On relations between γ-operations

Let expX be the power set of a non-empty set?X. A function γ:?expX→expX is said to be monotonic iff A?B?X implies γA?γB. Császár?[2] investigated relations between the monotonic functions. The purpose of the paper is to investigate some results concerning particular monotonic functions.     

Espace des modules des variétés hyperboliquement plongéesThe moduli space of hyperbolically imbedded manifolds

In this Note, we construct the moduli space of hyperbolically imbedded manifolds. We recall that the moduli space of compact hyperbolic manifolds has been constructed by Brody and Wright. To construct our moduli space, we use a general criterion to represent analytic functors by coarse moduli spaces due to Schumacher. The objects to deform are couples (X,D) where X is a compact manifold and D is a normal crossing divisor in X such that X?D is hyperbolically imbedded in X. This criterion is based on two ingredients: in our case, the first is the existence of semi-universal logarithmic deformation due to Kawamata. The second is a consequence of a theorem of stability of hyperbolically imbedded spaces through logarithmic deformations. We use the relative-distance of Kobayashi to simplify the proof. To cite this article: A. Khalfallah, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 237–242.     

Minimax estimators that shift towards a hypersphere for location vectors of spherically symmetric distributions

Let X be a p-dimensional random vector with density f(6X?θ6) where θ is an unknown location vector. For p ≥ 3, conditions on f are given for which there exist minimax estimators θ?(X) satisfying 6Xt6 · 6θ?(X) ? X6 ≤ C, where C is a known constant depending on f. (The positive part estimator is among them.) The loss function is a nondecreasing concave function of 6θ?? θ6². If θ is assumed likely to lie in a ball in $\text{R}$^p, then minimax estimators are given which shrink from the observation X outside the ball in the direction of P(X) the closest point on the surface of the ball. The amount of shrinkage depends on the distance of X from the ball.     

Weak Compactness and Variational Characterization of the Convexity

We prove that if X is a Banach space and ${f : X \rightarrow \mathbb{R} \cup \{+\infty\}}$ is a proper function such that f ? ? attains its minimum for every ? ε X ^*, then the sublevels of f are all relatively weakly compact in X. As a consequence we show that a Banach space X where there exists a function ${f : X \rightarrow \mathbb{R}}$ such that f ? ? attains its minimum for every ? ε X ^* is reflexive. We also prove that if ${f : X \rightarrow \mathbb{R} \cup \{+\infty\}}$ is a weakly lower semicontinuous function on the Banach space X and if for every continuous linear functional ? on X the set where the function f ? ? attains its minimum is convex and non-empty then f is convex.     

Compact supported endographs and fuzzy sets

A metric is defined on a space of functions from a locally compact metric space X into the unit interval I in terms of the Hausdorff metric distance between their compact supported endographs in X × I. Convergence in this metric is shown to be equivalent to the conjunction of the Hausdorff metric convergence of supports in X and two conditions involving numerical values of the functions. The space of nonempty compact subsets of X with the Hausdorff metric is imbedded in the above function space by the characteristic function on subsets of X. Applications of these results to fuzzy subsets of X and fuzzy dynamical systems on X are indicated.     

Orbits and invariants of visible group actions

We consider actions G?×?X?→?X of the affine, algebraic group G on the irreducible, affine, variety X. If [k[X] ^G ]?=?[k[X]] ^G we call the action visible. Here [A] denotes the quotient field of the integral domain A. If the action is not visible we construct a G-invariant, birational morphism φ: Z?→?X such that G?×?Z?→?Z is a visible action. We use this to obtain visible open subsets U of X. We also discuss visibility in the presence of other desirable properties: What if G?×?X?→?X is stable? What if there is a semi-invariant f ∈ k[X] such that G?×?X _f ?→?X _f is visible? What if X is locally factorial? What if G is reductive?     

Boundaries and JSJ Decompositions of CAT(0)-Groups

Let G be a one-ended group acting discretely and co-compactly on a CAT(0) space X. We show that ?X has no cut points and that one can detect splittings of G over two-ended groups and recover its JSJ decomposition from ?X. We show that any discrete action of a group G on a CAT(0) space X satisfies a convergence type property. This is used in the proof of the results above but it is also of independent interest. In particular, if G acts co-compactly on X, then one obtains as a corollary that if the Tits diameter of ?X is bigger than 3π/2 then it is infinite and G contains a free subgroup of rank 2.     

Upper bounds for the order of an additive basis obtained by removing a finite subset of a given basis

Let A be an additive basis of order h and X be a finite nonempty subset of A such that the set A?X is still a basis. In this article, we give several upper bounds for the order of A?X in function of the order h of A and some parameters related to X and A. If the parameter in question is the cardinality of X, Nathanson and Nash already obtained some of such upper bounds, which can be seen as polynomials in h with degree (|X|+1). Here, by taking instead of the cardinality of X the parameter defined by , we show that the order of A?X is bounded above by . As a consequence, we deduce that if X is an arithmetic progression of length ?3, then the upper bounds of Nathanson and Nash are considerably improved. Further, by considering more complex parameters related to both X and A, we get upper bounds which are polynomials in h with degree only 2.     

Minimal kernels of weakly complete spaces

Let X be a weakly complete space i.e. X a complex space endowed with a C^k-smooth, k?0, plurisubharmonic exhaustion function. We give the notion of minimal kernelΣ¹=Σ¹(X) of X by the following property: x∈Σ¹ if no continuous plurisubharmonic exhaustion function is strictly plurisubharmonic near x. The study of the geometric properties of the minimal kernels is the aim of present paper. After stating that the minimal kernel Σ¹ of a weakly complete space can be defined by a single plurisubharmonic exhaustion function ?, called minimal, using the characterization in terms of Bremermann envelopes, we prove the following, crucial, result: if X is a weakly complete manifold and ? a minimal function for X, the nonempty level sets Σ_c¹=Σ¹∩{?=c} have the local maximum property. In the last section we discuss the special case of weakly complete surfaces. We prove that if dim_cX=2 and c is a regular value of a minimal function ? then the nonempty level sets Σ_c¹=Σ¹∩{?=c} are compact spaces foliated by holomorphic curves.     

Analytic functions whose range is dense in a ball

Denote by Δ(resp. $\text{Δ}$) the open (resp. closed) unit disc in C. Let E be a closed subset of the unit circle T and let F be a relatively closed subset of T ? E of Lesbesgue measure zero. The following result is proved. Given a complex Banach space X and a bounded continuous function f:F → X, there exists an extension f? of f, bounded and continuous on $\text{}\text{?}\text{gD ? E}$, analytic on Δ and satisfying sup$\text{{6}\text{f}\text{?}\text{(z)6:zε}\text{δ}\text{?E}$. This is applied to show that for any separable complex Banach space X there exists an analytic function from Δ to X whose range is contained and dense in the unit ball of X.     

Compact spaces, elementary submodels, and the countable chain condition, II

Given a space 〈X,T〉 in an elementary submodel of H(θ), define XM to be X∩M with the topology generated by . It is established that if XM is compact and satisfies the countable chain condition, while X is not scattered and has cardinality less than the first inaccessible cardinal, then X=XM. If the character of XM is a member of M, then “inaccessible” may be replaced by “1-extendible”.     

Glauber dynamics for the mean-field Ising model: cut-off, critical power law, and metastability

We study the Glauber dynamics for the Ising model on the complete graph, also known as the Curie–Weiss Model. For β < 1, we prove that the dynamics exhibits a cut-off: the distance to stationarity drops from near 1 to near 0 in a window of order n centered at [2(1 ? β)]^?1 n log n. For β = 1, we prove that the mixing time is of order n ^3/2. For β > 1, we study metastability. In particular, we show that the Glauber dynamics restricted to states of non-negative magnetization has mixing time O(n log n).     

Lipschitz stability of operators in Banach spaces

We consider approximations of an arbitrarymap F: X → Y between Banach spaces X and Y by an affine operator A: X → Y in the Lipschitz metric: the difference F — A has to be Lipschitz continuous with a small constant ? > 0. In the case Y = ? we show that if F can be affinely ?-approximated on any straight line in X, then it can be globally 2?-approximated by an affine operator on X. The constant 2? is sharp. Generalizations of this result to arbitrary dual Banach spaces Y are proved, and optimality of the conditions is shown in examples. As a corollary we obtain a solution to the problem stated by Zs. Páles in 2008. The relation of our results to the Ulam-Hyers-Rassias stability of the Cauchy type equations is discussed.     

A. D. Alexandrov's Problem for CAT(0)-Spaces

We solve the well-known problem by A. D. Alexandrov for nonpositively curved spaces. Let X be a geodesically complete locally compact space nonpositively curved in the sense of Alexandrov and connected at infinity. The main theorem reads as follows: Each bijection f : X → X such that f and the inverse f ^?1 of f preserve distance 1 is an isometry of X.     

Verdier specialization via weak factorization

Let X?V be a closed embedding, with V?X nonsingular. We define a constructible function ψ _X,V on X, agreeing with Verdier’s specialization of the constant function 1 _V when X is the zero-locus of a function on V. Our definition is given in terms of an embedded resolution of X; the independence of the choice of resolution is obtained as a consequence of the weak factorization theorem of Abramovich–Karu–Matsuki–W?odarczyk. The main property of ψ _X,V is a compatibility with the specialization of the Chern class of the complement V?X. With the definition adopted here, this is an easy consequence of standard intersection theory. It recovers Verdier’s result when X is the zero-locus of a function on V. Our definition has a straightforward counterpart Ψ _X,V in a motivic group. The function ψ _X,V and the corresponding Chern class c _SM(ψ _X,V ) and motivic aspect Ψ _X,V all have natural ‘monodromy’ decompositions, for any X?V as above. The definition also yields an expression for Kai Behrend’s constructible function when applied to (the singularity subscheme of) the zero-locus of a function on V.     

A new criterion for the logarithmic Sobolev inequality and two applications

We give a criterion for the logarithmic Sobolev inequality (LSI) on the product space X₁×?×XN. We have in mind an N-site lattice, unbounded continuous spin variables, and Glauber dynamics. The interactions are described by the Hamiltonian H of the Gibbs measure. The criterion for LSI is formulated in terms of the LSI constants of the single-site conditional measures and the size of the off-diagonal entries of the Hessian of H. It is optimal for Gaussians with positive covariance matrix. To illustrate, we give two applications: one with weak interactions and one with strong interactions and a decay of correlations condition.     

A Characterization of Orthogonality

Let X be a real inner product space of (finite or infinite) dimension greater than one. We proved (see Theorem 7, Chapter 1 of our book [1]) that if T is a separable translation group of X, and d an appropriate distance function of X which is supposed to be invariant under T and the orthogonal group O of X, then there are, up to isomorphism, exactly two solutions of geometries (X,G(T,O)), G the group generated by T ∪ O, namely euclidean and hyperbolic geometry over X. With the same geometrical definition for both geometries of arbitrary (finite or infinite) dimension > 1 we will characterize in this note the notion of orthogonality.     

An alternative definition of coarse structures

Roe [J. Roe, Lectures on Coarse Geometry, University Lecture Series, vol. 31, Amer. Math. Soc., Providence, RI, 2003] introduced coarse structures for arbitrary sets X by considering subsets of X×X. In this paper we introduce large scale structures on X via the notion of uniformly bounded families and we show their equivalence to coarse structures on X. That way all basic concepts of large scale geometry (asymptotic dimension, slowly oscillating functions, Higson compactification) have natural definitions and basic results from metric geometry carry over to coarse geometry.     

Approximating minimum-length-sequence metrics: a cautionary note

In numerical taxonomy one may wish to measure the dissimilarity of classifications S and T by computing the distance between them with an appropriate metric. A minimum-length-sequence (MLS) metric requires that the user identify a set X of meaningful transformations of classifications; the MLS metric μ_x is then defined by requiring that μ_x (S,T) be the length of a shortest sequence of transformations from X that carries S into T.For a given application it may be relatively easy to identify an appropriate set X of transformations, but it may be difficult or impossible to design an efficient algorithm to compute μ_x. In this case it is natural to restrict the definition to obtain an approximation ? to the original metric μ_x such that ? has an efficient algorithm for its computation. This restriction process must be performed carefully lest the approximation fail to satisfy the metric properties. We present a general result about this problem and apply it in two ways. First we prove that a published ‘metric’ on partitions of a set in fact violates the triangle inequality and so is merely a semimetric. Then we clarify the relationship between the nearest neighbor interchange metric on labeled binary trees and the closest partition distance measure proposed by Waterman and Smith (1978).     

Computability of a topological poset

For subspaces X and Y of Q the notation Xh_?Y means that X is homeomorphic to a subspace of Y and X∼Y means Xh_?Yh_?X. The resulting set P(Q)/∼ of equivalence classes is partially-ordered by the relation if Xh_?Y. In a previous paper by the author it was established that this poset is essentially determined by considering only the scattered X⊆Q of finite Cantor-Bendixson rank. Results from that paper are extended to show that this poset is computable.