Stability of a Pivoting Strategy for Parallel Gaussian Elimination

Gaussian elimination with partial pivoting achieved by adding the pivot row to the kth row at step k, was introduced by Onaga and Takechi in 1986 as means for reducing communications in parallel implementations. In this paper it is shown that the growth factor of this partial pivoting algorithm is bounded above by _n <#60; 3 ^n–1, as compared to 2 ^n–1 for the standard partial pivoting. This bound _n, close to 3 ^n–2, is attainable for class of near-singular matrices. Moreover, for the same matrices the growth factor is small under partial pivoting.This revised version was published online in October 2005 with corrections to the Cover Date.     

A Note on a Paper by P. Amodio and F. Mazzia

In 1999 Amodio and Mazzia presented a new backward error analysis for LU factorization and introduced a new growth factor _n . Their very interesting approach allowed them to obtain sharp error bounds. In particular, they derive nice results assuming that partial pivoting is used. However, the forward error bound for the solution of a linear system whose coefficient matrix A is an M-atrix given in Theorem 4.1 of that paper is not correct. They first obtain a bound for the condition number (U) assuming that one has the LU factorization of an M-matrix and then they apply the bounds obtained when partial pivoting is used. But if P is the permutation associated with partial pivoting then PA = LU can fail to be an M-atrix and the bound for (U) can be false, as shown in our Example 1.1. We also prove that, for a pivoting strategy presented in the paper, the growth factor of an M-matrix A is _n(A) = 1 and (U) (A), where U is the upper triangular matrix obtained after applying such a pivoting strategy.This revised version was published online in October 2005 with corrections to the Cover Date.     

用三角函数法获得非线性Boussinesq方程的广义孤子解

找到一个合适的代换——三角函数法，将非线性Boussinesq微分方程转换为非线性代数方程组.用吴消元法求解该非线性代数方程组，从而获得一般形式Boussinesq微分方程的广义孤子解. 关键词： Boussinesq方程 吴消元法 非线性代数方程组 孤子解     

Addition,elimination, exchange,and epimerization in nitro sulfones

Three reactions were studied in the diastereomers of 1‐(benzenesulfonyl)‐2‐nitro‐1‐phenylpropane ( 1A and 1B ) and briefly in related compounds: elimination of the benzenesulfonyl group, epimerization of one diastereomer to the other, and deuterium/hydrogen exchange at the methine group next to nitro in starting material. The two diastereomers showed quite different reactivity. The high melting diasteromer showed rapid elimination and some exchange. The low melting diastereomer (at approximately a half‐life) showed extensive epimerization, and elimination to the alkene, but little exchange. There is little effect of aromatic substituents on reaction course. The situation is complicated by re‐addition of benzenesulfinate to the alkene. The addition reaction was similar to elimination in agreement with the Principle of Microscopic Reversibility expectations. An electron transfer mechanism for addition is calculated to be comparatively favorable. Copyright © 2007 John Wiley & Sons, Ltd.     

Reparametrization of models of non-stationary chemical kinetics as a method of investigation of the inverse problem

A new approach to solve the inverse problem of the non-stationary chemical kinetics by a reparametrization of the initial model is suggested. This approach permits to obtain all solutions of IP in the case of a finite number or some parametric functions in the case of the infinite number of solutions.     

On the Robustness of Gaussian Elimination with Partial Pivoting

It has been recently shown that large growth factors might occur in Gaussian Elimination with Partial Pivoting (GEPP) also when solving some plausibly natural systems. In this note we argue that this potential problem could be easily solved, with much smaller risk of failure, by very small (and low cost) modifications of the basic algorithm, thus confirming its inherent robustness. To this end, we first propose an informal model with the goal of providing further support to the comprehension of the stability properties of GEPP. We then report the results of numerical experiments that confirm the viewpoint embedded in the model. Basing on the previous observations, we finally propose a simple scheme that could be turned into (even more) accurate software for the solution of linear systems.     

Solid-Phase Organic Synthesis of Methyl Isoxazole-5-carboxylates Based on Wang Resin–Bound 1-Phenylselenoacrylate

A novel Wang resin–bound 1-phenylselenoacrylate reagent has been developed and used as dipolarophile for the 1,3-dipolar cycloaddition with nitrile oxides to the regioselective synthesis of methyl 3-substituted isoxazole-5-carboxylates in good yields and purities.     

The Onsager-Casimir relations revisited

We study the fate of the Onsager-Casimir reciprocity relations for a continuous system when some of its variables are eliminated adiabatically. Just as for discrete systems, deviations appear in correction terms to the reduced evolution equation that are of higher order in the time scale ratio. The deviations are not removed by including correction terms to the coarse-grained thermodynamic potential. However, via a reformulation of the theory, in which the central role of the thermodynamic potential is taken over by an associated Lagrangian-type expression, we arrive at a modified form of the Onsager-Casimir relations that survives the adiabatic elimination procedure. There is a simple relation between the time evolution of the redefined thermodynamic forces and that of the basic thermodynamic variables; this relation also survives the adiabatic elimination. The formalism is illustrated by explicit calculations for the Klein-Kramers equation, which describes the phase space distribution of Brownian particles, and for the corrected Smoluchowski equation derived from it by adiabatic elimination of the velocity variable. The symmetry relation for the latter leads to a simple proof that the reality of the eigenvalues of the simple Smoluchowski equation is not destroyed by the addition of higher order corrections, at least not within the framework of a formal perturbation expansion in the time scale ratio.     

Regio-selective cyanation of (Z)-(1,2-dibromo-2-arylvinyl)triisopropylsilane with suppression of halogen elimination

Highly regio-selective cyanation of vicinal (Z)-dibromoalkenyl silanes was achieved by a vinylic Rosenmund-von Braun reaction, significantly suppressing side-production of alkyne. The alkyne was generated by a halogen elimination side-reaction that is an intrinsic problem in metal-activation of vicinal dihaloalkenes. We have studied to overcome the problem, and finally found the combination of CuCN and O = PPh₃ in toluene solvent effectively controlled the production of byproducts. The resultant single isomer has significance in potentially application as a multi-tunable synthetic scaffold.