Decomposing the essential spectrum

We use C^*-algebra theory to provide a new method of decomposing the essential spectra of self-adjoint and non-self-adjoint Schrödinger operators in one or more space dimensions.     

Approximate and pseudo-amenability of various classes of Banach algebras

We continue the investigation of notions of approximate amenability that were introduced in work of the second and third authors together with R.J. Loy. It is shown that every boundedly approximately contractible Banach algebra has a bounded approximate identity, and that the Fourier algebra of the free group on two generators is not operator approximately amenable. Further examples are obtained of ?¹-semigroup algebras which are approximately amenable but not amenable; using these, we show that bounded approximate contractibility need not imply sequential approximate amenability. Results are also given for Segal algebras on locally compact groups, and algebras of p-pseudo-functions on discrete groups.     

Radial rapid decay property for cocompact lattices

We study Haagerup inequality for radial functions on uniform lattices in semisimple Lie groups, with respect to Riemannian metrics and, in some case, to word metrics. In particular we extend the Swiatkowski-Valette results to any lattice acting properly and essentially transitively on classical buildings.     

Uniqueness results for nonlocal Hamilton-Jacobi equations

We are interested in nonlocal eikonal equations describing the evolution of interfaces moving with a nonlocal, non-monotone velocity. For these equations, only the existence of global-in-time weak solutions is available in some particular cases. In this paper, we propose a new approach for proving uniqueness of the solution when the front is expanding. This approach simplifies and extends existing results for dislocation dynamics. It also provides the first uniqueness result for a Fitzhugh-Nagumo system. The key ingredients are some new perimeter estimates for the evolving fronts as well as some uniform interior cone property for these fronts.     

Saturated packings and reduced coverings obtained by perturbing tilings

Let a normed space X possess a tiling T consisting of unit balls. We show that any packing P of X obtained by a small perturbation of T is completely translatively saturated; that is, one cannot replace finitely many elements of P by a larger number of unit balls such that the resulting arrangement is still a packing.In contrast with that, given a tiling T of Rⁿ with images of a convex body C under Euclidean isometries, there may exist packings P consisting of isometric images of C obtained from T by arbitrarily small perturbations which are no longer completely saturated. This means that there exists some positive integer k such that one can replace k−1 members of P by k isometric copies of C without violating the packing property. However, we quantify a tradeoff between the size of the perturbation and the minimal k such that the above phenomenon occurs.Analogous results are obtained for coverings.     

On a deterministic linear partial differential system

We develop an Lp theory for the Cauchy problem of linear partial differential systems of the form

     

Multiplicity of characteristics with Lagrangian boundary values on symmetric star-shaped hypersurfaces

In this paper, the multiplicity of Lagrangian orbits on C² smooth compact symmetric star-shaped hypersurfaces with respect to the origin in R2n is studied. These Lagrangian orbits begin from one Lagrangian subspace and end on another. An infinitely many existence result is proved via Z₂-index theory. This is a multiplicity result about the Arnold Chord Conjecture in some sense, and is a generalization of the problem about the multiplicity of Lagrangian orbits beginning from and ending on the same Lagrangian subspace which was considered in the authors' previous paper [F. Guo, C. Liu, Multiplicity of Lagrangian orbits on symmetric star-shaped hypersurfaces, Nonlinear Anal. 69 (4) (2008) 1425-1436].     

Automated structure verification based on 1H NMR prediction

A unique opportunity exists when an experimental NMR spectrum is obtained for which a specific chemical structure is anticipated. A process of Verification--the confirmation of a postulated structure--is now possible, as opposed to Elucidation-the de novo determination of a structure. A method for automated structure verification is suggested, which compares the chemical shifts, intensities and multiplicities of signals in an experimental 1H NMR spectrum with those from a predicted spectrum for the proposed structure. A match factor (MF) is produced and used to classify the spectrum-structure match into one of three categories, correct, ambiguous, or incorrect. The verification result is also augmented by the spectrum assignment obtained as part of the verification process. This method was tested on a set of synthetic spectra and several sets of experimental spectra, all of which were automatically prepared from raw data. Taking into account even the most problematic structures, with many labile protons present and poor prediction accuracy, 50% of all spectra can still be automatically verified without any false positives or negatives. In a blind test on a typical set of data, it is shown that fewer than 31% of the structures would need manual evaluation. This means that a system is possible whereby 69% of the spectra are prepared and evaluated automatically, and never need to be seen or evaluated by a human.     

An algorithm for three-dimensional Voronoi S-network

The paper presents an algorithm for calculating the three-dimensional Voronoi-Delaunay tessellation for an ensemble of spheres of different radii (additively-weighted Voronoi diagram). Data structure and output of the algorithm is oriented toward the exploration of the voids between the spheres. The main geometric construct that we develop is the Voronoi S-network (the network of vertices and edges of the Voronoi regions determined in relation to the surfaces of the spheres). General scheme of the algorithm and the key points of its realization are discussed. The principle of the algorithm is that for each determined site of the network we find its neighbor sites. Thus, starting from a known site of the network, we sequentially find the whole network. The starting site of the network is easily determined based on certain considerations. Geometric properties of ensembles of spheres of different radii are discussed, the conditions of applicability and limitations of the algorithm are indicated. The algorithm is capable of working with a wide variety of physical models, which may be represented as sets of spheres, including computer models of complex molecular systems. Emphasis was placed on the issue of increasing the efficiency of algorithm to work with large models (tens of thousands of atoms). It was demonstrated that the experimental CPU time increases linearly with the number of atoms in the system, O(n).     

Stereospecific assignments of protein NMR resonances based on the tertiary structure and 2D/3D NOE data

In many cases of protein structure determination by NMR a high-quality structure is required. An important contribution to structural precision is stereospecific assignment of magnetically nonequivalent prochiral methylene and methyl groups, eliminating the need for introducing pseudoatoms and pseudoatom corrections in distance restraint lists. Here, we introduce the stereospecific assignment program that uses the resonance assignment, a preliminary 3D structure and 2D and/or 3D nuclear Overhauser effect spectroscopy peak lists for stereospecific assignment. For each prochiral group the algorithm automatically calculates a score for the two different stereospecific assignment possibilities, taking into account the presence and intensity of the nuclear Overhauser effect (NOE) peaks that are expected from the local environment of each prochiral group (i.e., the close neighbors). The performance of the algorithm has been tested and used on NMR data of alpha-helical and beta-sheet proteins using homology models and/or X-ray structures. The program produced no erroneous stereospecific assignments provided the NOEs were carefully picked and the 3D model was sufficiently accurate. The set of NOE distance restraints produced by nmr2st using the results of the SSA module was superior in generating good-quality ensembles of NMR structures (low deviations from upper limits in conjunction with low root-mean-square-deviation values) in the first round of structure calculations. The program uses a novel approach that employs the entire 3D structure of the protein to obtain stereospecific assignment; it can be used to speed up the NMR structure refinement and to increase the quality of the final NMR ensemble even when no scalar or residual dipolar coupling information is available.