Some quantitative results in singularity theory

Y. Yomdin. "Some quantitative results in singularity theory." Annales Polonici Mathematici 87.1 (2005): 277-299. <http://eudml.org/doc/280641>.

@article{Y2005,
abstract = {The classical singularity theory deals with singularities of various mathematical objects: curves and surfaces, mappings, solutions of differential equations, etc. In particular, singularity theory treats the tasks of recognition, description and classification of singularities in each of these cases. In many applications of singularity theory it is important to sharpen its basic results, making them "quantitative", i.e. providing explicit and effectively computable estimates for all the important parameters involved. This opens new possibilities for applications in analysis, geometry, differential equations, dynamics, and, last not least, in computations. Application of the results of singularity theory in numerical data processing with finite accuracy stresses another important requirement: the "normalizing transformations" must be explicitly computable. The most natural interpretation of this requirement is in terms of the "jet calculus": given the Taylor polynomials of the input data, we should be able to produce explicitly the Taylor polynomials of the output normalizing transformations. This papers provides a sample of initial results in these directions.},
author = {Y. Yomdin},
journal = {Annales Polonici Mathematici},
keywords = {singularity; normal form; normalizing transformation; Taylor polynomial; jet calculus; explicit bounds},
language = {eng},
number = {1},
pages = {277-299},
title = {Some quantitative results in singularity theory},
url = {http://eudml.org/doc/280641},
volume = {87},
year = {2005},
}

TY - JOUR
AU - Y. Yomdin
TI - Some quantitative results in singularity theory
JO - Annales Polonici Mathematici
PY - 2005
VL - 87
IS - 1
SP - 277
EP - 299
AB - The classical singularity theory deals with singularities of various mathematical objects: curves and surfaces, mappings, solutions of differential equations, etc. In particular, singularity theory treats the tasks of recognition, description and classification of singularities in each of these cases. In many applications of singularity theory it is important to sharpen its basic results, making them "quantitative", i.e. providing explicit and effectively computable estimates for all the important parameters involved. This opens new possibilities for applications in analysis, geometry, differential equations, dynamics, and, last not least, in computations. Application of the results of singularity theory in numerical data processing with finite accuracy stresses another important requirement: the "normalizing transformations" must be explicitly computable. The most natural interpretation of this requirement is in terms of the "jet calculus": given the Taylor polynomials of the input data, we should be able to produce explicitly the Taylor polynomials of the output normalizing transformations. This papers provides a sample of initial results in these directions.
LA - eng
KW - singularity; normal form; normalizing transformation; Taylor polynomial; jet calculus; explicit bounds
UR - http://eudml.org/doc/280641
ER -