A criterion of global equivalence of linear homogeneous differential equations of the n-th order, n ≧ 3, is derived, which is in general effective, i.e. expressible in terms of coefficients and quadratures.

Let A and B be commutative Noetherian local rings such that B contains A and B is flat and integral over A. It is shown that if M is a balanced big Cohen-Macaulay A-module (that is, every system of parameters for A is an M-sequence), then M⊗AB is a balanced big Cohen-Macaulay B-module. An example of a ring A is given such that, if B is the completion of A, then the analogous result is false in this case. This answers a question posed by Riley in the negative.

The inequality (0·1) below is naturally associated with the equation −(pu′)′ + qu = λu. By assuming that one end-point of the interval (a, b) is regular and the other limit-point for this equation, Everitt characterized the best constant K in tems of spectral properties of the equation. This paper sketches a theory for more general inequalities (0·2), (0·3) similarly related to the equation Su = λTu. Here S and T are ordinary, symmetric differential expressions. A characterization of the best constants in (0·2), (0·3) is given which generalises that of Everitt.

For the case when S is of order 1 and T is multiplication by a positive function, all possible inequalities are given together with the best constants and cases of equality. Furthermore, an example is given of a valid inequality (0·1) on an interval with both end-points regular for the corresponding differential equation. This contradicts a conjecture by Everitt and Evans. Finally, the general theory for the left-definite inequality (0·3) is specialised to the case when S is a Sturm-Liouville expression. A family of examples is given for which the best constants can be explicitly calculated.

We consider classes of functions satisfying certain simple criteria of sign and smoothness and a decomposition property. It is known that these properties are possessed by Chebysheffian B-splines and it is shown here that they are also possessed by certain trigonometric B-splines. For such a class of functions, we derive a variation-diminishing property and analyse interpolation both on a finite set of nodes and on an infinite, periodically spaced set of nodes. The results are also applied to interpolation by complex polynomial splines on the circle.

In the general theory of ordinary linear quasi-differential equations, the set of Shin–Zettl matrices plays an important role. This paper displays certain properties of these matrices and their behaviour under a special form of transformation. Essentially, the problems can be considered within the framework of linear algebra.

The simple 2q-knots, q ≧ 5, for which contains no ℤ-torsion, have been classified in terms of Hermitian duality pairings on their homology and homotopy modules. In this paper, a necessary and sufficient condition is given for such a knot to be doubly-null-concordant.

Let X be a set with infinite cardinality m and let B be the Baer-Levi semigroup, consisting of all one-one mappings a:X→X for which ∣X/Xα∣ = m. Let Km=<B 1B>, the inverse subsemigroup of the symmetric inverse semigroup ℐ(X) generated by all products β−γ, with β,γ∈B. Then Km = <N2>, where N2 is the subset of ℐ(X) consisting of all nilpotent elements of index 2. Moreover, Km has 2-nilpotent-depth 3, in the sense that

Let Pm be the ideal {α∈Km: ∣dom α∣<m} in Km and let Lm be the Rees quotient Km/Pm. Then Lm is a 0-bisimple, 2-nilpotent-generated inverse semigroup with 2-nilpotent-depth 3. The minimum non-trivial homomorphic image of Lm also has these properties and is congruence-free.

Let , with each pi a finite sum of real multiples of real powers of x, and with D = d/dx. Suppose that pi≧0, pN > 0, and p0≧ε > 0 on the interval [1, ∞). Suppose that there exists a j such that degree pi − 2j > degree pi−2i for all i ≠ j. We compute the deficiency index d(Ln) in L2[l, ∞) of any power of L. It is known that d(L) = N. We give an exact formula for d(Ln) which, if degree pi − 2j > 0, shows that d(Ln) = nN for all n if and only if j = 0, and that the limit as n approaches infinity of d(Ln)/n is N + j. We show in addition that if degree pi − 2i ≦ 0 for all i, d(Ln) = nN for all n.

Let L0, M0 be closed densely defined linear operators in a Hilbert space H which form an adjoint pair, i.e. . In this paper, we study closed operators S which satisfy and are regularly solvable in the sense of Višik. The abstract results obtained are applied to operators generated by second-order linear differential expressions in a weighted space L2(a, b; w).

It is shown that if A is an algebra over a field, then the regularity of the semigroup algebra A[G] implies that the semigroup G is periodic. This enables us to characterize regular semigroup algebras of semigroups with d.c.c. on principal ideals. Also, regular self-injective semigroup algebras are described.

Semilinear parabolic equations of the form u1 = ∇2u + δf(u), where f is positive and is finite, are known to exhibit the phenomenon of blow-up, i.e. for sufficiently large S, u becomes infinite after a finite time t*. We consider one-dimensional problems in the semi-infinite region x>0 and find the time to blow-up (t*). Also, the limiting behaviour of u as t→t*- and x→∞ is determined; in particular, it is seen that u blows up at infinity, i.e. for any given finite x, u is bounded as t→t*. The results are extended to problems with convection.

The modified equation xu, = uxx +f(u) is discussed. This shows the possibility of blow-up at x =0 even if u(0, f) = 0. The manner of blow-up is estimated.

Finally, bounds on the time to blow-up for problems in finite regions are obtained by comparing u with upper and lower solutions.

The dynamical behaviour of a slender rod is analyzed here in terms of a generalization of Euler's elastica theory. The model includes a linear stress-strain relation but nonlinear geometric terms. Properties of the rod may vary along its length and various boundary conditions are considered. A rotational inertia term that is neglected in many theories is retained, and is essential to the analysis. By use of the equivalence of an energy and a Sobolev norm, and by reformulation of the equations as a semilinear system, global existence of solutions is proved for any smooth initial data. Equilibrium solutions that are stable in the static sense of minimizing the potential energy are then proved to be stable in the dynamic sense due to Liapounov.

Explicit orthogonality relations are found for the associated Laguerre and Hermite polynomials. One consequence is the construction of the [n − 1/n] Padé approximation to Ψ(a + 1, b; x)/Ψ(a, b; x), where Ψ(a, b; x) is the second solution to the confluent hypergeometric differential equation that does not grow rapidly at infinity.

Critical Rayleigh numbers are obtained for the onset of convection when the Maxwell–Cattaneo heat flux law is employed. It is found that convection is possible in both heated above and below cases.

For each ring R, we construct a topological space pt (R) which includes as a subspace both the classical spectrum specR and the torsion theoretic spectrum R-sp. For many rings (e.g. rings with Krull dimension), spec R is a retract of pt (R) and the retraction map θ generalizes the Gabriel correspondence for noetherian rings. There is a natural decomposition theory on MOD-R which extends the Goldman theory in the same way that the tertiary theory extends the primary theory. The map θ provides a direct comparison between this new decomposition theory and the tertiary theory. The space pt (R) is closely connected with the lattice of hereditary torsion theories on R, and for fully bounded (not necessarily noetherian) R, this connection is very tight.

We prove that classical solutions of the perturbed wave equation in ℝn × ℝ (n = odd ≧ 3) do not satisfy Huygens' principle in the presence of symmetries. The difficulties arising from the singularities of the Riemann function (for large space dimensions) are overcome by considering a class of potentials and initial data which are radial and smooth. Our method is elementary and based on energy estimates.

We give conditions on pairs of non-negative weight functions U and V which are sufficient that for 1<p≤<∞

where Hλ is the Hankel transformation.

The technique of proof is a variant of Muckenhoupt's recent proof for the boundedness of the Fourier transformation between weighted Lp spaces, and we can also use this variant to prove a somewhat different boundedness theorem for the Fourier transformation.

The existence of solutions to equations in normed spaces is proved when the nonlinear part of the equation satisfies growth and asymptotic conditions, whether the linear part is invertible or not. For this, we use the coincidence degree theory developed by Mawhin. We apply our abstract results to boundary value problems for nonlinear vector ordinary differential equations. In particular, we consider the Picard boundary value problem at the first eigenvalue and the periodic boundary value problem at resonance. In both cases, the nonlinear term can be of superlinear type. Also, necessary and sufficient conditions of Landesman-Lazer type are obtained.

Let E be the set of idempotents in Sn, the semigroup of all singular selfmaps of {1,…, n}. For each α in Sn, there is a unique (κ(α)≧1 such that αψEκ,(α). It is known that κ(α)≦ n + cycl α -fix α, where cyclα is the number of cyclic orbits of a and fix α is the number of fixed points. Equality holds only in the case where a is of rank n – 1. An improved upper bound is obtained for κ(α), applying to elements of arbitrary rank. A lower bound is obtained also.