Ricardo A. Sáenz

Fecha de entrega: 2 de marzo Problema 1 Demuestra las identidades $latex \displaystyle\binom{n}{0}\binom{m}{k} + \binom{n}{1}\binom{m}{k-1} + \cdots + \binom{n}{k-1}\binom{m}{1} + \binom{n}{k}\binom{m}{0} = \binom{n+m}{k}.$ $latex \displaystyle\binom{n}{0} - \binom{n}{1} + \binom{n}{2} - \binom{n}{3} + \cdots + (-1)^m \binom{n}{m} = (-1)^m\binom{n-1}{m}.$ $latex \displaystyle\binom{n}{0}\binom{0}{m} + \binom{n}{1}\binom{1}{m} + \binom{n}{2}\binom{2}{m} + \binom{n}{3}\binom{3}{m} + \cdots + \binom{n}{n}\binom{n}{m} = \binom{n}{m} 2^{n-m}.$ En 3, suponemos que $latex m\le n$ y, en caso que $latex k<m$, entonces $latex \displaystyle\binom{k}{m} = 0$. Problema 2 Muestra las desigualdades $latex \displaystyle \frac{n^k}{k^k} \le \binom{n}{k} \le \frac{n^k}{k!}.$ Problema 3 Muestra que $latex \displaystyle \binom{n}{10} \sim \frac{n^{10}}{10!}$. Problema 4 Muestra que, si los eventos A y B son excluyentes, entonces $latex P(A) + P(B) = P(A\cup B).$ Muestra que, para cualquiera...

Due March 2 Problem 1 Let f be integrable on $latex [0,b]$ and define, on $latex [0,b]$, $latex \displaystyle g(x) = \int_x^b \frac{f(t)}{t} dt.$ Then g is integrable on $latex [a,b]$ and $latex \displaystyle \int_0^b g(x) dx = \int_0^b f(t) dt$. Problem 2 Let $latex F\subset\R$ be a closed set such that $latex m(\R\setminus F) <\infty$, and let $latex \delta(x) = d(x,F)$ be the distance from x to F . Then $latex \delta(x)$ is a Lipschitz function. Let $latex \displaystyle I(x) = \int_\R \frac{\delta(x)}{|x-y|^2} dy.$ Then $latex I(x) = \infty$ for any $latex x\in\R\setminus F$, and $latex I(x) < \infty$ for a.e $latex x\in F$. Problem 3 There exists $latex f\in L^1(\R^d)$ and a sequence $latex f_n\in L^1(\R^d)$ such that $latex f_n\to f$ in $latex L^1$, but $latex f_n(x) \not\to f(x)$ for every x . Problem 4 Consider the function defined on $latex \R$ by $latex f(x) = \begin{cases} x^{-1/2} & 0 < x < 1,\\0 & \text{otherwise.} \end{cases}$ For a fixed enumaer...

Due February 23 Probl For any two Cantor sets $latex \mathcal C_1, \mathcal C_2$, as constructed in HW2, Problem 3, there exists a continuous, bijective and increasing function $latex F:[0,1]\to[0,1]$ that maps $latex \mathcal C_1$ surjectively onto $latex \mathcal C_2$. There exists a measurable function f and a continous function $latex \Phi$ such that $latex f\circ\Phi$ is non-measurable. Problem 2 Given a collection of sets $latex E_1, E_2, \ldots, E_n$, there exists a disjoint collection $latex F_1, F_2, \ldots, F_N$, $latex N=2^n - 1$, such that $latex \bigcup E_j = \bigcup F_k$ and each $latex \displaystyle E_j = \bigcup_{F_k\subset E_j} F_k.$ Problem 3 (Tchebychev inequality) Let $latex f\ge0$ be integrable, $latex \alpha > 0$ and $latex E_\alpha = \{x:f(x)>\alpha\}$. Then $latex \displaystyle m(E_\alpha) \le \frac{1}{\alpha}\int f.$ Problem 4 Let $latex f\ge 0$ be finite almost everywhere, $latex E_k = \{ x:f(x) > 2^k\}$ and $latex F_k = \{x: 2^k < f(x) \le ...

Fecha de entrega: 23 de febrero Problema 1 Experimenta, haz una conjetura y demuestra, tanto por inducción como combinatóricamente, el valor de la suma $latex \displaystyle 0\cdot \binom{n}{0} +1\cdot \binom{n}{1} +2\cdot \binom{n}{2} + \ldots (n-1)\cdot \binom{n}{n-1} +n\cdot \binom{n}{n}.$ Problema 2 Demuestra la identidad $latex \displaystyle\binom{n}{0} - \binom{n}{1} + \binom{n}{2} - \binom{n}{3} +\cdots + (-1)^n\binom{n}{n} = 0$ interpretando combinatóricamente los términos positivos y negativos. Problema 3 Demuestra combinatóricamente que $latex \displaystyle\binom{n}{0} + \binom{n}{1}2 + \binom{n}{2}4 + \cdots + \binom{n}{n-1} + \binom{n}{n}2^n = 3^n.$ Problema 4 ¿De cuántas formas puedes acodomodar 8 torres (iguales) en un tablero de ajedrez de tal forma que no se ataquen entre sí? Responde la pregunta anterior, pero si tenemos 4 torres blancas y 4 torres negras. Repite la pregunta, pero en el caso en que las 8 torres son distintas. Problema 5 Las palabras ERRATA y BARBA...