Fangqi Lou

A Lovász Theta Bound for a Graph Saturation Parameter

For a finite simple graph G on n vertices, let μ(G) be the minimum trace of a real symmetric matrix X satisfying 0 ≼ X ≼ I, Xᵢⱼ = 0 on the edges of G, and ⟨J,X⟩ = n. We prove that μ(G) ≤ ϑ(complement G) for every G, answering affirmatively question (iii) in Section 7 of Carli Sil...

Shadow-flow rigidity of D_n polytopes and quartic descent at the 24-cell

For every n >= 4, we prove that the D_n root polytope and its polar have only globally linear odd admissible vertex speeds in every nonzero direction. Admissibility means affine compatibility on every facet transverse to that direction. For the regular 24-cell, every fixed nontri...

Shadow-flow rigidity of D_n polytopes and quartic descent at the 24-cell

For every n >= 4, we prove that the D_n root polytope and its polar have only globally linear odd admissible vertex speeds in every nonzero direction. Admissibility means affine compatibility on every facet transverse to that direction. For the regular 24-cell, every fixed nontri...

Double-star labellings of defect one can require arbitrarily many edge-difference changes

For a bijective labelling of an n-vertex tree by 0,...,n-1, define its defect as n-1 minus the number of distinct absolute edge differences. We compare two labellings by counting the fixed edges on which their differences disagree. For every integer s >= 2, we construct a double...

A Lovász Theta Bound for a Graph Saturation Parameter

For a finite simple graph G on n vertices, let μ(G) be the minimum trace of a real symmetric matrix X satisfying 0 ≼ X ≼ I, Xᵢⱼ = 0 on the edges of G, and ⟨J,X⟩ = n. We prove that μ(G) ≤ ϑ(complement G) for every G, answering affirmatively question (iii) in Section 7 of Carli Sil...

Bruhat Monotonicity of the Pan–Skandera–Wang Bijection

Pan, Skandera, and Wang defined a bijection from permutations preserving two balanced consecutive blocks to permutations preserving the parity of every position, and conjectured that each input lies below its image in the strong Bruhat order. We prove Conjecture 7.8 of arXiv:2606...

Double-star labellings of defect one can require arbitrarily many edge-difference changes

For a bijective labelling of an n-vertex tree by 0,...,n-1, define its defect as n-1 minus the number of distinct absolute edge differences. We compare two labellings by counting the fixed edges on which their differences disagree. For every integer s >= 2, we construct a double...

Bruhat Monotonicity of the Pan–Skandera–Wang Bijection

Pan, Skandera, and Wang defined a bijection from permutations preserving two balanced consecutive blocks to permutations preserving the parity of every position, and conjectured that each input lies below its image in the strong Bruhat order. We prove Conjecture 7.8 of arXiv:2606...

The Mean-Field Limit of Online Rademacher Vector Balancing

Let X₁, …, Xₙ be independent uniform vectors in {−1, 1}ⁿ. At time k, a sign is chosen after observing X₁, …, Xₖ. We identify the limit of the minimum expected terminal maximum norm of n^(−1/2) ∑ₖ₌₁ⁿ εₖXₖ with the mean-field control value introduced by Fiedler, Jackson, Lacker, an...

Positive Lower Natural Density Does Not Ensure Carleson Frame Sampling

We construct a locally finite, strictly increasing real sampling sequence with lower natural density 1/3 and upper natural density 4/3 for which a sampled Carleson orbit is Bessel but has no positive frame lower bound. The diagonal spectrum is fixed explicitly as z_j = exp(-2^(-j...

The Mean-Field Limit of Online Rademacher Vector Balancing

Let X₁, …, Xₙ be independent uniform vectors in {−1, 1}ⁿ. At time k, a sign is chosen after observing X₁, …, Xₖ. We identify the limit of the minimum expected terminal maximum norm of n^(−1/2) ∑ₖ₌₁ⁿ εₖXₖ with the mean-field control value introduced by Fiedler, Jackson, Lacker, an...

Positive Lower Natural Density Does Not Ensure Carleson Frame Sampling

We construct a locally finite, strictly increasing real sampling sequence with lower natural density 1/3 and upper natural density 4/3 for which a sampled Carleson orbit is Bessel but has no positive frame lower bound. The diagonal spectrum is fixed explicitly as z_j = exp(-2^(-j...