On sums of degrees of the partial quotients in continued fraction expansions of Laurent series

For any formal Laurent series with coefficients cn lying in some given finite field, let x=[a₀(x);a₁(x),a₂(x),…] be its continued fraction expansion. It is known that, with respect to the Haar measure, almost surely, the sum of degrees of partial quotients grows linearly. In this note, we quantify the exceptional sets of points with faster growth orders than linear ones by their Hausdorff dimension, which covers an earlier result by J. Wu.     

Large deviation principles for sequences of logarithmically weighted means

In this paper we consider several examples of sequences of partial sums of triangular arrays of random variables {Xn:n?1}; in each case Xn converges weakly to an infinitely divisible distribution (a Poisson distribution or a centered Normal distribution). For each sequence we prove large deviation results for the logarithmically weighted means with speed function . We also prove a sample path large deviation principle for {Xn:n?1} defined by , where σ²∈(0,∞) and {Un:n?1} is a sequence of independent standard Brownian motions.     

On some geometric parameters in Banach spaces

Let X be a normed linear space and be the unit sphere of X. Let , , and J(X)=sup{‖x+y‖∧‖x−y‖}, x and y∈S(X) be the modulus of convexity, the modulus of smoothness, and the modulus of squareness of X, respectively. Let . In this paper we proved some sufficient conditions on δ(?), ρX(?), J(X), E(X), and , where the supremum is taken over all the weakly null sequence xn in X and all the elements x of X for the uniform normal structure.     

Quasi-convex density and determining subgroups of compact abelian groups

For an abelian topological group G, let denote the dual group of all continuous characters endowed with the compact open topology. Given a closed subset X of an infinite compact abelian group G such that w(X)<w(G), and an open neighborhood U of 0 in T, we show that . (Here, w(G) denotes the weight of G.) A subgroup D of G determines G if the map defined by r(χ)=χ?D for , is an isomorphism between and . We prove that

     

Operators on the space of vector-valued totally measurable functions

Let L(X,Y) stand for the space of all bounded linear operators between real Banach spaces X and Y, and let Σ be a σ-algebra of sets. A bounded linear operator T from the Banach space B(Σ,X) of X-valued Σ-totally measurable functions to Y is said to be σ-smooth if ‖T(fn)Y_‖→0 whenever a sequence of scalar functions (‖fn(⋅)X_‖) is order convergent to 0 in B(Σ). It is shown that a bounded linear operator is σ-smooth if and only if its representing measure is variationally semi-regular, i.e., as An↓∅ (here stands for the semivariation of m on A∈Σ). As an application, we show that the space Lσs(B(Σ,X),Y) of all σ-smooth operators from B(Σ,X) to Y provided with the strong operator topology is sequentially complete. We derive a Banach-Steinhaus type theorem for σ-smooth operators from B(Σ,X) to Y. Moreover, we characterize countable additivity of measures in terms of continuity of the corresponding operators .     

Well-posedness of the fourth-order perturbed Schrödinger type equation in non-isotropic Sobolev spaces

In this paper we study the Cauchy problem of the non-isotropically perturbed fourth-order nonlinear Schrödinger type equation: ((x₁,x₂,…,xn)∈Rn, t?0), where a is a real constant, 1?d<n is an integer, g(x,|u|)u is a nonlinear function which behaves like α|u|u for some constant α>0. By using Kato method, we prove that this perturbed fourth-order Schrödinger type equation is locally well-posed with initial data belonging to the non-isotropic Sobolev spaces provided that s₁,s₂ satisfy the conditions: s₁?0, s₂?0 for or for with some additional conditions. Furthermore, by using non-isotropic Sobolev inequality and energy method, we obtain some global well-posedness results for initial data belonging to non-isotropic Sobolev spaces .     

A remark on Ricceri's conjecture for a class of nonlinear eigenvalue problems

Consider the eigenvalue problem : −Δu=λf(x,u) in Ω, u=0 on ∂Ω, where Ω is a bounded smooth domain in RN. Denote by the set of all Carathéodory functions f:Ω×R→R such that for a.e. x∈Ω, f(x,⋅) is Lipschitzian with Lipschitz constant L, f(x,0)=0 and , and denote by (resp. ) the set of λ>0 such that has at least one nonzero classical (resp. weak) solution. Let λ₁ be the first eigenvalue for the Laplacian-Dirichlet problem. We prove that and . Our result is a positive answer to Ricceri's conjecture if use f(x,u) instead of f(u) in the conjecture.     

Haj?asz-Sobolev imbedding and extension

The author establishes some geometric criteria for a Haj?asz-Sobolev -extension (resp. -imbedding) domain of Rn with n?2, s∈(0,1] and p∈[n/s,∞] (resp. p∈(n/s,∞]). In particular, the author proves that a bounded finitely connected planar domain Ω is a weak α-cigar domain with α∈(0,1) if and only if for some/all s∈[α,1) and p=(2−α)/(s−α), where denotes the restriction of the Triebel-Lizorkin space on Ω.     

A bilinear version of Orlicz-Pettis theorem

Given three Banach spaces X, Y and Z and a bounded bilinear map , a sequence x=n(xn)⊆X is called B-absolutely summable if is finite for any y∈Y. Connections of this space with are presented. A sequence x=n(xn)⊆X is called B-unconditionally summable if is finite for any y∈Y and z^∗∈Z^∗ and for any M⊆N there exists xM∈X for which _∑n∈M〈B(xn,y),z^∗〉=〈B(xM,y),z^∗〉 for all y∈Y and z^∗∈Z^∗. A bilinear version of Orlicz-Pettis theorem is given in this setting and some applications are presented.     

Rearrangement of conditionally convergent series on a small set

We consider ideals I of subsets of the set of natural numbers such that for every conditionally convergent series _∑n∈ωan and every there is a permutation such that _∑n∈ωaπr(n)=r and

     

A K-theoretical invariant and bifurcation for a parameterized family of functionals

Let F:={fx:x∈X} be a family of functionals defined on a Hilbert manifold and smoothly parameterized by a compact connected orientable n-dimensional manifold X, and let be a smooth section of critical points of F. The aim of this paper is to give a sufficient topological condition on the parameter space X which detects bifurcation of critical points for F from the trivial branch. Finally we are able to give some quantitative properties of the bifurcation set for perturbed geodesics on semi-Riemannian manifolds.     

Isochronicity for a class of reversible systems

This paper deals with the relation between isochronicity and first integral for a class of reversible systems: , , which associates to the first integral of the form H(x,y)=F(x)y²+G(x). Two necessary and sufficient conditions are given to characterize isochronicity for these systems. Moreover, we apply these results to show that there exists a class of polynomial reversible systems of degree n with isochronous center for any n.     

The oscillation of differential transforms - entire Jacobi expansions

Let L=(1−x²)D²−((β−α)−(α+β+2)x)D with , and . Let f∈C^∞[−1,1], , with normalized Jacobi polynomials and the Cn decrease sufficiently fast. Set Lk=L(Lk−1), k?2. Let ρ>1. If the number of sign changes of (Lkf)(x) in (−1,1) is O(k1/(ρ+1)), then f extends to be an entire function of logarithmic order . For Legendre expansions, the result holds with replaced with .     

An almost sure central limit theorem for products of sums of partial sums under association

Let be a strictly stationary positively or negatively associated sequence of positive random variables with EX₁=μ>0, and VarX₁=σ²<∞. Denote , and γ=σ/μ the coefficient of variation. Under suitable conditions, we show that

     

Systems of orthogonal polynomials arising from the modular j-function

Let be the polynomial whose zeros are the j-invariants of supersingular elliptic curves over . Generalizing a construction of Atkin described in a recent paper by Kaneko and Zagier (Computational Perspectives on Number Theory (Chicago, IL, 1995), AMS/IP 7 (1998) 97-126), we define an inner product on for every . Suppose a system of orthogonal polynomials {P_n,ψ(x)}_n=0^∞ with respect to exists. We prove that if n is sufficiently large and ψ(x)P_n,ψ(x) is p-integral, then over . Further, we obtain an interpretation of these orthogonal polynomials as a p-adic limit of polynomials associated to p-adic modular forms.     

Large deviations for martingales and derivatives

Fix a sequence of positive integers (mn) and a sequence of positive real numbers (wn). Two closely related sequences of linear operators (Tn) are considered. One sequence has given by the Lebesgue derivatives . The other sequence has given by the dyadic martingale when (l−1)/n²?x<l/n² for l=1,…,n². We prove both positive and negative results concerning the convergence of .     

A hypercyclicity criterion with applications

A continuous linear operator on a topological vector space X is called hypercyclic if there is x∈X such that the orbit {Tnx}_n?0 is dense in X. We establish a criterion for hypercyclicity, and study some applications. In particular, we establish hypercyclic left-multipliers on the space L(X,Y) of continuous linear operators between X and Y, provided with the topology of uniform convergence on bounded sets, for some spaces X,Y of holomorphic functions.     

A tripling construction for overlarge sets of KTS

An overlarge set of , denoted by , is a collection {(X?{x},B_x):x∈X}, where X is a (v+1)-set, each (X?{x},B_x) is a and {B_x:x∈X} forms a partition of all triples on X. In this paper, we give a tripling construction for overlarge sets of KTS. Our main result is that: If there exists an with a special property, then there exists an . It is obtained that there exists an for u=2²ⁿ⁻¹−1 or u=qⁿ, where prime power q≡7 (mod 12) and m≥0,n≥1.     

Exact rates in log law for positively associated random variables

Let be a strictly stationary sequence of positively associated random variables with mean zero and finite variance. Set , Mn=maxk?n|Sk|, n?1. Suppose . In this paper, we study the exact convergence rates of a kind of weighted infinite series of , and as ε↘0, respectively.     

Carleson measure problems for parabolic Bergman spaces and homogeneous Sobolev spaces

Let be the space of solutions to the parabolic equation having finite norm. We characterize nonnegative Radon measures μ on having the property , 1≤p≤q<∞, whenever . Meanwhile, denoting by v(t,x) the solution of the above equation with Cauchy data v₀(x), we characterize nonnegative Radon measures μ on satisfying , β∈(0,n), p∈[1,n/β], q∈(0,∞). Moreover, we obtain the decay of v(t,x), an isocapacitary inequality and a trace inequality.