Inhabitants of interesting subsets of the Bousfield lattice

In 1979, Bousfield defined an equivalence relation on the stable homotopy category. The set of Bousfield classes has some important subsets such as the distributive lattice DL of all classes $〈E〉$ which are smash idempotent and the complete Boolean algebra cBA of closed classes. We provide examples of spectra that are in DL, but not in cBA; in particular, for every prime p, the Bousfield class of the Eilenberg–MacLane spectrum $〈H{\mathbb{F}}_p〉$ is in $\mathbf{DL}?\mathbf{cBA}$.     

On the Hochschild homology of quantum SL(N)

We show that the quantized coordinate ring $A:=k_q[\mathit{SL}(N)]$ satisfies van den Bergh's analogue of Poincaré duality for Hochschild (co)homology with dualizing bimodule being $A_{\sigma }$, the A-bimodule which is A as k-vector space with right multiplication twisted by the modular automorphism σ of the Haar functional. This implies that ${\mathrm{H}}_{N^2?1}(A,A_{\sigma })?k$, generalizing our previous result for $k_q[\mathit{SL}(2)]$. To cite this article: T. Hadfield, U. Krähmer, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Degree Conditions and Degree Bounded Trees

In this paper, we give sufficient conditions for a graph to have degree bounded trees. Let G be a connected graph and $A\subseteq V(G)$. We denote by ${\sigma }_k(A)$ the minimum value of the degree sum in G of any k pairwise nonadjacent vertices of A, and by $w(G-A)$ the number of components of the subgraph $G-A$ of G induced by $V(G)-A$. Our main results are the following: (i) If ${\sigma }_k(A)⩾|G|-1$, then G contains a tree T with maximum degree ⩽k and $A\subseteq V(T)$. (ii) If ${\sigma }_{k-w(G-A)}(A)⩾|A|-1$, then G contains a spanning tree T with $d_T(x)⩽k$ for any $x\in A$. These are generalizations of the result by S. Win [S. Win, Existenz von Gerüsten mit Vorgeschriebenem Maximalgrad in Graphen, Abh. Math. Seminar Univ. Humburg 43 (1975) 263–267] and degree conditions are sharp.     

Reconstruction and subgaussian processes

This Note presents a randomized method to approximate any vector v from some set $T\subset {\mathbb{R}}^n$. The data one is given is the set T, and k scalar products ${(〈X_i,v〉)}_{i=1}^k$, where ${(X_i)}_{i=1}^k$ are i.i.d. isotropic subgaussian random vectors in ${\mathbb{R}}^n$, and $k\ll n$. We show that with high probability any $y\in T$ for which ${(〈X_i,y〉)}_{i=1}^k$ is close to the data vector ${(〈X_i,v〉)}_{i=1}^k$ will be a good approximation of v, and that the degree of approximation is determined by a natural geometric parameter associated with the set T. This extends and improves recent results by Candes and Tao. To cite this article: S. Mendelson et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Noethérianité et privilège en géométrique analytique p-adique

Let k be a non-Archimedean field, X a k-affinoid space and f an analytic function over X. We describe precisely how the geometric connected components of the spaces $\{x\in X||f(x)|?\epsilon \}$ behave with regards to ε. We also obtain a result concerning privileged neighbourhoods and adapt a theorem from complex analytic geometry about Noetherianity for germs of analytic functions. To cite this article: J. Poineau, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Extended Picard complexes for algebraic groups and homogeneous spaces

For a smooth geometrically integral algebraic variety X over a field k of characteristic 0, we define the extended Picard complex $\mathrm{UPic}(\overline{X})$. It is a complex of length 2 which combines the Picard group $\mathrm{Pic}(\overline{X})$ and the group $U(\overline{X}):=\overline{k}{[\overline{X}]}^{\times }/{\overline{k}}^{\times }$, where $\overline{k}$ is a fixed algebraic closure of k and $\overline{X}=X{\times }_k\overline{k}$. For a connected linear k-group G we compute the complex $\mathrm{UPic}(\overline{G})$ (up to a quasi-isomorphism) in terms of the algebraic fundamental group ${\pi }_1(\overline{G})$. We obtain similar results for a homogeneous space X of a connected k-group G. To cite this article: M. Borovoi, J. van Hamel, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Generating all the Steiner trees and computing Steiner intervals for a fixed number of terminals

In this work we present an enumeration algorithm for the generation of all Steiner trees containing a given set W of terminals of an unweighted graph G such that $|W|=k$, for a fixed positive integer k. The enumeration is performed within $O(n)$ delay, where $n=|V(G)|$ consequence of the algorithm is that the Steiner interval and the strong Steiner interval of a subset $W\subseteq V(G)$ can be computed in polynomial time, provided that the size of W is bounded by a constant.     

Monochromatic progressions in random colorings

Let $N^{+}(k)=2^{k/2}{k}^{3/2}f(k)$ and $N^{?}(k)=2^{k/2}{k}^{1/2}g(k)$ where $f(k)\to \infty$ and $g(k)\to 0$ arbitrarily slowly as $k\to \infty$. We show that the probability of a random 2-coloring of $\{1,2,\dots ,N^{+}(k)\}$ containing a monochromatic k-term arithmetic progression approaches 1, and the probability of a random 2-coloring of $\{1,2,\dots ,N^{?}(k)\}$ containing a monochromatic k-term arithmetic progression approaches 0, as $k\to \infty$. This improves an upper bound due to Brown, who had established an analogous result for $N^{+}(k)=2^k\log kf(k)$.     

Continuity in H1-norms of surfaces in terms of the L1-norms of their fundamental forms

The main purpose of this Note is to show how a ‘nonlinear Korn's inequality on a surface’ can be established. This inequality implies in particular the following interesting per se sequential continuity property for a sequence of surfaces. Let ω be a domain in ${\mathbb{R}}^2$, let $\mathit{\theta }:\overline{\omega }\to {\mathbb{R}}^3$ be a smooth immersion, and let ${\mathit{\theta }}^k:\overline{\omega }\to {\mathbb{R}}^3$, $k?1$, be mappings with the following properties: They belong to the space ${\mathit{H}}^1(\omega )$; the vector fields normal to the surfaces ${\mathit{\theta }}^k(\omega )$, $k?1$, are well defined a.e. in ω and they also belong to the space ${\mathit{H}}^1(\omega )$; the principal radii of curvature of the surfaces ${\mathit{\theta }}^k(\omega )$ stay uniformly away from zero; and finally, the three fundamental forms of the surfaces ${\mathit{\theta }}^k(\omega )$ converge in ${\mathit{L}}^1(\omega )$ toward the three fundamental forms of the surface $\mathit{\theta }(\omega )$ as $k\to \infty$. Then, up to proper isometries of ${\mathbb{R}}^3$, the surfaces ${\mathit{\theta }}^k(\omega )$ converge in ${\mathit{H}}^1(\omega )$ toward the surface $\mathit{\theta }(\omega )$ as $k\to \infty$. To cite this article: P.G. Ciarlet et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

A counterexample to the local–global principle of linear dependence for Abelian varieties

Let A be an Abelian variety defined over a number field k. Let P be a point in $A(k)$ and let X be a subgroup of $A(k)$. Gajda and Kowalski asked in 2002 whether it is true that the point P belongs to X if and only if the point $(P\mathrm{mod}\mathfrak{p})$ belongs to $(X\mathrm{mod}\mathfrak{p})$ for all but finitely many primes $\mathfrak{p}$ of k. We provide a counterexample.