bluuebunny/crossref_metadata_2025 huggingface dataset:
Contact problem of an elastic plug in an elastic region
Authors: Steven G. P.
Published: February 2009

The contact problem investigated in this paper may be more fully described as a three dimensional elastic body with a circular hole through it; inside this tunnel is press fitted a solid elastic plug of finite length. Shear stresses are taken to be absent along the contact interface.

An influence coefficient technique is used to model the governing integral equation. For the elastic region the displacement influence coefficients due to bands of constant pressure are determined using a numerical quadrature on Fourier integrals. However, the plug, being of finite length, requires the superposition of two separate solutions to boundary value problems before the displacement influence coefficients can be determined.

Contact pressure distributions are presented for a sample of parameter variations and also for a case where hydrostatic pressure is present in the tunnel in the elastic region. Despite both components being elastic the imposition of a constant interference displacement along the interface still gives rise to the characteristic singularity in contact pressure at the edges of contact due to the strain discontinuity at these points.
On the solution of a structured nonlinear programme
Authors: Berry L. T. M.
Published: February 2009

Explicit formulae are derived for the projected gradient vector and trial dual variables required in the application of Rosen's method [4] to the solution of a Minimum Cost Network problem.
A transformation of Cooke's treatment of some triple integral equations
Authors: Love E. R. , Clements D. L.
Published: February 2009

The reduction of an important class of triple integral equations to a pair of simultaneous Fredholm equations has been carried out by Cooke [1]. In this paper, Cooke's equations are transformed to new uncoupled Fredholm equations which, for certain important cases, are shown to be simpler than Cooke's and also superior for the purposes of solution by iteration.
A programming problem with anLpnorm in the objective function
Authors: Mond Bertram , Schechter Murray
Published: February 2009

We consider a programming problem in which the objective function is the sum of a differentiable function and thepnorm ofSx, whereSis a matrix andp> 1. The constraints are inequality constraints defined by differentiable functions. With the aid of a recent transposition theorem of Schechter we get a duality theorem and also a converse duality theorem for this problem. This result generalizes a result of Mond in which the objective function contains the square root of a positive semi-definite quadratic function.
Nonlinear least squares — the Levenberg algorithm revisited
Authors: Osborne M. R.
Published: February 2009

One of the most succesful algorithims for nonlinear least squares calculations is that associated with the names of Levenberg, Marquardt, and Morrison. This algorithim gives a method which depends nonlinearly on a parameter γ for computing the correction to the current point. In this paper an attempt is made to give a rule for choosing γ which (a) permits a satisfactory convergence theorem to be proved, and (b) is capable of satisfactory computer implementation. It is beleieved that the stated aims have been met with reasonable success. The convergence theorem is both simple and global in character, and a computer code is given which appears to be at least competitive with existing alternatives.
On the stability of differential-difference equations
Authors: Braddock R. D. , Van Den Driessche P.
Published: February 2009

The local properties of non-linear differential-difference equations are investigated by considering the location of the roots of the eigen-equation derived from the lineraised approximation of the original model. A general linear system incorporating one time delay is considered and local stability results are obtained for cases in which the coefficient matrices satisfy certain assumptions. The results have applications to recent Biological and Economic models incorporating time lags.
Effect of static magnetic field on the helical flow of incompressible cholesteric liquid crystal between two coxial circular cylinders having rotational and axial velocities
Authors: Paria G. , Sharma A. K.
Published: February 2009

The paper investigates the effect of a static magnetic field on the helical flow of an incompressible cholestenc liquid crystal with director of unit magnitude between two coaxial circular cylinders rotating with different angular velocities about their common axis and moving with different axial velocities. At low shear rates with a weak magnetic field in the axial direction, the axial velocity, the angular velocity and the orientation of molecules between the two cylinders have been obtained. It is found that the magnetic field has influenced the orientation of molecules while the axial velocity and the angular velocity remain uneffected by the magnetic field.
Properties of propagation of quantum spin systems
Authors: Robinson Derek W.
Published: February 2009

Using an estimate on the group velocity we give an independent proof of the existence of time translations for a large class of short range interactions. We demonstrate that these systems satisfy a strong form of causal propagation and that space-time algebras in suitable space-like directions are disjoint. Finally we derive criteria for dispersion of the interaction in terms of the algebraic density of the orbit of local subalgebras under the evolution or under the associated group of shifts. In this sense the Heisenberg and X-Y models are dispersive but the Ising model is not.
Convergence of degenerate-kernel methods
Authors: Sloan Ian H.
Published: February 2009

Additional convergence results are given for the approximate solution in the space L2(a, b) of Fredholm integral equations of the second kind, y = f + Ky, by the degenerate-kernel methods of Sloan, Burn and Datyner. Convergence to the exact solution is provided for a class of these methods (including ‘method 2’), under suitable conditions on the kernel K, and error bounds are obtained. In every case the convergence is faster than that of the best approximate solution of the form yn = Σnan1u1, where u1, …, un are the appropriate functions used in the rank-n degenerate-kernel approximation. In addition, the error for method 2 is shown to be relatively unaffected if the integral equation has an eigenvalue near 1.
Similarity solutions of the steady state cosmic-ray equation of transport
Authors: Webb G. M.
Published: February 2009

Similarity solutions of the steady-state equation of transport for the distribution functionF0of cosmic rays in the interplanetary region are obtained by theuse of transformation groups. The solutions are derived in detail for a spherically-symmetric model of the interplanetary region with an effective radial diffusion coefficient κ = κ0(p)rbwithrthe heliocentric radial distance.pthe particle momentum, κ0(p) an arbitary function ofp, and the solar wind velocity is radial and of constant speed V. Solutions for which the similarity variable η is a function ofronly are also derived; these are of particular impoartance when theF0is specified on a boundary of given radius. Non spherically-symmetric solutions can also be obtained by group methods and examples of such solutions are listed, without derivation, for the equation of transport incorporating the effects of anisotropic diffusion (diffusion coefficient κ1in the radial direction and κ2normal to it). The solutions are the most extensive steady-state analytic solutions yet obtained, and contain previous analytic solutions as special cases.
On the asymptotic solution of an elliptic interior layer problem
Authors: Barton N. G.
Published: February 2009

An interior layer problem posed by an elliptic partial differential equation of the type ε∇2φ - x∂φ/∂y = f(x, y, ε), 0 < ε ≪ 1, is investigated. This equation arises, for example, in the theory of rotating fluids and the important feature of the problem is an interior layer of width O(ε1/3) in which the solution has a relatively large magnitude.

The paper considers the simplest case which involves an interior layer, that is, where the domain is rectangular and f(x, y, ε) = εA for A constant. A leading approximation is derived and it is shown to be asymptotic to the exact solution in nearly all of the domain as ε → 0. The error estimates are derived using an a priori estimate for the solution of elliptic equations and a technique which optimizes the estimates is introduced. The applicability and limitations of the estimation technique are discussed briefly.
The numerical solution of stochastic differential equations
Authors: Kloeden P. E. , Pearson R. A.
Published: February 2009

A method is proposed for the numerical solution of Itô stochastic differential equations by means of a second-order Runge–Kutta iterative scheme rather than the less efficient Euler iterative scheme. It requires the Runge–Kutta iterative scheme to be applied to a different stochastic differential equation obtained by subtraction of a correction term from the given one.
The stability of continental shelf waves I. Side band instability and long wave resonance
Authors: Grimshaw R.
Published: February 2009

Continental shelf waves are examined for side band instability. It is shown that a modulated shelf wave is described by a nonlinear Schrödinger equation, from which the stability criterion is derived. Long shelf waves are stable to side band modulations, but as the wavenumber is increased there are regions of instability (in wavenumber space). A change of stability occurs at each long wave resonance, defined by the condition that the group velocity of the shelf wave equals a long wave speed. Equations describing the long wave resonance are derived.
On the existence of piecewise continuous optimal controls
Authors: Noussair E. S.
Published: February 2009

Existence of piecewise optimal control is proved when the cost function includes one or both of (a) a cost of sudden switching (discontinuity) of control variables, and (b) a cost associated with the maximum rate of variation of the control over segments of the path for which the control is continuous.
Translations as gauge transformations
Authors: Smrz P. K.
Published: February 2009

A local description of space and time in which translations are included in the group of gauge transformations is studied using the formalism of fibre bundles. It is shown that the flat Minkowski space–time may be obtained from a non-flat connection in a de Sitter structured fibre bundle by choosing at least two different cross-sections. The interaction terms in the covariant derivative of a Dirac wave function that correspond to translations may be interpreted as the mass term of the Dirac equation, and then the two cross-sections (gauges) correspond to the description of a fermion and antifermion respectively.
Evaluation of the acoustic impedance of a screen
Authors: Macaskill C. , Tuck E. O.
Published: February 2009

A direct numerical computation is provided for the impedance of a screen consisting of a regular array of slits in a plane wall. The problem is solved within the framework of oscillatory Stokes flow, and results presented as a function of porosity, frequency and viscosity.
The deformation of rubber cylinders and tubes by rotation
Authors: Chadwick P. , Creasy C. F. M. , Hart V. G.
Published: February 2009

A detailed analytical and numerical study is made of the deformation of highly elastic circular cylinders and tubes produced by steady rotation about the axis of symmetry. Explicit results are obtained through the use of Ogden's strain–energy function for incompressible isotropic elastic materials which, as well as being analytically convenient, is capable of reproducing accurately the observed isothermal behaviour of vulcanized rubber over a wide range of deformations. The three problems of steady rotation considered here concern (i) a tube shrink-fitted to a rigid spindle, (ii)an unconstrained tube, and (iii) a solid cylinder. In each case a set of restictions on the material constans appearing in the strain–energy function is stated which ensures that a tubular of cylindrical shape-preserving deformation exists for all angular spees and that, for problems (i) and (iii), there is no other solution. In connection with problems (ii) and (iii) values of the material constans are also given which correspond to the bifuraction or non-existence of soultions. Enegry consideration are used to determine the local stability of the various solutions obtained.
On a direct method for the determination of an exact invariant for the time-dependent harmonic oscillator
Authors: Leach P. G. L.
Published: February 2009

An exact invariant is found for the one-dimensional oscillator with equation of motion . The method used is that of linear canonical transformations with time-dependent coeffcients. This is a new approach to the problem and has the advantage of simplicity. When f(t) and g(t) are zero, the invariant is related to the well-known Lewis invariant. The significance of extension to higher dimension of these results is indicated, in particular for the existence of non-invariance dynamical symmetry groups.
Generalized shear deformations for isotropic incompressible hyperelastic materials
Authors: Hill James M.
Published: February 2009

For isotropic incompressible hyperelastic materials the single function characterizing generalized shear deformations or as they are sometimes called anti-plane strain deformations must satisfy two distinct partial differential equations. Knowles [5] has recently given a necessary and sufficient condition for the strain–energy function of the material which if satisfied ensures that the two equations have consistent solutions. It is shown here for the general material not satisfying Knowles' criterion that the only possible consistent solution of the two partial differential equations are those which are already known to exist for all strain–energy functions. More general types of generalized shear deformations for such meterials are shown to exist only for special or restricted form ot the strain-energy function. In derving these results we also obtain an alternative derivation of Knowles' criterion.
Supercritical flow of an ideal fluid over a spillway
Authors: Collings I. L. , Grimshaw R.
Published: February 2009

The irrotational flow of an incompressible, inviscid fluid over a spiliway is considered. The reciprocal ε of the Froude number is taken to be small and the method of matched asymptotic expansions is applied. The bed of the spillway is horizontal far upstream and makes an angle α with the horizontal far downstream. The inner expansion is valid upstream and over the spillway, but is invalid far downstream. The outer expansion which is valid downstream fails to satisfy the upstream conditions. Unknown constants in the outer expansion are determined by the matching and composite expansions obtained.
A nonlinear singular perturbation problem on a semi-infinite interval
Authors: Shepherd J. J.
Published: February 2009

We consider a nonlinear singular perturbation problem on a semi-infinite interval, that is a generalization of the well-known Lagerstrom model equation intended to model low Reynolds number flow. By applying a Green's function method and the contraction mapping principle, we are able to obtain existence, uniqueness and asymptoticity results for this problem.
Inverse power law potentials about polygonal prisms and in polygonal cavities
Authors: Philip J. R.
Published: February 2009

The effects on adsorption of the geometry of the solid may be studied through calculations based on a (distance)−ε (ε> 3) intermolecular potential. This paper establishes the result that the potential due to an infinitely long polygonal homogeneous solid prism, at position r in the plane of its right section, is – . Here ρi = ∣ r − ri ∣, where the ri are the position vectors of the n vertices of the polygon, and θij are the angles r − ri makes with the two sides of the polygon which meet at vertex ri. The g's are exact functions of θij. They are, in general, integrals of associated Legendre functions, but they are elementary for ε an even integer. A similar result holds for the potential within an infinitely long polygonal prismatic cavity. The analysis involves a systematic superposition schema and the concept of a supplementary potential with datum within the solid at infinity. The cases ε = 6 and ε = 4 are treated in detail and illustrative solutions given for the following configurations: semi-infinite laminae, deep rectangular cracks, square prisms, square prismatic cavities and regular n-gonal prismatic cavities.
Fast local convergence with single and multistep methods for nonlinear equations
Authors: Abbott James P. , Brent Richard P.
Published: February 2009

The authors have noticed an oversight in Section 3.1 of this paper. To correct this error it is necessary to assume a uniform differentiability condition onG(x,h). This is required, for example, to imply δ on line 4 of p. 178 can be chosen independent ofh. For brevity we note that this oversight has been corrected in [1], which is available from the authors. Also, the results of subsequent sections are unaffected.
On tensor operators and characteristic identities for semi-simple Lie algebras
Authors: Gould M. D.
Published: February 2009

Tensor identities for finite dimensional representations of arbitrary semi-simple Lie algebras are derived and are applied to the construction of left-projection operators which project out the shift components of tensor operators from the left. The corresponding adjoint identities are also derived and are used for the construction of right-projection operators. It is also shown that, on a finite dimensional irreducible representation, these identities may be considerably reduced. Commutation relations between the shift tensors of a tensor operator are also computed in terms of the roots appearing in the tensor identities.
Existence of optimal controls for systems governed by second order linear parabolic partial delay-differential equations with first boundary conditions
Authors: Teo K. L.
Published: February 2009

In this paper, we consider a class of systems governed by second order linear parabolic delay-partial differential equations with first boundary conditions. Our main results are reported in Theorems 3.1 and 3.2. As in [9, Theorems 4.1 and 4.2], the coefficients and forcing terms of the system considered in Theorem 3.1 are linear in the control variables. On the other hand, the forcing terms of the system considered in Theorem 3.2 are allowed to be nonlinear in the control variables at the expense of dropping the control variables in the cost integrand.
The method of the hypercircle for a class of nonlinear equations
Authors: Arthurs A. M. , Hart V. G.
Published: February 2009

We present a geometrical method for the solution of a certain class of non-linear boundary value problems. The results generalize those of the standard hypercircle method for linear problems. Two illustrative examples are described.
Series operations using minimum storage
Authors: Stokes A. N.
Published: February 2009

An algorithm is given for transforming a polynomial with n coefficients to a continued fraction accurate to the same order. Only n numbers are held in storage at each stage. An extension to produce an inverse polynomial, also accurate to order n, is described.
Complementary variational principles for a class of nonlinear boundary value problems
Authors: Anderson N. , Arthurs A. M.
Published: February 2009

Complementary variational principles are presented for a class of nonlinear boundary value problems S* Sφ = g(φ) in which g is not necessarily monotone. The results are illustrated by two examples, accurate variational solutions being obtained in both cases.
Multivariable adaptive parameter and state estimators with convergence analysis
Authors: Moore J. B. , Ledwich G.
Published: February 2009

The convergence properties of a very general class of adaptive recursive algorithms for the identification of discrete-time linear signal models are studied for the stochastic case using martingale convergence theorems. The class of algorithms specializes to a number of known output error algorithms (also called model reference adaptive schemes) and equation error schemes including extended (and standard) least squares schemes. They also specialize to novel adaptive Kalman filters, adaptive predictors and adaptive regulator algorithms. An algorithm is derived for identification of uniquely parameterized multivariable linear systems.

A passivity condition (positive real condition in the time invariant model case) emerges as the crucial condition ensuring convergence in the noise-free cases. The passivity condition and persistently exciting conditions on the noise and state estimates are then shown to guarantee almost sure convergence results for the more general adaptive schemes.

Of significance is that, apart from the stability assumptions inherent in the passivity condition, there are no stability assumptions required as in an alternative theory using convergence of ordinary differential equations.
The stability of Bishop's warfare strategy
Authors: Hathaway Neville J. , Murray Ian W. , Rickard Johan A.
Published: February 2009

The paper analyes the dynamics of duopoly output game involving a warfare strategy proposed by Robert Bishop. Necessary and sufficient conditions are obtained for the stability of a duopoly warfare game.
Exclusion regions for eigenvalues of linear operators
Authors: Hennagin Stephen C. , Linz Peter
Published: February 2009

The quesiton of the location of the eigenvalues of a linear operator is considered. In particular, a numerical technique is developed which can be used to demonstrate the absence of eigenvalues in certain segements of the real line.
Doubling strategies for backgammon-like games
Authors: Tuck E. O.
Published: February 2009

Optimal strategies are obtained for two-player games with an alternating staek doubling option. A complete two-parameter analysis is provided for games that must end within two moves, and a recursive procedure then enables a solution for games of any number of moves. Examples are given of relevance to extureme end games in backgammon.
Some pure radiation filds in general relativity
Authors: Akabari R. P. , Dave U. K. , Patel L. K.
Published: February 2009

A Demianski-type metric investigated in connection with Einstein's field equations corresponding to pure radiation fields. With aid of complex vectorical formalism, a general solution of these fiel equations is obtained. The solution is algebraically spcial. A particular case of the solution is considered which includes many known solutions; among them are the raiationg versions of some of Kinnersley's solutions.
A scheduling problem with a simple graphical solution
Authors: Beecham A. F. , Hurley A. C.
Published: February 2009

It is shown that a problem which arose in the scheduling of two simultaneous competitions between a number of golf clubs may be reduced to that of 4- colouring the edges of a certain bipartite graph which has 4 edges meeting at each vertex. This colouring problem is solved by an analysis in terms of directed cycles, which is simple to carry through in a practical case and is easily extended to the problem with 4 replaced by 2m. The more general colouring problem with 4 replaced by any positive integer is solved by relating it to the marriage problem enunciated by Philip Hall and to the latin multiplication technique of Kaufmann but, in practical applications, this approach involves severe computational difficulties.
On a nonlinear hydromagnetic convection under a rotational constraint
Authors: Riahi N.
Published: February 2009

Nonlinear hydromagnetic convection in a horizontal layer of fluid rotating about the vertical axis is investigated using the mean field approximation. The boundary layer method is used assuming large Rayleigh number R for different ranges of the Chandrasekhar number Q and the Taylor number Ta. The heat flux F is determined for the value of the horizontal wave number α which maximizes F. It is shown that, for certain regions of the parameter space (R, Q, Ta), F and α change discontinuously for Ta greater than some critical value (given R and Q). Thus, for Ta about this critical value, wave numbers and heat fluxes of two different values will be predicted simultaneously. Also, for certian regions of the parameter space, the field can facilitate convection, but rotation can facilitate convection only for sufficiently large Ta.
On the influence of the initial data in a combustion problem
Authors: Tam K. K.
Published: February 2009

The combustion of a material can be modelled by two coupled parabolic partial differential equations for the temperature and concentration of the material. This paper deals with properties of the solution of these equations inside a cylinder or a sphere and under given initial conditions. Bounds for the variation of the temperature with the initial conditions are first established by considering a decoupled form of the equations. Then the coupled system is used to obtain approximate expressions for the temporal evolution of temperature and concentration.
A boundary integral equation method for the numerical solution of a second order elliptic equation with variable coefficients
Authors: Clements D. L.
Published: February 2009

A method is derived for the solution of boundary value problems governed by a second-order elliptic partial differential equation with variable coefficients. The method is obtained by expressing the solution to a particular problem in terms of an integral taken round the boundary of the region under consideration.
Geometric programming with probabilistic decision variables
Authors: Jefferson T. R. , Scott C. H.
Published: February 2009

Here we consider a particular class of stochastic geometric programs in which the randomness occurs in the decision variables. Specifically we analyse a program in which we specify a joint normal probability for the dicision variables and require the constraint set to be satisfied in the chance constrained mode. A numerical example is given to illustrate the approach.
Poisoning-induced exchange of steady states in a catalytic chemical reactor
Authors: Do D. D. , Weiland R. H.
Published: February 2009

Slow catalyst poisoning can result in the sudden failure of a chemical reactor operating isothermally with substrate-inhibited kinetics. At failure, a satisfactory steady state is exchanged for one of low conversion. The method of matched asymptotic expansions is used to give a detailed description of the exchange process in the phase plane. The structure of the jump is ascertained by separate asymptotic expansions across two adjoining transition regions in which the independent variables contain unknown shifts.
Solution of a general homogeneous linear difference equation
Authors: Singh V. N.
Published: February 2009

In Section 3, since relation (6) is valid only for n ≥ 2r, the condition n ≥ r in relation (9) should be replaced by n ≥ 2r. When r ≤ n ≤ 2r − 1, relation (9) still holds but now relations (6) and (7) should be replaced, respectively, by the relations

and

the asterisk again denoting omission of the last column. The proof of (9) for r ≤ n ≤ 2r − 1 is exactly similar to its proof for n ≥ 2r.
The kernel of a rule of approximate integration
Authors: Loxton J. H. , Sanders J. W.
Published: February 2009

It is well known that the trapezoidal rule of quadrature is exact for linear functions on [0, 1], and easy to see that it is exact for functions of the form f = l+g where l is linear and g is odd about ½. Not so well known is an example of a continuous function for which the trapezoidal rule is exact but which does not have this form. In this paper we show that if the trapezoidal rule is exact for f then f has the form above provided it has either absolutely convergent Fourier series or continuous second derivative. We consider one-sided versions in which the approximate integrals are non-negative, and also characterize those sequences arising as the approximate integrals of a function with absolutely convergent Fourier series.
Invariance properties of Schoenberg's tone row system
Authors: Fill James A. , Izenman Alan J.
Published: February 2009

This paper organizes in a systematic manner the major features of a general theory ofm-tone rows. A special case of this development is the twelve-tone row system of musical composition as introduced by Arnold Schoenberg and his Viennese school. The theory as outlined here applies to tone rows of arbitrary length, and can be applied to microtonal composition for electronic media.
On the solution of a partially structured nonlinear program
Authors: Simms P. D.
Published: February 2009

A class of partially structured nonlinear programming problems, containing the capacitated nonlinear minimum cost multicommodity flow problem, is considered. Such problems, although large, can often be solved efficiently and with minimal computational storage by gradient projection methods.
A modified barrier function method with improved rate of convergence for degenerate problems
Authors: Jittorntrum Krisorn , Osborne M. R.
Published: February 2009

In a previous paper the authors have shown that the classical barrier function has anO(r) rate of convergence unless the problem is degenerate when it reducesO(r½). In this paper a modified barrier function algorithm is suggested which does not suffer from this problem. It turns out to have superior scaling properties which make it preferable to the classical algorithm, even in the nondegenerate case, if extrapolation is to be used to accelerate convergence.
An algorithm for solving the restricted least squares problem
Authors: Clark David
Published: February 2009

This paper presents an algorithm to solve the least squares problem when the parameters are restricted to be non-negative. The algorithm is based on the branch and bound method which has been suggested for this problem, and shares with it the property that an unrestricted least squares subproblem is solved at each step. However, improvements have been made to the branching rules by making use of the convexity of the problem, and the Kuhn–Tucker conditions are used to test for optimality. The resulting algorithm becomes essentially iterative in nature, and linearity of the number of subproblems solved can be shown under assumptions which have always been observed in practice.
Uniform estimation of the eigenvalues of Sturm–Liouville problems
Authors: Paine John , de Hoog Frank
Published: February 2009

The perturbation of the eigenvalues of a regular Sturm–Liouville problem in normal form which results from a small perturbation of the coefficient function is known to be uniformly bounded. For numerical methods based on approximating the coefficients of the differential equation, this result is used to show that a better bound on the error is obtained when the problem is in normal form. A method having a uniform error bound is presented, and an extension of this method for general Sturm–Liouville problems is proposed and examined.
Small strain oscillatory squeeze film flow of simple fluids
Authors: Phan-Thien N.
Published: February 2009

The oscillatory squeeze film problem is solved for the simple fluid in the sense of Noll. It is shown that dynamic properties of polymeric liquids can be measured on the plastometer in the oscillatory mode. This should be useful to food and other technologists who have to deal with awkward, highly viscous materials.
Solution of a general homogeneous linear difference equation
Authors: Singh V. N.
Published: February 2009

Solutions of a homogeneous (r + 1)-term linear difference equation are given in two different forms. One involves the elements of a certain matrix, while the other is in terms of certain lower Hessenberg determinants. The results generalize some earlier results of Brown [1] for the solution of a 3-term linear difference equation.
The one-dimensional gas-lubricated slider bearing
Authors: Shepherd J. J.
Published: February 2009

Under the appropriate physical hypotheses, the problem of determining the pressure distribution in a gas-filled slider bearing becomes a singular perturbation problem as Λ, the bearing number, tends to infinity. This paper extends the results of an earlier one by the author to consider the case where the film profile has jump discontinuities in slope at points interior to the bearing. Application of the methods of general singular perturbation theory establishes the appropriate existence-uniqueness results for this problem, and a means is devised by which uniformly valid asymptotic approximations to the pressure distribution may be obtained for large values of Λ.
The asymptotic response of a calorimeter
Authors: McNabb A.
Published: February 2009

An algorithm is given for calculating the asymptotic behaviour of the temperature of the fluid in an adiabatic calorimeter, and used to derive the asymptote for a finite cylinder.
Singularity subtraction in the numerical solution of integral equations
Authors: Anselone P. M.
Published: February 2009

The singularity subtraction technique described by Kantorovich and Krylov in [11] is designed to reduce or overcome the effect of a weakly singular kernel in the numerical solution of integral equations. First, the equation is rearranged in such a way that the singularity of the kernel is at least partially cancelled by the smoothness of the solution, and then numerical integration is applied. We present convergence results and error bounds under general conditions on the nature of the singularity and the numerical integration procedure. Numerical examples demonstrate the benefit of the singularity subtraction technique.
The properties of solutions of weakly singular integral equations
Authors: Vainikko G. , Pedas A.
Published: February 2009

We examine the differential properties of the solution of the linear integral equation of the second kind, whose kernel depends on the difference of arguments and has an integrable singularity at the point zero. The derivatives of the solution of the equation have singularities at the end points of the domain of integration, and we derive precise estimates for these singularities.
A piecewise polynomial approximation to the solution of an integral equation with weakly singular kernel
Authors: Vainikko G. , Uba P.
Published: February 2009

We construct collocation methods with an arbitrary degree of accuracy for integral equations with logarithmically or algebraically singular kernels. Superconvergence at collocation points is obtained. A grid is used, the degree of non-uniformity of which is in good conformity with the smoothness of the solution and the desired accuracy of the method.
The iterated projection solution for the Fredholm integral equation of second kind
Authors: Chatelin Françoise , Lebbar Rachid
Published: February 2009

We are concerned with the solution of the second kind Fredholm equation (and eigenvalue problem) by a projection method, where the projection is either an orthogonal projection on a set of piecewise polynomials or an interpolatory projection at the Gauss points of subintervals.

We study these cases of superconvergence of the Sloan iterated solution: global superconvergence for a smooth kernel, and superconvergence at the partition points for a kernel of “Green's function” type. The mathematical analysis applies for the solution of the inhomogeneous equation as well as for an eigenvector.
Collocation methods for two dimensional weakly singular integral equations
Authors: Graham Ivan G.
Published: February 2009

Given a Fredhoim integral equation of the second kind, which is defined over a certain region ⊆ R2, we define and , two different numerical approximations to its solution, using the collocation and iterated collocation methods respectively. We describe without proof some known results concerning the general convergence properties of and when the kernel and solution of the integral equation are smooth. Then, we prove rigorously order of convergence estimates for and which are applicable in the practically siginificant case when is a rectangle, and the kernel of the integral equation is weakly singular. These estimates are illustrated by the numerical solution of a two dimensional weakly singular equation which arises in electrical engineering.
Deferred correction for the integral equation eigenvalue problem
Authors: Chu K. W. , Spence A.
Published: February 2009

This paper considers the improvement of approximate eigenvalues and eigenfunctions of integral equations using the method of deferred correction. A convergence theorem is proved and a numerical example illustrating the theory is given.
The extension of generalized cross-validation to a multi-parameter class of estimators
Authors: O'Brien D. M. , Holt J. N.
Published: February 2009

The method of generalized cross-validation (GCV) provides a good value for the “ridge” regularization parameter for an ill-conditioned linear system, such as the system produced by discretization of a Fredholm integral equation of the first kind. In this note we apply GCV to a wider class of estimators than the one parameter ridge estimators. We observe that the expected values of the parameter mean-square error, the predictive mean-square error, and the GCV function are simultaneously minimized over this new class, so we accept the minimizer of the GCV function as the best computable estimator. We present a simple algorithm for computing this estimator from the data, so that a numerical search is not needed.
Stability analysis of Runge–Kutta methods applied to a basic Volterra integral equation
Authors: Baker Christopher T. H. , Wilkinson Joan C.
Published: February 2009

Our purpose in this paper is to display the stability analysis of Runge–Kutta methods applied to a Volterra integral equation of a simple form. As prerequisite we define, and then develop the structure of, the class of Runge–Kutta methods considered. The test equation is taken as the “;basic” equation ; the simple form of this equation permits ready insight into features which are more obscure when considering (as elsewhere [1], [2], [6]) equations of a more complicated form. Due to the structure of the methods and the nature of the test equation, the stability analysis reduces to the study of recurrence relations of the form Фk + 1 = MФ k + γk (k = 0, 1, 2, …) which are common in stability discussions in numerical analysis.
A convergence theorem for singular integral equations
Authors: Elliott David
Published: February 2009

The principal result of this paper states sufficient conditions for the convergence of the solutions of certain linear algebraic equations to the solution of a (linear) singular integral equation with Cauchy kernel. The motivation for this study has been the need to provide a convergence theory for a collocation method applied to the singular integral equation taken over the arc (−1, 1). However, much of the analysis will be applicable both to other approximation methods and to singular integral equations taken over other arcs or contours. An estimate for the rate of convergence is also given.
The optimum processing of clipped signals: an approach based an a likelihood ratio statistic
Authors: Keats R. G. , Frost Winifred , Dobson Annette
Published: February 2009

The likelihood ratio approach to the detection of small signals in the presence of noise is investigated in the case where the available data have been clipped. The statistic obtained is the ratio of orthant probabilities and appears intractable; accordingly an approximation to this statistic is developed by truncating an appropriate Taylor expansion. Approximations are obtained for the mean and variance of this modified statistic and compared with those obtained from computer simulations.
A tournament problem
Authors: Wallis W. D.
Published: February 2009

It is shown that, for any round-robin tournament, one can find a pairing of the teams and allocate home and away matches so that only one member of each pair plays at home in each round.
The existence of Hadley convective regimes of atmospheric motion
Authors: Dutton John A. , Kloeden Peter E.
Published: February 2009

The solutions of the equations describing deep global convection on a rotating planet are discussed. The existence of generalized steady axisymmetric solutions is established. It is then shown that these are classical solutions when the heat source is sufficiently smooth. The solutions are shown to be unique when the heating is sufficiently weak and asymptotically stable when the shear is sufficiently small. Finally, the application of these results to Earth's atmosphere is discussed, with eddy viscosity replacing molecular viscosity.
Static plane-symmetric solution of a scalar-tensor theory of gravitation
Authors: Reddy D. R. K. , Rao V. U. M.
Published: February 2009

Vacuum field equations in a scalar-tensor theory of gravitation, proposed by Ross, are obtained with the aid of a static plane-symmetric metric. A closed form exact solution to the field equations in this theory is presented which can be considered as an analogue of Taub's empty space-time in Einstein's theory.
The effect of an enclosed air cavity on a rectangular drum
Authors: Gottlieb H. P. W.
Published: February 2009

The effect of an enclosed air cavity on the natural vibration frequencies of a rectangular membrane is investigated. The modes specified by an even integer are not affected. For the odd-odd modes, the frequency equation is found via a Green's function formulation and is solved to first order in a parameter representing the effect of the cavity of the rectangular drum. The frequencies are raised, with the fundamental being most affected. In the case of degeneracies, each degenerate mode contributes to the frequency shift, but the degeneracy itself is not broken to first order.
Stability of plane Couette flow for high Reynolds number
Authors: Davis G. B. , Morris A. G.
Published: February 2009

Experimental evidence shows that plane Couette flow becomes unstable when the Reynolds number R reaches certain critical values. Linear stability theory does not predict these observations and has been unable to locate these instabilities. A Chebyshev/QR numerical technique is used to investigate much higher values of R than those previously tested. In particular, values of R up to 108 are confidently tested, whereas previously values of R up to only 2 × 104 have been considered.
Computation of critical parameters for a problem in combustion theory
Authors: Tam K. K.
Published: February 2009

An iteration scheme previously obtained by the author is used to study the dependence of criticality on initial data and the parameters in a combustion problem. Numerical results are presented for a slab, a cylinder and a sphere. These are compared with the results of previous workers.
Rapidly-convergent methods for evaluating elliptic integrals and theta and elliptic functions
Authors: Fenton J. D. , Gardiner-Garden R. S.
Published: February 2009

The expressions for elliptic integrals, elliptic functions and theta functions given in standard reference books are slowly convergent as the parameter m approaches unity, and in the limit do not converge. In this paper we use Jacobi's imaginary transformation to obtain alternative expressions which converge most rapidly in the limit as m → 1. With the freedom to use the traditional formulae for m ≤ ½ and those obtained here for m ≥ ½, extraordinarily rapidly-convergent methods may be used for all values of m; no more than three terms of any series need be used to ensure eight-figure accuracy.
Monotonicity properties of the zeros of Bessel functions
Authors: McCann Roger C. , Love E. R.
Published: February 2009

Let jν, denote the first positive zero of Jν. It is shown that jν/(ν + α) is a strictly decreasing function of ν for each positive α provided ν is sufficiently large. For each α lowe bounds on ν are given to assure the monotonicity of jν/(ν + α). From this it is shown that jν > ν + j0 for all ν > 0, which is both simpler and an improvement on the well known inequality Jν ≥ (ν (ν + 2))1/2.
Approximate solutions for coupled moment equations
Authors: Winley Graham , Tognetti Keith
Published: February 2009

In studying the coupled differential equations for the moments of a stochastic process it is often found that the equation for thejth moment involves higher moments. The usual methods of “decoupling” such a system of equations to obtain estimates of the moments are surveyed and shown generally to result in a system of nonlinear simultaneous differential equations which may be readily solved by numerical methods.

Often, estimates of the first and second moments are the main concern. In this case, two further assumptions reported in the literature can be used to simplify the system and avoid the expense of solving the nonlinear equations. These two techniques are evaluated and compared with a new technique. Two processes are analysed, one representing a chemical reaction and the other population growth.
Ray analysis and punching problems for stretched elastic plates
Authors: Moodie T. Bryant , Tait R. J. , Barclay D. W.
Published: February 2009

The present paper presents a ray analysis for a problem of technical importance in fragmentation studies. The problem is that of suddenly punching a circular hole in either isotropic or transversely isotropic plates subjected to a uniaxial tension field. The ray method, which involves only differentiation, integration, and simple algebra, is shown to be particularly useful in clarifying the propagation process of the resulting unloading waves and obtaining the attendant discontinuities of the various quantities involved. Numerical results obtained from the ray analysis are presented in graphical form and compared with those obtained by more elaborate schemes.
A note on the random walk model arising in double diffusion
Authors: Liyanage L. H. , Hill J. M. , Gulati C. M.
Published: February 2009

The discrete random walk problem for the unrestricted particle formulated in the double diffusion model given in Hill [2] is solved explicitly. In this model it is assumed that a particle moves along two distinct horizontal paths, say the upper path I and lower path 2. For i = 1, 2, when the particle is in path i, it can move at each jump in one of four possible ways, one step to the right with probability pi, one step to the left with probability qi, remains in the same position with probability ri, or exchanges paths but remains in the same horizontal position with probability si (pi + qi + ri + si = 1). Using generating functions, the probability distribution of the position of an unrestricted particle is derived. Finally some special cases are discussed to illustrate the general result.
Saddle point and duality in the optimization theory of convex set functions
Authors: Lai Hang-Chin , Yang Shu-Shih
Published: February 2009

For a set function G on an atomless finite measure space (X, , m), we define the subgradient, conjugate set of and conjugate functional of G. It is proved that a minimization problem of set function G has an optimal solution if and only if the Lagrangian on × L1(X, , m)has a saddle point (Ω0, f0) such that

where f0 is an element of the conjugate set (for the definition, see the later context).