Approximation on Nash sets with monomial singularities
... monomial singularities. Indeed, we start in Section 6 with a result concerning extension of Sν functions, which is well-known for Nash submanifolds (see Proposition 6.2): If X ⊂ M is a Nash set with monomial singularities, then there are continuous extension linear maps θ : Sν (X) → Sν (M ) such tha ...
... monomial singularities. Indeed, we start in Section 6 with a result concerning extension of Sν functions, which is well-known for Nash submanifolds (see Proposition 6.2): If X ⊂ M is a Nash set with monomial singularities, then there are continuous extension linear maps θ : Sν (X) → Sν (M ) such tha ...
on the ubiquity of simplicial objects
... In the second chapter we specialise to simplicial sets. We will develop some simplicial homotopy theory and then establish an equivalence between simplicial homotopy theory and the classical homotopy theory of topological spaces. Most of our results in this section come from May’s Simplicial Objects ...
... In the second chapter we specialise to simplicial sets. We will develop some simplicial homotopy theory and then establish an equivalence between simplicial homotopy theory and the classical homotopy theory of topological spaces. Most of our results in this section come from May’s Simplicial Objects ...
On Nano Regular Generalized and Nano Generalized
... That is NCl(NInt (A)) = A. Since A is nano regular open, NInt(A) = A. Every Nano open set is nano regular open. ...
... That is NCl(NInt (A)) = A. Since A is nano regular open, NInt(A) = A. Every Nano open set is nano regular open. ...
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.