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In the present paper, we expand the domain of work on theconcept of semiderivations in 3-prime near-rings through the study ofstructure and commutativity of near-rings admitting semiderivations sat-isfying certain differential identities. Moreover, several examples havebeen provided at places which show that the assumptions in the hypothe-ses of various theorems are not altogether superfluous. 1. IntroductionThroughout this paper, N is a zero-symmetric left near ring. A near ringN is called zero symmetric if 0x= 0 for all x∈ N (recall that in a left nearring x0 = 0 for all x∈ N). N is called 3-prime if xNy = {0} implies x= 0or y = 0. The symbol Z(N) will represent the multiplicative center of N,that is, Z(N) = {x∈ N | xy= yxfor all y∈ N}.For any x,y∈ N; as usual[x,y] = xy−yxand x◦y= xy+yxwill denote the well-known Lie product andJordan product, respectively. Recall that N is called 2-torsion free if 2x= 0implies x= 0 for all x∈ N. For terminologies concerning near-rings we referto G. Pilz [7].An additive mapping d: N → N is said to be a derivation if d(xy) = xd(y)+d(x)yforall x,y∈ N, orequivalently, asnotedin [8], that d(xy) = d(x)y+xd(y)for all x,y ∈ N. An additive mapping d: N → N is called semiderivation ifthere exists a function g : N → N such that d(xy) = xd(y) + d(x)g(y) =g(x)d(y)+d(x)yand d(g(x)) = g(d(x)) for all x,y∈ N.Obviously, any deriva-tion is a semiderivation, but the converse is not true in general (see [6]). Therehas been a greatdeal of workconcerning derivations in near-rings(see [1, 2, 4, 5]where further references can be found). In this paper, we study the commuta-tivity of addition and multiplication of near-rings. Two well-known results forderivations in near-rings have been generalized for semiderivation. In fact, ourresults generalize some theorems obtained by the authors together with Rajiin [1].

The paper deals with the problem of meromorphic functionssharing a small function with its differential polynomials and improvesthe results of Liu and Gu [9], Lahiri and Sarkar [8], Zhang [13] and Zhangand Yang [14] and also answer some open questions posed by Kit-Wing Yu[16]. In this paper we provide some examples to show that the conditionsin our results are the best possible. 1. Introduction, definition and resultsIn this paper by meromorphic functions we will always mean meromorphicfunctions in the complex plane.Let f and g be two non-constant meromorphic functions and let a be a finitecomplex number. We say that f and g share a CM, provided that f − a andg − a have the same zeros with the same multiplicities. Similarly, we say thatf and g share a IM, provided that f −a and g−a have the same zeros ignoringmultiplicities. In addition we say that f and g share ∞ CM, if 1/f and 1/gshare 0 CM, and we say that f and g share ∞ IM, if 1/f and 1/g share 0 IM.We adopt the standard notations in Nevanlinna’s value distribution theoryof meromorphic functions such as the characteristic function T(r,f), the count-ing function of the poles N(r,∞;f) and the proximity function m(r,∞;f) (see[10]). For a non-constant meromorphic function f we denote by S(r,f) anyquantity satisfying S(r,f) = o(T(r,f)) as r → ∞, outside of a possible ex-ceptional set of finite linear measure. Let k ∈ N and a ∈ C∪ {∞}. We useN

The aim of this paper is to present some properties of sT- continuous functions. Moreover, we obtain a characterization and pre- serving theorems of semi-compact, S-closed and s-closed spaces. The study of semi-open sets and semi-continuity in topological spaces was initiated by Levine (10). In 2009, Noiri et al. (13) defined the notion T-open sets and deduced some results. Quite recently, Al-omari et al. (1) have obtained some properties of T-open sets and characterizations of S-closed spaces. In this paper, we present some properties of sT-continuous functions. Moreover, we obtain characterizations and preserving theorems of semi-compact, S-closed and s-closed spaces. 2. Preliminaries Throughout this paper, (X,�) and (Y,�) stand for topological spaces on which no separation axiom is assumed unless otherwise stated. For a subset A of X, the closure of A and the interior of A will be denoted by Cl(A) and Int(A), respectively. Let (X,�) be a space and S a subset of X. A subset S of X is said to be semi-open (10) if there exists an open set U of X such that U ⊆ S ⊆ Cl(U), or equivalently if S ⊆ Cl(Int(S)). The complement of a semi-open set is said to be semi-closed. The intersection of all semi-closed sets containing S is called the semi-closure of S and is denoted by sCl(S). The semi-interior of S, denoted by sInt(S), is defined by the union of all semi-open sets contained in S. It is verified in (2) that sCl(A) = A ∪ Int(Cl(A)) and sInt(A) = A ∩ Cl(Int(A)) for any subset A ⊆ X. A point x ∈ X is said to be in the �-semiclosure of A, denoted by x ∈ �-sCl(A), if A ∩ Cl(V ) 6 � for each semi-open set V containing x. A subset A ⊆ X is said to be �-semiclosed (8) if A = �-sCl(A). The complement of a �-semiclosed set is called a �-semiopen

Let G be a cyclic group of prime power order. There is anatural embedding of Z[G] into a product of rings of integers of cyclotomicfields. In this paper the image of the embedding is determined, and wealso compute the index of the image. 1. IntroductionFor a finite abelian group Glet Z[G] be the integral group ring of G, andlet I G be the augmentation ideal. For each complex character χof Glet Q(χ)be the cyclotomic field generated by the values of χand Z[χ] be its ring ofintegers.ConsiderΦ : Z[G] −→Y χ∈Gb Z[χ]Φ(α) = (...,χ(α),...),where the domain of χis extended to Z[G] by linearity. The map Φ is aninjective ring homomorphism. The goal of this paper is to determine Φ(Z[G])when Gis cyclic of prime power order.For an elementβ= (...,β χ ,...) ∈Y χ∈Gb Z[χ],let us refer to its components β χ as the character values of β. We find that,when Gis cyclic of prime power order, we can express the necessary and suf-ficient condition for β∈ Φ(Z[G]) as congruence relations among the charactervalues of β. As a byproduct, we also compute the index of Φ(Z[G]) inQZ[χ].There are refined type of conjectures on the values of L-functions (cf. [1],[2], [4], [5]) which predict (among others) that certain elements ofQ

Let Λ be a robustly transitive set of a diffeomorphism f ona closed C ∞ manifold. In this paper, we characterize hyperbolicity of Λin C 1 -generic sense. 1. IntroductionA fundamental problem in differentiable dynamical systems is to understandhow a robust dynamic property (that is, a property that holds for a system andall C 1 nearby ones) on the underlying manifold would influences the behaviorof the tangent map on the tangent bundle. In this paper, we study the robustdynamic property for a transitive set. Let M be a closed C ∞ manifold, and letDiff(M) be the space of diffeomorphisms of M endowed with the C 1 -topology.Denote by d the distance on M induced from a Riemannian metric k·k on thetangent bundle TM. Let f ∈ Diff(M) and Λ be a closed f-invariant set. Theset Λ is transitive if there is a point x ∈ Λ such that ω(x) = Λ. Here ω(x) isthe forward limit set of x. Denote by f| Λ the restriction of f to the set Λ. Amaximal invariant set of f in an open set U, denoted by Λ f (U), is the set ofpoints whose whole orbit contained in U, that is,Λ

Let G =(V,E)beagraphwithchromaticnumber χ(G). Adominating set D of G is called a chromatic transversal dominating set(ctd-set) if D intersects every color class of every χ-partitionof G. Theminimumcardinalityofactd-setofG iscalledthechromatictransversaldomination number of G and is denoted by γ ct (G). In this paper wecharacterizetheclassoftrees,unicyclicgraphsandcubicgraphsforwhichthe chromatic transversal domination number is equal to the connecteddominationnumber. 1. IntroductionAll the graphs considered in this paper unless otherwise specifically statedare finite, connected and simple and are consistent with the terminology usedin Harary [4]. Let G = (V,E) be a simple graph of order p. For a subset Sof V, N(S) denotes the set of all vertices adjacent to some vertex in S andN[S] = N(S) ∪S.A vertex v of G is called a support if it is adjacent to a pendant vertex. Anyvertex of degree greater than one is called an internal vertex. A graph G iscalled a unicyclic graph, if G contains exactly one cycle.A subset D ⊆ V is a dominating set, if every v ∈ V − D is adjacent tosome u ∈ D. The domination number γ = γ(G) is the minimum cardinality ofa dominating set of G. A dominating set D is called a connected dominatingset if the induced subgraph hDi is connected. The minimum cardinality of aconnected dominating set is called the connected domination number and isdenoted by γ

In this paper we introduce the notion of P-strongly regu-lar near-ring. We have shown that a zero-symmetric near-ring N is P-strongly regular if and only if N is P-regular and P is a completelysemiprime ideal. We have also shown that in a P-strongly regular near-ring N, the following holds: (i) Na+ P is an ideal of N for any a∈N.(ii) Every P-prime ideal of N containing P is maximal. (iii) Every idealI of N fulfills I+ P = I 2 + P. 1. IntroductionThroughout this paper, N denotes a zero-symmetric right near-ring. Aright N-subgroup (left N-subgroup) of N is a subgroup I of (N,+) such thatIN ⊆ I(NI ⊆ I). A quasi-ideal of N is a subgroup Q of (N,+) such thatQN ∩ NQ ⊆ Q. Right N-subgroups and left N-subgroups are quasi-ideals.The intersection of a family of quasi-ideals is again a quasi-ideal.Nis called regular, if for every element aof Nthere exists an element x∈ Nsuch that a= axa. Let P be an ideal of N. Then the near-ring N is said tobe a P-regular near-ring if for each a ∈ N, there exists an element x ∈ Nsuch that a= axa+ p for some p∈ P. If P = 0, then a P-regular near-ringis a regular near-ring. Here the notion of P-regularity is a generalization ofregularity. There are near-rings which are P-regular but not regular.V. A. Andrunakievich [1] defined P-regular rings and S. J. Choi [3] extendedthe P-regularity of a ring to the P-regularity of a near-ring. In this paper weintroduce the notion of P-strongly regular near-ring and obtain equivalent con-ditions for a near-ring to be P-strongly regular. We also introduce the notionsof P-prime ideals and P-near-ring in this paper. I. Yakabe [7] characterizedregular zero-symmetricnear-rings without non-zero nilpotent elements in termsof quasi-ideals. In this paper we characterize P-strongly regular near-ring interms of quasi-ideals. For the basic terminology and notation we refer to [6].

The aim of this paper is to establish the well-known and veryuseful classical Saalschutz’s theorem for the series 3 F 2 (1) by following afft method. In addition to this, two summation formulas closelyrelated to the Saalschutz’s theorem have also been obtained. The resultsestablished in this paper are further utilized to show how one can obtaincertain known and useful hypergeometric identities for the series 3 F 2 (1)and 4 F 3 (1) already available in the literature. 1. Introduction and results requiredWe start with the following well-known and useful classical Saalschutz’stheorem [4, p. 87, Section 51] for the series 3 F 2 (1). If n is a non-negativeinteger and if a , b , c are independent of n ,(1.1) 3 F 2 [ n; a; bc; 1+ a + b c n ; 1]=( c a ) n ( c b ) n ( c ) n ( c a b ) n : As mentioned in almost all the standard books on generalized hypergeomet-ric series that this theorem can be established with the help of the followingEuler’s transformation formula [4, p. 60, Eq.(5)]. If

Even though there is a formula for h 0 ( X,O X ( nK X )) for acanonical threefold X , it is not easy to compute h 0 ( X,O X ( nK X )) be-cause the formula has a term due to singularities. In this paper, we finda way to control the term due to singularities. We show nonvanishing ofplurigenus for the case when the index r in the singularity type 1 r (1 ,− 1 ,b )is sufficiently large. Throughout this paper X is assumed to be a projective threefold with onlycanonical singularities and an ample canonical divisor K X over the complexnumber field C, i.e., a canonical threefold.It is well known that H 0 ( X,O X ( mK X )) does not vanish and generates abirational map for a sufficiently large m . When X is a surface of general type, H 0 ( X,O X ( mK X )) does not vanish for m ≥ 2 and H 0 ( X,O X ( mK X )) generatesa birational map for m ≥ 5.In a case of a threefold of general type, M. Reid and A. R. Fletcher de-scribed the formula for χ ( O X ( nK X )) (see [1]). Combining the formula for χ

This paper introduces a local non-positivity of two set-valued mappings (F,Q) and considers the existences and properties of solutions for set-valued mixed vector FQ-implicit variational inequality problems and set-valued mixed vector FQ-complementarity problems in the neigh- borhood of a point belonging to an underlined domain K of the set-valued mappings, where the neighborhood is contained in K. This paper generalizes and extends many results in (1, 3-7).

Every non-associative algebra Lcorresponds to its sym- metric semi-Lie algebra L(;) with respect to its commutator. It is an interesting problem whether the equality Autnon(L) = AutsemiiLie(L) holds or not (2), (13). We flnd the non-associative algebra au- tomorphism groups Autnon (WN0;0;1(0;1;r1;:::;rp) ) and AutsemiiLie (WN0;0;1(0;1;r1;:::;rp)), where every automorphism of the automor- phism groups is the composition of elementary maps (3), (4), (7), (8), (9), (10), (11). The results of the paper show that the F-algebra au- tomorphism groups of a polynomial ring and its Laurent extension make easy to flnd the automorphism groups of the algebras in the paper.

In this paper, we introduce the SP-integral and the SPfi-integral deflned on an interval in the n-dimensional Euclidean space R n . We also investigate the relationship between these two integrals. It is well known (3) that the Perron integral deflned on an interval of the real line R by major and minor functions which are not assumed to be continuous is equivalent to the one deflned by continuous major and minor functions and that the strong Perron integral deflned on an interval of R by strong major and minor functions is equivalent to the McShane integral. In this paper, we introduce Perron-type integrals deflned on an inter- val of the n-dimensional Euclidean spaceR n using the strong major and minor functions, and investigate the relationship between these integrals. We shall call it the strong Perron integral, or brie∞y SP-integral. For a subset E of the n-dimensional Euclidean spaceR n , the Lebesgue measure of E is denoted by jEj. For a point x = (x1;x2;¢¢¢ ;xn) 2R n , the norm of x is kxk = max1•in jxij and the --neighborhood U(x;-) of x is an open cube centered at x with sides equal to 2-. For an interval I = (a1;b1)£(a2;b2)£¢¢¢(an;bn) ofR n with ai fi(fi 2 (0;1)), then the interval I is said to be fi-regular.

In this paper we explicitly compute the orders of am- biguous ideal class groups of layers of the basic Z3-extension of certain cubic flelds and give an example for Greenberg's conjecture. n +"p(k) for su-ciently large n: Greenberg's conjecture (3) claims that both "p(k);‚p(k) vanishes for the cyclotomic Zp-extension, contained in the fleld obtained by adjoining all p-power roots of unity to k; of any totally real number fleld k: In this paper we explicitly compute the orders of ambiguous ideal class groups of layers of the basic Z3- extension of certain cubic flelds and give an example for Greenberg's conjecture.

In this paper, we introduce the notion of symmetric bi-derivations of a B-algebra and investigate some related properties. We study the notion of symmetric bi-derivations of a 0-commutative B-algebra and state some related properties. © 2016 Korean Mathem Department of Mathematics, Faculty of Science, Yaşar University, Izmir, 35100, Turkey

Let R be a ring, (S,�) a strictly ordered monoid and ! : S ! End(R) a monoid homomorphism. The skew generalized power se- ries ring R((S,!)) is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal'cev-Neumann Laurent series rings. In this paper, we in- vestigate the interplay between the ring-theoretical properties of R((S,!)) and the graph-theoretical properties of its zero-divisor graph ( R((S,!))). Furthermore, we examine the preservation of diameter and girth of the zero-divisor graph under extension to skew generalized power series rings.

In this paper, we provide semi-convergence results of theparameterized inexact Uzawa method with singular preconditioners forsolving singular saddle point problems. We also provide numerical exper-iments to examine the effectiveness of the parameterized inexact Uzawamethod with singular preconditioners. 1. IntroductionWe consider the singular saddle point problem of the form(1) A B−B T 0 xy = f−g ,where A ∈ R m×m is a symmetric positive definite matrix, B ∈ R m×n is arank-deficient matrix of rank(B) = r < n with m ≥ n, f ∈ R m and g ∈ R n .The singular saddle point problem (1) is important and arises in many differentapplications of scientific computing and engineering, such as the mixed finiteelement methods for Navier-Stokes equations, computational fluid dynamics,constrained optimization, the weighted least squares problems, electronic net-works, linear elasticity, and so forth [1, 10, 14, 15].When B is of full column rank, the linear system (1) is nonsingular. Forthis case, many relaxation iterative methods based on matrix splittings andtheir convergence properties have been proposed and analyzed, e.g., SOR-likemethod [11], GSOR (Generalized SOR) method [2], PIU (Parameterized Inex-act Uzawa) method [3], SSOR-like method [8, 20], GSSOR (Generalized SSOR)method [7, 18], several variants of Uzawa method [16, 17], and so on.

An H-magic labeling of a H-decomposable graph G is a bijection f : V ( G ) ∪ E ( G ) → { 1 , 2 , … , p + q } such that for every copy H in the decomposition, ∑ v ∈ V ( H ) f ( v ) + ∑ e ∈ E ( H ) f ( e ) is constant. The labeling f is said to be H-E-super magic if f ( E ( G ) ) = { 1 , 2 , … , q } . In this paper, we prove that a complete bipartite graph is H-E-super magic decomposable where H ≅ K 1 , n with n ≥ 1 .

The atom-bond connectivity index of a graph G(ABC indexfor short) is defined as the summation of quantitiesq d(u)+d(v)−2d(u)d(v) overall edges of G. A cactus graph is a connected graph in which every blockis an edge or a cycle. The aim of this paper is to obtain the first andsecond maximum values of the ABC index among all n vertex cactusgraphs. 1. IntroductionSuppose G is a simple connected graph with vertex and edge sets V (G) andE(G), respectively. A block of G is a maximal connected subgraphof G withoutcut-vertex. A cactus is a connected graph in which every block is an edge or acycle [18, p. 160]. These are connected graphs in which each edge belongs toat most one cycle. An example of a cactus graph is depicted in Figure 1.Figure 1. Examples of cactus graphs.Cactus graphs have several applications in computer science and biologyand so it is a topic of interest among many researchers in different scientificdisciplines. In [1, 6], it is proved that some graph problems which are NP-hardfor general graphs can be solved in polynomial time for cacti. On the otherhand, in [15] a number of combinatorial optimization problems are presented

This paper concerns with exploiting an oblique projection technique to solve a general class of large and sparse least squares prob- lem over symmetric arrowhead matrices. As a matter of fact, we de- velop the conjugate gradient least squares (CGLS) algorithm to obtain the minimum norm symmetric arrowhead least squares solution of the general coupled matrix equations. Furthermore, an approach is offered for computing the optimal approximate symmetric arrowhead solution of the mentioned least squares problem corresponding to a given arbitrary matrix group. In addition, the minimization property of the proposed algorithm is established by utilizing the feature of approximate solutions derived by the projection method. Finally, some numerical experiments are examined which reveal the applicability and feasibility of the handled algorithm.

We give the characterization of H¨older differentiability pointsand non-differentiability points of the Riesz-N´agy-Taka´cs (RNT) singularfunction Ψ a,p satisfying Ψ a,p (a) = p. It generalizes recent multifractaland metric number theoretical results associated with the RNT function.Besides, we classify the singular functions using the singularity order de-duced from the H¨older derivative giving the information that a strictlyincreasing smooth function having a positive derivative Lebesgue almosteverywhere has the singularity order 1 and the RNT function Ψ a,p hasthe singularity order g(a,p) = alogp+(1−a)log(1−p)aloga+(1−a)log(1−a) ≥1. 1. IntroductionRecently many ([8, 9, 10, 16]) studied the Cantor function, a singular func-tion which is not strictly increasing and the Minkowski’s Question Mark func-tion which is a strictly increasing singular function. Also J. Parad´isetal. ([17])studied some conditions of the null and infinite derivatives of the RNT strictlyincreasing singular function using metric number theory.Recently we ([4]) also studied multifractal characterization of the null andinfinite derivative sets and the non-differentiability set of the RNT singularfunction, which is the typical singular function related to mutifractal theory.In this paper, we employ the Ho¨lder derivative, which is a generalized form ofthe usual derivative, of the RNT function on the unit interval. This definitionextends the concept of the singularity to the general singularity for a strictlyincreasing continuous function. For every point in the unit interval, we ([4])can give some code or dyadic expansion using digit 0 and 1, generating thedistribution set determined by the frequency of the zero in its expansion. Thedistribution sets in the unit interval are the local dimension sets by a self-similar

We consider a condition under which the projectivization P (Ek) of a complex k-bundle Ek → M over an even-dimensional manifold M can have the hard Lefschetz property, affected by [4]. It depends strongly on the rank k of the bundle Ek. Our approach is purely algebraic by using rational Sullivan minimal models [2]. We will give some examples. The contents of this paper is according to [6]. A Poincare duality space Y of the formal dimension fd(Y ) = max{i;H (Y ;Q) = 0} = 2m is said to be cohomologically symplectic (c-symplectic) if u = 0 for some u ∈ H(Y ;Q) and, furthermore, is said to have the hard Lefschetz property (or simply the Lefschetz property) with respect to the c-symplectic class u, if the maps ∪u : Hm−j(Y ;Q) → H(Y ;Q) 0 ≤ j ≤ m are monomorphisms (then called the Lefschetz maps) [7]. (Then we often say that H∗(Y ;Q) is a Lefschetz algebra [4].) For example, a compact Kahler manifold has the hard Lefschetz property [7], [3, Theorem 4.35]. Let M be an even-dimensional manifold and ξ : E → M be a complex k-bundle over M . The projectivization of the bundle ξ P (ξ) : CP k−1 j → P (E) → M satisfies the rational cohomology algebra condition (∗) : H∗(P (E);Q) = H∗(M ;Q)[x] (xk + c1xk−1 + · · ·+ cjxk−j + · · ·+ ck−1x+ ck) where ci are the Chern classes of ξ and x is a degree 2 class generating the cohomology of the complex projective space fiber (Leray-Hirsch theorem) [1], [4], [7, p.122]. The manifold P (E) appears as the exceptional divisor in blow-up construction for a certain embedding of M [5], [7, Chap.4]. When M is a non-toral symplectic nilmanifold of dimension 2n, there is a bundle E such that P (E) is not Lefschetz [8], [4, Example 4.4]. In general, for a 2k-dimensional manifold M and a fibration

In this paper, we consider characters of SL 2 (Z) and thenapply them to newforms of integral weight, level 4 and of trivial character.More precisely, we prove that all of them are actually level 1 forms ofsome nontrivial character. As a byproduct, we prove that they all areeigenfunctions of the Fricke involution with eigenvalue −1. IntroductionThe Fricke involution W N of level N, also known as the canonical involution,actsonthe spaceofnewformsoflevelN, integralweightk, and trivialcharacter.Here k is necessarily even and positive. It is well-known that Hecke eigenformsbehave well under the Fricke involution. More specifically, if f is a normalizedHecke eigenform of some level N, weight k and of trivial character, then wehave f| k W N = cg with c ∈ C × and g another normalized Hecke eigenform inthe same space (see Lemma 1.1 below or Theorem 4.6.16 in [6]). The Fouriercoefficients of g can be explicitly determined by that of f but the scalar c isleft mysterious in general.Question 1. Can we explicitly determine c with the information on f?Let f =P

In this paper, we study some quantity equivalent to the normof Bloch to A pα composition operator where A pα is the weighted Bergmanspace on the unit ball of C n (0 < p < ∞ and −1 < α < ∞). 1. IntroductionLet n ≥ 1 be an integer. Let H(B) denote the space of all holomorphicfunctions in the unit ball B ≡ B n of the complex n-dimensional Euclideanspace C n . Let U stand for B 1 . The Bloch space B = B(U) is defined byB = {f ∈ H(U) : sup z∈U |f ′ (z)|(1 − |z| 2 ) < ∞}. With the norm kfk B =|f(0)|+sup z∈U |f ′ (z)|(1−|z| 2 ), B is a Banach space. Let ν denote the Lebesguemeasure on C n , so normalized that ν(B) = 1. For −1 < α < ∞, we setc α = Γ(n+α+1)/{Γ(n+1)Γ(α+1)} and dν α (z) = c α (1−|z| 2 ) α dν(z), z ∈ B.Note that ν α (B) = 1. For 0 < p < ∞ and −1 < α < ∞, the weighted Bergmanspace A pα ≡ A pα (B) is defined by A pα = {f ∈ H(B) :R B |f(z)| dν (z) < ∞}.For f ∈ A pα , we write kfk A pα =R B |f(z)| p dν α (z) 1p .In 2003, Kwon and Lee proved the following theorem. This result relates tothe pull-back property (see [1], [2], [5], [7] for example).Theorem 1 ([6]). Let f : U → U be a holomorphic function with f(0) = 0.For 1 ≤ p < ∞ and −1 < α < ∞, the bounded operator C

A three-valued function f defined on the vertices of a di-graph D = (V,A), f : V → {−1,0,+1} is a minus total dominatingfunction(MTDF) if f(N − (v)) ≥ 1 for each vertex v ∈ V. The minustotal domination number of a digraph D equals the minimum weight ofan MTDF of D. In this paper, we discuss some properties of the minustotal domination number and obtain a few lower bounds of the minustotal domination number on a digraph D. 1. IntroductionFor terminology and notation on graph theory not given here we follow [1].Let G = (V,E) be a simple graph (digraph or undirected graph). Let v be avertex in V . The open neighborhood of v is the set N(v) = {u ∈ V |uv ∈ E}and the degree of v is d G (v) = |N(v)|. The maximum degree and minimumdegree of G are denoted by ∆(G) and δ(G), respectively. When no ambiguitycan occur, we often simply write d(v), ∆, and δ instead of d G (v), ∆(G), andδ(G), respectively. For any S ⊆ V, G[S] is the subgraph induced by S. Wedenote χ(G) by the chromatic number of G. Let D = (V,A) be a digraph. Foreach vertex v ∈ V , let N

In this paper, we first introduce a generalized concept of Berge strong equilibrium for a generalized game G = (Xi;Ti,fi)i∈I of normal form, and using a fixed point theorem for compact acyclic maps in admissible convex sets, we establish the existence theorem of generalized Berge strong equilibrium for the game G with acyclic values. Also, we have demonstrated by examples that our new approach is useful to produce generalized Berge strong equilibria.

In this paper, under some suitable integrability and smooth-ness conditions on f, we establish the central limit theorems forX k≤N k −1 f(S k /σ√k),where S k is the partial sums of strictly stationary mixing random vari-ables with EX 1 = 0 and σ 2 = EX 21 +2P ∞k=1 EX 1 X 1+k . We also estab-lish an almost sure limit behaviors of the above sums. 1. IntroductionLet {X n ,n≥ 1} be a sequence of random variables on some probabilityspace (Ω,F,P). Let F ba denote the σ-field generated by the random variablesX a ,X a+1 ,...,X b . For any two σ-field A,B ⊂ F putρ(A,B) = supˆCov(X,Y)k Xk 2 k Y k 2 : X∈ L 2 (A),Y ∈ L 2 (B)˙,φ(A,B) = sup{| P(B| A) −P(B) |: A∈ A,B∈ B},α(A,B) = sup{| P(AB) −P(A)P(B) |: A∈ A,B∈ B},where and in the sequel k X k p = (E|X| p ) 1/p for 1 ≤ p

In this paper, a boundary version of the Schwarz and Carath-´eodory inequality are investigated. New inequalities of the Carath´eodory’sinequality and Schwarz lemma at boundary are obtained by taking intoaccount zeros of f(z) function which are different from zero. The sharp-ness of these inequalities is also proved. 1. IntroductionLet f be a function which is holomorphic on the D : {z : |z| < 1} and vanishat z = 0, and suppose that |f| < 1 for all z ∈ D. Then the inequality(1.1) |f(z)| ≤ |z|holds for all z ∈ D, and moreover(1.2) |f ′ (0)| ≤ 1.Equality is achieved in (1.1) (for some nonzero z ∈ D) or in (1.2) if and onlyif f is an entire linear function of the form f(z) = e iα z, where α is a realnumber([2], p. 381).Let the zeros of f be z 1 ,z 2 ,...,z n . If we apply inequality (1.1) to the func-tion f(z)Q nk=1 h 1−z k zz+z k i, we can conclude in the following Schwarz’s inequality:(1.3) |f(z)| ≤ |z|Y nk=1 z −z k 1−z k z and|f ′ (0)| ≤Y nk=1 |z k |. Received May 3, 2013.2010 Mathematics Subject Classification. Primary 30C80.Key words and phrases. Schwarz lemma on the boundary, holomorphic function, Julia-Wolff-Lemma.

J. Malkevitch dened the coded path in r-valent polytopal graphs of uniform face structure and showed many interesting properties of the coded paths. In this paper, we study the simplicity of coded paths in an m-valent planar multigraph which is not a polytopal graph. and theorems briey. A graph G is an ordered triple (V (G), E(G), G) consisting of a nonempty set V (G) of vertices, a set E(G) of edges, and an incidence function G that associates with each edge of G an unordered pair of (not necessarily distinct) elements ofV (G). Ife is an edge andu andv are vertices such that G(e) = uv, then e is said to join two vertices u and v; the vertices u and v are called the endpoints (or endvertices) of the edge e. The endpoints of an edge are said to be incident to an edge and two vertices which are incident with the same edge, are said to be adjacent. Two or more edges that join the same pair of distinct vertices are called parallel edges (or multiple edges). An edge joining a vertex to itself is called a loop. A graph with no loops or no parallel edges is called a simple graph. A graph which is not simple is said to be a multigraph. The number of edges at the vertex v is called the valence of v (or the degree of v) and is denoted by d(v). If every vertex of a graph G has the same valence r, then G is called an r-valent (or r-regular) graph. A graphG is called planar if it can be drawn in the plane so that the edges of the graph intersect only at vertices. When a connected planar graph is drawn in the plane, the regions bounded by edges of the graph which do not contain neither vertices nor edges in their interiors are called faces. There will always be precisely one face which is unbounded in a planar graph and this will be called the innite face . The edges bounding a face are called its sides.

In this paper we investigate (n+1)(n ≥ 3)-dimensional con-tact CR-submanifolds M of (n−1) contact CR-dimension in a completesimply connected Sasakian space form of constant φ-holomorphic sec-tional curvature c 6= −3 which satisfy the condition h(FX,Y )+h(X,FY )= 0 for any vector fields X,Y tangent to M, where h and F denote thesecond fundamental form and a skew-symmetric endomorphism (definedby (2.3)) acting on tangent space of M, respectively. 1. IntroductionLet S 2m+1 be a (2m + 1)-unit sphere in the complex (m + 1)-space C m+1 ,i.e.,S 2m+1 := {(z 1 ,...,z m+1 ) ∈ C m+1 | m X +1j=1 |z j | 2 = 1}.For any point z ∈ S 2m+1 we put ξ = Jz, where J denotes the complex structureof C m+1 . Denoting by π the orthogonal projection : T z C m+1 → T z S 2m+1 andputting φ = π◦J, we can see that the set (φ,ξ,η,g) defines a Sasakian structureon S 2m+1 , where g is the standard metric on S 2m+1 induced from that of C m+1 and η is a 1-form dual to ξ. Hence S 2m+1 can be considered as a Sasakianmanifold of constant curvature 1 (cf. [2, 4, 5, 6, 7, 8, 9]).Let M be an (n+1)-dimensional submanifold tangent to the structure vectorfield ξ of S

In this paper, we prove that the total domination number of a P6-free graph of order n � 3 and minimum degree at least one which is not the cycle of length 6 is at most n+1 , and the bound is sharp. 1. Introduction A total dominating set of a graph G with no isolated vertex is a set S of vertices of G such that every vertex is adjacent to a vertex in S. The total dom- ination number of G, denoted by t(G), is the minimum cardinality of a total dominating set of G. Total domination in graphs was introduced by Cockayne, Dawes, and Hedetniemi (3). For notation and graph theory terminology we in general follow (3). Let G = (V,E) be a graph with vertex set V of order n. The degree, neighborhood and closed neighborhood of a vertex v in the graph G are denoted by d(v), N(v) and N(v) = N(v)∪{v}, respectively. The minimum degree and maximum degree of the graph G are denoted by �(G) and �(G), respectively. For any S ⊆ V , N(S) = S v∈S N(v). Let G(S) denote the graph induced by S. Let Cn, Pn and K1,n−1 denote the cycle, the path and star of order n, respectively. A graph is Pn-free if it does not contain Pn as an induced subgraph.

In this paper we give an explicit formula for the total number of subgroups of a finite abelian -group up to rank three.

In this paper, Mond-Weir type duality results for a uncertainmultiobjective robust optimization problem are given under generalizedinvexity assumptions. Also, weak vector saddle-point theorems are ob-tained under convexity assumptions. 1. IntroductionConsider an uncertain multiobjective robust optimization problem:(MRP) minimize (f 1 (x),...,f l (x))subject to g j (x,v j )

In this paper we present a robust duality theory for gener-alized convex programming problems under data uncertainty. Recently,Jeyakumar, Li and Lee [Nonlinear Analysis 75 (2012), no. 3, 1362–1373]established a robust duality theory for generalized convex programmingproblems in the face of data uncertainty. Furthermore, we extend re-sults of Jeyakumar, Li and Lee for an uncertain multiobjective robustoptimization problem. 1. IntroductionConsider the standard nonlinear programming problem with inequality con-straints(P) inf x∈R n {f(x) : g i (x)

According to the paper in [3], the kernel of an ordered semigroup S is a completely regular subsemigroup of S. In this note we show that the kernel of an ordered semigroup S is not a completely regular subsemigroup of S, in general.

In this paper, we investigate a stability of the functionalequationX 3i=03 C i (−1) 3−i f(ix+ y) = 0by using the fixed point theory in the sense of L. Ca˘dariu and V. Radu. 1. IntroductionIn 1940, S. M. Ulam [24] raised a question concerning the stability of homo-morphisms: Given a group G 1 , a metric group G 2 with the metric d(·,·), and apositive number e, does there exist a δ>0 such that if a mapping f: G 1 → G 2 satisfies the inequalityd(f(xy),f(x)f(y))

In the paper we study the finite-time ruin probability in a general risk model with constant interest force, in which the claim sizes are pairwise quasi-asymptotically independent and arrive according to an arbitrary counting process, and the premium process is a general sto- chastic process. For the case that the claim-size distribution belongs to the consistent variation class, we obtain an asymptotic formula for the finite-time ruin probability, which holds uniformly for all time horizons varying in a relevant infinite interval. The obtained result also includes an asymptotic formula for the infinite-time ruin probability.

In this paper, we first talk about a more wide class of non-linear mappings, Then, we deal with weak convergence theorems for gen-eralized hybrid mappings in a Hilbert space. 1. IntroductionLet C be a nonempty closed convex subset of a real Hilbert space H. Thena mapping T :C → C is said to be nonexpansive if kTx − Tyk ≤ kx − ykfor all x,y ∈ C. The set of fixed points of T is denoted by F(T). A mappingT : C → C with F(T) 6= ∅ is called quasi-nonexpansive if kx −Tyk ≤ kx −ykfor all x ∈ F(T) and y ∈ C. It is well-known that the set F(T) of fixed pointsof a quasi-nonexpansive mapping T is closed and convex; see [10].A important example of nonexpansivemappings in a Hilbert space is a firmlynonexpansive mapping ifkFx −Fyk 2 ≤ hx −y,Fx−Fyifor all x,y ∈ C; see for instance, [3, 5]. It is known that a mapping F : C → Cis firmly nonexpansive if and only ifkFx−Fyk 2 +k(I −F)x −(I −F)yk 2 ≤ kx−yk 2 for all x,y ∈ C, where I is the identity mapping on H. It is also known that afirmly nonexpansive mapping F can be deduced from an equilibrium problemin a Hilbert space; see, for instance, [2, 4].Recently, Kohsaka and Takahashi [11] introduced the following nonlinearmapping: Let E be a smooth, strictly convex and reflexive Banach space, letJ be the duality mapping of E and let C be a nonempty closed convex subsetof E. Then, a mapping S : C → C is said to be nonspreading ifφ(Sx,Sy)+φ(Sy,Sx) ≤ φ(Sx,y)+φ(Sy,x)

The structure of a poset P with smallest element 0 is looked at from two view points. Firstly, with respect to the Zariski topology, it is shown that Spec(P), the set of all prime semi-ideals of P, is a compact space and Max(P), the set of all maximal semi-ideals of P, is a compact T1 subspace. Various other topological properties are derived. Secondly, we study the semi-ideal-based zero-divisor graph structure of poset P, denoted by GI(P), and characterize its diameter. 1. Preliminaries Throughout this paper, (P,≤) denotes a poset with a least element 0, and all prime and maximal semi-ideals of P are assumed to be proper. For M ⊆ P, let (M) l := {x ∈ P : x ≤ m for all m ∈ M} denote the lower cone of M in P, and dually let (M) u := {x ∈ P : m ≤ x for all m ∈ M} be the upper cone of M in P. For A,B ⊆ P, we write (A,B) l instead of (A ∪ B) l and

In [5], Johnson introduced the primitivity of subgroups and proved that a finite group G is supersolvable if every primitive subgroup of G has a prime power index in G. In that paper, he also posed an interesting problem: what a group looks like if all of its primitive subgroups are maximal. In this note, we give the detail structure of such groups in solvable case. Finally, we use the primitivity of some subgroups to characterize T-group and the solvable -groups.

In this paper, we characterize surgery presentations for -homology 3-spheres and -homology 3-spheres obtained from by Dehn surgery along a knot or link which admits an even net diagram and show that the Casson invariant for -homology spheres and the -invariant for -homology spheres can be directly read from the net diagram. We also construct oriented 4-manifolds bounding such homology spheres and find their some properties.

We prove some sharp extremal distance results for functionsin weighted Bergman spaces on the upper halfplane. We also prove newanalogous results in the context of bounded strictly pseudoconvex do-mains with smooth boundary. 1. IntroductionIf Y is a normed space and X ⊂ Y , then we set dist Y (f,X) = inf g∈X kf −gk Y . If the space Y is clear from the context, we write simply dist(f,X). Inthe problems we are going to consider, X itself is going to be a (quasi)-Banachspace.We denote by H(Ω) the space of all holomorphic functions on an open setΩ ⊂ C n . In this paper we consider distance problems in weighted Bergmanspaces over the upper half-plane C + = {x + iy : y > 0} and over a boundedstrictly pseudoconvex domain Ω ⊂ C n with smooth boundary.The weighted Bergman space A pα (C + ) consists of all functions f ∈ H(C )such thatkfk p,α = Z ∞0 Z ∞−∞ |f(x +iy)| p y α dxdy 1p −1 and 0 < p < ∞ (see [5] and [6]). The above spaces are Banachspaces for p ≥ 1 and complete metric spaces for 0 < p < 1. It is naturalto consider the space A

The notion of an enlarged p-ideal and a fuzzy p-ideal in BCI-algebras with degree are introduced. Related properties of them are in-vestigated. 1. IntroductionThe concept of a fuzzy set is applied to generalize some of the basic conceptsof general topology ([1]). Rosenfeld ([6]) constituted a similar application tothe elementary theory of groupoids and groups. Xi ([7]) applied to the conceptof fuzzy set to BCK-algebras. Y. B. Jun and J. Meng ([4]) introduced of fuzzyp-ideals in BCI-algebras and studied their properties.In this paper, we introduce the notion of an enlarged p-ideal and a fuzzyp-ideal in BCI-algebras with degree. We study related properties of them.2. PreliminariesWe review some definitions and properties that will be useful in our results.By a BCI-algebra we mean an algebra (X,∗,0) of type (2,0) satisfying thefollowing conditions:(a1) (∀x,y,z ∈ X)(((x ∗y)∗(x ∗z))∗(z ∗y) = 0),(a2) (∀x,y ∈ X)((x ∗(x ∗y))∗y = 0),(a3) (∀x ∈ X)(x ∗x = 0),(a4) (∀x,y ∈ X)(x∗y = 0, y ∗x = 0 ⇒ x = y).If a BCI-algebra X satisfies the following identity:(a5) (∀x ∈ X)(0∗x = 0),then X is called a BCK-algebra.In any BCI-algebra X one can define a partial order “≤” by putting x ≤ yif and only if x ∗y = 0.A BCI-algebra X has the following properties

In the present paper, we show that a continuous linear opera-tor T on a Frechet space satisfies the Hypercyclic Criterion with respect toa syndetic sequence must satisfy the Kitai Criterion. On the other hand,an operator, hereditarily hypercyclic with respect to a syndetic sequencemust be mixing. We also construct weighted shift operators satisfyingthe Hypercyclicity Criterion which do not satisfy the Kitai Criterion. Inother words, hereditarily hypercyclic operators without being mixing. 1. IntroductionLet X denote a separableinfinite dimensional Frechet space and L(X) standsfor the space of continuous linear operators on X. The operator T in L(X)is said to be hypercyclic when there exists a vector x in X such that its or-bit under T, i.e., orb(T,x) = {T n x : n ∈ N}, is dense in X. The study ofhypercyclic operators on a Banach space was initiated in 1969 when Rolewiczproved that any multiple λB,|λ| > 1, of the standard backward shift B onl p , 1 ≤ p < +∞, is hypercyclic [12]. It is interesting to know what type ofoperators can actually be hypercyclic: Backward and bilateral shifts [12, 15],translation and differentiation operators [4, 8], adjoint of multiplication opera-tors [8], composition operators [4] and weighted composition operators [11, 17].For a complete survey of hypercyclicity, see book [1].Kitai [9] stated in particular, a simple, useful and general criterion for hy-percyclicity of operators which was isolated in a restricted form. Then it wasindependently rediscovered by R. Gethner and J. H. Shapiro in a general formand with a weakened assumptions [6]. This criterion referred to as the Hyper-cyclicity Criterion and has been used to determin the hypercyclicity of someclasses of operators . The following version appears in [3].Definition 1.1. Let T ∈ L(X) and let (n

A new Hardy-Hilberttype integral inequality for double se-rieswithweights canbeestablishedbyintroducingaparameter λ (withλ > 1− 2pq )andaweight function ofthe formx 1− 2r (with r > 1). Andthe constant factors of new inequalities established areproved tobe thebestpossible. Inparticular,forcaser =2,anewHilberttypeinequalityisobtained. Asapplications,anequivalentformisconsidered. 1. IntroductionLet {a n } and {b n } be non-negative sequences of real numbers, 1p + 1q = 1and p > 1. IfP ∞n=1 a pn < + ∞ andP ∞n=1 b q < + ∞, then(1.1)X ∞m=1 X ∞n=1 ln mn a m b n m−n≤ πsin πp ! 2 X ∞n=1 a pn ! 1p X ∞n=1 b qn ! q and(1.2)X ∞m=1 X ∞n=1 a m b n m+n≤πsin πp X ∞n=1 a pn ! 1p X ∞n=1 b qn ! 1q ,where the constant factors ( πsin πp ) 2 in (1.1) and πsin p in (1.2) are the bestpossible. And the equalities in (1.1) and (1.2) hold if and only if {a n }, or {b n }is a zero-sequence. They are the famous Hardy-Hilbert inequalities (see [6]),Owing to the importance of the Hardy-Hilbert inequality in analysis andapplications, some mathematicians have been studying them. In particular,some excellent results of (1.2) appear in a lot of the articles (such as [1, 2, 3, 4, 8]etc.). However, the research articles of (1.1) are few. The purpose of thepresent paper is to establish an extension of (1.1), and to prove the constantfactor to be the best possible. And then some important and especial resultsare enumerated, and an equivalent form is considered.

In this paper, we introduce the notions of Smarandache weakBE-algebra, Q-Smarandache filters and Q-Smarandache ideals. We showthat a nonempty subset F of a BE-algebra X is a Q-Smarandache filterif and only if A(x,y) ⊆F, which A(x,y) is a Q-Smarandache upper set.The relationship between these notions are stated and proved. 1. IntroductionThe Smarandache algebraic structures theory was introduced in 1998 byR. Padilla [9]. In [5], Kandasamy studied of Smarandache groupoids, sub-groupoids, ideal of groupoids, seminormal sub-groupoids, Smarandache Bolgroupoids, and strong Bol groupoids and obtained many interesting resultsabout them. Smarandache semigroups are very important for the study ofcongruences, and they were studied by Padilla [9].A Smarandache weak structure on a set S means a structure on S that hasa proper subset P with a weaker structure. By proper subset of a set S, wemean a subset P of S, different from the empty set, from the original set S,and from the idempotent elements if any.In [4], Borumand Saeid et al. studied the concept of Smarandache BCH-algebrasand obtainedmanyinterestingresultsabout Smarandache(fresh, cleanand fantastic) ideal in a BCH-algebras. Smarandache BL-algebras have beeninvented by Borumand Saeid et al. [3], and they dealed with Smarandache idealstructures in Smarandache BL-algebras.Recently, H. S. Kim and Y. H. Kim defined a BE-algebra [6]. S. S. Ahnand K. S. So defined the notion of ideals in BE-algebras, and then stated andproved several characterizationsof such ideals [2]. In [8], B. L. Meng introducedthe notion of an CI-algebra as a generalization of a BE-algebra.

In this paper, we introduce the notions of minimal semicon-tinuity, strongly -semiclosed graph, -semiclosed graph, -semi-, -semicompact and investigate some properties for such notions.

Let Pn be the set of all monic integral self-reciprocal poly- nomials of degree n whose all zeros lie on the unit circle. In this paper we study the following question: For P(z), Q(z) 2 Pn, does there exist a continuous mapping r ! Gr(z) 2 Pn on (0;1) such that G0(z) = P(z) and G1(z) = Q(z)?

The commuting graph of an arbitrary ring R, denoted by ( R), is a graph whose vertices are all non-central elements of R, and two distinct vertices a and b are adjacent if and only if ab = ba. In this paper, we investigate the connectivity, the diameter, the maximum degree and the minimum degree of the commuting graph of group ring ZnQ8. The main result is that ( ZnQ8) is connected if and only if n is not a prime. If ( ZnQ8) is connected, then diam(ZnQ8)= 3, while ( ZnQ8) is disconnected then every connected component of ( ZnQ8) must be a complete graph with a same size. Further, we obtain the degree of every vertex in ( ZnQ8), the maximum degree and the minimum degree of ( ZnQ8).

In [6, 8], we showed that any ideal of is generated by at most two polynomials of the `standard' forms when l is even. The purpose of this paper is to find the `standard' generators of the cyclic codes over of length a multiple of p, namely the ideals of with an integer l which is a multiple of p. We also find an explicit description of their duals in terms of the generators when a = 2.