A Hermeneutic Introduction to Maps

Introduction to Combinatorial Maps

Preface Maps as a mathematical main topic arose probably from the four color problem[Ore1] and the more general map coloring problem[Rin1] in the mid of the nineteenth century. I could not list even main references on them because it is well known for mathematicians and beyond the scope of this lecture notes. Here, I only intend to present a comprehensive theory of combinatorial maps as a rigor...

An Introduction to Diffusion Maps

This paper describes a mathematical technique [1] for dealing with dimensionality reduction. Given data in a high-dimensional space, we show how to find parameters that describe the lower-dimensional structures of which it is comprised. Unlike other popular methods such as Principle Component Analysis and Multi-dimensional Scaling, diffusion maps are non-linear and focus on discovering the unde...

An Introduction to Exposure Maps

The counts image of an extended X-ray source invariably contains instrumental artifacts which complicate surface brightness measurements. These artifacts are caused by imperfections in the mirror and the detector and are both energy and position dependent. Often, one would like to generate an image in physical units (e.g. photon sec cm arcsec), with these artifacts removed. A common approach is...

An Introduction to Sociolinguistics

Introduction to circular data

In many diverse scientific fields, the measurements are directions. For instance, a biologist may be measuring the direction of flight of a bird or the orientation of an animal. A series of such observations is called ”directional data”. Since a direction has no magnitude, these can be conveniently represented as points on the circumference of a unit circle centered at the origin or as unit ...