Kernel density estimation is the most widely-used practical method for accurate nonparametric density estimation. However, long-standing worst-case theoretical results showing that its performance worsens exponentially with the dimension of the data have quashed its application to modern high-dimensional datasets for decades. In practice, it has been recognized that often such data have a much ...

We give a unified description of tetrahedra with lightlike faces in 3d anti-de Sitter, de Sitter and Minkowski spaces their duals hyperbolic half-pipe spaces. show that both types are determined by generalized cross-ratio values commutative 2d real algebra generalizes the complex numbers. Equivalently, pair edge lengths dihedral angles. prove dual precisely ideal introduced Danciger. Finally, w...

We give asymptotically tight estimate of tangent space variation on a Riemannian submanifold of Euclidean space with respect to the local feature size of the submanifold. We show that the result is a consequence of structural properties of local feature size function of the Riemannian submanifold.

A 6-component “wave function” (not field, but S-matrix interpretable) for a massive spin-1 particle parallels the Dirac “chiralitydoubled” 4-component wave function for a spin-1/2 particle, by pairing two wave functions for same spin but opposite “handedness”. The correlated “opposite-parity” pair of complex 3-vectors defines a fluctuating spin-correlated lightlike “internal velocity” as well a...

We introduce the process of symplectic reduction along a submanifold as uniform approach to taking quotients in geometry. This construction holds categories smooth manifolds, complex analytic spaces, and algebraic varieties, has an interpretation terms derived stacks shifted It also encompasses Marsden–Weinstein–Meyer reduction, Mikami–Weinstein pre-images Poisson transversals under moment maps...

Using the submanifold quantum mechanical scheme, the restricted Dirac operator in a submanifold is defined. Then it is shown that the zero mode of the Dirac operator expresses the local properties of the submanifold, such as the Frenet-Serret and generalized Weierstrass relations. In other words this article gives a representation of a further generalized Weierstrass relations for a general k-s...

Certain criteria are demonstrated for a spatial derivation of a von Neumann algebra to generate a one-parameter semigroup of endomorphisms of that algebra. These are then used to establish a converse to recent results of Borchers and of Wiesbrock on certain one-parameter semigroups of endomorphisms of von Neumann algebras (specifically, Type III1 factors) that appear as lightlike translations i...