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Homework 13, Real Analysis 2

Due May 25 Problem 1 Suppose is measure-preserving, with . If E is invariant, then there exists a set E’ so that , and E and E’ differ by a set of measure zero. Problem 2 Let be measure-preserving, with . Then is ergodic if and only if whenever is absolutely continuous with respect to and is invariant (that […]

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Tarea 14, Matemáticas discretas

Fecha de entrega: 25 de mayo Problema 1 Sea G el grafo cuyos vértices corresponden a las aristas de , y en el cual son adyacentes si dichas aristas tienen un vértice en común. Calcula el número cromático de G.  Problema 2 Muestra que las regiones formadas por rectas en el plano son 2-coloreables. Muestra que las […]

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Homework 12, Real Analysis 2

Due May 18 Problem 1 The purpose of the following exercises is to prove the following statement: If is a translation-invariant Borel measure on that is finite on compact sets, then is a multiple of Lebesgue measure. Let be a translate of the cube If , then for each integer n. is absolutely continuous with respect to m, […]

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Tarea 13, Matemáticas discretas

Fecha de entrega: 18 de mayo Problema 1 ¿Es planar el grafo que resulta de eliminar una arista de ? ¿Es planar el complemento de un ciclo de longitud 6? ¿Es planar el grafo que resulta de agregar a un hexágono sus tres diagonales principales? Problema 2 Supón que queremos unir tres casas a tres […]

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Homework 11, Real Analysis 2

Due May 11 Problem 1 Let be a rotation. Then it induces a measure-preserving map of the sphere with its measure . Problem 2 Use the polar coordinate formula to prove the following statements. for any d. . Problem 3 If is a finite Borel measure on the interval , then is a linear functional on […]

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