Global Invertibility Research Articles

A global invertibility theorem for manifolds with boundary

SynopsisA local homeomorphism from a compact, connected manifold with boundary to a simply connected manifold without boundary is shown to be one-to-one if it is one-to-one on each component of the boundary.

Global invertibility of Sobolev functions and the interpenetration of matter

SynopsisA global inverse function theorem is established for mappingsu: Ω → ℝn, Ω ⊂ ℝnbounded and open, belonging to the Sobolev spaceW1.p(Ω),p>n. The theorem is applied to the pure displacement boundary value problem of nonlinear elastostatics, the conclusion being that there is no interpenetration of matter for the energy-minimizing displacement field.

Invertibility of Lipschitz Continuous Mappings and Its Application to Electrical Network Equations

Conditions for unique solvability of nonlinear simultaneous equations satisfying Lipschitz conditions and an application to nonlinear network equations are proposed. It is shown that the global invertibility of a Lipschitz continuous mapping f of $\mathbb{R}^n $ into itself and the Lipschitz continuity of $f^{ - 1} $ are verified by investigating the positivity of some principal minors of the Jacobian matrix of f in spite of the existence of nondifferentiable points of f. This is a generalization of previous works, especially Fujisawa and Kuh’s theorem, obtained for continuously differentiable or piecewise-linear mappings. This generalization is given by employing the Lebesgue integration of the Jacobian matrix over an open interval of $\mathbb{R}^n $. This result is applied to network equations, both resistive and dynamical; especially to the latter, the Lipschitz continuity of such inverse mappings is of great importance to guarantee the uniqueness of the solutions. In addition, network examples for which the above result is useful are given, and it is demonstrated that simple conditions for the unique solvability are obtained in terms of differential coefficients of network element characteristics.

A formulation of nonlinear dynamic networks

Some properties on the structure of dynamic equations and a method of systematic formulations of a fairly general class of nonlinear RLC networks and lumped parameter active networks are presented. It is shown that by a differential and integral transformation such that each range of network variables becomes a standard function space, the equation and solution are described by synthetic operators of a certain standard type. The solution is then, definitely computable. Conditions for the unique solvability, which are given by global invertibility of the synthetic operator describing the equation, are derived from the contraction mapping theorem in a Banach space. These conditions are finally expressed in terms of differential coefficients of branch characteristics.

Conditions for the Existence of a Global Inverse of Semiconductor-Device Nonlinear-Network Operators

Necessary and sufficient conditions are presented for the global invertibility of a certain general operator that arises in the analysis of realistically modeled nonlinear networks containing transistors and diodes. It is often possible to determine by inspection whether or not a certain key condition is satisfied; whenever the conditions are satisfied there are convergent algorithms for computing the solution of the associated network problem.

Exponential representations for linear systems

Two aspects of the theory of exponential representation for the solution of linear differential equation systems are investigated. The first result for (2; 2)-order systems develops a relationship between the exponents of the system representation and a solution of an appropriately defined Riccati equation. A similar result is known for classical representations, thus providing a direct connection between the two theories. Second, as a collateral result, it is shown that the global invertibility of the defining exponential relationships does not necessarily imply that the representation is global.

Necessary and sufficient conditions for the global invertibility of certain nonlinear operators that arise in the analysis of networks

For a large class of operators of the form <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[F(\cdot)+A],</tex> in which <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</tex> is a not necessarily nonsingular real <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n \times n</tex> matrix, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F(\cdot)</tex> is a diagonal strictly monotone-increasing mapping of the set of all real <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> vectors <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E^{n}</tex> onto an open subset of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E^{n}</tex> , we give necessary and sufficient conditions under which <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F(\cdot)+A]</tex> possesses a global inverse on <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E^{n}</tex> . Operators of the type <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[F(\cdot)+A]</tex> frequently arise in the analysis of nonlinear networks and are encountered in other areas as well. In particular, for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</tex> the short-circuit conductance matrix of a resistance network, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F(x)</tex> the transpose of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(f_{1}(x_{1}), f_{2}(x_{2}), \cdots , f_{n}(x_{n}))</tex> for all <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x \in E^{n}</tex> in which the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f_{j}(\cdot)</tex> are the usual exponential diode functions, we give a complete solution to the problem of determining whether or not <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[F(\cdot)+A]</tex> possesses an inverse on <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E^{n}</tex> .

On Solving Nonlinear Equations with a One-Parameter Operator Imbedding

One parameter operator imbedding to modify Newton method for solution of nonlinear equations