Electronic structure, bonding, spectra, and linear and nonlinear electric properties of Ti@C28

The potential energy surface (PES) of Ti@C(28) has been revisited, and the stationary points have been carefully characterized. In particular, the C(2v) symmetry structure considered previously turns out to be a transition state lying 2.3 kcal/mol above the ground state of C(3v) symmetry at the MP2/6-31G(d) level. A large binding energy of 181.3 kcal/mol is found at the ROMP2/6-31G(d) level. Topological analysis of the generalized Ti@C(28) density reveals four bond paths between Ti and carbon atoms of the host. The character of all four contacts corresponds to a partially covalent closed shell interaction. UV-vis, IR, and Raman spectra are calculated and compared with C(28)H(4). The dipole moment and the static electronic and double harmonic vibrational (hyper)polarizabilities have been obtained. Distortion of the fullerene cage due to encapsulation leads to nonzero diagonal components of the electronic first hyperpolarizability β, and to an increase in the diagonal components of the electronic polarizability α and second hyperpolarizability γ. However, introduction of the Ti atom causes a comparable or larger reduction in most cases due to localized bonding interactions. At the double harmonic level, the average vibrational β is much larger than its electronic counterpart, but the opposite is true for α and for the contribution to γ that has been calculated. There is also a very large anharmonic (nuclear relaxation) contribution to β which results from a shallow PES with four minima separated by very low barriers. Thus, the vibrational γ (and α) may, likewise, become much larger when anharmonicity is taken into account.     

The Funk and Hilbert geometries for spaces of constant curvature

The goal of this paper is to introduce and to study analogues of the Euclidean Funk and Hilbert metrics on open convex subsets of the hyperbolic space $\mathbb H ^n$ H n and of the sphere $S^n$ S n . We highlight some striking similarities among the three cases (Euclidean, spherical and hyperbolic) which hold at least at a formal level. The proofs of the basic properties of the classical Funk metric on subsets of $\mathbb R ^n$ R n use similarity properties of Euclidean triangles which of course do not hold in the non-Euclidean cases. Transforming the side lengths of triangles using hyperbolic and circular functions and using some non-Euclidean trigonometric formulae, the Euclidean similarity techniques are transported into the non-Euclidean worlds. We start by giving three representations of the Funk metric in each of the non-Euclidean cases, which parallel known representations for the Euclidean case. The non-Euclidean Funk metrics are shown to be Finslerian, and the associated Finsler norms are described. We then study their geodesics. The Hilbert geometry of convex sets in the non-Euclidean constant curvature spaces $S^n$ S n and $\mathbb H ^n$ H n is then developed by using the properties of the Funk metric and by introducing a non-Euclidean cross ratio. In the case of Euclidean (respectively spherical, hyperbolic) geometry, the Euclidean (respectively spherical, hyperbolic) geodesics are Funk and Hilbert geodesics. This leads to a formulation and a discussion of Hilbert’s Problem IV in the non-Euclidean settings. Projection maps between the spaces $\mathbb R ^n, \mathbb H ^n$ R n , H n and the upper hemisphere establish equivalences between the Hilbert geometries of convex sets in the three spaces of constant curvature, but such an equivalence does not hold for Funk geometries.     

Regulation of catalytic behaviour of hydrolases through interactions with functionalized carbon-based nanomaterials

The interaction of enzymes with carbon-based nanomaterials (CBNs) is crucial for the function of biomolecules and therefore for the design and development of effective nanobiocatalytic systems. In this study, the effect of functionalized CBNs, such as graphene oxide (GO) and multi-wall carbon nanotubes (CNTs), on the catalytic behaviour of various hydrolases of biotechnological interest was monitored and the interactions between CBNs and proteins were investigated. The enzyme–nanomaterial interactions significantly affect the catalytic behaviour of enzymes, resulting in an increase up to 60 % of the catalytic efficiency of lipases and a decrease up to 30 % of the esterase. Moreover, the use of CNTs and GO derivatives, especially those that are amine-functionalized, led to increased thermal stability of most the hydrolases tested. Fluorescence and circular dichroism studies indicated that the altered catalytic behaviour of enzymes in the presence of CBNs arises from specific enzyme–nanomaterial interactions, which can lead to significant conformational changes. In the case of lipases, the conformational changes led to a more active and rigid structure, while in the case of esterases this led to destabilization and unfolding. Kinetic and spectroscopic studies indicated that the extent of the interactions between CBNs and hydrolases can be mainly controlled by the functionalization of nanomaterials than by their geometry.     

Harmonic symmetrization of convex sets and of Finsler structures,with applications to Hilbert geometry

David Hilbert discovered in 1895 an important metric that is canonically associated to an arbitrary convex domain Ω$\Omega$ in the Euclidean (or projective) space. This metric is known to be Finslerian, and the usual proof of this fact assumes a certain degree of smoothness of the boundary of Ω$\Omega$, and refers to a theorem by Busemann and Mayer that produces the norm of a tangent vector from the distance function. In this paper, we develop a new approach for the study of the Hilbert metric where no differentiability is assumed. The approach exhibits the Hilbert metric on a domain as a symmetrization of a natural weak metric, known as the Funk metric. The Funk metric is described as a tautological   weak Finsler metric, in which the unit ball in each tangent space is naturally identified with the domain Ω$\Omega$ itself. The Hilbert metric is then identified with the reversible tautological weak Finsler structure   on Ω$\Omega$, and the unit ball of the Hilbert metric at each point is described as the harmonic symmetrization of the unit ball of the Funk metric. Properties of the Hilbert metric then follow from general properties of harmonic symmetrizations of weak Finsler structures.     

Length spectra and the Teichmüller metric for surfaces with boundary

We consider some metrics and weak metrics defined on the Teichmüller space of a surface of finite type with nonempty boundary, that are defined using the hyperbolic length spectrum of simple closed curves and of properly embedded arcs, and we compare these metrics and weak metrics with the Teichmüller metric. The comparison is on subsets of Teichmüller space which we call “ε ₀-relative \({\epsilon}\)-thick parts”, and whose definition depends on the choice of some positive constants ε ₀ and \({\epsilon}\). Meanwhile, we give a formula for the Teichmüller metric of a surface with boundary in terms of extremal lengths of families of arcs.     

Shift coordinates,stretch lines and polyhedral structures for Teichmüller space

This paper has two parts. In the first part, we study shift coordinates on a sphere S equipped with three distinguished points and a triangulation whose vertices are the distinguished points. These coordinates parametrize a space that we call an unfolded Teichmüller space. This space contains Teichmüller spaces of the sphere with boundary components and cusps (which we call generalized pairs of pants), for all possible values of and satisfying . The parametrization of by shift coordinates equips this space with a natural polyhedral structure, which we describe more precisely as a cone over an octahedron in . Each cone over a simplex of this octahedron is interpreted as a Teichmüller space of the sphere with boundary components and cusps, for fixed and , the sphere being furthermore equipped with an orientation on each boundary component. There is a natural linear action of a finite group on whose quotient is an augmented Teichmüller space in the usual sense. We describe several aspects of the geometry of the space . Stretch lines and earthquakes can be defined on this space. In the second part of the paper, we use the shift coordinates to obtain estimates on the behaviour of stretch lines in the Teichmüller space of a surface obtained by gluing hyperbolic pairs of pants. We also use the shift coordinates to give formulae that express stretch lines in terms of Fenchel-Nielsen coordinates. We deduce the disjointness of some stretch lines in Teichmüller space. We study in more detail the case of a closed surface of genus 2. Authors’ addresses: A. Papadopoulos, Institut de Recherche Mathématique Avancée, Université Louis Pasteur and CNRS, 7 rue René Descartes, 67084 Strasbourg Cedex, France and Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany; G. Théret, Institut de Recherche Mathématique Avancée, Université Louis Pasteur and CNRS, 7 rue René Descartes, 67084 Strasbourg Cedex, France and Department of Mathematical Sciences, University of Aarhus, Ny Munkegade, Building 1530, DK-8000 Aarhus C, Denmark     

Optimization Method for the Selection of the Appropriate Fuzzy Implication

In this paper, we introduce a method allowing us to choose the most suitable fuzzy implication in an inference system application. We introduce also a similarity measure, which we call degree of sameness of two fuzzy implications in an inference system application.     

广义Bethe树上奇偶马尔可夫链场上的若干极限性质

通过构造两个非负鞅证明了一个强极限定理,然后把它应用到本文所定义的广义Bethe树上的奇偶马尔可夫链场上,从而获得了此马氏链场上的一类强极限定理.