Computing Puiseux-Series Solutions to Determinantal Equations via Combinatorial Relaxation

1990 ◽  
Vol 19 (6) ◽  
pp. 1132-1161 ◽  
Author(s):  
Kazuo Murota
Keyword(s):  
Series Solutions ◽  
Puiseux Series ◽  
Combinatorial Relaxation

scholarly journals PUISEUX POWER SERIES SOLUTIONS FOR SYSTEMS OF EQUATIONS

2010 ◽  
Vol 21 (11) ◽  
pp. 1439-1459 ◽  
Author(s):  
FUENSANTA AROCA ◽  
GIOVANNA ILARDI ◽  
LUCÍA LÓPEZ DE MEDRANO
Keyword(s):  
Algebraic Curves ◽  
Newton Polygon ◽  
Series Solutions ◽  
Systems Of Equations ◽  
Algebraic Equations ◽  
Puiseux Series ◽  
Tropical Variety ◽  
Power Series Solutions ◽  
Tropical Varieties ◽  
The Ideal

We give an algorithm to compute term-by-term multivariate Puiseux series expansions of series arising as local parametrizations of zeroes of systems of algebraic equations at singular points. The algorithm is an extension of Newton's method for plane algebraic curves, replacing the Newton polygon by the tropical variety of the ideal generated by the system. As a corollary we deduce a property of tropical varieties of quasi-ordinary singularities.

How switching the substituent of a pyrene derivative from a methyl to a butyl affects the fluorescence response of polystyrene randomly labeled with pyrene

2010 ◽  
Vol 88 (3) ◽  
pp. 217-227 ◽  
Author(s):  
Mark Ingratta ◽  
Manoj Mathew ◽  
Jean Duhamel
Keyword(s):  
Organic Solvents ◽  
Excellent Agreement ◽  
Fluorescence Spectra ◽  
Series Solutions ◽  
Specific Nature ◽  
Polymer Backbone ◽  
Excimer Fluorescence ◽  
Fluorescence Response ◽  
Different Types ◽  
Blob Model

A series of polystyrenes randomly labeled with 1-pyrenebutanol were prepared by copolymerizing styrene and 1-pyrenebutylacrylate yielding the CoBuE–PS series. Solutions of CoBuE–PS were prepared in nine organic solvents having viscosities ranging from 0.36 to 5.5 mPa·s and the fluorescence spectra and pyrene monomer and excimer fluorescence decays were acquired. Analysis of the fluorescence spectra yielded the IE/IM ratio, whereas analysis of the fluorescence decays with the fluorescence blob model (FBM) yielded the parameters N blobo , <kblob × Nblob> , and k blobo . These parameters were compared to those obtained with two other series of pyrene-labeled polystyrenes, which had been studied earlier, namely CoA–PS and CoE–PS where pyrene was attached to the polymer backbone via a methylamide and benzyl methylether linker, respectively. Although the parameters IE/IM, N blobo , <kblob × Nblob>, and k blobo took different values according to the specific nature of the linker connecting pyrene to the polystyrene backbone, they exhibited trends that were quite similar for all the pyrene-labeled polystyrene constructs. The excellent agreement between the parameters retrieved for the three different types of pyrene-labeled polystyrenes suggests that the FBM accounts satisfyingly for differences in the nature of the label used, while still retrieving information pertinent to the polymer of interest.

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