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.     

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.     

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.     

Selective Nascent Polymer Catch‐and‐Release Enables Scalable Isolation of Multi‐Kilobase Single‐Stranded DNA

Scalable methods currently are lacking for isolation of long ssDNA, an important material for numerous biotechnological applications. Conventional biomolecule purification strategies achieve target capture using solid supports, which are limited in scale and susceptible to contamination owing to nonspecific adsorption and desorption on the substrate surface. We herein disclose selective nascent polymer catch and release (SNAPCAR), a method that utilizes the reactivity of growing poly(acrylamide‐co‐acrylate) chains to capture acrylamide‐labeled molecules in free solution. The copolymer acts as a stimuli‐responsive anchor that can be precipitated on demand to pull down the target from solution. SNAPCAR enabled scalable isolation of multi‐kilobase ssDNA with high purity and 50–70 % yield. The ssDNA products were used to fold various DNA origami. SNAPCAR‐produced ssDNA will expand the scope of applications in nanotechnology, gene editing, and DNA library construction.     

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.     

Analysis of the Symmetry of Crystal Packing Forces by Methyl Proton Tunneling: A Strategy for the Unambiguous Assignment of the Magnetic Jahn – Teller Effect in Molecules

Intrinsic and extrinsic forces behind the distortion in metal atom clusters can be readily distinguished provided that the clusters are embedded in a suitable ligand environment and that the tunneling of the protons in the peripheral ligands is then analyzed by inelastic neutron scattering. For the [Cr₃O(OOCCH₃)₆(H₂O)₃]Cl⋅6 H₂O model system studied, the tunneling process is very sensitive to the local environment. Thus a tool is available to allow a better assessment of the cause of structural distortions.     

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.     

Constructing spherical CR manifolds by gluing tetrahedra

We propose a general method of constructing spherical CR manifolds by gluing tetrahedra adapted to CR geometry. We obtain spherical CR structures on the complement of the figure eight knot and the Whitehead link complement with holonomy in $\mathrm{PU}(2,1,\mathbb{Z}[\omega ])$ and $\mathrm{PU}(2,1,\mathbb{Z}[\mathrm{i}])$ respectively (the same integer rings appearing in real hyperbolic geometry). To cite this article: E. Falbel, C. R. Acad. Sci. Paris, Ser. I 340 (2005).