The size‐Ramsey number of trees

Given a graph G, the size‐Ramsey number $\hat r(G)$ is the minimum number m for which there exists a graph F on m edges such that any two‐coloring of the edges of F admits a monochromatic copy of G. In 1983, J. Beck introduced an invariant β(·) for trees and showed that $\hat r(T) = \Omega (\beta (T))$ . Moreover he conjectured that $\hat r(T) = \Theta (\beta (T))$ . We settle this conjecture by providing a family of graphs and an embedding scheme for trees. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011     

Asymptotics of solutions for sub critical non‐convective type equations

We study the Cauchy problem for non‐linear dissipative evolution equations (1) where ?? is the linear pseudodifferential operator and the non‐linearity is a quadratic pseudodifferential operator (2) û ≡ ?_x→ξ u is the Fourier transformation. We consider non‐convective type non‐linearity, that is we suppose that a(t,0,y) ≠ 0. Let the initial data , are sufficiently small and have a non‐zero total mass , where is the weighted Sobolev space. Then we give the main term of the large time asymptotics of solutions in the sub critical case. Copyright © 2004 John Wiley & Sons, Ltd.     

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.     

Families of pairs of graphs with a large number of common cards

The vertex‐deleted subgraph G?v, obtained from the graph G by deleting the vertex v and all edges incident to v, is called a card of G. The deck of G is the multiset of its unlabelled vertex‐deleted subgraphs. The number of common cards of G and H (or between G and H) is the cardinality of the multiset intersection of the decks of G and H. In this article, we present infinite families of pairs of graphs of order n ≥ 4 that have at least \begin{eqnarray*}2\lfloor\frac{1}{3}(n-1)\rfloor\end{eqnarray*} common cards; we conjecture that these, along with a small number of other families constructed from them, are the only pairs of graphs having this many common cards, for sufficiently large n. This leads us to propose a new stronger version of the Reconstruction Conjecture. In addition, we present an infinite family of pairs of graphs with the same degree sequence that have \begin{eqnarray*}\frac{2}{3}(n+5-2\sqrt{3n+6})\end{eqnarray*} common cards, for appropriate values of n, from which we can construct pairs having slightly fewer common cards for all other values of n≥10. We also present infinite families of pairs of forests and pairs of trees with \begin{eqnarray*}2\lfloor\frac{1}{3}(n-4)\rfloor\end{eqnarray*} and \begin{eqnarray*}2\lfloor\frac{1}{3}(n-5)\rfloor\end{eqnarray*} common cards, respectively. We then present new families that have the maximum number of common cards when one graph is connected and the other disconnected. Finally, we present a family with a large number of common cards, where one graph is a tree and the other unicyclic, and discuss how many cards are required to determine whether a graph is a tree. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 146–163, 2010     

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     

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     

The spectrum of 2‐idempotent 3‐quasigroups with conjugate invariant subgroups

A ternary quasigroup (or 3‐quasigroup) is a pair (N, q) where N is an n‐set and q(x, y, z) is a ternary operation on N with unique solvability. A 3‐quasigroup is called 2‐idempotent if it satisfies the generalized idempotent law: q(x, x, y) = q(x, y, x) = q(y, x, x)=y. A conjugation of a 3‐quasigroup, considered as an OA(3, 4, n), $({{N}},{\mathcal{B}})$, is a permutation of the coordinate positions applied to the 4‐tuples of ${\mathcal{B}}$. The subgroup of conjugations under which $({{N}},{\mathcal{B}})$ is invariant is called the conjugate invariant subgroup of $({{N}},{\mathcal{B}})$. In this article, we determined the existence of 2‐idempotent 3‐quasigroups of order n, n≡7 or 11 (mod 12) and n≥11, with conjugate invariant subgroup consisting of a single cycle of length three. This result completely determined the spectrum of 2‐idempotent 3‐quasigroups with conjugate invariant subgroups. As a corollary, we proved that an overlarge set of Mendelsohn triple system of order n exists if and only if n≡0, 1 (mod 3) and n≠6. © 2010 Wiley Periodicals, Inc. J Combin Designs 18: 292–304, 2010     

Fundamental solutions of homogeneous fully nonlinear elliptic equations

We prove the existence of two fundamental solutions Φ and of the PDE \input amssym $F(D^2\Phi) = 0 \quad {\rm in} \ {\Bbb{R}}^n \setminus \{ 0 \}$ for any positively homogeneous, uniformly elliptic operator F. Corresponding to F are two unique scaling exponents α*, > −1 that describe the homogeneity of Φ and . We give a sharp characterization of the isolated singularities and the behavior at infinity of a solution of the equation F(D²u) = 0, which is bounded on one side. A Liouville‐type result demonstrates that the two fundamental solutions are the unique nontrivial solutions of F(D²u) = 0 in \input amssym ${\Bbb{R}}^n \setminus \{ 0 \}$ that are bounded on one side in both a neighborhood of the origin as well as at infinity. Finally, we show that the sign of each scaling exponent is related to the recurrence or transience of a stochastic process for a two‐player differential game. © 2010 Wiley Periodicals, Inc.     

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     

Spectral theory for the fractal Laplacian in the context of h‐sets

An h‐set is a nonempty compact subset of the Euclidean n‐space which supports a finite Radon measure for which the measure of balls centered on the subset is essentially given by the image of their radius by a suitable function h. In most cases of interest such a subset has Lebesgue measure zero and has a fractal structure. Let Ω be a bounded C^∞ domain in with Γ ? Ω. Let where (?Δ)^?1 is the inverse of the Dirichlet Laplacian in Ω and tr^Γ is, say, trace type operator. The operator B, acting in convenient function spaces in Ω, is studied. Estimations for the eigenvalues of B are presented, and generally shown to be dependent on h, and the smoothness of the associated eigenfunctions is discussed. Some results on Besov spaces of generalised smoothness on and on domains which were obtained in the course of this work are also presented, namely pointwise multipliers, the existence of a universal extension operator, interpolation with function parameter and mapping properties of the Dirichlet Laplacian. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

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      

Tangency sets in PG(3, q)

A tangency set of PG (d,q) is a set Q of points with the property that every point P of Q lies on a hyperplane that meets Q only in P. It is known that a tangency set of PG (3,q) has at most points with equality only if it is an ovoid. We show that a tangency set of PG (3,q) with , or points is contained in an ovoid. This implies the non‐existence of minimal blocking sets of size , , and of with respect to planes in PG (3,q), and implies the extendability of partial 1‐systems of size , , or to 1‐systems on the hyperbolic quadric . © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 462–476, 2008     

Arc‐chromatic number of digraphs in which every vertex has bounded outdegree or bounded indegree

A k‐digraph is a digraph in which every vertex has outdegree at most k. A ‐digraph is a digraph in which a vertex has either outdegree at most k or indegree at most l. Motivated by function theory, we study the maximum value Φ (k) (resp. ) of the arc‐chromatic number over the k‐digraphs (resp. ‐digraphs). El‐Sahili [3] showed that . After giving a simple proof of this result, we show some better bounds. We show and where θ is the function defined by . We then study in more detail properties of Φ and . Finally, we give the exact values of and for . © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 315–332, 2006     

Littlewood‐Paley functionals on graphs

Let Γ be a graph equipped with a Markov operator P. We introduce discrete fractional Littlewood‐Paley square functionals and prove their ‐boundedness under various geometric assumptions on Γ.     

Compactness of the Canonical Solution Operator of $ \bar \partial $ on Bounded Pseudoconvex Domains

On bounded pseudoconvex domains Ω the orthogonal projection P_q : L²_(p,q) (Ω) → ker _q is given by P_q = Id – S_{q+1 q} = Id – ^*_q+1N_{q+1 q}, where S_q is the canonical solution operator of the ‐equation and N_q is the ‐Neumann operator. We prove a formula for the solution operator S_q restricted on (0, q)‐forms with holomorphic coefficients. And as an application we get a characterization of compactness of the solution operator restricted on (0, q)‐forms with holomorphic coefficients. On general (0, q)‐forms we show that this condition is necessary for compactness of the solution operator.     

Small degree out‐branchings

Using a suitable orientation, we give a short proof of a strengthening of a result of Czumaj and Strothmann 4 : Every 2‐edge‐connected graph G contains a spanning tree T with the property that for every vertex v. As an analogue of this result in the directed case, we prove that every 2‐arc‐strong digraph D has an out‐branching B such that . A corollary of this is that every k‐arc‐strong digraph D has an out‐branching B such that , where . We conjecture that in this case would be the right (and best possible) answer. If true, this would again imply a strengthening of a result from 4 concerning spanning trees with small degrees in k‐connected graphs when k ≥ 2. We prove that for acyclic digraphs the existence of an out‐branching satisfying prescribed bounds on the out‐degrees of each vertex can be checked in polynomial time. A corollary of this is that the existence of arc‐disjoint branchings , , where the first is an out‐branching rooted at s and the second an in‐branching rooted at t, can be checked in polynomial time for the class of acyclic digraphs © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 297–307, 2003     

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.     

On a free boundary problem in electrostatics

Let Ω_i ? ?^N, i = 0, 1, be two bounded separately star-shaped domains such that $ \Omega _0 \supset \bar \Omega _1 $. We consider the electrostatic potential u defined in $ \Omega : = \Omega _0 \backslash \bar \Omega _1 $: The geometry of the two boundary components Γ₀ and Γ₁ is not given, but instead the electrostatic potential u is supposed to satisfy the further boundary conditions Using a best possible maximum principle, we show that this free boundary problem has a unique solution which is radially symmetric.     

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.