ON A THEOREM OF A.A. LYAPUNOV

Abstract

In this note we obtain some extensions and an approximation of the Lyapunov convexity theorem by means of the bilinear integration of a set-valued function. The integration is performed successively with respect to a non-atomic, a direct sum and a Darboux vector measure. The necessary counterexamples are provided.     

A question of L. A. Shemetkov

Translated fromAlgebra i Logika, Vol. 31, No. 6, pp. 624–636, November–December, 1992.     

A problem of A. I. Mal'tsev

Erevan. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 29, No. 6, pp. 3–11, November–December, 1988.     

A theorem of A. G. Vitushkin

An important condition for the coincidence of the algebras R(E) and C(E) on a compactum was found by A. G. Vitushkin. In this note we give a simple proof of Vitushkin's theorem.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 135, pp. 178–181, 1984.     

A theorem of Yu. A. Aminov

We give a simple proof of a somewhat stronger theorem of Yu. A. Aminov concerning a lower estimate of the diameter of a surface immersed in R, this estimate being a function of the mean curvature of the surface and of its Ricci curvature. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 45, pp. 53–55, 1974.     

A Few Words about V. A. Zalgaller

A short version of a lecture given at the conference in the Weizmann Institute (Rehovot, Israel) devoted to V. A. Zalgaller's 80th birthday.     

A V. A. Markov type inequality in Lp

It is proved that for any algebraic polynomial P of degree at most n we have for 1 p ≤ + t8, x ≥ 1 the inequality For p ≥1 and x ≥ 1 we construct a polynomial P_* of degree n for which      

Lessons for O.R. from A.I.: A Scheduling Case Study

This paper compares operational research (O.R.) with the currently fashionable topic of artificial intelligence (A.I.) at a detailed level. A.I. concepts, techniques and history are summarized, and O.R. and A.I. approaches compared. A case study-the development of a potential aid to scheduling repair jobs on RAF squadrons—is described. The paper shows where A.I. concepts and techniques were used in the prototype program, known as the Fault Identification & Expediting Repair (FIXER) system. Finally, some lessons for O.R. are drawn.