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This short paper is motivated by the paper of Bu et al. [C. Bu, C. Feng, P. Dong, A note on computational formulas for the Drazin inverse of certain block matrices, J. Appl. Math. Comput.(38) (2012) 631-640], where the authors gave additive formula for Drazin inverse for matrices under new conditions, and two representations under some specific conditions. Here is shown that the additive formula is not valid for all matrices which satisfy given conditions. Also, here is proved that the representations which were given in mentioned paper do not extend the results given by Hartwig et al. [R. Hartwig, X. Li, Y. Wei, Representations for the Drazin inverse of a 2 _ 2 block matrix, SIAM J. Matrix. Anal. Appl. (27)(2006) 757-771 ], in fact they are equivalent.

In this short paper, we will show that the space of real valued uniformly continuous functions defined on a metric space (X,d) is a ring if and only if every subset A ? X has one of the following properties: ? A is Bourbaki-bounded, i.e., every uniformly continuous function on X is bounded on A. ? A contains an infinite uniformly isolated subset, i.e., there exist ? > 0 and an infinite subset F ? A such that d(a,x) ? ? for every a ? F, x ? X n \{a}.

In this short paper, generally, we define a family of functions fk depends on the Eisenstein series with weight 2k, for k ( N. More detail, by considering the function fk, we define a derivative formula for generating weakly modular forms with weight 12. As a result for this, we claim that this formula gives an advantage to find the special solutions of some differential equations.

© 2022, University of Nis. All rights reserved.In this paper the authors gave an iteration identity for the generalized Laplace transform L2n and the generalized Glasser transform G2n. Using this identity a Parseval-Goldstein type theorem for the L2n-transform and the G2n-transform is given. By making use of these results a number of new Parseval-Goldstein type identities are obtained for these and many other well-known integral transforms.The identities proven in this paper are shown to give rise to useful corollaries for evaluating infinite integrals of special functions. Some examples are also given.

In this paper, we present the necessary and sufficient conditions for a poset to be a poset of the union of join and meet irreducible elements of the slim lattice. Slim lattices are special finite lattices that are intensively investigated recently. The problem that we solved in this paper is a generalization of the problem proposed very recently by Cz?dli.

In this paper, a new form of separation axioms called r-fuzzy soft Ti;(i = 0,1,2,3,4), r-fuzzy soft regular and r-fuzzy soft normal axioms are introduced in a fuzzy soft topological space based on the paper Ayg?no?lu et al. [7]. Also, the relations of these axioms with each other are investigated with the help of examples. Furthermore, some fuzzy soft invariance properties, namely fuzzy soft topological property and hereditary property are specified.

In this paper, we point out that the proof of Theorem 2.4(5), Proposition 2.6(1) and Proposition 2.8(1) in the paper titled ?On (L,M)-fuzzy convex structures? (Filomat 33(13): 4151-4163, 2019) are not true in general. Then, we give three correct proofs of these results

The subject of this paper is an analytic approximate method for a class of stochastic differential equations with coefficients that do not necessarily satisfy the Lipschitz and linear growth conditions but behave like a polynomials. More precisely, equations from the observed class have unique solutions with bounded moments and their coefficients satisfy polynomial condition. Approximate equations are defined on partitions of a time interval, and their coefficients are Taylor approximations of the coefficients of the initial equation. The rate of Lp convergence increases when degrees in Taylor approximations of coefficients increase. At the end of the paper, an example is provided to support the main theoretical result.

In this paper, we define and introduce some new concepts of the higher order strongly generalized preinvex functions and higher order strongly monotone operators involving the arbitrary bifunction and function. Some new relationships among various concepts of higher order strongly general preinvex functions have been established. It is shown that the optimality conditions of the higher order strongly general preinvex functions are characterized by a class of variational inequalities, which is called the higher order strongly generalized variational-like inequality. Auxiliary principle technique is used to suggest an implicit method for solving higher order strongly generalized variational-like inequalities. Convergence analysis of the proposed method is investigated using the pseudo-monotonicity of the operator. It is shown that the new parallelogram laws for Banach spaces can be obtained as applications of higher order strongly affine generalized preinvex functions, which is itself a novel application. Some special cases also discussed. Results obtained in this paper can be viewed as refinement and improvement of previously known results.

The aim of this paper is to correct results from the published paper: A companion of Ostrowski Type Integral Inequality Using a 5-Step Kernel with Some Applications, Filomat 30:13 (2016), 3601-3614.

We pointed to 4-dimensional generalized Riemannian spaces important for applications in some parts of physics here. Complete metric tensors and its possibilities to be applied in cosmology are analyzed in this paper. We used the results of N. O. Vesic, presented in [14]. At the end of the paper, we studied the diagonal symmetric metric tensor in the cosmological context.

In the paper, we study the value distribution of the differential polynomial Afn f(k) + Bf n+1 -1, where f is a transcendental meromorphic function and n(? 2),k(?2) are positive integers. We prove an inequality for the Nevanlinna characteristic function T(r,f) in terms of reduced counting function only. The result of the paper not only improves the result due to Q.D. Zhang [J. Chengdu Ins. Meteor., 20(1992), 12-20], also partially improves a recent result of H. Karmakar and P. Sahoo [Results Math., (2018),73:98].

The aim of the paper is to obtain a new version of Serret-Frenet formulae for a quaternionic curve in R4 by using the method given by Bharathi and Nagaraj. Then, we define quaternionic helices in H named as quaternionic right and left X-helix with the help of given a unit vector field X. Since the quaternion product is not commutative, the authors ([4], [7]) have used by one-sided multiplication to find a space curve related to a given quaternionic curve in previous studies. Firstly, we obtain new expressions by using the right product and the left product for quaternions. Then, we generalized the construction of Serret-Frenet formulae of quaternionic curves. Finally, as an application, we obtain an example that supports the theory of this paper.

In this paper, we introduce a new iterative scheme for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed points of a finite family of nonspreading mappings and a finite family of nonexpansive multivalued mappings in Hadamard space. We state and prove strong and ? convergence theorems of the proposed iterative process. The results obtained in this paper extend and improve some recent known results.

This paper introduces and studies a new class of multidimensional numerical integration, which we call ?strongly positive definite cubature formulas?. We establish, among others, a characterization theorem providing necessary and sufficient conditions for the approximation error (based on such cubature formulas) to be bounded by the approximation error of the quadratic function. This result is derived as a consequence of two characterization results, which are of independent interest, for linear functionals obtained in a more general seeting. Thus, this paper extends some result previously reported in [2, 3] when convexity in the classical sense is only assumed. We also show that the centroidal Voronoi Tesselations provide an efficient way for constructing a class of optimal of cubature formulas. Numerical results for the two-dimensional test functions are given to illustrate the efficiency of our resulting cubature formulas.

The aim of this paper is to study the difference gap (in short, D-gap) function and error bounds for a class of the random mixed equilibrium problems in real Hilbert spaces. Firstly, we consider regularized gap functions of the Fukushima type and Moreau-Yosida type. Then difference gap functions are established by using these terms of regularized gap functions. Finally, the global error bounds for random mixed equilibrium problems are also developed. The results obtained in this paper are new and extend some corresponding known results in literatures. Some examples are given for the illustration of our results.

Multisplitting methods are useful to solve differential-algebraic equations. In this connection, we discuss the theory of matrix splittings and multisplittings, which can be used for finding the iterative solution of a large class of rectangular (singular) linear system of equations of the form Ax = b. In this direction, many convergence results are proposed for different subclasses of proper splittings in the literature. But, in some practical cases, the convergence speed of the iterative scheme is very slow. To overcome this issue, several comparison results are obtained for different subclasses of proper splittings. This paper also presents a few such results. However, this idea fails to accelerate the speed of the iterative scheme in finding the iterative solution. In this regard, Climent and Perea [J. Comput. Appl. Math. 158 (2003), 43-48: MR2013603] introduced the notion of proper multisplittings to solve the system Ax = b on parallel and vector machines, and established convergence theory for a subclass of proper multisplittings. With the aim to extend the convergence theory of proper multisplittings, this paper further adds a few results. Some of the results obtained in this paper are even new for the iterative theory of nonsingular linear systems.

In this paper we construct and investigate stability features of two stochastic hepatitis C models with an isolation stage which are obtained by an introduction of stochastic perturbations into the deterministic model for hepatitis C with an isolation stage. One of the stochastic models has only disease- free equilibrium and the other endemic equilibrium state. Aforementioned equilibriums belong to the equilibriums of corresponding deterministic system. For both of models, first of all, we prove the existence and uniqueness of global positive stochastic solution. Thereafter, by using suitable Lyapunov functions, we investigate stability properties of both models. We close the paper with numerical simulation with reliable data of hepatitis C transmission to illustrate our theoretical results.

In this paper, we apply the Faber polynomial expansions to find upper bounds for the general coefficients |amk+1| (k >= 3) of functions in the subclass N?m (?,?;?). The results presented in this paper would generalize and improve some recent works.

The problems considered in this paper are described in polyhedral multi-valued mappings for higher order(s-th) discrete (PDSIs) and differential inclusions (PDFIs). The present paper focuses on the necessary and sufficient conditions of optimality for optimization of these problems. By converting the PDSIs problem into a geometric constraint problem, we formulate the necessary and sufficient conditions of optimality for a convex minimization problem with linear inequality constraints. Then, in terms of the Euler-Lagrange type PDSIs and the specially formulated transversality conditions, we are able to obtain conditions of optimality for the PDSIs. In order to obtain the necessary and sufficient conditions of optimality for the discrete-approximation problem PDSIs, we reduce this problem to the form of a problem with higher order discrete inclusions. Finally, by formally passing to the limit, we establish the sufficient conditions of optimality for the problem with higher order PDFIs. Numerical approach is developed to solve a polyhedral problem with second order polyhedral discrete inclusions.

In this paper, we initiated new types of near open and closed sets called higher order sets. These sets generated by much iteration of topological interior and closure operations for a given set. These new sets are the terms of many sequences of topological near open and near closed sets. We studied many generalizations of the classical near open sets to these sequences. This paper is the starting point of a new way for researchers to study the high order topology.

In this paper the Alexandroff one point compactification of the 2-dimensional Khalimsky (K-, for brevity) plane (resp. the 1-dimensional Khalimsky line) is called the infinite K-sphere (resp. the infinite K-circle). The present paper initially proves that the infinite K-circle has the fixed point property (FPP, for short) in the set Con(Z*), where Con(Z*) means the set of all continuous self-maps f of the infinite K-circle. Next, we address the following query which remains open: Under what condition does the infinite K-sphere have the FPP? Regarding this issue, we prove that the infinite K-sphere has the FPP in the set Con*((Z2)*) (see Definition 1.1). Finally, we compare the FPP of the infinite K-sphere and that of the infinite M-sphere, where the infinite M-sphere means the one point compactification of the Marcus-Wyse topological plane.

The purpose of the paper is to investigate the problems of unique range sets in the most general setting. Accordingly, we have studied sufficient conditions for a general polynomial to generate a unique range set which put all the variants of unique range sets into one structure. Most importantly, as an application of the main result we have been able to accommodate not only examples of critically injective polynomials but also examples of non-critically injective polynomials generating unique range sets which are for the first time being exemplified in the literature. Furthermore, some of these examples show that characterization of unique range sets generated by non-critically injective polynomials does not always demand gap polynomials which also complements the recent results by An and Banerjee-Lahiri in this direction. Moreover, one of the lemmas proved in this paper improves and generalizes some results due to Frank-Reinders and Lahiri respectively.

Zeroing neural networks (ZNN) approach, has been presented to solve a lot of time-varying problems activated by monotonically increasing functions. However, the existing ZNN models for timevarying quadratic programming based on ZNN approach may be different from each other in structures, but share two common restrictions, i.e., the function must be convex and unbounded. In order to relax the above restrictions in solving time-varying quadratic programming (TVQP) problems, this paper proposes a saturation-allowed zeroing neural networks (SAZNN) model based on the ZNN approach. Comparing with existing models, the activation function (AF) of SAZNN model tolerates more kinds of functions, e.g., saturation function, non-convex function and unbounded function. Finally, this paper provides simulation results synthesized by the proposed SAZNN model activated by various AFs and verifies the superiority of the proposed SAZNN model in terms of convergence, efficiency and stability.

In this paper, with the help of measure of noncompactness together with Darbo-type fixed point theorem, we focus on the infinite system of third-order differential equations u???i + au??i + bu?i + cui = fi(t, u1(t), u2(t),...) where fi ? C(R x R?,R) is ?-periodic with respect to the first coordinate and a,b,c ? R are constants. The aim of this paper is to obtain the results with respect to the existence of ?-periodic solutions of the aforementioned system in the Banach sequence space lp (1 ? p < ?) utilizing the respective Green?s function. Furthermore, some examples are provided to support our main results.

In this paper, we analyzed the physical meaning of scalar curvatures for a generalized Riemannian space. It is developed the Madsen's formulae for pressures and energy-densities with respect to the corresponding energy-momentum tensors. After that, the energy-momentum tensors, pressures, energy-densities and state-parameters are analyzed with respect to different concepts of generalized Riemannian spaces. At the end of this paper, linearities of the energy-momentum tensor, pressure, energy-density and the state-parameter are examined., Comment: 17 pages, 0 figures, manuscript

In this paper, we introduce the notion of exceptional family for the system of implicit generalized order complementarity problems in vector lattice. We present some alternative existence results of the solutions for the system of implicit generalized order complementarity problems via topological degree aspects. The new developments in this paper generalize and improve some known results in the literature.

Regarding the study of digital topological rough set structures, the present paper explores some mathematical and systemical structures of the Marcus-Wyse (MW-, for brevity) topological rough set structures induced by the locally finite covering approximation (LFC-, for brevity) space (R2,C) (see Proposition 3.4 in this paper), where R2 is the 2-dimensional Euclidean space. More precisely, given the LFC-space (R2,C), based on the set of adhesions of points in R2 inducing certain LFC-rough concept approximations, we systematically investigate various properties of the MW-topological rough concept approximations (D -M, D+M) derived from this LFC-space (R2,C). These approaches can facilitate the study of an estimation of roughness in terms of an MW-topological rough set. In the present paper each of a universe U and a target set X(? U) need not be finite and further, a covering C is locally finite. In addition, when regarding both an M-rough set and an MW-topological rough set in Sections 3, 4, and 5, the universe U(? R2) is assumed to be the set R2 or a compact subset of R2 or a certain set containing the union of all adhesions of x ? X (see Remark 3.6).

In this paper, we define the quaternionic Fock spaces $\mathfrak{F}_{\alpha}^p$ of entire slice hyperholomorphic functions in a quaternionic unit ball $\mathbb{B}$ in $\mathbb{H}.$ We also study growth estimate and various results of entire slice regular functions in these spaces. The work of this paper is motivated by the recent work of \cite{alpa12} and \cite{marco}.

Measuring and managing project quality is one of the fundamental problems in project management. In this paper, once project is completed, we define the measure of task?s quality and the measure of overall project quality. We construct mathematical model of the process of quality improvement as the sequence of mutually dependent projects, where every project in the sequence is the revision of its previous one. We prove that it is possible, at certain point, to obtain a project with the highest quality measure and with failures less than the initially given level. The purpose of this paper is, to help companies to achieve the satisfying level of project quality by using the proposed model. According to our knowledge, paper offers an original connection between project management and measure theory potentially interesting to a reader for further research.

We present an efficient method from Computational geometry, a branch of computer science devoted to the study of algorithms, for mathematical visualization of a third order root solver. For many decades the quality of iterative methods for solving nonlinear equations were analyzed only by using numerical experiments. The disadvantage of this approach is the inconvenient fact that convergence behavior strictly depends on the choice of initial approximations and the structure of functions whose zeros are sought, which often makes the convergence analysis very hard and incomplete. For this reason in this paper we apply dynamic study of iterative processes relied on basins of attraction, a new and powerful methodology developed at the beginning of the 21th century. This approach provides graphic visualization of the behavior of convergent sequences and, consequently, offers considerably better insight into the quality of applied root solvers, especially into the domain of convergence. For demonstration, we present dynamic study of one parameter family of Halley?s type introduced in the first part of the paper. Characteristics of this family are discussed by basins of attractions for various values of the involved parameter. Special attention is devoted to clusters of polynomial roots, one of the most difficult problems in the topic. The analysis of the methods and presentation of basins of attractions are performed by the computer algebra system Mathematica.

This paper adds in our hands a different analytic numeric method to solve a class of fuzzy fractional differential equations (FFDEs) based on the residual power series method (RPSM) under strongly generalized differentiability. The analytic and approximate solutions are provided with the series form according to their parametric form. The new method explained in the current paper has a lot of advantages as follows: First, its nature is global according to the obtainable solutions along with being able to solve numerous problems such as mathematical, physical and engineering ones. Second. It is easily noted that it is precise, needs few efforts to have the required results achieved, alongside being developed for nonlinear problems and cases. As for the third advantage, it can be said that any point in the interval of interest will be possibly picked, in addition, to have the approximate solutions applied. Fourth, the method does not need the variables discretization, also it is not implemented by computational round of errors. At last, the results reached in the current paper show several features concerning the new method such as potentiality, generality and superiority to handle such problems arising in physics and engineering as well.

Main goal of this paper is to establish various basic formulas for the generalized integral transform involving the generalized convolution product. In order to establish these formulas, we use the translation operator which was introduced in [9]. It was not easy to establish basic formulas for the generalized integral transforms because the generalized Brownian motion process used in this paper has the nonzero mean function. In this paper, we can easily establish various basic formulas for the generalized integral transform involving the generalized convolution product via the translation operator.

The study concerning the classification of the fuzzy subgroups of finite groups is a significant aspect of fuzzy group theory. In early papers, the number of distinct fuzzy subgroups of some nonabelian groups is calculated by the natural equivalence relation. In this paper, we treat to classifying fuzzy subgroups of some groups by a new equivalence relation which has a consistent group theoretical foundation. In fact, we determine exact number of fuzzy subgroups of finite non-abelian groups of order p3 and special classes of dihedral groups.

In this paper, weprove existence of fixed and coincidence points for a general class of multivalued mappings satisfying a new generalized contractive condition in incomplete metric spaces which generalize a number of published results in the last decades. In addition, this article not only brings a new approaches on the subject and but also involves several non-trivial examples which demonstrate the significance of the motivation. Finally, the obtained results of this paper provide a result on the convergence of successive approximations for certain operators (not necessarily linear) on a norm space (not necessarily a Banach space). In particular, we conclude that the renowned Kelisky-Rivlin theorem works on iterates of the Bernstein operators on an incomplete subspace of C[0,1].

WOS: 000500749700020 In the present paper, we consider non-trivial warped product pseudo slant submanifolds of type M-perpendicular to X-f M-theta and M-theta X-f M-perpendicular to of Kenmotsu f-manifold (M) over bar. Firstly, we get some basic properties of these type warped product submanifolds. Then, we prove the general sharp inequalities for mixed totally geodesic warped product pseudo slant submanifolds and also we consider equality cases. Also generalizes some previous inequalities as well. King Khalid University, Saudi Arabia [R. G. P. 1/65/40] The authors would like to express their sincere thanks to the Dr Akram Ali for his valuable suggestions towards to improvement of the paper. Finally, the authors would like to express their gratitude to Deanship of Scientific Research at King Khalid University, Saudi Arabia for providing funding research groups under the research grant R. G. P. 1/65/40.

In this paper, we use the idea of normal family to investigate the problem of entire functions that share two entire functions with one of their derivatives. In particular, we solve an open problem posed in the last section of [11]. Some examples have been exhibited to show that the conditions used in the paper are sharp.

The present paper studies certain low-level separation axioms of a topological space, denoted by A(X), induced by a geometric AC-complex X. After proving that whereas A(X) is an Alexandroff space satisfying the semi-T1 2 -separation axiom, we observe that it does neither satisfy the pre T1 2 -separation axiom nor is a Hausdorff space. These are main motivations of the present work. Although not every A(X) is a semi-T1 space, after proceeding with an edge to edge tiling (or a face to face crystallization) of Rn, n ? N, denoted by T(Rn) as an AC complex, we prove that A(T(Rn)) is a semi-T1 space. Furthermore, we prove that A(En), induced by an nD Cartesian AC complex Cn = (En,N,dim), is also a semi-T1 space, n ? N. The paper deals with AC-complexes with the locally finite (LF-, for brevity) property, which can be used in the fields of pure and applied mathematics as well as digital topology, computational topology, and digital geometry.

For a Tychonoff space $X$, we denote by $(C(X), \tau_k, \tau_p)$ the bitopological space of all real-valued continuous functions on $X$ where $\tau_k$ is the compact-open topology and $\tau_p$ is the topology of pointwise convergence. In papers [5, 6, 13] variations of selective separability and tightness in $(C(X), \tau_k, \tau_p)$ were investigated. In this paper we continued to study the selective properties and the corresponding topological games in the space $(C(X), \tau_k, \tau_p)$., Comment: 10 pages. arXiv admin note: text overlap with arXiv:1805.04363

Extensions of polygroups such as direct hyper product and wreath product of polygroups have been introduced and studied earlier by Comer. In this paper, we define and study some other extensions. We first define regularly normal subpolygroups and prove that they induce quotient polygroup extensions. In addition, we prove that the kernel of every strong homomorphism is a regularly normal subpolygroup. This leads to new versions of the isomorphism theorems with respect to such subpolygroups. The main objective of the paper is to present a new extension K of a polygroup L by a polygroup H via the quotient polygroup H=I where I is regularly normal in H. This extension is a generalization of both the direct hyper product and the wreath product of polygroups.

Let H be the real quaternion algebra and Hmxn denote the set of all m x n matrices over H. For A ? Hm x n, we denote by A? the n x m matrix obtained by applying ? entrywise to the transposed matrix At, where ? is a nonstandard involution of H. A ? Hnxn is said to be ?-Hermitian if A = A?. In this paper, we construct a simultaneous decomposition of four real quaternion matrices with the same row number (A,B,C,D), where A is ?-Hermitian, and B,C,D are general matrices. Using this simultaneous matrix decomposition, we derive necessary and sufficient conditions for the existence of a solution to some real quaternion matrix equations involving ?-Hermicity in terms of ranks of the given real quaternion matrices. We also present the general solutions to these real quaternion matrix equations when they are solvable. Finally some numerical examples are presented to illustrate the results of this paper.

In this paper we study the pointwise convergence and convergence in L1-norm of double trigonometric series whose coefficients form a null sequence of bounded variation of order (p,0),(0,p) and (p,p) with the weight (jk)p-1 for some integer p > 1. The double trigonometric series in this paper represents double cosine series, double sine series and double cosine sine series. Our results extend the results of Young [9], Kolmogorov [4] in the sense of single trigonometric series to double trigonometric series and of M?ricz [6,7] in the sense of higher values of p.

The Cauchy problem with a rapidly oscillating initial condition for the homogeneous Schr?dinger equation was studied in [5]. Continuing the research ideas of this work and [3], in this paper we construct the asymptotic solution to the following mixed problem for the nonstationary Schr?dinger equation: Lhu ? ih?tu + h2?2xu-b(x,t)u = f(x,t), (x,t) ? ??= (0,1) x (0,T], u|t=0 = g(x), u|x=0 = u|x=1 = 0, (1) where h > 0 is a Planck constant, u = u(x,t,h). b(x,t), f(x,t) ? C??(??), g(x) ? C? [0,1] are given functions. The similar problem was studied in [7, 8] when the Plank constant is absent in the first term of the equation and asymptotics of solution of any order with respect to a parameter was constructed. In this paper, we use a generalization of the method used in [7].

This paper is concerned with the problem of finding common fixed points for a family of Bregman relatively weak nonexpansive mappings. The motivation is due to our finding of some gaps in a paper of K. S. Kim (Nonlinear Analysis, 73 (2010), 3413-3419), where the author was developing a hybrid iterative scheme for locating common fixed points of a nonlinear representation of a left reversible semigroup. After a brief discussion about the gaps and why they are fatal, we present a new approach by using Bergman type nonexpansive mappings. A correct version of Kim’s convergence theorem is given as a consequence of our new results, which also improve and extend some recent results in the literature.

The largest class of hyperstructures is the one which satisfies the weak properties and they are called Hv-structures. In this paper, the concept of Hv-S-act is introduced and some of their properties are investigated. The present paper establishes a possible connection between S-act, GHS-act and Hv-S-act. It is shown that the quotient of GHS-act with any equivalence relation is Hv-S-act. The main tool to study all hyperstructures is the fundamental relations. The study of fundamental relations in Hv-S-act reveals some interesting results. Specifically, these relations connect weak hyperactions with the corresponding classical actions.

Zagreb indices and their modified versions of a molecular graph originate from many practical problems such as two dimensional quantitative structure-activity (2D QSAR) and molecular chirality. Nowadays, they have become important invariants which can be used to characterize the properties of graphs from different aspects. LetVk n (or Ek n respectively) be a set of graphs of n vertices with vertex connectivity (or edge connectivity respectively) at most k. In this paper, we explore some properties of the modified first and second multiplicative Zagreb indices of graphs in Vkn and Ekn. By using analytic and combinatorial tools, we obtain some sharp lower and upper bounds for these indices of graphs in Vk n and Ekn. In addition, the corresponding extremal graphs which attain the lower or upper bounds are characterized. Our results enrich outcomes on studying Zagreb indices and the methods developed in this paper may provide some new tools for investigating the values on modified multiplicative Zagreb indices of other classes of graphs.

In this paper, we introduce the class of disturbed convex functions defined by means of distance perturbations in two dimensions on co-ordinates. Some quantum trapezoidal estimations are obtained for functions having two dimensional distance-disturbed convexity properties. Refined bounds of the quantum integrals of distance-disturbed convex functions on coordinates are deduced by using the rectangular finite elements technique. These approximations are as best as possible from the sharpness point of view. The sharpness of few results from the literature follows as consequence of the new results in this paper.

In this paper, we introduce and study a class of generalized implicit operator equilibrium problems (In short, GIOEP) and derive some propositions for this class of problems. We also prove some new existence results for the solution of this problem by using Fan KKM theorems in Hausdorff topological vector spaces. The results presented in this paper generalize and unify the corresponding results of several authors which are extensions of previously known results.

In this paper, using the new and improved form of H?lder?s integral inequality called H?lder-??can integral inequality, some new inequalities of the right-hand side of Hermite-Hadamard type inequality for prequasiinvex functions are established. The results obtained are compared with the known results. It is shown that the results obtained in this paper are better than those known ones.

Our study is developed in a general framework, namely a manifold M endowed with a (1,1)- tensor field ?, which is integrable. The present paper solves the following two problems: how many linear connections with torsion and without torsion exist, having the property of being parallel with respect to ?. To count all these connections with the given properties, certain algebraic techniques and results are used throughout the paper.