Consider a differential game of survival governed by the differential equation

in , with pay-off

where tF is the entry time of the trajectory (t, x(t)) into a given terminal set F. Under suitable conditions on f, g, h and the terminal set F, it was shown in (3) that the question of existence of value of such a game can be approached by considering a certain pair of partial differential equations called the Isaacs-Bellman equations.

The flow of a rarefied gas between parallel plates has been studied via the linearized Boltzmann transport equation. Using a general boundary condition, which includes an arbitrary ratio of specular to diffuse reflection from the wall, we have derived an integral equation for the mass flow velocity. The integral equation is solved by using a replication property of the kernel and application of the method of Muskelishvili.

The total volumetric flow rate is obtained and a slip boundary condition is deduced for use with the hydrodynamic equations.

Certain aspects of the eigenvalue spectrum associated with the Boltzmann equation are discussed.

In the course of a series of signal intensity measurements made at Cambridge on the electric waves received from broadcasting stations a marked difference between day and night conditions has been observed. The day-time signals are sensibly constant but marked fluctuations of intensity become apparent about sunset and continue throughout the night. These variations are detectable at Cambridge on the signals from London, where they represent a change in strength of about 5 per cent.

We present a class of approximately finite dimensional C*-algebras whose local multiplier algebras are simple with real rank zero and stable rank one.

In this paper we describe the relationship between characters of finitely generated torsion-free nilpotent groups of class 2 and their completions. In particular we give a detailed description of this connection for the discrete Heisenberg group.

The aim of the present note is to point to a strange fact, through an example, in connexion with the asymptotic distribution of sums of a random number of independent and identically distributed (i.i.d.) random variables. We shall formulate our conclusion after the introduction of some notations.

The purpose of this paper is to study finite simple groups whose Sylow 2-subgroups are isomorphic to those of PSL(5, q) for some odd q. This work originally appeared in the author's Ph.D. Thesis, (14), and at that time, the structure of the centralizers of involutions was worked out – see below for details. Subsequently, in view of the fact that if q ≡ – 1 (mod 4), the Sylow 2-subgroup of PSL(5, q) has sectional 2-rank 4 (being, in fact, isomorphic to the wreath product of a semidihedral group by a cyclic group of order 2), the complete classification, at least for this case, was needed for Gorenstein and Harada's monumental work on sectional 2-rank at most 4, (8), and therefore Collins and Solomon completed the characterization of PSL(5, q) and PSU(5, q) (for all odd q) in a neat paper, (5). Combining our result with theirs, we are able to state the following Theorem.

Dense immersions occur frequently in Lie group theory. Suppose that exp: g → G denotes the exponential function of a Lie group and a is a Lie subalgebra of g. Then there is a unique Lie group ALie with exponential function exp:a → ALie and an immersion f:ALie→G whose induced morphism L(j) on the Lie algebra level is the inclusion a → g and which has as image an analytic subgroup A of G. The group Ā is a connected Lie group in which A is normal and dense and the corestriction

is a dense immersion. Unless A is closed, in which case f' is an isomorphism of Lie groups, dim a = dim ALie is strictly smaller than dim h = dim H.

In this paper we investigate the Hausdorff dimension of the limit set of an infinitely generated discrete subgroup of hyperbolic isometries and obtain conditions for the limit set to have full dimension.

Following the method outlined in a previous paper, exact solutions in finite terms are obtained for the problem of an infinitely large plate with outer edge free and an inner free circular boundary, the plate being supported at any number of interior or boundary points and normally loaded over a circle. The load considered is symmetrical with respect to the centre of the circle.

1. There seems to be little doubt that the detailed mechanism of ionisation or excitation by electron impact will not admit of description, even to a first approximation, by the system of classical mechanics. However the experimental work of Franck and Horton and their collaborators, has shown that, within the limits of the error of experiment, an electron is able to impart the whole or part of its kinetic energy to the atom it excites. When its initial energy is greater than the excitation energy, the electron, instead of being brought to rest, retains the excess as kinetic energy.

Let Zn = n−½(X1 + X2 + … + Xn), where {Xn} is a sequence of independent and identically distributed random variables with EX1 = 0, and a common distribution function F and characteristic function ω. Suppose |ω|r is integrable for some integer r ≥ 1. For all n ≥ r, then Zn has a probability density function fn obtained by using the inversion formula.

The following generalization of a theorem of Weyl appeared in part I of this series of papers. Let G be a locally compact Abelian group with dual group ĝ. Let be a sequence in ĝ, not too slowly growing in a certain precise sense. Then, provided ĝ has ‘not too many’ elements of finite order, the sequences

are uniformly distributed on the circle, for almost all x in G.

In an investigation of the disintegration of by protons, resulting in the emission of three α-particles, Dee and Gilbert* found that the observed distributions of ranges and angles of these particles could be explained by assuming that one α-particle is first ejected, leaving a short-lived, excited nucleus which then decomposes into two more α-particles. The half-life of this excited beryllium nucleus was estimated to be about 10–21 sec.

Let GL2(R) be the general linear group of 2 × 2 invertible matrices in M2(R) over a commutative ring R with 1 and SL2(R) be the special linear group consisting of 2 × 2 matrices over R with determinant 1. In this paper we determine the homomorphisms from GL2 and SL2, as well as their projective groups, over Laurent polynomial rings to those groups over Gaussian domains, i.e. unique factorization domains (cf. Theorems 1, 2, 3 below). We also consider more generally the homomorphisms of non-projective groups over commutative rings containing a field which are generated by their units (cf. Theorems 4 and 5). So far the homomorphisms of two-dimensional linear groups over commutative rings have only been studied in some specific cases. Landin and Reiner[7] obtained the automorphisms of GL2(R), where R is a Euclidean domain generated by its units. When R is a type of generalized Euclidean domain with a degree function and with units of R and 0 forming a field, Cohn[3] described the automorphisms of GL2(R). Later, Cohn[4] applied his methods to the case of certain rings of quadratic integers. Dull[6] has considered the automorphisms of GL2(R) and SL2(R), along with their projective groups, provided that R is a GE-ring and 2 is a unit in R. McDonald [9] examined the automorphisms of GL2(R) when R has a large unit group. The most recent work of which we are aware is that of Li and Ren[8] where the automorphisms of E2(R) and GE2(R) were determined for any commutative ring R in which 2, 3 and 5 are units.

The propagator for a particle in the presence of a one-dimensional square-well potential is calculated. The related t-function is written down in Fourier transform. The analytic properties of the propagator and t-function are discussed.

It is shown that the functions which are analytic in tubular radial domains and satisfy certain growth conditions have distributional boundary values in the weak topology of (WΩ)′-space. Representation of analytic functions in terms of distributional boundary values are given. Converse results are also obtained. An analytic decomposition theorem is proved. The main theorems are established by means of a number of lemmas concerning WM, WΩ spaces and their dual spaces. Several new lemmas are proved for K{Mp} spaces from which results for WM-spaces can be easily deduced.

We generalize a theorem of R. Thomas, which sometimes allows one to tell by inspection that a finitely presented group G is infinite. Groups to which his theorem applies have presentations with not too many more relators than generators, with at least some of the relators being proper powers. Our generalization provides lower bounds for the ranks of the abelianizations of certain normal subgroups of G in terms of their indices. We derive Thomas's theorem as a special case.

Let F be a closed connected and non-orientable surface smoothly embedded in the 4-sphere S4 with normal Euler number e(F) = 0. We note that if e(F) = 0, then the non-orientable genus n is even (ef. [7]) and the tubular neighbourhood N(F) of F in S4 which is a D2-bundle over F has a trivial I-subbundle. Let τ be a trivial I-subbundle of N(F) and let τ* = F × I ⊂ N(F) be its orthogonal I-subbundle which is twisted. Then is a closed connected genus n – 1 orientable surface smoothly embedded in S4 and doubly covers F. We call this surface a doubled surface of F in S4 (associated with τ). If a trivial I-subbundle τ is given, then we see that the knot type of F* ⊂ S4 is uniquely determined.

We prove some elementary intuitive estimates on moving boundaries hitting times by one-dimensional Brownian motion (in ℝ and on the circle). These results give an alternative approach to Beurling's radial projection theorem on harmonic measure in a disc.