bluuebunny/crossref_metadata_2025 huggingface dataset:
On Plane Dilatational Plasticity
Authors: Berg C. A.
Published: September 2015

The plane plastic deformation of a generally anisotropic rigid‐plastic material which possesses a yield condition dependent upon mean triaxial stress and which, through the classical associated flow rule, exhibits plastic dilatation, is considered. This model is used to represent the behavior of micro‐porous ductile metals in which the micro‐cavities may be strongly aligned due to large prior plastic strains, as for example the material surrounding the tip of an extending notch in a ductile metal. It is shown that the stress and velocity fields are hyperbolic where a line of vanishing extension rate may be found in the plane of deformation, and that the characteristics of both the stress and velocity fields coincide with the lines of vanishing extension rate. Coincidence of the characteristics of stress and velocity fields in general anisotropic plastic bodies seems not to have been expected in earlier writings, but is a natural consequence of the associated flow rule. Simple means of determining whether a given stress state at yield lies in a hyperbolic or elliptic field are discussed. The role of characteristics in providing ductile fracture nuclei is discussed.
On the Spectrum and Decay of Random Two‐Dimensional Vorticity Distributions at Large Reynolds Number
Authors: Saffman P. G.
Published: September 2015

The hypothesis is made that the vorticity distribution in a field of random two‐dimensional vorticity (two‐dimensional turbulence) develops discontinuities. The energy spectrum is then calculated and shown to behave like k−4 for large values of the wave‐number k. The rates of decay of mean square vorticity and velocity are estimated. An expression for the growth of length scale is obtained and it is noted that the size of turbulent trailing vortices is apparently well fitted by the formula.
Construction of the Equivalent Plastic Strain Increment
Authors: Berg C. A.
Published: September 2015

Strain hardening plastic deformation of a material possessing a yield locus (f(σij)) which may be written as a homogeneous function of the stress components (σij), and which obeys the classical associated flow rule for metals (e.g. Bishop and Hill, 1951) is considered. The material may be anisotropic and may display plastic dilatation. A method is given for constructing the equivalent plastic strain increment in such a way that the increment of plastic work is always equal to the product of the equivalent plastic strain increment and the equivalent yield stress. Construction of the equivalent plastic strain at a corner in the yield locus is given. The method given here is implied in classical treatments of hardening (e.g. Hill, 1950) but seems not to have been given explicitly heretofore.
Models for the Growth of a Solid Tumor by Diffusion
Authors: Greenspan H. P.
Published: September 2015

A simple mathematical model of tumor growth by diffusion is constructed in order to examine and evaluate different hypotheses concerning the evolution of a solid carcinoma. A primary objective is to infer the chemical source of growth inhibition from the most easily obtained data, namely, the outer radius of the nodule as a function of time and a histological examination of the final dormant state. In section 6 some of the conclusions of this study relating to a prototype experiment and described with as little mathematics as possible.
Composite Approximations to the Solutions of the Orr‐Sommerfeld Equation
Authors: Reid W.H.
Published: September 2015

The method of matched asymptotic expansions is used to derive composite approximations to the solutions of the Orr‐Sommerfeld equation which satisfy Olver's completeness requirement. It is shown that the inner expansions can be obtained to all orders in terms of a certain class of generalized Airy functions, and these expansions are then used to derive approximations to the connection formulae. Because of the linearity of the problem it is possible and convenient to fix the normalization of the inner and outer expansions separately and then to relate them through the central matching coefficients. The Stokes multipliers can then be expressed in terms of the central matching coefficients and the coefficients which appear in the connection formulae. Once the inner and outer expansions have been matched they can be combined, if desired, to form composite approximations of either the additive or multiplicative type. For example, the ‘modified’ viscous solutions of Tollmien emerge in a natural way as first‐order composite approximations obtained by multiplicative composition; similarly, the form of the ‘viscous correction’ to the singular inviscid solutions which I conjectured some years ago emerges as a first‐order additive composite approximation. Because of the completeness requirement, however, these composite approximations are valid only in certain wedge shaped domains; approximations which are valid in the complementary sectors can then be obtained by the use of the connections formulae. The theory thus provides a relatively simple and explicit method of obtaining higher approximations, and its structure permits a direct comparison of the present results not only with the older heuristic theories but also with the comparison equation method.
The Algebra of the Riemann Curvature Tensor in General Relativity: The Relation of the Invariants of the Einstein Curvature Tensor to the Invariants Describing Matter
Authors: Greenberg Phillip
Published: September 2015

In this paper, the second in a series of papers on the algebra of the Riemann curvature tensor, we relate the algebraic invariants of the Einstein curvature tensor to the algebraic invariants of the trace‐free part of the Ricci tensor, and, consequently, to the trace‐free part of the stress‐energy tensor for the physical system. We also show explicitly how all of the components of the trace‐free part of the Ricci tensor transform under a Lorentz transformation.
On the Stability of Compressible Swirling Flow
Authors: Howard Louis N.
Published: September 2015

The linear stability to axisymmetric perturbations of compressible non‐dissipative swirling flow is shown to be insured if a suitably defined “Richardson number” depending on the basic velocity, temperature and density fields everywhere exceeds 1/4. This result combines and generalizes in a natural way some known results for incompressible swirling flow and stratified parallel flow.
A Unified Boundary Controllability Theory for Hyperbolic and Parabolic Partial Differential Equations
Authors: Russell David L.
Published: September 2015

With use of the method of spherical means we are able to show that control processes modelled by the wave equation in a domain Ω ⊆ Rn are exactly controllable in finite time by control forces applied at the boundary of the spatial region Ω. The introduction of certain concepts from harmonic analysis together with use of the Fourier transform enables us to apply this result to obtain finite time exact controllability theorems for processes modelled by the heat equation in the same region Ω with controls of the same type.
Contour Enhancement, Short Term Memory, and Constancies in Reverberating Neural Networks
Authors: Grossberg Stephen
Published: September 2015

A model of the nonlinear dynamics of reverberating on‐center off‐surround networks of nerve cells, or of cell populations, is analysed. The on‐center off‐surround anatomy allows patterns to be processed across populations without saturating the populations' response to large inputs. The signals between populations are made sigmoid functions of population activity in order to quench network noise, and yet store sufficiently intense patterns in short term memory (STM). There exists a quenching threshold: a population's activity will be quenched along with network' noise if it falls below the threshold; the pattern of supra threshold population activities is contour enhanced and stored in STM. Varying arousal level can therefore influence which pattern features will be stored. The total suprathreshold activity of the network is carefully regulated. Applications to seizure and hallucinatory phenomena, to position codes for motor control, to pattern discrimination, to influences of novel events on storage of redundant relevant cues, and to the construction of a sensory‐drive heterarchy are mentioned, along with possible anatomical substrates in neocortex, hypothalamus, and hippocampus.
Normalized Matching in Direct Products of Partial Orders
Authors: Hsieh W N. , Kleitman D. J.
Published: September 2015

In this paper we investigate the question: When does the direct product of partial orders each satisfying the normalized matching condition also satisfy it? A proof of Harper's result that a sufficient condition for this is that factor partial orders have rank populations that are multiplicatively convex, is presented. A general necessary and sufficient condition is described, and conditions which occur when one factor is a chain are obtained. In particular we show that under these circumstances it is necessary that the product order have multiplicatively convex population rank.
Uniform Asymptotic Approximations to the Solutions of the Orr‐Sommerfeld Equation. Part 1. Plane Couette Flow
Authors: Reid W. H.
Published: September 2015

This paper is concerned with the derivation of asymptotic approximations to the solutions of the Orr‐Sommerfeld equation and its adjoint which are uniformly valid in a full neighborhood of the critical point. In the case of plane Couette flow, it is shown that the expansions can be expressed to all orders in terms of a restricted class of generalized Airy functions and that the slowly varying coefficients in these expansions can be determined from simple recurrence relations. The expansions can easily be related to the Laplace integral representation of the solutions but this is not essential in their derivation. The structure of the theory also suggests how it can be immediately generalized to provide “first approximations” for a more general class of velocity profiles. Detailed results are given in the special case when the wave number α → 0. In this limit, the characteristic equation can be reduced to an especially simple form which permits a convenient global description of the modal structure.
Note on the Equations of Finite‐Strain Force and Moment Stress Elasticity
Authors: Reissner E.
Published: September 2015

Given a system of vectorial equilibrium equations for force and moment stress in a finitely deforming medium, we derive a system of force and moment strain‐displacement relations, consistent with the former via application of the principle of virtual work. We then obtain a system of vectorial compatibility equations, and we establish a connection between the present description of force stresses and strains, for the limiting case of absent moment stresses, with the classical description involving Kirchhoff stresses and Green strains, including the statement of constitutive equations corresponding to a given set of such equations for the measures of Kirchhoff and Green.
Heuristic Methods for Solving Large Scale Network Routing Problems: The Telpaking Problem
Authors: Claus Armin , Kleitman Daniel J.
Published: September 2015

An approach to a large scale network routing problem with nonlinear cost function is described, along with an example of its application. The approach to the problem involves a multistage construction process. This approach is applied to the telpaking problem. Results are obtained in applying this method to a 53 node sample problem.
The Evolution in Space and Time of Nonlinear Waves in Parallel Shear Flows
Authors: Benney D. J. , Maslowe S. A.
Published: September 2015

It is shown, using a quite general formulation, that the amplitude evolution equation for slowly varying finite amplitude waves is usually first order in both space and time. One advantage of the present formulation is that it becomes possible to easily identify, from their linear eigensolutions, interesting exceptional cases in which the amplitude evolves according to a partial differential equation that is second order in either space or time. The theory is applied to a number of specific problems, including flows with broken line profiles, and inviscid shear flows having nonlinear critical layers.
Heuristic Algorithms for Solving a Large Scale Multicommodity Flow Problem on a Network with a Step Function Cost
Authors: Ulrich Lionel Yves
Published: September 2015

This thesis presents a multicommodity flow problem on a telephone network, consisting in allocating telephone lines from one switching center to another to physical routes represented by edges. The criterion is the minimum of the total investment cost, in which the cost of each edge is a step function. Representing each group of lines between two vertices as a flow gives a very large model; decomposing these flows along paths permits the use of generalized programming; defining the total cost from the lines themselves exhibits a solution method by approximation. Three heuristic methods are presented. Solving by successive approximations to the actual costs of the lines gives a good solution. Rerouting lines from one path to another presents several pathologies which can be corrected, and shows the possibility of improving the solution yielded by the approximation method.
Some Higher‐Order Nonlinear Evolution Equations
Authors: Haberman Richard
Published: September 2015

Nonlinear evolution equations are generated which correspond to isospectral differential‐matrix eigenvalue problems. Although not discussed here, this procedure is appropriate for the analysis of the initial‐value solution by the inverse‐scattering method. In the 2 × 2 case the results obtained are consistent with other more general procedures. It yields the sine‐Gordon equation, the reduced Maxwell‐Bloch equations for self‐induced transparency, and also a type of PBBM equation for long nonlinear water waves. In the n × n case, equations are derived corresponding to a new class of triad and multitriad resonant nonlinear interaction of wave envelopes.
A Note on Some Subset Identities
Authors: Kleitman Daniel J.
Published: September 2015

A mapping is exhibited that provides a combinatorial proof of a well‐known identity.
The Three‐Wave Interaction—A Nondispersive Phenomenon
Authors: Kaup D. J.
Published: September 2015

The one‐dimensional form of the three‐wave resonant interaction, for bounded envelopes, is solved exactly by an inverse scattering transform, which was originally suggested by Zakharov and Manakov. In addition to solving a third‐order inverse scattering problem, we show how to solve this more complicated third‐order three‐wave problem in terms of three simpler second‐order problems. The general result is that during the interaction, all solitons contained in the envelope of the wave with the middle group velocity will “split,” and be added to the original number of solitons contained in both the fast and slow envelopes. These results are then applied to the “decay” and “explosive” instabilities of plasmas, and necessary and sufficient conditions are found for exciting the explosive instability in the asymptotic limit of t→+∞.
Weight Enumeration and the Geometry of Linear Codes
Authors: Greene Curtis
Published: September 2015

The problem of computing the weight distribution of a linear code is proved to be a special case of the problem of finding the Tutte polynomial (or generalized dichromatic polynomial) of a combinatorial geometry. Many other connections between the two fields are discussed.
Asymptotic Solutions of the Field‐Noyes Model for the Belousov Reaction—I. Homogeneous Oscillations
Authors: Stanshine J. A. , Howard L. N.
Published: September 2015

Multiple‐time‐scale techniques are used to solve the non‐linear autonomous system used by Field and Noyes to model the chemical oscillations of the Belousov reaction. An asymptotic representation, valid for a wide range of parameters, is found for a spatially homogeneous limit‐cycle solution. For certain values of the parameters, two limit‐cycle solutions are shown (asymptotically) to exist. For parameter values for which the limit cycle appears to be unique, it is shown to be linearly stable. The asymptotic solution is shown to correspond excellently to the numerical solution calculated by Field and Noyes for one set of parameters.
Secondary Bifurcation in Nonlinear Diffusion Reaction Equations
Authors: Keener J. P.
Published: September 2015

A set of two coupled nonlinear diffusion reaction equations is studied and the existence of secondary bifurcation is shown. Using the method of two‐timing, it is found that diffusion reaction equations of this type can exhibit an exchange of stability between distinct nontrivial solutions. This exchange can provide either a smooth or discontinuous transition between stable solutions, and the nontrivial solutions can be either steady or temporally periodic. This analysis is applied to the model biochemical reaction of Prigogine and the types of secondary bifurcation which occur in this model are classified.
Some Nonlinear Multiphase Interactions
Authors: Roskes G. J.
Published: September 2015

The propagation of multiphase modes is considered for the case when group‐velocity projections overlap. Criteria are developed for wave instabilities and for the existence of multiphase solitary envelope solutions.
Periodic Plane Waves for the Oregonator
Authors: Hastings S. P.
Published: September 2015

The Oregonator is a set of differential equations proposed by R. J. Field and R. M. Noyes as a model for the oscillating chemical reaction first studied by B. P. Belousov and A. M. Zhabotinskii. In this paper it is shown that the associated diffusion equations have periodic plane waves for parameter values not covered in earlier work. This amounts to studying a singularly perturbed system when nothing is known about the stability of periodic solutions for the reduced system.
Solutions of the Differential Equation u‴+λ2zu+(α−1)λ2u=0
Authors: Hershenov Joseph
Published: September 2015

This paper deals with the solutions of the differential equation u‴+λ2zu+(α−1)λ2u=0, in which λ is a complex parameter of large absolute value and α is an arbitrary constant, real or complex. After a discussion of the structure of the solutions of the differential equation, an integral representation of the solution is given, from which the series solutions and their asymptotic representations are derived. A third independent solution is needed for the special case when α−1 is a positive integer, and two derivations for this are given. Finally, a comparison is made with the results obtained by R. E. Langer.
On the Fourier Transform of Causal Distributions
Authors: Trione Susana Elena
Published: September 2015

We extend the distributional Bochner formula [1, p. 72, Theorem 26] to certain kinds of distributions. Theorem I.1 gives a formula [Eq. (I.1.14)] which makes it possible to obtain easily the Fourier transform of distributions of the form
urn:x-wiley:00222526:media:sapm1976554315:sapm1976554315-math-0001
As applications of the formula (I.1.14) we evaluate the Fourier transforms of the distributions Gα(P±i0, m, n) [Eq. (I.4.1)] and Hα(P±i0,n) [Eq. (II.1.1)]. It follows from Theorem II.3 that Hzk(P±i0,n) is, for 2K≠n+2r, r=0,1..., an elementary solution of the n‐dimensional ultrahyperbolic operator iterated k times.
Exchange of Stabilities in Autonomous Systems—II. Vertical Bifurcation
Authors: Lebovitz N. R. , Schaar R. J.
Published: September 2015

In the theory of singularly perturbed initial‐value problems, the principal assumption concerns a certain Jacobian matrix: all its eigenvalues should have negative real parts at each point of the reduced (or degenerate) path. If the reduced path contains a point of bifurcation, this assumption is violated. The simplest kind of bifurcation with exchange of stabilities involves just two smooth curves intersecting at a single point. The analysis of the singular perturbation theory in the case when bifurcation is present depends on whether or not both curves have finite slopes at the point of bifurcation. The case when both slopes are finite was treated in [1]; the case when the bifurcating curve has a vertical tangent is treated in the present paper.
A Priori Estimates for Solutions of Singular Perturbations with a Turning Point
Authors: Abrahamson Leif R.
Published: September 2015

A priori estimates are derived for the solutions of the boundary value problem εy″ + a(x)y′ + b(x)y = f(x), c ⩽ x ⩽ d, y(c) = α, y(d) = β. Here 0 < ε ≪ 1 is a small parameter and a(x) has a single simple zero in [c,d] (the turning point). It is shown that the solutions of this problem are uniformly bounded for ε→0 by the norms of f, α and β if and only if b(x)<0 at the turning point. However, in certain cases there are weak a priori estimates for the solutions even if this condition is not fulfilled.
Maximal Rectilinear Crossing of Cycles
Authors: Furry W. H. , Kleitman D. J.
Published: September 2015

We address the following question: In drawing a cycle on n vertices (or a graph all of whose degrees are 2) in the plane with straight line arcs, how many crossings can there be? A complete answer is given; namely, if n is odd, the number of crossings can be anything up to n(n−3)/2 except n(n−3)/2−1. For n even, the number of crossings in one cycle can be any integer up to n(n−4)/2+1; general bivalent even graphs can achieve any integer up to n(2n−7)/4 as the number of crossings.
Multitime Methods for Systems of Difference Equations
Authors: Hoppensteadt Frank C. , Miranker Willard L.
Published: September 2015

Systems of difference equations containing small parameters are studied by a constructive perturbation scheme analogous to the one developed by the authors for the study of differential equations. The method results in an averaging procedure for difference equations, and it is particularly well suited to certain highly oscillatory, nonlinear systems. The method is applied to problems from population genetics, pattern recognition, and the numerical analysis of stiff differential equations
Asymptotic Solutions of the Korteweg‐deVries Equation
Authors: Ablowitz M. J. , Segur H.
Published: September 2015

The long‐time asymptotic solution of the Korteweg‐deVries equation, corresponding to initial data which decay rapidly as |x|→∞ and produce no solitons, is found to be considerably more complicated than previously reported. In general, the asymptotic solution consists of exponential decay, similarity, rapid oscillations and a “collisionless shock” layer. The wave amplitude in this layer decays as [(lnt)/t]2/3. Only for very special initial conditions is the shock layer absent from the solution.
Another Three Part Sperner Theorem
Authors: Griggs J. R.
Published: September 2015

We show that a result of Katona can be made into a three part Sperner theorem which is independent of the best previously known such theorem, in that neither hypothesis implies the other. These three part theorems are stated in terms of a three dimensional rectangular integer lattice L, and give sufficient conditions for F ⊆ L, containing no two points on a line, to be no larger in size than the set of points of middle rank in L. The theorems apply to the more general problem in which L is the product of three symmetric chain orders and F ⊆ L contains no two points equal in two components and ordered in the third.
Existence and Uniqueness of the Minimizing Solution of Choquard's Nonlinear Equation
Authors: Lieb Elliott H.
Published: September 2015

The equation dealt with in this paper is
urn:x-wiley:00222526:media:sapm197757293:sapm197757293-math-0001
in three dimensions. It comes from minimizing the functional
urn:x-wiley:00222526:media:sapm197757293:sapm197757293-math-0002
which, in turn, comes from an approximation to the Hartree‐Fock theory of a plasma. It describes an electron trapped in its own hole. The interesting mathematical aspect of the problem is that
is not convex, and usual methods to show existence and uniqueness of the minimum do not apply. By using symmetrie decreasing re arrangement inequalities we are able to prove existence and uniqueness (modulo translations) of a minimizing ϕ. To prove uniqueness a strict form of the inequality, which we believe is new, is employed.
The Uniqueness of Odd Pair Designs
Authors: Griggs J. R.
Published: September 2015

A pair design is an ordering of all pairs in {1, 2,…, n} such that pairs that share an element have at least [(n − 3)/2] pairs separating them. Such designs can be constructed for all n. We present a shorter proof of a result of Simmons and Davis, that pair designs for odd n are unique up to permuting the numbers or reversing the order of the pairs.
Lectures on the Inverse Scattering Transform
Authors: Ablowitz Mark J.
Published: September 2015

In 1967 a new method of mathematical physics was discovered by Gardner, Greene, Kruskal, and Miura. Their significant results have received much attention, and numerous developments have occurred within the decade. In this paper a survey of some of these developments is made. By and large the emphasis is on relating the key ideas via typical examples.
Frequency Entrainment of a Forced van der Pol Oscillator
Authors: Flaherty J. E. , Hoppensteadt F. C.
Published: September 2015

A van der Pol relaxation oscillator that is subjected to external sinusoidal forcing can exhibit stable and unstable periodic and almost periodic responses. For some forcing amplitudes it even happens that two stable subharmonics having different periods may coexist. We investigate here the stable responses of such forced oscillators. By numerically computing the rotation number of stable oscillations for various values of the forcing amplitude and oscillator tuning, we obtain descriptions of regions of phase locking, successive bifurcation of stable subharmonic and almost periodic oscillations, and overlap regions where two distinct stable oscillations can coexist.
Flow of a Newtonian Fluid between Eccentric Rotating Cylinders and Related Problems
Authors: Kazaki J. Y. , Rivlin R. S.
Published: September 2015

An incompressible Newtonian fluid is contained in the annular region between two infinite cylinders, one or both of which rotate with constant angular velocities about their respective axes. The first‐order inertial correction to the forces exerted by the fluid on the cylinders is obtained in explicit algebraic form. The results are applied to the related problem in which the inner cylinder executes a planetary motion about the axis of the outer cylinder. They are also applied to the problem of the transverse sedimentation of a long cylinder in a half space of fluid bounded by a rigid wall. Certain anomalies which arise in this case are noted.
Exponential Structures
Authors: Stanley Richard P.
Published: September 2015

Generating functions of the type occur throughout enumerative combinatorics. We give a general setting for obtaining such generating functions in which the numbers M (n) have an explicit combinatorial meaning. Applications are given to the computation of Möbius functions and related invariants of certain posets.
The Propagation of Long Large Amplitude Internal Waves
Authors: Benney D. J. , Ko D. R. S.
Published: September 2015

This study is concerned with long internal gravity waves in a stratified fluid contained between rigid horizontal boundaries. For a general stratification, long waves of finite amplitude will tend to distort, and no permanent wave shape will result. In certain important cases, however, steady waveforms are found to be possible. The properties of such waves are investigated, and their relationship to the solutions provided by the weakly nonlinear theory is studied.
The Complete Solution of the Long‐Wave–Short‐Wave Resonance Equations
Authors: Ma Yan‐Chow
Published: September 2015

We give a complete analysis of the long‐wave–short‐wave resonance equations which appear in fluid mechanics as well as plasma physics. Using the inverse‐scattering technique, these equations can be reduced to a pair of linear integral equations (Marchenko equations), with the N‐soliton solutions intimately related to the asymptotic state of the evolution equations. The interaction of solitons and the conserved quantities are discussed.
Large Amplitude Rossby Waves
Authors: Benney D. J.
Published: September 2015

An investigation is made of the properties of periodic and solitary waves for a simple atmospheric model.
The Asymptotic Solution of the Korteweg‐deVries Equation in the Absence of Solitons
Authors: Miles John W.
Published: September 2015

The asymptotic solution of the Korteweg‐de Vries equation uτ + ⅓uxxx + 2uux = 0 for initial conditions from which no solitons evolve is obtained as a slowly varying similarity solution of the form τ−2/3(Vz−V2, where V = V(z/τ) and z = τ−1/3x. The results are consistent with, but go somewhat beyond, those recently obtained by Ablowitz and Segur [2] through a rather different approach.
Linearly Coupled, Slowly Varying Oscillators
Authors: Grimshaw R. , Allen J. S.
Published: September 2015

A dynamical system is considered whose normal frequencies and normal modes vary slowly with time in such a way that two frequencies come into close coincidence. When this occurs the corresponding normal modes undergo a drastic change in their physical properties. Away from coincidence, each normal mode conserves its action. A multiple‐time‐scale asymptotic procedure is employed to derive equations which describe the mode coupling at coincidence. These equations are solved exactly using parabolic cylinder functions. It is found that in general, action is exchanged between modes at coincidence, but that except for very strong coupling the amount of action exchanged is quite small.
Coalgebras and Bialgebras in Combinatorics
Authors: Joni S. A. , Rota G.‐C.
Published: September 2015

The following material is discussed in this paper: Incidence Coalgebras for PO sets; Reduced Boolean Coalgebras; Divided Powers Coalgebra; Dirichlet Coalgebra; Eulerian Coalgebra; Faà di Bruno Bialgebra; Incidence Coalgebras for Categories; The Umbral Calculus; Infinitesimal Coalgebras; Creation and Annihilation Operators; Point Lattice Coalgebras; Restricted Placements; Cleavages; and Hereditary Bialgebras.
Optimal Spreading in an n Dimensional Rectilinear Grid
Authors: Peck G. W.
Published: September 2015

A disturbance spreads in a rectilinear n dimensional grid moving from each affected point to at most one neighbor in each unit of time. The question how many points can be affected in N units of time is answered to the two leading orders of N.
A Method for Solvithe Separable Initial‐Value Problem of the Full Three‐Dimensional Three‐Wave Interaction
Authors: Kaup D. J.
Published: September 2015

Based on the recent solution of the three‐dimensional three‐wave interaction by H. Cornille, we have found a means for solving the separable initial‐value problem of this interaction, which is when the envelopes are initially nonoverlapping. Explicit means for calculating the kernels directly from the initial pulse profiles are given.
Long Internal Waves in Fluids of Great Depth
Authors: Ablowitz M. J. , Segur H.
Published: September 2015

An equation is derived that governs the evolution in two spatial dimensions of long internal waves in fluids of great depth. The equation is a natural generalization of Benjamin's (1967) one‐dimensional equation, and relates to it in the same way that the equation of Kadomtsev and Petviashvili relates to the Kortewegde‐Vries equation. The stability of one‐dimensional solitons with respect to long transverse disturbances is studied in the context of this equation. Solitons are found to be unstable with respect to such perturbations in any system in which the phase speed is a minimum (rather than a maximum) for the longest waves. Internal waves do not have this property, and are not unstable with respect to such perturbations.
On Time Discretizations for Spectral Methods
Authors: Gottlieb David , Turkel Eli
Published: September 2015

New methods are introduced for the time integration of the Fourier and Chebyshev methods of solution for dynamic differential equations. These methods are unconditionally stable, even though no matrix inversions are required. Time steps are chosen by accuracy requirements alone. For the Fourier method both leapfrog and Runge‐Kutta methods are considered. For the Chebyshev method only Runge‐Kutta schemes are tested. Numerical calculations are presented to verify the analytic results. Applications to the shallow water equations are presented.
Transformation and Reduction Formulas for Two‐Variable Hypergeometric Functions on the Sphere S2
Authors: Kalnins E. G. , Manocha H. L. , Miller Willard
Published: September 2015

We classify the two‐variable hypergeometric functions that arise as eigenfunctions of the Laplace‐Beltrami operator on S2 and characterize these functions in terms of elements in the enveloping algebra of so(3). This operator characterization is used to derive transformation and reduction formulas for the functions.
Multiple Solutions of Nonlinear Boundary‐Value Problems
Authors: Rosenblat S. , Szeto R.
Published: September 2015

A class of nonlinear boundary‐value problems containing a parameter is studied analytically and numerically. It is shown that under certain circumstances there are two families of solutions when the parameter tends to zero; one family comprises small solutions and is obtained by regular perturbations, while the other family comprises finite solutions incorporating boundary and interior layers. It is shown by numerical integration that the two families are smooth continuations of each other when the parameter passes through finite values.
Target Patterns and Horseshoes from a Perturbed Central‐Force Problem: Some Temporally Periodic Solutions to Reaction‐Diffusion Equations
Authors: Kopell N. , Howard L. N.
Published: September 2015

Existence and construction of some time‐periodic solutions of a class of reaction‐diffusion equations is described. These give examples of center‐structures from which emanate trains of waves with outward directed group velocity, in an infinite one‐dimensional spatial domain. A discrete set of distinct types of center‐structures is found. Similar results are found for sufficiently large finite regions with impermeable boundaries. The existence of many other time periodic solutions corresponding to spatially infinite irregular arrays of center‐structures of different types is also demonstrated for these systems. Some numerical examples are presented.
The Eigenvalues of Mathieu's Equation and their Branch Points
Authors: Hunter C. , Guerrieri B.
Published: September 2015

A comprehensive account is given of the behavior of the eigenvalues of Mathieu's equation as functions of the complex variable q. The convergence of their small‐q expansions is limited by an infinite sequence of rings of branch points of square‐root type at which adjacent eigenvalues of the same type become equal. New asymptotic formulae are derived that account for how and where the eigenvalues become equal. Known asymptotic series for the eigenvalues apply beyond the rings of branch points; we show how they can now be identified with specific eigenvalues.
Nonlinear Waves in a Self‐Gravitating Medium
Authors: Yueh T. Y.
Published: September 2015

To investigate the effects of finite amplitudes on the behavior of waves in a self‐gravitating medium, exact solutions of nonlinear waves are obtained in a simple situation. The waves are traveling in the direction of the axis of rotation of a homogeneous and uniformly rotating self‐gravitating medium. The nonlinear wave profile and the dispersion relation are obtained and compared with Jeans's linear theory. It is found that the wavelength‐amplitude plane can be divided into three regions: (1) a region where the shock wave is expected to form, (2) a region where the neutral wave exists, and (3) a region where there is gravitational instability. In the third case, it is expected that the unstable wave will eventually evolve to a neutral wave with finite amplitude. Based on this result, speculations are made on the evolution of the unstable density wave in a galaxy. It is established that the propagation of the nonlinear wave in a self‐gravitating medium is always “subsonic,” just as in the linear theory. The difference is that the maximum ratio of the phase speed to the sound speed of the basic state is 1 − A2 for the isothermal case and (1 − A2)(γ + 1)/2 for the polytropic case instead of 1 for the linear case, where A2 is the dimensionless depth of the valley of the density profile, and γ is the polytropic index. It is also shown that no solitary wave can be found in a homogeneous self‐gravitating medium. Comparison is made with nonlinear waves in a homogeneous plasma.
The Periodic Cubic Schrõdinger Equation
Authors: Ma Yan‐Chow , Ablowitz Mark J.
Published: September 2015

Solutions to the cubic nonlinear Schrödinger equation, periodic in space, are developed.
Evolution Equations for Long, Nonlinear Internal Waves in Stratified Shear Flows
Authors: Grimshaw R.
Published: September 2015

Evolution equations for long, nonlinear internal waves are derived when the basic stratified shear flow has a slow temporal and spatial variation as well as the usual dependence on the vertical coordinate. When the horizontal waveguide has a limited vertical extent the evolution equation is a variable coefficient Korteweg‐deVries equation, while in the deep fluid case the evolution equation is a variable coefficient Benjamin‐Davis‐Ono equation. Explicit expressions are obtained for the coefficients of these equations.
Weakly Nonlinear Internal Waves in Shear
Authors: Tung Ka‐Kit , Ko Denny R. S. , Chang Jerry J.
Published: September 2015

An evolution equation in a finite depth fluid for weakly nonlinear long internal waves is derived in a stratified and sheared medium. The equation reduces to the Korteweg‐deVries equation when the depth is small compared to the wavelength, and to the Benjamin‐Ono equation when the depth is large compared to the wavelength. Both the cases with and without critical levels are investigated. Numerical solutions to the evolution equation are presented to illustrate the effect of shear on the evolution of a waveform.
Large Amplitude Internal Waves of Permanent Form
Authors: Tung, Ka‐Kit , Chan, Tony F. , Kubota Toshi
Published: September 2015

In this paper the weakly nonlinear theory of long internal gravity waves propagating in stratified media is extended to the fully nonlinear case by treating Long's nonlinear partial differential equation for steady inviscid flows without restriction to small amplitudes and long wavelengths. The existence of finite amplitude solutions of “permanent form” is established analytically for a large class of stratification profiles, and properties are calculated numerically for the case of a hyperbolic tangent density profile in a large range of fluid depths. The numerical results agree well with the experimental data of Davis and Acrivos over the full range of wave amplitudes measured; such agreement is not obtainable with existing weakly nonlinear theories.
The Steady Viscous Flow of Two Differentially Rotating Immiscible Fluids
Authors: Baker Gregory R. , Mardeusz Steven J.
Published: September 2015

We have computed the steady, axisymmetric viscous boundary layers on either side of an interface between two immiscible, incompressible fluids that are in rigid body rotation far from the interface. The internal rotational Froude number is assumed small so that the interface may be considered horizontal. An application of our results to the spinup from rest of two immiscible slightly viscous fluids in a vertically mounted cylinder is discussed.
The Effect of Dissipation on Linearly Coupled, Slowly Varying Oscillators
Authors: Grimshaw R. , Allen J. S.
Published: September 2015

A dynamical system is considered whose normal frequencies and normal modes vary slowly with time in such a way that two frequencies come into close coincidence. When this occurs the corresponding normal modes undergo a drastic change in their physical description. A previous paper by us (1979) considered a conservative dynamical system, and showed that in general action is exchanged between modes at coincidence, but that except for very strong coupling the amount of action exchanged is quite small. The present paper extends this analysis to dissipative, or nonconservative, dynamical systems. Using a multiple‐time‐scale asymptotic procedure, the equations which describe the mode coupling at coincidence are derived and solved exactly using parabolic cylinder functions. The solutions show that while action is exchanged between modes at coincidence in a manner similar to that described above for conservative dynamical systems, the effect of dissipation is to ensure that the mode which suffers the smaller dissipation will dominate after mode coupling.
The Motion with Slip of a Thin Viscous Droplet over a Solid Surface
Authors: Lacey A. A.
Published: September 2015

A thin droplet of a viscous liquid moves under the influence of surface tension across the contaminated surface of a solid. There is a slip velocity of the fluid along the surface proportional to the tangential stress. For small values of the constant of proportionality and for a significant distance from the contact line there is an asymptotic expansion for the droplet thickness. The speed of the contact line is found to depend nonlinearly on the normal derivative of the leading term.
The Inverse Scattering Transform for the Benjamin‐Ono Equation—A Pivot to Multidimensional Problems
Authors: Fokas A. S. , Ablowitz M. J.
Published: September 2015

The initial value problem associated with the Benjamin‐Ono equation is linearized by a suitable extension of the inverse scattering transform. Essential is the formulation and solution of an associated nonlocal Riemann‐Hilbert problem in terms of initial scattering data. Solitons are given a definitive spectral characterization. Pure soliton solutions are obtained by solving a linear algebraic system whose coefficients depend linearly on [INLINEEQUATION], [INLINEEQUATION].
Gravity Waves in a Channel with a Rough Bottom
Authors: Rosales Rodolfo R. , Papanicolaou George C.
Published: September 2015

We derive effective equations for the surface elevation of gravity waves in a shallow channel with a rough bottom.
Optimal Forest Harvesting with Ordered Site Access
Authors: Wan Frederic Y. M. , Anderson Keith
Published: September 2015

A long‐standing problem in forestry management is the optimal harvesting of a growing population of trees to maximize the resulting discounted aggregate net revenue. For an ongoing forest, the trees are harvested and replanted repeatedly; for a once‐and‐for‐all forest, there is no replanting after a single harvest. In this paper, we outline a new formulation for the optimal‐harvest problem which avoids difficulties associated with functional‐differential equations or partial differential equations of state in the relevant optimal‐control problem encountered in recent studies of the ongoing‐forest problem. Our new formulation is based on the observation that tree logging is necessarily ordered by practical and/or regulatory considerations (e.g., it is illegal to cut the younger trees first in some jurisdictions); random access to tree sites does not occur in practice. The new formulation is described here for the simpler problem of a once‐and‐for‐all forest. New results for nonuniform initial age distributions and variable unit harvest costs for this simpler problem are reported herein; results for an ongoing forest will be reported in [10]. The new model is also of interest from a control‐theoretic viewpoint, as it exhibits the unique feature of having time as a state variable, in contrast to its usual role as an independent variable in conventional control problems.
On Spurious Solutions of Singular Perturbation Problems
Authors: Lange Charles G.
Published: September 2015

A study is made of several nonlinear boundary‐value problems of singular perturbation type for which a straightforward application of boundary‐layer theory leads to spurious solutions. It is shown that these problems can be treated successfully by a slight modification of the method of matched asymptotic expansions. The analysis leads to several novel features which are not present in routine singular perturbation problems.
The Evolution of Large‐Horizontal‐Scale Disturbances in Marginally Stable, Inviscid, Shear Flows. I. Derivation of the Amplitude Evolution Equations
Authors: Hickernell Fred J.
Published: September 2015

Two inviscid, stratified, shear flow models are studied. The amplitude evolution equations for large‐horizontal‐scale disturbances are derived. Particular attention is given to the case where the flows are marginally stable. In this case the amplitude evolution equations are second order in time. A typical example is the Boussinesq equation.
Nonlinear Dispersive Systems: Theory and Examples
Authors: Howes F. A.
Published: September 2015

Some recent results in the asymptotic theory of differential equations are applied to certain third‐order scalar and vector boundary‐value problems that model various nonlinear physicochemical and dispersive wave phenomena. The key to our approach is the reduction of the third‐order problems to asymptotically equivalent second‐order ones which are more amenable to analysis. Many examples are discussed, including the problem of solitary‐wave solutions of generalized Korteweg‐de Vries equations and coupled systems of such equations.
The Evolution of Disturbances in Shear Flows at High Reynolds Numbers
Authors: Benney D. J.
Published: September 2015

This paper is concerned with the propagation of wavelike disturbances in shear flows. The analysis shows that three dimensionality may dominate the evolution process.
Lax Pair for the Free‐Fermion Eight‐Vertex Model
Authors: Barouch E.
Published: September 2015

An explicit result on the quantum Lax pair for the eight‐vertex free‐fermion model with external fields in both directions is exhibited. Its relation to the quantum spectral transform and its corresponding Hamiltonian—theXYmodel with a transverse field—is discussed. The Yang‐Baxter relation is computed, and the equations of motion are solved.
An Exactly Solvable Nonlinear Partial Differential Equation with Solitary‐Wave Solutions
Authors: Cheng Hung
Published: September 2015

A pair of nonlinear partial differential equations occurring in a physical problem are solved exactly. Under proper conditions, these equations have solitary‐wave solutions.
A Solvable Nonlinear Wave Equation
Authors: Calogero F.
Published: September 2015

It is pointed out that the nonlinear wave equation
urn:x-wiley:00222526:media:sapm1984703189:sapm1984703189-math-0001 can be solved by quadratures. Here a and c are constants, A(y) and B(y) (arbitrary) functions; a t‐dependence of all these quantities can also be accommodated. This wave equation can also be rewritten in the (purely differential) form
urn:x-wiley:00222526:media:sapm1984703189:sapm1984703189-math-0002 via the substitutions
.
An Integrodifferential Equation Arising in Thermo‐Viscoelasticity
Authors: Weinitschke H. J.
Published: September 2015

Viscoelastic material at high temperature is subjected to a cooling process. The stresses built up in the body are determined from a system of equations containing a strongly temperature‐dependent viscosity η(T), where the temperature T is given by the heat conduction equation. It is shown that for simple geometries such as infinite cylinders and spheres, the basic equations can be reduced to a single Volterra‐type integrodifferential equation, which is shown to have a unique solution.
Multidimensional Inverse Scattering for First‐Order Systems
Authors: Nachman Adrian I. , Ablowitz Mark J.
Published: September 2015

A method for solving the inverse problem for a class of multidimensional first‐order systems is given. The analysis yields equations which the scattering data must satisfy; these equations are natural candidates for characterizing admissible scattering data. The results are used to solve the multidimensional N‐wave resonant interaction equations.