The passage from ordinary to partial differential equations is a difficult one. While many properties of ordinary differential equations admit analogies in the framework of partial differential equations, the techniques used to establish them may bear little resemblance to those which suffice in the simpler setting.

For n real variables {xi} of time t we consider the following equations:

(1)

Throughout {aij} and {fi} are assumed to be continuous. For any given initial conditions there is assumed to be at least one trajectory solution.

The object of this paper is to consider the relationships between matrix rings and rings having classical quotient rings which are quasi-Frobenius X (QFX) rings. The main result of this paper is Theorem 12, which shows that if S is a ring with a QFX right classical quotient ring T, then T is isomorphic to a direct sum of a finite number of matrix rings over local rings Ui, while S is almost a direct sum of matrix rings over rings Ci, the Ui being right classical quotient rings of the Ci.

In this note we examine the question of the existence of a best possible N-net for a bounded set in a normed linear space. A sufficient condition for existence is given which leads to easy proofs of some of the standard results. The pertinent reference here is the paper by Garkavi [1].

Let E be a normed linear space and let M be a bounded set in E. Any system of N points in E will be called an N-net. For a given M and the net SN = {y1, y2,…, yN} define

and

Let K be a compact subset of the real Euclidean space En. We say that K has constant width if the distance between each pair of distinct parallel hyperplanes which support K is constant. The collection of all compact convex subsets of En which have constant width is denoted .

Let X = {Xt, t ≥ 0} be a stationary Markov process with values in a measurable space (S, ℬ), transition function p, and initial distribution concentrated at a point x ∊ S. The occupation times of a set A ∊ ℬ are defined for t ≥ 0 by

where 1A is the indicator function of A. The expected occupation times are given by

Dans cet article, nous nous intéressons à trouver des conditions suffisantes au problème d'approximation polynômiale généralisée dans certains espaces seminormes. Notre résultat principal est basé sur un théorème d'unicité de Malliavin et sur la théorie de la représentation de Choquet.

The aim of this paper is to generalize the main results of [1] to GLn(B) by means of proofs which are more conceptual and less computational. In addition, by means of the Stone space we will obtain results which are new even for the case n = 2. Finally we shall make some remarks of a categorical nature.

The author is especially interested in the subject because of the overlap here of many areas of mathematics. Concepts from topology, model theory, and category theory are all relevant.

A friendship graph is a graph in which every two distinct vertices have exactly one common adjacent vertex (called a neighbour). Finite friendship graphs have been characterized by Erdós, Rényi and Sós [2]: Each finite friendship graph Fn which consists of n edge disjoint triangles such that all n>1 triangles have one vertex in common (F1 is a triangle i.e. the complete graph with three vertices). Thus Fn has 2n+1 vertices, 2n of them being of degree two and the remaining one (the common vertex of n triangles if n>1) being of degree 2n.

Browder and Wermer in [2] give a method for constructing Dirichlet subalgebras of the disk algebra. In this note we show that these Dirichlet algebras do not contain any non-constant functions which satisfy a Lipschitz-one condition on a subinterval of the unit circle.

Let G be a finite abelian group of the order pr and type (p, …, p), where p is a prime. A necessary and sufficient condition is determined for the existence of subgroups G1, G2, ⋯, Gn, one of the order pa and the rest of the order pb, such that G = G1 ∪ G2 ∪ ⋯ ∪ Gn and Gi, ∩ Gj,= {θ} when i ≠ j.

The class of all combinatorial geometries of rank three shall coincide with the class of all pairs (V, S) such that V is a set and S is a collection of non-empty subsets of V such that each pair of distinct elements of V belong to exactly one member of S. (See [3].)

Consider a combinatorial geometry (V, S) of rank three.

It is shown that the Lorentz structure of a compact prespace- time M can be so chosen such that M can not satisfy the strong energy condition. Thus, combining both the causal and the strong energy conditions, a stronger case against the compact space-times as proper arenas of physics can be made.

A Lidstone series provides a (formal) two-point expansion of a given function f(x) in terms of its derivatives of even order at the nodes 0 and 1 and takes the form.

Let Ω be an open subset of R and H be a complex Hilbert space; (,) represents scalar product in H.Let also A be a closed linear operator with domain DA dense in H and A* with domain D*A be its adjoint. Under graph scalar product DA and D*A are also Hilbert spaces.

The purpose of the paper is to extend a tight set function on a lattice with values in a locally convex space of special type to a measure on the cr-ring generated by . This result generalizes the extension theorem of Thomas [12, p. 151], which in turn contains the extension theorems of Pauc [9, p. 710], Fox [4, p. 525] and J. J. Uhl, Jr. [14, Corollary 2].

This paper develops another canonical form for (0, 1)-matrices which may be used in the same spirit as the nearly decomposable matrix [5] or the k-nearly decomposable matrix [1], This form is intrinsic in each fully indecomposable matrix and does not require the replacement of any of its non-zero entries by 0's.

Let X be a compact Hausdorff space and C(X) the complexvalued continuous functions on X. We say A is a function algebra on X if A is a point separating, uniformly closed subalgebra of C(X) containing the constant functions. Equipped with the sup-norm ‖f‖ = sup{|f(x)|: x ∊ X} for f ∊ A, A is a Banach algebra. Let MA denote the maximal ideal space.

Let D be the closed unit disk in C and let U be the open unit disk. We call A(D)={f ∊ C(D):f is analytic on U} the disk algebra. Let T be the unit circle and set C1(T) = {f ∊ C(T): f'(t) ∊ C(T)}.

Soient R un anneau commutatif unitaire, S une R-algèbre associative unitaire, A et B des sous-R-modules de S. Nous étudions des situations dans lesquelles HomR(A, B) peut être emboîté dans S. Les définitions et le résultat de base se trouvent dans la section 1; dans la section 2, nous examinons des applications à l'étude des structures d'algèbre que l'on peut définir sur A; la section 3 contient des exemples.

Our objective in this note is to prove a theorem about the spectral behavior of certain multiplication operators. It was motivated by and yields an extension of part of a theorem of Lumer [3] concerning the possibility of imbedding the classical Hardy spaces in abstract ones.