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     

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.     

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.     

Belohorec‐type oscillation theorem for second order sublinear dynamic equations on time scales

Consider the Emden‐Fowler sublinear dynamic equation (0.1) where $p\in C(\mathbb{T},R)$, where $\mathbb{T}$ is a time scale, 0 < α < 1. When p(t) is allowed to take on negative values, we obtain a Belohorec‐type oscillation theorem for (0.1). As an application, we get that the sublinear difference equation (0.2) is oscillatory, if and the sublinear q‐difference equation (0.3) where $t\in q^{\mathbb{N}_0}, q>1$, is oscillatory, if      

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     

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     

Infinitely Many Positive Solutions to Some Scalar Field Equations with Nonsymmetric Coefficients

In this paper the equation $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}} - \Delta u + a(x)u = |u|^{p - 1} u\;{\rm in }\;{\R}^N $ is considered, when N ≥ 2, p > 1, and $p < {{N + 2} \over {N - 2}}$ if N ≥ 3. Assuming that the potential a(x) is a positive function belonging to $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}L_{{\rm loc}}^{N/2} ({\R}^N )$ such that a(x) → a_∞ > 0 as |x|→∞ and satisfies slow decay assumptions but does not need to fulfill any symmetry property, the existence of infinitely many positive solutions, by purely variational methods, is proved. The shape of the solutions is described as is, and furthermore, their asymptotic behavior when $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}|a(x) - a_\infty |_{L_{{\rm loc}}^{N/2} ({\R}^N )} \to 0$ . © 2012 Wiley Periodicals, Inc.     

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     

Bubbling Solutions for the SU(3) Chern‐Simons Model on a Torus

We consider the following nonlinear system derived from the SU(3) Chern‐Simons models on a torus Ω: where $\delta_p$ denotes the Dirac measure at $p\in\Omega$ . When $\{p_j^1\}_1^{N_1}= \{p_j^2\}_1^{N_2}$ , if we look for a solution with $u_1=u_2=u$ , then (0.1) is reduced to the Chern‐Simons‐Higgs equation: The existence of bubbling solutions to (0.1) has been a longstanding problem. In this paper, we prove the existence of such solutions such that $u_1\ne u_2$ even if $\{p_j^1\}_1^{N_1}=\{p_j^2\}_1^{N_2}$ . © 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.     

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     

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     

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.     

Path homomorphisms,graph colorings,and boolean matrices

We investigate bounds on the chromatic number of a graph G derived from the nonexistence of homomorphisms from some path \begin{eqnarray*}\vec{P}\end{eqnarray*} into some orientation \begin{eqnarray*}\vec{G}\end{eqnarray*} of G. The condition is often efficiently verifiable using boolean matrix multiplications. However, the bound associated to a path \begin{eqnarray*}\vec{P}\end{eqnarray*} depends on the relation between the “algebraic length” and “derived algebraic length” of \begin{eqnarray*}\vec{P}\end{eqnarray*}. This suggests that paths yielding efficient bounds may be exponentially large with respect to G, and the corresponding heuristic may not be constructive. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 198–209, 2010     

Weak type (p,q)‐inequalities for the Haar system and differentially subordinated martingales

For any $1\leq p,\,q<\infty$, we determine the optimal constant $C_{p,q}$ such that the following holds. If $(h_k)_{k\geq 0}$ is the Haar system on [0,1], then for any vectors a_k from a separable Hilbert space $\mathcal{H}$ and $\varepsilon_k\in \{-1,1\}$, $k=0,\,1,\,2,\ldots,$ we have This is generalized to the sharp weak‐type inequality where X, Y stand for $\mathcal{H}$‐valued martingales such that Y is differentially subordinate to X.     

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.     

Generalized surface quasi‐geostrophic equations with singular velocities

This paper establishes several existence and uniqueness results for two families of active scalar equations with velocity fields determined by the scalars through very singular integrals. The first family is a generalized surface quasigeostrophic (SQG) equation with the velocity field u related to the scalar θ by $u=\nabla^\perp\Lambda^{\beta-2}\theta$ , where $1<\beta\le 2$ and $\Lambda=(-\Delta)^{1/2}$ is the Zygmund operator. The borderline case β = 1 corresponds to the SQG equation and the situation is more singular for β > 1. We obtain the local existence and uniqueness of classical solutions, the global existence of weak solutions, and the local existence of patch‐type solutions. The second family is a dissipative active scalar equation with $u=\nabla^\perp (\log(I-\Delta))^\mu\theta\ {\rm for}\ \mu>0$ , which is at least logarithmically more singular than the velocity in the first family. We prove that this family with any fractional dissipation possesses a unique local smooth solution for any given smooth data. This result for the second family constitutes a first step towards resolving the global regularity issue recently proposed by K. Ohkitani. © 2012 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.     

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     

Universality for the Focusing Nonlinear Schrödinger Equation at the Gradient Catastrophe Point: Rational Breathers and Poles of the Tritronquée Solution to Painlevé I

The semiclassical (zero‐dispersion) limit of solutions $q=q(x,t,\epsilon)$ to the one‐dimensional focusing nonlinear Schrödinger equation (NLS) is studied in a scaling neighborhood D of a point of gradient catastrophe ($x_0,t_0$) . We consider a class of solutions, specified in the text, that decay as $|x| \rightarrow \infty$ . The neighborhood D contains the region of modulated plane wave (with rapid phase oscillations), as well as the region of fast‐amplitude oscillations (spikes). In this paper we establish the following universal behaviors of the NLS solutions q near the point of gradient catastrophe: (i) each spike has height $3|q{_0}(x_0,t_0)|$ and uniform shape of the rational breather solution to the NLS, scaled to the size ${\cal O}(\epsilon)$ ; (ii) the location of the spikes is determined by the poles of the tritronquée solution of the Painlevé I (P1) equation through an explicit map between D and a region of the Painlevé independent variable; (iii) if $(x,t)\in D$ but lies away from the spikes, the asymptotics of the NLS solution $q(x,t, \epsilon)$ is given by the plane wave approximation $q_0(x,t, \epsilon)$ , with the correction term being expressed in terms of the tritronquée solution of P1. The relation with the conjecture of Dubrovin, Grava, and Klein about the behavior of solutions to the focusing NLS near a point of gradient catastrophe is discussed. We conjecture that the P1 hierarchy occurs at higher degenerate catastrophe points and that the amplitudes of the spikes are odd multiples of the amplitude at the corresponding catastrophe point. Our technique is based on the nonlinear steepest‐descent method for matrix Riemann‐Hilbert problems and discrete Schlesinger isomonodromic transformations. © 2013 Wiley Periodicals, Inc.