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This paper presents a reliable approach for solving linear systems of ordinary and fractional differential equations. First, the FDEs or ODEs of a system with initial conditions to be solved are transformed to Volterra integral equations. Then Taylor expansion for the unknown function and integration method are employed to reduce the resulting integral equations to a new system of linear equations for the unknown and its derivatives. The fractional derivatives are considered in the Riemann-Liouville sense. Some numerical illustrations are given to demonstrate the effectiveness of the proposed method in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we give an existence theorem about positive solutions for the Dirichlet boundary value problem of one dimensional Minkowski curvature equations. We apply the theorem to one parameter family of problems to investigate a constructive method for numerical range of parameters where positive solutions exist. Moreover, we establish a nonexistence theorem of positive solutions for the corresponding one parameter family of problems. The coefficient function may be singular at the boundary and nonlinear term satisfies a sublinear growth condition. Main argument for the proof of existence theorem is employed by Krasnoselskii's theorem of cone expansion and compression. We give a numerical algorithm and various examples to illustrate numerical information about ranges of the existence and nonexistence parameters which have been given only in a theoretical manner so far. [ABSTRACT FROM AUTHOR]

The Fibonacci cube of dimension n, denoted as Γ n , is the subgraph of the hypercube induced by vertices with no consecutive 1s. The Lucas cube Λ n is the cyclic version of Γ n . The irregularity of a graph G is the sum of | d (x) - d (y) | over all edges { x , y } of G. In two recent papers based on the recursive structure of Γ n , it is proved that the irregularity of Γ n and Λ n is two times the number of edges of Γ n - 1 and 2n times the number of vertices of Γ n - 4 , respectively. Using an interpretation of the irregularity in terms of couples of incident edges of a special kind, we give a bijective proof of both results. For these two graphs, we deduce also a constant time algorithm for computing the imbalance of an edge. In the last section using the same approach, we determine the number of edges and the sequence of degrees of the cube complement of Γ n . [ABSTRACT FROM AUTHOR]

The paper proposes a new explicit extragradient algorithm for solving an equilibrium problem over the solution set of another generalized equilibrium problem which usually is called bilevel equilibrium problems in a real Hilbert space. The algorithm combines extragradient methods with auxiliary problem techniques. Theorem of the strong convergence of the algorithm is established under standard assumptions imposed on the cost bifunctions. Finally, several of fundamental experiments are provided to illustrate the numerical behavior of the algorithm and also to compare with others. [ABSTRACT FROM AUTHOR]

The power graph $${\fancyscript{P}}(G)$$ of a group $$G$$ is the graph which has a vertex set of the group elements and two elements are adjacent if one is a power of the other. Chakrabarty, Ghosh, and Sen proved the main properties of the undirected power graph of a finite group. The aim of this paper is to generalize some results of their work and presenting some counterexamples for one of the problems raised by these authors. It is also proved that the power graph of a $$p$$ -group is $$2$$ -connected if and only if the group is a cyclic or generalized quaternion group and if $$G$$ is a nilpotent group which is not of prime power order then the power graph $${\fancyscript{P}}(G)$$ is $$2$$ -connected. We also prove that the number of edges of the power graph of the simple groups is less than or equal to the number of edges in the power graph of the cyclic group of the same order. This partially answers to a question in an earlier paper. Finally, we give a complete classification of groups in which the power graph is a union of complete graphs sharing a common vertex. [ABSTRACT FROM AUTHOR]

In 2011, Gross derived an O (n 2) -time algorithm to calculate the genus distribution of a given cubic Halin graph. In this paper, with the help of the overlap matrix, we obtain a recurrence relation for the Euler-genus polynomials of cubic caterpillar-Halin graphs. Explicit formulas for the embeddings of the cubic caterpillar-Halin graphs into surfaces with Euler-genus 0, 1 and 2 are also obtained. [ABSTRACT FROM AUTHOR]

In this paper, we study the metric dimension problem in maximal outerplanar graphs. Concretely, if β (G) denotes the metric dimension of a maximal outerplanar graph G of order n, we prove that 2 ≤ β (G) ≤ ⌈ 2 n 5 ⌉ and that the bounds are tight. We also provide linear algorithms to decide whether the metric dimension of G is 2 and to build a resolving set S of size ⌈ 2 n 5 ⌉ for G. Moreover, we characterize all maximal outerplanar graphs with metric dimension 2. [ABSTRACT FROM AUTHOR]

In this paper, we consider a differential inclusion governed by a set-valued nonexpansive mapping and study the asymptotic behavior (weak and strong convergence) of its solutions with various assumptions on this mapping. Then for a set-valued nonexpansive mapping, we define the corresponding resolvent (proximal) operator as a set-valued mapping and study some of its elementary properties. Subsequently, we apply the resolvent operator to state the implicit discretization of the differential inclusion and study the asymptotic behavior of its solutions which yields similar convergence results as in the continuous case. This provides an algorithm for approximating a fixed point of a set-valued nonexpansive mapping which extends the classical proximal point algorithm. An application to variational inequalities and a numerical comparison with another iterative method for approximating a fixed point of set-valued nonexpansive mappings are also presented. [ABSTRACT FROM AUTHOR]

For η ∈ { i , j , k } , a real quaternion matrix A is said to be η -Hermitian if A = A η ∗ , where A η ∗ = - η A ∗ η , and A ∗ stands for the conjugate transpose of A. In this paper, we present some practical necessary and sufficient conditions for the existence of an η -Hermitian solution to a system of constrained two-sided coupled real quaternion matrix equations and provide the general η -Hermitian solution to the system when it is solvable. Moreover, we present an algorithm and a numerical example to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, a meshless numerical scheme for solving an inverse source problem is considered. The proposed scheme is based on approximating the solution employing the thin plate spline (TPS) radial basis function (RBF). Applying this radial basis function results in a badly ill-condition system of equations. The Tikhonov regularization method is employed for solving this system of equations. Determination of regularization parameter is based on generalized cross-validation (GCV) technique. Some numerical examples are presented to demonstrate the accuracy and ability of this method. [ABSTRACT FROM AUTHOR]

In this paper, we propose a CQ-type algorithm for solving the split feasibility problem (SFP) in real Hilbert spaces. The algorithm is designed such that the step-sizes are directly computed at each iteration. We will show that the sequence generated by the proposed algorithm converges in norm to the minimum-norm solution of the SFP under appropriate conditions. In addition, we give some numerical examples to verify the implementation of our method. Our result improves and complements many known related results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we characterize the congruence lattice of a symmetric extended De Morgan algebra $$L$$ . We show that the congruence lattice of the algebra $$L$$ is a pseudocomplemented lattice, and that such a congruence lattice is a Stone lattice if and only if the lattice of the compact congruences on $$L$$ forms a complete Boolean lattice. In particular, we prove that the congruence lattice of $$L$$ is a Boolean lattice if and only if, it is a relative Stone lattice, which is the case, if and only if $$L$$ is finite. [ABSTRACT FROM AUTHOR]

In this paper, we present an algorithmic method to describe and classify filiform Lie algebras by considering the maximal dimension of abelian ideals and studying the relation between them and some known invariants of filiform Lie algebras. To do so, we have proved some new results about structure coefficients in the general law of filiform Lie algebras. Moreover, some mistakes are corrected in the already-known classification of filiform Lie algebras of dimension 8. [ABSTRACT FROM AUTHOR]

Björner, Lovász and Shor introduced a chip-firing game on a finite graph $$G$$ as follows. We put some chips on each vertex of $$G$$ , we say that a vertex is ready if it has at least as many chips as its degree, in which case we can fire it and the result is that it distributes one chip to each of its neighbors, this may cause other vertices to be ready, and so on. This game continues until no vertex can be fired. In this paper, we study chip-firing games on complete graphs. We obtain a sufficient and necessary condition for chip-firing games on complete graphs to be finite. [ABSTRACT FROM AUTHOR]

Let $${\mathcal {F}}$$ denote the class of all functions univalent in the unit disk $$\Delta \equiv \{\zeta \in {\mathbb {C}}\,:\,\left| \zeta \right| <1\}$$ and convex in the direction of the real axis. The paper deals with the subclass $${\mathcal {F}}^{(n)}$$ of these functions $$f$$ which satisfy the property $$f(\varepsilon z)=\varepsilon f(z)$$ for all $$z\in \Delta $$ , where $$\varepsilon =e^{2\pi i/n}$$ . The functions of this subclass are called $$n$$ -fold symmetric. For $${\mathcal {F}}^{(n)}$$ , where $$n$$ is odd positive integer, the following sets, $$\bigcap _{f\in {\mathcal {F}}^{(n)}} f(\Delta )$$ -the Koebe set and $$\bigcup _{f\in {\mathcal {F}}^{(n)}} f(\Delta )$$ -the covering set, are discussed. As corollaries, we derive the Koebe and the covering constants for $${\mathcal {F}}^{(n)}$$ . [ABSTRACT FROM AUTHOR]

In the present paper, we study the extrinsic and intrinsic geometry of submanifolds of an almost contact metric manifold admitting a quarter-symmetric metric connection. We deduce Gauss, Codazzi and Ricci equations corresponding to the quarter-symmetric metric connection and show some applications of these equations. Finally, we give an example verifying the results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a concept of graph convergence for the $$H(\cdot ,\cdot )$$ -co-accretive mapping in Banach spaces and prove an equivalence theorem between graph convergence and resolvent operator convergence for the $$H(\cdot ,\cdot )$$ -co-accretive mapping. Further, we consider a system of generalized variational inclusions involving $$H(\cdot ,\cdot )$$ -co-accretive mapping in real $$q$$ -uniformly smooth Banach spaces. Using resolvent operator technique, we prove the existence and uniqueness of solution and suggest an iterative algorithm for the system of generalized variational inclusions under some suitable conditions. Further, we discuss the convergence of iterative algorithm using the concept of graph convergence. Our results can be viewed as a refinement and generalization of some known results in the literature. [ABSTRACT FROM AUTHOR]

There are many articles in the literature dealing with the first-order and the second-order differential subordination and differential superordination problems for analytic functions in the unit disk, but there are only a few articles dealing with the third-order differential subordination problems. The concept of third-order differential subordination in the unit disk was introduced by Antonino and Miller, and studied recently by Tang and Deniz. Let $$\Omega $$ be a set in the complex plane $$\mathbb {C}$$ , let $$\mathfrak {p}(z)$$ be analytic in the unit disk $$\mathbb {U}=\{z:z\in \mathbb {C}\quad \text {and} \quad |z|<1\}$$ , and let $$\psi : \mathbb {C}^4\times \mathbb {U}\rightarrow \mathbb {C}$$ . In this paper, we investigate the problem of determining properties of functions $$\mathfrak {p}(z)$$ that satisfy the following third-order differential superordination: As applications, we derive some third-order differential superordination results for analytic functions in $$\mathbb {U}$$ , which are associated with a family of generalized Bessel functions. The results are obtained by considering suitable classes of admissible functions. New third-order differential sandwich-type results are also obtained. [ABSTRACT FROM AUTHOR]

Let $$G$$ be a graph with degree sequence $$d_1,d_2,\ldots ,d_n$$ . Given a positive integer $$p$$ , denote by $$e_p(G)=\sum _{i=1}^n d_i^p$$ . Caro and Yuster introduced a Turán-type problem for $$e_p(G)$$ : given an integer $$p$$ , how large can $$e_p(G)$$ be if $$G$$ has no subgraph of a particular type. They got some results for the subgraph of particular type to be a clique of order $$r+1$$ and a cycle of even length, respectively. Denote by $$ex_p(n,H)$$ the maximum value of $$e_p(G)$$ taken over all graphs with $$n$$ vertices that do not contain $$H$$ as a subgraph. Clearly, $$ex_1(n,H)=2ex(n,H)$$ , where $$ex(n,H)$$ denotes the classical Turán number. In this paper, we consider $$ex_p(n, C_5)$$ and prove that for any positive integer $$p$$ and sufficiently large $$n$$ , there exists a constant $$c=c(p)$$ such that the following holds: if $$ex_p(n, C_5)=e_p(G)$$ for some $$C_5$$ -free graph $$G$$ of order $$n$$ , then $$G$$ is a complete bipartite graph having one vertex class of size $$cn+o(n)$$ and the other $$(1-c)n+o(n)$$ . [ABSTRACT FROM AUTHOR]

In this paper, we use a result of N. S. Feldman to show that there are no supercyclic subnormal tuples in infinite dimensions. Also, we investigate some spectral properties of hypercyclic tuples of operators. Besides, we prove that if $$T$$ is a supercyclic $$\ell $$ -tuple of commuting $$n\times n$$ complex matrices, then $$\ell \ge n$$ and also there exists a supercyclic $$n$$ -tuple of commuting diagonal $$n\times n$$ matrices. Furthermore, we see that if $$T=(T_{1},\ldots ,T_{n})$$ is a supercyclic $$n$$ -tuple of commuting $$n\times n$$ complex matrices, then $$T_{j}$$ 's are simultaneously diagonalizable. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the central limit theorem and the law of the iterated logarithm for Galton-Watson branching processes with time-dependent immigration in varying environments. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the Morgenstern type bivariate-generalized exponential distribution. This distribution is an extension of Morgenstern type bivariate exponential distribution (MTBED), and the marginal distributions are generalized exponential distribution. We study some properties of this bivariate distribution. Also, some distributional properties of concomitants of order statistics as well as record values for the MTBED are studied. Recurrence relations between moments of concomitants are also obtained. [ABSTRACT FROM AUTHOR]

In this paper we introduce and study the concepts of graded second (gr-second) and graded coprimary (gr-coprimary) modules which are different from second and coprimary modules over arbitrary-graded rings. We list some properties and characterizations of gr-second and gr-coprimary modules and also study graded prime submodules of modules with gr-coprimary decompositions. We also deal with graded secondary representations for graded injective modules over commutative-graded rings. By using the concept of $$\sigma $$ -suspension $$(\sigma )M$$ of a graded module $$M,$$ we prove that a graded injective module over a commutative graded Noetherian ring has a graded secondary representation. [ABSTRACT FROM AUTHOR]

Take three integers $$m\ge 0,\,k\ge 1$$ , and $$n\ge 2$$ . Let $$a\ (\not \equiv 0)$$ be a holomorphic function in a domain $$D$$ of $$\mathbb {C}$$ such that multiplicities of zeros of $$a$$ are at most $$m$$ and divisible by $$n+1$$ . In this paper, we mainly obtain the following normality criterion: Let $${{{\fancyscript{F}}}}$$ be the family of meromorphic functions on $$D$$ such that multiplicities of zeros of each $$f\in {{\fancyscript{F}}}$$ are at least $$k+m$$ and such that multiplicities of poles of $$f$$ are at least $$m+1$$ . If each pair $$(f,g)$$ of $${{\fancyscript{F}}}$$ satisfies that $$f^{n}f^{(k)}$$ and $$g^{n}g^{(k)}$$ share $$a$$ (ignoring multiplicity), then $${{\fancyscript{F}}}$$ is normal. [ABSTRACT FROM AUTHOR]

A connected graph $$G$$ is said to be factor-critical if $$G-v$$ has a perfect matching for every vertex $$v$$ of $$G$$ . Lovász proved that every factor-critical graph has an ear decomposition. In this paper, the ear decomposition of the factor-critical graphs $$G$$ satisfying that $$G-v$$ has a unique perfect matching for any vertex $$v$$ of $$G$$ with degree at least 3 is characterized. From this, the number of maximum matchings of factor-critical graphs with the special ear decomposition is obtained. [ABSTRACT FROM AUTHOR]

The b-chromatic index $$\varphi '(G)$$ of a graph $$G$$ is the largest integer $$k$$ such that $$G$$ admits a proper $$k$$ -edge coloring in which every color class contains at least one edge incident to some edge in all the other color classes. The b-chromatic index of trees is determined and equals either to a natural upper bound $$m'(T)$$ or one less, where $$m'(T)$$ is connected with the number of edges of high degree. Some conditions are given for which graphs have the b-chromatic index strictly less than $$m'(G)$$ , and for which conditions it is exactly $$m'(G)$$ . In the last part of the paper, regular graphs are considered. It is proved that with four exceptions, the b-chromatic index of cubic graphs is $$5$$ . The exceptions are $$K_4$$ , $$K_{3,3}$$ , the prism over $$K_3$$ , and the cube $$Q_3$$ . [ABSTRACT FROM AUTHOR]

In this paper, we present some results on the existence of random coincidence points of expansive type completely random operators. Some applications to random fixed point theorems and random equations are given. [ABSTRACT FROM AUTHOR]

The energy of a simple graph $$G$$ , denoted by $$E(G)$$ , is defined as the sum of the absolute values of eigenvalues of $$G$$ . In this paper, we show that, among all subdivisions of graphs with $$n$$ vertices and chromatic number $$k$$ , the subdivision of the Turán graph $$T(n,k)$$ has the maximal energy. [ABSTRACT FROM AUTHOR]

An $$m\times n\,\{0,1\}$$ -matrix $$A$$ is said to be totally convertible if there exists a matrix $$B$$ obtained from $$A$$ by changing some 1's in $$A$$ to $$-1$$ 's such that for any submatrix $$A^{\prime }$$ of $$A$$ of order $$m$$ , the corresponding submatrix $$B^{\prime }$$ of $$B$$ satisfies $$\mathrm{per}(xI-A^{\prime })=\det (xI-B^{\prime })$$ . In this paper, motivated by the well-known Pólya's problem, our object is to characterize those totally convertible matrices. Associate a matrix $$A$$ with a bipartite graph $$G^{*}_A$$ . We first prove that a square matrix $$A$$ is totally convertible if and only if $$G^{*}_A$$ is Pfaffian, and then we generalize this result to an $$m\times n$$ $$\{0,1\}$$ -matrix. Moreover, the characterization of a totally convertible matrix provides an equivalent condition to compute the permanental polynomial of a bipartite graph by the characteristic polynomial of the skew adjacency matrix of its orientation graph. As applications, we give some explicit expressions of the permanental polynomials of two totally convertible matrices by the technique of Pfaffian orientation. [ABSTRACT FROM AUTHOR]

Cohen-Macaulayness of bipartite graphs is investigated by several mathematicians and has been characterized combinatorially. In this paper, we give some different combinatorial conditions for a bipartite graph which are equivalent to Cohen-Macaulayness of the graph. We prove that a bipartite graph is Cohen-Macaulay if and only if it is well covered and has a unique perfect matching. We also provide a fast algorithm to check Cohen-Macaulayness of a given bipartite graph. [ABSTRACT FROM AUTHOR]

The k-distance domination problem is to find a minimum vertex set D of a graph such that every vertex of the graph is either in D or within distance k from some vertex of D, where k is a positive integer. In the present paper, by using labeling method, a linear-time algorithm for k-distance domination problem on block graphs is designed. [ABSTRACT FROM AUTHOR]

Cardiorespiratory fitness (CRF) is not only an objective measure of physical activity, but also a useful diagnostic and prognostic health indicator for patients in clinical settings. There is a well-established inverse relationship between cardiorespiratory fitness (CRF) and mortality. However, the effect of CRF on mortality status might be different on subgroups of individuals and could be higher or lower than the estimated average effect of CRF. Thus, the objective of the study is to identify subgroups with higher or lower impact of CRF on mortality status. In addition, we evaluate and compare both tree-based and non-tree-based algorithms for identifying predictive features and subgroups. A penalized logistic regression with least absolute shrinkage and selection operator (LASSO) penalty is performed to identify the features that may be associated with low CRF and all-cause mortality. The algorithms considered are: virtual twins classification (VT(C)), generalized unbiased interaction detection and estimation (GUIDE) classification (Gc), GUIDE sum (Gs), GUIDE interaction (Gi) to find subgroups of participants where CRF exerts positive or negative association with all-cause mortality from the Ball State Adult Fitness Longitudinal Lifestyle Study (BALL ST) data. The overall result suggests that tree-based (VT and GUIDE) methods naturally define subgroups with fewer predictors and the non-tree-based method (logistic-LASSO) fails to find subgroups, only identify predictors that have impact on mortality status. In terms of predictive variable selection and subgroup identification, Gi is the best method compared to other tree-based and non-tree-based algorithms. Our study identifies subgroups that may be benefited from higher CRF. [ABSTRACT FROM AUTHOR]

In this article, we consider numerical methods for solving Allen–Cahn equation involving strong nonlinearities. A fast algorithm based on time two-mesh (TT-M) finite element (FE) scheme is proposed to overcome the time-consuming computation of nonlinear terms. The TT-M FE algorithm includes the following three main steps: Firstly, a nonlinear FE scheme is solved on a coarse time mesh τ c . Here, the FE method is used for spatial discretization and the implicit second-order backward difference scheme is used for temporal discretization. Secondly, the Lagrange's interpolation is used to obtain the interpolation result on the fine grid. Finally, a linearized FE system is solved on a fine time mesh τ (τ < τ c) . The stability analysis and priori error estimates are provided in detail. Numerical examples are given to demonstrate the validity of the proposed scheme. The TT-M FE method is compared with the traditional Galerkin FE method, and it is evident that the TT-M FE method can save the calculation time. [ABSTRACT FROM AUTHOR]

A theta graph is the union of three internally disjoint paths that have the same two distinct end vertices. We show that every graph of order $$n\ge 9$$ and size at least $$\lfloor \frac{7n-13}{2}\rfloor $$ contains two disjoint theta graphs. We also show that every 2-edge-connected graph of order $$n\ge 6$$ and size at least $$3n-5$$ contains two disjoint cycles, such that any specified vertex with degree at least three belongs to one of them. The lower bound on size in both is sharp in general. [ABSTRACT FROM AUTHOR]

Let $$D$$ be a simple digraph with vertex set $$V(D)$$ , and let $$f:V(D)\rightarrow \{-1,1\}$$ be a two-valued function. If $$k\ge 1$$ is an integer and $$\sum _{x\in N^-[v]}f(x)\ge k$$ for each $$v\in V(D)$$ , where $$N^-[v]$$ consists of $$v$$ and all vertices of $$D$$ from which arcs go into $$v$$ , then $$f$$ is a signed $$k$$ -dominating function on $$D$$ . A set $$\{f_1,f_2,\ldots ,f_d\}$$ of distinct signed $$k$$ -dominating functions on $$D$$ with the property that $$\sum _{i=1}^df_i(x)\le k$$ for each $$x\in V(D)$$ , is called a signed $$(k,k)$$ -dominating family (of functions) on $$D$$ . The maximum number of functions in a signed $$(k,k)$$ -dominating family on $$D$$ is the signed $$(k,k)$$ -domatic number of $$D$$ . In this article, we mainly present upper bounds on the signed $$(k,k)$$ -domatic number, in particular for regular digraphs. [ABSTRACT FROM AUTHOR]

In this note, we establish a distortion theorem for locally biholomorphic Bloch mappings $$f$$ satisfying $$||f||_{0}=1$$ and $$\det f'(0)=\alpha \in (0,1],$$ where $$\Vert f\Vert _{0}=\mathrm {sup}\{(1-|z|^{2})^\frac{n+1}{2n}|\det f'(z)| ^\frac{1}{n}:z\in \mathcal {B}^{n}\}.$$ This result extends the result of Bonk, Minda, and Yanagihara of one complex variable to higher dimensions. Moreover, a lower estimate for the radius of the largest univalent ball in the image of $$f$$ centered at $$f(0)$$ is given. [ABSTRACT FROM AUTHOR]

Let $$\mathcal {U}$$ denote the class of normalized analytic functions $$f$$ in the open unit disk $$\mathbb {D}$$ satisfying The $$\mathcal {U}$$ -radius is obtained for several classes of functions. These include the class of normalized analytic functions $$f$$ satisfying the inequality $${{\mathrm{Re}}}\, f(z)/g(z)>0$$ or $$\left| f(z)/g(z)-1\right| < 1$$ in $$\mathbb {D},$$ where $$g$$ belongs to a certain class of functions, the class of functions $$f$$ satisfying $$|f'(z)-1|<1$$ in $$\mathbb {D},$$ and functions $$f$$ satisfying $${{\mathrm{Re}}}\, f(z)/z>\alpha , 0 \le \alpha < 1,$$ in $$\mathbb {D}.$$ A recent conjecture by Obradović and Ponnusamy concerning the radius of univalence for a product involving univalent functions is also shown to hold true. [ABSTRACT FROM AUTHOR]

Given a starlike function $$g\in \mathcal S^*,$$ an analytic standardly normalized function $$f$$ in the unit disk $$\mathbb {D}$$ is called close-to-convex with respect to $$g$$ if there exists $$\delta \in (-\pi /2,\pi /2)$$ such that For the class $$\mathcal C(h)$$ of all close-to-convex functions with respect to $$h(z):=z/(1-z),\ z\in \mathbb {D},$$ a Fekete-Szegö problem is examined. [ABSTRACT FROM AUTHOR]

A ring $$R$$ is $$CN$$ if and only if for any $$x\in N(R)$$ and $$y\in R$$ , $$((1+x)y)^{n+k}=(1+x)^{n+k}y^{n+k}$$ , where $$n$$ is a fixed positive integer and $$k=0, 1,2$$ ; (2) Let $$R$$ be a $$CN$$ ring and $$n\ge 1$$ . If for any $$x, y\in R\backslash N(R)$$ , $$(xy)^{n+k}=x^{n+k}y^{n+k}$$ , where $$k=0, 1,2$$ , then $$R$$ is commutative; (3) Let $$R$$ be a ring and $$n\ge 1$$ . If for any $$x\in R\backslash N(R)$$ and $$y\in R$$ , $$(xy)^k=x^ky^k$$ , $$k=n, n+1, n+2$$ , then $$R$$ is commutative; (4) $$NLI $$ exchange rings are clean. [ABSTRACT FROM AUTHOR]

We define a more general type of integral on time scales. The new diamond integral is a refined version of the diamond-alpha integral introduced in 2006 by Sheng et al. A mean value theorem for the diamond integral is proved, as well as versions of Holder's, Cauchy-Schwarz's, and Minkowski's inequalities. [ABSTRACT FROM AUTHOR]