Boundedness Assumption Research Articles

Convergence to Global Equilibrium for a Kinetic Fermion Model

We study the long-time asymptotics of a kinetic model for fermions in a box with periodic boundary conditions. An entropy dissipation approach is used to prove decay to the global equilibrium for this nonlinear equation, that lacks dissipation in the position variable. We prove convergence at an algebraic rate depending on the smoothness of the solution. The result relies on some initial bounds and a uniform boundedness assumption for spatial derivatives of the solution.

Model selection for Gaussian regression with random design

assumption, it may happen that there is a large distortion between the two distances, resulting in some unusual rates of convergence for the squared L2-risk, as noticed by Baraud. We explain this phenomenon and then show that the use of the Hellinger distance instead of the L2-distance allows us to recover the usual rates and to carry out model selection in great generality. An extension to the L2risk is given under a boundedness assumption similar to that given by Wegkamp and by Yang.

Dynamics of Local Automorphisms of Embedded CR-Manifolds

A classical theorem due to Wong [1] states that the ball is the unique strictly pseudoconvex bounded domain having a noncompact automorphism group. Rosay [2] extended this theorem to bounded domains such that an orbit accumulates at a strictly pseudoconvex domain at the boundary. Later, Efimov [3] got rid of the boundedness assumption. Schoen [4] and, independently, Spiro [5] proved the CR-version of this result, where the strict pseudoconvexity of the accumulation point of an orbit also plays a crucial role. In the present note, we suggest two local CR-versions of this result for hypersurfaces. These versions manifest different behavior depending on whether the Levi form is definite or not. Let M be a real-analytic Levi-nondegenerate hypersurface in some neighborhood U of the origin in C , n ≥ 1 . Denote by Aut0 M the group of local diffeomorphisms of M , that is, the group of germs of biholomorphisms taking M locally into itself and preserving the origin. A point q = 0 (treated as its own radius-vector −→ 0q) is said to be contractible if there is a sequence of local automorphisms Φn ∈ Aut0 M such that

On the behaviour of ∞-harmonic functions near isolated points

In this work we study the behaviour of ∞-harmonic functions near isolated exceptional points. We discuss two situations: (i) when points are in the interior and (ii) when points are on a flat portion of the boundary, on which a solution vanishes. We consider both bounded and unbounded solutions. Under the assumptions of boundedness, solutions are well-behaved in either case. We show that unbounded solutions in (i) change sign, and provide some lower bounds for growth rates valid for both (i) and (ii). In (ii), if the solution has one sign then we can also find an upper bound. We also look at the case of the ball.

On some nonlocal evolution equations in Banach spaces

This note is concerned with the initial value problem for the abstract nonlocal equation \( (Au)' + (Bu) \ni f \) where A is a maximal monotone operator from a reflexive Banach space E to its dual E *, while B is a nonlocal maximal monotone operator from \( L^p (0, T; E)$ to $L^q (0, T; E^*)$ with $p^{-1} + q^{-1}=1, p \in (1,\infty)\). Under proper boundedness and coercivity assumptions on the operators, a solution is achieved by means of a discretization argument. Uniqueness and continuous dependence are also discussed and we prove some estimates for the discretization error. Finally, we deal with the approximation of linear Volterra integrodifferential operators.

On a discrete Boltzmann-Smoluchowski equation with rates bounded in the velocity variables

In this paper, the authors introduce a model of coagulation for macroscopic particles described by their mass and velocity. This model is related to the theory of sprays. Under an assumption of boundedness on the kernel of coagulation (in the space of velocities), they are able to prove that the equation of coagulation is related to a stochastic differential equation. They show that this stochastic equation admits a solution (which can be described in an explicit way). Then, some qualitative properties of the solution are presented: it admits a density, and the mass tends to infinity almost surely.

Discounting long run average growth in stochastic dynamic programs

Finding solutions to the Bellman equation often relies on restrictive boundedness assumptions. In this paper we develop a method of proof that allows to dispense with the assumption that returns are bounded from above. In applications our assumptions only imply that long run average (expected) growth is sufficiently discounted, in sharp contrast with classical assumptions either absolutely bounding growth or bounding each period (instead of long run) maximum (instead of average) growth. We discuss our work in relation to the literature and provide several examples.

Existence of solutions to an anisotropic phase‐field model

AbstractWe consider an anisotropic phase‐field model for the isothermal solidification of a binary alloy due to Warren–Boettinger ( Acta. Metall. Mater. 1995; 43(2):689). Existence of weak solutions is established under a certain convexity condition on the strongly non‐linear second‐order anisotropic operator and Lipschitz and boundedness assumptions for the non‐linearities. A maximum principle holds that guarantees the existence of a solution under physical assumptions on the non‐linearities. The qualitative properties of the solutions are illustrated by a numerical example. Copyright © 2003 John Wiley & Sons, Ltd.

Discrete‐Time Risk‐Sensitive Filters with Non‐Gaussian Initial Conditions and Their Ergodic Properties

ABSTRACTIn this paper, we study asymptotic stability properties of risk‐sensitive filters with respect to their initial conditions. In particular, we consider a linear time‐invariant systems with initial conditions that are not necessarily Gaussian. We show that in the case of Gaussian initial conditions, the optimal risk‐sensitive filter asymptotically converges to a suboptimal filter initialized with an incorrect covariance matrix for the initial state vector in the mean square sense provided the incorrect initializing value for the covariance matrix results in a risk‐sensitive filter that is asymptotically stable, that is, results in a solution for a Riccati equation that is asymptotically stabilizing. For non‐Gaussian initial conditions, we derive the expression for the risk‐sensitive filter in terms of a finite number of parameters. Under a boundedness assumption satisfied by the fourth order absolute moment of the initial state variable and a slow growth condition satisfied by a certain Radon‐Nikodym derivative, we show that a suboptimal risk‐sensitive filter initialized with Gaussian initial conditions asymptotically approaches the optimal risk‐sensitive filter for non‐Gaussian initial conditions in the mean square sense. Some examples are also given to substantiate our claims.

On the Finiteness of Light Rays Between a Source and an Observer on Conformally Stationary Space-Times

We use a convexity condition to prove the finiteness of the number of light rays joining a pointlike source with a pointlike observer in a stationary relativistic spacetime. The result is extended to the case of conformally stationary metrics under suitable boundedness assumptions on the conformal factor. We discuss our hypotheses in some explicit examples of standard static and stationary Lorentzian manifolds.

Representations of Starshaped Sets in Normed Linear Spaces

If S is a closed connected nonconvex locally compact and bounded subset of a real normed linear space or a closed connected nonconvex and bounded subset of a real reflexive Banach space, then kerS=∩{clconvSz:z∈D∩regS}, where regS denotes the set of regular points of S, D is a relatively open subset of S containing the set lncS of local nonconvexity points of S, and Sz={s∈S:z is visible from s via S}. An analogous intersection formula, with the set sphS of spherical points of S in place of regS is shown to hold for a closed connected nonconvex and bounded subset S of a real Banach space which is uniformly convex and uniformly smooth. If the assumption of boundedness of S is dropped, then in all specified settings the above representations hold with intersections of convSz in place of clconvSz. This strengthens and complements results of Borwein and Strojwas, Stavrakas, and the author. Finally, the possibility of generating similar intersection formulae in other configuration set-space is discussed.

Existence of solutions to a phase-field model for the isothermal solidification process of a binary alloy

We investigate the well-posedness of a phase-field model for the isothermal solidification of a binary alloy due to Warren-Boettinger [12]. Existence of weak solution as well as regularity and uniqueness results are established under Lipschitz and boundedness assumptions for the non-linearities. A maximum principle holds that guarantees the existence of a solution under physical assumptions on the non-linearities. Copyright (C) 2000 John Wiley & Sons, Ltd.

Existence of solutions to a phase‐field model for the isothermal solidification process of a binary alloy

We investigate the well-posedness of a phase-field model for the isothermal solidification of a binary alloy due to Warren–Boettinger [12]. Existence of weak solution as well as regularity and uniqueness results are established under Lipschitz and boundedness assumptions for the non-linearities. A maximum principle holds that guarantees the existence of a solution under physical assumptions on the non-linearities. Copyright © 2000 John Wiley & Sons, Ltd.

An optimal nonparametric adaptive control without external excitation

For discrete-time nonlinear stochastic systems with unknown nonparametric structure, a kernel estimation-based nonparametric adaptive controller is constructed by using a truncated certainty equivalence principle. Global stability and asymptotic optimality of the closed-loop systems are established without resorting to either external excitations or the boundedness assumptions of the system noises.

The fundamental theorem of asset pricing for unbounded stochastic processes

The Fundamental Theorem of Asset Pricing states - roughly speaking - that the absence of possibilities for a stochastic process S is equivalent to the existence of an equivalent martingale measure for S. It turns out that it is quite hard to give precise and sharp versions of this theorem in proper generality, if one insists on modifying the concept of no arbitrage as little as possible. It was shown in [DS94] that for a locally bounded R^d-valued semi-martingale S the condition of No Free Lunch with Vanishing Risk is equivalent to the existence of an equivalent local martingale measure for the process S. It was asked whether the local boundedness assumption on S may be dropped. In the present paper we show that if we drop in this theorem the local boundedness assumption on S the theorem remains true if we replace the term equivalent local martingale measure by the term equivalent sigma-martingale measure. The concept of sigma-martingales was introduced by Chou and Emery - under the name of semimartingales de la classe (Sigma_m). We provide an example which shows that for the validity of the theorem in the non locally bounded case it is indeed necessary to pass to the concept of sigma-martingales. On the other hand, we also observe that for the applications in Mathematical Finance the notion of sigma-martingales provides a natural framework when working with non locally bounded processes S. The duality results which we obtained earlier are also extended to the non locally bounded case. As an application we characterize the hedgeable elements. (author's abstract)

On nonlinear detectability

It is shown that the steady state property of the unobservable part of a given nonlinear dynamics is equivalent, under some boundedness assumptions, to the existence of a state detector (detectability). This property is illustrated, discussed and related to the existence of partial observers; local and global aspects are considered.

Iterative feedback tuning: theory and applications

We have examined an optimization approach to iterative control design. The important ingredient is that the gradient of the design criterion is computed from measured closed loop data. The approach is thus not model-based. The scheme converges to a stationary point of the design criterion under the assumption of boundedness of the signals in the loop. From a practical viewpoint, the scheme offers several advantages. It is straightforward to apply. It is possible to control the rate of change of the controller in each iteration. The objective can be manipulated between iterations in order to tighten or loosen performance requirements. Certain frequency regions can be emphasized if desired. This direct optimal tuning algorithm is particularly well suited for the tuning of the basic control loops in the process industry, which are typically PID loops. These primary loops are often very badly tuned, making the application of more advanced (for example, multivariable) techniques rather useless. A first requirement in the successful application of advanced control techniques is that the primary loops be tuned properly. This new technique appears to be a very practical way of doing this, with an almost automatic procedure.

A New Algorithm for State-Constrained Separated Continuous Linear Programs

During the last few decades, significant progress has been made in solving large-scale finite-dimensional and semi-infinite linear programming problems. In contrast, little progress has been made in solving linear programs in infinite-dimensional spaces despite their importance as models in manufacturing and communication systems. Inspired by the research on separated continuous linear programs, we propose a new class of continuous linear programming problems that has a variety of important applications in communications, manufacturing, and urban traffic control. This class of continuous linear programs contains the separated continuous linear programs as a subclass. Using ideas from quadratic programming, we propose an efficient algorithm for solving large-scale problems in this new class under mild assumptions on the form of the problem data. We prove algorithmically the absence of a duality gap for this class of problems without any boundedness assumptions on the solution set. We show this class of problems admits piecewise constant optimal control when the optimal solution exists. We give conditions for the existence of an optimal solution. We also report computational results which illustrate that the new algorithm is effective in solving large-scale realistic problems (with several hundred continuous variables) arising in manufacturing systems.

Closedness of Sum Spaces andthe Generalized Schrödinger Problem

Some general closedness properties of sum spaces of measurable functions are established. As application one obtains existence and uniqueness results for solutions of the generalized Schr\"odinger problem under an integrability condition but without any topological or boundedness assumptions. They also allow to prove an interesting structural result for distributions with multivariate marginals, to prove existence results for optimal approximations in additive statistical models, and to give an extension of Kolmogorov's representation theorem for continuous functions of severalvariables. This extension implies that any locally bounded measurable function has an exact representation by a neural network with one hidden layer.

Some remarks on a boundedness assumption for monotone dynamical systems

We study the consequences of the assumption that all forward orbits are bounded for monotone dynamical systems. In particular, it turns out that this assumption has more implications than is immediately apparent.