We present a new formulation of some basic differential geometric notions on a smooth manifold M , in the setting of nonstandard analysis. In place of classical vector fields, for which one needs to construct the tangent bundle of M , we define a prevector field, which is an internal map from ∗M to itself, implementing the intuitive notion of vectors as infinitesimal displacements. We introduce...

Essentially, there exists just the dimension segregating (square) matrix subspaces. In view of algebraic operations, this quantity is not particularly descriptive. For differential geometric information on matrix inversion, the second fundamental form is found for the set of inverses of the invertible elements of a matrix subspace. Several conditions for this form to vanish are given, such as b...

be the operator of mean value over a radius r sphere centered at y ∈ R. The integral transform I is clearly injective. Let C be a compact hypersurface in R isotopic to a sphere. Theorem 1.1 Let f(x) be a smooth function vanishing near C. Then one can recover f from its mean values along the spheres tangent to C, and the inversion is given by an explicit formula. In fact we will show that this t...

We present an augmented reality application which introduces differential geometry in educational dynamic geometry software. New functionality such as a Frenet frame, center and circle of curvature in arbitrary curve points, and others were implemented. Dynamic geometry allows to study differential geometric properties under movement. Using this tool we developed examples which enable teachers ...

We use geometric properties of Gromov-Hausdorff-convergence to present a way to construct rough but natural invariants of metric geometry.

This book aims to introduce the reader to the geometry of surfaces and submanifolds in the conformal n-sphere. Various models for Möbius geometry are presented: the classical projective model, the quaternionic approach, and an approach that uses the Clifford algebra of the space of homogeneous coordinates of the classical model — the use of 2-by-2 matrices in this context is elaborated. For eac...

Article history: Received 19 June 2010 Available online 22 April 2011 Communicated by Z. Shen

Autoregressive fractionally integrated moving average ARFIMA pro cesses are widely used for modeling time series exhibiting both long memory and short memory behavior Properties of Toeplitz matrices associated with the spectral density functions of Gaussian ARFIMA processes are used to compute di erential geometric quantities INTRODUCTION Time series data occurring in several areas such as geol...

Projective differential geometry was initiated in the 1920s, especially by Élie Cartan and Tracey Thomas. Nowadays, the subject is not so well-known. These notes aim to remedy this deficit and present several reasons why this should be done at this time. The deeper underlying reason is that projective differential geometry provides the most basic application of what has come to be known as the ...

The matters which I shall discuss today will be largely concerned with the general problem of imbedding a coordinate manifold of class C in a euclidean space of sufficiently many dimensions. I imagine you all have a fair idea as to what is meant by an w-dimensional manifold of class C. Briefly this may be described as a Hausdorff space each point of which admits a neighborhood homeomorphic to t...