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Let G = (V,E) be a graph and k be a positive integer. A k-dominating set of G is a subset S ⊆ V such that each vertex in V \S has atleast k neighbors in S. A Roman k-dominating function on G is a functionf : V → {0,1,2} such that every vertex v with f(v) = 0 is adjacent toat least k vertices v 1 ,v 2 ,...,v k with f(v i ) = 2 for i = 1,2,...,k. In thepaper titled “Roman k-domination in graphs” (J. Korean Math. Soc. 46(2009), no. 6, 1309–1318) K. Kammerling and L. Volkmann showed thatfor any graph G with n vertices, γ kR (G) + γ kR (G) ≥ min {2n,4k + 1},and the equality holds if and only if n ≤ 2k or k ≥ 2 and n = 2k + 1or k = 1 and G or G has a vertex of degree n − 1 and its complementhas a vertex of degree n − 2. In this paper we find a counterexampleof Kammerling and Volkmann’s result and then give a correction to theresult. 1. IntroductionLet G = (V,E) be agraphwith vertexset V = V (G) andedge set E = E(G).A k-dominatingsetof G is a subset S ⊆ V such that every vertex in V \S has atleast k neighbors in S. The k-domination numberγ

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

In this paper, we propose a new incremental gradient methodfor solving a regularized minimization problem whose objective is thesum of msmooth functions and a (possibly nonsmooth) convex function.This method uses an adaptive stepsize. Recently proposed incrementalgradient methods for a regularized minimization problem need O(mn)storage, where n is the number of variables. This is the drawback ofthem. But, the proposed new incremental gradient method requires onlyO(n) storage. 1. IntroductionIn this paper, we consider the regularized minimization problem whose formis(1) min x∈ℜ n F λ (x) := f(x) +λP(x),where λ > 0, P : ℜ n → (−∞,∞] is a proper, convex, lower semicontinuous(lsc) function [20], and(2) f(x) :=X mi=1 f i (x),where each function f i is real-valued and smooth (i.e., continuously differen-tiable) on an open subset of ℜ n containing domP = {x | P(x) < ∞}.The minimization problem (1) we consider arises in many applications suchas (supervised) learning [7, 12, 27], regression [17, 23], neural network training[11, 21, 29], and data mining/classification [5, 15, 22, 28]. For the l

In this paper, the effect of absolute values on the behaviorof functions f in the spaces Q K is investigated. It is clear that g ∈Q K (∂D) ⇒ |g| ∈ Q K (∂D), but the converse is not always true. For fin the Hardy space H 2 , we give a condition involving the modulus ofthe function only, such that the condition together with |f|∈Q K (∂D) isequivalent to f ∈Q K . As an application, a new criterion for inner-outerfactorisation of Q K spaces is given. These results are also new for Q p spaces. 1. IntroductionDenote by ∂D the boundary of the unit disk D in the complex plane C.Let H(D) be the space of functions analytic in D. Throughout this paper, weassume that K: [0,∞) → [0,∞) is a right-continuous and increasing function.A function f∈ H(D) belongs to the space Q K ifkfk 2Q K = sup a∈D Z D |f ′ (z)| 2 K(g(a,z))dA(z)

Let (s α (n)) n≥1 be the lexicographically greatest Sturmianword of slope α > 0. For a fixed σ > 1, we consider Dirichlet seriesof the form ν σ (α) :=P ∞n=1 s α (n)n −σ . This paper studies the singularproperties of the real function ν σ , and the Lebesgue-Stieltjes measurewhose distribution is given by ν σ . 1. IntroductionThroughout the paper, N(resp. N 0 ) denotes the set of positive (resp. non-negative) integers. We mean by ⌊·⌋ (resp. ⌈·⌉) the floor (resp. ceiling) function,and by A ∗ the set of finite words over the alphabet A, i.e., the free monoid gen-erated by A.For α ≥ 0, an arithmetic function s α : N→ N 0 is defined bys α (n) := ⌈αn⌉ −⌈α(n −1)⌉.Then s α := s α (1)s α (2)··· is an infinite word over the alphabet {⌈α⌉−1,⌈α⌉}.Now we set, for a fixed σ > 1,(1) ν σ (α) :=X ∞n=1 s α (n)n σ ,i.e., Dirichlet series whose coefficients are given by s α . From now on, we assumeσ > 1 unless otherwise stated explicitly. This real function ν σ : [0,∞) → Rwasfirstly considered in [3], and shown to be continuous at every irrational, whereasleft-continuous but not right-continuous at every rational. Furthermore, ν

In this paper, we characterize some PGL(2,q) by their orders and maximum element orders. We also prove that PSL(2,p) with p > 3 a prime can be determined by their orders and maximum element orders. Moreover, we show that, in general, if q = p n with p a prime and n > 1, PGL(2,q) can not be uniquely determined by their orders and maximum element orders. Several known results are generalized. All groups considered in this paper are finite. For a group G, as in (27), we construct its prime graph ( G) as follows: the vertices are the primes dividing the order of G and two vertices p and r are joined by an edge if and only if G contains an element of order pr. Denote by T(G) = {�i(G)|1 6 i 6 t(G)} the set of all connected components of the graph ( G), where t(G) is the number of the connected components of ( G). If the order of G is even, we always assume that 2 ∈ �1(G). Let

This paper determines the structures of one-homogeneousweight codes over finite chain rings and studies the algebraic propertiesof these codes. We present explicit constructions of one-homogeneousweight codes over finite chain rings. By taking advantage of the distance-preserving Gray map defined in [7] from the finite chain ring to its residuefield, we obtain a family of optimal one-Hamming weight codes over theresidue field. Further, we propose a generalized method that also includesthe examples of optimal codes obtained by Shi et al. in [17]. 1. IntroductionConstant-weight codes represent an important class of codes within the fam-ily of error-correcting codes [11]. A linear code having constant-weight meansthat every nonzero codeword has the same weight. In the literature there aremany papers on binary constant-weight codes which have several applicationssuch as the design of demultiplexers for nano-scale memories [8] and the con-struction of frequency hopping lists for use in GSM networks [12]. Especially,considerable research has been done on the central problem regarding constant-weight codes which is the determination of A(n,d,w), the largest possible sizeof a constant-weight code of length n, Hamming distance at least d, and con-stant weight w. Due to the difficulty in finding good constant-weight codes,various upper and lower bounds on A(n,d,w) have been developed [1, 3, 15, 18].Moreover, there are further studies in this direction including nonbinary finitefields in [2, 10]. It has been shown that there exists a unique one-weight binarylinear code of dimension k such that any two columns in its generator matrixare linearly independent for every positive integer k. Later, this result has beenextended to the ring Z

In this paper, we investigate the insertion-of-factors-proper- ty (simply, IFP) on skew polynomial rings, introducing the concept of strongly �-IFP for a ring endomorphism �. A ring R is said to have strongly �-IFP if the skew polynomial ring R(x;�) has IFP. We examine some characterizations and extensions of strongly �-IFP rings in relation with several ring theoretic properties which have important roles in ring theory. We also extend many of related basic results to the wider classes, and so several known results follow as consequences of our results. Throughout this paper, all rings are associative with identity. We denote by R(x) the polynomial ring with an indeterminate x over R. degf(x) denotes the degree of f(x) ∈ R(x). Let Z and Zn denote the ring of integers and the ring of integers modulo n, respectively. Due to Bell (3), a ring R is called to satisfy the insertion-of-factors-property (simply, an IFP ring) if ab = 0 implies aRb = 0 for a,b ∈ R. Narbonne (22) and Shin (26) used the terms semicommutative and SI for the IFP, respectively. Commutative rings have clearly IFP, and any reduced ring (i.e., a ring without nonzero nilpotent elements) has IFP by a simple computation. A ring is called Abelian if every idempotent is central. IFP rings are Abelian by a simple computation. Another generalization of a reduced ring is an Armendariz ring. Rege and Chhawchharia (25) called a ring R Armendariz if whenever the product of any two polynomials over a ring is zero, then so is the product of any pair of coefficients from the two polynomials. The class of IFP rings and the c of Armendariz rings do not imply each other by (25, Example 3.2) and (13, Example 14). Note that the polynomial rings over IFP rings need not have IFP by (13, Example 2), but for an Armendariz ring R, R has IFP if and only if

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

It is well known that an integral domain D is a UFD if andonly if every nonzero prime ideal of D contains a nonzero principal prime.This is the so-called Kaplansky’s theorem. In this paper, we give this typeof characterizations of a graded PvMD (resp., G-GCD domain, GCDdomain, B´ezout domain, valuation domain, Krull domain, π-domain). 0. IntroductionThis is a continuation of our works on Kaplansky-type theorems [13, 20].It is well known that an integral domain D is a UFD if and only if everynonzero prime ideal of D contains a nonzero principal prime [19, Theorem 5].This is the so-called Kaplansky’s theorem. A generalization of this type oftheorems was first studied by Anderson and Zafrullah in [5], where they gaveseveral characterizations of this type for GCD domains, valuation domains, andPru¨fer domains. Also, characterizations for PvMDs and Krull domains weregiven in [15] and [6], respectively.Later, in [20], the second-named author gave a Kaplansky-type charac-terization of G-GCD domains and PvMDs and gave an ideal-wise version ofKaplansky-type theorems. This ideal-wise version is then used to give char-acterizations of UFDs, π-domains, and Krull domains. Let D be an integraldomain with quotient field K, X be an indeterminate over D, and D[X] bethe polynomial ring over D. A nonzero prime ideal Q of D[X] is called anupper to zero in D[X] if Q ∩ D = (0). Clearly, Q is an upper to zero in D[X]if and only if Q = fK[X] ∩ D[X] for some nonzero polynomial f ∈ D[X].For f ∈ D[X], let c(f) be the ideal of D generated by the coefficients of f.In [13], the first two authors of this paper studied an integral domain D suchthat every upper to zero in D[X] contains a prime (resp., primary) element

In [6], it is proved that the lengths of periods occurring inthe β-expansion of a rational series r noted by Per β (r) depend onlyon the denominator of the reduced form of r for quadratic Pisot unitseries. In this paper, we will show first that every rational r in theunit disk has strictly periodic β-expansion for Pisot or Salem unit basisunder some condition. Second, for this basis, if r = PQ is written inreduced form with |P| < |Q|, we will generalize the curious property“Per β ( PQ ) = Per β ( 1Q )”. 1. IntroductionThe notion of β-expansions of real numbers was introduced by A. R´enyi[11]. Since then, their arithmetic, diophantine and ergodic properties havebeen extensively studied by several researchers. In this paper, we consider ananalogue of this concept in algebraic function fields over finite fields. There arestriking analogies between these digit systems and the classical β-expansionsof real numbers. In order to pursue this analogy, we recall the definition of realβ-expansions and survey the problems corresponding to our results.Let β be a fixed real number greater than 1 and let x be a positive realnumber. A convergent seriesP

It is well-known that the area of parabolic region between aparabola and any chord P 1 P 2 on the parabola is four thirds of the area oftriangle ∆P 1 P 2 P. Here we denote by P the point on the parabola wherethe tangent is parallel to the chord P 1 P 2 . In the previous works, thefirst and third authors of the present paper proved that this property isa characteristic one of parabolas. In this paper, with respect to triangles∆P 1 P 2 Q where Q is the intersection point of two tangents to X at P 1 and P 2 we establish some characterization theorems for parabolas. 1. IntroductionFor a smooth function f : I → Rdefined on an open interval, usually we saythat f is strictly convex if the graph of f has positive curvature κ with respectto the upward unit normal N. This condition is equivalent to f ′′ (x) > 0 on theinterval I.In this article, we study strictly locally convex plane curves. A regular planecurve X : I → R 2 defined on an open interval I, is called convex if, for all s ∈ Ithe trace X(I) of X lies entirely on one side of the closed half-plane determinedby the tangent line to X at s ([4]). A regular plane curve X : I → R

Tomova (8) gave an upper bound for the distance of a bridge surface for a knot with two different bridge positions in a 3-manifold. In this paper, we show that the result of Tomova (8, Theorem 10.3) can be improved in the case when there are two different bridge spheres for a link in S 3 . Hempel (4) introduced the concept of distance of a Heegaard surface, and it is shown by many authors that it well represents various complexities of 3- manifolds. For example, Hartshorn (2) showed that the Euler characteristic of an incompressible surface in a 3-manifold bounds the distance of its Heegaard splittings, and Scharlemann and Tomova (7) showed that the Euler character- istic of any Heegaard splitting of a 3-manifold similarly bounds the distance of any non-isotopic Heegaard splitting. The above concept and results have been extended to bridge surfaces for knots and links in closed 3-manifolds, and have been studied by several authors. For example, Bachman and Schleimer (1) proved that Hartshorn's results can be extended to the distance of a bridge surface for a knot in a closed orientable 3-manifold, and also Tomova (8) proved that Scharlemann and Tomova's results can be extended to the distance of a bridge surface for a knot in a closed ori- entable 3-manifold. Moreover, Johnson and Tomova (6) proved that Tomova's result can be extended to a bridge surface for a tangle in a compact 3-manifold. Recently Jang (5) showed that for a link in a closed orientable 3-manifold, the result of Bachman and Schleimer (1) can be improved in the case when there exist essential meridional spheres. In this paper, we find a property of essential simple closed curves disjoint from the disk complex of a 3-ball containing trivial arcs (for detail, see Lemma 2.1). This allows us to improve the result of Tomova (8, Theorem 10.3) in the case when there are two different bridge spheres for a link in S 3 .

In this paper we introduce and study the new properties (W �), (UWa), (UW E) and (UW �). The main goal of this paper is to study relationship between these new properties and other Weyl type theorems. Moreover, we reconsider several earlier results obtained respec- tively in (11), (18), (14), (1) and (13) for which we give stronger versions. by σa(T) the approximate point spectrum of T. If the range R(T) of T is closed and α(T) < ∞ (resp. β(T) < ∞), then T is called an upper semi-Fredholm (resp. a lower semi-Fredholm) operator. If T ∈ L(X) is either upper or lower semi Fredholm, then T is called a semi-Fredholm operator, and the index of T is defined by ind(T) = α(T) − β(T). If both of α(T) and β(T) are finite, then T is called a Fredholm operator. An operator T ∈ L(X) is called a Weyl operator if it is a Fredholm operator of index zero. The Weyl spectrum σW(T) of T is defined by σW(T) = {λ ∈ C | T − λI is not a Weyl operator}. For a bounded linear operator T and a nonnegative integer n, define T(n) to be the restriction of T to R(T n ), viewed as a map from R(T n ) into R(T n ) (in particular T(0) = T). If for some integer n the range space R(T n ) is closed and T(n) is an upper (resp. a lower) semi-Fredholm operator, then T is called an upper (resp. a lower) semi-B-Fredholm operator. A semi-B-Fredholm operator T is an upper or a lower semi-B-Fredholm operator, and in this case the index of T is defined as the index of the semi-Fredholm operator T(n), see (12). Moreover if T(n) is a Fredholm operator, then T is called a B-Fredholm operator, see

In this paper, we classified finite p-groups Gsuch thatC G (x)/hxiis cyclic for all non-central elements x∈G. This solved a problem pro-posed By Y. Berkovoch. 1. IntroductionIf G is a finite group, then 1 ≤ hxi ≤ C G (x) ≤ G for all elements x ∈ G.In particular, G is abelian if and only if |G : C G (x)| = 1 for all x ∈ G.Moreover, K. Ishikawa [5] classified finite p-groups with |G : C G (x)| ≤ p 2 .Along another line, X. H. Li and J. Q. Zhang [6] classified finite p-groups with|C G (x) : hxi| ≤ p k , where k = 1,2 and p > 2. Y. Berkovich [1] proposed thefollowing:Problem 116(ii). Classify the p-groups G such that C G (H)/H is cyclic forall noncentral cyclic H < G.In other words, Problem 116(ii) requires to classify finite p-groups G suchthat C G (x)/hxi is cyclic for all non-central elements x ∈ G.For convenience, the groups in Problem 116(ii) are called P-groups. LetS = {G G is a P-group}. In this paper, S is determined, and hence theProblem 116(ii) is solved.2. PreliminariesAssume G is a finite p-group. Let r(G) = max{log

In this paper, by using the modular forms of weight nk (2 ≤n ∈ Nand k ∈ Z), we construct a formula which generates modular formsof weight 2nk+4. This formula consist of some known results in [14] and[4]. Moreover, we obtain Fourier expansion of these modular forms. Wealso give some properties of an operator related to the derivative formula.Finally, by using the function j 4 , we obtain the Fourier coefficients ofmodular forms with weight 4. 1. Introduction, definitions and notationsThroughout our paper, we use the following standard notations:N= {1,2,...}, as usual, Zdenotes the set of integers and Cdenotes the setof complex numbers. Let k ∈ ZandH= {z ∈ C: Im(z) > 0}.The set of all Mo¨bius transformations of the formz ′ =az +bcz +d,where a,b,c,d ∈ Zwith ad−bc = 1, is called the modular group and is denotedby Γ (cf. [1]).A function f saidto be amodular form of weight 2k if it satisfiesthe followingconditions (cf. [1], [14]):(i) f is analytic in H.(ii)f az +bcz +d = (cz + d) 2k

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 ≥ 12 and size at least ⌊ 11n−18 2 ⌋ contains three disjoint theta graphs. As a corollary, every graph of order n ≥ 12 and size at least ⌊ 11n−18 2 ⌋ contains three disjoint cycles of even length. 1. Terminology and introduction In this paper, we only consider finite undirected graphs, without loops or multiple edges. We use (1) for the notation and terminology not defined here. A theta graph is the union of three internally disjoint paths that have the same two distinct end vertices. Let n be a positive integer, let Kn denote the complete graph of order n and K − be the graph obtained by removing exactly one edge from K4. For a graph G, we denote its vertex set, edge set, minimum degree by V (G), E(G) and �(G), respectively. The order and size of a graph G, are defined by |V (G)| and |E(G)|, respectively. A set of subgraphs is said to be vertex-disjoint or independent, if no two of them have any common vertex in G, and we use disjoint to stand for vertex-disjoint throughout this paper. If u is a vertex of G and H is either a subgraph of G or a subset of V (G), we define NH(u) to be the set of neighbors of u contained in H, and dH(u) = |NH(u)|. For a subset U of V (G), G(U) denotes the subgraph of G induced by U. In particular, we often use (U) to stand for G(U). If S is a set of subgraphs of G, we write G ⊇ S, it means that S is isomorphic to a subgraph of G, in particular, we use mS to represent a set of m vertex-disjoint copies of S. When S = {x1,x2,...,xt}, we may also use (x1,x2,...,xt) to denote ({x1,x2,...,xt}). Let V1,V2 be two disjoint subsets or subgraphs of G, we use E(V1,V2) to denote the set of edges in G with one end-vertex in V1, while the other in V2, for simplicity, let E(x,V2) stand for E({x},V2), E(V1,x)

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

In this paper, we study the relation between two dynamicalsystems (V,f) and (V,g) with f◦g= g◦f. As an application, we show thatan endomorphism (respectively a polynomial map with Zariski dense, ofbounded Preper(f)) has only finitely many endomorphisms (respectivelypolynomial maps) of bounded degree which are commutable with f. 1. IntroductionA dynamical system (V,f) consists of a set V and a self map f: V → V.If V is a subset of a projective space P n defined over a finitely generated fieldKover Q, then we have arithmetic height functions on V, which make a hugecontribution to the study of (V,f). In this paper, we show the following resultsby studying arithmetic relations between two dynamical systems (V,f) and(V,g) with f◦g= g◦f.Main Theorem (Theorems 3.3 and 4.2). (1) Let φ: P nC → P nC be an en-domorphism on P nC , of degree at least 2. Then there are only finitelymany endomorphisms of degree dwhich are commutable with φ:Com(φ,d) := {ψ∈ End(P nC ) | φ◦ψ= ψ◦φ, degψ= d}is a finite set.(2) Let f : A

In this paper we show that any chain transitive set of a dif-feomorphism on a compact C ∞ -manifold which is C 1 -stably limit shad-owable is hyperbolic. Moreover, it is proved that a locally maximal chaintransitive set of a C 1 -generic diffeomorphism is hyperbolic if and only ifit is limit shadowable. Transitivesets, homoclinicclassesand chaincomponentsofadiffeomorphismare natural candidates to replace the hyperbolic basic sets in nonhyperbolictheory of differentiable dynamical systems, and many recent papers exploredtheir “hyperbolic-like” properties (for more details, see [2, 6, 8, 9, 14, 15]).In this paper we study the chain transitive sets which are limit shadowable.Let us be more precise. Let M be a compact C ∞ -manifold, and let Diff(M) bethe space of diffeomorphisms of M endowed with the C 1 -topology. Denote byd the distance on M induced from a Riemannian metric on the tangent bundleTM. For δ > 0, a sequence {x n } n∈Z in Λ is called a δ-limit chain iflim |n|→∞ d(f(x

The Euler zeta function is defined by ζ E (s)=P ∞n=1(−1) n−1s .The purpose of this paper is to find formulas of the Euler zeta func-tion’s values. In this paper, for s ∈ N we find the recurrence formula ofζ E (2s) using the Fourier series. Also we find the recurrence formula ofP ∞n=1(−1) n−1 (2n−1) 2s−1 , where s ≥ 2(∈ N). 1. IntroductionThe Euler zeta function is defined by ζ E (s) =P ∞n=1(−1) n n s (see [3, 4]). Inthis paper we investigate the recurrence formula of the Euler zeta function fors = 2n with Fourier series. By this result we can find ζ E (2n) for all n ∈ N.For s ∈ C, the Riemann zeta function or the Euler-Riemann zeta function,ζ(s) is defined byζ(s) =X ∞n=1 1n s (s ∈ C), (see [5, 6])which converges when the real part of s is greater than 1. R. Ap´ery provedthat the number ζ(3) is irrational. But it is still an open problem to proveirrationality of ζ(2k +1) for the long time.As well known special values, for any positive even number 2n,ζ(2n) = (−1) n+1 B 2n (2π) 2n 2(2n)!, (see [1])where B

In a recent systematic study, C. Sandon and F. Zanello of- fered 30 conjectured identities for partitions. As a consequence of their study of partition identities arising from Ramanujan's formulas for mul- tipliers in the theory of modular equations, the present authors in an earlier paper proved three of these conjectures. In this paper, we provide proofs for the remaining 27 conjectures of Sandon and Zanello. Most of our proofs depend upon known modular equations and formulas of Ra- manujan for theta functions, while for the remainder of our proofs it was necessary to derive new modular equations and to employ the process of duplication to extend Ramanujan's catalogue of theta function formulas.

The study of the identities of symmetry for the Bernoulli polynomials arises from the study of Gauss's multiplication formula for the gamma function. There are many works in this direction. In the sense of p-adic analysis, the q-Bernoulli polynomials are natural extensions of the Bernoulli and Apostol-Bernoulli polynomials (see the introduction of this paper). By using the N-fold iterated Volkenborn integral, we derive serval identities of symmetry related to the q-extension power sums and the higher order q-Bernoulli polynomials. Many previous results are spe- cial cases of the results presented in this paper, including Tuenter's classi- cal results on the symmetry relation between the power sum polynomials and the Bernoulli numbers in (A symmetry of power sum polynomials and Bernoulli numbers, Amer. Math. Monthly 108 (2001), no. 3, 258- 261) and D. S. Kim's eight basic identities of symmetry in three variables related to the q-analogue power sums and the q-Bernoulli polynomials in (Identities of symmetry for q-Bernoulli polynomials, Comput. Math. Appl. 60 (2010), no. 8, 2350-2359).

The intersection of all two-sided ideals of an ordered semi-group, if it is non-empty, is called the kernel of the ordered semigroup. Aleft ideal Lof an ordered semigroup (S,·,≤) having a kernel Iis said to besimple if I is properly contained in Land for any left ideal L ′ of (S,·,≤),I is properly contained in L ′ and L ′ is contained in Limply L ′ = L. Thenotions of simple right and two-sided ideals are defined similarly. In thispaper, the author characterize when an ordered semigroup having a ker-nel is the class sum of its simple left, right and two-sided ideals. Further,the structure of simple two-sided ideals will be discussed. 1. IntroductionThe intersection of all two-sided ideals of a semigroup, if it is non-empty, iscalled the kernel [5] of the semigroup.Let S be a semigrouphavinga kernelI. A left ideal L of S is said to be simpleif I ⊂ L, (The symbol ⊂ stands for proper inclusion of sets.) and for any leftideal L ′ of S, if I ⊂ L ′ ⊆ L, then L ′ = L. The notions of simple right and two-sided ideals are defined similarly. These concepts were introduced and studiedby Schwarz [6]. Indeed, the author characterized when a semigroup having akernel is the class sum of its simple left, right or two-sided ideals. Further, thestructure of simple two-sided ideals of some semigroups was investigated.The purpose of this paper is to extend Schwarz’s results to ordered semi-groups. Section 2 recalls some certain definitions and results used throughoutthis paper. In Section 3, we introduce the concepts of simple left, right and two-sided ideals of an ordered semigroup having a kernel. Necessary and sufficientconditions for an ordered semigroup having a kernel to be the class sum of itssimple left, right or two-sided ideals are given (Theorems 3.9-3.11). Further,we obtain important corollaries using annihilators of an ordered semigroup.Sections 4-5 investigate the structure of simple two-sided ideals of an orderedsemigroup having a kernel.

In this paper, we obtain new generating functions involvingfamilies of pairs of inverse functions by using a generalization of the Sri-vastava’s theorem [H. M. Srivastava, Some generalizations of Carlitz’stheorem, Pacific J. Math. 85 (1979), 471–477] obtained by Tremblay andFug`ere [Generating functions related to pairs of inverse functions, Trans-form methods and special functions, Varna ’96, Bulgarian Acad. Sci.,Sofia (1998), 484–495]. Special cases are given. These can be seen as gen-eralizations of the generalized Bernoulli polynomials and the generalizeddegenerate Bernoulli polynomials. 1. IntroductionIn 1977, motivated by the work of Srivastava and Singhal [27] on the Ja-cobi polynomials, Carlitz obtained the following generating function for theLaguerre polynomials [2, p. 525, Eq.(5.5)]:X ∞n=0 L (α+λn)n (x +ny)t n =(1+ω) α+1 e −xω 1−λω +ω(1+ω)y(1.1) ,where α,λ are arbitrary complex numbers and ω is a function of t defined by(1.2) ω = t(1+ω) λ+1 e −yωwith ω(0) = 0.In the same paper, Carlitz [2, p. 521, Theorem 1 and Eq.(2.10)] extendedthese results to the formsX

Let R be an I-semiring and S(R) be the set of all identity-summand elements of R. In this paper we introduce the total graphof R with respect to identity-summand elements, denoted by T(Γ(R)),and investigate basic properties of S(R) which help us to gain interestingresults about T(Γ(R)) and its subgraphs. 1. IntroductionAssociating a graph to an algebraic structure is a research subject and hasattracted considerable attention. In fact, the research in this subject aims atexposing the relationship between algebra and graph theory and at advancingthe application of one to the other.In 1988, Beck [11] introduced the idea of a zero-divisor graph of a commu-tative ring R with identity. This notion was later redefined by Anderson andLivingston in [6]. Since then, there has been a lot of interest in this subject andvarious papers were published establishing different properties of these graphsas well as relations between graphs of various extensions. The total graph ofa commutative ring was introduced by Anderson and Badawi in [3], as thegraph with all elements of R as vertices, and two distinct vertices x,y∈ Rareadjacent if and only if x+ y∈ Z(R) where Z(R) is the set of all zero divisorof R. In [4], Anderson and Badawi studied the subgraph T

In this paper, we prove that the 3-cycle is light in the familyof 1-planar graphs with minimum vertex degree at least 5 and minimumedge degree at least 12. This generates a known result of Fabrici andMadaras [8]. 1. IntroductionAll graphs considered in the paper are finite, simple and undirected. Weuse V (G), E(G), F(G), δ(G) and ∆(G) to denote the set of vertices, the setof edges, the set of faces, the minimum degree and the maximum degree ofa plane graph G, respectively. The degree of an edge uv in G is the value ofd G (u) + d G (v). A k-, k + - and k − -vertex(resp.face) is a vertex (resp.face) ofdegree k, at least k and at most k, respectively. For other undefined conceptswe refer the reader to [2].A graph is 1-planar if it can be drawn on a plane so that each edge is crossedby at most one other edge. The notion of 1-planarity was introduced by Ringel[14], who proved that each 1-planar graph is vertex 7-colorable. This is thefirst result on the colorings of 1-planar graphs, and from then on, many authorsstartedto investigatethe coloringproblems(see[1, 3, 4, 6, 15, 16, 17, 21, 23, 24])and the structural properties (see [5, 8, 9, 10, 11, 13, 18, 19, 20, 22]) of 1-planargraphs. One of possible approaches in the study of local graph structures canbe formalized in the following way (see [12]):Let H be a connected graph and G be a family of graphs. If for any graphG ∈ G, G contains a subgraph K ≃ H such that max

In this paper, we give a necessary and sufficient condition for the hyponormality of the block Toeplitz operators T�, where � = F +G∗, F(z), G(z) are some matrix valued polynomials on the vector valued Bergman space L2(D,Cn). We also show some necessary conditions for the hyponormality of TF+Gwith F + G∗ 2 h∞ Mn×n on L2(D,Cn). Let D and T be the open unit disk and unit circle in the complex plane C re- spectively and dA be the normalized Lebesgue area measure on D. h ∞ denotes the space of all bounded harmonic functions on D. L ∞ (D,dA) and L 2 (D,dA) denote the space of essential bounded measurable functions and the space of the square integrable functions on D with respect to dA, respectively. The Bergman space L 2 consists of all analytic functions in L 2 (D,dA). We denote the space of vector valued square integrable functions on D by L 2 (D,C n ) = L 2 (D,dA)⊗C n and the vector valued Bergman space on D by L 2(D,C n ) = L 2 ⊗ C n , respec- tively, where ⊗ denotes the Hilbert space tensor product. In this paper F T denotes the transpose of the matrix F and G ∗ denotes the adjoint of the matrix G.

An element r of a commutative semiring R with identity issaid to be identity-summand if there exists 1 6= a∈Rsuch that r+a= 1.In this paper, we introduce and investigate the identity-summand graphof R, denoted by Γ(R). It is the (undirected) graph whose vertices arethe non-identity identity-summands of Rwith two distinct vertices jointby an edge when the sum of the vertices is 1. The basic properties andpossible structures of the graph Γ(R) are studied. 1. IntroductionOne of the associated graphs to a ring R is the zero-divisor graph; it is asimple graph with vertex set Z(R)\{0}, and two vertices x and y are adjacentif and only if xy = 0 which is due to Anderson and Livingston [8]. This graphwas first introduced by Beck, in [11], where all the elements of R are consideredas the vertices. Since then, there has been a lot of interest in this subject andvarious papers were published establishing different properties of these graphsas well as relations between graphs of various extensions [3, 8-9, 18, 20-21].Recently, the study of graphs of rings are extended to include semirings as in[16].As a generalization of rings, structure of semirings have proven to be usefultools in various disciplines. They have important applications in mathematicsand theoretical computer science [19, 22]. From now on let R be a commutativesemiring with identity. We define another graph on R, Γ(R), with vertices aselements of S

A graph G is called a fractional (g,f,n)-critical graph if any n vertices are removed from G, then the resulting graph admits a fractional (g,f)-factor. In this paper, we determine the new toughness condition for fractional (g,f,n)-critical graphs. It is proved that G is fractional (g,f,n)-critical if t(G) � b 2 −1+bn a . This bound is sharp in some sense. Furthermore, the best toughness condition for fractional (a, b,n)-critical graphs is given. All graphs considered in this paper are finite, loopless, and without multiple edges. The notation and terminology used but undefined in this paper can be found in (2). Let G be a graph with the vertex set V (G) and the edge set E(G). For a vertex x ∈ V (G), we use dG(x) and NG(x) to denote the degree and the neighborhood of x in G, respectively. Let �(G) denote the minimum degree of G. For any S ⊆ V (G), the subgraph of G induced by S is denoted by G(S). Suppose that g and f are two integer-valued functions on V (G) such that 0 ≤ g(x) ≤ f(x) for all x ∈ V (G). A spanning subgraph F of G is called a (g,f)-factor if g(x) ≤ dF(x) ≤ f(x) for each x ∈ V (G). A fractional (g,f)- factor is a function h that assigns to each edge of a graph G a number in (0,1) so that for each vertex x we have g(x) ≤ P e∈E(x) h(e) ≤ f(x). If g(x) = a, f(x) = b for all x ∈ V (G), then a fractional (g,f)-factor is a fractional (a,b)- factor. Moreover, if g(x) = f(x) = k (k ≥ 1 is an integer throughout this paper, and we will not reiterate it again) for all x ∈ V (G), then a fractional (g,f)-factor is just a fractional k-factor.

Let R be a commutative ring with identity and M an R-module. In this paper, we associate a graph to M, say Γ(M), such thatwhen M = R, Γ(M) is exactly the classic zero-divisor graph. Many well-known results by D. F. Anderson and P. S. Livingston, in [5], and by D. F.Anderson and S. B. Mulay, in [6], have been generalized for Γ(M) in thepresent article. We show that Γ(M) is connected with diam(Γ(M)) ≤ 3.We also show that for a reduced module M with Z(M) ∗ 6= M \ {0},gr(Γ(M)) = ∞ if and only if Γ(M) is a star graph. Furthermore, weshow that for a finitely generated semisimple R-module M such that itshomogeneous components are simple, x,y∈ M\ {0} are adjacent if andonly if xRTyR = (0). Among other things, it is also observed thatΓ(M) = ∅ if and only if M is uniform, ann(M) is a radical ideal, andZ(M) ∗ 6= M\{0}, if and only if ann(M) is prime and Z(M) ∗ 6= M\{0}. 1. IntroductionAll rings in this paper are commutative with identity and all modules areunitary right modules. Let G be an undirected graph. We say that G isconnected if there is a path between any two distinct vertices. For distinctvertices x and y in G, the distance between x and y, denoted by d(x,y), is thelength of a shortest path connecting x and y (d(x,x) = 0 and d(x,y) = ∞ ifno such path exists). The diameter of G isdiam(G) = sup{d(x,y) | x and y are vertices of G}.A cycle of length n in G is a path of the form x

An SQNS-group G is a group in which every proper sub-group of G is either s-quasinormal or self-normalizing and a minimalnon-SQNS-group is a group which is not an SQNS-group but all ofwhose proper subgroups are SQNS-groups. In this note all the finiteminimal non-SQNS-groups are determined. 1. IntroductionThroughout this paper, only finite groups are considered.Given a group theoretical property P, a group belonging to a class of groupsP is called a P-group and the other groups are called non-P-groups. A minimalnon-P-groupis a non-P-groupall of whose propersubgroupsare P-groups. Theproblem of determining all the finite minimal non-P-groups has been studiedby several authors and there are many remarkable examples about the minimalnon-P-groups: minimal non-abelian groups (Miller and Moreno [7]), minimalnon-nilpotent groups (Schmidt), minimal non-supersolvable groups ([1]) andminimal non-p-nilpotent groups (Itˆo).Recall that a subgroup H of a group G is said to be s-quasinormal in G ifHK = KH for any Sylow subgroup K of G. Let SQNS denotes the classof all groups in which every proper subgroup is either s-quasinormal or self-normalizing. Our principal object here is the classification of all the finiteminimal non-SQNS-groups. Combining Theorems 3.3 and 3.4 in this paper,we get the following:Main Theorem. Let G be a minimal non-SQNS-group. Then G is solvableand is isomorphic to one of the following groups

An injective coloring of a graph G is an assignment of colorsto the vertices of G so that any two vertices with a common neighborreceive distinct colors. A graph G is said to be injectively k-choosableif any list L(v) of size at least k for every vertex v allows an injectivecoloring φ(v) such that φ(v) ∈ L(v) for every v ∈ V (G). The least k forwhich G is injectively k-choosable is the injective choosability number ofG, denoted by χ li (G). In this paper, we obtain new sufficient conditionsto be χ li (G) = ∆(G). Maximum average degree, mad(G), is defined bymad(G) = max{2e(H)/n(H) : H is a subgraph of G}. We prove that ifmad(G) < 8k−33k , then χ li (G) = ∆(G) where k = ∆(G) and ∆(G) ≥ 6. Inaddition, when ∆(G) = 5 we prove that χ li (G) = ∆(G) if mad(G) < 177 ,and when ∆(G) = 4 we prove that χ li (G) = ∆(G) if mad(G) < 73 . Theseresults generalize some of previous results in [1, 4]. 1. IntroductionAll graphsconsidered in this paper aresimple, finite, and undirected. We useV (G),E(G) and ∆(G) to denote the vertex set, the edge set and the maximumdegree of G, respectively. Here we introduce some notation. A k-vertex is avertex of degree k, and a k

In this paper, by using the q-difference analogue of lemmaon the logarithmic derivative lemma to re-establish some estimates ofNevanlinna characteristics of f(qz), we deal with the value distributionand uniqueness of certain types of q-difference polynomials. 1. IntroductionIn this paper, we assume that the reader is familiar with the standard sym-bols and fundamental results of Nevanlinna theory, such as the proximity func-tion m(r,f), counting function N(r,f), characteristic function T(r,f) for ameromorphic function f(z) in the complex plane (see e.g. [7, 14]). We also useN (2 (r, 1f ) to denote the counting function of zeros of f(z) such that the multiplezeros are counted once and the simple zeros are not counted in {z : |z| ≤ r}. Wenow recall that a meromorphic function a(z) is said to be a small function off(z) if T(r,a) = S(r,f), where S(r,f) is used to denote any quantity satisfyingS(r,f) = o({T(r,f)} as r → ∞, possibly outside of a set of finite logarithmicmeasure, furthermore, possibly outside of a set of logarithmic density 0, i.e.,outside of a set E such that lim

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

In this paper, we introduce the q-analogue of p-adic log gamma functions with weight alpha. Moreover, we give a relationship be- tween weighted p-adic q-log gamma functions and q-extension of Genocchi and Euler numbers with weight alpha. Assume that p is a fixed odd prime number. Throughout this paper Z, Zp, Qp and Cp will denote the ring of integers, the field of p-adic rational numbers and the completion of the algebraic closure of Qp, respectively. Also we denote N � = N ( {0} and exp(x) = e x . Let vp : Cp ! Q ( {1} (Q is the field of rational numbers) denote the p-adic valuation of Cp normalized so that vp (p) = 1. The absolute value on Cp will be denoted as |·|, and |x| p = p v p(x)

A graph is symmetric if its automorphism group acts transi-tively on the set of arcs of the graph. In this paper, we classify tetravalentsymmetric graphs of order 9pfor each prime p. 1. IntroductionLet G be a permutation group on a set Ω and α ∈ Ω. Denote by G α thestabilizer of α in G, that is, the subgroup of G fixing the point α. We saythat G is semiregular on Ω if G α = 1 for every α ∈ Ω and regular if G istransitive and semiregular. Throughout this paper, we consider undirectedfinite connected graphs without loops or multiple edges. For a graph X we useV (X), E(X) and Aut(X) to denote its vertex set, edge set, and automorphismgroup, respectively. For u,v ∈ V (X), denote by {u,v} the edge incident to uand v in X.A graphX is said to be vertex-transitive if Aut(X) acts transitivelyon V (X).An s-arc in a graph is an ordered (s +1)-tuple (v 0 ,v 1 ,...,v s−1 ,v s ) of verticesof the graph X such that v i−1 is adjacent to v i for 1 ≤ i ≤ s, and v i−1 6= v i+1 for 1 ≤ i ≤ s − 1. In particular, a 1-arc is called an arc for short and a 0-arcis a vertex. For a subgroup G ≤ Aut(X), a graph X is said to be (G,s)-arc-transitive and (G,s)-regular if G is transitive and regular on the set of s-arcsin X, respectively. A (G,s)-arc-transitive graph is said to be (G,s)-transitiveif it is not (G,s+1)-arc-transitive. In particular, a (G,1)-arc-transitive graphis simply called G-symmetric. A graph X is simply called s-arc-transitive, s-regular and s-transitive if it is (Aut(X),s)-arc-transitive, (Aut(X),s)-regularand (Aut(X),s)-transitive, respectively.Arc-transitive or s-transitive graphs have received considerable attentionin the literature. For example, s-transitive graphs of order np was classifiedin [3, 4, 23] depending on n=1, 2 or 3, where p is a prime. Li [13] showedthat there exists an s-transitive graph of odd order if and only if s ≤ 3. Forthe case of valency 4, Gardiner and Praeger [8, 9] characterized tetravalent

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

Let H be a complex separable infinite dimensional Hilbert space. In this paper, a necessary and sufficient condition is g for an operator T on H to satisfy that f(T) obeys generalized Weyl's the- orem for each function f analytic on some neighborhood of �(T). Also we investigate the stability of generalized Weyl's theorem under (small) compact perturbations. Throughout this paper, H will always denote a complex separable infinite dimensional Hilbert space. We denote by B(H) the algebra of all bounded linear operators on H, and by K(H) the ideal of compact operators in B(H). Let T 2 B (H). We denote by �(T) andp(T) the spectrum of T and the point spectrum of T respectively. Denote by kerT and ranT the kernel of T and the range of T respectively. T is called a semi-Fredholm operator, if ranT is closed and either dimkerT or dimkerTis finite; in this case, indT := dimkerT −dimkerTis called the index of T. In particular, if −1 < indT <

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].

A (1 ; )-embedded graph is a graph that can be embedded on a surface with Euler characteristic so that each edge is crossed by at most one other edge. A graph G is called -linear if there exists an integral constant such that e( G ' ) v ( G ' ) + for each GG. In this paper, it is shown that every (1 ; )-embedded graph G is 4-linear for all possible , and is acyclicly edge-(3∆( G) + 70)-choosable for = 1 ; 2. In this paper, all graphs considered arenite, simple and undirected. We use V ( G), E( G), ( G) and ∆( G) to denote the vertex set, the edge set, the minimum degree and the maximum degree of a graph G, respectively. Let e( G) = j E( G) j and v( G) = j V ( G) j . Moreover, for an embedded graph G (i.e., a graph that can be embedded on a surface), by F ( G) we denote the face set of G. Let f ( G) = j F ( G) j . The girth g( G) of a graph G is the length of the shortest cycle of G. A k-, k + - and k -vertex (or face) is a vertex (or face) of

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

Assume that X, partitioned into 2 2 block form, is a solution of the system of quaternion matrix equations A1XB1 = C1;A2XB2 = C2: We in this paper give the maximal and minimal ranks of the sub- matrices in X; and establish necessary and sufficient conditions for the submatrices to be zero, unique as well as independent. As applications, we consider the common inner inverse G; partitioned into 2 2 block form, of two quaternion matrices M and N: We present the formulas of the maximal and minimal ranks of the submatrices of G; and describe the properties of the submatrices of G as well. Thendings of this paper generalize some known results in the literature.

Let k 1 be an integer, and let G be a 2-connected graph of order n with n maxf7;4k +1g, and the minimum degree (G) k +1. In this paper, it is proved that G has a fractional k-factor excluding any given edge if G satises max fdG(x);dG(y)g n for each pair of nonadjacent vertices x;y of G. Furthermore, it is showed that the result in this paper is best possible in some sense.

In this paper, several properties of endomorphism rings of modules are investigated. A multiplication module M over a commutative ring R induces a commutative ring M ⁄ of endomorphisms of M and hence the relation between the prime (maximal) submodules of M and the prime (maximal) ideals of M ⁄ can be found. In particular, two classes of ideals of M⁄ are discussed in this paper: one is of the form GM⁄(M,N) = {f 2 M⁄ | f(M) µ N} and the other is of the form GM⁄(N,0) = {f 2 M⁄ | f(N) = 0} for a submodule N of M.

Y. Hirano introduced the concept of a quasi-Armendariz ring which extends both Armendariz rings and semiprime rings. A ring R is called quasi-Armendariz if aiRbj = 0 for each i,j whenever polynomials f(x) = Pm=0 aix i ,g(x) = Pn=0 bjx j 2 R(x) satisfy f(x)R(x)g(x) = 0. In this paper, we first extend the quasi-Armendariz property of semiprime rings to the skew polynomial rings, that is, we show that if R is a semiprime ring with an epimorphism ae, then f(x)R(x;ae)g(x) = 0 im- plies aiRae i+k (bj) = 0 for any integer k ‚ 0 and i,j, where f(x) = Pm i=0 aix i ,g(x) = Pn j=0 bjx j 2 R(x;ae). Moreover, we extend this prop- erty to the skew monoid rings, the Ore extensions of several types, and skew power series ring, etc. Next we define ae-skew quasi-Armendariz rings for an endomorphism ae of a ring R. Then we study several exten- sions of ae-skew quasi-Armendariz rings which extend known results for quasi-Armendariz rings and ae-skew Armendariz rings. Throughout this paper R denotes an associative ring with identity. We denote by R(x) the polynomial ring with an indeterminate x over R. Rege and Chhawchharia (18) introduced the notion of an Armendariz ring. A ring R is called Armendariz if whenever polynomials f(x) = Pm=0 aix i ,g(x) = Pn j=0 bjx j 2 R(x) satisfy f(x)g(x) = 0, then aibj = 0 for each i,j. The name "Armendariz ring" was chosen from the fact that Armendariz (2, Lemma 1) had showed that a reduced ring (i.e., a ring without nonzero nilpotent elements) sat- isfies this condition. Many properties of Armendariz rings have been studied by several authors (1, 8, 10, 11, 12). Hirano (5) introduced a quasi-Armendariz ring which is generalizing an Armendariz ring. A ring R is called quasi-Armendariz if whenever polynomials f(x) = P m=0 aix i ,g(x) = P n=0 bjx j 2 R(x) satisfy f(x)R(x)g(x) = 0, then aiRbj = 0 for each i,j. Hirano (5, Corollary 3.8) proved that semiprime rings are quasi-Armendariz rings. Moreover, he showed that the class of quasi-Armendariz rings is Morita stable (4, Theorem 3.12 and Proposition 3.13), and that if R is a quasi-Armendariz ring, then some exten- sions of R (e.g., the n-by-n upper triangular matrix ring, the polynomial ring) are also quasi-Armendariz rings. But most of these properties are not stable in Armendariz rings (for example, (10, Examples 1 and 3, etc.)).

In this paper, more generalized q-factorial coecients are examined by a natural extension of the q-factorial on a sequence of any numbers. This immediately leads to the notions of the extended q-Stirling numbers of both kinds and the extended q-Lah numbers. All results described in this paper may be reduced to well-known results when we set q = 1 or use special sequences.

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