Magnetic flows on Sol-manifolds: dynamical and symplectic aspects

We consider magnetic flows on compact quotients of the 3-dimensional solvable geometry Sol determined by the usual left-invariant metric and the distinguished monopole. We show that these flows have positive Liouville entropy and therefore are never completely integrable. This should be compared with the known fact that the underlying geodesic flow is completely integrable in spite of having positive topological entropy. We also show that for a large class of twisted cotangent bundles of solvable manifolds every compact set is displaceable.Comment: Final version to appear in CMP. Two new remarks have been added as well as some numerical calculations for metric entrop

Tzitz\'eica transformation is a dressing action

We classify the simplest rational elements in a twisted loop group, and prove that dressing actions of them on proper indefinite affine spheres give the classical Tzitz\'eica transformation and its dual. We also give the group point of view of the Permutability Theorem, construct complex Tzitz\'eica transformations, and discuss the group structure for these transformations

On Rank Problems for Planar Webs and Projective Structures

We present old and recent results on rank problems and linearizability of geodesic planar webs.Comment: 31 pages; LaTeX; corrected the abstract and Introduction; added reference

Structure-activity study of a laminin α1 chain active peptide segment Ile-Lys-Val-Ala-Val (IKVAV)

AbstractThe IKVAV sequence, one of the most potent active sites of laminin-1, has been shown to promote cell adhesion, neurite outgrowth, and tumor growth. Here we have determined the structural requirements of the IKVAV sequence for cell attachment and neurite outgrowth using various 12-mer synthetic peptide analogs. All-l- and all-d-IKVAV peptides showed cell attachment and neurite outgrowth activities. In contrast, all-l- and all-d-reverse-sequence peptides were not active. Some of the analogs, in which the lysine and isoleucine residues of the IKVAV peptide were substituted with different amino acids, promoted cell attachment, but none of the analog peptides showed neurite outgrowth activity comparable to that of the IKVAV peptide. These results suggest that the lysine and isoleucine residues are critical for the biological functions of the IKVAV peptide

Strominger--Yau--Zaslow geometry, Affine Spheres and Painlev\'e III

We give a gauge invariant characterisation of the elliptic affine sphere equation and the closely related Tzitz\'eica equation as reductions of real forms of SL(3, \C) anti--self--dual Yang--Mills equations by two translations, or equivalently as a special case of the Hitchin equation. We use the Loftin--Yau--Zaslow construction to give an explicit expression for a six--real dimensional semi--flat Calabi--Yau metric in terms of a solution to the affine-sphere equation and show how a subclass of such metrics arises from 3rd Painlev\'e transcendents.Comment: 38 pages. Final version. To appear in Communications in Mathematical Physic

G-Brownian Motion as Rough Paths and Differential Equations Driven by G-Brownian Motion

The present paper is devoted to the study of sample paths of G-Brownian motion and stochastic differential equations (SDEs) driven by G-Brownian motion from the view of rough path theory. As the starting point, we show that quasi-surely, sample paths of G-Brownian motion can be enhanced to the second level in a canonical way so that they become geometric rough paths of roughness 2 < p < 3. This result enables us to introduce the notion of rough differential equations (RDEs) driven by G-Brownian motion in the pathwise sense under the general framework of rough paths. Next we establish the fundamental relation between SDEs and RDEs driven by G-Brownian motion. As an application, we introduce the notion of SDEs on a differentiable manifold driven by GBrownian motion and construct solutions from the RDE point of view by using pathwise localization technique. This is the starting point of introducing G-Brownian motion on a Riemannian manifold, based on the idea of Eells-Elworthy-Malliavin. The last part of this paper is devoted to such construction for a wide and interesting class of G-functions whose invariant group is the orthogonal group. We also develop the Euler-Maruyama approximation for SDEs driven by G-Brownian motion of independent interest

Bundles over Nearly-Kahler Homogeneous Spaces in Heterotic String Theory

We construct heterotic vacua based on six-dimensional nearly-Kahler homogeneous manifolds and non-trivial vector bundles thereon. Our examples are based on three specific group coset spaces. It is shown how to construct line bundles over these spaces, compute their properties and build up vector bundles consistent with supersymmetry and anomaly cancelation. It turns out that the most interesting coset is $SU(3)/U(1)^2$. This space supports a large number of vector bundles which lead to consistent heterotic vacua, some of them with three chiral families.Comment: 32 pages, reference adde