Connections on the Total Space of a Holomorphic Lie Algebroid

The main purpose of the paper is the study of the total space of a holomorphic Lie algebroid E. The paper is structured in three parts. In the first section, we briefly introduce basic notions on holomorphic Lie algebroids. The local expressions are written and the complexified holomorphic bundle is introduced. The second section presents two approaches on the study of the geometry of the complex manifold E. The first part contains the study of the tangent bundle \(T_{\mathbb {C}}E=T'E\oplus T''E\) and its link, via the tangent anchor map, with the complexified tangent bundle \(T_{\mathbb {C}}(T'M)=T'(T'M)\oplus T''(T'M)\). A holomorphic Lie algebroid structure is emphasized on \(T'E\). A special study is made for integral curves of a spray on \(T'E\). Theorem 2.8 gives the coefficients of a spray, called canonical, obtained from a complex Lagrangian on \(T'E\). In the second part of section two, we study the holomorphic prolongation \(\mathcal {T}'E\) of the Lie algebroid E. In the third section, we study how a complex Lagrange (Finsler) structure on \(T'M\) induces a Lagrangian structure on E. Three particular cases are analysed by the rank of the anchor map, the dimensions of manifold M, and those of the fibres. We obtain the correspondent on E of the Chern–Lagrange nonlinear connection from \(T'M\).     

Interpolation operators on a triangle with one curved side

We construct certain Lagrange, Hermite and Birkhoff-type operators, which interpolate a given function and some of its derivatives on the border of a triangle with one curved side, as well as some of their product and Boolean sum operators. We study the interpolation properties and the order of accuracy (degree of exactness and precision set) of the constructed operators, respectively the remainders of the corresponding interpolation formulas. Finally, we give some numerical examples.     

Strong and Weak Solutions to Second Order Differential Inclusions Governed by Monotone Operators

In this paper we introduce the concept of a weak solution for second order differential inclusions of the form u″(t) ∈ Au(t) + f(t), where A is a maximal monotone operator in a Hilbert space H. We prove existence and uniqueness of weak solutions to two point boundary value problems associated with such kind of equations. Furthermore, existence of (strong and weak) solutions to the equation above which are bounded on the positive half axis is proved under the optimal condition tf(t) ∈ L ¹(0, ∞; H), thus solving a long-standing open problem (for details, see our comments in Section 3 of the paper). Our treatment regarding weak solutions is similar to the corresponding theory related to the first order differential inclusions of the form f(t) ∈ u′(t) + Au(t) which has already been well developed.     

On the Construction of Composite Wannier Functions

We give a constructive proof for the existence of a Bloch basis of rank \({N}\) which is both smooth (real analytic) and periodic with respect to its \({d}\)-dimensional quasi-momenta, when \({1\leq d\leq 2}\) and \({N\geq 1}\). The constructed Bloch basis is conjugation symmetric when the underlying projection has this symmetry, hence the corresponding exponentially localized composite Wannier functions are real. In the second part of the paper, we show that by adding a weak, globally bounded but not necessarily constant magnetic field, the existence of a localized basis is preserved.     

An existence result for a nonhomogeneous problem in R2 related to nonlinear Hencky-type materials

This paper investigates a nonlinear and non-homogeneous system of partial differential equations. The motivation comes from the fact that in a particular case the problem discussed here can be used in modeling the behavior of nonlinear Hencky-type materials. The main result of the paper establishes the existence of a nontrivial solution in an adequate functional space of Orlicz–Sobolev type by using Schauder’s fixed point theorem combined with adequate variational techniques.     

Geometrical and dynamical properties of general Euler top system

In this paper we present various geometrical and dynamical properties of the Euler top system in general form. Also the stability problem for the equilibrium states of them are discussed.     

THE MAIN INVARIANTS OF A COMPLEX FINSLER SPACE

In this paper we extend the results obtained in [3], where are investigated the general settings of the two-dimensional complex Finsler manifolds, with respect to a local complex Berwald frame. The geometry of such manifolds is controlled by three real invariants which live on T ′M : two horizontal curvature invariants K and W and one vertical curvature invariant I. By means of these invariants are defined both the horizontal and the vertical holomorphic sectional curvatures. The complex Landsberg and Berwald spaces are of particular interest. Complex Berwald spaces coincide with K¨ahler spaces, in the two-dimensional case. We establish the necessary and sufficient condition under which K is a constant and we obtain a characterization for the K¨ahler purely Hermitian spaces by the fact K = W = constant and I = 0. For the class of complex Berwald spaces we have K = W = 0.Finally, a classification of two-dimensional complex Finsler spaces for which the horizontal curvature satisfies a special property is obtained.     

The equations of a holomorphic subspace in a complex Finsler space

In [Mu1] we underlined the motifs of holomorphic subspaces in a complex Finsler space: induced nonlinear connection, coupling connections, and the induced tangent and normal connections. In the present paper we investigate the equations of Gauss, H-and A-Codazzi, and Ricci equations of a holomorphic subspace. We deduce the link between the holomorphic curvatures of the Chern-Finsler connection and its induced tangent connection. Conditions for totally geodesic holomorphic subspaces are obtained. Communicated by János Szenthe     

Critical field for complete vortex expulsion from narrow superconducting strips

We have measured the maximum field for which vortices are completely expelled from a thin-film superconducting strip. Niobium strips of width W were field cooled and imaged with a scanning Hall-probe microscope. Below a critical field B(m) approximately Phi(0)/W(2) all flux was expelled; above this field vortices were observed with a density increasing approximately linearly with field. The small value of the critical field, which is orders of magnitude less than in the bulk, implies that superconducting devices should be designed with narrow wires to eliminate the generation of noise from vortex motion.