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Programs are linguistic structures which contain identifications of individuals: memory locations, data types, classes, objects, relations, functions etc. must be identified selectively or definingly. The first part of the essay which deals with identification by showing and designating is rather short, whereas the remaining part dealing with paraphrasing is rather long. The reason is that for an identification by showing or designating no linguistic compositions are needed, in contrast to the case of identification by paraphrasing. The different types of functional paraphrasing are covered here in great detail because the concept of functional paraphrasing is the foundation of functional programming. The author had to decide whether to cover this subject here or in his essay Purpose versus Form of Programs where the concept of functional programming is presented. Finally, the author came to the conclusion that this essay on identification is the more appropriate place.
In system theory, state is a key concept. Here, the word state refers to condition, as in the example Since he went into the hospital, his state of health worsened daily. This colloquial meaning was the starting point for defining the concept of state in system theory. System theory describes the relationship between input X and output Y, that is, between influence and reaction. In system theory, a system is something that shows an observable behavior that may be influenced. Therefore, apart from the system, there must be something else influencing and observing the reaction of the system. This is called the environment of the system.
Bekanntlich gibt es keinen befriedigenden unendlich dimensionalen Ersatz für das Lebesgue-Mass. Andererseits lassen sich viele Techniken klassischer Analysis auch auf unendlich dimensionale Situationen übertragen. Eine Möglichkeit hierzu gibt die Theorie differenzierbarer Masse. Man definiert Richtungsableitungen für Masse ähnlich wie für Funktionen. Eines der zentralen Beispiele ist das Wiener-Mass. Stochastische Integration bezüglich der Brownschen Bewegung, insbesondere das Skorokhod-Integral ergeben sich in natürlicher Weise durch diesen Ansatz und auch die Grundideen des MalliavinKalküls lassen sich in diesem Rahmen einfach erläutern. Die Vorträge geben die meisten Beweise.
In this paper we study a particular class of \(n\)-node recurrent neural networks (RNNs).In the \(3\)-node case we use monotone dynamical systems theory to show,for a well-defined set of parameters, that,generically, every orbit of the RNN is asymptotic to a periodic orbit.Then, within the usual 'learning' context of NeuralNetworks, we investigate whether RNNs of this class can adapt their internal parameters soas to 'learn' and then replicate autonomously certain external periodic signals.Our learning algorithm is similar to identification algorithms in adaptivecontrol theory. The main feature of the adaptation algorithm is that global exponential convergenceof parameters is guaranteed. We also obtain partial convergence results in the \(n\)-node case.
In this paper we present a domain decomposition approach for the coupling of Boltzmann and Euler equations. Particle methods are used for both equations. This leads to a simple implementation of the coupling procedure and to natural interface conditions between the two domains. Adaptive time and space discretizations and a direct coupling procedure leads to considerable gains in CPU time compared to a solution of the full Boltzmann equation. Several test cases involving a large range of Knudsen numbers are numerically investigated.
Application of Moment Realizability Criteria for Coupling of the Boltzmann and Euler Equations
(1998)
The moment realizability criteria have been used to test the domains of validity of the Boltzmann and Euler Equations. With the help of this criteria teh coupling of the Boltzmann and Euler equations have been performed in two dimensional spatial space. The time evolution of domain decompositions for such equations have been presented in different time steps. The numerical resulta obtained from the coupling code have been compared with those from the pure Boltzmann one.