Energy Estimate for the Type-Ⅱ Superconducting Film

We study the minimizers of the Ginzburg-Landau model for variable thickness, superconducting, thin films with high k, placed in an applied magnetic field hex, when hex is of the order of the ＂first critical field＂, i.e. of the order of ｜lnε｜. We obtain the asymptotic estimates of minimal energy and describe the possible locations of the vortices.     

Derivation of lower critical magnetic field for anisotropic Ginzburg-Landau model in superconductivity

In this paper, the authors discuss the vortex structure of an anisotropic Ginzburg-Landau model for superconducting thin film proposed by Du. We obtain the estimate for the lower critical magnetic field $ H_{C_1 } $ H_{C_1 } which is the first critical value of h _ex corresponding to the first phase transition in which vortices appear in the superconductor. We also find local minimizers of the anisotropic superconducting thin film with a large parameter κ, and for the applied magnetic field near the critical field we discuss the asymptotic behavior of the local minimizers.     

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.     

Uniqueness of weak solutions of time-dependent 3-D Ginzburg-Landau model for superconductivity

We prove the uniqueness of weak solutions of the time-dependent 3-D Ginzburg-Landau model for superconductivity with (Ψ ₀, A ₀) ∈ L ²(Ω) initial data under the hypothesis that (Ψ, A) ∈ C([0, T]; L ³(Ω)) using the Lorentz gauge.      

Error estimates of a linear decoupled Euler–FEM scheme for a mass diffusion model

We present error estimates of a linear fully discrete scheme for a three-dimensional mass diffusion model for incompressible fluids (also called Kazhikhov–Smagulov model). All unknowns of the model (velocity, pressure and density) are approximated in space by C ⁰-finite elements and in time an Euler type scheme is used decoupling the density from the velocity–pressure pair. If we assume that the velocity and pressure finite-element spaces satisfy the inf–sup condition and the density finite-element space contains the products of any two discrete velocities, we first obtain point-wise stability estimates for the density, under the constraint lim_(h,k)→0 h/k = 0 (h and k being the space and time discrete parameters, respectively), and error estimates for the velocity and density in energy type norms, at the same time. Afterwards, error estimates for the density in stronger norms are deduced. All these error estimates will be optimal (of order O(h+k){\mathcal{O}(h+k)}) for regular enough solutions without imposing nonlocal compatibility conditions at the initial time. Finally, we also study two convergent iterative methods for the two problems to solve at each time step, which hold constant matrices (independent of iterations).     

On estimating the derivatives of symmetric diffusions in stationary random environment,with applications to ∇<Emphasis Type="Italic">ϕ</Emphasis> interface model

We consider diffusions on ℝ^d or random walks on ℤ^d in a random environment which is stationary in space and in time and with symmetric and uniformly elliptic coefficients. We show existence and H?lder continuity of second space derivatives and time derivatives for the annealed kernels of such diffusions and give estimates for these derivatives. In the case of random walks, these estimates are applied to the Ginzburg-Landau ∇ϕ interface model.     

Strict Semimonotonicity Property of Linear Transformations on Euclidean Jordan Algebras

Motivated by the equivalence of the strict semimonotonicity property of the matrix A and the uniqueness of the solution to the linear complementarity problem LCP(A,q) for q∈R ₊ ⁿ , we study the strict semimonotonicity (SSM) property of linear transformations on Euclidean Jordan algebras. Specifically, we show that, under the copositive condition, the SSM property is equivalent to the uniqueness of the solution to LCP(L,q) for all q in the symmetric cone K. We give a characterization of the uniqueness of the solution to LCP(L,q) for a Z transformation on the Lorentz cone ℒ₊ ⁿ . We study also a matrix-induced transformation on the Lorentz space ℒ ⁿ .     

Two-dimensional incompressible viscous flow around a small obstacle

In this work we study the asymptotic behavior of viscous incompressible 2D flow in the exterior of a small material obstacle. We fix the initial vorticity ω₀ and the circulation γ of the initial flow around the obstacle. We prove that, if γ is sufficiently small, the limit flow satisfies the full-plane Navier–Stokes system, with initial vorticity ω₀ + γδ, where δ is the standard Dirac measure. The result should be contrasted with the corresponding inviscid result obtained by the authors in Iftimie et al. (Comm. Part. Differ. Eqn. 28, 349–379 (2003)), where the effect of the small obstacle appears in the coefficients of the PDE and not only in the initial data. The main ingredients of the proof are L ^p − L ^q estimates for the Stokes operator in an exterior domain, a priori estimates inspired on Kato’s fixed point method, energy estimates, renormalization and interpolation.     

Vortices and pinning effects for the Ginzburg‐Landau model in multiply connected domains

We consider the two‐dimensional Ginzburg‐Landau model with magnetic field for a superconductor with a multiply connected cross section. We study energy minimizers in the London limit as the Ginzburg‐Landau parameter κ = 1/? → ∞ to determine the number and asymptotic location of vortices. We show that the holes act as pinning sites, acquiring nonzero winding for bounded fields and attracting all vortices away from the interior for fields up to a critical value h_ex = O(|1n?|). At the critical level the pinning effect breaks down, and vortices appear in the interior of the superconductor at locations that we identify explicitly in terms of the solutions of an elliptic boundary value problem. The method involves sharp upper and lower energy estimates, and a careful analysis of the limiting problem that captures the interaction between the vortices and the holes. © 2005 Wiley Periodicals, Inc.     

Γ-Convergence of 2D Ginzburg-Landau functionals with vortex concentration along curves

We study the variational convergence of a family of twodimensional Ginzburg-Landau functionals arising in the study of superfluidity or thin-film superconductivity as the Ginzburg-Landau parameter ε tends to 0. In this regime and for large enough applied rotations (for superfluids) or magnetic fields (for superconductors), the minimizers acquire quantized point singularities (vortices). We focus on situations in which an unbounded number of vortices accumulate along a prescribed Jordan curve or a simple arc in the domain. This is known to occur in a circular annulus under uniform rotation, or in a simply connected domain with an appropriately chosen rotational vector field. We prove that if suitably normalized, the energy functionals Γ-converge to a classical energy from potential theory. Applied to global minimizers, our results describe the limiting distribution of vortices along the curve in terms of Green equilibrium measures.     

Formulae for the distance in some quasi-Banach spaces

Let (A _{0, A 1} ) be a compatible pair of quasi-Banach spaces and 1etA be a corresponding space of real interpolation type such thatA ₀ ∩A ₁ is not dense inA. Upper and lower estimates are obtained for the distance of any elementf ofA fromA ₀ ∩A ₁ . These lead to formulae for the distance in a large number of concrete situations, such as whenA ₀ ∩A ₁ =L ^∞ andA is either weak-L ^q, a ‘grand’ Lebesgue space or an Orlicz space of exponential type.     

Hanf numbers for fragments of ℶ h (A)

We describe an ordinalh(A) which plays a key role in the model theory of the admissible fragmentL _A . In particular, the Hanf number ofL _A is ℶh(A) IfL _A isL _κ=ω wherecf(k)>ω thenh(A) can be characterized as the least ordinal which is notH(k ⁺)-recursive. Financial support received from NSF Grant GP-23114 Financial support received from NSF Grant GP-23114 and the A. P. Sloan Foundation     

The cluster-galaxy cross-correlation and the average infall velocity around cluster in the modified CDM models

The cluster-galaxy cross-correlation and the average infall velocity around cluster for the tilted CDM models, the hot-cold mixed dark matter models (MDM) and theA CDM models are calculated. And it is found that the prediction of the cluster-galaxy cross-correlation of theA CDM models with Ω_A=0.9, Ω_C=0.1,h=1,b = 1 or 2 is much higher than that of observation for the radius larger than 10 h⁻¹ Mpc (H₀ = 100 h.kms⁻¹Mpc⁻¹), and the other models are compatible with the observation on the scale of (3–20) h⁻ Mpc. The results can be used to compare with new observations to restrict the nature of the dark matter and the biasing factor. Project supported by the National Natural Science Foundation of China (Grant No. 19733001).     

Inversion of a theorem on action of analytic functions and multiplicative properties of some subclasses of the hardy space H∞

In this paper, function spaces V∩l _A ^p (w) are considered in the context of their multiplicative structure. The space V is determined by conditions on the values of a function in a disk (for example, C_A,Lip _Aα). We denote by l _A ^p (w) the space of power series such that their Taylor coefficients are p-summable with weight w. For an analytic function Φ acting in a space of this type, we prove the following alternative: either Φ″(z)≡0, or the space is a Banach algebra with respect to pointwise multiplication. For a wide class of weights w, we establish the continuity of the identity embeddingmult(V∩l _A ^p (w))↪multl _A ^p . An estimate for the l^p-multiplicative norm of random polynomials is found. This estimate can be considered as an extension of the known result by Salem-Zygmund. Bibliography: 10 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 232, 1996, pp. 50–72. Translated by S. Shimorin.     

Complete classification of parallel Lorentz surfaces in neutral pseudo hyperbolic 4-space

A Lorentz surface of an indefinite space form is called a parallel surface if its second fundamental form is parallel with respect to the Van der Waerden-Bortolotti connection. Such surfaces are locally invariant under the reflection with respect to the normal space at each point. Parallel surfaces are important in geometry as well as in general relativity since extrinsic invariants of such surfaces do not change from point to point. Recently, parallel Lorentz surfaces in 4D neutral pseudo Euclidean 4-space $ \mathbb{E}_2^4 $ \mathbb{E}_2^4 and in neutral pseudo 4-sphere S ₂⁴ (1) were classified in [14] and in [10], respectively. In this paper, we completely classify parallel Lorentz surfaces in neutral pseudo hyperbolic 4-space H ₂⁴ (−1). Our main result states that there are 53 families of parallel Lorentz surfaces in H ₂⁴ (−1). Conversely, every parallel Lorentz surface in H ₂⁴ (−1) is obtained from the 53 families. As an immediate by-product, we achieve the complete classification of all parallel Lorentz surfaces in 4D neutral indefinite space forms.     

Pseudo-dimension and entropy of manifolds formed by affine-invariant dictionary

We consider the manifolds H _n(φ) formed by all possible linear combinations of n functions from the set {φ(A⋅+b)}, where x→Ax+b is arbitrary affine mapping in the space ℝ^d. For example, neural networks and radial basis functions are the manifolds of type H _n(φ). We obtain estimates for pseudo-dimension of the manifold H _n(φ) for wide collection of the generator function φ. The estimates have the order O(d ² n) in degree scale, that is the order is proportional to number of parameters of the manifold H _n(φ). Moreover the estimates for ɛ-entropy of the manifold H _n(φ) are obtained. Mathematics subject classifications (2000) 41A46, 41A50, 42A61, 42C10 V. Maiorov: Supported by the Center for Absorption in Science, Ministry of Immigrant Absorption, State of Israel.     

Lorentz space estimates for the Ginzburg-Landau energy

In this paper we prove novel lower bounds for the Ginzburg-Landau energy with or without magnetic field. These bounds rely on an improvement of the “vortex-balls construction” estimates by extracting a new positive term in the energy lower bounds. This extra term can be conveniently estimated through a Lorentz space norm, on which it thus provides an upper bound. The Lorentz space L2,∞ we use is critical with respect to the expected vortex profiles and can serve to estimate the total number of vortices and get improved convergence results.     

A Probabilistic Technique for Finding Almost-Periods of Convolutions

We introduce a new probabilistic technique for finding ‘almost-periods’ of convolutions of subsets of groups. This gives results similar to the Bogolyubovtype estimates established by Fourier analysis on abelian groups but without the need for a nice Fourier transform to exist. We also present applications, some of which are new even in the abelian setting. These include a probabilistic proof of Roth’s theorem on three-term arithmetic progressions and a proof of a variant of the Bourgain–Green theorem on the existence of long arithmetic progressions in sumsets A+B that works with sparser subsets of {1, . . . , N} than previously possible. In the non-abelian setting we exhibit analogues of the Bogolyubov–Freiman–Halberstam–Ruzsa-type results of additive combinatorics, showing that product sets A ₁ · A ₂ · A ₃ and A ² · A ⁻² are rather structured, in the sense that they contain very large iterated product sets. This is particularly so when the sets in question satisfy small-doubling conditions or high multiplicative energy conditions. We also present results on structures in A · B.     

Some Simple Estimates for the Singular Values of Matrices

Abstract We first provide a simple estimate for ||A~(-1)||_∞ and ||A~(-1)||_1 of a strictly diagonally dominant matrixA. On the Basis of the result, we obtain an estimate for the smallest singular value of A. Secondly, by scalingwith a positive diagonal matrix D, we obtain some simple estimates for the smallest singular value of an H-matrix, which is not necessarily positive definite. Finally, we give some examples to show the effectiveness ofthe new bounds.     

The Yosida class is universal

We discuss families of meromorphic functions f _h obtained from single functions f by the re-scaling process f _h (z) = h ^−α f (h + h ^−β z), generalising Yosida’s process f _h (z) = f (h + z). The main objective is to obtain information about the value distribution of the generating functions f themselves. Among the most prominent (generalised) Yosida functions are the elliptic functions and also some first, second and fourth Painlevé transcendents. The Yosida class A ₀ contains all limit functions of generalised Yosida functions-the Yosida class is universal.