bluuebunny/crossref_metadata_2025 huggingface dataset:
Clusters on a Thin Quadratic Lattice (Transfer Matrix Technique)
Authors: Lieb Elliott H. , Beyer W. A.
Published: September 2015

Formulae for the distribution of clusters of independently occupied sites on an n × 1, n × 2 and n × 3 quadratic lattice are derived by use of the transfer matrix technique. In principle, the formulae can be extended to an n × r lattice for arbitrary r. Both plane and cylindrical lattices are considered.
A Note on the Laboratory Simulation of Planetary Flows
Authors: Greenspan H. P.
Published: September 2015

The depth‐averaged equations used frequently as a model for large scale motions in the ocean are shown to be the exact theory for quasi‐geostrophic flows in special laboratory configurations. This analogy permits the simulation of certain planetary phenomena.
A New Class of Nonlinear Waves in Parallel Flows
Authors: Benney D. J. , Bergeron R. F.
Published: September 2015

Waves in parallel shear flows are found to have different characteristics depending on whether nonlinear or viscous effects dominate near the critical layer. In this paper a nonlinear theory is developed which gives rise to a class of disturbances not found in the classical viscous theory. It is suggested that the modes found from such an analysis may be of importance in the breakdown of laminar flow due to free stream disturbances.
Phase Transition and Eigenvalue Degeneracy of a One Dimensional Anharmonic Oscillator
Authors: Thompson Colin J. , Kac Mark
Published: September 2015

For a one dimensional lattice model with long range exponential interaction the problem of deriving the thermodynamic properties of the system reduces to the problem of determining the spectrum of a differential equation for a one dimensional anharmonic oscillator with potential αx2 + βγx4 where γ is the inverse of the range of the interaction and α is a function of temperature which is positive at high temperatures and negative at low temperatures. It is proved that at low temperatures, α < 0, and in the limit γ → 0, (corresponding to an infinitely long range interaction) the spectrum becomes asymptotically degenerate. The relation of degeneracy to the occurrence of a phase transition is discussed and the generalization to higher dimensions is mentioned briefly.
Representation of Isotropic Turbulence by Scalar Functions
Authors: Orszag Steven A.
Published: September 2015

The maximum number N(n) of scalar functions of wavevector needed to represent the general cumulant of order n of isotropic turbulence is determined. In addition to reproducing the known results that N(2) = 1, N(3) = 2, it is shown that N(4) = 4, N(5) = 4, N(6) = 9, etc.
Asymptotic Behavior of Solutions of Non‐linear Differential Equations
Authors: Levinson N.
Published: September 2015

A method is given for showing that formal “approximate solutions” of non‐linear differential equations are in fact the leading terms in an asymptotic representation of actual solutions. Examples are given to show how the method is applied. Because of the great diversity in the form of solutions the presentation of a method is much simpler than the inevitably complicated statement of theorems.
Eigenvalue Degeneracy and Existence and Non‐existence of a Phase Transition in One Dimension
Authors: Thompson Colin J.
Published: September 2015

We consider a one dimensional Ising chain with interaction potential J(k) such that J(k) = 0 when k > n. By a perturbation argument we show that long range order exists at sufficiently low temperatures if and only if
urn:x-wiley:00222526:media:sapm1969484299:sapm1969484299-math-0001
This is consistent with Dyson's recent theorems and in addition predicts that when J(k) = k−2 there is no long range order.
The Asymptotics of Nonlinear Diffusion
Authors: Benney D. J. , Lange C. G.
Published: September 2015

The asymptotic behavior of linear diffusion processes may be significantly altered by weak nonlinear effects. A method is developed for investigating such situations. Examples of both deterministic and statistical initial value problems are discussed in the one dimensional case.
Proportions of Irreducible Diagrams
Authors: Kleitman Daniel J.
Published: September 2015

We consider diagrams consisting of pairings among 2n integers, a1, & , a2n. Certain classes of such diagrams called “reducible diagrams of type k” are defined. We investigate the proportion of diagrams lying in these classes, obtaining asymptotic results for large values of n, and fixed k.
Promotion Phenomenon for Three Particle Exchange
Authors: Blau Irwin
Published: September 2015

In the two particle exchange, it has been shown [1] that the asymptotic form of the scattering amplitude at high energies behaves differently for t = 4µ2 and t ≠ 4µ2. In this paper, the three particle exchange is considered for t = 9µ2 and t ≠ 9µ2. For t near but not equal to 9µ2 in the three particle exchange, the asymptotic behavior of the scattering amplitude is also discussed.
Generalized Ferroelectric Model on a Square Lattice
Authors: Baxter R. J.
Published: September 2015

A completely general square‐lattice ferroelectric model is considered, in which the configuration energies have arbitrary values at each vertex. It is found that the partition function can be evaluated by straightforward extensions of the method used for the regular ice, F‐ and KDP‐models, provided three sets of conditions are satisfied. These ensure that the two ordered F‐model states are degenerate (in the limit of a large lattice), so we conclude that the application of any staggered field which breaks this degeneracy also renders the problem insoluble by the present methods.

The partition function for these ordered states is obtained explicitly and found to be given quite simply as a product of certain parameters associated with each vertex. Finally some conjectures are made concerning the spontaneous staggered polarization.
On the Transfer Matrix for the Two‐Dimensional Ising Model
Authors: Abraham D. B.
Published: September 2015

The eigenvalue problem for the transfer matrix of the two‐dimensional Ising lattice, with free boundaries perpendicular to the transfer direction, is solved. The mathematical mechanism which produces the asymptotic degeneracy of the eigenvalue spectrum below the bulk transition temperature is different from the case with cyclic boundary conditions. The results are used to calculate the two‐spin correlation function between two spins in the same edge of the lattice, and between two spins in opposite edges.
On Mean Convergence of Fourier‐Bessel Series of Negative Order
Authors: Benedek A. , Panzone R.
Published: September 2015

The expansion of f ∈ Lp(0, 1) Fourier series of Bessel functions of order

converges to f in Lp whenever
urn:x-wiley:00222526:media:sapm1971503281:sapm1971503281-math-0001

Let

be the space of p‐integrable functions with respect to the measure t dt and
where {sn}, n = 1, 2, …, is the set of positive zeros of Jv. Then, the expansion of
in a Fourier series of functions ψn, −1 < ν < −½, converges to
in
whenever
urn:x-wiley:00222526:media:sapm1971503281:sapm1971503281-math-0002
A Distribution‐ Theoretic Approach to a Stability Problem of Orr
Authors: Willke H. L.
Published: September 2015

In a classic paper [1] of 1907, W. M'Farr Orr discovered, among other things, the “infinitesimal” instability of inviscid plane Couette flow. Surprisingly, although Orr's paper remains a standard reference in the field, later investigators [2, 3] have been able to call inviscid plane Couette flow stable without finding it necessary to controvert Orr's result. What has happened is that, at least in problems governed by linear (or linearized) equations with time‐independent coefficients, the term “instability” has come to be identified with the presence of solutions exhibiting exponential time‐growth. Orr found instability indeed: a class of solutions certain members of which grow in time by more than each preassigned factor. Unlike the exponential instabilities, however, Orr's solutions die away like 1/t after achieving their greatest growth. This ephemerality probably accounts for the discounting of Orr's result. Orr did not look into the general initial value problem. This is done in the sequel, with the result that the situation becomes clear. Under general disturbances, Couette flow turns out to be neither stable nor quasi‐asymptotically stable*. The rate of growth depends on the smoothness of the initial data: classical solutions grow no faster than t, but sufficiently rough distribution‐valued initial data leads to growth matching any power of t. Before presenting detailed results, we briefly review Orr's fundamental work on the problem.
A Second Order Differential Operator with Eigenvalue Degeneracy
Authors: Hochstadt Harry
Published: September 2015

The second order differential operator Lu = −u″ + (M − |x|)u defined on ( −∞, ∞) is analyzed. It is shown that the pair of successive eigenvalues λ2n, λ2n+1 is asymptotically degenerate as M becomes large, for all n.
Ray Method Expansions for Surface and Internal Waves in Inhomogeneous Oceans of Variable Depth
Authors: Hector D. , Cohen J. , Bleistein N.
Published: September 2015

A ray method formalism is developed for the analysis of surface and internal waves in an inhomogeneous ocean of variable depth. In this method, we deduce from the governing system of equations a system of first order ordinary differential equations, for the group lines (rays of the ray method) and the propagation of phase and amplitude on them. The dispersion relation for these waves arises as an eigen‐condition on an eigen‐value problem involving an ordinary differential equation in the depth variable. The deduced equation for amplitude propagation has the interpretation of a statement of conservation of action.
The Algebra of the Riemann Curvature Tensor in General Relativity: Preliminaries
Authors: Greenberg Phillip J.
Published: September 2015

In a four‐dimensional curved space‐time it is well‐known that the Riemann curvature tensor has twenty independent components; ten of these components appear in the Weyl tensor, and nine of these components appear in the Einstein curvature tensor. It is also known that there are fourteen combinations of these components which are invariant under local Lorentz transformations. In this paper, we derive explicitly closed form expressions which contain these twenty independent components in a manifest way. We also write the fourteen invariants in two ways; firstly, we write them in terms of the components; and, secondly, we write them in a covariant fashion, and we further derive the appropriate characteristic value equations and the corresponding Cayley‐Hamilton equations for these invariants. We also show explicitly how all of the relevant components transform under a Lorentz transformation. We shall follow the very general and powerful methods of Sachs [1]. We shall not point out at every stage of the calculation which equations are due to Sachs, and which equations are new; this is easily ascertained. Generally speaking, however, the equations depending on the Einstein curvature tensor, and the emphasis placed on this tensor, appear to be new.
Semi‐Resonant Interactions and Frequency Dividers
Authors: Ablowitz, M. J. , Funk B. A. , Newell A. C.
Published: September 2015

A theory describing the behavior of a system as it evolves slowly through internal nonlinear resonances is presented. The energy sharing process is seen to be quite complex as it depends crucially on both nonlinear and frequency detuning effects. Two phenomena are discussed in detail although the general ideas are applicable to many situations. Firstly we examine the interaction between the quadratically coupled oscillators whose natural frequencies are in the ratio 2:1 for a limited period of time. Such a system is shown to be an extremely useful switching device. Secondly we examine the time dependent Duffing equation and find that smooth forward and reverse transitions occur without the presence of dissipation.
Hyperspheric Integer Programming
Authors: Claus A.
Published: September 2015

This paper develops an efficient method for finding the optimal solution to linear mathematical programs on 0–1 variables.

It is shown that the lattice (0–1) points satisfying some linear constraint of dimension n can equally be represented by those lying in a hypersphere of the same dimension.

The lattice points satisfying two linear constraints can be represented by a hypersphere which contains the intersection of the hyperspheres of the two constraints.

The method for finding the optimal solution consists of enumerating lattice points which are close to the center of the hypersphere corresponding to the constraints. As soon as a better value of the objective function has been found, than some lower bound, we find a new hypersphere which contains the lattice points of the constraints at which the objective function remains higher than the best known value. We continue in this manner until we have at some stage enumerated all lattice points within a given hypersphere and found none which give a better value.
Efficient Communication by Phone Calls
Authors: Lebensold Kenneth
Published: September 2015

We are given the question: how many phone calls are needed for n people to pool all their information in a succession of k‐person party line phone calls? The question was proposed by Erdös for the special case K = 2. We prove here the result that [n − 2/K − 1] + [n − 1 /k] + 1 calls are required if 1 ≤ n ≤ k2, while 2[n − 2/k − 1] are required for n > k2.
Conservation Laws for the Fully Nonlinear Long Wave Equations
Authors: Miura Robert M.
Published: September 2015

The fully nonlinear long wave equations describe the motion over a flat bottom of a two‐dimensional inviscid fluid with a free surface in a gravitational field in the long wave approximation. These equations are shown to possess an infinite number of conservation laws (in two space dimensions) in the form
urn:x-wiley:00222526:media:sapm197453145:sapm197453145-math-0001
The conserved densities T and the fluxes −X and −Y are polynomials in the height h and the horizontal and vertical components of velocity, u and v, and also in integrals of powers of u. The method of proof is a modification of the method recently devised by D. J. Benney to prove that these same equations possess an infinite number of conservation laws (in one space dimension) in the form
urn:x-wiley:00222526:media:sapm197453145:sapm197453145-math-0002
where T and X are polynomials in the height h and integrals of powers of u. Conservation laws which explicitly contain x and t are also given.
Equilibrium Arrival Rates at a Partially‐Obstructed Intersection
Authors: Hellinger Steven J.
Published: September 2015

A lane of traffic approaches an intersection where a traffic signal mediates between two possibilities: continued travel on that lane, or a turn onto a one‐way street. Movement stops temporarily whenever a vehicle arrives at the intersection during a period permitting the alternative that is not the driver's intention. A simple model is presented which mathematically verifies the intuition that this type of signal cannot allow traffic to move efficiently during the rush‐hour.
The Upper Bound Conjecture and Cohen‐Macaulay Rings
Authors: Stanley Richard P.
Published: September 2015

Let Δ be a triangulation of a (d − 1)‐dimensional sphere with n vertices. The Upper Bound Conjecture states that the number of i‐dimensional faces of Δ is less than or equal to a certain explicit number ci(n, d). A proof is given of a more general result. The proof uses the result, proved by G. Reisner, that a certain commutative ring associated with Δ is a Cohen‐Macaulay ring.
Method for Solving the Sine‐Gordon Equation in Laboratory Coordinates
Authors: Kaup D. J.
Published: September 2015

By solving the direct and inverse scattering problems for a rather ‘unconventional’ eigenvalue problem, we can solve the initial value problem for the sine‐Gordon equation entirely in laboratory coordinates. This allows us to extend and accurately define the class of initial value problems which can be solved by ‘inverse scattering transforms.’ Simple examples are given to illustrate both the direct and the inverse scattering methods.
Exchange of Stabilities in Autonomous Systems
Authors: Lebovitz N. R. , Schaar R. J.
Published: September 2015

Equilibrium solutions y = ϕ(x) of an autonomous system of differential equations, depending on a parameter x, are considered. Bifurcation of a second family of solutions y = ψ(x) and exchange of stabilities is supposed to occur at (x,y) = (0,0). Considering x as slowly varying leads to a singularly perturbed initial‐value problem whose reduced path encounters a point of bifurcation. Rigorous asymptotic estimates are found for the difference between the (unique) solution of the full problem and that solution of the reduced problem which proceeds along stable segments of the reduced path.
On the Existence of Discrete Frequencies of Oscillation in a Rotating Fluid
Authors: Schaeffer David G.
Published: September 2015

We give a necessary and sufficient condtion for there to exist a discrete frequency of oscillation in an inviscid, axially symmetric rotating ring of fluid, a problem introduced by Barcilon. It turns out that for most cross sections of the ring there are no discrete frequencies, only a continuous spectrum.
Asymptotic Approximations for Prolate Spheroidal Wave Functions
Authors: Miles John W.
Published: September 2015

Uniformly valid (with respect to the independent variable) asymptotic approximations to the radial, prolate spheroidal wave functions are constructed from Bessel‐function and Coulomb‐wave‐function models for large values of the wave number c. The prolate angular functions also are considered, but more briefly. The emphasis is on qualitative accuracy (such as might be useful to the physicist), rather than on efficient algorithms for very accurate numerical computation, and the error factor for most of the approximations is 1 + O (1/c) as c↑∞.
Significant Interactions Between Small and Large Scale Surface Waves
Authors: Benney D. J.
Published: September 2015

Under certain conditions it is found that the interactions between short and long waves are especially important. The type of analysis presented has applications to many nonlinear dispersive wave systems, but the detailed discussion is restricted to gravity‐capillary wave interactions.
A Nonlinear Difference Scheme and Inverse Scattering
Authors: Ablowitz M. J. , Ladik J. F.
Published: September 2015

A nonlinear partial difference equation is obtained and solved by the method of inverse scattering. In a certain continuum limit it is shown how this equation approximates the nonlinear Schrodinger equation and a related nonlinear differential‐difference equation. At all times the solutions can be compared, and the scheme is shown to be convergent. These ideas apply to other nonlinear evolution equations as well.
A Note on the Theory of Low‐Frequency Waves in a Rotating Stratified Channel
Authors: Vermersch John A. , Beardsley Robert C.
Published: September 2015

It is found that the possible low‐frequency, quasigeostrophic motions in a rotating, stratified channel with a wave‐maker at one end include: (i) “standing waves” whose amplitudes are damped exponentially away from the forcing, and (ii) baroclinic internal Kelvin waves, trapped to the right‐hand wall when facing in the direction of phase propagation. The Kelvin waves are excited only if the wave‐maker transfers mean energy to the fluid. The standing waves, on the other hand, carry no energy and thus serve mainly to provide continuity between the wave‐maker and the fluid. When the bottom of the channel is inclined to the horizontal by a small angle θ, topographic oscillations are possible. These waves behave like topographic Rossby waves if the forcing frequency is greater than sN and if the ratio HN/fL is small, where s=tanθ, θ is the angle of the bottom slope, L is the width of the channel, H is the mean depth, f is the Coriolis parameter, and N is the Brunt‐Vaisala (or buoyancy) frequency. It is determined that topographic Rossby waves cannot exist in the channel if HN/fL⪞0.65. If the wave‐maker frequency is smaller than sN, and if HN/fL~1, the topographic oscillations become bottom‐trapped, decaying away from the bottom boundary in a distance ~λf/N, where λ is the horizontal wavelength. The phase and energy of the bottom‐trapped wave both move to the left of an observer who is facing shallow water. The Kelvin waves are basically unchanged when the bottom is inclined if their down‐channel wavelength is large relative to the width of the channel. The standing oscillations of the flat‐bottom case exist as complex‐horizontal‐wave‐number solutions to the topographic wave dispersion relation. Although these waves have propagating phase when s≠0, they are still trapped to the forcing, and do not transfer net energy from the wave‐maker to the fluid. All three modes are required to solve the general matching conditions for an arbitrary wave‐maker when the channel has a sloping bottom.
Asymptotic Solutions of the Field‐Noyes Model for the Belousov Reaction—II. Plane Waves
Authors: Stanshine J. A.
Published: September 2015

The spatially homogeneous limit cycle solutions found by Stanshine and Howard [7] for the Field‐Noyes model of the Belousov reaction are used to obtain plane wave solutions for the related reaction‐diffusion equations. Perturbation techniques are implemented first to find long wavelength waves and then successively to find shorter wavelength waves. Plane wave solutions are then shown to exist for sufficiently short wavelengths for parametric values which are physically plausible but for which spatially homogeneous oscillations do not (apparently) exist.
A Note on Uniquely lntersectable Graphs
Authors: Wang Da‐Lun
Published: September 2015

The following conjecture of Alter and Wang is proven. Consider the intersection graph Gn,m,n⩾2m, determined by the family of all m‐element subsets of an n‐element set. Then any realization of Gn,m as an intersection graph by a family of sets satisfies |∪iAi|⩾n; and if |∪iAi|=n, then F must be the family of all m‐element subsets of ∪iAi.
A General Theory for Interactions Between Short and Long Waves
Authors: Benney D. J.
Published: September 2015

Nonlinear partial differential equations which permit both short and long wave solutions are studied. The theory reveals a variety of phenomena relevant to a broad class of physical problems. Among the properties investigated are resonances, instabilities and steady state solutions.
A Theorem Concerning the Integer Lattice
Authors: Bell David E.
Published: September 2015

This note shows that if an integer linear program in n variables has more than 2n linear inequality constraints, then either some of the constraints are unnecessary or there is at least one feasible integer point.
The Modulation of an Internal Gravity‐Wave Packet, and the Resonance with the Mean Motion
Authors: Grimshaw R. H. J.
Published: September 2015

An inviscid, incompressible, stably stratified fluid occupies a horizontal channel, along which an internal gravity‐wave packet is propagating. The wave induced mean motions are calculated, and the equations describing the evolution of the wave amplitude derived. When the group velocity of the wave packet coincides with a long‐wave speed there is a resonance, and the equations describing this resonance are derived.
On Bifurcating Periodic Solutions at Low Frequency
Authors: Davis Stephen H. , Rosenblat S.
Published: September 2015

A study is made of differential equations which have low‐frequency periodic bifurcating solutions. It is shown that the classical Poincaré‐Hopf technique for constructing periodic solutions encounters difficulties in the low‐frequency limit, and that the branches are determined by nonlinear, rather than linear, balances. Two types of models are investigated: one is autonomous and the other non‐autonomous, with a forcing term of small, fixed frequency. The latter model is relevant to several fluid‐dynamical situations.
Flow over a Wing with an Attached Free Vortex
Authors: Saffman P. G. , Sheffield J. S.
Published: September 2015

Exact solutions are constructed for two‐dimensional inviscid potential flow over a wing with a free line vortex standing over the wing. The loci of positions of the free vortex are found, and the lift is calculated. It is found that the lift on the wing can be significantly increased by the free vortex.
Multivariate Sequences of Binomial Type
Authors: Reiner David L.
Published: September 2015

Using the notion of convolution of binomial sequences it is possible to show that Sheffer sequences, cross sequences, and Steffensen sequences are only mild generalizations of ordinary sequences of binomial type.
On Thermal Convection in a Large Box
Authors: Brown S. N. , Stewartson K.
Published: September 2015

Daniels [4] has shown that the presence of imperfectly insulating sidewalls transforms the onset of Bénard convection in a layer of fluid confined by one pair of lateral boundaries from a bifurcation to a continuous transition as the Rayleigh number increases through the critical value. In this paper these ideas are extended to accommodate rectangular containers with large, horizontal dimensions. The appropriate boundary conditions for the amplitude equations are derived, and a limited discussion of the properties of the solutions is undertaken. In particular it is shown that, in agreement with earlier theories and experiments, there is a preference for rolls parallel to the shorter side.
Permanent Wave Structures on an Open Two Layer Fluid
Authors: Bryant P. J.
Published: September 2015

There is a significant nonlinear interaction between fast free surface waves and slow interface waves when the group velocity of the free surface waves is the same as the phase velocity of the interface waves. This interaction leads to permanent wave structures consisting of a wave group of permanent envelope on the free surface and a wave of permanent shape on the interface. A theory is developed for periodic permanent wave structures of this type, from which solutions are found numerically. The theory includes all significant quadratic nonlinear interactions between free surface harmonics and interface harmonics, as well as between the interface harmonics themselves. It is found that there is a sequence of forms of differing free surface group structure.
The Equation of Motion of a Vortex Layer of Small Thickness
Authors: Moore D. W.
Published: September 2015

The equation governing the evolution of a vortex layer whose thickness is small compared to its radius of curvature and in which the vorticity is everywhere the same is obtained, viscous and compressibility effects being neglected. The method of matched asymptotic expansions is applied and results in an extension, which is unexpectedly simple, of Birkhoff's integro‐differential equation for vortex sheets. The equation is applied to long waves on a straight vortex layer of uniform thickness.
Buckling of a Complete Spherical Shell Under Uniform External Pressure
Authors: Rauch Harry E. , Jacobs Neal H. , Marz Jonathan L.
Published: September 2015

Axisymmetric buckling of a complete spherical shell under uniform external pressure is treated by the spectral method. Among other results the lower critical pressures of 7% and 5% of classical linear buckling pressures are computed for radius‐thickness ratios of 100 and 200 respectively. Both symmetric (non‐physical) and non‐symmetric buckling are computed. The non‐symmetric buckled shapes at the two lower critical pressures mentioned are virtually identical.
A Model Equation to Study the Effects of Nonlinearity, Surface Tension and Viscosity in Water Waves
Authors: Ferguson Warren , Saffman Philip , Yuen Henry
Published: September 2015

The formation of short capillary waves on long, finite amplitude gravity waves is studied by solving numerically a non‐linear partial differential equation which models effects of surface tension, viscosity, unsteadiness and finite amplitude.
Simple Harmonic Generation: an Exact Method of Solution
Authors: Kaup D. J.
Published: September 2015

The general solution for simple harmonic generation is obtained by inverse scattering techniques. The time evolution of the scattering data is found to be much more complicated than in all other cases which are solvable by inverse scattering. It is also pointed out that the instability of the real solutions of simple harmonic generation is related to the singular nature of the scattering data about ζ = 0.
On the Location of the Maximum Stirling Number(s) of the Second Kind
Authors: Canfield E. Rodney
Published: September 2015

Let S(n, k) denote Stirling numbers of the second kind, and Kn be the integer(s) such that S(n, Kn) ⩾ S(n, k) for all k. We determine the value(s) of Kn to within a maximum error of 1.
Exact Solutions of Some Nonlinear Evolution Equations
Authors: Rosales Rodoljo R.
Published: September 2015

A different approach to finding solutions of certain diffusive‐dispersive nonlinear evolution equations is introduced. The method consists of a straightforward iteration procedure, which is carried to all terms, followed by a summation of the resulting infinite series. Sometimes this is done directly and other times in terms of inverses of operators in an appropriate space. We first illustrate the method with Burgers's and Thomas's equations, and show how it quickly leads to the Cole‐Hopf and Thomas transformations which linearize these equations. The method is described in detail with the Korteweg‐de Vries equation and then applied to the modified KdV, sine‐Gordon, nonlinear (cubic) Schrödinger, complex modified KdV and Boussinesq equations. In all these cases the multisoliton solutions are easily obtained, and new expressions for some of them follow. More generally, the Marčenko integral equations, together with the inverse problem that originates them, follow naturally from the approach. A method for modifying known solutions (in a way different from the known Backlund transformations) is also developed. Thus, for example, formulas for the interaction of solitons with an arbitrary given solution are obtained. Other equations tractable by this approach are presented. These include the vector‐valued cubic Schrödinger equation and a two‐dimensional nonlinear Schrödinger equation. Higher‐order and matrix‐valued equations with nonscalar dispersion functions are also included.
Permanent Wave Structures and Resonant Triads in a Layered Fluid
Authors: Bryant P. J. , Laing A. K.
Published: September 2015

A closed three layer fluid with small density differences between the layers has two closely related modes of gravity wave propagation. The nonlinear interactions between the wave modes are investigated, particularly the nearly resonant or significant interactions. Permanent wave solutions are calculated, and it is shown that a permanent wave of the slower mode can generate resonantly a wave harmonic of the faster mode. The equations governing resonant triads of the two modes are derived, and solutions having a permanent structure are calculated from them. It is found that some resonant triad solutions vanish when the triad is embedded in the set of all harmonics with wavenumbers in its neighborhood
Pairs of Adjacent Hamiltonian Circuits with Small Intersection
Authors: West Douglas B.
Published: September 2015

Consider the question: When can the edges in a pair of Hamiltonian circuits be redistributed to form another pair of circuits with the same union and intersection? A class of pairs is exhibited which intersect in two edges and cannot be rearranged in this way. A connection to algorithms for the traveling salesman problem is explained using the convex polytope of Hamiltonian circuits in . The exhibited pair is shown to be an edge of that polytope.
On the Multi‐soliton Solutions of Some Nonlinear Evolution Equations
Authors: Ma Yan‐Chow
Published: September 2015

We use the Hirota's technique to find the multi‐soliton solutions of some nonlinear evolution equations. The interaction of the solitons is studied.
Moments of Absorption Time for a Conditioned Random Walk
Authors: Beyer W. A. , Waterman M.S.
Published: September 2015

A random walk on the set of integers {0,1,2,...,a} with absorbing barriers at 0 and a is considered. The transition times from the integers z (0<z<a) are random variables with finite moments. The nth moment of the time to absorption at a, Dz,n, conditioned on the walk starting at z and being absorbed at a, is discussed, and a difference equation with boundary values and initial values for Dz,n is given. It is solved in several special cases. The problem is motivated by questions from biology about tumor growth and multigene evolution which are discussed.
Short Proof of a General Weight Burnside Lemma
Authors: Peck G. W.
Published: September 2015

A trace evaluation is shown to supply an elementary proof of a weighted Burnside lemma for weights not necessarily constant on orbits.
An Example of the Effect of Rescaling of the Reflection Coefficient on the Scattering Potential for the One‐Dimensional Schrodinger Equation
Authors: Moses H. E.
Published: September 2015

Let us consider the one‐dimensional Schrödinger equation with the scattering potential V(x)=2δ(x) and corresponding reflection coefficient b(k)=−i/(k + i). The potential satisfies a theorem of Deift and Trubowitz which states that non‐negative measurable potentials V(x) satisfying a certain range condition have reflection coefficients b(k) such that b(0)=−1. We rescale the reflection coefficient for V(x)=2δ(x) by writing b(k)=−iv/(k + 1) where 0<v<1. It is shown how V(x) changes with v, through the use of the Gelfand‐Levitan equation. This example illustrates how sensitive the potential is to rescaling of the reflection coefficient. In particular, the rescaling leads to a negative portion of V(x), as is expected from the Deift‐Trubowitz theorem. The example of this paper will be used in a later paper to illustrate the nature of bounds on potentials obtained through the use of variational principles.
Combinatorics of RNA Hairpins and Cloverleaves
Authors: Waterman Michael S.
Published: September 2015

For an abstract single‐stranded RNA, a combinatorial analysis is given for two important structures, hairpins and cloverleaves. The number of possible hairpins is O(2n), while the number of possible three‐petal cloverleaves is O(n32n), and the number of possible general cloverleaves is O((2.21)n).
Steady Gravity‐Capillary Waves On Deep Water—1. Weakly Nonlinear Waves
Authors: Chen B. , Saffman P. G.
Published: September 2015

The form of weakly nonlinear waves of permanent form on deep water is considered under the effect of both surface tension and gravity. The Wilton ripple phenomenon is reexamined and shown to be associated with a bifurcation in which a wave of permanent form can double its period. It is shown further that Wilton's ripples are a special case of a more general phenomenon in which bifurcation into subharmonics and fractional higher harmonics can occur.
Probabilities of Independent Choices Being Ordered
Authors: Shearer James , Kleitman Daniel J.
Published: September 2015

A simple proof of the following result of Baumert et al. is given. If A and B are chosen independently according to a probability distribution defined on subsets of an n element set, then the probability that A contains B is at least . It's shown to lead to a stronger conclusion, and to suggest definition of concepts of “partition number” and of “orthogonality of partitions into chains” as well as several conjectures.
Singularly Perturbed Semilinear Systems
Authors: Howes F. A.
Published: September 2015

Solutions of a singularly perturbed vector boundary‐value problem are studied under the principal assumption that the trivial solution of the unperturbed equation is stable in certain senses. This is accomplished by constructing special invariant regions in which solutions display the kind of nonuniformity known as boundary‐layer behavior.
Resonance in a Weakly Nonlinear System with Slowly Varying Parameters
Authors: Kevorkian J.
Published: September 2015

We survey multiple‐variable expansion procedures appropriate for nonlinear systems in resonance using the model of two coupled weakly nonlinear oscillators with either constant or slowly varying frequencies. In the autonomous problem we show that an n‐variable expansion (where n depends on the order of accuracy desired) yields uniformly valid results. We also consider the problem of passage through resonance for the nonautonomous problem and describe the solution by constructing a sequence of three expansions. The solution before resonance is developed as a generalized multiple‐variable expansion and is matched with an inner expansion valid during resonance. This latter is then matched with a postresonance solution and determines it completely. Numerical integrations are used to substantiate the theoretical results. The dominant effect of passage through resonance is shown to be the excitation of a higher‐order oscillation beyond resonance. Contrary to the claim in a recent work, the total action of the system does not remain constant if one accounts for the leading perturbation terms in the postresonance solution. Instead, the total action goes from one constant value to another.
On the Inverse Scattering Problem for Cubic Eigenvalue Problems of the Class ψxxx + 6Qψx + 6Rψ = λψ
Authors: Kaup D. J.
Published: September 2015

The inverse scattering problem for cubic eigenvalue equations of the form ψxxx + 6Qψx + 6Rψ = λψ is outlined and formally solved. Many properties of the scattering data are obtained, the continuous spectrum is briefly discussed, special one soliton solutions are obtained, and the infinity of conserved quantities are determined in terms of the scattering data.
A Birkhoff Theorem for Doubly Stochastic Matrices with Vector Entries
Authors: Clapp M. H. , Shiflett R. C.
Published: September 2015

The concept of a doubly stochastic matrix whose entries come from a convex subset of the unit square is defined. It is proved that the only convex subsets of the unit square which contain (0,0) and (1, 1) and allow an extension of Birkhoff's characterization of the extreme points of the set of doubly stochastic matrices are parallelograms. A sufficient condition is given for a matrix to be extreme when the convex subset is not a parallelogram.
Periodically Perturbed Bifurcation—1. Simple Bifurcation
Authors: Rosenblat S. , Cohen Donald S.
Published: September 2015

We consider the effect of a periodic perturbation on the bifurcation behavior of a system of differential equations. It is shown that constant solutions are modified into periodic solutions, and that the perturbations leads to the separation of solution branches. Under certain conditions the system may have no solution in the neighborhood of the bifurcation point.
Constant Sustainable Consumption Rate in Optimal Growth with Exhaustible Resources
Authors: Wan Frederic Y. M.
Published: September 2015

The problem of optimal growth with an exhaustible resource deposit under R. M. Solow's criterion of maximum sustainable consumption rate, previously formulated as a minimum‐resource‐extraction problem, is shown to be a Mayer‐type optimal‐control problem. The exact solution of the relevant firstorder necessary conditions for optimality is derived for a Cobb‐Douglas production function, whether or not the constant unit resource extraction cost vanishes. The related problem of maximizing the terminal capital stock over an unspecified finite planning period is investigated for the development of more efficient numerical schemes for the solution of multigrade‐resource deposit problems. The results for this finite‐horizon planning problem are also important from a theoretical viewpoint, since they elucidate the economic content of the optimal growth paths for infinite‐horizon problems.
A Note on Nonbreaking Waves in Hyperelastic Materials
Authors: Lustman Liviu
Published: September 2015

This note treats the evolution of waves on hyperelastic materials, due to initial jump discontinuities in the gradient of strain. In general, these discontinuities become unbounded in finite time, leading to discontinuous strain. There are, however, certain cases in which the gradient jump remains finite for all times. We show here that the class of materials admitting such exceptional waves is fairly large, including Hadamard materials and generalized Hooke materials. An earlier example of Jeffrey and Teymur [6] is also discussed and reset in a more general framework.
The Rolling Motion of A Viscous Fluid On and Off a Rigid Surface
Authors: Benney D. J. , Timson W. J.
Published: September 2015

An analytical solution is obtained for the local flow field near a point of steady attachment to or detachment from a rigid boundary. The possible free surface shapes depend on the capillary number.
Transient Solutions to Some Weakly Diffusive Nonlinear Diffusion Equations
Authors: Lange C. G. , Larson D. A.
Published: September 2015

A multiple time scale procedure is developed and used to describe the full transient behavior of solutions to certain classes of nonlinear diffusion initial value problems where the diffusion coefficients are in a certain sense “weak.” Specific examples are presented which demonstrate the explicit manner in which various traveling wave and oscillation patterns develop from arbitrary initial data for certain well‐known model systems in population biology and chemical kinetics.
On the Partial Sums of the Rearrangements of a Complex Series
Authors: Kleitman D. J. , Li Shuo-Yen Robert
Published: September 2015

Let {zj} be a set of complex numbers, all with magnitude ⩽ 1, which sum to a real number ε⩾1. Then the zj's can be rearranged so that all partial sums are no greater in magnitude than (ε2 + 1)1/2, and this bound is best possible. Bergström showed that if , the best bound is .
On the Wetting of a Surface by a Very Viscous Fluid
Authors: Greenspan H. P. , McCay B. M.
Published: September 2015

The movement of a small viscous droplet on a flat surface is considered in self‐spreading, in roundup, and as a result of an adhesion gradient. Edge conditions based on the dynamic contact angle are discussed within the context of lubrication theory. Tangential slip at the contact line is shown to be small compared to the normal velocity. The shapes of the droplet and its contact line are calculated, and nonlinear and inertial effects assessed.
Asymptotic Limits for a Two‐Dimensional Recursion
Authors: Laquer H. Turner
Published: September 2015

In this paper we consider the general two‐dimensional recursion g(n + 1, r + 1) = g(n + 1, r) + bg(n, r) + g(n, r + 1) with boundary conditions g(n, 0) = g(0, r) = 1 We develop various exact and asymptotic formulas for the g(n, r).
The Theory of Constructive Signal Analysis
Authors: Benedetto John J.
Published: September 2015

Let h(x) = e−αxk(x), where and λ0=0. The closure theorem, Vh = L1(ℝ), is proved for various α and k (Vh is the L1‐closed variety generated by h). The Tauberian condition, |ĥ| > 0, is not used, since generally this condition is difficult to compute directly. The functions h arise naturally in time series and analytic number theory. The technique of proof is constructive and depends on the semigroup {γj} generated by {λj}. The semigroup theory which consolidates and completes the results herein will be developed separately as “A closure problem for signals in semigroup invariant systems.”
A Monotonicity Property of Partial Orders
Authors: Kleitman D. J. , Shearer J. B.
Published: September 2015

A proof using the FKG inequalities of the following result is obtained. Let P be a partially ordered set on a1 ⩽ a2 ⩽ ⋯ ⩽ am and b1 ⩽ b2 ⩽ ⋯ ⩽ bn. Let P(x) be the proportion of linear extentions of P for which x holds. If x and y are disjunctions of conjunctions of additional inequalities of the form ai ⩾ bj, then P(x and y) ⩾ P(x)P(y). An example is provided that shows the result can be false if we don't assume the {ai} and {bj} are linearly ordered in P.
Harmonic Analysis and Expansion Formulas for Two‐Variable Hypergeometric Functions
Authors: Kalnins E. G. , Manocha H. L. , Miller Willard
Published: September 2015

The authors show how their Lie‐theoretic characterization of two‐variable hypergeometric functions can be employed to derive expansion theorems involving these functions. In particular, the functions arise as solutions of the Laplace, wave, heat, Helmholtz, and Schrooinger equations, and new bases can be constructed from the functions with which to expand general solutions of these physically important equations.
Note on Asymptotic Solutions of the Korteweg‐de Vries Equation with Solitons
Authors: J. Ablowitz Mark , Kodama Yuji
Published: September 2015

The long time asymptotic solution of the Korteweg‐de Vries equation containing both solitons and the dispersive wavetrain is described. It is shown that a soliton interacts elastically both with the dispersive wavetrain and with other solitons. An explicit formula for the phase shift produced upon interaction is given. These ideas also apply to other nonlinear evolution equations solvable by the inverse scattering transform.
The Eigenvalues of the Angular Spheroidal Wave Equation
Authors: Hunter C. , Guerrieri B.
Published: September 2015

Approximate relations are obtained between the eigenvalues λ and the ellipticity parameter c2 of the angular spheroidal wave equation. Although based on WKBJ methods and the assumption that λ is large, the relations are useful throughout the complex c2‐plane. They are exact at c2 = 0, and reproduce the standard asymptotic formulas for λ when c2 is large. At intermediate values of c2, they provide approximations for the square‐root branch points of the multivalued function λ(c2) in the complex c2‐plane at which adjacent eigenvalues of the same class become equal in pairs. These branch points lie on an infinite sequence of distorted circular rings. Their exact locations have been computed for the first four rings for angular wavenumbers m = 0,…,4.
A Nonlinear Theory for the Generation of Water Waves by Wind
Authors: Akylas T. R.
Published: September 2015

A viscous boundary‐layer flow, first proposed by Lock as a model for the initiation of water waves by wind, is studied. A stability analysis of the flow indicates that the linear instability present is relatively weak. However, under certain conditions, the system can support coalescing normal modes, and thus, direct resonance between such modes is possible. A nonlinear perturbative analysis at direct resonance is performed, and the equation governing the slow temporal evolution of a finite‐amplitude oscillatory disturbance is derived. The appropriate evolution equation is of second order in time. The nonlinear effects are of crucial importance in the development of a resonant disturbance. Comparison of the theoretical predictions with experiments suggests that direct resonance may play a significant role in explaining the observed large initial growth rates of gravity water waves of wavelength longer than about 12 cm.
Barring the Z Pentomino from the Infinite Checkerboard
Authors: Shearer James B.
Published: September 2015

We show that the minimum fraction of squares which must be blocked off to bar the Z pentomino from the infinite checkerboard is ⅓.
Waiting‐Time Behavior in a Nonlinear Diffusion Equation
Authors: Kath William L. , Cohen Donald S.
Published: September 2015

We study the nonlinear diffusion equation ut*=(unux)x, which occurs in the study of a number of problems. Using singular‐perturbation techniques, we construct approximate solutions of this equation in the limit of small n. These approximate solutions reveal simply the consequences of this variable diffusion coefficient, such as the finite propagation speed of interfaces and waiting‐time behavior (when interfaces wait a finite time before beginning to move), and allow us to extend previous results for this equation.
Barring Hexominoes from the Infinite Checkerboard
Authors: Shearer James B.
Published: September 2015

In this paper we find the minimum fraction of squares which must be blocked off to bar each of the 35 hexominoes from the infinite checkerboard.