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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 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.

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 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

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 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.】

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.

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 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 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.

M. Junger, G. Reinelt, and W. R. Pulleyblank asked the following questions ([2]). (1) Is it true that every simple planar 2-edge connected bipartite graph has a 3-partition in which each component consists of the edge set of a simple path? (2) Does every simple planar 2-edge connected graph have a 3-partition in which every component consists of the edge set of simple paths and triangles? The purpose of this paper is to provide a positive answer to the second question for simple outerplanar 2-vertex connected graphs and a positive answer to the first question for simple planar 2-edge connected bipartite graphs one set of whose bipartition has at most 4 vertices.

In this paper we establish limsup and liminf theorems for the increments of partial sum processes of a dependent station- ary Gaussian sequence.

In this paper we obtained the following : Let H be a Hilbert space and L be a subspace lattice on H. Let X and Y be operators acting on H. If the range of X is dense in H, then the following are equivalent: (1) there exists an operator A in AlgL such that AX = Y , (2) sup ‰ kE ? Y fk kE ? Xfk : f 2 H; E 2 L ae = K < 1: Moreover, if condition (2) holds, we may choose the operator A such that kAk = K.