Ces dernières années, de nombreuses recherches ont été consacrées à l’étude de la structure extrémale de la somme de deux convexes; signalons, par exemple, les travaux de Bair-Fourneau-Jongmans [3], Edelstein-Fesmire [5], Husain-Tweddle [6], Jongmans [7], Klee [8] et Roy [9]. Dans cette direction, Klee [8] a obtenu ce résultat très intéressant, qui a d’ailleurs été généralisé plus tard par Jongmans [7]: dans un espace euclidien ℝd, tout point extrême z de la somme d’un convexe fermé A et d’un convexe compact B s’écrit de façon unique comme la somme d’un point extrême xz de A et d’un point extrême yz de B; de plus, tout point extrême x de A livre un point extrême y de B tel que z = x + y soit un point extrême de A + B. Par contre, il peut exister des points extrêmes de B dont la somme avec un point extrême de A n’est jamais un point extrême de A + B; cette constatation conduit au problème: est-il possible de caractériser l’ensemble des yz?

In what follows we shall mean by a plane convex body K a compact convex subset of the Euclidean plane having nonempty interior. We shall denote by h (K, θ) the supporting function of K restricted to the unit circle. This measures the signed distances from the origin to the supporting line of K with outward normal (cos θ, sin θ). The right hand and left hand derivatives of h (K, θ) exist everywhere and are equal except on a countable set.

A continuous real vector function is said to be nonoscillatory on an interval if at least one of its components is of constant positive or negative sign there. In this note, various existence criteria for nonoscillatory solutions of the system x(m) = (-l)mQ(t)x are established. In some cases, additional monotonicity properties for these solutions are also given.

Let Tn denote a tournament with vertices labelled 1, …, n. Any undefined terms can be found in [5]. The converse of Tn is the tournament obtained by reversing the orientation of all the arcs in Tn. A tournament is called self-converse (s.c.) if . The transitive tournaments are examples of s.c. tournaments. In this paper we provide a structural characterization of s.c. tournaments and we also characterize the score vectors of s.c. tournaments.

I [1] have shown that

are the only square pseudo-Lucas numbers in the set of pseudo-Lucas numbers defined by

1

In this paper, it is shown that none of the pseudo-Lucas numbers are of the form 2S2, where S is an integer.

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.

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].

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].

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.

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.

A purely combinatorial and elementary proof of Johnson-Dulmage-Mendelsohn's theorem, which gives a quite sharp upper bound on the number of permutation matrices needed for representing a doubly stochastic matrix by their convex combination, is given.

Almost from the day that B. L. van der Waerden [10] proved his now famous theorem on arithmetic progressions, mathematicians have been working to find a new or an improved constructive proof of that result, but without much success.

All rings R considered here are commutative with identity and all the modules are unital right modules. As defined by Mehdi [6] a module MR is said to be a multiplication module if for every pair of submodules K and N of M, K ⊂ N implies K=NA for some ideal A of R. This concept generalizes the well known concept of a multiplication ring.

This note answers a question raised by Lee Mohler in 1970, by exhibiting a finite topological space X which is the union of closed subspaces Y, Z, such that Y, Z, and Y ⋂ Z, but not X, have the fixed point property. The example is a triangulation △ of S3, the points of X being the simplices of Δ and the closed sets the subcomplexes of △.

For a bounded linear operator T on a Hilbert space , denote by Lat0T the lattice of all linear submanifolds of such that , and by (resp. Lat T) the sublattice consisting of operator ranges (resp. closed subspaces).

We show that under mild conditions the Central Limit Theorem holds for general stochastic processes.

In this note we present an array of results which deals with the question "When is the product of two [a, b]-compact spaces an [a, b]-compact space".

In section 1, we give some essential terminology. In section 2, we define some new classes of functions and then obtain some product theorems. In section 3, we give some applications of the product theorems obtained in section 2.

In [4], Niven proved that the set A of integers for all s ≥ l and all n ≥ 1 has density zero, being the sum of the sth powers of all positive divisors of n. However his argument contains a mistake (see Remark 1). In this paper we give a proof of Niven's result and establish several related results, one of which generalizes a result of Dressier (See Theorem 3 and Remark 2).

A series ∑ an is said to be summable (I) to the limit A if

(*)

Clearly the limit is the same whether x→∞ through all real values or only positive integer values, and the expression whose limit is being taken can also be expressed in the two equivalent forms

where [x] is the greatest integer ≤x.