bluuebunny/crossref_metadata_2025 huggingface dataset:
Some Properties of Long Nonlinear Waves
Authors: Benney D. J.
Published: September 2015

This paper is concerned with certain mathematical aspects of long nonlinear wave propagation on a free surface. Some special solutions are noted and the equations are shown to have an infinite number of conservation laws.
Ising Chain with Random Magnetic Moments
Authors: Barouch E. , Rogers S. R.
Published: September 2015

The Ising chain with random magnetic moments is discussed, and the thermodynamic functions are computed.
Proportions of Irreducible Diagrams
Authors: Hsieh W N.
Published: September 2015

We investigate the proportion of graphs on n‐vertices each of which is m‐valent, that are “k‐reducible”. An asymptotic formula is developed for large n and fixed values of the other variables, in terms of an n‐independent parameter that is evaluated for a number of (m, k) values.
Separation of Eigenvalues of the Wave Equation for the Unit Ball in RN
Authors: Graham Keith D.
Published: September 2015

It is shown that π is the infinium gap between the consecutive square roots of the eigenvalues of the wave equation in a hypespherical domain for both the Neumann (free) and the full range of mixed (elastic) homogeneous boundary conditions. Previous literature contains the same information apparently only for the Dirichlet (fixed) boundary condition. These square roots of the eigenvalues are the zeros of solutions of a differential equation in Bessel functions (first kind) and their first derivatives. The infinium gap is uniform for Bessel functions of orders x ≥ ½ (as well as for x = 0). The intervals between the roots are actually monotone decreasing in length. These results are obtained by interlacing zeros of Bessel and associated functions and comparing their relative displacements with oscillation theory. If Wl denotes the lth positive root for some fixed order x, the minimum gap property assures that {exp(±iwlt|l = 1, 2,...} form a Riesz basis in L2(0, τ) for τ > 2. This has application to the problem of controlling solutions of the wave equation by controlling the boundary values.
On α‐dynamos
Authors: Greenspan H. P.
Published: September 2015

The function α that describes the generation of large scale magnetic fields from small scale motions is assumed to be non‐zero only in a thin boundary layer. As a consequence, the problems for the poloidal and meridional fields in a symmetric rotating configuration decouple and may be solved sequentially. Analysis of the boundary layer yields equivalent boundary conditions on interior and exterior field components. The general eigenvalue problem is formulated but only the case of neutral stability is examined.
Spin‐Down in a Rotating Stratified Fluid: Part I
Authors: Friedlander Susan
Published: September 2015

Stress driven flow of a stratified rotating fluid is studied in the context of a linearized Boussinesq cylindrical model, with particular interest in the long term quasi‐steady state. The importance of the non‐linear terms is estimated by considering a numerical model for the full non‐linear equations.
On the Computation of Powers of Sparse Polynomials
Authors: Fateman Richard J.
Published: September 2015

This paper examines the time for computing integer powers of a class of polynomials in one or several variables. A member f of the class of “sparse” polynomials can be characterized by the number of nonzero terms, t, in the representation of f. Where, for power 4, the most obvious methods result in

multiplications, a method is presented which requires fewer than

multiplications.

This algorithm requires, for t » n » 1 about tn/n! + O(tn − 1 /(2(n − 2)!)) multiplications, which is proved to be a lower‐bound on the computation.
Dynamo Action of Isotropically Driven Motions of a Rotating Fluid
Authors: Gubbins David
Published: September 2015

Fluid motions are driven by a random isotropic body force in a rotating system. The fluid is incompressible and infinite in extent. The dynamo action of the resulting fluid motions is investigated by Fourier decomposition. The global effects of the turbulent motions are expressed in terms of gradients of the local mean magnetic field, Bo. Two aspects of the problem are novel. The spectral approach is used to solve for the turbulent quantities while gradients of Bo are retained, and secondly a non‐zero mean emf. is obtained from an entirely isotropic forcing.
On a Lorentz Transformation Suitable for Local Problems in General Relativity
Authors: Greenberg Philip J. , Knauer James P.
Published: September 2015

In this paper we derive a six‐parameter Lorentz transformation which is suitable for investigating local problems in general relativity. These six parameters appear manifestly in the equations describing the transformation. In addition, the six parameters are related to the conventional parameters of the Lorentz transformation, namely, three components of the relative velocity between two observers, and three angles which describe a rotation of axes, and which we take to be Eulerian angles.
Uniform Asymptotic Approximations to the Solutions of the Orr‐Sommerfeld Equation Part 2. The General Theory
Authors: Reid W. H.
Published: September 2015

This paper deals with the derivation of “first approximations” to the solutions of the Orr‐Sommerfeld equation which are uniformly valid in a full neighborhood of the critical point. To this order the theory is remarkably simple. The essential elements in the theory are all well known from the older heuristic theories but its general structure is substantially different. The uniform approximations are also vastly simpler than the composite approximations obtained recently by the method of matched asymptotic expansions.
The Inverse Scattering Transform‐Fourier Analysis for Nonlinear Problems
Authors: Ablowitz Mark J. , Kaup David J. , Newell Alan C. , Segur Harvey
Published: September 2015

A systematic method is developed which allows one to identify certain important classes of evolution equations which can be solved by the method of inverse scattering. The form of each evolution equation is characterized by the dispersion relation of its associated linearized version and an integro‐differential operator. A comprehensive presentation of the inverse scattering method is given and general features of the solution are discussed. The relationship of the scattering theory and Backlund transformations is brought out. In view of the role of the dispersion relation, the comparatively simple asymptotic states, and the similarity of the method itself to Fourier transforms, this theory can be considered a natural extension of Fourier analysis to nonlinear problems.
On the Modulations and Nonlinear Interactions of Waves and the Nonlinear Stability of a Near‐Critical System
Authors: Kennett Rosemary G.
Published: September 2015

The nonlinear interactions and modulations of an n‐dimensional wave and of a disturbance to a near‐critical system governed by a general (n + 1)‐dimensional system of equations are studied by perturbation methods. It is found that these modulations are governed by an evolution equation which is either by itself or coupled to a second equation, depending on the nature of the long wave solutions of the corresponding linearized system. When a single evolution equation exists, its leading terms are shown to give the nonlinear Schrödinger equation. Water waves and near‐critical plane Poiseuille flow are discussed as examples.
On the Existence and Formulation of Variational Principles for Nonlinear Differential Equations
Authors: Atherton R. W. , Homsy G. M.
Published: September 2015

The inverse problem of the calculus of variations, i.e., the existence and formulation of variational principles for systems of nonlinear partial differential equations, is treated by application of a theorem due to Vainberg. Three types of principles are discussed: potential, alternate potential, and composite principles. Potential principles are those for which the equations, as written, admit a variational formulation. Alternate potential principles are those for which the equations admit a variational formulation only after a differential transformation of variables. Composite principles are those in which, in addition to the original variables, a set of adjoint variables is defined. A composite principle may always be formulated. Consistency relations for the existence of a potential are formulated for general systems of nonlinear partial differential equations of any order. These conditions are quite stringent and severely restrict the class of operators which admit a variational formulation. The results of the development are used to treat many well‐known equations in nonlinear wave mechanics. These include the Burgers, Korteweg‐deVries, and sine‐Gordon equations and evolution and model equations for interfacial waves. The Navier‐Stokes equations and the Boussinesq equations for an infinite Prandtl number fluid are briefly treated; these latter equations do not possess a potential principle.
An Inequality for the Möbius Function of a Geometric Lattice
Authors: Greene Curtis
Published: September 2015

An inequality is derived for the Möbius function of a finite geometric lattice L. Equality is characterized by the modularity of certain elements of L. Applications are given to other inequalities involving Whitney numbers, ordered bases, and maximal chains.
Multiple Steady States in a Model Biochemical Reaction
Authors: Boa James A.
Published: September 2015

Phase plane techniques are used to find finite amplitude solutions of a nonlinear second order boundary value problem. Each of these solutions represents a steady state of a model biochemical reaction proposed by Prigogine, in a limiting value of a parameter.
A Universal Measure
Authors: Graves William H. , Molnar Suzanne M.
Published: September 2015

It is shown that for any ring R of sets there is a locally convex Hausdorff topological vector space M(R) and a measure χ : R → M(R) such that for any other locally convex Hausdorff topological vector space W and any measure µ : R → W, there exists a unique continuous linear map µ : M(R) → W such that
. Measure is defined in such a way that M(R)′, the space of all continuous linear functionals on M(R), corresponds to all finite, signed measures in the usual sense when R is a σ‐ring. The problem of extending measures from R to the σ‐ring generated by R is formulted in this setting. Properties of the universal measure χ are described. The topology of M(R) is studied via the pairing . For example, the weak compact subsets of M(R)′ are characterized. This characterization gives rise to a simple proof of the Orlicz‐Pettis theorem.
An Affine Representation for Transversal Geometries
Authors: Brylawski Thomas H.
Published: September 2015

Pregeometries (matroids) whose independent sets are the partial matchings of a relation (transversal pregeometries) can be canonically imbedded in a free‐simplicial pregeometry (one whose points lie freely on flats spanned by a simplex). Conversely, all subgeometries of such free‐simplicial pregeometries are transversal. Free‐simplicial pregeometries are counted and their duals are naturally constructed and shown to be free‐simplicial (showing self‐dual free‐simplexes corrspond to quasisymmetric relations). For more general transversal pregeometries, modular flats are characterized and transversal contractions are exemplified. Binary transversal pregeometries and their contractions (the class of binary gammoids) are shown to be the class of series‐parallel networks, providing insight for further characterizations of (coordinatized) gammoids by excluded minors. Theorem. All principal transversal pregeometries and their truncations have critical exponent at most 2.
The Wiener Spectrum in Spectral Synthesis
Authors: Benedetto John J.
Published: September 2015

Let E be a closed subset of a locally compact Abelian group Г and let k(E) be the space of absolutely convergent Fourier transforms on Г that vanish on E. (1) k(E) is characterized as an ideal of arithmetic means defined by the behavior of pseudo‐measures at infinity; and so spectral synthesis holds for an ideal I if I and its corresponding ideal of arithmetic means coincide. (2) We then prove that it is possible to synthesize pseudomeasures at infinity in a manner that runs parallel to Beurling's synthesis of weighted spaces. These two results are closely related by the notion of Wiener's spectrum.
On the Multiplicity of Solutions Of a Differential Equation Arising in Chemical Reactor Theory
Authors: Parter Seymour V. , Stein Myron L. , Stein Paul R.
Published: September 2015

Consider the boundary value problem
urn:x-wiley:00222526:media:sapm1975544293:sapm1975544293-math-0001
where β ⩾ 0, τ ⩾ 0. We are concerned with a mathematically rigorous numerical study of the number of solutions in any bounded portion of the positive quadrant (τ ⩾ 0, β ⩾ 0) of the τ, β plane. These correct computational results may then be matched with asymptotic (β→∞, τ ⩾ 0) results developed earlier. These numerical results are based on the development of a posteriori error estimates for the numerical solution of an associated initial‐value problem and a priori bounds on
.
Extensions of the Erdös‐Ko‐Rado Theorem
Authors: Greene C. , Katona G. O. H. , Kleitman D. J.
Published: September 2015

An antichain of subsets of 1,2,...,n has the Erdös‐Ko‐Rado property if |Ai|⩽n/2 and Ai∩Aj≠Ø(i=j). This paper contains a number of results concerning the distribution of sizes of sets in such a family, and also in families where the restriction |Ai|⩽n/2 is removed.
Turbulence Model Predictions for the Inhomogeneous Mixing Layer
Authors: Milinazzo F. , Saffman P. G.
Published: September 2015

The plane mixing layer between two streams of different densities and velocities is studied using a complete, phenomenological model of turbulence. The model is an extension of that described by Saffman [11]. The properties of similarity solutions are computed, and compared with experimental data. The qualitative dependence on velocity ratio, density ratio, compressibility and adverse pressure gradient is predicted correctly. The quantitative agreement is satisfactory.
A Model for Magnetohydrodynamic Convection Relevant to the Solar Dynamo Problem
Authors: Kennett Rosemary G.
Published: September 2015

An extension of the minimal convective system of Lorenz, Howard and Malkus to three‐dimensional motions in an electrically conducting fluid is presented. It is shown that at conditions relevant to the sun this system exhibits linearly stable periodic solutions with a non‐zero magnetic field, which are suggestive of dynamo action.
Construction of a Maximally Edge‐Connected Graph with Prescribed Degrees
Authors: Wang D. L.
Published: September 2015

Let 〈d1,d2,...,dp,〉 be a realizable degree sequence, di⩾2; then a graph G can be constructed so that deg(vi and so that for i≠j, the number of edge‐disjoint paths between vi and vj is (di,dj).
Nonlinear Theory of a Caustic Region
Authors: Bobbitt J. I. , Cumberbatch E.
Published: September 2015

The caustic formed when a water wave is propagating at incidence into increasing depth is considered first in the linear approximation. A scheme for a nonlinear approach is indicated by this analysis, and the nonlinear equations valid in the caustic region are obtained. Comparison with the gas‐dynamics case shows differences from the equations adopted for the sonic‐boom caustic.
Asymptotically Equivalent Singular Perturbation Problems
Authors: Rosenblat S.
Published: September 2015

A study is made of a two‐point nonlinear boundary‐value problem with a small parameter multiplying the highest derivative. It is shown that under certain circumstances the asymptotic solution to the problem is expressible in terms of the solution to a linear boundary‐value problem—in which case the two problems are said to be asymptotically equivalent. The coefficients of the linear problem necessarily satisfy certain conditions, and these conditions are shown to bear a close relationship to the equations obtained in constructing a solution to the nonlinear problem by standard matching methods.
On Internal Resonance of Two Damped Oscillators
Authors: Miles John W.
Published: September 2015

The damped motion of two oscillators with natural frequencies ω1 and ω2 is studied on the assumptions that: the oscillators are uncoupled for infinitesimal displacements and quadratically coupled for finite displacements; the frequencies are approximately in the ratio 2:1, such that ω2−2ω1=O(εω1), where ε is a dimensionless measure of the displacements; the logarithmic decrements of the two modes, λ1 and λ1, are O(ε). The motion may be described by modulated sine waves with carrier frequencies ω1,2 and slowly varying energies and phases that satisfy four first‐order, nonlinear differential equations. These equations admit one invariant and may be reduced to two first‐order equations if ω2=2ω1 and λ1,2<0; they admit two invariants and can be completely integrated in terms of elliptic functions if ω2=2ω1 and 2λ2=λ1. Numerical results are presented for typical parametric combinations.
On the Geometry of Dual Pairs
Authors: Cain B. E. , Saunders B. D. , Schneider Hans
Published: September 2015

The set of dual pairs of any norm v equivalent to a Hilbert norm is shown to be naturally homeomorphic to the sphere of the Hilbert space. The proof begins with a known result showing the representability of every vector as a sum of two orthogonal vectors, one coming from a cone and the other from its dual (a generalization of representation by orthogonal subspaces). The key theorem, showing that every non‐zero vector has a positive multiple which is the sum of two v‐dual vectors, follows from this and in turn provides the required homeomorphism. One consequence of this topological equivalence is the arc‐connectedness of the numerical range determined by v.
A Boolean Analysis of Addition and Multiplication
Authors: Faltin F. , Metropolis N. , Ross B. , Rota G.-C.
Published: September 2015

The notions of binary string and binary symmetric function are introduced and basic results presented. Boolean algorithms are given for binary addition and multiplication. An analysis of the redundancies involved is straightforward. The examination of carry propagation which arises in the Boolean analysis of functions may lead to a new interpretation of the notion of computational complexity.
Resonance in a Quantum Chain
Authors: Barouch Eytan
Published: September 2015

A resonance of a magnetic impurity with a quantum spin‐chain is studied. A Fresnel diffraction pattern is found.
Slowly Varying Waves and Shock Structures in Reaction‐Diffusion Equations
Authors: Howard L. N. , Kopell N.
Published: September 2015

Solutions of reaction‐diffusion equations which, locally in space and time, are close to plane wave trains are investigated. Apart from certain exceptional (and localized) regions, such solutions are approximately described by solutions of a Hamilton‐Jacobi equation obtained from the properties of ideal plane waves. This provides a fairly simple description of many features of the initial value problem, but implies that transition layers in which the approximate description is invalid will generally arise. The structure of these layers, which are mathematically analogous to gas‐dynamic shocks, is investigated on the basis of the full equations.
On Asymptotic Expansions and Error Bounds in the Derivation of Two‐Dimensional Shell Theory
Authors: Nair S. , Reissner E.
Published: September 2015

Known results on asymptotic two‐dimensional equations for circular cylindrical shells, including the effects of transverse shear and normal stress deformation, are supplemented by upper‐ and lower‐bound determinations of influence coefficients, using minimum‐potential and complementary energy principles in conjunction with asymptotic‐expansion results. The new bound analysis shows that the consequences of the asymptotic two‐dimensional theory are in exact agreement, except for terms which are small of higher order, with the corresponding consequences of three‐dimensional theory, for some classes of edge conditions. The analysis also shows the nature of the differences between results of two‐ and three‐dimensional theory, as a function of geometrical and elastic parameters, where this difference is of importance because of the effect of a St. Venant boundary layer.
Perturbation and Asymptotic Solutions for Problems in the Theory of Urban Land Rent
Authors: Wan Frederic Y. M.
Published: September 2015

Simple but accurate approximate solutions for R. M. Solow's theory of urban land rent with congest cost of transportation are obtained by perturbation and asymptotic methods. Only the case of an absentee landlord is discussed though the techniques used can be modified in a straightforward manner to handle the more interesting (and mathematically more complicated) case of municipal ownership. Solow's model is also extended to have expenditures for housing and space as separate items in the budget equation. This extension eliminates the ambiguity in the value for the fraction of income after travel cost spent on ground rent, without affecting the structure of the mathematical problems associated with Solow's original theory.
Rook Theory—IV. Orthogonal Sequences of Rook Polynomials
Authors: Goldman Jay R. , Joichi J. T. , White Dennis E.
Published: September 2015

We explain combinatorially the occurrence of certain classical sequences of orthogonal polynomials as sequences of rook polynomials, and we give some new examples related to general stairstep boards.
A Divisibility Problem
Authors: Lebensold Kenneth
Published: September 2015

We discuss the question of the size and composition of the largest subset Q of N = f 1, 2, ..., nf so that no one divides two others. We find that, as n→, 0.6725n ⩽ |Q| ⩽ 0.6736n, with more accurate bounds achievable by additional computation.
On the Solution of a Class of Nonlinear Partial Difference Equations
Authors: Ablowitz M. J. , Ladik J. F.
Published: September 2015

A systematic method for isolating certain nonlinear partial difference equations which are solvable by the method of inverse scattering is discussed. The equations are characterized by the dispersion relation (amplification factor) of the associated linearized equation. Corresponding to every neutrally stable linear scheme is a nonlinear difference equation which converges to an exactly solvable nonlinear evolution equation in the continuum limit. Explicit solition solutions are given.
Singularly Perturbed Nonlinear Boundary‐Value Problems Whose Reduced Equations Have Singular Points
Authors: Howes F. A.
Published: September 2015

Using results on implicit first‐order differential equations, the existence and the asymptotic behavior of solutions of singularly perturbed second‐order boundary‐value problems whose reduced equations have singular points are studied. The variety of possible nonuniform behavior is investigated by means of a stability analysis of the reduced solutions and is illustrated with several model problems which are canonical in terms of the first‐order singular‐point theory.
Nonlinear Transition Layers—The Second Painleve Transcendent
Authors: Haberman Richard
Published: September 2015

We investigate a large class of weakly nonlinear second‐order ordinary differential equations with slowly varying coefficients. We show that the standard two‐timing perturbation solution is not valid during the transition from oscillatory to exponentially decaying behavior. In all cases this difficulty is remedied by a nonlinear transition layer, whose leading‐order character is described by one special nonlinear differential equation known as the second Painlevé transcendent (in essence a nonlinear Airy equation). The method of matched asymptotic expansions yields the desired connection formula. The second Painlevé transcendent also provides two other types of transitions: (1) between weakly nonlinear solutions (either oscillatory or exponentially decaying) and special fully nonlinear solutions, and (2) between two of these special nonlinear solutions. These special solutions are of three: different kinds: (a) slowly varying stable equilibrium solutions, (b) “exploding” solutions, and (c) solutions depending on both the fast and slow scales (which emerge from the unstable zero equilibrium solution).
A Simultaneous Enumeration Approach to the Traveling Salesman Problem
Authors: Claus A.
Published: September 2015

The solution procedure proposed in this paper uses certain principles of analog computers. The idea of using analog rather than digital computers to solve mathematical programming problems is not new—various methods have been proposed to solve linear programming, network flows, as well as shortest path problems (Dennis, 1959; Stern, 1965). These problems can be more efficiently solved with digital computers. To find a solution to the traveling salesman problem as well as other integer programming problems is difficult with existing hardware, especially if the number of variables is large. The question thus arises whether different hardware configurations make it possible to solve integer problems more efficiently. One such configuration is proposed below for the traveling salesman problem.
A Model Nonlinear Problem Having a Continuous Locus of Singular Points
Authors: Howes F. A. , Parter S. V.
Published: September 2015

The asymptotic behavior of solutions of a singularly perturbed boundary value problem whose reduced equation has continuous bran,ches of singular points is investigated. In particular, unexpected limiting values of solutions for certain choices of the boundary conditions are found by means of a priori estimates.
The Alpha Function of a Matroid—I. Transversal Matroids
Authors: Kung Joseph. P. S.
Published: September 2015

Let n be the nullity function of the matroid G(S). The Mason alpha function is defined on subsets A of S by the recursion α(A)=n(A)−∑F⊂Aα(F), the summation being over all flats F strictly contained in A. The alpha function may be viewed as the first difference of the nullity. We study the behavior of a under strong maps, and apply our results to proving Mason's alpha criterion: a matroid is the dual of a transversal matroid if and only if its alpha function is non‐negative.
Some Optimization Problems with Bulk‐Service Queues
Authors: Barnett Arnold , Kleitman Daniel J.
Published: September 2015

An optimal control problem for a single bulk‐service queue is considered. Theorems that characterize the optimal control strategy in some circumstances are proved; they complement earlier results by Deb and Serfozo for a comparable problem. Most of the analysis uses direct probabilistic reasoning rather than specialized methods such as dynamic programming. Some illustrative examples are presented.
Activators and Inhibitors in Pattern Formation
Authors: Keener James P.
Published: September 2015

A wide class of diffusion reaction equations with activator‐inhibitor reaction dynamics are studied analytically. It is shown that when the inhibitor diffuses much faster than the activator, patterns with large amplitude nonuniformities may exist. It is further shown that with a slightly nonuniform source of activator, the patterns which emerge from the time dependent process have a directional preference. The results given coincide with the numerical results of Gierer and Meinhardt [6].
The Evolution of a Nonlinear Critical Level
Authors: Warn T. , Warn H.
Published: September 2015

The evolution of inviscid forced Rossby waves on a parallel flow in the presence of a critical layer is discussed. It is shown that the transient critical layer becomes nonlinear after sufficient time has elapsed and that the “− π” logarithmic phase shift is modified accordingly. All harmonics are found to be important in the nonlinear range, even outside the critical layer.
Invariances and Conservation Laws of the Korteweg‐de Vries Equation
Authors: Wadati Miki
Published: September 2015

Taking the Korteweg‐de Vries equation as an example of a soliton system, a connection between conservation laws and infinitesimal transformations (symmetries) is investigated. It is shown that there exist an infinite number of form‐invariant infinitesimal transformations. A one‐to‐one correspondence between conservation laws and infinitesimal transformations is established explicitly. Then it is concluded that the Korteweg‐de Vries equation has an infinite number of conservation laws, each of which corresponds to the symmetry of the system. Two methods of weaving an infinitesimal transformation into a finite one are presented; one of them is the extension of the Lie transformation to the nonlinear case, and the other is a transformation reducible to the Backlund transformation. Furthermore, it is found that Hamiltonian formalism naturally leads to the quantization of the wave field. During the discussions, a number of conjectures and theorems which have been proposed in earlier works are confirmed by using the infinitesimal transformation extensively.
Hermitian Forms and Eigenvalue Bounds
Authors: Warren F. W.
Published: September 2015

A method of quadratic forms for eigenvalue bounds described in an earlier paper is developed and extended. In particular it is shown how restrictions upon eigenvalues may be found for a class of linear partial differential equations whose coefficients depend on one space variable and which are also periodic in time. An application is made to the problem of linear stability of periodic helical flows in a pipe. It is shown that the motion is stable if a criterion of the form
urn:x-wiley:00222526:media:sapm1978593249:sapm1978593249-math-0001
is satisfied. Ωn(r) is the Fourier coefficient of the nth harmonic of the periodic angular velocity Ω(r), and Wn(r) is the corresponding coefficient of the axial velocity. b is the radius of the pipe, and R0 is about 4.4. Stronger but less simple results are possible. Bounds for steady inviscid flows are also given.
Global Dispersion Relation for Density Waves In a Certain Simplified Model of a Disk Shaped Galaxy
Authors: Nishimoto Toshihiko
Published: September 2015

For density waves in a certain simplified model of a disk shaped galaxy, the dominant term of the basic equation (governing density waves) may be represented by a cubic polynomial, in which the stability parameter Q is allowed to be somewhat less than unity near corotation. For such a differential equation, an asymptotic form of the global dispersion relation is presented. It is shown that there exist discrete complex roots of the dispersion relation with small negative imaginary parts. The real parts and the imaginary parts of these roots represent approximately the angular speeds and the growth rate of the amplitudes of the density waves, respectively. It is proved that there exist only a finite number of unstable normal modes of density waves.
Path‐Connected Partially Ordered Sets
Authors: Belding Russell
Published: September 2015

The graph of a partially ordered set (X, ⩽) has X as its set of vertices and (x,y) is an edge if and only if x covers y or y covers x. The poset is path‐connected if its graph is connected. Two integer‐valued metrics, distance and fence, are defined for path‐connected posets. Together the values of these metrics determine a path‐connected poset to within isomorphism and duality. The result holds for path‐connected preordered sets where distance and fence are pseudometrics. The result fails for non‐path‐connected posets.
Asymptotic Methods for Bifurcation and Stability Problems
Authors: Rosenblat S.
Published: September 2015

A method of asymptotically determining the bifurcating solutions of a nonlinear eigenvalue problem is described. The method is based on the smallness of a parameter which is different from the usual parameter used in the LyapunovSchmidt procedure. A discussion of the stability and evolution of bifurcating solutions is included. It is shown how the method may be useful for determining secondary bifurcations and turning points on bifurcation curves.
A New Model for Nonlinear Wind Waves. Part 2. Theoretical Model
Authors: Yuen Henry C. , Lake Bruce M.
Published: September 2015

A first‐order theory for the development of a wind‐driven wave field is proposed. This takes advantage of the physical interpretation presented by Lake and Yuen (1978) that a nonlinear wind‐wave system can be completely characterized, to first approximation, by a single nonlinear wave train with carrier frequency coinciding with the local dominant frequency. The result is the nonlinear Schrodinger equation for the wave envelope, supplemented by an equation describing the change of dominant‐wave frequency (or wavenumber) with fetch (or duration). The latter equation is a consequence of a form‐drag model proposed by Deardorff (1967) for the input of wind energy into the dominant wave. The extension of the model to a random wave field is briefly discussed.
Nonlinear Wave Groups in Deep Water
Authors: Bryant P. J.
Published: September 2015

Nonlinear wave groups in deep water consist of wave modes for which nonlinear interactions and dispersion are in balance. The evolution equations for the wave modes are derived, and properties of nonlinear wave groups are found from these equations. It is shown that the nonlinear wave groups are linearly unstable to sideband modulations in the sense that the linearized perturbation theory, in providing a good fit over the initial time interval, predicts that the growth of the modulations is exponential. Instead the perturbed wave group is shown to return cyclically to a state close to its initial state. The cyclic recurrence is demonstrated analytically for the simpler wave groups and numerically otherwise. The interactions between nonlinear wave groups of the same and of nearly the same central wavenumbers are calculated.
Rational Solutions of Painlevé Equations
Authors: Airault H.
Published: September 2015

With Bäcklund transformations, we construct explicit solutions of Painlevé equations 2 and 4. Independently, we find solutions of degenerate cases of equations 3 and 5. The six Painlevé transcendents are referred to as 1–6.
An Exact Solution of the Orr‐Sommerfeld Equation for Plane Couette Flow
Authors: Reid W. H.
Published: September 2015

The equation which governs the stability of plane Couette flow admits solutions of either well‐balanced or dominant‐recessive type. As is well known, the solutions of well‐balanced type can be expressed in terms of simple exponential or hyperbolic functions. The main aim of this paper, therefore, is to show that the solutions of dominant‐recessive type can be expressed exactly as the sum of three products which involve certain rapidly and slowly varying functions. The rapidly varying functions are simply Airy functions, together with their first integrals and first derivatives. Of the three slowly varying functions, one is expressible in terms of a hyperbolic function and the other two have simple integral representations which, for bounded values of the wavenumber α, have convergent expansions in powers of (iαR)−1. An application of these results is also made to the problem of semibounded plane Couette flow, and it is shown that the eigenvalue relation for this problem can be expressed in an extremely simple form.
A Formal Calculus for the Enumerative System of Sequences—I. Combinatorial Theorems
Authors: Jackson D. M. , Goulden I. P.
Published: September 2015

The enumeration of sequences plays an important part in combinatorial enumeration since sequences may be used as a means for encoding combinatorial structures. In the main, the enumeration of sequences has been treated by a collection of methods individually applied in particular cases. We present here a uniform constructive method for dealing with a large class of sequence problems. We obtain the generating function as the trace of an expression involving certain incidence matrices, expressions which are obtainable immediately from the combinatorial structure of a problem. The method relies on certain properties of matrices whose rank is equal to one. We demonstrate that these properties may be used to derive the required generating functions explicitly or, in more complex cases, as solutions to systems of linear equations. Results for permutations may be derived from those for sequences, and in certain frequently encountered special cases this operation may be carried out by a straightforward substitution. Applications of the method presented here will be given in Part II.
Optimal Joint Distributions of Several Random Variables with Given Marginals
Authors: Schay Geza
Published: September 2015

Solutions of a singularly perturbed vector boundary‐value problem are studied under the principle 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.
A Formal Calculus for the Enumerative System of Sequences—II. Applications
Authors: Jackson D. M. , Goulden l. P.
Published: September 2015

We apply the theory of sequence enumeration presented in Part I to a variety of problems. The problems have been classified according to the scheme given in Part I and include specializations to permutations for a number of bipartitions. Further extensions will be given in Part III.
Frobenius Endomorphisms in the Umbral Calculus
Authors: Morris Robert A.
Published: September 2015

Frobenius operators Fn are introduced on sequences of binomial type. The Laguerre polynomials are essentially characterized by the property that Fn coincides with n‐fold binomial convolution.
A Formal Calculus for the Enumerative System of Sequences—III. Further Developments
Authors: Jackson D. M. , Goulden I. P.
Published: September 2015

We use the material of Parts I and II to obtain further results in sequence enumeration. These include the enumeration of sequences with respect to both maximal π1 and π2‐paths, and the enumeration of sequences with distinguished substrings. We use the material to derive an enumerative proof of a generalization of a result of Foata and Schützenberger. Finally, we reconsider the enumeration of permutations with periodic pattern and show that the required generating function may be exhibited as the solution of a set of first‐order nonlinear differential equations. Subsequent work has shown these to be matrix Riccati equations, although we refer to them here as Volterra equations.
The Lie Theory of Two‐Variable Hypergeometric Functions
Authors: Kalnins E. G. , Manocha H. L. , Miller Willard
Published: September 2015

We develop a Lie‐algebraic method that associates with each of the 34 distinct second‐order hypergeometric functions in two variables a canonical system of partial differential equations. The special functions arise by partial separation of variables in these simple systems. Some consequences are a demonstration that all such functions appear as solutions of the 4‐variable wave equation and a classification of the possible imbeddings. In each case the functions are characterized by first‐ and second‐order operators in the enveloping algebra of the conformal symmetry algebra for the wave equation. In some cases the 3‐variable wave and heat equations and the 2‐variable Helmholtz equation also arise. This intimate relationship between Horn functions and some fundamental equations of mathematical physics shows that these functions are more interesting than was previously recognized and permits use of the powerful tools of Lie theory and separation of variables to obtain properties of the functions.
On Specializations of Matroids
Authors: Kung Joseph P. S.
Published: September 2015

In this paper, we give an internal description of a weak map between two matroids on the same set. This description is similar to the description of an elementary strong map in terms of a modular cut.
Steady Gravity‐Capillary Waves on Deep Water—II. Numerical Results for Finite Amplitude
Authors: Chen B. , Saffman P. G.
Published: September 2015

Finite amplitude capillary‐gravity waves of permanent form on deep water are studied numerically. Bifurcation and limit lines are calculated. Pure and combination waves are continued to maximum amplitude. It is found that the height is limited in all cases by the surface enclosing one or more bubbles.
On Maclaurin and Jacobi Objects Embedded in Halos: The Shift in the Point of Bifurcation
Authors: Waxman Allen M.
Published: September 2015

This paper concerns the equilibrium structure of Maclaurin spheroids and Jacobi ellipsoids embedded in nonrotating halos of uniform density. The halo is assumed unresponsive to the embedded object, whereas the embedded object is allowed to respond to the gravitational field of the halo. We also ignore the effects of the halo pressure field on the embedded object. It is shown how the halo modifies the classical Maclaurin and Jacobi sequences. In particular, we locate the intersection of these two sequences, i.e., the point of bifurcation, and present a formula for the eccentricity at bifurcation (eb) as a function of the ratio of halo density to density of rotating matter (ρH/ρB). We find that the halo increases the eccentricity at bifurcation; thus, it has a stabilizing influence. However, secular instability is never entirely suppressed, since eb→1 only for ρH/ρB→∞. It is seen that the Ostrike‐Peebles conjecture does not apply to the case of ρH/ρB≫1.
Resonant Transfer of Energy between Nonlinear Waves in Neighboring Pycnoclines
Authors: Liu A. K. , Kubota T. , Ko D. R. S.
Published: September 2015

The interaction of weakly nonlinear, long internal gravity waves in neighboring pycnoclines is studied. Two coupled equations which describe the evolution of the wave amplitudes are derived. These equations are shown to possess three conserved quantities. The numerical results demonstrate the existence of two types of periodic nonlinear wave solutions when mode‐two waves propagate along each pycnocline with nearly equal speeds. The energy transfer between these resonant waves is discussed for two pycnocline separations.
Erdös Conjecture on Sums of Distinct Numbers
Authors: Peck G. W.
Published: September 2015

A conjecture of Erdös that a set of n distinct numbers having the most linear combinations with coefficients 0,1 all equal are n integers of smallest magnitude is here proven. The result follows from a theorem of Stanley that implies that the integers from n have the most such linear combinations having k distinct values for every k. The same result is shown to hold for complex numbers and vectors in Hilbert space. It is shown that the number of linear combinations taking on k distinct values is maximized by the same configuration, for every k. Generalization to the case in which irregular distinctness restrictions are imposed is also given.
The Solutions of a Model Nonlinear Singular Perturbation Problem Having A Continuous Locus of Singular Points
Authors: Kedem Gershon , Parter Seymour V. , Steuerwalt Michael
Published: September 2015

Consider the boundary value problem εy″ =(y2 − t2)y′, −1 ⩽t⩽0, y(−1) = A, y(0) = B. We discuss the multiplicity of solutions and their limiting behavior as ε→+0+ for certain choices of A and B. In particular, when A = 1, B = 0, a bifurcation analysis gives a detailed and fairly complete analysis. The interest here arises from the complexity of the set of "turning points."
A Partition Theory of Planar Animals
Authors: Holladay Kenneth
Published: September 2015

A dissection of an animal is a partition of its cells into blocks that are themselves animals. The n‐rule allows only those dissections with the area of every block a multiple of n. If an animal with area divisible by n has no dissection allowed by the n‐rule, then the animal is said to be an n‐irreducible. The partition meet of all allowed dissections of a given animal is called the n‐dissection residue of that animal. This paper considers only planar animals. All 2‐irreducibles are found, and the problem of computing 2‐dissection residues is solved. Two theorems on n‐irreducibles are proved. One of them states that large n‐irreducibles always arise by adding cells to smaller n‐irreducibles.
Direct Resonance In Nonlinear Wave Systems
Authors: Akylas T. R. , Benney D. J.
Published: September 2015

In certain wave systems it is possible for the amplitudes associated with special wave numbers to be larger than expected. Such may be the case if two frequencies are close together. Direct resonances of this type are studied and some possible applications are discussed.
A Hierarchy of Inequalities
Authors: Daykin David E.
Published: September 2015

New methods give inequalities containing work of Ahlswede, Anderson, Chebychev, Daykin, Fortuin, Ginibre, Holley, Kasteleyn, Kleitman, Seymour and others.
The Instability of Nonmonotonic Wave Solutions of Parabolic Equations
Authors: Hagan Patrick S.
Published: September 2015

We consider the class of equations ut=f(uxx, ux, u) under the restriction that for all a,b,c. We first consider this equation over the unbounded domain − ∞ < x < + ∞, and we show that very nearly every bounded nonmonotonic solution of the form u(t, x)=ϕ(x−ct) is unstable to all nonnegative and all nonpositive perturbations. We then extend these results to nonmonotonic plane wave solutions u(t, x, y)=ϕ(x−ct) of ut = F(uxx, uxy, ux, uy, u). Finally, we consider the class of equations ut=f(uxx, ux, u) over the bounded domain 0 < x < 1 with the boundary conditions u(t, x)=A at x=0 and u(t, x)=B at x=1, and we find the stability of all steady solutions u(t, x)=ϕ(x).
Eulerian Binomial Type Revisited
Authors: Reiner David L.
Published: September 2015

George Andrew's theory of Eulerian binomial type [1] is shown to be a slight variation of Mullin and Rota's binomial type [4] by using ideas of Fillmore and Williamson [3]. Finally, a hint at some combinatorial interpretations of this theory is given.
A New Mechanism For Linear and Nonlinear Hydrodynamic Instability
Authors: Benney D. J. , Gustavsson L. Håkan
Published: September 2015

A certain class of three dimensional disturbances is studied which exhibit a strong resonance and consequently lead to the rapid development of relatively large amplitudes. For parallel flows a nonlinear theory is developed and results in evolution equations quite different from those usually found. It is believed that the selection mechanism explored here is relevant to a variety of nonlinear wave and instability problems.
An Experimental Study of Flow over an Impounding Dike
Authors: Greenspan H. P. , Johansson Arne V.
Published: September 2015

The containment capability of an impounding dike is examined for a dynamic spill in which a large volume of liquid is discharged from the rupture of a storage tank. In this situation, dikes as presently constructed and positioned would allow substantial overflow. Designs are advanced for more efficient safety dams of conventional scale and cost.
The Evolution of Nonlinear Wave Trains in Stratified Shear Flows
Authors: Liu A. K. , Benney D. J.
Published: September 2015

The propagation of an internal wave train in a stratified shear flow is investigated for a Boussinesq fluid in a horizontal channel. Linear effects are primarily reflected in the dispersion relation for the various modes. The phenomenon of Eckart resonance occurs for more realistic stratification profiles. The evolution of nonlinear internal wave packets is studied through a systematic perturbation analysis. A nonlinear Schrodinger equation for the envelope of the internal wave train is derived. Depending on the relative sign of the dispersive and nonlinear terms, a wave train may disperse or form an envelope soliton. The analysis demonstrates the existence of two types of critical layers: one the ordinary critical point where ū=c, while the other occurs where ū=cg. In order to calculate the coefficients of the nonlinear Schrodinger equation a numerical code has been developed which computes the second‐harmonic and induced mean motions. The existence of these envelope solitons and their dependence on environmental conditions are discussed.
On Bifurcation and Symmetry in Benard Convection and Taylor Vortices
Authors: Shearer M. , Walton I. C.
Published: September 2015

Symmetries of the nonlinear boundary value problems governing Bénard convection and Taylor vortices are described. Their effect on the corresponding bifurcation equation is deduced from the general analysis of Shearer (1978).