We consider the ability of deep neural networks to represent data that lies near a low-dimensional manifold in a high-dimensional space. We show that deep networks can efficiently extract the intrinsic, low-dimensional coordinates of such data. Specifically we show that the first two layers of a deep network can exactly embed points lying on a monotonic chain, a special type of piecewise linear...

In this paper we develop a straightforward procedure to construct higher dimensional isochronous Hamiltonian systems. We first show that class of singular systems obtained through the $\Omega$-modified is equivalent constrained Newtonian Even though such admit oscillations, they are effectively one degree freedom due constraints. Then generalize in terms $\Omega_i$-modified Hamiltonians and ide...

We describe a solitonic solution-generating technique for the five-dimensional General Relativity. Reducing the five-dimensional problem to the four-dimensional one, we can systematically obtain single-rotational axially symmetric vacuum solutions. Applying the technique for a simple seed solution, we have previously obtained the series of stationary solutions which includes S-rotating black ri...

X iv :0 70 8. 11 71 v1 [ m at h. D G ] 8 A ug 2 00 7 ON THE SPINOR STRUCTURE OF THE HOMOGENEOUS AND ISOTROPIC UNIVERSE IN CLOSED MODEL. R. A. Sharipov Abstract. The closed homogeneous and isotropic universe is considered. The bundles of Weyl and Dirac spinors for this universe are explicitly described. Some explicit formulas for the basic fields and for the connection components in stereographi...

This letter proposes a novel robust interval observer for two-dimensional (treated as synonym double-indexed system) linear time-invariant discrete-time system described by the Fornasini-Marchesini second model. is subject to unknown but bounded state disturbances and measurement noise. Built on recent estimation strategies designed one-dimensional systems, proposed based introduction of weight...

A bstract We study the evaporation of two-dimensional black holes in JT gravity from a three-dimensional point view. partial dimensional reduction AdS 3 Poincaré coordinates leads to an extremal 2D hole coupled ‘bath’: holographic dual remainder 3D spacetime. Partially reducing BTZ gives us finite temperature version. compute entropy radiation using geodesics then focus on case and describe dyn...

Parallel coordinates is one of the most popular and widely used visualization techniques for large, high dimensional data. Often, data attributes are visualized on individual axes with polylines joining them. However, some data attributes are more naturally represented with a spherical coordinate system. We present a novel coupling of parallel coordinates with spherical coordinates, enabling th...