Search results
Results from the WOW.Com Content Network
However, this map is two-to-one, so we want to identify s ~ −s to yield P 1 (R) ≅ S 1 /~ where the topology on this space is the quotient topology induced by the quotient map S 1 → P 1 (R). Thus, when we consider P 1 ( R ) as a moduli space of lines that intersect the origin in R 2 , we capture the ways in which the members (lines in this ...
In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers.In particular, complex geometry is concerned with the study of spaces such as complex manifolds and complex algebraic varieties, functions of several complex variables, and holomorphic constructions such as holomorphic vector bundles and coherent sheaves.
AlphaGeometry is an artificial intelligence (AI) program that can solve hard problems in Euclidean geometry.It was developed by DeepMind, a subsidiary of Google.The program solved 25 geometry problems out of 30 from the International Mathematical Olympiad (IMO) under competition time limits—a performance almost as good as the average human gold medallist.
In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an idealized ruler and a pair of compasses.
In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same distance in a given direction. A translation can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system .
A module is called flat if taking the tensor product of it with any exact sequence of R-modules preserves exactness. Torsionless A module is called torsionless if it embeds into its algebraic dual. Simple A simple module S is a module that is not {0} and whose only submodules are {0} and S. Simple modules are sometimes called irreducible. [5 ...
Absolute geometry is a geometry based on an axiom system consisting of all the axioms giving Euclidean geometry except for the parallel postulate or any of its alternatives. [69] The term was introduced by János Bolyai in 1832. [70] It is sometimes referred to as neutral geometry, [71] as it is neutral with respect to the parallel postulate.
In addition, the Euclidean geometry (which has zero curvature everywhere) is not the only geometry that the plane may have. The plane may be given a spherical geometry by using the stereographic projection. This can be thought of as placing a sphere tangent to the plane (just like a ball on the floor), removing the top point, and projecting the ...