bluuebunny/crossref_metadata_2025 huggingface dataset:
A Statistical Procedure for Cluster Recognition with Application to Atlanta Leukemia‐Lymphoma Data
Authors: Rosenbaum D. M. , Kleitman D. J. , Saks M. E.
Published: September 2015

This paper contains a description of a simple technique for analyzing a multivariable data set to detect significant clustering of data points. The method is illustrated by application to geographic leukemia and lymphoma incidence data in the Atlanta and Providence areas. Unusual clustering patterns are observed for myeloma cases when compared with those of other lymphomas.
The Evolution of Large‐Horizontal‐Scale Disturbances in Marginally Stable, Inviscid, Shear Flows. II. Solutions of the Boussinesq Equation
Authors: Hickernell Fred J.
Published: September 2015

In part I the Boussinesq equation was found to describe large‐horizontal‐scale disturbances in certain marginally stable flows. In this paper the properties of the solutions of this equation are explored. Solutions that are bounded for all time and those that blow up in finite time are observed analytically and numerically. Sufficient conditions for both boundedness and blowup are given.
On the Crenation of a Compound Liquid Droplet
Authors: Landman K. A.
Published: September 2015

The dynamic response of a small, very viscous liquid droplet composed of a core fluid surrounded by a thin fluid shell is examined as additional fluid is deposited into this incompressible shell. At early times, the shell incorporates the extra mass by ruffling its external surface, and a number of crenations form. These protuberances decrease in size and number over a longer time period, and eventually the droplet again becomes spherical, with an increased radius. This sequence of events and its dependence on the rheological properties of the fluids are studied. These effects compare well qualitatively with those obtained using a surface fluid rather than a finite thickness shell of fluid. The possible implications of this fluid model for the surface ruffling effects observed in cell biology are discussed.
On the Inverse Scattering Transform for the Kadomtsev‐Petviashvili Equation
Authors: Ablowitz M. J. , Yaacov D. Bar , Fokas A. S.
Published: September 2015

The initial value problem of the Kadomtsev‐Petviashvili equation for one choice of sign in the equation has been recently investigated in the literature. Here we consider the other choice of sign. We introduce suitable eigenfunctions which though bounded are not analytic in the spectral parameter. This, in contrast to the known case, prevents us from formulating the inverse problem as a nonlocal Riemann‐Hilbert boundary value problem. Nevertheless a suitable formulation is given and a formal solution is constructed via a linear integral equation.
On Two‐Phase Flow in a Rotating Boundary Layer
Authors: Ungarish M. , Greenspan H. P.
Published: September 2015

The two‐phase flow induced by a rotating disk in a stationary unbounded mixture is considered. The generalized similarity assumption of von Karman reduces the averaged equations of motion with a linear drag between the phases to a system of ordinary differential equations. These are investigated by asymptotic and numerical techniques. The equations display a nontrivial behavior in a sublayer near the boundary, whose thickness is of the order of the particle size. The volume fraction of the dispersed phase is singular unless a small suction is applied on the disk or a small diffusion term is added to the continuity equations. Outside this sublayer, the velocity field is quite similar to a rescaled classical von Karman flow. Good agreement between asymptotic and numerical solution is obtained, although there is considerable stiffness in the equations. The motion of a solid particle in a von Karman flow is also discussed, but the present investigation is restricted to small radii because the shear‐lift force is neglected.
Nonlinear Wave Packets in Flows with Critical Layers
Authors: Benney D. J.
Published: September 2015

This paper is a theoretical study of nonlinear wave propagation in homogeneous and stratified shear flows. For a wave packet, it is known that levels where the flow velocity is equal to the phase or group velocity are regions where nonlinear effects are especially significant. In this study certain aspects of the flow configuration are examined in the neighborhood of each of these regions.
Impurity Dynamics of the XY‐Model
Authors: Barouch Eytan
Published: September 2015

The time dependent expectation value of a magnetic perturbation of the XY‐Hamiltonian is computed exactly. Thermalization information is obtained.
Collapse in the n‐Dimensional Nonlinear Schrödinger Equation—A Parallel with Sundman's
Authors: Berkshire F. H. , Gibbon J. D.
Published: September 2015

Collapse of solutions of the n‐dimensional nonlinear Schrödinger equation are studied using the integrals of the motion and an equation corresponding to the Lagrange‐Jacobi virial equation of classical mechanics. There are strong parallels with collapse in the classical N‐body problem and in particular with the results of K. F. Sundman. Collapse occurs when the amplitude of the solution becomes singular as the initial data collapse to the center of mass in finite time. In some cases the singularity is inevitable (for negative energy), but in others only a necessary condition for collapse can be derived, involving the angular momentum.
Nonlinear Amplitude Evolution of Baroclinic Wave Trains and Wave Packets
Authors: Moroz I. M. , Brindley J.
Published: September 2015

The weakly nonlinear theory of baroclinic wave trains and wave packets is examined by the use of systematic expansion procedures in appropriate powers of a small parameter measuring the supercriticality according to linear theory; well‐known multiple scaling techniques are employed. Crucial importance is ascribed to the magnitude of parameters measuring dissipation and dispersion relative to each other and to the supercriticality, and equations describing the slow evolution in space and time of the wave amplitude are established for a range of parameter values. For vanishingly small dissipation the wave train equations have straightforward oscillatory solutions, dependent on initial conditions, and for large dissipation steady equilibration, independent of initial conditions, is predicted. For moderately small dissipation, however, a wide variety of behaviors is possible—including steady equilibration, single and multiple periodicity, and aperiodicity—in the solutions of the equations, which are recognizable as generalisations of the well‐known Lorenz attractor equations. Equations describing the evolution of wave packets take a variety of forms; for vanishingly small dissipation or for large dissipation, they are essentially parabolic and of nonlinear Schrödinger type, whilst for moderate dissipation they are of Lorenz type, modified by spatial variations. Solutions of a number of these equations are discussed and compared, where appropriate, with experimental results.
A Logarithmic Connection for Circular Permutation Enumeration
Authors: Goulden I. P. , Jackson D. M.
Published: September 2015

A general theorem is obtained for the enumeration of permutations equivalent under cyclic rotation. This result gives the generating function as the logarithm of a determinant which arises in the enumeration of a related linear permutation enumeration. Applications of this theorem are given to a number of classical enumerative problems.
Stability and Structure of Stretched Vortices
Authors: Robinson A. C. , Saffman P. G.
Published: September 2015

The radially symmetric Burgers vortex is an example of a solution to the Navier‐Stokes equations in which the intensification of vorticity due to vortex stretching is balanced by the diffusion of vorticity through viscosity. The linear stability of the symmetric Burgers vortex to a class of two‐dimensional perturbations is demonstrated. Analytical solutions obtained from a perturbation analysis and numerical computations are presented of nonsymmetric Burgers vortices in which the radial flow field in a plane perpendicular to the vorticity is nonsymmetric.
On Nonlinear Kelvin‐Helmholtz Waves Near Direct Resonance
Authors: Ma Yan-Chow
Published: September 2015

The evolution equation for the nonlinear Kelvin‐Helmholtz wave envelope with its carrier wavenumber near direct resonance is formulated directly by using the nonlinear dispersion relation. The stability of a wavetrain is examined, and the long‐time evolution for an arbitrary initial condition is studied through inverse scattering transforms.
Resonant Modal Interactions and Adiabatic Invariance for a Nonlinear Wave Equation in a Variable Domain
Authors: Kevorkian J. , Li H. K.
Published: September 2015

We extend the technique of isolating and order reducing transformations for computing adiabatic invariants in finite‐degree‐of‐freedom Hamiltonian systems to the case of the non‐Hamiltonian modal representation of a wave equation with weak nonlinearities in a slowly varying domain. We exhibit the mechanism of resonant interactions for two or more normal modes whereby the associated actions change rapidly in a short period. In the Hamiltonian problem there are a number of global adiabatic invariants associated with each resonance. We derive conditions for which similar adiabatic invariants can be found for the non‐Hamiltonian case. The results are then verified by extensive numerical computations.
A Theory for Spontaneous Mach‐Stem Formation in Reacting Shock Fronts. II. Steady‐Wave Bifurcations and the Evidence for Breakdown
Authors: Majda Andrew , Rosales Rodolfo
Published: September 2015

This paper continues earlier work of the authors on a theory for spontaneous Mach‐stem formation. Shock formation in smooth solutions of the scalar integrodifferential conservation law from paper I is demonstrated through detailed numerical experiments—this completes the basic argument from paper I. The steady‐state bifurcation of planar detonation waves into “shallow‐angle” reactive Mach stem structures is analyzed. The conclusions of this analysis agree with those predicted through the time‐dependent asymptotics in paper I and provide a completely independent confirmation of that theory.
On Random Representable Matroids
Authors: Kelly D. G. , Oxley J. G.
Published: September 2015

Results are obtained on the likely connectivity properties and sizes of circuits in the column dependence matroid of a random r × n matrix over a finite field, for large r and n. In a sense made precise in the paper, it is shown to be highly probable that when n is less than r such a matroid is the free matroid on n points, while if n exceeds r it is a connected matroid of rank r. Moreover, the connectivity can be strengthened under additional hypotheses on the growth of n and r, using the notion of vertical connectivity; and the values of k for which circuits of size k exist can be determined in terms of n and r.
Some Remarks on the Number of Solutions to the Equation f(X1) + … + f(Xn) = 0
Authors: Richman David R.
Published: September 2015

Let S, T be finite sets, and let f be a function from S to T. Fix an element t in T, and let cn denote the number of n‐tuples (X1,…,Xn) satisfying f(X1) + … + f(Xn) = t here + denotes any binary operation on T. The sequence c1, c2,… satisfies a linear recurrence relation of degree at most |T|.
The Painlevé Property and Hirota's Method
Authors: Gibbon J. D. , Radmore P. , Tabor M. , Wood D.
Published: September 2015

The connection between the Painlevé property for partial differential equations, proposed by Weiss, Tabor, and Carnevale, and Hirota's method for calculating N‐soliton solutions is investigated for a variety of equations including the nonlinear Schrödinger and mKdV equations. Those equations which do not possess the Painlevé property are easily seen not to have self‐truncating Hirota expansions. The Bäcklund transformations derived from the Painlevé analysis and those determined by Hirota's method are shown to be directly related. This provides a simple route for demonstrating the connection between the singular manifolds used in the Painlevé analysis and the eigenfunctions of the AKNS inverse scattering transform.
Three‐Dimensional Stability and Bifurcation of Capillary and Gravity Waves on Deep Water
Authors: Chen B. , Saffman P. G.
Published: September 2015

The stability to three‐dimensional disturbances of finite‐amplitude capillary waves on deep water is studied. Two families of steady finite‐amplitude three‐dimensional capillary waves, which result from the bifurcation associated with stationary disturbances, are calculated. Earlier work on three‐dimensional gravity waves is extended, and a new family of solutions if found.
Secular Cooling in Convection
Authors: Fowler A. C.
Published: September 2015

We show how long‐term cooling effects in convection can be analyzed in terms of the underlying boundary‐layer structure, by a method analogous to averaging in nonlinear oscillations.
Vortex Induced Lift on a Flat Plate with a Curved Forward‐Facing Flap
Authors: Tanveer S.
Published: September 2015

Free streamline solutions are obtained for two dimensional inviscid incompressible flow past a flat plate with a forward‐facing curved flap. It is shown that it is possible to shape the curved flap to make the adverse pressure gradient on top of the flap less severe than for a straight flap and thus increase the prospects of making the flow experimentally realizable.
Strong Interactions between Solitary Waves Belonging to Different Wave Modes
Authors: Gear J. A.
Published: September 2015

Strong interactions between weakly nonlinear long waves are studied. Strong interactions occur when the linear long wave phase speeds are nearly equal although the waves belong to different modes. Specifically we study this situation in the context of internal wave modes propagating in a density stratified fluid. The interaction is described by two coupled Korteweg‐deVries equations, which possess both dispersive and nonlinear coupling terms. It is shown that the coupled equations possess an exact analytical solution involving the characteristic “sech2” profile of the Korteweg‐deVries equation. It is also shown that when the coefficients satisfy some special conditions, the coupled equations possess an n‐solition solution analogous to the Korteweg‐deVries n‐solition solution. In general though the coupled equations are found not to be amenable to solution by the inverse scattering transform technique, and thus a numerical method has been employed in order to find solutions. This method is described in detail in Appendix A. Several numerical solutions of the coupled equations are presented.
Vortex Induced Lift on Two Dimensional Low Speed Wings
Authors: Weinitschke H. J.
Published: September 2015

Cooling of viscoelastic material from high temperature to room temperature is reconsidered for a set of constitutive equations different from those used in a recent paper. For simple geometries such as plates, cylinders and spheres, the basic equations reduce again to a Volterra‐type integrodifferential equation, to which the results of the earlier paper apply.
Thermal Runaway in the Earth's Mantle
Authors: Fowler A. C.
Published: September 2015

We examine the possibility that thermal runaway may occur locally in the earth's asthenosphere, due to a coupling between the velocity and temperature fields due to a strongly temperature‐dependent viscosity. The analysis is based on a partial description of convection in the earth, in which the boundary‐layer nature of the motion is taken into account. We find that realistic parameter values are consistent with thermal runaway occurring on length scales of thousands of kilometers, or time scales of order 108 years. If thermal runaway does occur, one would expect partial melting, and probably consequent volcanism.
Some Stability Results for Advection‐Diffusion Equations
Authors: Howes F. A.
Published: September 2015

The stability of steady‐state solutions of reaction‐advection‐diffusion equations to perturbations in the initial data is examined by means of a comparison theorem. In particular, conditions on the reactive terms and the advective terms are used to prove stability or asymptotic stability of steady boundary‐layer and shock‐layer solutions and to estimate the maximum allowable size of the initial perturbation. Many examples are discussed, including the time‐dependent Lagerstrom‐Cole problem and a problem modeling transonic flow in a nonuniform duct.
An Example of Stability Exchange in a Hamiltonian Wave System
Authors: Zufiria J. A. , Saffman P. G.
Published: September 2015

A simple analytical model has been constructed to demonstrate Saffman's (1985) proof of Tanaka's (1983, 1985) results on the superharmonic stability of deep water waves. The model shows the change of geometrical and algebraic multiplicity of the eigenvalues and eigenvectors of the stability equation at the critical points. It confirms the existence of Hamiltonian systems with limit points at which there is no change of stability.
Resonantly Interacting Weakly Nonlinear Hyperbolic Waves in the Presence of Shocks: A Single Space Variable in a Homogeneous, Time Independent Medium
Authors: Cehelsky Priscilla , Rosales Rodolfo R.
Published: September 2015

We present a systematic asymptotic theory for resonantly interacting weakly nonlinear hyperbolic waves for a single space variable in a homogeneous, time independent medium. This theory extends the results previously presented by A. Majda and R. Rosales, under similar hypotheses, to the case where waves break and shocks form. Similarly the theory of nonresonant interacting waves for general hyperbolic systems developed recently by J. Hunter and J. B. Keller, when specialized to a single space variable, is included as a special case. However, we are also able to treat the general situation when resonances occur in the hyperbolic system. Such resonances are the typical case when the hyperbolic system has at least three equations and when, for example, small‐amplitude periodic initial data are prescribed. In the important physical example of the 3 ✕ 3 system describing compressible fluid flow in a single space variable, the resonant asymptotic theory developed by the authors yields, as limit equations, a pair of inviscid Burgers equations coupled through a linear integral operator with known kernel defined through the initial data for the entropy wave. These are the same equations derived by Majda and Rosales previously. However, the waves are displaced relative to the positions prescribed by them.
A Steadily Translating Pair of Equal and Opposite Vortices with Vortex Sheets on Their Boundaries
Authors: Tanveer S.
Published: September 2015

The shapes of a steadily translating pair of equal and opposite vortices with vortex sheets on their boundaries is determined. A continuum of solutions is found to exist between the two extremes: a hollow vortex pair and a vortex pair with no vortex sheets.
Sound Radiation by Instability Wavepackets in a Boundary Layer
Authors: Haj‐Hariri H. , Akylas T. R.
Published: September 2015

Sound radiation by instability wavepackets evolving from general initial conditionsin a low‐Mach‐number, slightly unstable boundary layer is studied. The formulation can be viewed as an exact instance of Lighthill's [13] acoustic analogy. The original disturbance may be line‐ or point‐centered with an infinitesimal amplitude. The sound field is confined within an expanding sphere. The directivity of sound is beamed and points upstream. This “superdirectivity” is slowly enhanced in time whereas the sound level is slowly attenuated. Nonlinear processes, once they come into play, reverse the temporal behavior of the sound level and sound directivity. For the boundary‐layer profiles studied, nonlinearity causes a “burst,” in a finite time, of the sound field in the form of a dipole.
Analytic Representations of Thermonuclear Reaction Rates
Authors: Haubold H. J. , Mathai A. M.
Published: September 2015

The basic physical principles common to all nonresonant and resonant thermonuclear reaction rates are taken as a basis to derive the standard forms of both types of reaction rates. Results are given for the closed‐form representations of the standard nonresonant and resonant thermonuclear reaction rates in terms of a certain type of Meijer's G‐function. Then, for physical reasons the nonresonant reaction rate with modified Maxwell‐Boltzmann distribution is considered. The mathematical method for the derivation of closed‐form representations for the modified nonresonant reaction‐rate integral, which is also suitable for numerical computations, is given in detail. Some new techniques developed by the authors are used for deriving the results. The reaction‐rate systematics thus obtained in terms of Meijer's G‐function is a contribution to Fowler's methods for the numerical computation of thermonuclear reaction rates.
Excitation of Direct Resonances in Plane Poiseuille Flow
Authors: Gustavsson L. Håkan
Published: September 2015

The direct resonance mechanism between vertical vorticity and vertical velocity is studied for plane Poiseuille flow. The resonance term, obtained from the initialvalue problem, has been determined numerically for various initial disturbances. The amplitude, in wave‐number space, is found to decay with time for all resonances. Because of its appearance in the Laplace‐transform plane as a double pole, it is shown that a resonance term will have an initial amplitude which grows with Reynolds number, thus indicating that resonances will dominate the initial development of a disturbance at large Reynolds number.
On the Computation of Nonlinear Planetary Waves
Authors: Eydeland Alexander , Turkington Bruce
Published: September 2015

A numerical method is developed to solve a class of nonlinear, nonlocal eigenvalue problems defined in an infinite strip, and is applied to compute solitary planetary waves in a sheared zonal current on the beta‐plane. This method, an iterative procedure derived from the natural variational structure of these problems, is implemented in the physical case when the ambient parallel flow has a linear or a quadratic velocity profile. The results of the numerical experiments establish rigorous limits on the range of validity of the formal asymptotic theory of weakly nonlinear long waves, and also reveal some new phenomena involving strongly nonlinear waves. The iterative procedure is analyzed in a general setting, and is shown to be globally convergent without restriction on the wave amplitude.
The Propagation of Finite‐Amplitude Waves in a Model Boundary Layer
Authors: Balagondar P. M. , Maslowe S. A. , Melkonian S.
Published: September 2015

Finite‐amplitude wave propagation is considered in flows of boundary‐layer type when the wavelength is long compared to the boundary layer thickness. In this limit, the evolution of the amplitude is governed by the Benjamin‐Ono equation and we have computed the coefficients of its nonlinear and dispersive terms for the specific case of Tietjens's model. The propagation of wave packets is also considered, and it is found that for packets centered about an O(1) wavenumber questions again arise relative to long waves, except that now the packet‐induced mean flow is the “long wave.” By introducing an appropriate scaling for the far field and employing multiple scales in the direction transverse to the flow, it is shown how the mean‐flow distortion can be made to vanish at infinity.
Asymptotic Analysis of Forced Nonlinear Sturm‐Liouville Systems
Authors: Lange Charles G.
Published: September 2015

A generalized WKB method is used to construct formal asymptotic approximations of solutions of certain forced nonlinear Sturm‐Liouville systems. By means of three connected expansions it is possible to obtain a fairly complete picture of the global behavior of the small‐norm solution branches. Results are presented for both slowly varying and rapidly varying forcing functions.
A Model Equation Illustrating Subcritical Instability to Long Waves in Shear Flows
Authors: Herron I. H. , Maslowe S. A. , Melkonian S.
Published: September 2015

A model equation somewhat more general than Burger's equation has been employed by Herron [1] to gain insight into the stability characteristics of parallel shear flows. This equation, namely, ut + uuy = uxx + uyy, has an exact solution U(y) = −2tanh y. It was shown in [1] that this solution is linearly stable, and more recently, Galdi and Herron [3] have proved conditional stability to finite perturbations of sufficiently small initial amplitude using energy methods. The present study utilizes multiple‐scaling methods to derive a nonlinear evolution equation for a long‐wave perturbation whose amplitude varies slowly in space and time. A transformation to the heat‐conduction equation has been found which enables this amplitude equation to be solved exactly. Although all disturbances ultimately decay due to diffusion, it is found that subcritical instability is possible in that realistic disturbances of finite initial amplitude can amplify substantially before finally decaying. This behavior is probably typical of perturbations to shear flows of practical interest, and the results illustrate deficiencies of the energy method.
Solutions of Multidimensional Extensions of the Anti‐Self‐Dual Yang‐Mills Equation
Authors: Ablowitz Mark J. , Costa David G. , Tenenblat Keti
Published: September 2015

Motivated by recent work on the generalized wave and Sine‐Gordon equatons, various multidimensional extensions of the classical self‐dual Yang‐Mills equation are developed. A method to obtain a broad class of solutions is given.
Unimodal Polynomials Arising from Lie Superalgebras
Authors: Van der Jeugt Joris
Published: September 2015

Using the existence of a principal five‐dimensional subalgebra for several types of basic Lie superalgebras, the unimodality of the principally specialized character of a typical finite‐dimensional module with highest weight Λ is derived. This gives rise to some remarkable unimodal partition polynomials.
A Two‐Dimensional Ising Model in a Magnetic Field—A Scalar Representation of the Partition Function
Authors: Chitlaru‐Briggs Sanda , Barouch Eytan
Published: September 2015

The partition function of the two‐dimensional Ising model in the presence of a magnetic field is expressed as a fourfold integral of a double product of elementary functions. The correlation functions and the magnetization per site are obtained as well.
Orthogonality via Transforms
Authors: Freeman J. M.
Published: September 2015

Let e(x, t) = ∑pn(x)tn be the generating function of a polynomial sequence, and
the transform of multiplication by x relative to e(x, t). We show that the sequence pn(x) is orthogonal precisely when
is a t‐variable, i.e., maps K[t] into itself and increases degree by 1. We also show how transform techniques can shed light on the recursion relations and differential equations for pn(x).
Polynomials of Binomial Type and Renewal Sequences
Authors: Stam A. J.
Published: September 2015

We study polynomials of binomial type that have an exponential generating function of the form { 1 − f(u)−x}. They have a close connection with renewal sequences. The asymptotic behavior as n − ∞ is studied.
Positive Solutions of Annular Elastic Membrane Problems with Finite Rotations
Authors: Grabmüller Hans , Pirner Robert
Published: September 2015

Following the study of axisymmetric deformations of annular membranes under a normal surface load on the basis of the Föppl‐Hencky small‐finite‐deflection theory, which has been completed in recent works, the corresponding methods are here generalized as to apply to finite‐deformation problems of the Simmonds‐Libai simplified Reissner equations of finite rotations. The question of uniqueness of positive regular solutions is solved in general, and an integrale‐quation technique is developed which yields existence and nonexistence results for an extensive range of the boundary data where previous investigations failed. In contrast to the Föppl theory, some of the boundary conditions are nonlinear, and hence an extension to the finite‐rotation case is not straightforward. Moreover there are significant differences between the small‐ and the finite‐rotation theory.
The Ballot Problem Revisited
Authors: Olive Gloria
Published: September 2015

An alternative approach to the classical ballot problem leads to some classes of special functions.
The Inviscid Initial Value Problem for a Piecewise Linear Mean Flow
Authors: Henningson Dan S.
Published: September 2015

The time evolution of a small disturbance on a piecewise linear mean flow, approximating a parabolic profile, is calculated using Fourier transform methods. The solution is found to consist of two parts: one dispersive, incorporating the spreading of waves; one convective, characterized by a convection of the disturbance with the local mean velocity. Two dispersive modes are found: one symmetric with respect to the channel center line and one antisymmetric. The dispersivity of the symmetric mode is in fair agreement with the symmetric mode obtained for inviscid parabolic flow, whereas the antisymmetric mode is misrepresented. One of the parts of the solution to the horizontal velocities is found to be purely three‐dimensional. This results from fluid elements retaining part of their horizontal momentum as they are lifted up by the time integrated effect of the vertical velocity. Calculations of the development of a particular disturbance modeling two vortex pairs are also made. The results show that the dispersive part, although decaying, is largest for the vertical velocity. For the horizontal velocity the three‐dimensional lift‐up effect provides the largest amplitudes. This part does not show any sign of decay, in agreement with earlier analysis by Gustavsson [8] and Landahl [16]. This last effect partly explains the sensitivity to three‐dimensional disturbances seen in transition experiments and calculations. Comparison of the solution to a full numerical simulation using the Navier‐Stokes equations shows good agreement for short times.
Long Nonlinear Water Waves in a Channel Near a Cut‐off Frequency
Authors: Kantzios Yorgos D. , Akylas T. R.
Published: September 2015

A theoretical study is made of the free‐surface flow induced by a wavemaker, performing torsional oscillations about a vertical axis, in a shallow rectangular channel near a cut‐off frequency. Exactly at cut‐off, linearized water‐wave theory predicts a temporally unbounded response due to a resonance phenomenon. It is shown, through a perturbation analysis using characteristic variables, that the nonlinear response is governed by a forced Kadomtsev—Petviashvili (KP) equation with periodic boundary conditions across the channel. This nonlinear initial‐boundary‐value problem is investigated analytically and numerically. When surface‐tension effects are negligible, the nonlinear response reaches a steady state and exhibits jump phenomena. On the other hand, in the high‐surface‐tension regime, no steady state is obtained. These results are discussed in connection with similar forced wave phenomena studied previously in a deepwater channel and related laboratory experiments.
The Modulated Phase Shift for Weakly Dissipated Nonlinear Oscillatory Waves of the Korteweg‐deVries Type
Authors: Haberman Richard
Published: September 2015

Nonlinear dispersive oscillatory waves are analyzed for Korteweg‐deVries type partial differential equations with slowly varying coefficients and arbitrary small perturbations. Spatial and temporal evolution of the amplitude parameters are determined in the usual way by the possible dissipation of the wave actions for both momentum and energy. For dissipative perturbations, both wave actions are shown to be valid to a higher order. Thus, the first variation of the wave action equations is used to derive equations for the slow modulations of the phase shift. It is shown that the phase shift satisfies a set of two coupled linear equations.
Explicit Approximations for Strictly Nonlinear Oscillators with Slowly Varying Parameters with Applications to Free‐Electron Lasers
Authors: Kevorkian J. , Li Y. P.
Published: September 2015

The first part of this paper summarizes the mathematical modeling of free‐electron lasers (FEL), and the remainder concerns general perturbation methods for solving FEL and other strictly nonlinear oscillatory problems with slowly varying parameters and small perturbations. We review and compare the Kuzmak‐Luke method and that of near‐identity averaging transformations. In order to implement the calculation of explicit solutions we develop two approximation schemes. The first involves use of finite Fourier series to represent either the leading approximation of the solution or the transformation of the governing equations to a standard form appropriate for the method of averaging. In the second scheme we fit a cubic polynomial to the potential such that the leading approximation is expressible in terms of elliptic functions. The ideas are illustrated with a number of examples, which are also solved numerically to assess the accuracy of the various approximations.
Asymptotic Expansions and Influence Coefficients for Edge‐Loaded Conical Shells
Authors: Nair S. , Ahmadshahi A. M.
Published: September 2015

The equations of linear elasticity for rotationally symmetric deformations are expanded using a small parameter related to the thickness to radius of curvature ratio of the shell to obtain the classical thin shell equations of conical shells as a first approximation. These classical equations with variable coefficients permit further asymptotic expansions in the cases of steep as well as shallow cones, yielding systems of equations with constant coefficients. Solutions of these equations are used to compute the influence coefficients relating edge loads and edge displacements.
The Determinants of Matrices Whose Elements Decrease Geometrically in Diagonal Direction
Authors: Kato Yoshifumi
Published: September 2015

In this article, by taking the famous two Roger—Ramanujan identities as an example, the author considered which type of functions admit an infinite production expression in the sense of q‐analogue theory. The key observation is that the series
urn:x-wiley:21905991:media:sapm1988783167:sapm1988783167-math-0001 can be expressed as the determinant of an infinite matrix whose elements decrease geometrically in diagonal direction. In cases a = 1 and a = q, they coincide with the two R‐R identities. As a result, it is shown that the series has already an infinite product expression. The proof is rather elementary. In fact, first, we decompose the given matrix A(a, q) as
urn:x-wiley:21905991:media:sapm1988783167:sapm1988783167-math-0002 P is an infinite matrix having a decreasing geometric sequence on the semidiagonal away from the main diagonal by one row under. Q is a similar matrix but above.

Secondly, we interchange determinant and trace through exponential map. Then we need to calculate the trace of matrices which are products of P and Q. In short, we attribute the problem to calculate the sums of geometric sequences. The remaining question is to investigate I.T.[B1B2 ⋯ B2k] where Bi = P or Q in a calculable style. For the meaning of I.T., refer to the infinite product expression which the author obtained. The merit of this argument exists in the fact that the determinant of any matrix stated in the title is always expressible in an infinite product form. That is to say, a great many functions have an infinite product expression.
Singular Perturbation Analysis of Integral Equations
Authors: Lange Charles G. , Smith Donald R.
Published: September 2015

A study is given of certain linear singularly perturbed Volterra or Fredholm vector integral equations, including also certain integrodifferential equations. Several examples are discussed that illustrate the rich diversity of (in some cases surprising) behavior that is possible for such equations. A special class of equations is studied further, using a version of the additive multivariable technique, and the results are justified by appropriate error estimates.
On the Solutions of a Second Order Differential Equation Arising in the Theory of Resonant Oscillations in a Tank
Authors: Byatt‐Smith J. G.
Published: September 2015

The properties of the solutions of the differential equation y″ = y2 − x2 − c subject to the condition that y is bounded for all finite x discussed. The arguments of Holmes and Spence have been used by Ockendon, Ockendon, and Johnson to show that there are no solutions if c is large and negative. Numerically we find that solutions exist provided c is greater than a critical value c* and estimate this value to be c* = −…. As x tends to + ∞ the solutions are asymptotic to
. The relation between A+ and ϕ+ are found analytically as A+ → ∞. This problem arises as a connection problem in the theory of resonant oscillations of water waves.
The Temporal Evolution of a System in Combustion Theory. II
Authors: Tam K. K.
Published: September 2015

A model consisting of two nonlinear coupled parabolic equations governing the combustion of a material is considered. Under certain assumptions, the concentration of the combustible material is shown to be largely spatially homogeneous. Construction of upper and lower solutions for the equation governing the temperature is then given in terms of the solution of an associated ordinary integrodifferential equation. Estimates for the critical parameters for the three type‐A geometries are obtained.
A Rule for Fast Computation and Analysis of Soliton Automata
Authors: Papatheodorou T. S. , Ablowitz M. J. , Saridakis Y G.
Published: September 2015

Cellular automata that exhibit soliton behavior are studied. A simple rule—the fast‐rule theorem (FRT)—is introduced, which allows immediate calculation of the evolution process. The FRT is suitable for the development of parallel algorithms, for the analysis and prediction of the behavior of the automata, and for hand calculation as well. The FRT consists of first selecting a finite set of sites and then obtaining the next state by simply inverting one bit in these sites, while leaving the remaining bits unchanged. This finite set can be detected by inspection. The distinction between single and multiple particles is made precise and shown to depend not only on the number of consecutive zeros separating particles but also on their position relative to the sites in the finite set mentioned above. Three applications are given. The first is a demonstration of the FRT as an analytical tool and settles the (up to now) open question of stability. The other two applications demonstrate the use of the FRT in the construction of a periodic particle and in obtaining soliton collisions by hand calculations.
Some Explicit Resonating Waves in Weakly Nonlinear Gas Dynamics
Authors: Pego Robert L.
Published: September 2015

Equations governing leading order wave amplitudes of resonating almost periodic wave trains in weakly nonlinear acoustics have been obtained by Majda and Rosales [Stud. Appl. Math. 71:149–179 (1984)]. These equations consist of a pair of Burgers equations coupled through an integral term with a known kernel. Numerical experiments reported by Majda, Rosales, and Schonbek have suggested the existence of smooth solutions of this system whose components consist of traveling waves moving in opposite directions. For the simplest cosine kernel, explicit formulae are given here for such resonating wave solutions. There is a wave of maximum amplitude with a “peak.” For more general kernels, small amplitude resonating waves are constructed via bifurcation.
The Combinatorics of Laguerre, Charlier, and Hermite Polynomials
Authors: Labelle Jacques , Yeh Yeong Nan
Published: September 2015

This paper describes several combinatorial models for Laguerre, Charlier, and Hermite polynomials, and uses them to prove combinatorially some classical formulas. The so‐called “Italian limit formula” (from Laguerre to Hermite), the Appel identity for Hermite polynomials, and the two Sheffer identities for Laguerre and Charlier polynomials are proved. We also give bijective proofs of the three‐term recurrences. These three families form the bottom triangle in R. Askey's chart classifying hypergeometric orthogonal polynomials.
On a System of Differential Equations Connected with the Gravitational Instability of a Multicomponent Medium in Newtonian Cosmology
Authors: Mathai A. M.
Published: September 2015

The gravitational clustering of a multicomponent medium in an expanding universe has been considered by many authors in recent years. The system of differential equations associated with such a multicomponent medium is generalized in this article by incorporating arbitrary parameters. Explicit analytic solutions of this generalized system are given for various situations. Physical interpretations of the solutions are not considered, but the various solutions of physical problems obtained by various authors can be seen to be special cases of the general solutions given in this article.
Multiple Instabilities in a Triply Diffusive System
Authors: Moroz Irene M.
Published: September 2015

The model equations describing two‐dimensional convection in a fluid system driven by three diffusing components are studied. For such a model Griffiths found the state of pure conduction could become unstable to simultaneous steady and oscillatory convection. When the diffusing agents are temperature, salt, and angular velocity, Arneodo et al. found five instabilities of the rest state, including three multiple instabilities. In this paper we return to the model introduced by Griffiths, and, identifying his multiple instability as one of those found by Arneodo et al., we use dynamical systems theory to derive and study the evolution equations for the amplitudes of convection close to bifurcation.
Direct Resonance of Nonaxisymmetric Disturbances in Pipe Flow
Authors: Gustavsson L. Håkan
Published: September 2015

A mechanism which may lead to algebraic growth, followed by exponential decay, of small nonaxisymmetric disturbances in pipe flow is considered. The mechanism is interpreted as a direct resonance between the perturbations of the pressure and the streamwise velocity. The eigenvalue problems for the pressure and the velocity modes have been solved numerically for complex streamwise wave number, and 36 resonances have been investigated. A plot of the propagation speed versus the damping rate shows that the resonances follow certain sequences as the azimuthal wave number increases. The largest propagation speed is found to be ≈ 0.69 times the centerline velocity. No lowest speed is obtained, and as the azimuthal wave number increases the propagation speed decreases. The effects of changing the Reynolds number have also been investigated. It is found that the streamwise wave number and the damping rate are proportional to 1/R as R → ∞. The complex phase speed and the propagation speed become independent of R in the same limit.
Stability of the Ekman Layer in a Mixture
Authors: Greenspan H. P.
Published: September 2015

The mixture Ekman layer is shown to be unstable to infinitesimal perturbations when the ambient volume‐fraction distribution is not constant. An approximate theory indicates that the minimum critical Reynolds number is zero.
Instability of Three Dimensional Waves in Shear Flows
Authors: Benney D. J. , Zhou Y. F
Published: September 2015

The instability of a shear flow is greatly affected by the presence of one or more preexisting waves. This problem is considered for an oblique wave on a parallel shear flow with a free surface. The analysis uses a mean flow first harmonic theory.
A New Class of Brittle Graphs
Authors: Jamison B. , Olariu S.
Published: September 2015

A graph G is perfectly orderable in the sense of Chvátal if there exists a linear order on the set of vertices of G such that no induced path with vertices a, b, c, d and edges ab, bc, cd has a < b and d < c. A perfectly orderable graph G is brittle if every induced subgraph of G contains a vertex which is either endpoint or midpoint of no induced path with three edges in G. We present a new class of brittle graphs by forbidden configurations.
Traveling Wave Solutions for a Combustion Problem
Authors: Tam K. K.
Published: September 2015

A simple model for combustion consisting of two parabolic equations is considered. It is shown that plane wave solutions exist. Certain invariant integrals are obtained, from which a nonexistence criterion is derived. The effect of the wave speed on the shape of the wave is deduced.
On the Generation of Mean Flows by the Interaction of Görtler Vortices and Tollmien‐Schlichting Waves in Curved Channel Flows
Authors: Bassom Andrew P. , Hall Philip
Published: September 2015

There are many fluid flows where the onset of transition can be caused by different instability mechanisms which compete in the nonlinear regime. Here the interaction of a centrifugal instability mechanism with the viscous mechanism which causes Tollmien‐Schlichting waves is discussed. The interaction between these modes can be strong enough to drive the mean state; here the interaction is investigated in the context of curved channel flows so as to avoid difficulties associated with boundary layer growth. Essentially it is found that the mean state adjusts itself so that any modes present are neutrally stable even at finite amplitude. In the first instance the mean state driven by a vortex of short wavelength in the absence of a Tollmien‐Schlichting wave is considered. It is shown that for a given channel curvature and vortex wavelength there is an upper limit to the mass flow rate which the channel can support as the pressure gradient is increased. When Tollmien‐Schlichting waves are present then the nonlinear differential equation to determine the mean state is modified. At sufficiently high Tollmien‐Schlichting amplitudes it is found that the vortex flows are destroyed, but there is a range of amplitudes where a fully nonlinear mixed vortex‐wave state exists and indeed drives a mean state having little similarity with the flow which occurs without the instability modes. The vortex and Tollmien‐Schlichting wave structure in the nonlinear regime has viscous wall layers and internal shear layers; the thickness of the internal layers is found to be a function of the Tollmien‐Schlichting wave amplitude.
A Note on the Closed‐Form Determination of Zeros and Poles of Generalized Analytic Functions
Authors: Ioakimidis N. I.
Published: September 2015

The generalized method of Burniston and Siewert for the derivation of closed‐form formulae for the zeros (and/or poles) of analytic functions inside a closed contour in the complex plane is further extended to the case of generalized analytic functions with real and imaginary parts satisfying homogeneous generalized Cauchy‐Riemann equations. Two special cases and one generalization of this approach are also considered in brief.
On Optimum Propulsion by Means of Small Periodic Motions of a Rigid Profile. I. Properties of Optimum Motions
Authors: Urbach H. P.
Published: September 2015

The problem of optimum thrust generation by means of a rigid profile performing small arbitrarily periodic motions in an inviscid incompressible fluid is studied. The motions considered have to generate a prescribed mean value of thrust and must be such that the contribution to this mean thrust by the suction at the leading edge does not exceed a certain given value. Furthermore, the motions are in general subjected to a maximum type constraint on their amplitude. For this infinite dimensional, nonconvex and nonsmooth optimization problem, a generalized Lagrange multiplier rule is derived. In case the constraint on the amplitude is omitted, the optimum motions are calculated analytically; for the general case a number of properties of the solutions are derived from the Lagrange multiplier rule.