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In a recent paper [4], Xu studied some alternating multiple zeta values. In particular, he gave two recurrence formulas of alternating multiple zeta values $ \zeta^\star({\bar 1}, \{1\}_{m-1}, {\bar 1}) $ and $ \zeta^\star(2, \{1\}_{m-1}, {\bar 1}) $. In this paper, we will give the closed forms representations of $ \zeta^\star({\bar 1}, \{1\}_{m-1}, {\bar 1}) $ and $ \zeta^\star(2, \{1\}_{m-1}, {\bar 1}) $ in terms of single zeta values and polylogarithms.

In this paper, we prove some new Simpson's type inequalities for partial $ (p, q) $-differentiable convex functions of two variables in the context of $ (p, q) $-calculus. We also show that the findings in this paper are generalizations of comparable findings in the literature.

In this paper, we derive the explicit analytical solution of incommensurate fractional differential equation systems with fractional order $ 1 < \alpha, \beta < 2 $. The derivation is extended from a recently published paper by Huseynov et al. in [1], which is limited for incommensurate fractional order $ 0 < \alpha, \beta < 1 $. The incommensurate fractional differential equation systems were first converted to Volterra integral equations. Then, the Mittag-Leffler function and Picard's successive approximations were used to obtain the analytical solution of incommensurate fractional order systems with $ 1 < \alpha, \beta < 2 $. The solution will be simplified via some combinatorial concepts and bivariate Mittag-Leffler function. Some special cases will be discussed, while some examples will be given at the end of this paper.

In this paper, we investigate the Hyers-Ulam stability of additive and Fréchet functional equations on restricted domains. We improve the bounds and thus the results obtained by S. M. Jung and J. M. Rassias. As a consequence, we obtain asymptotic behaviors of functional equations of different types. One of the objectives of this paper is to bring out the involvement of functional equations in various characterizations of inner product spaces.

The paper is devoted to the integration of the loaded modified Kortewegde Vries equation in the class of rapidly decreasing functions. It is well known that loaded differential equations in the literature are usually called equations containing in the coefficients or in the right-hand side any functionals of the solution, in particular, the values of the solution or its derivatives on manifolds of lower dimension. In this paper, we consider the Cauchy problem for the loaded modified Korteweg-de Vries equation. The problem is solved using the inverse scattering method, i.e. the evolution of the scattering data of a non-self-adjoint Dirac operator is derived, the potential of which is a solution to the loaded modified Korteweg-de Vries equation in the class of rapidly decreasing functions. A specific example is given to illustrate the application of the results obtained.

This paper has been designed to introduce the method for solving the Bi-level Multi-objective (BL-MO) Fully Quadratic Fractional Optimization Model through Fuzzy Goal Programming (FGP) approach by utilising non-linear programming. In Fully Quadratic Fractional Optimization Model, the objective functions are in fractional form, having quadratic functions in both numerator and denominator subject to quadratic constraints set. The motive behind this paper is to provide a solution to solve the BL-MO optimization model in which number of decision-makers (DM) exists at two levels in the hierarchy. First, the fractional functions with fuzzy demand, which are in the form of fuzzy numbers, are converted into crisp models by applying the concept of α-cuts. After that, membership functions are developed which are corresponding to each decision-maker’s objective and converted into simpler form to avoid complications due to calculations. Finally, the model is simplified by applying FGP approach, and a compromised solution to the initial model is obtained. An algorithm, flowchart and example are also given at the end to explain the study of the proposed model.

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, it is derived a rich hierarchy for the Toda lattice with a selfconsistent source in the class of periodic functions. We discuss the complete integrability of the constructed systems that is based on the transformation to the spectral data of an associated discrete Hill‘s equation with periodic coefficients. In particular, Dubrovintype equations are derived for the time-evolution of the spectral data corresponding to the solutions of any system in the hierarchy. At the end of the paper, we illustrate our theory on concrete example with analytical and numerical results.

The dynamic transitions of the Brusselator model has been recently analyzed in Y. Choi et’al (2021) and T. Ma, S. Wang (2011). Our aim in this paper is to address the relation between the pattern formation and dynamic transition results left open in those papers. We consider the problem in the setting of a 2D rectangular box where an instability of the homogeneous steady state occurs due to the perturbations in the direction of several modes becoming critical simultaneously. Our main results are two folds: (1) a rigorous characterization of the types and structure of the dynamic transitions of the model from basic homogeneous states and (2) the relation between the dynamic transitions and the pattern formations. We observe that the Brusselator model exhibits different transition types and patterns depending on the nonlinear interactions of the pattern of the critical modes.

When the reduced twisted $C^*$-algebra $C^*_r(\mathcal{G}, c)$ of a non-principal groupoid $\mathcal{G}$ admits a Cartan subalgebra, Renault's work on Cartan subalgebras implies the existence of another groupoid description of $C^*_r(\mathcal{G}, c)$. In an earlier paper, joint with Reznikoff and Wright, we identified situations where such a Cartan subalgebra arises from a subgroupoid $\mathcal{S}$ of $\mathcal{G}$. In this paper, we study the relationship between the original groupoids $\mathcal{S}, \mathcal{G}$ and the Weyl groupoid and twist associated to the Cartan pair. We first identify the spectrum $\mathfrak{B}$ of the Cartan subalgebra $C^*_r(\mathcal{S}, c)$. We then show that the quotient groupoid $\mathcal{G}/\mathcal{S}$ acts on $\mathfrak{B}$, and that the corresponding action groupoid is exactly the Weyl groupoid of the Cartan pair. Lastly we show that, if the quotient map $\mathcal{G}\to\mathcal{G}/\mathcal{S}$ admits a continuous section, then the Weyl twist is also given by an explicit continuous $2$-cocycle on $\mathcal{G}/\mathcal{S} \ltimes \mathfrak{B}$., 32 pages

This paper considers a continuous-time quasi birth-death (qbd) process, which informally can be seen as a birth-death process of which the parameters are modulated by an external continuous-time Markov chain. The aim is to numerically approximate the time-dependent distribution of the resulting bivariate Markov process in an accurate and efficient way. An approach based on the Erlangization principle is proposed and formally justified. Its performance is investigated and compared with two existing approaches: one based on numerical evaluation of the matrix exponential underlying the qbd process, and one based on the uniformization technique. It is shown that in many settings the approach based on Erlangization is faster than the other approaches, while still being highly accurate. In the last part of the paper, we demonstrate the use of the developed technique in the context of the evaluation of the likelihood pertaining to a time series, which can then be optimized over its parameters to obtain the maximum likelihood estimator. More specifically, through a series of examples with simulated and real-life data, we show how it can be deployed in model selection problems that involve the choice between a qbd and its non-modulated counterpart.

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.

In this paper, we introduce a new type of a generalized split monotone variational inclusion (GSMVI) problem in the framework of real Hilbert spaces. By incorporating an inertial extrapolation method and an Halpern iterative technique, we establish a strong convergence result for approximating a solution of GSMVI and fixed point problems of certain nonlinear mappings in the framework of real Hilbert spaces. Many existing results are derived as corollaries to our main result. Furthermore, we present a numerical example to support our main result and propose an open problem for interested researchers in this area. The result obtained in this paper improves and generalizes many existing results in the literature.

so-metrizable spaces are a class of important generalized metric spaces between metric spaces and s n sn -metrizable spaces where a space is called an so-metrizable space if it has a σ \sigma -locally finite so-network. As the further work that attaches to the celebrated Alexandrov conjecture, it is interesting to characterize so-metrizable spaces by images of metric spaces. This paper gives such characterizations for so-metrizable spaces. More precisely, this paper introduces so-open mappings and uses the “Pomomarev’s method” to prove that a space X X is an so-metrizable space if and only if it is an so-open, compact-covering, σ \sigma -image of a metric space, if and only if it is an so-open, σ \sigma -image of a metric space. In addition, it is shown that so-open mapping is a simplified form of s n sn -open mapping (resp. 2-sequence-covering mapping if the domain is metrizable). Results of this paper give some new characterizations of so-metrizable spaces and establish some equivalent relations among so-open mapping, s n sn -open mapping and 2-sequence-covering mapping, which further enrich and deepen generalized metric space theory.

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.

Fractal is a common feature of many deterministic complex networks. The complex networks with fractal features have interesting structure and good performance. The network based on hypergraph is named hypernetwork. In this paper, we construct a hypernetwork model with fractal properties, and obtain its topological properties. Moreover, according to the exact controllability theory, we obtain the node controllability and the hyperedge controllability of the fractal hypernetwork. The simulation results show that the measure of hyperedge controllability is smaller than that of node in the fractal hypernetwork. In addition, We compare the controllability of three types of hypernetwork, which are easier to control by their hyperedges. It is shown the fractal hypernetwork constructed in this paper has the best controllability. Because of the good controllability of our fractal hypernetwork model, it is suitable for the topology structure of many real systems.

The mean curvature-based image deblurring model is widely used to enhance the quality of the deblurred images. However, the discretization of the associated Euler-Lagrange equations produce a nonlinear ill-conditioned system which affect the convergence of the numerical algorithms like Krylov subspace methods. To overcome this difficulty, in this paper, we present two new symmetric positive definite (SPD) preconditioners. An efficient algorithm is presented for the mean curvature-based image deblurring problem which combines a fixed point iteration (FPI) with new preconditioned matrices to handle the nonlinearity and ill-conditioned nature of the large system. The eigenvalues analysis is also presented in the paper. Fast convergence has shown in the numerical results by using the proposed new preconditioners.

The main objective of this work is to explore and characterize the idea of $ s $-type preinvex function and related inequalities. Some interesting algebraic properties and logical examples are given to support the newly introduced idea. In addition, we attain the novel version of Hermite-Hadamard type inequality utilizing the introduced preinvexity. Furthermore, we establish two new identities, and employing these, we present some refinements of Hermite-Hadamard-type inequality. Some special cases of the presented results for different preinvex functions are deduced as well. Finally, as applications, some new inequalities for the arithmetic, geometric and harmonic means are established. Results obtained in this paper can be viewed as a significant improvement of previously known results. The awe-inspiring concepts and formidable tools of this paper may invigorate and revitalize for additional research in this worthy and absorbing field.

In this paper we consider an optimal control problem governed by a space-time fractional diffusion equation with non-homogeneous initial conditions. A spectral method is proposed to discretize the problem in both time and space directions. The contribution of the paper is threefold: (1) A discussion and better understanding of the initial conditions for fractional differential equations with Riemann-Liouville and Caputo derivatives are presented. (2) A posteriori error estimates are obtained for both the state and the control approximations. (3) Numerical experiments are performed to verify that the obtained a posteriori error estimates are reliable.

The inverse Frobenius-Perron problem (IFPP) is a collective term for a family of problems that requires the construction of an ergodic dynamical system model with prescribed statistical characteristics. Solutions to this problem draw upon concepts from ergodic theory and are scattered throughout the literature across domains such as physics, engineering, biology and economics. This paper presents a survey of the original formulation of the IFPP, wherein the invariant probability density function of the system state is prescribed. The paper also reviews different strategies for solving this problem and demonstrates several of the techniques using examples. The purpose of this survey is to provide a unified source of information on the original formulation of the IFPP and its solutions, thereby improving accessibility to the associated modeling techniques and promoting their practical application. The paper is concluded by discussing possible avenues for future work.

The quantum calculus has emerged as a connection between mathematics and physics. It has wide applications, particularly in quantum mechanics, analytic number theory, combinatorial analysis, operation theory etc. The $ q $-calculus, which serves as a powerful tool to model quantum computing, has drawn attention of many researchers in the field of special functions and as a result the $ q $-analogues of certain special functions, especially hypergeometric function, 1-variable Hermite polynomials, Appell polynomials etc., are introduced and studied. In this paper, we introduce the 2-variable $ q $-Hermite polynomials by means of generating function. Also, its certain properties like series definition, recurrence relations, $ q $-differential equation and summation formulas are established. The operational definition and some integral representations of these polynomials are obtained. Some examples are also considerd to show the efficacy of the proposed method. Some concluding remarks are given. At the end of this paper, the graphical representations of these polynomials of certain degrees with specified values of $ q $ are given.

The main goal of this paper is to develop the significance of generalized fractional integral inequalities via convex functions. We obtain the new version of fractional integral inequalities with the extended Wright generalized Bessel function acting as a kernel for the convex function, which deals with the Hermite-Hadamard type and trapezoid type inequalities. Moreover, we establish new mid-point type and trapezoid type integral inequalities for $ (\eta_{1}, \eta_{2}) $-convex function related to Hermite-Hadamard type inequality. We establish new version of integral inequality for $ (\eta_{1}, \eta_{2}) $-convex function related to Fejer type. The results discussed in this paper are a generalized version of many inequalities in literature.

Lot sizing models involve operational and tactical decisions. These decisions may entail multi-level production processes such as assembly operations with multiple plants and limited capacities. Lot sizing problems are widely recognized as NP-hard problems therefore difficult to solve, especially the ones with multiple plants and capacity constraints. The level of complexity rises to an even higher level when there is an interplant transfer between the plants. This paper presents a Genetic Algorithm (GA) based solution methodology applied to large scale multi-plant capacitated lot sizing problem with interplant transfer (MPCLSP-IT). Although the GA has been a very effective and widely accepted meta-heuristic approach used to solve large scale complex problems, it has not been employed for MPCLSP-IT problem. This paper solves the MPCLSP-IT problem in large scale instances by using a genetic algorithm, and in doing so successfully obtains a better solution in terms of computation time when compared to the results obtained by the other methods such as Lagrangian relaxation, greedy randomized adaptive search procedure (GRASP) heuristics, and GRASP-path relinking techniques used in extant literature.

The present paper aims to find some new midpoint-type inequalities for twice quantum differentiable convex functions. The consequences derived in this paper are unification and generalization of the comparable consequences in the literature on midpoint inequalities. © 2021 Muhammad Aamir Ali et al., published by De Gruyter. National Natural Science Foundation of China, NSFC: 11301127, 11601485, 11626101, 11701176, 11971241, 61673169 Funding information : The work was supported by the Natural Science Foundation of China (Grant Nos. 61673169, 11301127, 11701176, 11626101, 11601485, 11971241). 2-s2.0-85105011594

In this paper we give Hadamard inequalities for exponentially $ (\alpha, h-m) $-convex functions using Riemann-Liouville fractional integrals for strictly increasing function. Results for Riemann-Liouville fractional integrals of convex, $ m $-convex, $ s $-convex, $ (\alpha, m) $-convex, $ (s, m) $-convex, $ (h-m) $-convex, $ (\alpha, h-m) $-convex, exponentially convex, exponentially $ m $-convex, exponentially $ s $-convex, exponentially $ (s, m) $-convex, exponentially $ (h-m) $-convex, exponentially $ (\alpha, h-m) $-convex functions are particular cases of the results of this paper. The error estimations of these inequalities by using two fractional integral identities are also given.

This paper argues that any economic phenomena should be observed by two different scales, and any economic laws are scale-dependent. A one-scale law arising in either macroeconomics or microeconomics might be mathematically correct and economically relevant, however, sparking debates might arise for a different scale. This paper re-analyzes the basic assumptions of the Evans model for dynamic economics, and it concludes that they are quite reasonable on a large time-scale, but the assumptions become totally invalid on a smaller scale, and a fractal modification has to be adopted. A two-scale price dynamics is suggested and a fractal variational theory is established to maximize the profit at a given period. Furthermore Evans 1924 variational principle for the maximal profit is easy to be solved for a quadratic cost function using the Lagrange multiplier method. Here a quadratic-cubic cost function and a nonlinear demand function are used, and the stationary condition of the variational formulation is derived step by step, and a more complex dynamic system is obtained. The present derivation process can be extended to a more complex cost function and a more complex demand function, and the paper sheds a promising light on mathematics treatment of complex economic problems.

The aim of this paper is to study the solvability of solutions of non-local problems with integral conditions in spatial variables for high-order linear parabolic equations in the classes of regular solutions (which have all the squared derivatives generalized by S. L. Sobolev that are included in the corresponding equation) . Previously, similar problems were studied for high-order parabolic equations, either in the one-dimensional case, or when certain conditions of smallness on the coefficients are met equations. In this paper, we present new results on the solvability of non-local problems with integral spatial variables for high-order parabolic equations a) in the multidimensional case with respect to spatial variables; b) in the absence of smallness conditions. The research method is based on the transition from a problem with non-local integral conditions to a problem with classical homogeneous conditions of the first or second kind on the side boundary for a loaded integro-differential equation. At the end of the paper, some generalizations of the obtained results will be described.

In a seminal paper, Stallings introduced folding of morphisms of graphs. One consequence of folding is the representation of finitely-generated subgroups of a finite-rank free group as immersions of finite graphs. Stallings's methods allow one to construct this representation algorithmically, giving effective, algorithmic answers and proofs to classical questions about subgroups of free groups. Recently Dani--Levcovitz used Stallings-like methods to study subgroups of right-angled Coxeter groups, which act geometrically on CAT(0) cube complexes. In this paper we extend their techniques to fundamental groups of non-positively curved cube complexes., 12 pages

In this paper a thorough analysis is carried out of the type of order reduction that Lawson methods exhibit when used to integrate nonlinear initial boundary value problems. In particular, we focus on nonlinear reaction-diffusion problems, and therefore, this study is important in a large number of practical applications modeled by this type of nonlinear equations. A theoretical study of the local and global error of the total discretization of the problem is carried out, taking into account both, the error coming from the space discretization and that due to the integration in time. These results are also corroborated by the numerical experiments performed in this paper., Ministerio de Ciencia e Innovación and Regional Development European Funds, Grant/Award Number: PGC2018-101443-B-I00; Junta de Castilla y León and Feder, Grant/Award Number: VA169P20

The B-spline curves have been grasped tremendous achievements inside the widely identified field of Computer Aided Geometric Design (CAGD). In CAGD, spline functions have been used for the designing of various objects. In this paper, new Quadratic Trigonometric B-spline (QTBS) functions with two shape parameters are introduced. The proposed QTBS functions inherit the basic properties of classical B-spline and have been proved in this paper. The proposed scheme associated with two shape parameters where the classical B-spline functions do not have. The QTBS has been used for designing of different parts of airplane like winglet, airfoil, turbo-machinery blades and vertical stabilizer. The designed part can be controlled or changed using free parameters. The effect of shape parameters is also expressed.

This paper proposes a simple, accurate and effective empirical formula to determine the number of supporting nodes in a newly-developed method, the localized method of fundamental solutions (LMFS). The LMFS has the merits of meshless, high-accuracy and easy-to-simulation in large-scale problems, but the number of supporting nodes has a certain impact on the accuracy and stability of the scheme. By using the curve fitting technique, this study established a simple formula between the number of supporting nodes and the node spacing. Based on the developed formula, the reasonable number of supporting nodes can be determined according to the node spacing. Numerical experiments confirmed the validity of the proposed methodology. This paper perfected the theory of the LMFS, and provided a quantitative selection strategy of method parameters.

This paper focuses on the finite-time anti-synchronization for a class of delayed master-slave inertial neural networks. By means of using the property of quadratic inequality of one variable and designing the fractional and polynomial controllers of time variable, two sufficient conditions to assure the finite-time anti-synchronization for the master-slave delayed inertial neural networks are established. Our controllers designed related to time variable t and the study method on the finite-time anti-synchronization are different from these in the existing papers.

This paper studies uniform convergence rate of Hermite\,--\,Pad\'eapproximants (simultaneous Pad\'e approximants)$\{\pi^j_{n,\overrightarrow{m}}(z)\}_{j=1}^k$for a system of exponential functions $\{e^{\lambda_jz}\}_{j=1}^k$,where $\{\lambda_j\}_{j=1}^k$ are different nonzerocomplex numbers. In the general case a research of theasymptotic properties of Hermite\,--\,Pad\'e approximants is arather complicated problem. This is due to the fact that in theirstudy mainly asymptotic methods are used, in particular,the saddle-point method. An important phase in the applicationof this method is to find a special saddle contour (the Cauchyintegral theorem allows to choose an integration contour ratherarbitrarily), according to which integration should be carriedout. Moreover, as a rule, one has to repy only on intuition.In this paper, we propose a new method to studying the asymptotic properties of Hermite\,--\,Pad\'eapproximants, that is based on the Taylor theorem and heuristicconsiderations underlying the Laplace and saddle-point methods,as well as on the multidimensional analogue of the Van Rossumidentity that we obtained. The proved theorems complement andgeneralize the known results by other authors.

Venture capital exit strategy is a key condition of realizing venture capital appreciation and circular operation. Based on the equity sale method of venture capital exit, this paper explores strategic choices of venture capitalists and venture entrepreneurs for external investment, and constructs a venture capital exit strategy model with the paradigm of classical game theory and quantum game theory, respectively. A series of experiments demonstrated the proposed method can achieve the unification of Nash equilibrium and Pareto equilibrium. Therefore, this paper expands the basic theoretical support and provide practical support for the choice of venture capital exit strategies.

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.

This paper presents a simplified model of the population migration problem, addressed to first-year engineering students in order to show them the use of linear algebra tools. The study consists of predicting the census in the city centre and in the suburbs, determining the city population equilibrium point, and making a sociological interpretation of population flows. This practical problem is part of the seminar “Applications of Linear Algebra in Engineering”, which is being held at the Universitat Politècnica de Catalunya-BarcelonaTech (UPC). This seminar consists in the learning of linear algebra by the implementation of real applications where mathematical tools are required to resolve them. This paper presents an application of linear algebra to the population migration problem and analyses students’ appreciation through anonymous surveys and personal interviews. The surveys assessed students’ motivation towards the subject of linear algebra and their learning of mathematical concepts. Personal interviews were conducted for students in order to let them express in detail their opinion about the seminar. The results confirm that the introduction of real applications in the learning of mathematics increases students’ motivation and involvement, which implies an improvement in students’ performance in the first courses of STEM degrees. Peer Reviewed Objectius de Desenvolupament Sostenible::4 - Educació de Qualitat

The present paper is devoted to studying the maps corresponding to the suborbital graphs for the normalizer ΓB(N) of Γ0(N) modulo N, where N denotes a positive integer. We reveal the complete structure of these maps, finding their vertices, edges, darts, and faces explicitly. The maps we investigated in the present paper were all regular maps of large genus except for some low values of N.

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 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.

The motivation of this paper is to study the uniqueness problems of meromorphic functions concerning differential polynomials that share a small function. The results of the paper improve and generalize the recent results due to Fengrong Zhang and Linlin Wu [13]. We also solve an open problem as posed in the last section of [13].

In this paper, we present a new modelling method to create 3D models. First, characteristic cross section curves are generated and approximated by generalized elliptic curves. Then, a vector-valued sixth-order partial differential equation is proposed, and its closed form solution is derived to create PDE surface patches from cross section curves where two adjacent PDE-surface patches are automatically stitched together. With the approach presented in this paper, C2 continuity between adjacent surface patches is well-maintained. Since surface creation of the model is transformed into the generation of cross sectional curves and few undetermined constants are required to describe cross sectional curves accurately, the proposed approach can save manual operations, reduce information storage, and generate 3D models quickly. This research is supported by the PDE-GIR project, which has received funding from the European Union Horizon 2020 Research and Innovation Programme under the Marie SkodowskaCurie grant agreement No 778035. Andres Iglesias also thanks the project TIN2017-89275-R funded by MCIN/AEI/10.13039/501100011033/FEDER “Una manera de hacer Europa”.

This paper studies the solution procedure of a bus transit system with a rectangular service area that had been cited more than two hundred times. We will point out that they applied relations suitable for continuous variables, which are not held for a discrete variable and will result in invalid results. We provide our solution procedure to the same example proposed by the original paper to illustrate that their results are less accurate. Our findings will help researchers understand this kind of bus transit system.

In this paper, the Painlevé integrable property of the (1 + 1)-dimensional generalized Broer–Kaup (gBK) equations is first proven. Then, the Bäcklund transformations for the gBK equations are derived by using the Painlevé truncation. Based on a special case of the derived Bäcklund transformations, the gBK equations are linearized into the heat conduction equation. Inspired by the derived Bäcklund transformations, the gBK equations are reduced into the Burgers equation. Starting from the linear heat conduction equation, two forms of N-soliton solutions and rational solutions with a singularity condition of the gBK equations are constructed. In addition, the rational solutions with two singularity conditions of the gBK equation are obtained by considering the non-uniqueness and generality of a resonance function embedded into the Painlevé test. In order to understand the nonlinear dynamic evolution dominated by the gBK equations, some of the obtained exact solutions, including one-soliton solutions, two-soliton solutions, three-soliton solutions, and two pairs of rational solutions, are shown by three-dimensional images. This paper shows that when the Painlevé test deals with the coupled nonlinear equations, the highest negative power of the coupled variables should be comprehensively considered in the leading term analysis rather than the formal balance between the highest-order derivative term and the highest-order nonlinear term.

The video images captured at long range usually have low contrast floating objects of interest on a sea surface. A comparative experimental study of the statistical characteristics of reflections from floating objects and from the agitated sea surface showed the difference in the correlation and spectral characteristics of these reflections. The functioning of the recently proposed modified matched subspace detector (MMSD) is based on the separation of the observed data spectrum on two subspaces: relatively low and relatively high frequencies. In the literature the MMSD performance has been evaluated in generally and moreover using only a sea model (additive Gaussian background clutter). This paper extends the performance evaluating methodology for low contrast object detection and moreover using only the real sea dataset. This methodology assumes an object of low contrast if the mean and variance of the object and the surrounding background are the same. The paper assumes that the energy spectrum of the object and the sea are different. The paper investigates a scenario in which an artificially created model of a floating object with specified statistical parameters is placed on the surface of a real sea image. The paper compares the efficiency of the classical Matched Subspace Detector (MSD) and MMSD for detecting low-contrast objects on the sea surface. The article analyzes the dependence of the detection probability at a fixed false alarm probability on the difference between the statistical means and variances of a floating object and the surrounding sea.

The Coronavirus herd immunity optimizer (CHIO) is a new human-based optimization algorithm that imitates the herd immunity strategy to eliminate of the COVID-19 disease. In this paper, the coronavirus herd immunity optimizer (CHIO) is modified to tackle a discrete power scheduling problem in a smart home (PSPSH). PSPSH is a combinatorial optimization problem with NP-hard features. It is a highly constrained discrete scheduling problem concerned with assigning the operation time for smart home appliances based on a dynamic pricing scheme(s) and several other constraints. The primary objective when solving PSPSH is to maintain the stability of the power system by reducing the ratio between average and highest power demand (peak-to-average ratio (PAR)) and reducing electricity bill (EB) with considering the comfort level of users (UC). This paper modifies and adapts the CHIO algorithm to deal with such discrete optimization problems, particularly PSPSH. The adaptation and modification include embedding PSPSH problem-specific operators to CHIO operations to meet the discrete search space requirements. PSPSH is modeled as a multi-objective problem considering all objectives, including PAR, EB, and UC. The proposed method is examined using a dataset that contains 36 home appliances and seven consumption scenarios. The main CHIO parameters are tuned to find their best values. These best values are used to evaluate the proposed method by comparing its results with comparative five metaheuristic algorithms. The proposed method shows encouraging results and almost obtains the best results in all consumption scenarios.

This paper proposes a modification of the imperialist competitive algorithm to solve multi-objective optimization problems with hybrid methods (MOHMICA) based on a modification of the imperialist competitive algorithm with hybrid methods (HMICA). The rationale for this is that there is an obvious disadvantage of HMICA in that it can only solve single-objective optimization problems but cannot solve multi-objective optimization problems. In order to adapt to the characteristics of multi-objective optimization problems, this paper improves the establishment of the initial empires and colony allocation mechanism and empire competition in HMICA, and introduces an external archiving strategy. A total of 12 benchmark functions are calculated, including 10 bi-objective and 2 tri-objective benchmarks. Four metrics are used to verify the quality of MOHMICA. Then, a new comprehensive evaluation method is proposed, called “radar map method”, which could comprehensively evaluate the convergence and distribution performance of multi-objective optimization algorithm. It can be seen from the four coordinate axes of the radar maps that this is a symmetrical evaluation method. For this evaluation method, the larger the radar map area is, the better the calculation result of the algorithm. Using this new evaluation method, the algorithm proposed in this paper is compared with seven other high-quality algorithms. The radar map area of MOHMICA is at least 14.06% larger than that of other algorithms. Therefore, it is proven that MOHMICA has advantages as a whole.

In this paper we propose an approximation method for solving second kind Volterra integral equation systems by radial basis functions. It is based on the minimization of a suitable functional in a discrete space generated by compactly supported radial basis functions of Wendland type. We prove two convergence results, and we highlight this because most recent published papers in the literature do not include any. We present some numerical examples in order to show and justify the validity of the proposed method. Our proposed technique gives an acceptable accuracy with small use of the data, resulting also in a low computational cost.

Techniques and methods of linear optimization underwent a significant improvement in the 20th century which led to the development of reliable mixed integer linear programming (MILP) solvers. It would be useful if these solvers could handle mixed integer nonlinear programming (MINLP) problems. Piecewise linear approximation (PLA) is one of most popular methods used to transform nonlinear problems into linear ones. This paper will introduce PLA with brief a background and literature review, followed by describing our contribution before presenting the results of computational experiments and our findings. The goals of this paper are (a) improving PLA models by using nonuniform domain partitioning, and (b) proposing an idea of applying PLA partially on MINLP problems, making them easier to handle. The computational experiments were done using quadratically constrained quadratic programming (QCQP) and MIQCQP and they showed that problems under PLA with nonuniform partition resulted in more accurate solutions and required less time compared to PLA with uniform partition.