Pairs of Convex Bodies with Centrally Symmetric Intersections of Translates

For a pair of convex bodies $K$ and $K$ in $E^d$, the $d$-dimensional intersections $K \cap (x + K),$ $x \in E^d,$ are centrally symmetric if and only if $K$ and $K$ are represented as direct sums $K = R \oplus P$ and $K = R \oplus P$ such that: (i) $R$ is a compact convex set of some dimension $m$, $0 \le m \le d,$ and $R = z - R$ for a suitable vector $z \in E^d$, (ii) $P$ and $P$ are isothetic parallelotopes, both of dimension $d-m$.     

Optimal transformations for prediction in continuous-time stochastic processes: finite past and future

In the classical Wiener-Kolmogorov linear prediction problem, one fixes a linear functional in the future of a stochastic process, and seeks its best predictor (in the L²-sense). In this paper we treat a variant of the prediction problem, whereby we seek the most predictable non-trivial functional of the future and its best predictor; we refer to such a pair (if it exists) as an optimal transformation for prediction. In contrast to the Wiener-Kolmogorov problem, an optimal transformation for prediction may not exist, and if it exists, it may not be unique. We prove the existence of optimal transformations for finite past and future intervals, under appropriate conditions on the spectral density of a weakly stationary, continuous-time stochastic process. For rational spectral densities, we provide an explicit construction of the transformations via differential equations with boundary conditions and an associated eigenvalue problem of a finite matrix.This research was partially supported by ARO (MURI grant) DAAH04-96-1-0445, NSF grant DMS-0074276, and CNPq grant 301179/00-0.     

Some New Transversality Properties

The theorems of Ceva and Menelaus are concerned with cyclic products of ratios of lengths of collinear segments of triangles or more general polygons. These segments have one endpoint at a vertex of the polygon and one at the intersection point of a side with a suitable line. To these classical results we have recently added a selftransversality theorem in which the suitable line is determined by two other vertices. Here we present additional transversality properties in which the suitable line is determined either by a vertex and the intersection point of two diagonals, or by the intersection points of two pairs of such diagonals. Unexpectedly it turns out that besides several infinite families of systematic cases there are also a few sporadic cases.     

Explicit Formulas for Green’s Functions on the Annulus and on the Möbius Strip

Given a fixed point free antianalytic involution k of a domain G in thecomplex plane, bounded by a finite number of analytic curves, k-invariant Greensfunctions are defined on G. The Lindelöfs principle is extended to k-invariantGreens functions. When G is the annulus, k-invariant Greens functions areobtained in the explicit form. Since the factorization of the annulus by the group kgenerated by k produces a Möbius strip, the respective result helped us to obtain explicitforms for Greens functions on the Möbius strip.     

d≤1 ∪d≥25 and constrained KP hierarchy from BRST invariance in thec≠3 topological algebra

The BRST invariance condition in a highest-weight representation of the topological ( twisted N=2) algebra captures the invariant content of two-dimensional gravity coupled to matter. The topological algebra allows reductions to either the DDK-dressed matter or the Kontsevich-Miwa-dressed matter related to Virasoro-constrained KP hierarchy. The standard DDK formulation is recovered by splitting the topological generators into c=–26 reparametrization ghosts + matter + Liouville, while a similar splitting involving c=–2 ghosts gives rise to the matter dressed in exactly the way required in order that the theory be equivalent to Virasoro constraints on the KP hierarchy. The two dressings of matter with the Liouville differ also by their ghost numbers, which is similar to the existence of representatives of BRST cohomologies with different ghost numbers. The topological central charge c3 provides a two-fold covering of the allowed regiond1 d25 of the matter central charge d via d=(c+1)(c+6)/(c–3). The Liouville field is identified as the ghost-free part of the topological U(1) current. The construction thus allows one to establish a direct relation (presumably an equivalence) between the Virasoro-constrained KP hierarchies, minimal models, and the BRST invariance condition for highest-weight states of the topological algebra.Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 95, No. 2, pp. 239–250, May, 1993.     

A Time Domain Characterization of 2-Microlocal Spaces

In Kolwankar and Lévy Véhel, new functional spaces, denoted $K^{s,s}_{x_0}$, were introduced. These spaces characterize the fine local regularity of functions, much in the spirit of 2-microlocal spaces $C^{s,s}_{x_0}$. In contrast with $\C^{s,s}_{x_0}$ spaces, however, $K^{s,s}_{x_0}$ spaces are defined through simple estimations on the pointwise values of the functions. In this work, we generalize the definition of $K^{s,s}_{x_0}$ spaces and prove the equality $C^{s,s}_{x_0}=K^{s,s}_{x_0}$ for $s+s>0$, $s>0$. Using this result, we propose an algorithm able to estimate a part of the 2-microlocal frontier. Experiments on sampled data show that reasonable accuracy is achieved even for difficult functions such as continuous but nowhere differentiable ones. As a by-product, robust estimators of both the pointwise and the local exponents are obtained.     

‘Outside’ as a primitive notion in constructive projective geometry

In intuitionistic (or constructive) geometry there are positive counterparts, apart and outside, of the relations = and incident. In this paper it is shown that the relation outside suffices to define incident, apart and equality. The equivalence of the new system with Heyting's system is shown and as a simple corollary one obtains duality for intuitionistic projective geometry.     

Augmented non-quadratic penalty algorithms

Auslender, Cominetti and Haddou have studied, in the convex case, a new family of penalty/barrier functions. In this paper, we analyze the asymptotic behavior of augmented penalty algorithms using those penalty functions under the usual second order sufficient optimality conditions, and present order of convergence results (superlinear convergence with order of convergence 4/3). Those results are related to the analysis of pure penalty algorithms, as well as augmented penalty using a quadratic penalty function. Limited numerical examples are presented to appreciate the practical impact of this local asymptotic analysis.This research was partially supported by NSERC grant OGP0005491     

Real entire functions of infinite order and a conjecture of Wiman

We prove that if f is a real entire function of infinite order, then ff has infinitely many non-real zeros. In conjunction with the result of Sheil-Small for functions of finite order this implies that if f is a real entire function such that ff has only real zeros, then f is in the Laguerre-Pólya class, the closure of the set of real polynomials with real zeros. This result completes a long line of development originating from a conjecture of Wiman of 1911.     

Invariant Subspaces for Positive Operators Acting on a Banach Space with Markushevich Basis

We introduce weak quasinilpotence for operators. Then, by substituting Markushevich basis and weak quasinilpotence at a nonzero vector for Schauder basis and quasinilpotence at a nonzero vector, respectively, we answer a question on the invariant subspaces of positive operators in [3].     

Conformal spaces

A conformal space is a non-singular metric vector space to which has been adjoined a null-cone of points at infinity. We define a conformal space in terms of a higher dimensional coordinate space, and then state and prove a fundamental theorem of conformal geometry.     

The Hyperbell Algorithm for Global Optimization: A Random Walk Using Cauchy Densities

This article presents a new algorithm, called theHyperbell Algorithm, that searches for the global extrema ofnumerical functions of numerical variables. The algorithm relies on theprinciple of a monotone improving random walk whose steps aregenerated around the current position according to a gradually scaleddown Cauchy distribution. The convergence of the algorithm is provenand its rate of convergence is discussed. Its performance is tested onsome hard test functions and compared to that of other recentalgorithms and possible variants. An experimental study of complexityis also provided, and simple tuning procedures for applications areproposed.     

Generalized Markov fields and Dirichlet forms

We prove that Gaussian measure-indexed random fields, of which the covariance functional is given by the dual form of a transient Dirichlet form, have the global Markov property (where global here means w.r.t. arbitrary, not necessarily open sets), if and only if the Dirichlet form has the local property. Applications to Nelson's free Euclidean field of quantum theory and to Rozanov's generalized random functions are given.     

Das Verfahren der Treppeniteration und verwandte Verfahren zur Lösung algebraischer Eigenwertprobleme

Summary We have described two closely related numerical methods for the solution of eigenvalue problems called stabilized iteration and mutually stabilized biiteration. The iteration process shows numerical stability, has no loss of leading figures and is self-correcting, givingsome eigenvectors immediately in the second case, in triangular decomposition in an important special case of the stabilized iteration, called Treppeniteration.The processes are mathematically closely related toRutishauser's LR-Transformation, but differ in numerical aspect. A certain mixed process, based onRutishauser and the author's ideas, seems to be the optimum way with regard to both security and numerical labor.     

On Nonconvexity of Graphs of Polynomials of Several Real Variables

We consider transversal (orthogonal) perturbations of finite-dimensional convex sets and estimate the degree of nonconvexity of resulting sets, i.e. we estimate the nonconvexity of graphs of continuous functions. We prove that a suitable estimate of nonconvexity of graphs over all lines induces a nice estimate of the nonconvexity of graphs of the entire function. Here, the term nice means that in the well-known Michael selection theorem it is possible to replace convex sets of a multivalued mapping by such nonconvex sets. As a corollary, we obtain positive results for polynomials of degree two under some restrictions on coefficients. Our previous results concerned the polynomials of degree one and Lipschitz functions. We show that for a family of polynomials of degree three such estimate of convexity in general does not exist. Moreover, for degree 9 we show that the nonconvexity of the unique polynomial P(x,y)=x⁹+x³y realizes the worst possible case.     

Minimaxtheoreme und das Integraldarstellungsproblem

In the present paper conditions for the strict determinateness of two-person zero-sum games are considered. In order to get such minimax theorems we first study games with concave-convex pay-off function. If a game does not have this convexity property one usually passes to a mixed extension where both players are allowed to use probability measures (-additive randomizations) or, more generally, probability contents (finitely additive randomizations) as mixed strategies. By means of a very general minimax theorem for such finitely additive randomizations it can be shown that the problem of strict determinateness of -additive randomizations is equivalent to an integral representation problem. The latter is investigated in the last paragraph.

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Diese Arbeit enthält einen Teil der Ergebnisse der Habilitationsschrift des Verfassers.     

On pure equilibria for bimatrix games

In Shapley (1964) several conditions are given for the existence of pure saddlepoints for a matrix game. In this paper we show that only a few of these conditions, when translated to the situation of a bimatrix game guarantee the existence of pure equilibria. Further, we associate with a bimatrix game a directed graph as well as a so-called binary game. If this graph has no cycles, then the bimatrix game in question has a pure equilibrium. It is shown that the binary game for a bimatrix game without a pure equilibrium possesses a fundamental subgame, which can be characterized by means of minimal cycles.     

Experiments with an interactive paired comparison simplex method for molp problems

In this paper, an interactive paired comparison simplex based method formultipleobjectivelinearprogramming (MOLP) problems is developed and compared to other interactive MOLP methods. Thedecisionmaker (DM)'s utility function is assumed to be unknown, but is an additive function of his known linearized objective functions. A test for utilityefficiency for MOLP problems is developed to reduce the number of efficient extreme points generated and the number of questions posed to the DM. The notion of strength ofpreference is developed for the assessment of the DM's unknown utility function where he can express his preference for a pair of extreme points as strong, weak, or almost indifferent. The problem of inconsistency of the DM is formalized and its resolution is discussed. An example of the method and detailed computational results comparing it with other interactive MOLP methods are presented. Several performance measures for comparative evaluations of interactive multiple objective programming methods are also discussed.All rights reserved. This study, or parts thereof, may not be reproduced in any form without written permission of the authors.     

Imperfect and Nonideal Clutters: A Common Approach

We prove three theorems. First, Lovászs theorem about minimal imperfect clutters, including also Padbergs corollaries. Second, Lehmans result on minimal nonideal clutters. Third, a common generalization of these two. The endeavor of working out a common denominator for Lovászs and Lehmans theorems leads, besides the common generalization, to a better understanding and simple polyhedral proofs of both.* Visiting of the French Ministry of Research and Technology, laboratoire LEIBNIZ, Grenoble, November 1995—April 1996.     

Elliptic polylogarithms: An analytic theory

In this article we introduce a natural elliptic generalization of theclassical polylogarithms, study the properties of these functions and theirrelations with Eisenstein series.