bluuebunny/crossref_metadata_2025_split huggingface dataset:
Stability criteria for linear systems and realizability criteria for RC networks
Authors: Fuller A. T. 
Published: November 2008

ABSTRACTA new set of stability criteria for linear systems is derived. This shows that about half of the Hurwitz criteria are redundant when certain of the coefficients of the characteristic equation are known to be positive. The theory is applied to obtain a very short derivation of the known aperiodicity criteria. The conditions for realizability of RC networks are shown to be closely related to the stability and aperiodicity criteria, and are stated as sets of criteria in terms of the polynomial coefficients. Two basic theorems are involved which give the necessary and sufficient conditions for the roots of two polynomial equations to be real and separated.
Classical electrodynamics as a distribution theory. II
Authors: Taylor J. G. 
Published: November 2008

ABSTRACTIn a previous paper by the author (3) it was shown how the theory of distributions of L. Schwartz enables a mathematically consistent formalism to be given for a system composed of point charges interacting through their classical electromagnetic field. In the present paper a definition of the energy and momentum of the field lying in a space-like surface is given, and it is shown that from this four-vector it is possible to derive the usual equation of conservation of total energy and momentum.
Analytic functions of finite dimensional linear transformations
Authors: Afriat S. N. 
Published: November 2008

Since the first introduction of the concept of a matrix, questions about functions of matrices have had the attention of many writers, starting with Cayley(i) in 1858, and Laguerre(2) in 1867. In 1883, Sylvester(3) defined a general function φ(a) of a matrix a with simple characteristic roots, by use of Lagrange's interpolation formula, and Buchheim (4), in 1886, extended his definition to the case of multiple characteristic roots. Then Weyr(5) showed in 1887 that, for a matrix a with characteristic roots lying inside the circle of convergence of a power series φ(ζ), the power series φ(a) is convergent; and in 1900 Poincaré (6) obtained the formulaefor the sum, whereCis a circle lying in and concentric with the circle of convergence, and containing all the characteristic roots in its ulterior, such a formula having effectively been suggested by Frobenius(7) in 1896 for defining a general function of a matrix. Phillips (8), in 1919, discovered the analogue, for power series in matrices, of Taylor's theorem. In 1926 Hensel(9) completed the result of Weyr by showing that a necessary and sufficient condition for the convergence of φ(a) is the convergence of the derived series φ(r)(α) (0 ≼r<mα; α) at each characteristic root α of a, of orderrat most the multiplicitymαof α. In 1928 Giorgi(10) gave a definition, depending on the classical canonical decomposition of a matrix, which is equivalent to the contour integral formula, and Fantappie (11) developed the theory of this formula, and obtained the expressionfor the characteristic projectors.
A theorem on the cumulative product of independent random variables
Authors: Ruben Harold 
Published: November 2008

1. Introductory discussion and summary. Consider a sequence {ui} of independent real or complex-valued random variables such that E(ui) = 1, and a sequence of mutually exclusive events S1, S2,…, such that Si depends only on u1, u2, …,ui, with ΣP(Sj) = 1. Define the random variable n = n(u1, u2,…) = m when Sm occurs. We shall obtain the necessary and sufficient conditions under whichreferred to as the product theorem.
Hilbert transforms and almost periodic functions
Authors: Cossar J. 
Published: November 2008

The Hilbert transform, Hf, of a function f is defined by Hf = g, whereP denoting the Cauchy principal value and the integral being assumed to exist in some sense. When f is suitably restricted, Hf exists andIn the first part of Theorem 1 sufficient conditions are given for the validity of (1·2) rather more general than those of Wood ((6), p. 31). The present proof is based on the well-known condition of Riesz for the validity of (1·2), namely, that f is Lp(−∞, ∞) for some p > 1, and on the ‘Parseval’ relation (Lemma 3, (b)), which was used in a similar way by Hardy ((3), p. 110).
A random walk problem
Authors: Gillis J. 
Published: November 2008

We consider a random walk on a two-dimensional rectangular lattice in which steps are strictly between nearest neighbour points. The conditions of the walk are that the walker must, at each step, turn either to the right or to the left of his previous step with respective probabilities ½(1+α), ½(1−α), (≤ α ≤ 1). To fix the ideas it is assumed that he starts from the origin and the probability of each of the four possible starting directions is ¼. If Ar denotes the probability of return to the origin after r steps we shall show thatwhere β = ½(α + α−1) and Pn is the nth Legendre polynomial. It is clear that Ar is zero for r ≢ 0(mod 4).
Non-linear Lagrangians and Palatine's device
Authors: Buchdahl H. A. 
Published: November 2008

ABSTRACTField equations in general relativity theory have sometimes been generated by subjecting, in an invariant action integral, the components of linear connexion and the components of a covariant tensor of valence 2 to independent variation. The conceptual objections to this process, and some of the manifold formal difficulties inherent in it, are discussed in some detail. At the same time certain results obtained elsewhere are strengthened and in part corrected.
The Motion of a Biased Bowl with Perturbing Projection Conditions
Authors: Brearley M. N. 
Published: November 2008

ABSTRACTThe equations of motion of a biased bowl rolling on a plane horizontal grass green are derived by Lagrange's method, using an appropriate representation of the effect of the green. The equations are solved approximately under general initial conditions which include the presence of a perturbing wobble.An expression is derived for the total angle of precession of the bowl, and the equation of the path is found in parametric form. The theory predicts that (contrary to universal belief) the effect of an initial wobble on the bowl path is negligible under normal playing conditions.A detailed comparison is made between the predictions of the theory and experimental results obtained on several greens of different speeds.
Note on the Jacobi identity for Whitehead products
Authors: Hilton P. J. 
Published: November 2008

The Jacobi identity for Whitehead products,where α ɛ πl(X), β ɛ πm(M), γ ɛ πn (X), l, m, n ≥ 2 is now well known. In this note we extend the identity to the case l, m, n > 1; the extension is easy and we are merely filling a small gap in the literature. However, the extension does enable us to improve a result in (2).
Inequalities for functions regular in the unit circle
Authors: Linden C. N. 
Published: November 2008

Let f(z) = u(z) + iv(z) be a function regular for |z| < 1 with real and imaginary parts u(z) and u(z) respectively. The following theorems have been proved by Miss M. L. Cartwright(1) and (2).Theorem A. if v(0) = 0, 0 < α < ∞ andfor 0 ≤ r < 1, 0 ≤ θ < 2π thenfor 0 ≤ r < 1, 0 ≤ θ < 2π
A cosmological model of light propagation
Authors: Prokhovnik S. J. 
Published: November 2008

AbstractA theory of light propagation consistent with our present cosmological views is investigated in the context of a uniformly expanding universe. It leads to the relativistic equivalence of fundamental observers and also provides a natural substratum associated with an observable fundamental reference frame for every locality of the universe. Special Relativity emerges as a theory satisfying the observable universe.The model provides a quantitative interpretation of the recent findings in radio and optical astronomy. It has the advantage of simplicity and its approach may assist in an understanding of what is happening in the mathematics of more sophisticated systems.
The hydromagnetics of an ellipsoid moving in a cross-field
Authors: Ludford G. S. S. , Singh M. P. 
Published: November 2008

AbstractOur previous discussion ((4), (5)) is here extended to an ellipsoid of arbitrary conductivity. Again, the flow is ultimately cylindrical and the magnetic field undisturbed. An ellipsoidal insulator allows fluid to pass through its circumscribing cylinder, but also forces some to go around. An ellipsoidal conductor behaves like a very tall insulator, irrespective of its own height or conductivity: all the fluid goes round. If it is very wide an unexpectedly large amount of fluid is carried with it.
Short surface waves in the presence of a finite dock. II
Authors: Holford R. L. 
Published: November 2008

AbstractThe generalized wave-making problem for the forced high-frequency oscillations of a finite dock is solved through use of the Green's function obtained in the first part of this paper. The specific cases of heave and roll are considered with particular reference to the forces on the dock and the amplitude and phase of the radiated waves. These results are then utilized to solve the problem of transmission of short waves under a fixed dock.
Quadratic forms as Liapunov functions for linear differential equations with real constant coefficients
Authors: Lehnigk Siegfried H. 
Published: November 2008

AbstractThis paper deals with the existence of a quadratic form as a Liapunov function for a linear homogeneous vector differential equation with constant coefficient matrix such that the total derivative of the Liapunov function is strictly negative semi-definite and not identically equal to 0 for every non-trivial solution of the given equation. A theorem is proved which guarantees the existence of a strictly semi-definite quadratic form which is not identically equal to 0 for every non-trivial solution if and only if the coefficient matrix of the equation is not proportional to the unit matrix. Furthermore, it is proved that if the equilibrium of the given linear equation is asymptotically stable and if the coefficient matrix is not proportional to the unit matrix, then there exists a positive definite quadratic form as a Liapunov function whose total derivative is strictly negative semi-definite and not identically equal to 0 for every non-trivial solution of the given equation. It is also shown how this theorem fits into the system of already known theorems concerned with quadratic forms as Liapunov functions for a linear equation with constant coefficient matrix.
A note on Milne-Thomson's general solution of Navier–Stokes equations
Authors: Nigam G. D. 
Published: November 2008

The Navier–Stokes equations governing the motion of viscous compressible fluids arewhere ρ is the density, q is the velocity vector, V is the scalar potential of the external force field, I is the idemfactor and Φ is the stress-tensorMilne-Thomson ((l)) has given a general solution of (1) and (2) in the form
Minimal verbal subgroups
Authors: Kovács L. G. , Newman M. F. 
Published: November 2008

Introduction. In this note we obtain a description of the structure of the minimal verbal subgroups of a finite group which has, among others, the following consequences:Theorem 1. A finite group which belongs to the variety generated by its proper subgroups and proper factor-groups belongs either to the variety generated by its proper subgroups or to the variety generated by its proper factor-groups.
Singular measures with absolutely continuous convolution squares
Authors: Hewitt Edwin , Zuckerman Herbert S. 
Published: November 2008

Introduction. A famous construction of Wiener and Wintner ((13)), later refined by Salem ((11)) and extended by Schaeffer ((12)) and Ivašev-Musatov ((8)), produces a non-negative, singular, continuous measure μ on [ − π,π[ such thatfor every ∈ > 0. It is plain that the convolution μ * μ is absolutely continuous and in fact has Lebesgue–Radon–Nikodým derivative f such that For general locally compact Abelian groups, no exact analogue of (1 · 1) seems possible, as the character group may admit no natural order. However, it makes good sense to ask if μ* μ is absolutely continuous and has pth power integrable derivative. We will construct continuous singular measures μ on all non-discrete locally compact Abelian groups G such that μ * μ is a absolutely continuous and for which the Lebesgue–Radon–Nikodým derivative of μ * μ is in, for all real p > 1.
The secular variation of the earth's magnetic field
Authors: Allan D. W. , Bullard E. C. 
Published: November 2008

AbstractThe magnetic field observable outside a body of conducting fluid in which field is imbedded may be considerably altered by convection currents in the fluid. One possible explanation of the geomagnetic secular variation foci is that localized convection cells in the earth's core disturb the main field present. An analytic solution for such a process is readily obtained by assuming the form and dimensions for such a cell, and shows that the magnitude of the secular variation cannot easily be explained on these lines without the presence of a subsurface toroidal magnetic field of some hundreds of gauss which is ‘convected through’ the surface of the core.
Strong summability of infinite series on a scale of Abel type summability methods
Authors: Mishra Babban Prasad 
Published: November 2008

Introduction. In a recent paper, Borwein(1) constructed a new method of summability which would read: Letand let {sn} be any sequence of numbers. If, for λ > − 1,is convergent for all x in the open interval (0,1) and tends to a finite limit s as x → 1 in (0,1), we say that the sequence {sn} is Aλ convergent to s and write sn → s(Aλ). The A0 method is the ordinary Abel method.
Summability by Dirichlet convolutions
Authors: Segal S. L. 
Published: November 2008

Ingham (3) discusses the following summation method:A series ∑an will be said to be summable to s ifwhere, as usual, [x] indicates the greatest integer ≤ x. (An equivalent method was introduced somewhat earlier by Wintner (8), but the notation (I) for the above method and the attachment to Ingham's name seem to have become usual following [(1), Appendix IV].) The method (I) is intimately connected with the prime number theorem and the fact that the Riemann zeta-function ζ(s) has no zeros on the line σ = 1. Ingham proved, among other results, that (I) is not comparable with convergence but, nevertheless, for every δ > 0, (I) ⇒ (C, δ) and for every δ, 0 < δ < 1, (C, −δ) ⇒ (I), where the (C, k) are Cesàro means of order k.
Finite amplitude convection in a self-gravitating fluid sphere containing heat sources
Authors: Baldwin P. 
Published: November 2008

AbstractThe non-linear equations governing the convective flow of heat, beyond marginal stability, in a fluid sphere containing a uniform distribution of heat sources, and cooled at its surface, are expressed in the Boussinesq approximation as a perturbation of the steady-state conduction solution. A steady-state axisymmetric poloidal flow is assumed, and the equations for an approximate solution, involving the first spherical harmonic only, are obtained using a special case of the general evolution criterion of Glansdorff and Prigogine. The solution is found numerically using an iterative procedure, the results showing good agreement with those obtained by Stuart's ‘shape assumption’ for values of the Rayleigh number just above the critical value for the onset of marginal stability.
Coupled wave equations for propagation in inhomogeneous compressible plasmas
Authors: Burman R. 
Published: November 2008

AbstractThis paper deals with wave propagation in inhomogeneous compressible electron plasmas. The waves are described by Maxwell's equations coupled to the linearized single-fluid equations of hydrodynamics. Coupled wave equations are derived which describe the propagation of coupled electromagnetic and electron acoustic waves. Results are obtained for generally inhomogeneous plasmas and are specialized to planar and cylindrically stratified media. Particular attention is given to the planar case and several approximate techniques for treating the equations are discussed. The fields in a region of coupling are investigated.
The multiplicative process in semiconductor radiation detectors. A practical application of Good's theorem in the theory of branching processes
Authors: Ogawa Iwao , Furukawa Masao 
Published: November 2008

AbstractThe electron avalanche that takes place within an internally amplifying semiconductor radiation detector has been analysed statistically, treating the process as an example of multi-dimensional branching processes. By extending the result obtained from Good's theorem to the case of continuous variable, an integral equation is derived for the generating function of the probability that a single primary electron-hole pair, formed at a depth x, induces an avalanche consisting of n pairs. On the basis of this equation, expressions are deduced for the mean value and the standard deviation σ of the total number n of pairs in an avalanche induced by n0 primary pairs produced at x, as well as the probability w(x, n0, n) of getting an avalanche of ‘size’ n. In particular cases these expressions lead to various simple relations of practical importance.
Double Meijer transformations of certain hypergeometric functions
Authors: Srivastava H. M. , Singhal J. P. 
Published: November 2008

Following the usual notation for generalized hypergeometric functions we let(a) denotes the sequence of A parametersthat is, there are A of the a parameters and B of the b parameters. Thus ((a))m has the interpretationwith a similar interpretation for ((b))m; Δ(k; α) stands for the set of k parametersand for the sake of brevity, the pair of parameters like α + β, α − β will be written as α ± β, the gamma product Γ(α + β) Γ(α − β) as Γ(α ± β), and so on.
Note on the isotropic scattering of radiation from a point source in a finite spherical atmosphere
Authors: Cassell J. S. 
Published: November 2008

1. In (1) M. G. Smith solved the equationwherefor R > 0, ø( − R) = − ø(R) and 0 < ω ≤ 1. In order to determine a constant, A(R1), appearing in the solution, two expressions for q(R1) were obtained, namelyand
Complementary variational principles for ∇2φ = f(φ) with applications to the Thomas–Fermi and Liouville equations
Authors: Arthurs A. M. , Robinson P. D. 
Published: November 2008

AbstractComplementary variational principles associated with the boundary-value problem ∇2φ = f(φ) in ∇,φ = φ0 on the boundary of ∇, are presented. The theory is applied to the Thomas–Fermi equation in quantum mechanics and the Liouville equation in equilibrium statistical mechanics.
On ‘translated quasi-Cesàro’ summability. II
Authors: Kuttner B. 
Published: November 2008

In a recent paper (1), I considered the summability method (D, α) defined, for α > 0, by the sequence-to-sequence transformationWe note that, as is easily verified (and as was pointed out in (1)) a necessary and sufficient condition for the convergence of (1), and thus for the applicability of (D, α), is thatshould converge. It was proved in (1) that, provided that (2) converges, a sequence summable (C, r) for any r > − 1 is necessarily summable (D, α). We now show that we can strengthen this result by replacing Cesàro by Abel summability. Moreover, we can omit the hypothesis that (2) converges provided that we interpret (1) as an Abel sum.
Some results on generalized hypergeometric polynomials
Authors: Shah Manilal 
Published: November 2008

AbstractIn this paper, using a generalized hypergeometric polynomial defined bywhere Δ(m, − n) denotes the set of m-parameters:and m, n are positive integers, we have established some infinite series, transformations, integrals and expansion formulae for generalized hypergeometric polynomials. The polynomial is in a generalized form which yields many known polynomials with proper choice of parameters. Special cases have also been given.
Polynomial functors
Authors: Passi I. B. S. 
Published: November 2008

1. Introduction: If G is a group, Z(G) its integral group-ring and AG the augmentation ideal, then we can form the Abelian groupsIn (5) we have studied the structure of these Abelian groups which we called polynomial grouups. If C denotes the category of Abelian groups, then Pn and Qn are functors from C into C. We call these functors polynomial functors. The object of this work is to study the nature of these funtors. Except for n = 1, these functors are non-additive. In fact, in the sense of Eilenberg–Maclane (4) these are functors of degree exactly n (Theorem 2·3). Because of their non-additive nature, their derived functors cannot be calculated in the traditional Cartan–Eilenberg(1) method. We have to make use of the more recent theory of Dold–Puppe (3).
Absolute convexity in certain topological linear spaces
Authors: Maddox I. J. , Roles J. W. 
Published: November 2008

For r > 0 a non-empty subset U of a linear space is said to be absolutely r-convex if x, y ∈ U and |λ|r + |μ|r ≤ 1 together imply λx + μy∈ U, or, equivalently, xl, …, xn∈ U andIt is clear that if U is absolutely r-convex, then it is absolutely s-convex whenever s < r. A topological linear space is said to be r-convex if every neighbourhood of the origin θ contains an absolutely r-convex neighbourhood of the origin. For the case 0 < r ≤ 1, these concepts were introduced and discussed by Landsberg(2).
On Mill's Ratio
Authors: Gupta Baikunth N. 
Published: November 2008

AbstractThe inequalities related to Mill's Ratio were conjectured true by Birnbaum (2) and were proved by Murty (4) for sufficiently large x > 0. In the present paper, these are proved for all finite x and have been used to calculate an upper limit for Mill's Ratio over the range x > – 1.
On the short-wave asymptotic motion due to a cylinder heaving on water of finite depth. II
Authors: Rhodes-Robinson P. F. 
Published: November 2008

AbstractThis paper continues the investigation into the short-wave asymptotic motion due to a cylinder heaving on water of finite constant depth, and we now present a rigorous method for the special case of a half-immersed circular cylinder. The aim is to justify, in one particular case, those assumptions made in a previous paper giving a non-rigorous method for an arbitrary smooth cylinder which intersects the free surface normally, concerning the asymptotic values on the cylinder of the two related auxiliary potentials which were introduced to enable an evaluation of the coefficients describing wave-making and virtual mass. This is done by constructing an integral equation of the second kind whose kernel is small and deducing the leading term in the iteration solution. The method is an extension of the method for infinite depth and it is found that there is a simple relationship between the kernels. In conclusion, computations for these asymptotic results are presented and compared with the trend of known non-asymptotic results.
Sets of synthesis for general algebras
Authors: Körner T. W. 
Published: November 2008

AbstractFollowing a program of Reiter, sets of synthesis in Banach algebras with certain Ditkin conditions are discussed. It is shown that the Ditkin Shilov lemma is in a general context best possible. Various methods for the construction of algebras are discussed. These are applied to show that very strong homogeneity conditions are needed to ensure that star sets are of synthesis for regular Ditkin algebras with Rn as carrier space.
On the geodesic completeness of causal space-times
Authors: Clarke C. J. S. 
Published: November 2008

AbstractA geodesic in space-time is complete if it can be extended to infinite values of its affine parameter: it is shown that all strongly causal spaces are conformal to space-times in which all null geodesies are complete, and that a wide class of space-times are conformal to ones in which almost no null geodesies are complete.