bluuebunny/crossref_metadata_2025_split huggingface dataset:
The formation and growth of shock waves in the one-dimensional motion of a gas
Authors: Pillow A. F. 
Published: November 2008

In many compressible fluid flow problems the classical solution breaks down completely owing to the formation of regions of infinite acceleration in the flow field. The actual behaviour of the fluid in such cases does not seem to have been investigated mathematically and this is largely due to the difficulties which enter with non-uniform shocks. It is with these difficulties that this paper is principally concerned.The way in which discontinuities may arise mathematically in a flow field is first discussed. The equations governing the one-dimensional motion of a gas due to an accelerating piston are then set up. It is shown that when allowance is made for varying entropy conditions due to the presence of non-uniform shocks these differential equations reduce (outside the shock wave) to three first order quasilinear ones.The initial solution breaks down when a point of infinite acceleration occurs in the flow field. From this point onwards a shock wave grows in the fluid and behind it three different sets of characteristics are required to describe the flow. By working in the plane of two quantities that are constant along two different sets of characteristics, we can use the shock-jump conditions to determine the equation of the shock-line in this plane and to reduce the equations of the characteristics to three differential equations, which would be linear if the relation between the entropy and other flow variables were known.In the case of a constantly accelerating piston a first approximation is found by neglecting reflexions and entropy variations behind the shock. Using this as a basis we then find a second approximation for the entropy in the neighbourhood of the initial portion of the shock-line and show that the problem reduces to the solution of a second order linear partial differential equation. The introduction of a Riemann function and the satisfaction of the boundary conditions at the shock lead to an integral equation whose solution enables us to determine the position of the shock as a function of the time. The solution is in the form of a power series and is valid provided the shock wave does not become too strong.Finally, it is shown that if the piston is given a constant terminal velocity a reflected wave from the shock is reflected again from the piston and eventually overtakes the shock and reduces its velocity to a final steady value which is in agreement with the value arising from an impulsive start.
Surface waves on deep water in the presence of a submerged circular cylinder. II
Authors: Ursell F. 
Published: November 2008

ABSTRACTIn the first part of this paper a method was given for constructing a wave potential when the normal velocity is a prescribed function of the angular variable on a submerged circular cylinder. It was shown that the method breaks down for values of the parameters Ka and Kf for which a certain infinite determinant vanishes. The vanishing of this determinant implies the existence of a non-trivial velocity potential, such that the normal velocity vanishes on the cylinder and both velocity components vanish at infinity. In this part of the paper it is shown that there can be no non-trivial solution of this kind; in other words the infinite determinant does not vanish. In the absence of a general uniqueness theory for surface waves it seems worth while to establish this particular result.
An extension of the law of the iterated logarithm
Authors: Cassels J. W. S. 
Published: November 2008

1. Let x1, x2, …, xn, … be a set of independent variables each with a uniform probability distribution in 0 ≤ x ≤ 1. If 0 ≤ α < β ≤ 1 we denote by FN (α, β) the number of x1, …, xN which satisfy α < x ≤ β,and put RN(α, β) = FN(α, β) − N(β − α).
Triple velocity correlations in isotropic turbulence
Authors: Stewart R. W. 
Published: November 2008

AbstractThe triple velocity correlation, in turbulence produced by inserting a square-mesh grid near the beginning of the working section of a wind tunnel, has been measured for mesh Reynolds numbers of RM = 5300, 21,200 and 42,400 (RM = UM/ν, where U is the mean wind speed in the working section of the tunnel and M is the centre to centre spacing of the rods making up the grid; ν is the kinematic viscosity of air). At the lowest Reynolds number the correlation has been measured at distances downstream of the grid varying from 20 to 120M. This range covers practically all of the initial period of the decay of turbulence, where the turbulent intensity varies as t−1.
The theoretical behaviour of a magnetic monopole in a wilson cloud chamber
Authors: Cole H. J. D. 
Published: November 2008

AbstractDirac has suggested that the quantization of electric charge could be explained by the existence of magnetic monopoles. In view of this hypothesis, this paper investigates what theoretically would be the behaviour of such monopoles in a Wilson cloud chamber. The treatment, which for simplicity is basically classical, closely follows Bohr's work on the decrease of velocity and ionization properties of α- and β-particles, and expressions are derived for the rate of decrease of energy and the number of ion-pairs produced per centimetre by a monopole passing through a gas. These expressions are then discussed with particular reference to the case of heavy particles, and the main differences between them and the corresponding expressions for α-particles both as to range and ionization are indicated; these differences can be summarized by saying that monopoles have much shorter paths, but create many more ion-pairs per centimetre than α-particles. Also, the very sharp increase in the ionization at the end of the path of an electric particle is missing, the ionization for the monopole decreasing to a small amount near the end of the path.
A note on the correspondence between the x, t-plane and the characteristic plane in a problem of interaction of plane waves of finite amplitude
Authors: Stocker P. M. , Meyer R. E. 
Published: November 2008

AbstractThe head-on interaction, in the one-dimensional, unsteady isentropic flow of a perfect gas, of a simple compression wave and a simple expansion wave is studied by considering typical examples. The physical aspect of the problem is discussed in (1); in this note the possibility of shock formation is ignored, and the correspondence defined by the complete mathematical solution of the equations of isentropic flow between the x, t-plane and the plane of the characteristic variables is elucidated.The solution is distinguished by the appearance of two limit lines and a second-order limit point where they meet. It is found that the image of the characteristic plane in the x, t-plane is four-sheeted; all sheets overlap each other, but each covers only part of the plane, and the only point common to all sheets is the second-order limit point, where both limit lines are cusped (§ 3·1).The solution also contains an edge of regression, and a discussion of the properties of this type of singularity will be found in §§ 2 and 2·1.
A further method for the evaluation of zeros of Bessel functions and some new asymptotic expansions for zeros of functions of large order
Authors: Olver F. W. J. 
Published: November 2008

In a recent paper (1) I described a method for the numerical evaluation of zeros of the Bessel functions Jn(z) and Yn(z), which was independent of computed values of these functions. The essence of the method was to regard the zeros ρ of the cylinder functionas a function of t and to solve numerically the third-order non-linear differential equation satisfied by ρ(t). It has since been successfully used to compute ten-decimal values of jn, s, yn, s, the sth positive zeros* of Jn(z), Yn(z) respectively, in the ranges n = 10 (1) 20, s = 1(1) 20. During the course of this work it was realized that the least satisfactory feature of the new method was the time taken for the evaluation of the first three or four zeros in comparison with that required for the higher zeros; the direct numerical technique for integrating the differential equation satisfied by ρ(t) becomes unwieldy for the small zeros and a different technique (described in the same paper) must be employed. It was also apparent that no mere refinement of the existing methods would remove this defect and that a new approach was required if it was to be eliminated. The outcome has been the development of the method to which the first part (§§ 2–6) of this paper is devoted.
An inequality for real positive functions
Authors: Hayman W. K. 
Published: November 2008

In the theory of functions of a complex variable a positive quantity g(r), denned in terms of a circle |z| = r, can often be estimated by means of another similar quantity f(R) denned in terms of a circle |z| = R with R > r. Inequalities of the formarise in this connexion. Putting R = r + h we have for every h > 0.
Calculation of S-matrix elements
Authors: Chisholm J. S. R. 
Published: November 2008

AbstractA method of calculating S-matrix elements is described. All factors corresponding to lines in Feynman-Dyson graphs are put in the form of scalar boson line factors acted on by certain operators and involving only the energy-momenta of external lines and particle masses. A general formula is given for the integral of a product of scalar boson factors, and the effect of a product of operators and on this result is described. The infinities appear in a different form, and it is shown how subtraction procedure can be applied to eliminate them.
An irreducible non-convex region
Authors: Ollerenshaw Kathleen , Mordell L. J. 
Published: November 2008

Some years ago I showed ((4), § 6, pp. 88–91) that the star domain K defined by the inequalitieshas the minimum determinant Δ(K) = 2 and has an infinity of singular critical lattices. In this note I show that there is a unique irreducible star domain . That ís to say, there is just one star domain H contained in but different from K for which Δ(H) = Δ(K) = 2, and such that Δ(H′) < 2 for every star domain H′ contained in but different from H.
The Parseval formulae for monotonic functions. IV
Authors: Edmonds Sheila M. 
Published: November 2008

This note contains two counter-examples which have a bearing on work published in earlier papers (1, 2, 3) of the sequence. It has been shown that the Parseval formulae for Fourier cosine and sine transforms,hold in a number of cases when two of the functions concerned are monotonic, and that there are corresponding results for Fourier series. In some of these theorems, the existence of one side of the relevant equation (in the ordinary finite sense) is postulated; others contain no such existence condition, and show that either both sides are finite and equal, or both sides are ‘ + ∞’. In the case of sine transforms, we always obtain theorems of the second type (which is clearly the stronger); but cosine transforms are more troublesome. The first of the examples which we shall give is designed to show that the cosine formula is not always true in the ‘finite or infinite’ sense when two of the functions are monotonic.
Hydrodynamic stability of the flow in the laminar boundary layer between parallel streams
Authors: Lock R. C. 
Published: November 2008

ABSTRACTA method is given for determining the stability of small sinusoidal oscillations in the steady laminar flow of a horizontal wind over the surface of a liquid at rest (with particular reference to the flow of air over water), taking into account viscosity, gravity and surface tension. It is shown that there are two fundamental types of oscillation of the system, which may be called ‘water’ waves and ‘air’ waves, and curves showing the conditions for neutral stability of these two types of wave are given for a range of wind speeds from 100 to 300 cm./sec.
The consistency of cardinal series
Authors: Noble M. E. 
Published: November 2008

1. Whittaker (7), generalizing a result of Ferrar(3), showed that the cardinal series based on the positive and negative integers is consistent in the sense that, ifand an integral function f (x) is denned bythen provided 0 < λ > 1In this note I show that results of Paley- Wiener, Levinson and others on biorthogonal series can be made to yield a consistency theorem for cardinal series based on sequences λn, wherethat is, serieswhereWe use Fourier transform technique and need hypotheses, a little more restrictive than Whittaker's, which would reduce in the case .
An invariant characterization of pseudo-valuations on a field
Authors: Cohn P. M. 
Published: November 2008

This note is concerned with pseudo-valuations on a (commutative) fieldF. Apseudo-valuation on F† is defined as a real-valued functionWonFsuch that(1)W(x) ≥ 0for all x∈FandW(0) = 0,but W does not vanish identically on F,(2)W(xy) ≤W(x)W(y) for allx, y∈F,(3)W(x−y) ≤W(x) +W(y)for all x, y∈F.Wisnon-Archimedean, if it satisfies the ultrametric inequality(3′)W(x−y) ≤ max {W(x),W(y)}.
Coupled forms of the differential equations governing radio propagation in the ionosphere
Authors: Clemmow P. C. , Heading J. 
Published: November 2008

ABSTRACTIt is shown that the equations governing oblique propagation in a horizontally stratified ionosphere with an oblique magnetic field can be cast into a form suitable for solution by successive approximations. In the general case this ‘coupled’ form consists of four first order differential equations, each being associated with one characteristic wave. In the special cases (a) horizontal magnetic field, (b) plane of incidence perpendicular to the magnetic meridian, two second order coupled equations of a particular type can be derived, each of which is associated with a pair of corresponding upgoing and downgoing characteristic waves. These latter equations are similar to, and include, those already given by Försterling(7) for vertical incidence.The cases in which there is no coupling are briefly considered from the point of view of the first order equations.The general formulation provides a basis for assessing the validity of the standard ‘ray’ approximation, alternative to that developed by Booker (2), and brings out the nature of its breakdown in coupling and reflexion regions.Specific applications and extensions of the theory are left for later consideration.
The statistics of scaled random events
Authors: Davies R. O. , Leech J. W. 
Published: November 2008

ABSTRACTWhen independent random events are scaled down by a factor n, the number of counts in a given time interval no longer obeys the Poisson Law. The correct distribution function is derived together with its moment-generating function and explicit formulae (n = 2, 4) for the first four moments about the mean. The theory is in agreement with experiments carried out with a scale-of-four counter. A more general treatment is given and is applied to find the variance for all even values of n.
A reciprocity theorem on the propagation of radio waves via the ionosphere
Authors: Budden K. G. 
Published: November 2008

ABSTRACTThe reciprocity theorem for electrical systems which include a radiation link was believed to be true only when the media within the system have dielectric constants which are symmetric tensors. This condition is not fulfilled by the ionosphere, so that the reciprocity theorem is not generally applicable when the radiation link includes one or more reflexions from the ionosphere. It is here proved that, when the ionosphere is horizontally stratified, and when the path from transmitter to receiver is in the magnetic meridian (north-south and south-north transmission), the reciprocity theorem applies (a) when the transmitting and receiving aerials both radiate or receive waves whose electric vector is in the plane of incidence, and (b) when both aerials radiate or receive waves whose electric vector is horizontal. Further, (c) if the electric vector radiated or received is horizontal for one aerial and in the plane of incidence for the other, then there is reciprocity in signal amplitude, but the phase changes for transmission in the two directions differ by 180°. These results are valid for any law of variation of electron density and collision frequency with height. They are based on a ‘full-wave’ theory, and therefore apply to all frequencies. They are unaffected if the path includes multiple reflexions, and if allowance is made for the curvature of the earth.
The retardation of alpha particles when capture and loss occur
Authors: Scott J. M. C. 
Published: November 2008

ABSTRACTThe usual formula for the rate of loss of energy by a fast particle contains a factor z2, the square of the charge, which might suggest that He+ should lose energy at a quarter of the rate that He++ does. It is shown that when Born's approximation is applicable the energy-loss is reduced instead by a factor of about ⅝.
A non-nilpotent Lie ring satisfying the Engel condition and a non-nilpotent Engel group
Authors: Cohn P. M. 
Published: November 2008

Let L be a Lie ring and denote the product of x and y in L by [x, y]. The ring L is said to satisfy the Engel condition (cf. (1)), if for every pair of elements x, yεL there is an integer k = k(x, y)such thatIf k(x, y) can be taken equal to a fixed integer n for all x, y ε L then L is said to satisfy the n-th Engel condition.
Centre-of-mass state vectors
Authors: Hamilton J. 
Published: November 2008

ABSTRACTThe properties of centre-of-mass systems at high energies are discussed, and the single time form of field theory is used to set up state vectors that describe the simpler centre-of-mass states of fields. These states, which are labelled by the eigenvalues of total angular momentum, are expanded in terms of the spherical harmonic resolution of the fields about a fixed point. A discussion of the application of the method to pion-nucleon scattering is given.
Molecules With tetrahedral and octahedral symmetry. IV. A modified Thomas–Fermi scheme for molecules with heavy atoms in the outer positions
Authors: Coulson C. A. , March N. H. 
Published: November 2008

ABSTRACTIn Part I of this series it was shown that for CCl4 the approximation of smoothing out the Cl nuclei over the surface of a sphere did not lead to reasonable results for the Thomas–Fermi distribution of electrons in the molecule. An alternative method in which the inner-shell Cl electrons were first compressed into the nuclei was not theoretically satisfactory because of the neglect of the exclusion principle. In this paper therefore a general scheme is presented in which we describe the inner-shell electrons of the outer atoms wave mechanically, assuming them to be undisturbed by molecular binding, but still use a Thomas–Fermi description for the valence electrons. A method by means of which the exclusion principle can be imposed is formulated. The following paper of the series deals with the application of the method described here to obtain an improved valence electron distribution in CCl4.
The numerical evaluation of a class of integrals. II
Authors: Das S. C. 
Published: November 2008

Consider the integralwhere x1, x2, …, xN are jointly distributed in a multivariate normal distribution f(x1, x2, …, xN) with (pij) as the correlation matrix. The integral has been expressed in an infinite series of tetrachoric functions for N≥2. The infinite series is not only complicated, but also is very slowly convergent and is consequently not of much practical use. Plackett (8) obtains a reduction formula for expressing normal integrals in four variates as a finite sum of single integrals of tabulated functions. These integrals have then to be evaluated by a rather awkward numerical quadrature.
On a probability problem arising in the theory of counters
Authors: Takács L. 
Published: November 2008

ABSTRACTThe paper contains some remarks on the problems treated by Hammersley(9). With the aid of a new and simple method the exact solutions of Hammersley's problems and the asymptotic behaviour of the solutions are found. The problems in question are: A Poisson process is transformed into a new process by a Type II counter, and this new process is transformed again by a Type I counter. The distribution of the number of events occurring in each transformed sequence is to be determined.
The G-values of various games
Authors: Guy Richard K. , Smith Cedric A. B. 
Published: November 2008

A disjunctive combination of a finite set of two-person games Γ1, Γ2, …, Γk may be defined thus: The players play alternately, each in turn making a move in one and only one of the individual games. If, in addition, the conditions are imposed that(i) a player loses if unable to move (in any game),(ii) the games are impartial, i.e. the allowable moves from any position do not depend on which player is about to play (or on the previous moves, though these can be ‘built in’ to the position if necessary),(iii) the games are of bounded play, i.e. for each game Γi corresponding to any initial position Pj there is an integer bij such that the game must terminate after at most bij moves, then Grundy (6) has shown that there is a function G(P) (called by him Ω(P)) of the positions P with the following properties:(1a) G(P) = 0 for a terminal position, from which no move is possible; for other positions G(P) is the smallest non-negative integer different from all values of G(Qi), where there is a permissible move from P to Qi,(1b) for a disjunctive combination of games, G for the combined position is the nim-sum of the G's the individual positions. By the nim-sum, we mean that each separate G is to be written in the scale of 2, as Σar 2r, and then in forming the sum, the ar's are to be added mod 2 for each value of r, as in the theory of Nim((1), (7), (8), pp. 36–8).
A theorem of homological algebra
Authors: Rees D. 
Published: November 2008

The present note is concerned with the proof and applications of the following theorem: Let A be a commutative ring, N be an A-module, and g1, …, gk be elements of A such that (g1,…, gi−1) A:gi = (g1,…, gi−1) N:gi = (g1,…, gi−1) N (i = 1,…, k), so that 0:g1 = 0 in N. Let g denote the ideal (g1 …, gk), let B = A/g and let M be any A-module such that gM = 0. thenwhere, on the right-hand side of (ii), M, N/gN are considered as B-modules.