Blow Analytic Mappings and Functions

1993 ◽  
Vol 36 (4) ◽  
pp. 497-506
Author(s):  
Etsuo Yoshinaga
Keyword(s):  
Analytic Functions ◽  
Classical Theorem ◽  
Analytic Family ◽  
Isolated Singularities ◽  
Inverse Mapping ◽  
Continuous Map ◽  
Inverse Mapping Theorem ◽  
Blowing Up ◽  
Analytic Mappings

AbstractLet π: M —> Rn be the blowing-up of Rn at the origin. Then a continuous map-germ f: (Rn — 0,0) —> Rm is called blow analytic if there exists an analytic map-germ such that Then an inverse mapping theorem for blow analytic mappings as a generalization of classical theorem is shown. And the following is shown. Theorem: The analytic family of blow analytic functions with isolated singularities admits an analytic trivialization after blowing-up.

A concept of inner prederivative for set-valued mappings and its applications

2018 ◽  
Vol 24 (3) ◽  
pp. 1059-1074
Author(s):  
Michel H. Geoffroy ◽  
Yvesner Marcelin
Keyword(s):  
Optimality Conditions ◽  
Necessary Optimality Conditions ◽  
Welfare Economics ◽  
Inverse Mapping ◽  
First Order ◽  
Inverse Mapping Theorem ◽  
Homogeneous Set ◽  
Optimal Allocations ◽  
Variational Perturbations ◽  
Set Valued Mappings

We introduce a class of positively homogeneous set-valued mappings, called inner prederivatives, serving as first order approximants to set-valued mappings. We prove an inverse mapping theorem involving such prederivatives and study their stability with respect to variational perturbations. Then, taking advantage of their properties we establish necessary optimality conditions for the existence of several kind of minimizers in set-valued optimization. As an application of these last results, we consider the problem of finding optimal allocations in welfare economics. Finally, to emphasize the interest of our approach, we compare the notion of inner prederivative to the related concepts of set-valued differentiation commonly used in the literature.

scholarly journals TOWARDS DIFFERENTIAL CALCULUS IN STRATIFIED GROUPS

2013 ◽  
Vol 95 (1) ◽  
pp. 76-128 ◽  
Author(s):  
VALENTINO MAGNANI
Keyword(s):  
Implicit Function Theorem ◽  
Group Structure ◽  
Mean Value ◽  
Implicit Function ◽  
Inverse Mapping ◽  
Natural Group ◽  
Stratified Groups ◽  
First Order ◽  
Inverse Mapping Theorem ◽  
Rank Theorem

AbstractWe study graded group-valued continuously differentiable mappings defined on stratified groups, where differentiability is understood with respect to the group structure. We characterize these mappings by a system of nonlinear first-order PDEs, establishing a quantitative estimate for their difference quotient. This provides us with a mean value estimate that allows us to prove both the inverse mapping theorem and the implicit function theorem. The latter theorem also relies on the fact that the differential admits a proper factorization of the domain into a suitable inner semidirect product. When this splitting property of the differential holds in the target group, then the inverse mapping theorem leads us to the rank theorem. Both implicit function theorem and rank theorem naturally introduce the classes of image sets and level sets. For commutative groups, these two classes of sets coincide and correspond to the usual submanifolds. In noncommutative groups, we have two distinct classes of intrinsic submanifolds. They constitute the so-called intrinsic graphs, that are defined with respect to the algebraic splitting and everywhere possess a unique metric tangent cone equipped with a natural group structure.

The Weak Inverse Mapping Theorem

2015 ◽  
Vol 34 (3) ◽  
pp. 321-342 ◽  
Author(s):  
Daniel Campbell ◽  
Stanislav Hencl ◽  
František Konopecký
Keyword(s):  
Inverse Mapping ◽  
Inverse Mapping Theorem

scholarly journals Differential calculus in Fréchet spaces

1981 ◽  
Vol 24 (1) ◽  
pp. 93-122 ◽  
Author(s):  
Duong Minh Duc
Keyword(s):  
Differential Calculus ◽  
Fréchet Spaces ◽  
Inverse Mapping ◽  
Inverse Mapping Theorem ◽  
Special Case

We apply Keller's method to the study of differential calculus in Frechet spaces and establish an inverse mapping theorem. A special case of this theorem is similar to a theorem of Yamamuro.

Notes on the inverse mapping theorem in locally convex spaces

1980 ◽  
Vol 21 (3) ◽  
pp. 419-461 ◽  
Author(s):  
Sadayuki Yamamuro
Keyword(s):  
Analytic Study ◽  
Locally Convex Spaces ◽  
Inverse Mapping ◽  
Inverse Mapping Theorem ◽  
Locally Convex ◽  
Convex Spaces

Several problems arising from a functional analytic study on Omori's inverse mapping theorem are considered arriving at an inverse mapping theorem in locally convex spaces.

scholarly journals On Mean Distortion for Analytic Functions with Positive Real Part in a Circle

1967 ◽  
Vol 29 ◽  
pp. 221-228
Author(s):  
Yûsaku Komatu
Keyword(s):  
Unit Circle ◽  
Analytic Functions ◽  
Positive Real Part ◽  
Classical Theorem ◽  
Positive Real ◽  
Distortion Theorems

Let be the class of analytic functions Ф(z) which are regular and of positive real part in the unit circle | z | <1 and normalized by Ф(0) = 1. Several distortion theorems have been obtained on various functionals in this class. In a previous paper [4] we have dealt with mean distortion which generalizes a classical theorem of Rogosinski [6].

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