Showing papers in "The Journal of The Australian Mathematical Society. Series B. Applied Mathematics in 1986"
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TL;DR: In this article, it was shown that invexity can be substituted for convexity in the saddle point problem and in the Slater constraint qualification for both constrained and unconstrained problems.
Abstract: 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.
526 citations
TL;DR: In this article, a derivative-free line search in the range of g is used to establish superlinear convergence from within any compact level set of γ on which g has a differentiable inverse function g−1.
Abstract: 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.
100 citations
TL;DR: In this article, the authors consider the Volterra-lotka equations for two competing species in which the right-hand sides are periodic in time and show that conditions recently given by K. Gopalsamy imply the existence of a periodic solution with positive components, also imply the uniqueness and asymptotic stability of the solution.
Abstract: We consider the Volterra-Lotka equations for two competing species in which the right-hand sides are periodic in time. Using topological degree, we show that conditions recently given by K. Gopalsamy, which imply the existence of a periodic solution with positive components, also imply the uniqueness and asymptotic stability of the solution. We also give optimal upper and lower bounds for the components of the solution.
94 citations
TL;DR: In this paper, sufficient conditions are obtained for the existence of a globally asymptotically stable strictly positive almost-periodic solution of a Lotka-Volterra system with almost periodic coefficients.
Abstract: Sufficient conditions are obtained for the existence of a globally asymptotically stable strictly positive (componentwise) almost-periodic solution of a Lotka-Volterra system with almost periodic coefficients.
76 citations
TL;DR: In this article, sufficient conditions are given for the occurrence of various types of asymptotic behavior in the solution of a class of n th order neutral delay differential equations, in the form of certain inequalities amongst the constants involved in the definition of the differential equations.
Abstract: 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.
56 citations
TL;DR: In this article, the stability and stationary points of the constrained minimization problem were shown to be stable under the Mangasarian-Fromovitz constraint qualification (MFCQ) condition.
Abstract: This paper presents three theorems concerning stability and stationary points of the constrained minimization problem:In summary, we prove that, given the Mangasarian-Fromovitz constraint qualification (MFCQ), the feasible set M[H, G] is a topological manifold with boundary, with specified dimension; (ℬ) a compact feasible set M[ H, G] is stable (perturbations of H and G produce homeomorphic feasible sets) if and only if MFCQ holds; under a stability condition, two lower level sets of f with a Kuhn-Tucker point between them are homotopically related by attachment of a k-cell (k being the stationary index in the sense of Kojima).
52 citations
TL;DR: In this article, a ratio game approach to the generalized fractional programming problem is presented and duality relations established, which suggests certain solution procedures for solving fractional programs involving several ratios in the objective function.
Abstract: A ratio game approach to the generalized fractional programming problem is presented and duality relations established. This approach suggests certain solution procedures for solving fractional programs involving several ratios in the objective function.
52 citations
TL;DR: In this paper, the role played by optimization in the choice of parameters for Tikhonov regularization of first-kind integral equations is surveyed and asymptotic analyses are presented for a selection of practical optimizing methods applied to a model deconvolution problem.
Abstract: 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.
29 citations
TL;DR: In this paper, the duality between the nonlinear programming problem and the Mond-Weir dual using Hanson-Mond generalized convexity conditions was established using a modified version of the Wolfe dual.
Abstract: Recently, Hanson and Mond formulated a type of generalized convexity and used it to establish duality between the nonlinear programming problem and the Wolfe dual. Elsewhere, Mond and Weir gave an alternate dual, different from the Wolfe dual, that allowed the weakening of the convexity requirements. Here we establish duality between the nonlinear programming problem and the Mond-Weir dual using Hanson-Mond generalized convexity conditions.
25 citations
TL;DR: In this article, a solution to two cusp-like free-surface flow problems involving the steady motion of an ideal fluid under the infinite-Froude-number approximation was found.
Abstract: Solutions are found to two cusp-like free-surface flow problems involving the steady motion of an ideal fluid under the infinite-Froude-number approximation. The flow in each case is due to a submerged line source or sink, in the presence of a solid horizontal base.
20 citations
TL;DR: In this article, the authors established conditions on the geometry of the bow in order that it should be splash-free at zero gravity, by solving the mathematical problem exactly and obtained solutions for finite non-zero gravity by solving a non-linear integral equation numerically.
Abstract: 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.
TL;DR: On demontre des criteres d'oscillation pour les solutions d'equations differentielles fonctionnelles du type: X (n) (t)+Λx n (t-τ)+p(t)f(x(t)-τ))=0 ou λ, τ>0 as mentioned in this paper.
Abstract: On demontre des criteres d'oscillation pour les solutions d'equations differentielles fonctionnelles du type: X (n) (t)+Λx (n) (t-τ)+p(t)f(x(t-τ))=0 ou λ, τ>0
TL;DR: In this article, the results of an analysis of equations aux differences ordinaires d'ordre 2 and 3 for the fonctions trigonometriques, elliptiques de Jacobi and hyperboliques are presented.
Abstract: On obtient des equations aux differences ordinaires d'ordre 2 et 3 pour les fonctions trigonometriques, elliptiques de Jacobi et hyperboliques. On applique les resultats a l'etablissement d'equations aux differences partielles pour les solutions simples de l'equation des ondes et de trois equations aux derivees partielles non lineaires d'evolution
TL;DR: In this paper, the derivation of velocity potentials describing the generation of infinitesimal gravity waves in a motionless liquid with an inertial surface composed of uniformly distributed floating particles, due to fundamental line and point sources with time-dependent strengths submerged in a liquid of finite constant depth, is discussed.
Abstract: This note is concerned with the derivation of velocity potentials describing the generation of infinitesimal gravity waves in a motionless liquid with an inertial surface composed of uniformly distributed floating particles, due to fundamental line and point sources with time-dependent strengths submerged in a liquid of finite constant depth.
TL;DR: In this article, a theory of generating functions with partial derivatives with respect to one of the variables of the generating relations is proposed. But this theory is restricted to bilateral and unilateral generating functions and does not include the generating functions which are partly bilateral and partly unilateral.
Abstract: 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.
TL;DR: In this paper, the line distribution of Stokes flow singularities is used to model the flow around a slender body which is straddling a flat interface between two viscous fluids.
Abstract: Line distributions of Stokes flow singularities are used to model the flow around a slender body which is straddling a flat interface between two viscous fluids. Motion of the slender body parallel to the interface and normal to the interface is considered where the axis of symmetry of the slender body is always perpendicular to the undisturbed interface. Asymptotic approximations to the force distributions on the slender body are evaluated and the relative contributions of that part of the slender body in one fluid to the force distribution in the other fluid and of the interface interaction to the force distribution are examined. It is observed that a shielding region exists about the interface which is due to the interaction with that part of the slender body in the other fluid. Finally, for parallel motion, the first order interface deformation is calculated.
TL;DR: In this article, a linear model reference adaptive control (MRAC) technique is extended to cover nonlinear and nonlinearizable systems (several equilibria, etc) and used to stabilize the system about a model.
Abstract: The known linear model reference adaptive control (MRAC) technique is extended to cover nonlinear and nonlinearizable systems (several equilibria, etc) and used to stabilize the system about a model. The method proposed applies the same Liapunov Design Technique but avoids the classical error equation. Instead it operates in the product of the state spaces of plant and model, aiming at convergence to a diagonal set. Control program, Liapunov functions and adaptive law are specified. The case is illustrated on a two-degrees of freedom robotic manipulator.
TL;DR: In this paper, the asymptotic properties of solutions of the non-linear eigenvalue problem, associated with the homogeneoud Dirichlet problem for are investigated, and 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.
Abstract: The asymptotic properties of solutions of the non-linear eigenvalue problem, associated with the homogeneoud Dirichlet problem forare 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 functionalwhich 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.
TL;DR: In this paper, a methode de linearisation a deux pas, which permet de ramener l'etude a la solution d'une equation differentielle non lineaire ordinaire, is proposed.
Abstract: On considere un probleme essentiel de la theorie de la combustion comprenant une equation parabolique non lineaire avec des conditions initiales et aux limites. On developpe une methode de linearisation a deux pas, qui permet de ramener l'etude a la solution d'une equation differentielle non lineaire ordinaire
TL;DR: In this paper, a thin wing, flying in air above a dynamic water surface, is analyzed in the asymptotic limit as the clearance-to-length ratio tends to zero, leading to a nonlinear integral equation for the one-dimensional pressure distribution beneath the wing, which is solved numerically.
Abstract: Steady potential flow about a thin wing, flying in air above a dynamic water surface, is analysed in the asymptotic limit as the clearance-to-length ratio tends to zero. This leads to a non-linear integral equation for the one-dimensional pressure distribution beneath the wing, which is solved numerically. Results are compared with established “rigid-ground” and “hydrostatic” theories. Short waves lead to complications, including non-uniqueness, in some parameter ranges.
TL;DR: In this paper, the dual integral equations describing heat flow about a circular heat flux sensor on the surface of a layered medium are derived and discussed, together with the extent to which the heat flow which would occur in the absence of a Heat Flux Sensor.
Abstract: The dual integral equations describing heat flow about a circular Heat Flux Sensor on the surface of a layered medium are derived and discussed, together with the extent to which the Heat Flux Sensor measures the heat flow which would occur in the absence of a Heat Flux Sensor. An asymptotic analysis provides new analytical results supporting those derived previously by numerical methods. It is suggested that some properties of the general problem of a Heat Flux Sensor on the surface of a multiply-layered medium can be approximated by a lumped-parameter model depending on only four non-dimensional numbers: namely, two non-dimensional linear heat transfer coefficients, and essentially two non-dimensional thermal resistances. Some support for the lumped parameter model is provided.
TL;DR: In this article, the authors give sufficient conditions for order-bounded convex operators to be generically differentiable (Gâteaux or Frechet) when the range space is a countably order-complete Banach lattice.
Abstract: We give sufficient conditions for order-bounded convex operators to be generically differentiable (Gâteaux or Frechet). 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 Frechet 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.
TL;DR: In this paper, exact wave-height solutions for trapped waves over two new three-parameter depth topographies are presented for both a semi-infinite and a truncated convex exponential profile, as well as for a semiinfinite concave profile.
Abstract: Abstract 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).
TL;DR: In this article, a class of convex optimal control problems involving linear hereditary systems with linear control constraints and nonlinear terminal constraints is considered, and a result on the existence of an optimal control is proved and a necessary condition for optimality is given.
Abstract: A class of convex optimal control problems involving linear hereditary systems with linear control constraints and nonlinear terminal constraints is considered. A result on the existence of an optimal control is proved and a necessary condition for optimality is given. An iterative algorithm is presented for solving the optimal control problem under consideration. The convergence property of the algorithm is also investigated. To test the algorithm, an example is solved.
TL;DR: In this article, a representation du gradient generalise, which is appropriee du point de vue analytique and du point of vue du calculière, is presented.
Abstract: On cherche a trouver une representation du gradient generalise qui est appropriee du point de vue analytique et du point de vue du calcul. Pour cela on s'appuie sur une famille particuliere de fonctions non differentiables (actuellement lineaires par morceau) qui contient des exemples convexes et non convexes et qui est interessante pour developper des procedures d'estimation statistique
TL;DR: In this article, the relation between semi-infinite programs and optimisation problems with finitely many variables and constraints is reviewed, and two classes of convex semidefinite programs are defined, one based on the fact that a convex set may be represented as the intersection of closed halfspaces, while the other class is defined using the representation of the elements of the convex sets as convex combinations of points and directions.
Abstract: In this paper the relations between semi-infinite programs and optimisation problems with finitely many variables and constraints are reviewed. Two classes of convex semi-infinite programs are defined, one based on the fact that a convex set may be represented as the intersection of closed halfspaces, while the other class is defined using the representation of the elements of a convex set as convex combinations of points and directions. Extension to nonconvex problems is given. A common technique of solving a semi-infinite program computationally is to derive necessary conditions for optimality in the form of a nonlinear system of equations with finitely many equations and unknowns. In the three-phase algorithm, this system is constructed from the optimal solution of a discretised version of the given semi-infinite program. i.e. a problem with finitely many variables and constraints. The system is solved numerically, often by means of some linearisation method. One option is to use a direct analog of the familiar SOLVER method.
TL;DR: In this paper, a resultat de dualite symetrique bien connu a une classe de problemes non differentiables, en posant certaines hypotheses de convexite.
Abstract: On etend un resultat de dualite symetrique bien connu a une classe de problemes non differentiables, en posant certaines hypotheses de convexite
TL;DR: In this article, the authors demontre l'existence des equations de Maxwell dans un milieu conducteur, in which les parametres constitutifs sont constants par morceaux sur R 3.
Abstract: On demontre l'existence des equations de Maxwell dans un milieu conducteur dont les parametres constitutifs sont constants par morceaux sur R 3 . On etudie la convergence de ces solutions a la suite quasi statique ou l'on neglige les courants de deplacement. On examine la regularite de la solution limite et les conditions aux limites classiques (continuite du champ electrique tangentiel et de la densite de courant normale)
TL;DR: In this article, the form of the foliage density equation when f is a trigonometric polynomial, extending earlier results due to J. R. Philip, is given.
Abstract: The foliage density equation is the means by which the foliage density g in a leaf canopy, as a function of the angle of inclination of the leaves, is to be estimated from discrete data gathered using photometric methods or point quadrats. It is an integral equation relating f, a function of angle estimated from measurements, to the unknown function g. The explicit formula for g is known and depends upon f and its first three derivatives; the operator f →, g is unbounded, and the problem is ill posed.In this paper we give the form of g when f is a trigonometric polynomial, extending earlier results due to J. R. Philip. This provides a means of estimating g without directly estimating the derivatives of f from numerical data. To assess the reliability of the method we discuss the convergence of Fourier series representations of f and g.
TL;DR: In this article, a finite sum of Kamps de Feriet's double hypergeometric polynomial in sterms of differen t o kind of singl se hypergeometri oc polynomials in order of order is obtained.
Abstract: (Received 11 July 1984; revised 27 March 1985)AbstractThe main objec otf present paper t iso obtain a finite summatio onf Srivastava's generaltriple hypergeometric serie in terms s of Kampe de Feriet's double hypergeometric series.A numbe or f finite sum of Kamps dee Feriet's double hypergeometric polynomial in sterms of differen t o kindf singl se hypergeometri oc polynomialf highe r order ares ,obtained. Some known result of Manochs a and Sharm [9]a [10], , Munot [11], Pathan [12],Qureshi [15], Quresh ani d Pathan [16] an d Snvastava [26] ar e deduced as special cases. Aresult of Pathan [13, page 316 (1.2) is als]o corrected here.1. IntroductionA unification of Lauricella's fourteen triple hypergeometri F