bluuebunny/crossref_metadata_2025 huggingface dataset:
Iterative solution of quasilinear parabolic equations by parabolic equations with constant coefficients
Authors: Lavery John E.
Published: February 2009

A method for solving quasilinear parabolic equations of the types

that differs radically from previously known methods is proposed. For each initial-boundary-value problem of one of these types that has boundary conditions of the first kind (second kind), a conjugate initial-boundary-value problem of the other type that has boundary conditions of the second kind (first kind) is defined. Based on the relations connecting the solutions of a pair of conjugate problems, a series of parabolic equations with constant coefficients that do not change step to step is constructed. The method proposed consists in calculating the solutions of the equations of this series. It is shown to have linear convergence. Results of a series of numerical experiments in a finite-difference setting show that one particular implementation of the proposed method has a smaller domain of convergence than Newton's method but that it sometimes converges faster within that domain.
Insulating circular rugs
Authors: Weir G. J.
Published: February 2009

A thin, partially insulating circular rug is placed on a uniform half space up through which a steady heat flow passes. The corresponding dual integral equations are solved using Tranter's method, finite Legendre transforms and Mellin-Bames contour integrals. An untabulated Bessel (or Stieltjes) transform similar to the discontinuous WeberSchafheitlin integral is evaluated, and a simple expression derived for the rug's surface temperature.
Pointwise bounds for a solution of a nonlinear boundary value problem in partial differential equations
Authors: Anderson N. , Arthurs A. M.
Published: February 2009

Pointwise bounds are obtained for the solution of a Dirichlet problem involving the nonlinear Liouville equation in the plane, Illustrative calculations are performed for a square domain.
Steady states of the reaction-diffusion equations. Part II: Uniqueness of solutions and some special cases
Authors: Burnell J. G. , Lacey A. A. , Wake G. C.
Published: February 2009

In an earlier paper (Part I) the existence and some related properties of the solution to a coupled pair of nonlinear elliptic partial differential equations was considered. These equations arise when material is undergoing an exothermic chemical reaction which is sustained by the diffusion of a reactant. In this paper we consider the range of parameters for which the uniqueness of solution is assured and we also investigate the converse question of multiple solutions. The special case of the one dimensional shape of the infinite slab is investigated in full as this case admits to solution by integration.
On maximum principles for diffusion in the presence of three diffusion paths
Authors: Lee Alexander I. , Hill James M.
Published: February 2009

This note examines maximum principles for systems of parabolic partial differential equations describing diffusion in the presence of three diffusion paths. The particular system under consideration arises from a random walk model. For a more general system constraints on the various constants are given which guarantee maximum principles. Remarkably, the physical system arising from the random walk model automatically satisfies these constraints.
Non-linear optimization of the material constants in Ogden's stress-deformation function for incompressinle isotropic elastic materials
Authors: Twizell E. H. , Ogden R. W.
Published: February 2009

In previous papers, three terms have been included in Ogden's stress-deformation function for incompressible isotropic elastic materials. The material constants have been calculated by elementary methods and the resulting fits to sets of experimental data have been moderately good.

The purpose of the present paper is to improve upon established correlation between theory and experiment by means of a systematic optimization procedure for calculating material constants. For purposes of illustration the Levenberg-Marquardt non-linear least squares optimization algorithm is adapted to determine the material constants in Ogden's stress-deformation function.

The use of this algorithm for three-term stress-deformation functions improves somewhat on previous results. Calculations are also carried out in respect of a four-term stress-deformation function and further improvement in the fit is achieved over a large range of deformation.
Hearing the shape of an annular drum
Authors: Gottlieb H. P. W.
Published: February 2009

The asymptotic expansion for a spectral function of the Laplacian operator, involving geometrical properties of the domain, is demonstrated by direct calculation for the case of a doubly-connected region in the form of a narrow annular membrane. By utilizing a known formula for the zeros of the eigenvalue equation containing Bessel functions, the area, total perimeter and connectivity are all extracted explicitly.
Existence theorems for Segal quantization via spectral theory in Krein space
Authors: Broadbridge P.
Published: February 2009

Segal's unitarizing complex structure J is shown, in the Fermi-Dirac case, to be the orthogonal component in the polar decomposition of the real skew adjoint generator of classical dynamics. It is proven that in the Bose-Einstein case, the classical symplectic dynamics cannot be unitarized unless the generator is similar to a real skew adjoint operator.

With the classical Hamiltonian strictly positive, J is the pseudo-orthogonal component in the polar decomposition of the generator, using spectral theory in Krein space with indefinite metric. Thus, J can be expressed simply in terms of the projection E(0) onto the subspace of classical solutions with negative frequency. This complements the physicists' experience that conceptual difficulties arise when dynamically invariant separation of positive and negative frequency solutions is impossible.
Field of a charged particle in the presence of scalar meson fields in general relativity
Authors: Reddy D. R. K. , Rao V. U. M.
Published: February 2009

Field equations for coupled gravitational and zero mass scalar fields in the presence of a point charge are obtained with the aid of a static spherically symmetric conformally flat metric. A closed from exact solution of the field equations is presented which may be considered as describing the field of a charged particle at the origin surrounded by the scalar meson field in a flat space-time.
Scattering analysis and synthesis of wave trains
Authors: Sobey R. J. , Colman E. J.
Published: February 2009

The Zakharov-Shabat scattering transform is an exact solution technique for the nonlinear Schrödinger equation, which describes the time evolution of weakly nonlinear wave trains. Envelope soliton and uniform wave train solutions of the nonlinear Schrödinger equation are separable in scattering transform space. The scattering transform is a potential analysis and synthesis technique for natural wave trains. Discrete versions of the direct and inverse scattering transform are presented, together with proven algorithms for their numerical computation from typical ocean wave records. The consequences of discrete resolution are considered.
The generation of surface waves by an intense cyclone
Authors: Viera F. , Buchwald V. T.
Published: February 2009

Assuming a travelling oscillating pressure source model, this paper sets out to investigate the observation of surface gravity waves generated by a cyclone moving with constant speed v. It is shown that when the source frequency is near the critical resonant value g/4ν, large amplitude waves may be generated. There is some agreement with observations of waves from cyclone Pam of February, 1974.
Surface-wave interaction with a deeply submerged circular duct
Authors: Miles John W.
Published: February 2009

The interaction of a surface wave of angular frequency ω with a deeply submerged, vertical open-mouthed, circular duct of radius a is considered. The resulting boundary- value problem is solved by the Wiener-Hopf technique. The pressure-amplification factor (the ratio of the complex amplitude of the pressure in the depths of the duct to that of the incident wave in the plane of the mouth) is determined in closed form as a function of the dimensionless wave number K = ω2a/g.
Interaction of internal waves in a continuous thermocline model
Authors: Hung N. T. , Maslowe S. A.
Published: February 2009

Weak nonlinear interactions are studied for systems of internal waves when the Brunt-Väisälä frequency is proportional to sech z, where z = 0 is the centre of the thermocline. Explicit results expressed in terms of gamma functions have been obtained for the interaction coefficients appearing in the amplitude evolution equations. The cases considered include resonant triads as well as second and third harmonic resonance. In the non-resonant situation, the Stokes frequency correction due to finite-amplitude effects has been computed and the extension to wave packets is outlined. Finally, the effect of a mean shear on resonant interactions is discussed.
Low Reynolds number flow in a heated tube of varying section
Authors: Bestman A. R.
Published: February 2009

In this paper we study the effect of forced and free convection heat transfer on flow in an axisymmctric tube whose radius varies slowly in the axial direction. Asymptotic series expansions in terms of a small parameter ∈, which is a measure of the radius variation, are obtained for the velocity components, pressure and temperature on the assumption that the Reynolds number (R) is of order one. The effect of the free convection parameter or Grashof number (G) on the axial velocity, temperature distribution, shear stress and heat flux at the wall are discussed quantitatively for a locally constricted tube.
The propagation of small amplitude long waves on the surface of superfluid helium
Authors: Johnson R. S.
Published: February 2009

Various problems related to the propagation of small amplitude long waves on the surface of superfluid helium (helium II), usually called third sound, are studied on the basis of the appropriate governing equations. The two-fluid continuum model due to Landau is considered, with the effects of healing and relaxation incorporated, and viscosity, heat conduction and compressibility terms retained. The helium vapour is treated as a classical (Newtonian) compressible gas and the exact jump conditions across the liquid/vapour interface are employed. These liquid, vapour and jump equations constitute the exact problem although, in an effort to reduce the complexity of the equations, a simplified set of ‘model’ surface boundaiy conditions is also introduced. This full set of equations is non-dimensionalised taking care that all physical parameters are defined using only the undisturbed depth of the layer as the appropriate length scale. The ratio depth/wave-lenght (δ) is then a separate parameter as is the wave amplitude/depth ratio (ɛ). The limit which corresponds to the wave under discussion is then ɛ, δ → 0 with all the other parameters fixed.

A number of analyses are presented, four of which describe various aspects of the linearised theory and two examine the nature of the far-field nonlinear problem. Using the simplified surface boundary conditions we discuss in turn: the wave motion in the absence of healing: the rôle of a second wave speed leading to a wave hierarchy; and the effects of healing. The final linearised problem makes use of the full vapour model, but again the healing terms are ignored. This latter analysis suggests that if the upper boundary of the vapour is sufficiently close to the liquid surface then third sound is suppressed.

The complexity of the equations, particularly when the nonlinear terms are to be examined, is such that the incompressible limit is now taken in the absence of both healing and relaxation. Imposing the physically realistic limiting process (ɛ, δ → 0) we show that the only equation valid in the far-field is the Burgers equation. However, we also demonstrate that allowing the other parameters to be functions of e (which is not physically realisable in practice) it is easy to derive, for example, the Korteweg-de Vries equation.
On a complementarity problem associated with nondifferentiable programming
Authors: Craven B. D. , Mond B. , Parida J.
Published: February 2009

This paper deals with the question of the existence of a solution to the stationary-point problem corresponding to a given nonlinear nondifferentiable program. An existence theorem for the stationary-point problem is presented under some convexity and regularity conditions on the functions involved, which also guarantee an optimal solution to the nonlinear program.
A cusp-like free-surface flow due to a submerged source or sink
Authors: Tuck E. O. , Broeck J.-M. Vanden
Published: February 2009

A solution is found for a line source or sink beneath a free surface, at a unique squared Froude number of 12.622.
solutions of period three for a non-linear difference equation
Authors: Brown A.
Published: February 2009

The paper uses the factorisation method to discuss solutions of period three for the difference equation

which has been proposed as a simple mathematical model for the effect of frequency dependent selection in genetics. Numerical values are obtained for the critical values of a at which solutions of period three first appear. In addition, the interval in which stable solutions are possible has been determined. Exact solutions are given for the case a = 4 and these have been used to check the results.
Delayed responses and stability in two-species systems
Authors: Gopalsamy K.
Published: February 2009

It is shown that if intraspecific self-regulating negative feedback effects are strong enough such that a nontrivial steady state of a two species system is locally asymptotically stable, then time delays in the positive feedback as well as in other interspecific interactions cannot destabilise the system and hence delay induced instability leading to persistent oscillations is impossible whatever the magnitude of the time delays. A method is also proposed for an estimate of decay rate of perturbations.
On the design of mortgages and the need for indexation
Authors: Gair F. C.
Published: February 2009

There is a well-recognized need for a mortgage instrument that will operate satisfactorily in the presence of a volatile inflation. This paper analyses a large class of mortgages—the ‘continuous mortgage’ (CM)—as a basis for such design. In particular, it is shown that the real (inflation-adjusted) payment stream is exponentially sensitive to changes in the real interest rate. Consequently to realize a satisfactory design, the mortgage must be indexed by assigning the real interest rate. Then, under appropriate restraints, the CM offers a continuum of satisfactory mortgage designs.
An observation problem for the Bessel differential operator
Authors: Werner K.-D.
Published: February 2009

In this paper, the parabolic partial differential equation ut = urr + (1/r)ur − (v2/r2)u, where v ≥ 0 is a parameter, with Dirichlet, Neumann, and mixed boundary conditions is considered. The final state observability for such problems is investigated.
The diffraction of long elastic waves by elliptic cylindrical cavities
Authors: Buchwald V. T. , Cong T. Tran
Published: February 2009

The method of asymptotic matching introduced by Buchwald [I] is adapted to the case of the diffraction of plane longitudinal and shear waves by cylindrical cavities with elliptic cross-sections. It is assumed that the dimensions of the cross section are small compared with the wavelength of the incident waves. Asymptotic formulae for the scattered wave potentials are obtained.

The method is valid when the cavity reduces to a two-dimensional stress free crack whose length is small compared with the wavelength. Formulae for the scattered waves, and for the stress-concentrations at the crack tips are obtained.
New algorithms for discrete-time optimal control problems
Authors: Nedeljković Nikola B.
Published: February 2009

The paper presents new demonstrably convergent first-order iterative algorithms for unconstrained discrete-time optimal control problems. The algorithms, which solve the linear-quadratic problem in one iterative step, are particularly suited for solving nonlinear problems with linear constraints via penalty function methods. The proofs of the reduction of cost at each iteration and convergence of the algorithms are provided.
Solutions of period four for a non-linear difference equation
Authors: Brown A.
Published: February 2009

The paper extends earlier work by using the factorisation method to discuss solutions of period four for the difference equation

This equation was suggested by R. M. May as a simple mathematical model for the effect of frequency-dependent selection in genetics. It is shown that for a given value of the parameter, a, the identification of solutions of period four can be reduced to finding real roots for a polynomial equation of degree eight. The appropriate values of xn follow from a quartic equation. By splitting up the problem in this way it becomes relatively straightforward to determine the critical values of a at which the various solutions of period four first appear and to discuss the stability of these solutions. Intervals of stability are tabulated in the paper.
Unsteady waves on an open two layer fluid
Authors: Sharp P. W.
Published: February 2009

The evolution of small amplitude waves on an open two layer fluid is investigated. The spatially periodic surface and interface displacements are represented as Fourier series with time dependent coefficients, for which evolution equations with all significant quadratic interactions included, are derived. Solutions to these equations are found analytically for a small number of harmonics, and numerically for a larger number of harmonics. Two numerical solutions are given to illustrate the evolution properties.
Notes on Sraffa's fixed capital model
Authors: Woods J. E.
Published: February 2009

This paper considers a single-product industries, fixed capital model discussed briefly by Sraffa in Chaapter X if his book. The analysis is elementary, being based on the direct calculation of a certain martrix inverse. This generalises the approach adopted for circulating capital models. It also demonstrates that common economic and mathematical frameworks exist for both circulating and fixed capital models.
On the approximation order from certain multivariate spline spaces
Authors: Dahmen Wolfgang , Micchelli Charles A.
Published: February 2009

In this paper, we determine the optimal controlled approximation rates from certain bivariate splines on regular meshes.
Eigenvalues of the Laplacian with Neumann boundary conditions
Authors: Gottlieb H. P. W.
Published: February 2009

Various grometrical properties of a domain may be elicited from the asymptotic expansion of a spectral function of the Laplacian operator for that region with apporpriate boundary conditions. Explicit calculations, using analytical formulae for the eigenvalues, are performed for the cases fo Neumann and mixed boundary conditions, extending earlier work involving Dirichet boundary conditions. Two- and three-dimensional cases are considered. Simply-connected regions dealt with are the rectangle, annular sector, and cuboid. Evaluations are carried out for doubly-connected regions, including the narrow annulus, annular cylinder, and thin concentric spherical cavity. The main summation tool is the Poission summation formula. The calculations utilize asymptotic expansions of the zeros of the eigenvalue equations involving Bessel and related functions, in the cases of curved boundaries with radius ratio near unity. Conjectures concerning the form of the contributions due to corners, edges and vertices in the case of Neumann and mixed boundary conditions are presented.
The lumped mass finite element method for a parabolic problem
Authors: Chen C. M. , Thomée V.
Published: February 2009

For the heat equation in two space dimensions we consider semidiscrete and totally discrete variants of the lumped mass modification of the standard Galerkin method, using piecewise linear approximating functions, and demonstrate error estimates of optimal order in L2 and of almost optimal order in L∞.
A canonical form and solution for the matrix Riccati differential equation
Authors: Leipnik R. B.
Published: February 2009

A canonical form of the self-adjoint Matrix Riccati Differential Equation with constant coefficients is obtained in terms of extremal solutions of the self-adjoint Matrix Riccati Algebraic (steady-state) Equations. This form is exploited in order to obtain a convenient explicit solution of the transient problem. Estimates of the convergence rate to the steady state are derived.
Families of three-stage third order Runge-Kutta-Nyström methods for y″ = f (x, y, y′)
Authors: Chawla M. M. , Sharma S. R.
Published: February 2009

In this paper all three-stage third order (explicit) Runge-Kutta-Nyström (R-K-N) methods for y″ = f (x, y, y′) are presented. While determining particular methods we require that when these methods are applied to the test equation: y″ − (α + β) y ′ + αβy = 0, the measure of the relative error F, introduced by Rutishauser [4], should not deteriorate in the case of equal eigenvalues (β → α). Further, we require that when these methods are applied to special differential equations y″ = f(x, y) they should possess either of the two properties: (P1) a method remains of order three but is two-stage, (P2) a method remains three-stage but attains order four. We present new R-K-N methods which are stabilized in the sense of Ruthishauser [4] and which possess the property (P1). (There does not exist any three-stage third order R-K-N method which is stabilized and which possesses the property (P2).)
Nonlinear programming duality and matrix game equivalence
Authors: Chandra S. , Craven B. D. , Mond B.
Published: February 2009

Certain well known results on linear programming duality and matrix game equivalence are extended to nonlinear and fractional programming problems.
Reflection of oblique shock waves in incompressible elastic solids
Authors: Duszczyk B. , Kosinski S. , Wesolowski Z.
Published: February 2009

Using a semi-inverse method proposed by Wright the reflection of a finite elastic plane shock wave at a plane boundary of a special elastic incompressible material is examined. Three types of boundary conditions are considered. In the case of frictionless-rigid boundary the reflected wave is a single simple wave. For clamped boundary the solution indicates a possibility of irregular reflection as well. There is no reflection solution in the case of a free boundary.
Global asymptotic stability in a periodic Lotka-Volterra system
Authors: Gopalsamy K.
Published: February 2009

A set of easily verifiable sufficient conditions are obtained for the existence of a globally asymptotically stable periodic solution in a Lotka-Volterra system with periodic coefficients.
On the general random walk formulation for diffusion in media with Diffusivities
Authors: Hill James M. , Hughes Barry D.
Published: February 2009

A general discrete multi-dimensional and multi-state random walk model is proposed to describe the phenomena of diffusion in media with multiple diffusivities. The model is a generalization of a two-state one-dimensional discrete random walk model (Hill [8]) which gives rise to the partial differential equations of double diffusion. The same partial differential equations are shown to emerge as a special case of the continuous version of the present general model. For two states a particular generalization of the model given in [8] is presented which is not restricted to nearest neighbour transitions. Under appropriate circumstances this two-state model still yields the partial differential equations of double diffusion in the continuum limit, but an example of circumstances leading to a radically different continuum limit is presented.
The generalized Mayer theorem in the approximating Hamiltonian method
Authors: Bakulev A. P. , Bogolubov N. N. , Kurbatov A. M.
Published: February 2009

With the help of the generalized Mayer theorem we obtain an improved inequality for free energies of model and approximating systems, where only “connected parts” over the approximating Hamiltonian are taken into account. For a concrete system we discuss the problems of convergence of appropriate series of “connected parts ”.
Simple nonlinear dual control problems
Authors: Murray J. M.
Published: February 2009

In this paper we consider a simple, nonlinear optimal control problem with sufficient convexity to enable us to formulate its dual problem. Both primal and dual problems will include constraints on both the states and controls. The constraints in one problem may cause the “optimal” dual states to be discontinuous. However, we will look at conditions under which the presence of constraints does not force discontinuities and the optimal states and costates are absolutely continuous.
A non-linear difference equation with two parameters. II
Authors: Brown A.
Published: February 2009

The paper discusses solutions of period 4 for the difference equation

where k and m are real parameters, with k > 0. For given values of k and m there are at most three solutions with period 4 and equations are set up to determine the elements of these solutions and the stability of each solution. Only real solutions are considered. The procedure that is used to find these solutions allows unstable solutions to be identified as well as stable solutions.

In a previous paper, solutions of period 2 and period 3 were examined for this equation and there was evidence of anomalous behaviour in the way the stability intervals occurred. Some preliminary information about solutions of period 4 was mentioned in the discussion. The present paper provides more complete results, which confirm the anomalous behaviour and give a better idea of how the stability criterion changes for different families of solutions. These results are used to indicate the variety of behaviour that can be found for one-parameter systems by imposing suitable conditions on m and k.
Strong system equivalence (I)
Authors: Anderson B. D. O. , Coppel W. A. , Cullen D. J.
Published: February 2009

Strong system equivalence is defined for polynomial realizations of a rational matrix. It is shown that any polynomial realization is strongly system equivalent to a generalized state-space realization, and two generalized state-space realizations are strongly system equivalent if and only if they are constant system equivalent.
Strong pseudo-convexity and symmetric duality in nonlinear programming
Authors: Mishra M.S. , Nanda S. , Acharya D.
Published: February 2009

In this note, the weak duality theorem of symmetric duality in nonlinear programming and some related results are established under weaker (strongly Pseudo-convex/strongly Pseudo-concave) assumptions. These results were obtained by Bazaraa and Goode [1] under (stronger) convex/concave assumptions on the function.
Surface mounted heat flux sensors
Authors: Weir G. J.
Published: February 2009

The dual integral equations describing heat flow about a circular Heat Flux Sensor on the surface of a layered medium are derived and discussed, together with the extent to which the Heat Flux Sensor measures the heat flow which would occur in the absence of a Heat Flux Sensor. An asymptotic analysis provides new analytical results supporting those derived previously by numerical methods.

It is suggested that some properties of the general problem of a Heat Flux Sensor on the surface of a multiply-layered medium can be approximated by a lumped-parameter model depending on only four non-dimensional numbers: namely, two non-dimensional linear heat transfer coefficients, and essentially two non-dimensional thermal resistances. Some support for the lumped parameter model is provided.
Force distribution along a slender body straddling an interface
Authors: Fulford G. R. , Blake J. R.
Published: February 2009

Line distributions of Stokes flow singularities are used to model the flow around a slender body which is straddling a flat interface between two viscous fluids. Motion of the slender body parallel to the interface and normal to the interface is considered where the axis of symmetry of the slender body is always perpendicular to the undisturbed interface. Asymptotic approximations to the force distributions on the slender body are evaluated and the relative contributions of that part of the slender body in one fluid to the force distribution in the other fluid and of the interface interaction to the force distribution are examined. It is observed that a shielding region exists about the interface which is due to the interaction with that part of the slender body in the other fluid. Finally, for parallel motion, the first order interface deformation is calculated.
Airfoils moving in air close to a dynamic water surface
Authors: Grundy I. H.
Published: February 2009

Steady potential flow about a thin wing, flying in air above a dynamic water surface, is analysed in the asymptotic limit as the clearance-to-length ratio tends to zero. This leads to a non-linear integral equation for the one-dimensional pressure distribution beneath the wing, which is solved numerically. Results are compared with established “rigid-ground” and “hydrostatic” theories. Short waves lead to complications, including non-uniqueness, in some parameter ranges.
Global asymptotic stability in an almost-periodic Lotka-Volterra system
Authors: Gopalsamy K.
Published: February 2009

Sufficient conditions are obtained for the existence of a globally asymptotically stable strictly positive (componentwise) almost-periodic solution of a Lotka-Volterra system with almost periodic coefficients.
Parseval's integral and the Jacobi expansions in series of Bessel fuinctions
Authors: Lekner John
Published: February 2009

The sums

are shown to approximate J0 (z); the error terms are series in higher order Bessel functions, leading with J2M (z). Similar sums approximate J1(z). These sums may be looked on as extensions of the Jacobi expansions for cos z and sin z in series of Bessel functions. They become numerically useful for M > |z|.
Expansion formulae for general triple hypergeometric series
Authors: Mohd Ch. Wali , Qureshi M. I.
Published: February 2009

The main object of present paper is to obtain a finite summation of Srivastava's general triple hypergeometric series in terms of Kampé de Fériet's double hypergeometric series. A number of finite sums of Kampé de Fériet's double hypergeometric polynomials in terms of different kinds of single hypergeometric polynomials of higher order, are obtained. Some known results of Manocha and Sharma [9], [10], Munot [11], Pathan [12], Qureshi [15], Qureshi and Pathan [16] and Srivastava [26] are deduced as special cases. A result of Pathan [13, page 316 (1.2)] is also corrected here.
The foliage density equation revisited
Authors: Miller John Boris
Published: February 2009

The foliage density equation is the means by which the foliage density g in a leaf canopy, as a function of the angle of inclination of the leaves, is to be estimated from discrete data gathered using photometric methods or point quadrats. It is an integral equation relating f, a function of angle estimated from measurements, to the unknown function g. The explicit formula for g is known and depends upon f and its first three derivatives; the operator f →, g is unbounded, and the problem is ill posed.

In this paper we give the form of g when f is a trigonometric polynomial, extending earlier results due to J. R. Philip. This provides a means of estimating g without directly estimating the derivatives of f from numerical data. To assess the reliability of the method we discuss the convergence of Fourier series representations of f and g.
Criticality dependence on data and parameters for a problem in combustion theory
Authors: Tam K. K.
Published: February 2009

A central problem in the theory of combustion, consisting of a nonlinear parbolic equation together with initial and boundary conditions, is considered. The influence of the initial and boundary data examined. In the main part of the study, a two-step linearization is developed such that the interesting features of the original problem are given by the solution of a non-liner and ordinary differential equation. Approximate solutions are obtained and upper and lower solutions are used to assess the validity of the approximations. Whenever possible, results are compared with those obtained previously and there is good agreement in all cases.
Duality with generalized convexity
Authors: Egudo R. R. , Mond B.
Published: February 2009

Recently, Hanson and Mond formulated a type of generalized convexity and used it to establish duality between the nonlinear programming problem and the Wolfe dual. Elsewhere, Mond and Weir gave an alternate dual, different from the Wolfe dual, that allowed the weakening of the convexity requirements. Here we establish duality between the nonlinear programming problem and the Mond-Weir dual using Hanson-Mond generalized convexity conditions.
A note on nondifferentiable symmetric duality
Authors: Craven B. D.
Published: February 2009

Under suitable hypotheses on the function f, the two constrained minimization problems:

are well known each to be dual to the other. This symmetric duality result is now extended to a class of nonsmooth problems, assuming some convexity hypotheses. The first problem is generalized to:

in which T and S are convex cones, S* is the dual cone of S, and ∂y denotes the subdifferential with respect to y. The usual method of proof uses second derivatives, which are no longer available. Therefore a different method is used, where a nonsmooth problem is approximated by a sequence of smooth problems. This duality result confirms a conjecture by Chandra, which had previously been proved only in special cases.
An optimal control problem involving a class of linear time-lag systems
Authors: Teo K. L. , Wong K. H. , Wu Z. S.
Published: February 2009

A class of convex optimal control problems involving linear hereditary systems with linear control constraints and nonlinear terminal constraints is considered. A result on the existence of an optimal control is proved and a necessary condition for optimality is given. An iterative algorithm is presented for solving the optimal control problem under consideration. The convergence property of the algorithm is also investigated. To test the algorithm, an example is solved.
Nonlinear model reference adaptive control
Authors: Skowronski J. M.
Published: February 2009

The known linear model reference adaptive control (MRAC) technique is extended to cover nonlinear and nonlinearizable systems (several equilibria, etc) and used to stabilize the system about a model. The method proposed applies the same Liapunov Design Technique but avoids the classical error equation. Instead it operates in the product of the state spaces of plant and model, aiming at convergence to a diagonal set. Control program, Liapunov functions and adaptive law are specified. The case is illustrated on a two-degrees of freedom robotic manipulator.
Ziebur's matrix equation for population growth
Authors: Brown A.
Published: February 2009

The paper examines a matrix equation given by Ziebur [6] for the growth of a population in which the birth-rate and death-rate are age-dependent. For convenience the population was sub-divided into four age groups, with the same birth-rate and death-rate for individuals in a particular group, and the matrix equation relates the numbers in each sub-division in consecutive years. This avoids delay terms and makes it easier to modify the growth equation but it is shown that the form suggested by Ziebur for the transition matrix leads to some difficulties.
On some oscillation criteria for a class of neutral type functional differential equations
Authors: Zahariev A.I. , Bainov D. D.
Published: February 2009

The present paper proves criteria for oscillation of the soultions of functional differential equations of the type

where λ, τ > 0.
Two infinite-Froude-number cusped free-surface flows due to a submerged line source or sink
Authors: Collings I. L.
Published: February 2009

Solutions are found to two cusp-like free-surface flow problems involving the steady motion of an ideal fluid under the infinite-Froude-number approximation. The flow in each case is due to a submerged line source or sink, in the presence of a solid horizontal base.
A note on the singularities in the theory of water waves with an inertial surface
Authors: Mandal B. N. , Kundu Krishna
Published: February 2009

This note is concerned with the derivation of velocity potentials describing the generation of infinitesimal gravity waves in a motionless liquid with an inertial surface composed of uniformly distributed floating particles, due to fundamental line and point sources with time-dependent strengths submerged in a liquid of finite constant depth.
Polynomial identities for simple Lie superalgebras
Authors: Gould M. D.
Published: February 2009

Polynomial identities for the generators of a simple basic classical Lie superalgebra are derived in arbitrary representations generated by a maximal (or minimal) weight vector. The infinitesimal characters occurring in the tensor product of two finite dimensional irreducible representations are also determined.
The finite-section approximation for integral equations on the half-line
Authors: de Hoog Frank , Sloan Ian H.
Published: February 2009

Integral equations on the half line are commonly approximated by the finite-section approximation, in which the infinite upper limit is replaced by a positive number β. A novel technique is used here to rederive a number of classical results on the existence and uniqueness of the solution of the Wiener-Hopf and related equations, and is then extended to obtain existence, uniqueness and convergence results for the corresponding finite-section equations. Unlike the methods used in the recent work of Anselone and Sloan, the present methods are constructive, and result in explicit asymptotic bounds for the error introduced by the finite-section approximation.
On the continuity of the Wiener-Hopf factorization operation
Authors: Green Michael , Anderson Brian D.O.
Published: February 2009

The problem of passing from an L∞ function to a Wiener-Hopf factorization is considered. It is shown that a small L∞ perturbation which does not change the factorization indices will lead to small Lp (1 < p < ∞) perturbations in the Wiener-Hopf factors, but can lead to large L∞ perturbations, unless the derivatives are controlled during the perturbation.
On the uniqueness of solitary Rossby waves
Authors: Kloeden P. E.
Published: February 2009

The explicit solitary Rossby wave solutions found by Larichev, Reznik and Berestov are shown to be unique for the model equations considered, in the sense that there are no other antisymmetric wave solutions which are not of these forms. This is done by adapting arguments used by Amick and Fraenkel to show the uniqueness of the Hill's vortex solution. It is based on the maximum principle and the domain folding method of Gidas, Ni and Nirenberg, and involves showing that the function ψ/y is radially symmetric, where ψ is the streamfunction of a solitary wave and y the horizontal cartesian coordinate perpendicular to the x-axis, along which the waves move at steady positive speed. This argument is also used to show the uniqueness of the well-known explicit solutions for cylindrical vortices. The result does not apply directly to rider solutions of Flierl et al., which are not antisymmetric, but it does restrict the possible rider solutions that can form because of their association with particular antisymmetric solutions.
Finite element error estimates for subsonic flow
Authors: Chow S. S. , Carey G. F.
Published: February 2009

Error estimates are derived for a finite element analysis of plane steady subsonic flows described by the full potential equation. The analysis is based on the use of the theory of variational inequalities to accomodate the subsonic flow constraint and leads to a suboptimal estimate relative to that obtained for linear potential flow. We then consider an alternative dual formulation of the problem and obtain an optimal estimate subject to reasonable regularity assumptions.
The solution of some integral equations
Authors: Ahner John F. , Lowndes John S.
Published: February 2009

Algorithms are developed by means of which certain connected pairs of Fredholm integral equations of the first and second kinds can be converted into Fredholm integral equations of the second kind. The methods are then used to obtain the solutions of two different sets of triple integral equations tht occur in mixed boundary value problems involving Laplace' equation and the wave equation respectively.
Notes on a paper by J. B. Miller
Authors: Lavoie J. L.
Published: February 2009

Two sums given by J. B. Miller are evaluated in terms of classical hypergeometric results.
Observables in modular field thoey
Authors: Kleeman R.
Published: February 2009

The observables of modular quantisation are studied from the point of view of locality. Such a study allows identification of possible Hamiltonians and also enables us to generalize the fundamental trilinear commutation relations of parafield theory. A comparison of modular field theory with a normal U(m) gauge theory, begun in an earlier publication, is completed with the conclusion that the two are equivalent except that the former has certain restrictions on its observables.
Determining statistical dominance by linear programming
Authors: Drynan Ross G.
Published: February 2009

This note generalises the necessary and sufficient conditions for one act to be dominated by another when the two acts available to the decision maker have outcomes contingent on discrete states of nature whose probabilities of occurrence are known only to the extent of linear partial information. The generalisation relates to the dominance of an act by a set of acts. The presentation is in terms of general vector dominance, of which statistical dominance is only a particular case.
Eigenvalues of the Laplacian for rectilinear regions
Authors: Gottlieb H. P. W.
Published: February 2009

From a knowledge of the eigenvalue spectrum of the Laplacian on a domain, one may extract information on the geometry and boundary conditions by analysing the asymptotic expansion of a spectral function. Explicit calculations are performed for isosceles right-angle triangles with Dirichlet or Neumann boundary conditions, yielding in particular the corner angle terms. In three dimensions, right prisms are dealt with, including the solid vertex terms.
Stable continuous orthonormalisation techniques for linear boundary value problems
Authors: van Loon P. M. , Mattheij R. M. M.
Published: February 2009

An investigation is made of a hybrid method inspired by Riccati transformations and marching algorithms employing (parts of) orthogonal matrices, both being decoupling algorithms. It is shown that this so-called continuous orthonormalisation is stable and practical as well. Nevertheless, if the problem is stiff and many output points are required the method does not give much gain over, say, multiple shooting.
Necessary conditions for a mathematical programming problem with set and cone constraints
Authors: Nagahisa Youji
Published: February 2009

This paper is devoted to the derivation of a necessary condition of F. John type which must be satisfied by a solution of a mathematical programming problem with set and cone constraints. The necessary condition is applied to an optimisation problem defined on functional spaces with inequality state constraints. Furthermore a pseudo open mapping theorem is developed in the course of proving the main theorem.
On algorithms for generalised smoothing splines
Authors: Osborne M. R. , Prvan Tania
Published: February 2009

Recently, considerable interest has been shown in the connection between smoothing splines and a particular class of stochastic processes. Here the connection with an equivalent class of least squares problems is used to develop algorithms, and properties of the solution are examined. We give an estimate of the condition number of the solution process and compare this with an estimate for the condition number of the Reinsch algorithm in its conventional implementation.
Potential of several arbitrarily located disks
Authors: Fabrikant V. I.
Published: February 2009

The electrostatic field of a set of arbitrarily located circular disks is considered. A set of governing integral equations is derived by a new method. It is shown that some integral characteristics can be found without solving the integral equations. The upper and lower bounds for the total charge are found from a set of linear algebraic equations whose coefficients are defined by simple geometric characteristics of the system. Examples considered show sufficient sharpness of the estimations.
Mean flow induced by the viscous critical layer in a stratified fluid
Authors: Smyth Noel
Published: February 2009

The evolution of the critical layer in a viscous, stratified fluid is examined in the limit of large Richardson and Reynolds numbers. A source far above the critical layer and of amplitude ɛ is turned on at t = 0 and the behaviour of both the steady state and transients is found. Viscosity dominates over nonlinearity in the critical layer for , Re being an appropriately defined Reynolds number. Wave amplitudes are found to grow as the critical layer is approached, then decay rapidly due to the action of viscosity in a critical layer of O((Re)−1/3) around the critical level. The critical layer acts as a source of vorticity, which diffuses into the outer flow, resulting in an induced mean flow of . This induced mean flow causes the critical level to move towards the incoming wave.
Two-phase flow in Hele-Shaw cells: numberical studies of sweep efficiency in a five-spot pattern
Authors: Sahwartiz Leonard W. , Degregoria Anthony J.
Published: February 2009

The unsteady Hele-Shaw problem is a model nonlinear system that, for a certain parameter ranger, exhibits the phenomenon known as viscous fingering. While not directly applicable to multiphase porous-media flow, it does prove to be an adequate mathematical model for unstable dieplacement in laboratory parallel-plate devices. We seek here to determine, by use of an accurate boundary-integral frount-tracking scheme, the extent to which the simplified system captures the canonical nonlinear behavior of displacement flows and, in particular, to ascertain the role of noise in such systems. We choose to study a particular pattern of injection and production “wells.” The pattern chosen is the isolated “five-spot,” that is a single source surrounded by four symmetrically-placed sinks in an infinite two-dimensional “reservoir.” In cases where the “pusher” fluid has negligible viscosity, sweep efficiency is calculated for a range of values of the single dimensionless parameter τ, an inverse capillary number. As this parameter is reduced, corresponding to increased flow rate or reduced interfacial tension, this efficiency decreases continuously. For small values of τ, these stable displacements change abruptly to a regime characterized by unstable competing fingers and a significant reduction in sweep efficiency. A simple stability argument appears to correctly predict the noise level required to transit from the stable to the competing-finger regimes. Published compilations of experimental results for sweep efficiency as a function of viscosity ratio showed an unexplained divergence when the pusher fluid is less viscous. Our simulations produce a similar divergence when, for a given viscosity ratio, the parameter τ is varied.