bluuebunny/crossref_metadata_2025 huggingface dataset:
Swimming of slender fish in a non-uniform velocity field
Authors: Newman J. N.
Published: February 2009

The hydrodynamic pressure forces acting upon a slender fish are derived for the case of a fish swimming in a non-uniform velocity field. Possible applications are the effects on fish propulsion of swimming in waves, in turbulent eddies, and in the presence of other fish or a moving ship. The fish is assumed to be a slender body, with no vorticity shed into the fluid except at a single abrupt trailing edge located at the posterior end of the fish, and to be performing small lateral swimming undulations of its body. The non-uniform field through which the fish swims is assumed to be irrotational, and this field as well as the body undulations must be slowly-varying on the length-scale of the lateral fish dimensions. Expressions are derived for the local force and the time-averaged total thrust force. These are applied to the study of steady-state bow-riding and wave-riding of porpoises.
Some remarks on relative stability
Authors: Kloeden Peter E.
Published: February 2009

In 1962 Lakshmikantham ([1], [2]) extended the concept of extreme stability (e.g. [4]) of a system described by an ordinary differential equation, not necessarily with uniqueness, to relative stability of two such systems. Here we show the restrictiveness of his definition of relative stability in that it implies not only are the solutions of two systems unique for each initial condition, they are in fact identical. We then introduce and give an example of a weaker version of relative stability which is of some interest for control systems. For greater simplicity and generality we use Roxin's attainability set defined General Control Systems [3] to describe the dynamics of our systems, as they subsume both ordinary differential equations without uniqueness and ordinary differential control equations.
Thermodynamic limit for a system with dipole-dipole interactions
Authors: Smith E. R. , Perram J. W.
Published: February 2009

It is shown that for the three dimensional Ising model with dipole-dipole interactions, the thermodynamic limit of the free energy with simple boundary conditions is not the same as the thermodynamic limit of the free energy with periodic boundary conditions. A variational principle is developed to connect the two free energies.
Some axially symmetric zero mass meson solutions of Einstein's equations
Authors: Patel L. K. , Trivedi V. M.
Published: February 2009

An axially symmetric metric in oblate spheroidal co-ordinates is considered. Two exact solutions of the field equations corresponding to zero mass meson fields are obtained. The details of the solutions are also discussed. These solutions are also generalized to include electromagnetic fields.
The effect of viscous dissipation on non-linear convection at high Rayleigh number
Authors: Van der Borght R.
Published: February 2009

When studying deep convection in a compressible medium the effects of viscous dissipation can become important and must be taken into account in any realistic model. But even in shallow convection, for which the Boussinesq approximation is valid, the viscous dissipation effects will become important at high Rayleigh numbers. These effects are estimated with the help of asymptotic methods and the results are compared with those obtained by numerical integration.
A formula for summing divergent series
Authors: Drummond J. E.
Published: February 2009

A formula is given for assigning sums to divergent series where the ratio of adjacent terms varies slowly along the series. This formula consistş of a weighted average of partial sums and is shown to be a general formula which can be easily calculated using a simple recurrence relation. It appears to be more powerful than a repeated Aitken or Shankse1process as long as the transformed series remains divergent and it is also compared with the Padé approximants. It is demonstrated on a factorial series, on a nearly geometric divergent series and for the extrapolation of a velocity formula for small amplitudes of motion.
Aggregation–decomposition and equi-ultimate boundedness
Authors: Kloeden P. E.
Published: February 2009

The aggregation–decomposition method is used to derive a sufficient condition for the equi-ultimate boundedness of large-scale systems governed by nonlinear ordinary differential equations.
A generalisation of the study of sum and square law signal processors with multiple clipped inputs
Authors: Keats R. G. , Yu V. K-K.
Published: February 2009

The recent work of Cheng and Stokes on the processing of clipped signals from two or three receivers is extended and generalised by removing a number of restrictions. In particular, there is no restriction on the number of receivers and the restrictions on the statistical properties of the signal and noise processes have been considerably relaxed.

Mathematically—Plackett's result is used to expand the orthant proabilities involved in increasing powers of the input signal to noise ratio.
On the variational derivation of Einstein's strong field equations
Authors: Gregory L. J. , Klotz A. H.
Published: February 2009

It is shown that the necessary and sufficient condition for the transposition invariance of the field equations derivable from an Einstein-Kaufman variational action principle is the vanishing of xythe vector Γλ. When this condition is satisfied, the field equations become the so-called strong field equations of Einstein. In this sense, the latter can be claimed to follow from the same action principle.
Summing a common type of slowly convergent series of positive terms
Authors: Drummond J. E.
Published: February 2009

If the terms of a series behave like n−k where k is an exactly known constant, a formula using two terms transforms the series into a series of terms like n−k −2 provided k ≠ 1. The multiple use of this transformation is demonstrated in summing three series.
A Boltzmann-type equation in the kinetic theory of vehicular traffic
Authors: Pao C. V.
Published: February 2009

In the theory of vehicular traffic flow on a highway the traffic interaction process is often considered as a collision similar to the particles' interaction in the kinetic theory of gases. This concept leads to a Boltzmann-type nonlinear intergro-differential equation which governs the traffic density function. The purpose of this paper is to present a constructive method for the determintation of a solution for this type of equation under certain boundary and initial conditions. Our method is by successive approximation which yields existence of both global and local solutions of the problem.
Sufficient Fritz John optimality conditions for nondifferentiable convex programming
Authors: Craven B. D. , Mond B.
Published: February 2009

The Fritz John necessary conditions for optimality of a differentiable nonlinear programming problem have been shown, given additional convexity hypotheses, to be also sufficient (by Gulati, Craven, and others). This sufficiency theorem is now extended to minimization (suitably defined) of a function taking values in a partially ordered space, and to (convex) objective and constraint functions which are not always differentiable. The results are expressed in terms of subgradients.
A dam with general release rule
Authors: Yeo Geoffrey
Published: February 2009

A dam is considered with independently and identically distributed inputs occurring in a renewal process, and in particular a Poisson process, with a general release rate r(·) depending on the content. This is related to a GI/G/1 queue with service times dependent on the waiting time. Some results are obtained for the limiting content distribution when it exists; these are more complete for some special release rates, such as r(x) = μxα and r(x) = a + μx, and particular input size distributions.
Energy flux for plane waves in linear conservative systems
Authors: Hayes M.
Published: February 2009

The point form of the conservation of energy equation is used to give simple and direct proof of results concerning the mean energy flux vector for systems of sinusoidal small amplitude waves in linear conservative systems. No constitutive equation is used explicitly.
Homogeneous Newtonian cosmologies and their perturbations
Authors: Szekeres Peter , Rankin John R.
Published: February 2009

A review of Heckmann and Schücking's formulation of Newtonian cosmology is presented, which permits the discussion of models more general than those possessing both homogeneity and isotropy. In particular it is shown that all homogeneous cosmologies may be uniquely specified by the rate of shear tensor as an arbitrary function of time and specifying arbitrary initial values for expansion, rotation and density. Perturbations of these models are now discussed, with a view to their possible implications for galaxy formation. The Jeans criterion is shown to hold in all these models, even in the presence of viscosity; this generalizes a result of Bonnor which only applied to the isotropic case. Furthermore, Bonnor's analysis is considerably simplified in the present paper. Finally, a WKB-type of approximation procedure is described which appears to be successful in estimating the growth rate of unstable fluctuations.
An elementary derivation of Pontrayagin's maximum principle of optimal control theory
Authors: Blatt J. M. , Gray J. D.
Published: February 2009

Pontryagin's maximum principle is derived by elementary mathematical techniques. The conditions on the functions which enter are generally somewhat more stringent than in Pontryagin's derivation, but one (practically very awkward) condition of Pontryagin can be relaxed: continuity in the time variable can be replaced by a much weaker condition.
A note on continuation methods for the solution of nonlinear equations
Authors: Abbott James P. , Brent Richard P.
Published: February 2009

In this note we present a variable order continuation method for the solution of nonlinear equations when only a poor estimate of a solution is known. The method changes continuously from one which improves the global convergence characteristics to one which attains rapid convergence to a solution and proves to be more efficient than methods previously presented in [2].
A monoenergetic-source solution of the steady-state cosmic-ray equation of transport
Authors: Webb G. M.
Published: February 2009

A monoenergetic point-source solution of the steady-state cosmic-ray equation of transport for cosmic-rays in the interplanetary region in which monoenergetic particles are released isotropically and continuously from a fixed heliocentric position is derived by Lie Theory. A spherically-symmetric model of the propagation region is assumed incorporating anisotropic diffusion with a diffusion tensor symmetric about the radial direction, and the solar wind velocity is radial and of constant speed V. Because of the point release the solution is non-spherically-symmetric.
Green's function of the clamped punctured disk
Authors: Nakai Mitsuru , Sario Leo
Published: February 2009

If a thin elastic circular plate B: ∣z∣ < 1 is clamped (simply supported, respectively) along its edge ∣z∣ = 1, its deflection at z ∈ B under a point load at ζ ∈ B, measured positively in the direction of the gravitational pull, is the biharmonic Green's function β(z, ζ) of the clamped plate (γ(z, ζ) of the simply supported plate, respectively). We ask: how do β(z, ζ) and γ(z, ζ) compare with the corresponding deflections β0(z, ζ) and γ0(z, ζ) of the punctured circular plate B0: 0 < ∣ z ∣ < 1 that is “clamped” or “simply supported”, respectively, also at the origin? We shall show that γ(z, ζ) is not affected by the puncturing, that is, γ(·, ζ) = γ0(·, ζ), whereas β(·, ζ) is:

on B0 × B0. Moreover, while β((·, ζ) is of constant sign, β0(·, ζ) is not. This gives a simple counterexmple to the conjecture of Hadamard [6] that the deflection of a clampled thin elastic plate be always of constant sign:

The biharmonic Gree's function of a clampled concentric circular annulus is not of constant sign if the radius of the inner boundary circle is sufficiently small.

Earlier counterexamples to Hadamard's conjecture were given by Duffin [2], Garabedian [4], Loewner [7], and Szegö [9]. Interest in the problem was recently revived by the invited address of Duffin [3] before the Annual Meeting of the American Mathematical Society in 1974. The drawback of the counterexample we will present is that, whereas the classical examples are all simply connected, ours is not. In the simplicity of the proof, however, there is no comparison.
Are wave functions uniquely determined by their position and momentum distributions?
Authors: Corbett J. V. , Hurst C. A.
Published: February 2009

The problem of determining a square integrable function from both its modulus and the modulus of its Fourier transform is studied. It is shown that for a large class of real functions the function is uniquely determined from this data. We also construct fundamental subsets of functions that are not uniquely determined. In quantum mechanical language, bound states are uniquely determined by their position and momentum distributions but, in general, scattering states are not.
Two related estimation problems
Authors: Brisley W.
Published: February 2009

Two problems involving the “best” solution X for a matrix equation V×V* = M are discussed, together with methods for their solution, and a generalization of one of the methods beyond matrix equations.
A duality approach ot discrete time control theory
Authors: Scott C.H. , Jefferson T.R.
Published: February 2009

In this paper we use the theory of generalized geometric programming to develop a dual for a discrete time convex optimal control problem. This has interesting interpretational implications. Further it is shown that the variables in the dual problem are intimately related to the costate vector in the usual Maximum Principle approach.
Discrete Liapunov functions with Δ2V>0
Authors: Diamond Phil
Published: February 2009

Consideration of functions whose second difference along the trajectories of a difference equation is positive gives a stability theorem for autonomous discrete-time systems. Such functions can be used to estimate domains of nonglobal stability.
Maximum and minimum principles for capillary surface problems with prescribed contact angle
Authors: Anderson N. , Arthurs A. M.
Published: February 2009

Maximum and minimum principles for capillary surface problems with prescribed contact angle are derived in a unified manner from canonical variational theory. The results are illustrated by calculations for a liquid in a cylindrical container with circular cross-section.
On a computational algorithm for a class of optimal control problems involving discrete time delayed arguments
Authors: Murray J. M. , Teo K. L.
Published: February 2009

In this paper, a computational algorithm for solving a class of optimal control problems involving discrete time-delayed arguments is presented. By way of example, a simple model of a production firm is devised for which the algorithm is used to solve a decision-making problem.
A class of axially symmetric stationary exact solutions of Einstein's vacuum field equations
Authors: Patel M. D.
Published: February 2009

Einstein's vacuum field equations of an axially symmetric stationary rotating source are studied. Using the oblate spheroidal coordinate system, a class of asymptotically fiat solutions representing the exterior gravitational field of a stationary rotating oblate spheroidal source is obtained. Also it is proved that an analytic axisymmetric and stationary distribution of dust cannot be the source for the gravitational field described by the axisymmetric stationary metric.
Trajectory analysis and extrapolation in barrier function methods
Authors: Jittorntrum Krisorn , Osborne M. R.
Published: February 2009

It has been known for some time that if a certain non-degeneracy condition is satisfied then the successive solution estimates x(r) produced by barrier function techniques lie on a smooth trajectory. Accordingly, extrapolation methods can be used to calculate x(0). In this paper we analyse the situation further treating the special case of the log barrier function. If the non-degeneracy assumption is not satisfied then the approach to x(0) is like r½ rather than like r which would be expected in the non-degenerate case. A measure of sensitivity is introduced which becomes large when the non- degeneracy assumption is close to violation, and it is shown that this sensitivity measure is related to the growth of dix/dri with respect to i for fixed r small enough on the solution trajectory. With this information it is possible to analyse the extrapolation procedure and to predict the number of stages of extrapolation which are useful.
Asymptotic behaviour of a class of discontinuous difference equations
Authors: Fisher M. E. , Kloeden P. E.
Published: February 2009

Sufficient conditions for an equilibrium point to be an attractor or a global attractor are derived for a class of first-order difference equations which need not be continuous at the equilibrium point. These conditions involve Lyapunovlike functions which need not be continuous and are applied to the logistic equation with a piecewise continuous control.
A one-dimensional random walk with repulsion
Authors: Hines D. F. , Thompson C. J.
Published: February 2009

A one-sided one-dimensional random walk with repulsion from the origin is solved exactly. The walk imitates the self-avoiding walk problem insofar as the mean end-to-end distance of an n-step walk tends asymptotically to n as n tends to infinity.
Bifurcation and stability of periodic solutions from a zero eigenvalue
Authors: Landman K. A.
Published: February 2009

A study is made of the branching of time periodic solutions of a system of differential equations in R2 in the case of a double zero eigenvalue. It is shown that the solution need not be unique and the period of the solution is large. The stability of these solutions is analysed. Examples are given and generalizations to larger systems are discussed.
A note on the solution of the one-dimensional unsteady equations of arterial blood flow by the method of characteristics
Authors: Forbes L. K.
Published: February 2009

The “Hartree hybrid method” has recently been employed in one-dimensional non-linear aortic blood-flow models, and the results obtained appear to indicate that shock-waves could only form in distances which exceed physiologically meaningful values. However, when the same method is applied with greater numerical accuracy to these models, the existence of a shock-wave in the vicinity of the heart is predicted. This appears to be contrary to present belief.
On the unsteady squeezing of a viscous fluid from a tube
Authors: Skalak F. M. , Wang C. Y.
Published: February 2009

Viscous fluid is squeezed out from a shrinking (or expanding) tube whose radius varies with time as (1 – βt)½. The full Navier–Stokes equations reduce to a non-linear ordinary differential equation governed by a non-dimensional parameterSrepresenting the relative importance of unsteadiness to viscosity. This paper studies the analytic solutions for large |S| through the method of matched asymptotic expansions. A simple numerical scheme for integration is presented. It is found that boundary layers exist near the walls for large |S|. In addition, flow reversals and oscillations of the velocity profile occur for large negativeS(fast expansion of the tube).
Existence theorems for optimal control of quasilinear parabolic partial differential equations
Authors: Nababan S. , Noussair E. S.
Published: February 2009

The question on existence of optimal controls for a system governed by quasilinear parabolic partial differential equations which is linear in the control variables is considered. It is shown that whenever the controls converge in the weak * topology of L∞, the corresponding solutions converge uniformly. Using this result and results on lower semi-continuity of integral functionals, existence theorems for optimal controls are proved.
A representation result for two-input two-output decentralized control systems
Authors: Clements David J.
Published: February 2009

This paper studies the decentralized control and stabilization of two-input, two-output finite dimensional linear systems. A representation result for the system and a characterization of all stabilizing controllers are given in terms of certain fixed polynomial matrices and a stability constraint.
Optimal diameter assignment for gas pipeline networks
Authors: Bhaskaran Sita , Salzborn Franz J. M.
Published: February 2009

A linear programming model for optimally assigning diameters to a gas pipeline network is discussed. Computational results for a real life situation are presented, and certain properties that have to be satisfied by an optimal assignment are derived.
Aspects of the kinetic equations for a special one-dimensional system
Authors: Evans J. W.
Published: February 2009

Some initial value problems are considered which arise in the treatment of a one-dimensional gas of point particles interacting with a “hard-core” potential.

Two basic types of initial conditions are considered. For the first, one particle is specified to be at the origin with a given velocity. The positions in phase space of the remaining background of particles are represented by continuous distribution functions. The second problem is a periodic analogue of the first.

Exact equations for the delta-function part of the single particle distribution functions are derived for the non-periodic case and approximate equations for the periodic case. These take the form of differential operator equations. The spectral and asymptotic properties of the operators associated with the two cases are examined and compared. The behaviour of the solutions is also considered.
The problem of a rigid punch moving on a viscoelastic half-plane with inertial effects approximately included
Authors: Golden J. M.
Published: February 2009

The problem of an infinitely long rigid punch of uniform cross-section moving across a viscoelastic half-space at constant velocity, large enough so that inertial effects cannot be neglected, is examined and solved in various approximations. Frictional shear is assumed to exist between the punch and the half-space. The method, which is an extension of that developed in previous papers [6, 7], is applicable for any form of viscoelastic behaviour in the half-space. For the special case of discrete spectrum behaviour the method is described in detail. For the case where the punch is cylindrical and viscoelastic effects are small compared with elastic effects, explicit expressions are given for all quantities of interest, in particular the coefficient of hysteretic friction. A general Hilbert transform formula is derived in the appendix.
A note on least-squares approximation with range constraints
Authors: Evans R. J. , Cantoni A.
Published: February 2009

In this note we consider various theoretical aspects of the problem of least-squares approximation subject to constraints on the range of the approximating polynomial. The problem is treated from an optimization theory viewpoint. Rice's parameter space procedure is discussed.
A thermoelastic crack problem for an anisotropic slab
Authors: Clements D. L. , Tauchert T. R.
Published: February 2009

The problem of determining the temperature, displacement and stress fields around a single crack in an anisotropic slab is considered. The problem is reduced to Fredholm integral equations which may be solved numerically.
On a computational algorithm for time-laǵ optimal control problems with restricted phase coordinates
Authors: Teo K. L. , Craven B. D.
Published: February 2009

In this paper we present a computational method for solving a class of time-lag optimal control problems with restricted phase coordinates.
Fractional programming duality via geometric programming duality
Authors: Scott C. H. , Jefferson T. R.
Published: February 2009

A duality theory for a class of fractional programs is developed. A fractionalprogram which is non-convex is convexified using a one-to-one transformation. The resulting convex equivalent is then dualized with generalized geometric programming duality.
The structure of RI-invariant tweleve-tone rows
Authors: Fill James A. , Izenman Alan J.
Published: February 2009

This paper presents an efficient method for generating the class of all twelve-tone rows which are transpositions of their own retrograde-inversions. It is shown here that the members of this class can be obtained from a subclass of those rows whose first six notes are ascending and whose first note is C. The number of twelve-tone rows in this subclass is 192, and a complete listing is given in an appendix to this paper. The theory as developed here can be applied to tone rows having any even number of notes.
Stability of non-linear systems with time-varying delays
Authors: Lewis R. M.
Published: February 2009

A condition guranteeing the stability of linear systems with time delays in the interactions among elements is generalized to cover non-linear systems and discontinuous, unbounded delays.
Stability of large-scale systems with lagged interconnections
Authors: Kloeden P. E.
Published: February 2009

The aggregation-decomposition method is used to derive sufficient conditions for the uniform stability, uniform asymptotic stability and exponential stability of the null solution of large-scale systems described by functional differential equations with lags appearing only in the interconnections. The free subsystems are described by ordinary differential equations for which converse theorems involving Lyapunov functions exist and thus enable the sufficient conditions to be expressed in terms of Lyapunov functions rather than the more complicated Lyapunov functionals.
Equilibrium states of a Bose gas with repulsive interactions
Authors: Bratteli Ola , Robinson Derek W.
Published: February 2009

We consider infinite volume limit Gibbs states of a nonrelativistic quantum Bose gas consisting of one species of spinless particles with positive interaction potentials. The finite volume reduced density matrices are dominated by the corresponding matrices for the noninteracting gas, and as a consequence all infinite volume limit states are regular, locally normal, and analytic on the appropriate CCR algebra. For sufficiently short range repulsive two-body interactions, the cyclic vector associated with the limit state is separating for the σ-weak closure of the algebra in the associated representation.
Limit cycles in two spacies competition with time delays
Authors: Gopalsamy K. , Aggarwala B. D.
Published: February 2009

The existence of stable periodic oscillatory solutions in a two species competition model with time delays is established using a combination of Hopf-bifurcation theory and the asymptotic method of Krylov, Bogoliuboff and Mitropoisky.
The optimum processing of clipped signals: An approach based on minimum signal distortion
Authors: Keats R. G. , Cooper Joan
Published: February 2009

The minimization of signal distortion was one approach applied successfully in the theory of optimum signal detection for arrays [3]. The processors considered operated on the input as received.

In some applications it is desirable to clip the received signal before processing and the problem of optimum processing of such clipped signals then arises. Several approaches to this problem are being studied, but the present paper is concerned with that based on minimum signal distortion.
About a local approximation theorem and an inverse function theorem
Authors: Nieuwenhuis J. W.
Published: February 2009

This paper gives a theorem by which it is possible to derive in an easy way a local approximation theorem and an inverse function theorem. The latter theorems are not new. The main advantage of our paper is in giving a relatively short route to these results.
Solution of homogeneous linear difference equations
Authors: Love J. D.
Published: February 2009

When the first two elements of a sequence satisfying a second order difference equation are prescribed, the remaining elements are evaluated from a continued fraction and a simple product.
Hypercircle estimates for nonlinear problems
Authors: Arthurs A. M.
Published: February 2009

Recent hypercircie estimates for non-linear equations are extended to include a new class of boundary value problems of monotone type. The results are illustrated by the boundary value problem for the equilibrium-free surface of a liquid with prescribed contact angle.
Scale covariant Lagrangians and spaces reciprocal to static Einstein spaces
Authors: Buchdahl H. A.
Published: February 2009

The main aim of this paper is to derive a condition whose satisfaction ensures that the Euler–Lagrange equations of a non-linear scale covariant Lagrangian are satisfied by the metric of a space reciprocal to an arbitrary static Einstein space.
A note on maximal and minimal causal spaces
Authors: Nanda Sribatsa
Published: February 2009

It is shown in this note that the supremum and infimum of all causal topologies on Minkowski space are not causal. It is further shown that maximal and minimal elements exist in the set of all principal causal spaces.
A theoretical investigation of an asymmetric planing hull at infinite Froude number
Authors: Casling E. M.
Published: February 2009

The extent to which an asymmetric low-aspect-ratio flat ship is wetted when planing at infinite Froude number is investigated, with emphasis placed on its relationship with the shape of the hull. Two cases are considered. First the hull is assumed to have two laterally-asymmetric leading edges and, secondly, the hull is assumed to be yawed sufficiently for one of the leading edges to become a trailing edge. In the first case, the relationship involves a pair of coupled integral equations, but in the second case there is a complication by the occurrence of hull-wake interaction.
On the evolution of shock-waves in mathematical models of the aorta
Authors: Forbes L. K.
Published: February 2009

The one-dimensional, non-linear theory of pulse propagation in large arteries is examined in the light of the analogy which exists with gas dynamics. Numerical evidence for the existence of shock-waves in current one-dimensional blood-flow models is presented. Some methods of suppressing shock-wave development in these models are indicated.
On error bounds in strong approximations for eigenvalue problems
Authors: Kulkarani R. P. , Limaye B. V.
Published: February 2009

Some corrections of error bounds obtained by Chatelin and Lemordant for the first three terms of the asymptotic case of a strong approximation are given. The error bounds for the approximations of order 2 in the Galerkin method are compared with the Rayleigh quotients constructed with the eigenvectors in the Sloan method. A numerical experiment is also carried out.
Solution of a general liner difference equation
Authors: Rizvi S. A. H.
Published: February 2009

A matrix solution and a determinantal solution are obtained for a general linear recurrence relation.
Mean flows generated by a progressing water wave packert
Authors: Grimshaw R.
Published: February 2009

Equations are derived which describe the evolution of the mean flow generated by a progressing water wave packet. The effect of friction is included, and so the equations are subject to the boundary conditions first derived by Longuet-Higgins [10]. Solutions of the equations are obtained for a wave packet of finite length, and also for a uniform wave train. The latter solution is compared with experiments.
Love's integral and other relations between solutions to mixed boundary-value problems in potential theory
Authors: England A. H.
Published: February 2009

In a paper published in 1949, E. R. Love [10] found an integral relation between a separated solution (in oblate spheroidal coordinates) to a particular mixed boundary-value problem and a solution to the same problem using an integral representation. This note examines further representations of the same type which occur in some simple two and three-dimensional potential problems.
The boundary integral equation method for the solution of a class of problems in anisotropic elasticity
Authors: Clements David L. , Jones Oscar A. C.
Published: February 2009

A boundary integral procedure for the solution of an important class of problems in anisotropic elasticity is outlined. Specific numerical examples are considered in order to provide a comparison with the standard boundary integral method.
Deferred corrections for equations of the second kind
Authors: Qun Lin
Published: February 2009

A deferred correction procedure for the approximate solution of the second-kind equation is introduced, compared with an extrapolation procedure, and illustrated for integral and differential equations.
A formal comparison of methods proposed for the numerical solution of first kind integral equations
Authors: Anderssen R. S. , Prenter P. M.
Published: February 2009

A formal framework is constructed for the comparison of different stabilization techniques, such as Wiener filtering, regularization, Courant's method and Landweber–Strand iterations, for the solution of first kind integral equations. It is shown that, when they are applied to convolution equations, all these methods can be reinterpreted as Wiener filters. This equivalence is then used to derive some specific results about regularization, Courant's method and Landweber–Strand iteration.
Subsonic potential flow past a circle and the transonic controversy
Authors: Van Dyke M. D. , Guttmann A. J.
Published: February 2009

The Mach-number series expansion of the potential function for the two-dimensional flow of an inviscid, compressible, perfect, diatomic gas past a circular cylinder is obtained to 29 terms. Analysis of this expansion allows the critical Mach number, at which flow first becomes locally sonic, to be estimated as M* = 0.39823780 ± 0.00000001. Analysis also permits the following estimate of the radius of convergence of the series for the maximum velocity to be made: Mc = 0.402667605 ± 0.00000005, though we have been unable to determine the nature of the singularity of M = Mc. Since Mc exceeds M* by some 1.1%, it follows that this particular “airfoil” can possess a continuous range of shock-free potential flows above the critical Mach number. This result hopefully resolves a 70-year old controversy.
Thermal ignition in a reactive slab with unsymmetric boundary temperatures
Authors: Tam K. K. , Chapman P. B.
Published: February 2009

The problem of thermal ignition in a reactive slab with unsymmetric temperatures equal to 0 and T is considered. Steady state upper and lower solutions are constructed. It is found that T plays a critical role. Results similar to the case with symmetric boundary temperatures are expected when T is small. When T is sufficiently large, there is only one steady state upper or lower solution. The time dependent problem is then considered. Phenomena suggested by studying the upper and lower steady state solutions are confirmed.
On a two lag differential delay equation
Authors: Braddock R. D. , van den Driessche P.
Published: February 2009

The non-linear differential difference equation of the form

is investigated. This equation, with constant coefficients, is used to model the population level,N, of a single species, and incorporates two constant time lagsT2>T1> 0; for example, regeneration and reproductive lags. The linear equation is investigated analytically, and some linear stability regions are described. The special case in which the two delay terms are equally important in self damping,B = C, is investigated in detail. Numerical solutions for this case show stable limit cycles, with multiple loops appearing whenT2/T1is large. These may correspond to splitting of major peaks in population density observations.
On interpreting the sums of asymptotic series of positive terms
Authors: Drummond J. E.
Published: February 2009

Four different kinds of positive asymptotic series are identified by the limiting ratio of successive terms. When the limiting ratio is 1 the series is unsummable. When the ratio tends rapidly to a constant, whether greater or less than 1, the series is easily summed. When the ratio tends slowly to a constant not equal to 1 the series is compared with a binomial model which is then used to speed the convergence. When the ratio increases linearly, a limiting binomial and an exponential integral model are both used to speed convergence. The two resulting model sums are consistent and in this case are complex numbers. Truncation at the smallest term is found to be unreliable in the second case, invalid in the third case, and the exponential integral is used to produce a significantly improved truncation in the third case. A divergent series from quantum mechanics is also examined.
Entry flow in a straight circular pipe
Authors: Sinha P. C. , Aggarwal Meena
Published: February 2009

Here we discuss the development of the laminar flow of a viscous incompressible fluid from the entry to the fully developed situation in a straigbt circular pipe. Uniform entry conditions are considered and the analysis is based on the method of matched asymptotic expansions.
Exchange of equilibria in two species Lotka-Volterra competition models
Authors: Gopalsamy K.
Published: February 2009

Sufficient conditions are obtained for the existence of a unique asymptotically stable periodic solution for the Lotka-Volterra two species competition system of equations when the intrinsic growth rates are periodic functions of time.
Moving boundary problems in the flow of liquid through porous media
Authors: Lacey A. A.
Published: February 2009

The movement of the interface between two immiscible fluids flowing through a porous medium is discussed. It is assumed that one of the fluids, which is a liquid, is much more viscous than the other. The problem is transformed by replacing the pressure with an integral of pressure with respect to time. Singularities of pressure and the transformed variable are seen to be related.

Some two-dimensional problems may be solved by comparing the singularities of certain analytic functions, one of which is derived from the new variable. The implications of the approach of a singularity to the moving boundary are examined.
On the explosion of chain-thermal reactions
Authors: Ayeni R. O.
Published: February 2009

A chain reaction of oxygen (reactant) and hydrogen (active intermediary) with mtrosyl chloride (sensitizer) as a catalyst may be modelled mathematically as a non-isothermal reaction. In this paper we present an asymptotic analysis of a spatially homogeneous model of a non-isothermal branched-chain reaction. Of particular interest is the so-called explosion time and we provide an upper bound for it as a function of the activation energy which can vary over all positive values. We also establish a bound on the temperature when the activation energy is finite.
Converging factors for some asymptotic moment series that arise in numerical quadrature
Authors: Sidi Avram
Published: February 2009

In this work the asymptotic behavior of the partial sums of the divergent asymptotic moment series, where μiare the moments of the weight functionsw(x) =xαe−x, α > −1, andw(x) =xαEm(x), α > −1,m+ α > 0, on the interval [0, ∞), is analyzed. Expressions for the converging factors are derived by the author for the infinite range integraswithw(x) as given above.
Steady states of the reaction-diffusion equations. Part 1: Questions of existence and continuity of solution branches
Authors: Burnell J. G. , Lacey A. A. , Wake G. C.
Published: February 2009

When material is undergoing an exothermic chemical reaction which is sustained by the diffusion of a reactant, the steady-state regime is governed by a coupled pair of nonlinear elliptic partial differential equations with linear boundary conditions. In this paper we consider questions of existence of solutions to these equations. It is shown that, with the exception of the special case in which the mass-transfer is uninhibited on the boundary, a solution always exists, whereas in this special case a solution exists only for sufficiently low values of the exothermicity. Bounds are established for the solutions and the occurrence of minimal and maximal solutions is shown for some cases. Finally the behaviour of the solution set with respect to one of the parameters is studied.
Solution of general inhomogeneous linear difference equations
Authors: Van Hoa H.
Published: February 2009

The general solution of the rth inhomogeneous linear difference equation is given in the form

The coefficients , i = 2, …, r, and b(n−r)(n) can be evaluated from n values , k = 0, …, n − 1, which santisfy an rth order homogenous linear difference equation. In the rth order homogeneous case and if n ≥ 2r, the method requires the evaluation of r determinants of successive orders n − 2r + 1, n − 2r + 2, …, n − r. If r ≤ n ≤ 2r − 1, only n − r determinants are required, with orders varying from 1 to n − r. In the second order ihnomogenous case, can be evaluated from a continued fraction amd a simple product.
Some elementary exact channel flows
Authors: de Mestre Neville , Pakes Trevor
Published: February 2009

New polynomial solutions of the Navier-Stokes equations for steady uni-directional flow of viscous incompressible fluid, with a free surface, down inclined channels of specialized cross-section are considered. An inverse method is uded to obtain the geometrical shape of the channel by equating the polynomials solution to zero (i.e. the no-slip condition) and thence determining the boundary shape.
Cyclic gravity waves in deep water
Authors: Bryant P. J.
Published: February 2009

Numerical evidence is presented for the existence of unsteady periodic gravity waves of large height in deep water whose shape changes cyclically as they propagate. It is found that, for a given wavelength and maximum wave height, cyclic waves with a range of cyclic periods exist, with a steady wave of permanent shape being an extreme member of the range. The method of solution, using Fourier transforms of the nonlinear surface boundary conditions, determines the irrotational velocity field in the water and the water surface displacement as functions of space and time, from which properties of the waves are demonstrated. In particular, it is shown that cyclic waves are closer to the point of wave breaking than are steady permanent waves of the same wave height and wavelength.
Water waves, nonlinear Schrödinger equations and their solutions
Authors: Peregrine D. H.
Published: February 2009

Equations governing modulations of weakly nonlinear water waves are described. The modulations are coupled with wave-induced mean flows except in the case of water deeper than the modulation length scale. Equations suitable for water depths of the order the modulation length scale are deduced from those derived by Davey and Stewartson [5] and Dysthe [6]. A number of ases in which these equations reduce to a one dimensional nonlinear Schrödinger (NLS) equation are enumerated.

Several analytical solutions of NLS equations are presented, with discussion of some of their implications for describing the propagation of water waves. Some of the solutions have not been presented in detail, or in convenient form before. One is new, a “rational” solution describing an “amplitude peak” which is isolated in space-time. Ma's [13] soli ton is particularly relevant to the recurrence of uniform wave trains in the experiment of Lake et al.[10].

In further discussion it is pointed out that although water waves are unstable to three-dimensional disturbances, an effective description of weakly nonlinear two-dimensional waves would be a useful step towards describing ocean wave propagation.
The anti-cyclonic shear wave: a new geophysical wave
Authors: Hughes Roger L.
Published: February 2009

A new type of first baroclinic mode wave which propagates on an anti-cyclonic vorticity field in identified. It is of the vorticity class of waves which contains Rossby waves amongst others. This anti-cyclonic shear wave is produced by pressure variations distorting the vertical stratification in such a manner that the associated vortex stretching generates the velocity variation required for Bernoulli compatibility with the initial pressure variation. The wave travels at a speed characteristic of particles within the undisturbed shear flow and is a low frequency and low wavenumber wave, In the present study this wave is considered in the presence of a wark anti-cyclonic shear.
Stationary waves forced by topography in a vertically sheared, stratified, rotating fulid
Authors: Fandry C. B. , Hughes R. L. , Leslie L. M.
Published: February 2009

The effect of an isolated topographic bump in a two-layer fluid on a β-plane is investigated. An analytical solution is derived in terms of the appropriate Green's function for arbitrary topography of finite horizontal extent. It is found that the disturbances generated by the bump are composed of two fundamental modes which may be wave-like or evanescent. The wave-like modes are topographically induced Rossby waves which occur only when there is eastward flow in at least one of the layers. These waves are always confined to the downstream (eastward) side of the bump. Whereas previous studies of this type have concentrated on eastward flow over topography, the theory has been extended here to include a wide range of vertically sheared flows. Particularly important is the case of low level westward flow combined with upper level eastward flow, as it has direct application, for example, to the summertime atmospheric circulation over the sub-tropical regions of the continental land.masses. In this case a wave-like disturbance extends far downstream from the bump for sufficiently large shear, and is of smaller amplitude in the upper layer than in the lower layer because of the effects of the stratification. For small shears, the wave-like mode in the lower layer is small and the character of the disturbance is evanescent, confining it to the immediate neighbourhood of the bump. A stability analysis of the solutions shows that the disturbances may be baroclinically unstable for sufficiently large mean shear.
Inversion of the Struve transform of half integer order
Authors: McKellar B. H. J. , Box M. A. , Love E. R.
Published: February 2009

Defining a spherical Struve function we show that the Struve transform of half integer order, or spherical Struve transform,

where n is a non-negative integer, may under suitable conditions be solved for f(t):

where is the sum of the first n + 1 terms in the asymptotic expansion of φn(x) as x → ∞. The coefficients in the asymptotic expansion are identified as

It is further shown that functions φn (x) which are representable as spherical Struve transforms satisfy n + 1 integral constraints, which in turn allow the construction of many equivalent inversion formulae.
A discrete Fourier analysis of coarse mesh rebalancing and some associated iterative methods
Authors: Barry J. M. , Jenkinson J. H. , Pollard J. P.
Published: February 2009

Iterative methods for solving systems of linear equations may be accelerated by coarse mesh rebalance techniques. The iterative technique, the Method of Implicit Non-stationary Iteration (MINI), is examined through a local-mode Fourier analysis and compared to relaxation techniques as a potential candidate for such acceleration. Results of a global-mode Fourier analysis for MINI, relaxation methods, and the conjugate gradient method are reported for two test problems.