Near boundary vortices in a magnetic Ginzburg-Landau model: Their locations via tight energy bounds

Given a bounded doubly connected domain G⊂R², we consider a minimization problem for the Ginzburg-Landau energy functional when the order parameter is constrained to take S¹-values on ∂G and have degrees zero and one on the inner and outer connected components of ∂G, correspondingly. We show that minimizers always exist for 0<λ<1 and never exist for λ?1, where λ is the coupling constant ( is the Ginzburg-Landau parameter). When λ→1−0 minimizers develop vortices located near the boundary, this results in the limiting currents with δ-like singularities on the boundary. We identify the limiting positions of vortices (that correspond to the singularities of the limiting currents) by deriving tight upper and lower energy bounds. The key ingredient of our approach is the study of various terms in the Bogomol'nyi's representation of the energy functional.     

Uniqueness of vortexless Ginzburg-Landau type minimizers in two dimensions

In a simply connected two dimensional domain Ω, we consider Ginzburg-Landau minimizers u with zero degree Dirichlet boundary condition ${g \in H^{1/2}(\partial \Omega; \mathbb{S}^1)}$ . We prove uniqueness of u whenever either the energy or the Ginzburg-Landau parameter are small. This generalizes a result of Ye and Zhou requiring smoothness of g. We also obtain uniqueness when Ω is multiply connected and the degrees of the vortexless minimizer u are prescribed on the components of the boundary, generalizing a result of Golovaty and Berlyand for annular domains. The proofs rely on new global estimates connecting the variation of |u| to the Ginzburg-Landau energy of u. These estimates replace the usual global pointwise estimates satisfied by ${\nabla u}$ when g is smooth, and apply to fairly general potentials. In a related direction, we establish new uniqueness results for critical points of the Ginzburg-Landau energy.     

Curvelets and Fourier Integral Operators

A recent body of work introduced new tight-frames of curvelets E. Candès, D. Donoho, in: (i) Curvelets – a suprisingly effective nonadaptive representation for objects with edges (A. Cohen, C. Rabut, L. Schumaker (Eds.)), Vanderbilt University Press, Nashville, 2000, pp. 105–120; (ii) http://www.acm.caltech.edu/~emmanuel/publications.html, 2002 to address key problems in approximation theory and image processing. This paper shows that curvelets essentially provide optimally sparse representations of Fourier Integral Operators. To cite this article: E. Candès, L. Demanet, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

EKR type inequalities for 4-wise intersecting families

Let 1?t?7 be an integer and let F be a k-uniform hypergraph on n vertices. Suppose that |A∩B∩C∩D|?t holds for all A,B,C,D∈F. Then we have if holds for some ε>0 and all n>n₀(ε). We apply this result to get EKR type inequalities for “intersecting and union families” and “intersecting Sperner families.”     

A free analogue of Shannon's problem on monotonicity of entropy

We prove a free probability analog of a result of [S. Artstein, K. Ball, F. Barthe, A. Naor, Solution of Shannon's problem on monotonicity of entropy, J. Amer. Math. Soc. 17 (2004) 975-982]. In particular, we prove that if X₁,X₂,… are freely independent identically distributed random variables, then the function

     

On spaces which are D, linearly D and transitively D

In this note, we comment on D-spaces, linearly D-spaces and transitively D-spaces. We show that every meta-Lindelöf space is transitively D. If X is a weak -refinable TD-scattered space, then X is transitively D, where TD is the class of all transitively D-spaces. If X is a weak -refinable -scattered space, then X is a D-space, where is the class of all D-spaces, and hence every weak -refinable (or submetacompact) scattered space is a D-space. This gives a positive answer to a question mentioned by Martínez and Soukup. In the last part of this note, we show that if X is a weak -refinable space then X is linearly D.     

Two results on entire solutions of Ginzburg-Landau system in higher dimensions

In this short article we prove two results on the Ginzburg-Landau system of equations Δu=u(|u|²−1), where . First we prove a Liouville-type theorem which asserts that every solution u, satisfying , is constant (and of unit norm), provided N?4 (here M?1). In our second result, we give an answer to a question raised by Brézis (open problem 3 of (Proceedings of the Symposium on Pure Mathematics, vol. 65, American Mathematical Society, Providence, RI, 1999), about the symmetry for the Ginzburg-Landau system in the case N=M?3. We also formulate three open problems concerning the classification of entire solutions of the Ginzburg-Landau system in any dimension.     

W versus C local minimizers and multiplicity results for quasilinear elliptic equations

In this paper we show that the local minimizers of a class of functionals in the C¹-topology are still their local minimizers in . Using this fact, we study the multiplicity of solutions for a class of quasilinear elliptic equations via critical point theory.     

On the prime power factorization of n!, II

Let p, q be primes and m be a positive integer. For a positive integer n, let ep(n) be the nonnegative integer with pep(n)|n and pep(n)+1?n. The following results are proved: (1) For any positive integer m, any prime p and any ε∈Zm, there are infinitely many positive integers n such that ; (2) For any positive integer m, there exists a constant D(m) such that if ε,δ∈Zm and p, q are two distinct primes with max{p,q}?D(m), then there exist infinitely many positive integers n such that , . Finally we pose four open problems.     

Lipschitz and piecewise-C regularity for scalar minimizers of affine simple integrals

Lipschitz, piecewise-C¹ and piecewise affine regularity is proved for AC minimizers of the “affine” integral , under general hypotheses on , , and with superlinear growth at infinity.The hypotheses assumed to obtain Lipschitz continuity of minimizers are unusual: ρ(·) and ?(·) are lsc and may be both locally unbounded (e.g., not in L_loc¹), provided their quotient ?/ρ(·) is locally bounded. As to h(·), it is assumed lsc and may take +∞ values freely.     

Inducing regulation of any digraphs

For a given structure D (digraph, multidigraph, or pseudodigraph) and an integer r large enough, a smallest inducing r-regularization of D is constructed. This regularization is an r-regular superstructure of the smallest possible order with bounded arc multiplicity, and containing D as an induced substructure. The sharp upper bound on the number, ρ, of necessary new vertices among such superstructures for n-vertex general digraphs D is determined, ρ being called the inducing regulation number of D. For being the maximum among semi-degrees in D, simple n-vertex digraphs D with largest possible ρ are characterized if either or (where the case is not a trivial subcase of ).     

A general regularity theorem for functionals with φ-growth

We prove C1,α-regularity for local minimizers of functionals

     

On a singular perturbation problem involving a ``circular-well' potential

We study the asymptotic behavior, as a small parameter goes to 0, of the minimizers for a variational problem which involves a ``circular-well' potential, i.e., a potential vanishing on a closed smooth curve in . We thus generalize previous results obtained for the special case of the Ginzburg-Landau potential.

     

Équidistribution presque partout modulo 1 de suites oscillantes perturbées III : cas liouvillien multidimensionnel

We extend the results of uniform distribution modulo 1 given in [B. Rittaud, Équidistribution presque partout modulo 1 de suites oscillantes perturbées, Bull. Soc. Math. France 128 (2000) 451-471; B. Rittaud, Équidistribution presque partout modulo 1 de suites oscillantes perturbées, II: Cas Liouvillien unidimensionnel, Colloq. Math. 96 (1) (2003) 55-73], which deal with sequences of the form , where n(hn), and are polynomially increasing sequences, n(εn) a bounded sequence, essentially a C³-function Zd-periodic, Θ an element of Rd and t a real number. We remove the Diophantine hypothesis on Θ needed in [the first of above mentioned articles], and add a technical hypothesis on hn. We apply this result to the convergence of diagonal averages for d×d matrices.     

Global Jacobian and Γ-convergence in a two-dimensional Ginzburg-Landau model for boundary vortices

In the theory of 2D Ginzburg-Landau vortices, the Jacobian plays a crucial role for the detection of topological singularities. We introduce a related distributional quantity, called the global Jacobian that can detect both interior and boundary vortices for a 2D map u. We point out several features of the global Jacobian, in particular, we prove an important stability property. This property allows us to study boundary vortices in a 2D Ginzburg-Landau model arising in thin ferromagnetic films, where a weak anchoring boundary energy penalising the normal component of u at the boundary competes with the usual bulk potential energy. We prove an asymptotic expansion by Γ-convergence at the second order for this mixed boundary/interior energy in a regime where boundary vortices are preferred. More precisely, at the first order of the limiting expansion, the energy is quantised and determined by the number of boundary vortices detected by the global Jacobian, while the second order term in the limiting energy expansion accounts for the interaction between the boundary vortices.     

Smallest defining sets of directed triple systems

A directed triple system of order v, , is a pair (V,B) where V is a set of v elements and B is a collection of ordered triples of distinct elements of V with the property that every ordered pair of distinct elements of V occurs in exactly one triple as a subsequence. A set of triples in a D is a defining set for D if it occurs in no other on the same set of points. A defining set for D is a smallest defining set for D if D has no defining set of smaller cardinality. In this paper we are interested in the quantity

     

A remark on continued fractions with sequences of partial quotients

Given any infinite set B of positive integers , let τ(B) denote the exponent of convergence of the series . Let E(B) be the set . Hirst [K.E. Hirst, Continued fractions with sequences of partial quotients, Proc. Amer. Math. Soc. 38 (1973) 221-227] proved the inequality and conjectured (see Hirst [K.E. Hirst, Continued fractions with sequences of partial quotients, Proc. Amer. Math. Soc. 38 (1973), p. 225] and Cusick [T.W. Cusick, Hausdorff dimension of sets of continued fractions, Quart. J. Math. Oxford Ser. (2) 41 (1990), p. 278]) that equality holds in general. In [Bao-Wei Wang, Jun Wu, A problem of Hirst on continued fractions with sequences of partial quotients, Bull. London Math. Soc., in press], we gave a positive answer to this conjecture. In this note, we further show that the result in [Bao-Wei Wang, Jun Wu, A problem of Hirst on continued fractions with sequences of partial quotients, Bull. London Math. Soc., in press] is sharp.