A longstanding problem in mathematical physics is the rigorous derivation of the incompressible Euler equation from Newtonian mechanics. Recently, HanKwan and Iacobelli (Proc Am Math Soc 149:3045–3061, 2021) showed that in the monokinetic regime, one can directly obtain the Euler equation from a system of
We improve the previously best known upper bounds on the sizes of
 NSFPAR ID:
 10469879
 Publisher / Repository:
 Springer Science + Business Media
 Date Published:
 Journal Name:
 Mathematische Annalen
 ISSN:
 00255831
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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Abstract N particles interacting in ,$${\mathbb {T}}^d$$ ${T}^{d}$ , via Newton’s second law through a$$d\ge 2$$ $d\ge 2$supercritical meanfield limit . Namely, the coupling constant in front of the pair potential, which is Coulombic, scales like$$\lambda $$ $\lambda $ for some$$N^{\theta }$$ ${N}^{\theta}$ , in contrast to the usual meanfield scaling$$\theta \in (0,1)$$ $\theta \in (0,1)$ . Assuming$$\lambda \sim N^{1}$$ $\lambda \sim {N}^{1}$ , they showed that the empirical measure of the system is effectively described by the solution to the Euler equation as$$\theta \in (1\frac{2}{d(d+1)},1)$$ $\theta \in (1\frac{2}{d(d+1)},1)$ . HanKwan and Iacobelli asked if their range for$$N\rightarrow \infty $$ $N\to \infty $ was optimal. We answer this question in the negative by showing the validity of the incompressible Euler equation in the limit$$\theta $$ $\theta $ for$$N\rightarrow \infty $$ $N\to \infty $ . Our proof is based on Serfaty’s modulatedenergy method, but compared to that of HanKwan and Iacobelli, crucially uses an improved “renormalized commutator” estimate to obtain the larger range for$$\theta \in (1\frac{2}{d},1)$$ $\theta \in (1\frac{2}{d},1)$ . Additionally, we show that for$$\theta $$ $\theta $ , one cannot, in general, expect convergence in the modulated energy notion of distance.$$\theta \le 1\frac{2}{d}$$ $\theta \le 1\frac{2}{d}$ 
Abstract The free multiplicative Brownian motion
is the large$$b_{t}$$ ${b}_{t}$N limit of the Brownian motion on in the sense of$$\mathsf {GL}(N;\mathbb {C}),$$ $\mathrm{GL}(N\u037eC),$ distributions. The natural candidate for the large$$*$$ $\ast $N limit of the empirical distribution of eigenvalues is thus the Brown measure of . In previous work, the second and third authors showed that this Brown measure is supported in the closure of a region$$b_{t}$$ ${b}_{t}$ that appeared in the work of Biane. In the present paper, we compute the Brown measure completely. It has a continuous density$$\Sigma _{t}$$ ${\Sigma}_{t}$ on$$W_{t}$$ ${W}_{t}$ which is strictly positive and real analytic on$$\overline{\Sigma }_{t},$$ ${\overline{\Sigma}}_{t},$ . This density has a simple form in polar coordinates:$$\Sigma _{t}$$ ${\Sigma}_{t}$ where$$\begin{aligned} W_{t}(r,\theta )=\frac{1}{r^{2}}w_{t}(\theta ), \end{aligned}$$ $\begin{array}{c}{W}_{t}(r,\theta )=\frac{1}{{r}^{2}}{w}_{t}\left(\theta \right),\end{array}$ is an analytic function determined by the geometry of the region$$w_{t}$$ ${w}_{t}$ . We show also that the spectral measure of free unitary Brownian motion$$\Sigma _{t}$$ ${\Sigma}_{t}$ is a “shadow” of the Brown measure of$$u_{t}$$ ${u}_{t}$ , precisely mirroring the relationship between the circular and semicircular laws. We develop several new methods, based on stochastic differential equations and PDE, to prove these results.$$b_{t}$$ ${b}_{t}$ 
Abstract Sequence mappability is an important task in genome resequencing. In the (
k ,m )mappability problem, for a given sequenceT of lengthn , the goal is to compute a table whosei th entry is the number of indices such that the length$$j \ne i$$ $j\ne i$m substrings ofT starting at positionsi andj have at mostk mismatches. Previous works on this problem focused on heuristics computing a rough approximation of the result or on the case of . We present several efficient algorithms for the general case of the problem. Our main result is an algorithm that, for$$k=1$$ $k=1$ , works in$$k=O(1)$$ $k=O\left(1\right)$ space and, with high probability, in$$O(n)$$ $O\left(n\right)$ time. Our algorithm requires a careful adaptation of the$$O(n \cdot \min \{m^k,\log ^k n\})$$ $O(n\xb7min\{{m}^{k},{log}^{k}n\})$k errata trees of Cole et al. [STOC 2004] to avoid multiple counting of pairs of substrings. Our technique can also be applied to solve the allpairs Hamming distance problem introduced by Crochemore et al. [WABI 2017]. We further develop time algorithms to compute$$O(n^2)$$ $O\left({n}^{2}\right)$all (k ,m )mappability tables for a fixedm and all or a fixed$$k\in \{0,\ldots ,m\}$$ $k\in \{0,\dots ,m\}$k and all . Finally, we show that, for$$m\in \{k,\ldots ,n\}$$ $m\in \{k,\dots ,n\}$ , the ($$k,m = \Theta (\log n)$$ $k,m=\Theta (logn)$k ,m )mappability problem cannot be solved in strongly subquadratic time unless the Strong Exponential Time Hypothesis fails. This is an improved and extended version of a paper presented at SPIRE 2018. 
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