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.     

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.     

Cusp structures of alternating links

Summary An alternating link is canonically associated with every finite, connected, planar graph . The natural ideal polyhedral decomposition of the complement of is investigated. Natural singular geometric structures exist onS ³– , with respect to which the geometry of the cusp has a shape reflecting the combinatorics of the underlying link projection. For the class of balanced graphs, this induces a flat structure on peripheral tori modelled on the tessellation of the plane by equilateral triangles. Examples of links containing immersed, closed ₁-injective surfaces in their complements are given. These surfaces persist after most surgeries on the link, the resulting closed 3-manifolds consequently being determined by their fundamental groups.Oblatum 20-V-1991     

Ergodic Theorems and Ergodic Decomposition for Markov Chains

This paper considers Markov chains on a locally compact separable metricspace, which have an invariant probability measure but with no otherassumption on the transition kernel. Within this context, the limit providedby several ergodic theorems is explicitly identified in terms of the limitof the expected occupation measures. We also extend Yosidasergodic decomposition for Feller-like kernels to arbitrarykernels, and present ergodic results for empirical occupation measures, aswell as for additive-noise systems.     

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.     

On certain imbedding and extension theorems for a weighted class of functions

n- , S n–1 m (0m<n), () S . () , ().     

New elliptic potentials

We construct all tangential covers as zeroes of a particular set of polynomials. The resulting pointed curves give rise, through the Krichever dictionary, to elliptic KP solitons. We then give a criterion to detect the so-called hyperelliptic-tangential covers, which give rise to elliptic KdV solitons. Finally we construct new examples of the latter covers and write down the corresponding (doubly periodic finite-gap) source potentials.Dedicated to the memory of J.-L. Verdier     

Local symmetries and conservation laws

Starting with Lie's classical theory, we carefully explain the basic notions of the higher symmetries theory for arbitrary systems of partial differential equations as well as the necessary calculation procedures. Roughly speaking, we explain what analogs of higher KdV equations are for an arbitrary system of partial differential equations and also how one can find and use them. The cohomological nature of conservation laws is shown and some basic results are exposed which allow one to calculate, in principle, all conservation laws for a given system of partial differential equations. In particular, it is shown that symmetry and conservation law are, in some sense, the dual conceptions which coincides in the self-dual case, namely, for Euler-Lagrange equations. Training examples are also given.Translated from the Russian by B. A. Kuperschmidt.     

Central Limit Theorems for dependent variables. I

Summary This paper gives a flexible approach to proving the Central Limit Theorem (C.L.T.) for triangular arrays of dependent random variables (r.v.s) which satisfy a weak mixing condition called -mixing. Roughly speaking, an array of real r.v.s is said to be -mixing if linear combinations of its past and future are asymptotically independent. All the usual mixing conditions (such as strong mixing, absolute regularity, uniform mixing, -mixing and -mixing) are special cases of -mixing. Linear processes are shown to be -mixing under weak conditions. The main result makes no assumption of stationarity. A secondary result generalises a C.L.T. that Rosenblatt gave for strong mixing samples which are nearly second order stationary.     

‘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.     

Mutational equations in metric spaces

This paper summarizes an extension of differential calculus to a mutational calculus for maps from one metric space to another. The simple idea is to replace half-lines allowing to define difference quotients of maps and their various limits in the case of vector space by transitions with which we can also define differential quotients of a map. Their various limits are called mutations of a map. Many results of differential calculus and set-valued analysis, including the Inverse Function Theorem, do not really rely on the linear structure and can be adapted to the nonlinear case of metric spaces and exploited. Furthermore, the concept of differential equation can be extended tomutational equation governing the evolution in metric spaces. Basic Theorems as the Nagumo Theorem, the Cauchy-Lipschitz Theorem, the Center Manifold Theorem and the second Lyapunov Method hold true for mutational equations.This work was motivated by evolution equations of tubes in visual servoing on one hand, mathematical morphology on the other, when the metric spaces are power spaces. This paper begins by listing some consequences of general theorems concerning mutational equations for tubes.     

A Hubert space characterization among function spaces

— [0,1] ,E — - e=1 [0,1]. I — _E =1, E=L ₂ x _e =xL ₂ x E.

---

---

This work was prepared when the second author was a visiting professor of the CNR at the University of Firenze. He was supported by the Soros International Fund.     

On the Rubin–Stark Conjecture for a special class of CM extensions of totally real number fields

We use Greithers recent results on Brumers Conjecture to prove Rubins integral version of Starks Conjecture, up to a power of 2, for an infinite class of CM extensions of totally real number fields, called nice extensions under the assumption that a certain Iwasawa –invariant vanishes. As a consequence and under the same assumption, we show that the Brumer–Stark Conjecture is true for nice extensions, up to a power of 2.Mathematical Subject Classification (1991): 11R42, 11R58, 11R27Research on this project was partially supported by NSF grants DMS–9801267 and DMS–0196340.in final form: 16 July 2003     

Darboux transformations for higher-rank Kadomtsev-Petviashvili and Krichever-Novikov equations

It is shown that the action of a special rank 2 or rank 3 Darboux transformation, calledtransference, upon a pair of commuting ordinary differential operators of orders 4 and 6 implements the Abelian sum on their elliptic joint spectrum. A consequence of this is that, under the deformation of these commuting operators by the KP flow, every rank 2 KP solution corresponds to a solution of the Krichever-Novikov (KN) equation, and vice versa, with the transference process providing the correspondence between (2+1) and (1+1) dimensions. For a singular joint spectrum, it is further shown that transference at the singular point produces a correspondence between solutions of the singular KN equation and those of the KdV equation. These correspondences are illustrated by considering examples of a nondecaying rational KdV or Boussinesq solutions and the corresponding rational, singular-KN and rational KP solutions which the transference process generates.     

An approach to infinitary temporal proof theory

Aim of this work is to investigate from a proof-theoretic viewpoint a propositional and a predicate sequent calculus with an –type schema of inference that naturally interpret the propositional and the predicate until–free fragments of Linear Time Logic LTL respectively. The two calculi are based on a natural extension of ordinary sequents and of standard modal rules. We examine the pure propositional case (no extralogical axioms), the propositional and the first order predicate cases (both with a possibly infinite set of extralogical axioms). For each system we provide a syntactic proof of cut elimination and a proof of completeness.Supported by MIUR COFIN 02 Teoria dei Modelli e Teoria degli Insiemi, loro interazioni ed applicazioni.Supported by MIUR COFIN 02 PROTOCOLLO.Mathematics Subject Classification (2000):03B22, 03B45, 03F05     

On Multi-Period Market Extension with the Uniformly Equivalent Martingale Measure

For a discrete time, a certain abstract version of the Cox–Ross–Rubinstein Market is equipped with a portfolio basis, which secures a certain portfolio stability. Having in mind that the true probability measure for the fair market should be risk-neutral, it seems natural to condition the market model with the uniformly equivalent Martingale measure, for example, and such a condition is given as a kind of sandwich with the portfolio price; this condition also justifies that the market model admits its complete extension which seems to be important for analysis of possible new enterprizes.     

Risikominimierung bei der Portfolioplanung unter besonderer Berücksichtigung singulärer Kovarianzmatrizen

Zusammenfassung Bisher waren die Untersuchungen über Erwartungswert-Streuungs-effiziente Wertpapiermischungen auf den Fall nichtsingulärer Kovarianzmatrizen beschränkt. Der Fall singulärer Kovarianzmatrizen war bisher nicht konstruktiv behandelt worden. Es wird in dem vorstehenden Papier in einem allgemeinen Ansatz, der auch den Fall singulärer Kovarianzmatrizen zuläßt, die allgemeine Gültigkeit des Separationstheorems und des Zwei-Fonds-Theorems nachgewiesen.

---

Summary Previous studies of mean-variance-efficient portfolios got constructive results in the case of non-singular covariance matrices only, singular covariance matrices were not treated in a constructional way. The present paper proves the Separation Theorem and the Two-Fonds-Theorem within a general framework including the case of singular covariance matrices.