Fixed Point
... Theorem 2 (Banach’s fixed-point theorem). Assume that K ⊂ Rn is closed and that Φ : K → K is a contraction. That is, there exists 0 ≤ C < 1 such that kΦ(x) − Φ(y)k ≤ Ckx − yk for all x, y ∈ K, where k·k is any norm on Rn . Then the function Φ has a unique fixed point x̂ ∈ K. Now let x(0) ∈ K be arbi ...
... Theorem 2 (Banach’s fixed-point theorem). Assume that K ⊂ Rn is closed and that Φ : K → K is a contraction. That is, there exists 0 ≤ C < 1 such that kΦ(x) − Φ(y)k ≤ Ckx − yk for all x, y ∈ K, where k·k is any norm on Rn . Then the function Φ has a unique fixed point x̂ ∈ K. Now let x(0) ∈ K be arbi ...
MANIFOLDS AND CONNECTEDNESS Proposition 1. Let X be a
... Proposition 1. Let X be a topological manifold. Then X is locally connected. In other words, for every x ∈ X, x ∈ U , U open in X, there is a connected open set V so that x ∈ V ⊂ U . Remark. This is not true for all topological spaces Y . An example is Y = {0, 1, 1/2, 1/3, 1/4, . . . } with the subs ...
... Proposition 1. Let X be a topological manifold. Then X is locally connected. In other words, for every x ∈ X, x ∈ U , U open in X, there is a connected open set V so that x ∈ V ⊂ U . Remark. This is not true for all topological spaces Y . An example is Y = {0, 1, 1/2, 1/3, 1/4, . . . } with the subs ...
Linearly Ordered and Generalized Ordered Spaces
... hUn i of open subsets of X, and a sequence hDn i where Dn is a relatively closeddiscrete subset of Un , such that whenever G is open and p ∈ G, then for some n ≥ 1, p ∈ Un and Dn ∩ G 6= ∅. See [3] for more details. As noted above, the basic metrization theorem for GO-spaces is due to Faber. However, ...
... hUn i of open subsets of X, and a sequence hDn i where Dn is a relatively closeddiscrete subset of Un , such that whenever G is open and p ∈ G, then for some n ≥ 1, p ∈ Un and Dn ∩ G 6= ∅. See [3] for more details. As noted above, the basic metrization theorem for GO-spaces is due to Faber. However, ...
A note on two linear forms
... In 1967 H. Davenport and W. Schmidt [2] (see also Ch. 8 from Schmidt’s book [11]) proved that for any two independent linear forms L, P there exist infinitely many integer points x such that |L(x)| 6 C|P (x)| |x|−3, with a positive constant C depending on the coefficients of forms L, P . From this resu ...
... In 1967 H. Davenport and W. Schmidt [2] (see also Ch. 8 from Schmidt’s book [11]) proved that for any two independent linear forms L, P there exist infinitely many integer points x such that |L(x)| 6 C|P (x)| |x|−3, with a positive constant C depending on the coefficients of forms L, P . From this resu ...
Notes 2.4 - Haiku Learning : Login
... Each group is given one proof. Draw a picture and complete the proof. Write the theorem in conditional form. 2 people from your group(s) will be randomly selected to Write the proof on the board Explain the proof. As people explain their proof. Write the conclusion in your toolbox! ...
... Each group is given one proof. Draw a picture and complete the proof. Write the theorem in conditional form. 2 people from your group(s) will be randomly selected to Write the proof on the board Explain the proof. As people explain their proof. Write the conclusion in your toolbox! ...
5-6 Inequalities in One Triangle
... prove. 2. Show that this temporary assumption leads to a contradiction. 3. Conclude that the temporary assumption must be false and that what you want to prove must be true. ...
... prove. 2. Show that this temporary assumption leads to a contradiction. 3. Conclude that the temporary assumption must be false and that what you want to prove must be true. ...
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.