A new method for studying inequalities of the type ‖y(r)‖2<ε‖Sk−ry(k)‖2 + K(ε)‖S−ry‖2 and ‖y′‖2≦ Kp(S)‖Sy″‖ ‖S−1y‖ is presented here. With this new approach we obtain new and far reaching extensions of previously known inequalities of this sort as well as simpler proofs of the known cases. In addition we obtain an inequality of type ‖Sy′‖<ε‖(Sy′)′‖ + K(ε)‖y‖ for a general class of functions S. Also we give an elementary operator-theoretic proof of Everitt's characterization of the best constant as well as all cases of equality for

This paper deals with the oscillatory and asymptotic behaviour of all solutions of a class of nth order (n > 1) non-linear differential equations with deviating arguments involving the so called nth order r-derivative of the unknown function x defined by

where r1, (i = 0,1,…, n – 1) are positive continuous functions on [t0, ∞). The results obtained extend and improve previous ones in [7 and 15] even in the usual case where r0 = r1 = … = rn–1 = 1.

This paper obtains, under certain general conditions on the coefficient q, a best-possible upper bound on the real parameter λ for the differential equation

to have a non-trivial solution in the integrable-square space L2 (a, ∞).

Let Vrs, s = 1 …, K be Hermitian operators on Hilbert spaces Hr, r = 1, …, k. For x = x1⊗…⊗xk ∊ H1⊗…⊗Hk we define Δx by the formal determinantal expansion Δx = ⊗det{Vrsxr}. Δ is extended to all of H by linearity and continuity. The paper presents results concerning positivity properties of Δ on decomposable tensors.

It is shown that the fine structure of the asymptotic estimates for the eigenvalues of a large class of ordinary differential operators, can be described in terms of the Fourier coefficients of a function of class L2.

This paper deals with conditions which guarantee that a meromorphic function on the plane cannot satisfy any algebraic differential equation having coefficients in a given field of meromorphic functions. Some of the conditions are of growth type, while others depend on a representation for the function.

Uniformly valid asymptotic approximations are presented for solutions of the angular equations associated with the problem of diffraction by a plane angular sector. Error estimates are provided for all approximations. The asymptotic variable is related to the number of zeros of the solutions of the angular equations and expressions for the eigenvalues of the equations are presented in decoupled form.

Formulae are derived for the numbers of completely 0-simple and completely simple semigroups with m ℒ-classes, n ℜ-classes and underlying finite group G.

In this paper we study the stabilization problem for non autonomous control processes in Hilbert spaces. We prove that a stabilizing feedback exists if and only if an associated Riccati equation has a bounded solution which is symmetric and positive definite.

An application to control processes with delays in control is presented.

When a sheet of paper is crumpled in the hands and then crushed flat against a desk-top, the pattern of creases so formed is governed by certain simple rules. These rules generalize to theorems on folding Riemannian manifolds isometrically into one another. The most interesting results apply to the case in which domain and codomain have the same dimension. The main technique of proof combines the notion of volume with Hopf's concept of the degree of a map.

Let S be a Clifford semigroup with identity. The weak containment question is posed for S, and answered affirmatively when each of the maximal groups Se in S is amenable. The amenability of S itself is characterised in terms of PL(S), the set of normalised positive definite functions on S arising from the left regular representation of S. A type of mean associated with PL(S) and satisfying a condition weaker than left invariance is introduced.

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.