Borel fibrations and G-spaces

Considering discrete groups G only, we present an elementary proof of the familiar equivalence of the category of G-spaces (with maps equivariant up to homotopy) and the category of Borel fibrations over BG.     

A new class of cutting planes for the symmetric travelling salesman problem

A comprehensive class of cutting planes for the symmetric travelling salesman problem (TSP) is proposed which contains the known comb inequalities, the path inequalities and the 3-star constraints as special cases. Its relation to the clique tree inequalities is discussed. The cutting planes are shown to be valid for a relaxed version of the TSP, the travelling salesman problem on a road network, and—under certain conditions—to define facets of the polyhedron associated with this problem.     

Stellare Abänderungen und Schälbarkeit von Komplexen und Polytopen

Brugesser and Mani proved that the boundary-complex of a convex polytope can be shelled. This result lead to McMullen's proof of the Upper-bound-conjecture. We show that the shellability of complexes has a close connection to the theory of stellar operations. Several results on special shelling procedures and on non-shellable complexes are obtained.     

On affine scaling algorithms for nonconvex quadratic programming

We investigate the use of interior algorithms, especially the affine-scaling algorithm, to solve nonconvex — indefinite or negative definite — quadratic programming (QP) problems. Although the nonconvex QP with a polytope constraint is a hard problem, we show that the problem with an ellipsoidal constraint is easy. When the hard QP is solved by successively solving the easy QP, the sequence of points monotonically converge to a feasible point satisfying both the first and the second order optimality conditions.Research supported in part by NSF Grant DDM-8922636 and the College Summer Grant, College of Business Administration, The University of Iowa.     

Previsible sets for hyperfinite filtrations

Summary Previsible (or predictable) stochastic processes are defined for any filtration over a probability space (Dellacherie and Meyer (1978), IV. 61). This technical definition gives previsible processes certain predictability properties such as not being able to oscillate in unison with martingale differentials. Thus previsibility has become one essential ingredient in The General Theory of Stochastic Processes.We show that previsible sets for Keisler's (1984) special hyperfinite filtration are given both combinatorially and by a left filtration. Keisler's scheme has many other interesting features.Our main technical tool is an extension of Henson's (1979) analysis of analytic sets and the standard part map.     

Representation of text strings in binary computers

The optimal internal representation of text strings in binary computers is discussed. The average storage space required can be reduced if shift signals are used. Under certain conditions a text, composed of appr. 100 different symbols, can be economically stored using 32 different 5-bit code signals, including a set of special shift signals which specify the significance of the 5-bit configurations in much the same way as upper and lower case signals remove the ambiguity of 5-bit teletype signals.     

On Dirichlet series satisfying Riemann's functional equation

Summary According to convention, Hamburger's theorem (1921) says-roughly-that Riemann's (s) is uniquely determined by its functional equation. In 1944 Hecke pointed out that there are two distinct versions of Hamburger's theorem. Hecke's remark has led me, in examining just how rough the convention is, to prove that, with a weakening of certain auxiliary conditions, there are infinitely many linearly independent solutions of Riemann's functional equation (Theorem 1). In Theorem 1, as in Hamburger's theorem, the weight parameter is 1/2. In Theorem 2 we obtain stronger results when this parameter is 2: a Mittag-Leffler theorem for Dirichlet series with functional equations.Oblatum 23-XII-1992 & 9-IX-1993Research supported in part by NSA/MSP Grant MDA 90-H-4025 To the memory of Martin Eichler     

Chapter 1 Introduction: Extensions and new developments in DEA

The extensions, new developments and new interpretations for DEA covered in this paper include: (1) new measures of efficiency, (2) new models and (3) new ways of implementing established models with new results and interpretations presented that include treatments of congestion, returns-to-scale and mix and technical inefficiencies and measures of efficiency that can be used to reflect all pertinent properties. Previously used models, such as those used to identify allocative inefficiencies, are extended by means of assurance region approaches which are less demanding in their information requirements and underlying assumptions. New opportunities for research are identified in each section of this chapter. Sources of further developments and possible sources for further help are also suggested with references supplied to other papers that appear in this volume and which are summarily described in this introductory chapter.     

A Kantorovich-type convergence analysis for the Gauss-Newton-Method

Summary We present a (semilocal) Kantorovich-type convergence analysis for the Gauss-Newton-Method which reduces to the wellknown Newton-Kantorovich-Theorem for the Newton-Method in a natural way. Additionnally a classification of the nonlinear regression problem into adequate and not-adequate models is obtained.     

The Wiener-Hopf method for systems of pseudodifferential equations with an application to crack problems

The aim of this paper is to develop the Wiener-Hopf method for systems of pseudo-differential equations with non-constant coefficients and to apply it to the describtion of the asymptotic behaviour of solutions to boundary integral equations for crack problems when a crack occurs in a linear anisotropic elastic medium. The method was suggested in [15] for scalar pseudo-differential equations with constant coefficients and applied in [7] to the crack problems in the isotropic case. The existence and a-priori smoothness of solutions for the anisotropic case has been proved in [11, 12], while the isotropic case has been treated earlier in [7, 25, 41, 50]. Our results improve even those for the isotropic case obtained in [7, 50]. Asymptotic estimates for the behaviour of solutions in the anisotropic case have been obtained in [28] by a different method.In memoriam, dedicated to Professor Dr. V.D. Kupradze on the occasion of the 90th anniversary of his birthThis work was carried out during the first author's visit in Stuttgart in 1992 and supported by the DFG priority research programme Boundary Element Methods within the guest-programme We-659/19-2.     

Orthonormal systems satisfying an inequality of S. M. Nikol'skii

N- (p, q) (1 pN-, L _p - L _q -. , , , L L _q - , , .     

Large transversals to small families of unit disks

Summary It is proved that if the nonempty intersection of bounded closed convex sets AnB is contained in (A + F)U(B+F) and one of the following holds true: (i) the space X is less-than-three dimensional, (ii) AUB is convex, (iii) F is a one-point set, then AnBCA+F or AnBCB+F (Theorems 2 and 3). Moreover, under some hypotheses the characterization of A and B such that AnB is a summand of AUB is given (Theorem 3).     

A Niche Width Model of Optimal Specialization

Niche width theory, a part of organizational ecology, predicts whether specialist or generalist forms of organizations have higher fitness, in a continually changing environment. To this end, niche width theory uses a mathematical model borrowed from biology. In this paper, we first loosen the specialist-generalist dichotomy, so that we can predict the optimal degree of specialization. Second, we generalize the model to a larger class of environmental conditions, on the basis of the model's underlying assumptions. Third, we criticize the way the biological model is treated in sociological theory. Two of the model's dimensions seem to be confused, i.e., that of trait and environment; the predicted optimal specialization is a property of individual organizations, not of populations; and, the distinction between fine and coarse grained environments is superfluous.     

On the Mechanism for Nonlinear Representations of the Lorentz Group Arising in Quantum Field Theory

A mechanism is identified that leads to the correct law for the relativistic Wigner function transformation with respect to the Lorentz group as long as the corresponding relativistic wave functions have special transformation properties.     

A result from diophantine approximation which has applications in economics

We say that a real number allows poor approximations if we can find 0<=()<1 and a sequence of integers n₁2<... such that for all rationals p/q with qn. we have |–.p/q| > Kn _j ^–l– where K is a constant depending only on .In this note we prove that the set of numbers which allow poor approximations are precisely the very well-approximable numbers.The existence of numbers with poor approximations has been used by Cheng [1] to show the existence of a dense set of economies whose cone converges to the Walras equilibrium as slowly as 0(n^–1/2–) after n replications.     

Hysteresis in discrete systems of possibly interacting elements with a double-well energy

Summary We study in this paper the hysteretic behavior of a discrete system constituted by a finite number of elements (snap-springs) whose energy has two parabolic wells. The guideline idea is that, in many circumstances, hysteresis can be due to the presence of relative minimizers of a potential (metastable states) in which the system might get locked during its quasistatic evolution. A careful investigation is thus carried out of the relative minimizers of the total energy of our system of snap-springs under imposed total displacement, and of the barriers separating them. This is done both in the case of noninteracting elements and in the case in which some interaction is present that gives rise in the energy to an extra coherence term of special form. The results allow discussion of various hysteretic phenomena, also in the presence of vibrational motion of the elements. This study of a simple but suggestive discrete system will hopefully prove itself of help in understanding the implications regarding hysteresis of certain continuum theories recently proposed to model phase transitions in the solid state, in which the energy density is assumed, as here, to be biparabolic, and in which the coherence energy term we adopt arises in a natural way when equilibria involving mixtures of kinematically noncoherent phases are considered. In these cases the optimal microstructures are known to be layered, and physically this gives a good basis to our discrete calculation.     

Applications of the zero–one law to problems connected with Cauchy's functional equation

Summary Marek Kuczma's book, entitled An Introduction To The Theory Of Functional Equations And Inequalities, mentions a certain setV ₀ in several places and presents references as to where this set is discussed in the literature. The main result of this paper is a proof of the fact that the setA _M (V ₀)={xV ₀ f(x)>M} is saturated non-measurable for each additive discontinuous functionf and each real numberM. Other results aboutV ₀ are also presented. Connections between measure and category are stressed. The main tool in our proofs is a certain so-called zero–one law and its topological analogue. In addition it is shown that the zero–one law is equivalent to Smital's lemma.     

Invarianz gewisser Operatorenklassen unter nichtkommutativen ergodischen Mitteln

Let be a group of *-automorphisms on the algebra of bounded linear operators on a complex Hilbert space H. Then the strongly closed convex hull of the orbit of any compact operator under consists of compact operators. The same is true if one replaces compact by nuclear, Hilbert-Schmidt or positive Fredholm. We further discuss these results in the framework of the noncommutative mean ergodic theorem of KOVACS and SZ#x00FC;CS and formulate an analogous theorem for the algebra of compact operators on a complex Hilbert space.

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Gefördert von der Deutschen Forschungsgemeinschaft im Rahmen des Forschungsvorhabens Ko 506/1.     

Bergman-Vekua theory in differential form

We develop Bergman-Vekua integral operator theory on the basis of a new kind of series suggested by polynomial operators (classP _j operators); in a sense, this is an analog of the Weierstrass approach as opposed to the Cauchy-Riemann approach in classical complex analysis, which also has advantages over the original form in boundary value problems and other applications.Dedicated to the memory of my friend, Peter Henrici     

Role of information in the stochastic zero-sum differential game

In this paper, a linear-quadratic Gaussian zero-sum differential game is studied. Maneuverability is defined to measure players' strength. It is shown that a more maneuverable player would prefer a more observable information system. An example is given to show that a more controllable player might not prefer more observable measurements in the stochastic environment.The research reported in this paper was made possible through support extended to the Division of Engineering and Applied Physics, Harvard University, by the US Office of Naval Research under the Joint Services Electronics Program by Contract No. N00014-75-c-0648 and by the National Science Foundation under Grant No. GK31511.