Core compactness and diagonality in spaces of open sets

We investigate when the space $\mathcal O_X$ of open subsets of a topological space $X$ endowed with the Scott topology is core compact. Such conditions turn out to be related to infraconsonance of $X$, which in turn is characterized in terms of coincidence of the Scott topology of $\mathcal O_X\times\mathcal O_X$ with the product of the Scott topologies of $\mathcal O_X$ at $(X,X)$. On the other hand, we characterize diagonality of $\mathcal O_X$ endowed with the Scott convergence and show that this space can be diagonal without being pretopological. New examples are provided to clarify the relationship between pretopologicity, topologicity and diagonality of this important convergence space.Comment: revised version 12/06/10: example of a $T$-core compact convergence space that is not $T$-dual adde

Statistical analysis of tree parity machine synchronization time

The article describes the features of architecture and the principles of interaction of two (artificial neural networks. Synchronization status of networks allows to use the relevant information as a key to encrypt further communications. With the help of specially developed software tools the extensive statistical data on the networks synchronization time are collected and processed

Convergence of Functions: Equi-Semicontinuity

The ever increasing complexity of the systems to be modeled and analyzed, taxes the existing mathematical and numerical techniques far beyond our present day capabilities. By their intrinsic nature, some problems are so difficult to solve that at best we may hope to find a solution to an approximation of the original problem. Stochastic optimization problems, except in a few special cases, are typical examples of this class. This however raises the question of what is a valid "approximate" to the original problem. The design of the approximation , must be such that (i) the solution to the approximate provides approximate solutions to the original problem and (ii) a refinement of the approximation yields a better approximate solution. The classical techniques for approximating functions are of little use in this setting. In fact very simple examples show that classical approximation techniques dramatically fail in meeting the objectives laid out above. What is needed, at least at a theoretical level, is to design the approximates to the original problem in such a way that they satisfy an epi-convergence criterion. The convergence of the functions defining the problem is to be replaced by the convergence of the sets defined by these functions. That type of convergence has many properties but for our purpose the main one is that it implies the convergence of the (optimal) solutions. This article is devoted to the relationship between the epi-convergence and the classical notion of pointwise-convergence. A strong semicontinuity condition is introduced and it is shown to be the link between these two types of convergences. It provides a number of useful criteria which can be used in the design of approximates to difficult problems

Controllability and observabiliy of an artificial advection-diffusion problem

In this paper we study the controllability of an artificial advection-diffusion system through the boundary. Suitable Carleman estimates give us the observability on the adjoint system in the one dimensional case. We also study some basic properties of our problem such as backward uniqueness and we get an intuitive result on the control cost for vanishing viscosity.Comment: 20 pages, accepted for publication in MCSS. DOI: 10.1007/s00498-012-0076-

Bibliografia

Abstract