If f is a suitable meromorphic function then, by a classical technique in the calculus of residues, one can evaluate in closed form series of the form

1.1

A star is a connected graph in which every node but possibly one has valency 1. Let G be a graph and C a spanning subgraph of G in which every component is a star. With each component α of C let us associate a weight wα. Let Пα wα be the weight associated with the entire subgraph G the star polynomial of G is ΣПα wα where the summation is taken over all spanning subgraphs of G consisting of stars. In this paper an algorithm for finding star polynomials of graphs is given. The star polynomials of various classes of graphs are then found, and some results about node-disjoint decomposition of complete graphs and complete bipartite graphs are deduced.

We consider the second-order operator

1

where the coefficients are real continuous functions on an interval ℐ with w and p positive. The operator is assumed singular at only one endpoint which we take to be either 0 (finite singularity) or ∞ (infinite singularity). Let be the Hilbert space of all complex-valued, measurable functions f satisfying

In [A1] is defined a class of Markov operators on C(X) (X compact T2), called Generalized Averaging Operators (g.a.o.) which yield an easy solution to the following problem: given a fixed Markov operator T, find necessary and sufficient conditions on any other Markov operator R for the relation ker T ⊂ker R to hold. The main application of this is to inclusion relations between matrix summability methods.

Dans cet article, nous établirons un théorème sur la représentation de certaines séries de Dirichlet (ceci généralise un résultat de [3]). Avec ce nouveau résultat nous donnerons seulement quelques applications qui évidemment pourraient être augmentées de beaucoup. Par exemple, nous donnerons une expression asymptotique pour

1

In this paper we are concerned with power series of the type

1

which admit unique analytic extension onto a domain containing the negative real axis. Our primary object is to establish a general theorem giving a lower estimate for the number of different zeros of (1) on the negative real axis. W. Jurkat and A. Peyerimhoff showed that for a certain class of coefficient functions a(z) the number of negative zeros of (1) is closely related to the behaviour of a(z) at z = 0. In particular they proved the following theorem [4, p. 219, Theorem 4].

In classical information theory, the amount of information provided by an experiment is measured by a function of the probability distribution of the outcomes of the experiment. In this paper, information measures are functions of sequences of elements of a monoid (S, ∘) with identity e. It is assumed that the measures {μn: Sn → ℝ} of information are branching.

The notion of functionality for an internal horn functor H in a concrete category K was introduced in Banaschewski and Nelson [1], formalizing the condition that the structure on the H(A, B) is "pointwise" structure on sets of functions.

In this note, we have shown that the set of conditions given by Gel'fand and Vilenkin for the positive definiteness of a functional L(ϕ) [1, Theorem 7, p. 285] though sufficient is not necessary, thereby providing an answer to an open problem posed by them [1, Theorem 7, p. 285].

Singh introduced two conditions on a module MR in [7]. The author introduced the concept of h-neat submodule of such module in [3] and generalized some of the well known results of neat subgroups. A theorem of Erdelyi was also shown to be true for such modules in [4]. The main purpose of this paper is to generalize a well known result of K. M. Benabdallah and J. M. Irwin and M. Rafiq [2, Theorem 10].

The main result in this note is the following

Theorem: Let G be a finite solvable group. There exists a permutation σ of the set G such that {g • σ(g); g∈G} = G if and only if the Sylow 2-subgroup of G is non-cyclic or trivial

Recently H. Marubayashi [1,2] and S. Singh [10,11,12] generalized some results of torsion abelian groups for modules over some restricted rings, like bounded Dedekind prime rings, bounded hereditary Noetherian prime rings. Singh [12] introduced the concept of h-purity for a module MR satisfying the following conditions:

(I) Every finitely generated submodule of every homomorphic image of M is a direct sum of uniserial modules.

In the study of particular categories of integral domains, it has proved useful to know which overrings of the domains of interest lie within the category, and indeed whether all such overrings do. (Recall: an overring of R is a ring T with R ⊆ T ⊆ quotient field of R.) Two classes of domains classically studied in this setting are Prüfer domains and one-dimensional Noetherian domains. Since both of these classes are contained in the category of coherent domains, it is natural to investigate this category in this setting.

We define a square-reduced residue system (mod r) as the set of integers a (mod r) such that the greatest common divisor of a and r, denoted by (a, r), is a perfect square ≥ 1 and contained in a residue system (mod r). This leads to a Class-division of integers (mod r) based on the 'square-free' divisors of r. The number of elements in a square-reduced residue system (mod r) is denoted by b(r). It is shown that

(1)

(2)

In view of (2), b(r) is said to be 'specially multiplicative'. The exponential sum associated with a square-reduced residue system (mod r) is defined by

where the summation is over a square-reduced residue system (mod r).

B(n, r) belongs to a new class of multiplicative functions known as 'Quasi-symmetric functions' and

(3)

As an application, the sum is considered in terms of the Cauchy-composition of even functions (mod r). It is found to be multiplicative in r. The evaluation of the above sum gives an identity involving Pillai's arithmetic function

Every Banach space with a non-shrinking (unconditional) basis (Xi) can be renormed so that the biorthogonal sequence has a much smaller (unconditional) basis constant than (xi). On the other hand, if the unconditional constant of is C < 2 then the unconditional constant of (xi) is at most C/(2—C). This estimate is sharp.

Une fonction arithmétique f est une fonction définie sur l'ensemble des entiers naturels et à valeurs dans ℂ. On dit qu'une telle fonction f est additive si f(mn)=f(m) + f(n) si(m, n) = l et par ailleurs qu'elle est multiplicative si f(mn)=f(m)f(n) si (m, n) = l. Les sommes de la forme Σn≤xf(n) ont été largement étudiées.

Equivalence problems in Riesz-summability lead to the problem of finding the zeros of special power series in the unit circle, i.e.

In [2], Herstein proves the following result:

Theorem. Let R be a prime ring, d≠0 a derivation of R such that d(x) d(y) = d(y) d(x) for all x, y ∈ R. Then, if char r≠2, R is commutative, and if char R = 2, R is commutative or an order in a simple algebra which is 4-dimensional over its center.

Let (X, p) and (Y, d) be metric spaces with at least two points. It is usual for introductory courses in topology to study the set Yx of all functions mapping X to Y with the pointwise, compact-open, uniform convergence, and uniform convergence on compacta topologies. Some care is taken to show sufficient conditions for these topologies to be equivalent [1, 2]. However, the question of necessary conditions are dismissed with examples showing that the topologies are not in general equivalent.

Beineke and Harary gave an example of a family of tournaments Tn such that every subtournament of Tn is irreducible or transitive. We characterize all tournaments with this property.