We prove that there are no bounded domains with smooth boundaries in even-dimensional Euclidean spaces, such that the volumes cut off from them by affine hyperplanes depend algebraically on these hyperplanes. For convex ovals in R2, this is the Newton’s Lemma XXVIII, see [11], [14], [2], [3].

Article history: Received 4 August 2011 Available online 11 November 2013

Let {(Zi,Wi) : i = 1, . . . , n} be uniformly distributed in [0, 1]d × G(k, d), where G(k, d) denotes the space of k-dimensional linear subspaces of Rd. For a differentiable function f : [0, 1]k → [0, 1]d, we say that f interpolates (z,w) ∈ [0, 1]d × G(k, d) if there exists x ∈ [0, 1]k such that f(x) = z and ~ f(x) = w, where ~ f(x) denotes the tangent space at x defined by f . For a smoothness...

Locality-sensitive hashing converts high-dimensional feature vectors, such as image and speech, into bit arrays and allows high-speed similarity calculation with the Hamming distance. There is a hashing scheme that maps feature vectors to bit arrays depending on the signs of the inner products between feature vectors and the normal vectors of hyperplanes placed in the feature space. This hashin...

We consider the quantum hyperplane x ' x J = q , j x J x ' (i, j = 1..n) and define and consider = 'J and fl'J are complex numbers. We deformations of the form x ' x J q,~ x J x ~ + Zk ~ J X k + fl'J, where ~k prove that for generic q,j no nontrivial deformations exist for n/> 3 Mathematics Subject Classifications (1991). 17B37, 16E40.

simulators [9]. There are two main ways of obtaining a DR of a real world object. One is by acquisition with a This paper presents the properties of the discrete analytical hyperplanes. They are defined analytically in the discrete dophysical device like a camera for a 2D DR, a CT or MR main by Diophantine equations. We show that the discrete scanner for a 3D DR, a PET scanner for a 4D DR, etc....