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Main Title: A projection-based formulation of the Implicit Function Theorem and its application to time-varying manifolds
Author(s): Baum, Ann-Kristin
Type: Research Paper
Abstract: In this paper, we derive a projection-based formulation of the Implicit Function Theorem. We give conditions, when an algebraic, time-parameterized equation G(t,x) = 0 is solvable for components P^c x that are selected by a projection P^c and we derive an implicit function g that specializes P^c x in terms of the complementary components P x, where P = I - P^c. We apply this result to construct a projection-based parametric description of time-varying submanifolds and to generalize the concept of projections to these sets. We illustrate our results by several examples. The results are motivated by the positivity analysis of differential-algebraic equations (DAEs). These are implicit systems F(t,x,\dot x)=0 whose solutions x are supposed to remain componentwise nonnegative whenever the initial value is nonnegative. To entangle the differential and algebraic components in F(t,x,\dot x)=0 without changing the coordinate system, we pursue the presented projection-based solution of implicit algebraic equations.
Subject(s): algebraic equations
implicit function theorem
embedded submanifolds
Issue Date: 22-Oct-2014
Date Available: 17-Dec-2021
Language Code: en
DDC Class: 510 Mathematik
MSC 2000: 26B10 Implicit function theorems, Jacobians, transformations with several variables
57R40 Embeddings
Series: Preprint-Reihe des Instituts für Mathematik, Technische Universität Berlin
Series Number: 2014, 15
ISSN: 2197-8085
TU Affiliation(s): Fak. 2 Mathematik und Naturwissenschaften » Inst. Mathematik
Appears in Collections:Technische Universität Berlin » Publications

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