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This paper is concerned with the convergence of Rayleigh-Ritz approximations to the solution of an elliptic boundary value problem. Although the work arose in connection with the aerofoil problem (and it is to this problem that the results obtained are immediately applied), the methods here employed are suitable for use on the wider class of problem mentioned above.
Coburn [1] has derived the intrinsic form of the characteristic relations, for the steady, supersonic, three-dimensional motion of a polytropic gas. The purpose of this paper is to obtain a generalized form of these relations and to apply them to obtain two classes of complex-screw motions [2].
Although there is no need for a ‘distinguished’ submodule to be given a formal definition in the present paper, we like to indicate the meaning attached to this concept here. Perhaps the shortest way of doing so is to say that a distinguished submodule is a (covariant idempotent) functor from the category of (left) R-modules into itself mapping each R-module into its R-submodule specified by a family of left ideals of R. If is a family left ideals of R, then all elements of an R-module M of orders belonging to , do not, of course, in general form a submodule of M; but, there are certain families such that all the elements of orders from form a submodule in any R-module (distinguished submodules defined by ). Consequently, no particular structural properties of the R-module are involved in the definition of such submodules. In this way we can define radicals (in the sense of Kuroš [4]) of a module. In particular, we feel that an application of this method is an appropriate way in defining the (maximal) torsion submodule of a module.
Let Zn be the numer of individuals in the nth generation of a discrete branching process, descended from a single a singel ancestor, for which we put It is well known that the probability generating function of Zn is Fn(s), the n-th functional iterate of F(s), and that if m = EZ1 does not exceed unity, then lim (Harris [1], Chapter 1). In particular, extinction is certain.
In a stationary GI/G/1 queueing system in which the waiting time variance is finite, it can be shown that the serial correlation coefficients {ρn} of a (stationary) sequence of waiting times are non-negative and decrease monotonically to zero. By means of renewal theory we find a representation for Σ∞0 ρn from which necessary and sufficient condition for its finiteness can be found. In M/G/1 rather more can be said: {ρn} is convex sequence, the asymptotic form of ρ n can be given in a nearly saturated queue, and a simple explicit expression for Σ∞0 ρn exists. For the stationary M/M/1 queue we find the ρn's explicitly, illustrate them numerically, and derive a representation which shows that {ρn} is completely monotonic.
If (X, ) is a set X with topology we shall say that is connected if (X, ) is a connected topological space. We shall investigate the existence of and the properties of maximal connected topologies.
We consider a queueing system with k identical servers in parallel, the services being negative exponential with parameter μ. The input is a natural generalisation of the usual general recurrent input. If we denote the sequence of arrival points by {An, n ≧ 0} then the inter-arrival intervals are given by where the ƒi: are (integrable) non-negative functions and {Ui} is a sequence of identically and independently distributed random variables. In the simplest case, p = 0, this is just a general recurrent input. We write U(·) for the probability distribution function of the Un.
The development of a population over time can often be simulated by the behavior of a birth and death process, whose transition probability matrix P(t) = (Pij(t), where X(t) denotes the number of individuals at time t, satisfies the differential equations and the initial condition
In the past a number of papers have appeared which give representations of abstract lattices as rings of sets of various kinds. We refer particularly to authors who have given necessary and sufficient conditions for an abstract lattice to be lattice isomorphic to a complete ring of sets, to the lattice of all closed sets of a topological space, or to the lattice of all open sets of a topological space. Most papers on these subjects give the conditions in terms of special elements of the lattice. We thus have completely join-irreducible elements — G. N. Raney [7]; join prime, completely join prime, and supercompact elements — V. K. Balachandran [1], [2]; N-sub-irreducible elements — J. R. Büchi [5]; and lattice bisectors — P. D. Finch [6]. Also meet-irreducible and completely meet-irreducible dual ideals play a part in some representations of G. Birkhoff & 0. Frink [4].
In 1962, O. Frink [2] showed that in a pseudo-complemented semilattice 〈P; ∧, *, 0〉, the closed elements form a Boolean algebra. We shall consider an extension of this result to arbitrary commutative semigroups with zero.
Perturbation expansions are sought for the flow variables associated with the diffraction of a plane weak shock wave around convex-angled corners in a polytropic, inviscid, thermally-nonconducting gas. Lighthill's method of strained co-ordinates [4] produces a uniformly valid expansion for most of the diffracted front, while the remainder of this front is treated by a modification of the shock-ray theory of Whitham [6]. The solutions from these approaches are patched just inside the ‘shadow’ region yielding a plausible description of the entire diffracted shock front.
A subset of a topological space which is both closed and open is referred to as a clopen subset. Here, a 0-dimensional space is a Hausdorff space which has a basis of clopen sets. Here, a 0-dimensional space is a Hausdorff space which has a basis of clopen sets. By a compactification αX of a completely regular Hausdorff space X, we mean any compact space which contains X as a dense subspace. Two compactifications αX and γX are regarded as being equivalent if there exists a homeomorphism from αX onto γX which keeps X pointwise fixed. We will not distinguish between equivalent compactifications. With this convention, we can partially order any family of compactifications of X by defining αX ≧ γX if there exists a continuous mapping from γX onto αX which leaves X pointwise fixed. This paper is concerned with the study of the partially ordered family [X] of all 0-dimensional compactifications of a 0-dimensional space X.