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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

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

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).

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

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

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

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.

As a sequel to the papers (2, 3), we will complete our iden- tification of the groups of units of the finite local rings Z4(X)/(X k + 2t(X),2X r ) which is the most general type of finite local rings with a single nilpotent generator over Z4. with u(0) = 1, and deg(u) 0, then the elements of the form 1 + Xf(X), where f 2Z4(X) form a subgroup of U(R) of the group of units which we denote by U1(R) and we call such an element a 1-unit. In (2, XVIII.2) the group U1(R) is called the one group of R. If a = 0, then the set of 1-units do not form a subgroup and in that case we will consider the group of units U(R) of R. In (2) any finite Z4-algebra which is generated by a single element is of the form R =Z4(X)/(X k +2u(X)X a ,2X r ) with a < r < k and a polynomial u(X) such that u(0) = 1 and deg(u) < r i a. In this paper, we will identify the group of units U(R) of R which is a finite abelian 2-group by decomposing into a direct sum of cyclic subgroups thereby completing the identification of the groups of units of the finite rings which is generated by a single nilpotent element. For this we need to find the "natural" generators of the cyclic subgroups. As in (3), there is a natural surjective ring homomorphism ` :Z4(X)/(X k + 2u(X)X a ,2X r ) !F2(X)/(X k ) which induces a surjective group homomorphism which we still call `

Let G ⊗ G be the tensor square of a group G. The set of allelements a in G such that a ⊗ g = 1 ⊗ , for all g in G, is called the tensorcentre of G and denoted by Z ⊗ (G). In this paper some properties of thetensor centre of G are obtained and the capability of the pair of groups(G,G ′ ) is determined. Finally, the structure of J 2 (G) will be described,where J 2 (G) is the kernel of the map κ : G ⊗ G → G ′ . 1. IntroductionLet G and H be two groups which act on themselves by conjugation g g ′ =gg ′ g −1 and on each other in such a way that the following compatibility con-ditions are satisfied: ( g h) g ′ = g ( h ( g −1 g ′ )), ( h g) h ′ = h ( g ( h −1 h ′ ))for all g,g ′ in G and h,h ′ in H.The non-abelian tensor product G ⊗ H was introduced by R. Brown andJ.-L. Loday in [2] as the group generated by all the symbols g ⊗ h subject tothe following relationsgg ′ ⊗h = ( g g ′ ⊗ g h)(g ⊗h), g ⊗hh ′ = (g ⊗h)( h g ⊗ h h ′ )for all g,g ′ in G, h,h in H.If G = H, then G ⊗ G is considered to be the non-abelian tensor square,so that it will be focused in this paper. There exists an action of G on G ⊗Gsatisfying

For convex bodies, chord power integrals were introduced and studied in several papers (see [3], [6], [14], [15], etc.). The aim of this article is to study them further, that is, we establish the BrunnMinkowski-type inequalities and get the upper bound for chord power integrals of convex bodies. Finally, we get the famous Zhang projection inequality as a corollary. Here, it is deserved to mention that we make use of a completely distinct method, that is using the theory of inclusion measure, to establish the inequality. 1. Preliminaries The setting for this paper is n-dimensional Euclidean space R. We will denote by convex figure a compact convex subset of R, and by convex body a convex figure with nonempty interior. Let Sn−1 denote the unit sphere centered at the origin o in R, and write αn−1 for the (n − 1)-dimensional volume of Sn−1. Let Bn be the closed unit ball in R, write ωn for the n-dimensional volume of Bn. Note that ωn = 2π n 2 nΓ(n2 ) , αn−1 = nωn. By a direction, we mean a unit vector, that is, an element of Sn−1. If u is a direction, we denote by u⊥ the (n − 1)-dimensional subspace orthogonal to u and by lu the line through the origin parallel to u. Denote by AGi,n the affine Grassmann manifold of i-dimensional planes in R. It is a homogeneous space under the action of the motion group G(n) (see [7], p.199). Let dξk be the normalized invariant measure of AGi,n whose restriction to the Grassmann manifold Gi,n is the invariant probability measure. Let ξ1 be a random line intersecting K. Then vol1(K ∩ ξ1) is the chord length of the intersection K ∩ ξ1. The chord power integrals of K are defined by Iλ(K) = 2αn−1 n ∫ ξ1∈AG1,n vol1(K ∩ ξ1)dξ1, 0 ≤ λ < ∞. Received November 30, 2006; Revised April 5, 2007. 2000 Mathematics Subject Classification. 52A40.

Let $\mathcal{A}$ and $\mathcal{B}$ be two $C^{*}$-algebras such that $\mathcal{A}$ is prime. In this paper, we investigate the additivity of maps $\Phi$ from $\mathcal{A}$ onto $\mathcal{B}$ that are bijective and satisfy $$\Phi(A^{*}B+\eta BA^{*})=\Phi(A)^{*}\Phi(B)+\eta \Phi(B)\Phi(A)^{*}$$ for all $A, B\in \mathcal{A}$ where $\eta$ is a non-zero scalar such that $\eta\neq \pm1$. Moreover, if $\Phi(I)$ is a projection, then $\Phi$ is a $\ast$-isomorphism.

Let C(0,t) denote a generalized Wiener space, the space of real-valued continuous functions on the interval (0,t) and define a random vector Zn : C(0,t) → R n by Zn(x) = ( R t 1 0 h(s)dx(s),..., R tn 0 h(s)dx(s)), where 0 < t1 < ··· < tn < t is a partition of (0,t) and h ∈ L2(0,t) with h 6 0 a.e. In this paper we will introduce a simple formula for a generalized conditional Wiener integral on C(0,t) with the conditioning function Zn and then evaluate the generalized analytic conditional Wiener and Feynman integrals of the cylinder function F(x) = f( R t 0 e(s)dx(s)) for x ∈ C(0,t), where f ∈ Lp(R)(1 ≤ p ≤ ∞) and e is a unit element in L2(0,t). Finally we express the generalized analytic conditional Feynman integral of F as two kinds of limits of non-conditional generalized Wiener integrals of polygonal functions and of cylinder functions using a change of scale transformation for which a normal density is the kernel. The choice of a complete orthonormal subset of L2(0,t) used in the transformation is independent of e and the conditioning function Zn does not contain the present positions of the generalized Wiener paths.

All rings are commutative with $1$ and $n$ is a positive integer. Let $��: J(R)\to J(R)\cup{\emptyset}$ be a function where $J(R)$ denotes the set of all ideals of $R$. We say that a proper ideal $I$ of $R$ is $��$-$n$-absorbing primary if whenever $a_1,a_2,...,a_{n+1}\in R$ and $a_1a_2\cdots a_{n+1}\in I\backslash��(I)$, either $a_1a_2\cdots a_n\in I$ or the product of $a_{n+1}$ with $(n-1)$ of $a_1,...,a_n$ is in $\sqrt{I}$. The aim of this paper is to investigate the concept of $��$-$n$-absorbing primary ideals., 29 pages

According to Jacobson (31), a right ideal is bounded if it con- tains a non-zero ideal, and Faith (15) called a ring strongly right bounded if every non-zero right ideal is bounded. From (30), a ring is strongly right AB if every non-zero right annihilator is bounded. In this paper, we introduce and investigate a particular class of McCoy rings which sat- isfy Property (A) and the conditions asked by Nielsen (42). It is shown that for a u.p.-monoid M and � : M ! End(R) a compatible monoid homomorphism, if R is reversible, then the skew monoid ring RM is strongly right AB. If R is a strongly right AB ring, M is a u.p.-monoid and � : M ! End(R) is a weakly rigid monoid homomorphism, then the skew monoid ring RM has right Property (A).

In this paper, we give several sufficient conditions ensuringthat any positive radial solution (u,v) of the following γ-Laplacian sys-tems in the whole space R n has the components symmetry property u ≡ vˆ−div(|∇u| γ−2 ∇u) = f(u,v) in R n ,−div(|∇v| γ−2 ∇v) = g(u,v) in R n .Here n > γ, γ > 1.Thus, the systems will be reduced to a single γ-Laplacian equation:−div(|∇u| γ−2 ∇u) = f(u) in R n .Our proofs are based on suitable comparation principle arguments, com-bined with properties of radial solutions. 1. IntroductionIn 2008, Li and Ma [10] studied the stationary Schr¨odinger system(1.1)ˆ−∆u = u p v q in R n ,−∆v = u q v p in R n ,and obtained a components symmetry result:Proposition 1.1. Assume n > γ, 1 ≤ p,q ≤ n+2n−2 and p + q = n−2 . Thenany (L n2−n2 (R n )) 2 -positive solution pair (u,v) to (1.1) is radial symmetric, andhence u ≡ v = a(b 2 +|x −x 0 | 2 ) (2−n)/2 with a,b > 0 and x 0 ∈ R n .The proof was achieved by the classification result in [4] and the methodof moving planes based on the conformal invariant property. Afterwards, Leiand Li ([8]) studied the asymptotic radial symmetry and decay estimates ofpositive integrable solutions of(1.2)ˆ−div(|∇u

In this paper, we consider the problem of existence of confor- mal metrics with prescribed mean curvature on the unit ball of Rn, n � 3. Under the assumption that the order of flatness at critical points of pre- scribed mean curvature function H(x) is � 2)1,n 2), we give precise estimates on the losses of the compactness and we prove new existence result through an Euler-Hopf type formula. 1. Introduction and main result In this article, we consider the problem of existence of conformal scalar flat metric with prescribed boundary mean curvature on the standard n-dim- ensional ball. Let B n be the unit ball in R n , n ≥ 3, with Euclidean metric g0. Its boundary will be denoted by S n−1 and will be endowed with the standard metric still denoted by g0. Let H : S n−1 → R be a given function, we study the problem of finding a conformal metric g = u 4 n−2g 0 such that Rg = 0 in B n and hg = H on S n−1 . Here Rg is the scalar curvature of the metric g in B n and hg is the mean curvature of g on S n−1 . This problem is equivalent to solving the following nonlinear boundary value equation: (P) ( �u = 0 in B n ∂u ∂ν + n − 2 2 u = n − 2 2 Hu n n−2 on S n−1

In this paper, some omissions in Mishra and Lai [13], havebeen pointed out and their corrective measures have been discussed briefly. 1. IntroductionA pair of primal and dual problems in mathematical programming is calledsymmetric if the dual of the dual is the primal problem. Dorn [6] introducedthe concept of symmetric duality in quadratic programming. His results wereextended to nonlinear convex programming problems by Dantzig et al. [4] andlater by Bazaraa and Goode [3] over arbitrary cones.Mangasarian [11] introduced the concept of second-order duality for non-linear problems. Since then, many authors [1, 2, 7, 9, 15, 16] have workedon second-order symmetric duality. Mishra and Lai [13] studied Mond-Weirtype second-order multiobjective symmetric duality for the following pair ofproblems:(P) K−minimize f(x,y)−12p T ∇ yy f(x,y)psubject to −∇ y (λ T f)(x,y) −∇ yy (λ T f)(x,y) ∈ C ∗2 ,y T [∇ y (λ T f)(x,y) +∇ yy (λ T f)(x,y)] > 0,λ ∈ K ∗ , x ∈ C 1 .(D) K−maximize f(u,v)−12q

In this paper, we investigate p-biharmonic maps u : (M,g) →(N,h) from a Riemannian manifold into a Riemannian manifold with non-positive sectional curvature. We obtain that ifR M |τ(u)| a+p dv g < ∞ andR M |d(u)| 2 dv g < ∞, then u is harmonic, where a ≥ 0 is a nonnegativeconstant and p ≥ 2. We also obtain that any weakly convex p-biharmonichypersurfaces in space formN(c) with c ≤ 0 is minimal. These results giveaffirmative partial answer to Conjecture 2 (generalized Chen’s conjecturefor p-biharmonic submanifolds). 1. IntroductionHarmonic maps play a central roll in geometry. They are critical pointsof the energy E(u) =R M|du| 2 2 dv g for smooth maps between manifolds u :(M,g) → (N,h) and the Euler-Lagrange equation is that tension field τ(u)vanishes. Extensions to the notions of p-harmonic maps, F-harmonic mapsand f-harmonic maps were introduced and many results have been carried out(for instance, see [1, 2, 3, 8, 23]). In 1983, J. Eells and L. Lemaire [10] proposedthe problem to consider the biharmonic maps: they are critical maps of thefunctionalE

Let λ f (n) denote the n-th normalized Fourier coefficient ofa primitive holomorphic form f for the full modular group Γ = SL 2 (Z).In this paper, we are concerned with Ω-result on the summatorPy function n6x λ 2f (n 2 ), and establish the following resultX n6x λ 2f (n 2 ) = c 1 x + Ω(x 49 ),where c 1 is a suitable constant. 1. Introduction and main resultsAccording to the Langlands program, there are many hidden structures un-derlying the Fourier coefficients of an automorphic form. Thus it is very impor-tant and essential to investigateits summatoryfunction overa certain sequence.Let H ∗ k be the set of all normalized Hecke eigencuspforms of even integralweight kfor the full modular group Γ = SL 2 (Z). For f(z) ∈ H ∗ k , f(z) has thefollowing Fourier expansion at the cusp ∞f(z) =X ∞n=1 λ f (n)n k−12 e 2πinz ,where λ f (n) is real and satisfies the multiplicative property(1.1) λ f (m)λ f (n) =X d|(m,n) λ f mnd 2 for any integers m≥ 1 and n≥ 1.The size and oscillations of λ f (n) deserve deep research. In 1974, Deligne

In this note we study the structures of power-serieswise Ar- mendariz rings and IFP rings when they are skewed by ring endomor- phisms (or automorphisms). We call such rings skew power-serieswise Armendariz rings and skew IFP rings, respectively. We also investigate relationships among them and construct necessary examples in the pro- cess. The results argued in this note can be extended to the ordinary ring theoretic properties of power-serieswise Armendariz rings, IFP rings, and near-related rings. Throughout this paper every ring is an associative ring with identity. Let σ be an endomorphism of a ring R. We use R((x;σ)) (resp. R(x;σ)) to denote the skew power series ring (resp. skew polynomial ring) with an indeterminate x over a ring R, subject to the relation xr = σ(r)x for r ∈ R. Note that σ(1) = 1 for any skew power series ring (skew polynomial ring) R((x;σ)) (R(x;σ)), since 1x n = x n = x1x n−1 = σ(1)x n for any n ≥ 1 where 1 is the identity of R. Armendariz (2, Lemma 1) proved that whenever polynomials f(x) = m X i=0 aix i , g(x) = n X j=0 bjx j

For a field K, a square-free monomial ideal I of K[, . . ., ] is called an f-ideal, if both its facet complex and Stanley-Reisner complex have the same f-vector. Furthermore, for an f-ideal I, if all monomials in the minimal generating set G(I) have the same degree d, then I is called an f-ideal. In this paper, we prove the existence of f-ideal for and , and we also give some algorithms to construct f-ideals.

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.

Let R be a semiring, I a strong co-ideal of R and S(I) the set of all elements of R which are not prime to I. In this paper we investigate some interesting properties of S(I) and introduce the total identity-summand graph of a semiring R with respect to a co-ideal I. It is the graph with all elements of R as vertices and for distinct x,y 2 R, the vertices x and y are adjacent if and only if xy 2 S(I).

The intersection graph of a group G is an undirected graphwithout loops and multiple edges defined as follows: the vertex set is theset of all proper non-trivial subgroups of G, and there is an edge betweentwo distinct vertices Hand Kif and only if H∩K6= 1 where 1 denotes thetrivial subgroup of G.In this paper we characterize all finite groups whoseintersection graphs are planar. Our methods are elementary. Among thegraphs similar to the intersection graphs, we may count the subgrouplattice and the subgroup graph of a group, each of whose planarity wasalready considered before in [2, 10, 11, 12]. 1. Introduction and preliminariesA graph is called planar if it can be drawn on the plane in such a way thatits edges intersect only at their endpoints. There are interesting graphs con-structed from algebraic objects such as the subgroup lattice and the subgroupgraph of a group. Planarity of the subgroup lattice and the subgroup graph ofa group were studied by Bohanon and Reid in [2] and by Schmidt in [10, 11]and by Starr and Turner III in [12], and planarity of the intersection graph ofa module over any ring was studied in [13].Here we study planarity of the intersection graph of a finite group. Let G bea group. By the intersection graph of G we mean an undirected graph withoutloops and multiple edges defined as follows: the vertex set is the set of allproper non-trivial subgroups of G, and there is an edge between two distinctvertices H and K if and only if H∩K 6= 1 where 1 denotes the trivial subgroupof G.We call a group planar if its intersection graph is planar. For any naturalnumbers m and n, we use C

In 2005 Kadiri proved that the Riemann zeta function ζ(s)does not vanish in the regionRe(s) ≥ 1 −1R 0 log|Im(s)|, |Im(s)| ≥ 2with R 0 = 5.69693. In this paper we will show that R 0 can be taken R 0 =5.68371 using Kadiri’smethod together with Platt’s numerical verificationof Riemann Hypothesis. 1. IntroductionIt is well known that the distribution of prime numbers is deeply relatedwith the zeros of the Riemann zeta function ζ(s). Let π(x) be the number ofprimes up to x. Hadamard and de la Vall´ee Poussin proved the prime numbertheorem which states that π(x) is asymptotic to Li(x) =R x2dtlogt . In 1899, dela Vall´ee Poussin proved that ζ(s) does not vanish in the regionRe(s) ≥ 1−1R 0 log|Im(s)|, |Im(s)| ≥ 2with R 0 = 34.82. From this zero-free region for ζ(s) the error term π(x)−Li(x)can be estimated:π(x) −Li(x) = O x exp −rlogxR 0 when x → ∞.See [18] and Section 1 of [7]. The constant R 0 has been improved by manyauthors, particularly by Rosser [13, 14, 15], Schoenfeld [14, 15], Stechkin [16],Ford [1, 2], and Kadiri [7]. Recently Kadiri improved significantly on R

A k-hypertournament H on n vertices, where 2 � kn, is a pair H = (V,A), where V is the vertex set of H and A is a set of k-tuples of vertices, called arcs, such that for all subsets SV with |S| = k, A contains exactly one permutation of S as an arc. Recently, Li et al. showed that any strong k-hypertournament H on n vertices, where 3 � kn 2, is vertex-pancyclic, an extension of Moon's theorem for tournaments. In this paper, we prove the following generalization of another of Moon's theorems: If H is a strong k-hypertournament on n vertices, where 3 � kn 2, and C is a Hamiltonian cycle in H, then C contains at least three pancyclic arcs.

Constacyclic codes of length p s over R = Fpm + uFpm + u2Fpm are precisely the ideals of the ring R(x) hxp s −1i . In this paper, we investigate constacyclic codes of length p s over R. The units of the ring R are of the forms , �+u�, �+u� +u 2 and �+u 2 , where �, � and are nonzero elements of Fpm. We obtain the structures and Hamming distances of all (�+u�)-constacyclic codes and (�+u�+u 2 )-constacyclic codes of length p s over R. Furthermore, we classify all cyclic codes of length p s over R, and by using the ring isomorphism we characterize -constacyclic codes of length p s over R.

We study the structure of idempotents in polynomial rings, power series rings, concentrating in the case of rings without identity. In the procedure we introduce right Insertion-of-Idempotents-Property (simply, right IIP) and right Idempotent-Reversible (simply, right IR) as generalizations of Abelian rings. It is proved that these two ring proper- ties pass to power series rings and polynomial rings. It is also shown that �-regular rings are strongly �-regular when they are right IIP or right IR. Next the noncommutative right IR rings, right IIP rings, and Abelian rings of minimal order are completely determined up to isomorphism. These results lead to methods to construct such kinds of noncommuta- tive rings appropriate for the situations occurred naturally in studying standard ring theoretic properties. 1. Definitions and notations Throughout this paper, R denotes an associative ring without identity, unless otherwise stated. Denote the n by n full (resp., upper triangular) matrix ring over R by Matn(R) (resp., Un(R)). We let eij denote the usual matrix units with 1 in the (i,j)-position and zeros elsewhere, if the base ring has identity 1. Denote {(aij) ∈ Un(R) | the diagonal entries of (aij) are all equal} by Dn(R). Zn denotes the ring of integers modulo n. GF(p n ) denotes the Galois field of order p n . J(R) denotes the Jacobson radical of R. | | denotes the cardinality. The characteristic of R is denoted by charR, and |a| denotes the order of a ∈ R in the additive subgroup of R generated by a. R + means the additive Abelian group (R,+). The polynomial ring with an indeterminate x over R is denoted by R(x). While speaking about minimal ring in a certain class of rings, we refer to a ring with minimal order for rings in that class, due to Xue (21). The notation (S) stands for the two-sided ideal of R generated by ∅ 6 S ⊆ R, and we also write (a1,...,an) in place of (S) for simplicity when S = {a1,...,an}. A ring is called Abelian if every idempotent is central. The zero in a nil ring is the only idempotent and so every nil ring is Abelian. The class of Abelian

Let M be a module over a commutative ring R, and let T(M)be its set of torsion elements. The total torsion element graph of Mover R is the graph T(Γ(M)) with vertices all elements of M, and twodistinct vertices m and n are adjacent if and only if m+ n ∈ T(M).In this paper, we study the basic properties and possible structures oftwo (induced) subgraphs Tor 0 (Γ(M)) and T 0 (Γ(M)) of T(Γ(M)), withvertices T(M) \{0}and M\{0}, respectively. The main purpose of thispaper is to extend the definitions and some results given in [6] to a moregeneral total torsion element graph case. 1. IntroductionThe concept of the graph of the zero-divisors of a ring was first introducedby Beck in [12] when discussing the coloring of a commutative ring. For thevertices of the graph, he takes all elements of a commutative ring R and twodistinct vertices a,b ∈ R are adjacent if ab = 0. There are many ways toassociate a graph to a given ring R. The most well-known is certainly thezero-divisor graph Γ(R) introduced in [9] whose vertices are the nonzero zero-divisors of R. Some properties of this graph may be found in [5] and [10]. In [8],Anderson and Badawi define, for a commutative ring R with nonzero identity,its total graph Γ(R). The set of vertices of this graph is R and two differentelements x,y ∈ R are adjacent if and only if x + y ∈ Z(R) which Z(R) is theset of all zero-divisors of R. For a recent generalization of this type of graphsee [7] and [11]. Let M be a module over a commutative ring R and let T(M)be the set of all torsion elements of M. In [14], the notion of the total torsionelement graph of a module over a commutative ring is introduced and denotedby T(Γ(M)), as the graph with all elements of M as vertices and for distinctm,n ∈ M, the vertices m and n are adjacent if and only if m+n ∈ T(M). Theycharacterize the girths and diameters of T(Γ(M)) and two (induced) subgraphs

Nielsen and Rege-Chhawchharia called a ring R right McCoy if given nonzero polynomials f(x), g(x) over R with f(x)g(x) = 0, there exists a nonzero element r 2 R with f(x)r = 0. Hong et al. called a ring R strongly right McCoy if given nonzero polynomials f(x),g(x) over R with f(x)g(x) = 0, f(x)r = 0 for some nonzero r in the right ideal of R generated by the coefficients ofg(x). Subsequently, Kim et al. observed similar conditions on linear polynomials by finding nonzero r's in various kinds of one-sided ideals generated by coefficients. But almost all results obtained by Kim et al. are concerned with the case of products of linear polynomials. In this paper we examine the nonzero annihilators in the products of general polynomials. A ring is usually called reduced if it has no nonzero nilpotent elements. Cohn (5) called a ring R reversible if ab = 0 implies ba = 0 for a,b ∈ R. Due to Nar- bonne (21), a ring R is called semicommutative if ab = 0 implies aRb = 0 for a,b ∈ R. Reduced rings are reversible and reversible rings are semicommuta- tive, but not conversely in general. Rege and Chhawchharia called R an Ar- mendariz ring (24, Definition 1.1) if whenever any polynomials f(x),g(x) ∈ R(x) satisfy f(x)g(x) = 0, then ab = 0 for each coefficienta of f(x) and b of g(x). Any reduced ring is Armendariz by (3, Lemma 1), but the class of semicom- mutative rings and the class of Armendariz rings don't imply each other by (24, Example 3.2) and (8, Example 14). McCoy (20) showed that if two poly- nomials annihilate each other over a commutative ring then each polynomial has a nonzero annihilator in the base ring. In (10), Weiner showed this fact fails in noncommutative rings. Based on this result, Nielsen (22) and Rege and Chhawchharia (24) called a noncommutative ring R right McCoy (resp., left McCoy) if whenever any nonzero polynomials f(x),g(x) ∈ R(x) satisfy f(x)g(x) = 0, then f(x)c = 0 (resp., cg(x) = 0) for some nonzero c ∈ R

【In this paper we first characterize the hereditarily hypercyclicity of the unilateral (or bilateral) weighted shifts on the spaces $L^2(\mathbb{N},\mathcal{K})$ (or $L^2(\mathbb{Z},\mathcal{K})$ ) with weight sequence { $A_n$ } of positive invertible diagonal operators on a separable complex Hilbert space $\mathcal{K}$ . Then we give the necessary and sufficient conditions for the supercyclicity of those weighted shifts, which extends some previous results of H. Salas. At last, we give some conditions for the supercyclicity of three different weighted shifts.】

Let R be a commutative Noetherian (not necessarily local) ring, I an ideal of R and M a finitely generated R-module. In this paper, by computing the local cohomology modules and Ext-modules via the injective resolution of M, we proved that, if for an integer t > 0, dim for for and >0. This shows that> is a finite set for . Also, we prove that if is M-sequences in dimension > k and are some positive integers. Here, for a subset T of Spec(R), set .

The term rank of a matrix A over a semiring S is the least number of lines (rows or columns) needed to include all the nonzero entries in A. In this paper, we characterize linear operators that preserve the sets of matrix ordered pairs which satisfy extremal properties with respect to term rank inequalities of matrices over nonbinary Boolean semirings.

It is known that being hierarchical is a necessary and suffi- cient condition for a poset to admit MacWilliams identity. In this paper, we completely characterize the structures of posets which have an associ- ation scheme structure whose relations are indexed by the poset distance between the points in the space. We also derive an explicit formula for the eigenmatrices of association schemes induced by such posets. By using the result of Delsarte which generalizes the MacWilliams identity for linear codes, we give a new proof of the MacWilliams identity for hierarchical linear poset codes.

In this paper, a finite difference scheme for a two-point boun-dary value problem of 2nth-order ordinary differential equations is pre-sented. The convergence and uniqueness of the solution for the schemeare proved by means of theories on matrix eigenvalues and norm. Nu-merical examples show that our method is very simple and effective, andthat this method can be used effectively for other types of boundary valueproblems. 1. IntroductionAssume that a 2nth-order (n ≥ 2) two-point boundary valve problem (BVP)is given in the formy (2n) = f(x)y +g(x), 0 < x < 1,y (2α) (0) = a α , α = 0,1,...,n−1,y (2α) (1) = b α ,(1.1)where f(x) and g(x) are continuous functions on the closed interval [0,1] anda α ,b α are all real constants. The problem (1.1) frequently occurs in structuralengineering, astrophysics and other branches of physical sciences. The problem(1.1) has a unique analytic solution for a fixed positive integer n ≥ 2 under thegeneral condition [5](1.2) f(x) 6= ( −1) n j 2n

A finite group Gis called an NSN-group if every proper sub-group of G is either normal in G or self-normalizing. In this paper, thenon-NSN-groups whose proper subgroups are all NSN-groups are deter-mined. 1. IntroductionThestructureofthe groupwhosesubgroupsareallnormal(calledaDedekindgroup or a Hamiltonian group) has been completely classified by R. Dedekind,E. Wendt and R. Bare (see [9, Theorem 5.3.7]). Since then, many authors havedealt with generalizations of such kind of groups. We mention some of themhere. Pic [8] considered finite groups in which every subgroup S is quasinor-mal, that is, S satisfies SH = HS for all subgroups H of G, and Walls [11]studied groups with maximal subgroups of Sylow subgroups that are normalin G. Buckley et al. [2] dealt with groups in which all subgroups form at mosttwo conjugate classes and Brandl [1] classified groups all of whose non-normalsubgroups are conjugates.If N is a normal subgroup of G, then N is normalized by all elements of G.For a normal subgroup, the number of elements of G normalizing N is up tomaximum. On the other hand, if N

A Krawtchouk polynomial is introduced as the classical Mac- Williams identity, which can be expressed in weight-enumerator-free form of a linear code and its dual code over a Hamming scheme. In this paper we find a new explicit expression for the p-number and the q- number, which are more generalized notions of the Krawtchouk poly- nomial in the P-polynomial schemes by using an extended version of a discrete Green's function. As corollaries, we obtain a new expression of the Krawtchouk polynomial over the Hamming scheme and the Eber- lein polynomial over the Johnson scheme. Furthermore, we find another version of the MacWilliams identity over a Hamming scheme.

For two positive integers m and n, we let ${\mathcal P}_n$ be the open convex cone in ${\mathbb R}^{n(n+1)/2}$ consisting of positive definite n x n real symmetric matrices and let ${\mathbb R}^{(m,n)}$ be the set of all m x n real matrices. In this article, we investigate differential operators on the non-reductive manifold ${\mathcal P}_n \times {\mathbb R}^{(m,n)}$ that are invariant under the natural action of the semidirect product group GL(n,R) $\ltimes {\mathbb R}^{(m,n)}$ on the Minkowski-Euclid space ${\mathcal P}_n \times {\mathbb R}^{(m,n)}$. These invariant differential operators play an important role in the theory of automorphic forms on GL(n,R) $\ltimes {\mathbb R}^{(m,n)}$ generlaizing that of automorphic forms on GL(n,R)., Comment: revised version : more new results and more open problems. page 3 : Correction of the Killing form and the inner product page 6 : Correction of the Killing form Several new materials added page 13 : This This paper has been withdrawn by the author due to a new revised version. Correction of typographical errors in Example 4.2 (pp. 11-12) page 14 : A reference added

Given an integer n ≥ 1 and a compact group G, we findsome restrictions for the probability that two randomly picked elementsx n and yof Gcommute. In the case n= 1 this notion was investigated byW. H. Gustafson in 1973 and its influence on the structure of the grouphas been studied in the researches of several authors in last years. 1. Generalizations of the commutativity degreeP. Erdo˝s and P. Tur´an introduced in [6] the ratio(1.1) d(G) =|{(x,y) ∈ G×G : [x,y] = 1}||G| 2 ,known as the probability that two randomly chosen elements of a finite group Gcommute. It is called the commutativity degree of G(see for instance [5, 7, 8,10, 16]). Several authors have successively investigated whether restrictions on(1.1) originate restrictions on the group structure. The answer is positive andthere are many results of classification in the quoted papers in the finite casebut few in the infinite case. The short note of W. H. Gustafson [11] shows thatit is meaningful to consider the infinite case, when we move in the categoryof compact groups. However the situation becomes different, since we needof tools which are not common in Group Theory but in Abstract HarmonicAnalysis and Topology. There are several works deal with word equations overcompact groups which show the influence that a word equation may have on thestructure of a group (see [1, 17, 18, 23]). Moreover, [2, 15] are recent advanceson the topic, where character theory involved.From this moment, we will always assume (up to explicit alert) to deal withcompact groups satisfying the Hausdorff separability axiom, since this is usualin the context of topological groups.

The term rank of a matrix A is the least number of lines (rows or columns) needed to include all the nonzero entries in A. In this paper, we obtain a characterization of linear transformations that preserve term ranks of matrices over antinegative semirings. That is, we show that a linear transformation T from a matrix space into another matrix space over antinegative semirings preserves term rank if and only if T preserves any two term ranks k and l.

In this paper we establish some limsup results and a gener- alized uniform law of the iterated logarithm (LIL) for the increments of partial sums of a strictly stationary and linearly positive quadrant depen- dent (LPQD) sequence of random variables.

A CW complex B is said to be I-trivial if there does not exist a Z2-map from Si 1 to S( ) for any vector bundle over B and any integer i with i > dim . In this paper, we consider the question of determining whether kRPn is I-trivial or not, and to this question we give complete answers when k 1;3;8 and partial answers when k = 1;3;8. A CW complex B is I-trivial if it is \W-trivial", that is, if for every vector bundle over B, all the Stiefel{Whitney classes vanish. We nd, as a result, that k RP n is a counterexample to the converse of this statement when k = 2;4 or 8 and n � 2k.

In this paper, we consider the regularity theory and the exis- tence of smooth solution of a degenerate fully nonlinear equation describ- ing the evolution of the rolling stones with nonconvex sides: { M ( h) = ht F ( t;z;z hzz) in f 0 < z 1 g (0 ;T ) ht( z;t) = H( hz( z;t) ;h) on fz = 0 g: We establish the Schauder theory for C 2; -regularity of h.

In the present paper we describe a class of Gorenstein, �nite and at morphism ϱ : X ! Y of degree 6 of algebraic varieties, called Anglo{American covers. We prove a general Bertini theorem for them and we give some evidence that the cover ϱ : A ! P2 associated general (1 ; 3){polarized abelian surface is Anglo{American.

In this paper we give asymptotic formulas for the number of l-cyclic extensions of the rational function eld k = F q(T ) with pre- scribed l-class numbers inside some cyclotomic function elds, and den- sity results for l-cyclic extensions of k with certain properties on the ideal class groups.

Assume G is a finite p-group and i is a fixed positive integer. In this paper, finite p-groups G with for all nonnormal subgroups H are classified up to isomorphism. As a corollary, this answer Problem 116(i) proposed by Y. Berkovich in his book "Groups of Prime Power Order Vol. I" in 2008.