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Classification in Active Dimension 2 for Weighted Residual Dynamics

James Tian

Abstract

We study weighted residual dynamics associated with a rank-one projection in finite dimension. The iteration reduces, after finitely many steps, to a nonlinear recursion on a stabilized active subspace. We prove that this recursion can be classified when the active dimension is two: either a transverse reducing direction persists unchanged, or the coupled part collapses completely. As a consequence, we obtain a description of the limit in the active two-dimensional case and identify the threshold beyond which higher-dimensional behavior becomes more flexible.

Classification in Active Dimension 2 for Weighted Residual Dynamics

Abstract

We study weighted residual dynamics associated with a rank-one projection in finite dimension. The iteration reduces, after finitely many steps, to a nonlinear recursion on a stabilized active subspace. We prove that this recursion can be classified when the active dimension is two: either a transverse reducing direction persists unchanged, or the coupled part collapses completely. As a consequence, we obtain a description of the limit in the active two-dimensional case and identify the threshold beyond which higher-dimensional behavior becomes more flexible.
Paper Structure (4 sections, 9 theorems, 132 equations)

This paper contains 4 sections, 9 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Reduction to the Stabilized Active Block
  3. The Induced Weighted Recursion
  4. Classification in Active Dimension 2

Key Result

Lemma 2.1

Assume $\dim H=2$ and $P$ has rank-$1$. Then $R_{n}$ converges in norm to

Theorems & Definitions (22)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Example 2.4
  • proof
  • Proposition 3.1: Asymptotic block decoupling relative to the defect direction
  • proof
  • ...and 12 more