Let K be a bounded open convex set in euclidean n-space Rn symmetric in the origin 0. Further let L be a discrete point set in Rn containing 0 and at least n linearly independent points of Rn. Put mi = inf ui extended over all positive real numbers ui for which uiK contains i linearly independent points of L, i = 1, 2, …, n.

We study the absolute continuity of the convolution ${\it\delta}_{e^{X}}^{\natural }\star {\it\delta}_{e^{Y}}^{\natural }$ of two orbital measures on the symmetric spaces $\mathbf{SO}_{0}(p,p)/\mathbf{SO}(p)\times \mathbf{SO}(p)$, $\mathbf{SU}(p,p)/\mathbf{S}(\mathbf{U}(p)\times \mathbf{U}(p))$ and $\mathbf{Sp}(p,p)/\mathbf{Sp}(p)\times \mathbf{Sp}(p)$. We prove sharp conditions on $X$, $Y\in \mathfrak{a}$ for the existence of the density of the convolution measure. This measure intervenes in the product formula for the spherical functions.

In this paper first we prove the uniqueness theorem for an integral transform whose kernel is H-function. Later on we establish a new and interesting theorem concerning this transform and a generalized Laplace transform whose kernel is Meijer's G-function.

In addition to the results of the paper (Bachturin (1974)) we give the precise form of the necessary and sufficient conditions ensuring that all irreducible representations of a Lie algebra were of finite bounded degree.

In [3] and [4], the near-rings R with no zero divisors are studied. In particular, a near-ring R is a near-field if it has a non-zero right distributive element ([4], Theorem 1.2.). Also, (R, +) is a nilpotent group if not all non-zero elements of R are left identities of R ([3], Theorem 2). The purpose of the present paper is to extend the above results to a class of near-rings with zero divisors; that is, the set of annihilators of an element x in R, T(x) = {g/xg = 0} is either {0} or R. The examples of such near-rings are those R with (R, +) simple groups and those R with no zero divisors as given in [1], [2], [3] and [4]. For this r, we can easily see that R = A∪S where A = {x/T(x) = R} and S = {x/T(x) = {0}}. Then the second part of this paper will give a structural theorem on the semi-group (S, · ), and more properties on R can be derived.

Finite matrices with entries pij Fij (x1,…, xk), where {pij} is stochastic and Fij(.) is a k-variate probability distribution are discussed. It is shown that the matrix of k-variate Laplace-Stieltjes transforms of the Pij Fij(x1, …, xk) has a Perron-Frobenius eigenvalue which is a convex function in k variables in a suitably defined region. The values of the partial derivatives near the origin of this maximal eigenvalue are exhibited. They are quantities of interest in a variety of applications in Probability theory.

Let B be an associative ring with identity, A a subring of B containing the identity of B. If B is commutative then it is customary to define an element b of B to be integral over A if it satisties an equation of the form for some a1, a2, …, an A. This definition does not generalize readily to the case when B is non-commutative. Van der Waerden ([11], p. 75) defines b ∈ B to be integral over A if all powers of b belong to a finite A-module. This definition is quite satisfactory when A satisfies the ascending chain condition for left ideals, but in the general case this type of integrity is not necessarily transitive, even when B is commutative. Krull [6] calls an element b ∈ B which satisfies the above condition almost integral over A (but he only considers the commutative case). The subset Ā of B consisting of all almost integral elements over A is called the complete integral closure of A in B. If Ā = A, A is said to be completely integrally closed in B. More recently (in [3]), Gilmer and Heinzer (see also Bourbaki, [1]) have discussed these properties in the commutative case and have shown that the complete integral closure of A in B need not be completely integrally closed in B. If B is not commutative, the set A of elements of B almost integral over A, may not even form a ring. In [5] p. 122, Jacobson uses a definition equivalent to Van der Waerden's for the non-commutative case but the definition applies only for a very restricted class of rings.

Associated with each uniform space (S, U) there is a uniform lattice (P, V, ≦) called the scale of (S, U). The scale was first introduced by D. Bushaw (see [2] and [3]) for the purpose of studying stability in topological dynamics. Further properties of the scale were investigated in [4], and this paper pursues one of the research problems suggested in the latter paper.

Let n1 < n2 < … be the sequence of positive integers whose base b representations involve the digit t ≦ b —1 at most d—1 times. B. D. Craven [2] shows that Σi1/ni converges by giving an upper bound for this sum as a function of b and d.

Let Γ be a graph with finite vertex set V. Γ is homogeneous if whenever U1, U2 ⊆ V are such that the vertex subgraphs (U1), (U2) are isomorphic, then every isomorphism from (U1) to (U2) extends to an automorphism of Γ; homogeneous graphs were studied by Sheehan (1974) and were classified by the author. Γ is locally homogeneous if whenever U ⊆ V, then every automorphism of (U) extends to an automorphism of Γ. We prove that every locally homogeneous graph is homogeneous.

We establish two results in the pointwise convergence problem of a trigonometric series for some nonnegative integer m. These results not only generalize Hardy's theorem, the Jordan test theorem and Fatou's theorem, but also complement the results on pointwise convergence of those Fourier series associated with known L1-convergence classes. A similar result is also established for the case that , where {ln} satisfies certain conditions.

A complete characterization of Boolean algebras which admit nonatomic charges (i.e. finitely additive measures) is obtained. This also gives rise to a characterization of superatomic Boolean algebras. We also consider the problem of denseness of the set of all nonatomic charges in the space of all charges on a given Boolean algebra, equipped with a suitable topology.

The duality properties of the integration map associated with a vector measure m are used to obtain a representation of the (pre)dual space of the space Lp(m) of p-integrable functions (where 1<p<∞) with respect to the measure m. For this, we provide suitable topologies for the tensor product of the space of q-integrable functions with respect to m (where p and q are conjugate real numbers) and the dual of the Banach space where m takes its values. Our main result asserts that under the assumption of compactness of the unit ball with respect to a particular topology, the space Lp(m) can be written as the dual of a suitable normed space.

The four colour conjecture is well known to be equivalent to the proposition that every trivalent planar graph without an isthmus (i.e. an edge whose removal disconnects the graph) has an edge colouring in three colours ([1], p. 121). By an edge colouring we mean an assignment of colours to the edges of the graph so that no two edges of the same colour meet at a common vertex, and the graph is n-valent if n edges meet at each vertex. An edge colouring by three colours is called a Tait colouring; a trivalent graph which has a Tait colouring can be split in three edge-disjoint 1-factors, i.e. spanning monovalent subgraphs.

Inspired by a statement of W. Luh asserting the existence of entire functions having together with all their derivatives and antiderivatives some kind of additive universality or multiplicative universality on certain compact subsets of the complex plane or of, respectively, the punctured complex plane, we introduce in this paper the new concept of U-operators, which are defined on the space of entire functions. Concrete examples, including differential and antidifferential operators, composition, multiplication and shift operators, are studied. A result due to Luh, Martirosian and Müller about the existence of universal entire functions with gap power series is also strengthened.

We consider the effect on the stability properties of a graph G, of the presence in the automorphism group of G of automorphisms (uv)h, where u and v are vertices of G, and h is a permutation of vertices of G excluding u and v. We find sufficient conditions for an arbitrary graph and a cartesian product to have stability index one, and conjecture in the latter case that they are necessary. Finally we exhibit explicitly a large class of graphs which have stability index one.

The formal power series (fps) f(z) = Σ∞i=1ai zi is homozygous mod k if ai ≠ 0 implies i ≡ j mod k. This generalizes even and odd fps. If f is homozygous mod k then all iterates of f (fn = f ο fn−1) are also homozygous mod k, but the converse is false–there are many non-odd fps f for which f(f(z)) = z. It is shown that if f is not homozygous mod k but fn is homozygous, then fnr(z) = z for some r. If all coefficienrs ar real then, in fact, f(f(z)) = z.

Subject classification (Amer. Math Soc. (MOS) 1970): 39 A 05.

A graph H decomposes into a graph G if one can write H as an edge-disjoint union of graphs isomorphic to G. H decomposes into D, where D is a family of graphs, when H can be written as a union of graphs each isomorphic to some member of D, and every member of D is represented at least once. In this paper it is shown that the d-dimensional cube Qd decomposes into any graph G of size d each of whose blocks is either an even cycle or an edge. Furthennore, Qd decomposes into D, where D is any set of six trees of size d.

Staring form a probability σ on the half-line moments of any order A. G. Pakes has defined probabilities σr, by length biasing order r and gr, by the stationary-excess operation of order r, r = 1, 2,…Examples are given to show that σ can bt determined in the Stieltjes sence while σ1 and g1 are indeterminate in the Stieltjes sence. This shows that a statement in a recent paper by Pakes does not hold.

The BMO norm of f is equivalent to where Pt is the Poisson kernel. In this note, we show that Pt can be replaced by a nonnegative radial function h, which is positive in a neighbourhood of 0, with and , where h is the least decreasing radial majorant of h.