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Abstract We obtain new optimal estimates for the$$L^{2}(M)\to L^{q}(M)$$ $L^2\left(M\right)\to L^q\left(M\right)$,$$q\in (2,q_{c}]$$ $q\in (2,q_c]$,$$q_{c}=2(n+1)/(n-1)$$ $q_c=2(n+1)/(n-1)$, operator norms of spectral projection operators associated with spectral windows$$[\lambda ,\lambda +\delta (\lambda )]$$ $[\lambda ,\lambda +\delta (\lambda \left)\right]$, with$$\delta (\lambda )=O((\log \lambda )^{-1})$$ $\delta \left(\lambda \right)=O\left({(log\lambda )}^{-1}\right)$on compact Riemannian manifolds$$(M,g)$$ $(M,g)$of dimension$$n\ge 2$$ $n\ge 2$all of whose sectional curvatures are nonpositive or negative. We show that these two different types of estimates are saturated on flat manifolds or manifolds all of whose sectional curvatures are negative. This allows us to classify compact space forms in terms of the size of$$L^{q}$$ $L^q$-norms of quasimodes for each Lebesgue exponent$$q\in (2,q_{c}]$$ $q\in (2,q_c]$, even though it is impossible to distinguish between ones of negative or zero curvature sectional curvature for any$$q>q_{c}$$ $q>q_c$. 

The joint analysis of imaging‐genetics data facilitates the systematic investigation of genetic effects on brain structures and functions with spatial specificity. We focus on voxel‐wise genome‐wide association analysis, which may involve trillions of single nucleotide polymorphism (SNP)‐voxel pairs. We attempt to identify underlying organized association patterns of SNP‐voxel pairs and understand the polygenic and pleiotropic networks on brain imaging traits. We propose abi‐cliquegraph structure (ie, a set of SNPs highly correlated with a cluster of voxels) for the systematic association pattern. Next, we develop computational strategies to detect latent SNP‐voxelbi‐cliquesand an inference model for statistical testing. We further provide theoretical results to guarantee the accuracy of our computational algorithms and statistical inference. We validate our method by extensive simulation studies, and then apply it to the whole genome genetic and voxel‐level white matter integrity data collected from 1052 participants of the human connectome project. The results demonstrate multiple genetic loci influencing white matter integrity measures on splenium and genu of the corpus callosum. 

Abstract In this paper, we prove Strichartz estimates for many body Schrödinger equations in the periodic setting, specifically on tori ${𝕋}^d${\mathbb{T}^{d}}, where $d\ge 3${d\geq 3}. The results hold for both rational and irrational tori, and for small interacting potentials in a certain sense. Our work is based on the standard Strichartz estimate for Schrödinger operators on periodic domains, as developed in [J. Bourgain and C. Demeter,The proof of the $l^2$l^{2}decoupling conjecture,Ann. of Math. (2) 182 2015, 1, 351–389]. As a comparison, this result can be regarded as a periodic analogue of [Y. Hong,Strichartz estimates forN-body Schrödinger operators with small potential interactions,Discrete Contin. Dyn. Syst. 37 2017, 10, 5355–5365] though we do not use the same perturbation method. We also note that the perturbation method fails due to the derivative loss property of the periodic Strichartz estimate.