Some inequalities for partial orders

We establish some inequalities connecting natural parameters of a partial order P. For example, if every interval [a,b] contains at most maximal chains, if some antichain has cardinality v, and if there are ₁ chains whose union is cofinal and coinitial in P, then the chain decomposition number for P is ₁v (Theorem 2.2), and the inequality is sharp in a certain sense (Section 3).This paper was written while the authors were visitors at the Laboratoire d'algèbre ordinale, Département de Mathématiques, Université Claude Bernard, Lyon 1, France.Research supported by NSERC grant # A5198.     

On the second eigenfunctions of the Laplacian in R2

A conjecture about the nodal line of a second eigenfunction states that the nodal line of a second eigenfunction divides the domain by intersecting with the boundary of transversely, where is a bounded convex domain ofR ². We prove this conjecture provided has a symmetry. Also, we prove the multiplicity of the second eigenvalue is two at most provided is a bounded convex domain ofR ².Supported in part by NSF DMS 84-09447Home Institution: Department of Mathematics, University of California, San Diego, La Jolla, CA 92093, USA     

Density of the set of attractive compacta

A compact set K in a smooth closed manifold M is said to be attractive, if on M there exists a system of differential equations, for which K is an asymptotically stable invariant set. It is proved that the set of attractive compacta is dense and its complement contains a dense set of type G in the space of all compacta of the manifold M endowed with two natural topologies.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 143, pp. 170–175, 1985.     

Errors of solutions of linear algebraic systems

Error estimates are derived for the following methods: the sweepout method for tridiagonal systems, the method of square roots, the bordering method, and the method of reflection matrices. The book of S. K. Godunov is devoted to the last method; he altered the method so that it takes any matrix into a bidiagonal matrix; a considerable part of that book is devoted to the error of this alteration. In the present paper the method of reflection matrices is studied in the form in which it is expounded in the familiar book of D. K. Faddeev and V. N. Faddeeva. Recurrent formulas are obtained for the sweepout method which make it possible to successively estimate errors of the components of the solution vector. In the method of square roots the error of the solution vector is estimated by the quantity Here _i and are small quantities; the first characterizes the accuracy of small arithmetics effects, and the second the round-off error in the reverse step. Further, A is the matrix of the system, m, is its order, f is the vector of free terms, and C and are constants with 0 <1. We shall not present here the rather involved estimates for the bordering method. The error of the solution vector obtained by the method of reflection matrices is estimated by the quantity (p_A is the conditioning number of the matrix A) All estimates are obtained up to terms of higher order of smallness than and ₁. The estimates themselves are related to the classification of errors of computing processes proposed by the author in recent years.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 139, pp. 94–110, 1984.     

Method for measuring the electron distribution function in the low temperature plasma with a high time resolution

A hybrid method for measuring the electron distribution function in the low temperature plasma is described. The time resolution of the measurement is of the order of several sec.     

Vibrational modes and infrared absorption of interstitial oxygen in silicon

A quasi-linear Si₂O molecule model (QLMM) is suggested from an analysis of the configuration and the interactions of an isolated oxygen atom with its neighbor silicon atoms. The vibrational modes are assigned and the infrared absorption spectra are calculated in detail with the model. The theoretical results are in reasonably good agreement with reported experimental values. This agreement shows that for the analysis of the vibrational modes of the interstitial oxygen atom in silicon crystals it is not necessary to consider the coupling of the molecule with the rest of the lattice. The interaction of the oxygen atom with its six second-nearest silicon atoms only causes the level separation of the ₂ mode and the formation of the fine structure.     

Asymptotic behavior of parameter estimators of a multivariate discontinuous density

We study the asymptotic behavior of Bayesian and maximum likelihood estimators of the parameter when the density function f(x, ) for each fixed x has discontinuities in on some surface.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 74, pp. 83–107, 1977.