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"Eigenfunctions."
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Separation of variables for partial differential equations : an eigenfunction approach
\"Written at the advanced undergraduate level, the book will serve equally well as a text for students and as a reference for instructors and users of separation of variables. It requires a background in engineering mathematics, but no prior exposure to separation of variables. The abundant worked examples provide guidance for deciding whether and how to apply the method to any given problem, help in interpreting computed solutions, and give insight into cases in which formal answers may be useless\"--Jacket.
Book
Tunneling estimates and approximate controllability for hypoelliptic equations
This memoir is concerned with quantitative unique continuation estimates for equations involving a “sum of squares” operator
The first result is the tunneling estimate
The main
result is a stability estimate for solutions to the hypoelliptic wave equation
We then prove the approximate controllability of the
hypoelliptic heat equation
We also explain how the analyticity
assumption can be relaxed, and a boundary
Most results turn out to be optimal on a family of Grushin-type operators.
The main proof relies on the
general strategy to produce quantitative unique continuation estimates, developed by the authors in Laurent-Léautaud (2019).
eBook
Nodal sets of Laplace eigenfunctions: polynomial upper estimates of the Hausdorff measure
Let 𝕄 be a compact C∞-smooth Riemannian manifold of dimension n, n ≥ 3, and let 𝜑λ : ΔM𝜑λ + λ𝜑λ denote the Laplace eigenfunction on 𝕄 corresponding to the eigenvalue λ. We show that Hⁿ⁻¹({𝜑λ = 0}) ≤ Cλα, where α > 1/2 is a constant, which depends on n only, and C > 0 depends on 𝕄. This result is a consequence of our study of zero sets of harmonic functions on C∞-smooth Riemannian manifolds. We develop a technique of propagation of smallness for solutions of elliptic PDE that allows us to obtain local bounds from above for the volume of the nodal sets in terms of the frequency and the doubling index.
Journal Article
Nodal sets of Laplace eigenfunctions: proof of Nadirashvili's conjecture and of the lower bound in Yau's conjecture
Let u be a harmonic function in the unit ball B(0,1) ⊂ ℝⁿ, n ≥ 3, such that u(0) = 0. Nadirashvili conjectured that there exists a positive constant c, depending on the dimension n only, such that Hⁿ⁻¹({u = 0} ⋂ B) ≥ c. We prove Nadirashvili's conjecture as well as its counterpart on C∞-smooth Riemannian manifolds. The latter yields the lower bound in Yau's conjecture. Namely, we show that for any compact C∞-smooth Riemannian manifold M (without boundary) of dimension n, there exists c > 0 such that for any Laplace eigenfunction 𝜑λ on M, which corresponds to the eigenvalue λ, the following inequality holds: $\\mathrm{c}\\sqrt{\\mathrm{\\lambda }}\\le {\\mathrm{H}}^{\\mathrm{n}-1}\\left(\\{{\\mathrm{\\phi }}_{\\mathrm{\\lambda }}=0\\}\\right)$.
Journal Article
Statistical methods for temporal and space–time analysis of community composition data
This review focuses on the analysis of temporal beta diversity, which is the variation in community composition along time in a study area. Temporal beta diversity is measured by the variance of the multivariate community composition time series and that variance can be partitioned using appropriate statistical methods. Some of these methods are classical, such as simple or canonical ordination, whereas others are recent, including the methods of temporal eigenfunction analysis developed for multiscale exploration (i.e. addressing several scales of variation) of univariate or multivariate response data, reviewed, to our knowledge for the first time in this review. These methods are illustrated with ecological data from 13 years of benthic surveys in Chesapeake Bay, USA. The following methods are applied to the Chesapeake data: distance-based Moran's eigenvector maps, asymmetric eigenvector maps, scalogram, variation partitioning, multivariate correlogram, multivariate regression tree, and two-way MANOVA to study temporal and space–time variability. Local (temporal) contributions to beta diversity (LCBD indices) are computed and analysed graphically and by regression against environmental variables, and the role of species in determining the LCBD values is analysed by correlation analysis. A tutorial detailing the analyses in the R language is provided in an appendix.
Journal Article
Generic mean curvature flow I; generic singularities
It has long been conjectured that starting at a generic smooth closed embedded surface in R 3 , the mean curvature flow remains smooth until it arrives at a singularity in a neighborhood of which the flow looks like concentric spheres or cylinders. That is, the only singularities of a generic flow are spherical or cylindrical. We will address this conjecture here and in a sequel. The higher dimensional case will be addressed elsewhere. The key to showing this conjecture is to show that shrinking spheres, cylinders, and planes are the only stable self-shrinkers under the mean curvature flow. We prove this here in all dimensions. An easy consequence of this is that every singularity other than spheres and cylinders can be perturbed away.
Journal Article
The sphere packing problem in dimension 24
Building on Viazovska's recent solution of the sphere packing problem in eight dimensions, we prove that the Leech lattice is the densest packing of congruent spheres in twenty-four dimensions and that it is the unique optimal periodic packing. In particular, we find an optimal auxiliary function for the linear programming bounds, which is an analogue of Viazovska's function for the eight-dimensional case.
Journal Article
Small gaps between primes
We introduce a refinement of the GPY sieve method for studying prime k-tuples and small gaps between primes. This refinement avoids previous limitations of the method and allows us to show that for each k, the prime k-tuples conjecture holds for a positive proportion of admissible k-tuples. In particular, lim infn(pn+m − pn) < ∞ for every integer m. We also show that lim inf(pn+1 − pn) ≤ 600 and, if we assume the Elliott-Halberstam conjecture, that lim infn(pn+1 − pn) ≤ 12 and lim infn(pn+2 − pn) ≤ 600.
Journal Article
Euclidean triangles have no hot spots
We show that a second Neumann eigenfunction u of a Euclidean triangle has at most one (non-vertex) critical point p, and if p exists, then it is a non-degenerate critical point of Morse index 1. Using this we deduce that (1) the extremal values of u are only achieved at a vertex of the triangle, and (2) a generic acute triangle has exactly one (non-vertex) critical point and that each obtuse triangle has no (non-vertex) critical points. This settles the “hot spots” conjecture for triangles in the plane.
Journal Article