Cayley-Menger determinants


Complex Projective 4-Space

Niccolo Fontana Tartaglia, whom you may recognise from having discovered Cardano’s solution to the general cubic equation, also discovered a generalisation of Heron’s formula to compute the volume of a tetrahedron:


As you may expect, this can be generalised to compute the volume of any n-simplex (n = 2 reducing to Heron’s formula for the area of a triangle). I wondered how one would go about proving this identity, and then realised it can be accomplished by elementary facts about determinants. Firstly, it is easy to show the following result:

  • The volume of an n-simplex S with vertices at {, e_1, …, e_n} is equal to 1/n!, where e_i is the ith standard basis vector.

This can be proved, for instance, by subdividing a unit cube into n! simplices, each of which is congruent to S. Now…

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Notes from Ring Theory

A list of notes, by date Nov 8 – Densidad…
Nov 15

  • Pere Menal (\Cross {\sim} 1992) Ver si un módulo artiniano tiene {\End()} semilocal. Rosa Camp & W. Ricki {\rightarrow} Sí.
  • Krull ’34 {\rightarrow} Módulos Artinianos satisfacen el Teorema de Krull-Schmidt? No es difícil ver que si {M_R} es artiniano {\Rightarrow} \footnotesize(se forma una cadena descendiente)\normalsize {M_R=N_1\oplus\cdots\oplus N_s}, {N_i} indescomponibles. Es la única descomposición bajo isomorfía y reordenación de los sumandos? (Teorema de Krull-Schmidt). Facchini, H Levy, Vamos ’96 {\rightarrow} No.
  • Un {R}-módulo {M} generado por {r} elementos – {M} es imágen homomórfica de {R^r}

Ene 13

  • Módulos proyectivos finitamente generados {\leadsto} submonoides…
  • “full affine”
  • Monoids are tricky… why? No they’re not. They just have associativity and the presence of an identity
  • Fundamental theorem of abelian groups {\left(\mathds{N}^k\subseteq\mathds{Z}^k\right)}
  • {P/PJ(R)} finitely generated {\nRightarrow} {P} finitely generated. Counterexample given by Geramnov, Sakhaev
  • Fair-sized projective modules by Pavel ({A_5})
  • The trace ideal is a bi-lateral ideal: {Tr(P)=\sum_{f\in P^*}{f(P)}}{(P\rightarrow R)\in P^*=\Hom_R(P,R)}. Verify that {\left[Tr(P)\right]^2=Tr(P)}
  • Whitehead, as cited in Pavel’s article: {I} a bi-lateral idempotent ideal of {R} such that {_RI} is finitely generated {\Rightarrow I=Tr(P_R)}, with {P_R} a projective ideal
  • Hyman Bass {\rightarrow} Big proyective ideals

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    The Fredholm Alternative

    I came across this theorem while reading basic algebraic properties of Petri Nets and I wanted to know how many of you are acquainted with it. Please post in the comments!

    Scientific Clearing House

    One of the most useful theorems in applied mathematics is the Fredholm Alternative.  However, because the theorem has several parts and gets expressed in different ways, many people don’t know why it has “alternative” in the name.  For them, the theorem is a means of constructing solvability conditions for linear equations used in perturbation theory.

    The Fredholm Alternative Theorem can be easily understood if you consider solutions to the matrix equation  $latex A v = b$, for a matrix $latex A$ and vectors $latex v$ and $latex b$.  Everything that applies to matrices can then be generalized to infinite dimensional linear operators that occur in differential or integral equations.  The theorem is:  Exactly one of the two following alternatives hold

    1. $latex A v = b$ has one and only one solution
    2. $latex A^* w = 0$ has a nontrivial solution

    where $latex A^*$ is the transpose or adjoint of A. …

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    What riding my bike has taught me about white privilege

    trending topic in the USA right now


    The phrase “white privilege” is one that rubs a lot of white people the wrong way. It can trigger something in them that shuts down conversation or at least makes them very defensive. (Especially those who grew up relatively less privileged than other folks around them). And I’ve seen more than once where this happens and the next move in the conversation is for the person who brought up white privilege to say, “The reason you’re getting defensive is because you’re feeling the discomfort of having your privilege exposed.”

    I’m sure that’s true sometimes. And I’m sure there are a lot of people, white and otherwise, who can attest to a kind of a-ha moment or paradigm shift where they “got” what privilege means and they did realize they had been getting defensive because they were uncomfortable at having their privilege exposed. But I would guess that more often than…

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    Alexander Grothendieck 1928–2014

    Time to read a little bit about Grothendieck.

    Gödel's Lost Letter and P=NP


    Alexander Grothendieck, who signed his works in French “Alexandre” but otherwise kept the spelling of his German-Jewish heritage, passed away Thursday in southwestern France.

    Today we mourn his passing, and try to describe some of his vision.

    Part of the story of this amazing mathematician is that in 1970 he renounced his central position at the Institut des Hautes tudes Scientifiques (IHES) in Paris, and made himself so remote shortly after formally retiring from the University of Montpellier in 1988 that not even family and friends could track him. He boycotted his 1966 Fields Medal ceremony in Moscow to protest the Red Army’s presence in eastern Europe, and declined the Crafoord Prize in 1988.

    As captured by this obituary, he had left to seek a society kinder and more just than the ones that killed his father at Auschwitz and convicted him in 1977 of violating a French law…

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    Reconstructing Gödel

    Gödel's Lost Letter and P=NP


    Kurt Gdel left a large amount of unpublished writings and notebooks and preserved correspondence. Called his Nachlass, German for “after-leavings” or bequest, these writings were catalogued and organized by several—including his first biographer, John Dawson, for a heroic two years. Those of highest scientific and general interest were published in volumes III, IV, and V of KurtGdel:CollectedWorks. Among them was a list of 14 numbered assertions titled “My philosophical viewpoint” but without elaboration. They are believed associated to a lecture Gdel started preparing in the early 1960s but never gave, whose draft is in the Nachlass.

    Today we are delighted to have new communications from Gdel, as we have previously received around Halloween and All Saints’ Day, so we can continue our series of interviewswithhim.

    What the Nachlass shows clearly is a perfectionist at work. Dawson’s biography relates that a two-year…

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    Intro a la Teoría de Juegos, parte 3

    3. Juegos con un Valor

    El juego {G=(X,Y,M)} se dice que tiene un valor si

    \displaystyle  v_1=v_2  \ \ \ \ \ (9)

    o sea, si

    \displaystyle \sup_{x\in X}\inf_{y\in Y} M(x,y)=\inf_{y\in Y}\sup_{x\in X} M(x,y),

    y en este caso se llama valor del juego al número {v=v_1=v_2}.

    Cuando el juego tiene un valor {v}, las estrategias maximin y minimax (si existen) se llaman estrategias óptimas. Las estrategias maximin y minimax {\bar{x},\bar{y}} cumplen la condición (8) vista anteriormente, y, si ademas el juego tiene un valor {v}, entonces evidentemente vale

    \displaystyle  M(x,\bar{y})\leq v\leq M(\bar{x},y),\quad\forall x\in X,\forall y\in Y.  \ \ \ \ \ (10)

    A estas relaciones (10) se hace referencia diciendo que {(\bar{x},\bar{y})} es un punto de silla de la función {M}, o que {M} tiene un punto de silla en {(\bar{x},\bar{y})}.

    De (10) se sigue fácilmente que {v=M(\bar{x},\bar{y})}; así pues (10) se puede escribir así

    \displaystyle  M(x,\bar{y})\leq M(\bar{x},\bar{y})\leq M(\bar{x},y),\quad\forall x\in X,\forall y\in Y.

    Se ha visto que la existencia de un valor (9) y de estrategias maximin y minimax {\bar{x},\bar{y}} implica la existencia de un punto de silla (10). Recíprocamente, si valen las relaciones (10) de punto de silla, el juego tiene un valor {v}, y {\bar{x},\bar{y}} son estrategias óptimas para J1 y J2 respectivamente.

    En este caso en que la función {M} posee un punto de silla se dice que el valor del juego {v} y las estrategias óptimas {\bar{x},\bar{y}} constituyen la solución del juego. Resolver un juego es, pues, encontrar {v,\bar{x},\bar{y}} si es que existen.

    De las propiedades expuestas resulta que si hubiese dos puntos de silla, por ejemplo {(\bar{x},\bar{y}),(\tilde{x},\tilde{y})} sucedería que

    \displaystyle  M(x,\bar{y})\leq v\leq M(\bar{x},y),\quad\forall x\in X,\forall y\in Y \ \ \ \ \ (11)


    \displaystyle  M(x,\tilde{y})\leq v^\prime\leq M(\tilde{x},y),\quad\forall x\in X,\forall y\in Y \ \ \ \ \ (12)

    Sustituyendo en las primeras relaciones las variables {x,y} por {\tilde{x},\tilde{y}} y en la segunda por {\bar{x},\bar{y}} resulta

    \displaystyle  M(\tilde{x},\bar{y})\leq v\leq M(\bar{x},\tilde{y}),\ M(\bar{x},\tilde{y})\leq v^\prime\leq M(\tilde{x},\bar{y})

    de donde resulta {v=v^\prime}. Además los cuatro puntos {(\bar{x},\bar{y}),(\tilde{x},\tilde{y}),(\bar{x},\tilde{y}),(\tilde{x},\bar{y})} son puntos de silla por lo que es indiferente para cada jugador elegir una cualquiera de sus estrategias óptimas. Es fácil ver que el punto de silla definido para los juegos de dos personas es un punto de equilibrio con la definición que se dio para los juegos de {n} personas. Debe notarse que muchas propiedades enunciadas para los jugadores 1 y 2 siguen siendo válidas con ciertos cambios permutando entre sí los jugadores. Esto se debe a que el juego {G=(X,Y,M)} se corresponde con el juego {G^\prime=(Y,X,M^\prime)}, donde {M^\prime(y,x)=-M(x,y)} ya que ambos son el mismo juego cambiando de nombre a los dos jugadores. Este hecho permite omitir las demostraciones de las propiedades que resulten análogas en este tipo de correspondencia.

    Posted in Teoría de Juegos | Tagged , | 2 Comments

    Very interesting article on recent developments concerning solving open problems.

    Gödel's Lost Letter and P=NP


    Richard Hamilton is the mathematician who laid out the route that eventually led to the positive solution to the three-dimensional Poincaré Conjecture by Grigori Perelman. He is the Davies Professor of Mathematics at Columbia University. While Perelman famously declined both the Fields Medal in 2006 and the official Clay Millennium Prize recognition in 2010, citing among other factors the lack of concomitant recognition for Hamilton, Hamilton was awarded the Leroy Steele prize in 2009, shared the Shaw Prize in 2011, and had earlier won the 2003 Clay Research Award alongside Terence Tao.

    Today Ken and I wish to talk about programs in mathematics, not C++ programs, but programs of attack on a hard open problem. Ken likes the British form “programme” for this.

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    Teorema del punto fijo de Brouwer

    Nos damos un pequeño break de la teoría de juegos para hablar un poco del teorema del punto fijo de Brouwer. Como vimos en el post sobre teoría de juegos y el teorema del punto fijo éste es un teorema que fundamenta la teoría de juegos y un teorema muy importante en topología y topología algebraica. Veremos el teorema y su prueba en inglés.

    The theorem states the following: Let {f:D^2\rightarrow D^2} be a continuous map, where

    \displaystyle D^2 = \{(x, y)\in\mathbb{R}^2 : x^2 + y^2\leq 1\}

    Then {f} has a fixed point, i.e., there is some point {(x, y)\in D^2} with the property that {f(x, y) = (x, y)}

    Proof Suppose that {f:D^2\rightarrow D^2} does not have a fixed point, so that {f(x, y) \neq (x, y)} for all {(x, y) \in D^2}. So, for each point {(x, y) \in D^2} we get two points {(x, y)} and {f(x, y)}, and we can draw a line through them both. Extend this line beyond {(x, y)} until it meets the boundary of {D^2} (i.e., {\mathbb{S}^1}), and let {g(x, y)} be the point where this happens. So we get a function {g : D^2\rightarrow \mathbb{S}^1} as in the picture.

    This map {g} is continuous, essentially because if {(x', y')} is sufficiently close to {(x, y)}, then {f(x', y')} will be close to {f(x, y)} (since {f} is continuous) and, hence, {g(x', y')} will be reasonably close to {g(x, y)}. More rigorously, if {A} is an open arc around {g(x, y)}, then there is some radius {r} such that whenever {(x', y')} is in the open ball {B_r(x, y)} and {f(x', y')} is in the open ball {B_r(f(x, y))}, then {g(x', y')} is in {A}, as depicted below, where {A} is indicated by a bold line, and the balls around {(x, y)} and {f(x, y)} are indicated by the dotted circles of their perimeters. Any straight line which passes through both balls will hit the circle in the region {A}.

    Since {f} is continuous, there is some radius {\delta} such that {f(x', y')\in B_r(f(x, y))} whenever {(x', y')\in B_\delta(x, y)}. Hence the preimage {g^{-1}(A)} contains {B_{\min(\delta,r)}(x, y)}. The same argument can be applied to any point in the preimage, so {g^{-1}(A)} is open, i.e., {g} is continuous. If {(x, y)} is on the boundary of {D^2}, then {g(x, y) = (x, y)} no matter what {f(x, y)} is. Now define a map

    \displaystyle F : \mathbb{S}^1\times I \rightarrow \mathbb{S}^1

    by {F((x, y), t) = g(tx, ty)}. This map {F} is continuous, so we can think of it as a homotopy between the map {h : \mathbb{S}^1\rightarrow \mathbb{S}^1} defined by { h(x, y) = F((x, y), 0)} and {j : \mathbb{S}^1\rightarrow \mathbb{S}^1} defined by {j(x, y) = F((x, y), 1)}. Now {h(x, y) = g(0, 0)} for all {(x, y)}, so {h} is the constant map and thus {\deg(h) = 0}. On the other hand, however, {j(x, y) = g(x, y) = (x, y)} for all {(x, y)}, so {j} is the identity map and {\deg(j) = 1}. If {F} is a homotopy between {h} and {j}, then these degrees must be equal. Since they are not, the map {F} cannot exist. Hence nor can {g}, showing in turn that the map {f} must have had a fixed point in the first place.

    El teorema de Brouwer es un teorema de punto fijo que dice que una aplicación continua de un conjunto convexo y compacto en si mismo tiene un punto fijo. Schauder y Tychonoff ademas probaron que el teorema sigue siendo valido para espacios normados; y también para espacios localmente compactos. Luitzen Egbertus Jan Brouwer fue el principal teórico del Intuicionismo Matemático y el fundador de la topologia moderna.

    -Mathematics – The Harper Collins Dictionary. Borowski & Borwein 1991
    -Essential Topology – Martin Crossley 2005
    Links de interes:
    Brouwer fixed point theorem by Palmieri
    The Brouwer fixed point theorem and the game of Hex

    Posted in Compacidad y punto fijo, Teoría de Juegos | Tagged , , | 1 Comment

    Intro a la Teoría de Juegos, parte 2

    2. Juegos Rectangulares

    Adaptando las definiciones que se dieron anteriormente para los juegos rectangulares al caso de juegos de dos personas y de suma cero, resulta la definición que veremos a continuación en la que omitimos los calificativos “de dos personas” y “de suma cero”, que quedarán sobreentendidos en lo sucesivo. Por cierto, el juego se llama “rectangular” por la facilidad de acomodar los datos de una manera rectangular, tipo matriz.

    Despliegue rectangular de un juego

    Un juego rectangular {G} está determinado por una terna

    \displaystyle G=(X,Y,M)\ \ \ \ \ (1)

    donde {X,Y} son conjuntos cualesquiera y {M} una función que tiene por dominio el producto cartesiano {X\times Y} y que toma valores reales

    \displaystyle M:X\times Y\rightarrow\mathbb{R}.

    Mientras no se diga nada en contra se supone que {M} es una función acotada. Para una realización del juego {G} se supone que existen dos jugadores que llamaremos jugador 1, J1, y jugador 2, J2. El J1 elige un elemento {x\in X} y el J2 elige un elemento {y\in Y}, estas elecciones las hacen los dos jugadores ignorando cuál ha sido la elección del otro jugador.

    Una vez hechas estas elecciones, el J2 paga al J1 la cantidad {M(x,y)}. Así pues la ganancia del J1 es {M(x,y)} y la del J2 es el valor opuesto {\left(-M(x,y)\right)}. Por lo tanto el objetivo del J1 es conseguir el mayor valor posible de su ganancia {M(x,y)}; mientras que, por su parte, el J2 tratará de minimizar su pago {M(x,y)}.

    Los elementos {x\in X} se llaman estrategias (o estrategias puras) del J1 y los elementos {y\in Y} se llaman estrategias (o estrategias puras) del J2. La función {M} se llama función de pago del juego {G}.

    En un juego rectangular {G=(X,Y,M)} desempeñan un importante papel las dos funciones

    \displaystyle  V_1:X\rightarrow \mathbb{R},\quad V_2:Y\rightarrow \mathbb{R}

    definidas por

    \displaystyle  V_1(x)=\inf_{y\in Y} M(x,y),\quad V_2(y)=\sup_{x\in X} M(x,y)  \ \ \ \ \ (2)

    y los dos números

    \displaystyle  v_1=\sup_{x\in X} V_1(x),\quad v_2=\inf_{y\in Y} V_2(y). \ \ \ \ \ (3)

    entre estos elementos del juego siempre vale

    \displaystyle  V_1(x)\leq v_1,\quad v_2\leq V_2(y). \ \ \ \ \ (4)

    La interpretación intuitiva de estos datos es inmediata. El valor {V_1(x)} es la ganancia que tiene asegurada el J1 si elige la estrategia {x}; el número {v_1} es lo máximo que puede asegurarse si la estrategia la elige convenientemente. Análogas interpretaciones valen para {V_2(y)} y {v_2}. Un teorema importante que relaciona los valores {v_1,v_2} es el siguiente:

    Sea {G=(X,Y,M)} un juego rectangular. Entonces se verifica que

    \displaystyle v_1\leq v_2.\ \ \ \ \ (5)

    Cuando existe una estrategia {\bar{x}\in X} tal que

    \displaystyle  V_1(\bar{x})=\sup_{x\in X}V_1(x)=v_1  \ \ \ \ \ (6)

    entonces a esta estrategia {\bar{x}} se le llama estrategia maximin para el J1. Esta estrategia existe si el supremo de {V_1(x)} es accesible. De modo análogo, una estrategia minimax es una estrategia {\bar{y}\in Y} tal que

    \displaystyle  V_2(\bar{y})=\inf_{y\in Y}V_2(y)=v_2  \ \ \ \ \ (7)

    y su existencia equivale a decir que el extremo inferior de {V_2(y)} es accesible.

    Para estas estrategias se tiene, de (2), (6) y (7), que

    \displaystyle  v_1\leq M(\bar{x},y),\quad M(x,\bar{y})\leq v_2,\quad \forall x\in X,\forall y\in Y.  \ \ \ \ \ (8)

    Si existe la estrategia maximin {\bar{x}}, {v_1} es la ganancia que puede asegurarse el J1 jugando con ella. Del mismo modo, eligiendo la estrategia minimax (si existe), el J2 se asegura de que su pago no supere a {v_2} (o bien, que su ganancia no quede por debajo de {-v_2})

    proxima semana: Juegos con valor.

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