Sharp decay rates for the fastest conservative diffusions

In many diffusive settings, initial disturbances will gradually disappear and all but their crudest features — such as size and location — will eventually be forgotten. Quantifying the rate at which this information is lost is sometimes a question of central interest. Here this rate is addressed for the fastest conservative nonlinearities in the singular diffusion equation

$u_t=\mathrm{\Delta }(u^m),{(n?2)}_{+}/n<m?n/(n+2),u,t?0,\mathbf{x}\in {\mathbf{R}}^n,$

which governs the decay of any integrable, compactly supported initial density towards a characteristically spreading self-similar profile. A potential theoretic comparison technique is outlined below which establishes the sharp $1/t$ conjectured power law rate of decay uniformly in relative error, and in weaker norms such as $L^1({\mathbf{R}}^n)$. To cite this article: Y.J. Kim, R.J. McCann, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Level sets of β-expansions

Let ${\{{\mathrm{\epsilon }}_n(x)\}}_{n?1}$ be the sequence of β-digits of a real number $x\in (0\text{,}1)$, with the golden number $\beta =(\sqrt{5}+1)/2$ as basis. For any $0?p?1/2$, any $0<\tau ?1$ and any real number a, we consider the level set consisting of numbers x such that ${\sum }_{n=1}^{\infty }({\mathrm{\epsilon }}_n(x)?p)/n^{\tau }=a$. We prove that the Hausdorff dimension of this set is independent of a and τ, and that it is equal to $\log f(p)/\log \beta$ where $f(p)={(1?p)}^{1?p}/({(1?2p)}^{1?2p}{p}^p)$. To cite this article: A. Fan, H. Zhu, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Improved Cotlar's inequality in the context of local Tb theorems

In the context of local Tb theorems with $L^p$ testing conditions we prove an enhanced Cotlar's inequality. This is related to the problem of removing the so called buffer assumption of Hytönen–Nazarov, which is the final barrier for the full solution of S. Hofmann's problem. We also investigate the problem of extending the Hytönen–Nazarov result to non-homogeneous measures. We work not just with the Lebesgue measure but with measures μ in ${\mathbb{R}}^d$ satisfying $\mu (B(x,r))\le C{r}^n$, $n\in (0,d]$. The range of exponents in the Cotlar type inequality depend on n. Without assuming buffer we get the full range of exponents $p,q\in (1,2]$ for measures with $n\le 1$, and in general we get $p,q\in [2??(n),2]$, $?(n)>0$. Consequences for (non-homogeneous) local Tb theorems are discussed.     

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.     

On a p-Laplace equation with multiple critical nonlinearities

Using the Mountain-Pass Theorem of Ambrosetti and Rabinowitz we prove that $?{\mathrm{\Delta }}_{p}u?\mu {|x|}^{?p}{u}^{p?1}={|x|}^{?s}{u}^{p^{?}(s)?1}+u^{p^{?}?1}$ admits a positive weak solution in ${\mathbb{R}}^n$ of class $D_1^p({\mathbb{R}}^n)\cap C^1({\mathbb{R}}^n?\{0\})$, whenever $\mu <{\mu }_1$, and ${\mu }_1={[(n?p)/p]}^p$. The technique is based on the existence of extremals of some Hardy–Sobolev type embeddings of independent interest. We also show that if $u\in D_1^p({\mathbb{R}}^n)$ is a weak solution in ${\mathbb{R}}^n$ of $?{\mathrm{\Delta }}_{p}u?\mu {|x|}^{?p}{|u|}^{p?2}u={|x|}^{?s}{|u|}^{p^{?}(s)?2}u+{|u|}^{q?2}u$, then $u\equiv 0$ when either $1<q<p^{?}$, or $q>p^{?}$ and u is also of class $L_{\mathrm{loc}}^{\infty }({\mathbb{R}}^n?\{0\})$.     

Bounds on an exponential sum arising in Boolean circuit complexity

We study exponential sums of the form $S=2^{?n}{\sum }_{x\in {\{0,1\}}^n}{e}_m(h(x))e_q(p(x))$, where $m,q\in {\mathbf{Z}}^{+}$ are relatively prime, p is a polynomial with coefficients in ${\mathbf{Z}}_q$, and $h(x)=a(x_1+?+x_n)$ for some $1?a<m$. We prove an upper bound of the form $2^{?\Omega (n)}$ on $|S|$. This generalizes a result of J. Bourgain, who establishes this bound in the case where q is odd. This bound has consequences in Boolean circuit complexity. To cite this article: F. Green et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

New constructions of SSPDs and their applications

We present a new optimal construction of a semi-separated pair decomposition (i.e., SSPD) for a set of n points in ${\mathbb{R}}^d$. In the new construction each point participates in a few pairs, and it extends easily to spaces with low doubling dimension. This is the first optimal construction with these properties.As an application of the new construction, for a fixed $t>1$, we present a new construction of a t-spanner with $O(n)$ edges and maximum degree $O({\log }^{2}n)$ that has a separator of size $O(n^{1?1/d})$.     

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.     

Number of Crossing-Free Geometric Graphs vs. Triangulations

We show that there is a constant $\alpha >0$ such that, for any set P of n⩾ 5 points in general position in the plane, a crossing-free geometric graph on P that is chosen uniformly at random contains, in expectation, at least $(\frac{1}{2}+\alpha )M$ edges, where M denotes the number of edges in any triangulation of P. From this we derive (to our knowledge) the first non-trivial upper bound of the form $c^n\cdot \mathsf{tr}(P)$ on the number of crossing-free geometric graphs on P; that is, at most a fixed exponential in n times the number of triangulations of P. (The trivial upper bound of $2^M\cdot \mathsf{tr}(P)$, or $c=2^{M/n}$, follows by taking subsets of edges of each triangulation.) If the convex hull of P is triangular, then $M=3n-6$, and we obtain $c<7.98$.Upper bounds for the number of crossing-free geometric graphs on planar point sets have so far applied the trivial $8^n$ factor to the bound for triangulations; we slightly decrease this bound to $O({343.11}^n)$.