The object of this paper is to extend the range of validity of Ingham's Tauberian theorem for partitions (2). Ingham's theorem deals with many types of partitions, but it cannot be used in some cases of importance. With a slight modification of Ingham's notation we define

where P(u) is the number of solutions of

in integers ni ≤ 0, and the λi are a given set of numbers, not necessarily integers, such that 0 < λ1 < λ2 <.… If N(u), the number of λi not exceeding u, is given by

where B > 0, β > 0, and as u → ∞

then, according to Ingham's theorem, when u → ∞

where

A numerical solution of

with general linear boundary conditions along x = ±1, is described where at any time t the Chebyshev expansion of θ(x, t) in –1 ≤ x ≤ 1 is computed directly. Compared with the more usual finite difference methods, this method requires much less computation and there are no stability problems. Two cases are considered in detail.

The class of fundamental groups of non-singular complex projective varieties is an interesting, but as yet imperfectly understood, class of finitely presented groups. Membership of is known to be extremely restricted (see [22, 23]). In this paper, we employ geometrical rigidity properties to realize some group extensions as elements of as in our previous papers, we find it convenient to work simultaneously with the class ℋ of fundamental groups of compact Kähler manifolds.

Theorem. Let K be a bounded star-bodyin (n + 1)-dimensional space symmetric about the origin and of volume V < 2ζ(n +1). Then there is a lattice of unit determinant with a point at the origin and having no other points in K. (See (1, 2,3,4).)

Throughout this paper, k denotes a commutative ring. We will develop a theory of homological finiteness conditions for modules over certain graded k-algebras which generalizes known theory for group algebras. The simplest of our results, Theorem A below, generalizes certain results of Aljadeff and Yi on crossed products of polycyclic-by-finite groups (cf. [1, 11]), but also applies to many other crossed products in cases where little was previously known. Before stating the results, we recall definitions of graded and strongly graded rings. Let G be a monoid. Naively, a G-graded k-algebra is a k-algebra R which admits a k-module decomposition,

in such a way that Rg Rh ⊆ for all g, h ∈ G. If R is a G-graded k-algebra and X is any subset of G, then we write Rx for the k-submodule of R supported on X; that is

Note that if H is a submonoid of G then RH is a subalgebra of R.

We prove that for homogeneous spaces of isotropic Grassmannians the Borel map sends the basis of a truncated algebra of Schur Q-functions consisting of Q-functions or P-functions (depending on a case) onto the basis dual to the basis of Schubert cycles.

A transform method for the solution of linear difference equations, analogous to the method of the Laplace transform in the field of linear differential equations, has been described by Stone (1). The transform u(z) of a sequence un is defined by the equation

In this paper we consider the problem of distribution of stress in an infinite wedge of homogeneous elastic isotropic solid under the usual assumptions pertaining to plane strain in classical (infinitesimal) theory of elasticity. The wedge is supposed to occupy the region 0 ≤ ρ ≤ ∞, − α ≤ θ ≤ α (in plane polar coordinates) where the pole is taken on the apex of the wedge with the line bisecting the wedge angle as the initial line. We report here the investigations of two types of stress fields: (i) the stress field which is set up by the application of known pressure to inner surfaces of a crack situated on the bisector of the wedge angle, and (ii) the stress field generated by the indentation of the plane faces of the wedge by the rigid punch. The corresponding boundary-value problems are shown to be equivalent to the problem of solving dual integral equations involving inverse Mellin-transforms. Similar equations have been discussed by Srivastav ((l)) in a recent paper. The method of (1) is easily modified to reduce the dual equations to single Fredholm equations of the second kind which are best solved numerically. The boundary-value problems discussed here are of the mixed type and appear to have been considered only recently by Matczynski ((2)) who has investigated a contact problem. He identifies the problem with a Wiener-Hopf integral equation and solves the integral equation approximately using a method due to Koiter ((3)). The method presented here seems to have the advantage that it requires no elaborate tools of analysis which are necessary for resolving the problem of Wiener-Hopf and numerical results may be obtained comparatively easily.

A parametric bootstrap estimate (PB) may be more accurate than its non-parametric version (NB) if the parametric model upon which it is based is, at least approximately, correct. Construction of an optimal estimator based on both PB and NB is pursued with the aim of minimizing the mean squared error. Our approach is to pick an empirical estimate of the optimal tuning parameter ε∈[0, 1] which minimizes the mean square error of εNB+(1−ε) PB. The resulting hybrid estimator is shown to be more reliable than either PB or NB uniformly over a rich class of distributions. Theoretical asymptotic results show that the asymptotic error of this hybrid estimator is quite close in distribution to the smaller of the errors of PB and NB. All these errors typically have the same convergence rate of order . A particular example is also presented to illustrate the fact that this hybrid estimate can indeed be strictly better than either of the pure bootstrap estimates in terms of minimizing mean squared error. Two simulation studies were conducted to verify the theoretical results and demonstrate the good practical performance of the hybrid method.

The invariants of 3-manifolds defined by Kuperberg for involutory Hopf algebras and those defined by the authors for spherical Hopf algebras are the same for Hopf algebras on which they are both defined.

A composite number n is called a pseudoprime if n|2n− 2.

Theorem 1. If a and b are natural numbers such that (a, b) = 1, then there exist infinitely many pseudoprimes of the form ax + b, where x is a natural number.

The proof of this theorem is given by the author in (5). This proof is based on the following two lemmas.

In [1], Asimov and Franks give conditions under which a collection of periodic orbits of a diffeomorphism f:M→M of a compact manifold persists under arbitrary isotopy of f. Together with the Nielsen–Thurston theory, their result has been of pivotal importance in recent work on the periodic orbit structure of surface automorphisms (for example [3, 4, 7, 8, 9, 12, 13]). However, their proof uses bifurcation theory and as such depends crucially upon the differentiability of f. The periodic orbit results which make use of the Asimov–Franks theorem are therefore applicable only in the differentiable case, a limitation which belies their topological character. In this paper we shall use classical Nielsen-theoretic methods to prove the analogue of the Asimov–Franks result for homeomorphisms.

We study locally constant coefficients. We first study the theory of homotopy Kan extensions with locally constant coefficients in model categories, and explain how it characterizes the homotopy theory of small categories. We explain how to interpret this in terms of left Bousfield localization of categories of diagrams with values in a combinatorial model category. Finally, we explain how the theory of homotopy Kan extensions in derivators can be used to understand locally constant functors.

Kendall and Le [4] gave an exact algorithm for the computation of a polygonally generated (auxiliary) shape-density m̃(x, y) which we used to obtain ‘exact’ numerical densities for given (x, y). Here I derive an explicit formula for m̃(x, y), for an arbitrary convex plane polygon K, valid when the shape-point (x, y) lies in what will be called the ‘(upper) basic tile’ of the associated singular tessellation. In most circumstances this is all that is needed for the statistical analysis of collinearities.

The calculations of the velocity and temperature distributions for the turbulent flow in axially symmetrical and plane jets are described.

For axially symmetrical jets Tollmien's results derived from calculations based on the momentum theory are stated; the calculations on the basis of the modified vorticity transfer theory and on the vorticity transfer theory with symmetrical turbulence are carried out. Unfortunately it seems to be true that none of these theories provides results in good agreement with experiment for both the velocity and temperature distributions.

For plane jets the results obtained by Tollmien and based on the momentum transfer theory are stated; the calculations on the basis of the vorticity transfer theory are carried out. The velocity distributions on both theories are identical and in fair agreement with experiment. In order to test the validity of the theories it is necessary to compare the theoretical results for the temperature with experiment. No results for the latter seem to have been published.

Fix a prime integer p. We show that a closed orientable 3-manifold M admits an action of Zp with the fixed point set S1 if and only if M can be obtained as the result of surgery on a p-periodic framed link L and Zp acts freely on the components of L. We prove a similar theorem for free Zp-actions. As an interesting application, we prove the following, rather unexpected result: for any M as above and for any odd prime p, H1(M, Zp) ≠ Zp. We also prove a similar criterion of 2-periodicity for rational homology 3-spheres.

A group G is defined to be a B-group if any primitive permutation group which contains G as a regular subgroup is doubly transitive. In the case where G is finite the existence of families of B-groups has been established by Burnside, Schur, Wielandt and others and led to the investigation of S-rings. A survey of this work is given in [3], sections 13·7–13·12. In this paper the possibility of the existence of countable B-groups is discussed. Three distinct methods are given to embed a countable group as a regular subgroup of a simply primitive permutation group, and in each case a condition on the square root sets of elements of the group is necessary for the embedding to be carried out. It is easy to demonstrate that this condition is not sufficient, and the general question remains open.

In this paper we study the symplectic relations appearing as the generalized cotangent bundle liftings of smooth stable mappings. Using this class of symplectic relations the classification theorem for generic (pre) images of lagrangian submanifolds is proved. The normal forms for the respective classified puilbacks and pushforwards are provided and the connections between the singularity types of symplectic relation, mapped lagrangian submanifold and singular image, are established. The notion of special symplectic triplet is introduced and the generic local models of such triplets are studied. We show that the open swallowtails are canonically represented as pushforwards of the appropriate regular lagrangian submanifolds. Using the SL2(ℝ) invariant symplectic structure of the space of binary forms of n appropriate dimension we derive the generating families for the open swallowtails and the respective generating functions for its regular resolutions.