The fixed point property via dual space properties

A Banach space has the weak fixed point property if its dual space has a weak^∗ sequentially compact unit ball and the dual space satisfies the weak^∗ uniform Kadec-Klee property; and it has the fixed point property if there exists ε>0 such that, for every infinite subset A of the unit sphere of the dual space, A∪(−A) fails to be (2−ε)-separated. In particular, E-convex Banach spaces, a class of spaces that includes the uniformly nonsquare spaces, have the fixed point property.     

A W-transform-based criterion for the existence of bounded extensions of E-operators

Let Γ(H) be the symmetric Fock space over a Hilbert space H and ε:H→Γ(H) the exponential mapping. By an E-operator we mean an operator defined on ε(H). For an E-operator A, the composition mapping Φ=A°ε is called its W-transform. In this paper, we obtain a criterion based on the W-transform for checking whether or not an E-operator becomes a bounded linear operator on the Fock space.     

Using Lagrangians of hypergraphs to find non-jumping numbers(II)

Let r?2 be an integer. A real number α∈[0,1) is a jump for r if for any ε>0 and any integer m?r, any r-uniform graph with n>n₀(ε,m) vertices and density at least α+ε contains a subgraph with m vertices and density at least α+c, where c=c(α)>0 does not depend on ε and m. A result of Erd?s, Stone and Simonovits implies that every α∈[0,1) is a jump for r=2. Erd?s asked whether the same is true for r?3. Frankl and Rödl gave a negative answer by showing an infinite sequence of non-jumping numbers for every r?3. However, there are a lot of unknowns on determining whether or not a number is a jump for r?3. In this paper, we find two infinite sequences of non-jumping numbers for r=4, and extend one of the results to every r?4. Our approach is still based on the approach developed by Frankl and Rödl.     

Beta-expansion and continued fraction expansion

For any real number β>1, let ε(1,β)=(ε₁(1),ε₂(1),…,εn(1),…) be the infinite β-expansion of 1. Define . Let x∈[0,1) be an irrational number. We denote by kn(x) the exact number of partial quotients in the continued fraction expansion of x given by the first n digits in the β-expansion of x. If is bounded, we obtain that for all x∈[0,1)?Q,

     

Clique partitions of complements of forests and bounded degree graphs

In this paper, we prove that for any forest F⊂K_n, the edges of E(K_n)?E(F) can be partitioned into O(nlogn) cliques. This extends earlier results on clique partitions of the complement of a perfect matching and of a hamiltonian path in K_n.In the second part of the paper, we show that for n sufficiently large and any ε∈(0,1], if a graph G has maximum degree O(n^1-ε), then the edges of E(K_n)?E(G) can be partitioned into cliques provided there exist certain Steiner systems. Furthermore, we show that there are such graphs G for which Ω(ε²n^2-2ε) cliques are required in every clique partition of E(K_n)?E(G).     

Continuous approximation of random function

Modifying the methods of Lee [J. Math. Anal. Appl.61 (1977), 1–6], we show that each μ-measurable mapping f on a normal space T into a separable linear metric space E is almost continuous, where μ is a Radon probability measure. It is shown that for every ε > 0 there exists a compact subset K_ε ? T with μ(K_ε) > 1 ? ε and an elementary function g(t) = ∑ⁿ_{i = 1}h_i(t) x_i such that μ(t?K_ε; f(t) ≠ g(t)) < ε, where x_i?E and h_i(t) are real bounded continuous functions with disjoint supports.     

Bounding multiplicative energy by the sumset

We prove that the sumset or the productset of any finite set of real numbers, A, is at least |A|^4/3−ε, improving earlier bounds. Our main tool is a new upper bound on the multiplicative energy, E(A,A).     

Strong edge colorings of uniform graphs

For a graph G=(V(G),E(G)), a strong edge coloring of G is an edge coloring in which every color class is an induced matching. The strong chromatic index of G, χ_s(G), is the smallest number of colors in a strong edge coloring of G. The strong chromatic index of the random graph G(n,p) was considered in Discrete Math. 281 (2004) 129, Austral. J. Combin. 10 (1994) 97, Austral. J. Combin. 18 (1998) 219 and Combin. Probab. Comput. 11 (1) (2002) 103. In this paper, we consider χ_s(G) for a related class of graphs G known as uniform or ε-regular graphs. In particular, we prove that for 0<ε?d<1, all (d,ε)-regular bipartite graphs G=(U∪V,E) with |U|=|V|?n₀(d,ε) satisfy χ_s(G)?ζ(ε)Δ(G)², where ζ(ε)→0 as ε→0 (this order of magnitude is easily seen to be best possible). Our main tool in proving this statement is a powerful packing result of Pippenger and Spencer (Combin. Theory Ser. A 51(1) (1989) 24).     

On the unconditional convergence of series with respect to the Faber-Schauder system

We prove that for every 0 < ? < 1, there exists a measurable set E _ε ? [0, 1] with measure |E _ε | > 1??, such that the Faber-Schauder system is an unconditional convergence system for the representation of functions of the class C(E).     

On the value of a random minimum spanning tree problem

Suppose we are given a complete graph on n vertices in which the lenghts of the edges are independent identically distributed non-negative random variables. Suppose that their common distribution function F is differentiable at zero and D = F′ (0) > 0 and each edge length has a finite mean and variance. Let L_n be the random variable whose value is the length of the minimum spanning tree in such a graph. Then we will prove the following: lim_{n → ∞}E(L_n) = ζ(3)/D where ζ(3) = Σ_{k = 1}^∞ 1/k³ = 1.202… and for any ε > 0 lim_{n → ∞} Pr(|L_n^?ζ(3)/D|) > ε) = 0.     

Strong approximation in the Apollonian group

The Apollonian group is a finitely generated, infinite index subgroup of the orthogonal group OQ(Z) fixing the Descartes quadratic form Q. For nonzero v∈Z⁴ satisfying Q(v)=0, the orbits Pv=Av correspond to Apollonian circle packings in which every circle has integer curvature. In this paper, we specify the reduction of primitive orbits Pv mod any integer d>1. We show that this reduction has a multiplicative structure, and that mod primes p?5 it is the full cone of integer solutions to Q(v)≡0 for v?0. This analysis is an essential ingredient in applications of the affine linear sieve as developed by Bourgain, Gamburd and Sarnak.     

Multiple positive solutions of nonhomogeneous elliptic systems with strongly indefinite structure and critical Sobolev exponents

In this paper, we study the existence of multiple positive solutions to some Hamiltonian elliptic systems −Δv=λu+u^p+εf(x), −Δu=μv+v^q+δg(x) in Ω;u,v>0 in Ω; u=v=0 on ∂Ω, where Ω is a bounded domain in R^N (N?3); 0?f, g∈L∞(Ω); 1/(p+1)+1/(q+1)=(N−2)/N, p,q>1; λ,μ>0. Using sub- and supersolution method and based on an adaptation of the dual variational approach, we prove the existence of at least two nontrivial positive solutions for all λ,μ∈(0,λ₁) and ε,δ∈(0,δ₀), where λ₁ is the first eigenvalue of the Laplace operator −Δ with zero Dirichlet boundary conditions and δ₀ is a positive number.     

A note on ball-covering property of Banach spaces

By a ball-covering B of a Banach space X, we mean that B is a collection of open (or closed) balls off the origin whose union contains the unit sphere SX of X; and X is said to have the ball-covering property (BCP) provided it admits a ball-covering by countably many balls. In this note we give a natural example showing that the ball-covering property of a Banach space is not inherited by its subspaces; and we present a sharp quantitative version of the recent Fonf and Zanco renorming result saying that if the dual X^∗ of X is w^∗ separable, then for every ε>0 there exist a (1+ε)-equivalent norm on X, and an R>0 such that in this new norm SX admits a ball-covering by countably many balls of radius R. Namely, we show that R=R(ε) can be taken arbitrarily close to (1+ε)/ε, and that for X=?₁[0,1] the corresponding R cannot be equal to 1/ε. This gives the sharp order of magnitude for R(ε) as ε→0.     

Spectral reciprocity and matrix representations of unbounded operators

We study a family of unbounded Hermitian operators in Hilbert space which generalize the usual graph-theoretic discrete Laplacian. For an infinite discrete set X, we consider operators acting on Hilbert spaces of functions on X, and their representations as infinite matrices; the focus is on ?²(X), and the energy space HE. In particular, we prove that these operators are always essentially self-adjoint on ?²(X), but may fail to be essentially self-adjoint on HE. In the general case, we examine the von Neumann deficiency indices of these operators and explore their relevance in mathematical physics. Finally we study the spectra of the HE operators with the use of a new approximation scheme.     

Two-Phase Stefan Problem as the Limit Case of Two-Phase Stefan Problem with Kinetic Condition

Both one-dimensional two-phase Stefan problem with the thermodynamic equilibrium condition u(R(t),t)=0 and with the kinetic rule u_ε(R_ε(t),t)=εR_ε′(t) at the moving boundary are considered. We prove, when ε approaches zero, R_ε(t) converges to R(t) in C^1+δ/2[0,T] for any finite T>0, 0<δ<1.     

Factorization of delta-monotone linear mappings

For a delta-monotone linear mapping we prove that the factors in the polar decomposition are delta-monotone. Also, we prove that every delta-monotone linear mapping can be factored into a product of (1-ε)-monotone mappings for any ε∈(0,1). As an application in nonlinear case, we give a new proof of the following fact: the quasiconformality constant K(δ,n) of a δ-monotone mapping can be chosen such that K(δ,n) tends to 1 as δ tends to 1.     

Partitioning infinite trees

A graph G is bisectable if its edges can be colored by two colors so that the resulting monochromatic subgraphs are isomorphic. We show that any infinite tree of maximum degree Δ with infinitely many vertices of degree at least Δ −1 is bisectable as is any infinite tree of maximum degree Δ ≤ 4. Further, it is proved that every infinite tree T of finite maximum degree contains a finite subset E of its edges so that the graph T − E is bisectable. To measure how “far” a graph G is from being bisectable, we define c(G) to be the smallest number k > 1 so that there is a coloring of the edges of G by k colors with the property that any two monochromatic subgraphs are isomorphic. An upper bound on c(G), which is in a sense best possible, is presented. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 113–127, 2000     

Coincidence sets in quasilinear elliptic problems of monostable type

This paper concerns the formation of a coincidence set for the positive solution of the boundary value problem: −εΔpu=uq−1f(a(x)−u) in Ω with u=0 on ∂Ω, where ε is a positive parameter, Δpu=div(|∇u|^p−2∇u), 1<q?p<∞, f(s)∼|s|^θ−1s(s→0) for some θ>0 and a(x) is a positive smooth function satisfying Δpa=0 in Ω with infΩ|∇a|>0. It is proved in this paper that if 0<θ<1 the coincidence set Oε={x∈Ω:uε(x)=a(x)} has a positive measure for small ε and converges to Ω with order O(ε1/p) as ε→0. Moreover, it is also shown that if θ?1, then Oε is empty for any ε>0. The proofs rely on comparison theorems and the energy method for obtaining local comparison functions.     

On oscillatory solutions of certain forced Emden-Fowler like equations

We give a constructive proof of existence to oscillatory solutions for the differential equations x^″(t)+a(t)λ|x(t)|sign[x(t)]=e(t), where t?t₀?1 and λ>1, that decay to 0 when t→+∞ as O(t−μ) for μ>0 as close as desired to the “critical quantity” . For this class of equations, we have limt→+∞E(t)=0, where E(t)<0 and E^″(t)=e(t) throughout [t₀,+∞). We also establish that for any μ>μ^? and any negative-valued E(t)=o(t−μ) as t→+∞ the differential equation has a negative-valued solution decaying to 0 at + ∞ as o(t−μ). In this way, we are not in the reach of any of the developments from the recent paper [C.H. Ou, J.S.W. Wong, Forced oscillation of nth-order functional differential equations, J. Math. Anal. Appl. 262 (2001) 722-732].     

On solutions to the Degasperis-Procesi equation

In this paper we consider a new integrable equation (the Degasperis-Procesi equation) derived recently by Degasperis and Procesi (1999) [3]. Analogous to the Camassa-Holm equation, this new equation admits blow-up phenomenon and infinite propagation speed. First, we give a proof for the blow-up criterion established by Zhou (2004) in [12]. Then, infinite propagation speed for the Degasperis-Procesi equation is proved in the following sense: the corresponding solution u(x,t) with compactly supported initial datum u₀(x) does not have compact x-support any longer in its lifespan. Moreover, we show that for any fixed time t>0 in its lifespan, the corresponding solution u(x,t) behaves as: u(x,t)=L(t)e−x for x?1, and u(x,t)=l(t)ex for x?−1, with a strictly increasing function L(t)>0 and a strictly decreasing function l(t)<0 respectively.