Explicit sparse almost‐universal graphs for ${\bf {{\cal G}(n, {k \over n})}}$

For any integer n, let be a probability distribution on the family of graphs on n vertices (where every such graph has nonzero probability associated with it). A graph Γ is ‐almost‐universal if Γ satisifies the following: If G is chosen according to the probability distribution , then G is isomorphic to a subgraph of Γ with probability 1 ‐ . For any p ∈ [0,1], let (n,p) denote the probability distribution on the family of graphs on n vertices, where two vertices u and v form an edge with probability p, and the events {u and v form an edge}; u,v ∈ V (G) are mutually independent. For k ≥ 4 and n sufficiently large we construct a ‐almost‐universal‐graph on n vertices and with O(n)polylog(n) edges, where q = ? ? for such k ≤ 6, and where q = ? ? for k ≥ 7. The number of edges is close to the lower bound of Ω( ) for the number of edges in a universal graph for the family of graphs with n vertices and maximum degree k. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

Strengths and Weaknesses of LH Arithmetic

In this paper we provide a new arithmetic characterization of the levels of the og‐time hierarchy (LH). We define arithmetic classes and that correspond to ‐LOGTIME and ‐LOGTIME, respectively. We break and into natural hierarchies of subclasses and . We then define bounded arithmetic deduction systems ′ whose ‐definable functions are precisely B( ‐LOGTIME). We show these theories are quite strong in that (1) LIOpen proves for any fixed m that , (2) TAC, a theory that is slightly stronger than ′ whose (LH)‐definable functions are LH, proves LH is not equal to ‐TIME(s) for any m> 0, where 2^s ∈L, s(n) ∈ ω(log n), and (3) TAC proves LH ≠ for all k and m. We then show that the theory TAC cannot prove the collapse of the polynomial hierarchy. Thus any such proof, if it exists, must be argued in a stronger systems than ours.     

Phase transition for potentials of high‐dimensional wells

For a potential function that attains its global minimum value at two disjoint compact connected submanifolds N^± in , we discuss the asymptotics, as ? → 0, of minimizers u_? of the singular perturbed functional under suitable Dirichlet boundary data . In the expansion of E _? (u_?) with respect to , we identify the first‐order term by the area of the sharp interface between the two phases, an area‐minimizing hypersurface Γ, and the energy c of minimal connecting orbits between N⁺ and N^?, and the zeroth‐order term by the energy of minimizing harmonic maps into N^± both under the Dirichlet boundary condition on ?Ω and a very interesting partially constrained boundary condition on the sharp interface Γ. © 2012 Wiley Periodicals, Inc.     

Multiplicity results near the principal eigenvalue for boundary‐value problems with periodic nonlinearity

Let us consider the boundary‐value problem where g: ? → ? is a continuous and T ‐periodic function with zero mean value, not identically zero, (λ, a) ∈ ?² and ∈ C [0, π ] with ∫^π ₀ (x) sin x dx = 0. If λ ₁ denotes the first eigenvalue of the associated eigenvalue problem, we prove that if (λ, a) → (λ ₁, 0), then the number of solutions increases to infinity. The proof combines Liapunov–Schmidt reduction together with a careful analysis of the oscillatory behavior of the bifurcation equation. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On (ε,k)‐min‐wise independent permutations

A family of permutations of [n] = {1,2,…,n} is (ε,k)‐min‐wise independent if for every nonempty subset X of at most k elements of [n], and for any x ∈ X, the probability that in a random element π of , π(x) is the minimum element of π(X), deviates from 1/∣X∣ by at most ε/∣X∣. This notion can be defined for the uniform case, when the elements of are picked according to a uniform distribution, or for the more general, biased case, in which the elements of are chosen according to a given distribution D. It is known that this notion is a useful tool for indexing replicated documents on the web. We show that even in the more general, biased case, for all admissible k and ε and all large n, the size of must satisfy as well as This improves the best known previous estimates even for the uniform case. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

Topological degree for a mean field equation on Riemann surfaces

We consider the following mean field equations: (0.1) where M is a compact Riemann surface with volume 1, h is a positive continuous function on M, ρ is a real number, (0.2) and where Ω is a bounded smooth domain, h is a C¹ positive function on Ω, and ρ ∈ ?. Based on our previous analytic work [14], we prove, among other things, that the degree‐counting formula for ( 0.1 ) is given by () for ρ ∈ (8mπ, 8(m + 1)π). © 2003 Wiley Periodicals, Inc.     

On the convergence rate of vanishing viscosity approximations

Given a strictly hyperbolic, genuinely nonlinear system of conservation laws, we prove the a priori bound ‖u(t, ·) ? u^?(t, ·)‖ = O(1)(1 + t) · |ln ?| on the distance between an exact BV solution u and a viscous approximation u^?, letting the viscosity coefficient ? → 0. In the proof, starting from u we construct an approximation of the viscous solution u^? by taking a mollification u * and inserting viscous shock profiles at the locations of finitely many large shocks for each fixed ?. Error estimates are then obtained by introducing new Lyapunov functionals that control interactions of shock waves in the same family and also interactions of waves in different families. © 2004 Wiley Periodicals, Inc.     

Uniqueness of weak solutions in critical space of the 3‐D time‐dependent Ginzburg‐Landau equations for superconductivity

We prove the uniqueness of weak solutions of the 3‐D time‐dependent Ginzburg‐Landau equations for super‐conductivity with initial data (ψ₀, A₀)∈ L² under the hypothesis that (ψ, A) ∈ L^s(0, T; L^r,∞) × (0, T; with Coulomb gauge for any (r, s) and satisfying + = 1, + = 1, ≥ , ≥ and 3 < r ≤ 6, 3 < ≤ ∞. Here L^r,∞ ≡ is the Lorentz space. As an application, we prove a uniqueness result with periodic boundary condition when ψ₀ ∈ , A₀ ∈ L³ (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Tauberian theorem for m‐spherical transforms on the Heisenberg group

In this paper we prove a Tauberian type theorem for the space L ( H _n ). This theorem gives sufficient conditions for a L ( H _n ) submodule J ? L ( H _n ) to make up all of L ( H _n ). As a consequence of this theorem, we are able to improve previous results on the Pompeiu problem with moments on the Heisenberg group for the space L^∞( H _n ). In connection with the Pompeiu problem, given the vanishing of integrals ∫ z ^m L _g f ( z , 0) dσ ( z ) = 0 for all g ∈ H _n and i = 1, 2 for appropriate radii r₁ and r₂, we now have the (improved) conclusion f ≡ 0, where = · · · and form the standard basis for T^(0,1)( H _n ). (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Circular choosability via combinatorial Nullstellensatz

A p‐list assignment L of a graph G assigns to each vertex v of G a set of permissible colors. We say G is L‐(P, q)‐colorable if G has a (P, q)‐coloring h such that h(v) ? L(v) for each vertex v. The circular list chromatic number of a graph G is the infimum of those real numbers t for which the following holds: For any P, q, for any P‐list assignment L with , G is L‐(P, q)‐colorable. We prove that if G has an orientation D which has no odd directed cycles, and L is a P‐list assignment of G such that for each vertex v, , then G is L‐(P, q)‐colorable. This implies that if G is a bipartite graph, then , where is the maximum average degree of a subgraph of G. We further prove that if G is a connected bipartite graph which is not a tree, then . © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 190–204, 2008     

Blowup for nonlinear wave equations describing boson stars

We consider the nonlinear wave equation modeling the dynamics of (pseudorelativistic) boson stars. For spherically symmetric initial data, u₀(x) ∈ C (?³), with negative energy, we prove blowup of u(t, x) in the H^1/2‐norm within a finite time. Physically this phenomenon describes the onset of “gravitational collapse” of a boson star. We also study blowup in external, spherically symmetric potentials, and we consider more general Hartree‐type nonlinearities. As an application, we exhibit instability of ground state solitary waves at rest if m = 0. © 2007 Wiley Periodicals, Inc.     

Infinite games in the Cantor space and subsystems of second order arithmetic

In this paper we study the determinacy strength of infinite games in the Cantor space and compare them with their counterparts in the Baire space. We show the following theorems: 1. RCA₀ ? ‐Det* ? ‐Det* ? WKL₀. 2. RCA₀ ? ( )2‐Det* ? ACA₀. 3. RCA₀ ? ‐Det* ? ‐Det* ? ‐Det ? ‐Det ? ATR₀. 4. For 1 < k < ω, RCA₀ ? ( )_k ‐Det* ? ( )_{k –1}‐Det. 5. RCA₀ ? ‐Det* ? ‐Det. Here, Det* (respectively Det) stands for the determinacy of infinite games in the Cantor space (respectively the Baire space), and ( )_k is the collection of formulas built from formulas by applying the difference operator k – 1 times. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability of Cahn‐Hilliard fronts

We prove stability of the kink solution of the Cahn‐Hilliard equation ∂_tu = ∂( ∂u − u/2 + u³/2), x ∈ ℝ. The proof is based on an inductive renormalization group method, and we obtain detailed asymptotics of the solution as t → ∞. We prove stability of the kink solution of the Cahn‐Hilliard equation ∂_tu = ∂( ∂u − u/2 + u³/2), x ∈ ℝ. The proof is based on an inductive renormalization group method, and we obtain detailed asymptotics of the solution as t → ∞. © 1999 John Wiley & Sons, Inc.     

On Local Invertible Operators in L2 (ℝ1, H)

We study operators of the form Lu = — G(t) u(t) in L²([t₀ — δ, t₀ + δ], H) with = L² ([t₀— δ, t₀ + δ], H ) in the neighbourhood [t₀— δ, t₀ + δ] of a point t₀ ∈ ℝ¹. Such problems arise in questions on local solvability of partial differential equations (see [6] and [7]). For these operators,one of the major questions is if they are invertible in a neighbourhood of a point t ∈ ℝ¹. To solve this problem we establish needed commutator estimates. Using the commutator estimates and factorization theorems for nonanalytic operator-functions we give additional conditions for the nonanalytic operator -function G(t) and show that the operator L (or ) with some boundary conditions is local invertible.     

Blow‐up estimates for a quasi‐linear reaction–diffusion system

In this paper, some sufficient conditions under which the quasilinear elliptic system ‐div(∣?_u∣^p‐2?_u) = uv, ‐div(∣?_u∣^q‐2?_u) = uv in ?^N(N≥3) has no radially symmetric positive solution is derived. Then by using this non‐existence result, blow‐up estimates for a class of quasilinear reaction–diffusion systems u_t = div (∣?_u∣^p‐2?_u)+uv,vt = div(∣?_v∣^q‐2?_v) +uv with the homogeneous Dirichlet boundary value conditions are obtained. Copyright © 2003 John Wiley & Sons, Ltd.     

Random coloring method in the combinatorial problem of Erdős and Lovász

The work deals with a combinatorial problem of P. Erd?s and L. Lovász concerning simple hypergraphs. Let denote the minimum number of edges in an n‐uniform simple hypergraph with chromatic number at least . The main result of the work is a new asymptotic lower bound for . We prove that for large n and r satisfying the following inequality holds where . This bound improves previously known bounds for . The proof is based on a method of random coloring. We have also obtained results concerning colorings of h‐simple hypergraphs. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

Quasilinear degenerate parabolic equations of Kirchhoff type

We investigate the evolution problem where H is a Hilbert space, A is a self‐adjoint linear non‐negative operator on H with domain D(A), and is a continuous function. We prove that if , and , then there exists at least one global solution, which is unique if either m never vanishes, or m is locally Lipschitz continuous. Moreover, we prove that if for all , then this problem is well posed in H. On the contrary, if for some it happens that for all , then this problem has no solution if with β small enough. We apply these results to degenerate parabolic PDEs with non‐local non‐linearities. Copyright © 1999 John Wiley & Sons, Ltd.     

Ryser's embedding problem for Hadamard matrices

What is the minimum order of a Hadamard matrix that contains an a by b submatrix of all 1's? Newman showed that where c^? denotes the smallest order greater than or equal to c for which a Hadamard matrix exists. It follows that if 4 divides both a and b, and if the Hadamard conjecture is true, then . We establish the improved bounds for min {a,b} ≥ 2. The Hadamard conjecture therefore implies that if 4 divides both 2ab and ?a/2? ?b/2?, then (a, b) = 2 · max {?a/2?b, ?b/2?a}. Our lower bound comes from a counting argument, while our upper bound follows from a sub‐multiplicative property of : Improvements in our upper bound occur when suitable conference matrices or Bush‐type Hadamard matrices exist. We conjecture that any (1,?1)‐matrix of size a by b occurs as a submatrix of some Hadamard matrix of order at most . © 2005 Wiley Periodicals, Inc. J Combin Designs     

Blow‐up analysis for a system of heat equations coupled via nonlinear boundary conditions

In this paper, we study a system of heat equations coupled via nonlinear boundary conditions (1) Here p, q>0. We prove that the solutions always blow up in finite time for non‐trivial and non‐negative initial values. We also prove that the blow‐up occurs only on S_R = ?B_R for Ω = B_R = {x ? ?ⁿ:|x|<R}and under some assumptions on the initial values. Copyright © 2007 John Wiley & Sons, Ltd.     

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.