The Banach algebra generated by a C0-semigroup

Let $\mathbf{T}={\{T(t)\}}_{t?0}$ be a bounded $C_0$-semigroup on a Banach space with generator A. We define $A_{\mathbf{T}}$ as the closure with respect to the operator-norm topology of the set $\{\stackrel{?}{f}(\mathbf{T}):f\in L^1({\mathbb{R}}_{+})\}$, where $\stackrel{?}{f}(\mathbf{T})={\int }_0^{\infty }f(t)T(t)\mathrm{d}t$ is the Laplace transform of $f\in L^1({\mathbb{R}}_{+})$ with respect to the semigroup T. Then $A_{\mathbf{T}}$ is a commutative Banach algebra. It is shown that if the unitary spectrum $\sigma (A)\cap \mathrm{i}\mathbb{R}$ of A is at most countable, then the Gelfand transform of $S\in A_{\mathbf{T}}$ vanishes on $\sigma (A)\cap \mathrm{i}\mathbb{R}$ if and only if, ${\lim }_{t\to \infty }6T(t)S6=0$. Some applications to the semisimplicity problem are given. To cite this article: H. Mustafayev, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

The cyclic and simplicial cohomology of the Cuntz semigroup algebra

The main objective of this paper is to determine the simplicial and cyclic cohomology groups of the Cuntz semigroup algebra ${?}^1({\mathbf{S}}_m)$. We also determine the simplicial and cyclic cohomology of the tensor algebra of a Banach space, a class which includes the algebra on the free semigroup on m-generators ${?}^1({\mathbf{FS}}_m)$. In order to do so, we first establish some general results which can be used when studying simplicial and cyclic cohomology of Banach algebras in general. We then turn our attention to ${?}^1({\mathbf{S}}_m)$, showing that the cyclic cohomology groups of degree n vanish when n is odd and are one-dimensional when n is even ($n?2$). Using the Connes–Tzygan exact sequence, these results are used to show that the simplicial cohomology groups of degree n vanish for $n?1$. A similar strategy is used for the tensor algebra of a Banach space.     

Recovery of sparsest signals via ℓq-minimization

In this paper, it is proved that every s-sparse vector $\mathbf{x}\in {\mathbb{R}}^n$ can be exactly recovered from the measurement vector $\mathbf{z}=\mathbf{Ax}\in {\mathbb{R}}^m$ via some ${?}^q$-minimization with $0<q?1$, as soon as each s-sparse vector $\mathbf{x}\in {\mathbb{R}}^n$ is uniquely determined by the measurement z. Moreover it is shown that the exponent q in the ${?}^q$-minimization can be so chosen to be about $0.6796\times (1?{\delta }_{2s}(\mathbf{A}))$, where ${\delta }_{2s}(\mathbf{A})$ is the restricted isometry constant of order 2s for the measurement matrix A.     

(Prefix) reversal distance for (signed) strings with few blocks or small alphabets

We study the String Reversal Distance problem, an extension of the well-known Sorting by Reversals problem. String Reversal Distance takes two strings S and T built on an alphabet Σ as input, and asks for a minimum number of reversals to obtain T from S. We consider four variants: String Reversal Distance, String Prefix Reversal Distance (a constrained version of the previous problem, in which any reversal must include the first letter of the string), and the signed variants of these problems, namely Signed String Reversal Distance and Signed String Prefix Reversal Distance. We study algorithmic properties of these four problems, in connection with two parameters of the input strings: the number of blocks they contain (a block being a maximal substring such that all letters in the substring are equal), and the alphabet size $|\mathrm{\Sigma }|$. Concerning the number of blocks, we show that the four problems are fixed-parameter tractable (FPT) when the considered parameter is the maximum number of blocks among the two input strings. Concerning the alphabet size, we first show that String Reversal Distance and String Prefix Reversal Distance are NP-hard even if the input strings are built on a binary alphabet $\mathrm{\Sigma }=\{0,1\}$, each 0-block has length at most two and each 1-block has length one. We also show that Signed String Reversal Distance and Signed String Prefix Reversal Distance are NP-hard even if the input strings have only one letter. Finally, when $|\mathrm{\Sigma }|=O(1)$, we provide a singly-exponential algorithm that computes the exact distance between any pair of strings, for a large family of distances that we call well-formed, which includes the four distances we study here.     

On functors preserving skeletal maps and skeletally generated compacta

A map $f:X\to Y$ between topological spaces is skeletal if the preimage $f^{?1}(A)$ of each nowhere dense subset $A?Y$ is nowhere dense in X. We prove that a normal functor $F:\mathbf{Comp}\to \mathbf{Comp}$ is skeletal (which means that F preserves skeletal epimorphisms) if and only if for any open surjective map $f:X\to Y$ between metrizable zero-dimensional compacta with two-element non-degeneracy set $N^f=\{x\in X:|f^{?1}(f(x))|>1\}$ the map $Ff:FX\to FY$ is skeletal. This characterization implies that each open normal functor is skeletal. The converse is not true even for normal functors of finite degree. The other main result of the paper says that each normal functor $F:\mathbf{Comp}\to \mathbf{Comp}$ preserves the class of skeletally generated compacta. This contrasts with the known ??epin?s result saying that a normal functor is open if and only if it preserves the class of openly generated compacta.     

Estimation par ondelettes dans les systèmes dynamiques

Let $\{X_t,t\in \mathbb{Z}\}$ be a stochastic process valued on $[0,1]$ where $X_{t+1}=\phi (X_t)$, φ a piecewise expanding map, preserving measure μ with density f. We give consistent estimates of the invariant measure f and the map φ with the linear wavelets method using the base defined by Cohen, DeVore and Daubechies. We obtain the optimal rate of convergence of the MISE. To cite this article: M.-L. Vanharen, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Existence of a solution ‘in the large’ for the 3D large-scale ocean dynamics equations

For the 3D system of equations describing large-scale ocean dynamics in the Cartesian coordinate system existence and uniqueness of a solution on an arbitrary time interval $[0,T]$ is proved and the norm $6{\stackrel{?}{\mathbf{u}}}_{x}6$ is shown to be continuous in time on $[0,T]$. To cite this article: G.M. Kobelkov, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

A topological nullstellensatz for tensor-triangulated categories

Let $\mathrm{Spec}(\mathbb{T})$ be the spectrum of a tensor-triangulated category $(\mathbb{T},?,\mathbf{1})$. We show that there is a homeomorphism between the spectral space of radical thick tensor ideals in $(\mathbb{T},?,\mathbf{1})$ and the collection of open subsets of $\mathrm{Spec}(\mathbb{T})$ in inverse topology. In fact, we prove a more general result in terms of supports on $(\mathbb{T},?,\mathbf{1})$ and work by combining methods from commutative algebra, topology and tensor triangular geometry.     

The D+E[Γ⁎] construction from Prüfer domains and GCD-domains

Let $D?E$ denote an extension of integral domains, Γ be a nonzero torsion-free grading monoid with $\Gamma \cap ?\Gamma =\{0\}$, ${\Gamma }^{?}=\Gamma ?\{0\}$ and $D+E[{\Gamma }^{?}]=\{f\in E[\Gamma ]|f(0)\in D\}$. In this paper, we give a necessary and sufficient criteria for $D+E[{\Gamma }^{?}]$ to be a Prüfer domain or a GCD-domain.     

A continuum of periodic solutions to the planar four-body problem with two pairs of equal masses

In this paper, we apply the variational method with Structural Prescribed Boundary Conditions (SPBC) to prove the existence of periodic and quasi-periodic solutions for the planar four-body problem with two pairs of equal masses $m_1=m_3$ and $m_2=m_4$. A path $q(t)$ on $[0,T]$ satisfies the SPBC if the boundaries $q(0)\in \mathbf{A}$ and $q(T)\in \mathbf{B}$, where A and B are two structural configuration spaces in ${({\mathbf{R}}^2)}^4$ and they depend on a rotation angle $\theta \in (0,2\pi )$ and the mass ratio $\mu =\frac{m_2}{m_1}\in {\mathbf{R}}^{+}$.We show that there is a region $\mathrm{\Omega }?(0,2\pi )\times R^{+}$ such that there exists at least one local minimizer of the Lagrangian action functional on the path space satisfying the SPBC $\{q(t)\in H^1([0,T],{({\mathbf{R}}^2)}^4)|q(0)\in \mathbf{A},q(T)\in \mathbf{B}\}$ for any $(\theta ,\mu )\in \mathrm{\Omega }$. The corresponding minimizing path of the minimizer can be extended to a non-homographic periodic solution if θ is commensurable with π or a quasi-periodic solution if θ is not commensurable with π. In the variational method with the SPBC, we only impose constraints on the boundary and we do not impose any symmetry constraint on solutions. Instead, we prove that our solutions that are extended from the initial minimizing paths possess certain symmetries.The periodic solutions can be further classified as simple choreographic solutions, double choreographic solutions and non-choreographic solutions. Among the many stable simple choreographic orbits, the most extraordinary one is the stable star pentagon choreographic solution when $(\theta ,\mu )=(\frac{4\pi }{5},1)$. Remarkably the unequal-mass variants of the stable star pentagon are just as stable as the equal mass choreographies.     

Étude en temps petit des solutions d'EDS conduites par des mouvements browniens fractionnaires

We study, in small times, the properties of the operator ${\mathbf{P}}_t(f)(x)=\mathbb{E}(f(X_t^x))$, where ${(X_t^x)}_{t?0}$ is the solution of a stochastic differential equation driven by fractional Brownian motions with the same Hurst parameter $H>\frac{1}{4}$. To cite this article: F. Baudoin, L. Coutin, C. R. Acad. Sci. Paris, Ser. I 341 (2005).