We consider the nonconvolution initial value problem

where μ is a small positive parameter, b(t, s) is a given real kernel, and F, g are given real functions. For the convolution case b(t,s) = a(t − s). Lodge, McLeod, and Nohel recently established many qualitative properties of the solution of (+); we extend their results to the general nonconvolution problem. In particular, conditions are given that ensure that the solution of (+) decreases to a limiting value α(μ) > 1 as t → ∞.

The publication of the results in this paper has been delayed, for non-mathematical reasons. However, the author has given lectures on these results in Smolenice [2], Gainesville, Tulane. Riverside (1971), Milan (1972) and in Paris. The main consideration in this paper is the notion of a fragmented ring and its multiplicative semigroup. A fragmented ring is a ring with an identity having a finite set of idempotents, these commuting and therefore being central. In a subsequent paper we shall consider associated ideas in a purely semigroup-theoretic context and, in so doing, point out some differences between general semigroups and those semigroups that are the multiplicative semigroups of rings.

The connection between the structure of a near-ring and that of the group on which it acts is used to obtain results concerning the structure of near-rings. A generalized R series is defined for an R module, where R is a zero-symmetric left near-ring, and it is shown that all R modules have maximal R series. The idea of a near-ring which annihilates a series is introduced and some easy consequences of the definition are pointed out. Semi-primitive near-rings are introduced and a general structural result connecting the last two ideas is given. Some special cases which generalize earlier results on endomorphism near-rings are stated. Finally some of the limitations of the idea of semi-primitive near-rings are shown, and some applications are given, in particular to the endomorphism near-rings of soluble groups and of the symmetric groups.

We study the initial value problem for the nonlinear Volterra integrodifferential equation

where μ > 0 is a small parameter, a is a given real kernel, and F, g are given real functions; (+) models the elongation ratio of a homogeneous filament of a certain polyethylene which is stretched on the time interval (— ∞ 0], then released and allowed to undergo elastic recovery for t > 0. Under assumptions which include physically interesting cases of the given functions a, F, g, we discuss qualitative properties of the solution of (+) and of the corresponding reduced problem when μ = 0, and the relation between them as μ → 0+, both for t near zero (where a boundary layer occurs) and for large t. In particular, we show that in general the filament does not recover its original length, and that the Newtonian term —μy′ in (+) has little effect on the ultimate recovery but significant effect during the early part of the recovery.

Techniques from the theory of partial differential equations are employed to prove the uniform convergence of the eigenfunction expansion associated with a two-parameter system of ordinary differential equations of the second order.

This paper studies a linked system of second order ordinary differential equations

where xx ∈ [ar, br] and the coefficients qrars are continuous, real valued and periodic of period (br − ar), 1 ≤ r,s ≤ k. We assume the definiteness condition det{ars(xr)} > 0 and 2k possible multiparameter eigenvalue problems are then formulated according as periodic or semi-periodic boundary conditions are imposed on each of the equations of (*). The main result describes the interlacing of the 2k possible sets of eigentuples thus extending to the multiparameter case the well known theorem concerning 1-parameter periodic equation.

The paper is concerned with giving sufficient conditions that in the non-linear boundary-value problem

there should be no secondary bifurcation, i.e. that, given a branch of solutions (u, λ) bifurcating from the trivial solution, there should be no further bifurcation on that branch. Sufficient conditions on G are given which include, for example, Kolodner's problem of the motion of a heavy rotating string.

A Neumann boundary value problem for the equation rot μ − λμ = u is considered. The approach is by an integral equation method based on Cauchy's integral formula for generalized harmonic vector fields. Results on existence and uniqueness are obtained in terms of the familiar Fredholm alternative.

In the coordinatization of lattices by Baer semigroups, two notable gaps that remain to be filled concern the coordinatization of modular and distributive lattices. In this paper we present coordinatizations of modular lattices. In a subsequent paper we shall deal with the distributive case. Here we show that a bounded lattice L is modular if and only if L can be coordinatized by a Baer semigroup S such that if eS, fS ∊ R(S) then there exist idempotents ē, ∊ S such that ēS = eS, S = fS and e¯, commute; equivalently, if and only if L can be coordinatized by a Baer semigroup S such that if eS, fS ∊ R(S) with e idempotent then there is an idempotent such that S = fS and e = ee.

This paper considers semilinear elliptic boundary value problems of the form

where the partial derivative ∂f/∂u is bounded above by the least eigenvalue of the linear elliptic operator L. Existence and uniqueness of solutions is proved by using monotone operator theory and sub and supersolution techniques.

A new proof of the Holley-Preston generalisation of the Fortuin-Kastelyn-Ginibre inequalities is given, and Batty's extension to the case of infinite products is discussed briefly. An application of the theorems in combinatorial probability theory is described.

We consider the Friedrichs extension A of a minimal Sturm-Liouville operator L0 and show that A admits a Schrödinger factorization, i.e. that one can find first order differential operators Bk with where the μk are suitable numbers which optimally chosen are just the lower eigenvalues of A (if any exist). With the help of this theorem we derive for the special case L0u = −u″ + q(x)u with q(x) → 0 (|x| → ∞) the inequality

σd(A) being the discrete spectrum of A. This inequality is seen to be sharp to some extent.

In this paper a combination is given of the two methods mentioned in the title with the aim of optimising the definition of the absolute temperature with respect to physical transparency and mathematical simplicity. The mathematical tools used are the elements of vector analysis, i.e. “usual” mathematics (“normale” Mathematik) in the sense of Born [1, p. 162, cf. also 22, p. 579].

The existence of periodic solutions is proved for first order vector ordinary and functional differential equations when the right-hand side satisfies a one-sided growth restriction of Wintner type together with some conditions of asymptotic nature. Special cases in the line of Landesman-Lazer and of Winston are explicited.

In this paper we study diffusion-reaction equations arising in the theory of chemical reactions. We prove the global existence and uniqueness of a solution without any restriction for the Lewis number and the Biot numbers. In addition, it is shown that there is at least one stationary state which is globally asymptotically stable (in a rather strong sense) provided the Thiele number is sufficiently small. These results are obtained as special cases from much more general results on semilinear parabolic systems, derived below.

It has been known for some time that certain least-squares problems are “ill-conditioned”, and that it is therefore difficult to compute an accurate solution. The degree of ill-conditioning depends on the basis chosen for the subspace in which it is desired to find an approximation. This paper characterizes the degree of ill-conditioning, for a general inner-product space, in terms of the basis.

The results are applied to least-squares polynomial approximation. It is shown, for example, that the powers {1, z, z2,…} are a universally bad choice of basis. In this case, the condition numbers of the associated matrices of the normal equations grow at least as fast as 4n, where n is the degree of the approximating polynomial.

Analogous results are given for the problem of finite interpolation, which is closely related to the least-squares problem.

Applications of the results are given to two algorithms—the Method of Moments for solving linear equations and Krylov's Method for computing the characteristic polynomial of a matrix.

Existence and uniqueness theorems are established for dual trigonometric equations having right-hand sides that are given functions of bounded variation. The first equation in each pair has coefficients, say {Jn (n + h)} or (jn (n + h – ½)}, and the second equation coefficients {jn )}, where h is a nonnegative constant. A potential problem involving mixed boundary conditions of first and third kind is associated with each dual series. The potential problem is analysed using a stepwise perturbation procedure involving solutions in powers of h. The analysis demonstrates that the present dual series problem can be resolved if the dual series problem associated with the case h = 0 is solvable, the latter being a result obtained earlier.