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Zhiqiang Xu 《Applied and Computational Harmonic Analysis》2018,44(2):497-508
The paper presents several results that address a fundamental question in low-rank matrix recovery: how many measurements are needed to recover low-rank matrices? We begin by investigating the complex matrices case and show that generic measurements are both necessary and sufficient for the recovery of rank-r matrices in . Thus, we confirm a conjecture which is raised by Eldar, Needell and Plan for the complex case. We next consider the real case and prove that the bound is tight provided . Motivated by Vinzant's work [19], we construct 11 matrices in by computer random search and prove they define injective measurements on rank-1 matrices in . This disproves the conjecture raised by Eldar, Needell and Plan for the real case. Finally, we use the results in this paper to investigate the phase retrieval by projection and show fewer than orthogonal projections are possible for the recovery of from the norm of them, which gives a negative answer for a question raised in [1]. 相似文献
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Masoud Hassani 《Comptes Rendus Mathematique》2017,355(11):1133-1137
In this paper, we study the irreducible representation of in . This action preserves a quadratic form with signature . Thus, it acts conformally on the 3-dimensional Einstein universe . We describe the orbits induced in and its complement in . This work completes the study in [2], and is one element of the classification of cohomogeneity one actions on [5]. 相似文献
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Let be a Lipschitz domain and Γ be a relatively open and non-empty subset of its boundary ?Ω. We show that the solution to the linear first-order system:(1) vanishes if and . In particular, square-integrable solutions ζ of (1) with vanish. As a consequence, we prove that: is a norm if with , for some with as well as . We also give a new and different proof for the so-called ‘infinitesimal rigid displacement lemma’ in curvilinear coordinates: Let , , satisfy for some with . Then there exists a constant translation vector and a constant skew-symmetric matrix , such that . 相似文献
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We investigate the regularity of random attractors for the non-autonomous non-local fractional stochastic reaction–diffusion equations in with . We prove the existence and uniqueness of the tempered random attractor that is compact in and attracts all tempered random subsets of with respect to the norm of . The main difficulty is to show the pullback asymptotic compactness of solutions in due to the noncompactness of Sobolev embeddings on unbounded domains and the almost sure nondifferentiability of the sample paths of the Wiener process. We establish such compactness by the ideas of uniform tail-estimates and the spectral decomposition of solutions in bounded domains. 相似文献
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Andrea Luigi Tironi 《Discrete Mathematics》2018,341(11):3152-3158
Let be a hypersurface in with defined over a finite field of elements. In this note, we classify, up to projective equivalence, hypersurfaces as above which reach two elementary upper bounds for the number of -points on which involve a Thas’ invariant. 相似文献
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We compare the isoperimetric profiles of and of with that of a round 5-sphere (of appropriate radius). Then we use this comparison to obtain lower bounds for the Yamabe constants of and . Explicitly we show that and . We also obtain explicit lower bounds in higher dimensions and for products of Euclidean space with a closed manifold of positive Ricci curvature. The techniques are a more general version of those used by the same authors in Petean and Ruiz (2011) [15] and the results are a complement to the work developed by B. Ammann, M. Dahl and E. Humbert to obtain explicit gap theorems for the Yamabe invariants in low dimensions. 相似文献
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Let denote the unit sphere in the Euclidean space . We develop LeVeque type inequalities for the discrepancy between the rotationally invariant probability measure and the normalized counting measures on . We obtain both upper bound and lower bound estimates. We then use these inequalities to estimate the discrepancy of the normalized counting measures associated with minimal energy configurations on . 相似文献
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We study the dynamics of infinitely many Cucker–Smale (C–S) flocking particles under the interplay of random communication and incompressible fluids. For the dynamics of an ensemble of flocking particles, we use the kinetic Cucker–Smale–Fokker–Planck (CS–FP) equation with a degenerate diffusion, whereas for the fluid component, we use the incompressible Navier–Stokes (N–S) equations. These two subsystems are coupled via the drag force. For this coupled model, we present the global existence of weak and strong solutions in . Under the extra regularity assumptions of the initial data, the unique solvability of strong solutions is also established in . In a large coupling regime and periodic spatial domain , we show that the velocities of C–S particles and fluids are asymptotically aligned to two constant velocities which may be different. 相似文献