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In this paper we prove that no packing of unit balls in Euclidean space ℝ⁸ has density greater than that of the E₈-lattice packing.

We propose a new technique, called wild binary segmentation (WBS), for consistent estimation of the number and locations of multiple change-points in data. We assume that the number of change-points can increase to infinity with the sample size. Due to a certain random localisation mechanism, WBS works even for very short spacings between the change-points and/or very small jump magnitudes, unlike standard binary segmentation. On the other hand, despite its use of localisation, WBS does not require the choice of a window or span parameter, and does not lead to a significant increase in computational complexity. WBS is also easy to code. We propose two stopping criteria for WBS: one based on thresholding and the other based on what we term the 'strengthened Schwarz information criterion'. We provide default recommended values of the parameters of the procedure and show that it offers very good practical performance in comparison with the state of the art. The WBS methodology is implemented in the R package wbs, available on CRAN. In addition, we provide a new proof of consistency of binary segmentation with improved rates of convergence, as well as a corresponding result for WBS.

In this paper we investigate the curvature of conformal deformations by noncommutative Weyl factors of a flat metric on a noncommutative 2-torus, by analyzing in the framework of spectral triples functionals associated to perturbed Dolbeault operators. The analogue of Gaussian curvature turns out to be a sum of two functions in the modular operator corresponding to the non-tracial weight defined by the conformal factor, applied to expressions involving the derivatives of the same factor. The first is a generating function for the Bernoulli numbers and is applied to the noncommutative Laplacian of the conformal factor, while the second is a two-variable function and is applied to a quadratic form in the first derivatives of the factor. Further outcomes of the paper include a variational proof of the Gauss-Bonnet theorem for noncommutative 2-tori, the modular analogue of Polyakov’s conformal anomaly formula for regularized determinants of Laplacians, a conceptual understanding of the modular curvature as gradient of the Ray-Singer analytic torsion, and the proof using operator positivity that the scale invariant version of the latter assumes its extreme value only at the flat metric.

Consider a system of N bosons in three dimensions interacting via a repulsive short range pair potential N²V(N(x i — x j )), where x = (x₁,…,x N ) denotes the positions of the particles. Let H N denote the Hamiltonian of the system and let ψ N,t be the solution to the Schrödinger equation. Suppose that the initial data ψ N,0 satisfies the energy condition $\\langle \\psi _{N,0},H_{N}^{k}\\psi _{N,0}\\rangle \\leq C^{k}N^{k}$ for k = 1,2,…. We also assume that the k-particle density matrices of the initial state are asymptotically factorized as N → ∞. We prove that the k-particle density matrices of ψ N,t are also asymptotically factorized and the one particle orbital wave function solves the Gross-Pitaevskii equation, a cubic nonlinear Schrödinger equation with the coupling constant given by the scattering length of the potential V. We also prove the same conclusion if the energy condition holds only for k = 1 but the factorization of ψ N,0 is assumed in a stronger sense.

The fundamental \"two-fluid\" model for describing plasma dynamics is given by the Euler–Maxwell system, in which compressible ion and electron fluids interact with their own self-consistent electromagnetic field. We prove global stability of a constant neutral background, in the sense that irrotational, smooth and localized perturbations of a constant background with small amplitude lead to global smooth solutions in three space dimensions for the Euler–Maxwell system. Our construction is robust in dimension 3 and applies equally well to other plasma models such as the Euler-Poisson system for two-fluids and a relativistic Euler–Maxwell system for two fluids. Our solutions appear to be the first nontrivial global smooth solutions in all of these models.

In this paper, we study the Cauchy problem for a quasilinear degenerate parabolic stochastic partial differential equation driven by a cylindrical Wiener process. In particular, we adapt the notion of kinetic formulation and kinetic solution and develop a well-posedness theory that includes also an L1-contraction property. In comparison to the previous works of the authors concerning stochastic hyperbolic conservation laws [J. Funct. Anal. 259 (2010) 1014–1042] and semilinear degenerate parabolic SPDEs [Stochastic Process. Appl. 123 (2013) 4294–4336], the present result contains two new ingredients that provide simpler and more effective method of the proof: a generalized Itô formula that permits a rigorous derivation of the kinetic formulation even in the case of weak solutions of certain nondegenerate approximations and a direct proof of strong convergence of these approximations to the desired kinetic solution of the degenerate problem.

We consider the 2-dimensional focusing mass critical NLS with an inhomogeneous nonlinearity: i∂tu+Δu+k(x)|u|2u=0i\\partial _tu+\\Delta u+k(x)|u|^{2}u=0. From a standard argument, there exists a threshold Mk>0M_k>0 such that H1H^1 solutions with ‖u‖L2>Mk\\|u\\|_{L^2}>M_k are global in time while a finite time blow-up singularity formation may occur for ‖u‖L2>Mk\\|u\\|_{L^2}>M_k. In this paper, we consider the dynamics at threshold ‖u0‖L2=Mk\\|u_0\\|_{L^2}=M_k and give a necessary and sufficient condition on kk to ensure the existence of critical mass finite time blow-up elements. Moreover, we give a complete classification in the energy class of the minimal finite time blow-up elements at a nondegenerate point, hence extending the pioneering work by Merle who treated the pseudoconformal invariant case k≡1k\\equiv 1.

We consider nonparametric estimation of the mean and covariance functions for functional/longitudinal data. Strong uniform convergence rates are developed for estimators that are local-linear smoothers. Our results are obtained in a unified framework in which the number of observations within each curve/cluster can be of any rate relative to the sample size. We show that the convergence rates for the procedures depend on both the number of sample curves and the number of observations on each curve. For sparse functional data, these rates are equivalent to the optimal rates in nonparametric regression. For dense functional data, root-n rates of convergence can be achieved with proper choices of bandwidths. We further derive almost sure rates of convergence for principal component analysis using the estimated covariance function. The results are illustrated with simulation studies.

A fundamental assumption usually made in causal inference is that of no interference between individuals (or units); that is, the potential outcomes of one individual are assumed to be unaffected by the treatment assignment of other individuals. However, in many settings, this assumption obviously does not hold. For example, in the dependent happenings of infectious diseases, whether one person becomes infected depends on who else in the population is vaccinated. In this article, we consider a population of groups of individuals where interference is possible between individuals within the same group. We propose estimands for direct, indirect, total, and overall causal effects of treatment strategies in this setting. Relations among the estimands are established; for example, the total causal effect is shown to equal the sum of direct and indirect causal effects. Using an experimental design with a two-stage randomization procedure (first at the group level, then at the individual level within groups), unbiased estimators of the proposed estimands are presented. Variances of the estimators are also developed. The methodology is illustrated in two different settings where interference is likely: assessing causal effects of housing vouchers and of vaccines.

The Kashiwara-Vergne (KV) conjecture is a property of the Campbell-Hausdorff series put forward in 1978. It has been settled in the positive by E. Meinrenken and the first author in 2006. In this paper, we study the uniqueness issue for the KV problem. To this end, we introduce a family of infinite-dimensional groups $KRV^0_n$ , and a group KRV 2 which contains $KRV^0_2$ as a normal subgroup. We show that KRV 2 also contains the Grothendieck-Teichmüller group GRT 1 as a subgroup, and that it acts freely and transitively on the set of solutions of the KV problem SolKV. Furthermore, we prove that SolKV is isomorphic to a direct product of affine line 𝔸 1 and the set of solutions of the pentagon equation with values in the group $KRV^0_3$ . The latter contains the set of Drinfeld's associators as a subset. As a by-product of our construction, we obtain a new proof of the Kashiwara-Vergne conjecture based on the Drinfeld's theorem on existence of associators.