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. 
Abstract It has been recently established in David and Mayboroda (Approximation of green functions and domains with uniformly rectifiable boundaries of all dimensions.
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Abstract Finite volume, weighted essentially nonoscillatory (WENO) schemes require the computation of a smoothness indicator. This can be expensive, especially in multiple space dimensions. We consider the use of the simple smoothness indicator
, where$$\sigma ^{\textrm{S}}= \frac{1}{N_{\textrm{S}}1}\sum _{j} ({\bar{u}}_{j}  {\bar{u}}_{m})^2$$ ${\sigma}^{\text{S}}=\frac{1}{{N}_{\text{S}}1}{\sum}_{j}{({\overline{u}}_{j}{\overline{u}}_{m})}^{2}$ is the number of mesh elements in the stencil,$$N_{\textrm{S}}$$ ${N}_{\text{S}}$ is the local function average over mesh element$${\bar{u}}_j$$ ${\overline{u}}_{j}$j , and indexm gives the target element. Reconstructions utilizing standard WENO weighting fail with this smoothness indicator. We develop a modification of WENOZ weighting that gives a reliable and accurate reconstruction of adaptive order, which we denote as SWENOZAO. We prove that it attains the order of accuracy of the large stencil polynomial approximation when the solution is smooth, and drops to the order of the small stencil polynomial approximations when there is a jump discontinuity in the solution. Numerical examples in one and two space dimensions on general meshes verify the approximation properties of the reconstruction. They also show it to be about 10 times faster in two space dimensions than reconstructions using the classic smoothness indicator. The new reconstruction is applied to define finite volume schemes to approximate the solution of hyperbolic conservation laws. Numerical tests show results of the same quality as standard WENO schemes using the classic smoothness indicator, but with an overall speedup in the computation time of about 3.5–5 times in 2D tests. Moreover, the computational efficiency (CPU time versus error) is noticeably improved.