Subsets of the Real Line
... If α and β are ordinals, then we write α < β if α ∈ β. It is easy to check that α ⊆ β ←→ (α ∈ β ∨ α = β). It is clear that the empty set ∅ is an ordinal which is also denoted by the usual symbol 0. By the method of transfinite induction it is not difficult to prove that for any ordinal α the equalit ...
... If α and β are ordinals, then we write α < β if α ∈ β. It is easy to check that α ⊆ β ←→ (α ∈ β ∨ α = β). It is clear that the empty set ∅ is an ordinal which is also denoted by the usual symbol 0. By the method of transfinite induction it is not difficult to prove that for any ordinal α the equalit ...
arXiv:1205.2342v1 [math.GR] 10 May 2012 Homogeneous compact
... We classify compact homogeneous geometries which look locally like compact spherical buildings. Geometries which look locally like buildings arise naturally in various recognition problems in group theory. Tits’ seminal paper A local approach to buildings [51] is devoted to them. Among other things, ...
... We classify compact homogeneous geometries which look locally like compact spherical buildings. Geometries which look locally like buildings arise naturally in various recognition problems in group theory. Tits’ seminal paper A local approach to buildings [51] is devoted to them. Among other things, ...
Decomposition of Generalized Closed Sets in Supra Topological
... Now we recall some definitions and results which are useful in the sequel. ...
... Now we recall some definitions and results which are useful in the sequel. ...
Affine Decomposition of Isometries in Nilpotent Lie Groups
... have more assumptions on the algebraic structure of the space, but Riemannian Lie groups have more assumptions on the metric structure than subRiemannian Carnot groups. Our setting of metric Lie groups is a generalization of the both. But to stress the logic, our proof of the general result actually ...
... have more assumptions on the algebraic structure of the space, but Riemannian Lie groups have more assumptions on the metric structure than subRiemannian Carnot groups. Our setting of metric Lie groups is a generalization of the both. But to stress the logic, our proof of the general result actually ...
THE CHINESE REMAINDER THEOREM INTRODUCED IN A
... asked the girl how many eggs there were. The girl said she did not know, but she remembered that when she counted them by twos, there was one left over; when she counted them by threes, there were two left over; when she counted them by fours, there were three left over; when she counted them by ve ...
... asked the girl how many eggs there were. The girl said she did not know, but she remembered that when she counted them by twos, there was one left over; when she counted them by threes, there were two left over; when she counted them by fours, there were three left over; when she counted them by ve ...
Using Congruence Theorems - IHS Math
... (B) determine the image or pre-image of a given two-dimensional figure under a composition of rigid transformations, a composition of non-rigid transformations, and a composition of both, including dilations where the center can be any point in the plane (5) Logical argument and constructions. The s ...
... (B) determine the image or pre-image of a given two-dimensional figure under a composition of rigid transformations, a composition of non-rigid transformations, and a composition of both, including dilations where the center can be any point in the plane (5) Logical argument and constructions. The s ...
A May-type spectral sequence for higher topological Hochschild
... simplicial objects in C is a Reedy cofibration between Reedy cofibrant objects whenever the following all hold: (1) Each object Xn and Yn of C is cofibrant. (2) Each degeneracy map si : Xn Ñ Xn`1 and si : Yn Ñ Yn`1 is a cofibration in C (3) Each map Xn Ñ Yn is a cofibration in C . A consequence of t ...
... simplicial objects in C is a Reedy cofibration between Reedy cofibrant objects whenever the following all hold: (1) Each object Xn and Yn of C is cofibrant. (2) Each degeneracy map si : Xn Ñ Xn`1 and si : Yn Ñ Yn`1 is a cofibration in C (3) Each map Xn Ñ Yn is a cofibration in C . A consequence of t ...
Brouwer fixed-point theorem
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after Luitzen Brouwer. It states that for any continuous function f mapping a compact convex set into itself there is a point x0 such that f(x0) = x0. The simplest forms of Brouwer's theorem are for continuous functions f from a closed interval I in the real numbers to itself or from a closed disk D to itself. A more general form than the latter is for continuous functions from a convex compact subset K of Euclidean space to itself.Among hundreds of fixed-point theorems, Brouwer's is particularly well known, due in part to its use across numerous fields of mathematics.In its original field, this result is one of the key theorems characterizing the topology of Euclidean spaces, along with the Jordan curve theorem, the hairy ball theorem and the Borsuk–Ulam theorem.This gives it a place among the fundamental theorems of topology. The theorem is also used for proving deep results about differential equations and is covered in most introductory courses on differential geometry.It appears in unlikely fields such as game theory. In economics, Brouwer's fixed-point theorem and its extension, the Kakutani fixed-point theorem, play a central role in the proof of existence of general equilibrium in market economies as developed in the 1950s by economics Nobel prize winners Kenneth Arrow and Gérard Debreu.The theorem was first studied in view of work on differential equations by the French mathematicians around Poincaré and Picard.Proving results such as the Poincaré–Bendixson theorem requires the use of topological methods.This work at the end of the 19th century opened into several successive versions of the theorem. The general case was first proved in 1910 by Jacques Hadamard and by Luitzen Egbertus Jan Brouwer.