Ricardo A. Sáenz

Due date: December 7 Problem 1 Let $latex f_n(x) = a_n x^2 + b_n x + c_n$ be a sequence of quadratic polynomials such that $latex \displaystyle \int_0^1 |f_n(x)| dx \to 0.$ Then the coefficient sequences $latex a_n, b_n, c_n$ all converge to zero. Problem 2 For $latex r\in\mathbb Z_+$, let $latex \mathscr P_r$ be the space of polynomials of degree at most $latex r$. If $latex f_n\in\mathscr P_r$ converge uniformly to f  in [0,1], then $latex f\in\mathscr P_r$. The polynomials $latex \displaystyle f_n(x) = 1 + \frac{1}{2}x + \frac{1}{2^2}x^2 + \ldots + \frac{1}{2^n} x^n$ converge uniformly on [0,1], but their limit is not a polynomial function. Problem 3 Let $latex \mathscr H$ be the subspace of functions $latex f\in C([0,1])$ that satisfy $latex f(1 - x) = f(x)$. Then $latex \mathscr H$ is an infinite dimensional closed subspace of $latex C([0,1])$. Problem 4 Let $latex \mathscr I: C([0,1])\to C([0,1])$ be the operator given by $latex \disp...

Due November 30 Problem 1 Let  Y be a finite dimensional subspace of the Hilbert space  X . Then  Y is closed in  X . Problem 2 For each $latex n\in\Z$, let $latex e_n(x) = e^{2\pi i nx}$. Then $latex \displaystyle \int_0^1 e_n(x) \overline{e_m(x)} dx = \begin{cases} 1 & n=m\\ 0 & n\not=m. \end{cases}$   Problem 3 Let  Y be a close subspace of the Hilbert space  X and $latex T:X\to Y$ the orthogonal projection onto  Y , $latex Tx = \text{Proj}_Y x.$ T is continuous. Problem 4 Let  Y be a closed subspace of the Hilbert space  X , and let $latex Y^\perp = \{ x\in X: x\perp Y\}.$ $latex Y^\perp$ is a closed subspace of  X . $latex X\cong Y\oplus Y^\perp$.

Problem 1 Let $latex \Omega\subset\R^n$ be a bounded domain and $latex u_0\in C(\bar\Omega)$. Suppose that $latex u\in C^{2,1}(\Omega\times(0,\infty))\cap C(\bar\Omega\times[0,\infty))$ is a solution of $latex \begin{cases} \partial_t u - \Delta u=0 & \text{in }\Omega\times(0,\infty)\\ u(\cdot,0) = u_0 & \text{on } \Omega \\ u=0 & \text{on } \partial\Omega\times(0,\infty).\end{cases}$ Then there exist constants $latex \mu, C>0$, depending only on $latex n,\Omega$, such that $latex \displaystyle \sup_\Omega |u(\cdot,t)| \le C e^{-\mu t} \sup_\Omega |u_0|$ for any $latex t>0$. Problem 2 Let $latex \Omega\subset\R^n$ be a bounded domain, $latex c\in C(\bar\Omega\times[0,T])$ with $latex c\ge -c_0$ for a constant $latex c_0\ge 0$, and $latex u_0\in C(\Omega)$ nonnegative. Suppose $latex u\in C^{2,1}(\Omega\times(0,T])\cap C(\bar\Omega\times[0,T])$ is a solution of $latex \begin{cases} \partial_t u - \Delta u + cu = -u^2 & \text{in }\Omega\times(0,T]\\ u(\cdot,0) = u_0...

Due November 23 Problem 1 Let  X be a metric space, $latex \mathcal C_X$ the set of its nonempty compact subsets and $latex d_H$ the Hausdorff metric. If  X is discrete, then $latex (\mathcal C_X,d_H)$ is discrete. Let $latex A\subset X$ be a finite set of isolated points of  X . Then  A is an isolated point in $latex \mathcal C_X$. Problem 2 Let $latex X=C([-1,1])$ with the inner product $latex \displaystyle (f,g) = \int_{-1}^1 f\bar g.$ Apply the Gram-Schmidt process to the sequence $latex f_n(x) = x^n$ to obtain the orthonormal polynomials $latex p_0, p_1, p_2, p_3$, such that each $latex p_n$ is of degree  n . These are the first  Legendre polynomials . Problem 3 If $latex ||\cdot||_1, ||\cdot||_2$ are norms induced by inner products in $latex \R^l$, then they are equivalent: there exist constants $latex c,C>0$ such that $latex c||x||_1 \le ||x||_2 \le C||x||_1$ for all $latex x\in\R^l$. ( Hint: use the Gram-Schmidt process to construct orthonrmal bases for each inner ...

Problem 1 $latex \displaystyle \int_{-\infty}^\infty e^{-\pi x^2} e^{-2\pi i x\xi} dx = e^{-\pi\xi^2}$ for any $latex \xi\in\R$.  Hint:  For each N,  calculate the contour integral $latex \displaystyle \int_\gamma e^{-\pi (z+i\xi)^2} dz$ over the contour $latex \gamma$ around the rectangle with vertices $latex -N, N, N-i\xi, -N-i\xi$. Take $latex N\to\infty$. Problem 2 $latex \displaystyle \int_{\R^n} e^{-\pi|x|^2} e^{-2\pi i x\cdot \xi} dx = e^{-\pi|\xi|^2}$ for every $latex \xi\in\R^n$. Problem 3 $latex K(x,t) = t^{-n/2} e^{-|x|^2/4t}$ satisfies the heat equation. $latex G(x,t) = (1 - 4\alpha t)^{-n/2} e^{\alpha|x|^2/(1-4\alpha t)}$, for any $latex \alpha>0$, also satisfies the heat equation. Problem 4 Let $latex u_0:[0,\infty)\to\R$ be bounded and continuous, with $latex u_0(0)=0$. Find an integral representation for the solution of the problem $latex \displaystyle \begin{cases} \partial_t u - \partial^2_x u = 0 & x>0, t>0 \\ u(x,0) = u_0(x) & x>0 \\ u(0,t) =...

Due November 16 Problem 1 If $latex A=\begin{pmatrix} 1/12 & 5/8\\ 5/8 & 1/12\end{pmatrix}$, then the map $latex x\mapsto Ax$ is a contraction in $latex \R^2$. Problem 2 Let $latex F(x,t) = \dfrac{tx}{x^2+1}$. Then, for all $latex t\in\R$, $latex |F(x,t) - F(y,t)| \le |t||x-y|$. Problem 3 The function $latex f(x) = \sqrt x$ on $latex [0,\infty)$ is uniformly continuous but not Lipschitz. Problem 4 Consider the operator $latex \Phi:C([-1,1])\to C([-1,1])$ given by $latex \displaystyle \Phi(x)(t) = 1 + 2 \int_0^t s x(s) ds,$ for any $latex x(t)\in C([-1,1])$. Starting from the constant function $latex x_0(s)=1$, verify explicitly that the  n th iteration of $latex x_{n+1} = \Phi(x_n)$ is the  n th Taylor polynomial of $latex t\mapsto e^{t^2}$ around $latex t=0$.

In all problems, $latex \Omega\subset\R^n$ is a $latex C^1$ bounded domain, and derivatives are understood as weak derivatives. Problem 1 The space $latex H^1(\Omega)$ is a Hilbert space with respect to the inner product $latex \displaystyle \langle u, v \rangle_{H^1} = \int_\Omega \big( u\cdot v + \nabla u \cdot \nabla v \big).$ Problem 2 The $latex H_0^1$ and $latex H^1$ norms are equivalent in the space $latex H_0^1(\Omega)$. Problem 3 The restriction $latex u\mapsto u|_{\mathbb \partial\Omega}$ is bounded from $latex H^1(\Omega)$ into $latex L^2(\partial\Omega)$. ( Hint: Extend the normal field $latex \nu$ from $latex \partial\Omega$ to $latex \bar\Omega$ and use the divergence theorem, together with the Cauchy inequality.) Problem 4 Consider, in the disk $latex \mathbb D$, the functions $latex u_k(r,\theta) = r \cos k\theta$, $latex k\in\N.$ The $latex u_k$ are orthogonal in $latex H^1(\mathbb D)$, with respect to the $latex H^1$ product. ( Hint:  Use integration in polar coord...

Due November 9 Problem 1 The fixed points of a continuous $latex f:\mathbb B^n\to\mathbb B^n$ might not be interior. Problem 2 The Brouwer fixed point theorem is false for the open ball. Problem 3 Let $latex K\subset\R^n$ be compact and convex, and $latex f:K\to K$ continuous. Then  f has a fixed point. Problem 4 Let $latex K\subset\R^n$ be compact and convex with $latex C^1$ boundary, $latex x_0\in K$, and $latex b:K\setminus\{x_0\}\to\partial K$ given by the intersection point of the line from $latex x_0$ to $latex x$, on the side of $latex x$. Then $latex b\in C^1$. Note that, if $latex x_0$ is an interior point of $latex K$, then $latex b$ is a retraction from $latex K\setminus\{x_0\}$ onto $latex \partial K$.  

Problem 1 Identify $latex \R^2$ with $latex \C$. The 2-dimensional zonal harmonics are then given by $latex Z_0^{e^{i\theta}}(e^{i\varphi}) = \dfrac{1}{2\pi},\qquad Z_k^{i\theta}(e^{i\varphi}) = \dfrac{1}{\pi}\cos k(\varphi - \theta),\quad k>0.$ Verify explicitly the properties of zonal harmonics seen in class. The polynomial $latex F_k$ given by $latex F_k(x\cdot y) = Z_k^x(y)$ is equal to $latex \dfrac{1}{\pi}T_k$ for $latex k>0$, where $latex T_k$ is the Chebyshev polynomial given by $latex R_k(\cos\theta) = \cos k\theta$. Problem 2 The Gegenbauer polynomials $latex C_k^\lambda$ are given by the generating function $latex \displaystyle \sum_{k=0}^\infty C_k^\lambda(t) r^k = (1 - 2rt + r^2)^{-\lambda}.$ The polynomials $latex F_k$, for $latex n\ge 3$, are given by $latex \displaystyle F_k(t) = \frac{n-2+2k}{\omega_n (n-2)}C_k^{(n-2)/2}(t).$ ( Hint: Apply the operator $latex D=1 + \dfrac{r}{\lambda}\dfrac{d}{dr}$ to the generating function above, and consider the expansio...

Due November 2 Problem 1 If $latex A\subset X$ is connected, then so is its closure $latex \bar A$. If A is connected and $latex A \subset B \subset \bar A$, then so is B . If $latex A\subset \R^n$ is convex, then $latex \bar A$ is convex. Problem 2 If $latex f:[0,1]\to[0,1]$ is continuous, then there exists $latex c\in[0,1]$ such that $latex f(c) = c$. If $latex f:\mathbb S^1\to\R$ is continuous, then there exists $latex \xi\in\mathbb S^1$ such that $latex f(\xi) = f(-\xi)$. Problem 3 State whether the following are true or false. If $latex A,B$ are path connected, then $latex A\cap B$ is path connected. If $latex A, B\subset\R^n$ are convex, then $latex A\cap B$ is convex. Problem 4 Let $latex A\cap B\not=\emptyset$ in some metric space. State whether the following are true or false. If $latex A,B$ are path connected, then $latex A\cup B$ is path connected. If $latex A,B\subset\R^n$ are convex, then $latex A\cup B$ is convex.  

Problem 1 Let $latex u\in C^2$ be a solution of $latex \begin{cases} \Delta u = 0 & \text{in } \R^n\setminus B_R\\ u=0 & \text{on } \partial B_R.\end{cases}$ Then $latex u\equiv 0$ if $latex \displaystyle \lim_{|x|\to\infty}\frac{u(x)}{\log |x|} = 0,\quad n=2;\qquad \lim_{|x|\to\infty} u(x) = 0,\quad n=3.$ Problem 2 Let $latex \Omega\subset\R^n$ be a bounded $latex C^1$-domain that satisfies the inner sphere condition, and $latex f\in C(\Omega)$ bounded. If $latex u\in C^2(\Omega)\cap C^1(\bar \Omega)$ is a solution of $latex \begin{cases} \Delta u = f & \text{in } \Omega\\ u=0 & \text{on } \partial\Omega,\end{cases}$ then $latex \displaystyle \sup_{\partial\Omega}\Big|\frac{\partial u}{\partial\nu}\Big| \le C \sup_\Omega|f|,$ where $latex C>0$ depends only on $latex n,\Omega$. Problem 3 A polynomial $latex p\in\mathscr P_k$ if and only if $latex \sum x_j \partial_j p = kp.$ Let $latex p\in\mathscr P_k$, and consider its orthogonal projection $latex |x|^2q$ onto ...

Due October 19 Problem 1 Let  X be a complete metric space. The countable intersection of dense $latex G_\delta$ sets in  X is a dense $latex G_\delta$ set in  X . If a set and its complement are dense subsets of  X , at most one can be $latex G_\delta$. If X doesn't have isolated points, a countable dense subset of  X cannot be $latex G_\delta$. Problem 2 Let  X be a complete metric space. If $latex O\subset X$ is open, then  O is a metric subspace of the second category. If $latex \{F_n\}$ are closed subsets of  X with $latex X = \bigcup_n F_n$, then $latex \bigcup_n \text{int}(F_n)$ is dense in  X . Problem 3 Let  X be a complete metric space. The set of points of discontinuity of the characteristic function $latex \chi_O$ of an open set  O is a nowhere dense subset of  X . Given open sets $latex \{O_n\}$, there exists $latex x\in X$ such that $latex \chi_{O_n}$ is continuous at  x for each  n . Problem 4 Let $latex d(x) = d(x,\Z)$ denote the dista...

Problem 1 Find the Green's function for the Laplace equation in the upper half-space $latex \R^{n+1}_+ = \{x_{n+1}>0\}$ and then derive a formal integral representation for a solution of the Dirichlet problem $latex \begin{cases} \Delta u = 0 &\text{in } \R^{n+1}_+\\ u = \phi & \text{on }\{x_{n+1}=0\}.\end{cases}$ Problem 2 Use the Poisson integration formula to prove the Harnack inequality $latex \displaystyle\Big(\frac{R}{R+r}\Big)^{n-2} \frac{R-r}{R+r} u(x_0) \le u(x) \le \Big(\frac{R}{R-r}\Big)^{n-2} \frac{R+r}{R-r}u(x_0),$ for a harmonic function  u in $latex B_R(x_0)\subset\R^n$, where $latex r=|x-x_0|<R$. Problem 3 Use Problem 2 to give another proof of Liuville's theorem: a harmonic function in $latex \R^n$ bounded from above, or from below, is constant.

Due October 12 Problem 1 Let  X be a metric space. We say that a collection $latex \{F_\alpha\}$ of subsets of  X has the finite intersection property (FIP) if any finite subcollection $latex \{F_{\alpha_1}, F_{\alpha_2},\ldots,F_{\alpha_k}\}$ of them has nonempty intersection: $latex F_{\alpha_1}\cap F_{\alpha_2}\cap\ldots\cap F_{\alpha_k}\not=\emptyset.$ X is compact if and only if the intersection of any collection of closed sets that has the FIP is nonempty. Give an example of a decreasing sequence of nonempty closed sets in a metric space with empty intersection. Problem 2 The closed ball $latex \bar B_r(x_0) = \{ x\in X: d(x,x_0)\le r\}$ is a closed set in  X . Is $latex \bar B_r(x_0) = \overline{B_r(x_0)}$ in every metric space? Problem 3 If $latex f:X\to Y$ is continuous, its  graph  $latex G=\{(x,f(x)): x\in X\}$ is closed in $latex X\times Y$. Problem 4 If X is a metric space, then $latex \diam \bar A = \diam A$ for any $latex A\subset X$.    

Problem 1 If $latex u$ is harmonic in the connected domain $latex \Omega$ and is not constant, then $latex u(\Omega)$ is open in $latex \R$. Suppose $latex \Omega$ is bounded and that its boundary $latex \partial\Omega$ is connected. If $latex u$ is harmonic in $latex \Omega$, then $latex u(\Omega)\subset u(\partial\Omega)$. Problem 2 A radial harmonic function on $latex \mathbb B$ is constant. A positive harmonic function on $latex \R^d$ is constant. Problem 3 Suppose $latex u(x)$ is harmonic in some domain in $latex \R^n$. Then $latex v(x) = |x|^{2-n} u\Big(\dfrac{x}{|x|^2}\Big)$ is also harmonic in a suitable domain. Problem 4 For $latex n=2$, find the Green's function for the Laplace operator on the first quadrant.  

Due October 5 Problem 1 Let $latex L^1([a,b])$ be the space of real valued continuous functions with the $latex d_1$ metric. The polynomials are dense in $latex L^1([a,b])$. Is $latex L^1([a,b])$ separable? Problem 2 Let $latex f:[a,b]\to\R$ be a continuous function such that $latex \displaystyle \int_a^b f(x) x^n dx = 0$ for all $latex n=0,1,2,\ldots$. Then $latex f(x)=0$ for all $latex x\in[a,b].$ Problem 3 If $latex X,Y$ are compact metric spaces, then the tensor space $latex \displaystyle C(X)\otimes C(Y) = \{ (x,y)\mapsto \sum_{k=1}^n f_k(x)g_k(y): f_k\in C(X), g_k\in C(Y), n\ge1\}$ is dense in $latex C(X\times Y)$. Note:  The product space $latex X\times Y$ has the metric $latex d_{X\times Y} \big( (x_1,y_1), (x_2,y_2) \big) = d_X(x_1,x_2) + d_Y(y_1,y_2).$ Problem 4 State whether the following are true: $latex \overline{A\cup B} \subset \overline{A}\cup \overline{B}$; $latex \overline{A\cup B} \supset \overline{A}\cup \overline{B}$; $latex \overline{A\cap B} \subset \ov...

Problem 1 Let $latex u_0\in L^2(0,\pi)$ be the solution of the heat equation seen in class. Then, for all $latex i,j\in\N$, $latex \displaystyle \sup_{x\in[0,\pi]}|\partial_x^i \partial_t^j u(x,t)|\to 0$ as $latex t\to\infty$. Problem 2 For any $latex u_0\in L^2(0,\pi), f\in L^2((0,\pi)\times(0,\infty))$, find a formal expression of a solution of the problem $latex \displaystyle \begin{cases} \partial_t u - \partial^2_x u = f & \text{in }(0,\pi)\times(0,\infty)\\ u(x,0) = u_0(x) & x\in (0,\pi)\\ u(0,t)=u(\pi,t)=0 & t\in (0,\infty). \end{cases}$ Problem 3 The wave operator $latex \partial_t^2 - \partial_x^2$ on $latex \R^2$ commutes with the Lorentz transformations (hyperbolic rotations) $latex \displaystyle T_\theta = \begin{pmatrix}\cosh\theta & \sinh\theta\\ \sinh\theta & \cosh\theta\end{pmatrix},$ for $latex \theta\in\R$.

Due September 28 Problem 1 Let $latex f_n:[a,b]\to\R$ a monotone sequence of continuous functions which converges pointwise to the continuous function $latex f:[a,b]\to\R$. Then $latex f_n\rightrightarrows f$ on $latex [a,b]$. Problem 2 Let $latex K:[0,1]\times[0,1]\to[0,1]$ be a continuous function and define the operator $latex \mathscr L:C([0,1])\to C([0,1])$ by $latex \displaystyle \mathscr Lf(x) = \int_0^1 K(x,y) f(y) dy$. Then, the image of the closed ball $latex \bar B_1(0)$ in $latex C([0,1])$ under $latex \mathscr L$ is compact. Such operator is called a  compact operator . Let $latex w:[0,1]\to\R$ be continuous. Then the operator $latex \displaystyle \mathscr Lf(x) = \int_0^x f(t) w(t)dt$ is compact. Problem 3 Let $latex F_1\subset F_2\subset \ldots$ an increasing sequence of compact subsets of a metric space, and let $latex K \subset\subset \bigcup F_n$. Then there exists  p  such that $latex K\subset F_p$. Problem 4 Let $latex \Omega\subset\R^m$ be open and $latex f_n:\...

Problem 1 Classify the following second-order PDE. $latex \displaystyle \sum_{i=1}^n \partial^2_{x_i x_i}u + \sum_{1\le i<j\le n} \partial^2_{x_i x_j}u = 0$ $latex \displaystyle \sum_{1\le i<j \le n} \partial^2_{x_i x_j}u = 0$ Problem 2 Use energy methods to discuss the uniqueness of the problem $latex \begin{cases} \Delta u - u^3 = f & \text{in }\Omega\\u = \phi & \text{on }\partial\Omega.\end{cases}$ Problem 3 Let $latex \Omega$ be a bounded $latex C^1$-domain in $latex \R^n$ and  u be a $latex C^2$-function in $latex \bar\Omega\times[0,T]$ satisfying $latex \begin{cases} u_t - \Delta u = f & \text{in }\Omega\times(0,\infty)\\ u(\cdot, 0) = u_0 & \text{in }\Omega\\ u=0 & \text{on }\partial\Omega\times(0,\infty).\end{cases}$ Then $latex \displaystyle \sup_{0\le t \le T} \int_\Omega |\nabla u(\cdot,t)|^2 dx + \int_0^T\int_\Omega |\partial_t u|^2 dx dt \le C\Big( \int_\Omega |\nabla u_0|^2 dx + \int_0^T\int_\Omega |f|^2 dxdt \Big),$ where  C is a positive...

Due September 21 Problem 1 Let  X  be a compact space and $latex f:X\to Y$ a continuous bijection. Then $latex f^{-1}:Y\to X$ is continuous. Give an example of a continuous bijection $latex f:X\to Y$, for a noncompact X , whose inverse is not continuous. Problem 2 Let  X be a discrete metric space. Any function on  X is uniformly continuos. Is  X always compact? If not, give necessary and sufficient conditions for a discrete space to be compact. Problem 3 Let X be a  subspace of  Y ,  i.e. a subset of the metric space  Y with the same metric. We say  X is closed in  Y if  X contains all its limits: if $latex x_n\in X$ is a sequence that converges to $latex x_0\in Y$, then $latex x_0\in X$. If  Y is complete, then  X is complete if and only if  X is closed in  Y . Problem 4 Every bounded set in $latex \R^l$ is totally bounded. Every closed and bounded set in $latex \R^l$ is compact.

Due September 14 Problem 1 If the measurable $latex f_n\searrow f\ge 0$ with $latex \int f_1 < \infty$, then $latex \int f_n \to \int f.$ Explain the condition $latex \int f_1 < \infty$. Problem 2 There exists a positive continuous $latex f \in L^1(\R)$ such that $latex \limsup_{|x|\to\infty} f(x) = \infty.$ If $latex f\in L^1(\R)$ is uniformly continuous, then $latex \lim_{|x|\to\infty}f(x) = 0.$ Problem 3 If $latex f\in L^1(\R)$ and $latex F(x) = \int_{-\infty}^x f$. Then  F is uniformly continuous. Problem 4 Let $latex f:D\to\R$ be uniformly continuous, with $latex D\subset\R$. If $latex x_0$ is a limit point of D , then  f has limit at $latex x_0$. f has a continuous extension to $latex \bar D$, the closure of  D.

Problem 1 Let $latex a = (a_1, \ldots, a_n), b, f, u_0$ continuous functions with $latex |a|\le 1/\kappa$, and $latex u\in C^1(\bar\R^{n+1}_+)$ a solution to the IVP $latex \displaystyle \partial_t u + \sum_{i=1}^n a_i \partial_{x_i}u + bu = f \qquad \text{in  }\R^{n+1}_+,$ $latex u(x,0) = u_0(x) \qquad \text{on } \R^n.$ Then, for any $latex P = (X,T)\in\R^{n+1}_+$, $latex \displaystyle \sup_{C_\kappa}(P)|e^{-\alpha t}u| \le \sup_{\partial_{-}C_\kappa(P)}|u_0| + \frac{1}{\alpha + \inf_{C_\kappa(P)}b} \sup_{C_\kappa(P)}|e^{-\alpha t}f|,$ where $latex \alpha>0$ is a constant such that $latex \displaystyle \alpha + \inf_{C_\kappa(P)}b > 0.$ Problem 2 Let $latex a = (a_1, \ldots,a_n), b, f, u_0$ be $latex C^1$ functions with $latex |a|\le 1/\kappa$, and $latex u$ a $latex C^2(\bar\R^{n+1}_+)$ solution of the IVP of the previous problem. Then, for any $latex P=(X,T)\in\R^{n+1}_+$, $latex \displaystyle ||u||_{C^1(C_\kappa(P))} \le C \big( ||u_0||_{C^1(\partial_-C_\kappa(P))} + ||f||_{C...

Problem 1 Find solutions of the following initial-value problems in $latex \R^2$. $latex \partial_y u + (1+x^2)\partial_x u - u = 0; u(x,0) = \tan x$ $latex \partial_y u + \partial_x u = u^2; u(x,0) = h(x)$ Problem 2 Find a smooth function $latex a(x,y)$ in $latex \R^2$ such that the equation $latex \partial_y u + a(x,y) \partial_x u = 0$ does not have global solutions for any Cauchy data $latex \{y=0\}$. Problem 3 Let $latex \alpha\in\R$ and $latex h(x)$ a continuous function in $latex \R$, and consider que problem $latex y\partial_x u + x \partial_y u = \alpha u; \quad u(x,0) = h(x).$ Find all points where $latex \{y=0\}$ is characteristic. What is the compatibility condition on  h on those points? Find the solution of the initial-value problem away of the point in (1). What is the domain of this solution? For the cases $latex \alpha=1, h(x) = x$ and $latex \alpha=3, h(x) = x$, check whether this solution can be extended over the characteristic points. Find the chara...

Due Septembre 7 Problem 1 If $latex E, F\subset\R$ are a closed and a compact disjoint sets, then $latex \text{dist}(E,F)>0$. Problem 2 If $latex E\subset\R$ is closed, then it is measurable. Follow the next steps. Prove that it is sufficient to assume that  E is compact, and thus $latex |E|_* < \infty.$ Given $latex \e>0$, choose an open $latex U\supset E$ with $latex |U|_* < |E|_* + \e.$ Prove that we can write $latex U\setminus E = \bigcup I_j$, where the $latex I_j$ are disjoint open intervals. If $latex I,J$ are disjoint open intervals, then $latex |I\cup J|_* = |I| + |J|.$ For each  N , $latex |U|_* \ge |E|_* + \sum_{j=1}^N |I_j|$. Conclude $latex |U\setminus E|_* < \e$. Problem 3 Find a sequence of measurable sets $latex E_1 \supset E_2 \supset ...$ such that, for $latex E = \bigcap E_j$, $latex |E| \not= \lim |E_j|$. Problem 4 For $latex E\subset\R$, let $latex U_n=\{x\in\R: \text{dist}(x,E) < 1/n\}$. If  E is compact, then $latex |E| = \lim |U_n...

Problem 1 If, for $latex k=1, 2, \ldots, N$, $latex y_k(x_1, \ldots, x_{n-1},t) = u(x_1 + \ldots + x_{n-1},t)$, then $latex Y = (y_k)$ solves the system $latex \displaystyle \partial_t y_k = \frac{Mr}{r - (x_1 + \ldots + x_{n-1}) - (y_1 + \ldots + y_N)} \Big( \sum_{i=1}^{n-1}\sum_{j=1}^N \partial_{x_i}y_j + 1 \Big)$ with $latex y_k(x,0)=0$ if and only if $latex u(s,t)$ solves the equation $latex \displaystyle \partial_t u = \frac{Mr}{r - s - Nu} \big( N(n-1)\partial_su + 1\big)$ with $latex u(x,0)=0$. Problem 2 The function $latex \displaystyle u(s,t) = \frac{r - s - \sqrt{(r-s)^2 - 2MNnrt}}{Nn}$ solves the equation of the previous problem near $latex (0,0)$. Problem 3 Find a solution as a power series expansion of the initial-value problem $latex \displaystyle \begin{cases} u_{tt} - u_{xx} - u=0,\quad (x,t)\in\R\times(0,\infty)\\ u(x,0) = x,\quad \partial_t u(x,0) = -x. \end{cases}$ Identify this solution.

Due August 31 Problem 1 Let $latex f_n:[a,b]\to\R$ be Riemann-integrable and $latex f_n\rightrightarrows f$. f is Riemann-integrable on $latex [a,b]$. ( Hint:  Given $latex \e > 0$, find a partition $latex \mathscr P$ of $latex [a,b]$ such that $latex U(f,\mathscr P) - L(f,\mathscr P) < \e,$ where $latex U(f,\mathscr P), L(f,\mathscr P)$ are the upper and lower sums of  f  with respect to $latex \mathscr P$, respectively.) $latex \displaystyle \int_a^b f_n \to \int_a^b f$. Problem 2 Consider the functions $latex \displaystyle f_n(x) = \frac{x}{1 + nx^2}.$ Then $latex f_n\rightrightarrows 0$, but $latex f_n'(0)\not\to 0$. Problem 3 Let $latex f\in C^k(\mathbb S)$, a  k -continuously differentiable periodic function, with period $latex 2\pi$, and let $latex a_n$ be its  n th Fourier coefficient. There exists $latex C>0$ such that $latex |a_n| \le \dfrac{C}{|n|^k}$. The series $latex \sum a_n e^{inx}$ converges uniformly if $latex k\ge 2$. Problem 4 Let  X be a metr...

Problem 1 Let  u be a $latex C^1$ function defined on a neighborhood of the domain $latex \Omega$ such that $latex \Delta u = 0$ in $latex \Omega$. Then $latex \displaystyle \int_{\partial \Omega} \partial_\nu u \; d\sigma = 0.$ Problem 2 If the power series $latex \sum a_\alpha x^\alpha$ converges at $latex x^0$, then it converges on the cube $latex |x_j|<r$ where $latex r = \min_i |x_i^0|$. Problem 3 If  u satisfies the Cauchy problem $latex \begin{cases} \partial_t^mu = G\big(x,t,(\partial_x^\alpha\partial_t^ju)_{|\alpha|+j\le m, j<m}\big)\\ \partial_t^ju(x,0) = \phi_j(x), \quad 0\le j < m,\end{cases}$ then the functions $latex y_{\alpha j} = \partial_x^\alpha\partial_t^ju$ satisfy the system $latex \partial_t y_{\alpha j} = y_{\alpha(j+1)} \quad |\alpha|+j < m,$                              $latex \partial_t y_{\alpha j} = \partial_{x_i}y_{(\alpha-e_i)(j+1)} \quad |\alpha|+j=m, j<m,$ $latex \displaystyle \partial_t y_{0m} = \frac{\partial G}{\partial t} + \sum_{|\a...

Due August 24 Problem 1 Two metrics $latex d_1, d_2$ on a space  X are equivalent if there exist constants $latex c, C>0$ such that $latex c d_1(x,y) \le d_2(x,y) \le C d_1(x,y)$ for all $latex x,y\in X$. Suppose $latex (X,d_1), (X,d_2)$ are equivalent. They have the same convergent sequences. They have the same Cauchy sequences. $latex (X,d_1)$ is complete if and only if $latex (X,d_2)$ is complete. Problem 2 Are $latex (C([0,1]), d_u)$ and $latex (C([0,1]), d_1)$ equivalent? Problem 3 Are $latex (\R,|\cdot|)$ and $latex (\R,d_B)$ equivalent? Problem 4 Suppose $latex (X,d_1)$ and $latex (X,d_2)$ have the same convergent sequences. Are they equivalent? Is one of complete if and only if the other is complete?

Las notas del curso que impartí en la Escuela de análisis ya están en la página de la escuela. Pueden verlas aquí:  Notas del curso Operadores de multiplicación La idea del curso fue introducir el Análisis armónico, tomando como pretexto la teoría de operadores de multiplicación, empezando con operadores diagonalizados y su relación con el teorema espectral de álgebra lineal, además de métodos de sumabilidad de series. El objetivo final es la demostración del teorema de Marcinkiewicz de multiplicadores de Fourier, que utiliza el teorema de Calderón y Zygmund de integrales singulares. Como “paréntesis cultural”, presento un bosquejo del contraejemplo de Fefferman a la conjetura del disco, que muestra que el operador de la bola no puede ser acotado en el espacio euclideano de dimensión mayor a 1.

Escuela de Análisis matemático Análisis de Fourier y operadores de multiplicación Bibliografía Clase 1: métodos de sumabilidad, teorema de Fejér Elias M. Stein, Rami Shakarchi,  Fourier Analysis: An Introduction , Princeton books.google.com.mx/books?id=FAOc24bTfGkC T. W. Körner, Fourier Analysis, Cambridge books.google.com.mx/books?id=OcZ5iKsGrmoC Clase 2: transformada de Fourier, distribuciones Elias M. Stein, Guido Weiss,  Introduction to Fourier Analysis on Euclidean Spaces , Princeton books.google.com.mx/books?id=xnIwDAAAQBAJ Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, John Wiley & Sons books.google.com.mx/books?id=wI4fAwAAQBAJ Clase 3: multiplicador de la bola Elias M. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, Princeton books.google.com.mx/books?id=ljcOSMK7t0EC Javier Enrique Sáenz Casas, La conjetura de Kakeya, Tesis e...

Escuela de Análisis matemático Análisis de Fourier y operadores de multiplicación Construye una función explícita $latex \eta$ como la que se necesita en la demostración del teorema de Marcinkiewicz: $latex \eta\in C^\infty(\mathbb R^d)$ tal que $latex \eta(\xi)=1$ si $latex |\xi|\le 1$ y $latex \eta(\xi)=0$ si $latex |\xi|\ge 2$. ( Sugerencia: considera la función en $latex \mathbb R$ dada por $latex \phi(t) = \begin{cases} e^{-\frac{1}{(1+t)^2}}e^{-\frac{1}{(1-t)^2}} & |t|<1\\ 0 & |t|\ge 1.) \end{cases}$ La hipótesis del teorema de Marcinkiewicz puede ser reemplazada por la condición $latex \displaystyle \sup_{R>0} R^{-d + 2|\alpha|} \int_{R\le|\xi|\le 2R} |\partial_\xi^\alpha m(\xi)|^2 d\xi \le A_\alpha$ para todo $latex 0\le|\alpha|\le l$, donde  l es el menor entero mayor que $latex d/2$. El multiplicador de Böchner-Riesz $latex m(\xi) = (1 - |\xi|^2)^\delta \chi_{\mathbb B}(\xi)$ satisface las hipótesis del teorema de Marcinkiewicz cuando $latex \delt...

Escuela de Análisis matemático Análisis de Fourier y operadores de multiplicación Sea $latex m\in L^\infty(\mathbb R^d)$, $latex m_R(\xi) = m(\xi/R)$ para $latex R>0$, y considera el operador de multiplicación dado por $latex \widehat{T_Rf}(\xi) = m_R(\xi)\hat f(\xi)$. Si $latex ||T_1f||_{L^p} \le M ||f||_{L^p}$, entonces $latex ||T_Rf||_{L^p} \le M||f||_{L^p}$ para todo $latex R>0$. Sean $latex T_1,T_2$ dos triángulos contiguos como en la figura, cada uno de base  b y altura  h , y sea $latex \tilde T_2$ el triángulo que resulta de trasladar a la izquierda el triángulo $latex T_2$ una distancia $latex 2(1-\alpha)b$, para $latex 1/2 < \alpha < 1$. Entonces $latex |T_1\cup\tilde T_2| = (\alpha^2 + 2(1-\alpha)^2)|T_1\cup T_2|$. $latex f(\alpha) = \alpha^2 -2 (1-\alpha)^2 < 1$ para todo $latex \alpha\in(1/2,1)$, y tiene mínimo 2/3. Para todo $latex n\in\mathbb N$, $latex A_n = \alpha^{2n} + 2(1-\alpha)^2 + 2(1-\alpha)^2\alpha^2 + \ldots + 2(1-\a...

Escuela de Análisis matemático Análisis de Fourier y operadores de multiplicación Si $latex f\in L^p(\mathbb R^d)$ y $latex g\in L^1(\mathbb R^d)$, entonces $latex f*g\in L^p(\mathbb R^d)$ y $latex ||f*g||_{L^p} \le ||f||_{L^p} ||g||_{L^1}.$ Si $latex f\in L^1(\mathbb R^d)\cap L^2(\mathbb R^d)$, entonces $latex ||\hat f||_{L^2} = ||f||_{L^2}.$ ( Sugerencia: Considera $latex h=f*g$, con $latex g(x) = \overline{f(-x)}$, y nota que $latex h(0) = \int \hat h$.) El operador $latex f\mapsto \hat f$ es sobreyectivo en $latex L^2(\mathbb R^d)$. ( Sugerencia: Si no, existiría $latex g\in L^2$ tal que $latex \int \hat f g = 0$ para toda $latex f\in L^2$. Utiliza el pasito del sombrero.) Sea $latex h(x) = \begin{cases} 0 & x<0\\1 & x\ge 0\end{cases}$ la función de Heaviside. Como distribuciones, $latex h' = \delta,$ la función delta de Dirac. Sea $latex \mathbf 1$ la función constante igual a 1. Como distribuciones, $latex \hat{\mathbf 1} = \delt...

Escuela de Análisis matemático Análisis de Fourier y operadores de multiplicación Si $latex T:V\to V$ es simétrico y $latex \lambda_i\not=\lambda_j$ son eigenvalores reales distintos de $latex T$, con eigenvectores $latex u_i, u_j$ correspondientes, entonces $latex u_i\perp u_j$. Sea $latex T:\mathscr H\to\mathscr H$ un operador acotado diagonalizado con sucesión multiplicadora $latex \lambda_k$. T es unitario si, y solo si, $latex |\lambda_k|=1$ para todo k . T es una proyección ortogonal si, y solo si, todo $latex \lambda_k = 0\text{ o } 1$. T es un operador compacto si, y solo si, $latex \lambda_k \to 0$. Si $latex \sum a_n$ es una serie convergente, entonces es Cesàro-sumable. Si $latex \sum a_n$ es Cesàro-sumable, entonces es Abel-sumable.

Park City Mathematics Institute Undergraduate Summer School 2018 Introduction to Harmonic Analysis Prove by induction, for the interval case, that $latex \min\{ \mathscr E_m(u): u|_{\{0,1\}}=v\} = \mathscr E_0(v) = (v(0)-v(1))^2,$ with the minimizer satisfying $latex \displaystyle u\Big(\frac{2k+1}{2^m}\Big) = \frac{1}{2}\Big(u\Big(\frac{k}{2^{m-1}}\Big) + u\Big(\frac{k+1}{2^{m-1}}\Big)\Big).$ The minimum of $latex f(x,y,z) = (a-x)^2 + (x-y)^2 + (y-a)^2 + (x-b)^2 + (b-z)^2 + (z-x)^2 + (y-z)^2 + (z-c)^2 + (c-y)^2$ is attained at $latex \displaystyle x^* = \frac{2a+2b+c}{5},\; y^* = \frac{2a+b+2c}{5},\; z^* = \frac{a+2b+2c}{5},$ with $latex f(x^*, y^*, z^*) = \dfrac{3}{5}\big((a-b)^2 + (b-c)^2 + (c-a)^2\big).$ Prove that one can obtain the values $latex u(F_2(q_1)) = x, u(F_3(q_1)) = y$ of a harmonic function in terms of the values $latex a,b,c$ at the points $latex p_2, q_1, p_3$, respectively  (as in the figure below). Use the previous problem to show that, if $latex u$ is a ha...

Park City Mathematics Institute Undergraduate Summer School 2018 Introduction to Harmonic Analysis Show that $latex ||\cdot||_{H^1}$ is a norm, by showing it is induced by an inner product. Consider, for a connected domain $latex \Omega$, the energy form $latex \mathscr E(u,v) = \int_\Omega \nabla u\cdot \nabla v$. $latex \mathscr E(u,v)$ is an inner product on $latex H^1$ modulo constants. $latex \mathscr E(u,v)$ is an inner product on $latex H_0^1$. Show the equivalences of the Dirichlet principle.

Park City Mathematics Institute Undergraduate Summer School 2018 Introduction to Harmonic Analysis Extend the result that the Hilbert transform is of weak-type $latex L^1$ to any operator of the form $latex Tf = K*f$ where $latex K$ satisfies: $latex \hat K\in L^\infty(\R)$ (say, taking the Fourier transform in the $latex L^2$ sense); there exists a constant $latex A>0$ such that $latex |K'(x)| \le A/x^2$ for any $latex x\in\R, \; x\not=0$. Show that we can replace (2) above by the condition 2'. There exists a constant $latex A>0$ such that $latex \displaystyle \int_{|x|\ge 2|t|} |K(x-t) - K(x)| dx \le A$ for any $latex t\in\R$. (Chebyshev Inequality) If $latex f\in L^p$, for some $latex 1 < p <\infty$, then $latex |\{x : |f(x)|>\alpha\}| < \dfrac{1}{\alpha^p} ||f||_{L^p}^p$ Prove that we can substitute the condition $latex \hat K\in L^\infty$ with the boundedness of $latex T$ on any $latex L^q(\R), \; q>1$, that is, 1'. there exists a...

Park City Mathematics Institute Undergraduate Summer School 2018 Introduction to Harmonic Analysis Let $latex \{\Phi_t\}_{t>0}$ be a collection that satisfies, for some constants $latex c_0, c_1, c_2$, and all $latex x\in\R^d$ and $latex t>0$, $latex \int \Phi_t = c_0$, $latex |\Phi_t(x)| \le c_1 t^{-d}$, $latex |\Phi_t(x)| \le c_2 t/|x|^{d+1}$. Then there exists $latex A>0$ such that $latex |\Phi_t*f(x)|\le A Mf(x)$, for all $latex x\in\R^d$, and thus $latex \Phi_t*f(x)\to f(x)$ as $latex t\to 0$ for almost every $latex x\in\R^d.$ (Needs complex analysis) $latex \displaystyle \hat Q _y(\xi) = \lim_{N\to\infty}\frac{1}{\pi} \int_{-N}^N \frac{x e^{-2\pi ix\xi}}{x^2+y^2} dx = -i\text{sgn}\xi e^{-2\pi y |\xi|}$. ( Hint: Use the residue theorem from complex analysis.) (Needs complex analysis) Give another proof that the Hilbert transform is bounded on $latex L^2$ by considering the Cauchy integral $latex \displaystyle Cf(z) = \frac{1}{i\pi} \int_{-\infty}^\infty \frac...

Park City Mathematics Institute Undergraduate Summer School 2018 Introduction to Harmonic Analysis Write the details of the proof of Vitalli's Lemma Use the Hardy-Littlewood maximal theorem to prove the Lebesgue differentiation theorem: if $latex f\in L^1(\R^d)$, then $latex \displaystyle \lim_{r\to 0} \frac{1}{|B_r(x)|} \int_{B_r(x)} f(y) dy = f(x)$ for almost every $latex x\in\R^d.$ State conditions on $latex \Phi$ such that, for $latex f\in L^1(\R^d)$, $latex \displaystyle \lim_{t\to 0} \Phi_t*f(x) = f(x)$ for almost every $latex x\in\R^d.$ Let $latex f(x) = \begin{cases} \dfrac{1}{|x|(\log|x|)^2} & 0<|x|<1/2\\0&\text{otherwise.}\end{cases}$ Then $latex f\in L^1(\R)$, but $latex Mf$ is not locally integrable.

Park City Mathematics Institute Undergraduate Summer School 2018 Introduction to Harmonic Analysis Verify the integral $latex \displaystyle \int_{\R^d}\frac{dx}{(|x|^2+1)^\frac{d+1}{2}} = \frac{\pi^\frac{d+1}{2}}{\Gamma(\frac{d+1}{2})}.$ Use Minkowski inequality to prove that, if $latex K\in L^1(\R^d)$ and $latex f\in L^p(\R^d)$, then $latex ||K*f||_{L^p}\le ||K||_{L^1}||f||_{L^p}.$ Prove that, if $latex f\in C_c(\R^d)$ and $latex y\in\R^d$, then $latex ||f(\cdot - ty) - f||_{L^p} \to 0$ as $latex t\to 0$. Let $latex \Phi\in L^1(\R^d)$ with $latex \int \Phi = 1$, and $latex \Phi_t(x) = t^{-d}\Phi(x/t).$ Then $latex \int \Phi_t = 1$ for all $latex t>0.$ There exists some $latex M>0$ such that $latex \int |\Phi_t| \le M$ for all $latex t > 0.$ For each $latex \delta>0$, $latex \displaystyle \int_{|x|\ge\delta} |\Phi_t(x)|dx \to 0$ as $latex t\to 0.$ If $latex f\in C_c(\R^d)$, then $latex \Phi_t*f(x) \to f(x)$ uniformly. State conditions on $latex \Phi$ (as i...

Park City Mathematics Institute Undergraduate Summer School 2018 Introduction to Harmonic Analysis A polynomial $latex p\in\mathscr P_k$ if and only if $latex \sum x_j \partial_j p = kp.$ $latex \dim \mathscr P_k = \binom{k+d-1}{k}$ Let $latex p\in\mathscr P_k$, and consider its orthogonal projection $latex |x|^2q$ onto the space $latex |x|^2\mathscr P_{k-2}$ with respect to the inner product seen in class. Then $latex p - |x|^2q$ is harmonic. ( Hint : Prove $latex \langle r,\Delta(p - |x|^2q\rangle = 0$ for every $latex r\in\mathscr P_{k-2}$.) If $latex p\in\mathscr H_k$, then $latex \dfrac{p}{|x|^{2k+d-2}}$ is harmonic on $latex \R^d\setminus\{0\}$. The spaces are invariant under rotations ,  i.e. if $latex f\in H_k$ then $latex f\circ R\in H_k$ for any rotation $latex R$.

Park City Mathematics Institute Undergraduate Summer School 2018 Introduction to Harmonic Analysis The function $latex x\mapsto P(x,\xi)$ is harmonic for each $latex \xi$, where $latex P(x,\xi)$ is the Poisson kernel for the ball. (Symmetry Lemma) For any $latex x\in\mathbb B$ and $latex \xi\in\mathbb S$, $latex \displaystyle \Big| |x|\xi - \frac{x}{|x|}\Big| = |x-\xi|$. (Harnack's inequality) If $latex u$ is a positive harmonic function on $latex \bar{\mathbb B}$, then $latex \displaystyle \frac{1-|x|}{(1+|x|)^{d-1}} u(0) \le u(x) \le \frac{1 + |x|}{(1-|x|)^{d-1}}u(0).$ (Hopf Lemma) If $latex u$ is a nonconstant harmonic function on $latex \bar{\mathbb B}$ and attains its maximum at $latex \zeta\in\mathbb S$, there exists $latex c>0$ such that $latex u(\zeta) - u(r\zeta) \ge c(1-r)$ for any $latex 0 < r < 1.$ If $latex u$ is harmonic on $latex \bar{\mathbb B}$ and its normal derivative is 0 everywhere on $latex \mathbb S$, then $latex u$ is constant.

Park City Mathematics Institute Undergraduate Summer School 2018 Introduction to Harmonic Analysis If $latex u$ is harmonic in the connected domain $latex \Omega$ and is not constant, then $latex u(\Omega)$ is open in $latex \R$. Suppose $latex \Omega$ is bounded and that its boundary $latex \partial\Omega$ is connected. If $latex u$ is harmonic in $latex \Omega$, then $latex u(\Omega)\subset u(\partial\Omega)$. A radial harmonic function on $latex \mathbb B$ is constant. A positive harmonic function on $latex \R^d$ is constant.

Park City Mathematics Institute Undergraduate Summer School 2018 Introduction to Harmonic Analysis Let $latex P:\R^d\to\R^d$ be orthogonal. $latex \Delta(u\circ P) = (\Delta u)\circ P$ If $latex u$ is harmonic, then $latex u\circ P$ is also harmonic. If $latex T$ is a translation and $latex u$ is harmonic, then $latex u\circ T$ is harmonic. Use polar coordinates to verify $latex \displaystyle \int_{\R^2}e^{-\pi |x|^2} dx = 1.$ Use (3.1) to prove $latex \displaystyle \int_{\R^d} e^{-\pi |x|^2} dx = 1$, for any $latex d\ge 1$. Use spherical coordinates and the previous result to prove $latex \omega_d = \dfrac{2\pi^{d/2}}{\Gamma(d/2)}$, where $latex \displaystyle \Gamma(s) = \int_0^\infty t^{s-1} e^{-t} dt$ is the Gamma function. Use the divergence theorem $latex \displaystyle \int_\Omega \nabla\cdot \vec F dx = \int_{\partial\Omega} \vec F \cdot \hat{\mathbf n} \; d\sigma$ to prove Green's identity. ( Hint: Consider $latex \vec F = u \nabla v - v\nabla u$.) The f...

Park City Mathematics Institute Undergraduate Summer School 2018 Introduction to Harmonic Analysis Let $latex R$ be a rotation in the plane. Consider the change of variables $latex (\xi, \eta) = R(x,y)$. Then $latex \displaystyle \frac{\partial^2 u}{\partial\xi^2} + \frac{\partial^2 u}{\partial\eta^2} = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}$. If $latex u$ is harmonic, then $latex u\circ R$ is also harmonic. Let $latex (r,\theta)$ be the polar coordinates of the plane. Then $latex \displaystyle \Delta u = \frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} + \frac{1}{r^2} \frac{\partial^2 u}{\partial\theta^2}$ Let $latex u$ be a harmonic function on $latex \R^2$. Then there exists a harmonic function $latex v$ that is conjugate to $latex u$, so $latex f = u + iv$ is holomorphic. ( Hint: Consider a line integral of the 1-form $latex \displaystyle - \frac{\partial u}{\partial y} dx + \frac{\partial u}{\partial x} dy$.) ...

La calificación ordinaria del curso Matemáticas discretas está distribuida de la siguiente forma: Examen escrito: 50% Proyecto final: 50% El examen escrito evaluará la totalidad del material cubierto en la clase. Tendrá una duración de dos horas, y será presentado el miércoles 27 de junio, a las 4:00 pm. El proyecto final consiste de una serie de problemas con un tema común, que puede tener como objetivo el desarrollo de un tema o la demostración de un teorema, no visto en clase. A cada estudiante le es asignado un proyecto distinto, y los proyectos no son transferibles. Para resolver los problemas del proyecto se permite revisar las notas de clase o las referencias bibliográficas del curso. No está permitido recibir la ayuda de algún compañero o alguien ajeno a la clase. Sin embargo, sí está permitido hacerme preguntas sobre el proyecto, y habrá horas de oficina para preguntas los días 14, 20 y 21 de junio, de 5:00 a 6:00 pm, en mi oficina. El proyecto debe ser entregado a más tar...

The final grade is distributed as follows: Final exam: 50% Final project: 50% The final exam is scheduled on June 27, 10:00 am, and will evaluate all the content of the course. The final projects extend the results from the course. Each student is assigned one project, and the projects cannot be changed. Each student must work alone, although any references may be consulted. There will be office hours for any questions related to the projects (or the material from the class) on June 14, 20 and 21, 11:00am - 12:00pm. The final projects are due June 27, 10:00 am. Yair Antonio Castillo:  Convex functions Jaime Daniel Hernández:  Sturm-Liouville operators Manuel Irán Torres:  The maximal function on L 2

Due June 8 Problem 1 Consider the Koch-type curve $latex K^l$, for $latex 1/4 < l < 1/2$, described by the diagram The function $latex t\mapsto K^l(t)$ satisfies a Hölder condition of exponent $latex \gamma = -\log l/\log 4$. $latex t\mapsto K^l(t)$ is a simple curve. $latex t\mapsto K^l(t)$ is continuous and nowhere differentiable. $latex \dim K^l = 1/\gamma$. Problem 2 If we take $latex l=1/2$ in the previous definition, we obtain a space-filling curve. Problem 3 On $latex \R^d$, define the functions, for $latex \alpha > 0$, $latex f_0(x) = \begin{cases} 1/|x|^{\alpha} & |x|<1\\0 & |x|\ge 1;\end{cases}\qquad f_\infty(x) = \begin{cases} 0 & |x|<1\\1/|x|^{\alpha} & |x|\ge 1.\end{cases}$ $latex f_0\in L^p$ if and only if $latex p\alpha < d$. $latex f_\infty \in L^p$ if and only if $latex p\alpha > d$. What happens if we replace $latex |x|^\alpha$ with $latex |x|^\alpha \big|\log 2|x|\big|$ in the previous definitions? Problem 4 Suppose ...

Fecha de entrega: 8 de junio Problema 1 Muestra que un código es capaz de corregir  d errores si y solo si es capaz de detectar  2d errores. Problema 2 Muestra que cualquier cadena de 0 y 1 de longitud 7 es ya sea una palabra en el código de Fano, o proviene de una única palabra del código de Fano cambiando un bit. Problema 3 El siguiente texto fue encriptado con un algoritmo de sustitución simple (no fueron reemplazados los signos de puntuación). Descífralo. xgqykakuar, pgtkuar, mratkuar, qra yovq arwlovq tra dka wuprpkvayv. qka vwluonr pu nvayv evtvayv, yvwv wuq vp ppuwuoqv hvotgpuar. qvo ga mortgpr ju vq ga evqtuor, mgvq ar qupv evp tkotgpr utver. vp bgv qupv dvpks vq vzmver, mvor umvqyu u evwrakrq j u oujrq. qka vwluonr mu' hutvopr lrakyr, ga kwlvtkp qv mgqr vzmvekyr. ¡bgv evpktku, qvcro, ev tokuygou, huqyu va vqr qv iv qg dkagou! vq mro vqr bgv vp lgva hvotgpuar dgv u ga xgsnuer u tuwlkuoqv ev arwlov. j upvnr tra vp xgvs, vp dgpuar, puq ivongvasuq bgv muqu qkvaer hrwlov. ...

Due June 1 Problem 1 Let $latex f:[0,1]\to\R$ satisfy a Hölder condition of exponent $latex \gamma > 1$. Then  f is constant. Is $latex f:[0,1]\to[0,1]\times[0,1]$ is a surjective Hölder function of exponent $latex \gamma$, then $latex \gamma \le 1/2$. (Prove directly, without using Lemma 2.2 from the text.) Problem 2 Let $latex K\subset\R$ be the set $latex \displaystyle K = \Big\{ \sum_{k=1}^\infty \frac{a_k}{4^k} \in\R : a_k=0\text{ or }2\Big\}$. Then $latex \dim K = 1/2$ and $latex 0 < \mathscr H^{1/2}(K) < \infty$. Problem 3 Let $latex 2N+1$ be an odd integer and consider the "middle $latex 1/(2N+1)$th" set  K , that is, the result of the Cantor process when removing the middle interval of length $latex 1/(2N+1)$ of the previous interval. Calculate $latex \dim K$ Prove that for any $latex 0 < \alpha < 1$, there exists a totally disconnected perfect set in $latex \R$ whose dimension is larger then $latex \alpha$. Problem 4 There exists a Cantor-lik...

Fecha de entrega: 1 de junio Problema 1 Muestra que, en un diseño de bloques, la hipótesis que cada individuo pertenece al mismo número de bloques es superflua; es decir, se sigue del resto de las hipótesis. Problema 2 Encuentra cinco números $latex v, b, k, r, \lambda$ que satisfagan las ecuaciones vistas en clase, pero $latex b<v$. Para cada $latex v>1$, construye un diseño de bloques con $latex b=v$. Problema 3 Muestra que el plano de Fano es el único sistema de Steiner con $latex v=7$. Problema 4 Supón que un sistema de Steiner tiene un subconjunto  S  de $latex (v-1)/2$ individuos tales que forman un sistema de Steiner por sí mismos considerando los bloques que pertenecen a  S . Muestra que  S es una muestra representativa de clubes. Problema 5 Muestra que el plano de Fano y $latex \mathbb F_3^2$ pueden ser coloreados con 3 colores, tal que cada bloque usa al menos dos colores (aunque no necesariamente los tres de ellos). Problema 6 ¿Cuántos cuadrados latinos hay de $...

Due May 25 Problem 1 Suppose $latex \tau$ is measure-preserving, with $latex \mu(X) = 1$. If  E is invariant, then there exists a set  E' so that $latex E' = \tau^{-1}(E')$, and  E and  E' differ by a set of measure zero. Problem 2 Let $latex \tau$ be measure-preserving, with $latex \mu(X)=1$. Then $latex \tau$ is ergodic if and only if whenever $latex \nu$ is absolutely continuous with respect to $latex \mu$ and $latex \nu$ is invariant (that is $latex \nu(\tau^{-1}(E) = \nu(E)$ for all measurable  E ), then $latex \nu = c\mu$), then $latex \nu = c\mu$ for some constant  c . Problem 3 The Hausdorff measure $latex \mathscr H^\alpha$ is not $latex \sigma$-finite on $latex \R^d$ if $latex \alpha < d$. Problem 4 Let $latex \{E_k\}$ be a sequence of Borel sets in $latex \R^d$. If $latex \dim E_k\le\alpha$ for all  k , then $latex \displaystyle \dim \bigcup E_k \le \alpha$.

Fecha de entrega: 25 de mayo Problema 1 Sea  G el grafo cuyos vértices corresponden a las aristas de $latex K_5$, 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 regiones formadas por una curva cerrada en el plano (que posiblemente se interseca a sí misma) son 2-coloreables. Problema 3 Da un ejemplo de un mapa, con países no necesarimente conexos, que no sea 100-colorable. Problema 4 Si cada cara de un mapa planar tiene un número par de aristas, entonces el grafo es bipartito. Si cada vértice de un mapa planar tiene grado par, entonces las caras son 2-coloreables. Problema 5 Considera el plano de Fano  $latex \mathcal F$ visto en clase. Representa cada recta en el plano de Fano por un punto, y cada punto x  de $latex \mathcal F$ como una recta que contiene, como puntos, a las rectas en $latex \mathcal F$ que pas...

Due May 18 Problem 1 The purpose of the following exercises is to prove the following statement:  If $latex \mu$ is a translation-invariant Borel measure on $latex \R^d$ that is finite on compact sets, then $latex \mu$ is a multiple of Lebesgue measure. Let $latex Q_r$ be a translate of the cube $latex \{x\in\R^d: 0 < x_j \le r, j=1,\ldots,d\}.$ If $latex \mu(Q_1) = c$, then $latex \mu(Q_{1/n}) = c/n^d$ for each integer  n . $latex \mu$ is absolutely continuous with respect to  m , and there is a locally integrable function  f such that $latex \displaystyle \mu(E) = \int_E f dx.$ By the differentiation theorem, is follows that $latex f(x) = c$ a.e., and hence $latex \mu = cm$. Problem 2 Suppose $latex \nu, \nu_1, \nu_2$ are signed measures on $latex (X,\mathscr M)$ and $latex \mu$ a positive measure. If $latex \nu_1\perp\mu$ and $latex \nu_2\perp\mu$, then $latex \nu_1+\nu_2 \perp \mu$. If $latex \nu_1\ll \mu$ and $latex \nu_2\ll\mu$, then $latex \nu_1 + \nu_2 \ll\mu$. ...

Fecha de entrega: 18 de mayo Problema 1 ¿Es planar el grafo que resulta de eliminar una arista de $latex K_5$? ¿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 pozos. ¿Es posible hacerlo sin que los caminos se crucen? Problema 3 Muestra que un grafo planar bipartito, con n vértices, puede tener a lo más 2 n -4 aristas. Problema 4 Muestra que un polihedro convexo, que solo tiene caras pentagonales y hexagonales, debe tener exactamente 12 caras pentagonales. Problema 5 Muestra que los siguientes grafos no son 3-coloreables. Problema 6 Considera  n  rectas genéricas en el plano, y considera el grafo formado por sus puntos de intersección y los segmentos de recta entre ellos. Muestra que este grafo es 3-coloreable. Problema 7 Muestra el corolario visto en clase:  si G es un grafo tal que cada subgrafo de G tiene al menos un vértice de gra...

Due May 11 Problem 1 Let $latex \rho:\R^d\to\R^d$ be a rotation. Then it induces a measure-preserving map of the sphere $latex \mathbb S^{d-1}$ with its measure $latex \sigma$. Problem 2 Use the polar coordinate formula to prove the following statements. $latex \displaystyle \int_{\R^2} e^{-\pi |x|^2} dx = 1$ $latex \displaystyle \int_{\R^d} e^{-\pi |x|^2} dx = 1$ for any  d . $latex \sigma(\mathbb S^{d-1}) = \dfrac{2\pi^{d/2}}{\Gamma(d/2)}$. $latex \displaystyle m(\mathbb B^d) = \frac{\pi^{d/2}}{\Gamma(d/2+1)}.$ Problem 3 If $latex \mu$ is a finite Borel measure on the interval $latex [a,b]$, then $latex \displaystyle f\mapsto l(f) = \int_a^b f d\mu$ is a linear functional on $latex C([a,b])$, positive in the sense that $latex l(f)\ge 0$ if $latex f\ge 0$. Conversely, if  l is a positive linear functional on $latex C([a,b])$, there exists a unique finite Borel measure $latex \mu$ on $latex [a,b]$ such that $latex l(f) = \int f d\mu$ for every $latex f\in C([a,b])$.

Fecha de entrega: 11 de mayo Problema 1 ¿Es planar el grafo que resulta de eliminar una arista de $latex K_5$? ¿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 pozos. ¿Es posible hacerlo sin que los caminos se crucen? Problema 3 Muestra que un grafo planar bipartito, con n vértices, puede tener a lo más 2 n -4 aristas. Problema 4 Muestra que un polihedro convexo, que solo tiene caras pentagonales y hexagonales, debe tener exactamente 12 caras pentagonales.

Fecha de entrega: 4 de mayo Problema 1 Muestra que, si  G tiene un apareamiento perfecto, entonces todo apareamiento maximal (en el sentido en que no existen aristas entre U y W libres) usa al menos la mitad de los vértices de  G . Problema 2 Utiliza el algoritmo de trayectorias de aumento para, si es posible, obtener un apareamiento perfecto del siguiente grafo. Problema 3 Averigua si el siguiente grafo tiene un apareamiento perfecto. Problem 4 Dadas al menos 3 rectas genéricas en el plano, muestra que entre las regiones en que dividen al plano se encuentra al menos un triángulo. Problema 5 ¿En cuántas regiones dividen al plano dos  n -ágonos convexos? Problema 6 ¿Cuál es el mínimo y el máximo número de regiones en que dividen al plano  n círculos? Problema 7 Muestra que 6 puntos genéricos en el plano forman al menos 3 cuadriláteros convexos. Encuentra 8 puntos genéricos en el plano que no contienen un pentágono convexo.

Due May 4 Problem 1 Let  X be a set and $latex \mathcal M$ a nonempty collection of subsets of  X  closed under complements and countable unions of disjoint sets. Then $latex \mathcal M$ is a $latex \sigma$-algebra. Problem 2 Let $latex (X,\mathcal M, \mu)$ be a measure space. Its  completion is defined as the collection $latex \overline{\mathcal M}$ of sets of the form $latex E\cup N$, where $latex E\in\mathcal M$ and $latex N\subset F$ for some $latex F\in\mathcal M$ with $latex \mu(F)=0,$ and $latex \bar\mu(E\cup N) = \mu(E).$ $latex \overline{\mathcal M}$ is the smallest $latex \sigma$-algebra containing $latex \mathcal M$ and all subsets of its elements of measure 0. The function $latex \bar\mu$ is a complete measure on $latex \overline{\mathcal M}$. Problem 3 Consider the Lebesgue exterior measure. Then a set is Caratheodory measurable if and only if is Lebesgue measurable. Problem 4 Let $latex (X,\mathcal M,\mu)$ be a measure space, $latex A,B,C$ subsets of  X such that $...

Due April 27 Problem 1 Let $latex f\in L^2(\R^d), k\in L^1(\R^d)$. $latex \displaystyle (f*k)(x) = \int_{\R^d} f(x-y)k(y) dy$ converges for a.e.  x . $latex ||f*k||_{L^2} \le ||f||_{L^2} ||k||_{L^1}$. $latex \widehat{(f*k)}(\xi) = \hat k(\xi) \hat f(\xi)$ for a.e. $latex \xi$. The operator $latex Tf = f*k$ is a Fourier multiplier operator with multiplier $latex m(\xi) = \hat k(\xi)$. Problem 2 Let $latex \Omega\subset\C$ be open, and $latex \mathscr H\subset L^2(\Omega)$ be the subspace of holomorphic functions on $latex \Omega$. $latex \mathscr H$ is a closed subspace of $latex L^2(\Omega)$. If $latex \{\phi_k\}$ is an orthonormal basis of $latex \mathscr H$, then $latex \displaystyle \sum_k |\phi_k(z)|^2 \le \frac{1}{\pi d(z,\C\setminus\Omega)}$ for $latex z\in\Omega$, where $latex d(z,\C\setminus\Omega)$ is the distance from  z to the complement of $latex \Omega$. The sum $latex B(z,w) = \sum_k \phi_k(z)\overline{\phi_k(w)}$ converges absolutely for $latex z,w\in\Omeg...

Fecha de entrega: 27 de abril Problema 1 Muestra que, si los costos de todas las aristas son distintos, entonces hay un único árbol más barato. Problema 2 Describe cómo construir árboles para los cuales: el producto del costo de sus aristas es mínimo; el máximo costo de sus aristas es mínimo. Problema 3 Si la capital de un gobierno se encuentra en el vértice  r , la primer línea construida será la línea más barata que sale de  r , digamos, a  s . Después, construiremos la línea más barata que sale de  r o de  s , y así sucesivamente. Muestra que el árbol que resulta es, de nuevo, el más barato. Problema 4 Muestra que un árbol es un grafo bipartito. ¿Es cierto que todo grafo bipartito es un árbol? Problema 5 ¿Existe un grafo bipartito con vértices de grados 3, 3, 3, 3, 3, 3, 3, 3, 3, 5, 6 y 6? Un grafo bipartito tiene 16 vértices de grado 5 y cierto número de vértices de grado 8. Si sabemos que los vértices de grado 8 se encuentran en el mismo lado, ¿cuántos vértices de gra...

Fecha de entrega: 20 de abril Problema 1 Encuentra el número de árboles no etiquetados con 6 vértices. Encuentra el número de estrellas etiquetadas con  n vértices Problema 2 Considera una matriz de $latex 2\times n-1$, cuyo primer renglón está formado por los números del 1 al $latex n-1$, y el segundo por números arbitrarios entre 0 y $latex n-1$. Construye el grafo con vértices $latex 0,1,2\ldots,n-1$ y aristas descritas por las columnas de esta matriz. Muestra que este grafo no siempre es un árbol. Demuestra que, si este grafo es conexo, entonces es un árbol. Muestra que cada componente conexa tiene a lo más un ciclo. Problema 3 Construye los árboles descritos por las siguientes codificaciones de Prüfer . 4330 105200 444440 169767620 4266410 Problema 4 Construye los árboles descritos por las siguientes codificaciones planares 11110000 11100100 11010100 11010010 11001010  

Due April 20 Problem 1 Let $latex \{\phi_k\}_{1\le k<\infty}$ be a complete orthonormal system for $latex L^2(\R^d)$. Then $latex \{\phi_{k,j}\}_{1\le k,j < \infty}$ a complete orthonormal system for $latex L^2(\R^d\times\R^d)$, where $latex \phi_{k,j}(x,y) = \phi_k(x)\phi(y)$. Problem 2 Let  P be the orthogonal projection onto a closed subspace  S of a Hilbert space. Then $latex P^2 = P\text{ and } P^*=P$. Conversely, if  P is a bounded operator such that $latex P^2 = P$ and $latex P^*=P$, then it is the orthogonal projection onto some closed subspace. Let $latex P_1\text{ and }P_2$ be the orthogonal projections onto the closed subspaces $latex S_1\text{ and }S_2$, respectively. Then $latex P_1P_2$ is an orthogonal projection if and only if they conmute, in that case, it projects onto $latex S_1\cap S_2$. Problem 3 Let $latex \{u_k\}$ be a complete orthonormal system for a Hilbert space  H , and $latex (a_k)$ a sequence of positive numbers such that $latex \sum_k a_k^2 ...

Fecha de entrega: 13 de abril Problema 1 Demuestra que al conectar dos vértices  u  y  v  en un grafo  G  con una nueva arista, se crea un nuevo ciclo si y solo si  u  y  v  se encuentran en la misma componente conexa de  G . Problema 2 Sea  G  un grafo conexo y  e  una arista de  G . Muestra que  e  no es una arista de corte si y solo si  e  es parte de un ciclo en  G . Problema 3 Determina cuáles de los siguientes 4 grafos tienen una caminata euleriana, y cuáles una caminata euleriana cerrada. Dibújala, si es el caso. Problema 4 Determina cuáles de los siguientes 2 grafos tienen un ciclo hamiltoniano. Problema 5 Muestra que, en un árbol, cualquiera dos vértices pueden ser conectados por una única trayectoria. De manera inversa, muestra que si en un grafo cualquiera dos vértices pueden ser conectados por una única trayectoria, entonces el grafo es un árbol.   Problema 6 Muestra que un árbol con un vértice de grado  d  tiene al menos  d hojas.

Fecha de entrega: 23 de marzo Problema 1 Calcula la tabla de diferencias para la sucesión $latex x_n = 2n^2-n+3$, y encuentra una fórmula para $latex \sum_{k=0}^n x_k$. Si la sucesión $latex x_n$ está determinada por un polinomio cúbico, y los primeros términos del renglón 0 de su tabla de diferencias son 1, -1, 3, 10, determina $latex x_n$ y encuentra una fórmula para $latex \sum_{k=0}^n x_k$. Encuentra la suma $latex 1^5 + 2^5 + \ldots + n^5$. Problema 2 Sea $latex A$ un conjunto con $latex n$ elementos y $latex B$ un conjuntos de $latex k$ elementos. Muestra que el número de de funciones $latex f:A\to B$ sobreyectivas es $latex k!S(n,k)$. Problema 3 Formula y demuestra el siguiente enunciado como un teorema de grafos: " En un grupo de personas existen dos de ellas que conocen al mismo número de personas cada uno ". Problema 4 Por medio de un ejemplo, muestra que si eliminamos una arista de un grafo conexo  G , el resultado puede ser un grafo disconexo. Muestra qu...

Due March 23 Problem 1 If $latex \{K_\delta\}$ is a family of better kernels, there exists a constant $latex A>0$ such that $latex \sup_{\delta>0} |f*K_\delta(x)| \le A f^*(x)$ for all $latex f\in L^1$. Problem 2 For $latex a,b>0$, let $latex \begin{cases} x^a\sin x^{-b} & 0 < x \le 1, \\ 0 & x=0.\end{cases}$ f is of bounded variation iff $latex a>b$. For each $latex 0<\alpha<1$, construct an $latex \alpha$-Hölder continuos function that is not of bounded variation. If $latex a=b=2$,  f' exists at every point but is not integrable. Problem 3 Define the  one-sided maximal function  for locally integrable functions on $latex \R$ as $latex \displaystyle f_+^*(x) = \sup_{h>0} \frac{1}{h} \int_x^{x+h} |f|.$ If $latex E_\alpha^+ = \{x\in\R: f_+^*(x)>\alpha\}$, then $latex \displaystyle m(E_\alpha^+) = \frac{1}{\alpha} \int_{E_\alpha^+} |f|.$ Problem 4 Let $latex f:\R\to\R$ be absolutely continuous. f maps sets of measure zero to sets of measu...

Due March 16 Problem 1 Consider the function on $latex \R$ given by $latex f(x) = \begin{cases}\dfrac{1}{|x|(\log |x|)^2} & |x|\le 1/2\\0 & \text{otherwise.}\end{cases}$ f is integrable. $latex f^*(x) \ge \dfrac{c}{|x|\log 1/|x|}$ for some $latex c>0$ and all $latex |x|\le 1/2$. $latex f^*$ is not locally integrable. Problem 2 Let $latex E\subset [0,1]$ be a measurable set such that there exists $latex \alpha > 0$ such that $latex m(E\cap I) \ge \alpha m(I)$ for all intervals $latex I\subset[0,1]$. Then $latex m(E)=1$. Problem 3 Let $latex F\subset\R$   be closed and $latex \delta(x)$ the distance from  x to  F . Then $latex \delta(x+y) =o(|y|)$ for almost every $latex x\in F$. Problem 4 Suppose $latex \{K_\delta\}$ is a family of kernels that satisfies $latex \displaystyle \int_\R K_\delta = 0$ for all $latex \delta>0$. For some $latex A>0$, $latex |K_\delta(x)| \le A \min\{\delta^{-d}, \delta/|x|^{d+1}\}$ for all $latex \delta > 0, x\in\R^d$. If $la...

Fecha de entrega: 16 de marzo Problema 1 Decimos que un subconjunto de $latex \{1, 2, \ldots, n\}$ es  extraordinario si satisface que su mínimo es igual a su número de elementos, o sea $latex \min S = |S|.$ Por ejemplo, $latex \{3,5,8\}$ es extraordinario. Muestra que el número de subconjuntos extraordinarios de $latex \{1, 2, \ldots, n\}$ es igual al número de Fibonacci $latex F_n$. Problema 2 Sean $latex 2n$ puntos equidistantes en un círculo, y $latex f_n$ el número de formas en que podemos unir estos puntos en pares, de tal manera que los $latex n$ segmentos no se crucen. Encuentra una fórmula recursiva para $latex f_n$. Problema 3 Muestra que el número de arreglos de $latex 2\times n$ $latex \begin{pmatrix}x_{11}& x_{12}&\ldots& x_{1n}\\x_{21}& x_{22}&\ldots& x_{2n}\end{pmatrix}$ con los números $latex 1, 2, \ldots, 2n$ de tal forma que cada renglón y cada columna es creciente, es igual a $latex C_n$. Problema 4 Determina la división en diagonales del pol...