Leveraging recent advances in additive combinatorics, we exhibit explicit matrices satisfying the Restricted Isometry Property with better parameters. Namely, for $$\varepsilon=3.26\cdot 10^{-7}$$ , large $$k$$ and $$k^{2-\varepsilon} \le N\le k^{2+\varepsilon}$$ construct $$n \times N$$ RIP of order $$k = \Omega( n^{1/2+\varepsilon/4)}$$ .

The paper deals with the homogenization of a magneto-elastodynamics equation satisfied by displacement $$u_\varepsilon $$ an elastic body which is subjected to oscillating magnetic field $$B_\varepsilon generating Lorentz force $$\partial _t u_\varepsilon \times B_\varepsilon . When only depends on time or space, oscillations induce increase mass in homogenized equation. More generally, when ti...

We consider a conservation law model of traffic flow, where the velocity each car depends on weighted average density $\rho$ ahead. The averaging kernel is exponential type: $w_\varepsilon (s)=\varepsilon^{-1} e^{-s / \varepsilon}$. For any decreasing function $v$, we prove that, as $\varepsilon \to 0$, limit solutions to nonlocal equation coincides with unique entropy-admissible solution scala...

In 1991 De Giorgi conjectured that, given $\lambda >0$, if $\mu_\varepsilon$ stands for the density of Allen-Cahn energy and $v_\varepsilon$ represents its first variation, then $\int [v_\varepsilon^2 + \lambda] d\mu_\varepsilon$ should $\Gamma$-converge to $c\lambda \mathrm{Per}(E) k \mathcal{W}(\Sigma)$ some real constant $k$, where $\mathrm{Per}(E)$ is perimeter set $E$, $\Sigma=\partial E$,...

This paper investigates the propreties of persistence diagrams stemming from almost surely continuous random processes on [0, t]. We focus our study two variables which together characterize barcode: number points diagram inside a rectangle $$]\!-\!\infty ,x]\times [x+\varepsilon ,\infty [$$ , $$N^{x,x+\varepsilon }$$ and bars length $$\ge \varepsilon $$ $$N^\varepsilon . For with strong Markov...

Fix a word $w$ in free group $F$ on $r$ generators. A $w$-random permutation the symmetric $S_N$ is obtained by sampling independent uniformly random permutations $\sigma_{1},\ldots,\sigma_{r}\in S_{N}$ and evaluating $w\left(\sigma_{1},\ldots,\sigma_{r}\right)$. In [arXiv:1104.3991, arXiv:1202.3269] it was shown that average number of fixed points $1+\theta\left(N^{1-\pi\left(w\right)}\right)$...

List-decodability of Reed–Solomon codes has received a lot attention, but the best-possible dependence between parameters is still not well-understood. In this work, we focus on case where list-decoding radius form $r=1-\varepsilon $ ...