Recent progress in natural language processing has been driven by advances in both model architecture and model pretraining. Transformer architectures have facilitated building higher-capacity models and pretraining has made it possible to effectively utilize this capacity for a wide variety of tasks. Transformers is an open-source library with the goal of opening up these advances to the wider machine learning community. The library consists of carefully engineered stateof-the art Transformer architectures under a unified API. Backing this library is a curated collection of pretrained models made by and available for the community. Transformers is designed to be extensible by researchers, simple for practitioners, and fast and robust in industrial deployments. The library is available at https://github.com/ huggingface/transformers.

ABSTRACT. We study the isoperimetric problem for Euclidean space endowed with a continuous density. In dimension one, we characterize isoperimetric regions for a unimodal density. In higher dimensions, we prove existence results and we derive stability conditions, which lead to the conjecture that for a radial log-convex density, balls about the origin are isoperimetric regions. Finally, we prove this conjecture and the uniqueness of minimizers for the density exp(|x| 2 ) by using symmetrization techniques.

We prove that the standard double bubble provides the least-area way to enclose and separate two regions of prescribed volume in R 3 . * 2000 Mathematics Subject Classification. Primary: 53A10. Secondary: 53C42. ing curves no longer meet at 120 degrees.) Unfortunately, for general costs, even for equal volumes, (7.1) does not imply both regions connected. We thank undergraduate Ken Dennison for raising this question.

In this paper we introduce a new method for proving area-minimization which we call "paired calibrations." We begin with the simplest application, the cone over the tetrahedron, which appears in soap films. We then discuss immiscible fluid interfaces, crystal surfaces, and one-dimensional networks minimizing other norms.

Abstract. We add to the literature the well-known fact that an isoperimetric hypersurface S of dimension at most six in a smooth Riemannian manifold M is a smooth submanifold. If the metric is merely Lipschitz, then S is still Hölder differentiable.

We prove that a region of small prescribed volume in a smooth, compact Riemannian manifold has at least as much perimeter as a round ball in the model space form, using differential inequalities and the Gauss-Bonnet-Chern theorem with boundary term. First we show that a minimizer is a nearly round sphere. We also provide some new isoperimetric inequalities in surfaces.. P ≥ (1 − C K 0 V 2/(n+1))P *. 1.2 Small isoperimetric regions are spheres. Theorem 2.2 proves that a least-perimeter enclosure S of small volume is a (nearly round) sphere. (This result was known only in the relatively trivial case when the ambient M is two dimensional.) This is of course well known at the infinitesimal level, i.e., in Euclidean space. Our proof is a compactness argument. The main difficulties are to show that S lies in a small ball and to show that the convergence to the limit is smooth. These results follow by so-called "monotonicity of mass ratio" and the Allard regularity theorem [A], given bounds on the mean curvature. Such bounds follow from the Heintze-Karcher [HK] estimate on enclosed volume in terms of mean curvature. 1.3 Sharp upper bounds on least perimeter. If the sectional curvature K of the ambient manifold M satisfies an opposite inequality K ≥ K 0 , Theorem 3.4 shows that a least-perimeter enclosure of volume V has at most as much perimeter P as a round sphere of the same volume in the model space form of curvature K 0 , with equality only if K = K 0. It comes from integrating a second order differential inequality [BP]:

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We prove the existence of isoperimetric regions in R n with density under various hypotheses on the growth of the density. Along the way, we prove results on the boundedness of isoperimetric regions.