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Hyperbolic geometry is a non-Euclidean geometry where the first four axioms of Euclidean geometry are kept but the fifth axiom, the parallel postulate, is changed.The fifth axiom of hyperbolic geometry says that given a line L and a point P not on that line, there are at least two lines passing through P that are parallel to L. [1]
CAP theorem (theoretical computer science) CPCTC (triangle geometry) Cameron–Erdős theorem (discrete mathematics) Cameron–Martin theorem (measure theory) Cantor–Bernstein–Schroeder theorem (set theory, cardinal numbers) Cantor's intersection theorem (real analysis) Cantor's isomorphism theorem (order theory)
A convex cap, also known as a convex floating body [1] or just floating body, [2] is a well defined structure in mathematics commonly used in convex analysis for approximating convex shapes. In general it can be thought of as the intersection of a convex Polytope with a half-space .
The PACELC theorem, introduced in 2010, [8] builds on CAP by stating that even in the absence of partitioning, there is another trade-off between latency and consistency. PACELC means, if partition (P) happens, the trade-off is between availability (A) and consistency (C); Else (E), the trade-off is between latency (L) and consistency (C).
They are used particularly in algebraic geometry and related fields. A key result known as Chevalley's theorem in algebraic geometry shows that the image of a constructible set is constructible for an important class of mappings (more specifically morphisms) of algebraic varieties (or more generally schemes). In addition, a large number of ...
An illustration of Carathéodory's theorem for a square in R 2. Carathéodory's theorem in 2 dimensions states that we can construct a triangle consisting of points from P that encloses any point in the convex hull of P. For example, let P = {(0,0), (0,1), (1,0), (1,1)}. The convex hull of this set is a square.
Gabriel–Rosenberg reconstruction theorem; Gram's theorem; Grauert–Riemenschneider vanishing theorem; Grothendieck existence theorem; Grothendieck trace formula; Grothendieck–Riemann–Roch theorem; Grothendieck's connectedness theorem; Gudkov's conjecture
There is some minor argument on whether Saccheri really meant that, as he published his work in the final year of his life, came extremely close to discovering non-Euclidean geometry and was a logician. Some believe Saccheri concluded as he did only to avoid the criticism that might come from seemingly-illogical aspects of hyperbolic geometry.