The effect of heat transfer on boundary-layer growth is considered for the problem of a main stream of small Mach number which is instantaneously released past a heated cylinder, or alternatively the cylinder is accelerated impulsively into a stream at rest. In particular the effect of different values of the given wall temperature on the interval to separation and the thickness of the boundary layer is investigated. It is found that the former is decreased and the latter increased by the presence of heat transfer.

The interaction of conduction electrons with a vibrating metallic lattice has been considered by Ziman in terms of non-adiabatic perturbations to the adiabatic approximation of the Born-Oppenheimer theory. This programme is continued to calculate the self-energy of the electrons. When account is taken of the dependence of the energy levels of the adiabatic Hamiltonian itself upon the electron configuration, the final result is the same as the standard one.

Let S denote a compact, connected, orientable surface with genus g and h boundary components. We refer to S as a surface of genus g with h holes. Let [Mscr ]S denote the mapping class group of S, the group of isotopy classes of orientation-preserving homeomorphisms S → S.

Let G be a group. G is hopfian if every homomorphism from G onto itself is an automorphism. G is residually finite if for every g ∈ G with g ≠ 1 there exists a normal subgroup of finite index in G which does not contain g. Every finitely generated residually finite group is hopfian ([11, 12]). A group G is hyperhopfian ([2, 3]) if every homomorphism ψ G → G with ψ(G) normal in G and G/ψ(G) cyclic is an automorphism. As observed in [14], examples of hopfian groups which are not hyperhopfian are afforded by the fundamental groups of torus knots.

By a result of Grossman [5], [Mscr ]S is residually finite. Since [Mscr ]S is also finitely generated, it is hopfian. It is a natural question to ask whether [Mscr ]S is hyperhopfian. In this paper, we shall answer a more general question. We say that a group G is ultrahopfian if every homomorphism ψ: G → G with ψ(G) normal in G and G/ψ(G) abelian is an automorphism. Note that an ultrahopfian group is hyperhopfian. We shall prove the following result.

If f is a real, indefinite, binary quadratic form of discriminant d and if κ(f) is the minimum of | f | taken over all integer values of x, y, not both zero, then it is well known that and that this is a ‘best possible’ result.

Let S(ℝn) be the space of Schwartz class functions and S'(ℝn) be the dual space of S(ℝn). Given a k-dimensional manifold M in ℝn with area (in case k = 1, M is a curve with length), then for 1 ≤ p ≤ ∞ we say that M has the p-approximate property of for each TεS'(ℝn) with supp T⊆M, and εLp(ℝn), we can find a sequence of measures {Tj,j = 1,2,…} on M, absolutely continuous with respect to the area measure on M, such that ∥j-∥p → 0 as j → ∞.

In his paper 10 the author investigated the structure of m-full ideals by analysing their syzygies and, as one special case, showed how the Betti numbers of Borel stable ideals over polynomial rings can be computed. The same result, among other things, was also obtained by Eliahou and Kervaire1 by a different method.

The complete (or “extended”) symmetry groups, investigated in Part I, are certain groups of orthogonal transformations, generated by reflections. Every such group has a subgroup of index two, consisting of those transformations which are of positive determinant (i.e., “movements” or “displacements”). The positive subgroup (in this sense) of [k1, k2, …, kn−1] is denoted by [k1, k2, …, kn−1]′, and is “the rotation group” (or, briefly, “the group”) of either of the regular polytopes {k1, k2, …, kn−1}, {kn−1, kn−2, …, k1}; e.g., [3, 4]′ is the octahedral group.

Let F a totally real cubic field. For any number α in F, let α, α′, α″ denote the conjugates of α. Define the function T(α) by

.

In [11], Partington proved that if λ is a Banach sequence space with a monotone basis having the Banach-Saks property, and (Xn) is a sequence of Banach spaces each having the Banach-Saks property, then the vector sequence space ΣλXn has this same property. In addition, Partington gave an example showing that if λ and each Xn, have the weak Banach-Saks property, then ΣλXn need not have the weak Banach-Saks property.

In this paper we define two order relations between every two partizan games with possible draw positions. We use these relations to prove two ‘simplest form’ theorems, and to describe an algorithm for computing the simplest form of the ‘finite games’ which are an extension of Conway's short games.

This paper is concerned with the small transverse displacement of an infinite thin plane isotropic plate due to the application of a transverse force applied at an arbitrary point of the plate. The plate has its outer edge free, and is clamped along and bounded internally by a closed curve that can be mapped onto the unit circle by means of a polynomial. Three polynomials are considered and in each of these cases the deflexion is obtained in finite terms. Circular and elliptic holes as well as curvilinear polygonal holes are included as special cases.

Let $f$ be an entire function. For $n \in {\bb N}$, let $f^n$ denote the $n$th iterate of $f$. The set \[ F(f)=\{z:(f^n)\hbox{ is normal in some neighbourhood of }z\} \] is the Fatou set or the set of normality and its complement $J(f)$ is the Julia set. If $U$ is a component of $F$$(f)$, then $f$$(U)$ lies in some component $V$ of $F$$(f)$. If $U_n\cap U_m=\phi$ for $n \ne m$ where $U_n$ denotes the component of $F$$(f)$ which contains $f^n(U)$, then $U$ is called a wandering domain, else $U$ is called a pre-periodic domain, and if $U_n = U$ for some $n \in {\bb N}$ then $U$ is called periodic domain.

It is known that for entire functions $f$ and $g$, $f$$(g)$ has wandering domain if and only if $f$$(g)$ has wandering domain. In this paper we discuss the existence of wandering domains of composite entire functions with regards to its factors $f$ and $g$.

Fine topologies are discussed for Minkowski space. The conformal group and projective group have representations as C1 groups of homeomorphisms of various topologies.

An expansion of the Whittaker function Wk, m(z) in terms of Bessel functions is given for 2m an integer or zero. Simple formulae are derived for the cases of m = 0, ½, 1 and . The case m = 0 provides expansions for the exponential, sine, cosine and similar integrals. The cases m = ½ and provide expansions for the irregular Coulomb wave functions having angular momenta zero and one.

Let (R, 𝔪) be a Noetherian local ring and I an arbitrary ideal of R with analytic spread s. In [3] the authors proved the existence of a chain of ideals I ⊆ I[s] ⊆ ⋅⋅⋅ ⊆ I[1] such that deg(PI[k]/I) < s − k. In this article we obtain a structure theorem for this ideals which is similar to that of K. Shah in [10] for 𝔪-primary ideals.

A class of regular semigroups closed under taking direct products, regular subsemigroups and homomorphic images is an e(xistence) -variety of regular semigroups. The class of all combinatorial strict regular semigroups is the e-variety generated by the five element non-orthodox completely 0-simple semigroup and consists of all regular subdirect products of combinatorial completely 0-simple semigroups and/or rectangular bands. The bifree object on the set X in is the natural concept of a ‘free object’ in the class . is generated by the set X and the set of formal inverses X′ under the two binary operations of multiplication · and forming the sandwich element ∧A. Hence is a homomorphic image of the absolutely free algebra of type 〈2, 2〉 generated by X ∪X′. In this paper we shall describe the associated congruence on F〈2, 2〉(X∪X′) and construct a model of in terms of sets and binary relations. As an application, a model of the free strict pseudosemilattice on a set X is obtained.

One method of measuring frequency depends upon resonance. If a body of unit mass, subject to elastic constraints and experiencing a damping resistance proportional to its speed, be acted on by a periodic force F sin pt, the equation of motion may be written

The amplitude of the steady vibration ultimately set up is

the maximum value occurring when p2 = q2 − 2s2. If the motion, in the absence of the dri ving force F sin pt, be only slightly damped, s2 is very small compared with q2, and little error will be made if the value of p giving maximum amplitude be taken as equal to q. When s/q is small, the resonance is “sharp,” i.e. a small discrepancy between p and q, or, more strictly, between p and (q2 − 2s2)½, causes the amphtude to fall far below its maxìmum value. Thus, when the damping is small, maximum amplitude, f or variation of p, indicates that p is very nearly equal to q.

We determine the Thurston–Bennequin invariant of graph divide links, which include all closed positive braids, all divide links and certain negative twist knots. As a corollary of this and a result of P. Lisca and A. I. Stipsicz, we prove that the 3-manifold obtained from $S^3$ by Dehn surgery along a non-trivial graph divide knot K with coefficient r carries positive, tight contact structures for every r except the Thurston-Bennequin invariant of K.

Suppose that we have a set of observed frequencies classified in groups, the number in a typical group being nr, and that we wish to estimate the expectations in the groups according to some law with adjustable parameters. If these expectations are mr we take

Σ denoting summation over the groups, and according to the law in question the probability that one observation should come in the rth group is mr/N. The joint probability, for given mr, that there should be n1 observations in the first group, n2 in the second and so on, is therefore

1. In a paper “Beiträge zur Inversionsgeometric,” which will appear in the forthcoming volume of the Science Reports of the Tôhoku Imperial University, I have treated the problem of determining the necessary and sufficient condition in order that two plane curves which are given by natural equations should be transformable into each other by inversion. In connection with that paper I propose here to treat the analogous problem for Laguerre transformations:

Having given two plane curves, by natural equations, in which the functional relations are all supposed to be analytic, it is required to determine the necessary and sufficient condition in order that the two plane curves should be transformable into each other by a Laguerre transformation.