Équations du filtrage non linéaire de la prédiction et du lissage

We establish equations of non linear filtering, prediction (extrapolation) and smoothing (interpolation) in the case where the signal is a non degenerate diffusion process, and the observation is a noisy functional of the signal. We consider both the case of observation noise correlated with the signal, and the opposite case where we establish “robust” form of the equations. We study finally the case of unbounded coefficients, and the case where there is a feedback from the observation to the signal.     

On the weak discontinuity surfaces and tapes of equation systems in fluid mechanics

The weak discontinuity surfaces for a system of quasi-linear differential equations of higher order are developed. The classification of equation systems in fluid mechanics is based on the propagative weak discontinuity surfaces. Types of equations for different flow models are discussed. The conclusion is as follows:(a) For incompressible nonviscous flow, incompressible viscous flow and compressible viscous flow, the types of equations are all parabolic in the unsteady case and elliptic in the steady case.(b) For compressible nonviscous flow, the type of equations is hyperbolic in the unsteady case or steady supersonic case, and the type is elliptic in the steady subsonic case.     

Interpolation error estimates of a modified 8-node serendipity finite element

Summary. Interpolation error estimates for a modified 8-node serendipity finite element are derived in both regular and degenerate cases, the latter of which includes the case when the element is of triangular shape. For defined over a quadrilateral K, the error for the interpolant is estimated as , where in the regular case and in the degenerate case, respectively. Thus, the obtained error estimate in the degenerate case is of the same quality as in the regular case at least for . Results for some related elements are also given. Received June 2, 1997 / Published online March 16, 2000     

Asymptotic inference for AR(1) processes with (nonnormal) stable errors. IV. A note on the case of a negative unit root

In this note, we show that the limit distribution of the least-squares estimator in the case of a negative unit root is different from the limit distribution in the case of a positive unit root. Thus, the infinite-variance case is different from the finite-variance case. Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló, Hungary, 1997, Part II.     

A note on semilinear heat equation in hyperbolic space

Bandle et al. [1] obtained a quite interesting result about a semilinear heat equation that the Fujita exponent relative to the whole hyperbolic space is just the same as that relative to bounded domain in Euclidean space, and, in addition, the properties of solutions are different in the critical exponent case. Our purpose is to answer an open problem proposed by Bandle et al. for the critical exponent case, and it, together with the one obtained by them, shows that the critical exponent case does belong to the non-blow-up case, which is completely different from the case in Euclidean space.     

An approximation algorithm for the hierarchical median problem

The hierarchical median problem asks for a hierarchical sequence of solutions to the k-median problems of growing cardinality. The best algorithm known for this problem in the general metric case has competitive ratio 20.71. In the paper, the case is under study that the clients and facilities lie on the real line, as well as the case of a Euclidean space. An algorithm is proposed with competitive ratio 8 in the case of the real line, and 8 + 4√2 (approximately 13.66), in the Euclidean case.     

Stability of superposed viscous-viscoelastic (Rivlin-Ericksen) fluids in the presence of suspended particles through a porous medium

The Rayleigh-Taylor instability of a Newtonian viscous fluid overlying a Rivlin-Ericksen viscoelastic fluid containing suspended particles in a porous medium is considered. As in both Newtonian viscous-viscoelastic fluids the system is stable in the potentially stable case and unstable in the potentially unstable case, this holds for the present problem also. The effects of a variable horizontal magnetic field and a uniform rotation are also considered. The presence of magnetic field stabilizes a certain wave-number band, whereas the system is unstable for all wave-numbers in the absence of the magnetic field for the potentially unstable configuration. However, the system is stable in the potentially stable case and unstable in the potentially unstable case for highly viscous fluids in the presence of a uniform rotation.     

On constructing periodic solutions of quasi-linear non-self-contained systems with one degree of freedom, when amplitude equations possess multiple roots

Construction of periodic solutions of quasilinear non-self-contained systems with one degree of freedom, was investigated in [1 and 2]. In [1] the case of simple roots of amplitude equations was considered together with the case of a double root when the solution could be expanded into a series in integral powers of μ. In [2] the case of a double root is investigated in more detail Including expansions of solutions into series in μ^1/2. In the present paper, the case of arbitrary multiple roots for non-self-contained systems is reduced to the corresponding case for self-contained systems, which simplifies computations.     

Discussion on the convergence of Rosen's gradient projection method

Since Rosen's gradient projection method was published in 1960, a rigorous convergence proof of the method has remained to be an open question. A convergence proof for the three dimensional case is given in this paper. The whole proof, except one lemma which we failed to prove for the general case, is applicable to the general case. For the general case a convergence condition is given in the main theorem.This research was supported in part by the National Science Foundation under the research grant MCS 81-01214.     

Semi-Classical Analysis for Hartree Equations in Some Supercritical Cases

We study the semi-classical limit of the Hartree equation, which has focusing at a point. There exists a critical index indicating nonlinear effect around the caustic, and it is known that the influence by the nonlinearity is negligible in subcritical case (called linear caustic case), and that it is not in critical case (nonlinear caustic case). We give the asymptotic behavior beyond caustic in some supercritical cases which give rise to very strong nonlinear effect. Submitted: August 25, 2006. Accepted: December 11, 2006.     

Non-collision periodic solutions of second order singular dynamical systems

We study the existence of non-collision periodic solutions for second order singular dynamical systems. The repulsive case and the attractive case are dealt with using a unified topological approach. The proof is based on a well-known fixed point theorem for completely continuous operators, involving a new type of cone. We do not need to consider so-called strong force conditions. Moreover, for the repulsive case, the critical case can be covered. Recent results in the literature, even in the scalar case, are complemented, generalized and improved.     

The compound decision problem in the opponent case

The compound decision problem is considered under the assumption that Nature plays the role of an opponent. The empirical Bayes approach is a particular case of this approach. It is shown that in many instances sequential compound rules with good asymptotic properties for the fixed case have similar good properties in the opponent case.     

MODELING EFFECTS OF TWO INTERACTING POPULATIONS ON A RENEWABLE RESOURCE

Abstract In this paper, a general mathematical model is presented to study the effect of two populations on a resource biomass. The interaction between two populations is assumed to be competition, predation, or cooperation. These two populations may depend on the resource biomass partially, wholly, or they may predate on the resource. In each case, criteria for local stability, instability, and global stability are obtained. Numerical simulation is presented to illustrate the theoretical results obtained in each case. It is shown that the depletion of resource biomass is maximum in the case of cooperation and is minimum in the case of competition.     

Stochastic integral equations applied to telecommunications traffic without delay

Modern telecommunication techniques case the problem of traffic handling in the framework of fairly general networks, as applied to traffic without delay but with virtually arbitrary service-time distributions. In this paper we use stochastic integral equations to deal with the case involving the most general input process and lost calls. For this purpose, Fortet's equation, unsolved so far in the general case, is solved to analyze the single trunk group model. The stationary case is then treated as a special case. Finally we study networks which satisfy a certain assumption of symmetry. The same general stochastic assumptions are maintained throughout the paper.     

Duality theorems for a class of nonconvex programming problems

The object of this paper is to prove duality theorems for quasiconvex programming problems. The principal tool used is the transformation introduced by Manas for reducing a nonconvex programming problem to a convex programming problem. Duality in the case of linear, quadratic, and linear-fractional programming is a particular case of this general case.The authors are grateful to the referees for their kind suggestions.     

Redundancy allocation at component level versus system level

In this paper, we treat the problem of stochastic comparison of standby [active] redundancy at component level versus system level. In the case of standby redundancy, we present some interesting comparison results of both series systems and parallel systems in the sense of various stochastic orderings for both the matching spares case and non-matching spares case, respectively. In the case of active redundancy, a likelihood ratio ordering result of series systems is presented for the matching spares case; and for the non-matching spares case, a counterexample is provided to show that there does not exist similar result even for the hazard rate ordering. The results established here strengthen and generalize some of those known in the literature. Some numerical examples are also provided to illustrate the theoretical results.     

On nonlinear nuclear operators

A class of operators is introduced and referred to as nonlinear nuclear operators. As in the case of linear nuclear operators this class of nonlinear operators is defined by means of a tensor product of two topological vector spaces. It is then shown that, as in the linear case, a series representation of the operator is valid. The deviation of the nonlinear theory from the linear case is discussed and an example in the framework of generalized stochastic processes is given.     

Ordered Site Access and Optimal Forest Rotation

The ordered-site-access model of forest harvesting formulated for once-and-for-all forests in [7] is extended to the case of ongoing forests. The economic content of the corresponding optimal harvest schedule is delineated. For an infinite harvest sequence, the optimal schedule is shown to include the classical Faustmann rotation as a special case, and the effect of net revenue functions changing with harvest is studied. For the practically more important case of planning for a finite sequence of [INLINEEQUATION] harvests, the optimal harvest schedule is determined for a Faustmann environment with limited, and unlimited harvesting capacity, and its rapid convergence to the Faustmann rotation is shown for the case of unlimited harvesting capacity. The case of harvest cost functions varying with harvest rate is discussed. The existence of a steady-state optimal harvesting schedule (involving a pathwise uniform age distribution) for the more realistic Heaps-Neher environment and its relation to the Faustmann rotation are analyzed. The evolution of the optimal harvest schedule for a finite harvest sequence in a Heaps-Neher environment toward this steady-state (Faustmann type) rotation is demonstrated.     

Lower bounds of principal eigenvalue in dimension one

For the principal eigenvalue with bilateral Dirichlet boundary condition, the so-called basic estimates were originally obtained by capacitary method. The Neumann case (i.e., the ergodic case) is even harder, and was deduced from the Dirichlet one plus a use of duality and the coupling method. In this paper, an alternative and more direct proof for the basic estimates is presented. The estimates in the Dirichlet case are then improved by a typical application of a recent variational formula. As a dual of the Dirichlet case, the refine problem for bilateral Neumann boundary condition is also treated. The paper starts with the continuous case (one-dimensional diffusions) and ends at the discrete one (birth-death processes). Possible generalization of the results studied here is discussed at the end of the paper.     

Possible numbers of ones in 0–1 matrices with a given rank

We determine the possible numbers of ones in a 0–1 matrix with given rank in the generic case and in the symmetric case. There are some unexpected phenomena. The rank 2 symmetric case is subtle.