Interdisciplinarity in Operational Research — in the Past and in the Future: An Invitation to IFORS '84 in Washington D.C.

In anticipation of the main theme of the IFORS '84 Conference "Co-operation—The Culture for O.R. Success", the traditional concept of interdisciplinarity in operational research is discussed. Particular emphasis is put on two conditions for practising interdisciplinarity: (i) mutual appreciation between the disciplines and (ii) the systems terminology as a means for interdisciplinary communication.     

Scattering problem for the Schrödinger equation in the case of a potential linear in time and coordinate. I. Asymptotics in the shadow zone

The formal asymptotics of the scattering problem for the Schrödinger equation with a linear potential as x+¦t¦+ is studied. In the shadow zone a formal asymptotic expansion is constructed which is matched with the known asymptotics as t– The expansion constructed loses asymptotic character near the curve x=1/6 t³ (in the so-called projector zone). An assumption regarding the analogous behavior of the asymptotic series in the projector zone makes it possible to construct an expansion in the post-projection zone which goes over into the formulas for creeping waves.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 140, pp. 6–17, 1984.In conclusion, the authors would like to bring to the reader's attention another approach to asymptotics in the projector zone proposed by M. M. Popov (see the present collection).     

Estimates in the Carleson corona theorem,ideals of the algebra H∞, a problem of S.-Nagy

Let E₁, E₂, be Hilbert spaces, H^∞(E₁,E₂) be the space of functions, bounded and analytic in the disk D, with values in the space of bounded linear operators from E₁ to E₂. Estimates are investigated for a solution of the problem of S.-Nagy of finding a left inverse element for a function F, FεH^∞(E₁,E₂). For dim E₁=1 this problem is a generalization of the corona problem. Let C_n(δ)= sup_¦G¦_∞∶FεH^∞(E₁,E₂),dim E₁=n, ¦F¦_∞?1, ¦F(z)a¦₂?δ¦a¦₂(zεD,aεE₁);Gε H^∞(E₂,E₁) is a function of minimal norm for which . Then where a_n, C_n are constants depending only on n. The behavior of the function C₁ as δ→1 is described. Other results are obtained also.     

On the uniform estimate in the Calabi-Yau theorem,Ⅱ

We show that a pluripotential proof of the uniform estimate in the Calabi-Yau theorem works also in the Hermitian case.     

A Berry–Esseen Bound for U-Statistics in the Non-I.I.D. Case

Let X ₁,..., X _n be independent, not necessarily identically distributed random variables. An optimal Berry–Esseen bound is derived for U-statistics of order 2, that is, statistics of the form T=_1i<jn g _ij(X _i, X _j), where the g _ij are measurable functions such that |g _ij(X _i, X _j)|<. An application is given concerning Wilcoxon's rank-sum test.     

Winning “Big” in the St. Petersburg Game

Mathematical Notes - We present a self-contained and easy to follow proof for the value of the probability of large gains of a player in the celebrated St. Petersburg game.     

Operational Research in the U.K. in 1977: The Causes and Consequences of a Myth?

Instead of maintaining the original ideal of having a broad scientific basis for practical Operational Research, it is suggested that the O.R. community has uncritically accepted a myth which views O.R. as a "Hard" (Physical or even Mathematical) Science. It is argued that this has had profound consequences which explain the present difficulties O.R. workers have with problems involving major Social or Behavioural features. It is concluded that the "Hard" Science myth must be replaced, and a return made to the original conception of O.R., if we are to assist effectively in the complex social issues we face today.     

Fractional differentiation in the self-affine case. IV — random measures

The concept of fractional differentiation of stochastic processes introduced in Part I is translated to fractional densities of random measures. Thereby we consider self-affine random measures in the constructive and the axiomatic sense. Occupation measures and local time measures of self-affine random fields are special examples.     

Zermelo in the mirror of the Baer correspondence, 1930–1931

Around 1931 Zermelo had an extended correspondence with the young Reinhold Baer concerning the edition of Cantor's collected works. Some of the letters also deal with Skolem's paradox and Gödel's first incompleteness theorem. Whereas Zermelo's letters are lost, most of Baer's letters are contained in the Zermelo Nachlass. Besides giving insight into Zermelo's reaction to Skolem's and Gödel's results, the letters also demonstrate Baer's clear understanding of the behavior of models of set theory and of the relevance of Gödel's first incompleteness theorem.     

Carathéodory, Helly and the Others in the Max-Plus World

Carathéodory’s, Helly’s and Radon’s theorems are three basic results in discrete geometry. Their max-plus or tropical analogues have been proved by various authors. We show that more advanced results in discrete geometry also have max-plus analogues, namely, the colorful Carathéodory theorem and the Tverberg theorem. A conjecture connected to the Tverberg theorem—Sierksma’s conjecture—although still open for the usual convexity, is shown to be true in the max-plus setting.