Part II - Cornell Math
... (a) If α : I → X is a loop at x, then f ◦ α : I → Y is a loop at y. If β : I → X is a loop at x such that β p α, then f ◦ β is a loop at y and f ◦ β p f ◦ α. This path-homotopy is given by f ◦ H : I × [0, 1] → Y if H : I × [0, 1] → X is a path-homotopy from β to α. (b) Suppose that there is no s ...
... (a) If α : I → X is a loop at x, then f ◦ α : I → Y is a loop at y. If β : I → X is a loop at x such that β p α, then f ◦ β is a loop at y and f ◦ β p f ◦ α. This path-homotopy is given by f ◦ H : I × [0, 1] → Y if H : I × [0, 1] → X is a path-homotopy from β to α. (b) Suppose that there is no s ...
What is Zeckendorf`s Theorem?
... number has only finitely many representations in radix-F . Hence, we will eventually have (Z1), (Z2), and (Z3), and thus, a Zeckendorf representation for n + 1. We may further generalize the addition of two numbers in radix notation. To perform such an addition, we may simply add two numbers digit-w ...
... number has only finitely many representations in radix-F . Hence, we will eventually have (Z1), (Z2), and (Z3), and thus, a Zeckendorf representation for n + 1. We may further generalize the addition of two numbers in radix notation. To perform such an addition, we may simply add two numbers digit-w ...
SINGULAR CONTINUOUS SPECTRUM OF HALF
... Sparse potentials were also discussed by Gordon, Molchanov and Zagany in [14], where some results were given without proofs. Under some assumptions on the degree of sparseness of the potential one gets a purely singular continuous spectrum. An example of such a potential was constructed by Simon and ...
... Sparse potentials were also discussed by Gordon, Molchanov and Zagany in [14], where some results were given without proofs. Under some assumptions on the degree of sparseness of the potential one gets a purely singular continuous spectrum. An example of such a potential was constructed by Simon and ...
An Intuitive Introduction - University of Chicago Math Department
... h(s, 0) = f (s), h(s, 1) = g̃(s). Since the original homotopy h fixes the endpoints, t 7→ h̃(0, t) and t 7→ h̃(1, t) give paths from, respectively, x1 to a point in the image of g and from x2 to the point in the image of g. But both of these paths are entirely contained within the preimage of the po ...
... h(s, 0) = f (s), h(s, 1) = g̃(s). Since the original homotopy h fixes the endpoints, t 7→ h̃(0, t) and t 7→ h̃(1, t) give paths from, respectively, x1 to a point in the image of g and from x2 to the point in the image of g. But both of these paths are entirely contained within the preimage of the po ...
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.