The Assouad and lower dimensions dimension spectra quantify the regularity of a measure by considering relative concentric balls. On other hand, one can smoothness an absolutely continuous $L^p$ norms its density. We establish sharp relationships between these two notions. Roughly speaking, we show that smooth measures must be regular, but regular need not smooth.

In optimization, the notion of a partly smooth objective function is powerful for applications in algorithmic convergence and postoptimality analysis, yet complex to define. A shift focus first-order optimality conditions reduces concept simple constant-rank condition. this view, partial smoothness extends more general variational systems, encompassing particular saddlepoint operators underlyin...

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...