Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Parity and generalized multiplicity
HTML articles powered by AMS MathViewer

by P. M. Fitzpatrick and Jacobo Pejsachowicz PDF
Trans. Amer. Math. Soc. 326 (1991), 281-305 Request permission

Abstract:

Assuming that $X$ and $Y$ are Banach spaces and $\alpha :[a,b] \to \mathcal {L}(X,Y)$ is a path of linear Fredholm operators with invertible endpoints, in $[{\text {F}} - \text {P}1]$ we defined a homotopy invariant of $\alpha ,\sigma (\alpha ,I) \in {{\mathbf {Z}}_2}$, the parity of $\alpha$ on $I$. The parity plays a fundamental role in bifurcation problems, and in degree theory for nonlinear Fredholm-type mappings. Here we prove (a) that, generically, the parity is a $\bmod 2$ count of the number of transversal intersections of $\alpha (I)$ with the set of singular operators, (b) that if ${\lambda _0}$ is an isolated singular point of $\alpha$, then the local parity \[ \sigma (\alpha ,{\lambda _0}) \equiv \lim \limits _{\varepsilon \to 0} \sigma (\alpha ,[{\lambda _0} - \varepsilon ,{\lambda _0} + \varepsilon ])\] remains invariant under Lyapunov-Schmidt reduction, and (c) that $\sigma (\alpha ,{\lambda _0}) = {(- 1)^{{M_G}({\lambda _0})}}$, where ${M_G}({\lambda _0})$ is any one of the various concepts of generalized multiplicity which have been defined in the context of linearized bifurcation data.
References
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC: 58E07, 47H15, 58C99
  • Retrieve articles in all journals with MSC: 58E07, 47H15, 58C99
Additional Information
  • © Copyright 1991 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 326 (1991), 281-305
  • MSC: Primary 58E07; Secondary 47H15, 58C99
  • DOI: https://doi.org/10.1090/S0002-9947-1991-1030507-7
  • MathSciNet review: 1030507