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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 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)

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

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).

Approximate fixed point property problem for Mann itera-tion sequence of a nonexpansive mapping has been resolved on a Banachspace independent of uniform (strict) convexity by Ishikawa [Fixed pointsand iteration of a nonexpansive mapping in a Banach space, Proc. Amer.Math. Soc. 59 (1976), 65–71]. In this paper, we solve this problem fora class of mappings wider than the class of asymptotically nonexpan-sive mappings on an arbitrary normed space. Our results generalize andextend several known results. 1. IntroductionLet C be a nonempty subset of a normed space X and T : C → C be amapping. Then T is said to be(1) Lipschitzian if for each n ∈ N, there exists a positive real number k n such thatkT n x−T n yk ≤ k n kx−ykfor all x,y ∈ C,(2) uniformly k-Lipschitzian if T is Lipschitzian with sequence {k n } suchthat k n = k for all n ∈ N,(3) asymptotically nonexpansive ([8]) if T is Lipschitzian with sequence {k n }such that k n ≥ 1 for all n ∈ Nwith lim n→∞ k n = 1.Clearly, every nonexpansive mapping T (i.e., kTx − Tyk ≤ kx − yk forall x,y ∈ C) is asymptotically nonexpansive with sequence {1} and everyasymptotically nonexpansive mapping is uniformly k-Lipschitzian with k =sup

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

The aim of this paper is to study the Weyl type-theorems for the orthogonal direct sum S T, where S and T are bounded linear operators acting on a Banach space X. Among other results, we prove that if both T and S possesses property (gb) and if ( T) a(S), ( S) a(T), then S T possesses property (gb) if and only if SBF+ (S T) = SBF+ (S)( SBF+ (T). Moreover, we prove that if T and S both satises generalized Browder's theorem, then S T satises generalized Browder's theorem if and only if BW(S T) = BW(S) ( BW(T).

A globally convergent algorithm for solving equilibrium prob-lems is proposed. The algorithm is based on a proximal point algorithm(shortly (PPA)) with a positive definite matrix M which is not necessar-ily symmetric. The proximal function in existing (PPA) usually is thegradient of a quadratic function, namely, ▽(kxk 2M ). This leads to a prox-imal point-type algorithm. We first solve pseudomonotone equilibriumproblems without Lipschitzian assumption and prove the convergence ofalgorithms. Next, we couple this technique with the Banach contractionmethod for multivalued variational inequalities. Finally some computa-tional results are given. 1. IntroductionLet C be a nonempty closed convex subset in a real Euclidean space R n andf : C × C → R be a bifunction such that f(x,x) = 0 for every x ∈ C. Weconsider the following equilibrium problemshortly EP(f,C): Find x ∗ ∈ Csuch thatf(x ∗ ,y) ≥ 0 ∀y ∈ C.For the main results of this paper, we assume that:(A.1) f(·,·) is pseudomonotone on C,(A.2) f(x,·) is convex on C for all x ∈ C,(A.3) f is continuous on C ×C,(A.4) The solution set S of EP(f,C) is nonempty,(A.5) The matrix M is positive definite.Equilibrium problems appear frequently in many practical problems arising,for instance, physics, engineering, game theory, transportation, economics andnetwork (see [10, 13]). They become an attractive field for many researchersboth in theory and applications (see [2, 3, 5, 9, 15, 16, 17]). These problemsare models whose formulation includes optimization, variational inequalities,(vector) optimization problems, fixed point problems, saddle point problems

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

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

We present two kinds of construction methods for self-dualcodes over F 2 + u F 2 . Specially, the second construction (respectively,the rst one) preserves the types of codes, that is, the constructed codesfrom Type II (respectively, Type IV) is also Type II (respectively, TypeIV). Every Type II (respectively, Type IV) code over F 2 + u F 2 of freerank larger than three (respectively, one) can be obtained via the secondconstruction (respectively, the rst one). Using these constructions, weupdate the information on self-dual codes over F 2 + u F 2 of length 9 and 10,in terms of the highest minimum (Hamming, Lee, or Euclidean) weightand the number of inequivalent codes with the highest minimum weight. 1. IntroductionCodes over Z 4 have had great attention since the monumental paper [5],and self-dual codes over F 2 + u F 2 have also received some attention lately [1, 3]due to their importance. In fact, codes over Z 4 and F 2 + u F 2 are relatedto lattices, and they have Gray maps [3]. Even though they are similar toeach other, there are still some signi cant ff between them; one ofthe ff is that

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 establish a Marcinkiewicz-Zygmund type strong law for sequences of blockwise and pairwise m-dependent random variables. The sharpness of the results is illustrated by an example.

We improve some inequalities of Ostrowski-like type and further generalize them.Key words: Inequality, Error, Integral, Taylor, Ostrowski2000 MSC: 26D10, 41A55, 65D301. IntroductionIn 1938, Ostrowski [8] proved the following interesting integral inequality which has received con-siderable attention from many researchers.Theorem 1 (See [8]). Let f : [a,b] → R be continuous on [a,b] and differentiable on (a,b) whosederivative function f 0 : (a,b) → R is bounded on (a,b), i.e., kf 0 k ∞ = sup t∈(a,b) |f 0 (t)| < ∞. ThenZb f(x)−1b−a af(t)dt ∞5 14+x− a+b22 (b−a) 2 !(b−a)kf 0 k (1)for all x ∈ [a,b].This inequality gives an upper bound for the approximation of the integral average 1b−a R ba f(t)dtby the value f(x) at point x ∈ [a,b]. The first generalization of Ostrowskis inequality was given byG.V. Milovanovi´c and J.E. Peˇcari´c in [7]. However, note that estimate (1) can be applied only if f 0 is bounded. In the first part of this paper, we will improve (1) by assuming f 0 ∈ L p

Let {, } be a sequence of asymptotically almost negatively associated random variables and . In the paper, we get the precise results of Hjek-Rnyi type inequalities for the partial sums of asymptotically almost negatively associated sequence, which generalize and improve the results of Theorem 2.4-Theorem 2.6 in Ko et al. ([4]). In addition, the large deviation of for sequence of asymptotically almost negatively associated random variables is studied. At last, the Marcinkiewicz type strong law of large numbers is given.

In this paper we give a new characterization of real hyper- surfaces of type B, that is, a tube over a totally geodesicQP n in complex two-plane Grassmannians G2(C m+2 ), where m = 2n, with the Reeb vec- tor » belonging to the distribution D, where D denotes a subdistribution in the tangent space TxM such that TxM = D'D? for any point x 2 M and D? = Span{»1,»2,»3 }.

In this paper, we develop the idea of a generalized upper set in a BE-algebra. Furthermore, these sets are considered in the context of transitive and self distributive BE-algebras and their ideals, providing characterizations of one type, the generalized upper sets, in terms of the other type, ideals.

A new hard problem called the vector decomposition prob- lem (VDP) was recently proposed by Yoshida et al., and it was asserted that the VDP is at least as hard as the computational Di-e-Hellman problem (CDHP) under certain conditions. Kwon and Lee showed that the VDP can be solved in polynomial time in the length of the input for a certain basis even if it satisfles Yoshida's conditions. Extending our previous result, we provide the general condition of the weak instance for the VDP in this paper. However, when the VDP is practically used in cryptographic protocols, a basis of the vector space V is randomly chosen and publicly known assuming that the VDP with respect to the given ba- sis is hard for a random vector. Thus we suggest the type of strong bases on which the VDP can serve as an intractable problem in cryptographic protocols, and prove that the VDP with respect to such bases is di-cult for any random vector in V.

In this paper, we will deflne the Bergman projection type operator Pr and flnd conditions on which the operator Pr is bound-ed on L p (B;d"). By using the properties of the Bergman projection type operator Pr, we will show that if f 2 L p(B;d"), then (1i k w k 2 )rf(w) ¢ z 2 L p (B;d"). We will also show that if (1i k w k 2 ) rf(w)¢z hz;wi 2 L p (B;d"), then f 2 L p(B;d").

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.

In this paper, we obtain the general solution of a gen-eralized quadratic and additive type functional equation f(x + ay) + af(x - y) = f(x - ay) + af(x + y) for any integer a with a -1. 0, 1 in the class of functions between real vector spaces and investigate the generalized Hyers- Ulam stability problem for the equation.

In this paper we solve a generalized quadratic Jensen type functional equation m 2 f x + y + z m + f(x) + f(y) + f(z) = n 2 f x + y n + f y + z n + f z + x n and prove the stability of this equation in the spirit of Hyers, Ulam, Rassias, and G‚avruta.

In this paper, we prove the stability of a quadratic type functional equation a 2 f x + y + z a + a 2 f x i y + z a + a 2 f x + y i z a + a 2 f ix + y + z a = 4f(x) + 4f(y) + 4f(z)