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Connections between Bressan's Mixing Conjecture, the Branched Optimal Transport and Combinatorial Optimization

Bohan Zhou

Abstract

We investigate the 1D version of the notable Bressan's mixing conjecture, and introduce various formulation in the classical optimal transport setting, the branched optimal transport setting and a combinatorial optimization. In the discrete case of the combinatorial problem, we prove the number of admissible solutions is on the Catalan number. Our investigation sheds light on the intricate relationship between mixing problem in the fluid dynamics and many other popular fields, leaving many interesting open questions in both theoretical and practical applications across disciplines.

Connections between Bressan's Mixing Conjecture, the Branched Optimal Transport and Combinatorial Optimization

Abstract

We investigate the 1D version of the notable Bressan's mixing conjecture, and introduce various formulation in the classical optimal transport setting, the branched optimal transport setting and a combinatorial optimization. In the discrete case of the combinatorial problem, we prove the number of admissible solutions is on the Catalan number. Our investigation sheds light on the intricate relationship between mixing problem in the fluid dynamics and many other popular fields, leaving many interesting open questions in both theoretical and practical applications across disciplines.
Paper Structure (7 sections, 6 theorems, 32 equations, 9 figures)

This paper contains 7 sections, 6 theorems, 32 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Bressan's Mixing in 1D
  3. Optimal Transport Formulation
  4. Branched Optimal Transport
  5. Series representation of book-shifting problem
  6. Translation into branched transport model
  7. Combinatorics Formulation

Key Result

Lemma 2.1

For any sorting plan $\{\rho_0,\rho_1,\cdots,\rho_k,\cdots,\rho_n\}$, the cost of sorting satisfies:

Figures (9)

  • Figure 1: Mixing process in BV space
  • Figure 2: Bressan's mixing in 1D
  • Figure 3: $a_j$ permutes with $-b_j$
  • Figure 4: Permutations on the end terms
  • Figure 5: Branched transport graph
  • ...and 4 more figures

Theorems & Definitions (17)

  • Lemma 2.1: bressan2003lemma
  • proof
  • Remark 3.1
  • Remark 4.1
  • Example 4.1: Translate operations on series into a family tree
  • Remark 4.2
  • Remark 4.3
  • Example 5.1
  • Lemma 5.1
  • proof
  • ...and 7 more