bluuebunny/crossref_metadata_2025 huggingface dataset:
An Upper Bound on the Growth Rate of a Linear Instability in a Homogeneous Shear Flow
Authors: Hickernell Fred J.
Published: September 2015

Temporally growing modes of the linearized equations of motion for homogeneous shear flows in the beta‐plane are considered. A new upper bound on their rate of growth is derived. This bound is related to the necessary criterion for linear instability derived by Fjørtoft [1]. As a flow stabilizes due to increased beta‐effect or decreased basic‐state vorticity gradient, the upper bound on the growth rate decreases to zero. For more stable flows this newly derived bound is tighter than that derived by Høiland [2].
Fast Thermoviscous Convection
Authors: Fowler A. C.
Published: September 2015

This paper studies the asymptotic structure of convection in an infinite Prandtl number fluid with strongly temperature‐dependent viscosity, in the limit where the dimensionless activation energy 1/ε is large, and the Rayleigh number R, defined (essentially) with the basal viscosity and the prescribed temperature drop, is also large. We find that the Nusselt number N is given by N~CεR1/5, where C depends on the aspect ratio a. The relative error in this result is O(R−1/10ε−1/4, ε1/2, R−2/5ε−2, R−2/20ε−1/24), so that we cannot hope to find accurate confirmation of this result at moderate Rayleigh numbers, though it should serve as a useful indicator of the relative importance of R and ε. For the above result to be valid, we require R ≳ 1/ε5 ≫1. More important, however, is the asymptotic structure of the flow: there is a cold (hence rigid) lid with sloping base, beneath which a rapid, essentially isoviscous, convection takes place. This convection is driven by plumes at the sides, which generate vorticity due to thermal buoyancy, as in the constant viscosity case (Roberts, 1979). However, the slope of the lid base is sufficient to cause a large shear stress to be generated in the thermal boundary layer which joins the lid to the isoviscous region underneath (though a large velocity is not generated); consequently, the layer does not “see” the shear stress exerted by the interior flow (at leading order), and therefore the thermal boundary layer structure is totally self‐determined: it even has a similarity structure (as a consequence). This fact makes it easy to analyse the problem, since the boundary layer uncouples from the rest of the flow. In addition, we find an alternative scaling (in which the lid base is “almost” flat), but it seems that the resulting boundary layer equations have no solution, though this is certainly open to debate: the results quoted above are not for this case. When a free slip boundary condition is applied at the top surface, one finds that there exists a thin “skin” at the top of the lid which is a stress boundary layer. The shear stress changes rapidly to zero, and there exists a huge longitudinal stress (compressive/tensile) in this skin. For earthlike parameters, this stress far exceeds the fracture strength of silicate rocks.
Slowly Varying Phase Planes and Boundary‐Layer Theory
Authors: Kath William L.
Published: September 2015

A method is presented that combines phase‐plane techniques with the ideas of multiple scale and matched asymptotic expansions to explain the behavior of solutions to second‐order, nonlinear, nonautonomous, singular boundary‐value problems. It is shown that if one is willing to give up the detailed information provided by a procedure such as matched asymptotic expansions, then complete qualitative information can be obtained by the much simpler method given here. (“Complete” here means that the method provides a way of categorizing all possible solutions of such problems.) In addition, the similarities and differences between the present method and that of Melnikov, which has been useful in the study of dynamical systems, are noted.
Steady Deep‐Water Waves on a Linear Shear Current
Authors: Simmen J. A. , Saffman P. G.
Published: September 2015

The properties of steady, periodic, deep‐water gravity waves on a linear shear current are investigated. Numerical solutions for all waveheights, up to and including the limiting ones, are computed from a formulation which involves only the wave profile (parametrized in a natural way) and some constants of the motion. It is found that for some shear currents the highest waves are not necessarily those waves with sharp crests known as extreme waves. Furthermore a certain nonuniqueness in the sense of a fold is shown to exist, and a new type of limiting wave is discovered. For both small‐amplitude waves and extreme waves the numerical results are compared with theoretical predictions.
The Saffman‐Taylor Fingers in the Limit of Very Large Surface Tension
Authors: Pomeau Y.
Published: September 2015

In long rectangular Hele‐Shaw cells the Saffman‐Taylor instability generates steady advancing fingers of the less viscous fluid. The structure of these fingers is determined by the solution of a system of nonlinear integrodifferential equations for two unknown functions of the curvilinear coordinate defined along the boundary of the finger. The surface tension appears in these equations through a dimensionless coefficient called k. I analyze the solution when this coefficient tends to infinity. In this limit, the inviscid fluid tends to fill almost completely the cell, except for a thin layer of viscous fluid left on the sides of thickness of order k−½.
Ordered Site Access and Optimal Forest Rotation
Authors: Wan Frederic Y. M.
Published: September 2015

The ordered‐site‐access model of forest harvesting formulated for once‐and‐for‐all forests in [7] is extended to the case of ongoing forests. The economic content of the corresponding optimal harvest schedule is delineated. For an infinite harvest sequence, the optimal schedule is shown to include the classical Faustmann rotation as a special case, and the effect of net revenue functions changing with harvest is studied. For the practically more important case of planning for a finite sequence of [INLINEEQUATION] harvests, the optimal harvest schedule is determined for a Faustmann environment with limited, and unlimited harvesting capacity, and its rapid convergence to the Faustmann rotation is shown for the case of unlimited harvesting capacity. The case of harvest cost functions varying with harvest rate is discussed. The existence of a steady‐state optimal harvesting schedule (involving a pathwise uniform age distribution) for the more realistic Heaps‐Neher environment and its relation to the Faustmann rotation are analyzed. The evolution of the optimal harvest schedule for a finite harvest sequence in a Heaps‐Neher environment toward this steady‐state (Faustmann type) rotation is demonstrated.
Dynamics of Perturbed Wavetrain Solutions to the Ginzburg‐Landau Equation
Authors: Keefe Laurence R.
Published: September 2015

The bifurcation structure and asymptotic dynamics of even, spatially periodic solutions to the time‐dependent Ginzburg‐Landau equation are investigated analytically and numerically. All solutions spring from unstable periodic modulations of a uniform wavetrain. Asymptotic states include limit cycles, two‐tori, and chaotic attractors. Lyapunov exponents for some chaotic motions are obtained. These show the solution strange attractors to have a fractal dimension slightly greater than 3.
An Analytical Model of Periodic Waves in Shallow Water
Authors: Segur Harvey , Finkel Allan
Published: September 2015

An explicit, analytical model is presented of finite‐amplitude waves in shallow water. The waves in question have two independent spatial periods, in two independent horizontal directions. Both short‐crested and long‐crested waves are available from the model. Every wave pattern is an exact solution of the Kadomtsev‐Petviashvili equation, and is based on a Riemann theta function of genus 2. These biperiodic waves are direct generalizations of the well‐known (simply periodic) cnoidal waves. Just as cnoidal waves are often used as one‐dimensional models of “typical” nonlinear, periodic waves in shallow water, these biperiodic waves may be considered to represent “typical” nonlinear, periodic waves in shallow water without the assumption of one‐dimensionality.
Properties of Successive Sample Moment Estimators
Authors: Barouch E. , Chow S. , Kaufman G. M. , Wright T. H.
Published: September 2015

Moment type estimation of characteristics of a successively sampled finite population requires finding a solution to a pair of transcendental equations. Some global properties of two structurally distinct pairs—a symmetric and an asymmetric pair—of such equations are presented. Results of a Monte Carlo experiment designed to compare the performance of estimators computed by solving these transcendental equation pairs are reported.
An Alternative Approach to Nonlinear Instabilities in Hydrodynamics
Authors: Benney D. J. , Chow K.
Published: September 2015

The conventional theoretical methods used in the study of nonlinear hydrodynamic instabilities are known to suffer from severe amplitude limitations. In this paper a modified asymptotic theory is developed which is more general and less restrictive. Such an approach may be relevant to many nonlinear instability phenomena.
Amplitude Expansions and Normal Forms in a Model tor Thermohaline Convection
Authors: Moroz Irene M.
Published: September 2015

The thermohaline and other related problems in which competitiOn between stabilizing and destabilizing physical mechanisms can create oscillatory, steady, or multiple instabilities has been much studied in the literature. Various techniques have been used to find the finite‐amplitude description near the multiple bifurcation point, but there appear to be differences in the coefficients of the nonlinear evolution equation. We address this problem by presenting a comparative study of its derivation using the classical method of multiple scales and the lesser‐known (but more powerful) methods of center manifold with normal forms. The two approaches are shown to yield identical equations and provide insights into the source of the differences.
On the Solution of the Generalized Wave and Generalized Sine‐Gordon Equations
Authors: Ablowitz Mark J. , Beals Richard , Tenenblat Keti
Published: September 2015

The generalized wave equation and generalized sine‐Gordon equations are known to be natural multidimensional differential geometric generalizations of the classical two‐dimensional versions. In this paper we associate a system of linear differential equations with these equations and show how the direct and inverse problems can be solved for appropriately decaying data on suitable lines. An initial‐boundary‐value problem is solved for these equations.
Linearly Coupled, Slowly Varying Oscillators: The Interaction of a Positive‐Energy Mode With a Negative‐Energy Mode
Authors: Grimshaw R.
Published: September 2015

A linear dynamical system is considered whose normal frequencies and normal modes come into close coincidence. The case when both modes have positive energy has been discussed by Grimshaw and Allen (1979). Here the case when one mode has positive energy and the other mode has negative energy is discussed. Coupled equations are derived and solved exactly using parabolic cylinder functions. It is found that the action in both modes grows during the coupling.
Instabilities of Waves on a Free Surface
Authors: Benney D. J. , Chow K.
Published: September 2015

This paper is concerned with instabilities which develop on a class of parallel shear flows in the presence of a free surface. Implicit in this type of study is the existence of three dimensional instabilities on slightly perturbed two dimensional spatially periodic shear flows.
The Superharmonic Instability of Finite‐Amplitude Surface Waves on Water of Finite Depth
Authors: Zufiria J. A. , Saffman P. G.
Published: September 2015

Saffman's (1985) theory of the superharmonic stability of two‐dimensional irrotational waves on fluid of infinite depth has been generalized to solitary and periodic waves of permanent form on finite uniform depth. The frame of reference for the calculation of the Hamiltonian for periodic waves of finite depth is found to be the frame in which the mean horizontal velocity is zero.
On Sums of Lognormal Random Variables
Authors: Barouch E. , Kaufman G. M. , Glasser M. L.
Published: September 2015

Approximations to the characteristic function of the lognormal distribution are computed and used to calculate approximations to the density of sums of lognormal random variables.
Nonlinear Waves in a Shear Flow with a Vorticity Discontinuity
Authors: Pullin D. I. , Jacobs P. A.
Published: September 2015

We consider nonlinear finite‐amplitude progressive shear‐flow waves on a basic velocity profile consisting of two coflowing layers of inviscid equal‐density fluid, each of uniform but different vorticity. The problem is formulated as a nonlinear integral equation describing the shape of the vorticity discontinuity in a frame of reference in which the flow is steady. Numerical solutions to this equation are presented for a range of values of the vorticity ratio Ω. For 1 > © ≥ − 1 the theoretical maximum wave amplitude occurs when the wave crest forms a 90° corner which just touches the appropriate critical‐layer stagnation point. The linearized stability of the progressive wave states to arbitrary subharmonic isovortical disturbances is studied numerically. The results indicate stability at moderate values of the wave amplitude.
The Axisymmetric Laminar Plume: Asymptotic Solution for Large Prandtl Number
Authors: Worster M. Grae
Published: September 2015

The method of matched asymptotic expansions is applied to the axisymmetric boundary‐layer equations in order to determine approximate solutions for free convection from a point source of buoyancy in the limit of large Prandtl number (σ ≫ 1). In common with other types of free‐convection boundary layers at large Prandtl number, there is an inner region in which the temperature decays to its far‐field value, and a much wider, outer region in which the vorticity decays to zero. Unlike the other cases, the velocity is not of the same order of magnitude in the two regions, but is larger in the inner region by a factor of order 1n(1/∈2) in the inner region, where ∈ is a root of ∈41n(1/∈2)=1/σ.
Some Stability Results for Advection‐Diffusion Equations. II
Authors: Howes F. A.
Published: September 2015

Previous results on the stability of steady shock layer solutions of scalar reaction‐advection‐diffusion equations are strengthened by means of a more detailed treatment of the perturbation inside the shock. These sharper results verify a recent conjecture of Embid, Goodman, and Majda on the asymptotic stability of a standing shock in a diverging duct, and they suggest that a study of analogous questions for systems of conservation laws with dissipation is possible.
The Motion of a Free Asymmetric Top in n Dimensions
Authors: Julian William H. , Zund Joseph D.
Published: September 2015

The motion and dynamical stability of a free asymmetric top about its center of mass is investigated in a Euclidean space of dimension n. It is shown that each stationary rotation in which only two principal axes of inertia of the top move is stable if and only if no other principal axis has a coefficient of inertia intermediate to those of the two moving axes.
Resonantly Interacting, Weakly Nonlinear Hyperbolic Waves. II. Several Space Variables
Authors: Hunter J. K. , Majda A. , Rosales R.
Published: September 2015

A uniformly valid asymptotic theory of resonantly interacting high‐frequency waves for nonlinear hyperbolic systems in several space dimensions is developed. When applied to the equations of two‐dimensional compressible fluid flow, this theory both predicts the geometric location of the new sound waves produced from the resonant interaction of sound waves and vorticity waves as well as yielding a simplified system which governs the evolution of the amplitudes. In this important special case, this system is two Burgers equations coupled by a linear integral operator with known kernel given by the vortex strength of the shear wave. Several inherently multidimensional assumptions for the phases are needed in this theory, and theoretical examples are given which delineate these assumptions. Furthermore, explicit necessary and sufficient conditions for the validity of the earlier noninteracting wave theory of Hunter and Keller are derived; these explicit conditions indicate that generally waves resonate and interact in several dimensions.
On the Eigenvalue Problem for a Certain
Authors: Babarsky Richard , Wood Houston G.
Published: September 2015

This paper studies the spectral properties of the partial differential operator
urn:x-wiley:00222526:media:sapm1986753249:sapm1986753249-math-0001
over a finite region Ω. This operator, which arises in the analysis of nonaxisymmetric, rapidly rotating compressible flows, is treated as a perturbation
of the operator
which is generated by the terms
urn:x-wiley:00222526:media:sapm1986753249:sapm1986753249-math-0002
Using the fact that
, when defined on a suitable domain, is closed and self‐adjoint, it is shown that
[when acting on elements of
] is an operator with compact resolvent whose generalized eigenvectors are complete in ℒ2 (Ω).
On the completeness of the functions e−np(x) cos nx, e−np(x) sin nx for n ≥ 0 and p(x) a 2π periodic function
Authors: Minzoni A. A.
Published: September 2015

It is shown how a conjecture of G. B. Whitham on the completeness of the functions { ψ n } = { e−np(x) cos nx, e−np(x) sin nx } in the interval [0, 2π] is indeed true provided p has a Hölder continuous first derivative, i.e., p ∈ C1, α[0, 2π]. Also an elementary proof of Whitham's conjecture is given for p ∈ C2, α[0, 2π].
A Variational Approach to Nonlinear Singularly Perturbed Boundary‐Value Problems
Authors: Kath W. L. , Knessl C. , Matkowsky B. J.
Published: September 2015

Carrier and Pearson introduced a nonlinear singularly perturbed boundary value problem that has served as a paradigm for problems where the method of matched asymptotic expansions (MAE) apparently fails. The “failure” of MAE is its inability to select the location of possible internal layers, though their structure is determined. Thus, a straightforward application of MAE leaves the positions of any internal layers arbitrary, though the asymptotic expansion of the exact solution to the problem exhibits internal layers only at specific locations. For this reason the solutions produced by MAE have been referred to as spurious solutions. We resolve the question of finding the positions of the interior layers by employing the variational approach of Grasman and Matkowsky. In addition, we show that this method tells how solutions bifurcate as the boundary values are varied, and give an alternative motivation for the variational approach via Newton”s method.
Properties of a Time‐Dependent Nonlinear Viscous Critical Layer
Authors: Smith F. T. , Bodonyi R. J.
Published: September 2015

The time‐dependent, nonlinear, viscous critical‐layer problem posed by Gajjar and Smith is studied analytically and numerically, for various values of the time‐dependence parameter µ and the viscous parameter γc, to help to complete the understanding of the solution. The analysis concerns the nearly inviscid limit γc → 0 which corresponds to increasing disturbance amplitude, other limits for weaker disturbances having been studied previously. On the numerical side, computational results obtained for order‐1 values of µ, γc are described and are compared with analytical predictions. Particular attention is given to the phase shift produced across the critical layer.
An Exact Solution for the Optical Absorbance of Thin Films
Authors: Babu S. V. , Barouch E.
Published: September 2015

The electric and magnetic fields within optical thin films are obtained exactly in closed form as solutions of a system of difference equations presented by Berning and utilized by Dill, Neureuther, Tuttle, and Walker in their sample program for modeling optical lithography. The solution allows an explicit closed form expression for the absorbance.
Symmetries and Integrability
Authors: Fokas A. S.
Published: September 2015

Integrable nonlinear evolution equations in one‐spatial and one‐temporal dimensions possess a remarkably rich algebraic structure: Infinitely many symmetries and conserved quantities, existence of a bi‐Hamiltonian formulation etc. A certain operator Φ, called recursion operator, plays a central role in investigating the above algebraic properties. Recently the above theory has been extended to equations in two spatial and one temporal dimensions. In particular, the multidimensional analogue of the operator Φ has been found and its bi‐Hamiltonian factorization has been established. The above and other aspects of integrable equations, including mastersymmetries, are reviewed in this manuscript. Furthermore, a definition of integrability is proposed based on symmetry considerations.
Interpolation and Combinatorial Functions
Authors: Verde‐Star Luis
Published: September 2015

Using some basic results about polynomial interpolation, divided differences, and Newton polynomial sequences we develop a theory of generalized binomial coefficients that permits the unified study of the usual binomial coefficients, the Stirling numbers of the second kind, the q‐Gaussian coefficients, and other combinatorial functions. We obtain a large number of combinatorial identities as special cases of general formulas. For example, Leibniz's rule for divided differences becomes a Chu‐Vandermonde convolution formula for each particular family of generalized binomial coefficients.
On the Elementary Retarded, Ultrahyperbolic Solution of the Klein‐Gordon Operator, Iterated k Times
Authors: Trione Susana Elena
Published: September 2015

Let t = (t1,…,tn) be a point of ℝn. We shall write

. We put, by the definition, Wα(u, m) = (m−2u)(α − n)/4[π(n − 2)/22(α + n − 2)/2Г(α/2)]J(α − n)/2(m2u)1/2; here α is a complex parameter, m a real nonnegative number, and n the dimension of the space. Wα(u, m), which is an ordinary function if Re α ≥ n, is an entire distributional function of α. First we evaluate {□ + m2}Wα + 2(u, m) = Wα(u, m), where {□ + m2} is the ultrahyperbolic operator. Then we express Wα(u, m) as a linear combination of Rα(u) of differntial orders; Rα(u) is Marcel Riesz's ultrahyperbolic kernel. We also obtain the following results: W−2k(u, m) = {□ + m2}kδ, k = 0, 1,…; W0(u, m) = δ; and {□ + m2}kW2k(u, m) = δ. Finally we prove that Wα(u, m = 0) = Rα(u). Several of these results, in the particular case µ = 1, were proved earlier by a completely different method.
First Integrals and the Residual Solution for Orthotropic Plates in Plane Strain or Axisymmetric Deformations
Authors: Lin Yihan , Wan Frederic Y. M.
Published: September 2015

Two classes of exact solutions are derived for the equations of three dimensional linear orthotropic elasticity theory governing flat (plate) bodies in plane strain or axisymmetric deformations. One of these is the analogue of the Lévy solution for plane strain deformations of isotropic plates and is designated as the interior solutions. The other complementary class correspond to the Papkovich‐Fadle Eigenfunction solutions for isotropic rectangular strips and is designated as the residual solutions. For sufficiently thin plates, the latter exhibits rapid exponential decay away from the plate edges. A set of first integrals of the elasticity equations is also derived. These first integrals are then transformed into a set of exact necessary conditions for the elastostatic state of the body to be a residual state. The results effectively remove the asymptoticity restriction of rapid exponential decay of the residual state inherent in the corresponding necessary conditions for isotropic plate problems. The requirement of rapid exponential decay effectively limits their applicability to thin plates. The result of the present paper extend the known results to thick plate problems and to orthotropic plate problems. They enable us to formulate the correct edge conditions for two‐dimensional orthotropic thick plate theories with stress or mixed edge data.
A Canonical System of lntegrodifferential Equations Arising in Resonant Nonlinear Acoustics
Authors: Majda Andrew , Rosales Rodolfo , Schonbek Maria
Published: September 2015

In general, weakly nonlinear high frequency almost periodic wave trains for systems of hyperbolic conservation laws interact and resonate to leading order. In earlier work the first two authors and J. Hunter developed simplified asymptotic equations describing this resonant interaction. In the important special case of compressible fluid flow in one or several space dimensions, these simplified asymptotic equations are essentially two inviscid Burgers equations for the nonlinear sound waves, coupled by convolution with a known kernel given by the sum of the initial vortex strength and the derivative of the initial entropy. Here we develop some of the remarkable new properties of the solutions of this system for resonant acoustics. These new features include substantial almost periodic exchange of energy between the nonlinear sound waves, the existence of smooth periodic wave trains, and the role of such smooth wave patterns in eliminating or suppressing the strong temporal decay of sawtooth profile solutions of the decoupled inviscid Burgers equations. Our approach combines detailed numerical modeling to elucidate the new phenomena together with rigorous analysis to obtain exact solutions as well as other elementary properties of the solutions of this system.
The Asymptotic Solution of a Connection Problem of a Second Order Ordinary Differential Equation
Authors: Byatt‐Smith J. G. B.
Published: September 2015

The solutions of the equation
urn:x-wiley:-wiley:00222526:media:sapm1989802109-math-0001
are discussed in the limit ρ → 0. The solutions which oscillate about − |t| as t → ∞ have asymptotic expansions whose leading terms are
urn:x-wiley:-wiley:00222526:media:sapm1989802109-math-0002
where Ã+,
, Ã−, and
are constants. The connection problem is to determine the asymptotic expansion at + ∞. In other words, we wish to find (Ã+,
) as functions of Ã− and
The nonlinear solutions with Ã± not small are analyzed by using the method of averaging. It is shown that this method breaks down for small amplitudes. In this case a solution can be obtained on [0, ∞) as a small amplitude perturbation about the exact nonoscillating solution W(t) whose asymptotic expansion is
urn:x-wiley:-wiley:00222526:media:sapm1989802109-math-0005
This is a solution of (1) which corresponds to Ã+ ≡ 0 in (2). A quantity which determines the scale of the small amplitude response is −W'(0). This quantity is found to be exponentially small. The determination of this constant is shown to reduce to a solution of the equation for the first Painlevé transcendent. The asymptotic behavior of the required solution is determined by solving an integral equation.
Evolution Theory, Periodic Particles, and Solitons in Cellular Automata
Authors: Papatheodorou T. S. , Fokas A. S.
Published: September 2015

A theory for soliton automata is developed and applied to the analysis and prediction of patterns in their behavior. A complete characterization and method of construction of 1‐periodic particles is given. A general evolution theorem (GET) is obtained which provides significant information for a state in terms of preceding states. Application of this theorem yields several interesting results predicting periodicity and solitonic collisions. The GET explains and is based on a fundamental property of soliton automata, observed and analyzed in this paper, namely that pieces of information are lost on the left and reappear on the right.
On the Limiting Steady Configurations of a Model Plasma
Authors: Booty M. R.
Published: September 2015

A model equation describing the configuration of a simple plasma maintained by external radiation is studied. A branch of steady solutions of the equation was found by Eckhaus et al. to terminate at a finite critical value of the power of the external source, and this is attributed to the discontinuous nature of a nonlinear term in the governing equation. On introducing a small parameter to render the term continuous, a second branch of solutions is constructed in a neighborhood of the termination point of the original branch. This suggests that the termination point is formed as the limit of a subcritical fold in the surface of the steady solution branch.
Strong Coupling Limit of Certain Multidimensional Nonlinear Wave Equations
Authors: Ablowitz Mark J. , Schultz Cherie L.
Published: September 2015

A class of multidimensional nonlinear evolution equations of physical interest are considered in the limit of strong coupling. It turns out that the initial value solution is readily obtained. In the special case of the Davey‐Stewartson equation the inverse scattering transform is shown to reduce to the obtained solution via perturbation. A number of other features are discussed as well, such as action angle variables, periodic solutions, and quantum analogues.
Time Evolution of the Scattering Data for the Forced Toda Lattice
Authors: Wycoff D. , Kaup D. J.
Published: September 2015

Determination of the time evolution of the scattering data for an inverse scattering transform solution of the forced Toda lattice appears to require an overspecification of the boundary condition at the end of the lattice. This appears in the form of an apparent need to specify the values of two functions at the boundary rather than one. We present three different approaches to the resolution of this problem. One approach gives the Maclaurin series (in time) for the scattering data. The second approach gives the scattering data in terms of the solution to a nonlinear, nonlocal partial differential equation. The third approach gives the scattering data in terms of the solution to a linear integral equation. All three approaches reduce to one the number of functions which must be specified to determine a solution. The advantages and limitations of each approach are discussed.
P4‐Reducible Graphs—Class of Uniquely Tree‐Representable Graphs
Authors: Jamison B. , Olariu S.
Published: September 2015

We introduce and characterize a new class of graphs which has a unique tree representation, and which strictly contains the well‐known class of cographs. We define three graph operations and show that all the graphs in our class can be constructed from single‐vertex graphs by a finite sequence of these operations. Finally, we show that a number of computational problems, including the four classical optimization problems, can be solved efficiently for this new class of graphs.
Bäcklund Transformations and Inverse Scattering Solutions for Some Pseudospherical Surface Equations
Authors: Beals Richard , Rabelo Mauro , Tenenblat Keti
Published: September 2015

In is known that the equations [ut − g(u)ux]x = ±g′(u) describe pseudo‐spherical surfaces, i.e. that these equations are the integrability conditions for the structural equations of such surfaces, provided g satisfies g′ + µg = θ. In this paper we obtain self‐Bäcklund transformations for these equations by a geometric method, and show how the inverse scattering method generates global solutions.
Free Modules of Relative Invariants of Finite Groups
Authors: Reiner Victor
Published: September 2015

This paper addresses questions about when modules of relative invariants
of a finite group G acting on a polynomial ring R are free over the ring of invariant polynomials RG. A converse (first obtained by Shchvartsman) is proven of a result asserting that these modules are always free when the group is generated by pseudoreflections. We also re‐prove the characterization given by Shchvartsman of which characters χ of degree one have the above property, and deduce from this a characterization of which G have the above property for all their degree one characters.
Existence of Optimum Propulsion by Means of Periodic Motions of a Rigid Profile
Authors: Urbach H. P.
Published: September 2015

The optimum generation of thrust by means of a rigid profile which performs periodic motions in an inviscid incompressible fluid is discussed. For the description of the flow a linear theory is used. Apart from the demand that a certain mean value of thrust has to be generated, the motions considered have to satisfy an inequality constraint on the contribution of the mean suction force at the leading edge to the total thrust and are furthermore subject to a constraint on their amplitude. This leads to an infinite‐dimensional nonconvex optimization problem. It is proved that for most values of the parameters involved there exists at least one optimum motion.
On Some Self‐Similar Flows of Non‐Newtonian Fluids through a Porous Medium
Authors: Pascal H. , Pascal F.
Published: September 2015

This paper addresses the question of implications of the effects of fluid compressibility on the equations governing unsteady flows of power law fluids through a porous medium. A generalized Burgers equation is shown to arise in the flow description. Some self‐similar flows of practical interest are presented and discussed, from which the nonlinear effects associated with fluid compressibility are analyzed with regard to a better interpretation of transient pressure response in a well of non‐Newtonian displacing fluid tests.
Exact Traveling‐Wave Solutions for Model Geophysical Systems
Authors: Sachdev P. L. , Gupta Neelam
Published: September 2015

Exact traveling‐wave solutions of time‐dependent nonlinear inhomogeneous PDEs, describing several model systems in geophysical fluid dynamics, are found. The reduced nonlinear ODEs are treated as systems of linear algebraic equations in the derivatives. A variety of solutions are found, depending on the rank of the algebraic systems. The geophysical systems include acoustic gravity waves, inertial waves, and Rossby waves. The solutions describe waves which are, in general, either periodic or monoclinic. The present approach is compared with the earlier one due to Grundland (1974) for finding exact solutions of inhomogeneous systems of nonlinear PDEs.
The Determinants of Matrices Whose Elements Decrease Geometrically in the Diagonal Direction. II
Authors: Kato Yoshifumi , Koike Masao
Published: September 2015

We will show that a generalization of the well‐known Rogers‐Ramanujan identities by use of special determinants can be restated as a counting problem of Young diagrams under a cyclic equivalent relation. And we find a very strange observation.
Denert's Permutation Statistic Is Indeed Euler‐Mahonian
Authors: Foata Dominique , Zeilberger Doron
Published: September 2015

A conjecture by Marleen Denert concerning a bivariate statistic on the permutation group is proved. The statistic has the same distribution as the pair consisting of the number of descents and the major index.
The Evolution of Linearized Perturbations of Parallel Flows
Authors: Criminale W. O. , Drazin P. G.
Published: September 2015

The equations of an incompressible fluid are linearized for small perturbations of a basic parallel flow. The initial‐value problem is then posed by use of Fourier transforms in space. Previous results are systematized in a general framework and used to solve a series of problems for prototypical examples of basic shear flow and of initial disturbance. The prototypes of shear flow are (a) plane Couette flow bounded by rigid parallel walls, (b) plane Couette flow bounded by rigid walls at constant pressure, (c) unbounded two‐layer flow with linear velocity profile in each layer, (d) a piecewise linear profile of a boundary layer on a rigid wall. The prototypes of initial perturbation are the fundamental ones: (i) a point source of the field of the transverse velocity (represented by delta functions), (ii) an unbounded sinusoidal field of the transverse velocity, (iii) a point source of the lateral component of vorticity, (iv) a sinusoidal field of the lateral vorticity. Detailed solutions for an inviscid fluid are presented, but the problem for a viscous fluid is only broached.
Limits on the Transport of Heat and Momentum by Turbulent Convection with Large‐Scale Flow
Authors: Howard Louis N.
Published: September 2015

Under certain conditions, high‐Rayleigh‐number convection in a horizontal layer exhibits an organized pattern of large‐scale flow superposed on an irregular structure of turbulent plumes. Bounds on the heat and momentum fluxes in such a flow, which provide upper estimates of the possible level of the large‐scale flow, are presented.
Möbius Inversion of Random Acyclic Directed Graphs
Authors: Cohen Joel E.
Published: September 2015

Suppose a random acyclic digraph has adjacency matrix A with independent columns or independent rows. Then the mean Möbius inverse of the zeta matrix I + A is the Möbius inverse of the mean zeta matrix, i.e., E[(I + A)−1]=[I + E(A)]−1.
Asymptotic Solution of the Weakly Nonlinear Schrödinger Equation with Variable Coefficients
Authors: Srinivasan Radhakrishnan
Published: September 2015

A perturbation method based on Fourier analysis and multiple scales is introduced for solving weakly nonlinear, dispersive wave propagation problems with Fourier‐transformable initial conditions. Asymptotic solutions are derived for the weakly nonlinear cubic Schrödinger equation with variable coefficients, and verified by comparison with numerical solutions. In the special case of constant coefficients, the asymptotic solution agrees to leading order with previously derived results in the literature; in general, this is not true to higher orders. Therefore previous asymptotic results for the strongly nonlinear Schrödinger equation can be valid only for restricted initial conditions.
The DBAR Problem and the Thirring Model
Authors: Villarroel Javier
Published: September 2015

We study the Thirring model by means of a DBAR‐Riemann‐Hilbert problem. We undertake a rigorous study for both the direct and the inverse problem on the class of rapidly decaying potentials. This study includes a discussion of the existence and uniqueness of solutions of the direct and inverse problems, a characterization of the solutions, and an explicit expression for the scattering data in terms of initial scattering functions.
The Motion of Small Particles and Droplets in Quadratic Flows
Authors: Nadim Ali , Stone Howard A.
Published: September 2015

We consider the low‐Reynolds‐number motion of small particles and droplets in the most general unbounded quadratic flows. The fluid velocity and pressure fields are obtained by means of a vectorial formulation of Lamb's general solution, providing a compact representation for the requisite flow field. The leading‐order deformation of the droplet from its initial spherical shape is also calculated.
Divided Differences and Combinatorial Identities
Authors: Verde‐Star Luis
Published: September 2015

We present an algebraic theory of divided differences which includes confluent differences, interpolation formulas, Liebniz's rule, the chain rule, and Lagrange inversion. Our approach uses only basic linear algebra. We also show that the general results about divided differences yield interesting combinatorial identities when we consider some suitable particular cases. For example, the chain rule gives us generalizations of the identity used by Good in his famous proof of Dyson's conjecture. We also obtain identities involving binomial coefficients, Stirling numbers, Gaussian coefficients, and harmonic numbers.
General Three‐Dimensional Disturbances to Inviscid Couette Flow
Authors: Criminale W. O. , Long Bruce , Zhu Mei
Published: September 2015

The general solution to the linearized equations governing three‐dimensional disturbances to inviscid Couette flow has been obtained. This result extends the Orr solution to initial conditions that do not consist of a single Fourier sine component in the cross‐stream coordinate and a plane wave in the streamwise/spanwise coordinates. The time evolution of a measure of disturbance energy for some specific pulsed initial conditions is examined, and it is concluded that, while the rapid algebraic growth to large amplitude followed by decay exemplified by the Orr solution can be of importance for individual cross‐stream Fourier components, more realistic initial conditions, which in general consist of the sum of an infinite number of components, often display uniform decay to zero amplitude. However, an interesting example is described in which one positive definite measure of disturbance amplitude remains constant, yet the streamwise/spanwise velocity components grow linearly in time if the initial disturbance is three‐dimensional.
On the Connection Constants
Authors: Damiani E. , D'Antona O. , Naldi G.
Published: September 2015

We derive explicit formulas for the connection constants between sequences of polynomials in terms of their roots. As an application, we give some properties about the inversion of triangular matrices. By means of the concept of punctured partition of an integer, here introduced to compute connection constants, we also obtain a new expression of the relationship between elementary and complete homogeneous symmetric functions.
On the Nonlinear Development of Görtler Vortices in a Compressible Boundary Layer
Authors: Wadey P. D.
Published: September 2015

The nonlinear development of the Görtler instability in compressible boundary layers on curved walls is considered for vortices of asymptotically large wavenumber. The starting point for our calculations lies in the work of Hall and Lakin (Proc. Roy. Soc. London Ser. A 415:421–444), where the incompressible results were formulated. Without neglecting downstream partial derivatives, the initial development of a vortex from the point where it first starts to grow is calculated. It is shown how the same basic structure that occurs in incompressible flow exists, where the disturbance is confined to a core region bounded above and below by thin shear layers, but that the flow in the core region is of more complicated form than that for incompressible flow.
Instability of Three Dimensional Waves in Two Layered Flows
Authors: Zhou Y. F. , Benney D. J.
Published: September 2015

This paper is concerned with instabilities which may develop on a class of two layered parallel shear flows. The mean flow first harmonic theory is applied. Analysis shows that a preexisting oblique neutral wave is unstable to more general perturbations.
Interaction of Elastic Waves
Authors: Hunter John K.
Published: September 2015

We derive and analyze asymptotic equations for the interaction of weakly nonlinear elastic waves. We show that there are resonant triads involving two transverse and one longitudinal wave provided the wave speeds satisfy a certain irrationality condition. We study initial value and signaling problems, and the interaction of sawtooth wave packets.
Initial Algebraic Growth of Small Angular Dependent Disturbances in Pipe Poiseuille Flow
Authors: Bergström Lars
Published: September 2015

The evolution of small, angular dependent velocity disturbances in laminar pipe flow is studied. In particular, streamwise independent perturbations are considered. To fully describe the flow field, two equations are required, one for the radial and the other for the streamwise velocity perturbation. Whereas the former is homogeneous, the latter has the radial velocity component as a forcing term. First, the normal modes of the system are determined and analytical solutions for eigenfunctions, damping rates, and phase velocities are calculated. As the azimuthal wave number (n) increases, the damping rate increases and the phase velocities decrease. Particularly interesting are results showing that the phase velocities associated with the streamwise eigenfunctions are independent of the radial mode index when n = 1, and when n = 5 the same is obtained for phase velocities associated with the eigenfunctions of the radial component. Then, the initial value problem is treated and the time development of the disturbances is determined. The radial and the azimuthal velocity components always decay but, owing to the forcing, the streamwise component shows an initial algebraic growth, followed by a decay. The kinetic energy density is used to characterize the induced streamwise disturbance. Its dependence on the Reynolds number, the radial mode, and the azimuthal wave number is investigated. With a normalized initial disturbance, n = 1 gives the largest amplification, followed by n = 2 etc. However, for small times, higher values of n are associated with the largest energy density. As n increases, the distribution of the streamwise velocity perturbation becomes more concentrated to the region near the pipe wall.
On Polynomials with Low Peak Signal to Power (L∞ to L2 Norm) Ratios and Theorems of Kashin and Spencer
Authors: Hajela D. J.
Published: September 2015

In many engineering problems such as optical communications, radar and sonar problems, the electronic synthesis of speech, etc., as well as mathematical applications, a problem that arises is that of finding a waveform (trigonometric polynomial) with a specified spectrum, such that its crest factor is minimum, where the crest factor is the ratio of the peak signal energy (L∞ norm) to the power (L2 norm) of the waveform. The mathematical formulation of the problem is as follows: Given 1 ≤ m1 < m2 < … mk ≤ n and arbitrary complex numbers aj j = 1,…, k, we want to find signs εj = ±1 such that the growth of the crest factor
urn:x-wiley:00222526:media:sapm199287181:sapm199287181-math-0001 is minimized as k and n are allowed to become arbitrarily large. Recently B. Kashin has solved special cases of this general problem using some deep geometrical results. We solve the general problem just stated and in particular obtain Kashin's results as a special case. The analogous problem where the εj are replaced by αj and where the αj are complex numbers with | αj | = 1 is also solved. The results obtained are best possible in the sense that the asymptotic growth of the crest factors of the polynomials found is optimal. The preceding results are obtained by a solution of the following geometrical problem of independent interest: Given vectors υ1,…, υk in complex n‐dimensional space, where k ≤ n, we want to find signs εj = ±1 such that the growth of ‖ε1υ1 + … + εkυk‖∞ is minimized as k and n go to infinity. As a special case of this geometrical result we obtain some combinatorial discrepancy results of J. Spencer.
Exact Solutions of Linear Partial Differential Equations with Variable Coefficients
Authors: Sachdev P. L. , Vaganan B. Mayil
Published: September 2015

Bäcklund transformations relating the solutions of linear PDE with variable coefficients to those of PDE with constant coefficients are found, generalizing the study of Varley and Seymour [2]. Auto‐Bäcklund transformations are also determined. To facilitate the generation of new solutions via Bäcklund transformation, explicit solutions of both classes of the PDE just mentioned are found using invariance properties of these equations and other methods. Some of these solutions are new.
Particles in a Singular Asymptotic Limit of a Classical Field Theory
Authors: Stuart David M.
Published: September 2015

We give a detailed discussion of the singular asymptotic limit of a classical field theory as a small parameter goes to zero. The phenomenon under investigation is the appearance of particles in the limit. The particles are rescaled kinks of a sine‐Gordon field acted on by a second field. As the parameter goes to zero, these kinks shrink while their energy is invariant, leaving point‐like particles accelerated by the second field. The validity of this picture of the asymptotics hinges on the dispersive nature of the radiative correction to the kink produced by the perturbation. This is analysed using a perturbation theory based on complete integrability.
On the Strong L1 Trend to Equilibrium for the Boltzmann Equation
Authors: Arkeryd Leif
Published: September 2015

This paper considers the Boltzmann equation for hard and soft (k > 2) forces having an angular cutoff. The main result is strong L1 convergence to
global Maxwellians when time tends to infinity..