Nilpotence and generation in the stable module category

Nilpotence has been studied in stable homotopy theory and algebraic geometry. We study the corresponding notion in modular representation theory of finite groups, and apply the discussion to the study of ghosts, and generation of the stable module category. In particular, we show that for a finitely generated kG-module M, the tensor M-generation number and the tensor M-ghost number are both equal to the degree of tensor nilpotence of a certain map associated with M.     

Cubic regularization in symmetric rank-1 quasi-Newton methods

Quasi-Newton methods based on the symmetric rank-one (SR1) update have been known to be fast and provide better approximations of the true Hessian than popular rank-two approaches, but these properties are guaranteed under certain conditions which frequently do not hold. Additionally, SR1 is plagued by the lack of guarantee of positive definiteness for the Hessian estimate. In this paper, we propose cubic regularization as a remedy to relax the conditions on the proofs of convergence for both speed and accuracy and to provide a positive definite approximation at each step. We show that the n-step convergence property for strictly convex quadratic programs is retained by the proposed approach. Extensive numerical results on unconstrained problems from the CUTEr test set are provided to demonstrate the computational efficiency and robustness of the approach.     

Structure related beam propagation

The general theory of a class of structure related (SR) non-orthogonal coordinate systems is introduced for the first time. Special cases of this type of coordinate system used separately or in combination can automatically follow the local geometry of many common optical waveguide designs, thereby facilitating the fast and accurate simulation of photonic circuits, when incorporated within a FD-BPM algorithm. Substantial savings in run time and memory are possible in comparison with standard FD-BPM in rectangular coordinates.     

RNA-RNA noncovalent interactions investigated by microspray ionization mass spectrometry.

Electrospray ionization mass spectrometry is playing an increasing role in the study of noncovalent interactions involving biomolecules. RNA-RNA complexes are important in many areas of biology, including RNA catalysis, RNA splicing, ribosome function, and gene regulation. Here, microelectrospray mass spectrometry (microESI-MS) is used to study noncovalent base-pairing interactions between RNA oligonucleotides, an area not previously explored by this technique. Using a set of complementary RNA oligonucleotides, we demonstrate the formation of the expected double-helical RNA complexes composed of three distinct oligonucleotides. The ability to study specific RNA noncovalent interactions by microESI-MS has the potential to provide a unique method by which to analyze and assign precise molecular masses to RNA-RNA complexes.     

The orbit method and Gelfand pairs, associated with nilpotent Lie groups

Let K be a compact Lie group acting by automorphisms on a nilpotent Lie group N. One calls (K, N) a Gelfand pair when the integrable K-invariant functions on N form a commutative algebra under convolution. We prove that in this case the coadjoint orbits for G:= K × N which meet the annihilator of the Lie algebra of K do so in single K-orbits. This generalizes a result of the authors and R. Lipsman concerning Gelfand pairs associated with Heisenberg groups.     

Group actions on one-manifolds, II: Extensions of Hölder's Theorem

This self-contained paper is part of a series seeking to understand groups of homeomorphisms of manifolds in analogy with the theory of Lie groups and their discrete subgroups. In this paper we consider groups which act on with restrictions on the fixed point set of each element. One result is a topological characterization of affine groups in as those groups whose elements have at most one fixed point.