For Empedocles, a little over 2400 years ago, there were four chemical elements, fire, water, earth, and air, and they were continually brought together and torn apart by two opposing forces, harmony and discord. For Aristotle, a hundred years later, two binary classification schemes took the place of one: instead of harmony-discord, he had wet-dry and hot-cold. The serious alchemists of the middle ages found that nature was even more complicated than that; they classified matter by its luster, heaviness, combustilibity, solubility, and so on. Boyle, in the 1600's, offered a definition similar to that of a prime number – an element is a substance from which other substances can be made but which cannot be separated into different substances – and analytical chemistry was off and running. A hundred years later Lavoisier (inspired in part by Newton's insight into the paramount importance of weight) could formulate a modified definition that led for the first time to quantitative tables of chemical elements similar to our modern ones.

The familiar Fuglede-Putnam theorem asserts that AX = XB implies A*X = XB* when A and B are normal. We prove that A and B* be hyponormal operators and let C be a hyponormal commuting with A* and also let D* be a hyponormal operator commuting with B respectively, then for every Hilbert-Schmidt operator X, the Hilbert-Schmidt norm of AXD + CXB is greater than or equal to the Hilbert-Schmidt norm of A*XD* + C*XB*. In particular, AXD = CXB implies A*XD* = C*XB*. If we strengthen the hyponormality conditions on A, B*, C and D* to quasinormality, we can relax Hilbert-Schmidt operator of the hypothesis on X to be every operator and still retain the inequality under some suitable hypotheses.

A graph G, every vertex of which has degree at least three, is randomly 3-axial if for each vertex v of G, any ordered collection of three paths in G of length one with initial vertex v can be cyclically randomly extended to produce three internally disjoint paths which contain all the vertices of G. Randomly 3-axial graphs of order p > 4 are characterized for p ≢ 1 (mod 3), and are shown to be either complete graphs or certain regular complete bipartite graphs.

It is shown that for a solution of a Riccati equation with polynomial coefficients an expansion can be constructed as a Stieltjes continued fraction, with coefficients given by a recurrence relation, which is in general non-linear. Particular expansions associated with hypergeometric and confluent hypergeometric equations are given, and are shown to have a uniquely simple form.

It is shown that a transformation group with a locally compact Hausdorff phase space and an abelian phase group is locally weakly almost periodic if and only if it is P-locally weakly almost periodic for some replete semigroup P in the phase group.

Let D be a domain bounded by a Jordan curve. For 1 ≤ p ≤ ∞, let Lp(D) be the class of all functions f holomorphic in D such that where A is the area of D. For f ∈Lp(D), set

πn consists of all polynomials of degree at most n. Recently, Andre Giroux (J. Approx. Theory 28 (1980), 45–53) has obtained necessary and sufficient conditions, in terms of the rate of decrease of the approximation error , such that has an analytic continuation as an entire function having finite order and finite type. In the present paper we have considered the approximation error (*) on a Carathéodory domain and have extended the results of Giroux for the case 1 ≤ p < 2.

We introduce the Banach space of vector valued sequences lp, q(E), 1 ≤ p, q ≤ ∞, where E is a Banach space. Then we study the relation between lp, q(E) and the Schur multipliers of lpE, where E is taken to be some lr.

Let μ and ν be non-σ-finite pre-Radon measures on topological spaces X and Y respectively. Then there exists a unique pre-Radon measure λ on the product space X × Y which satisfies λ(A × B) = μ(A)ν(B) for all Borel sets A in X and B in Y such that μ(A) < ∞ and ν(B) < ∞.

This paper establishes a weak maximum principle for the difference u − v of solutions to nonlinear degenerate parabolic differential inequality

A graph is called point determining if distinct points have distinct neighborhoods. In this note we provide a sufficient condition for a point determining graph to have a 1-factor. This is an extension of some results obtained by David P. Sumner.

Sufficient conditions are obtained for the existence and linear stability of time independent age distributions in two species competition with age and time lagged density dependent mortality and fertility functions.

Let B be a commutative, semi-simple, regular, Tauberian Banach algebra with noncompact maximal ideal space Δ(B). Suppose B has the property that there is a constant C such that for every compact subset K of Δ(B) there exists a f ∈ B with = 1 on K, ‖f‖B ≤ C and has compact support. We prove that if A is a proper abstract Segal algebra over B then for every positive integer n there exists f ∈ B such that fn ∉ A but fn+1 ∈ A. As a consequence of this result we prove that if G is a nondiscrete locally compact abelian group, μ a positive unbounded Radon measure on Γ (the dual group of G), 1 ≤ p < q < ∞ and , then .

In the author's paper [7] it was proved that the fractional parts of the partial sums of an infinite series (of real terms) diverging to +∞ or −∞, in which the general term tends to zero, are everywhere dense in the closed unit interval. This result was extended to series of infinite oscillation (see Remark 4.1 of the said paper) on the argument that a sequence of partial sums having infinite oscillation has a subsequence that diverges to +∞ or −∞.

Let R be an associative ring with identity element which has Krull dimension on the left (see [3] for the relevant definitions). For any ordinal α, and any module RX, let τα(X) denote the sum in. X of all cyclic submodules of Krull dimension less than α. (The Krull dimension of a module X will be denoted by |x|.) It is clear that if X' is a submodule of X, then τα(X) = X' ∩ τα(X) and that f[τα(X)] ⊆ τα(Y) for any R-homomorphism f: X→Y. Thus τα defines a torsion preradical.