Sobolev quantities (norms, inner products, and distances) of probability density functions are important in the theory of nonparametric statistics, but have rarely been used in practice, due to a lack of practical estimators. They also include, as special cases, L quantities which are used in many applications. We propose and analyze a family of estimators for Sobolev quantities of unknown prob...

Fundamental Theorem of Analysis Theorem 1. Suppose that f ∈ H(0, 1). Then there exists a g ∈ L2(0, 1) such that f(x)− f(0) = ∫ x 0 g(t) dt, 0 < x < 1. We de ne the linear mapping (Sg)(x) := ∫ x 0 g(t) dt. Corollary 1. If f ∈ H(0, 1) then f − f(0) ∈ R(S). Thus, smoothness means that f is in the range of a smoothing operator. Theorem 2. Suppose that f ∈ H(0, 1) (periodic). For the Fourier coe cie...

The essential role played by differentiable structures in physics is reviewed in light of recent mathematical discoveries that topologically trivial space-time models, especially the simplest one, R , possess a rich multiplicity of such structures, no two of which are diffeomorphic to each other and thus to the standard one. This means that physics has available to it a new panoply of structure...

Real data tensors are typically high dimensional; however, their intrinsic information is preserved in low-dimensional space, which motivates the use of tensor decompositions such as Tucker decomposition. Frequently, real data tensors smooth in addition to being low dimensional, which implies that adjacent elements are similar or continuously changing. These elements typically appear as spatial...

The definition of an S-category is proposed by weakening the axioms of a Q-category introduced by Kontsevich and Rosenberg. Examples of Qand S-categories and (co)smooth objects in such categories are given.

Variable projection is a powerful technique in optimization. Over the last 30 years, it has been applied broadly, with empirical and theoretical results demonstrating both greater efficacy and greater stability than competing approaches. In this paper, we illustrate the technique on a large class of structured nonsmooth optimization problems, with numerical examples in sparse deconvolution and ...

Gompf’s end-sum techniques are used to establish the existence of an infinity of non-diffeomorphic manifolds, all having the same trivial R topology, but for which the exotic differentiable structure is confined to a region which is spatially limited. Thus, the smoothness is standard outside of a region which is topologically (but not smoothly) B ×R, where B is the compact three ball. The exter...