Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Symbolic Computations of the Two-Colored Diagrams for Central Configurations of the Planar N-vortex Problem

Xiang Yu, Shuqiang Zhu

TL;DR

The paper develops a symbolic-computation framework to enumerate two-colored diagrams arising in the singular-sequence approach to the planar $N$-vortex finiteness problem. By formulating the central-configuration equations as a complex, polynomial extended system and introducing $zw$-matrices, the authors translate diagrammatic rules into matrix constraints and outline an algorithm to exhaustively identify admissible diagrams. They perform a detailed $N=5$ case, obtaining 22 viable diagrams from 31 candidates and excluding 9 diagrams as impossible, illustrating the method’s feasibility and its potential to scale to larger $N$. The work advances finiteness results for vortex relative equilibria and collapse configurations by providing a practical, symbolic route to classify non-finite configurations via two-colored diagrams, building on and adapting prior N-body techniques. The approach has implications for rigorous finiteness proofs and could guide future investigations into higher-vortex systems.

Abstract

We apply the singular sequence method to investigate the finiteness problem for stationary configurations of the planar N-vortex problem. The initial step of the singular sequence method involves identifying all two-colored diagrams. These diagrams represent potential scenarios where finiteness may fail. We develop a symbolic computation algorithm to determine all two-colored diagrams for central configurations of the planar N-vortex problem.

Symbolic Computations of the Two-Colored Diagrams for Central Configurations of the Planar N-vortex Problem

TL;DR

The paper develops a symbolic-computation framework to enumerate two-colored diagrams arising in the singular-sequence approach to the planar -vortex finiteness problem. By formulating the central-configuration equations as a complex, polynomial extended system and introducing -matrices, the authors translate diagrammatic rules into matrix constraints and outline an algorithm to exhaustively identify admissible diagrams. They perform a detailed case, obtaining 22 viable diagrams from 31 candidates and excluding 9 diagrams as impossible, illustrating the method’s feasibility and its potential to scale to larger . The work advances finiteness results for vortex relative equilibria and collapse configurations by providing a practical, symbolic route to classify non-finite configurations via two-colored diagrams, building on and adapting prior N-body techniques. The approach has implications for rigorous finiteness proofs and could guide future investigations into higher-vortex systems.

Abstract

We apply the singular sequence method to investigate the finiteness problem for stationary configurations of the planar N-vortex problem. The initial step of the singular sequence method involves identifying all two-colored diagrams. These diagrams represent potential scenarios where finiteness may fail. We develop a symbolic computation algorithm to determine all two-colored diagrams for central configurations of the planar N-vortex problem.

Paper Structure

This paper contains 11 sections, 22 theorems, 62 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Basic notations
  3. Singular sequences for central configurations and coloring rules
  4. Central configurations of the planar N-vortex problem
  5. Singular sequences
  6. The two-colored diagrams
  7. Constraints when some sub-diagrams appear
  8. Algorithm for the two-colored diagrams
  9. Matrix rules
  10. Algorithm for determining $zw$-diagrams
  11. Example: $N=5$

Key Result

Lemma 3.4

Albouy2012Finiteness Let $\mathcal{X}$ be a closed algebraic subset of $\mathbb{C}^m$ and $f:\mathbb{C}^m\rightarrow \mathbb{C}$ be a polynomial. Either the image $F(\mathcal{X})\subset\mathbb{ C}$ is a finite set, or it is the complement of a finite set. In the second case one says that f is domina

Figures (3)

  • Figure 1: On the left, vertices 1,2 are $z$-circled, and a $z$-edge is between them; In the middle, vertices 1,2 are $z$- and $w$-circled, and a $zw$-edge is between them; On the right, vertices 1,2 are $w$-circled, and a $w$-edge is between them;
  • Figure 2: The nine impossible diagrams.
  • Figure 3: The 22 possible diagrams.

Theorems & Definitions (46)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3: Singular sequence
  • Lemma 3.4
  • Proposition 3.5: Estimate
  • Remark 3.6
  • ...and 36 more