Bounded Functions Research Articles

Structural aspects of the non-uniformly continuous functions and the unbounded functions within C(X)

We prove in this paper that if a metric space supports a real continuous function which is not uniformly continuous then, under appropriate mild assumptions, there exists in fact a plethora of such functions, in both topological and algebraical senses. Corresponding results are also obtained concerning unbounded continuous functions on a non-compact metrizable space.

On the Expected Total Reward with Unbounded Returns for Markov Decision Processes

We consider a discrete-time Markov decision process with Borel state and action spaces. The performance criterion is to maximize a total expected utility determined by unbounded return function. It is shown the existence of optimal strategies under general conditions allowing the reward function to be unbounded both from above and below and the action sets available at each step to the decision maker to be not necessarily compact. To deal with unbounded reward functions, a new characterization for the weak convergence of probability measures is derived. Our results are illustrated by examples.

Well-posedness for the continuity equation for vector fields with suitable modulus of continuity

We prove well-posedness of linear scalar conservation laws using only assumptions on the growth and the modulus of continuity of the velocity field, but not on its divergence. As an application, we obtain uniqueness of solutions in the atomic Hardy space, H1, for the scalar conservation law induced by a class of vector fields whose divergence is an unbounded BMO function.

Algebraic structure of continuous, unbounded and integrable functions

In this paper we study the large linear and algebraic size of the family of unbounded continuous and integrable functions in $[0,+\infty)$ and of the family of sequences of these functions converging to zero uniformly on compacta and in $L^1$-norm. In addition, we concentrate on the speed at which these functions grow, their smoothness and the strength of their convergence to zero.

On the heat integral identity for unbounded functions

On the heat integral identity for unbounded functions

Quaternionic Monge–Ampère operator for unbounded plurisubharmonic functions

In this paper, we generalize the definition of the quaternionic Monge–Ampere operator to some unbounded plurisubharmonic functions, and we prove that the quaternionic Monge–Ampere operator is continuous on the monotonically decreasing sequences of plurisubharmonic functions. After introducing the generalized Lelong number of a positive current, Demailly’s comparison theorems are showed. Moreover, we prove that the quaternionic Lelong–Jensen-type formula also holds for the unbounded plurisubharmonic function.

Stability Properties of the Value Function in an Infinite Horizon Optimal Control Problem

Properties of the value function are examined in an infinite horizon optimal control problem with an integrand index appearing in the quality functional with a discount factor. The properties are analyzed in the case when the payoff functional of the control system includes a quality index represented by an unbounded function. An upper estimate is given for the growth rate of the value function. Necessary and sufficient conditions are obtained to ensure that the value function satisfies the infinitesimal stability properties. The question of coincidence of the value function with the generalized minimax solution of the Hamilton–Jacobi–Bellman–Isaacs equation is discussed. The uniqueness of the corresponding minimax solution is shown. The growth asymptotic behavior of the value function is described for the logarithmic, power, and exponential quality functionals, which arise in economic and financial modeling. The obtained results can be used to construct grid approximation methods for the value function as the generalized minimax solution of the Hamilton–Jacobi–Bellman–Isaacs equation. These methods are effective tools in the modeling of economic growth processes.

Statistical Properties for Flows with Unbounded Roof Function, Including the Lorenz Attractor

For geometric Lorenz attractors (including the classical Lorenz attractor) we obtain a greatly simplified proof of the central limit theorem which applies also to the more general class of codimension two singular hyperbolic attractors. We also obtain the functional central limit theorem and moment estimates, as well as iterated versions of these results. A consequence is deterministic homogenisation (convergence to a stochastic differential equation) for fast-slow dynamical systems whenever the fast dynamics is singularly hyperbolic of codimension two.

Lyapunov Stability of the Generalized Stochastic Pantograph Equation

The purpose of the paper is to study stability properties of the generalized stochastic pantograph equation, the main feature of which is the presence of unbounded delay functions. This makes the stability analysis rather different from the classical one. Our approach consists in linking different kinds of stochastic Lyapunov stability to specially chosen functional spaces. To prove stability, we check that the solutions of the equation belong to a suitable space of stochastic processes, instead of searching for an appropriate Lyapunov functional. This gives us possibilities to study moment stability, stability with probability 1, and many other stability properties in an efficient way. We show by examples how this approach works in practice, putting emphasis on delay-independent stability conditions for the generalized stochastic pantograph equation. The framework can be applied to any stochastic functional differential equation with finite dimensional initial conditions.

Multiple-priors optimal investment in discrete time for unbounded utility function

This paper investigates the problem of maximizing expected terminal utility in a discrete-time financial market model with a finite horizon under nondominated model uncertainty. We use a dynamic programming framework together with measurable selection arguments to prove that under mild integrability conditions, an optimal portfolio exists for an unbounded utility function defined on the half-real line.

Times from Infection to Disease-Induced Death and their Influence on Final Population Sizes After Epidemic Outbreaks.

For epidemic models, it is shown that fatal infectious diseases cannot drive the host population into extinction if the incidence function is upper density-dependent. This finding holds even if a latency period is included and the time from infection to disease-induced death has an arbitrary length distribution. However, if the incidence function is also lower density-dependent, very infectious diseases can lead to a drastic decline of the host population. Further, the final population size after an epidemic outbreak can possibly be substantially affected by the infection-age distribution of the initial infectives if the life expectations of infected individuals are an unbounded function of infection age (time since infection). This is the case for lognormal distributions, which fit data from infection experiments involving tiger salamander larvae and ranavirus better than gamma distributions and Weibull distributions.

Robust Bayesian Regression Analysis Using Ramsay-Novick Distributed Errors with Student-t Prior

This paper investigates bayesian treatment of regression modelling with Ramsay - Novick (RN) distribution specifically developed for robust inferential procedures. It falls into the category of the so-called heavy-tailed distributions generally accepted as outlier resistant densities. RN is obtained by coverting the usual form of a non-robust density to a robust likelihood through the modification of its unbounded influence function. The resulting distributional form is quite complicated which is the reason for its limited applications in bayesian analyses of real problems. With the help of innovative Markov Chain Monte Carlo (MCMC) methods and softwares currently available, here we first suggested a random number generator for RN distribution. Then, we developed a robust bayesian modelling with RN distributed errors and Student-t prior. The prior with heavy-tailed properties is here chosen to provide a built-in protection against the misspecification of conflicting expert knowledge (i.e. prior robustness). This is particularly useful to avoid accusations of too much subjective bias in the prior specification. A simulation study conducted for performance assessment and a real-data application on the famously known "stack loss" data demonstrated that robust bayesian estimates with RN likelihood and heavy-tailed prior are robust against outliers in all directions and inaccurately specified priors.

Solutions of the average cost optimality equation for Markov decision processes with weakly continuous kernel: The fixed-point approach revisited

This paper shows the existence of lower semicontinuous solutions of the average cost optimality equation for Markov decision processes with Borel spaces, possible unbounded cost function and weakly continuous transition kernel. This is done imposing a growth condition on the cost function, a Lyapunov stability condition on the transition kernel and a set of standard compactness-continuity conditions. The solution of the average cost optimality equation is obtained by means of the Banach fixed-point theorem.

Quasicompact and Riesz Composition Endomorphisms of Lipschitz Algebras of Complex-Valued Bounded Functions and Their Spectra

In this paper, we study quasicompact and Riesz composition endomorphisms of Lipschitz algebras of complex-valued bounded functions on metric spaces, not necessarily compact. We give some necessary and some sufficient conditions that a composition endomorphism of these algebras to be quasicompact or Riesz. We also establish an upper bound and a formula for the essential spectral radius of a composition endomorphism T of these algebras under some conditions which implies that T is quasicompact or Riesz. Finally, we get a relation for the set of eigenvalues and the spectrum of a quasicompact and Riesz endomorphism of these algebras.

Correction to: Quasicompact and Riesz Composition Endomorphisms of Lipschitz Algebras of Complex-Valued Bounded Functions and Their Spectra

In the original publication of this article, the e-mail address of the corresponding author was inadvertently entered incorrectly.

Distribution-dependent concentration inequalities for tighter generalization bounds

Concentration inequalities are indispensable tools for studying the generalization capacity of learning models. Hoeffding's and McDiarmid's inequalities are commonly used, giving bounds independent of the data distribution. Although this makes them widely applicable, a drawback is that the bounds can be too loose in some specific cases. Although efforts have been devoted to improving the bounds, we find that the bounds can be further tightened in some distribution-dependent scenarios and conditions for the inequalities can be relaxed. In particular, we propose four types of conditions for probabilistic boundedness and bounded differences, and derive several distribution-dependent extensions of Hoeffding's and McDiarmid's inequalities. These extensions provide bounds for functions not satisfying the conditions of the existing inequalities, and in some special cases, tighter bounds. Furthermore, we obtain generalization bounds for unbounded and hierarchy-bounded loss functions. Finally we discuss the potential applications of our extensions to learning theory.

Applications of the Shorgin Identity to Bernstein Type Operators

By establishing an identity between a sequence of Bernstein-type operators and a sequence of Szasz–Mirakyan operators, we prove that the convergence of Bernstein-type operators is related to convergence with respect to Szasz–Mirakyan operators. As one application of this identity, we prove that whenever the parameters are conveniently chosen, if $$f\in C[0,\infty )$$ satisfies a growth condition of the form $$|f(t)|\le C e^{\alpha t}(C,\alpha \in \mathbb {R}^+)$$ , then the classical Bernstein operators $$B_{mn}(f(nu),x/n)$$ converge to the Szasz–Mirakyan operator $$S_m(f,x)$$ . This convergence generalizes the classical result of De la Cal and Liquin to unbounded functions; moreover, the rth derivative of $$B_{mn}(f(nu),x/n)$$ converges to the rth derivative of $$S_m(f,x)$$ . As another application of this identity, we derive Voronowskaja type result for the general Lototsky–Bernstein operators.

Local property of maximal unbounded plurifinely plurisubharmonic functions

ABSTRACTIn this paper, we give a sufficient condition such that -maximality is a -local notion for unbounded -plurisubharmonic functions. A characterization of -maximal functions is also shown.

Convergence and Robustness Analysis of the Exponential-Type Varying Gain Recurrent Neural Network for Solving Matrix-Type Linear Time-Varying Equation

To solve matrix-type linear time-varying equation more efficiently, a novel exponential-type varying gain recurrent neural network (EVG-RNN) is proposed in this paper. Being distinguished from the traditional fixed-parameter gain recurrent neural network (FG-RNN), the proposed EVG-RNN is derived from a vector- or matrix-based unbounded error function by a varying-parameter neural dynamic approach. With four different kinds of activation functions, the super-exponential convergence performance of EVG-RNN is proved theoretically in details, of which the error convergence rate is much faster than that of FG-RNN. In addition, mathematics proves that the computation errors of EVG-RNN can converge to zero, and it possesses the capability of restraining external interference. Finally, series of computer simulations verify and illustrate the better performance of convergence and robustness of EVG-RNN than that of FG-RNN and FTZNN when solving the identical linear time-varying equation.

ОБ ОБОБЩЕНИЯХ И ПРИЛОЖЕНИЯХ ВАРИАЦИОННЫХ ПРИНЦИПОВ НЕЛИНЕЙНОГО АНАЛИЗА

There are considered some classes of functions to which variational principles of nonlinear are applicable. In particular, it is shown that the Bishop-Phelps variational principle is applicable to some unbounded below functions. The properties of locally Lipschitzian mappings are investigated. Conditions for a mapping that is pseudo-Lipschitzian at every point of its graph to be Lipschitzian are derived.