A topological degree counting for some Liouville systems of mean field type

Let A = (a_ij)_{n × n} be an invertible matrix and A⁻¹ = (a^ij)_{n × n} be the inverse of A. In this paper, we consider the generalized Liouville system (0.1) where 0 < h_j ∈ C¹(M) and \input amssym $\rho_j \in \Bbb R^+$ , and prove that, under the assumptions of (H₁) and (H₂) (see Introduction), the Leray‐Schauder degree of (0.1) is equal to if ρ = (ρ₁, …, ρ_n) satisfies Equation (0.1) is a natural generalization of the classic Liouville equation and is the Euler‐Lagrangian equation of the nonlinear function Φ_ρ: The Liouville system (0.1) has arisen in many different research areas in mathematics and physics. Our counting formulas are the first result in degree theory for Liouville systems. © 2010 Wiley Periodicals, Inc.     

Sharp estimates for solutions of multi‐bubbles in compact Riemann surfaces

In this paper, we consider a sequence of multibubble solutions u_k of the equation (0.1) where h is a C^2,β positive function in a compact Riemann surface M, and ρ_k is a constant satisfying lim_k→+∞ ρ_k = 8mπ for some positive integer m ≥ 1. We prove among other things that where p_k,j are centers of the bubbles of u_k and λ_k,j are the local maxima of u_k after adding a constant. This yields a uniform bound of solutions as ρ_k converges to 8mπ from below provided that . It generalizes a previous result, due to Ding, Jost, Li, and Wang [18] and Nolasco and Tarantello [31], hich says that any sequence of minimizers u_k is uniformly bounded if ρ_k > 8π and h satisfies for any maximum point p of the sum of 2 log h and the regular part of the Green function, where K is the Gaussian curvature of M. The analytic work of this paper is the first step toward computing the topological degree of ( 0.1 ), which was initiated by Li [24]. © 2002 Wiley Periodicals, Inc.     

Obstacle Problems with Linear Growth: Hölder Regularity for the Dual Solution

For a strictly convex integrand f : ℝⁿ → ℝ with linear growth we discuss the variational problem among mappings u : ℝⁿ ⊃ Ω → ℝ of Sobolev class W¹₁ with zero trace satisfying in addition u ≥ ψ for a given function ψ such that ψ|_∂Ω < 0. We introduce a natural dual problem which admits a unique maximizer σ. In further sections the smoothness of σ is investigated using a special J-minimizing sequence with limit u* ∈ C^1,α (Ω) for which the duality relation holds.     

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.     

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)     

Classification of solutions for an integral equation

Let n be a positive integer and let 0 < α < n. Consider the integral equation We prove that every positive regular solution u(x) is radially symmetric and monotone about some point and therefore assumes the form with some constant c = c(n, α) and for some t > 0 and x₀ ? ?ⁿ. This solves an open problem posed by Lieb 12 . The technique we use is the method of moving planes in an integral form, which is quite different from those for differential equations. From the point of view of general methodology, this is another interesting part of the paper. Moreover, we show that the family of well‐known semilinear partial differential equations is equivalent to our integral equation (0.1), and we thus classify all the solutions of the PDEs. © 2005 Wiley Periodicals, Inc.     

A limiting distribution for the number of cuts needed to isolate the root of a random recursive tree

Let X_n be the number of cuts needed to isolate the root in a random recursive tree with n vertices. We provide a weak convergence result for X_n. The basic observation for its proof is that the probability distributions of are recursively defined by , where D_n is a discrete random variable with ? , which is independent of . This distributional recursion was not studied previously in the sense of weak convergence. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

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     

Mean Ramsey–Turán numbers

A ρ‐mean coloring of a graph is a coloring of the edges such that the average number of colors incident with each vertex is at most ρ. For a graph H and for ρ ≥ 1, the mean Ramsey–Turán number RT(n, H,ρ ? mean) is the maximum number of edges a ρ‐mean colored graph with n vertices can have under the condition it does not have a monochromatic copy of H. It is conjectured that where is the maximum number of edges a k edge‐colored graph with n vertices can have under the condition it does not have a monochromatic copy of H. We prove the conjecture holds for . We also prove that . This result is tight for graphs H whose clique number equals their chromatic number. In particular, we get that if H is a 3‐chromatic graph having a triangle then . © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 126–134, 2006     

On the Lp–Lq maximal regularity for the linear thermoelastic plate equation in a bounded domain

This paper is concerned with the thermoelastic plate equations in a domain Ω: subject to the boundary condition: u|=D_νu|=θ|=0 and initial condition: (u, u_t, θ)|_t=0=(u₀, v₀, θ₀). Here, Ω is a bounded domain in ?ⁿ(n≧2). We assume that the boundary ?Ω of Ω is a C⁴ hypersurface. We obtain an L_p–L_q maximal regularity theorem. Copyright © 2008 John Wiley & Sons, Ltd.     

A uniqueness condition for the polyharmonic equation in free space

Consider the polyharmonic wave equation ?u + (? Δ)^mu = f in ?ⁿ × (0, ∞) with time-independent right-hand side. We study the asymptotic behaviour of u ( x , t) as t → ∞ and show that u( x , t) either converges or increases with order t^α or In t as t → ∞. In the first case we study the limit $ u_0 \left({\bf x} \right) \colone \mathop {\lim }\limits_{t \to \infty } \,u\left({{\bf x},t} \right) $ and give a uniqueness condition that characterizes u₀ among the solutions of the polyharmonic equation ( ? Δ)^mu = f in ?ⁿ. Furthermore we prove in the case 2m ? n that the polyharmonic equation has a solution satisfying the uniqueness condition if and only if f is orthogonal to certain solutions of the homogeneous polyharmonic equation.     

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)     

Giant components in Kronecker graphs

Let \begin{align*}n\in\mathbb{N}\end{align*}, 0 <α,β,γ< 1. Define the random Kronecker graph K(n,α,γ,β) to be the graph with vertex set \begin{align*}\mathbb{Z}_2^n\end{align*}, where the probability that u is adjacent to v is given by p_u,v =α ^{u ? v} γ^{( 1‐u )?( 1‐v )}β^{n‐ u ? v ‐( 1‐u )?( 1‐v )}. This model has been shown to obey several useful properties of real‐world networks. We establish the asymptotic size of the giant component in the random Kronecker graph.© 2011 Wiley Periodicals, Inc. Random Struct. Alg.,2011     

Minors in lifts of graphs

We study here lifts and random lifts of graphs, as defined by Amit and Linial (Combinatorica 22 (2002), 1–18). We consider the Hadwiger number η and the Hajós number σ of ?‐lifts of Kⁿ and analyze their extremal as well as their typical values (that is, for random lifts). When ? = 2, we show that , and random lifts achieve the lower bound (as n → ∞). For bigger values of ?, we show . We do not know how tight these bounds are, and in fact, the most interesting question that remains open is whether it is possible for η to be o(n). When ? < O(log n), almost every ?‐lift of Kⁿ satisfies η = Θ(n) and for , almost surely . For bigger values of ?, almost always. The Hajós number satisfies , and random lifts achieve the lower bound for bounded ? and approach the upper bound when ? grows. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

Local approximations and intrinsic characterization of spaces of smooth functions on regular subsets of ℝn

We give an intrinsic characterization of the restrictions of Sobolev (?ⁿ ), Triebel–Lizorkin (?ⁿ ) and Besov (?ⁿ ) spaces to regular subsets of ?ⁿ via sharp maximal functions and local approximations. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Berezin transform in polynomial bergman spaces

Fix a smooth weight function Q in the plane, subject to a growth condition from below. Let K_m,n denote the reproducing kernel for the Hilbert space of analytic polynomials of degree at most n ? 1 of finite L²‐norm with respect to the measure e^?mQ dA. Here dA is normalized area measure, and m is a positive real scaling parameter. The (polynomial) Berezin measure for the point z₀ is a probability measure that defines the (polynomial) Berezin transform for continuous . We analyze the semiclassical limit of the Berezin measure (and transform) as m → + ∞ while n = m τ + o(1), where τ is fixed, positive, and real. We find that the Berezin measure for z₀ converges weak‐star to the unit point mass at the point z₀ provided that Δ Q(z₀) > 0 and that z₀ is contained in the interior of a compact set , defined as the coincidence set for an obstacle problem. As a refinement, we show that the appropriate local blowup of the Berezin measure converges to the standardized Gaussian measure in the plane. For points , the Berezin measure cannot converge to the point mass at z₀. In the model case Q(z) = |z|², when is a closed disk, we find that the Berezin measure instead converges to harmonic measure at z₀ relative to . Our results have applications to the study of the eigenvalues of random normal matrices. The auxiliary results include weighted L²‐estimates for the equation when f is a suitable test function and the solution u is restricted by a polynomial growth bound at ∞. © 2009 Wiley Periodicals, Inc.     

Analytic aspects of the Toda system: I. A Moser‐Trudinger inequality

In this paper, we analyze solutions of the open Toda system and establish an optimal Moser‐Trudinger type inequality for this system. Let Σ be a closed surface with area 1 and K = (a_ij)_{N × N} the Cartan matrix for SU(N + 1), i.e., We show that has a lower bound in (H¹(Σ))^N if and only if This inequality is optimal. As a direct consequence, if M_j < for 4π for j = 1, 2, …, N, Φ_M has a minimizer u that satisfies © 2001 John Wiley & Sons, 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.     

For most large underdetermined systems of linear equations the minimal 𝓁1‐norm solution is also the sparsest solution

We consider linear equations y = Φx where y is a given vector in ?ⁿ and Φ is a given n × m matrix with n < m ≤ τn, and we wish to solve for x ∈ ?^m. We suppose that the columns of Φ are normalized to the unit ??₂‐norm, and we place uniform measure on such Φ. We prove the existence of ρ = ρ(τ) > 0 so that for large n and for all Φ's except a negligible fraction, the following property holds: For every y having a representation y = Φx₀ by a coefficient vector x₀ ∈ ?^m with fewer than ρ · n nonzeros, the solution x₁ of the ??₁‐minimization problem is unique and equal to x₀. In contrast, heuristic attempts to sparsely solve such systems—greedy algorithms and thresholding—perform poorly in this challenging setting. The techniques include the use of random proportional embeddings and almost‐spherical sections in Banach space theory, and deviation bounds for the eigenvalues of random Wishart matrices. © 2006 Wiley Periodicals, Inc.     

Local existence for solutions of fully non-linear wave equations

Let Ω be a domain in ?ⁿ and let m? ?; be given. We study the initial-boundary value problem for the equation with a homogeneous Dirichlet boundary condition; here u is a scalar function, $ \bar D_x^m u: = (\partial _x^\alpha u)_{|\alpha | \le m} $ and certain restrictions are made on F guaranteeing that energy estimates are possible. We prove the existence of a value of T>0 such that a unique classical solution u exists on [0, T]×Ω. Furthermore, we show that T → ∞ if the data tend to zero.