A General Fractional Porous Medium Equation

We develop a theory of existence and uniqueness for the following porous medium equation with fractional diffusion: \input amssym $$\left\{ {\matrix{ {{{\partial u} \over {\partial t}} + \left( { ‐ \Delta } \right)^{\sigma /2} \left( {\left| u \right|^{m ‐ 1} u} \right) = 0,} \hfill & {x \in {\Bbb R} ^N ,\,\,t > 0,} \hfill \cr {u\left( {x,0} \right) = f\left( x \right),} \hfill & {x \in {\Bbb R} ^N .} \hfill \cr } } \right.$$ We consider data \input amssym $f\in L^1(\Bbb{R}^N)$ and all exponents $0<\sigma<2\;and\;m>0$ . Existence and uniqueness of a strong solution is established for $ m > {m_\ast}={(N-\sigma)_+}/N$ , giving rise to an L¹‐contraction semigroup. In addition, we obtain the main qualitative properties of these solutions. In the lower range ${0 < m} \le {m_\ast}$ existence and uniqueness happen under some restrictions, and the properties of the solutions are different from the ones for the case above m_*. We also study the dependence of solutions on f, m, and σ. Moreover, we consider the above questions for the problem posed in a bounded domain. © 2012 Wiley Periodicals, Inc.     

Global mild solutions of Navier‐Stokes equations

We establish a global well‐posedness of mild solutions to the three‐dimensional, incompressible Navier‐Stokes equations if the initial data are in the space ${\cal{X}}^{-1}$ defined by \input amssym ${\cal{X}}^{‐1} = \{f \in {\cal{D}}^\prime(R^3): \int_{{\Bbb{R}}^3}|\xi|^{‐1}|\widehat{f}|d\xi < \infty\}$ and if the norms of the initial data in ${\cal{X}}^{-1}$ are bounded exactly by the viscosity coefficient μ. © 2010 Wiley Periodicals, Inc.     

Group Invariant Scattering

This paper constructs translation‐invariant operators on $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}{\bf L}^2({{{\R}}}^d)$ , which are Lipschitz‐continuous to the action of diffeomorphisms. A scattering propagator is a path‐ordered product of nonlinear and noncommuting operators, each of which computes the modulus of a wavelet transform. A local integration defines a windowed scattering transform, which is proved to be Lipschitz‐continuous to the action of C ² diffeomorphisms. As the window size increases, it converges to a wavelet scattering transform that is translation invariant. Scattering coefficients also provide representations of stationary processes. Expected values depend upon high‐order moments and can discriminate processes having the same power spectrum. Scattering operators are extended on L ²(G), where G is a compact Lie group, and are invariant under the action of G. Combining a scattering on $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}{\bf L}^2({{{\R}}}^d)$ and on L ²(SO(d)) defines a translation‐ and rotation‐invariant scattering on $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}{\bf L}^2({{{\R}}}^d)$ . © 2012 Wiley Periodicals, Inc.     

C‐perfect hypergraphs

Let ${\mathcal{H}}=({{X}},{\mathcal{E}})Let ${\mathcal{H}}=({{X}},{\mathcal{E}})$ be a hypergraph with vertex set X and edge set ${\mathcal{E}}$. A C‐coloring of ${\mathcal{H}}$ is a mapping ?:X→? such that |?(E)|<|E| holds for all edges ${{E}}\in{\mathcal{E}}$ (i.e. no edge is multicolored). We denote by $\bar{\chi}({\mathcal{H}})$ the maximum number |?(X)| of colors in a C‐coloring. Let further $\alpha({\mathcal{H}})$ denote the largest cardinality of a vertex set S?X that contains no ${{E}}\in{\mathcal{E}}$, and $\tau({\mathcal{H}})=|{{X}}|-\alpha({\mathcal{H}})$ the minimum cardinality of a vertex set meeting all $E \in {\mathcal{E}}$. The hypergraph ${\mathcal{H}}$ is called C‐perfect if $\bar{\chi}({\mathcal{H}}\prime)=\alpha({\mathcal{H}}\prime)$ holds for every induced subhypergraph ${\mathcal{H}}\prime\subseteq{\mathcal{H}}$. If ${\mathcal{H}}$ is not C‐perfect but all of its proper induced subhypergraphs are, then we say that it is minimally C‐imperfect. We prove that for all r, k∈? there exists a finite upper bound h(r, k) on the number of minimally C‐imperfect hypergraphs ${\mathcal{H}}$ with $\tau({\mathcal{H}})\le {{k}}$ and without edges of more than r vertices. We give a characterization of minimally C‐imperfect hypergraphs that have τ=2, which also characterizes implicitly the C‐perfect ones with τ=2. From this result we derive an infinite family of new constructions that are minimally C‐imperfect. A characterization of minimally C‐imperfect circular hypergraphs is presented, too. © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 132–149, 2010     

Periodic problem for a nonlinear‐damped wave equation on the boundary

In this work we investigate the existence of periodic solutions in t for the following problem: We employ elliptic regularization and monotone method. We consider $\mbox{\boldmath{$\Omega$}}\mbox{\boldmath{$\subset$}}{\mathbb{R}}^{{{n}}} \ (n\geqslant 1)$ an open bounded set that has regular boundary Γ and Q=Ω ×(0,T), T>0, a cylinder of ${\mathbb{R}}^{n+1}$ with lateral boundary Σ = Γ × (0,T). Copyright © 2009 John Wiley & Sons, Ltd.     

A critical center‐stable manifold for Schrödinger's equation in three dimensions

Consider the focusing $\dot H^{1/2}$ ‐critical semilinear Schrödinger equation in $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}\R^3$ It admits an eight‐dimensional manifold of special solutions called ground state solitons. We exhibit a codimension‐1 critical real analytic manifold ${\cal N}$ of asymptotically stable solutions of (0.1) in a neighborhood of the soliton manifold. We then show that ${\cal N}$ is center‐stable, in the dynamical systems sense of Bates and Jones, and globally‐in‐time invariant. Solutions in ${\cal N}$ are asymptotically stable and separate into two asymptotically free parts that decouple in the limit—a soliton and radiation. Conversely, in a general setting, any solution that stays $\dot H^{1/2}$ ‐close to the soliton manifold for all time is in ${\cal N}$ . The proof uses the method of modulation. New elements include a different linearization and an endpoint Strichartz estimate for the time‐dependent linearized equation. The proof also uses the fact that the linearized Hamiltonian has no nonzero real eigenvalues or resonances. This has recently been established in the case treated here—of the focusing cubic NLS in $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}\R^3$ —by the work of Marzuola and Simpson and Costin, Huang, and Schlag. © 2012 Wiley Periodicals, Inc.     

Regularized trace of the inverse of the dirichlet laplacian

For the eigenvalues $( \lambda_{n}) _{n=1}^{\infty}$ of the Dirichlet Laplacian on a bounded convex domain $\font\open=msbm10 at 10pt\def\C{\hbox{\open C}}\Omega\subset{\C}$ , we find the sum of the series the regularized trace of the inverse of Dirichlet Laplacian. © 2011 Wiley Periodicals, Inc.     

The Eigenvalue Problem of Singular Ergodic Control

We consider the problem of finding a real number λ and a function u satisfying the PDE Here f is a convex, superlinear function. We prove that there is a unique λ* such that the above PDE has a viscosity solution u satisfying $\lim_{|x|\rightarrow \infty}u(x)/|x|=1$ . Moreover, we show that associated to λ* is a convex solution u* with D²u^*∈ $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}L^{\infty}(\R^N)$ and give two min‐max formulae for λ*. λ* has a probabilistic interpretation as being the least, long‐time averaged (ergodic) cost for a singular control problem involving f. © 2011 Wiley Periodicals, Inc.     

PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming

Suppose we wish to recover a signal \input amssym $\font\abc=cmmib10\def\bi#1{\hbox{\abc#1}} {\bi x} \in {\Bbb C}^n$ from m intensity measurements of the form $\font\abc=cmmib10\def\bi#1{\hbox{\abc#1}} |\langle \bi x,\bi z_i \rangle|^2$ , $i = 1, 2, \ldots, m$ ; that is, from data in which phase information is missing. We prove that if the vectors $\font\abc=cmmib10\def\bi#1{\hbox{\abc#1}}{\bi z}_i$ are sampled independently and uniformly at random on the unit sphere, then the signal x can be recovered exactly (up to a global phase factor) by solving a convenient semidefinite program–‐a trace‐norm minimization problem; this holds with large probability provided that m is on the order of $n {\log n}$ , and without any assumption about the signal whatsoever. This novel result demonstrates that in some instances, the combinatorial phase retrieval problem can be solved by convex programming techniques. Finally, we also prove that our methodology is robust vis‐à‐vis additive noise. © 2012 Wiley Periodicals, Inc.     

Piercing random boxes

Consider a set of n random axis parallel boxes in the unit hypercube in ${\bf R}^{d}$ , where d is fixed and n tends to infinity. We show that the minimum number of points one needs to pierce all these boxes is, with high probability, at least $\Omega_d(\sqrt{n}(\log n)^{d/2-1})$ and at most $O_d(\sqrt{n}(\log n)^{d/2-1}\log \log n)$ . © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 38, 365–380, 2011     

Increasing the output length of zero‐error dispersers

Let $\cal{C}$ be a class of probability distributions over a finite set Ω. A function $D : \Omega \mapsto\{0,1\}^{m}$ is a disperser for $\cal{C}$ with entropy threshold $k$ and error $\epsilon$ if for any distribution X in $\cal{C}$ such that X gives positive probability to at least $2^{k}$ elements we have that the distribution $D(X)$ gives positive probability to at least $(1-\epsilon)2^{m}$ elements. A long line of research is devoted to giving explicit (that is polynomial time computable) dispersers (and related objects called “extractors”) for various classes of distributions while trying to maximize m as a function of k. For several interesting classes of distributions there are explicit constructions in the literature of zero‐error dispersers with “small” output length m. In this paper we develop a general technique to improve the output length of zero‐error dispersers. This strategy works for several classes of sources and is inspired by a transformation that improves the output length of extractors (which was given by Shaltiel (CCC'06; Proceedings of the 21st Annual IEEE Conference on Computational Complexity, (2006) 46–60.) building on earlier work by Gabizon, Raz and Shaltiel (SIAM J Comput 36 (2006) 1072–1094). Our techniques are different than those of Shaltiel (CCC'06; Proceedings of the 21st Annual IEEE Conference on Computational Complexity (2006) 46–60) and in particular give non‐trivial results in the errorless case. Using our approach we construct improved zero‐error 2‐source dispersers. More precisely, we show that for any constant $\delta >0$ there is a constant $\eta >0$ such that for sufficiently large n there is a poly‐time computable function $D :\{0,1\}^{n}\times\{0,1\}^{n}\mapsto\{0,1\}^{\eta n}$ such that for every two independent distributions $X_1,X_2$ over $\{0,1\}^{n}$ each with support size at least $2^{\delta n}$ , the output distribution $D(X_1,X_2)$ has full support. This improves the output length of previous constructions by Barak, Kindler, Shaltiel, Sudakov and Wigderson (Proceedings of the 37th Annual ACM Symposium on Theory of Computing (2005) 1–10) and has applications in Ramsey theory and in improved constructions of certain data structures from the work of Fiat and Naor [SIAM J Comput 22 (1993)]. We also use our techniques to give explicit constructions of zero‐error dispersers for bit‐fixing sources and affine sources over polynomially large fields. These constructions improve the best known explicit constructions due to Rao (unpublished data) and Gabizon and Raz [Combinatorica 28 (2008)] and achieve $m=\Omega(k)$ for bit‐fixing sources and $m=k-o(k)$ for affine sources over polynomial size fields. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011     

Persistence of activity in threshold contact processes,an “Annealed approximation” of random Boolean networks

We consider a model for gene regulatory networks that is a modification of Kauffmann's J Theor Biol 22 (1969), 437–467 random Boolean networks. There are three parameters: $n = {\rm the}$ number of nodes, $r = {\rm the}$ number of inputs to each node, and $p = {\rm the}$ expected fraction of 1'sin the Boolean functions at each node. Following a standard practice in thephysics literature, we use a threshold contact process on a random graph on n nodes, in which each node has in degree r, to approximate its dynamics. We show that if $r\ge 3$ and $r \cdot 2p(1-p)>1$ , then the threshold contact process persists for a long time, which correspond to chaotic behavior of the Boolean network. Unfortunately, we are only able to prove the persistence time is $\ge \exp(cn^{b(p)})$ with $b(p)>0$ when $r\cdot 2p(1-p)> 1$ , and $b(p)=1$ when $(r-1)\cdot 2p(1-p)>1$ . © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011     

A Self‐Dual Polar Factorization for Vector Fields

We show that any nondegenerate vector field u in \begin{align*}L^{\infty}(\Omega, \mathbb{R}^N)\end{align*}, where Ω is a bounded domain in \begin{align*}\mathbb{R}^N\end{align*}, can be written as \begin{align*}u(x)= \nabla_1 H(S(x), x)\quad {\text for a.e.\ x \in \Omega}\end{align*}}, where S is a measure‐preserving point transformation on Ω such that \begin{align*}S^2=I\end{align*} a.e. (an involution), and \begin{align*}H: \mathbb{R}^N \times \mathbb{R}^N \to \mathbb{R}\end{align*} is a globally Lipschitz antisymmetric convex‐concave Hamiltonian. Moreover, u is a monotone map if and only if S can be taken to be the identity, which suggests that our result is a self‐dual version of Brenier's polar decomposition for the vector field as \begin{align*}u(x)=\nabla \phi (S(x))\end{align*}, where ? is convex and S is a measure‐preserving transformation. We also describe how our polar decomposition can be reformulated as a (self‐dual) mass transport problem. © 2012 Wiley Periodicals, Inc.     

Closed trail decompositions of complete equipartite graphs

The complete equipartite graph $K_m * {\overline{K_n}}$ has mn vertices partitioned into m parts of size n, with two vertices adjacent if and only if they are in different parts. In this paper, we determine necessary and sufficient conditions for the existence of a decomposition of $K_m * {\overline{K_n}}$ into closed trails of length k. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 374–403, 2009     

The cover time of random geometric graphs

We study the cover time of random geometric graphs. Let $I(d)=[0,1]^{d}$ denote the unit torus in d dimensions. Let $D(x,r)$ denote the ball (disc) of radius r. Let $\Upsilon_d$ be the volume of the unit ball $D(0,1)$ in d dimensions. A random geometric graph $G=G(d,r,n)$ in d dimensions is defined as follows: Sample n points V independently and uniformly at random from $I(d)$ . For each point x draw a ball $D(x,r)$ of radius r about x. The vertex set $V(G)=V$ and the edge set $E(G)=\{\{v,w\}: w\ne v,\,w\in D(v,r)\}$ . Let $G(d,r,n),\,d\geq 3$ be a random geometric graph. Let $C_G$ denote the cover time of a simple random walk on G. Let $c>1$ be constant, and let $r=(c\log n/(\Upsilon_dn))^{1/d}$ . Then whp the cover time satisfies © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 38, 324–349, 2011     

Acyclic dominating partitions

Given a graph G=(V, E), let ${\mathcal{P}}$ be a partition of V. We say that ${\mathcal{P}}$ is dominating if, for each part P of ${\mathcal{P}}$, the set V\P is a dominating set in G (equivalently, if every vertex has a neighbor of a different part from its own). We say that ${\mathcal{P}}$ is acyclic if for any parts P, P′ of ${\mathcal{P}}$, the bipartite subgraph G[P, P′] consisting of the edges between P and P′ in ${\mathcal{P}}$ contains no cycles. The acyclic dominating number ad(G) of G is the least number of parts in any partition of V that is both acyclic and dominating; and we shall denote by ad(d) the maximum over all graphs G of maximum degree at most d of ad(G). In this article, we prove that ad(3)=2, which establishes a conjecture of P. Boiron, É. Sopena, and L. Vignal, DIMACS/DIMATIA Conference “Contemporary Trends in Discrete Mathematics”, 1997, pp. 1–10. For general d, we prove the upper bound ad(d)=O(dlnd) and a lower bound of ad(d)=Ω(d). © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 292–311, 2010     

Vector diffusion maps and the connection Laplacian

We introduce vector diffusion maps (VDM), a new mathematical framework for organizing and analyzing massive high‐dimensional data sets, images, and shapes. VDMis a mathematical and algorithmic generalization of diffusion maps and other nonlinear dimensionality reduction methods, such as LLE, ISOMAP, and Laplacian eigenmaps. While existing methods are either directly or indirectly related to the heat kernel for functions over the data, VDM is based on the heat kernel for vector fields. VDM provides tools for organizing complex data sets, embedding them in a low‐dimensional space, and interpolating and regressing vector fields over the data. In particular, it equips the data with a metric, which we refer to as the vector diffusion distance. In the manifold learning setup, where the data set is distributed on a low‐dimensional manifold ${\cal M}^d$ embedded in $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}\R^p$ , we prove the relation between VDM and the connection Laplacian operator for vector fields over the manifold. © 2012 Wiley Periodicals, Inc.     

Regularity of renormalized solutions in the Boltzmann equation with long‐range interactions

It is well‐established that renormalized solutions of the Boltzmann equation enjoy some kind of regularity, or at least compactness, in the velocity variable when the angular collision kernel is nonintegrable. However, obtaining explicit estimates in convenient and natural functional settings proves rather difficult. In this work, we derive a velocity smoothness estimate from the a priori control of the renormalized dissipation. As a direct consequence of our result, we show that, in the presence of long‐range interactions, any renormalized solution F(t, x, v) to the Boltzmann equation satisfies locally ${\textstyle{F \over {1 + F}}} \in W_{t,x,v}^{s,p}$ for every $1 \le p \le {\textstyle{D \over {D - 1}}}$ and for some s > 0 depending on p. We also provide an application of this new estimate to the hydrodynamic limit of the Boltzmann equation without cutoff. © 2012 Wiley Periodicals, Inc.     

Classification of edge‐transitive rose window graphs

Given natural numbers n?3 and 1?a, r?n?1, the rose window graph R_n(a, r) is a quartic graph with vertex set $\{{{x}}_{{i}}|{{i}}\in {\mathbb{Z}}_{{n}}\} \cup \{{{y}}_{{i}}|{{i}}\in{\mathbb{Z}}_{{n}}\}Given natural numbers n?3 and 1?a, r?n?1, the rose window graph R_n(a, r) is a quartic graph with vertex set $\{{{x}}_{{i}}|{{i}}\in {\mathbb{Z}}_{{n}}\} \cup \{{{y}}_{{i}}|{{i}}\in{\mathbb{Z}}_{{n}}\}$ and edge set $\{\{{{x}}_{{i}},{{x}}_{{{i+1}}}\} \mid {{i}}\in {\mathbb{Z}}_n \} \cup \{\{{{y}}_{{{i}}},{{y}}_{{{i+r}}}\}\mid {{i}} \in{\mathbb{Z}}_{{n}}\}\cup \{\{{{x}}_{{{i}}},{{y}}_{{{i}}}\} \mid {{i}}\in {\mathbb{Z}}_{{{n}}}\}\cup \{\{{{x}}_{{{i+a}}},{{y}}_{{{i}}}\} \mid{{i}} \in {\mathbb{Z}}_{{{n}}}\}$. In this article a complete classification of edge‐transitive rose window graphs is given, thus solving one of the three open problems about these graphs posed by Steve Wilson in 2001. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 216–231, 2010     

Correlations for paths in random orientations of G(n,p) and G(n,m)

We study random graphs, both G( n,p) and G( n,m), with random orientations on the edges. For three fixed distinct vertices s,a,b we study the correlation, in the combine probability space, of the events $\{a\to s\}$ and $\{s\to b\}$ . For G(n,p), we prove that there is a $pc = 1/2$ such that for a fixed $p < pc$ the correlation is negative for large enough n and for $p > pc$ the correlation is positive for large enough n. We conjecture that for a fixed $n \ge 27$ the correlation changes sign three times for three critical values of p. For G(n,m) it is similarly proved that, with $p=m/({{n}\atop {2}})$ , there is a critical pc that is the solution to a certain equation and approximately equal to 0.7993. A lemma, which computes the probability of non existence of any $\ell$ directed edges in G(n,m), is thought to be of independent interest. We present exact recursions to compute \input amssym $\Bbb{P}(a\to s)$ and \input amssym $\Bbb{P}(a\to s, s\to b)$ . We also briefly discuss the corresponding question in the quenched version of the problem. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011