A structure theorem of Dirac-harmonic maps between spheres

For an arbitrary Dirac-harmonic map (φ,ψ) between compact oriented Riemannian surfaces, we shall study the zeros of |ψ|. With the aid of Bochner-type formulas, we explore the relationship between the order of the zeros of |ψ| and the genus of M and N. On the basis, we could clarify all of non-trivial Dirac-harmonic maps from S ² to S ².     

Semisummands and semiideals in banach spaces

Let |·| be a fixed absolute norm onR ². We introduce semi-|·|-summands (resp. |·|-summands) as a natural extension of semi-L-summands (resp.L-summands). We prove that the following statements are equivalent. (i) Every semi-|·|-summand is a |·|-summand, (ii) (1, 0) is not a vertex of the closed unit ball ofR ² with the norm |·|. In particular semi-L ^p-summands areL ^p-summands whenever 1<p≦∞. The concept of semi-|·|-ideal (resp. |·|-ideal) is introduced in order to extend the one of semi-M-ideal (resp.M-ideal). The following statements are shown to be equivalent. (i) Every semi-|·|-ideal is a |·|-ideal, (ii) every |·|-ideal is a |·|-summand, (iii) (0, 1) is an extreme point of the closed unit ball ofR ² with the norm |·|. From semi-|·|-ideals we define semi-|·|-idealoids in the same way as semi-|·|-ideals arise from semi-|·|-summands. Proper semi-|·|-idealoids are those which are neither semi-|·|-summands nor semi-|·|-ideals. We prove that there is a proper semi-|·|-idealoid if and only if (1, 0) is a vertex and (0, 1) is not an extreme point of the closed unit ball ofR ² with the norm |·|. So there are no proper semi-L ^p-idealoids. The paper concludes by showing thatw*-closed semi-|·|-idealoids in a dual Banach space are semi-|·|-summands, so no new concept appears by predualization of semi-|·|-idealoids.     

Bifurcations of resonant solutions and chaos in physical pendulum equation with suspension axis vibrations

This paper is a continuation of ＂Complex Dynamics in Physical Pendulum Equation with Suspension Axis Vibrations＂[1].In this paper,we investigate the existence and the bifurcations of resonant solution for ω0：ω：Ω ≈ 1：1：n,1：2：n,1：3：n,2：1：n and 3：1：n by using second-order averaging method,give a criterion for the existence of resonant solution for ω0：ω：Ω ≈ 1：m：n by using Melnikov＇s method and verify the theoretical analysis by numerical simulations.By numerical simulation,we expose some other interesting dynamical behaviors including the entire invariant torus region,the cascade of invariant torus behaviors,the entire chaos region without periodic windows,chaotic region with complex periodic windows,the entire period-one orbits region;the jumping behaviors including invariant torus behaviors converting to period-one orbits,from chaos to invariant torus behaviors or from invariant torus behaviors to chaos,from period-one to chaos,from invariant torus behaviors to another invariant torus behaviors;the interior crisis;and the different nice invariant torus attractors and chaotic attractors.The numerical results show the difference of dynamical behaviors for the physical pendulum equation with suspension axis vibrations between the cases under the three frequencies resonant condition and under the periodic/quasi-periodic perturbations.It exhibits many invariant torus behaviors under the resonant conditions.We find a lot of chaotic behaviors which are different from those under the periodic/quasi-periodic perturbations.However,we did not find the cascades of period-doubling bifurcation.     

Large centralizers in finite solvable groups

In 1955 R. Brauer and K. A. Fowler showed that ifG is a group of even order >2, and the order |Z(G)| of the center ofG is odd, then there exists a strongly real) elementx∈G−Z whose centralizer satisfies|C _G(x)|>|G|^1/3. In Theorem 1 we show that every non-abeliansolvable groupG contains an elementx∈G−Z such that|C _G(x)|>[G:G′∩Z]^1/2 (and thus|C _G(x)|>|G|^1/3). We also note that if non-abelianG is either metabelian, nilpotent or (more generally) supersolvable, or anA-group, or any Frobenius group, then|C _G(x)|>|G|^1/2 for somex∈G−Z. In Theorem 2 we prove that every non-abelian groupG of orderp ^mqⁿ (p, q primes) contains a proper centralizer of order >|G|^1/2. Finally, in Theorem 3 we show that theaverage |C(x)|, x∈G, is ≧c|G| ^1/3 for metabelian groups, wherec is constant and the exponent 1/3 is best possible.     

Dual space,carleson measure and sequence interpolation onA p (ϕ)(O<p<1)

LetD={z∈Σ:|z|<1} and ϕ be a normal function on [0,1). Forp∈(0,1) such a function ϕ is used to define a Bergman spaceA ^p (ϕ) onD with weight ϕ ^p (|·|)/(1-|·|²). In this paper, the dual space ofA ^p (ϕ) is given, four characteristics of Carleson measure onA ^p (ϕ) are obtained. Moreover, as an application, three sequence interpolation theorems inA ^p (ϕ) are derived. Supported by the Doctoral Program Foundation of Institute of Higher Education, P.R. China.     

Whaley's Theorem for Finite Lattices

Whaley's Theorem on the existence of large proper sublattices of infinite lattices is extended to ordered sets and finite lattices. As a corollary it is shown that every finite lattice L with |L|≥3 contains a proper sublattice S with |S|≥|L|^1/3. It is also shown that that every finite modular lattice L with |L|≥3 contains a proper sublattice S with |S|≥|L|^1/2, and every finite distributive lattice L with |L|≥4 contains a proper sublattice S with |S|≥3/4|L|. This revised version was published online in June 2006 with corrections to the Cover Date.     

The reconstruction theorem for endomorphisms

The reconstruction theorem deals with dynamical systems which are given by a map ψ : M → M together with a read out function 𝒻 : M → ℝ. Restricting to the cases where ψ is a diffeomorphism, it states that for generic (ψ, 𝒻 ) there is a bijection between elements x ∈ M and corresponding sequences (𝒻(x), 𝒻 (ψ(x)), . . . , 𝒻 (ψ ^{k -1}(x))) of k successive observations, at least for k sufficiently big. This statement turns out to be wrong in cases where ψ is an endomorphism. In the present paper we derive a version of this theorem for endomorphisms (and which is equivalent to the original theorem in the case of diffeomorphisms). It justifies, also for dynamical systems given by endomorphisms, the algorithms for estimating dimensions and entropies of attractors from obervations. Received: 20 June 2002     

Small prime solutions of a pair of linear equations in five prime variables

Assume an additional congruent condition on the coefficients. We prove that the pair 5 of linear equations ∑j=1^5 αλjpj = bλ （λ= 1, 2） has solutions in primes pj satisfying pj 〈〈 （｜b1｜＋｜b2｜＋1） maxλ,j ｜αλj｜^2318＋ε. This improves the exponent 79680 without assuming the additional condition of the second author＇s.     

Quantum chaos and fractals with atoms in cavities

We study the coupled translational, electronic, and field dynamics of the combined system “a two-level atom + a single-mode quantized field + a standing-wave ideal cavity”. In the semiclassical approximation with a point-like atom, interacting with the classical field, the dynamics is described by the Heisenberg equations for the atomic and field expectation values which are known to produce semiclassical chaos under appropriate conditions. We derive Hamilton–Schrödinger equations for probability amplitudes and averaged position and momentum of a point-like atom interacting with the quantized field in a standing-wave cavity. They constitute, in general, an infinite-dimensional set of equations with an infinite number of integrals of motion which may be reduced to a dynamical system with four degrees of freedom if the quantized field is supposed to be initially prepared in a Fock state. This system is found to produce semiquantum chaos with positive values of the maximal Lyapunov exponent. At exact resonance, the semiquantum dynamics is regular. At large values of detuning |δ|1, the Rabi atomic oscillations are usually shallow, and the dynamics is found to be almost regular. The Doppler–Rabi resonance, deep Rabi oscillations that may occur at any large value of |δ| to be equal to |αp₀|, is found numerically and described analytically (with α to be the normalized recoil frequency and p₀ the initial atomic momentum). Two gedanken experiments are proposed to detect manifestations of semiquantum chaos in real experiments. It is shown that in the chaotic regime values of the population inversion z_out, measured with atoms after transversing a cavity, are so sensitive to small changes in the initial inversion z_in that the probability of detecting any value of z_out in the admissible interval [−1,1] becomes almost unity in a short time. Chaotic wandering of a two-level atom in a quantized Fock field is shown to be fractal. Fractal-like structures, typical with chaotic scattering, are numerically found in the dependence of the time of exit of atoms from the cavity on their initial momenta.     

Two sided norm estimate of the Bergman projection on <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> spaces

We give some explicit values of the constants C ₁ and C ₂ in the inequality C ₁/sin(π/p) ⩽ |P| _p ⩽ C ₂/sin(π/p) where |P| _p denotes the norm of the Bergman projection on the L ^p space. Partially supported by MNZZS Grant No. ON144010.     

Homoclinic orbits and chaos in discretized perturbed NLS systems: Part II. Symbolic dynamics

Summary In Part I ([9], this journal), Li and McLaughlin proved the existence of homoclinic orbits in certain discrete NLS systems. In this paper, we will construct Smale horseshoes based on the existence of homoclinic orbits in these systems. First, we will construct Smale horseshoes for a general high dimensional dynamical system. As a result, a certain compact, invariant Cantor set Λ is constructed. The Poincaré map on Λ induced by the flow is shown to be topologically conjugate to the shift automorphism on two symbols, 0 and 1. This gives rise to deterministicchaos. We apply the general theory to the discrete NLS systems as concrete examples. Of particular interest is the fact that the discrete NLS systems possess a symmetric pair of homoclinic orbits. The Smale horseshoes and chaos created by the pair of homoclinic orbits are also studied using the general theory. As a consequence we can interpret certain numerical experiments on the discrete NLS systems as “chaotic center-wing jumping.”     

Additive Latin transversals and group rings

LetA={a ₁, …,a _k} and {b ₁, …,b _k} be two subsets of an abelian groupG, k≤|G|. Snevily conjectured that, when |G| is odd, there is a numbering of the elements ofB such thata _i+b _i,1≤i≤k are pairwise distinct. By using a polynomial method, Alon affirmed this conjecture for |G| prime, even whenA is a sequence ofk<|G| elements. With a new application of the polynomial method, Dasgupta, Károlyi, Serra and Szegedy extended Alon’s result to the groupsZ _p ^r andZ _p ^rin the casek<p and verified Snevily’s conjecture for every cyclic group. In this paper, by employing group rings as a tool, we prove that Alon’s result is true for any finite abelianp-group withk<√2p, and verify Snevily’s conjecture for every abelian group of odd order in the casek<√p, wherep is the smallest prime divisor of |G|. This work has been supported partly by NSFC grant number 19971058 and 10271080.     

The subalgebra systems of direct powers

A simple characterization of the subalgebra systems of direct powers of finitary universal algebras on a fixed infinite setA is given. For |I|≥|A| such subalgebra system of anI-power is precisely an algebraic closure systemS onA ^I closed under mutations ofI (which encompass both the precomposition by permutations ofI and allowing the values at specified elements ofI to become unrestricted) and such that each function in the intersection ofS is constant. For |I|<|A| the subalgebra systems ofI-powers are obtained as the restrictions toI of such closure systems on someA ^J withJ⊇I and |J|=|A|. Presented by J. D. Monk.     

Eventually areally meanp-valent functions

Theorems concerning areally meanp-valent functions are extended to eventually areally meanp-valent functions. In particular, suppose is eventually areally meanp-valent in the unit disc,b, c are positive integers,a≧max {p−1, 0}. If |a _n|≦Cn ^α for alln=bm+c,m=1, 2, …, then |a _n|≦C′n ^α for alln. This is a marked extension of results due to Goluzin and to Hayman.     

Systems of equations driven by stable processes

Let Z^j_t, j = 1, . . . , d, be independent one-dimensional symmetric stable processes of index α ∈ (0,2). We consider the system of stochastic differential equations where the matrix A(x)=(A_ij(x))1≤ i, j ≤ d is continuous and bounded in x and nondegenerate for each x. We prove existence and uniqueness of a weak solution to this system. The approach of this paper uses the martingale problem method. For this, we establish some estimates for pseudodifferential operators with singular state-dependent symbols. Let λ₂ > λ₁ > 0. We show that for any two vectors a, b∈ ℝ_d with |a|, |b| ∈ (λ₁, λ₂) and p sufficiently large, the L^p-norm of the operator whose Fourier multiplier is (|u · a|^α - |u · b|^α) / ∑_j=1^d |u_i|^α is bounded by a constant multiple of |a−b|^θ for some θ > 0, where u=(u₁ , . . . , u_d) ∈ ℝ^d. We deduce from this the L^p-boundedness of pseudodifferential operators with symbols of the form ψ(x,u)=|u · a(x)|^α / ∑_j=1^d |u_i|^α, where u=(u₁,...,u_d) and a is a continuous function on ℝ^d with |a(x)|∈ (λ₁, λ₂) for all x∈ ℝ^d. Research partially supported by NSF grant DMS-0244737. Research partially supported by NSF grant DMS-0303310.     

Functions with slowly-growing area and harmonic majorants

Letf be a function holomorphic inU={|z|<1}, and letA(R,f) be the area off(U)∩{|w|<R}, not counting multiplicities. IfA(R,f)=O(R ^γ) asR→∞ for a γ, 0≦γ<2, then the subharmonic function exp |f| ^p has a harmonic majorant inU for eachp, 0<p<2−γ. If 0≦γ<1 further, thene ^f is of Hardy classH ^p for eachp, 0<p<∞.     

Cocompactness and minimizers for inequalities of Hardy–Sobolev type involving N-Laplacian

The paper studies quasilinear elliptic problems in the Sobolev spaces W ^1,p (Ω), ${\Omega\subset{\mathbb R}^N}The paper studies quasilinear elliptic problems in the Sobolev spaces W ^1,p (Ω), W ì \mathbb R^N{\Omega\subset{\mathbb R}^N} , with p = N, that is, the case of Pohozhaev–Trudinger–Moser inequality. Similarly to the case p < N where the loss of compactness in W^1,p(\mathbb R^N){W^{1,p}({\mathbb R}^N)} occurs due to dilation operators u ?t^(N-p)/pu(tx){u {\mapsto}t^{(N-p)/p}u(tx)} , t > 0, and can be accounted for in decompositions of the type of Struwe’s “global compactness” and its later refinements, this paper presents a previously unknown group of isometric operators that leads to loss of compactness in W₀^1,N{W_0^{1,N}} over a ball in \mathbb R^N{{\mathbb R}^N} . We give a one-parameter scale of Hardy–Sobolev functionals, a “p = N”-counterpart of the H?lder interpolation scale, for p > N, between the Hardy functional ò\frac|u|^p|x|^p dx{\int \frac{|u|^p}{|x|^p}\,{\rm d}x} and the Sobolev functional ò|u|^pN/(N-mp)  dx{\int |u|^{pN/(N-mp)} \,{\rm d}x} . Like in the case p < N, these functionals are invariant with respect to the dilation operators above, and the respective concentration-compactness argument yields existence of minimizers for W ^1,N -norms under Hardy–Sobolev constraints.     

Construction of a class of multivariate compactly supported wavelet bases for L 2(? d )

In this paper, for a given d×d expansive matrix M with |detM| = 2, we investigate the compactly supported M-wavelets for L ²(ℝ ^d ). Starting with a pair of compactly supported refinable functions ϕ and [(j)\tilde]\tilde \varphi satisfying a mild condition, we obtain an explicit construction of a compactly supported wavelet ψ such that {2 ^j/2 ψ(M ^j · −k): j ∈ ℤ, k ∈ ℤd} forms a Riesz basis for L ²(ℝ ^d ). The (anti-)symmetry of such ψ is studied, and some examples are also provided.     

Growth and Hölder conditions for the sample paths of Feller processes

Let (A,D(A)) be the infinitesimal generator of a Feller semigroup such that C _c ^∞(ℝ ⁿ )⊂D(A) and A|C _c ^∞(ℝ ⁿ ) is a pseudo-differential operator with symbol −p(x,ξ) satisfying |p(•,ξ)|_∞≤c(1+|ξ|²) and |Imp(x,ξ)|≤c ₀Rep(x,ξ). We show that the associated Feller process {X _t } _{t ≥0} on ℝ ⁿ is a semimartingale, even a homogeneous diffusion with jumps (in the sense of [21]), and characterize the limiting behaviour of its trajectories as t→0 and ∞. To this end, we introduce various indices, e.g., β_∞ ^x :={λ>0:lim _{|ξ|→∞ | x − y |≤2/|ξ|}|p(y,ξ)|/|ξ|^λ=0} or δ_∞ ^x :={λ>0:liminf _{|ξ|→∞ | x − y |≤2/|ξ| |ε|≤1}|p(y,|ξ|ε)|/|ξ|^λ=0}, and obtain a.s. (ℙ ^x ) that lim _{t →0} t ^−1/λ _{s ≤ t} |X _s −x|=0 or ∞ according to λ>β_∞ ^x or λ<δ_∞ ^x . Similar statements hold for the limit inferior and superior, and also for t→∞. Our results extend the constant-coefficient (i.e., Lévy) case considered by W. Pruitt [27]. Received: 21 July 1997 / Revised version: 26 January 1998