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... exists a preopen set ρ in X containing xε such that f (ρ) ≤ β. Therefore, we obtain yν ∈ µ and f (ρ) ∧ µ = ∅. This shows that G(f ) is fuzzy pre-co-closed. Theorem 28. If f : X → Y is fuzzy precontinuous and Y is fuzzy co-T1 , then G(f ) is fuzzy pre-co-closed in X × Y . Proof. Let (xε , yν ) ∈ (X × ...

... exists a preopen set ρ in X containing xε such that f (ρ) ≤ β. Therefore, we obtain yν ∈ µ and f (ρ) ∧ µ = ∅. This shows that G(f ) is fuzzy pre-co-closed. Theorem 28. If f : X → Y is fuzzy precontinuous and Y is fuzzy co-T1 , then G(f ) is fuzzy pre-co-closed in X × Y . Proof. Let (xε , yν ) ∈ (X × ...

The sums of the reciprocals of Fibonacci polynomials and Lucas

... (x) are defined by Fn+2(x) = xFn+1(x) + Fn(x), n ≥ 0 with the initial values F0(x) = 0 and F1(x) = 1; Ln+2(x) = xLn+1(x) + Ln(x), n ≥ 0 with the initial values L0(x) = 2 and L1(x) = x. For x = 1 we obtain the usual Fibonacci numbers and Lucas numbers. Let ...

... (x) are defined by Fn+2(x) = xFn+1(x) + Fn(x), n ≥ 0 with the initial values F0(x) = 0 and F1(x) = 1; Ln+2(x) = xLn+1(x) + Ln(x), n ≥ 0 with the initial values L0(x) = 2 and L1(x) = x. For x = 1 we obtain the usual Fibonacci numbers and Lucas numbers. Let ...

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.