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In this paper, we show that Loewner spaces introduced by Heinonen and Koskela (Acta Math., 1998) are preserved under quasimöbius mappings between Ahlfors regular spaces. [ABSTRACT FROM AUTHOR]

This paper proves a theorem on convergence in mean of order p for double arrays of M-pairwise negatively dependent random variables, where 1 ≤ p < 2 . The main result extends some results in Thành (Int J Math Math Sci 27(8):1317–1320, 2005) and Hong and Hwang (Int J Math Math Sci 22(1):171–177, 1999). The proof is based on the von Bahr–Esseen inequality for M-pairwise negatively dependent random variables. The sharpness of the result is illustrated by two examples. [ABSTRACT FROM AUTHOR]

Let G be a simple graph with no isolated edges. The neighbor-distinguishing edge coloring of G is a proper edge coloring of G such that any pair of adjacent vertices have different sets consisting of colors assigned on their incident edges. The neighbor-distinguishing index of G , denoted by χ a ′ (G) , is the minimum number of colors in such an edge coloring of G . In this paper, we show that if G is a connected planar graph with maximum degree Δ ≥ 14 , then Δ ≤ χ a ′ (G) ≤ Δ + 1 , and χ a ′ (G) = Δ + 1 if and only if G contains a pair of adjacent vertices of maximum degree. This improves a result in [W. Wang, D. Huang, A characterization on the adjacent vertex distinguishing index of planar graphs with large maximum degree, SIAM J. Discrete Math. 29(2015), 2412–2431], which says that every connected planar graph G with Δ ≥ 16 has Δ ≤ χ a ′ (G) ≤ Δ + 1 , and χ a ′ (G) = Δ + 1 if and only if G contains a pair of adjacent vertices of maximum degree. [ABSTRACT FROM AUTHOR]

A graph G is (d 1 , d 2 , ... , d k) -colorable if the vertex set of G can be partitioned into subsets V 1 , V 2 , ... , V k such that the subgraph G [ V i ] induced by V i has maximum degree at most d i for i = 1 , 2 , ... , k . Novosibirsk's Conjecture (Sib lektron Mat Izv 3:428–440, 2006) says that every planar graph without 3-cycles adjacent to cycles of length 3 or 5 is 3-colorable. Borodin et al. (Discrete Math 310: 167–173, 2010) asked whether every planar graph without adjacent cycles of length at most 5 is 3-colorable. Cohen-Addad et al. (J Comb Theory Ser B 122:452–456, 2017) gave a negative answer to both Novosibirsk's conjecture and Borodin et al.'s problem. Zhang et al. (Discrete Math 339:3032–3042, 2016) asked whether every planar graph without adjacent cycles of length at most 5 is (1, 0, 0)-colorable. In this paper, we show that every planar graph without adjacent cycles of length at most five is (2, 0, 0)-colorable, which improves the result of Chen et al. (Discrete Math 339:886–905, 2016) who proved that every planar graph without cycles of length 4 and 5 is (2, 0, 0)-colorable. [ABSTRACT FROM AUTHOR]

A finite order P on a set V is reconstructible (respectively, reconstructible up to duality) by its comparability graph if each order on V which has the same comparability graph as P is isomorphic to P (respectively, is isomorphic to P or to its dual P ⋆ ). In this paper, we describe the finite orders which are reconstructible up to duality by their comparability graphs. This result is motivated by the characterization, obtained by Gallai (Acta Math Acad Sci Hungar 18:25–66, 1967), of the pairs of finite orders having the same comparability graph. Notice that a characterization of the finite orders which are reconstructible by their comparability graphs is easily deduced from Gallai's result. [ABSTRACT FROM AUTHOR]

Let A be an Archimedean f-algebra, let x ∈ A , and let π x : A → A be the linear map defined by π x y = x y , for all y ∈ A. The aim of our paper is to give necessary and sufficient conditions concerning the averaging property of (r.u) continuous projections on Archimedean f-algebras, with a range, R T , a vector sublattice of A, that maps weak order units into weak order units. As an application, we prove that if A is an Archimedean f-algebra with a unit element e, T is a positive projection on A, with a range, R T , a vector sublattice of A, such that T(e) is a weak order unit of A, then T is an averaging operator if and only if R T is π T e - invariant subspace. This improves considerably a result of Kuo et al. (J Math Anal Appl 303:509–521, 2005). [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the brush number of a graph-a concept introduced by McKeil, Messinger, Nowakowski and Pralat. Our main aim in this paper is to study the brush number of the Cartesian product of a tree with a path, and a tree with a cycle. Based on an optimal cleaning process of a tree, we describe cleaning processes that give upper bounds to the brush number of $$T\times P_n$$ and $$T\times C_n$$ . We also show that these bounds are tight if T is a star, a double star and some generalisation of a star. [ABSTRACT FROM AUTHOR]

In this paper, we studied a normality criterion concerning Hayman's question and proved: let n (≥ 2) , k (≥ 1) , m (≥ 0) be three integers, let h (z) (≢ 0) be a holomorphic function in a domain D with all zeros that have multiplicity at most m, and let F be a family of functions meromorphic in a domain D, all of whose zeros have multiplicity at least k + m . If, for any two functions f , g ∈ F , f n f (k) and g n g (k) share h(z) in D, then F is normal in D. The result gets rid of two conditions "all zeros of h(z) have multiplicity divisible by n + 1 " and "all poles of f(z) have multiplicity at least m + 1 " in the result due to Meng and Hu (Bull Malays Math Sci Soc 38:1331–1347, 2015). [ABSTRACT FROM AUTHOR]

In this paper, we study the further improvements of the reverse Young and Heinz inequalities for the wider range of v, namely v∈R. These modified inequalities are used to establish corresponding operator inequalities on a Hilbert space. [ABSTRACT FROM AUTHOR]

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]

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]

Let $$\mathcal {PO}_n$$ be the semigroup of all order-preserving partial transformations on the finite set $$X_n=\{1, 2,\ldots , n\}$$ . For $$1\le r\le n-1$$ , set $$\mathcal {PO}(n, r)=\{\alpha \in \mathcal {PO}_n: |\mathop {\text{ im }}\nolimits (\alpha )|\le r\}$$ . In this paper, we investigate the maximal regular subsemigroups and the maximal regular subsemibands of the semigroup $$\mathcal {PO}(n,r)$$ . First, we completely describe the maximal regular subsemigroups of the semigroup $$\mathcal {PO}(n,r)$$ , for $$1\le r\le n-1$$ . Secondly, we show that, for $$2\le r \le n-2$$ , any maximal regular subsemigroup of the semigroup $$\mathcal {PO}(n,r)$$ is a semiband and obtain that the maximal regular subsemigroups and the maximal regular subsemibands of the semigroup $$\mathcal {PO}(n,r)$$ coincide, for $$2\le r\le n-2$$ . Finally, we obtain the complete classification of maximal regular subsemibands of the semigroup $$\mathcal {PO}_n$$ . [ABSTRACT FROM AUTHOR]

In this paper, we unify previous definitions of weakened open subsets in a given topological space. We felt necessary to make this unification since we observed recently too many definitions, actually, more or less significant. We also show that our new framework is more general than the known supra-topological structure. Finally we show that, in our new framework, we can define well several standard topologies like concepts such as convergence, continuity, and compactness. We expect that our new framework could find application in images processing. [ABSTRACT FROM AUTHOR]

In this paper, we study a uniqueness question of meromorphic functions whose nonlinear differential polynomials share a nonzero small function. The results in this paper improve and extend many known results. [ABSTRACT FROM AUTHOR]

Let G be a finite group and π(G) be the set of prime divisors of the order of G. For t ϵ π(G) denote by nt (G) the order of a normalizer of t-Sylow subgroup of G and put n(G) = {nt (G) : t ϵ π(G)}. In this paper, we give an answer to the following problem, for the groups of Lie type Bn, Cn and Dn: "Let L be a finite non-abelian simple group and G be a finite group with n(L) = n(G). Is it true that L ≅= G?" In this paper, we find the first examples of non-abelian finite simple groups which are not isomorphic and they have the same set of orders of Sylow normalizers and hence, we show that the question above is not correct always. Let A be the set of prime numbers of order 2n, 2(n-1) and 2(n-2) mod q. The latter condition is necessary if n ≥ 5. Also, we show that Dn+1(q) is determined uniquely by its order and {nt (Dn+1(q)) : t ϵ A ∪ {2}} and if n = 2 or q ≢ ±1 (mod 8), then Bn(q) and Cn(q) are characterizable by their orders and orders of t-Sylow normalizers, where t ϵ A ∪{2}. If n≥3 and q ≡ ±1 (mod 8), then Bn(q) and Cn(q) are 2-characterizable by their orders and the orders of t-Sylow normalizers, where t ϵ A {2}. [ABSTRACT FROM AUTHOR]

This paper is concerned with the Lie superalgebra S( n, m) of special type over a field of prime characteristic. We construct the modular Lie superalgebra S( n, m) and discuss some properties of this algebra. Then the derivation superalgebra of S( n, m) is determined. [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]

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]

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 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]

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]

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]

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]

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]

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 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]

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 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]

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]

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]

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]

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]

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]

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]

It is known that the theory of Markov processes is a rapidly developing field with numerous applications to many branches of mathematics and physics, biology, and so on. But there are some physical models which cannot be described by such processes. One of such models is related to population genetics. These processes are called quadratic stochastic processes (q.s.p.). In the present paper, we associate to given q.s.p. two kind of processes, which call marginal processes. Note that one of them is Markov process. We prove that such kind of processes uniquely define q.s.p. Moreover, we provide a construction of nontrivial examples of q.s.p. Weak ergodicity of q.s.p. is also studied in terms of the marginal processes. [ABSTRACT FROM AUTHOR]

In this paper we study the relationship between g-frames for super Hilbert space H ⊕ K and g-frames for H and K and frames for H, K and H ⊕ K . Also the concepts of disjoint g-frames and strong complementary g-frame are introduced and studied. [ABSTRACT FROM AUTHOR]

Let Mn,m be the set of all n×m matrices with entries in F, where F is the field of real or complex numbers. For A,B ϵ Mn,m, we say that B is lgw-majorized (left generalized weakly majorized) by A if there exists an n×n g-row stochastic (generalized row stochastic) matrix R such that B = RA. In this paper, we characterize all linear operators that strongly preserve lgw-majorization on Mn,m and all linear operators that strongly preserve left weak matrix majorization on Mn. [ABSTRACT FROM AUTHOR]

In this paper we use shear construction to generate certain subclasses of harmonic univalent mappings with directional convexity because it allows us to study such functions by examining their related conformal univalent mappings. In this setting we find growth, distortion, and coefficient bounds for harmonic univalent mappings that are convex in both the directions of real axis and imaginary axis. [ABSTRACT FROM AUTHOR]

The aim of this paper is to prove existence results on fixed points for asymptotically regular uniformly expansive Kannan semigroup of selfmappings (with constant β < √2) defined on metric spaces equipped with uniform normal structure which further enjoys a kind of intersection property. As Banach spaces also fall in the class of metric spaces with uniform normal, therefore our results can be viewed as metric versions of some earlier results due to Kannan originally proved in reflexive Banach spaces besides generalizing certain previously known results due to Beg and Azam proved in convex metric spaces. [ABSTRACT FROM AUTHOR]

In this paper a(x)-convex functions are defined and Jensen, Hermite-Hadamard, Lah-Ribarić and other inequalities for them are derived. Related analogous of the Lagrange and the Cauchy mean value theorems are also obtained and means of the Cauchy type are generated. [ABSTRACT FROM AUTHOR]

A new subclass of multivalent harmonic functions defined by convolution is introduced in this paper. The subclass generates known subclasses of multivalent harmonic functions, and thus provides a unified treatment in the study of these subclasses. Sufficient coefficient conditions are obtained that are also shown to be necessary when the functions have negative coefficients. Growth estimates and extreme points are also determined. In addition conditions for starlikeness of the Dziok-Srivastava linear operator involving the generalized hypergeometric functions are discussed. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with oscillation of the nonlinear difference equation Δ(cn[Δ(dnΔxn)]γ) +qnf(xg(n))=0, n≥n0, where γ > 0 is the quotient of odd positive integers, cn, dn and qn are positive sequences of real numbers, g(n) is a sequence of nonnegative integers and f ϵ C(R,R) such that u f (u) > 0 for u ≠ 0. We establish some new sufficient conditions for oscillation by employing the Riccati substitution and the analysis of the associated Riccati difference inequality. Our results extend and improve some previously obtained ones. Some examples are considered to illustrate the main results. [ABSTRACT FROM AUTHOR]

In this paper, we determine the general solution of the functional equation f (kx+ y)+ f (kx-y) = g(x+y)+g(x-y)+h(x)+h(y) for fixed integers k with k ≠= 0,±1 without assuming any regularity condition on the unknown functions f ,g,h,h. The method used for solving these functional equations is elementary but exploits an important result due to Hosszú. The solution of this functional equation can also be determined in certain type of groups using two important results due to Székelyhidi. The results improve and extend some recent results. [ABSTRACT FROM AUTHOR]

Let R be a ring. Let m and n be positive integers, a right R-module M is called (m,n)-small injective, if every right R-homomorphism from an n-generated submodule of Jm to M extends to one from Rm to M. A ring R is called right (m,n)-small injective if the right R module RR is (m,n)-small injective. In this paper, we give some properties of (m,n)-small injective modules and right (m,n)-small injective rings. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce the concept of meromorphic starlike functions of reciprocal order a. Some sufficient conditions for functions belonging to this class are derived. [ABSTRACT FROM AUTHOR]

For an edge uv of a graph G, the weight of the edge e = uv is defined by w(e) = 1/ √d(u)d(v). Then R(G)= ΣuvϵE(G) 1/√d(u)d(v)= ΣeϵE(G)w(e) is called the Randić index of G. If G is a connected graph, then rad(G)=minx maxy d(x,y) is called the radius of G, where d(x;y) is the distance between two vertices x;y. In 2000, Caporossi and Hansen conjectured that for all connected graphs except the even paths, R(G) ≥ r(G). They proved the conjecture holds for all trees except the even paths. In this paper, it is proved that the conjecture holds for all unicyclic graphs, bicyclic graphs and some class of chemical graphs. [ABSTRACT FROM AUTHOR]

In this paper, first we characterize Cayley graphs of finite Brandt semigroups, and we give a criterion to check whether a finite digraph is a Cayley graph of a finite Brandt semigroup. Also Kelarev and Praeger gave necessary and sufficient conditions for Cayley graphs of semigroups to be vertex-transitive. Then, some authors gave descriptions for all vertex-transitive Cayley graphs of some special classes of semigroups. In this note similar descriptions for all vertex-transitive Cayley graphs of Brandt semigroups are given. [ABSTRACT FROM AUTHOR]

Let A denote the family of all analytic functions f in the unit disk D with the normalization f (0) = 0 = f′(0)=1. Define I = {f ϵ A : f is univalent in D}, U = {f ϵ A : ∣f′(z)(z/f(z))2-1∣ < 1 for z ϵ D}, and P(1/2)={f ϵ A : Re (f(z)/z) > 1/2 for z ϵ D}. In this paper, we determine the radius of univalency of F (z) = zf(z)/g(z) whenever f ϵ I or U , and g ϵ I or P (1/2). Based on our investigations, we conjecture that F is univalent in the disk ∣z∣ < 1/3 whenever f ∣ I and g ϵ P(1/2). We also conjecture that F is univalent in the disk ∣z∣ < √5-2 whenever both f and g are in I. [ABSTRACT FROM AUTHOR]

This paper is concerned with the integral type boundary value problems of the second order differential equations with one-dimensional p-Laplacian {[p(t)Φ(x'(t))]'+f(t,x(t),(t))=0, tϵ(0,1), ø1(x(0))=∫01 g(s)ø1(s(s))ds, ø2(x'(1))=∫01h(s)ø2(x'(s)) ds. Sufficient conditions to guarantee the existence of at least three positive solutions of this BVP are established. An example is presented to illustrate the main results. The emphasis is put on the one-dimensional p-Laplacian term [p(t)Φ(x'(t))]' involved with the function r, which makes the solutions un-concave. [ABSTRACT FROM AUTHOR]

In this paper the notion of simple and completely simple semi hypergroups are introduced. Basic properties of these algebraic structures are considered. Some methods for constructing new kinds of these hyperstructures are presented. The regularity of semi hypergroups is considered and three structural results are proved. [ABSTRACT FROM AUTHOR]