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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

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, 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

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

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

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

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 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.

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

In this paper, we 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

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

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

This paper is devoted to investigate the singular directionsof meromorphic functions in some angular domains. We will confirm theexistence of Hayman T directions in some angular domains. This is acontinuous work of Yang [8] and Zheng [10]. 1. Introduction and main resultsLet f(z) be a meromorphic function on the whole complex plane. We willuse the standard notation of the Nevanlinna theory of meromorphic functions,such as T(r,f), N(r,f), m(r,f), δ(a,f). For the detail, see [7]. The order andlower order of it are defined as followsλ(f) = limsup r→∞ logT(r,f)lograndµ(f) = liminf r→∞ logT(r,f)logr.In view of the second fundamental theorem of Nevanlinna, Zheng [11] intro-duced a new singular direction, which is named T direction.Definition 1.1. A direction L : argz = θ is called a T direction of f(z) if forany e > 0, we havelimsup r→∞ N(r,Z e (θ),f = a)T(r,f)> 0for all but at most two values of a in the extended complex plane Cb. HereN(r,Ω,f = a) =Z r1 n(t,Ω,f = a)tdt, Received December 23, 2011; Revised May 23, 2014.2010 Mathematics Subject Classification. Primary 30D10; Secondary 30D20, 30B10,34M05.Key words and phrases. Hayman T direction, angular domain, P´olya peaks, order.

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

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.

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

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

【In this paper 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.】

InthispaperweestablishsomenewresultsonsumsofHilbertspace frames and Riesz bases. We also provide a correction to somerecentlyestablishedresultsin[2]. 1. IntroductionThroughout this paper, H denotes a separable Hilbert space with the innerproduct h·,· . Recall that a sequence {f i } i∈I ⊆ H is a frame for H if there exist0 < A 6 B < ∞ such that(1.1) Akfk 2 6X i∈I | f,f i i| 2 6 Bkfk 2 for all f ∈ H. The constants A and B are called a lower and upper framebounds, respectively.We call a sequence {f i } i∈I ⊆ H a Bessel sequence for H, if the right handinequality in (1.1) holds for all f ∈ H.Let {f i } i∈I be a Bessel sequence for H. Then the bounded operatorT : H → l 2 , Tf = {hf,f i i} i∈I is called the analysisoperator of {f i } i∈I and its adjointT ∗ : l 2 → H, T ∗ ({c i } i∈I ) =X i∈I c i f i ,is called the synthesisoperator of {f i } i∈I . If {f i } i∈I is a frame for H, the frameoperator for {f i } i∈I is the operator S : H → H given by Sf =P i∈I hf,f i if i .It is clear that hSf,fi =P

In this paper, we consider several convolution sums, namely, Ai(m;n; N) (i = 1; 2; 3; 4), Bj(m;n;N) (j = 1; 2; 3), and Ck(m;n;N) (k = 1; 2; 3;:::, 12), and establish certain identities involving their nite products. Then we extend these types of product convolution identities to products involving Faulhaber sums. As an application, an identity in- volving the Weierstrass }-function, its derivative and certain linear com- bination of Eisenstein series is established.

Let (M 3 ,g) be a 3-dimensional closed Sasakian spin mani- fold. Let Smin denote the minimum of the scalar curvature of (M 3 ,g). Let � + > 0 be the first positive eigenvalue of the Dirac operator of (M 3 ,g). We proved in (13) that if � + belongs to the interval � + 2 1 , 5 � , then � + satisfies � + � Smin+6 8 . In this paper, we remove the restriction "if � + belongs to the interval � + 2 ( 1 , 5 )" and prove

In this note, the q-extension of the twisted Lerch Euler zetafunctions considered by Jang [Bull. Korean Math. Soc. 47 (2010), no.6, 1181–1188] is further investigated, and the generalized multiplicationtheorem for the q-extension of the twisted Lerch Euler zeta functionsis given. As applications, some well-known results in the references arededuced as special cases. 1. IntroductionThroughout this paper, Z p , Q p and C p will denote the ring of p-adic rationalintegers, the field of p-adic rational numbers and the completion of the algebraicclosure of Q p , respectively. Let v p be the normalized exponential valuation ofC p with |p| p = p −v p (p) = p −1 . When one talks of q-extension, q is variouslyconsidered as an indeterminate, a complex number q ∈ Cor a p-adic numberq ∈ C p . If q ∈ C, one normally assume |q| < 1. If q ∈ C p , then we normallyassume |q − 1| p < p 11−p , so that q x = exp(xlogq) for each x ∈ Z p . Forf ∈ UD(Z p ,C p ) = {f | f : Z p → C p is the uniformly differentiable function},the p-adic q-integral (also be called as q-Volkenborn integration) is defined by(see [6, 13])(1.1) I

The main purpose of this paper is using the analytic methods and the properties of Gauss sums to study the computational problem of one kind mean value of the generalized Kloosterman sums and Dirichlet character of polynomials, and give an interesting asymptotic formula for it.

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

In this paper, we find the coefficients for the Weierstrass and (, , )-functions in terms of the arithmetic identities appearing in divisor functions which are proved by Ramanujan ([23]). Finally, we reprove congruences for the functions and in Hahn's article [11, Theorems 6.1 and 6.2].

Let G be a planar graph with maximum degree ∆. The linear 2-arboricity la2 ( G) of G is the least integer k such that G can be partitioned into k edge-disjoint forests, whose component trees are paths of length at most 2. In this paper, we prove that (1) la2 ( G) � ⌈ ∆ 2 ⌉ + 8 if G has no adjacent 3-cycles; (2) la2 ( G) � ⌈ ∆ 2 ⌉ + 10 if G has no adjacent 4-cycles; (3) la2 ( G) � ⌈ ∆ 2 ⌉ + 6 if any 3-cycle is not adjacent to a 4-cycle of G.

In this paper, we dene an Lp analytic generalized Fourier- Feynman transform and a convolution product of functionals in a Ba- nach algebra F ( Ca;b(0 ;T )) which is called the Fresnel type class, and in more general class FA1;A2 of functionals dened on general function space Ca;b(0 ;T ) rather than on classical Wiener space. Also we obtain some relationships between the Lp analytic generalized Fourier-Feynman transform and convolution product for functionals in F ( Ca;b(0 ;T )) and in FA1;A2 .

In this paper, we flnd the maximum exponent of D £ E, the cartesian product of two digraphs D and E on n, n + 2 vertices, respectively for an even integer n ‚ 4. We also characterize the extremal cases.

Let D be an integrally closed domain with quotient field K, ⁄ be a star operation on D, X,Y be indeterminates over D, N⁄ = {f 2 D(X)| (cD(f)) ⁄ = D} and R = D(X)N⁄. Let b be the b-operation on R, and let ⁄c be the star operation on D defined by I ⁄c = (ID(X)N⁄) b \K. Finally, let Kr(R,b) (resp., Kr(D,⁄c)) be the Kronecker function ring of R (resp., D) with respect to Y (resp., X,Y ). In this paper, we show that Kr(R,b) µ Kr(D,⁄c) and Kr(R,b) is a kfr with respect to K(Y ) and X in the notion of (2). We also prove that Kr(R,b) = Kr(D,⁄c) if and only if D is a P⁄MD. As a corollary, we have that if D is not a P⁄MD, then Kr(R,b) is an example of a kfr with respect to K(Y ) and X but not a Kronecker function ring with respect to K(Y ) and X.

Let Z n be the Cartesian product of the set of integers Z and let (Z,T) and (Z n ,T n ) be the Khalimsky line topology on Z and the Khalimsky product topology onZ n , respectively. Then for a set X ‰Z n , consider the subspace (X,T n X) induced from (Z n ,T n ). Considering a k-adjacency on (X,T n X), we call it a (computer topological) space with k-adjacency and use the notation (X,k,Tn X ) := Xn,k. In this paper we

In (6) and (7), we introduced graph von Neumann algebras which are the (groupoid) crossed product algebras of von Neumann al- gebras and graph groupoids via groupoid actions. We showed that such crossed product algebras have the graph-depending amalgamated reduced free probabilistic properties. In this paper, we will consider a scalar- valued W ⁄ -probability on a given graph von Neumann algebra. We show that a diagonal graph W ⁄ -probability space (as a scalar-valued W ⁄ - probability space) and a graph W ⁄ -probability space (as an amalgamated W ⁄ -probability space) are compatible. By this compatibility, we can find the relation between amalgamated free distributions and scalar-valued free distributions on a graph von Neumann algebra. Under this compat- ibility, we observe the scalar-valued freeness on a graph von Neumann algebra.

For any integer k ‚ 2, let P(c,k + 1;q) be the number of all k+1-tuples with positive integer coordinates (a1,a2,...,ak+1) such that 1 • aiq, (ai,q) = 1, a1a2 ···ak+1 · c (modq) and 2 - (a1 + a2 + ···+ak+1), and E(c,k+1;q) = P(c,k+1;q)i ` k(q) 2 . The main purpose of this paper is using the properties of Gauss sums, primitive characters and the mean value theorems of Dirichlet L-functions to study the hybrid mean value of the r-th hyper-Kloosterman sums Kl(h,k + 1,r;q) and E(c,k + 1;q), and give an interesting mean value formula.

In this paper, we define generalized Fourier-Hermite func- tionals on a function space Ca,b(0, T) to obtain a complete orthonormal set in L2(Ca,b(0, T)) where Ca,b(0, T) is a very general function space. We then proceed to give a necessary and sufficient condition that a func- tional F in L2(Ca,b(0, T)) has a generalized Fourier-Wiener function space transform F√ 2,i (F) also belonging to L2(Ca,b(0, T)).

In this paper, by estimating small ball probabilities of l ∞ - valued Gaussian processes, we investigate Chung-type law of the iter- ated logarithm of l ∞ -valued Gaussian processes. As an application, the Chung-type law of the iterated logarithm of l∞-valued fractional Brown- ian motion is established.

Let m be a positive integer. We show that for any given real number ∈ (0; 1) and complex number with | | ≤ 1 which satisfy e2i m 1, there exists a Blaschke product B of degree 2m + 1 which has a fixed point of multiplier m at the point at infinity such that the restriction of the Blaschke product B on the unit circle is a critical circle map with rotation number . Moreover if the given real number is irrational of bounded type, then a modified Blaschke product of B is quasiconformally conjugate to some rational function of degree m + 1 which has a fixed point of multiplier m at the point at infinity and a Siegel disk whose boundary is a quasicircle containing its critical point. point of f and λ = f ' (z0) is called the multiplier of z0 if z0 ∈ C. The multiplier of z0 = ∞ is defined as the multiplier of the origin for ψ ◦ f ◦ ψ −1 , where ψ(z) = 1/z. The fixed point z0 is attracting, repelling or indifferent if its multiplier λ satisfies that |λ| 1 or |λ| = 1 respectively. Attracting fixed points belong to the Fatou set and repelling fixed points belong to the Julia set. In the case that z0 is indifferent, the classification is more complicated. The fixed point z0 is parabolic, a Siegel point or a Cremer point if its multiplier is a root of unity, z0 ∈ F (f ) or z0 ∈ J(f ) respectively. Parabolic fixed points belong to the Julia set. The Fatou component containing a Siegel point is called a Siegel disk centered at z0. Non-repelling fixed points "capture" at least one critical point of f , which is a solution of the equation f ' (z) = 0. In this paper, we investigate rational functions with Siegel disks. Let f

Main purpose of this paper is to combine the optimal form of Fan's best approximation theorem and Nash's equilibrium existence theorem into a single existence theorem simultaneously. For this, we first prove a general best proximity pair theorem which includes a number of known best proximity theorems. Next, we will introduce a new equilibrium concept for a generalized Nash game with normal form, and as applications, we will prove new existence theorems of Nash equilibrium pairs for generalized Nash games with normal form.

In this paper, we study the mean values of the homogeneous Dedekind sums and Cochrane sums in short intervals [ 1, p 3 ] and [ 1, p 4 ] , and give some asymptotic formulae by using the mean values of the Dirichlet L-functions. §

In this paper, we have characterizations of idempotent matrices over general Boolean algebras and chain semirings. As a consequence, we obtain that a fuzzy matrix A = [ai,j ] is idempotent if and only if all ai,j-patterns of A are idempotent matrices over the binary Boolean algebra B1 = {0, 1}. Furthermore, it turns out that a binary Boolean matrix is idempotent if and only if it can be represented as a sum of line parts and rectangle parts of the matrix.

The domination number γ(G) of a graph G = (V, E) is the minimum cardinality of a subset of V such that every vertex is either in the set or is adjacent to some vertex in the set. The bondage number of b(G) of a graph G is the cardinality of a smallest set of edges whose removal from G results in a graph with domination number greater than γ(G). In this paper, we calculate the bondage number of the Cartesian product of cycles C3 and Cn for all n.

The main purpose of this paper is to use the properties of primitive characters, Gauss sums and Ramanujan’s sum to study the hybrid mean value of generalized Bernoulli numbers, general Kloosterman sums and Gauss sums, and give two asymptotic formulae.

The main purpose of this paper is using the properties of Gauss sums, primitive characters and the mean value theorems of Dirichlet L-functions to study the hybrid mean value of the r-th hyper-Kloosterman sums Kl(h;k+1;r;q) and the hyper Cochrane sums C(h;q;m;k); and give an interesting mean value formula.

In this paper, we establish limit theorems containing both the moduli of continuity and the large incremental results for finite dimensional Gaussian processes with N parameters, via estimating upper bounds of large deviation probabilities on suprema of the Gaussian processes.

In this paper we establish some results on the increments of a d-dimensional Gaussian process with the usual Euclidean norm. In particular we obtain the law of iterated logarithm and the Book-Shore type theorem for the increments of ad-dimensional Gaussian process, via estimating upper bounds and lower bounds of large deviation probabilities on the suprema of the d-dimensional Gaussian process.

Let F be a non-trivial non-Archimedian field. The sequence spaces were defined and studied by Soma-sundaram[4], where these spaces denote the spaces of entire and analytic sequences defined over F, respectively. In 1997, these spaces were generalized by Mursaleen and Qamaruddin[1] by considering an arbitrary sequence . They characterized some classes of infinite matrices considering these new classes of sequences. In this paper, we further generalize the above mentioned spaces and define the spaces ), and ). We also study some matrix transformations on these new spaces.

In this paper, based on large deviation probabilities on Gaussian random vectors, we obtain Strassen's functional LIL for d-dimensional self-similar Gaussian process in Holder norm via estimating large deviation probabilities for d-dimensional self-similar Gaussian process in Holder norm.

In this paper, using Gauss multiplication formula, a recurrence relation for Bernoulli numbers, generalizing Namias’ results, is given.