $G$ is uniquely identified by its distances from the vertices of $W$, or equivalently if distinct vertices of $G$ have distinct representations with respect to $W$. A resolving set of minimum cardinality is called a $metric basis$ for $G$ and this cardinality is the $metric dimension$ of $G$, denoted by $\beta(G)$. The metric dimension of some wheel related graphs is studied recently by Siddiqui and Imran. In this paper, we study the metric dimension of wheels with $k$ consecutive missing spokes denoted by $W(n,k)$. We compute the exact value of the metric dimension of $W(n,k)$ which shows that wheels with consecutive missing spokes have unbounded metric dimensions. It is natural to ask for the characterization of graphs with an unbounded metric dimension. The exchange property for resolving a set of $W(n,k)$ has also been studied in this paper and it is shown that exchange property of the bases in a vector space does not hold for minimal resolving sets of wheels with $k$-consecutive missing spokes denoted by $W(n,k)$.

In this paper, we obtain the upper bounds for the n-th (n ≥ 1) coefficients for meromorphic bi-subordinate functions of complex order by using Faber polynomial expansions. The results, which are presented in this paper, would generalize those in related works of several earlier authors.

In this paper, we established some common fixed point theorems of four self-mappings in completed quaternion valued $G-$metric space. Moreover, we gave an example of completed quaternion valued $G-$metric space and example for supporting our main results. The results obtained in this paper extend and improve some recent results.

This paper is devoted to a partial differential equation approach to the problem of construction of Feller semigroups associated with one-dimensional diffusion processes with boundary conditions in theory of stochastic processes. In this paper we investigate the boundary-value problem for a one-dimensional linear parabolic equation of the second order (backward Kolmogorov equation) in curvilinear bounded domain with one of the variants of nonlocal Feller-Wentzell boundary condition. We restrict our attention to the case when the boundary condition has only one term and it is of the integral type. The classical solution of the last problem is obtained by the boundary integral equation method with the use of the fundamental solution of backward Kolmogorov equation and the associated parabolic potentials. This solution is used to construct the Feller semigroup corresponding to such a diffusion phenomenon that a Markovian particle leaves the boundary of the domain by jumps.

In the paper, we solve one extremal problem of the theory of approximation of functional classes by linear methods. Namely, questions are investigated concerning the approximation of classes of differentiable functions by $\lambda$-methods of summation for their Fourier series, that are defined by the set $\Lambda =\{{{\lambda }_{\delta }}(\cdot )\}$ of continuous on $\left[ 0,\infty \right)$ functions depending on a real parameter $\delta$. The Kolmogorov-Nikol'skii problem is considered, that is one of the special problems among the extremal problems of the theory of approximation. That is, the problem of finding of asymptotic equalities for the quantity $$\mathcal{E}{{\left( \mathfrak{N};{{U}_{\delta}} \right)}_{X}}=\underset{f\in \mathfrak{N}}{\mathop{\sup }}\,{{\left\| f\left( \cdot \right)-{{U}_{\delta }}\left( f;\cdot;\Lambda \right) \right\|}_{X}},$$ where $X$ is a normalized space, $\mathfrak{N}\subseteq X$ is a given function class, ${{U}_{\delta }}\left( f;x;\Lambda \right)$ is a specific method of summation of the Fourier series. In particular, in the paper we investigate approximative properties of the three-harmonic Poisson integrals on the Weyl-Nagy classes. The asymptotic formulas are obtained for the upper bounds of deviations of the three-harmonic Poisson integrals from functions from the classes $W^{r}_{\beta,\infty}$. These formulas provide a solution of the corresponding Kolmogorov-Nikol'skii problem. Methods of investigation for such extremal problems of the theory of approximation arised and got their development owing to the papers of A.N. Kolmogorov, S.M. Nikol'skii, S.B. Stechkin, N.P. Korneichuk, V.K. Dzyadyk, A.I. Stepanets and others. But these methods are used for the approximations by linear methods defined by triangular matrices. In this paper we modified the mentioned above methods in order to use them while dealing with the summation methods defined by a set of functions of a natural argument.

This paper aims to establish adequate conditions that are intended for the oscillation of every solution of the second order advanced type half-linear difference equations with noncanonical form. Initially, we derive a sufficient condition that ensures all solutions of the studied equation are either oscillatory or tending to zero. Secondly, we obtain a criteria for the oscillation of all solutions of the studied equation. These criteria are obtained by using Riccati transformation and summation averaging method. The results established in this paper in essence complement, extend and enhance the existing outcomes recorded in the literature. The improvement of our main results are illustrated through three examples.

This paper aims to present a new pathwise approximation method, which gives approximate solutions of order $\begin{array}{} \displaystyle \frac{3}{2} \end{array}$ for stochastic differential equations (SDEs) driven by multidimensional Brownian motions. The new method, which assumes the diffusion matrix non-degeneracy, employs the Runge-Kutta method and uses the Itô-Taylor expansion, but the generating of the approximation of the expansion is carried out as a whole rather than individual terms. The new idea we applied in this paper is to replace the iterated stochastic integrals Iα by random variables, so implementing this scheme does not require the computation of the iterated stochastic integrals Iα. Then, using a coupling which can be found by a technique from optimal transport theory would give a good approximation in a mean square. The results of implementing this new scheme by MATLAB confirms the validity of the method.

The presented paper is strictly connected, among others, with the paper On the sum of some alternating series, Comp. Math. Appl. (2011), written by Wituła and Słota. A problem concerning the form of determinants formulated in the cited paper is solved here. Next, the obtained result is adapted to solve some system of linear equations and the description of the sum of alternating series.

In this paper, we consider the improved estimation of average production of peppermint at block level of Barabanki district of Uttar Pradesh State (India). We suggest certain estimators for population-mean. Here, population refers to production population as study variable and auxiliary-variable refers to Area of field. We study the sampling properties naming bias and MSE of estimators, which are presently proposed by us in the paper. We compare our proposed estimators with other ones existing in literature. For the support of the theoretical findings, we carry out a numerical study for the natural population on primary data collected from Banikodar Block of Barabanki District situated in Uttar Pradesh State.

In this paper the linear representation of a graph is defined. A linear representation of a graph is a subgroup of $GL(p,\mathbb{R})$, the group of invertible matrices of order $ p $ and real coefficients. It will be demonstrated that every graph admits a linear representation. In this paper, simple and finite graphs will be used, framed in the graphs theory's area

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).

This paper describes a valuable linear yaw-roll tractor-semitrailer (TST) model with five-degree-of-freedom (DOFs) for control algorithm development when steering and braking. The key parameters, roll stiffness, axle cornering stiffness, and fifth-wheel stiffness, are identified by the genetic algorithm (GA) and multistage genetic algorithm (MGA) based on TruckSim outputs to increase the accuracy of the model. Thus, the key parameters of the simplified model can be modified according to the real-time vehicle states by online lookup table and interpolation. The TruckSim vehicle model is built referring to the real tractor (JAC-HFC4251P1K7E33ZTF6×2) and semitrailer (Luyue LHX9406) used in the field test later. The validation of the linear yaw-roll model of a tractor-semitrailer using field test data is presented in this paper. The field test in the performance testing ground is detailed, and the test data of roll angle, roll rate, and yaw rate are compared with the outputs of the model with maps of the key parameters. The results indicate that the error of the tractor’s roll angle and semitrailer’s roll angle between model data and test data is 1.13% and 1.24%, respectively. The roll rate and yaw rate of the tractor and semitrailer are also in good agreement.

The main goal of this paper is to introduce the notion of stabilizers in EQ-algebras and develop stabilizer theory in EQ-algebras. In the paper, we introduce (fuzzy) left and right stabilizers and investigate some related properties of them. Then, we discuss the relations among (fuzzy) stabilizers, (fuzzy) prefilters (filters) and (fuzzy) co-annihilators. Also, we obtain that the set of all prefilters in a good EQ-algebra forms a relative pseudo-complemented lattice, where Str(F, G) is the relative pseudo-complemented of F with respect to G. These results will provide a solid algebraic foundation for the consequence connectives in higher fuzzy logic.