Approximate integer programming is the following: For a given convex body
Using numerical integration, in 1969 Penston (Mon Not R Astr Soc 144:425–448, 1969) and Larson (Mon Not R Astr Soc 145:271–295, 1969) independently discovered a self-similar solution describing the collapse of a self-gravitating asymptotically flat fluid with the isothermal equation of state
- Award ID(s):
- 1810868
- NSF-PAR ID:
- 10283355
- Publisher / Repository:
- Springer Science + Business Media
- Date Published:
- Journal Name:
- Communications in Mathematical Physics
- Volume:
- 386
- Issue:
- 3
- ISSN:
- 0010-3616
- Page Range / eLocation ID:
- p. 1551-1601
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract A long-standing problem in mathematical physics is the rigorous derivation of the incompressible Euler equation from Newtonian mechanics. Recently, Han-Kwan 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
N particles interacting in ,$${\mathbb {T}}^d$$ , via Newton’s second law through a$$d\ge 2$$ supercritical mean-field limit . Namely, the coupling constant in front of the pair potential, which is Coulombic, scales like$$\lambda $$ for some$$N^{-\theta }$$ , in contrast to the usual mean-field scaling$$\theta \in (0,1)$$ . Assuming$$\lambda \sim N^{-1}$$ , they showed that the empirical measure of the system is effectively described by the solution to the Euler equation as$$\theta \in (1-\frac{2}{d(d+1)},1)$$ . Han-Kwan and Iacobelli asked if their range for$$N\rightarrow \infty $$ was optimal. We answer this question in the negative by showing the validity of the incompressible Euler equation in the limit$$\theta $$ for$$N\rightarrow \infty $$ . Our proof is based on Serfaty’s modulated-energy method, but compared to that of Han-Kwan and Iacobelli, crucially uses an improved “renormalized commutator” estimate to obtain the larger range for$$\theta \in (1-\frac{2}{d},1)$$ . Additionally, we show that for$$\theta $$ , one cannot, in general, expect convergence in the modulated energy notion of distance.$$\theta \le 1-\frac{2}{d}$$ -
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p and a homogeneous system ofm linear equations for$$a_{j,1}x_1+\dots +a_{j,k}x_k=0$$ with coefficients$$j=1,\dots ,m$$ . Suppose that$$a_{j,i}\in \mathbb {F}_p$$ , that$$k\ge 3m$$ for$$a_{j,1}+\dots +a_{j,k}=0$$ and that every$$j=1,\dots ,m$$ minor of the$$m\times m$$ matrix$$m\times k$$ is non-singular. Then we prove that for any (large)$$(a_{j,i})_{j,i}$$ n , any subset of size$$A\subseteq \mathbb {F}_p^n$$ contains a solution$$|A|> C\cdot \Gamma ^n$$ to the given system of equations such that the vectors$$(x_1,\dots ,x_k)\in A^k$$ are all distinct. Here,$$x_1,\dots ,x_k\in A$$ C and are constants only depending on$$\Gamma $$ p ,m andk such that . The crucial point here is the condition for the vectors$$\Gamma in the solution$$x_1,\dots ,x_k$$ to be distinct. If we relax this condition and only demand that$$(x_1,\dots ,x_k)\in A^k$$ are not all equal, then the statement would follow easily from Tao’s slice rank polynomial method. However, handling the distinctness condition is much harder, and requires a new approach. While all previous combinatorial applications of the slice rank polynomial method have relied on the slice rank of diagonal tensors, we use a slice rank argument for a non-diagonal tensor in combination with combinatorial and probabilistic arguments.$$x_1,\dots ,x_k$$ -
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