Solutions for a class of neutral differential equations with nonatomicD-operator

This paper is concerned with the existence and uniqueness of solutions to first-order linear differential equations of neutral type with a nonatomicD-operator. We use an algebraic approach, in the context of the theory of convolution operators. Necessary and sufficient conditions are given for the initial value problem to have unique solutions for a class of equations.This work was done while the author was at the Department of Mathematics, Virginia Polytechnic Institute and State University.Instituto de Desarrollo Tecnológico para la Industria Quimica (INTEC), dependent on Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET) and the Universidad Nacional del Litoral (UNL).     

Connections on tangent bundles of higher order associated to regular Lagrangians

Given a regular Lagrangian L of order k on M we show that there exists a canonical connection _L on T ^2k–1 M whose paths are the solutions of the Euler-Lagrange equations for L. If L is the kinetic energy defined by a Riemannian metric then _L is the Riemannian connection.Partially supported by Subdirección General de Cooperación Internacional, Programa de Cooperación Científica con Iberoamérica, SPAIN and CNPq-BRAZIL, Proc. 30.1115/79.     

The non-existence of symplectic multi-derivative Runge-Kutta methods

A sufficient condition for the symplecticness ofq-derivative Runge-Kutta methods has been derived by F. M. Lasagni. In the present note we prove that this condition can only be satisfied for methods withq1, i.e., for standard Runge-Kutta methods. We further show that the conditions of Lasagni are also necessary for symplecticness so that no symplectic multi-derivative Runge-Kutta method can exist.This research has been supported by project PB89-0351 (Dirección General de Investigación Científica y Técnica) and by project No. 20-32354.91 of Swiss National Science Foundation.     

Optimal control in coefficients of elliptic equations with state constraints

This paper is concerned with state constrained optimal control problems of elliptic equations, the control being a coefficient of the partial differential equation. Existence of an optimal control is proved and optimality conditions are derived. We perform finite-element approximations of optimal control problems and state some convergence results: we prove convergence of optimal controls and states as well as convergence of Lagrange multipliers.This research was partially supported by the Dirección General de Investigación Científica y Técnica (Madrid).     

Vandermonde matrices on the circle: Spectral properties and conditioning

Summary We study Vandermonde matrices whose nodes are given by a Van der Corput sequence on the unit circle. Our primary interest is in the singular values of these matrices and the respective (spectral) condition numbers. Detailed information about multiplicities and eigenvectors, however, is also obtained. Two applications are given to the theory of polynomials.Dedicated to R. S. Varga on the occasion of his sixtieth birthdayResearch of A. C. supported by the Fundación Andes, Chile, and by the German Academic Exchange Service (DAAD), Federal Republic of GermanyResearch of W. G. supported, in part, by the National Science Foundation, USA, (Grant CCR-8704404)Research of S. R. supported by the Fondo Nacional de Desarollo Cientßfico y Tecnológico (FONDECYT), Chile, (Grant 237/89), by the Universidad Técnica F. Santa Marßa, Valparaßso, Chile, (Grant 89.12.06), and by the German Academic Exchange Service (DAAD), Federal Republic of Germany     

The geometry of surfaces in 4-space from a contact viewpoint

We study the geometry of the surfaces embedded in ⁴ through their generic contacts with hyperplanes. The inflection points on them are shown to be the umbilic points of their families of height functions. As a consequence we prove that any generic convexly embedded 2-sphere in ⁴ has inflection points.The research of the second author was partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Brazil.The research of the third author was partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Brazil.     

Lower subdifferentiability of quadratic functions

In this paper we characterize those quadratic functions whose restrictions to a convex set are boundedly lower subdifferentiable and, for the case of closed hyperbolic convex sets, those which are lower subdifferentiable but not boundedly lower subdifferentiable.Once characterized, we will study the applicability of the cutting plane algorithm of Plastria to problems where the objective function is quadratic and boundedly lower subdifferentiable.Financial support from the Dirección General de Investigación Científica y Técnica (DGICYT), under project PS89-0058, is gratefully acknowledged.     

Boundary control problems for quasi-linear elliptic equations:

In this paper we prove some optimality conditions, in the form of a Pontryagin's principle, for boundary control problems governed by quasi-linear elliptic equations. Because of the presence of state constraints, we distinguish the cases of qualified and nonqualified conditions for optimality. Both cases are treated in the paper. Neither convexity of the control set nor differentiability of the functions involved in the control problem are assumed.This research was partially supported by Dirección General de Investigation Científica y Técnica (Madrid).     

Deterministic impulse control in native forest ecosystems management

This paper illustrates an application of impulse control systems analysis to the management of renewable resources. We obtain an explicit definition for the impulse optimal strategy in the case of a nonlinear natural system which models certain types of forests. The result is in agreement with an econo-ecological conservative action on the forest. The examples developed in this paper associate new theories on impulse or jump control with natural forest management. In addition, we compare our results with the classical ones. We also show that a nonoptimal managing action on the forest is frequently the cause for significant medium-term losses.This work was supported by CONICET, Consejo Nacional de Investigaciones Científicas y Técnicas, and by the Universidad Nacional del Litoral, Argentina.     

ARMA identification

In view of recent results on the asymptotic behavior of the prediction error covariance for a state variable system (see Ref. 1), an identification scheme for autoregressive moving average (ARMA) processes is proposed. The coefficients of thed-step predictor determine asymptotically the system momentsU ₀,...,U _d–1. These moments are also nonlinear functions of the coefficients of the successive 1-step predictors. Here, we estimate the state variable parameters by the following scheme. First, we use the Burg technique (see Ref. 2) to find the estimates of the coefficients of the successive 1-step predictors. Second, we compute the moments by substitution of the estimates provided by the Burg technique for the coefficients in the nonlinear functions relating the moments with the 1-step predictor coefficients. Finally, the Hankel matrix of moment estimates is used to determine the coefficients of the characteristic polynomial of the state transition matrix (see Refs. 3 and 4).A number of examples for the state variable systems corresponding to ARMA(2, 1) processes are given which show the efficiency of this technique when the zeros and poles are separated. Some of these examples are also studied with an alternative technique (see Ref. 5) which exploits the linear dependence between successive 1-step predictors and the coefficients of the transfer function numerator and denominator polynomials.In this paper, the problems of order determination are not considered; we assumed the order of the underlying system. We remark that the Burg algorithm is a robust statistical procedure. With the notable exception of Ref. 6 that uses canonical correlation methods, most identification procedures in control are based on a deterministic analysis and consequently are quite sensitive to errors. In general, spectral identification based on the windowing of data lacks the resolving power of the Burg technique, which is a super resolution method.This work was supported by NATO Research Grant No. 585/83, by University of Nice, by Thomson CSF-DTAS, by Instituto Nacional de Investigação Científica, and by CIRIT (Comissió Interdepartmental de Recerca i Innovació Technológica de Catalunya). The work of the third author was also partially supported by Army Research Office Contract DAAG-29-84-k-005.Simple ARMA(2, 1) Basic language analysis programs to construct random data were written by the second author and Dr. K. D. Senne, MIT Lincoln Laboratory. Lack of stability of the direct estimation was observed at TRW with the help of Dr. G. Butler. Analysis programs in FORTRAN for ARMA(p, q) were written and debugged at CAPS by the fifth author. The research was helped by access to VAXs at Thomson CSF-DTAS, Valbonne, France, at CAPS, Instituto Superior Técnico, and at Universitat Politécnica de Catalunya. In particular, the authors explicitly acknowledge Thomson CSF-DTAS and Dr. H. Gautier for extending the use of their facilities to the authors from September 1983 until June 1984, when the examples presented were simulated.on leave from University of Southern California, Los Angeles, California.formerly visiting at Massachusetts Institute of Technology and Laboratory for Information and Decision Systems, while on leave from Instituto Superior Técnico, Lisbon, Portugal.     

On Goppa Codes and Weierstrass Gaps at Several Points

We generalize results of Homma and Kim [J. Pure Appl. Algebra Vol. 162, (2001), pp. 273–290] concerning an improvement on the Goppa bound on the minimum distance of certain Goppa codes.This paper was written while the second author was visiting The University of Valladolid (Dpto. Algebra, Geometría y Topología; Fac. de Ciencias) supported by the Grant SB2000-0225 from the Secretaria de Estado de Educacíon y Universidades del Ministerio de Educacíon, Cultura y Deportes de España.     

Fractional programming by lower subdifferentiability techniques

The notion of lower subdifferentiability is applied to the analysis of convex fractional programming problems. In particular, duality results and optimality conditions are presented, and the applicability of a cutting-plane algorithm using lower subgradients is discussed. These methods are useful also in generalized fractional programming, where, in the linear case, the performance of the cutting-plane algorithm is compared with that of the most efficient version of the Dinkelbach method, which is based on the solution of a parametric linear programming problem.The authors wish to thank Mr. Jaume Timoneda for his help in the implementation of the numerical methods on the computer and the referees for valuable comments and suggestions; the present improved statement and proof of Proposition 2.1 is due to one of them. Financial support from the Dirección General de Investigación Científica y Técnica (DGICYT), under project PS89-0058, is gratefully acknowledged.     

Structural constrained controllers for discrete-time linear systems

In this paper, we present an approach for designing structural constrained controllers for discrete-time linear systems, based on a new stabilizability property of the Riccati equation solution. First, the feedback stabilization problem under a general structural constraint is considered and a simple numerical procedure to solve it is presented. Special attention is given to the output feedback stabilization problem, for which sufficient conditions for the existence and convergence toward a stabilizing matrix are provided. Some examples are solved and comparisons with other methods available in the literature are made.This research has been developed with the financial support of the Conselho Nacional de Desenvolvimento Científico e Tecnológico, CNPq, Brasília, Brazil.     

Second-order properties of the loss probability inM/M/s/s +c systems

The Erlang loss function, which gives the steady state loss probability in anM/M/s/s system, has been extensively studied in the literature. In this paper, we look at the similar loss probability inM/M/s/s + c systems and an extension of it to nonintegral number of servers and queue capacity. We study its monotonicity properties. We show that the loss probability is convex in the queue capacity, and that it is convex in the traffic intensity if is below some ^* and concave if is greater that ^*, for a broad range of number of servers and queue capacities. We prove that the one-server loss system is the onlyM/M/s/s +c system for which the loss probability is concave in the traffic intensity in all its range.Research supported by Grant BD/645/90-RM from Junta Nacional de Investigação Científica e Tecnológica.On leave from: Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1096 Lisboa Codex, Portugal.     

Non-connected toric Hilbert schemes

We construct small (50 and 26 points, respectively) point sets in dimension 5 whose graphs of triangulations are not connected. These examples improve our construction in J. Amer. Math. Soc. 13:3 (2000), 611–637 not only in size, but also in that the associated toric Hilbert schemes are not connected either, a question left open in that article. Additionally, the point sets can easily be put into convex position, providing examples of 5-dimensional polytopes with non-connected graph of triangulations.Mathematics Subject Classification (2000): Primary 52B11; Secondary 52B20The main result in this paper was obtained in the fall of 2001, while I was a visiting professor in the Department of Mathematics, U.C. Davis, supported by U. C. Davis, M.S.R.I. and the Spanish Government. I am also partially supported by grant BFM2001–1153 of the Spanish Dirección General de Enseñanza Superior e Investigación Científica. The paper is dedicated to Bernd Sturmfels on his 40th birthday.     

A general cone decomposition theory based on efficiency

LetK ₁ andK ₂ be two convex cones in some common vector space. This paper is concerned with the question of finding a good decomposition, with respect toK ₁ andK ₂, of a given element of the Minkowski sumK ₁ +K ₂. We propose the criterion of efficiency as a measure for the quality of a decomposition. This criterion allows us to set up a framework from which a general cone decomposition theory is then derived.Research supported by Dirección General de Investigación Científica y Técnica (DGICYT), under project PS89-0058.     

Integral curves of derivations

We integrate, by a constructive method, derivations of even degree on the sections of an exterior bundle by families of Z ₂-graded algebra automorphisms, dependent on a real parameter, and which satisfy a flow condition. We also study the case of local endomorphisms when their components of degree zero and derivations and with no component of negative degree, but then we have integral families of R-linear automorphisms. This integration method can be applied to the Frölicher—Nijenhuis derivations on the Cartan algebra of differential forms, and to the integration of superfields on graded manifolds.Research partially supported by the Comisión Asesora Investigación Científica y Técnica of the Spanish Ministry of Education, program 1985–87, number 120.     

Kelley-Morse+Types of well order is not a conservative extension of Kelley Morse

Summary Assuming the consistency ofZF + There is an inaccessible number of inaccessibles, we prove that Kelley Morse theory plus types is not a conservative extension of Kelley-Morse theory.This paper was partially supported by: Dirección de Investigación de la Pontificia Universidad Católica de Chile (DIUC); Fondo Nacional de Desarrollo Científico y Tecnológico (FONDECYT)     

Order conditions for canonical Runge-Kutta-Nyström methods

We are concerned with Runge-Kutta-Nyström methods for the integration of second order systems of the special formd ² y/dt ²=f(y). If the functionf is the gradient of a scalar field, then the system is Hamiltonian and it may be advantageous to integrate it by a so-called canonical Runge-Kutta-Nyström formula. We show that the equations that must be imposed on the coefficients of the method to ensure canonicity are simplifying assumptions that lower the number of independent order conditions. We count the number of order conditions, both for general and for canonical Runge-Kutta-Nyström formulae.This research has been supported by Junta de Castilla y León under project 1031-89 and by Dirección General de Investigación Científica y Técnica under project PB89-0351.     

Approximation of average cost optimal policies for general Markov decision processes with unbounded costs

The aim of the paper is to show that Lyapunov-like ergodicity conditions on Markov decision processes with Borel state space and possibly unbounded cost provide the approximation of an average cost optimal policy by solvingn-stage optimization problems (n = 1, 2, ...). The used approach ensures the exponential rate of convergence. The approximation of this type would be useful to find adaptive procedures of control and to estimate stability of an optimal control under disturbances of the transition probability.Research supported in part by Consejo Nacional de Ciencia y Tecnologia (CONACYT) under grant 0635P-E9506.Research supported by Fondo del Sistema de Investigatión del Mar de Cortés under Grant SIMAC/94/CT-005.