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## Weak Convergence

### Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems (1990-01-01): 1-16 , January 01, 1990

The basic questions dealt with in this book concern approximations of relatively complex processes by simpler and more tractable processes. These processes might be controlled or not controlled. The approximation might be of interest for either the purposes of engineering design, or for other analytical or numerical work. The theory of weak convergence of measures seems to be the fundamental and most widely used and successful tool for such purposes. See, for example, the techniques, examples and references in the books [B6, E2, K20]. In this chapter, we will discuss the basic ideas in the subject which will be useful for our work in the sequel. The detailed examples will not be dealt with in this chapter, but are left to the rest of the book. The basic techniques discussed in this chapter will all be further developed and worked out in detail in connection with the applications dealt with in the succeeding chapters. The concepts and criteria have a rather abstract flavor as they are initially introduced, but when applied in the succeeding chapters, they take on concrete and readily usable forms.

## The ternary discriminator function in universal algebra

### Mathematische Annalen (1971-09-01) 191: 167-180 , September 01, 1971

## Estimating the cardinality of a difference subset of the discrete multi-torus $\mathbb{Z}_{3}^{n}$

### Journal of Mathematical Sciences (2012-04-01) 182: 456-462 , April 01, 2012

In this paper, we obtain an upper estimate for the cardinality of a subset of the discrete torus over a field of three elements of which any four points do not form a nonsingular parallelogram.

## Quadrature methods for highly oscillatory linear and non-linear systems of ordinary differential equations: part II

### BIT Numerical Mathematics (2012-06-01) 52: 383-405 , June 01, 2012

In this paper we present efficient numerical approximation for systems of highly oscillatory ordinary differential equations with matrices of variable coefficients. We assume that the spectrum of the matrix is purely imaginary and the frequency of oscillation grows large. We develop the *asymptotic* and the *Filon*-type methods for linear systems with time dependent matrices and we integrate oscillatory quadrature rules with *waveform relaxation* methods employing the *WRF* method for non-linear systems. We solve matrix exponential in *Lie* groups employing *Magnus* expansion. The methods are illustrated in several numerical examples of interest.

## Using Types and Modes

### axịom™ (1992-01-01): 59-91 , January 01, 1992

In this chapter we look at the key notion of *type* and its generalization *mode*. We show that every AXIOM object has a type that determines what you can do with the object. In particular, we explain how to use types to call specific functions from particular parts of the library and how types and modes can be used to create new objects from old. We also look at Record and Union types and the special type Any. Finally, we give you an idea of how AXIOM manipulates types and modes internally to resolve ambiguities.

## Die Grundrechenoperationen

### Mathematik (1974-01-01): 13-24 , January 01, 1974

### Zusammenfassung

„Am Anfang war das Zählen“. Die benutzten Elemente, die natürlichen Zahlen, dienen uns zu zwei Zwecken: Wir fixieren mit ihnen Anzahlen und drücken Ordnungen aus.

## On the interior regularity of weak solutions to the 2-D incompressible Euler equations

### Calculus of Variations and Partial Differential Equations (2017-08-21) 56: 1-19 , August 21, 2017

We study whether some of the non-physical properties observed for weak solutions of the incompressible Euler equations can be ruled out by studying the vorticity formulation. Our main contribution is in developing an interior regularity method in the spirit of De Giorgi–Nash–Moser, showing that local weak solutions are exponentially integrable, uniformly in time, under minimal integrability conditions. This is a Serrin-type interior regularity result $$\begin{aligned} u \in L_\mathrm{loc}^{2+\varepsilon }(\Omega _T) \implies \mathrm{local\ regularity} \end{aligned}$$ for weak solutions in the energy space $$L_t^\infty L_x^2$$ , satisfying appropriate vorticity estimates. We also obtain improved integrability for the vorticity—which is to be compared with the DiPerna–Lions assumptions. The argument is completely local in nature as the result follows from the structural properties of the equation alone, while completely avoiding all sorts of boundary conditions and related gradient estimates. To the best of our knowledge, the approach we follow is new in the context of Euler equations and provides an alternative look at interior regularity issues. We also show how our method can be used to give a modified proof of the classical Serrin condition for the regularity of the Navier–Stokes equations in any dimension.

## Homogenization of Differential Operators

### Acta Mathematicae Applicatae Sinica (2002-03-01) 18: 9-14 , March 01, 2002

###
*Abstract*

In this note, we present a method of constructing the homogenized operator for a general sequence of differential operators. As an example, we construct the homogenized operator for a sequence of linear parabolic operators.

## A General Architecture for Decentralized Supervisory Control of Discrete-Event Systems

### Discrete Event Systems (2000-01-01) 569: 111-118 , January 01, 2000

We consider a generalized form of the conventional decentralized control architecture for discrete-event systems where the control actions of a set of supervisors can be “fused” using both union and intersection of enabled events. Namely, the supervisors agree a *priori* on choosing “fusion by union” for certain controllable events and “fusion by intersection” for certain other controllable events. We show that under this generalized architecture, a larger class of languages can be achieved than before since a relaxed version of the notion of co-observability appears in the necessary and sufficient conditions for the existence of supervisors. The computational complexity of verifying these new necessary and sufficient conditions is studied.Finally, a method of partitioning the controllable events between “fusion by union” and “fusion by intersection” is presented.