bluuebunny/crossref_metadata_2025_split huggingface dataset:
Complex homogeneous linear forms
Authors: Rogers K. 
Published: November 2008

LetZ, Q, Cdenote respectively the ring of rational integers, the field of rational numbers and the field of complex numbers. Minkowski (4) solved the problem of minimizingforx, y ∈ Z(i)orZ(ρ), wherea, b, c, d ∈ Chave fixed determinant Δ ≠ 0. Here ρ = exp 2/3πi, andZ(i)andZ(p)are the rings of integers inQ(i)andQ(ρ)respectively. In fact he found the best possible resultsforZ(i), andforZ(ρ), wherewhile Buchner (1) used Minkowski's method to show thatforZ(i√2). Hlawka(3) has also proved (1·2), and Cassels, Ledermann and Mahler (2) have proved both (1·2) and (1·3). In a paper being prepared jointly by H. P. F. Swinnerton-Dyer and the author, general problems of the geometry of numbers in complex space are discussed and a systematic method given for solving the above problem for all complex quadratic fields Q(ϑ). Here, ϑ is a non-real number satisfying. an irreduc7ible quadratic equation with rational coefficients. The above problem is solved in detail forQ(i√5), for whichand the ‘critical forms’ can be reduced to
Scattering by a semi-infinite resistive strip of dominant-mode propagation in an infinite rectangular wave-guide
Authors: Papadopoulos V. M. 
Published: November 2008

ABSTRACTThe scattering of the dominant transverse electric mode in an infinite perfectly conducting rectangular wave-guide by a semi-infinite resistive strip, centrally placed and parallel to the electric field, is calculated by the use of Laplace transforms. Formulae are derived for the amplitude of the scattered waves, and the numerical results, obtained using a digital computer, are given for various values of the surface resistivity of the strip.
On prime-power groups in which the derived group has two generators
Authors: Blackburn N. 
Published: November 2008

Burnside ((1), p. 241) has proved the following theorem:If G is a non-metabelian p-group, then the centre of the derived group of G cannot be cyclic. In particular, a non-Abelian group of order p3 cannot be the derived group of a p-group.
The asymptotic solution of differential equations with a turning point and singularities
Authors: Thorne R. C. 
Published: November 2008

ABSTRACTAsymptotic solutions of the differential equationfor large positive values of u, are examined; z is a complex variable in a domain Dz in which P1(z) and z2q(z) are regular and p1(z) does not vanish. In this paper it is shown that there exist Airy-type expansions of the solutions of this equation which are valid uniformly with respect to z in a domain in which z = 0 and z = z0 are interior points. If Dz is unbounded and the equation has a regular singularity at infinity, Airy-type expansions exist which are valid at z = 0, z = z0 and z = δ. If p(z) = constant + O (│z│-1) as │ z │ → ∞ in Dz, similar expansions also exist. The results given here are new.
A converse theorem on overconvergence of sequences of partial sums
Authors: Noble M. E. 
Published: November 2008

A number of authors ((2), (4), (5)) have recently considered problems relating to the convergence of specified sequences of partial sums of a gap Taylor series on its circle of convergence. For convenience in stating such results we shall standardize the Taylor series as Σanzn with the unit circle as circle of convergence, and denote the partial sums and the sum function by sn(z) and f(z) respectively. Following the notation of ((4)) and (5) we shall define φ(x) for 0 ≤ x < ∞ to be the least concave majorant of log (|an| + 2). One of the recent results in this field ((5), Theorem 1, and see also (2)) may be stated asTHEOREM A. Suppose that f(z) is regular on an arc α < arg z < β. Suppose also that there are sequences of integers nk, Nk → ∞ such that Nk – nk → ∞ and am = 0 if nk < m < Nk.
The cooperative behaviour of a two-dimensional defect crystal
Authors: Elcock E. W. 
Published: November 2008

ABSTRACTThe Ising model of a two-dimensional ferromagnetic crystal containing defects is considered. It is shown that the combinatorial method is particularly appropriate to the investigation of the two-dimensional defect crystal and how it may be readily modified to calculate the partition function for such a crystal. It is found that the cooperative behaviour of the infinite perfect crystal, as indicated by the occurrence of a logarithmic singularity in the specific heat as a function of temperature, is not destroyed in the defect crystal; the singularity persists, though at a progressively lower temperature as the density of defects is increased, until a high density of defects is reached. The value of this limiting density of defects, above which the cooperative behaviour is destroyed, is calculated.
A method for the numerical integration of the linear diffusion equation
Authors: Hartree D. R. 
Published: November 2008

ABSTRACTIn one numerical method for integration of the diffusion equation in one dimension, the time derivative is replaced by a finite difference in a time interval, and the space derivative by the mean of its values at the beginning and end of the interval. This leads to a set of ordinary differential equations, one for each interval, which have to be solved in succession. Each of these equations is second-order with two-point boundary conditions; the process of integration from one end is severely unstable, the more so the smaller the tune interval. This paper is concerned with a practical, direct and stable method for solving them by integration of two first-order equations, one being integrated inwards and the other outwards, one boundary condition being satisfied in each integration. The extension to axially symmetrical diffusion is briefly considered.
The existence of Bartlett-Rajalakshman goodness of fitG-tests for multivariate autoregressive processes with finitely dependent residuals
Authors: Walker A. M. , Lindley D. V. 
Published: November 2008

Bartlett and Rajalakshman (3) have derived two types of large sample goodness of fit tests for the hypothesis that a multivariate (or vector) time series {X(t)},t= 0, ± 1, ± 2, …, is generated by a linear autoregressive process. This may be defined as the stationary solution of an equation of the formwhere theAiare square matrices such that the roots of the determinantal equation ‖ zp+A1zp−1+ … +Ap‖ = 0 have moduli less than unity, and {U(t)} is a sequence of independent random vector variables with a common distribution.
On the decomposition of spaces in Cartesian products and unions
Authors: Ganea Tudor , Hilton Peter J. 
Published: November 2008

The present paper is concerned with particular cases, obtained by suitably restricting the spaces involved, of the following general problem.Given a topological space X, we ask whether there exist integers n ≥ 2 and non-contractible spaces X1, …, Xn such that X has the homotopy type of the Cartesian product X1, × … × Xn or of the union X1, v … v Xn.
On the existence of periodic solutions of a certain third-order differential equation
Authors: Ezeilo J. O. C. 
Published: November 2008

In this paper we shall be concerned with the differential equationin which a and b are constants, p(t) is a continuous periodic function of t with a least period ω, and dots indicate differentiation with respect to t. The function h(x) is assumed continuous for all x considered, so that solutions of (1) exist satisfying any assigned initial conditions. In an earlier paper (2) explicit hypotheses on (1) were established, in the two distinct cases:under which every solution x(t) of (1) satisfieswhere t0 depends on the particular x chosen, and D is a constant depending only on a, b, h and p. These hypotheses are, in the case (2),or, in the case (3),In what follows here we shall refer to (2) and (H1) collectively as the (boundedness) hypotheses (BH1), and to (3) and (H2) as the hypotheses (BH2). Our object is to examine whether periodic solutions of (1) exist under the hypotheses (BH1), (BH2).
On the 1-dimensional category of Cartesian products
Authors: Berstein I. 
Published: November 2008

The following example arose out of an unsuccessful attempt to improve a result by Ganea and Hilton (5), according to whichimplies that for each field of coefficients F one at least of the factors Xi is acyclic; here cat X is the Lusternik-Schnirelmann category of X. It was natural to ask whether this assumption also implies that one at least of the Xi is simply connected. This could be proved if it were possible to obtain spaces of arbitrarily large 1-dimensional category ((3), (4)) by forming Cartesian products of sufficiently many 2-dimensional pseudoprojective spaces (1) corresponding to different primes. However, we prove that this is impossible:Theorem. If P1, …, Pn are 2-dimensional pseudo-projective spaces corresponding to different primes p1, …, pn, thenBefore proving this theorem, we recall the definitions involved. The 1-dimensional category cat1X of a space X is the least number of open sets covering X and such that each closed path in any of them is contractible in X. A 2-dimensional pseudo-projective space P corresponding to the prime p is the complex obtained from the disk |z| ≤ 1 by identifying the points ei[θ+(2mπ|p)], m = 0, 1, …, p − 1. We have (1)for k ≥ 2 and cat1P = 3 since π1(P) is not free and cat1P ≤ 1 + dim P (4).
Stability of a Column of Rotating Viscous Liquid
Authors: Gillis J. 
Published: November 2008

ABSTRACTA long column of viscous liquid of radius a and uniform density p is rotating about its axis with angular velocity ω. It is shown that this motion is stable to plane perturbations of wave number 8 provided that the surface tension T satisfies . This critical value is higher than that required for stability of the similar motion of a nonviscous liquid, but is otherwise independent of the coefficient of viscosity. The rate of development of instability when T is less than the critical value is also studied. Some numerical results are given.The condition has recently been obtained by Hocking (2) for the special cases of very high and very low Reynolds numbers.
Representations of the n-dimensional rotation group
Authors: Stone A. P. 
Published: November 2008

ABSTRACTThe commutators of the infinitesimal operators of the n-dimensional rotation group Rn with vector operators under Rn are expressed in a vectorial notation. The infinitesimal operators for the representations (l 0…0) are treated in detail. Shift operators for l are constructed and are used to derive the branching rule for these representations. The energy levels and degeneracy of bound states of a particle under an inverse square force in n dimensions are found by wave mechanics and by expressing the Hamiltonian in terms of Casimir's operator for Rn+1. Differential operators which transform one radial wave function into another are obtained.
Simple reduction theorems for extension and torsion functors
Authors: Northcott D. G. 
Published: November 2008

Let Λ be a commutative ring with an identity element and c an element of Λ which is not a zero divisor Denote by Ω the residue class ring Λ/Λc. If now M is a Λ-module for which c is not a zero divisor, and A is an Ω-module, then a theorem of Rees (2) asserts that, for every non-negative integer n, we have a Λ-isomorphismThis reduction theorem has found a number of useful and interesting applications.
Equations for stochastic path integrals
Authors: Bartlett M. S. 
Published: November 2008

Apart from the study of the integralwhere {X(u)} is a stationary Gaussian process with autocorrelation function ρ(t), by Kac and Siegert(1), most stochastic functionals of the general typehave been considered for {X(u)} either additive or Markovian (see, for example, (2), (3)), and in the Markovian case only for diffusion-type processes (Darling and Siegert (4)). More general approaches exist (e.g. (5), (6)), but seem less concerned with the investigation of specific problems. Some preliminary remarks here are therefore aimed at examining the structure of integrals of type (2), or such further extensions of the formal Riemann sum typethat would be expected to have well-behaved distributional properties for {X(t)} Markovian, and associated equations for studying these properties. As an example, the sumsay, is considered (i) for the normal linear Markovian process, (ii) for simple birth-and-death and emigration—immigration processes.
Expansions of the Poiseuille and the irregular Coulomb functions
Authors: Meligy A. S. 
Published: November 2008

ABSTRACTAn expansion of the Whittaker function Mk, m(z) in terms of functions of the same kind is obtained. It generalizes previous results and is used to derive expansions of the second Whittaker function Wk,m (z) in terms of the M functions for the cases m = 0 and ½. The case m = 0 provides expansions for the Poiseuille and the exponential integral functions. The case m = ½ provides an expansion for the irregular Coulomb wave function having angular momentum zero in terms of the regular Coulomb functions.
Two inequalities for parabolic cylinder functions
Authors: Olver F. W. J. 
Published: November 2008

In this paper upper bounds are established for the principal solution of the differential equationand its derivative, for unrestricted values of the complex variable t and the complex parameter μ. The results may have little interest in their own right, but they are of great value in developing the asymptotic theory of linear second-order differential equations in a domain containing two turning points. Equation (1·1) is the simplest example of a differential equation of this type.
Correlation problems in evolutionary stochastic processess
Authors: Ramakrishnan Alladi , Radha T. K. 
Published: November 2008

ABSTRACTIn a previous contribution to theseProceedings(Ramakrishnan(1)) the concept of product density was introduced to describe the statistical distribution of a discrete number of particles in a continuous spaceE, corresponding to a single pointt, wheretis the parameter with respect to which the stochastic process evolves. This is extended to densities corresponding to n points on thetaxis and correlation problems associated with these density functions are studied with particular reference to electron-photon cascades.
On queues in which customers are served in random order
Authors: Kingman J. F. C. 
Published: November 2008

ABSTRACTThe queue M |G| l is considered in the case in which customers are served in random order. A formal solution is obtained for the waiting time distribution in the steady state, and is used to consider the exponential decay of the distribution. The moments of the waiting time are examined, and the asymptotic form of the distribution in heavy traffic is found. Finally, the problem is related to those of the busy period and the approach to the steady state.
On the Number of Real Roots of a Random Algebraic Equation
Authors: Samal G. 
Published: November 2008

ABSTRACTLet Nn be the number of real roots of a random algebraic equation The coefficients ξν are independent random variables identically distributed with expectation zero; the variance and third absolute moment are finite and non-zero. It is proved thatwhere εν tends to zero, but εν log n tends to infinity. The measure of the exceptional set tends to zero as n tends to infinity.
The stability of a queue with non-independent inter-arrival and service times
Authors: Loynes R. M. 
Published: November 2008

IntroductionHere we shall mention only the results referring to stability. The definitions of the various quantities Tn, Sn, SNn, and the basic hypotheses made concerning their structure will be found in §§ 2·1, 3·1 or 4·1. For convenience we shall introduce some further terminology in this section. The single-server queues {SNn, Tn} arising in connexion with queues in series will be called the component queues, and the queue {Sn, sTn} implicit in the discussion of many-server queues will be called the consolidated queue. We have already in § 2.33 called the single-server queue {Sn, Tn} critical if E(S0-T0) = 0. We shall now call it subcritical if E(S0 − To) > 0 and supercritical if E(S0 − T0) < 0. A system of queues in series is subcritical if each component queue is subcritical, critical if (at least) one component queue is critical and the rest are subcritical, and supercritical if (at least) one component queue is supercritical. A many-server queue will be described in these terms according to the character of its consolidated queue. Finally, a single-server queue {Sn, Tn} will be said to be of type M if it has the property considered in Corollary 1 to Theorem 5: the sequences {Sn} and {Tn} are independent of each other, and one is composed of mutually independent non-constant random variables.Single-server queues:(i) Subcritical: stable (Theorem 3).(ii) Supercritical: unstable (Theorem 2).(iii) Critical: stable, properly substable, or unstable (examples in §2·33, including one due to Lindley); unstable if type M (Theorem 5, Corollary 1).Queues in series:(i) Subcritical: stable (Theorem 7).(ii) Supercritical: unstable (Theorem 7).(iii) Critical: stable, properly substable, or unstable, if the component queues are substable (examples in § 3·2); unstable if any component queue is unstable (Theorem 7), and in particular if any critical component queue is of type M (Theorem 7, Corollary).Many-server queues:(i) Subcritical: stable or properly substable (Theorem 8, and example in § 4·3).(ii) Supercritical: unstable (Theorem 8).(iii) Critical: stable, properly substable, or unstable, if consolidated queue is substable (examples in § 4·3); unstable if consolidated queue unstable (Theorem 8), and in particular if this is of type M (Theorem 8, Corollary).From Lemma 1 it follows that none of these queues can be properly substable if all the servers are initially unoccupied.
A storage model with continuous infinitely divisible inputs
Authors: Gani† J. , Prabhu N. U. 
Published: November 2008

AbstractThis paper presents a general theory of storage for dams subject to a steady release, and with continuous inputs having infinitely divisible distributions. The paper extends some earlier work by Downton, Lindley, Smith and Kendall ((10), see discussion), and gives a detailed account of results previously sketched by the authors (6).
Isometric flows in Hilbert space
Authors: Sz.-Nagy Béla 
Published: November 2008

1. Let {Vi}i≥0 be a weakly (hence also strongly) continuous semigroup of (linear) contraction operators on a Hilbert spaceH, i.e. |Vt| ≤ 1 (t≥ 0). LetZandWdenote the corresponding infinitesimal generator and cogenerator, i.e.Zis in general non-bounded, but closed and densely defined, andWis a contraction operator (everywhere defined inH), such that 1 is not a proper value ofW. Conversely, every contraction operatorWnot having the proper value 1 is the infinitesimal cogenerator of exactly one semigroup {Vi} of the above type; one has namelyin the sense of the functional calculus for contraction operators (4).
A note on blocks of modular representations
Authors: Srinivasan Bhama 
Published: November 2008

1. The notation used will be that of (2).G is a finite group, is a ring which is either(1·1) a field of characteristic p, or(1·2) the ring of valuation integers of a field of characteristic 0 with a complete discrete valuation. In this case, is the maximal ideal of .
Diffraction of sound pulses by a fluid cylinder
Authors: Mishra S. K. 
Published: November 2008

AbstractIn this paper, we consider the problem of diffraction of two-dimensional sound pulses by a homogeneous fluid circular cylinder contained in another homogeneous fluid. The line source is situated outside the cylinder and is parallel to its axis. It is supposed that the velocity of sound inside the cylinder is less than the velocity of sound in the surrounding medium. We investigate the problem by the method of dual transformation as developed by Friedlander. The pulse propagation modes both inside and outside the cylinder are obtained, We interpret the modes as diffracted pulses in terms of Keller's Geometrical Theory of Diffraction. The results agree with Friedlander's conjecture.
Imbedding and isotopy of spheres in manifolds
Authors: Levine J. 
Published: November 2008

1. The following results are special cases of theorems of Irwin and Zeeman, respectively (see (4), (9), (10)).(a) Let V be a (2n − m + 1)−connected piecewise-linear m−manifold (bounded or not), where m − n ≥ 3. Then any element of πn(V) can be represented by a piecewise-linear imbedding of Sn in V.(b) Let M be a (2n − m + l)-connected closed piecewise-linear (m − 1)-manifold, where m − n ≥ 3. Then two piecewise linear imbeddings of Sn−1 in V are isotopic if and only if they are homotopic.
Conditions equivalent to central symmetry of convex curves
Authors: Hammer P. C. , Smith T. Jefferson 
Published: November 2008

In this paper we establish that a convex planar body C is centrally symmetric provided either one of the following conditions hold:(1) Each line halving the circumference of the boundary γ of C is a diametral line.(A diametral line is a line intersecting C in a chord of maximal length in the family of parallel chords.)(2) Each line halving the area of C is a diametral line.
Abelian curves of 2-power conductor
Authors: Ogg A. P. 
Published: November 2008

Let Q be the field of rational numbers, and let A be an Abelian curve (an Abelian variety of dimension one) defined over Q. Following Weil, the conductor of A iswhere p runs over all primes, and the exponent is explained below; the primes dividing N are exactly those for which A has degenerate reduction modulo p.
The Evaluation of Eigenvalues of a Differential Equation Arising in a Problem in Genetics
Authors: Miller G. F. , Goodwin E. T. 
Published: November 2008

ABSTRACTThis paper concerns the determination of the smallest eigenvalue of a second order differential equation containing two parameters which arises in problems concerning genic selection under random drift in a population of finite size. A table of values is given, the method of computation is described, and the asymptotic behaviour for large values of one of the parameters is investigated.
On the almost everywhere convergence of Fourier series
Authors: Yadav B. S. 
Published: November 2008

Let f be a 2π-periodic function of the class L(−π,π). PutWe call, with Žuk(6), the quantity L(p)(h, f) the L-modulus of smoothness of order p of the function f. Žuk has recently obtained, in (5) and (6), generalizations of a number of classical results on the absolute convergence of Fourier series, as also on the order of Fourier coefficients by employing the concept of the L-modulus of smoothness which is obviously a more general concept than that of the modulus of continuity. It is the purpose of this note to prove a theorem on the almost everywhere convergence of Fourier series of f involving the concept of L(p)(h, f).
Cohomology operations and duality
Authors: Maunder C. R. F. 
Published: November 2008

Since Thom first introduced the notion of the ‘dual’ of a Steenrod square, in (12), it has become apparent that calculation with such duals in the cohomology of, say, a simplicial complex X will often yield information about the impossibility of embedding X in Sn, for certain values of n. For example, the celebrated theorem that cannot be embedded in can easily be proved in this way. In this paper, we seek to generalize this method to any pair of extraordinary cohomology theories h* and k*, and natural stable cohomology operation θ: h* → k*. We show in section 3 that a simplicial embeddingf: X → Sn gives rise via the Alexander duality isomorphism to a homology operationwhich is independent of n, the particular embedding f, and even the particular triangulations of X and Sn. If h* and k* are multiplicative cohomology theories, there are Kronecker productsif h0(S0) = k0(S0) = G, a field, and the Kronecker products make h*, h* and k*, k* into dual vector spaces over G, then can be turned into a cohomology operation c(θ): k*(X)→h*(X), by using this duality. This is certainly true if h* = k* = H*(;Zp), p prime, and in this case we have the original situation considered by Thom, who showed, for example, that