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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.

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

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

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

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

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

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

In this paper, we present a numerical solution of a two-phase fractional Stefan problem with time derivative described in the Caputo sense. In the proposed algorithm, we use a special case of front-fixing method supplemented by the iterative procedure, which allows us to determine the position of the moving boundary. The presented method is an extension of a front-fixing method for the one-phase problem to the two-phase case. The novelty of the method is a new discretization of the partial differential equation dedicated to the second phase, which is carried out by introducing a new spatial variable immobilizing the moving boundary. Then, the partial differential equation is transformed to an equivalent integro-differential equation, which is discretized on a homogeneous mesh of nodes with a constant spatial and time step. A new convergence criterion is also proposed in the iterative algorithm determining the location of the moving boundary. The motivation for the development of the method is that the analytical solution of the considered problem is impossible to calculate in some cases, as can be seen in the figures in the paper. Moreover, the change of the boundary conditions makes obtaining a closed analytical solution very problematic. Therefore, creating new numerical methods is very valuable. In the final part, we also present some examples illustrating the comparison of the analytical solution with the results received by the proposed numerical method.

In 2006, A. Stakhov introduced a new coding/decoding process based on generating matrices of the Fibonacci p-numbers, which he called the Fibonacci coding/decoding method. Stakhov&rsquo, s papers have motivated many other scientists to seek certain generalizations by introducing new additional coefficients into recurrence of Fibonacci p-numbers. In 2013, I. Włoch et al. studied (2,q)-distance Fibonacci numbers F2(q,n) and found some of their combinatorial properties. In this paper, we state a new coding theory based on the sequence (Tq(n))n=&minus, &infin, which is an extension of Włoch&rsquo, s sequence (F2(q,n))n=0&infin

Let Ckn,l be a simple closed k-curves with l elements in Zn and W:=Ckn,l&or, ⋯&or, Ckn,l︷m-times be an m-iterated digital wedges of Ckn,l, and F(Conk(W)) be an alignment of fixed point sets of W. Then, the aim of the paper is devoted to investigating various properties of F(Conk(W)). Furthermore, when proceeding with this work, this paper addresses several unsolved problems. To be specific, we firstly formulate an alignment of fixed point sets of Ckn,l, denoted by F(Conk(Ckn,l)), where l(&ge, 7) is an odd natural number and k&ne, 2n. Secondly, given a digital image (X,k) with X♯=n, we find a certain condition that supports n&minus, 1,n&minus, 2&isin, F(Conk(X)). Thirdly, after finding some features of F(Conk(W)), we develop a method of making F(Conk(W)) perfect according to the (even or odd) number l of Ckn,l. Finally, we prove that the perfectness of F(Conk(W)) is equivalent to that of F(Conk(Ckn,l)). This can play an important role in studying fixed point theory and digital curve theory. This paper only deals with k-connected digital images (X,k) such that X♯&ge, 2.

The Łukasiewicz conjunction (sometimes also considered to be a logic of absolute comparison), which is used in multivalued logic and in fuzzy set theory, is one of the most important t-norms. In combination with the binary operation ‘maximum’, the Łukasiewicz t-norm forms the basis for the so-called max-Łuk algebra, with applications to the investigation of systems working in discrete steps (discrete events systems; DES, in short). Similar algebras describing the work of DES’s are based on other pairs of operations, such as max-min algebra, max-plus algebra, or max-T algebra (with a given t-norm, T). The investigation of the steady states in a DES leads to the study of the eigenvectors of the transition matrix in the corresponding max-algebra. In real systems, the input values are usually taken to be in some interval. Various types of interval eigenvectors of interval matrices in max-min and max-plus algebras have been described. This paper is oriented to the investigation of strong, strongly tolerable, and strongly universal interval eigenvectors in a max-Łuk algebra. The main method used in this paper is based on max-Ł linear combinations of matrices and vectors. Necessary and sufficient conditions for the recognition of strong, strongly tolerable, and strongly universal eigenvectors have been found. The theoretical results are illustrated by numerical examples.

In this paper a new method called the least squares differential quadrature method (LSDQM) is introduced as a straightforward and efficient method to compute analytical approximate polynomial solutions for nonlinear partial differential equations with fractional time derivatives. LSDQM is a combination of the differential quadrature method and the least squares method and in this paper it is employed to find approximate solutions for a very general class of nonlinear partial differential equations, wherein the fractional derivatives are described in the Caputo sense. The paper contains a clear, step-by-step presentation of the method and a convergence theorem. In order to emphasize the accuracy of LSDQM we included two test problems previously solved by means of other, well-known methods, and observed that our solutions present not only a smaller error but also a much simpler expression. We also included a problem with no known exact solution and the solutions computed by LSDQM are in good agreement with previous ones.

In this paper, a localized boundary knot method is proposed, based on the local concept in the localized method of fundamental solutions. The localized boundary knot method is formed by combining the classical boundary knot method and the localization approach. The localized boundary knot method is truly free from mesh and numerical quadrature, so it has great potential for solving complicated engineering applications, such as multiply connected problems. In the proposed localized boundary knot method, both of the boundary nodes and interior nodes are required, and the algebraic equations at each node represent the satisfaction of the boundary condition or governing equation, which can be derived by using the boundary knot method at every subdomain. A sparse system of linear algebraic equations can be yielded using the proposed localized boundary knot method, which can greatly reduce the computer time and memory required in computer calculations. In this paper, several cases of simply connected domains and multi-connected domains of the Laplace equation and bi-harmonic equation are demonstrated to evidently verify the accuracy, convergence and stability of this proposed meshless method.

The purpose of this paper is to introduce the split combination of variational inclusion problem which combines the concept of the modified variational inclusion problem introduced by Khuangsatung and Kangtunyakarn and the split variational inclusion problem introduced by Moudafi. Using a modified Halpern iterative method, we prove the strong convergence theorem for finding a common solution for the hierarchical fixed point problem and the split combination of variational inclusion problem. The result presented in this paper demonstrates the corresponding result for the split zero point problem and the split combination of variation inequality problem. Moreover, we discuss a numerical example for supporting our result and the numerical example shows that our result is not true if some conditions fail.

The paper presents an analysis and overview of vertex–frequency analysis, an emerging area in graph signal processing. A strong formal link of this area to classical time–frequency analysis is provided. Vertex–frequency localization-based approaches to analyzing signals on the graph emerged as a response to challenges of analysis of big data on irregular domains. Graph signals are either localized in the vertex domain before the spectral analysis is performed or are localized in the spectral domain prior to the inverse graph Fourier transform is applied. The latter approach is the spectral form of the vertex–frequency analysis, and it will be considered in this paper since the spectral domain for signal localization is well ordered and thus simpler for application to the graph signals. The localized graph Fourier transform is defined based on its counterpart, the short-time Fourier transform, in classical signal analysis. We consider various spectral window forms based on which these transforms can tackle the localized signal behavior. Conditions for the signal reconstruction, known as the overlap-and-add (OLA) and weighted overlap-and-add (WOLA) methods, are also considered. Since the graphs can be very large, the realizations of vertex–frequency representations using polynomial form localization have a particular significance. These forms use only very localized vertex domains, and do not require either the graph Fourier transform or the inverse graph Fourier transform, are computationally efficient. These kinds of implementations are then applied to classical time–frequency analysis since their simplicity can be very attractive for the implementation in the case of large time-domain signals. Spectral varying forms of the localization functions are presented as well. These spectral varying forms are related to the wavelet transform. For completeness, the inversion and signal reconstruction are discussed as well. The presented theory is illustrated and demonstrated on numerical examples.

Many problems of practical interest can be modeled and solved by using fuzzy graph (FG) algorithms. In general, fuzzy graph theory has a wide range of application in various fields. Since indeterminate information is an essential real-life problem and is often uncertain, modeling these problems based on FG is highly demanding for an expert. A vague graph (VG) can manage the uncertainty relevant to the inconsistent and indeterminate information of all real-world problems in which fuzzy graphs may not succeed in bringing about satisfactory results. Domination in FGs theory is one of the most widely used concepts in various sciences, including psychology, computer sciences, nervous systems, artificial intelligence, decision-making theory, etc. Many research studies today are trying to find other applications for domination in their field of interest. Hence, in this paper, we introduce different kinds of domination sets, such as the edge dominating set (EDS), the total edge dominating set (TEDS), the global dominating set (GDS), and the restrained dominating set (RDS), in product vague graphs (PVGs) and try to represent the properties of each by giving some examples. The relation between independent edge sets (IESs) and edge covering sets (ECSs) are established. Moreover, we derive the necessary and sufficient conditions for an edge dominating set to be minimal and show when a dominance set can be a global dominance set. Finally, we try to explain the relationship between a restrained dominating set and a restrained independent set with an example. Today, we see that there are still diseases that can only be treated in certain countries because they require a long treatment period with special medical devices. One of these diseases is leukemia, which severely affects the immune system and the body’s defenses, making it impossible for the patient to continue living a normal life. Therefore, in this paper, using a dominating set, we try to categorize countries that are in a more favorable position in terms of medical facilities, so that we can transfer the patients to a suitable hospital in the countries better suited in terms of both cost and distance.

The aim of this paper is to study, from an algebraic point of view, the properties of interdependencies between sets of elements (i.e., pieces of secrets, atmospheric variables, etc.) that appear in various natural models, by using the algebraic hyperstructure theory. Starting from specific examples, we first define the relation of dependence and study its properties, and then, we construct various hyperoperations based on this relation. We prove that two of the associated hypergroupoids are Hv-groups, while the other two are, in some particular cases, only partial hypergroupoids. Besides, the extensivity and idempotence property are studied and related to the cyclicity. The second goal of our paper is to provide a new interpretation of the dependence relation by using elements of the theory of algebraic hyperstructures. Cílem našeho článku je studovat, z algebraického pohledu, vlastnosti závislostí mezi množinami prvků (atmosférické veličiny, sdílená tajemství apod.), které se objevují v různých obvyklých kontextecb. K tomuto studiu používáme algebraickou teorii hyperstruktur. Na konkrétních příkadech ukazujeme, jaké vlastnosti jsou vyžadovány a jak jim musí odpovídat definice tohoto pojmu. Ukazujeme, že dva z hypergrupoidů, které lze tímto způsobem přirozeně zkonstruovat, jsou Hv-grupy, zatímco jiné dva jsou parciální hypergrupoidy. Studujeme také extenzivitu a idempotentnost příslušných vlastností a jejich vztah k cyklicitě. Dalším cílem našeho článku je poskytnout nový pohled na zkoumanou problematiku právě zapojením algebraické teorie hyperstruktur.

Recently, type 2 degenerate Euler polynomials and type 2 q-Euler polynomials were studied, respectively, as degenerate versions of the type 2 Euler polynomials as well as a q-analog of the type 2 Euler polynomials. In this paper, we consider the type 2 degenerate q-Euler polynomials, which are derived from the fermionic p-adic q-integrals on Z p , and investigate some properties and identities related to these polynomials and numbers. In detail, we give for these polynomials several expressions, generating function, relations with type 2 q-Euler polynomials and the expression corresponding to the representation of alternating integer power sums in terms of Euler polynomials. One novelty about this paper is that the type 2 degenerate q-Euler polynomials arise naturally by means of the fermionic p-adic q-integrals so that it is possible to easily find some identities of symmetry for those polynomials and numbers, as were done previously.

In the year 2004, Chang and Teng investigated an inventory model for deteriorating items in which the supplier not only provides a cash discount, but also allows a permissible delay in payments. The main purpose of the present investigation is three-fold, as follows. First, it is found herein that Theorem 1 of Chang and Teng (2004) has notable shortcomings in terms of their determination of the optimal solution of the annual total relevant cost Z ( T ) by adopting the Taylor-series approximation method. Theorem 1 in this paper does not make use of the Taylor-series approximation method in order to overcome the shortcomings in Chang and Teng (2004) and alternatively derives all the optimal solutions of the annual total relevant cost Z ( T ) . Secondly, this paper systematically revisits the annual total relevant cost Z ( T ) in Chang and Teng (2004) and presents in detail the mathematically correct ways for the derivations of Z ( T ) . Thirdly, this paper not only shows that Theorem 1 of Chang and Teng (2004) is not necessarily true for finding the optimal solution of the annual total relevant cost Z ( T ) , but it also demonstrates how Theorem 1 in this paper can locate all of the optimal solutions of Z ( T ) . The mathematical analytic investigation presented in this paper is believed to be useful for correct managerial considerations and managerial decisions.

In the paper, we introduce some subclasses of harmonic mapping, the analytic part of which is related to general starlike (or convex) functions with a symmetric conjecture point defined by subordination. Using the conditions satisfied by the analytic part, we obtain the integral expressions, the coefficient estimates, distortion estimates and the growth estimates of the co-analytic part g, and Jacobian estimates, the growth estimates and covering theorem of the harmonic function f. Through the above research, the geometric properties of the classes are obtained. In particular, we draw figures of extremum functions to better reflect the geometric properties of the classes. For the first time, we introduce and obtain the properties of harmonic univalent functions with respect to symmetric conjugate points. The conclusion of this paper extends the original research.

In the fields of continuous casting and the roll stepped cooling, the heat transfer coefficient is piecewise linear. However, few papers discuss the solution of the backward heat conduction problem in this situation. Therefore, the aim of this paper is to solve the backward heat conduction problem, which has the piecewise linear heat transfer coefficient. Firstly, the ill-posed of this problem is discussed and the truncated regularized optimization scheme is introduced to solve this problem. Secondly, because the regularization parameter is the key factor for the regularization method, this paper presents an improved method for choosing the regularization parameter to reduce the iterative number and proves the fourth-order convergence of this method. Furthermore, the numerical simulation experiments show that, compared with other methods, the improved method of fourth-order convergence effectively reduces the iterative number. Finally, the truncated regularized optimization scheme is used to estimate the initial temperature, and the results of numerical simulation experiments illustrate that the inverse values match the exact values very well.

In this paper, we consider fault-tolerant resolving sets in graphs. We characterize n-vertex graphs with fault-tolerant metric dimension n, n &minus, 1 , and 2, which are the lower and upper extremal cases. Furthermore, in the first part of the paper, a method is presented to locate fault-tolerant resolving sets by using classical resolving sets in graphs. The second part of the paper applies the proposed method to three infinite families of regular graphs and locates certain fault-tolerant resolving sets. By accumulating the obtained results with some known results in the literature, we present certain lower and upper bounds on the fault-tolerant metric dimension of these families of graphs. As a byproduct, it is shown that these families of graphs preserve a constant fault-tolerant resolvability structure.

This paper uses the viscosity implicit midpoint rule to find common points of the fixed point set of a nonexpansive mapping and the zero point set of an accretive operator in Banach space. Under certain conditions, this paper obtains the strong convergence results of the proposed algorithm and improves the relevant results of researchers in this field. In the end, this paper gives numerical examples to support the main results.

Given a digital image (or digital object) (X,k),X⊂Zn, this paper initially establishes a group structure of the set of self-k-isomorphisms of (X,k) with the function composition, denoted by Isok(X) or Autk(X). In particular, let Ckn,l be a simple closed k-curve with l elements in Zn. Then, the group Isok(Ckn,l) is proved to be isomorphic to the standard dihedral group Dl with order l. The calculation of this quantity Isok(Ckn,l) is a key step for obtaining many new results. Indeed, it is essential for exploring many features of Isok(X). Furthermore, this quantity is proved to be a digital topological invariant. After proceeding with an Isok(X)-action on (X,k), we investigate some properties of fixed point sets by this action. Finally, we explore various features of fixed point sets by this action from the viewpoint of digital k-curve theory. This paper only deals with k-connected digital images (X,k) whose cardinality is equal to or greater than 2. Besides, we discuss some errors that have appeared in the lilterature.

In this paper we investigate the unified theory for solutions of differential equations without impulses and with impulses, even at variable times, allowing the presence of beating phenomena, in the space of regulated functions. One of the aims of the paper is to give sufficient conditions to ensure that a regulated solution of an impulsive problem is globally defined.

The paper is concerned with complex fuzzy numbers and complex fuzzy inner product spaces. In the classical complex number set, a complex number can be expressed using the Cartesian form or polar form. Both expressions are needed because one expression is better than the other depending on the situation. Likewise, the Cartesian form and the polar form can be defined in a complex fuzzy number set. First, the complex fuzzy numbers (CFNs) are categorized into two types, the polar form and the Cartesian form, as type I and type II. The properties of the complex fuzzy number set of those two expressions are discussed, and how the expressions can be used practically is shown through an example. Second, we study the complex fuzzy inner product structure in each category and find the non-existence of an inner product on CFNs of type I. Several properties of the fuzzy inner product space for type II are proposed from the modulus that is newly defined. Specfically, the Cauchy-Schwartz inequality for type II is proven in a compact way, not only the one for fuzzy real numbers. In fact, it was already discussed by Hasanhani et al; however, they proved every case in a very complicated way. In this paper, we prove the Cauchy-Schwartz inequality in a much simpler way from a general point of view. Finally, we introduce a complex fuzzy scalar product for the generalization of a complex fuzzy inner product and propose to study the condition for its existence on CFNs of type I.

An arbitrary univariate polynomial of nth degree has n sequences. The sequences are systematized into classes. All the values of the first class sequence are obtained by Newton&rsquo, s polynomial of nth degree. Furthermore, the values of all sequences for each class are calculated by Newton&rsquo, s identities. In other words, the sequences are formed without calculation of polynomial roots. The New-nacci method is used for the calculation of the roots of an nth-degree univariate polynomial using radicals and limits of successive members of sequences. In such an approach as is presented in this paper, limit play a catalytic&ndash, theoretical role. Moreover, only four basic algebraic operations are sufficient to calculate real roots. Radicals are necessary for calculating conjugated complex roots. The partial limitations of the New-nacci method may appear from the decadal polynomial. In the case that an arbitrary univariate polynomial of nth degree (n &ge, 10) has five or more conjugated complex roots, the roots of the polynomial cannot be calculated due to Abel&rsquo, s impossibility theorem. The second phase of the New-nacci method solves this problem as well. This paper is focused on solving the roots of the quintic equation. The method is verified by applying it to the quintic polynomial with all real roots and the Degen&ndash, Abel polynomial, dating from 1821.

The convergence using the fuzzy semi-metric and dual fuzzy semi-metric is studied in this paper. The infimum type of dual fuzzy semi-metric and the supremum type of dual fuzzy semi-metric are proposed in this paper. Based on these two types of dual fuzzy semi-metrics, the different types of triangle inequalities can be obtained. We also study the convergence of these two types of dual fuzzy semi-metrics.

The present paper studies the fixed point property (FPP) for closed k-surfaces. We also intensively study Euler characteristics of a closed k-surface and a connected sum of closed k-surfaces. Furthermore, we explore some relationships between the FPP and Euler characteristics of closed k-surfaces. After explaining how to define the Euler characteristic of a closed k-surface more precisely, we confirm a certain consistency of the Euler characteristic of a closed k-surface and a continuous analog of it. In proceeding with this work, for a simple closed k-surface in Z 3 , say S k , we can see that both the minimal 26-adjacency neighborhood of a point x &isin, S k , denoted by M k ( x ) , and the geometric realization of it in R 3 , denoted by D k ( x ) , play important roles in both digital surface theory and fixed point theory. Moreover, we prove that the simple closed 18-surfaces M S S 18 and M S S 18 &prime, do not have the almost fixed point property (AFPP). Consequently, we conclude that the triviality or the non-triviality of the Euler characteristics of simple closed k-surfaces have no relationships with the FPP in digital topology. Using this fact, we correct many errors in many papers written by L. Boxer et al.

The hyperspace consists of all the subsets of a vector space. It is well-known that the hyperspace is not a vector space because it lacks the concept of inverse element. This also says that we cannot consider its normed structure, and some kinds of fixed point theorems cannot be established in this space. In this paper, we shall propose the concept of null set that will be used to endow a norm to the hyperspace. This normed hyperspace is clearly not a conventional normed space. Based on this norm, the concept of Cauchy sequence can be similarly defined. In addition, a Banach hyperspace can be defined according to the concept of Cauchy sequence. The main aim of this paper is to study and establish the so-called near fixed point theorems in Banach hyperspace.

In this paper, we consider a resolvent operator which depends on the composition of two mappings with &oplus, operation. We prove some of the properties of the resolvent operator, that is, that it is single-valued as well as Lipschitz-type-continuous. An existence and convergence result is proven for a generalized implicit set-valued variational inclusion problem with &oplus, operation. Some special cases of a generalized implicit set-valued variational inclusion problem with &oplus, operation are discussed. An example is constructed to illustrate some of the concepts used in this paper.

The Colebrook-White equation is often used for calculation of the friction factor in turbulent regimes; it has succeeded in attracting a great deal of attention from researchers. The Colebrook–White equation is a complex equation where the computation of the friction factor is not direct, and there is a need for trial-error methods or graphical solutions; on the other hand, several researchers have attempted to alter the Colebrook-White equation by explicit formulas with the hope of achieving zero-percent (0%) maximum deviation, among them Dejan Brkić and Pavel Praks. The goal of this paper is to discuss the results proposed by the authors in their paper:” Accurate and Efficient Explicit Approximations of the Colebrook Flow Friction Equation Based on the Wright ω-Function” and to propose more accurate formulas.

This paper focuses on the problem of finite-time stabilization of homogeneous, non-Lipschitz systems with dilations. A key contribution of this paper is the design of a virtual recursive Hölder, non-Lipschitz state feedback which renders the non-Lipschitz systems in the special case dominated by a lower-triangular nonlinear system, finite-time stable. The proof is based on a recursive design algorithm developed recently, to construct the virtual Hölder continuous, finite-time stabilizer as well as a C1 positive definite and proper Lyapunov function that guarantees finite-time stability of the non-Lipschitz non-linear systems.

In a series of papers, we discussed the solution of Laplace’s differential equation (DE) by using fractional calculus, operational calculus in the framework of distribution theory, and Laplace transform. The solutions of Kummer’s DE, which are expressed by the confluent hypergeometric functions, are obtained with the aid of the analytic continuation (AC) of Riemann–Liouville fractional derivative (fD) and the distribution theory in the space D′R or the AC of Laplace transform. We now obtain the solutions of the hypergeometric DE, which are expressed by the hypergeometric functions, with the aid of the AC of Riemann–Liouville fD, and the distribution theory in the space D′r,R, which is introduced in this paper, or by the term-by-term inverse Laplace transform of AC of Laplace transform of the solution expressed by a series.

The rate of change of any function versus its independent variables was defined as a derivative. The fundamentals of the derivative concept were constructed by Newton and l’Hôpital. The followers of Newton and l’Hôpital defined fractional order derivative concepts. We express the derivative defined by Newton and l’Hôpital as an ordinary derivative, and there are also fractional order derivatives. So, the derivative concept was handled in this paper, and a new definition for derivative based on indefinite limit and l’Hôpital’s rule was expressed. This new approach illustrated that a derivative operator may be non-linear. Based on this idea, the asymptotic behaviors of functions were analyzed and it was observed that the rates of changes of any function attain maximum value at inflection points in the positive direction and minimum value (negative) at inflection points in the negative direction. This case brought out the fact that the derivative operator does not have to be linear; it may be non-linear. Another important result of this paper is the relationships between complex numbers and derivative concepts, since both concepts have directions and magnitudes.

This paper presents the study of algebraic structures equipped with the inverted associativity axiom. Initially, the definition of the left and the right almost-groups is introduced and afterwards, the study is focused on the more general structures, which are the left and the right almost-hypergroups and on their enumeration in the cases of order 2 and 3. The outcomes of these enumerations compared with the corresponding in the hypergroups reveal interesting results. Next, fundamental properties of the left and right almost-hypergroups are proved. Subsequently, the almost hypergroups are enriched with more axioms, like the transposition axiom and the weak commutativity. This creates new hypercompositional structures, such as the transposition left/right almost-hypergroups, the left/right almost commutative hypergroups, the join left/right almost hypergroups, etc. The algebraic properties of these new structures are analyzed and studied as well. Especially, the existence of neutral elements leads to the separation of their elements into attractive and non-attractive ones. If the existence of the neutral element is accompanied with the existence of symmetric elements as well, then the fortified transposition left/right almost-hypergroups and the transposition polysymmetrical left/right almost-hypergroups come into being.

In this paper, we introduce a new class of multi-dimensional robust optimization problems (named (P)) with mixed constraints implying second-order partial differential equations (PDEs) and inequations (PDIs). Moreover, we define an auxiliary (modified) class of robust control problems (named (P)(b¯,c¯)), which is much easier to study, and provide some characterization results of (P) and (P)(b¯,c¯) by using the notions of normal weak robust optimal solution and robust saddle-point associated with a Lagrange functional corresponding to (P)(b¯,c¯). For this aim, we consider path-independent curvilinear integral cost functionals and the notion of convexity associated with a curvilinear integral functional generated by a controlled closed (complete integrable) Lagrange 1-form.

In this paper, the residual probability function is applied to analyze the survival probability of two used components relative to each other in the case when their lifetimes are dependent. The expression of the function by copulas has been derived along with some examples of particular copulas. The behaviour of the residual probability function in terms of the underlying dependence is also discussed. The residual probability order is also considered in the dependent case. In the class of Archimedean survival copulas, we prove that the residual probability order implies the usual stochastic order in the reversed direction, and the hazard rate order concludes the residual probability order.

Network representation learning aims to learn low-dimensional, compressible, and distributed representational vectors of nodes in networks. Due to the expensive costs of obtaining label information of nodes in networks, many unsupervised network representation learning methods have been proposed, where random walk strategy is one of the wildly utilized approaches. However, the existing random walk based methods have some challenges, including: 1. The insufficiency of explaining what network knowledge in the walking path-samplings, 2. The adverse effects caused by the mixture of different information in networks, 3. The poor generality of the methods with hyper-parameters on different networks. This paper proposes an information-explainable random walk based unsupervised network representation learning framework named Probabilistic Accepted Walk (PAW) to obtain network representation from the perspective of the stationary distribution of networks. In the framework, we design two stationary distributions based on nodes’ self-information and local-information of networks to guide our proposed random walk strategy to learn representational vectors of networks through sampling paths of nodes. Numerous experimental results demonstrated that the PAW could obtain more expressive representation than the other six widely used unsupervised network representation learning baselines on four real-world networks in single-label and multi-label node classification tasks.

In this paper, we find some inequalities which involve Euler’s function, extended Euler’s function, the function τ, and the generalized function τ in algebraic number fields.

Compared with the linear Black–Scholes model, nonlinear models are constructed through taking account of more practical factors, such as transaction cost, and so it is difficult to find an exact analytical solution. Combining the Haar wavelet integration method, which can transform the partial differential equation into the system of algebraic equations, the homotopy perturbation method, which can linearize the nonlinear problems, and the variational iteration method, which can solve the large system of algebraic equations efficiently, a novel numerical method for the nonlinear Black–Scholes model is proposed in this paper. Compared with the traditional methods, it has higher efficiency and calculation precision.

In this paper, the authors investigate two special families of series involving the reciprocal central binomial coefficients and Lucas numbers. Connections with several familiar sums representing the integer-valued Riemann zeta function are also pointed out.

In this paper, the notions of L-fuzzy subalgebra degree and L-subalgebras on an effect algebra are introduced and some characterizations are given. We use four kinds of cut sets of L-subsets to characterize the L-fuzzy subalgebra degree. We induce an L-fuzzy convexity by the L-fuzzy subalgebra degree, and we prove that a morphism between two effect algebras is an L-fuzzy convexity preserving mapping and a monomorphism is an L-fuzzy convex-to-convex mapping. Finally, it is proved that the set of all L-subalgebras on an effect algebra can form an L-convexity, and its L-convex hull formula is given.

The accurate description of the flow of water in porous media is of the greatest importance due to its numerous applications in several areas (groundwater, soil mechanics, etc.). The nonlinear Richards equation is often used as the governing equation that describes this phenomenon and a large number of research studies aimed to solve it numerically. However, due to the nonlinearity of the constitutive expressions for permeability, it remains a challenging modeling problem. In this paper, the stationary form of Richards’ equation used in saturated soils is solved by two numerical methods: generalized finite differences, an emerging method that has been successfully applied to the transient case, and a finite element method, for benchmarking. The nonlinearity of the solution in both cases is handled using a Newtonian iteration. The comparative results show that a generalized finite difference iteration yields satisfactory results in a standard test problem with a singularity at the boundary.

In our paper, we propose two new iterative algorithms with Meir–Keeler contractions that are based on Tseng’s method, the multi-step inertial method, the hybrid projection method, and the shrinking projection method to solve a monotone variational inclusion problem in Hilbert spaces. The strong convergence of the proposed iterative algorithms is proven. Using our results, we can solve convex minimization problems.

In this paper, we introduce a new geometric description of the manifolds of matrices of fixed rank. The starting point is a geometric description of the Grassmann manifold Gr(Rk) of linear subspaces of dimension r&lt, k in Rk, which avoids the use of equivalence classes. The set Gr(Rk) is equipped with an atlas, which provides it with the structure of an analytic manifold modeled on R(k−r)×r. Then, we define an atlas for the set Mr(Rk×r) of full rank matrices and prove that the resulting manifold is an analytic principal bundle with base Gr(Rk) and typical fibre GLr, the general linear group of invertible matrices in Rk×k. Finally, we define an atlas for the set Mr(Rn×m) of non-full rank matrices and prove that the resulting manifold is an analytic principal bundle with base Gr(Rn)×Gr(Rm) and typical fibre GLr. The atlas of Mr(Rn×m) is indexed on the manifold itself, which allows a natural definition of a neighbourhood for a given matrix, this neighbourhood being proved to possess the structure of a Lie group. Moreover, the set Mr(Rn×m) equipped with the topology induced by the atlas is proven to be an embedded submanifold of the matrix space Rn×m equipped with the subspace topology. The proposed geometric description then results in a description of the matrix space Rn×m, seen as the union of manifolds Mr(Rn×m), as an analytic manifold equipped with a topology for which the matrix rank is a continuous map.

In this paper, we introduce a new class of harmonic univalent functions with respect to k-symmetric points by using a newly-defined q-analog of the derivative operator for complex harmonic functions. For this harmonic univalent function class, we derive a sufficient condition, a representation theorem, and a distortion theorem. We also apply a generalized q-Bernardi–Libera–Livingston integral operator to examine the closure properties and coefficient bounds. Furthermore, we highlight some known consequences of our main results. In the concluding part of the article, we have finally reiterated the well-demonstrated fact that the results presented in this article can easily be rewritten as the so-called (p,q)-variations by making some straightforward simplifications, and it will be an inconsequential exercise, simply because the additional parameter p is obviously unnecessary.

Let G be a graph on n vertices and m edges, with maximum degree Δ(G) and minimum degree δ(G). Let A be the adjacency matrix of G, and let λ1≥λ2≥…≥λn be the eigenvalues of G. The energy of G, denoted by E(G), is defined as the sum of the absolute values of the eigenvalues of G, that is E(G)=|λ1|+…+|λn|. The energy of G is known to be at least twice the minimum degree of G, E(G)≥2δ(G). Akbari and Hosseinzadeh conjectured that the energy of a graph G whose adjacency matrix is nonsingular is in fact greater than or equal to the sum of the maximum and the minimum degrees of G, i.e., E(G)≥Δ(G)+δ(G). In this paper, we present a proof of this conjecture for hyperenergetic graphs, and we prove an inequality that appears to support the conjectured inequality. Additionally, we derive various lower and upper bounds for E(G). The results rely on elementary inequalities and their application.

In this paper, a penalty method is used together with a barrier method to transform a constrained nonlinear programming problem into an unconstrained nonlinear programming problem. In the proposed approach, Newton’s method is applied to the barrier Karush–Kuhn–Tucker conditions. To ensure global convergence from any starting point, a trust-region globalization strategy is used. A global convergence theory of the penalty–barrier trust-region (PBTR) algorithm is studied under four standard assumptions. The PBTR has new features, it is simpler, has rapid convergerce, and is easy to implement. Numerical simulation was performed on some benchmark problems. The proposed algorithm was implemented to find the optimal design of a canal section for minimum water loss for a triangle cross-section application. The results are promising when compared with well-known algorithms.

This year, we introduced the concept of infra soft topology as a new generalization of soft topology. To complete our analysis of this space, we devote this paper to presenting the concepts of infra soft connected and infra soft locally connected spaces. We provide some descriptions for infra soft connectedness and elucidate that there is no relationship between an infra soft topological space and its parametric infra topological spaces with respect to the property of infra soft connectedness. We discuss the behaviors of infra soft connected and infra soft locally connected spaces under infra soft homeomorphism maps and a finite product of soft spaces. We complete this manuscript by defining a component of a soft point and establishing its main properties. We determine the conditions under which the number of components is finite or countable, and we discuss under what conditions the infra soft connected subsets are components.