On open-loop and feedback attainability of a closed set for nonlinear control systems

In this note, we investigate the existence of controls which allow to reach a given closed set K through trajectories of a nonlinear control system. In the case where the set is sufficiently regular we give a condition allowing to find a feedback control law which ensures the existence of trajectories to reach the set. We also consider the case where all the trajectories reach K. When K is not necessarily attainable but only viable, we build a set-valued feedback for which the set is invariant. Our approach concerns continuous dynamics, possibly not C¹, so our methods do not come from geometric control theory. Furthermore, we do not require any regularity of the set K in order to obtain our results, except when we want to establish the existence of a feedback control law to achieve our goals.     

The lattice of strict completions of a finite poset

For a given finite poset , we construct strict completions of P which are models of all finite lattices L such that the set of join-irreducible elements of L is isomorphic to P. This family of lattices, , turns out to be itself a lattice, which is lower bounded and lower semimodular. We determine the join-irreducible elements of this lattice. We relate properties of the lattice to properties of our given poset P, and in particular we characterize the posets P for which . Finally we study the case where is distributive. Received October 13, 2000; accepted in final form June 13, 2001.     

Uniform Families and Count Matroids

Given an r-graph G on [n], we are allowed to add consecutively new edges to it provided that every time a new r-graph with at least l edges and at most m vertices appears. Suppose we have been able to add all edges. What is the minimal number of edges in the original graph? For all values of parameters, we present an example of G which we conjecture to be extremal and establish the validity of our conjecture for a range of parameters. Our proof utilises count matroids which is a new family of matroids naturally extending that of White and Whiteley. We characterise, in certain cases, the extremal graphs. In particular, we answer a question by Erdős, Füredi and Tuza. Received: May 6, 1998 Final version received: September 1, 1999     

Invariants de Maslov équivariants

We construct two cohomological invariants associated to pairs of Lagrangian sub-bundles of a symplectic bundle on a compact manifold upon which a compact Lie group is acting. The first invariant, which we call the classical equivariant Maslov H-invariant, provides an obstruction to Lagrangian transversality and lives in the Borel cohomology. The second invariant, which we call the equivariant Maslov U-invariant, generalises the author's results in K-Theory 13 (1998), 347–361 to the equivariant context and provides a necessary and sufficient condition for equivariant Lagrangian transversality, up to homotopic stability, and lives in the U-theory (intermediate between the real complex K-theories). As an application, we show that two Lagrangian sub-bundles of a symplectic bundle on a homogeneous space are always stably transverse.     

On m-Ary Overpartitions

Presently there are a lot of activities in the study of overpartitions, objects that were discussed by MacMahon, and which have recently proven useful in several combinatorial studies of basic hypergeometric series. In this paper we study some similar objects, which we name m-ary overpartitions. We consider divisibility properties of the number of m-ary overpartitions of a natural number, and we prove a theorem which is a lifting to general m of the well-known Churchhouse congruences for the binary partition function. Received October 11, 2004     

Motivic orthogonal two-dimensional representations of Gal(ℚ/ℚ)

Given a two-dimensional compatible family of ℓ-adic representations which is motivic and which respects an orthogonal form up to similitudes, we show how to express itsL-function in terms of a Hecke character. We give several examples and in particular we analyze a representation associated to a certainK3 surface which arose in the study of Kloosterman sums.     

New Order Relations in Set Optimization

In this paper we study a set optimization problem (SOP), i.e. we minimize a set-valued objective map F, which takes values on a real linear space Y equipped with a pre-order induced by a convex cone K. We introduce new order relations on the power set P(Y)\mathcal{P}(Y) of Y (or on a subset of it), which are more suitable from a practical point of view than the often used minimizers in set optimization. Next, we propose a simple two-steps unifying approach to studying (SOP) w.r.t. various order relations. Firstly, we extend in a unified scheme some basic concepts of vector optimization, which are defined on the space Y up to an arbitrary nonempty pre-ordered set (Q,\preccurlyeq)(\mathcal{Q},\preccurlyeq) without any topological or linear structure. Namely, we define the following concepts w.r.t. the pre-order \preccurlyeq\preccurlyeq: minimal elements, semicompactness, completeness, domination property of a subset of Q\mathcal{Q}, and semicontinuity of a set-valued map with values in Q\mathcal{Q} in a topological setting. Secondly, we establish existence results for optimal solutions of (SOP), when F takes values on (Q,\preccurlyeq)(\mathcal{Q},\preccurlyeq) from which one can easily derive similar results for the case, when F takes values on P(Y)\mathcal{P}(Y) equipped with various order relations.     

On the connection between the representation type of an algebra and its lattice of preradicals

In this paper, we study lattices of preradicals which are not small classes (in which case we say that the corresponding rings are p-large), and specially we consider some infinite representation type algebras. We construct an injective assignment between lattices of preradicals, using a full functor between the corresponding categories of modules, that satisfies certain conditions. We show that the polynomial ring over any field is p-large, and we use this fact to provide examples and some classes of algebras (both tame and wild) which are p-large.     

A filtering problem with a small nonlinear term

In this paper, we consider a filtering problem where the signal X _t satisfies a slightly nonlinear stochastic differential equation and we want to obtain estimates of X _t. To this end, we decompose the nonlinearity with two techniques—a deterministic one and a stochastic one—and this leads us to two sequences of estimates which can be computed by solving finite dimensional equations. We want to compare their performances: we solve this problem in most cases if we restrict ourselves to sufficiently small times t and we give conditions which permit to conclude also for larger times     

Applications of geometric means on symmetric cones

Let V be a Euclidean Jordan algebra, and let be the corresponding symmetric cone. The geometric mean of two elements a and b in is defined as a unique solution, which belongs to of the quadratic equation where P is the quadratic representation of V. In this paper, we show that for any a in the sequence of iterate of the function defined by converges to a. As applications, we obtain that the geometric mean of can be represented as a limit of successive iteration of arithmetic means and harmonic means, and we derive the L?wner-Heinz inequality on the symmetric cone Furthermore, we obtain a formula which leads a Golden-Thompson type inequality for the spectral norm on V. Received October 5, 1999 / Revised March 6, 2000 / Published online October 30, 2000     

Envelopes of commutative rings

Given a significative class ℱ of commutative rings, we study the precise conditions under which a commutative ring R has an ℱ-envelope. A full answer is obtained when ℱ is the class of fields, semisimple commutative rings or integral domains. When ℱ is the class of Noetherian rings, we give a full answer when the Krull dimension of R is zero and when the envelope is required to be epimorphic. The general problem is reduced to identifying the class of non-Noetherian rings having a monomorphic Noetherian envelope, which we conjecture is the empty class.     

Asymptotically optimal estimating equation with strongly consistent solutions for longitudinal data

In this article, we introduce a conditional marginal model for longitudinal data, in which the residuals form a martingale difference sequence. This model allows us to consider a rich class of estimating equations which contains several estimating equations proposed in the literature. A particular sequence of estimating equations in this class contains a random matrix R _i−1^*(β) as a replacement for the “true” conditional correlation matrix of the ith individual. Using the approach of [12], we identify some sufficient conditions under which this particular sequence of equations is asymptotically optimal (in our class). In the second part of the article, we identify a second set of conditions under which we prove the existence and strong consistency of a sequence of estimators of β defined as roots of estimation equations which are martingale transforms (in particular, roots of the sequence of asymptotically optimal equations).     

Multipliers and Modulus on Banach Algebras Related to Locally Compact Groups

LetGbe a locally compact group. In this paper we study moduli of products of elements and of multipliers of Banach algebras which are related to locally compact groups and which admit lattice structure. As a consequence, we obtain a characterization of operators onL^∞(G) which commute with convolutions whenGis amenable as discrete.     

Minimizing the jump number for partially ordered sets: A graph-theoretic approach

The purpose of this paper is to present a graph-theoretic approach to the jump number problem for N-free posets which is based on the observation that the Hasse diagram of an N-free poset is a line digraph. Therefore, to every N-free poset P we can assign another digraph which is the root digraph of the Hasse diagram of P. Using this representation we show that the jump number of an N-free poset is equal to the cyclomatic number of its root digraph and can be found (without producing any linear extension) by an algorithm which tests if a given poset is N-free. Moreover, we demonstrate that there exists a correspondence between optimal linear extensions of an N-free poset and spanning branchings of its root digraph. We provide also another proof of the fact that optimal linear extensions of N-free posets are exactly greedy linear extensions. In conclusion, we discuss some possible generalizations of these results to arbitrary posets.     

Finite Groups in Which Sylow Normalizers Admit Supplements

For a group G and a subgroup M of G, we say that a subgroup A of G is a supplement to M in G, if G = MA. We prove that a finite group in which every Sylow normalizer admits a nilpotent supplement is solvable. In particular we confirm a conjecture of Buchthal. Moreover we investigate finite groups in which Sylow normalizers admit solvable supplements. Supported by SRFDP(20010001046)     

Weighted Euclidean centers

This article addresses the derivation of a Euclidean center, which is defined as the point in decision space from which one can inscribe the largest sphere contained by the constraints. We extend this concept by introducing its weighted version, which we term the weighted Euclidean center. We show that by assigning weights to the different decision variables we can traverse the entire decision space. In addition, we show that the concept of a weighted Euclidean center and that of the achievement scalarizing function introduced by Wierzbicki are intimately related.     

Estimation of a Location Parameter with Restrictions or “vague information” for Spherically Symmetric Distributions

In this article we consider estimating a location parameter of a spherically symmetric distribution under restrictions on the parameter. First we consider a general theory for estimation on polyhedral cones which includes examples such as ordered parameters and general linear inequality restrictions. Next, we extend the theory to cones with piecewise smooth boundaries. Finally we consider shrinkage toward a closed convex set K where one has vague prior information that θ is in K but where θ is not restricted to be in K. In this latter case we give estimators which improve on the usual unbiased estimator while in the restricted parameter case we give estimators which improve on the projection onto the cone of the unbiased estimator. The class of estimators is somewhat non-standard as the nature of the constraint set may preclude weakly differentiable shrinkage functions. The technique of proof is novel in the sense that we first deduce the improvement results for the normal location problem and then extend them to the general spherically symmetric case by combining arguments about uniform distributions on the spheres, conditioning and completeness.     

Generalized mixed integer rounding inequalities: facets for infinite group polyhedra

We present a generalization of the mixed integer rounding (MIR) approach for generating valid inequalities for (mixed) integer programming (MIP) problems. For any positive integer n, we develop n facets for a certain (n + 1)-dimensional single-constraint polyhedron in a sequential manner. We then show that for any n, the last of these facets (which we call the n-step MIR facet) can be used to generate a family of valid inequalities for the feasible set of a general (mixed) IP constraint, which we refer to as the n-step MIR inequalities. The Gomory Mixed Integer Cut and the 2-step MIR inequality of Dash and günlük  (Math Program 105(1):29–53, 2006) are the first two families corresponding to n = 1,2, respectively. The n-step MIR inequalities are easily produced using periodic functions which we refer to as the n-step MIR functions. None of these functions dominates the other on its whole period. Finally, we prove that the n-step MIR inequalities generate two-slope facets for the infinite group polyhedra, and hence are potentially strong.      

On Subspaces and Quotients of Banach Spaces C(K, X)

 We study the relation of to the subspaces and quotients of Banach spaces of continuous vector-valued functions , where K is an arbitrary dispersed compact set. More precisely, we prove that every infinite dimensional closed subspace of totally incomparable to X contains a copy of complemented in . This is a natural continuation of results of Cembranos-Freniche and Lotz-Peck-Porta. We also improve our result when K is homeomorphic to an interval of ordinals. Next we show that complemented subspaces (resp., quotients) of which contain no copy of are isomorphic to complemented subspaces (resp., quotients) of some finite sum of X. As a consequence, we prove that every infinite dimensional quotient of which is quotient incomparable to X, contains a complemented copy of . Finally we present some more geometric properties of spaces. Received 8 November 2000; in revised form 7 December 2001