bluuebunny/crossref_metadata_2025 huggingface dataset:
Magnetic Susceptibility of the Random One‐ Dimensional Transverse Ising Model
Authors: Barouch E. , McCoy B. M.
Published: September 2015

We study the 1‐ dimensional Ising model with random coupling constants in the presence of a transverse magnetic field h. In the nonrandom (translationally invariant) case it is known that there is a value of h for which the magnetic susceptibility diverges logarithmically. We demonstrate that in the random case this logarithmic divergence is smoothed out into an infinitely differentiable essential singularity.
Dynamics of a Local Perturbation in the X‐Y Model II—Excitations
Authors: Abraham D.B. , Barouch E. , Gallavotti G. , Martin-Löf A.
Published: September 2015

Excitations of a single spin embedded in the X Y chain are studied. We find that the magnetization does not approach equilibrium, even if the external magnetic field approaches a non‐zero limit, and that oscillatory excitation resonates with the edge of the one‐particle spectrum band.
On Scheduling Some Transport Systems with Discrete‐Time Passenger Arrivals
Authors: Barnett Arnold
Published: September 2015

A transportation system is operated with one vehicle and N service points. At one terminal of the system passengers arrive in groups only at discrete points in time; at the other service points, riders enter steadily at constant rates. Vehicle travel times between adjacent service points are random variables. The problem addressed in this paper is that of devising the operating strategy which minimizes average waiting time for passengers of the system. Using different kinds of probabilistic analysis, we obtain optimal policy statements both when the discrete‐time arrivals occur at fixed instants of equal spacing, and when these arrivals occur in Poisson fashion. Some specific applications are discussed.
On the Settling Speed of Free and Fixed Suspensions
Authors: Saffman P. G.
Published: September 2015

Dilute suspensions of small spheres in a viscous fluid are examined for the three cases of: I, a regular periodic array; II, a random array of freely moving particles; and III a random array in which the particles are held rigidly. It is known that the difference between average force and average velocity for a particle is a different function of concentration for the three cases. A unified analytical treatment is given for the point particle approximation, which explains the different qualitative dependence on concentration. It is shown that the differences between I and II are kinematical in nature, depending entirely on the difference between the geometries of regular and random arrays. The different result for case III is due to the shielding effect of a dilute suspension when the velocities, but not the forces, of the particles are fixed.
Model Equations for Turbulent Shear Flow
Authors: Saffman P. G.
Published: September 2015

A set of model equations for the calculation of turbulent shear flows is presented. It is shown how the equations can be modified to allow for the inclusion of viscosity, compressibility and density variations. Some results of numerical computations are given. The constant in the law of the wall is predicted to within 10% for flow over a perfectly smooth wall, and a dependence on wall roughness is obtained. The empirical law of the wall for compressible flow is also shown to be a consequence of the equations.
A Note on Random Triangles
Authors: Halász Sylvia , Kleitman D. J.
Published: September 2015

In this paper we prove that the point in a triangle T maximizing the probability that it lies within a randomly chosen triangle within T is the center of T. A simple general closed expression for the indicated probability at any point P of T is obtained, though the proof proceeds by a symmetry argument on contributions to derivatives of the probability. It is also shown that the indicated probability is linearly related to the average over points in T of the product of the areas within T on either side of the line determined by P and

.

This gives the proof for n = 3 of the following conjecture of A. Prékopa:

Fix a point P in an n − 1 dimensional simplex S. Take n independent points, uniformly distributed in S, and consider the probability ρ(P) of P being inside the random simplex determined by these n points.

Conjecture: ρ(P) is maximal, when P is the center of gravity of S.
A Note on Resonance and Nonlinear Dispersive Waves
Authors: Ablowitz M. J.
Published: September 2015

The general theory for the slow dispersion of nonlinear wave trains, first studied by Whitham, is applied to a wave train, which, in the weakly nonlinear limit, exhibits resonant singularities. Numerical and perturbation methods are used to develop singly periodic solutions both away from and near all such critical values. Similarly, the equations governing the slow modulations of such a system are found by asymptotic analysis. The expansions are found to be valid so long as the wave train is sufficiently nonlinear. These ideas should be applicable to other problems where resonant singularities arise, in particular, multiphase modes.
A Note on Unigraphic Sequences
Authors: Kleitman Daniel J. , Li S.‐Y.
Published: September 2015

A characterization of unigraphic sequences is given.
A Theorem on the Convergence of Padé Approximants
Authors: Baker George A.
Published: September 2015

We prove a theorem which gives necessary and sufficient conditions on the distribution of poles and zeros of the Padé approximants for point‐by‐point convergence. The special case of convergence to a function meromorphic in a disk by a sequence of Padé approximants free of extraneous poles and zeros is proven.
Uniform Flow Past an Axially Uniform Viscous Vortex
Authors: Kroll John
Published: September 2015

It is believed that the flow past a tornado causes the formation of smaller vortices which produce the “suction spots” observed along the path of destruction. Here we develop a greatly simplified mathematical model to investigate this phenomenon. An axially uniform vortex is developed by visualizing a circular tube with uniform surface suction of fluid possessing circulation at infinity. This vortex is then perturbed by a uniform flow past it. An inner asymptotic expansion of an E1/3 radial boundary layer is matched to an outer expansion to obtain a solution. The results show that a stagnation point developing into a secondary vortex is formed in a free shear layer at critical flow conditions. However, it is difficult to apply our results quantitatively because of the difficulty in comparing the axially uniform vortex with a real tornado vortex.
The Combinatorics of Polynomial Sequences
Authors: Reiner David L.
Published: September 2015

We present a combinatorial model for the several kinds of polynomial sequences of binomial type and develop many of the theorems about them from this model. In the first section, we present a prefab model for the binomial formula and the generating‐function theorem. In Sec. 2, we introduce the notion of U‐graph and give examples of binomial prefabs of U‐graphs. The umbral composition of U‐graphs provides an interpretation of umbral composition of polynomial sequences in Sees. 3 and 5. Rota's interpretation of the Stirling numbers of the first kind as sums of the Mobius function in the partition lattice inspired our model for inverse sequences of binomial type in Sec. 4. Section 6 contains combinatorial proofs of several operator‐theoretic results. The actions of shift operators and delta operators are explained in set‐theoretic terms. Finally, in Sec. 6 we give a model for cross sequences and Sheffer sequences which is consistent with their decomposition into sequences of binomial type. This provides an interpretation of shift‐invariant operators. Of course, all of these interpretations require that the coefficients involved be integer and usually non‐negative as well.
A Generalization of the Direct and Inverse Problem for the Radial Schrödinger Equation
Authors: Moses Harry E.
Published: September 2015

We consider the operators H0= −d2/dr2 and H1 = −d2/dr2 + V(r) (0< r< ∞) acting on a Hilbert space of complex functions f(r) such that the subspaces in which the operators are defined consist of twice differentiable functions which satisfy the boundary condition (d/dr)f(0) = αf(0). H1 and H0 are Hermitian in this subspace. Assuming V(r)→0 as r→∞ sufficiently rapidly, the scattering operator formalism is set up for the direct scattering problem. Next we consider the inverse problem of determining V(r) from H0 and the spectral measure function for the spectrum of H1 through the use of an appropriate Gelfand‐Levitan equation. It is shown that generally the value of α associated with H1 differs from that for H0, i.e., H1 and H0 generally operate in different subspaces. Thus scattering cannot be defined. However, by changing the spectral measure function, one obtains a new Gelfand‐Levitan equation such that H1 is the same as before [i.e., α and V(r) are the same] from the operator H0, which uses the same value of α as H1. Thus H1 and the new H0 operate in the same subspace of Hilbert space, and scattering can be defined. The process of obtaining the new H0 after finding H1 from the old H0 is somewhat analogous to renormalization in field theory, where a new H0 is picked to have properties compatible with H1. A necessary and sufficient condition on the spectral data is given which makes the domains of H0 and H1 coincide and thus makes “renormalization” unnecessary. The direct problem is a generalization of the usual l=0 radial Schrödinger equation. The inverse problem is a generalization of the corresponding inverse problem. It is also a generalization of the case α=0 for H0 considered by Gelfand and Levitan in their early work on the inverse spectral problem. An incompletely understood connection of the inverse problem for the radial equation to solutions of the Korteweg‐deVries equation in the half space is discussed. The existence of such a connection is one of the motivations for studying the generalized radial Schrodinger equation.
Instabilities Associated with Forced Nonlinear Waves
Authors: Benney D. J.
Published: September 2015

This paper is concerned with the response of nonlinear wave systems to external forcing by a uniform train of sinusoidal waves. In particular the stability of the forced solution is studied and other resonant situations are investigated.
Counting Successions in Permutations
Authors: Reilly J. W. , Tanny S. M.
Published: September 2015

Let π = (π(1), π(2),...,π(n)) be a permutation of the arbitrary n‐set S of postive integers. A succession in π is any pair π(i), π(i + 1) with π(i + 1) = π(i) + 1, i = 1,2..., n −1. We show that the number of permutations of S which have precisely k successions depends only upon n, k and b, where b is the number of maximal disjoint intervals in the set [n + m]\S, and n + m is the largest element in S. We derive a linear recurrence relation for this number, which we call the succession number σ(n, k; b), as well as an explicit formula in terms of derangement numbers. The linear recurrence is used to derive the generating function for succession numbers. is also derived by formal power series methods from a well‐known generating function for succession in general integer sequences.
Intersection Theory for Embeddings of Matroids into Uniform Geometries
Authors: Brylawski Tom
Published: September 2015

A uniform combinatorial geometry G is a finite geometric lattice which has the same number, WG(i,j), of flats of corank j in every upper interval of rank i. Examples of these geometries include affine and projective geometries, boolean algebras, and partition lattices as well as any homogeneous geometry (those in which flats of the same rank have equal cardinalities). Matrix equations relate the matroid structure M of a set S embedded into G to the intersection numbers of the embedding. In particular, if and PM(m,n) equals the number of m‐elements subsets of S with corank n, then T·PM·WG=IM→G, where IM→G(p,q) counts the flats in G of corank q which contain p points of S. The intersection matrix of an embedding of M or its dual into another uniform geometry G′ can be computed from IM→G, while PG−M may be computed from PM, where M is a sub geometry of a homogeneous geometry G, and G − M is its set complement. Applications for linearly representable matroids include matrix generalizations to the critical exponent of Crapo and Rota and to the codeweight polynomial of a linear code.
Numerical Evidence for the Existence of New Types of Gravity Waves of Permanent Form on Deep Water
Authors: Chen B. , Sallman P. G.
Published: September 2015

Numerical results are presented for the shape of finite‐amplitude steady irrotational inviscid gravity waves obtained by solving an integrodifferential equation. It is found that the family of solutions giving the wave as a function of height or equivalent parameter has bifurcation points for h/λ≈0.13, where h is waveheight and λ is wavelength. Two bifurcation points and the branches emanating from them are found specifically, corresponding to a doubling and tripling of the wavelength. Solutions on the new branches are calculated.
Multivariate Exponential Operators
Authors: Joni S. A.
Published: September 2015

An unexpected connection between a certain class of exponential approximation operators and polynomial sequences of binomial type was discovered by Ismail. Building on this result, we present a multivariate analogue of these exponential operators.
Hamiltonian Cycles of Adjacent Triples
Authors: Peck G. W.
Published: September 2015

A construction is given for ordering triples chosen from an ordered set of elements, so that each triple agrees with each neighbor in two of its members and has third member that is a neighbor of its neighbor's third member. Neighbors here are adjacent in order, and also the first is neighbor to the last among both elements and triples. Joichi and White have given a different construction.
Periodically Perturbed Bifurcation. II. Hopf Bifurcation
Authors: Rosenblat S. , Cohen Donals 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 periodic solutions are, in general, modified into quasiperiodic solutions. Different phenomena are encountered in resonance and near‐resonance conditions, leading in some cases to separation of solution branches and in other cases to new solution branches.
Structure of a Linear Array of Uniform Vortices
Authors: Saffman P. G. , Szeto R.
Published: September 2015

The shape and properties of an infinite steady linear array of uniform vortices are calculated. A nonlinear singular integrodifferential equation is obtained for the shapes, which is solved numerically by Newton's method and Euler continuation to give a one parameter family of shapes as size over separation is varied. The kinematic properties and energy of the array are obtained. It is found that there exists an array of maximum area, for given separation, which also possesses minimum energy in accordance with a general argument of Kelvin. A simple model based on elliptical vortices is constructed, which reproduces the qualitative kinematic properties and is quantitatively quite accurate. Continuation of the numerical solution past the array of maximum area leads to a limit of finite, lens shaped, touching vortices. This array is also shown to be limit of a finite amplitude bifurcation of a vortex sheet of finite thickness. The stability of the array to two dimensional subharmonic and superharmonic disturbances is considered. General arguments, based on ideas of Kelvin, are given to show that the array is stable to superharmonic disturbances if the area is less than the maximum and otherwise unstable, and that it is always unstable to subharmonic disturbances, of which the pairing instability is a special case. It is verified by direct calculation in an Appendix that hollow vortices, whose shapes can be determined analytically in closed form, are unstable to the pairing instability whatever their size. Some speculations are made about the possible relevance of the results to the observed properties of organized structures in the turbulent mixing layer.
Large Amplitude Solitary Waves in Unbounded Stratified Fluids
Authors: Benney D. J. , Grimshaw R.
Published: September 2015

Solitary waves of arbitrary amplitude are found to exist for a class of unbounded stratified flow configurations. In special cases the solutions are similar to those obtained from the Benjamin‐Ono equation.
Adiabatic Invariance and Passage Through Resonance for Nearly Periodic Hamiltonian Systems
Authors: Kevorkian J.
Published: September 2015

This is a continuation of previous work on passage through resonance to nearly periodic Hamiltonian systems. We review the classical technique for calculating adiabatic invariants and exhibit the occurrence of zero divisors in the results as a certain critical term evolves slowly through a resonance condition. We then isolate the coordinate associated with the singularity and remove the remaining coordinate from the Hamiltonian to any desired order by successive canonical transformations. This is a variant of the von Zeipel procedure used extensively in celestial mechanics. The momentum conjugate to the cyclic coordinate is an adiabatic invariant, and the reduced Hamiltonian is then solved by constructing and matching three multiple variable expansions which describe the solution before, during, and after resonance passage.
Chaotic Behavior in Slowly Varying Systems of Nonlinear Differential Equations
Authors: Keener James P.
Published: September 2015

A technique is developed to find parameter regions of chaotic behavior in certain systems of nonlinear differential equations with slowly varying periodic coefficients. The technique combines previous results on how to find branches of periodic solutions which terminate with a homoclinic orbit and results on how to find chaotic trajectories in the neighborhood of homoclinic trajectories of the autonomous system. The technique is applied to the continuous stirred tank reaction A → B, for which it is shown that a slowly varying periodic flow rate can yield aperiodic temperature fluctuations.
The Evolution of Waves Near Direct‐Resonance Conditions
Authors: Akylas T. R. , Benney D. J.
Published: September 2015

An analytical and numerical study is made of the propagation of waves whose wavenumbers are close to conditions for direct resonance. These waves emerge from the linear initial value problem as the dominant far field signal. The nonlinear evolution characteristics are investigated. It is found that weakly nonlinear waves at direct resonance may rapidly amplify and tend to become fully nonlinear in a finite time.
Modulations of Sinh‐Gordon and Sine‐Gordon Wavetrains
Authors: Forest M. Gregory , McLaughlin David W.
Published: September 2015

An invariant representation of the modulation equations for the sinh‐ and sine‐Gordon wavetrains is derived. A simple derivation of the representation which makes fundamental use of squared eigenfunctions is presented. This representation is used to place the modulation equations in Riemann invariant form and to cast them in a Hamiltonian form. The multiphase sinh‐Gordon study is complete, while the sine‐Gordon theory for more than one phase possesses technical difficulties which are described in the text. Explicit results on real two‐phase sine‐Gordon waves are included in Section VI.
Vortex Induced Lift on Two Dimensional Low Speed Wings
Authors: Saffman P. G. , Tanveer S.
Published: September 2015

Exact free streamline solutions are found for two dimensional inviscid incompressible flow past a single and a multiple flap wing using a hodograph method. It is shown that solutions do not exist for arbitrary shapes, but that a geometrical constraint must be satisfied between the shape of the wing and the angle of attack. High lift coefficients are obtained for both cases. These solutions model the flow situation for a wing claimed in the past to give high lift at low speed.
The Self‐Focusing Singularity in the Nonlinear Schrödinger Equation
Authors: Wood David
Published: September 2015

Under certain circumstances, solutions of the cylindrically symmetric nonlinear Schrödinger equation collapse to a singularity in a finite time. An asymptotic series for the solution near the singularity is derived here. At leading order, the central amplitude of the spike grows like[(log Δt)/Δt]1/2, where Δt is the time remaining to the appearance of the singularity.
Direct Resonance in Double‐Diffusive Systems
Authors: Maslowe S. A.
Published: September 2015

In thermohaline convection and a number of other systems, both direct and oscillatory modes of instability are possible. Should these modes coalesce at some point in parameter space, nonlinear instabilities are likely to be more dramatic in the neighborhood of such a point than elsewhere. A finite‐amplitude evolution equation describing such events is derived here by the method of multiple scales. The results are compared with those obtained previously for model systems and by other methods. The direct resonance point of view is found to provide some new insights. A generalization to allow for slow spatial modulation of the amplitude is given and, among other possibilities, solitary‐wave solutions are obtainable.
Master Symmetries and Similarity Equations of the XYh Model
Authors: Barouch E. , Fuchssteiner B.
Published: September 2015

Master symmetries of the XYh model are derived and utilized to establish a matrix difference similarity equation.
The Propagation of Nonlinear Wave Packets in a Shear Flow with a Free Surface
Authors: Oikawa M. , Chow K. , Benney D. J.
Published: September 2015

The instability properties of weakly nonlinear wave packets show a strong directional dependence. The presence of shear is found to enhance the importance of three dimensional effects.
Explicit Series Formulae for the Evaluation of Integrals by the Method of Steepest Descents
Authors: Campbell John A. , Fröman Per Olof , Walles Erik
Published: September 2015

Explicit series formulae are given for use in the method of steepest descents, where it is required to find the contribution to an integral of the form ∫evw(z)φ(z)dz from a well‐isolated saddle point z0. Here v is a large positive parameter, and w(N)(z0) ≠ 0 while lower‐order derivatives at z0 are equal to zero. Examples where the formulae are helpful are presented.
A Class of Generating Functions for the Jacobi and Related Polynomials
Authors: Srivastava H. M. , Popov Blagoj S.
Published: September 2015

This paper aims at presenting a general class of mixed generating functions for the Jacobi polynomials. It is also shown how the main generating function can be suitably applied to yield numerous further results involving Jacobi polynomials and various other polynomials associated with them.
Exact Simple Waves on Shear Flows in a Compressible Barotropic Medium
Authors: Sachdev P. L. , Philip Varughese
Published: September 2015

Self‐similar solutions describing simple waves on shear flows in a finite compressible barotropic atmosphere are found. These include the simple waves on shear flows in water as a special case. By making use of a number of transformations it becomes possible to write these solutions in an exact form. This form, though not explicit, is similar to the incomplete Beta function which seems to characterize this class of nonlinear physical problems.
Forced Nonlinear Evolution Equations and the Inverse Scattering Transform
Authors: Fokas A. S. , Ablowitz M. J.
Published: September 2015

The Korteweg—de Vries and nonlinear Schrödinger equations with an external forcing of distribution type are considered. The reflection coefficient is found to satisfy a nonlinear equation of a certain characteristic form which also appears in the semi‐infinite problem.
The Third‐Order Singular‐Perturbation Expansion of the Planar Cold‐Fluid Magnetron Equations
Authors: Kaup D. J. , Thomas Gary E.
Published: September 2015

The singular‐perturbation expansion of the plasma cold‐fluid equations for crossed fields in a planar geometry is presented. The general expansion is carried out to third order. Various instabilities that occur in the first, second, and third orders are discussed.
Traveling Wave Solutions for Combustion in a Porous Medium
Authors: Tam K. K.
Published: September 2015

Traveling wave solutions are sought for a model of combustion in a porous medium. The problem is formulated as a nonlinear eigenvalue problem for a system of ordinary differential equations of order four, defined over an infinite interval. A shooting method is used to prove existence, and a priori bounds for the solution and parameters are obtained.
The Third‐Harmonic Resonance for Capillary‐Gravity Waves with O(2) Spatial Symmetry
Authors: Dias Frédéric , Bridges Thomas J.
Published: September 2015

It is well known that the addition of surface‐tension effects to the classic Stokes model for water waves results in a countable infinity of values of the surface tension coefficient at which two traveling waves of differing wavelength travel at the same speed. In this paper the third‐harmonic resonance (interaction of a one‐crested wave with a three‐crested wave) with O(2) spatial symmetry is considered. Nayfeh analyzed the third‐harmonic resonance for traveling waves and found two classes of solutions. It is shown that there are in fact six classes of periodic solutions when the O(2) symmetry is acknowledged. The additional solutions are standing waves, mixed waves and secondary branches of “Z‐waves.” The normal form and symmetry group for each of the solution classes are developed, and the coefficients in the normal form are formally computed using a perturbation method. The physical aspects of the most unusual class of waves (three‐mode mixed waves) are illustrated by plotting the wave height as a function of x for discrete values of t.
Analysis of a Nonlinear Diffusive Amplitude Equation for Waves on Thin Films
Authors: Melkonian S. , Maslowe S. A.
Published: September 2015

This paper presents analytical and numerical solutions of a new amplitude equation governing long waves on thin films. At lowest order in the long‐wave parameter, the equation is nondispersive and represents a balance between nonlinearity and cross‐stream diffusion. Numerical solutions tracing the temporal evolution of an initially localized disturbance indicate that the aforementioned diffusion partly mitigates the tendency of the wave to break. We have also obtained a closed‐form solution resembling an undular bore propagating in an oblique direction.
Asymptotic Theory of Steady Axisymmetrical Needlelike Crystal Growth
Authors: Xu Jian‐Jun
Published: September 2015

The present paper is concerned with the stationary needle crystal growth with arbitrary undercooling. We discuss two classes of asymptotic solutions: (1) the regular‐tip solutions; (2) the smooth‐root solutions. When the surface tension is nonzero, the regular‐tip solutions may not have smooth roots. Among the regular‐tip solutions, however, one can identify a “principal regular‐tip solution,” which has the best behavior in the far field and is physically acceptable.
O(2)‐lnvariant Hamiltonians on ℂ4
Authors: Bridges Thomas J.
Published: September 2015

A normal‐form theory and a group‐theoretic classification for periodic solutions of O(2)‐invariant Hamiltonians on ℂ4 is developed. The theory applies to Hamiltonian systems with an O(2) spatial symmetry that also have a linear‐mode interaction. Our motivation is the classic (m, n) mode‐interaction problem for capillary‐gravity waves. It is well known that the addition of surface‐tension effects to irrotational water waves results in a countable infinity of values of the surface‐tension coefficient at which two traveling waves of differing wavelength travel at the same speed. However, recognizing the reflectional symmetry in space, the linearized problem is actually spanned by four traveling waves. In other words there is an O(2) symmetry in space. A classification theorem for group‐invariant Hamiltonian systems, based on a listing of the isotropy subgroups and their fixed‐point spaces, is used to show that there are between seven and fourteen classes of periodic solutions in O(2)‐invariant Hamiltonian systems with a mode interaction. The results are used to interpret, from a group‐theoretic viewpoint, the classic Wilton ripple.
Inverse Spectral Transform for the Nonlinear Evolution Equation Generating the Davey‐Stewartson and Ishimori Equations
Authors: Konopelchenko B. G. , Matkarimov B. T.
Published: September 2015

A 2 + 1‐dimensional nonlinear differential equation integrable by the inverse‐spectral‐transform method with the quartet operator representation is proposed. This GL(2, C)‐valued chiral‐field‐type equation is the generating (prototype) equation for the Davey‐Stewartson and Ishimori equations. It coincides with the nonlinear equation for the Davey‐Stewartson eigenfunction ψDS. The initial‐value problem for this equation is solved by the techniques for the
and the nonlocal Riemann‐Hilbert problem. The classes of exact solutions with the functional parameters and exponential‐rational solutions are constructed by the
method. The static lump solution in the case α = i and the exponentially localized solution at α = i are found. Other similar examples of nonlinear integrable equations in 2 + 1 and 1 + 1 dimensions are discussed.
Sequences of Symmetric Functions of Binomial Type
Authors: loeb Daniel E.
Published: September 2015

We take advantage of the combinatorial interpretations of many sequences of polynomials of binomial type to define a sequence of symmetric functions corresponding to each sequence of polynomials of binomial type. We derive many of the results of umbra! calculus in this context, including a Taylor expansion and a binomial identity for symmetric functions. Surprisingly, the delta operators for all the sequences of binomial type correspond to the same operator on symmetric functions.
Degeneracies and Direct Resonances in Water‐Table Flow
Authors: Håkan Gustavsson L. , Shanthini R.
Published: September 2015

Degeneracies of the Orr‐Sommerfeld eigenmodes and direct resonances between the Orr‐Sommerfeld eigenmodes and vorticity eigenmodes are studied in water‐table flow. The sensitivity of the characteristics of these algebraic mechanisms to flow parameters, such as the Reynolds number (R), the slope of the table (θ), and a material parameter (γ), are investigated. It is found that the mechanisms become operative at subtransitional R, and their damping rates decrease with increasing R. When the mean flow profile is slightly distorted from the ultimate parabolic profile, the characteristics of the direct resonances show remarkable variations. Also, some of the algebraic mechanisms in watertable flow are shown to have the same characteristics and modal structures as some of those in plane Poiseuille flow.
A Note on Inviscid Secondary Instability in Shear Flows
Authors: Chow K. W.
Published: September 2015

In a recent series of papers, Benney and Chow have proposed to understand parallel flow instabilities in terms of a wave‐mean‐flow interaction. This note supplements the series by pointing out some rather surprising properties of the stability equations derived.
The Rotation‐Modified Kadomtsev‐Petviashvili Equation: An Analytical and Numerical Study
Authors: Grimshaw R. , Tang S.
Published: September 2015

The rotation‐modified Kadomtsev‐Petviashvili equation, derived by Grimshaw in 1985, is studied both analytically and numerically to determine the structure of solutions which are initially localized. It is shown that solitary‐like waves can be found, whose wavefronts are curved in a direction transverse to the propagation direction, which remain unsteady, and which are always accompanied by trailing Poincaré waves. These effects are more noticeable as the effects of rotation are increased.
On Combinatorial Structures Kept Fixed by the Action of a Given Permutation
Authors: Constantineau Ivan , Labelle Jacques
Published: September 2015

Let β be a given permutation of [n]={1,2,...,n} of type (β1, β2,...,βn) (i.e., β has β1 cycles of length i; ∑iβ1 = n. We find (in terms of the β1's and bijectively) the number of endofunctions, permutations, cyclic permutations, derangements, fixed point free involutions, assemblies of octopuses, octopuses, idempotent endofunctions, rooted trees (i.e. contractions), forests of rooted trees, trees, vertebrates, relations (digraphs), symmetric relations (simple graphs), partitions, and connected endofunctions on [n], kept fixed by the natural action (byconjugation) of β. This approach leads to algorithms generating these structures.
The β‐Extension of the Multivariable Lagrange Inversion Formula
Authors: Zeng Jiang
Published: September 2015

We show that Gessel's combinatorial proof of the multivariable Lagrange inversion formula can be given a ,β‐extension, which generalizes Foata and Zeilberger's, β‐extension of MacMahon's master theorem. Moreover, we show that there is no need to use Jacobi's identity in the derivation of the Lagrange formula. Finally, combining Gessel's method and ours, we obtain a new proof of Jacobi's identity.
Bifurcation of Solutions for an Inverse Problem in Potential Theory
Authors: Chen B. , Jorge M. C. , Minzoni A. A.
Published: September 2015

Local existence and global numerical continuation of some solution branches are studied in an inverse potential problem in the exterior of a sphere. It is shown how multiple solutions arise where the solution field is tangent to the sphere. This work provides a tractable example of bifurcation arising from the edge of a continuum of eigenvalues.
Hele Shaw Flows with Time‐Dependent Free Boundaries Involving Injection Through Slits
Authors: Richardson S.
Published: September 2015

Consider the classical Hele Shaw situation with two parallel planes separated by a narrow gap. Suppose a Newtonian fluid to be injected into this gap through straight‐line slits as well as, possibly, at isolated points. We are interested in predicting the plan‐view of the resultant expanding blob, or blobs, of fluid. We show that the method of solution developed earlier for dealing with such problems when the injection is only at isolated points can be adapted to yield solutions in this more general situation.
On the Temporal and the Spatial Kelvin‐Helmholtz Instability of a Hypersonic Jet Confined by a Longitudinal Magnetic Field
Authors: Yue Z. Y. , Winnewisser G.
Published: September 2015

Both the temporal and the spatial Kelvin‐Helmholtz instability of a hypersonic jet confined by a longitudinal magnetic field are investigated. The reflection modes are calculated for a set of typical jet parameters. Analytical treatments for both temporal and spatial resonant modes, i.e., the most important reflection modes with the local maximum growth rates, are given. Various properties of the resonant modes are plotted against the jet temperature. The growth rates of the temporal and the spatial resonant modes are connected by the group velocity. The astrophysical implications of the present investigation are discussed.
On the Existence and Parameter Dependence of Similarity Solutions for Free Convection on a Vertical Plate
Authors: Tam K. K.
Published: September 2015

The free convection on a heated vertical plate is considered, both for the case of prescribed plate temperature and the case of prescribed heat flux. The parameter dependence of the similarity solutions are investigated, and the existence proofs given for a range of the parameter values.
Fractional Calculus and Its Applications Involving Certain Classes of Functional Relations
Authors: Pandey R. N. , Srivastava H. M.
Published: September 2015

An interesting functional relation between Fox's H‐function and the Digamma function ψ(z) was derived recently by applying the Riemann‐Liouville fractional differintegral operator
of (real or complex) order µ The object of this paper is to present much simpler alternative derivations of substantially more general classes of functional relations without using fractional calculus. Some relevant historical details are also provided.
Wave Packet Critical Layers in Shear Flows
Authors: Maslowe S. A. , Benney D. J. , Mahoney D. J.
Published: September 2015

In the linear inviscid theory of shear flow stability, the eigenvalue problem for a neutral or weakly amplified mode revolves around possible discontinuities in the eigenfunction as the singular critical point is crossed. Extensions of the linear normal mode approach to include nonlinearity and/or wave packets lead to amplitude evolution equations where, again, critical point singularities are an issue because the coefficients of the amplitude equations generally involve singular integrals. In the past, viscosity, nonlinearity, or time dependence has been introduced in a critical layer centered upon the singular point to resolve these integrals. The form of the amplitude evolution equation is greatly influenced by which choice is made. In this paper, a new approach is proposed in which wave packet effects are dominant in the critical layer and it is argued that in many applications this is the appropriate choice. The theory is applied to two‐dimensional wave propagation in homogeneous shear flows and also to stratified shear flows. Other generalizations are indicated.
Scalar Conservation Laws and Spatially Dependent Flux Functions
Authors: He Yuanping , Moodie T. Bryant
Published: September 2015

In this paper, we carry out a detailed analysis of the signaling and initial value problems associated with a general scalar conservation law that admits spatial variation in the flux function. The approach adopted here is to introduce a nonlinear phase variable directly into the problem and carry out the analysis in the space and phase variables rather than in space‐time. This facilitates our efforts to create, through a bifurcation analysis, a clear picture of the process by which a smooth wave breaks to generate a propagating shock wave.
The Interaction of a Strong Bore with Small Disturbances in Shallow Water
Authors: Yu J. , Kevorkian J.
Published: September 2015

This paper gives an extension of previous work [2] on weakly nonlinear shallow water waves over a variable bottom to include the effects of strong bores and small surface disturbances. We first consider the interaction of a strong bore with quiescent water over an isolated bottom disturbance to highlight some of the modifications that are introduced in our results for both noncritical and transcritical Froude numbers. We also exhibit the secular effect on the bore trajectory of a bottom disturbance that has a nonzero average. In a second example, we consider the interaction of a strong bore with a small amplitude periodic surface disturbance upstream. We show that downstream of the bore, the wave length of this disturbance increases, whereas its amplitude increases (decreases) depending on whether the bore speed is larger (smaller) than a critical value. We also use this example to illustrate the derivation of the solution and bore trajectory to second order accuracy. All our asymptotic results, obtained in the form of multiple scale expansions, are compared with numerical solutions for a number of illustrative cases.
Characteristics Analysis of the Parabolized Stability Equations
Authors: Haj‐Hariri Hossein
Published: September 2015

Some mathematical details of the parabolized stability equations (PSE) [4] are investigated. In particular, the sources of (unwanted) ellipticity in these equations are identified and suggestions are made for their suppression. Both the compressible and incompressible equations in primitive‐variable formulation are discussed. Remarks are also made on the velocity‐vorticity formulation. A slight modification to the PSEs usually method of treatment of streamwise derivatives higher than first is proposed. Also, an expression is derived for limiting the streamwise gradient of the pressure shape function.
Asymptotic Analysis of a Nonlinear Problem in Thermoelasticity
Authors: Sabri Önc#x00FC; T. , Bryant Moodie T.
Published: September 2015

In this paper we employ our previously formulated constitutive relations for an elastic heat conductor to examine the nonlinear influence of the temperature on the propagation of second sound. Utilizing the method of geometric optics we ascertain that large values of second sound travel more slowly than do smaller ones. As well, numerical results for a signaling problem are displayed graphically in order to highlight the qualitative properties of the temperature and strain waves. The choice of signaling data was motivated by experimental studies.
On the Asymptotic Solution of Non‐Hamiltonian Systems Exhibiting Sustained Resonance
Authors: Bosley D. L. , Kevorkian J.
Published: September 2015

With the exception of some special examples, much of the literature on the formal construction of asymptotic solutions of systems exhibiting sustained resonance concerns Hamiltonian problems, for which the reduced problem is of order two when a single resonance is present. In the Hamiltonian case, the resonance manifold is a curve that is explicitly defined by the governing equations and is independent of the actual sustained resonance solution. When the basic standard form system is non‐Hamiltonian, with M slow and N fast variables, the corresponding reduced problem is of order M + 1; in general it involves all of the slow variables, P1,…, PM, plus the resonant phase Q. In this paper, the solution of a general non‐Hamiltonian system in standard form is formally constructed for the case of a single sustained resonance. First, a well‐known example is reviewed, for which the projection of the solutions on the resonance manifold can be derived a priori, independent of the evolution of Q. Then, the general case is solved, using a generalization of the multiple scale method of Kuzmak‐Luke, where knowledge of the asymptotic solution for Q (as well as higher‐order terms) is needed to define the projection of the solution on the resonance manifold. The results simplify significantly when initial conditions are chosen exactly on the resonance manifold; the modifications necessary for arbitrary initial conditions are also given. Two examples are discussed in detail to illustrate the procedure. The asymptotic results are confirmed for several test cases by comparison with numerical integrations of the exact equations.
Multi‐Duality in Minimal Surface—Type Problems
Authors: Gao David Yang , Yang Wei H.
Published: September 2015

The multi‐duality of the nonlinear variational problem inf J(u, Λu) is studied for minimal surfaces‐type problems. By using the method developed by Gao and Strang [1], the Fenchel‐Rockafellar's duality theory is generalized to the problems with affine operator Λ. Two dual variational principles are established for nonparametric surfaces with constant mean curvature. We show that for the same primal problem, there may exist different dual problems. The primal problem may or may not possess a solution, whereas each dual problem possesses a unique solution. An evolutionary method for solving the nonlinear optimal‐shape design problem is presented with numerical results.
Induced Surfaces and Their Integrable Dynamics
Authors: Konopelchenko B. G.
Published: September 2015

A method is considered to induce surfaces in three‐dimensional (pseudo) Euclidean space via the solutions to two‐dimensional linear problems (20 LPs) and their integrable dynamics (deformations) via the 2 + 1‐dimensional nonlinear integrable equations associated with these 2D LPs. Coordinates Xi of the induced surfaces are defined as integrals over certain bilinear combinations of the wave functions ψ of these 20 LPs. General formulation as well as three concrete examples are considered. Some properties and features of such induction are discussed. Three‐dimensional Riemann spaces associated with 2 + 1‐dimensional nonlinear integrable equations are considered also.
Solutions to the Plateau Problem for the Prescribed Mean Curvature Equation via the Mountain Pass Lemma
Authors: Lami Dozo E. , Mariani M. C.
Published: September 2015

Assuming that a function g solves the classical Plateau problem for a disc‐type surface, we give conditions on a non zero function H in ℝ3 with respect to g to obtain multiple weak solutions to the corresponding problem with mean curvature H.
An Asymptotic Analysis of Localized Solutions for Some Reaction‐Diffusion Models in Multidimensional Domains
Authors: Ward Michael J.
Published: September 2015

In the limit ∊ → 0, a spike‐layer solution is constructed for the reaction‐diffusion equation
urn:x-wiley:00222526:media:sapm1996972103:sapm1996972103-math-0001 where b > 0 and D is a bounded convex domain. Here Q(u) is such that there exists a unique radially symmetric function uc(∊−1 r) satisfying ∊2Δuc + Q(uc) = 0 in all of ℛN, with uc(ρ) decaying exponentially at infinity. The spike‐layer solution has the form u ~ uc [∊|x − x0|], where the spike‐layer location x0 ∊ D is to be determined subject to the condition that dist(x0, ∂D) as ∊ → D. The determination of x0 is shown to be exponentially ill conditioned and asymptotic estimates for the exponentially small eigenvalues and the corresponding eigenfunctions associated with the linearized problem are obtained. These spectral results are used together with a limiting solvability condition to derive an equation for x0. For a strictly convex domain, it is shown that there is an x0 that is located at an O(∊) distance away from the point in D that is furthest from ∂D. Finally, hot‐spot solutions to Bratu's equation are constructed asymptotically in a singularly perturbed limit.
Erbliche Stoffwechselkrankheiten – eine Übersicht
Authors: Zschocke Johannes
Published: September 2015

Klinisches Bild, Diagnose und Therapie der erblichen Stoffwechselkrankheiten lassen sich am besten aus den spezifischen betroffenen Stoffwechselwegen ableiten, welche durch die einzelne, meist enzymatische Störung betroffen sind. Dabei lassen sich die Störungen des Intermediärstoffwechsels, welche den Stoffwechsel der Aminosäuren, Kohlenhydrate und Fettsäuren sowie den mitochondrialen Energiestoffwechsel betreffen, von anderen Stoffwechselstörungen abgrenzen. Die Intermediärstoffwechselstörungen zeigen meist eine Manifestation erst nach der Geburt, nicht selten akute Stoffwechselentgleisungen, und können oft durch metabolische Interventionen behandelt werden. Sie werden durch die üblichen klinisch-chemischen Basisuntersuchungen und einige selektive Screeninganalysen erfasst, die allgemein unter dem Begriff „Stoffwechseldiagnostik“ subsumiert werden. Die anderen Stoffwechselkrankheiten verursachen sehr unterschiedliche, oft für den betroffenen Stoffwechselweg typische klinische Symptome und können vielfach durch Screeningtests nachgewiesen werden, die spezifisch angefordert werden müssen. Die verschiedenen Krankheitsgruppen mit den wichtigsten einzelnen Störungen werden in dem vorliegenden Artikel zusammenfassend dargestellt.