Exponential decay of scattering coefficients

We study an aspect of the following general question: which properties of a signal can be characterized by its scattering transform? We show that the energy contained in high order scattering coefficients is upper bounded by the energy contained in the high frequencies of the signal. This result links the decay of the scattering coefficients of a signal with the decay of its Fourier transform. Additionally, it allows to generalize some results of Mallat (2012), by relaxing the admissibility condition on the wavelet family

Le lemme fondamental pond\'er\'e pour le groupe m\'etaplectique

We state a variant of Arthur's weighted fundamental lemma for the metaplectic group of Weil, which will be an essential ingredient of the stable trace formula. Over a local field of large enough residual characteristic, we give a proof using the method of descent, which is conditional upon the weighted nonstandard fundamental lemma on Lie algebras. In view of the works of Chaudouard and Laumon, this condition is expected to hold.Comment: 41 page

Effects of Homework on American Elementary-School Students\u27 Subjective Well-Being

Homework has always been a hotly-debated issue in the US because of the significant role it plays in children’s lives. Despite the wealth of research conducted on the relationship between homework and academic achievement, there has been almost no research on the relationship between homework and student well-being. The purpose of this proposed study is to investigate how homework affects American elementary-school students’ subjective well-being. Utilizing a correlational longitudinal design, students aged 6 to 10 will respond to orally-administered subjective well-being measures at three timepoints throughout the school year, and parents will submit online weekly reports of how much time their child spends on homework throughout the school year. It is predicted that students who spend more time on homework will have lower levels of subjective well-being than their peers who spend less time on homework. It is also predicted that student socioeconomic status and learning disability status will moderate the relationship between time spent on homework and subjective well-being, respectively. The research findings may shed light on previously understudied effects of homework, and may help policymakers, school officials, and teachers make informed decisions regarding the assignment of homework at the elementary-school level

Phase retrieval with random Gaussian sensing vectors by alternating projections

We consider a phase retrieval problem, where we want to reconstruct a $n$-dimensional vector from its phaseless scalar products with $m$ sensing vectors. We assume the sensing vectors to be independently sampled from complex normal distributions. We propose to solve this problem with the classical non-convex method of alternating projections. We show that, when $m\geq Cn$ for $C$ large enough, alternating projections succeed with high probability, provided that they are carefully initialized. We also show that there is a regime in which the stagnation points of the alternating projections method disappear, and the initialization procedure becomes useless. However, in this regime, $m$ has to be of the order of $n^2$. Finally, we conjecture from our numerical experiments that, in the regime $m=O(n)$, there are stagnation points, but the size of their attraction basin is small if $m/n$ is large enough, so alternating projections can succeed with probability close to $1$ even with no special initialization