bluuebunny/crossref_metadata_2025 huggingface dataset:
Axisymmertic Stokes flow images in spherical free surfaces with applications to rising bubbles
Authors: Harper J. F.
Published: February 2009

A theorem is derived for the hydrodynanuc image of an axially symmetric slow viscous (Stokes) flow in a sphere which is impermeable and free of shear stress. A second theorem establishes a sense in which such a flow past an arbitrary rigid surface or shear-free sphere becomes, on inversion in an arbitrary sphere with its centre on the axis of symmetry, a flow past the rigid or shear-free inverse of that surface or sphere.

The theorems are used to simplify the proofs of a number of known results for images of point singularities in plane and spherical rigid and free boundaries, and for a pair of bubbles rising steadily in line in a viscous fluid. They also give for the first time accurate numerical solutions for the velocities of each of a larger number of spherical bubbles rising quasi-steadily in line. These enable one to assess the accuracy of simple approximations to those velocities.
Symmetries and the inverse problem of Lagrangian dynamics for linear systems
Authors: Sarlet W.
Published: February 2009

We discuss general, time-dependent, linear systems of second-order ordinary differential equations. A study is made of the similarities and discrepancies between the inverse problem of Lagrangian mechanics on the one hand, and the search for linear dynamical symmetries on the other hand.
A Galerkin-Petrov method for singular integral equations
Authors: Elliott David
Published: February 2009

A Galerkin-Petrov method for the approximate solution of the complete singular integral equation with Cauchy kernel, based upon the use of two sets of orthogonal polynomials, is considered. The principal result of this paper proves convergence of the approximate solutions to the exact solution making use of a convergence theorem previously given by the author. In conclusion, some related topics such as a first iterate of the approximate solution and a discretized Galerkin-Petrov method are considered. The paper extends to a much more general equation many results obtained by other authors in particular cases.
Bounds for the growth of a perturbation in some double-diffusive convection problems
Authors: Gupta J. R. , Sood S. K. , Shandil R. G. , Banerjee M. B. , Banerjee K.
Published: February 2009

Bounds are presented for the modulus of the complex growth rate p of an arbitrary oscillatory perturbation, neutral or unstable, in some double-diffusive problems of relevance in oceanography, astrophysics and non-Newtonian fluid mechanics.
An asymptotic theory for dispersion of reactive contaminants in parallel flow
Authors: Barton N. G.
Published: February 2009

The classic problem, first treated by Taylor [18], of the dispersion of inert soluble matter in fluid flow continues to attract attention from researchers describing the approach to the asymptotic state [5, 17]. The present article considers some of the complications caused when the solute is chemically active, Dispersing chemically active solutes occur in diverse fields such as chromatography, chemical engineering and environmental fluid mechanics.

The asymptotic large-time analysis of Chatwin [5] is re-worked to handle the case of reactive solutes dispersing in parallel flow. Matching between moderate and large-time solutions requires consideration of the integral moments of the reactive contaminant could, and the Aris method of moments is therefore invoked and modified for reaction effects. The results are applied in detail to the outstanding paractical example—the chemical flow reactor (a device used to measure reaction rates for chemical reactions taking place between fluids). For this case, the paper provides a practical alternative to recent variable diffusion coefficient studies [6, 7, 15], and presents further results for the concentration distribution and for the limiting behaviour under weak and vigorous recactions at the boundary of the flow.
Harmless delays in a periodic ecosystem
Authors: Gopalsamy K.
Published: February 2009

Sufficient conditions are obtained for the existence of a unique linearly asymptotically stable positive periodic solution of an ecosystem model of two species competition in a periodic environment with time lags in interspecific interactions. It is shown that if the self-regulating intraspecific interaction effects are strong enough and act without time delays then time delays of any length in the interspecific interactions cannot destabilise an otherwise stable ecosystem in a periodic environment.
On the application of a numerical algorithm for Hopf bifuraction to the hunting of a wheelset
Authors: Huilgol R. R. , Janus R. , Lohe M. A. , Sag T. W.
Published: February 2009

The main ideas of Hopf bifurcation theory and its relevance to the development of periodic motions of an autonomous system depending on a parameter are presented, and an algorithm for the computation of the orbits is described. It is then shown that a model system for the motion of a wheelset can be cast in the form amenable to Hopf bifurcation theory. Numerical results for the period and amplitudes of the lateral and yaw motions are obtained in terms of the forward speed of the wheelset, and the wheel-rail profile parameters.

It is found that the period of oscillation decreases while the lateral and yaw motion amplitudes increase as the forward speed increases, for any given rail and wheel profile. While the effect of wheel curvature on the lateral motions seems to be non-existent, its effect on the yaw motion amplitude and the period is to increase them very slightly as the wheel profile changes from a conical to a curved profile. On the other hand, the effect of rail curvature on the lateral amplitude, for instance, is significant; the larger the curvature the smaller the amplitude for a given forward speed.
On nonlinear overstable convection rolls in a rotating system
Authors: Riahi N.
Published: February 2009

Finite amplitude oscillatory convection rolls in the form of travelling waves are studied for a horizontal layer of a low Prandtl number fluid heated from below and rotating rapidly about a vertical axis. The results of the stability and nonlinear analyses indicate that there is no subcritical instability and that the oscillatory rolls are unstable for the ranges of the Prandtl number and the rotation rate considered in this paper.
Infinitely divisible random transition probabilities with application to dependent Markov chains
Authors: Chesson Peter L.
Published: February 2009

Random transition probability matrices with stationary independent factors define “white noise” environment processes for Markov chains. Two examples are considered in detail. Such environment processes can be used to construct several Markov chains which are dependent, have the same transition probabilities and are jointly a Markov chain. Transition rates for such processes are evaluated. These results have application to the study of animal movements.
A unified boundary integral equation method for a class of second order elliptic boundary value problems
Authors: Rezayat M. , Rizzo F. J. , Shippy D. J.
Published: February 2009

A generalized integral equation formulation and a systematic numerical solution procedure are presented for a class of boundary value problems governed by a general second-order differential equation of elliptic type. Diverse numerical examples include problems of plane-wave scattering, three-dimensional fluid flow, and plane heat transfer for a body with a moving flame boundary. The last example employs certain representation functions useful to increase solution effectiveness in problems with an isolated integrable singularity.
Minimisation problems for implicit functionals defined by differential equations of liver kinetics
Authors: Bass L. , Bracken A. J. , Vyborny R.
Published: February 2009

A substance carried convectively through the liver by the blood undergoes two successive enzymatic transformations. The resulting concentrations of the three forms of the substance are determined, as functions of position along the blood flow in the steady state, by coupled ordinary differential equations of the first order on a finite interval. The densities along the blood flow of the activities of the two (immobile) transforming enzymes are described by two non-negative and normalised functions of position.

The problem, suggested by recent experimental results, is to choose these two functions so as to minimise the concentration of the once-transformed (intermediate) form of the substance at one boundary (the liver outlet). That minimisation is particularly significant biologically when the intermediate form is toxic and the second transformation renders it harmless. In this problem of optimal control (exerted perhaps by natural selection), the classical approach through Euler's equations is inapplicable because of the constraints on the two density functions. Moreover, when the enzyme kinetics and hence the differential equations are non-linear, the functional to be minimised is not obtainable explicitly. Instead it appears, after some manipulation of the coupled equations, as the terminal boundary value of the solution of a non-linear Volterra integral equation, which involves the two density functions (one explicitly and one implicitly) as control variables.

Appropriate existence, uniqueness and boundedness results are obtained for the solution of this integral equation, and the problem is then solved rigorously for one class of non-linearities (including saturation kinetics). Some unanswered questions are posed for another class (including substrate-inhibition kinetics).
An integral equation for immiscible fluid isplacement in a two-dimensional porous medium or Hele-Shaw cell
Authors: Davidson M. R.
Published: February 2009

An integral equation for the normal velocity of the interface between two immiscible fluids flowing in a two-dimensional porous medium or Hele-Shaw cell (one fluid displaces the other) is derived in terms of the physical parameters (including interfacial tension), a Green's function and the given interface. When the displacement is unstable, ‘fingering’ of the interface occurs. The Saffman-Taylor interface solutions for the steady advance of a single parallel-sided finger in the absence of interfacial tension are seen to satisfy the integral equation, and the error incurred in that equation by the corresponding Pitts approximating profile, when interfacial tension is included, is shown. In addition, the numerical solution of the integral equation is illustrated for a sinusoidal and a semicircular interface and, in each case, the amplitude behaviour inferred from the velocity distribution is consistent with conclusions based on the stability of an initially flat interface.
A note on the instantaneous streamlines, pathlines and pressure contours for a cavitation bubble near a boundary
Authors: Cerone P. , Blake J. R.
Published: February 2009

Instantaneous streamlines, particle pathlines and pressure contours for a cavitation bubble in the vicinity of a free surface and near a rigid boundary are obtained. During the collapse phase of a bubble near a free surface, the streamlines show the existence of a stagnation point between the bubble and the free surface which occurs at a different location from the point of maximum pressure. This phenomenon exists when the initial distance of the bubble is sufficiently close to the free surface for the bubble and free surface to move in opposite directions during collapse of the bubble. Pressure calculations during the collapse of a cavitation bubble near a rigid boundary show that the maximum pressure is substantially larger than the equivalent Rayleigh bubble of the same volume.
Necessary optimality conditions for a problem with costs of rapid variation of control
Authors: Kibalczyc K. , Walczak S.
Published: February 2009

In this paper a control problem with a cost functional depending on the number of switchings and on the speed of alterations of control is considered. Necessary conditions for the existence of an optimal solution are given.
Buckling analysis of plates of arbitrary shape
Authors: Bucco D. , Mazumdar J.
Published: February 2009

A simple and efficient numerical technique for the buckling analysis of thin elastic plates of arbitrary shape is proposed. The approach is based upon the combination of the standard Finite Element Method with the constant deflection contour method. Several representative plate problems of irregular boundaries are treated and where possible, the obtained results are validated against corresponding results in the literature.
An exactly solvable two component classical Coulomb system
Authors: Forrester Peter J.
Published: February 2009

A two component classical Coulomb system is considered, in which particles of charge +q and + 2q are constrained to lie on a circle and interact via the two-dimensional Coulomb potential. At a special value of the coupling constant the correlation functions are calculated exactly and the asymptotic form of the truncated charge-charge correlation is found to obey Jancovici's sum rule.
Improved lower bounds for the motion of moving boundaries
Authors: Hill James M. , Dewynne Jeffrey N.
Published: February 2009

In a recent paper the authors give upper and lower bounds for the motion of the moving boundary for the classical Stefan problem for plane, cylindrical and spherical geometries. On comparison with the exact Neumann solution for the plane geometry and no surface radiation the bounds obtained are seen to be quite adequate for practical purposes except for the lower bound at small Stefan numbers. Here improved lower bounds are obtained which in some measure remove this inadequacy. Time dependent surface conditions are also examined and the new lower bounds obtained for the classical problem are illustrated numerically.
A note on hypergeometric polynomials
Authors: Qureshi M. I. , Pathan M. A.
Published: February 2009

In a paper which appeared in this journal, Manocha and Sharma [6] obtained some results of Carlitz [4], Halim and Salain [5] and generalized a few of them by using fractional derivatives. The present paper is concerned with some erroneous results of this paper [6]. Many more sums of the product of hypergeometric polynomials are also obtained.
On the rate of convergence of finite difference schemes on nonuniform grids
Authors: De Hoog Frank , Jackett David
Published: February 2009

Finite difference schemes for some two point boundary value problems are analysed. It is found that for schemes defined on nonuniform grids, the order of the local truncation error does not fully reflect the rate of convergence of the numerical approximation obtained. Numerical results are presented that indicate that this is also the case for higher dimensional problems.
Characteristic identities for semi-simple Lie algebras
Authors: Gould M. D.
Published: February 2009

We present a new derivation of the polynomial identities satisfied by certain matricesAwith entriesAij(i, j= 1,…,n) from the universal enveloping algebra of a semi-simple Lie algebra. These polynomial identities are exhibited in a representation-independent way asp(A)= 0 wherep(x)(herein called the characteristic polynomial ofA) is a polynomial with coefficients from the centreZof the universal enveloping algebra. The minimum polynomial identitym(A)= 0 of the matrixAoverZis also obtained and it is shown thatp(x)andm(x)possess properties analogous to the characteristic and minimum polynomials respectively of a matrix with numerical entries. Acting on a representation (finite or infinite dimensional) admitting an infinitesimal character these polynomial identities may be expressed in a useful factored form. Our results include the characteristic identities of Bracken and Green [1] as a special case and show that these latter identities hold also in infinite dimensional representations.
Second order optimality conditions for mathematical prograramming with set functions
Authors: Chou J. H. , Hsia Wei-Shen , Lee Tan-Yu
Published: February 2009

Second order necessary and sufficient conditions are given for a class of optimization problems involving optimal selection of a measurable subset from a given measure subspace subject to set function inequalities. Relations between twice-differentiability at Ω and local convexity at Ω are also discussed.
Generalized functions for applications
Authors: Craven B. D.
Published: February 2009

A simple rigorous approach is given to generalized functions, suitable for applications. Here, a generalized function is defined as a genuine function on a superset of the real line, so that multiplication is unrestricted and associative, and various manipulations retain their classical meanings. The superset is simply constructed, and does not require Robinson's nonstandard real line. The generalized functions go beyond the Schwartz distributions, enabling products and square roots of delta functions to be discussed.
A note on the boundary integral equation method for the solutions of second order elliptic equations
Authors: Clements D. L. , Haselgrove M. , Barnett D. M.
Published: February 2009

The boundary integral equation method is obtained by expressing a solution to a particular partial differential equation in terms of an integral taken round the boundary of the region under consideration. Various methods exist for the numerical solution of this integral equation and the purpose of this note is to outline an improvement to one of these procedures.
A nonlinear difference equation with two parameters
Authors: Brown A.
Published: February 2009

The paper is mainly concerned with the difference equation

where k and m are parameters, with k > 0. This equation arises from a method proposed for solving a cubic equation by iteration and represents a standardised form of the general problem. In using the above equation it is essential to know when the iteration process converges and this is discussed by means of the usual stability criterion. Critical values are obtained for the occurrence of solutions with period two and period three and the stability of these solutions is also examined. This was done by considering the changes as k increases, for a give value of m, which makes it effectively a one-parameter problem, and it turns out that the change with k can differ strongly from the usual behaviour for a one-parameter difference equation. For m = 2, for example it appears that the usual picture of stable 2-cycle solutions giving way to stable 4-cycle solutions is valid for smaller values of k but the situation is recersed for larger values of k where stable 4-cycle solutions precede stable 2-cycle solutions. Similar anomalies arise for the 3-cycle solutions.
Numerical calculation of unstable immiscible fluid displacement in a two-dimensional porous medium or Hele-Shaw cell
Authors: Davidson M. R.
Published: February 2009

A numerical procedure for calculating the evolution of a periodic interface between two immiscible fluids flowing in a two-dimensional porous medium or Hele-Shaw cell is described. The motion of the interface is determined in a stepwise manner with its new velocity at exach time step being derived as a numerical solution of a boundary integral equation. Attention is focused on the case of unstable displacement charaterised physically by the “fingering” of the interface and computationally by the growth of numerical errors regardless of the numerical method employed. Here the growth of such error is reduced and the usable part of the calculation extended to finite amplitudes. Numerical results are compared with an exact “finger” solution and the calculated behaviour of an initial sinusoidal displacement, as a function of interfacial tension, initial amplitude and wavelength, is discussed.
Asymptotic estimates for symmetric vortex streets
Authors: Keady G.
Published: February 2009

Steady plane inviscid symmetric vortex streets are flows defined in the stripR× (0,b) and periodic inxwith period 2ain which the flow in (−a,a) × (0,b) is irrotational outside a vortex core on which the vorticity takes a prescribed constant value. A family of such vortex street flows, characterised by a variational principle in which the area |Aα| and the centroidycof the vortex coreAαare fixed, will be considered. For such a family, indexed by a parameter α, suppose that the coresAαbecome small in the sense that

Asymptotic estimates on functionals such as flux constant and speed are obtained.
On duality for convex minimization with nested maxima
Authors: Scott C. H. , Jefferson T. R. , Sirri E.
Published: February 2009

In this paper, we consider convex programs with linear constraints where the objective function involves nested maxima of linear functions as well as a convex function. A dual program is constructed which has interpretational significance and may be easier to solve than the primal formulation. A numerical example is given to illustrate the method.
Transient electromagnetic response of a layered conducting medium at asymptotically late times
Authors: O'Brien D. M. , Smith R. S.
Published: February 2009

In this paper we consider a pair of horizontal conducting loops in the air above a horizontally layered ground. The transmitting loop is driven by a current source which rises from zero at time zero to a final constant value at time τ. We first compute the e.m.f. induced in the receiving loop and derive an asymptotic series for the e.m.f. at late times. Secondly, we estimate the error in truncating the asymptotic series at N terms and design a reliable numerical algorithm for summing the asymptotic series.
Simple examples of the derivation of amplitude equations for systems of equations possessing bifurcations
Authors: Roberts A. J.
Published: February 2009

The method of Coullet and Spiegel [3], which derives ordinary differential equations describing the time evolution of a system of partial differential equations when the system is near critical, is applied to some simple problems. These problems serve to illustrate simply many features of the method.
Spherical curves and quadratic relationships for special functions
Authors: Mehlum Even , Wimp Jet
Published: February 2009

We show that the position vector of any 3-space curve lying on a sphere satisfies a third-order linear (vector) differential equation whose coefficients involve a single arbitrary function A(s). By making various identifications of A(s), we are led to nonlinear identities for a number of higher transcendental functions: Bessel functions, Horn functions, generalized hypergeometric functions, etc. These can be considered natural geometrical generalizations of sin2t + cos2t = 1. We conclude with some applications to the theory of splines.
Splitting least squares and collocation procedures for two-point boundary value problems
Authors: Locker John , Prenter P. M.
Published: February 2009

Let L, T, S, and R be closed densely defined linear operators from a Hubert space X into X where L can be factored as L = TS + R. The equation Lu = f is equivalent to the linear system Tv + Ru = f and Su = v. If Lu = f is a two-point boundary value problem, numerical solution of the split system admits cruder approximations than the unsplit equations. This paper develops the theory of such splittings together with the theory of the Methods of Least Squares and of Collocation for the split system. Error estimates in both L2 and L∞ norms are obtained for both methods.
Strong system equivalence (II)
Authors: Coppel W. A. , Cullen D. J.
Published: February 2009

The notion of strong system equivalence, which was defined and studied in Anderson, Coppel and Cullen [1], is here given a module-theoretic characterization and a dynamical interpretation.
Critical point behaviour of the diffusion length for radiative transfer
Authors: Grant I. F. , McKellar B. H. J.
Published: February 2009

Critical point behaviour of the diffusion length γ for the solutions of the radiative transfer equation deep in a homogenous medium is studied. The Legendre expansion of the medium's phase function P(cos ψ) is taken to be an infinite series and is characterized by the parameters h0, h1h2,…. A characteristic equation for γ is given in terms of an infinite continued fraction. From this equation it is shown that as any one of the hn, say hp, approaches zero, the others being held constant, γ behaves as , where the critical exponent is found to be vp = ½ for all p = 0, 1, 2,….
Asymptotic location of nodal lines using geodesic theory
Authors: Mahony J. J. , Norbury J.
Published: February 2009

The asymptotic properties of solutions of the non-linear eigenvalue problem, associated with the homogeneoud Dirichlet problem for

are investigated. Here f and g are smooth functions of position in a finite plane region with a smooth boundary. The results for the positive solution are well established, but knowledge of other branches of solutions is scarce. Here positive solutions are pieced together across lines partitioning the domain, and variational arguments are framed, as an effective means of locating the lines, so that the composite function is everywhere a solution of *. Heuristic arguments suggest strongly that there is a close relationship between the nodal lines of * and certain classes of weighted geodesic lines defined by the classical variational problem for the functional

which provides an effective basis for computation. Some results are proved but others remain conjectures. Analogous results are proved for the associated ordinary differential equation. The geometry of the solutions is surprisingly restricted when the coefficients are spatially variable. The arguments are extended to a class of reactive, diffusive systems. It is possible to predict the pattern of domains of different outcomes in terms of properties of the surface on which the reactions occur, without a knowledge of the chemical kinetics. The results appear to provide a basis for stringent testing of the postulated role of reactive-diffusive mechanisms in the formation of complex patterns in biological species.
Exact edge wave solutions for some generalised exponential shelf topographies
Authors: Louis J. P. , Clarke D. J.
Published: February 2009

Exact wave-height solutions are presented for trapped waves over two new three-parameter depth topographies. Dispersive properties are calculated for both a semi-infinite and a truncated convex exponential profile, as well as for a semi-infinite concave profile. The analysis in all three cases is general in that both horizontal divergence and rotational effects are included. These solutions may be used for either high-frequency edge wave or low-frequency shelf wave studies by taking appropriate limits (f → 0 for edge wave and ε = f2L2/gH ≪ 1 for shelf waves).
Optimal control in liver kinetics
Authors: Fink A. M.
Published: February 2009

We solve a minimization problem in liver kinetics posed by Bass, et al., in this journal, (1984), pages 538–562. The problem is to choose the density functions for the location of two enzymes, in order to minimize the concentration of an intermediate form of a substance at the outlet of the liver. This form may be toxic to the rest of the body, but the second enzyme renders it harmless. It seems natural that the second enzyme should be downstream from the first. However, we can show that the minimum problem is sometimes solved by an overlap of the supports of the two density functions. Even more surprising is that, for certain forms of the kinetic functions and high levels of transformation of the first enzymatic reaction, some of the first enzyme should be located downstream from all the second enzyme. This suggests that the first reaction should be relatively slow.
Existence and regularity results for Maxwell's equations in the quasi-static limit
Authors: Carey A. L. , O'Brien D. M.
Published: February 2009

We prove the existence of solutions of Maxwell's equations for a conducting medium whose constitutive parameters are piecewise constant on R3, and then examine the convergence of these solutions in the quasi-static limit in which displacement currents are neglected. Secondly, we examine the regularity of the limiting solution and the sense in which the classical boundary conditions hold, namely, continuity of the tangential electric field and the normal current density.
Ship bows with contiunous and splashless flow attachment
Authors: Madurasinghe M. A. D. , Tuck E. O.
Published: February 2009

In two-dimensional bow-like flows past a semi-infinite body, one must in general expect a free-surface discontinuity, in the form of a splash or spray jet. However, there is numerical evidence that special body shapes do exist for which this splash is absent. In this study, we first establish conditions on the geometry of the bow in order that it should be splash-free at zero gravity, by solving the mathematical problem exactly. We then obtain solutions for finite non-zero gravity, by solving a non-linear integral equation numerically. A class of splashless body geometries with a downward directed segment at the extreme of the bow, to which the free surface attaches tangentially, is demonstrated in detail.
Boundary value control problems involving the bessel differential operator
Authors: Werner K.-D.
Published: February 2009

In this paper, we consider the hyperbolic partial differential equation wrr = wrr + 1/r wr − ν2 /r2w, where v ≥ 1/2 or ν = 0 is aprameter, with the Dirichlet, Neumann and mixed boundary conditions. The boundary controllability for such problems is investigated. The main resutl is that all “finite energy” intial states can be steered to the zero state in time T, using a control f ∈ L2 (0, T), provided T > 2. Furthermore, necessary conditions for controllability are also presented.
On the dynamic analysis of a Leontief model with fixed capitial
Authors: Woods J. E.
Published: February 2009

In Convex Structures and Economic Theory, Nikaideo analysed, inter alia, a circulating capital Leontief model where final demand could exhibit either proportional or non-proporaitonal growth. This paper extends his analsis to a fixed capital modee. Analogues of Nikaido's results are derived for the closed model and for the open model under blaanced grwoth. However, the results obtained here for the open model with unbalanced growth are weaker than Nikaido's.
Ordinary and partial difference equations
Authors: Potts Renfrey B.
Published: February 2009

Ordinary difference equations (OΔE's), mostly of order two and three, are derived for the trigonometric, Jacobian elliptic, and hyperbolic functions. The results are used to derive partial difference equations (PΔE's) for simple solutions of the wave equation and three nonlinear evolutionary partial differential equations.
Oscilations of higher-order neutral equations
Authors: Ladas G. , Sficas Y. G.
Published: February 2009

Sufficient conditions are given for the occurrence of various types of asymptotic behaviour in the solution of a class of n th order neutral delay differential equations. The conditions are in the form of certain inequalities amongst the constants involved in the definition of the differential equations, and specify either oscillatory behavior, or asymptotic divergence, or solutions which converge to zero.
What is invexity?
Authors: Ben-Israel A. , Mond B.
Published: February 2009

Recently it was shown that many results in Mathematical Programming involving convex functions actually hold for a wider class of functions, called invex. Here a simple characterization of invexity is given for both constrained and unconstrained problems. The relationship between invexity and other generalizations of convexity is illustrated. Finally, it is shown that invexity can be substituted for convexity in the saddle point problem and in the Slater constraint qualification.
Generic differentiability of order-bounded convex oparators
Authors: Borwein Jonathan M.
Published: February 2009

We give sufficient conditions for order-bounded convex operators to be generically differentiable (Gâteaux or Fréchet). When the range space is a countably order-complete Banach lattice, these conditions are also necessary. In particular, every order-bounded convex operator from an Asplund space into such a lattice is generically Fréchet differentiable, if and only if the lattice has weakly-compact order intervals, if and only if the lattice has strongly-exposed order intervals. Applications are given which indicate how such results relate to optimization theory.
On stability and stationary points in nonlinear optimization
Authors: Guddat J. , Jongen H. Th. , Rueckmann J.
Published: February 2009

This paper presents three theorems concerning stability and stationary points of the constrained minimization problem:

In summary, we provethat, given the Mangasarian-Fromovitz constraint qualification (MFCQ), the feasible setM[H, G] is a topological manifold with boundary, with specified dimension; (ℬ) a compact feasible setM[H, G] is stable (perturbations ofHandGproduce homeomorphic feasible sets) if and only if MFCQ holds;under a stability condition, two lower level sets offwith a Kuhn-Tucker point between them are homotopically related by attachment of ak-cell (kbeing the stationary index in the sense of Kojima).
The “global” convergence of Broyden-like methods with suitable line search
Authors: Griewank Anderas
Published: February 2009

Iterative methods for solving a square system of nonlinear equations g(x) = 0 often require that the sum of squares residual γ (x) ≡ ½∥g(x)∥2 be reduced at each step. Since the gradient of γ depends on the Jacobian ∇g, this stabilization strategy is not easily implemented if only approximations Bk to ∇g are available. Therefore most quasi-Newton algorithms either include special updating steps or reset Bk to a divided difference estimate of ∇g whenever no satisfactory progress is made. Here the need for such back-up devices is avoided by a derivative-free line search in the range of g. Assuming that the Bk are generated from an rbitrary B0 by fixed scale updates, we establish superlinear convergence from within any compact level set of γ on which g has a differentiable inverse function g−1.
Optimisation in the regularisation ill-posed problems
Authors: Davies A. R. , Anderssen R. S.
Published: February 2009

We survey the role played by optimization in the choice of parameters for Tikhonov regularization of first-kind integral equations. Asymptotic analyses are presented for a selection of practical optimizing methods applied to a model deconvolution problem. These methods include the discrepancy principle, cross-validation and maximum likelihood. The relationship between optimality and regularity is emphasized. New bounds on the constants appearing in asymptotic estimates are presented.
Apex duality for constrained optimization
Authors: Duffin R. J. , Karney D. F. , Prisman E. Z.
Published: February 2009

In this paper, we develop a duality theory for the Apex dual in the case of primal constraints. As suggested by Duffin in [4], the objective function in this framework is a weighted average of the Legendre-Lagrangian function evaluated at key points. We show that whenever this new dual is feasible there is no duality gap for this dual, and moreover, no duality gap for both the Lagrangian and Wolfe duals too. We conclude with an outline of an algorithm to solve constrained minimization problems in the Apex framework.
Green manure and nitrogenous fertilizer—a two-sector optimal-growth model
Authors: Mustard David
Published: February 2009

A continuous-time model of a two-sector trading economy with a finitely-saturating production function and constant population is constructed. To apply it to the farm-economy problem of optimal trading of cereal for chemical nitrogenous fertilizer and of optimal allocation of land for green manure the special assumptions are made of similar technologies in both sectors and of a nitrogen-capital loss proportional to the cereal production. Optimal-control theory is applied to get the pair of controls that maximizes an infinite-horizon integral of utility of consumption. The analysis of the model is shown to reduce to that of the Ramsey-Koopmans one-sector neoclassical model of optimal growth. Calculations of the feedback control and optimal time-paths for the standardized dimensionless model are tabulated for a range of 7 utility functions of constant elasticity of marginal utility of magnitude 1 to 4 and of production functions of degree 2 to 4. Two particular analytic solutions are given. The application of the results both to the farm-economy model and to the Ramsey-Koopmans model is illustrated.
On partly bilateral and partly unilateral generating functions
Authors: Pathan M. A. , Yasmeen
Published: February 2009

The purpose of this work is to begin the development of a theory of generating functions that will not only include the generating functions which are partly bilateral and partly unilateral but also provide a set of expansions, by taking successive partial derivatives with respect to one of the variables of the generating relations. Our starting point is a result of Exton [4] on associated Laguerre polynomials whose application gives certain generating functions of the polynomials of Jacobi and Appell, and functions of n variables of Lauricella.
The long jump record revisited
Authors: de mestre Neville
Published: February 2009

A mathematical model is presented in which the long Jump is treated as the motion of a projectile under gravity with slight drag. The first two terms of a perturbation solution are obtained and are shown to be more accurate than earlier approximate analytical solutions. Results from the perturbation analysis are just as accurate as results from various numerical schemes, and require far less computer time.

The model is modified to include the observation that a long-jumper's centre of mass is forward of his feet at take-off and behind his feet on landing.

The modified model is used to determine the take-off angle for the current world long jump record, resulting in several interesting observations for athletic coaches.
A hybrid boundary integral/Taylor series approach to some nonlinear equations from fluid mechanics
Authors: Coleman C. J.
Published: February 2009

We show that a combination of Taylor series and boundary integral methods can lead to an effective scheme for solving a class of nonlinear partial differential equations. The method is illustrated through its application to an equation from two dimensional fluid mechanics.
Closed form solution to some mixed boundary value problems for a charged sphere
Authors: Fabrikant V. I.
Published: February 2009

A new method is described which allows an exact solution in a closed form to the following non-axisymmetric mixed boundary-value problem for a charged sphere: arbitrary potential values are given at the surface of a spherical segment while an arbitrary charge distribution is prescribed on the rest of the sphere. The method is founded on a new integral representation of the kernel of the governing integral equation. Several examples are considered. All the results are expressed in elementary functions. Some further applications of the method are discussed. No similar result seems to have been published previously.
A heuristic method for the determination of a Hamiltonian circuit in a graph
Authors: Karmakar Sudhangshu B.
Published: February 2009

A heuristic methodology for the identification of a circuit passing through all the vertices only once in a graph is presented. The procedure is based upon defining a normal form of a matrix and then transforming the adjacency matrix into its normal form. For a class of graphs known to be Hamiltonian, it is conjectured that this methodology will identify circuits in a small number of steps and in many cases merely by observation.
Necessary and sufficient condition for oscillations of neutral differential equations
Authors: Kulenović M. R. S. , Ladas G. , Meimaridou A.
Published: February 2009

Consider the neutral delay differential equation

where p ∈ R, τ ≥ 0, q1 > 0, σ1 ≥ 0, for i = 1, 2, …, k. We prove the following result.

Theorem. A necessary and sufficient condition for the oscillation of all solutions of Eq. (1) is that the characteristic equation

has no real roots.
On an extremum problem
Authors: Matula Joanna
Published: February 2009

We consider an optimization problem in which the function being minimized is the sum of the integral functional and the full variation of control. For this problem, we prove the existence theorem, a necessary condition in an integral form and a local necessary condition in the case of monotonic controls.
A computational method for free time optimal control problems, with application to maximizing the range of an aircraft-like projectile
Authors: Teo K. L. , Jepps G. , Moore E. J. , Hayes S.
Published: February 2009

A class of non-standard optimal control problems is considered. The non-standard feature of these optimal control problems is that they are of neither fixed final time nor of fixed final state. A method of solution is devised which employs a computational algorithm based on control parametrization techniques. The method is applied to the problem of maximizing the range of an aircraft-like gliding projectile with angle of attack control.
A class of strongly stable operator approximations
Authors: Ahues Mario
Published: February 2009

We show the strongly stable convergence of some non-collectively-compact approximations of compact operators. Special attention is devoted to Anselone's singularity subtraction discretization of certain singular integral operators. Numerical experiments are provided.
Sparse matrix factorizations of transfer matrices
Authors: Batchelor Murray T.
Published: February 2009

Sparse matrix factorizations of transfer matrices for the interactions round a face model are reviewed. The sparse factors of a more general Ising model containing first, second and third nearest neighbour interactions are also presented. For both models the factorizations are achieved by considering the required auxiliary spin sets as a hierarchy of interacting spins.
On the analysis of joint production
Authors: Woods J. E.
Published: February 2009

It is well known that n-process, n-commodity (or square), models of productive single-product industries have positive solutions to their price and quantity systems if the rates of profit and growth lie in appropriate non-negative intervals. On the other hand, negative prices and quantities can occur in formal solutions of models of square, productive, multiple-product industries even when the rates of profit and growth are less than their respective maximum positive values. It is shown in this paper that these differences can be attributed to the presence in joint production of dominance, in either row or column versions. Results on positive solutions to the price (respectively, quantity) system are derived in terms of the absence of column (respectively, row) dominance of the net output matrix. As the concepts of row and column dominance are defined in terms of linear inequalities, the basic mathematical results to be applied are theorems of the alternative.
Eigenvalues of the Laplacian for the third boundary value problem
Authors: Zayed E. M. E.
Published: February 2009

The spectral function , where are the eigenvalues of the two-dimensional Laplacian, is studied for a variety of domains. The dependence of θ(t) on the connectivity of a domain and the impedance boundary conditions is analysed. Particular attention is given to a doubly-connected region together with the impedance boundary conditions on its boundaries.
Properties of short-crested waves in water of finite depth
Authors: Marchant T. R. , Roberts A. J.
Published: February 2009

Short-crested waves are defined as propagating surface gravity waves which are doubly-periodic in the horizontal plane. Linearly, the short-crested wave system we consider occurs when two progressive wavetrains of equal amplitude and frequency are propagating at an angle to each other.

Solutions are calculated via a computer-generated perturbation expansion in wave steepness. Harmonic resonance affects the solutions but Padé approximants can be used to estimate wave properties such as maximum wave steepness, frequency, kinetic energy and potential energy.

The force exerted by waves being reflected by a seawall is also calculated. Our results for the maximum depth-integrated onshore wave force in the standing wave limit are compared with experiment. The maximum force exerted on a seawail occurs for a steep wave in shallow water incident at an oblique angle. Results are given for this maximum force.
Reflection from a water waves from a vertical vortex sheet in water of finite depth
Authors: McKee W. D. , Tesoriero F.
Published: February 2009

The reflection-transmission properties of water waves obliquely incident upon a vortex sheet in water of finite depth are studied. The problem is reduced to that of solving two integral equation. An accurate Galerkin solution is obtained which supports the use of the “variational method” in water wave problems that has recently been questioned by Kirby and Dalyrmple.
Integro-differential equations for the self-organisation of liver zones by competitive exclusion of cell-types
Authors: Bass L. , Bracken A. J. , Holmåker K. , Jefferies B. R. F.
Published: February 2009

A model is developed for the seif-organisation of zones of enzymatic activity along a liver capillary (hepatic sinusoid) lined with cells of two types, which contain different enzymes and compete for sites on the wall of the sinusoid. An effectively non-local interaction between the cells arises from local consumption of oxygen from blood flowing throug1 the sinusoid, which gives rise to gradients of oxygen concentration in turn influencing rates of division and of death of the two cell-types. The process is modelled by a pair of coupled non-linear integro-differential equations for the cell-densities as functions of time and position along the sinusoid. Existence of a unique, bounded, non-negative solution of the equations is proved, for prescribed initial values. The equations admit infinitely many stationary solutions, but it is shown that all except one are unstable, for any given set of the model parameters. The remaining solution is shown to be asymptotically stable against a large class of perturbations. For certain ranges of the model parameters, the asymptotically stable stationaxy solution has a zonal structure, with cells of one type located entirely upstream of cells of the other type, and with jump discontinuities in the cell densities at a certain distance along the sinusoid. Such sinusoidal zones can account for zones of enzymatic activity observed in the intact liver. Exceptional cases are found for singular choices of model parameters, such that stationary cell-densities cannot be asymptotically stable individually, but together form an asymptotically stable set. Certain mathematical questions are left open, notably the behaviour of large deviations from stationary solutions, and the global stability of such solutions. Possible generalisations of the model are described.
Existence of optimal control for nonlinear systems with quadratic performance
Authors: Balachandran K. , Somasundaram D.
Published: February 2009

We prove the existence of optimal control for nonlinear systems having implicit derivative with quadratic performace criteria.
A spot welding reliability problem
Authors: Rumsewicz M. , Taylor P.
Published: February 2009

When multiple operators are connected to a single power source which is not large enough to simultaneously supply all users, interference can take place. This paper considers two models of a spot welding station which differ in the method of resolving interference between users. In the first model the system is allowed to become overloaded and the consequent deterioration in weld quality is accepted. In the second model any request which will overload the system is rejected. Expressions are found for (a) the proportion of poor quality welds in the first model and (b) the probability of operators being rejected in the second model. Numerical results are given which indicate how small the power supply can be made for a typical welding shop while keeping interference at a minimal level.