Tootfinder

A Simplified Proof for the Edge-Density of 4-Planar Graphs
Aaron B\"ungener
https://arxiv.org/abs/2509.20999 https://arxiv.org/pdf/2509.20999

A Simplified Proof for the Edge-Density of 4-Planar Graphs
A graph on $n \ge 3$ vertices drawn in the plane such that each edge is crossed at most four times has at most $6(n-2)$ edges -- this result proven by Ackerman is outstanding in the literature of beyond-planar graphs with regard to its tightness and the structural complexity of the graph class. We provide a much shorter proof while at the same time relaxing the conditions on the graph and its embedding, i.e., allowing multi-edges and non-simple drawings.

Tightness of SDP and Burer-Monteiro Factorization for Phase Synchronization in High-Noise Regime
Anderson Ye Zhang
https://arxiv.org/abs/2510.01522 https://

Tightness of SDP and Burer-Monteiro Factorization for Phase Synchronization in High-Noise Regime
We study the difference between the maximum likelihood estimation (MLE) and its semi-definite programming (SDP) relaxation for the phase synchronization problem, where $n$ latent phases are estimated based on pairwise observations corrupted by Gaussian noise at a level $σ$. While previous studies have established that SDP coincides with the MLE when $σ\lesssim \sqrt{n / \log n}$, the behavior in the high-noise regime $σ\gtrsim \sqrt{n / \log n}$ remains unclear. We address this gap by quanti…

Non-fibered strongly quasipositive links and tightness
Isacco Nonino, Miguel Orbegozo Rodriguez
https://arxiv.org/abs/2509.26183 https://arxiv.org/pdf/2509…

Non-fibered strongly quasipositive links and tightness
It is well known that for fibered links in $\mathbb{S}^3$ being strongly quasipositive and supporting a tight contact structure are equivalent notions (arXiv:math/0509499). In this note we analyze the relation between these two properties for non fibered links. A non fibered link (together with an incompressible Seifert surface) induces a natural partial open book (arXiv:2509.09615). We prove that strongly quasipositive links induce tight contact structures. We also show that, in contrast to th…

Thermodynamic Performance Limits for Score-Based Diffusion Models
Nathan X. Kodama, Michael Hinczewski
https://arxiv.org/abs/2510.06174 https://arxiv.org/p…

Thermodynamic Performance Limits for Score-Based Diffusion Models
We establish a fundamental connection between score-based diffusion models and non-equilibrium thermodynamics by deriving performance limits based on entropy rates. Our main theoretical contribution is a lower bound on the negative log-likelihood of the data that relates model performance to entropy rates of diffusion processes. We numerically validate this bound on a synthetic dataset and investigate its tightness. By building a bridge to entropy rates - system, intrinsic, and exchange entropy…

Certain results on selection principles associated with bornological structure in topological spaces
Debraj Chandra, Subhankar Das, Nur Alam
https://arxiv.org/abs/2511.04038 https://arxiv.org/pdf/2511.04038 https://arxiv.org/html/2511.04038
arXiv:2511.04038v1 Announce Type: new
Abstract: We study selection principles related to bornological covers in a topological space $X$ following the work of Aurichi et al., 2019, where selection principles have been investigated in the function space $C_\mathfrak{B}(X)$ endowed with the topology $\tau_\mathfrak{B}$ of uniform convergence on bornology $\mathfrak{B}$. We show equivalences among certain selection principles and present some game theoretic observations involving bornological covers. We investigate selection principles on the product space $X^n$ equipped with the product bornolgy $\mathfrak{B}^n$, $n\in \omega$. Considering the cardinal invariants such as the unbounding number ($\mathfrak{b}$), dominating numbers ($\mathfrak{d}$), pseudointersection numbers ($\mathfrak{p}$) etc., we establish connections between the cardinality of base of a bornology with certain selection principles. Finally, we investigate some variations of the tightness properties of $C_\mathfrak{B}(X)$ and present their characterizations in terms of selective bornological covering properties of $X$.
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Surgeries on knots and tight contact structures
Zhenkun Li, Shunyu Wan, Hugo Zhou
https://arxiv.org/abs/2510.05294 https://arxiv.org/pdf/2510.05294

Surgeries on knots and tight contact structures
For any knot $K$ in $S^3$ and any positive rational $r$, we show that smooth $(-r)$-surgery on $K$ always admits a tight contact structure. More specifically, the tightness is detected by the non-vanishing Heegaard Floer contact invariant.

Symmetry of concentration and scaling for self-bounding functions
George Crowley, I\~naki Esnaola
https://arxiv.org/abs/2509.22375 https://arxiv.org/pdf/25…

Symmetry of concentration and scaling for self-bounding functions
We prove generalised concentration inequalities for a class of scaled self-bounding functions of independent random variables, referred to as ${(M,a,b)}$ self-bounding. The scaling refers to the fact that the component-wise difference is upper bounded by an arbitrary positive real number $M$ instead of the case $M=1$ previously considered in the literature. Using the entropy method, we derive symmetric bounds for both the upper and lower tails, and study the tightness of the proposed bounds. Ou…