The geometry of the Gauss–Picard modular group

We give a construction of a fundamental domain for PU(2,1,\mathbbZ [i]){{\rm PU}(2,1,\mathbb{Z} [i])}, that is the group of holomorphic isometries of complex hyperbolic space with coefficients in the Gaussian ring of integers \mathbbZ [i]{\mathbb{Z} [i]}. We obtain from that construction a presentation of that lattice and relate it, in particular, to lattices constructed by Mostow.     

Spherical CR-manifolds of dimension 3

A sphericalCR-structure on a smooth (2n–1)-manifoldM is a maximal collection of distinguished charts modeled on the boundary H ⁿ of the complex hyperbolic space, where coordinate changes are restrictions of transformations from PU(n, 1). There exists a development map , where is the universal covering ofM, which is a local diffeomorphism. We study properties of the development maps and holonomy groups of sphericalCR-structures on compact 3-dimensional manifolds. We also give constructions of fundamental domains for some discrete subgroups of PU(2, 1).     

Fundamental Domains for Finite Subgroups in <Emphasis Type="Italic">U</Emphasis>(2) and Configurations of Lagrangians

We show that every finite subgroup of U(2) is contained with index two in a group generated by involutions fixing Lagrangian planes. We describe fundamental domains for their action on related to the configuration of these Lagrangian planes.     

A hyperchaotic system without equilibrium

This article introduces a new chaotic system of 4-D autonomous ordinary differential equations, which has no equilibrium. This system shows a hyper-chaotic attractor. There is no sink in this system as there is no equilibrium. The proposed system is investigated through numerical simulations and analyses including time phase portraits, Lyapunov exponents, and Poincaré maps. There is little difference between this chaotic system and other chaotic systems with one or several equilibria shown by phase portraits, Lyapunov exponents and time series methods, but the Poincaré maps show this system is a chaotic system with more complicated dynamics. Moreover, the circuit realization is also presented.     

Self‐Assembled Cyclic d,l‐α‐Peptides as Generic Conformational Inhibitors of the α‐Synuclein Aggregation and Toxicity: In Vitro and Mechanistic Studies

Many peptides and proteins with large sequences and structural differences self‐assemble into disease‐causing amyloids that share very similar biochemical and biophysical characteristics, which may contribute to their cross‐interaction. Here, we demonstrate how the self‐assembled, cyclic d,l ‐α‐peptide CP‐2 , which has similar structural and functional properties to those of amyloids, acts as a generic inhibitor of the Parkinson′s disease associated α‐synuclein (α‐syn) aggregation to toxic oligomers by an ?off‐pathway“ mechanism. We show that CP‐2 interacts with the N‐terminal and the non‐amyloid‐β component region of α‐syn, which are responsible for α‐syn′s membrane intercalation and self‐assembly, thus changing the overall conformation of α‐syn. CP‐2 also remodels α‐syn fibrils to nontoxic amorphous species and permeates cells through endosomes/lysosomes to reduce the accumulation and toxicity of intracellular α‐syn in neuronal cells overexpressing α‐syn. Our studies suggest that targeting the common structural conformation of amyloids may be a promising approach for developing new therapeutics for amyloidogenic diseases.     

On products of isometries of hyperbolic space

We show that for arbitrary fixed conjugacy classes C₁,…,Cl, l?3, of loxodromic isometries of the two-dimensional complex or quaternionic hyperbolic space there exist isometries g₁,…,gl, where each gi∈Ci, and whose product is the identity. The result follows from the properness, up to conjugation, of the multiplication map on a pair of conjugacy classes in rank 1 groups.     

Extended Systems with Deterministic Local Dynamics and Random Jumps

We consider particle systems on lattices with internal dynamics at each site and random jumps between sites. Models with simple chaotic local dynamics, namely expanding circle maps, are considered. Results on mean drift rates, central limit theorems and dependences on jump parameters are proved. A version of most of the results in this paper is contained in this author’s Ph.D. thesis [K]. This research is partially supported by a grant from the NSF.