The Constructive Implicit Function Theorem and Proof in Logistic Mixtures

The Constructive Implicit Function Theorem and Proof in Logistic Mixtures

There is the work by Bridges et al (1999) on the key features of a constructive proof of the implicit function theorem, including some applications to physics and mechanics. For mixtures of logistic distributions such information is lacking, although a special instance of the implicit function theorem prevails therein. The theorem is needed to see that the ridgeline function, which carries info...

Constructive Proof of the Carpenter’s Theorem

We give a constructive proof of Carpenter’s Theorem due to Kadison [14, 15]. Unlike the original proof our approach also yields the real case of this theorem. 1. Kadison’s theorem In [14] and [15] Kadison gave a complete characterization of the diagonals of orthogonal projections on a Hilbert space H. Theorem 1.1 (Kadison). Let {di}i∈I be a sequence in [0, 1]. Define a = ∑ di<1/2 di and b = ∑ d...

The Implicit Function Theorem and Implicit Parametrizations∗

We discuss a differential equations treatment of the implicit functions problem. Our approach allows a precise and complete description of the solution, of continuity and differentiability properties. The critical case is also considered. The investigation is devoted to dimension two and three, but extensions to higher dimension are possible. MSC: 26B10, 34A12, 53A05. keywords: implicit functio...

The Constructive Implicit Function Theorem and Applications in Mechanics Douglas Bridges University of Waikato

We examine some ways of proving the Implicit Function Theorem and the Inverse Function Theorem within Bishop's constructive mathematics. Section 2 contains a new, entirely constructive proof of the Implicit Function Theorem. The paper ends with some comments on the application of the Implicit Function Theorem in classical mechanics.

A Constructive Proof of Gleason's Theorem