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## Sequential analysis in large-scale problems in linear programming

### Cybernetics (1981-07-01) 17: 548-556 , July 01, 1981

## Counting elements in homotopy sets

### Mathematische Zeitschrift (1981-12-01) 178: 527-554 , December 01, 1981

## On the simplicity of the full group of ergodic transformations

### Israel Journal of Mathematics (1981-09-01) 40: 345-349 , September 01, 1981

Let*E* denote an invertible, non-singular, ergodic transformation of (0, 1). Then the full group of*E* is perfect. If*E* preserves the Lebesgue measure, then the full group is simple. If*E* preserves no measure equivalent to Lebesgue, then the full group is simple. If*E* preserves an infinite measure, then there exists a unique normal subgroup. If*T* is any invertible transformation preserving the Lebesgue measure, then the full group is simple if and only if*T* is ergodic on its support.

## Error estimates for interpolatory quadrature formulae

### Numerische Mathematik (1981-10-01) 37: 371-386 , October 01, 1981

### Summary

In this paper we study the remainder of interpolatory quadrature formulae. For this purpose we develop a simple but quite general comparison technique for linear functionals. Applied to quadrature formulae it allows to eliminate one of the nodes and to estimate the remainder of the old formula in terms of the new one. By repeated application we may compare with quadrature formulae having only a few nodes left or even no nodes at all. With the help of this method we obtain asymptotically best possible error bounds for the Clenshaw-Curtis quadrature and other Pólya type formulae.

Our comparison technique can also be applied to the problem of definiteness, i.e. the question whether the remainder*R[f]* of a formula of order*m* can be represented as*c·f*^{(m)}(ξ). By successive elimination of nodes we obtain a sequence of sufficient criteria for definiteness including all the criteria known to us as special cases.

Finally we ask for good and worst quadrature formulae within certain classes. We shall see that amongst all quadrature formulae with positive coefficients and fixed order*m* the Gauss type formulae are worst. Interpreted in terms of Peano kernels our theorem yields results on monosplines which may be of interest in themselves.

## Mordell integrals and Ramanujan's “lost” notebook

### Analytic Number Theory (1981-01-01) 899: 10-48 , January 01, 1981

## Coréduction algébrique d'un espace analytique faiblement Kählérien compact

### Inventiones mathematicae (1981-06-01) 63: 187-223 , June 01, 1981

## Structure of the essential spectrum of evolution operators of nonstationary neutron transport

### Functional Analysis and Its Applications (1981-07-01) 15: 229-231 , July 01, 1981

## Continuous Time Processes

### Applied Probability (1981-01-01) 23: 183-224 , January 01, 1981

There is a fundamental difference between the mathematical formulation of a discrete time stochastic process and a continuous time stochastic process. In discrete time, it is necessary to specify only the mechanism for transition from one state to another, and of course the initial state (distribution) of the system. For Markov chains, this consists of the transition matrix and the initial vector. Everything about the chain can, in principle, be deduced from this matrix and vector.

## Vitesses maximales de décroissance des erreurs et tests optimaux associés

### Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete (1981-01-01) 55: 261-273 , January 01, 1981

### Summary

Under reasonable conditions on *Θ*, both errors of the tests that the law of a sample of size n belongs to *Θ*, go to zero like exp[−*αn*] and exp[− *βn*]. We shall determine the best possible values for *α* and *β* and give a construction of sequences of tests for which the errors decrease with such an optimal rate.