On Ginzburg-Landau Vortices of Superconducting Thin Films

In this paper, we discuss the vortex structure of the superconducting thin films placed in a magnetic field. We show that the global minimizer of the functional modelling the superconducting thin films has a bounded number of vortices when the applied magnetic field hex 〈 Hc1 ＋ K log ｜ logε｜ where Hc1 is the lower critical field of the film obtained by Ding and Du in SIAM J. Math. Anal., 2002. The locations of the vortices are also given.     

ERROR ESTIMATION FOR NUMERICAL METHODS USING THE ULTRA WEAK VARIATIONAL FORMULATION IN MODEL OF NEAR FIELD SCATTERING PROBLEM

In this paper, we investigate the use of ultra weak variational formulation to solve a wave scattering problem in near field optics. In order to capture the sub-scale features of waves, we utilize evanescent wave functions together with plane wave functions to approximate the local properties of the field. We analyze the global convergence and give an error estimation of the method. Numerical examples are also presented to demonstrate the effectiveness of the strategy.     

n-Lie代数的Frattini子代数及非嵌入定理

In this paper,we prove the nonimbedding theorem in nilpotent n-Liealgebras which is an analogue to the nonimbedding theorem of Burnsids in groupsof prime power order.We also study the properties of Frattini subalgebras of n-Liealgebras over the field with characteristic zero,and prove that the Frattini subalgebraof any k-solvable(k≥2)n-Lie algebra is zero.     

On n-Dimensional Matrix Groups

An n×matrix A={a_(ij)} will be called a 2-dimensional matrix of order n. We shall study groups of 2m -dimensional matrices of order n over a field of characteristic O with respect to an associative matrix product and we obtain the dimensions of such matrix groups.     

ASYMPTOTIC ANALYSIS OF LINEARLY ELASTIC SHALLOWSHELLS WITH VARIABLE THICKNESS

g1. IntroductionIn this paper j we identify the two-dimensiona1 model of a shallow shell with variablethickness. More precisely we consider a family of linearly elastic shallow shells with variablethickness. We show that, if the aPplied forces are of specific order of magnitude, the covariaatcomponents of the scaled displacement field converge, as the thickness of the shell goesto zerQ, to a two dimensional problem that constitutes the model of a shallow shell withvariable thickness. The key …     

快速配置法中数值积分的误差控制策略

We propose two error control techniques for numerical integrations in fast multiscale collocation methods for solving Fredholm integral equations of the second kind with weakly singular kernels. Both techniques utilize quadratures for singular integrals using graded points. One has a polynomial order of accuracy if the integrand has a polynomial order of smoothness except at the singular point and the other has exponential order of accuracy if the integrand has an infinite order of smoothness except at the singular point. We estimate the order of convergence and computational complexity of the corresponding approximate solutions of the equation. We prove that the second technique preserves the order of convergence and computational complexity of the original collocation method. Numerical experiments are presented to illustrate the theoretical estimates.     

Complex and p-Adic Meromorphic Functions f′P′( f ),g′P′(g) Sharing a Small Function

Let K be a complete algebraically closed p-adic field of characteristic zero.We apply results in algebraic geometry and a new Nevanlinna theorem for p-adic meromorphic functions in order to prove results of uniqueness in value sharing problems, both on K and on C. Let P be a polynomial of uniqueness for meromorphic functions in K or C or in an open disk. Let f, g be two transcendental meromorphic functions in the whole field K or in C or meromorphic functions in an open disk of K that are not quotients of bounded analytic functions. We show that if f′P′( f) and g′P′(g) share a small function α counting multiplicity, then f = g, provided that the multiplicity order of zeros of P′satisfy certain inequalities. A breakthrough in this paper consists of replacing inequalities n ≥ k+2 or n ≥ k+3 used in previous papers by Hypothesis(G). In the p-adic context, another consists of giving a lower bound for a sum of q counting functions of zeros with(q-1) times the characteristic function of the considered meromorphic function.     

Counting isomorphism classes of pointed hyperelliptic curves of genus 4 over finite fields with even characteristic

This paper is devoted to counting the number of isomorphism classes of pointed hyperelliptic curves over finite fields. We deal with the genus 4 case and the finite fields are of even characteristics. The number of isomorphism classes is computed and the explicit formulae are given. This number can be represented as a polynomial in q of degree 7, where q is the order of the finite field. The result can be used in the classification problems and it is useful for further studies of hyperelliptic curve cryptosystems, e.g. it is of interest for research on implementing the arithmetics of curves of low genus for cryptographic purposes. It could also be of interest for point counting problems; both on moduli spaces of curves, and on finding the maximal number of points that a pointed hyperelliptic curve over a given finite field may have.     

OSCILLATION CRITERIA FOR STRONGLY SUPERLINEAR AND STRONGLY SUBLINEAR HIGHER ORDER DYNAMIC EQUATIONS ON TIME SCALES

We establish some new criteria for the oscillation of even order nonlinear dynamic equation. We study the case of strongly super-linear and the case of strongly sub-linear subject to various conditions.     

On a critical Neumann problem with a perturbation of lower order

We investigate the solvability of the Neumann problem （1.1） involving the critical Sobolev nonlinearity and a term of lower order. We allow a coefficient of u in equation （1.1） to be unbounded. We prove the existence of a solution in a weighted Sobolev space.     

Lorentz space estimates and Jacobian convergence for the Ginzburg-Landau energy with applied magnetic field

In this paper, we continue the study of Lorentz space estimates for the Ginzburg-Landau energy started in [15]. We focus on getting estimates for the Ginzburg-Landau energy with external magnetic field h _ex in certain interesting regimes of h _ex . This allows us to show that for configurations close to minimizers or local minimizers of the energy, the vorticity mass of the configuration (u, A) is comparable to the L ^{2, ∞} Lorentz space norm of ∇ _A u. We also establish convergence of the gauge-invariant Jacobians (vorticity measures) in the dual of a function space defined in terms of Lorentz spaces. Supported by an NSF Graduate Research Fellowship.     

New Results on EX Graphs

By the extremal number ex(n; t) = ex(n; {C ₃, C ₄, . . . , C _t }) we denote the maximum size (that is, number of edges) in a graph of order n > t and girth at least g ≥ t + 1. The set of all the graphs of order n, containing no cycles of length ≥ t, and of size ex(n; t), is denoted by EX(n; t) = EX(n; {C ₃, C ₄, . . . , C _t }), these graphs are called EX graphs. In 1975, Erdős proposed the problem of determining the extremal numbers ex(n; 4) of a graph of order n and girth at least 5. In this paper, we consider a generalized version of this problem, for t ≥ 5. In particular, we prove that ex(29; 6) = 45, also we improve some lower bounds and upper bounds of ex _u (n; t), for some particular values of n and t.     

Minimizers near the first critical field for the nonself-dual Chern–Simons–Higgs energy

The Chern–Simons–Higgs energy serves as a model for high temperature superconductivity. We show the existence of weak solutions to the CSH equations that are minimizers of the CSH energy. The solutions are vortexless for an applied magnetic field h _ex below the critical field strength, whereas vortices appear when h _ex exceeds the critical field strength. D. Spirn was supported in part by NSF grants DMS-0510121 and DMS-0707714. X. Yan was supported in part by NSF grants DMS-0700966 and DMS-0401048.     

On the height of the Kronecker product ofS n characters

Given any two heightsh ₁,h ₂, we can choose wide enough partitionsν, μ ∈ Par(n) such thath(ν)=h ₁,h(μ)=h ₂ andh(x _ν⊗xμ)=h ₁sdh ₂.     

Convergence of an explicit finite volume scheme for first order symmetric systems

Summary.  This paper is devoted to the derivation of a O(h ^1/2) error estimate for the classical upwind, explicit in time, finite volume scheme for linear first order symmetric systems. Such a result already existed for the corresponding implicit in time finite volume scheme, since it can be interpreted as a particular case of the space-time discontinuous Galerkin method but the technique of proof, used in that case, does not extend to explicit schemes. The general framework, recently developed to analyse the convergence rate of finite volume schemes for non linear scalar conservation laws, can not be used either, because it is not adapted for systems, even linear. In this article, we propose a new technique, which takes advantage of the linearity of the problem. The first step consists in controlling the approximation error ∥u−u _h ∥ _L2 by an expression of the form <ν _h , g>−2<μ _h , gu>, where u is the exact solution, g is a particular smooth function, and μ _h , ν _h are some linear forms depending on the approximate solution u _h . The second step consists in carefully estimating the error terms <μ _h , gu> and <ν _h , g>, by using uniform stability results for the discrete problem and regularity properties of the continuous solution. Received December 20, 2001 / Revised version received January 2, 2001 / Published online November 27, 2002 Mathematics Subject Classification (1991): 65N30     

Vorticity measure of type-II superconducting thin films

In this paper, we study the Ginzburg-Landau system of variable thickness superconducting thin films, placed in an applied magnetic field h _ex , when h _ex is of the order of the “first critical field”, i.e. of the order of |ln?ε|. We examine the asymptotic behavior of the “vorticity-measures” associated to the vortices of the solution and describe the distribution of vortices.     

The spectrum of the averaging operator for a finite homogeneous graph

We give an estimate for the spectrum of the averaging operator T₁(Γ, 1) over the radius 1 for the finite (q+1)-homogeneous quotient graph Γ/X, where X is an infinite (q+1)-homogeneous tree associated with the free group G over a finite set of generators S={x₁ ..., x_p} (2p=q+1), and Γ, a subgroup of finite index in G. T₁(Γ, 1) is defined on the subspace L²(Γ/G, 1) ⊖ E_ex, where E_ex is the subspace of eigenfunctions of T₁(Γ, 1) with eigenvalue λ such that |λ|=q+1. We present a construction of some finite homogeneous graphs such that the spectrum of their adjacency matrices can be calculated explicitly. Bibliography: 11 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 205, 1993, pp. 92–109. Translated by A. M. Nikitin.     

An answer to a question by J. Milnor

We consider two commuting automorphismsT ₁,T ₂ of the Lebesque space (M, M, μ) such thath _m,n=h(T ₁ ^m T ₂ ⁿ )<∞ whereh is the measure-theoretic entropy. Under additional assumptions we show the existence of the limits lim (1/m)h _m,n wherem→∞,n→∞,m/n→ω and ω is an irrational number.     

A local variational relation and applications

In [BGH] the authors show that for a given topological dynamical system (X,T) and an open coveru there is an invariant measure μ such that infh _μ(T,ℙ)≥h _top(T,U) where infimum is taken over all partitions finer thanu. We prove in this paper that if μ is an invariant measure andh _μ(T,ℙ) > 0 for each ℙ finer thanu, then infh _μ(T,ℙ > 0 andh _top(T,U) > 0. The results are applied to study the topological analogue of the Kolmogorov system in ergodic theory, namely uniform positive entropy (u.p.e.) of ordern (n≥2) or u.p.e. of all orders. We show that for eachn≥2 the set of all topological entropyn-tuples is the union of the set of entropyn-tuples for an invariant measure over all invariant measures. Characterizations of positive entropy, u.p.e. of ordern and u.p.e. of all orders are obtained. We could answer several open questions concerning the nature of u.p.e. and c.p.e.. Particularly, we show that u.p.e. of ordern does not imply u.p.e. of ordern+1 for eachn≥2. Applying the methods and results obtained in the paper, we show that u.p.e. (of order 2) system is weakly disjoint from all transitive systems, and the product of u.p.e. of ordern (resp. of all orders) systems is again u.p.e. of ordern (resp. of all orders). Project supported by one hundred talents plan and 973 plan.     

Q 2-free Families in the Boolean Lattice

For a family F{{\cal F}} of subsets of [n] = {1, 2, ..., n} ordered by inclusion, and a partially ordered set P, we say that F{{\cal F}} is P-free if it does not contain a subposet isomorphic to P. Let ex(n, P) be the largest size of a P-free family of subsets of [n]. Let Q ₂ be the poset with distinct elements a, b, c, d, a < b,c < d; i.e., the 2-dimensional Boolean lattice. We show that 2N − o(N) ≤ ex(n, Q ₂) ≤ 2.283261N + o(N), where N = \binomn?n/2 ?N = \binom{n}{\lfloor n/2 \rfloor}. We also prove that the largest Q ₂-free family of subsets of [n] having at most three different sizes has at most 2.20711N members.