bluuebunny/crossref_metadata_2025 huggingface dataset:
Voigt profile functions in the complex domain
Authors: Reichel A.
Published: April 2009

The functions where ξ > 0, are essentially Fourier convolutions of Gaussian and Cauchy type distributions, and have thus found application in many fields. In reactor theory, and no doubt elsewhere, the need has arisen to analyse complicated integrals and functional equations involving ψ and φ, the so-called Voigt profiles.
On the distribution of the time to first emptiness of a store with stochastic input
Authors: Hasofer A. M.
Published: April 2009

Kendall [4] has given for the distribution of the time to first emptiness in a store with an input process which is homogeneous and has non-negative independent increments and an output of one unit per unit time the formula. In this formula,zis the initial content of the store,g(t, z) is the density function of the time to first emptiness τ(z), defined byandk(t, x) is the density function of the input process ξ(t), defined by.
The joint distribution of occupation totals for a simple random walk
Authors: Barnett V. D.
Published: April 2009

A great deal of attention has been given in the literature to the various properties of the simple binomial random walk. Explicit expressions are available for first passage times, absorption probabilities, average duration of the walk up to absorption and other quantities of interest. One aspect of the behaviour of this work which has, however, attracted little attention is the form of the distribution of occupation totals. This paper is devoted to the derivation of an explict expression for the joint probility generating function of the occupation totals up to absorption, for the binomial random walk in the presence of two absorbing points. The appropriate marginal form of this p.g.f. yields the distribution of the occupation total, and expected occupation total, at any particular lattice point. The limiting forms of these results provide explicit expressions for the corresponding quatities in the case of a binomial random walk having a single absorbing point and, where relevent, in the case of the unrestricted binomial random walk.
A simple proof of Watson's partition congruences for powers of 7
Authors: Garvan F. G.
Published: April 2009

Ramanujan conjectured that ifnis of a specific form thenp(n), the number of unrestricted partitions ofn, is divisible by a high power of 7. A modified version of Ramanujan's conjecture was proved by G. N. Watson.

In this paper we establish appropriate generating formulae, from which Watson's results follow easily.

Our proofs are more straightforward than those of Watson. They are elementary, depending only on classical identities of Euler and Jacobi. Watson's proofs rely on the modular equation of seventh order. We also need the modular equation but we derive it using the elementary techniques of O. Kolberg.
(PVB) functions and integration
Authors: Sarkhel D. N. , Kar A. B.
Published: April 2009

We introduce the notion of functions of bounded proximal variation and the notion of orderly connected topology on the real line. Using these notions, we define in a novel way an integral of Perron type, including virtually all the known integrals of Perron and Denjoy types and admitting mean value theorems and integration by parts and the analog of Marcinkiewicz theorem for the ordinary Perron integral.
Epimorphisms preserving perfect radicals
Authors: Berrick A. J.
Published: April 2009

This is an investigation of whether a group epimorphism maps the maximal perfect subgroup of its domain onto that of its image. It is shown how the question arises naturally from considerations of algebraic K-theory and Quillen's plus-construction. Some sufficient conditions are obtained; these relate to the upper central series, or alternatively the derived series, of the domain. By means of topological/homological techniques, the results are then sharpened to provide, in certain circumstances, conditions which are necessary as well as sufficient.
A generalized Fourier transformation for L1(G)-Modules
Authors: Liu Teng-Sun , Rooij Arnoud C. M. Van , Wang Ju-Kwei
Published: April 2009

Let G be a compact abelian group with dual Ĝ and let K be a Banach L1 (G)-module. We introduce the notion of character convolution transformation of K which reduces to ordinary Fourier or Fourier-Stieltjes transformation when K is one of the spaces Lp(G), M(G). We show that the question of what maps Ĝ → K extend to multipliers of K is a question of asking for descriptions of the character convolution transforms. In this setting some results of Helson-Edward and Schoenberg-Eberlein find generalizations, as do some classical results, including the inversion formula and the Parseval relation. We then apply these results to transformation groups, obtaining a variant of a theorem of Bochner and an extension of a theorem of Ryan.
A free surface problem arising in the drainage of a uniformly irrigated field: Existence and uniqueness results
Authors: Hoek John Van Der , Barnes C. J. , Knight J. H.
Published: April 2009

A field comprising uniformly porous soil overlying an impervious subsoil is drained through equally spaced tile drains placed on the boundary between the two layers of soil. When this field is subject to uniform irrigation, a free boundary forms in the porous region above the zone of saturation. We study the free boundary value problem which thus arises using the theory of variational inequalities. Existence and uniqueness results are established.
Continuity properties of vector-valued convex functions
Authors: Trudzik L. I.
Published: April 2009

We extend various characterizations of scalar-valued lower semicontinuity and determine their relationship to the continuity of vector-valued convex functions. Order completeness of the range space is not assumed.
Some entire functions with fixpoints of every order
Authors: Baker I. N.
Published: April 2009

In this paper f(z) will always stand for an entire transcendental function of the complex variable z. For p= 1, 2, … the natural iterate fD(z) of f(z) is defined by These natural iterates are themselves entire transcendental functions; they have been studied by various writers, notably Fatou [3]. References to many papers on iterated will be found in [1].
On the theory of a cascade of stalled aerofoils
Authors: Woods L. C.
Published: April 2009

A mathematical theory of the separating flow past a cascade of aerofoils is developed. The flow is assumed to be inviscid, incompressible and twodimensional. The wakes are represented by regions of stationary fluid, which could, in the general case, maintain a pressure gradient, although much of the theory is developed for the case of constant wake pressure.
Unsteady flow through a cascade of aerofoils
Authors: Low A. H. , Woods L. C.
Published: April 2009

This paper extends some earlier work by Woods [8] on the transient forces caused by unsteady transverse motions of a rigid cascade of aerofoils to the case when the inlet velocity is varying in magnitude. The flow is incompressible and two-dimensional. Expressions for the growth of lift and moment on a member of the cascade with increasing inlet velocity are evaluated and shown to be generalizations of Wagner's classical results for an isolated aerofoil.
Quadratic forms in Poisson and multi-nomial variables
Authors: Whittle P.
Published: April 2009

Letrepresent the deviations from expectation of a set of multinomial or independent Poisson variables, and Η be a positive definite matrix. A lower bound is obtained forPrin terms ofPr, Whereis a vector of normal variables with the same mean and covariance matrix as Δ.
On Tests of independence in several dimensions
Authors: Lancaster H. O.
Published: April 2009

Introductory. This paper considers a canonical form, or rather a class of canonical forms, for three dimensional probability distributions subject to a rather mild restriction. These canonical forms are used to develop suitable tests of independence and lead to a consideration of the partition of χ2in the analysis of complex contingency tables. Where these methods and Bartlett's are both applicable it is shown that they give comparable results; but the partitioning methods are more general.
A characterization of semi-prime ideals in near-rings
Authors: Groenewald N. J.
Published: April 2009

It is well-known that in any near-ring, any intersection of prime ideals is a semi-prime ideal. The aim of this note is to prove that any ideal is a prime ideal if and only if it is equal to its prime radical. As a consequence of this we have any semi-prime ideal I in a near-ring N is the intersection of minimal prime ideals of I in N and that I is the intersection of all prime ideals containing I.
A note on a paper by S. Lal
Authors: Gutiérrez A.
Published: April 2009

In this paper, the questions about bitopological spaces proposed by S. Lal are solved and one of his counterexamples rectified.
Powers of chordal graphs
Authors: Balakrishnan R. , Paulraja P.
Published: April 2009

An undirected simple graph G is called chordal if every circle of G of length greater than 3 has a chord. For a chordal graph G, we prove the following: (i) If m is an odd positive integer, Gm is chordal. (ii) If m is an even positive integer and if Gm is not chordal, then none of the edges of any chordless cycle of Gm is an edge of Gr, r < m.
Linearity and weak convergence on the boundary of numerical range
Authors: Das K. C. , Craven B. D.
Published: April 2009

Stampfli and Embry have shown that a point of the numerical range of an operator is extreme if and only if a set of vectors corresponding to it is linear. This is generalized here to show that a point of the closure of the numerical range is extreme if and only if a corresponding set of sequences forms a linear space. A more geometric alternative proof is given for a theorem of Das and Garske concerning weak convergence to zero at the unattained extreme points of the closure of the numerical range.The result is shown to hold also for lone extreme points of the numerical range which lie on line segments on its boundary. Further, a bound is obtained on the norm of the weak limit of the weakly convergent sequences corresponding to points on a line segment on the boundary of numerical range.
The Burkill approximately continuous integral
Authors: Bullen P. S.
Published: April 2009

This paper defines descriptive, Riemann, and constructive integrals equivalent to the approximately continuous integral of Burkill.
Absolute regularity of the Nörlund mean
Authors: Kwee B.
Published: April 2009

Let {Pn} be any sequence of real or complex numbers subject to the sole restriction And let If tn → s, whenever sn → s we say that the sequence {sn} is summable Nörlund or summable (N, p) to s.
Arithmetic properties of lacunary power series with integral coefficients
Authors: Mahler K.
Published: April 2009

This note is concerned with arithmetic properties of power series with integral coefficients that are lacunary in the following sense. There are two infinite sequences of integers {rn} and {sn}, satisfying such that It is also assumed that f(z) has a positive radius of convergence, Rf say, where naturally . A power series with these properties will be called admissible.
Chains of congruences on a completely 0-simple semigroup
Authors: Preston G. B.
Published: April 2009

Let ρ and σ be two congruences on a completely 0-simple semigroup. Suppose that there is a maximal chain of congruences from ρ to σ which is of finite length. Then, as we shall show, any maximal chain of congruences from ρ to σ finite and of the same length.
Certain representation algebras
Authors: Conlon S. B.
Published: April 2009

Let Λ be the set of inequivalent representations of a finite group over a field . Λ is made the basis of an algebra over the complex numbers , called the representation algebra, in which multiplication corresponds to the tensor product of representations and addition to direct sum. Green [5] has shown that if char (the non-modular case) or if is cyclic, then is semi-simple, i.e. is a direct sum of copies of . Here we consider two modular, non-cyclic cases, viz, where is or 4 (alternating group) and is of characteristic 2.
Some asymptotic results for the periodogram of a stationary time series
Authors: Walker A. M.
Published: April 2009

Letx1,x2,…,xn, benconsecutive observations generated by a stationary time series {x}, t = 0, ±1, ±2,…, with E(xt2) < ∈. The periodogram of the set of observations, which may be defined as a function Inof angular frequency with range [0, π] that is proportional to, plays an important part in methods of making inferences about the structure of {xt}, particularly its spectral distribution function or spectral density.
Desargues' theorem in n-space
Authors: Mandan Sahib Ram
Published: April 2009

Two sets of r + 2 points, Pi, P'i, each spanning a projective space of r + 1 dimensions, [r + 1], which has no solid ([3]) common with that spanned by the other, are said to be projective from an [r — 1], if here is an [r — 1] which meets the r + 2 joins Pi ′i. It is to be proved that the two sets are projective, if and only if the r + 2 intersections Ai of their corresponding [r]s lie in a line a. Ai are said to be the arguesian points and a the arguesian line of the sets. When r= 1, the proposition becomes the well- known Desargues' two-triangle theorem (3) in a plane. Therefore in analogy with the same we name it as the Desargues' theorem in [2r]. Following Baker (1, pp. 8—39), we may prove this theorem in the same synthetic style by making use of the axioms and the corresponding proposition of incidence in [2r + 1] or with the aid of the Desargues' theorem in a plane and the axioms of [2r] only. But the use of symbols makes its proof more concise; the algebraic approach adopted here is due to the referee (Arts. 2, 3, 5, 6, 7). Pairs of sets of r + p points each projective from an [r— 1] are also introduced to serve as a basis for a much more thorough investigation.
Conditionally convergent spectral expansions
Authors: Smart D. R.
Published: April 2009

We will consider a reflexive Banach space 𝔅, with real or complex scalars, and a bounded operator in 𝔅 with a real spectrum.
On well-bounded operators
Authors: Ringrose J. R.
Published: April 2009

This note is concerned with a question arising from some work of D. R. Smart (2). In the introduction we indicate the nature of the problem. Notations will be explained only in as far as they differ from those of (2).
Nozzle flows found by the hodograph method. II
Authors: Cherry T. M.
Published: April 2009

This is a sequel to a recent paper [1] on the construction by the hodograph method of trans-sonic nozzle-flows of a perfect gas. At the end of that paper it was shown how we can obtain regular flows that are ultimately uniform (as required in the test section of a supersonic wind tunnel), and the object now is to give some quantitative examples of such flows. The gas is supposed to have the polytropic equation of state Pρ−γ = constant, and the calculations have been made for the case γ = 1.4, with the Mach number M = 2.25 at the test section. The results, which are exhibited graphically, are indicative of what may be expected for other supersonic values of M, and it is hoped that they may be significant for the design of wind tunnels.
A characterization of rational numbers by p-adic Ruban continued fractions
Authors: Laohakosol Vichian
Published: April 2009

A type of p–adic continued fraction first considered by A. Ruban is described, and is used to give a characterization of rational numbers.
On the Alexander polynomial of alternating algebraic knots
Authors: Murasugi Kunio
Published: April 2009

A conjecture of Fox about the coefficients of the Alexander polynomial of an alternating knot is proved for alternating algebraic (or arborescent) knots, which include two-bridge knots.
Indecomposable 1-factorizations of the complete multigraph
Authors: Colbourn Charles J. , Colbourn Marlene J. , Rosa Alexander
Published: April 2009

The existence of 1-factorizations of the complete multigraph λKn which cannot be decomposed into 1-factorizations with smaller λ is studied.
A few more balanced Room squares
Authors: Stinson D. R. , Vanstone S. A.
Published: April 2009

The existence problem for balanced Room squares is, in general, unsolved. Recently, B. A. Anderson gave a construction for a class of these designs with side 2n − 1, where n is odd and n ≥ 3. For n even, the existence has not yet been settled. In this paper, we use the affine geometry of dimension 2 k and order 2, and a hill-climbing algorithm, to construct a number of new balanced Room squares directly. Recursive techniques based on finite geometries then give new squares of side 22k − 1 for infinitely many values of k.
A generalization of Tong's theorem and properties of pairwise perfectly normal spaces
Authors: López-Pellicer Manuel , Gutiérrez Angel
Published: April 2009

In this paper we give some properties of the pairwise perfectly normal spaces defined by Lane. In particular we prove that a space (X, P, Q) is pairwise perfectly normal if and only if every P(Q)–closed set is the zero of a P(Q)–l.s.c. and Q(P)–u.s.c. function. Also we characterize the pairwise perfect normality in terms of sequences of semicontinuous functions by means of a result which contains the known Tong's characterization of perfectly normal topological spaces, whose proof we modify by using the technique of binary relations.
The range of an o-weakly compact mapping
Authors: Dodds P. G.
Published: April 2009

It is shown that a weakly compact convex set in a locally convex space is a zonoform if and only if it is the order continuous image of an order interval in a Dedekind complete Riesz space. While this result implies the Kluv´nek characterization of the range of a vector measure, the techniques of the present paper are purely order theoretic.
On the optimality of certain estimates for algebraic values of analytic functions
Authors: Wakabayashi Isao
Published: April 2009

We prove, by constructing a function with given parameters, that the estimate by G. V. Chudnovsky of the number of points at which a meromorphic function has algebraic Taylor coefficients is optimal. The construction is carried out by the use of interpolation series.
A note on generalised wreath product groups
Authors: Praeger Cheryl E. , Rowley C. A. , Speed T. P.
Published: April 2009

Generalised wreath products of permutation groups were discussed in a paper by Bailey and us. This note determines the orbits of the action of a generalised wreath product group on m–tuples (m ≥ 2) of elements of the product of the base sets on the assumption that the action on each component is m–transitive. Certain related results are also provided.
Scales of functions
Authors: Erdös P. , Rogers C. A. , Taylor S. J.
Published: April 2009

We consider real-valued functions ƒ(x) which are defined for all sufficiently large real numbers x. In discussing the behaviour of such functions as x → + ∞, it is useful to compare ƒ with the functions of some “comparison scale”. The early work in this field was due to Du Bois-Reymond (see, for example, (2), (3)). This was elaborated by Hardy, (6), who was mainly concerned with what he calls the “logarithmico-exponential” scale of functions. This “scale of Hardy” may be defined as the smallest class ℋ of functions ƒ with the following properties:

(i) if ƒ ∈ ℋ, ƒ, is defined and continuous for all sufficiently large values of x;

(ii) (a) the function ƒ(x) ≡ α, where α is any real constant, is in ℋ;

(b) the function ƒ(x) ≡ x is in ℋ;

(iii) if ƒ and g are in ℋ and g is non-zero for all sufficiently large values of x, then

are also in ℋ.
The existence of parametric surface integrals
Authors: Michael J. H.
Published: April 2009

In [2] we studied parametric n-surfaces (f, Mn), where Mn was a compact, oriented, topological n-manifold and f a continuous mapping of Mn into the real euclidean k-space Rn (k≧n). A definition of bounded variation was given and, for each surface with bounded variation and each projection P from Rk to Rn, a signed measure: Was constructed. This measure was used to define a linear type of surface integral: over a “measurable” subset A of Mn, as the Lebesgue-Stieltjes integral: .
The expansion of functions in ultraspherical polynomials
Authors: Elliott David
Published: April 2009

The ultraspherical polynomial (x) of degree n and order λ is defined by for n = 0, 1, 2, …. Of these polynomials, the most commonly used are the Chebyshev polynomials Tn(x) of the first kind, corresponding to λ = 0; the Legendre polynomials Pn(x) for which λ = ½; and the Chebyshev polynomials Un(x) of the second kind (λ = 1). In the first case the standardisation is different from that given in equation (1), since.
Expectation values of operators in the quasi chemical equilibrium theory
Authors: Blatt John M.
Published: April 2009

Expectation values of one-particle and two-particle operators are evaluated in the quasi-chemical equilibrium (pair correlation) approximation to statistical mechanics. Certain reductions, corresponding to the “quenching” of interactions by the Pauli exclusion principle, are carried out quite generally. More specific reductions, which lead to immediately useful expressions, are possible on the assumption of extreme Bose-Einstein condensation of the correlated pairs.
The survival of a mutant gene under selection. II
Authors: Moran P. A. P.
Published: April 2009

R. A. Fisher (1930) has obtained approximate expressions for the probability of survival of a new mutant in a finite population of haploid individuals in which the generations are non-overlapping. Suppose that we have M haploid individuals which are either of genotype a or A, and suppose that a has a small selective advantage over A so that the relative numbers of offspring have expectations proportional to 1 + s and 1 respectively, where s is small and positive. If each generation is produced by binomial sampling with probabilities proportional to the numbers of a and A individuals in the previous generation multiplied by their respective selective values, and if initially there is only one individual of type a, the probability of the population ultimately becoming entirely of this type is approximately so long as s2M is small. This also holds when s is small and negative.
On statistical independence and zero correlation in several dimensions
Authors: Lancaster H. O.
Published: April 2009

Bivariate distributions, subject to a condition of φ2 boundedness to be defined later, can be written in a canonical form. Sarmanov [4] used such a form to deduce that two random variables are independent if and only if the maximal correlation of any square summable function, ξ (x1), of the first variable with any square summable function, η(x2), of the second variable is zero. This is equivalent to the condition that the canonical correlations are all zero. The theorem of Sarmanov [4] was proved without any restriction in Lancaster [2] and the proof is now extended to an arbitrary number of dimensions.
On a problem of G. Golomb.
Authors: Erdös P.
Published: April 2009

In his paper on sets of primes with intermediate density Golomb1 proved the following theorem: Let 2 < p1 < P2 < … be any sequence of primes for whichfor every i and j. Denote by A (x) the number of P's not exceeding x. Then.
The inhomogeneous minima of indefinite quadratic forms
Authors: Barnes E. S.
Published: April 2009

If f(x) is a real indefinite quadratic form in n variables with determinant d ≠ 0, we set for any real α.
On the differential equations of H. Lewy
Authors: Smith-White W. B.
Published: April 2009

It is known that the theory of Cauchy's problem for differential equations with two independent variables is réducible to the corresponding problem for systems of quasi-linear equations. The reduction is carried further, by means of the theory of characteristics, to the case of systems of equations of the special form first considered by H. Lewy [1]. The simplest case is that of the pair of equationswhere the aii depend on z1 and z2. The problem to be considered is that of finding functions z1(x, y), z2(x, y) which satisfy (1) and which take prescribed values on x + y = 0.
Lattice embeddings of prime power groups
Authors: Barnes D. W.
Published: April 2009

Let G be a group of order pn where p is some prime. Denote by nr (G) the number of subgroups of G of order pr. If H is the elementary abelian group of order pn, then.
Some applications of the Eulerian functions of a finite group
Authors: Wall G. E.
Published: April 2009

This paper is chiefly concerned with inequalities for the numbers of subgroups of a finite p-group. The following are typical results. Let G be a p-group of order pa, and let nG(pk) denote the number of subgroups of G of order pk.
On the Clifford collineation, transform and similarity groups. II.
Authors: Bolt Beverley , Room T. G. , Wall G. E.
Published: April 2009

The present paper deals with the Clifford groups in the case p = 2. For the most part, it runs parallel to the previous paper I ([1]) on the case p > 2, and a number of proofs are therefore either given in outline or omitted. A general introduction to both papers is given in I, § 1.
Rational approximation with series
Authors: Fenna D.
Published: April 2009

The Siegel conjecture on the rational approximation to algebraic numbers was proved a few years ago by K. F. Roth [1] with the following theorem: Let α be any algebraic number, not rational. If has an infinity of solutions in integers h and q (q > 0) tehn k ≤ 2.
Acceleration and the “clock paradox”.
Authors: Hurst C. A.
Published: April 2009

In the controversy which has been continuing over the “clock paradox”, it has been claimed by supporters ([1]) of the orthodox interpretation that the difference in behaviour of clocks carried by observers in motion relative to each other can be traced to differencesin the motion relative to the universe. Whilst A remains unaccelerated relative to the universe, B for a portion of his journey does not. Professor Dingle, in reply ([2]), remarked that he would “not dispute a possible slight effect” but could not understand why, if the acceleration of B were so important, it did not appear explicitly in the answer.
Necessary and sufficient conditions for unitary similarity
Authors: Wiegmann N. A.
Published: April 2009

If A and B are two complex matrices and if U is a complex unitary matrix such that UAUCT = B (where UCT denotes the conjugate transpose of U), then A and B are said to be unitarily similar. Necessary and sufficient conditions that two matrices be unitarily similar have been dealt with in [5] (from the point of view of group representation theory) and in [2] (from the point of view of developing a canonical form under unitary similarity).
Cocycles and representations of groups of CAR type
Authors: Carey A. L. , Moran William
Published: April 2009

Representations of non-type I groups G which may be expressed as an increasing union of type I normal subgroups are considered. Groups with this structure are natural generalisations of the CAR algebra (viewed as a twisted group C*-algebra) and are also group theoretic analogues of AF algebras. This paper gives a systematic account of their representation theory based on a canonical construction of one-cocycles for the G-action on the dual of a normal subgroup. Some examples are considered showing how to construct inquivalent irreducible representations (non-cohomologous cocycles) and also factor representations by a method which generalises the well-known construction of non-isomorphic factors for the CAR algebra.
Cumulants and partition lattices II: generalised k-statistics
Authors: Speed T. P.
Published: April 2009

The role played by the Möbius function of the lattice of all partitions of a set in the theory of k-statistics and their generalisations is pointer out and the main results conscerning these statistics are drived. The definitions and formulae for the expansion of products of generalished k-statistics are presented from this viewpoint and applied to arrays of random variables whos moments satisfy stitable symmentry constraints. Applications of the theory are given including the calculation of (joint) cumulants of k-statistics, the minimum variace estimation of (generalised) moments and the asymptotic behaviour of generalised k-statistics viewed as (reversed) martingales.
Special series of unitary representations of groups acting on homogeneous trees
Authors: Mantero Anna Maria , Zappa Anna
Published: April 2009

Let G be a group acting faithfully on a homogeneous tree of order p + 1, p > 1. Let be the space of functions on the Poission boundary ω, of zero mean on ω. When p is a prime. G is a discrete subgroup of PGL2(Qp) of finite covolume. The representations of the special series of PGL2(Qp), Which are irreducible and unitary in an appropriate completion of , are shown to be reducible when restricted to G. It is proved that these representations of G are algebraically reducible on and topologically irreducible on endowed with the week topology.
Orbit entropy in noninvertible mappings
Authors: Burkart Uhland
Published: April 2009

Bassed on the intrinsic structure of a selfmapping T: S → S of an arbitrary set S, called the orbit-structure of T, a new entropy is defined. The idea is to use the number of preimages of an element x under the iterates of T to characterize the complexity of the transformation T and their orbit graphs. The fundamental properties of the orbit entropy related to iteration, iterative roots and iteration semigroups are studied. For continuous (differentiable) functions of Rn to Rn, the chaos of Li and Yorke is characterized by means of this entropy, mainly using the method of Straffingraphs.
Integral averaging techniques for the oscillation of second order sublinear ordinarry differential equations
Authors: Philos Ch. G.
Published: April 2009

New oscillation criteria are established for second order sublinear ordinary differential equations with alternating coefficients. These criteria are obtained by using an integral averaging technique and can be applied in some special cases in which other classical oscillation results are no applicable.
Dense scales of functions
Authors: Rogers C. A.
Published: April 2009

In a recent paper [1] on scales of functions (which will be called SF for short) P. Erdös C. A. Rogers and S. J. Taylor developed the theory of scales of functions and used the Continuum Hypothesis to establish the existence of scales of functions having certain desirable properties. One of these properties was that of being dense in a certain sense. In the course of some joint work Taylor and I have felt the need for scales which are dense in a rather stronger sense. The object of this note is to indicate how the methods of SF can be used to show that the Continuum Hypothesis implies the existence of scales with the required properties.
Dini's theorem for almost periodic functions
Authors: Greve W.
Published: April 2009

The object of this note is to extend Dini's theorem about (monotonic) sequences of continuous functions on a compact topological space to the case where the underlying domain is an abstract group which is free from topological restrictions. Continuous functions are replaced by almost periodic real-valued functions and the main result may be stated as follows: If a monotonically increasing sequence (fn) of almost periodic real-valued functions on a group G converges pointwise to an almost periodic function f on G, then the sequence converges to f uniformly. The basic idea of the present (elementary) proof is due to v. Kampen [2] and A. Weil [4], i.e., every almost periodic function on a group induces a kind of compact topology in it, relative to which the function is continuous. We modify this idea with the aid of the mean-value of an almost periodic function and obtain a pseudometric topology. This topology facilitates convergence proofs greatly. Moreover, it turns out to be equivalent with the previous one (Lemma 1). No use will be made of the theory of bounded matrix representations. This is significant as any use of the ‘Approximation Theorem’ [3, p. 66, see also p. 226] would violate the claim of an elementary proof.
A theoretical investigation of an aerofoil equipped with a split flap
Authors: Hurley D. G.
Published: April 2009

For a theory of the flow about an aerofoil equipped with a split flap to be satisfactory, account must be taken of the wake which is formed downstream of the combination. In this paper the wing and the flap are each represented by a flat plate and it is assumed that the pressure in the wake is constant and equal to the free stream value, i.e. (it is assumed that the wake may be represented by the region between two free stream-line which extend downstream from the trailing edges of the wi ng and the flap1) (see Fig. 1).
The order cycle system Of stock control
Authors: Finch P. D.
Published: April 2009

The order cycle system of stock control can be formulated as follows: demands for stock occur according to some pattern which we call the demand process and specify in detail later in this section. At fixed intervals of time orders are placed to replenish stock. Let orders for stock be placed at the instants jN, j = 1,2, …; the interval [(j — 1)(N, jN) which we suppose closed at its lower end-point and open at its upper end-point, is called the j-th order cycle and N the length of this interval is called the order cycle period. We suppose that an order placed at time jN is delivered into stock at time jN + li where {li} is a sequence of non-negative random variables independent of the demand process and the order cycle period, we suppose also that the li are mutually independent and identically distributed with common distribution function L(x) with L(0 +) = 0 and finite expectation . The quantity li is called the lead time of the j-th order, that is of the order placed at jN and is supposed independent of the amount ordered. That portion of a demand, if any, which cannot be satisfied immediately is satisfied from future deliveries, thus every demand is satisfied eventually and a negative inventory or back orders can be held. We shall suppose that all order cycle periods under consideration are multiples of some fixed time interval of length τ which we shall take as our unit of time. For example, the interval of length τ could be one day and we would consider order cycle periods which were integral multiples of days. For convenience we take τ = 1 and suppose that the order cycle peiod N is an integer. In this paper we shall consider the following two demand processes.
On the busy period in the queueing systemGI/G/1
Authors: Finch P. D.
Published: April 2009

A number of authors have studied busy period problems for particular cases of the general single-server queueing system. For example, using the now standard notation of Kendall [7],GI/M/1 was studied by Conolly [3] and Takacs [18]. Earlier work onM/M/1 includes that of Ledermann and Reuter [8] and Bailey [1]. Kendall [6], Takacs [17], and Prabhu [11] have consideredM/G/1.
The general theory of canonical correlation and its relation to functional analysis
Authors: Hannan E. J.
Published: April 2009

The classical theory of canonical correlation is concerned with a standard description of the relationship between any linear combination of ρ random variablesxs, and any linear combination ofqrandom variablesytinsofar as this relation can be described in terms of correlation. Lancaster [1] has extended this theory, forp=q= 1, to include a description of the correlation of any function of a random variablexand any function of a random variabley(both functions having finite variance) for a class of joint distributions ofxandywhich is very general. It is the purpose of this paper to derive Lancaster's results from general theorems concerning the spectral decomposition of operators on a Hilbert space. These theorems lend themselves easily to the generalisation of the theory to situations wherepandqare not finite. In the case of Gaussian, stationary, processes this generalisation is equivalent to the classical spectral theory and corresponds to a canonical reduction of a (finite) sample of data which is basic. The theory also then extends to any number of processes. In the Gaussian case, also, the present discussion-is connected with the results of Gelfand and Yaglom [2] relating to the amount of information in one random process about another.
Tests for discriminant functions
Authors: Williams E. J.
Published: April 2009

In previous papers (Bartlett 1951, Williams 1952a, b, c; 1953, 1955) certain exact tests for discriminant functions have been discussed. These papers have also indicated how the tests may usefully be applied to a wide range of problems: the significance of discriminant function coefficients, the interpretation of interactions in factorial experiments, the concurrence of regression lines with the same independent variables and the proportionality of regression lines with different independent variables, the comparison of sets of ratios, and the testing of arbitrary scores applied to frequency data.
Infinite linear systems with homogeneous kernel of degree –1
Authors: Cherry T. M.
Published: April 2009

The main concern of this paper is with the solution of infinite linear systems in which the kernel k is a continuous function of real positive variables m, n which is homogeneous with degree –1, so that If k is a rational algebraic function it is supposed further that the continuity extends up to the axes m = 0, n > 0 and n = 0, m > 0; the possibly additional restriction when k is not rational is discussed in § 1,2.
On the stability of linear canonical systems with periodic coefficients
Authors: Coppel W. A. , Howe A.
Published: April 2009

By a linear canonical system we mean a system of linear differential equations of the formwhereJis an invertible skew-Hermitian matrix andH(t) is a continuous Hermitian matrix valued function. We reserve the name Hami1tonia for real canonical systems withwhereIkdenotes thek×kunit matrix. In recent years the stability properties of Hamiltonian systems whose coefficient matrixH(t) is periodic have been deeply investigated, mainly by Russian authors ([2], [3], [5], [7]). An excellent survey of the literature is given in [6]. The purpose of the present paper is to extend this theory to canonical systems. The only work which we know of in this direction is a paper by Yakubovič [9].
A theorem in asymptotic number theory
Authors: Tull Jack P.
Published: April 2009

Let α(n) be a multiplicative arithmetic function. H. Delange [1] has proved that if |α(n)| ≦ 1 for all n and for a certain constant ρ, , where if ρ = 1 then then . He applied this result to several problems such as uniform distribution (mod 1) of certain types of sequences.
The stability of harrod's growth model of an economy
Authors: Lush P. E.
Published: April 2009

Nevile [2] has shown that if Rt is a certain measure of the rate of growth of the national income in Harrod's growth model of an economy, then Rt satisfies the non-linear recurrence relation , where 0 < k < and −1 < c < 1. The definition of Rt ([2] p. 369) is such that Rt > 0 for all t. Nevile has pointed out features of the model that indicate that it may be unstable. In this paper I propose to show that the model is, in general. unstable, but that proper choice of the initial values R0R1 apparently leads to stability. In order to do this, we require the conditions (if any) under which Rt converges.
On the radical of a ring with minimum condition
Authors: Barnes D. W.
Published: April 2009

The purpose of this note is to establish the following characterisation of the radical: Theorem. Let R be a ring with the minimum condition for left ideals. Then the radical of R is the intersection of the maximal nilpotent subrings of R.
Eigenfunctions of plane elastostatics III the Wedge
Authors: Buchwald V. T.
Published: April 2009

The boundary value problem of the infinite wedge in plane elastostatics is reduced to the solution of a differential-difference equation. The complementary function of this equation is determined in the form of a Fourier integral, which, on expansion by residue theory, gives the complete eigenfunction expansion for the wedge. The properties of the eigenfunctions are discussed in some detail, and orthogonality property is derived.
A Problem in shock wave decay
Authors: Burnside R. R. , Mackie A. G.
Published: April 2009

One of the elementary applications of the Rankine-Hugoniot shock relations which relate conditions on the two sides of a plane shock wave is that of determining the flow when a piston is pushed with constant velocity ū into a tube containing gas at rest. A shock wave races into the undisturbed gas at a constant speed Ū whose value can easily be found in terms of ū and the constants which specify the uniform condition of the gas at rest. If, however, the piston is suddenly brought to rest after a finite time the subsequent behaviour of the shock wave is very difficult to determine. A rarefaction wave is generated at the piston, and, as the velocity of the shock is subsonic relative to the gas behind it, this eventually overtakes the shock wave causing it to weaken. Since the energy supplied is finite the ultimate speed of the shock will tend to that of a sound wave. The analytical treatment of the flow behind the shock is made difficult by the entropy gradients which arise because of the variation in shock strength. It is further complicated by the disturbances which are reflected off the piston and give rise to a secondary interaction with the shock. Indeed, it seems safe to say that a complete description of the motion would certainly depend on some form of numerical integration.
Neo-Lorentzian relativity
Authors: Prokhovnik S. J.
Published: April 2009

Our present view of the universe suggest that the set of mutually receding galaxies may provide a natural substratum for the propagation of light. It is shown that this assumption leads to a consistent derivation and interpretation of special relativity, along the lines evvisaged by Lorentz but requiring also the employment of Einstein's measurement definitions. The time-dilatation and Fitzgerald contraction effects emerge as intelligible consequences of this approach, and their interaction with an associated anisotropy effect produces the relativity of simultaneity, the reciprocity phenomenon and the results described by Einstein's principles; the approach provides a definitive resolution of the “clock paradox” within the framework of Special Relativity.
A note on cumulative sums of markovian variables
Authors: Phatarfod R. M.
Published: April 2009

Consider a positive regular Markov chain X0, X1, X2,… with s(s finite) number of states E1, E2,… E8, and a transition probability matrix P = (pij) where = , and an initial probability distribution given by the vector p0. Let {Zr} be a sequence of random variables such that and consider the sum SN = Z1+Z2+ … ZN. It can easily be shown that (cf. Bartlett [1] p. 37), where λ1(t), λ2(t)…λ1(t) are the latent roots of P(t) ≡ (pijethij) and si(t) and t′i(t) are the column and row vectors corresponding to λi(t), and so constructed as to give t′i(t)Si(t) = 1 and t′i(t), si(o) = si where t′i(t) and si are the corresponding column and row vectors, considering the matrix .
An asymptotic expansion for a class of multivariate normal integrals
Authors: Ruben Harold
Published: April 2009

Let x = (x1, x2,…, xn) be a normal random vector with zero expectation vector and with a variance-covariance matrix which has 1 for its diagonal elements and ρ for its off-diagonal elements. Consider the quantity where.
Fractional iteration of exponentially growing functions
Authors: Szekeres G.
Published: April 2009

The fractional iteration ofexand solutions of the functional equationhave frequently been discussed in literature. G. H. Hardy has shown (in [3], and in greater detail in [4]) that the asymptotic behaviour of the solutions of (1) cannot be expressed in terms of the logarithmico-exponential scale, although they are comparable with each member of the scale.1Hence solutions of (1) provide a remarkably simple instance of functions whose manner of growth does not fit into the scale ofL-functions but requires non-elementary orders of infinity for an accurate representation. This raises quite naturally the question whether there exists a most regularly growing solution of equation (1) which might serve as a prototype for this kind of growth. In a slightly more general context we may ask whether there exists a ‘best’ family of fractional iteratesfσ(x), satisfying.
Tables of the logaritm of iteration of ex—1
Authors: Morris K. W. , Szekeres G.
Published: April 2009

In the preceding paper 1) one of us has proposed a definition for the “best” or most regularly growing fractional iterates of logarithmico-exponential type functions. The definition was essentially based on two observations. First, that the functional equation has exactly one solution with the property that is totally monotonic for every x > 0. Secondly, that if f(x) is a logarithmicoexponential function such that then the Abel equation has (apart from the arbitrariness of an additive constant) exactly one solution with the property that exists) and the same is true for any reasonably well-behaved function with property (3) whose manner of growth does not transcend Hardy's scale of L-functions. Thus every such function has a uniquely determined family of fractional iterates given by and these fσ(x) may be regarded as the most regularly growing iterates of f(x).
The use of Spitzer's identity in the investigation of the busy period and other quantities in the queueGI/G/1
Authors: Kingmán J. F. C.
Published: April 2009

As an illustration of the use of his identity [10], Spitzer [11] obtained the Pollaczek-Khintchine formula for the waiting time distribution of the queueM/G/1. The present paper develops this approach, using a generalised form of Spitzer's identity applied to a three-demensional random walk. This yields a number of results for the general queueGI/G/1, including Smith' solution for the stationary waiting time, which is established under less restrictive conditions that hitherto (§ 5). A soultion is obtained for the busy period distribution inGI/G/1 (§ 7) which can be evaluated when either of the distributions concerned has a rational characteristic function. This solution contains some recent results of Conolly on the queneGI/En/1, as well as well-known results forM/G/1.
The practical use of variation principles in non-linear mechanics
Authors: Blatt J. M. , Lyness J. N.
Published: April 2009

Contrary to the general impression that variation principles are of purely theoretical interest, we show by means of examples that they can lead to considerable practical advantages in the solution of non-linear vibration problems. In this paper, we develop a variation principle for the period of a free oscillation, as a function of the average value of the Lagrangian over one period. Even extremely simple-minded approximations to the true motion lead to excellent values for the period. The stability of such free oscillations against small disturbances of the initial conditions is treated in a previous paper.
The endomorphism ring of a locally free module
Authors: Franzsen W. N. , Schultz P.
Published: April 2009

We identify a large class of rings over which locally free modules are determined by their endomorphism rings. We characterize these endomorphism rings and consider under what circumstances the conditions on the locally free modules can be relaxed, for example by requiring that only one of the rings need be in the special class, or by replacing ‘free' by “projective”.
Two characterizations of quasi-pseudometrizable bitopological spaces
Authors: Romaguera S.
Published: April 2009

By using pairwise relatively complete families of functions, which we define here, we obtain two characterizations of quasi-pseudometrizable bitopological spaces from which some known theorems can be derived as easy corollaries.