Idempotent Probability Research Articles

Functor of Idempotent Probability Measures with Compact Support and Open Mappings

Functor of Idempotent Probability Measures with Compact Support and Open Mappings

Dugundji Compacta and the Space of Idempotent Probability Measures

Dugundji Compacta and the Space of Idempotent Probability Measures

Dugundji Compacta and the Space of Idempotent Probability Measures

Для заданной группы $(G,X,\alpha)$ топологических преобразований на тихоновском пространстве $X$ построена группа $(I(G, X), I(X), I(\alpha))$ топологических преобразований на пространстве $I(X)$ идемпотентных вероятностных мер. Показано, что если действие $\alpha$ группы $G$ открыто, то действие $I(\alpha)$ группы $I(G, X)$ также открыто; при этом приведен пример, показывающий, что открытость действия $\alpha$ существенна. Установлено, что если диагональное произведение $\Delta f_{p}$ заданного семейства $\{f_{p}, f_{pq}; A\}$ непрерывных отображений является вложением, то диагональное произведение $\Delta I(f_{p})$ семейства $\{I(f_{p}), I(f_{pq}); A\}$ непрерывных отображений также является вложением. Получен один из критериев компактности Дугунджи пространства идемпотентных вероятностных мер. Библиография: 18 названий.

On Some Properties of Infinite Iterations of the Functor of Idempotent Probability Measures

On Some Properties of Infinite Iterations of the Functor of Idempotent Probability Measures

Maximum entropy derived and generalized under idempotent probability to address Bayes-frequentist uncertainty and model revision uncertainty: An information-theoretic semantics for possibility theory

Typical statistical methods of data analysis only handle determinate uncertainty, the type of uncertainty that can be modeled under the Bayesian or confidence theories of inference. An example of indeterminate uncertainty is uncertainty about whether the Bayesian theory or the frequentist theory is better suited to the problem at hand. Another example is uncertainty about how to modify a Bayesian model upon learning that its prior is inadequate. Both problems of indeterminate uncertainty have solutions under the proposed framework. The framework is based on an information-theoretic definition of an incoherence function to be minimized. It generalizes the principle of choosing an estimate that minimizes the reverse relative entropy between it and a previous posterior distribution such as a confidence distribution. The simplest form of the incoherence function, called the incoherence distribution, is a min-plus probability distribution, which is equivalent to a possibility distribution rather than a measure-theoretic probability distribution. A simple case of minimizing the incoherence leads to a generalization of minimizing relative entropy and thus of maximizing entropy. The framework of minimum incoherence is applied to problems of Bayesian-confidence uncertainty and to parallel problems of indeterminate uncertainty about model revision.

Functor of Idempotent Probability Measures with Compact Support and Open Mappings

In this paper, we show that the functor of idempotent probability measures with compact support acting in the category of Tikhonov spaces and their continuous mappings is normal. It is found that this functor is monodic. Further, it is proved that the functor of idempotent probability measures with compact support preserves the openness of continuous mappings of Tikhonov spaces.

On quantization dimensions of idempotent probability measures

The concept of quantization dimensions (upper and lower) of idempotent probability measures on a metric compactum is introduced. These dimensions, which are defined using a general functorial scheme, are analogous to the quantization dimensions of classical probability distributions. It is proved that the quantization dimensions of an idempotent measure do not exceed the corresponding box dimensions of its support. At the same time, for any non-negative number b not exceeding the upper box dimension of a metric compactum X, there exists an idempotent probability measure μb with support equal to X and the upper quantization dimension equal to b. For the lower quantization dimension, a similar intermediate value theorem is valid under some additional restrictions on the compactum X.

Geometrical properties of the space of idempotent probability measures

Although traditional and idempotent mathematics are "parallel'', by an application of the category theory we show that objects obtained the similar rules over traditional and idempotent mathematics must not be "parallel''. At first we establish for a compact metric space X the spaces P(X) of probability measures and I(X) idempotent probability measures are homeomorphic ("parallelism''). Then we construct an example which shows that the constructions P and I form distinguished functors from each other ("parallelism'' negation). Further for a compact Hausdorff space X we establish that the hereditary normality of I<sub>3</sub>(X)\ X implies the metrizability of X.

On metrization of the functor of idempotent probability measures

In idempotent mathematics, an analogue of a probability measure on a compactum X is a normed functional µ : C(X) → R, linear with respect to idempotent arithmetic operations. For ordinary probability measures, a meaningful theory of quantization has long been available, which has a wide range of applications (quantization of a measure is called its approximation by measures with finite supports). The question naturally arises of constructing a similar theory for idempotent probability measures. Quantization presupposes the presence of a metric on the space I(X) of idempotent probability measures, compatible with the topology and defining a metrization of the functor I of idempotent measures sensu V. V. Fedorchuk. A version of the metric on the space I(X) was defined in a joint paper by L. Bazilevich, D. Repovs, and M. Zarichnyi when proving that this space is homeomorphic to the Hilbert cube for any infinite metric compactum X. However, the structure of the metric of Bazilevich et al. is too complicated for it to be used for estimating approximations. In this paper, we propose a modified version of the metrization of the functor I, which is more convenient for constructing a theory of quantization of idempotent probability measures.

The functor of idempotent probability measures and completeness index of uniform spaces

The functor of idempotent probability measures and completeness index of uniform spaces

I ФУНКТОРИНИ ТЕКИС ФАЗОЛАР КАТЕГОРИЯСИГА КЎТАРИШ

Uniform properties of the functor Iof idempotent probability measures with compact support are studied. It is proved that this functor can be lifted to the category Unif of uniform spaces and uniformly continuous maps

On a metric of the space of idempotent probability measures

<p>In this paper we introduce a metric on the space I(X) of idempotent probability measures on a given compact metric space (X; ρ), which extends the metric ρ. It is proven the introduced metric generates the pointwise convergence topology on I(X).</p>

Kawada-Itô-Kelley theorem for quantum semigroups

Idempotent states on locally compact quantum semigroups with weak cancellation properties are shown to be Haar states on a certain sub-object described by an operator system with comultiplication. We also give a characterization of the situation when this sub-object is actually a compact quantum subgroup. In particular we reproduce classical results on idempotent probability measures on locally compact semigroups with cancellation.

Homotopy Properties of the Space If(X) of Idempotent Probability Measures

A subspace If(X) of the space of idempotent probability measures on a given compact space X is constructed. It is proved that if the initial compact space X is contractible, then If(X) is a contractible compact space as well. It is shown that the shapes of the compact spaces X and If(X) are equal. It is also proved that, given a compact space X, the compact space If(X) is an absolute neighborhood retract if and only if so is X.

Некоторые применения пространства идемпотентных вероятностных мер

For idempotent probability measures, max–plus variant of Fubini theorem is established. Further, it is proven metrizability of a given compact if the space of idempotent probability measures defined on the compact, is hereditary normal.

О функторе If идемпотентных вероятностных мер

In the present paper, we proof the functor If of idempotent probability measures acting in the category of Hausdorff compact spaces and their continuous maps is normal.

Large deviations of the long term distribution of a non Markov process

We prove that the long term distribution of the queue length process in an ergodic generalised Jackson network obeys the Large Deviation Principle with a deviation function given by the quasipotential. The latter is related to the unique long term idempotent distribution, which is also a stationary idempotent distribution, of the large deviation limit of the queue length process. The proof draws on developments in queueing network stability and idempotent probability.

Max-Plus Stochastic Control and Risk-Sensitivity

In the Maslov idempotent probability calculus, expectations of random variables are defined so as to be linear with respect to max-plus addition and scalar multiplication. This paper considers control problems in which the objective is to minimize the max-plus expectation of some max-plus additive running cost. Such problems arise naturally as limits of some types of risk sensitive stochastic control problems. The value function is a viscosity solution to a quasivariational inequality (QVI) of dynamic programming. Equivalence of this QVI to a nonlinear parabolic PDE with discontinuous Hamiltonian is used to prove a comparison theorem for viscosity sub- and super-solutions. An example from mathematical finance is given, and an application in nonlinear H-infinity control is sketched.

Idempotent probability measures on ultrametric spaces

Following the construction due to Hartog and Vink we introduce a metric on the set of idempotent probability measures (Maslov measures) defined on an ultrametric space. This construction determines a functor on the category of ultrametric spaces and nonexpanding maps. We prove that this functor is the functorial part of a monad on this category. This monad turns out to contain the hyperspace monad.

Risk sensitive stochastic control and differential games

We give a concise introduction to risk sensitive control of Markov diffusion processes and related two-controller, zero-sum differential games. The method of dynamic programming for the risk sensitive control problem leads to a nonlinear partial differential equation of Hamilton-Jacobi-Bellman type. In the totally risk sensitive limit, this becomes the Isaacs equation for the differential game. There is another interpretation of the differential game using the Maslov idempotent probability calculus. We call this a max-plus stochastic control problem. These risk sensitive control/differential game methods are applied to problems of importance sampling for Markov diffusions.