High performance solution of the complex symmetric eigenproblem

Complex symmetric matrices often appear in quantum physics in the solution methods of partial differential equations such as the Schrödinger equation. We have recently introduced a new fast and efficient direct eigensolver for this problem in [4], and reported its performance in the eigenvalue calculation in [3]. In this paper, we further report on some benchmark tests for computing the full and partial eigenspectrum on a variety of super computing machines, i.e., the Cray J-932, the DEC Alfa 8400, and the SGI Power Challenge 8000 and 10000. We observe that in all cases the new algorithm is much faster than codes available in standard state of the art eigensolver packages such as LAPACK.     

Commuting extensions and cubature formulae

Based on a novel point of view on 1-dimensional Gaussian quadrature, we present a new approach to d-dimensional cubature formulae. It is well known that the nodes of 1-dimensional Gaussian quadrature can be computed as eigenvalues of the so-called Jacobi matrix. The d-dimensional analog is that cubature nodes can be obtained from the eigenvalues of certain mutually commuting matrices. These are obtained by extending (adding rows and columns to) certain noncommuting matrices A₁,...,A_d, related to the coordinate operators x₁,...,x_d, in R^d. We prove a correspondence between cubature formulae and “commuting extensions” of A₁,...,A_d, satisfying a compatibility condition which, in appropriate coordinates, constrains certain blocks in the extended matrices to be zero. Thus, the problem of finding cubature formulae can be transformed to the problem of computing (and then simultaneously diagonalizing) commuting extensions. We give a general discussion of existence and of the expected size of commuting extensions and briefly describe our attempts at computing them.     

Calculating multidimensional discrete variable representations from cubature formulas

Finding multidimensional nondirect product discrete variable representations (DVRs) of Hamiltonian operators is one of the long standing challenges in computational quantum mechanics. The concept of a "DVR set" was introduced as a general framework for treating this problem by R. G. Littlejohn, M. Cargo, T. Carrington, Jr., K. A. Mitchell, and B. Poirier (J. Chem. Phys. 2002, 116, 8691). We present a general solution of the problem of calculating multidimensional DVR sets whose points are those of a known cubature formula. As an illustration, we calculate several new nondirect product cubature DVRs on the plane and on the sphere with up to 110 points. We also discuss simple and potentially very useful finite basis representations (FBRs), based on general (nonproduct) cubatures. Connections are drawn to a novel view on cubature presented by I. Degani, J. Schiff, and D. J. Tannor (Num. Math. 2005, 101, 479), in which commuting extensions of coordinate matrices play a central role. Our construction of DVR sets answers a problem left unresolved in the latter paper, namely, the problem of interpreting as function spaces the vector spaces on which commuting extensions act.     

Real-space observation of spectral degeneracy breaking in a waveguide-coupled disk microresonator

We report on the real-space observation of resonant frequency splitting in a high-Q waveguide-coupled silicon-on-insulator microdisk resonator. Phase sensitive near-field analysis reveals the stationary nature of the two resonant states, and spectral investigations clearly show their orthogonality. These measurements emphasize the role of the coupling waveguide in this splitting phenomenon. The symmetry of the two stationary whispering gallery modes is clearly observed and is found to follow the axial symmetry of the waveguide-coupled microdisk as it has been reported by earlier theoretical predictions.     

Symmetrized general hopping current equation

We show that within the validity range of local thermal equilibrium--therein, however, irrespective of the magnitude of the driving force--a simple current equation can be formulated that expresses the current in terms of a product of a local nonequilibrium conductivity and a sinh function of half the electrochemical potential drop (normalized with respect to kBT) over the local hopping distance. This local current/driving force relation takes account of both electrical and compositional effects and can be generalized as to include interactions and structural variations.     

Intramolecular carbozincation of unactivated alkenes occurs through a zinc radical transfer mechanism

The cyclizations of a number of terminally unsaturated alkenyl zinc iodides to cyclopentylmethylzinc iodides, formerly believed to be nonradical in nature, have been revealed as radical chain cyclizations initiated by adventitious oxygen. Five cases are presented in which the published carbozincation cis/trans selectivities are essentially the same as those found for the cyclizations of the unsaturated alkyl iodide precursors of the alkylzinc iodides by the iodine atom transfer method at approximately the same temperatures. In addition, it has been found that one of the organozinc cyclizations does not occur in a system in which oxygen has been rigorously excluded. The combined findings strongly suggest that these organozinc cyclizations occur by a zinc radical transfer mechanism rather than by a conventional carbometallation that is thought to occur with the analogous organolithium and organomagnesium cyclizations.     

Phonomicrosurgical management of vocal fold polyps: The subepithelial microflap resection technique

Vocal fold polyps are typically caused by acute and chronic trauma to the microvasculature of the superficial lamina propria (SLP). Shearing stresses that are induced by hyperfunctional glottal sound production lead to bleeding into the SLP and malformed neo-vascularized masses. Because the primary process does not involve the epithelium, the authors designed a technique to resect hemorrhagic polyps by epithelial cordotomy with partial or complete preservation of the vocal fold epithelium. This approach is different from the traditional microsurgical resection of hemorrhagic polyps by amputation with or without the carbon dioxide laser. Forty patients who underwent microlaryngoscopic resection of hemorrhagic polyps from 1996 through 1998 were reviewed retrospectively. Thirty-six of the 40 procedures were by epithelial cordotomy and subepithelial removal of the polyp contents. Sixteen of 36 were assisted by a subepithelial infusion of saline and epinephrine, and all were 3 mm to 6 mm. Four of 40 polyps were amputated; all of these were less than 3 mm and were pedicled on a narrow base. Cold instruments were used exclusively in all 40 patients. Postoperative laryngeal stroboscopy within 2 weeks revealed improved mucosal wave propagation and improved glottal closure in all 33 patients in whom postoperative strobovideolaryngoscopy was available. The epithelial cordotomy technique was introduced to minimize disturbance of normal SLP and epithelium. Despite the hemorrhagic nature of these lesions, cold instruments could be used exclusively with facility due to careful microdissection between the polyp and the residual normal SLP and the enhanced hemostasis provided by the subepithelial infusion of saline and epinephrine. The rapid return to improved glottal function is the result of this ultra tissue-sparing technique.     

Reactions along the astrophysical s-process path and prospects for neutron radiotherapy with the Liquid-Lithium Target (LiLiT) at the Soreq Applied Research Accelerator Facility (SARAF)

The European Physical Journal A - An empirical formula is proposed for the two-proton decay half-lives. This study is an extension of the empirical formula reported recently by us for calculating...