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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 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

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

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)

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

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

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 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

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)

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 <

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

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.

A Cayley map is a 2-cell embedding of a Cayley graph into an orientable surface with the same local orientation induced by a cyclic permutation of generators at each vertex. In this paper, we provide classifications of prime-valent regular Cayley maps on abelian groups, dihedral groups and dicyclic groups. Consequently, we show that all prime-valent regular Cayley maps on dihedral groups are balanced and all prime-valent regular Cayley maps on abelian groups are either balanced or anti-balanced. Furthermore, we prove that there is no prime-valent regular Cayley map on any dicyclic group. In this paper, we only consider undirected finite connected graphs without loops and multiple edges. For a simple graph i, an arc of i is an ordered pair (x,y) of adjacent vertices of i. Thus, every edge of i gives rise to a pair of opposite arcs. By V (i), E(i), D(i) and Aut(i), we denote the vertex set, the edge set, the arc set and the automorphism group of i, respectively. A graph i is said to be vertex-transitive, edge-transitive and arc-transitive if Aut(i) acts transitively on the vertex set, the edge set and the arc set of i, respectively. A graph i is one-regular if Aut(i) is arc-transitive and the stabilizer of each arc in Aut(i) is trivial. We consider Aut(i) as an acting group on V (i), E(i) and D(i) according to the context. For a simple graph i with arc set D, an embedding of i or a map with the underlying graph i is a triple M = (D;R,L), where R is a permutation of D whose orbits coincide with the sets of arcs based at the same vertex and L is an involution of D whose orbits are the pairs of arcs induced by the same edge. The permutations R and L are called a rotation and an arc-reversing involution of M, respectively. Let Mon(M) be the permutation group hR,Li generated by R and L and call it a monodromy group of M. Then, Mon(M) acts transitively

There are two well known reduction formulae for structural constants of the cohomology ring of Grassmannians, i.e., LittlewoodRichardson coefficients. Two reduction formulae are a conjugate pair in the sense that indexing partitions of one formula are conjugate to those of the other formula. A nice bijective proof of the first reduction formula is given in the authors’ previous paper while a (combinatorial) proof for the second reduction formula in the paper depends on the identity between Littlewood-Richardson coefficients of conjugate shape. In this article, a direct bijective proof for the second reduction formula for Littlewood-Richardson coefficients is given. Our proof is independent of any previously known results (or bijections) on tableaux theory and supplements the arguments on bijective proofs of reduction formulae in the authors’ previous paper.

In [8], it is shown that the complete multipartite graph Kn(2t) having n partite sets of size 2t, where n ≥ 6 and t ≥ 1, has a decomposition into gregarious 6-cycles if n ≡ 0, 1, 3 or 4 (mod 6). Here, a cycle is called gregarious if it has at most one vertex from any particular partite set. In this paper, when n ≡ 0 or 3 (mod 6), another method using difference set is presented. Furthermore, when n ≡ 0 (mod 6), the decomposition obtained in this paper is ∞-circular, in the sense that it is invariant under the mapping which keeps the partite set which is indexed by ∞ fixed and permutes the remaining partite sets cyclically.

In this paper, we investigate bounded matrices over regular rings. We observe that every bounded matrix over a regular ring can be described by idempotent matrices and invertible matri- ces. Let A;B 2 Mn(R) be bounded matrices over a regular ring R. We prove that (AB) d = U(BA) d U i1 for some U 2 GLn(R). Let I be an ideal of a ring R. We say that I is a bounded ideal of R in case there exists a positive integer m such that x m = 0 for all nilpotent x 2 I. We say that A 2 Mn(R) is a bounded matrix provided that Mn(R)AMn(R) is a bounded ideal of Mn(R). An element x 2 R is regular if there exists y 2 R such that x = xyx. A ring R is said to be regular in case every element in R is regular. In this paper, we investigate bounded matrices over regular rings. We observe that every bounded matrix over a regular ring can be described by idempotent matrices and invertible matrices. Let A;B 2 Mn(R) be bounded matrices over a regular ring R. It is shown that (AB) d = U(BA) d U i1 for some

Suppose that an inflnite graph G of bounded degree has flnite number of ends, each of which is p-regular, where 1 < p < 1. Then we can identify all the positive (bounded, respectively) p-harmonic functions on G. In this paper, we study the Liouville type property on a graph. By the Liouville property on a graph G, we mean that every bounded har- monic function on G is constant. It immediately follows that the set of all bounded harmonic functions on G having Liouville property is in one to one correspondence with the real line R. It seems natural to regard the case that the set of all bounded harmonic functions on G is in one to one correspondence with R l for some positive integer l as a general- ized version of the Liouville property. In line with this view point, we consider the case of the p-Laplacian operator (1 < p < 1) and the posi- tive (bounded, respectively) p-harmonic functions on graphs of bounded degree. In the case that a graph G has a flnite number of ends, each of which is p-regular, we identify all the positive (bounded, respectively) p-harmonic functions on G in section 4. On the other hand, Kanai(5) proved that if G and H are roughly isometric graphs of bounded degree, then H is parabolic whenever G is. In (9), Soardi proved that if G and H are roughly isometric graphs of bounded degree, and if G has no nonconstant harmonic functions with flnite energy, then neither has H. Later, the second author (8) proved that the dimension of the space of harmonic functions with flnite energy is preserved under rough isometries between graphs of bounded degree. In this paper, we also discuss its rough isometric invariance in section 3

In this paper we will show that if there exist deriva- tions D, G on a n!-torsion free semi-prime ring R such that the mapping D 2 + G is n-commuting on R, then D and G are both commuting on R. And we shall give the algebraic conditions on a ring that a Jordan derivation is zero. Throughout this paper R will be represent an associative ring with center Z(R). The commutator xy i yx (resp. the Jordan product xy + yx) will be denoted by (x;y) (resp. hx;yi). We make extensive use of the basic identities (xy;z) = (x;z)y + x(y;z), (x;yz) = (x;y)z + y(x;z). Let rad(A) denote the (Jacobson) radical of an algebra A. Recall that R is prime if aRb = (0) implies that either a = 0 or b = 0, and is semi-prime if aRa = (0) implies a = 0. An additive mapping D from R to R is called a derivation if D(xy) = D(x)y + xD(y) holds for all x;y 2 R. A derivation D is inner if there exists a 2 R such that D(x) = (a;x) holds for all x 2 R. And also, an additive mapping D from R to R is called a Jordan derivation if D(x 2 ) = D(x)x + xD(x) holds for all x 2 R. An additive mapping F from R to R is said to be a commuting (resp. centralizing) if (F(x);x) = 0 (resp. (F(x);x) 2 Z(R)) holds for all x 2 R. More generally, for a positive integer n, we define a mapping F to be n-commuting if (F(x);x n ) = 0 for all x 2 R. The underlying idea of our research is Posner's second theorem (7, Theorem 2) which is the beginning of the study concerning centralizing and commuting mappings, which states that the existence of a nonzero

In this paper we prove that every Jordan ?? -derivation on aLie triple system is a ?? -derivation. Specially, we conclude that everyJordan derivation on a Lie triple system is a derivation. 1. IntroductionThe concept of Lie triple system was ???rst introduced by N. Jacobson [2, 3](see also [4]). We recall that a Lie triple system is a vector space J over a ???eldKwith a trilinear mapping J ??J ??J 3 ( x,y,z ) 7??? [ x,y,z ] ??? J satisfyingthe following axioms(i) [ x,y,z ] = ??? [ y,x,z ] , (ii) [ x,y,z ]+[ y,z,x ]+[ z,x,y ] = 0 , (iii) [ u,v, [ x,y,z ]] = [[ u,v,x ] ,y,z ]+[ x, [ u,v,y ] ,z ]+[ x,y, [ u,v,z ]]for all u,v,x,y,z ??? J. It follows from (i) that [ x,x,y ] = 0 for all x,y ??? J. It is clear that every Lie algebra with product [ .,. ] is a Lie triple systemwith respect to [ x,y,z ] := [[ x,y ] ,z ] . Conversely, any Lie triple system J canbe considered as a subspace of a Lie algebra (Bertram [1], Jacobson [3]).Throughout this paper, let Cbe the complex ???led and

In this paper we obtain the closed forms of some hypergeometric functions. As an application, we obtain the explicit forms of the Bergman kernel functions for intersection of two complex ellipsoids $\{z \in \mathbb{C}^3 \colon |z_1|^p + |z_2|^q < 1, \quad |z_1|^p + |z_3|^r < 1\}$. We consider cases $p=6, q= r= 2$ and $p=q=r=2$. We also investigate the Lu Qi-Keng problem for $p=q=r=2$., 19 pages. arXiv admin note: text overlap with arXiv:0810.2632 by other authors

In this paper we begin the study of Lk-2-type hypersurfaces of a hypersphere S n+1 � R n+2 for k � 1. Let : M 3 ! S 4 be an ori- entable Hk-hypersurface, which is not an open portion of a hypersphere. ThenM3 is ofLk-2-type if and only ifM3 is a Clifford tori S1(r1)×S2(r2), r2 +r 2 = 1, for appropriate radii, or a tube T r(V 2) of appropriate con- stant radiusr around the Veronese embedding of the real projective plane RP 2( p 3).

Let G be a finite group and nse(G) be the set of numbersof elements of G of the same order. In this paper, we prove that thesimple group Sz(2 2m+1 ), where 2 2m+1 −1 is a prime number, is uniquelydetermined by nse(Sz(2 2m+1 )) and |Sz(2 2m+1 )|. 1. IntroductionFor a finite group G, by π(G) we denote the set of primes q such that Gcontains an element of order q and by π e (G) we mean the set of element ordersof G. Let i ∈ π e (G) and m i = m i (G) be the number of elements of order i inG. Then we denote the set of numbers of elements of G of the same order bynse(G), that is; nse(G) = {m i | i ∈ π e (G)}. The prime graph of group G whichis denoted by Γ(G) is a graph with vertex-set π(G), two distinct vertices u andv are adjacent if and only if uv ∈ π e (G). The number of connected componentsof Γ(G) is denoted by t(G) and the set of the connected components is denotedby {π i | i = 1,2,...,t(G)}. If G is a group of even order, then we alwaysassume that 2 ∈ π 1 .The characterizationby nse is one of the problems related to the Thompson’sproblem (Problem 12.37 of [10]) that posed by Shao and et al.. In [12], theyproved that if G is a simple K

Let x : M n−1 → R n (n ≥ 4) be an umbilical free hyper-surface with non-zero principal curvatures. Then x is associated witha Laguerre metric g, a Laguerre tensor L, a Laguerre form C, and aLaguerre second fundamental form B, which are invariants of x underLaguerre transformation group. We denote the Laguerre scalar curvatureby R and the trace-free Laguerre tensor by L˜ := L− 1n−1 tr(L)g. Inthis paper, we prove a local classification result under the assumption ofparallel Laguerre form and an inequality of the typekL˜k ≤ cR,where c= 1(n−3) √ (n−2)(n−1) is appropriate real constant, depending onthe dimension. 1. IntroductionLet x : M n−1 → R n be an umbilical free hypersurface with non-zero princi-pal curvatures. Let ξ : M → S n−1 be its unit normal. Let {e 1 ,e 2 ,...,e n−1 }be the orthonormal basis for TM with respect to dx · dx, consisting of unitprincipal vectors. Let r i = 1k i , r = r 1 +r 2 +···+r n−1 n−1 be the curvature radius andmean curvature radius of x respectively, where k

A k-total-coloring of a graph G is a coloring of using k colors such that no two adjacent or incident elements receive the same color. The total chromatic number of G is the smallest integer k such that G has a k-total-coloring. Let G be a planar graph with maximum degree . In this paper, it`s proved that if and G does not contain adjacent 5-cycles, then the total chromatic number is .

In this paper, we classify all twisted torus knots which are doubly middle Seifert-fibered. Also we show that all of these knots possibly except a few admit Dehn surgery producing a non-Seifert-fibered graph manifold which consists of two Seifert-fibered spaces over the disk with two exceptional fibers, glued together along their boundaries. This provides another infinite family of knots in admitting Dehn surgery yielding such manifolds as done in [5].

In this paper, we give explicit and new identities for theBernoulli numbers of the second kind which are derived from a non-lineardifferential equation. 1. IntroductionFor r ∈ N, the Bernoulli polynomials of order r are defined by the generatingfunction to be te t −1 r e xt = te t −1 ×···× te t −1 | {z } r-times (1) e xt= X ∞n=0 B (r)n (x)t n n!, (see [1]-[16]).When x = 0, B (r)n = B (r)n (0) are called the Bernoulli numbers of order r.As is well known, the Bernoulli polynomials of the second kind are given bythe generating function to be(2)tlog(1+t)(1+t) x =X ∞n=0 b n (x)t n n!, (see [3, 5, 7, 14]).Indeed, b n (x) = B (n)n (x +1).When x = 0, b n = b n (0) are called the Bernoulli numbers of the secondkind.The first few Bernoulli numbers of the second kind are b 0 = 1, b 1 = 12 ,b 2 = − 16 , b 3 = 14 , b 4 = − 1930 , b 5 = 94 , .... Received October 8, 2014; Revised May 12, 2015.2010 Mathematics Subject Classification. 05A19,11B68,34A34.Key words and phrases. Bernoulli numbers of second kind, non-linear differentialequation.

A cyclotomic polynomial Φ n (x) is said to be ternary if n=pqr for three distinct odd primes p < q < r. Let A(n) be the largestabsolute value of the coefficients of Φ n (x). If A(n) = 1 we say that Φ n (x)is flat. In this paper, we classify all flat ternary cyclotomic polynomialsΦ pqr (x) in the case q≡±1 (mod p) and 4r≡±1 (mod pq). 1. IntroductionLetΦ n (x) =Y n k=1 (k,n)=1 (x −e 2πikn ) = φ X ( )j=0 a(n,j)x j be the n-th cyclotomic polynomial, where φ is the Euler totient function. Itcan be shown that a(n,j) ∈ Z. LetA(n) = max{|a(n,j)| | 0 ≤ j ≤ φ(n)}denote the largest absolute value of the coefficients of Φ n (x). If A(n) = 1 wesay that Φ n (x) is flat. It turns out that for the purpose of determining A(n),it suffices to consider squarefree and odd integers n. Clearly, if n has at mosttwo distinct odd prime factors, then A(n) = 1.However, the coefficients of ternary cyclotomic polynomials Φ pqr (x), wherep < q < r areodd primes, become much morecomplicated, such asa(3·5·7,7) =−2 and a(5·7·11,119) = −3. The investigation about the coefficients of ternarycyclotomic polynomials have a long history and there are many references onthis subject, see, for instance, [1-17, 19, 20, 21, 23, 24]. One interesting openproblem involving this topic is to give a complete characterization of all flatternary cyclotomic polynomials, but this appears very difficult. Throughout

The coset diagrams are composed of fragments, and the frag- ments are further composed of circuits at a certain common point. A condition for the existence of a certain fragment of a coset diagram in a coset diagram is a polynomial f in Z(z): In this paper, we answer the question: how many polynomials are obtained from the fragments, evolved by joining the circuits (n;n) and (m;m); where n < m, at all points.

For , let denote the class of locally univalent normalized analytic functions in the unit disk satisfying the condition >. In the present paper, we shall obtain the sharp upper bound for Fekete- functional for the complex parameter .

Let A and B be two unital C ∗ -algebras. Denote by W(a)the numerical range of an element a∈ A. We show that the conditionW(ax) = W(bx),∀x∈ A implies that a= b. Using this, among otherresults, it is proved that if φ : A → B is a surjective map such thatW(φ(a)φ(b)φ(c)) = W(abc) for all a,band c∈A, then φ(1) ∈Z(B) andthe map ψ= φ(1) 2 φis multiplicative. 1. IntroductionLet A be a C ∗ -algebra with unit 1 and let S(A) be the state space of A, i.e.,S(A) = {ϕ∈ A ′ : ϕ≥ 0,ϕ(1) = 1} (here A ′ is the topological dual of A). Foreach a∈ A, the algebraic numerical range V(a) and numerical radius v(a) aredefined byV(a) = {f(a) : f∈ S(A)} and v(a) = sup z∈V (a) |z|.By the Gelefand-Naimark theorem, every C ∗ -algebra may be viewed as a closed∗-subalgebra of B(H) where B(H) denotes the algebra of all bounded linearoperators acting on a Hilbert space H. It is well known that V(a) is the closureof W(a) and v(a) = w(a) = sup λ∈W(a) |λ|, where W(a) = {(at,t) : t∈ H,ktk = 1}and (,) denotes the inner product. Here W(a) is called the usual numericalrange of the operator a.In the last few decades, there has been a considerable interest in the problemof characterization of maps that preserves the numerical range or the numericalradius, see for instance the papers [4, 12, 13, 15] and the references therein.Notice that, based on the aforesaid, preserving the usual numerical range Wimplies the preservation of the spacial numerical range V. Therefore, we willconcentrate our study henceforth on W. Recently, Hou and Di described in [9]surjective maps on the algebra B(H) which preserves the numerical range of

Generalized Fibonacci and Lucas sequences (U-n) and (V-n) are defined by the recurrence relations Un+1 = PUn +QU(n-1) and Vn+1 = PVn + QV(n-1), n >= 1, with initial conditions U-0 = 0, U-1 = 1 and V-0 = 2, V-1 = P. This paper deals with Fibonacci and Lucas numbers of the form U-n (P, Q) and V-n (P, Q) with the special consideration that P >= 3 is odd and Q = -1. Under these consideration, we solve the equations V-n = 5kx(2), V-n = 7kx(2), V-n = 5kx(2)+/- 1, and V-n = 7kx(2)+/- 1 when k vertical bar P with k > 1. Moreover, we solve the equations V-n = 5x(2)+/- 1 and V-n = 7x(2 +/-)1.

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

Let D be a division ring and w() be a generalized group monomial over . In this paper, we investigate subnormal subgroups and maximal subgroups of which satisfy the identity .