A mean value theorem for non differentiable mappings in Banach
... that X satisfies (∗) and that there is a constant K such that F (x, u, p) > 0 for all (x, u) ∈ Ω × R and for all p ∈ X ∗ such that kpk > K. Then for all x, y ∈ Ω such that the line segment [x, y] is contained in Ω, |u(x) − u(y)| ≤ Kkx − yk. Results in the direction have been observed by G. Barles [1 ...
... that X satisfies (∗) and that there is a constant K such that F (x, u, p) > 0 for all (x, u) ∈ Ω × R and for all p ∈ X ∗ such that kpk > K. Then for all x, y ∈ Ω such that the line segment [x, y] is contained in Ω, |u(x) − u(y)| ≤ Kkx − yk. Results in the direction have been observed by G. Barles [1 ...
inequalities involving the mean and the standard deviation of
... Ω = V = R with the inner product x,y = k=1 xk yk , ϕi (x) = hi (x), 1 ≤ i ≤ n + 1, U = Ᏹ and J = F. The functions hi , 2 ≤ i ≤ n + 1, are linear. Therefore, they are convex and affine. In addition, the function h1 (x) = x1 − 1 is affine and convex and Ᏹ is nonempty. Consequently, the functions hi , 1 ...
... Ω = V = R with the inner product x,y = k=1 xk yk , ϕi (x) = hi (x), 1 ≤ i ≤ n + 1, U = Ᏹ and J = F. The functions hi , 2 ≤ i ≤ n + 1, are linear. Therefore, they are convex and affine. In addition, the function h1 (x) = x1 − 1 is affine and convex and Ᏹ is nonempty. Consequently, the functions hi , 1 ...
Embeddings vs. Homeomorphisms (Lecture 13)
... and so forth (this concatenation is well-defined since almost all of the isotopies hi are constant on any given compact subset of Rn ). To begin, we may assume without loss of generality that f (0) = 0 (otherwise, we can reduce to this case by conjugating by a relevant translation). Since f is an op ...
... and so forth (this concatenation is well-defined since almost all of the isotopies hi are constant on any given compact subset of Rn ). To begin, we may assume without loss of generality that f (0) = 0 (otherwise, we can reduce to this case by conjugating by a relevant translation). Since f is an op ...
topic 9 geometry proofs
... When you solve an equation in algebra, you are doing a step-by-step process to convince someone that an answer is valid. That is the process of writing a proof. So we will begin by looking at the steps in the solution of an equation, adding the algebra reasons for each step. These reasons will come ...
... When you solve an equation in algebra, you are doing a step-by-step process to convince someone that an answer is valid. That is the process of writing a proof. So we will begin by looking at the steps in the solution of an equation, adding the algebra reasons for each step. These reasons will come ...
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.