In this paper we propose a new technical tool for analyzing representations of Hilbert ${{C}^{*}}$- product systems. Using this tool, we give a new proof that every doubly commuting representation over ${{\mathbb{N}}^{k}}$ has a regular isometric dilation, and we also prove sufficient conditions for the existence of a regular isometric dilation of representations over more general subsemigroups of $\mathbb{R}_{+}^{k}$.

Let M be a free module of rank n over a commutative ring R with unit and let Σn denote the symmetric group acting on a fixed basis of M in the usual way. Let Mm denote the direct sum of m copies of M and let S be the symmetric ring of Mm over R. Then each element of Σn acts diagonally on Mm and consequently on S; denote by Xn the subgroup of Gl(Mm) so defined. The ring of invariants, SΣn, is called the ring of vector invariants by H. Weyl [ 3, Chapter II, p. 27] when R = Q. In this paper a set of generators valid over any ring R is given. This set of generators is somewhat larger than Weyl's. It is interesting to note that, over the integers, his algebra and SΣn have the same Hilbert-Poincaré series, are equal after tensoring with the rationals, and have the same fraction fields, although they are not equal.

Jackson type theorems are obtained for the comonotone approximation of piecewise monotone functions by polynomials.

We use the homological perturbation lemma to give an algebraic proof of the cyclic Eilenberg-Zilber theorem for cylindrical modules.

Any term or symbol undefined in this paper is defined in [5]. For graphs F and G, G > F means G contains a subgraph isomorphic to F and E(G) denotes the edge set of G. If E ⊆ E(G), 〈E〉 is the subgraph of G whose edge set is E and whose vertex set is that subset of vertices of G which are incident with edges in E.

Let $R$ be a ring and let $g$ be an endomorphism of $R$. The additive mapping $d:\,R\,\to \,R$ is called a Jordan semiderivation of $R$, associated with $g$, if

$$d\left( {{x}^{2}} \right)=d\left( x \right)x+g\left( x \right)d\left( x \right)=d\left( x \right)g\left( x \right)+xd\left( x \right)\,\text{and}\,d\left( g\left( x \right) \right)=g\left( d\left( x \right) \right)$$

for all $x\,\in \,R$. The additive mapping $F:\,R\,\to \,R$ is called a generalized Jordan semiderivation of $R$, related to the Jordan semiderivation $d$ and endomorphism $g$, if

$$F\left( {{x}^{2}} \right)=F\left( x \right)x+g\left( x \right)d\left( x \right)=F\left( x \right)g\left( x \right)+xd\left( x \right)\,\,and\,F\left( g\left( x \right) \right)=g\left( F\left( x \right) \right)$$

for all $x\,\in \,R$. In this paper we prove that if $R$ is a prime ring of characteristic different from 2, $g$ an endomorphism of $R,\,d$ a Jordan semiderivation associated with $g,\,F$ a generalized Jordan semiderivation associated with $d$ and $g$, then $F$ is a generalized semiderivation of $R$ and $d$ is a semiderivation of $R$. Moreover, if $R$ is commutative, then $F\,=\,d$.

We prove Wise's $W$-cycles conjecture. Consider a compact graph $\Gamma '$ immersing into another graph $\Gamma $. For any immersed cycle $\Lambda :{{S}^{1}}\to \Gamma $, we consider the map $\Lambda '$ from the circular components $\mathbb{S}$ of the pullback to $\Gamma '$. Unless $\Lambda '$ is reducible, the degree of the covering map $\mathbb{S}\to {{S}^{1}}$ is bounded above by minus the Euler characteristic of $\Gamma '$. As a corollary, any finitely generated subgroup of a one-relator group has a finitely generated Schur multiplier.

If G and H are two groups such that their integral group rings Z(G) and Z(H) are isomorphic, does it follow that G and H are isomorphic? This is the isomorphism problem and an affirmative answer is obtained in case G is a sub group of the group of order preserving permutations of a totally ordered set.

P. Erdös and R. Rado [1] proved that to each pair of positive integers n and k, with k ≥ 3, there corresponds a least positive integer φ(n, k) such that if is a family of more than φ(n, k) sets, each set with n elements, then some k of the sets have pair-wise the same intersection.

For any space X, DG(X, A) is an abelian subgroup of [X, A] when A is an H-group. DG(X, X) is a ring for any H-group X.

Let R be a ring and G a group. The group ring RG consists of all functions f: G → R with finite support. Addition is pointwise and multiplication is defined for f, h ∊ RG and g ∊ G, by

The support group of f is defined to be the subgroup of G generated by the support of f. The element f is idempotent if ff = f

We prove the following result.

In this paper, we study the ${{L}^{p}}$ mapping properties of a class of singular integral operators with rough kernels belonging to certain block spaces. We prove that our operators are bounded on ${{L}^{p}}$ provided that their kernels satisfy a size condition much weaker than that for the classical Calderón–Zygmund singular integral operators. Moreover, we present an example showing that our size condition is optimal. As a consequence of our results, we substantially improve a previously known result on certain maximal functions.

The purpose of this paper is to continue the study of certain “distinguished” subsets of the convergence domain of a matrix, as developed by A. Wilansky [6] and G. Bennett [1], We also consider continuous linear functionals on the domain, and the extent to which their representation is unique; this turns out to be connected with the behaviour of the subsets.

In this paper we extend a well-known theorem of M. Scheunert on skew-symmetric bicharacters of groups to the case of skew-symmetric bicharacters on arbitrary cocommutative Hopf algebras over a field of characteristic not 2. We also classify polycharacters on (restricted) enveloping algebras and bicharacters on divided power algebras.

In this note, we first give a characterization of super weakly compact convex sets of a Banach space $X$: a closed bounded convex set $K\,\subset \,X$ is super weakly compact if and only if there exists a ${{w}^{*}}$ lower semicontinuous seminorm $P$ with $P\,\ge \,{{\sigma }_{K}}\,\equiv \,{{\sup }_{x\in K}}\left\langle \,\cdot \,,\,x \right\rangle $ such that ${{P}^{2}}$ is uniformly Fréchet differentiable on each bounded set of ${{X}^{*}}$. Then we present a representation theoremfor the dual of the semigroup swcc$\left( X \right)$ consisting of all the nonempty super weakly compact convex sets of the space $X$.

The purpose of the present note is to give a combinatorial interpretation of the coefficients of expansion of the Ramanujan continued fraction ([1], p. 295)

The result is expressed by formula (12) below.

The enumeration of distinct score vectors of a tournament leads to the following problem: (Erdős and Moser, see Moon [2], p. 68). Given n ≥ 1, k ≥ 0, determine the number of distinct sequences of positive integers

In the first and second parts of this sequence we dealt with applications of graph theory to distance distribution in certain sets in euclidean spaces, to potential theory, to estimations of the transfinite diameter [1] and to value distribution of "triangle functional " (e.g. perimeter, area of triangles) [2]. The basic tool is provided in all these applications by the result formulated as Lemma 2. This, an essentially pure logical result, proves to be a very flexible and versatile instrument in applications.

Here the same method is used in an abstract setting. First we deduce certain results for the density of a given family of subsets of an abstract set S in another family of subsets of the same S. Then we apply the results obtained to distance distribution in certain (e.g. totally bounded or compact) sets in metric spaces, in particular in a normed linear function space. Applications of this method to functional on Hilbert spaces were given by Katona [3].

It is shown that, in characteristic zero, a finite subgroup of a general linear group is generated by pseudo-reflections if and only if its ring of coinvariants satisfies Poincaré duality.

Lehmer [3] and Reisel [7] have devised tests for determining the primality of integers of the form A2n—1. Tables of primes of these forms may be found in [7] and Williams and Zarnke [10]. Little work, however, seems to have been done on integers of the form N=2A3n—1. Lucas [6] gave conditions that were only sufficient for the primality of N. Recently Lehmer [4] has given a method for determining the primality of an integer N if the factorization of N+1 is known.

We find a lower bound on the absolute value of the discriminant of the minimal polynomial of an integral symmetric matrix and apply this result to find a lower bound on Mahler's measure of related polynomials and to disprove a conjecture of D. Estes and R. Guralnick.