Who doesn't remember the first integration formula one ever sees? \[\int u^n=\frac{u^{n+1}}{n+1}\]
Yes, for some reason letters $u$ and $v$ are famous for integration and derivation formulas. Probably somebody would complain at this stage, "what about the constant of integration?" \[\int u^n=\frac{u^{n+1}}{n+1}+K\]
Of course there should be a constant of integration! What was I thinking? But wait... there's something else missing! What am I integrating against?\[\int u^ndu=\frac{u^{n+1}}{n+1}\]
I don't pretend to explain integral calculus in this short text, there must be (literally) hundreds of text books on integral calculus. I'm just here to rant about something I don't like. I actually don't care if one uses $u$ and $v$ and other $x$ and $y$, the name of the variable is the least important thing. For formula memorising purposes I don't even see wrong in not writing down the constant of integration. However, something that really freaks me out is not writing the differential with respect to which we're integrating.
Just as in the joke just up here, my trauma becomes important when talking about multivariable calculus. It is important to know the order of integration, particularly if the region of integration is not a rectangle. \[\int_0^1\int_0^x f(x,y)dydx\neq\int_0^x\int_0^1 f(x,y)dxdy\]
This is one of the typical exercises I would ask my vector calculus students to reverse the order of integration. But Fubini's Theorem isn't just about rewriting everything in the reversed order, it is about parametrising the integrated region in such a way that the reversed order makes sense.
\[\int_0^1\int_0^x f(x,y)dydx=\int_0^1\int_y^1 f(x,y)dxdy\]
I totally understand that Fubini's Theorem might be tricky to get at the beginning.
However, something that I really dislike is when something like this appears: \[\int_a^b dxf(x)\]
WTF?! I mean, yes, in the Riemann sense, if $\{a=x_0 < x_1 < \ldots < x_n=b\}$ is a partition of $[a,b]$ then the value of the integral of an integrable function $f:[a,b]\rightarrow\mathbb R$ may be calculated as
\[\int_a^b f(x)dx=\lim_{n\rightarrow\infty}\sum_{k=1}^nf(x_k)(x_k-x_{k-1})\]
which is basically the sum of some products that may be interchanged, $xy=yx$. But I will just simply say it, this:
\[\int_a^b dxf(x)\] looks horrible. Do not do it.
I'm a time traveller and these are some of the stories I have encountered or I am going to.
Monday, 24 February 2014
Tuesday, 18 February 2014
Nach Deutschland
Nächste Woche gehe ich nach Deutschland. Ich gehe zu einer Konferenz/Workshop über Statistische Mechanik in Darmstadt. Du kannst die Website hier sehen. Dies ist das erste Mal dass ich nach Deutschland gehe, nachdem ich angefangen habe, Deutsch zu lernen. Natürlich, ich bin sehr froh und nervös weil es mir schwer fällt Deutsch zu sprechen und Deutsche Leute sprechen sehr schnell.
Diesmal ist es ernst! Ich weiß jeder kann da Englisch sprechen aber ich möchte mein Deutsch üben. Das letzte Mal war ich in Deutschland, ich kannte nichts. Ich bin zu der WGT 2009 (Wave Gotik Treffen) gegangen (vor 5 Jahren!). Es war sehr cool! Die erste Mal habe ich eine meine Lieblingsmusikbands gehört: Katra.
Natürlich wird nächste Woche anders sein: vielen Mathematiker und mein Talk über mein PhD (du kannst die Slides hier sehen)... aber es wird sehr cool sein!
Diesmal ist es ernst! Ich weiß jeder kann da Englisch sprechen aber ich möchte mein Deutsch üben. Das letzte Mal war ich in Deutschland, ich kannte nichts. Ich bin zu der WGT 2009 (Wave Gotik Treffen) gegangen (vor 5 Jahren!). Es war sehr cool! Die erste Mal habe ich eine meine Lieblingsmusikbands gehört: Katra.
Natürlich wird nächste Woche anders sein: vielen Mathematiker und mein Talk über mein PhD (du kannst die Slides hier sehen)... aber es wird sehr cool sein!
Monday, 17 February 2014
My Imaginary Friends
"Can you find two square roots of 4?” he asked. Most of us answered correctly with no surprise there. “But can you find three cubic roots of 8?” he continued… “maybe you’re not prepared for that yet”. I clearly wasn’t, for that question haunted me for the rest of my life.
I must have been 15 years old when that happened. Fortunately, the next year I was taught the secret “all numbers different from zero have exactly n different n-th roots”. In particular, the roots of $-1$:
\[z^n-1=0\quad\Leftrightarrow\quad z\in\left\{\exp\left(\frac{2\pi k i}{n}\right):0\leq k\leq n-1\right\}\]
where, of course, $i$ is the elementary component of the imaginary numbers, the square root of $-1$
\[i^2=-1\]
I must have been 15 years old when that happened. Fortunately, the next year I was taught the secret “all numbers different from zero have exactly n different n-th roots”. In particular, the roots of $-1$:
\[z^n-1=0\quad\Leftrightarrow\quad z\in\left\{\exp\left(\frac{2\pi k i}{n}\right):0\leq k\leq n-1\right\}\]
where, of course, $i$ is the elementary component of the imaginary numbers, the square root of $-1$
\[i^2=-1\]
At my 17 years old, I was happy. I had one last shot at imaginary operations in the last year of preparatory school. I thought I knew everything there was to know about complex numbers... I couldn’t have been more wrong!
Before I turned 18, I was completely sure I was going to study a degree in computing engineering. Fortunately, I found a course in Applied Maths. When I went home with the good news that I have found the first love of my life (that is, my degree) my mum freaked out. It is now clear to me why she freaked out and I understand her. She didn't know, and still doesn't know, what doing maths is about... but that's another story. The point here is that she proposed me to look at the syllabus of the Actuarial Sciences programme.
I only understood the word Calculus, and that was enough for me to take a leap of faith and study actuarial sciences. In my second Algebra course, while studying monoids, groups, rings, and fields... the complex numbers visited me again.
This time it was a fellow classmate who asked the question "can we raise a complex number to a power that is itself a complex number?" Wow! That was completely ridiculous! But what really blew my mind was the answer of the lecturer: "Yes, sure. But that will be seen only by those taking the subject on Complex Analysis".
I had to take that class! I realised only mathematicians took that class as an elective unit. My solution: change my degree from Actuarial Sciences to Applied Maths. However, while doing the transfer I had to pay some validations of my previous units and I couldn't afford it. The option to delay the payment was to actually sign in for both courses at the same time and make the payment when I finished both degrees... so I did. Anything to know what it even means something like $i^i$.
The moment arrived. I signed in to the Complex Analysis course and my favourite lecturer was giving the class. I was so happy! I didn't just learn how to evaluate $i^i$, but to find logarithms of complex numbers, trigonometric functions, etc. In fact, I even wrote an article for the maths uni mag that you should be able to read here (that is if you can read Spanish).
But I liked something even more! There were real integrals which could be easily evaluated if considering the complex version of it and a convenient path. I remember perfectly the problem on my final exam and I remember I was able to solve beautifully \[\int_0^{\infty}\frac{dx}{1+x^{2n}}\quad\text{for }n\in\mathbb N\]
This time the course finished and I knew I didn't know everything. But I knew what I wanted. By that time I discovered that my vocation was not with the Complex Analysis even though by then I had a fascination with the Riemann Hypothesis (which I bet it is true!) but with Probability.
Every single time I found an integral, I tried to apply Complex Analysis techniques. Most of the times without very good results. It was usually easier to just do the integrals by the regular means.
When I finished my degrees, I was already working in a market research company as a statistician. I really liked my job. But I got to believe that my life has got rid of the complex numbers for good... along with other things I used to like. The epsilons and the deltas. They were no more in my life. It was like that for 5 long years...
I managed to travel to the UK for an MSc in Applied Maths and I saw them again. Fourier transforms, inverse Fourier transforms, and oscillatory integrals. It was cool that after so long they keep visiting me. After all I don't like when friends just disappear... and the complex numbers and I see each other as we've been friends all the time regardless of the time we last saw each other. Those are the good friends.
As of today, the last time I saw them was when I tried a real integral via a saddle point approximation that if possible, I would have been able to find a large deviation principle on the boundary of a semi-infinite totally asymmetric simple exclusion process. I don't remember quite well what went wrong.
Chances are we will meet again this week. I hope this is not the last time...
Before I turned 18, I was completely sure I was going to study a degree in computing engineering. Fortunately, I found a course in Applied Maths. When I went home with the good news that I have found the first love of my life (that is, my degree) my mum freaked out. It is now clear to me why she freaked out and I understand her. She didn't know, and still doesn't know, what doing maths is about... but that's another story. The point here is that she proposed me to look at the syllabus of the Actuarial Sciences programme.
I only understood the word Calculus, and that was enough for me to take a leap of faith and study actuarial sciences. In my second Algebra course, while studying monoids, groups, rings, and fields... the complex numbers visited me again.
This time it was a fellow classmate who asked the question "can we raise a complex number to a power that is itself a complex number?" Wow! That was completely ridiculous! But what really blew my mind was the answer of the lecturer: "Yes, sure. But that will be seen only by those taking the subject on Complex Analysis".
I had to take that class! I realised only mathematicians took that class as an elective unit. My solution: change my degree from Actuarial Sciences to Applied Maths. However, while doing the transfer I had to pay some validations of my previous units and I couldn't afford it. The option to delay the payment was to actually sign in for both courses at the same time and make the payment when I finished both degrees... so I did. Anything to know what it even means something like $i^i$.
The moment arrived. I signed in to the Complex Analysis course and my favourite lecturer was giving the class. I was so happy! I didn't just learn how to evaluate $i^i$, but to find logarithms of complex numbers, trigonometric functions, etc. In fact, I even wrote an article for the maths uni mag that you should be able to read here (that is if you can read Spanish).
But I liked something even more! There were real integrals which could be easily evaluated if considering the complex version of it and a convenient path. I remember perfectly the problem on my final exam and I remember I was able to solve beautifully \[\int_0^{\infty}\frac{dx}{1+x^{2n}}\quad\text{for }n\in\mathbb N\]
This time the course finished and I knew I didn't know everything. But I knew what I wanted. By that time I discovered that my vocation was not with the Complex Analysis even though by then I had a fascination with the Riemann Hypothesis (which I bet it is true!) but with Probability.
Every single time I found an integral, I tried to apply Complex Analysis techniques. Most of the times without very good results. It was usually easier to just do the integrals by the regular means.
When I finished my degrees, I was already working in a market research company as a statistician. I really liked my job. But I got to believe that my life has got rid of the complex numbers for good... along with other things I used to like. The epsilons and the deltas. They were no more in my life. It was like that for 5 long years...
I managed to travel to the UK for an MSc in Applied Maths and I saw them again. Fourier transforms, inverse Fourier transforms, and oscillatory integrals. It was cool that after so long they keep visiting me. After all I don't like when friends just disappear... and the complex numbers and I see each other as we've been friends all the time regardless of the time we last saw each other. Those are the good friends.
As of today, the last time I saw them was when I tried a real integral via a saddle point approximation that if possible, I would have been able to find a large deviation principle on the boundary of a semi-infinite totally asymmetric simple exclusion process. I don't remember quite well what went wrong.
Chances are we will meet again this week. I hope this is not the last time...
Thursday, 13 February 2014
Resident Evil: The Markov Chain
Yesterday was the first time I taught on the course of Stochastic Processes. No kidding I have been dreaming about it like for 10 years now, and it was awesome. I came up with the following problem to which I don't have the whole solution yet, but as the term passes I'll make the students solve it for me and I'll eventually post the solution. Here it is!
-----------------------------------------------
Jill Valentine (who’s going to be the hero of our story because it’s about to be Valentine’s Day...) is forced to enter a mansion in the middle of the forest because outside the zombie apocalypse is going wild. However, the rooms in the mansion are also filled with zombies and she is compelled to be moving equally likely to any adjacent room to the one she’s in at any given moment in time.
What she doesn’t know is that there’s a room where Nemesis (the code name of a terrible bio-weapon...) is dormant until a member of S.T.A.R.S (Special Tactics and Rescue Squad, and yes, of course Jill is a member!) appears and it will kill her, no doubt about it.
Fortunately, there’s also a helipad (and obviously she can fly the chopper there...) with a helicopter fully functional there. Were she able to reach this room she would fly a way to a secure place after blowing the mansion up with all zombies inside it.
We have a map of the mansion:
1. The rooms in which Jill is in the mansion are a Markov chain. What is the state space? What is the initial distribution? What is the transition probability matrix?
2. What is more likely? Will our heroine survive or will she perish in a not very pleasant way at the hands of the bio-weapon?
3. How long are we expecting to see Jill in action before she reaches her destiny?
------------------------------------------------
If you solve it before I do, do let me know at my Twitter account :P
Follow me on Twitter:
-----------------------------------------------
Jill Valentine (who’s going to be the hero of our story because it’s about to be Valentine’s Day...) is forced to enter a mansion in the middle of the forest because outside the zombie apocalypse is going wild. However, the rooms in the mansion are also filled with zombies and she is compelled to be moving equally likely to any adjacent room to the one she’s in at any given moment in time.
What she doesn’t know is that there’s a room where Nemesis (the code name of a terrible bio-weapon...) is dormant until a member of S.T.A.R.S (Special Tactics and Rescue Squad, and yes, of course Jill is a member!) appears and it will kill her, no doubt about it.
Fortunately, there’s also a helipad (and obviously she can fly the chopper there...) with a helicopter fully functional there. Were she able to reach this room she would fly a way to a secure place after blowing the mansion up with all zombies inside it.
We have a map of the mansion:
1. The rooms in which Jill is in the mansion are a Markov chain. What is the state space? What is the initial distribution? What is the transition probability matrix?
2. What is more likely? Will our heroine survive or will she perish in a not very pleasant way at the hands of the bio-weapon?
3. How long are we expecting to see Jill in action before she reaches her destiny?
------------------------------------------------
If you solve it before I do, do let me know at my Twitter account :P
Follow me on Twitter:
Labels:
education,
gamer,
maths,
Probability,
Resident Evil
Wednesday, 12 February 2014
De cómo los videojuegos han tenido impacto en mi vida
Prepare the unexpected...
Si hago una analogía de mi vida como una partícula que se mueve en el espacio tiempo describiendo una trayectoria que ocurre debido a fuerzas que están en el camino. Definitivamente los videojuegos han sido, y no veo porque no seguirán así, una de esas fuerzas.
No recuerdo la fecha exacta, pero seguro fue a mediados de la década de los 80. Lo que sí recuerdo es el momento en el que me volví gamer. Mis padres nos llevaron a mi hermano y a mi (mi hermana todavía no nacía) a comer a casa de unos amigos suyos. El hijo menor de aquella pareja acababa de recibir, de un viaje que su padre había hecho a los United States, un Nintendo Entertainment System (1985), o NES como se conoce en el mundo gamer o lo que en México fue durante mucho tiempo el paradigma de videojuegos: “un Nintendo”.
El juego era, obviamente, Super Mario Bros (1985). El original y primero de toda la infinidad de juegos de Mario que ahora existen. Mi hermano y yo teníamos un Atari 2600 (1978) pero realmente sólo nos dedicábamos a jugar Othello (1980), cuyas reglas nunca supimos bien exactamente, y Fishing Derby (1980) que venía en un cartucho con como 60 juegos. Realmente no recuerdo ninguno de los otros.
Pero en la época del Atari los videojuegos para mi eran sólo un pasatiempo intrascendente. Fue en los primeros segundos de Super Mario, donde queriendo o no, el videojuego se volvió formativo… ¡Hay que sobrevivir al primer Goomba!
Afortunadamente no todos los juegos son igual de traumáticos, Resident Evil (1996) es de mis juegos favoritos de mediados de las siguiente década, pero afortunadamente tienes un tiempo para aprender a moverte a usar el control, de cierta manera un propedéutico antes de entrar al campo de batalla. Pero a la fecha en cada situación la filosofía correcta es siempre estar preparado para ese primer Goomba.
Never give up…
A principios de la década de los 90, mi madre incursionó en una aventura que cambiaría su vida, y la mía junto con la de mis hermanos por consecuencia. Se dedicó a dirigir el servicio de transporte escolar de la escuela a la que yo iba. El negocio empezó con dos microbuses que por la madrugada se movían alrededor de la más caótica ciudad del mundo para dejar alumnos en su escuela y en la tarde los regresaban en sus casas. Pero la historia que me interesa contar ahora es qué pasaba con los microbuses entre la tarde y las madrugadas en las que trabajaban.
A la fecha desconozco los detalles de la operación, pero justo en la esquina del parque en la calle donde yo vivía solía existir un terreno donde mi mamá consiguió (me imagino que) rentar para estacionar los microbuses. Siendo niños, yo el mayor de alrededor 12 años, a mi hermano, mi vecino y yo nos llamaba la atención ir a ver “a los camiones”. Pero justo en ese terreno, en el lado que daba a Avenida División del Norte se rentaba para una farmacia… en esta farmacia, había “una maquinita”... en esta maquinita había un juego que no sólo cambió la historia de los videojuegos, sino que también mi vida.
Para la fecha, yo ya era fan de Street Fighter II: The World Warrior (1991) incluso mi hermano y yo teníamos en casa la versión de SNES (1991) que apareció en 1992. Pero la maquinita de la que hablo no era esta... acompañando a mi mamá a la farmacia vi que un joven, que supongo en edad adolescente, jugaba un juego de pelea en esta maquinita cuyas gráficas a mi parecer eran simplemente impresionantes. Era como ver personas de verdad peleando. En eso, el fondo se oscureció y la maquinita con una voz penetrante e imperativa dijo “FINISH HIM!” y enfrente de mi vi cómo el ninja azul le arrancó la cabeza junto con la espina vertebral a su oponente… estaba claro, la batalla había terminado.
Sí, debo decir que fue impactante. Pero lejos de ser el que esté traumado por la violencia en los videojuegos, situación que puede o no ser cierta, definitivamente me enamoré del juego Mortal Kombat (1992).
Inmediatamente metí mi moneda para retar al que acababa de hacer semejante acto criminal. Obviamente perdí, pues él sabía y yo no. Me preguntó que si sabía jugar Street Fighter a lo que contesté que sí. Me dijo que debía hacer un “abuguet” y así aprendí a jugar con Sub-Zero en Mortal Kombat.
Los siguientes días, semanas y meses regresé, quería aprender a jugar bien y acabar, es decir entrar al torneo y matar a todos, el juego. Esta tarea demostró ser cara y difícil. Llegar a pelear con Goro, un sujeto nada amigable con 4 brazos, ganarle y luego pelear con Shang Tsung, un misterioso anciano que se transforma en cualquier otro combatiente, parecía imposible. Pero llegando a mi casa estudiaba las estrategias, veía que había fallado, me proponía intentar nuevas formas de atacarlo para poder realizarlo la siguiente ocasión.
Una vez incluso no supe qué hacer y hasta decidí que hacer planas de “Tengo que ganarle a Shang Tsung” sería buena idea. Hice las planas, lo escribí 50 veces. Supongo que en si haber escrito las planas no ayudaron a ganarle, pero definitivamente me ayudaron a desahogar mi frustración. El día de hoy estoy a medio doctorado tratando de resolver un problema, y quizá escribir esta historia no me ayude a resolver el problema. Pero estoy seguro que me desahogué y mañana regresaré con una nueva estrategia.
Tengo muchos más ejemplos de cómo ciertos juegos me han enseñado y aplico lo aprendido en la “vida real” aunque cierto es que cuando decido jugar un nuevo videojuego mi objetivo es distraerme precisamente de esa realidad, olvidar el mundo en el que vivo e involucrarme en la realidad virtual que mi mente crea con ayuda del videojuego para simplemente divertirme. Nunca me he puesto a jugar con la intención de que un juego me cambie la vida, sin embargo, siempre pasa así en menor o mayor medida.
Life is a game… play it!
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