What is the least integer $\text{sd}(n)$ such that every graph on $n$ vertices has fractional chromatic number $p /q$, where $p$ and $q$ are positive integers and $q \le \text{sd}(n)$? An upper bound on the determinants of Hadamard matrices implies that $\text{sd}(n)\le 2^{-n}(n+1)^{(n+1) /2}$. The only known lower bound on $\text{sd}(n)$ that is exponential in $n$ (asymptotically, roughly $1.346^n/\sqrt{\log n}$) was obtained using an iterated Mycielski construction [D. C. Fisher, J. Graph Theory 20 (1995), 403-409]. We improve on this bound by constructing a family of graphs which shows that $\text{sd}(n) \geq 2^{n/2}$.
A meander is a pair consisting of a straight line in the plane and of a smooth closed curve transversally intersecting the line, considered up to an isotopy preserving the straight line. The number of meanders with 2 N intersections grows exponentially with N , but asymptotics still remains conjectural. A meander defines a pair of transversally intersecting simple closed curves on a 2 -sphere. In this paper we consider such pairs on a closed oriented surface of arbitrary genus. The number of these higher genus meanders still admits exponential upper and lower bounds as N grows. Fixing the number n of bigons in the complement to the union of the two curves, we compute the precise asymptotics of genus g meanders with n bigons and with at most 2 N intersections and show that it grows polynomially with N . We obtain a similar result in the case of oriented curves.
In this paper, we study the sensitivity of the Lempel–Ziv 77 (LZ77) algorithm under multiple substitutions. Akagi, Funakoshi and Inenaga [JIC, 2023] showed that when applying a single edit the size z of the LZ77 compression of any string increases by a factor of at most 2. This raises the question of what happens under multiple edits. The result of Akagi et al. implies that the compression increases by a factor of at most 2^k. We conjectured that the optimal multiplicative factor for the sensitivity of LZ77 under k edits was Θ(k). Rather surprisingly, we show that this sensitivity is constant, with k appearing in a linear additive term. Namely, we show that if w' is at Hamming distance at most k from w, we have C_LZ77(w') ≤ 3 · C_LZ77(w) + 4k. This bound is almost tight: we exhibit words for which C_LZ77(w') ≥(3· C_LZ77(w)+k)(1-ε). This result contrasts with Lempel–Ziv 78, where a single edit can significantly deteriorate compressibility, a phenomenon known as a one-bit catastrophe. We further refine this bound, focusing on the coefficient 3 in front of C_LZ77(w), and establish a trichotomy based on the compressibility of w. More precisely, for n ≫ k^7, we prove the following bounds: - if C_LZ77(w) ≲ k^3/2√(n), the compression may increase by up to a factor of ≈ 3, - if k^3/2√(n)≲ C_LZ77(w) ≲ k^1/3n^2/3, this factor is at most ≈ 2, - if C_LZ77(w) ≳ k^1/3n^2/3, the factor is at most ≈ 1.
BACKGROUND:Improved Physics models for the elastic and inelastic interactions of electrons in liquid water have recently been developed and implemented in the Geant4-DNA toolkit, allowing detailed electron transport simulations up to MeV energies. PURPOSE:To present a preliminary performance evaluation of a new set of Geant4-DNA Physics models, which will become the default Physics list for liquid water in the upcoming Geant4 public releases. METHODS:A new set of Physics models (constructor), denoted as « Option4 X », is implemented in Geant4-DNA that combines a newly developed inelastic model with two alternative models for the simulation of elastic scattering, one based on the ELSEPA software and another one based on the analytic Uehara Screened Rutherford model. The inelastic model includes all the low-energy developments of the existing « Option4 » Physics constructor, upgrades its internal consistency (via sum rules), and extends its application from 10 keV to 10 MeV using the Relativistic Plane Wave Born Approximation (RPWBA). RESULTS:Preliminary results for electron dose point kernels (DPKs), frequency-mean (yF) and dose-mean (yD) lineal energies in nanometric spheres, S-values in spherical cells, and the W-value of liquid water are simulated, and a comparison is presented between the new Physics constructor, « Option4 X », the current default Physics constructor, « Option2 », and the low-energy Physics constructor, « Option4 ». Significant improvement is offered by « Option4 X » in the simulation of DPK and the W-value, which have been two long-lasting performance limitations of the current default Physics constructor, while the choice of elastic model has a relatively small influence. CONCLUSIONS:«Option4 X» improves and extends the current Geant4-DNA transport capabilities up to 10 MeV and will become the default Geant4-DNA Physics constructor for liquid water in the upcoming public releases of Geant4. This development will allow Geant4-DNA users, along with users of Monte Carlo software that are built on Geant4 (like GATE and TOPAS-nBio), to perform more accurate electron track structure simulations over a wider energy range (10 eV ‒ 10 MeV) and cover radiotherapeutic applications of track structure simulations not reachable until now.
