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In this paper we find a suitable bound for the number of commutators which is required to express every element of the derived group of a solvable group satisfying the maximal condition for normal subgroups. The precise formulas for expressing every element of the derived group to the minimal number of commutators are given.

In [2] it was shown that a 1-generator quasi-cyclic code C of length n = ml of index l over is free if C is generated by a polynomial which divides . In this paper, we prove that a necessary and sufficient condition for a cyclic code over of length m to be free is that it is generated by a polynomial which divides . We also show that this can be extended to finite local rings with a principal maximal ideal.

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.

As a continuation of computing the zeta function of a regular covering graph by Mizuno and Sato in [9], we derive in this paper computational formulae for the zeta functions of a graph bundle and of any (regular or irregular) covering of a graph. If the voltages to derive them lie in an abelian or dihedral group and its fibre is a regular graph, those formulae can be simplified. As a by-product, the zeta function of the cartesian product of a graph and a regular graph is obtained. The same work is also done for a discrete torus and for a discrete Klein bottle.

Let Xt be an m-dimensional linear process deflned by Xt = P 1=0 Aj Ztij; t = 1;2;:::, where fZtg is a sequence of m-dimensional random vectors with mean 0 : m £ 1 and positive deflnite covariance matrix i : m £ m and fAjg is a sequence of coe-cient matrices. In this paper we give su-cient conditions so that P(ns) t=1 Xt (properly normalized) converges weakly to Wiener measure if the corresponding result for P (ns) t=1 Zt is true.

Hypermaps are cellular embeddings of hypergraphs in compact and connected surfaces, and are a generalisation of maps, that is, 2-cellular decompositions of closed surfaces. There is a well known correspondence between hypermaps and co-compact subgroups of the free product . In this correspondence, hypermaps correspond to conjugacy classes of subgroups of , and hypermap coverings to subgroup inclusions. Towards the end of [9] the authors studied regular hypermaps with extra symmetries, namely, G-symmetric regular hypermaps for any subgroup G of the outer automorphism Out of the triangle group . This can be viewed as an extension of the theory of regularity. In this paper we move in the opposite direction and restrict regularity to normal subgroups of of finite index. This generalises the notion of regularity to some non-regular objects.

The Property P conjecture states that the 3-manifold Yr obtained by Dehn surgery on a non-trivial knot in S 3 with surgery coe-cient r 2 Q has the non-trivial fundamental group (so not simply connected). Recently Kronheimer and Mrowka pro- vided a proof of the Property P conjecture for the case r = §2 that was the only remaining case to be established for the conjecture. In particular, their results show that the two phenomena of having a cyclic fundamental group and having a homomorphism with non- cyclic image in SU(2) are quite difierent for 3-manifolds obtained by Dehn flllings. In this paper we extend their results to some other Dehn surgeries via the A-polynomial, and provide more evidence of the ubiquity of the above mentioned phenomena.

The purpose of this paper is to describe the structure of the rings with a monic polynomial and for some nonnegative integer . Especially we will see that any ideal of such rings can be generated by at most two elements of the special form and we will find the 'minimal' set of generators of the ideals. We indicate how to identify the isomorphism types of the ideals as by finding the isomorphism types of the ideals of some particular ring. Also we will find the annihilators of the ideals by finding the most 'economical' way of annihilating the generators of the ideal.

Sung et al. [13] obtained a WLLN (weak law of large numbers) for the array of random variables under a Cesaro type condition, where and large two sequences of integers. In this paper, we extend the result of Sung et al. [13] to a martingale type p Banach space

In this paper we construct q-Genocchi numbers and polynomials. By using these numbers and polynomials, we inves- tigate the q-analogue of alternating sums of powers of consecutive integers due to Euler.

In this paper we prove that if AX=b is a partial sign-solvable linear system with A being sign non-singular matrix and if , then is a sign-solvable linear system, where denotes the submatrix of A occupying rows and columns in o and xo and be are subvectors of x and b whose components lie in . For a sign non-singular matrix A, let be the fully indecomposable components of A and let denote the set of row numbers of . We also show that if is a partial sign-solvable linear system, then, for , if one of the components of xor is a fixed zero solution of Ax=b, then so are all the components of .

In this paper, we give an alternative proof that the number of doubly alternating Baxter permutations is Catalan.

A real matrix A is called a sign-central matrix if for, every matrix with the same sign pattern as A, the convex hull of columns of contains the zero vector. A sign-central matrix A is called a tight sign-central matrix if the Hadamard (entrywise) product of any two columns of A contains a negative component. A real vector x = is called stable if . A tight sign-central matrix is called a sign-central matrix if each of its columns is stable. In this paper, for a matrix B, we characterize those matrices C such that [B, C] is tight () sign-central. We also construct the matrix C with smallest number of columns among all matrices C such that [B, C] is sign-central.

In this paper, we extend the concept of the group of self homotopy equivalences of a space X to that of an object in the category of pairs. Mainly, we study the group of pair homotopy equivalences from a CW-pair (X, A) to itself which is the special case of the extended concept. For a CW-pair (X, A), we find an exact sequence where G is a subgroup of . Especially, for CW homotopy associative and inversive H-spaces X and Y, we obtain a split short exact sequence provided the two sets and [X, Y] are trivial.

John Harer conjectured that the canonical map from braid group to mapping class group induces zero homology homo- morphism. To prove the conjecture it su-ces to show that this map preserves the flrst Araki-Kudo-Dyer-Lashof operation. To get information on this homology operation we need to investigate the image of braiding under the Harer map. The main result of this paper is to give both algebraic and geometric interpretations of the image of braiding under the Harer map. For this we need to calculate long chains of consecutive actions of Dehn twists on the fundamental group of surface.

In this paper, we consider cooperative games arising from integer domination problem on graphs. We introduce two games, game and its monotonic relaxed game, and focus on their cores. We first give characterizations of the cores and the relationship between them. Furthermore, a common necessary and sufficient condition for the balancedness of both games is obtained by making use of the technique of linear programming and its duality.

In this paper, we prove that the topological entropy of a sequence of equi-continuous monotone maps on circles is . As applications, we give the estimation of the entropies for some skew products on annular and torus. We also show that a diffeomorphism f on a smooth 2-dimensional closed manifold and its extension on the unit tangent bundle have the same entropy.

Let G be a symmetric group. In this paper we describe a method that for a certain irreducible characterof G it flnds a subgroup H such that the restriction ofon H has a linear con- stituent with multiplicity one. Then using a well known algorithm we can construct a representation of G afiording ´. m ), l1 > ¢¢¢ > lm > 0; when we have ai parts of size li. Since the number of irreducible characters of a group is equal to the number of conjugacy classes, which in the case of Sn is the number of partitions of n, the irreducible characters of Sn are labelled by partitions of n. We denote the irreducible character labelled by the partition ‚ by (‚), so Irr(Sn) = f(‚) j ‚ ' ng. If G is a flnite group andis an irreducible character of G, an e-cient and simple method to construct representations of flnite groups has been presented in (2). This method is applicable whenever G has a subgroup H such thatH has a linear constituent with multiplicity 1. We call such a subgroup H a ´-subgroup. In practice this algorithm is quite fast when H has a small order, but can be very slow for a large H. For using this method to construct representations of G, we need to flnd a ´-subgroup for each irreducible characterof G. If G is a simple group or a covering group of a simple group, then a ´- subgroup for each nontrivial irreducible characterof G of degree < 32 has been found in (1). Also if ‚ is a partition of n and ‚ 0 is the conjugate

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").

We call the bipartite graph G is normal provided the reduced adjacency matrix A of G is normal. In this paper we give graph representations of normal matrices. In addition we shall have the characterization of signed bipartite normal graphs.

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.

Let R be a ring with identity, X the set of all nonzero, nonunits of R and G the group of all units of R. We will consider two group actions on X by G, the regular action and the conjugate action. In this paper, by investigating two group actions we can have some results as follows: First, if G is a flnitely generated abelian group, then the orbit O(x) under the regular action on X by G is flnite for all nilpotents x 2 X. Secondly, if F is a fleld in which 2 is a unit and F n f0g is a flnitley generated abelian group, then F is flnite. Finally, if G in a unit-regular ring R is a torsion group and 2 is a unit in R, then the conjugate action on X by G is trivial if and only if G is abelian if and only if R is commutative.

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.

The set of all matrices with entries in is de­noted by . We say that a linear operator T on is a (U, V)-operator if there exist invertible matrices and such that either T(X) = UXV for all X in , or m = n and T(X) = for all X in . In this paper we show that a linear operator T preserves the rank of matrices over the nonnegative integers if and only if T is a (U, V)­operator. We also obtain other characterizations of the linear operator that preserves rank of matrices over the nonnegative integers.

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.

Regarding all particles at a flxed site as a cluster, the size of the largest cluster under the zero range invariant measures is well studied by Jeon et al.(5) for the case of density one. Here, the density of the flnite zero-range process is given by the ratio between the number m of particles and the number n of sites. In this paper, we study the lower density case, i.e., the case m = o(n): Especially, when m » n fl ;0 < fl < 1, we show that there is an interesting cutofi point around fl = 1=2.

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 we prove that if T is a k-th root of a p­hyponormal operator when T is compact or T is normal for some integer n > k, then T is (generalized) scalar, and that if T is a k-th root of a semi-hyponormal operator and have the property (T) is contained in an angle /k with vertex in the origin, then T is subscalar.

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

In [2], Jahangiri studied the harmonic starlike functions of order , and he defined the class T() consisting of functions J = h + where hand g are the analytic and the co-analytic part of the function f, respectively. In this paper, we introduce the class T(, ) of analytic functions and prove various coefficient inequalities, growth and distortion theorems, radius of convexity for the function h, if the function J belongs to the classes T() and T(, ).

In this paper we present another characterization of (§1)-invariant sequences. We also introduce truncated Fibonacci and Lucas sequences of the second kind and show that a sequence x 2 R 1 is (i1)-invariant(1-invariant resp.) if and only if D £ 0 ⁄ is perpendicular to every truncated Fibonacci(truncated Lucas resp.) sequence of the second kind where D = diag((i1) 0 ;(i1) 1 ;(i1) 2 ;:::).

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.

A k-sized balanced sampling plan excluding contigu- ous units of order v and index ‚, denoted by BSEC(v;k;‚), is said to be bicyclic if it admits an automorphism consisting of two dis- joint cycles of length v . In this paper, we obtain a necessary and su-cient condition for the existence of bicyclic BSEC(v;3;2)s.

In this paper we derive the central limit theorem for , where is a triangular array of nonnegative numbers such that $sup_n{\sum}_{i=1}^n\;a_{ni}^2\;.

In this paper, we first find a necessary and sufficient condition for the existence of multiple orthogonal polynomials by the moments of a pair of measures and then give representations for multiple orthogonal polynomials. We also prove four term recurrence relations for multiple orthogonal polynomials of type II and several interesting relations for multiple orthogonal polynomials are given. A generalized recurrence relation for multiple orthogonal polynomials of type I is found and then four term recurrence relations are obtained as a special case.

Let (R, m) be a 2-dimensional regular local ring with algebraically closed residue field R/m. Let K be the quotient field of R and v be a prime divisor of R, i.e., a valuation of K which is birationally dominating R and residually transcendental over R. Zariski showed that there are finitely many simple v-ideals and all the other v-ideals are uniquely factored into a product of those simple ones. It then was also shown by Lipman that the predecessor of the smallest simple v-ideal P is either simple (P is free) or the product of two simple v-ideals (P is satellite), that the sequence of v-ideals between the maximal ideal and the smallest simple v-ideal P is saturated, and that the v-value of the maximal ideal is the m-adic order of P. Let m = (x, y) and denote the v-value difference |v(x) - v(y)| by . In this paper, if the m-adic order of P is 2, we show that . We also show that when w is the prime divisor associated to a simple v-ideal of order 2 and that w(R) = v(R) as well.

Let A be a Banach algebra, we say that A has the strongly double limit property (SDLP) if for each bounded net in A and each bounded net $(a^{\ast}\;_\beta)\;in\;A^{\ast},\;lim_\alpha\;lim_\beta=lim_\beta\;lim_\alpha$ whenever both iterated limits exist. In this paper among other results we show that if A has the SDLP and is (n - 2)-weakly amenable, then A is n-weakly amenable. In particular, it is shown that if is weakly amenable and A has the SDLP, then A is weakly amenable.

In this paper, we deflne the weak dimension and the chain-weak di† † † † . . . . . . † † † † . . . . . . . . . . . . . . . . ..... . .... ..... † † † † . . . . . . . . . . . . . . . ...... ...... ..... † † † . . . . . . . . . . . . . . . ...... ...... ...... † † † . . . . . . . . . . . . . . . ...... ...... ..... † † † † † † † † † † † † ......... ......... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ......... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

A pair of pants is a building block of oriented surfaces. The purpose of this paper is to formulate the matrix presentations of elements of the Teichmuller space of a pair of pants. In the level of the matrix group , we shall show that an odd number of traces of matrix presentations of the generators of the fundamental group of should be negative.

In this paper we consider the decompositions of subdirect sums and direct sums in bounded BCK-algebras. The main results are as follows. Given a bounded BCK-algebra X, if X can be decomposed as the subdirect sum of a nonzero ideal family of X, then I is finite, every is bounded, and X is embeddable in the direct sum ; if X is with condition (S), then it can be decomposed as the subdirect sum if and only if it can be decomposed as the direct sum ; if X can be decomposed as the direct sum , then it is isomorphic to the direct product .

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.

The notion of U-exact sequence (or quasi-exact sequence) of modules was introduced by Davvaz and Parnian-Garamaleky as a generalization of exact sequences. In this paper, we prove further results about quasi-exact sequences. In particular, we give a generalization of Schanuel's Lemma. Also we obtain some relation-ship between quasi-exact sequences and superfluous (or essential) submodules.

The aim of this paper is to study order-congruences on a S-poset A and to characterize the order-congruences by the concepts of pseudooreders on A and quasi-chains module a congruence p. Some homomorphism theorems of S-posets are given which is similar to the one of ordered semigroups. Finally, It is shown that there exists the non-trivial order-congruence on a S-poset by an example.

Let R be a ring R and be an endomorphism of R. R is called -rigid (resp. reduced) if for any implies a = 0. An ideal I of R is called a -ideal if . R is called -quasi-Baer (resp. right (or left) -p.q.-Baer) if the right annihilator of every -ideal (resp. right (or left) principal -ideal) of R is generated by an idempotent of R. In this paper, a skew polynomial ring A = R[] of a ring R is investigated as follows: For a -rigid ring R, (1) R is -quasi-Baer if and only if A is quasi-Baer if and only if A is -quasi-Baer for every extended endomorphism on A of (2) R is right -p.q.-Baer if and only if R is -p.q.-Baer if and only if A is right p.q.-Baer if and only if A is p.q.-Baer if and only if A is -p.q.-Baer if and only if A is right -p.q.-Baer for every extended endomorphism on A of .

Let R be a ring with left identity. Let G : be a symmetric biadditive mapping and g the trace of G. Let be an endomorphism and an epimorphism. In this paper we show the following: (i) Let R be 2-torsion-free. If g is ()-skew-commuting on R, then we have G = 0. (ii) If g is ()-skew-centralizing on R, then g is ()-commuting on R. (iii) Let . Let R be (n+1)!-torsion-free. If g is n-()-skew-commuting on R, then we have G = 0. (iv) Let R be 6-torsion-free. If g is 2-()-commuting on R, then g is ()-commuting on R.

In this paper, we investigate some properties of the entire function of the hyper order less than sharing one value CM with its k-th derivative.

We consider 2-colored digraphs of the primitive min- istrong digraphs having given exponents. In this paper we give bounds for 2-exponents of primitive extremal ministrong digraphs.

In this paper, we establish a number system in which arises from a Haar wavelet basis in connection with decompositions of certain Cuntz algebra representations on ( ). Number systems in are also of independent interest [9]. We study radix-representations of : : … ㆍ … as

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