bluuebunny/crossref_metadata_2025_split huggingface dataset:
An Analysis of Triode Valve Rectification
Authors: Landale S. E. A. 
Published: November 2008

It is shown that a quartic equation may be found which expresses the grid characteristic of a valve accurately over a considerable range, and that for modern dull filament valves the grid current curve is exponential over only a very restricted length.Expressions are given for calculating rectified current and the amplitude of harmonics produced by a simple grid rectifier when voltages of the following wave-forms are applied: E sin pt, (A + B sin nt), and (A + B cos mt + C cos nt)sin pt. Further, a formula is given from which we may calculate the rectified current produced by a cumulative grid rectifier when a voltage E sin pt is applied.All these formulae are applicable if the grid characteristic, is parabolic, cubic, quartic or quintic in form.It is also shown that when an acoustically modulated radio frequency voltage is applied to a cumulative grid rectifier, even if the valve has a parabolic grid characteristic, an infinite number of harmonics of the modulation frequency are produced. These harmonics are due to the. action of the grid leak and condenser. It appears, however, that in practice the amplitude of these harmonics will be very small compared with those produced, primarily, by the curvature of the grid characteristic.
The relative absorbing powers of the L-levels for radiation of varying wave-length
Authors: Skinner H. W. B. 
Published: November 2008

1. Changes in the relative intensities of the lines in the fluorescentL-spectrum of Cerium excited by radiation of various wave-lengths have been observed.2. These results imply a change in the relative absorbing powers of the threeL-levels as the wave-length of the absorbed radiation diminishes from a value just below the absorption wave-length of theL-levels to a value considerably below. The absorbing power of theLI-level becomes increased relative to the absorbing powers of the otherL-levels as the wave-length diminishes. The results agree with those published by H. Robinson in a recent paper.3. These results imply a breakdown of the law that μ/λ3is a constant (where μ is the absorption coefficient of X-rays of wave-length λ) as applied to theindividual L-levels of an element.4. A comparison is made between the above results, and some results on the relative absorbing process of theL-levels obtained by Ellis and Skinner from β-ray spectra.
The Spectra of some Lithium-like and Sodium-like Atoms
Authors: Hartree D. R. 
Published: November 2008

It is possible to obtain approximate theoretical relations between the spectra of different atoms ionised to such an extent that they possess the same electron structure. In this paper these relations are used to extrapolate values of the terms of the spectra of various lithium-like and sodium-like atoms.From the terms so estimated lines can be calculated, and in many cases correlated with lines observed by Millikan and Bowen in the hot spark spectrum of the corresponding element. Revised term values can then be calculated from the lines so identified.The spectra of lithium-like atoms treated in this way include those from Be II up to O VI, and those of sodium-like atoms, the spectra P V and S VI.
Extension of a Geometrical Porism and other Theorems
Authors: Grace J. H. 
Published: November 2008

It is a familiar fact in Solid Geometry that the problem of finding a twisted cubic curve with four given tangents is poristic. In fuller statement four prescribed tangents are apparently sufficient to determine such a curve yet(A) In general there is no twisted cubic touching four given lines,(B) If there is one there is an infinite number.
A percentile table of the relation between the true and the observed correlation coefficient from a sample of 4
Authors: Allan F. E. 
Published: November 2008

The exact distribution of the simple correlation coefficient was established by Dr R. A. Fisher in 1915 (Biometrika, X, p. 507). If ρ is the true correlation in the population, and r the sample value, then the frequency element is given bywhere θ = cos− (−ρr).
Notes on special systems of orthogonal Functions (III): A System of orthogonal polynomials
Authors: Hardy G. H. 
Published: November 2008

Generalities. Suppose thatis real and L2(0, ∞), that is its Mellin transform, and that Then K(1)(x) generates a “Watson transformation”, with a “Fourier theorem”valid for every F(x) of L2(0, ∞).
On a conjecture of Mordell concerning binary cubic forms
Authors: Davenport H. 
Published: November 2008

Let f(x, y) be a binary cubic form with real coefficients and determinant D ≠ 0. In a recent paper, Mordell has proved that there exist integral values of x, y, not both zero, for whichThese inequalities are best possible, since they cannot be satisfied with the sign of strict inequality when f(x, y) is equivalent tofor the case D < 0, or tofor the case D > 0.
The thermodynamics of crystal lattices
Authors: Gow Margaret M. 
Published: November 2008

SummaryUsing the results of the preceding paper of this series (by M. Bradburn), the equation of state and the elastic constants for a monatomic face-centred cubic lattice are calculated. Central forces with a potential of the form −ar−m+br−n are assumed to act between the particles. Numerical results are obtained for five sets of values (m, n) and represented in tables and diagrams. The general features of these are discussed and compared with previous computations.I take this opportunity of thanking Prof. Born who suggested this problem to me, for his interest in my work and his advice on many occasions. I am also indebted to Dr R. Fürth for his numerous helpful suggestions.
Note on the multiplication of series by Cauchy's rule
Authors: Hardy G. H. 
Published: November 2008

1. I proved in 1908(1) that if A = Σam and B = Σbn are convergent, andfor large m and n, thenis convergent (necessarily to AB); and this theorem has been extended in a number of directions both by other writers and by myself. Thus we may replace (1), when am and bn are real, bywe may use conditions unsymmetrical in am and bn; we may put the same problem for the product of any number of series; and we may consider modes of ‘Dirichlet multiplication’ based on sequences λm and μn, reducing to Cauchy's when λm = m and μn = n. The appropriate references will be found in(2).
Solution of a Moment Problem for Bounded Functions
Authors: Verblunsky S. 
Published: November 2008

1. Given a sequence of variables a1, a2, …, we define a sequence of polynomials B1, B2, … byFor every positive integer n, Bn is a polynomial in a1, …, an, with rational coefficients. We write Bn ≡ Bn (a1, …, an) to indicate the variables on which Bn depends. From the polynomials Bn, we form a new sequence of polynomials Dn in accordance with the following definitions.
On an equation occurring in Falkner and Skan's approximate treatment of the equations of the boundary layer
Authors: Hartree D. R. 
Published: November 2008

The differential analyser has been used to evaluate solutions of the equationwith boundary conditions y = y′ = 0 at x = 0, y′ → 1 as x → ∞, which occurs in Falkner and Skan's approximate treatment of the laminar boundary layer. A numerical iterative method has been used to improve the accuracy of the solutions, and the results show that the accuracy of the machine solutions is about 1 in 1000, or rather better.It is shown that the conditions are insufficient to specify a unique solution for negative values of β a discussion of this situation is given, and it is shown that for the application to be made of the solution the appropriate condition is that y′ → 1 from below, and as rapidly as possible, as x → ∞. The condition that y′ → 1 from below can be satisfied only for values of β0, greater than a limiting value β0, whose value is approximately − 0·199, and which is related to the point at which the laminar boundary layer breaks away from the boundary.
Methods of construction of Geiger-Mueller counters and their use in coincidence experiments
Authors: Curran S. C. , Petržílka V. 
Published: November 2008

The technique adopted in some experiments employing Geiger-Mueller counters is described. The behaviour of a counter in which the extinction of the discharge is external is contrasted with behaviour of a counter in which extinction is internal and dependent on the addition to the gas of some organic vapour such as alcohol. It is shown that the internally extinguished type of counter may be used with a resistance of the order 50,000 ohms in series with it while still retaining all its useful features.Methods of construction of internally extinguished counters are described, particular attention being paid to the question of developing a thin-walled type suitable for experiments employing the method of coincidences. The effect of using different wall materials and gases is investigated with a view to finding those most suitable.The measurement of the quantum energy of hard γ-radiation by the method of coincidences is discussed and details of a reliable two-stage amplifying circuit with a resolving time T of 10−6 sec. are given. Various factors which may influence the value of T are noted.
Canonical systems on a reducible variety
Authors: Macpherson R. E. 
Published: November 2008

The object of this note is to extend the proof, for the general canonical varieties of any dimension k < d on a Vd whose existence was established by Dr Todd in a recent paper, of the property of adjunction given by the relationwhere S is any non-singular variety of dimension d−1 lying on Vd, and Xk[S] denotes the canonical system of dimension k on S. This formula was proved by Todd in the case when Vd is a general non-singular variety, and S is non-singular and belongs to a sufficiently general linear system. Using this result, I show how to remove the latter restriction. (1·1) cannot be taken, without further proof, as defining canonical systems on virtual or isolated varieties, as is done by B. Segre, since the canonical systems on an effective isolated variety are already well defined.
The postulation of a multiple curve
Authors: Segre B. 
Published: November 2008

1. The postulation of a multiple curve for primals of sufficiently large order in space of any number of dimensions has been obtained recently by J. A. Todd, by a simple and elegant degeneration argument which, however, is not deemed to be a conclusive proof by the author himself. And, indeed, in order to make sure of the unconditional validity of such an argument, one should ascertain whether(i) the postulation θk of an irreducible non-singular curve ϲ, of order c and genus p, for the primals of sufficiently large order n of [r + 2] (r ≥ 1), required to go through it with multiplicity k (≥ 1), is a function of k, c, p, n, r only;(ii) it is possible, by means of a continuous variation of ϲ, to reduce this curve to connected polygon ϲ′ having the same virtual characters as ϲ, in such a way that each intermediate position of ϲ is still irreducible and non-singular;(iii) the postulation θk of ϲ equals the similarly defined postulation of ϲ′.
On the stability of crystal lattices VII. Long-wave and short-wave stability for the face-centred cubic lattice
Authors: Power S. C. 
Published: November 2008

It is shown that the theorem stated in Born's paper, and proved for the case of a linear lattice of N equal particles under certain restrictions concerning the forces between the particles, that macroscopic stability (stability for long waves) implies microscopic stability, may be extended to three dimensions for the particular case of a face-centred cubic lattice, where the effects of all neighbours, other than the first twelve neighbours, are neglected.I take this opportunity of expressing my sincere thanks to Prof. Born for much valuable advice.
An extension of a theorem of Mehler's on Hermite polynomials
Authors: Kibble W. F. 
Published: November 2008

It was shown by Mehler (1866) thatwhere Hk(x) denotes the Hermite polynomial(Hermite, 1864a, b), which can be expressed in terms of Weber's parabolic cylinder function (Whittaker, 1903). The series is convergent if | ρ | < 1, and divergent if | ρ | > 1. If ρ = 1 and x = y = 0 the series is divergent, and Hille's work (1938) shows that it will therefore be divergent for all real or complex values, except possibly real positive values, of x and y.
The radiations emitted by and its formation in the deuteron bombardment of uranium
Authors: Feather N. , Krishnan R. S. 
Published: November 2008

The production of by deuteron bombardment of uranium has been established and the radiations of this product have been examined by the absorption method. The normal mode of disintegration involves β-particles having an upper energy limit of 1·20 ± 0·02 MeV. and a low-energy γ-ray of 76 ± 3 keV. quantum energy. Evidence has been found for electrons of less energy (about 0·1 per disintegration) and more penetrating γ-rays of relatively low intensity. Quantum radiations corresponding broadly to the L X-radiations, to be expected to follow the internal absorption of the low-energy γ-ray, have also been detected.The excitation function for the process has been studied, and the conclusion reached that the Oppenheimer-Phillips mechanism is responsible for the larger part of the yield.
On the number of representations of an integer as the sum of three r-free integers
Authors: Mirsky L. 
Published: November 2008

Let r, s be two fixed integers greater than 1. A positive integer will be called r-free if it is not divisible by the rth power of any prime.In a series of papers ((l)–(5)) Evelyn and Linfoot considered the problem of determining an asymptotic formula for the number Qr, s(n) of representations of a large positive integer n as the sum of s r-free integers; for s ≥ 4 their results were subsequently sharpened by Barham and Estermann(6).
Expansions in terms of sets of functions with complex eigenvalues
Authors: Peierls R. 
Published: November 2008

In the following I discuss the properties, in particular the completeness of the set of eigenfunctions, of an eigenvalue problem which differs from the well-known Sturm-Liouville problem by the boundary condition being of a rather unusual type.The problem arises in the theory of nuclear collisions, and for our present purpose we take it in the simplified formwhere 0 ≤ x ≤ 1. V(x) is a given real function, which we assume to be integrable and to remain between the bounds ± M, and W is an eigenvalue. The eigenfunction ψ(x) is subject to the boundary conditionsand
On perturbation problems associated with finite boundaries
Authors: Wassermann G. D. 
Published: November 2008

The methods of a previous paper (9) are improved and extended. It is assumed that the eigenfunctions and eigenvalues of an eigenvalue problem given by an elliptic differential equation are known subject to given boundary conditions on a finite boundary. It is shown how the corresponding quantities can be obtained for a similar problem in which the original differential equation, boundary and boundary conditions are simultaneously perturbed. The introduction of a surface displacement vector allows of a Taylor expansion of all quantities and a subsequent separation of orders. The problem of finding the perturbed eigenfunctions for each order then reduces to the solution of an inhomogeneous differential equation subject to known boundary conditions. These equations are solved by a variational method. An application of Green's theorem at each stage enables us to find the perturbed eigenvalues. The method is applied to a problem of which an exact solution is known and good agreement is obtained.The author is greatly indebted to Dr H. Fröhlich for many interesting discussions and some valuable suggestions.
On the iteration of a continuous mapping of a compact space into itself
Authors: Friedlander F. G. 
Published: November 2008

1. The questions considered in this note are suggested by the elementary topology of the trajectories of systems of non-linear differential equations. Such a system may be assumed in the formand the values of the dependent variables x1, x2, …, xn at ‘time’ t can be represented by a point P(t) in a ‘phase space’ . As t varies, P(t) describes a curve in , which is a trajectory of (1). Now it often happens that contains a subspace E (usually of lower dimension) with the following properties: (i) by considering the trajectories generated by points P(t) which are, for t = 0, in E, all the trajectories of (1) are obtained; (ii) if P(0) is in E, then P(t) is not in E for 0 < t < c, where c is a constant independent of P(0) in E; (iii) if P(0) is in E, then the trajectory meets E again for some finite t at a point P(T) (T is not necessarily the same for all points of E). By considering P(T) as the image of P(O), a mapping of E into itself is defined which is associated with the system (1), and the topology of the trajectories of (1) can be studied conveniently by discussing this mapping. When the functions fi in (1) satisfy the continuity and Lipschitz conditions of the classical existence-and-uniqueness theorem, the mapping is one-one and continuous. The study of this ‘transformation theory’, initiated by Poincaré, has been developed chiefly by G. D. Birkhoff(l,2). His results have been applied to problems of ‘non-linear mechanics’ by N. Levinson(3).
Stochastic processes relating to particles distributed in a continuous infinity of states
Authors: Ramakrishnan Alladi 
Published: November 2008

Many stochastic problems arise in physics where we have to deal with a stochastic variable representing the number of particles distributed in a continuous infinity of states characterized by a parameterE, and this distribution varies with another parametert(which may be continuous or discrete; iftrepresents time or thickness it is of course continuous). This variation occurs because of transitions characteristic of the stochastic process under consideration. If theE-space were discrete and the states represented byE1,E2, …, then it would be possible to define a functionrepresenting the probability that there are ν1particles inE1, ν2particles inE2, …, att. The variation of π withtis governed by the transitions defined for the process; ν1, ν2, … are thus stochastic variables, and it is possible to study the moments or the distribution function of the sum of such stochastic variableswith the help of the π function which yields also the correlation between the stochastic variables νi.
A comparison of heisenberg's spectrum of turbulence with experiment
Authors: Proudman Ian , Batchelor G. K. 
Published: November 2008

AbstractIn this paper, the theoretical double and triple velocity correlation functions, f(r), g(r) and h(r), which correspond to Heisenberg's spectrum of isotropic turbulence, are obtained numerically for two Reynolds numbers. One set of these correlations is for the limiting case of infinite Reynolds number. In addition, a method is developed for deriving the approximate form of the double correlations for any Reynolds number, which is not too small, from the corresponding correlations for infinite Reynolds number. These theoretical correlations are then compared with the results of experiment.
The structure of the turbulent boundary layer
Authors: Townsend A. A. 
Published: November 2008

ABSTRACTThe nature of the turbulent motion in a boundary layer with zero longitudinal pressure gradient has been investigated with the techniques of hot-wire anemometry which have been developed for the study of shear flow in wakes. Measurements have been made of the intensities of the turbulent velocity components, the turbulent shear stresses, the rates of transport of turbulent energy by diffusive movements, the intensities and flattening factors of the down-stream spatial derivatives of the three velocity components, and spectra of the down-stream component of the velocity fluctuation, at traverses through the boundary layer at three stations where the Reynolds numbers (based on the displacement thickness and free stream velocity) were respectively 3630, 4360 and 5080. Over the range of measurement, which did not include the laminar sublayer, the turbulent motion was similar at all three stations, and could be described in terms of universal functions. By considering the turbulent energy balance, it is shown that, except in the outer part of the layer where the turbulence resembles strongly the turbulence in a wake, there is a strong flow of energy directed toward the wall and transported by the action of turbulent pressure gradients. It is concluded that, most probably in contact with the laminar sublayer, there must be a ‘dissipative layer’ within which most of the turbulent energy dissipation takes place, and that the bulk of the eddies are, in a sense, attached to the wall and have very high rates of shear in that region. In agreement with this view of the structure of the turbulence, length scales derived from the apparent eddy viscosity and the local turbulent intensity are found to be comparable with distance from the wall. Length scales in the direction of the mean stream are much larger, and it is believed that the typical eddy is very elongated in this direction, and has its vorticity directed roughly parallel to the direction of maximum positive mean rate of strain.
Isolated minima of the product of n linear forms
Authors: Barnes E. S. 
Published: November 2008

Letbe n linear forms with real coefficients and determinant Δ = ∥ aij∥ ≠ 0; and denote by M(X) the lower bound of | X1X2 … Xn| over all integer sets (u) ≠ (0). It is well known that γn, the upper bound of M(X)/|Δ| over all sets of forms Xi, is finite, and the value of γn has been determined when n = 2 and n = 3.
The solution of the Schrödinger equation for finite systems, with special reference to the motion of electrons in Coulomb electric fields and uniform magnetic fields
Authors: Dingle R. B. 
Published: November 2008

ABSTRACTA number of methods are formulated for solving the Schrödinger equation for systems of finite extent. The methods are developed in detail for the particular case of an electron moving in a Coulomb field (e.g. hydrogen-like atom), with a boundary consisting of a sphere of given radius. In the second part of the paper these results are transformed into those for an electron moving in a uniform magnetic field, the boundary of the system being cylindrical.
Stochastic processes associated with random divisions of a line
Authors: Ramakrishnan Alladi 
Published: November 2008

ABSTRACTA class of stochastic processes associated with points randomly distributed in a line of finite extensionL, is considered. A general integral equation for the function representing the probability distribution of the stochastic variable under consideration is derived and solved by using the Laplace transform technique. Examples of the above class of processes are cited. In particular, the problem of the fluctuations in brightness of the Milky Way is discussed in detail. The results of Chandrasekhar and Munch in regard to this astrophysical problem are derived in a simple and direct manner.
Lie rings satisfying the Engel condition
Authors: Higgins P. J. 
Published: November 2008

1. Let be a Lie ring in which the product of elements x and y is denoted by xy. The inner derivations of , i.e. the mappings X:a→ax for fixed elements x of , form a Lie ring under the product [X, Y] = XY – YX, and the mapping x→ X is a homo-morphism of onto . We shall say that satisfies the nth Engel condition if Xn = 0 for all X in , i.e. iffor all a, x; in . If satisfies the maximum condition on subrings, it is known (1) that this condition implies the nilpotence of ; indeed, must then be nilpotent even if the integer n is allowed to depend on the element X of . We consider here the case in which n is independent of X but does not necessarily satisfy the maximum condition, and inquire whether is then nilpotent.