Core foundations of abstract geometry

Core foundations of abstract geometry.

Human adults from diverse cultures share intuitions about the points, lines, and figures of Euclidean geometry. Do children develop these intuitions by drawing on phylogenetically ancient and developmentally precocious geometric representations that guide their navigation and their analysis of object shape? In what way might these early-arising representations support later-developing Euclidean...

Foundations of Abstract Interpretation

Semantics Similar to concrete semantics:  A complete lattice (L#, ≤) as the domain for abstract elements  A monotone function F# corresponding to the concrete function F  Then the abstract semantics is the least fixed point of F#, lfp F# If F# “correctly approximates” F, then lfp F# “correctly approximates” lfp F. An Example Abstract Domain for Values of Variables How to relate the two?  Co...

Foundations of Differential Geometry

These notes are from a lecture course Di erentialgeometrie und Lie Gruppen which has been held at the University of Vienna during the academic year again in and in WS It is not yet complete and will be enlarged during the year In this lecture course I give complete de nitions of manifolds in the beginning but beside spheres examples are treated extensively only later when the theory is develope...

Algebraic Geometry Foundations

Foundations of Euclidean Constructive Geometry

Euclidean geometry, as presented by Euclid, consists of straightedge-andcompass constructions and rigorous reasoning about the results of those constructions. A consideration of the relation of the Euclidean “constructions” to “constructive mathematics” leads to the development of a first-order theory ECG of “Euclidean Constructive Geometry”, which can serve as an axiomatization of Euclid rathe...