implicit function theorem

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implicit function theorem

[im′plis·ət ¦fəŋk·shən ‚thir·əm]

(mathematics)

A theorem that gives conditions under which an equation in variables x and y may be solved so as to express y directly as a function of x ; it states that if F (x,y) and ∂ F (x,y)/∂ y are continuous in a neighborhood of the point (x₀, y₀) and if F (x,y) = 0 and ∂ F (x,y)/∂ y ≠ 0, then there is a number ε > 0 such that there is one and only one function ƒf(x) that is continuous and satisfies F [x,ƒ(x)] = 0 for | x-x₀| < ε,="" and="" satisfies="">x₀) = y₀.

References in periodicals archive ?

By invoking the Implicit Function Theorem, there exists a function f such that the identity [y'.sub.1] - E'Qy = f(t,Qy) holds.

RUNGE-KUTTA METHODS REVISITED FOR A CLASS OF STRUCTURED STRANGENESS-FREE DIFFERENTIAL-ALGEBRAIC EQUATIONS

To illustrate that [GAMMA] (V, [[PHI].sup.*.sub.0], [[PHI].sup.*.sub.1]) satisfies the implicit function theorem, the following theorem is given.

Adaptive Neural Control of Hypersonic Vehicles with Actuator Constraints

and the implicit function theorem, there exists a subfamily of variations satisfying restriction equation (43).

Variational Problems with Partial Fractional Derivative: Optimal Conditions and Noether's Theorem

and using the analytic implicit function theorem, one solves w in terms of z, [bar.z], and [bar.w] getting a representation of M as

On Propagation of Sphericity of Real Analytic Hypersurfaces across Levi Degenerate Loci

By Propositions 3.1, 4.1 we can apply the implicit function theorem (see, e.g., [4, Theorem 1.20]) to conclude that F maps a sufficiently small neighborhood of a positive, constant function c in [C.sup.[alpha]][(0, 1).sub.0] homeomorphically onto a neighborhood of [square root of (2c')] in [C.sup.[alpha]+[1/2]] [(0, 1).sub.0].

Determination of a nonlinearity from blow-up time

Advanced topics included vector-valued functions, the implicit function theorem, extremal problems, matrix-valued holomorphic functions, matrix equations, realization theory, eigenvalue location and zero location problems, convexity, and some special results relating to matrices with nonnegative entries.

Linear Algebra in Action, 2nd Edition

The implicit function theorem is applied in a manner that requires further qualification because the optimization problem is of an unconstrained kind without any redundant variables.

Numerical reduced variable optimization methods via implicit functional dependence with applications

By implicit function theorem, there exists [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that (10) has a unique solution ([lambda], [M.sub.2], [M.sub.3]) satisfying [lambda], [M.sub.2] > 0 and [M.sub.3] > 0.

On the existence of central configurations of 2k + 2p + 2/-body problems

The continuity of the constant equilibria follows from the Implicit Function Theorem and the hypothesis of normal hyperbolicity.

Normal hyperbolicity and continuity of global attractors for a nonlocal evolution equations

We prove that the first-order condition is an implicit function [R.sub.i] = [R.sub.i]([R.sub.j]), except possibly at a point of measure zero using the Implicit Function Theorem. We first establish two lemmas where we define

Incorporating policymaker costs and political competition into rent-seeking games

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