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Preprint 15-2014

A projection-based formulation of the Implicit Function Theorem and its application to time-varying manifolds

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Author(s) : Ann-Kristin Baum

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 15-2014

MSC 2000

26B10 Implicit function theorems, Jacobians, transformations with several variables
57R40 Embeddings

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.

Keywords : Algebraic Equations, Implicit Function Theorem, Projections, Embeddings, Embedded Submanifolds

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