A remark on Hilbert's matrix

We exhibit a Jacobi matrix T which has simple spectrum and integer entries, and 0 commutes with Hilbert's matrix. As an application we replace the computation of the eigenvectors of Hilbert's matrix (a very ill-conditioned problem) by the computation of the eigenvectors of T (a nicely stable numerical problem).     

Strongly e-movable convergence and spaces of ANR's

Let X be a metric space and let ANR(X) denote the hyperspace of all compact ANR's in X. This paper introduces a notion of a strongly e-movable convergence for sequences in ANR(X) and proves several characterizations of strongly e-movable convergence. For a (complete) separable metric space X we show that ANR(X) with the topology induced by strongly e-movable convergence can be metrized as a (complete) separable metric space. Moreover, if X is a finite-dimensional compactum, then strongly e-movable convergence induces on ANR(X) the same topology as that induced by Borsuk's homotopy metric.For a separable Q-manifold M, ANR(M) is locally arcwise connected and A, B ? ANR(M) can be joined by an arc in ANR(M) iff there is a simple homotopy equivalence ?: A → B homotopic to the inclusion of A into M.     

A characterization of those categories whose internal logic is Hilbert's ε-calculus

We characterize categories whose internal logic is Hilbert's ε-calculus as those categories which have a proper factorization system satisfying the axiom of choice and in which every non-initial object is injective. We provide an example of such a category where the law of excluded middle is not valid.     

Strong metric subregularity of mappings in variational analysis and optimization

Although the property of strong metric subregularity of set-valued mappings has been present in the literature under various names and with various (equivalent) definitions for more than two decades, it has attracted much less attention than its older “siblings”, the metric regularity and the strong (metric) regularity. The purpose of this paper is to show that the strong metric subregularity shares the main features of these two most popular regularity properties and is not less instrumental in applications. We show that the strong metric subregularity of a mapping F acting between metric spaces is stable under perturbations of the form $f+F$, where f is a function with a small calmness constant. This result is parallel to the Lyusternik–Graves theorem for metric regularity and to the Robinson theorem for strong regularity, where the perturbations are represented by a function f with a small Lipschitz constant. Then we study perturbation stability of the same kind for mappings acting between Banach spaces, where f is not necessarily differentiable but admits a set-valued derivative-like approximation. Strong metric q-subregularity is also considered, where q is a positive real constant appearing as exponent in the definition. Rockafellar's criterion for strong metric subregularity involving injectivity of the graphical derivative is extended to mappings acting in infinite-dimensional spaces. A sufficient condition for strong metric subregularity is established in terms of surjectivity of the Fréchet coderivative, and it is shown by a counterexample that surjectivity of the limiting coderivative is not a sufficient condition for this property, in general. Then various versions of Newton's method for solving generalized equations are considered including inexact and semismooth methods, for which superlinear convergence is shown under strong metric subregularity. As applications to optimization, a characterization of the strong metric subregularity of the KKT mapping is obtained, as well as a radius theorem for the optimality mapping of a nonlinear programming problem. Finally, an error estimate is derived for a discrete approximation in optimal control under strong metric subregularity of the mapping involved in the Pontryagin principle.     

Functoriality and uniformity in Hrushovski's groupoid-cover correspondence

The correspondence between definable connected groupoids in a theory T and internal generalised imaginary sorts of T, established by Hrushovski in [“Groupoids, imaginaries and internal covers,” Turkish Journal of Mathematics, 2012], is here extended in two ways: First, it is shown that the correspondence is in fact an equivalence of categories, with respect to appropriate notions of morphism. Secondly, the equivalence of categories is shown to vary uniformly in definable families, with respect to an appropriate relativisation of these categories. Some elaborations on Hrushovki's original constructions are also included.     

Positive semi-definite matrices as sums of squares

If K is a field and char K ≠ 2, then an element α?K is a sum of squares in K if and only if α ? 0 for every ordering of K. This is the famous theorem of Artin and Landau. It has been generalized to symmetric matrices over K by D. Gondard and P. Ribenboim. They have also shown that Artin's theorem on positive definite rational functions has a suitable extension to positive definite matrix functions. In this paper we attain two goals. First, we show that similar theorems are valid for Hermitian matrices instead of symmetric ones. Second, we simplify D. Gondard and P. Ribenboim's proof of their second theorem and strengthen it.     

A classification of 4-connected graphs

Having observed Tutte's classification of 3-connected graphs as those attainable from wheels by line addition and point splitting and Hedetniemi's classification of 2-connected graphs as those obtainable from K₂ by line addition, subdivision and point addition, one hopes to find operations which classify n-connected graphs as those obtainable from, for example, K_n+1. In this paper I give several generalizations of the above operations and use Halin's theorem to obtain two variations of Tutte's theorem as well as a classification of 4-connected graphs.     

Properly efficient and efficient solutions for vector maximization problems in euclidean space

Recently Benson proposed a definition for extending Geoffrion's concept of proper efficiency to the vector maximization problem in which the domination cone K is any nontrivial, closed convex cone. We give an equivalent definition of his notion of proper efficiency. Our definition, by means of perturbation of the cone K, seems to offer another justification of Benson's choice above Borwein's extension of Geoffrion's concept. Our result enables one to prove some other theorems concerning properly efficient and efficient points. Among these is a connectedness result.     

Roth's theorems and decomposition of modules

It is shown that there is a connection between Roth's theorems on similarity and equivalence of block-triangular matrices and decomposition of modules. The module property is that if $\text{M?}\text{N⊕M}\text{N}$, then N is a summand of M. This holds for any commutative ring if M is finitely presented. New proofs of Roth's theorems are given for commutative rings. Some results are established in the noncommutative case.     

Stability of mappings with bounded distortion in the Sobolev norm on the John domains of Heisenberg groups

This article completes the authors’s series on stability in the Liouville theorem on the Heisenberg group. We show that every mapping with bounded distortion on a John domain of the Heisenberg group is approximated by a conformal mapping with order of closeness √K ? 1 in the uniform norm and with order of closeness K ? 1 in the Sobolev L _p ¹ -norm for all p < C/K?1. We construct two examples, demonstrating the asymptotic sharpness of our results.     

Compositional complexity of Boolean functions

We define two measures, γ and c, of complexity for Boolean functions. These measures are related to issues of functional decomposition which (for continuous functions) were studied by Arnol'd, Kolmogorov, Vitu?kin and others in connection with Hilbert's 13th Problem. This perspective was first applied to Boolean functions in [1]. Our complexity measures differ from those which were considered earlier [3, 5, 6, 9, 10] and which were used by Ehrenfeucht and others to demonstrate the great complexity of most decision procedures. In contrast to other measures, both γ and c (which range between 0 and 1) have a more combinatorial flavor and it is easy to show that both of them are close to 0 for literally all “meaningful” Boolean functions of many variables. It is not trivial to prove that there exist functions for which c is close to 1, and for γ the same question is still open. The same problem for all traditional measures of complexity is easily resolved by statistical considerations.     

Block designs and graph imbeddings

Heffter first observed that certain imbeddings of complete graphs give rise to BIBD's with k = 3 and λ = 2 (and sometimes λ = 1); Alpert established a one-to-one correspondence between BIBD's with k = 3 and λ = 2 and triangulation systems for complete graphs. In this paper we extend this correspondence to PBIBD's on two association classes with k = 3, λ₁ = 0 and λ₂ = 2, and triangulation systems for strongly regular graphs. The group divisible designs of Hanani are used to construct triangulations for the graphs K_n(m), in each case permitted by the euler formula. Conversely, triangular imbeddings of K_n(m) are constructed which lead to new group divisible designs. A process is developed for “doubling” a given PBIBD of an appropriate form. Various extensions of these ideas are discussed, as is an application to the construction of quasigroups.     

Finite dimensional convex structures II: the invariants

Various relations between the dimension and the classical invariants of a topological convex structure have been obtained, leading to an equivalence between Helly's and Carathéodory's theorem, and to the closedness of the hull of compact sets in finite-dimensional convexities. It is also shown that the Radon number of an n-dimensional binary convexity is in most cases equal to the Radon number of the n-cube, and a natural condition is presented under which the invariants are equal to dimension plus one.     

Character sums in algebraic number fields

The Pólya-Vinogradov inequality is generalized to arbitrary algebraic number fields K of finite degree over the rationals. The proof makes use of Siegel's summation formula and requires results about Hecke's zeta-functions with Grössencharacters. One application is to the problem of estimating a least totally positive primitive root modulo a prime ideal of K, least in the sense that its norm is minimal.     

Topological entropy for discontinuous semiflows

We study two variations of Bowen's definitions of topological entropy based on separated and spanning sets which can be applied to the study of discontinuous semiflows on compact metric spaces. We prove that these definitions reduce to Bowen's ones in the case of continuous semiflows. As a second result, we prove that our entropies give a lower bound for the τ-entropy defined by Alves, Carvalho and Vásquez (2015). Finally, we prove that for impulsive semiflows satisfying certain regularity condition, there exists a continuous semiflow defined on another compact metric space which is related to the first one by a semiconjugation, and whose topological entropy equals our extended notion of topological entropy by using separated sets for the original semiflow.     

Rigid cohomology via the tilting equivalence

We define a de Rham cohomology theory for analytic varieties over a valued field $K^{?}$ of equal characteristic p with coefficients in a chosen untilt of the perfection of $K^{?}$ by means of the motivic version of Scholze's tilting equivalence. We show that this definition generalizes the usual rigid cohomology in case the variety has good reduction. We also prove a conjecture of Ayoub yielding an equivalence between rigid analytic motives with good reduction and unipotent algebraic motives over the residue field, also in mixed characteristic.     

Uniform K-theory,and Poincaré duality for uniform K-homology

We revisit ?pakula's uniform K-homology, construct the external product for it and use this to deduce homotopy invariance of uniform K-homology.We define uniform K-theory and on manifolds of bounded geometry we give an interpretation of it via vector bundles of bounded geometry. We further construct a cap product with uniform K-homology and prove Poincaré duality between uniform K-theory and uniform K-homology on spin^c manifolds of bounded geometry.     

Stillman's question for exterior algebras and Herzog's conjecture on Betti numbers of syzygy modules

Let K be a field of characteristic 0 and consider exterior algebras of finite dimensional K-vector spaces. In this short paper we exhibit principal quadric ideals in a family whose Castelnuovo–Mumford regularity is unbounded. This negatively answers the analogue of Stillman's Question for exterior algebras posed by I. Peeva. We show that, via the Bernstein–Gel'fand–Gel'fand correspondence, these examples also yields counterexamples to a conjecture of J. Herzog on the Betti numbers in the linear strand of syzygy modules over polynomial rings.     

Une condition suffisante de minimalité pour les géodésiques de la métrique de HoferA sufficient minimality condition for geodesics of the Hofer's metric

We give a new sufficient condition for a Hamiltonian H to generate a length minimizing geodesic of the Hofer's metric on the group of Hamiltonian diffeomorphisms on $\text{R}{}^{\mathrm{2n}}$. This condition is related to the spectra of the linearized maps of the flow {φ^H_t} generated by H at the fixed points of the flow. In addition we show that if φ₀, φ₁ are two diffeomorphisms linked by such a geodesic, then the Hofer's distance between φ₀ and φ₁ is the same as Viterbo's one. To cite this article: J. Le Crapper, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 359–364.