The normal subgroup structure of the extended Hecke groups

We consider the extended Hecke groups generated by T(z) = −1/z, S(z) = −1/(z + λ) and R(z) = 1/z with λ ≥ 2. In this paper, firstly, we study the fundamental region of the extended Hecke groups . Then, we determine the abstract group structure of the commutator subgroups , the even subgroup , and the power subgroups of the extended Hecke groups . Also, finally, we give some relations between them.     

Lifting Unitary Elements with Application to Approximately Inner Automorphisms

Let A be a separable unital nuclear simple C＊-algebra with torsion K0 （A）, free K1 （A） and with the UCT. Let T ： A→M（K）/K be a unital homomorphism. We prove that every unitary element in the commutant of T（A） is an exponent, thus it is liftable. We also prove that each automorphism α on E with α ∈ Aut0（A） is approximately inner, where E is a unital essential extension of A by K and α is the automorphism on A induced by α.     

Wavelets and Geometric Structure for Function Spaces

Abstract With Littlewood–Paley analysis, Peetre and Triebel classified, systematically, almost all the usual function spaces into two classes of spaces: Besov spaces and Triebel–Lizorkin spaces ; but the structure of dual spaces of is very different from that of Besov spaces or that of Triebel–Lizorkin spaces, and their structure cannot be analysed easily in the Littlewood–Paley analysis. Our main goal is to characterize in tent spaces with wavelets. By the way, some applications are given: (i) Triebel–Lizorkin spaces for p = ∞ defined by Littlewood–Paley analysis cannot serve as the dual spaces of Triebel–Lizorkin spaces for p = 1; (ii) Some inclusion relations among these above spaces and some relations among and L ¹ are studied. Supported by NNSF of China (Grant No. 10001027)     

Hyperbolic Mapping Classes and Their Lifts on the Bers Fiber Space

Let S be a Riemann surface with genus p and n punctures. Assume that 3p - 3 n ＞ 0 and n ≥ 1. Let a be a puncture of S and let (～S) = S ∪ {a}. Then all mapping classes in the mapping class group Mods that fixes the puncture a can be projected to mapping classes of Mod(～S) under the forgetful map. In this paper the author studies the mapping classes in Mods that can be projected to a given hyperbolic mapping class in Mod(～S).     

Some generalizations of Knopp's identity*

For integers a, b and n > 0, define

Analytic Extension of Functions from Analytic Hilbert Spaces

Let M be an invariant subspace of Hv2. It is shown that for each f∈M⊥, f can be analytically extended across (?)Bd\σ(Sz1,…, Szd).     

Invariant Subspaces for Banach Space Operators with a Multiply Connected Spectrum

We consider a multiply connected domain where denotes the unit disk and denotes the closed disk centered at with radius r _j for j = 1, . . . , n. We show that if T is a bounded linear operator on a Banach space X whose spectrum contains ∂Ω and does not contain the points λ₁, λ₂, . . . , λ _n , and the operators T and r _j (T − λ _j I)⁻¹ are polynomially bounded, then there exists a nontrivial common invariant subspace for T ^* and (T − λ _j I)^*-1.     

Ultraproducts of real interpolation spaces between L p -spaces

Let , be a family of compatible couples of L^p-spaces. We show that, given a countably incomplete ultrafilter in , the ultraproduct of interpolation spaces defined by the real method is isomorphic to the direct sum of an interpolation space of type , an intermediate K?the space between and being a purely atomic measure space, and a K?the function space K(Ω₃) defined on some purely non atomic measure space (Ω₃, ν₃) in such a way that Ω₂ ∪ Ω₃ ≠∅. The research of first and third authors is partially supported by the MEC and FEDER project MTM2004-02262 and AVCIT group 03/050.     

Polar Sets and Relative Dimension for Generalized Brownian Sheet

Let{(t);t∈R_ ~N}be a d-dimensional N-parameter generalized Brownian sheet.Necessaryand sufficient conditions for a compact set E×F to be a polar set for(t,(t))are proved.It is also provedthat if 2N≤αd,then for any compact set ER_>~N,d-2/2 Dim E≤inf{dimF:F ∈ B(R~d),P{(E)∩F≠φ}>0}≤d-2/β DimE,and if 2N>αd,then for any compact set FR~d\{0},α/2(d-DimF)≤inf{dimE:E∈B(R_>~N),P{(E)∩F≠φ}>0}≤β/2(d-DimF),where B(R~d)and B(R_>~N)denote the Borel σ-algebra in R~d and in R_>~N respectively,dim and Dim are Hausdorffdimension and Packing dimension respectively.     

Existence and Multiplicity Results for a Degenerate Elliptic Equation

The goal of this paper is to study the multiplicity result of positive solutions of a class of degenerate elliptic equations. On the basis of the mountain pass theorems and the sub- and supersolutions argument for p-Laplacian operators, under suitable conditions on the nonlinearity f（x, s）, we show the following problem：-△pu=λu^α-a（x）u^q in Ω,u│δΩ=0 possesses at least two positive solutions for large λ, where Ω is a bounded open subset of R^N, N ≥ 2, with C^2 boundary, λ is a positive parameter, Ap is the p-Laplacian operator with p 〉 1, α, q are given constants such that p - 1 〈α 〈 q, and a（x） is a continuous positive function in Ω^-.     

Effective Reducibility of a Class of Linear Differential Equations with Quasiperiodic Coefficients

In this paper we consider the effective reducibility of the following linear differentialequation: x = (A ∈Q(t,∈))x, |∈| ≤ ∈0, where A is a constant matrix, Q(t,e) is quasiperiodic in t, and e is a small perturbation parameter. We prove that if the eigenvalues of A and the basic frequencies of Q satisfy some non-resonant conditions, the linear differential equation can be reduced to y = (A^*(∈) R^*(t, ∈))y, |∈| ≤ ∈o, where R^* is exponentially small in ∈.     

A Geometric Problem and the Hopf Lemma. Ⅱ

Abstract A classical result of A. D. Alexandrov states that a connected compact smooth n-dimensional manifold without boundary, embedded in ℝⁿ⁺¹, and such that its mean curvature is constant, is a sphere. Here we study the problem of symmetry of M in a hyperplane X_n+1 =constant in case M satisfies: for any two points (X′,X_n+1), on M, with , the mean curvature at the first is not greater than that at the second. Symmetry need not always hold, but in this paper, we establish it under some additional conditions. Some variations of the Hopf Lemma are also presented. Several open problems are described. Part I dealt with corresponding one dimensional problems. (Dedicated to the memory of Shiing-Shen Chern) * Partially supported by NSF grant DMS-0401118.     

Blow-up criterion for 2-D Boussinesq equations in bounded domain

We extend the results for 2-D Boussinesq equations from ℝ² to a bounded domain Ω. First, as for the existence of weak solutions, we transform Boussinesq equations to a nonlinear evolution equation U _t + A(t, U) = 0. In stead of using the methods of fundamental solutions in the case of entire ℝ², we study the qualities of F(u, υ) = (u · ▽)υ to get some useful estimates for A(t, U), which helps us to conclude the local-in-time existence and uniqueness of solutions. Second, as for blow-up criterions, we use energy methods, Sobolev inequalities and Gronwall inequality to control and by and . Furthermore, can control by using vorticity transportation equations. At last, can control . Thus, we can find a blow-up criterion in the form of .      

Multilinear Singular Integrals with Rough Kernel

For a class of multilinear singular integral operators T _A ,

Boundedness of Some Marcinkiewicz Integral Operators Related to Higher Order Commutators on Hardy Spaces

In this paper, the authors study the boundedness properties of μΩ↑m,b generated by the function b ∈Lipβ（R^n）（0 〈β≤ 1/m） and the Marcinkiewicz integrals operator μΩ. The boundednesses are established on the Hardy type spaces Hb^m^p,n（R^n） and the Herz Hardy type spaces Hbm Kq^α,p（R^b）.     

A Nontrivial Product of Filtration s ＋ 5 in the Stable Homotopy of Spheres

In this paper, some groups Ext A^s.t （Zp, Zp） with specialized s and t are first computed by the May spectrM sequence. Then we make use of the Adams spectral sequence to prove the existence of a new nontrivial family of filtration s＋5 in the stable homotopy groups of spheres πpnq＋（s＋3）pq＋（s＋1）q-5S which is represented （up to a nonzero scalar） by β＋2bohh∈ExtA^s＋5,P^nq＋（n＋3）pq＋（n＋1）q＋s（Zp, Zp） in the Adams spectral sequence, where p ≥ 5 is a prime number, n ≥3, 0≤ s 〈 p - 3, q = 2（p - 1）.     

Supersaturation For Ramsey-Turán Problems

For an l-graph , the Turán number is the maximum number of edges in an n-vertex l-graph containing no copy of . The limit is known to exist [8]. The Ramsey–Turán density is defined similarly to except that we restrict to only those with independence number o(n). A result of Erdős and Sós [3] states that as long as for every edge E of there is another edge E′of for which |E∩E′|≥2. Therefore a natural question is whether there exists for which . Another variant proposed in [3] requires the stronger condition that every set of vertices of of size at least εn (0<ε<1) has density bounded below by some threshold. By definition, for every . However, even is not known for very many l-graphs when l>2. We prove the existence of a phenomenon similar to supersaturation for Turán problems for hypergraphs. As a consequence, we construct, for each l≥3, infinitely many l-graphs for which . We also prove that the 3-graph with triples 12a, 12b, 12c, 13a, 13b, 13c, 23a, 23b, 23c, abc, satisfies . The existence of a hypergraph satisfying was conjectured by Erdős and Sós [3], proved by Frankl and R?dl [6], and later by Sidorenko [14]. Our short proof is based on different ideas and is simpler than these earlier proofs. * Research supported in part by the National Science Foundation under grants DMS-9970325 and DMS-0400812, and an Alfred P. Sloan Research Fellowship. † Research supported in part by the National Science Foundation under grants DMS-0071261 and DMS-0300529.     

The Asymptotics of the Heat Semigroup for a General Bounded Domain with Mixed Boundary Conditions

Abstract Small-time asymptotics of the trace of the heat semigroup where {μ _ν } are the eigenvalues of the negative Laplacian in the (x ¹, x ²)-plane, is studied for a general bunded domain Ω with a smooth boundary ∂Ω, where a finite number of Dirichlet, Neumann and Robin boundary conditions, on the piecewise smooth parts Γ _i (i = 1, ..., n) of ∂Ω such that , are considered. Some geometrical properties associated with Ω are determined.     

Hilbert-Schmidt Hankel Operators on the Bergman Space of Planar Domains

In this paper we study the problem of the membership of H _ϕ in the Hilbert-Schmidt class, when and Ω is a planar domain. We find a necessary and sufficient condition.We apply this result to the problem of joint membership of H _φ and in the Hilbert-Schmidt class. Using the notion of Berezin Transform and a result of K. Zhu we are able to give a necessary and sufficient condition. Finally, we recover a result of Arazy, Fisher and Peetre on the case with φ holomorphic.