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\"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.

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).

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.

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: $\\mathrm{c}\\sqrt{\\mathrm{\\lambda }}\\le {\\mathrm{H}}^{\\mathrm{n}-1}\\left(\\{{\\mathrm{\\phi }}_{\\mathrm{\\lambda }}=0\\}\\right)$.

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.

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.

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.

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.

The ground-state energy of small molecules is determined efficiently using six qubits of a superconducting quantum processor. Scalable quantum simulation Quantum simulation is currently the most promising application of quantum computers. However, only a few quantum simulations of very small systems have been performed experimentally. Here, researchers from IBM present quantum simulations of larger systems using a variational quantum eigenvalue solver (or eigensolver), a previously suggested method for quantum optimization. They perform quantum chemical calculations of LiH and BeH 2 and an energy minimization procedure on a four-qubit Heisenberg model. Their application of the variational quantum eigensolver is hardware-efficient, which means that it is optimized on the given architecture. Noise is a big problem in this implementation, but quantum error correction could eventually help this experimental set-up to yield a quantum simulation of chemically interesting systems on a quantum computer. Quantum computers can be used to address electronic-structure problems and problems in materials science and condensed matter physics that can be formulated as interacting fermionic problems, problems which stretch the limits of existing high-performance computers 1 . Finding exact solutions to such problems numerically has a computational cost that scales exponentially with the size of the system, and Monte Carlo methods are unsuitable owing to the fermionic sign problem. These limitations of classical computational methods have made solving even few-atom electronic-structure problems interesting for implementation using medium-sized quantum computers. Yet experimental implementations have so far been restricted to molecules involving only hydrogen and helium 2 , 3 , 4 , 5 , 6 , 7 , 8 . Here we demonstrate the experimental optimization of Hamiltonian problems with up to six qubits and more than one hundred Pauli terms, determining the ground-state energy for molecules of increasing size, up to BeH 2 . We achieve this result by using a variational quantum eigenvalue solver (eigensolver) with efficiently prepared trial states that are tailored specifically to the interactions that are available in our quantum processor, combined with a compact encoding of fermionic Hamiltonians 9 and a robust stochastic optimization routine 10 . We demonstrate the flexibility of our approach by applying it to a problem of quantum magnetism, an antiferromagnetic Heisenberg model in an external magnetic field. In all cases, we find agreement between our experiments and numerical simulations using a model of the device with noise. Our results help to elucidate the requirements for scaling the method to larger systems and for bridging the gap between key problems in high-performance computing and their implementation on quantum hardware.