On the regularity of a free boundary near contact points with a fixed boundary

We investigate the regularity of a free boundary near contact points with a fixed boundary, with C1,1 boundary data, for an obstacle-like free boundary problem. We will show that under certain assumptions on the solution, and the boundary function, the free boundary is uniformly C¹ up to the fixed boundary. We will also construct some examples of irregular free boundaries.     

Truncated Toeplitz Operators and Boundary Values in Nearly Invariant Subspaces

We consider truncated Toeplitz operator on nearly invariant subspaces of the Hardy space H ². Of some importance in this context is the boundary behavior of the functions in these spaces which we will discuss in some detail.     

On Meromorphically Normal Families of Meromorphic Mappings of Several Complex Variables into P(C)

We shall prove some meromorphic normality criteria for families of meromorphic mappings of several complex variables into P^N(C), the complex N-dimensional projective space, related to Nochka's Picard type theorems. Some related results (e.g., extension theorems, improved normality criteria, and quasi-normality criteria) will be obtained also. The technique in this paper mainly depends on Stoll's normality criteria for families of non-negative divisors on a domain of Cⁿ.     

Hyperelliptic functions solutions of some nonlinear partial differential equations using the direct method

In this paper, we will use a simple and direct method to obtain particular solutions of some (2+1) dimensional nonlinear partial differential equations expressed in terms of the Kleinian hyperelliptic functions for a given curve y²=f(x) whose genus is two. We observe that this method generalizes the auxiliary method.     

The L resolvents of elliptic operators with uniformly continuous coefficients

We consider the strongly elliptic operator A of order 2m in the divergence form with bounded measurable coefficients and assume that the coefficients of top order are uniformly continuous. For 1<p<∞, A is a bounded linear operator from the L^p Sobolev space H^m,p into H^−m,p. We will prove that (A−λ)⁻¹ exists in H^−m,p for some λ and estimate its operator norm.     

Dynamic behaviour for a nonautonomous heroin epidemic model with time delay

In this paper, we have modified the White and Comiskey heroin epidemic model (White and Comiskey in Math. Biosci. 208:312–324, 2007) into a nonautonomous heroin epidemic model with distributed time delay. We have introduced some new threshold values R _* and R ^* and further obtained that the heroin-using career will be permanent when R _*>1 and the heroin-using career will be going to extinct when R ^*<1. Using the method of Lyapunov functional, some sufficient conditions are derived for the global asymptotic stability of the system. The aim of this modification is to identify the parameters of interest for further study, with a view to informing and assisting policy-maker in targeting prevention and treatment resources for maximum effectiveness.     

On Parseval super-frame wavelets

Suppose that η1,...,ηn are measurable functions in L2(R).We call the n-tuple (η1,…,ηn) a Parseval super frame wavelet of length n if {2k/2η1(2kt-)（@）...(@)2k/2ηn(2kt-l):k,l∈Z}is a Parseval frame for L2...     

The Laplace operator on a hyperbolic manifold I. Spectral and scattering theory

Using techniques of stationary scattering theory for the Schrödinger equation, we show absence of singular spectrum and obtain incoming and outgoing spectral representations for the Laplace-Beltrami operator on manifolds Mⁿ arising as the quotient of hyperbolic n-dimensional space by a geometrically finite, discrete group of hyperbolic isometries. We consider manifolds Mⁿ of infinite volume. In subsequent papers, we will use the techniques developed here to analytically continue Eisenstein series for a large class of discrete groups, including some groups with parabolic elements.     

Characterization of some finite groups by order and length of one conjugacy class

We study the possibility of characterizing S ∈ {²D_n(2), ²D_n+1(2)} by simple conditions when 2ⁿ⁺¹ > 5 is a prime. Furthermore, we will show that Thompson’s conjecture is valid under some weak condition for these groups.     

Hyperbolicity of arborescent tangles and arborescent links

In this paper, we study the hyperbolicity of arborescent tangles and arborescent links. We will explicitly determine all essential surfaces in arborescent tangle complements with non-negative Euler characteristic, and show that given an arborescent tangle T, the complement X(T) is non-hyperbolic if and only if T is a rational tangle, T=Qm*T^′ for some m?1, or T contains Qn for some n?2. We use these results to prove a theorem of Bonahon and Siebenmann which says that a large arborescent link L is non-hyperbolic if and only if it contains Q₂.     

Hecke Eigenfunctions of Quantized Cat Maps Modulo Prime Powers

This paper continues the work done in Olofsson [Commun Math Phys 286(3):1051–1072, 2009] about the supremum norm of eigenfunctions of desymmetrized quantized cat maps. N will denote the inverse of Planck’s constant and we will see that the arithmetic properties of N play an important role. We prove the sharp estimate ||ψ||_∞ = O(N ^1/4) for all normalized eigenfunctions and all N outside of a small exceptional set. We are also able to calculate the value of the supremum norms for most of the so called newforms. For a given N = p ⁿ , with n > 2, the newforms can be divided in two parts (leaving out a small number of them in some cases), the first half all have supremum norm about ${2/\sqrt{1\pm 1/p}}This paper continues the work done in Olofsson [Commun Math Phys 286(3):1051–1072, 2009] about the supremum norm of eigenfunctions of desymmetrized quantized cat maps. N will denote the inverse of Planck’s constant and we will see that the arithmetic properties of N play an important role. We prove the sharp estimate ||ψ||_∞ = O(N ^1/4) for all normalized eigenfunctions and all N outside of a small exceptional set. We are also able to calculate the value of the supremum norms for most of the so called newforms. For a given N = p ⁿ , with n > 2, the newforms can be divided in two parts (leaving out a small number of them in some cases), the first half all have supremum norm about 2/?{1±1/p}{2/\sqrt{1\pm 1/p}} and the supremum norm of the newforms in the second half have at most three different values, all of the order N ^1/6. The only dependence of A is that the normalization factor is different if A has eigenvectors modulo p or not. We also calculate the joint value distribution of the absolute value of n different newforms.     

Asymptotic properties in semiparametric partially linear regression models for functional data

We consider the semiparametric partially linear regression models with mean function XTβ + g(z), where X and z are functional data. The new estimators of β and g(z) are presented and some asymptotic results are given. The strong convergence rates of the proposed estimators are obtained. In our estimation, the observation number of each subject will be completely flexible. Some simulation study is conducted to investigate the finite sample performance of the proposed estimators.     

The shadowing chain lemma for singular Hamiltonian systems involving strong forces

We consider a planar autonomous Hamiltonian system :q+?V(q) = 0, where the potential V: ?² \{??}?? ? has a single well of infinite depth at some point ?? and a strict global maximum 0at two distinct points a and b. Under a strong force condition around the singularity ?? we will prove a lemma on the existence and multiplicity of heteroclinic and homoclinic orbits ?? the shadowing chain lemma ?? via minimization of action integrals and using simple geometrical arguments.     

Chow groups of surfaces with h2,0⩽1

We will investigate the geometry of rational equivalence classes of points on a surface S. We will show that if S is a general projective K3 surface then these equivalence classes are dense in the complex topology. We will also show that if S has the property that these equivalence classes are Zariski dense, then h^2,0(S)?1. To cite this article: C. Maclean, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Restrictions on the zeros of a polynomial as a consequence of conditions on the coefficients of even powers and odd powers of the variable

The classical Eneström–Kakeya Theorem states that if p(z)=∑_v=0ⁿa_vz^v is a polynomial satisfying 0⩽a₀⩽a₁⩽⋯⩽a_n, then all of the zeros of p(z) lie in the region |z|⩽1 in the complex plane. Many generalizations of the Eneström–Kakeya theorem exist which put various conditions on the coefficients of the polynomial (such as monotonicity of the moduli of the coefficients). We will introduce several results which put conditions on the coefficients of even powers of z and on the coefficients of odd powers of z. As a consequence, our results will be applicable to some polynomials to which these related results are not applicable.     

Asymptotic dynamics of nonlinear Schrödinger equations with many bound states

We consider a nonlinear Schrödinger equation with a bounded localized potential in . The linear Hamiltonian is assumed to have three or more bound states with the eigenvalues satisfying some resonance conditions. Suppose that the initial data is localized and small of order n in H¹, and that its ground state component is larger than n^3−ε with ε>0 small. We prove that the solution will converge locally to a nonlinear ground state as the time tends to infinity.     

Asymptotic centers and best compact approximation of operators intoC(K)

The existence of best compact approximations for all bounded linear operators fromX intoC(K) is related to the behavior of asymptotic centers inX ^*. IfK is just one convergent sequence, the condition is that everyω ^*-convergent sequence inX ^* will have an asymptotic center. We first study this property, solving some open problems in the theory of asymptotic centers. IfK is more “complex,” the asymptotic centers should behave “continuously.” We use this observation to construct operators fromC[0,1] intoC(ω ²) and from ?₁ intoL ₁ without best compact approximation. We also construct spacesX ₁,X ₂, isomorphic to a Hilbert space, and operatorsT ₁,∶X ₁→C(ω ²),T ₂∶?₁→X ₂ without best compact approximations.     

Multipotent GBL-algebras

We investigate GBL-algebras with the property that for every element a there is a positive integer n such that a ⁿ = a ⁿ⁺¹. We show that varieties of GBL-algebras generated by such algebras are commutative. We also present examples showing that other conditions forcing commutativity will be hard to come by.     

The method of infinite ascent applied on A 4 ± nB 3 = C 2

Each of the Diophantine equations A ⁴ ± nB ³ = C ² has an infinite number of integral solutions (A,B,C) for any positive integer n. In this paper, we will show how the method of infinite ascent could be applied to generate these solutions. We will investigate the conditions when A, B and C are pair-wise co-prime. As a side result of this investigation, we will show a method of generating an infinite number of co-prime integral solutions (A,B,C) of the Diophantine equation aA ³ +cB ³ = C ² for any co-prime integer pair (a, c).     

Some weighted estimates for Littlewood-Paley functions and radial multipliers

We prove some weighted estimates for certain Littlewood-Paley operators on the weighted Hardy spaces H_w^p (0<p?1) and on the weighted L^p spaces. We also prove some weighted estimates for the Bochner-Riesz operators and the spherical means.