Trying out latex in WP v2.0

Found this blog

trying out some \LaTeX in wordpress.

How did it come out? Please comment below.

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Topological maps or topographic maps?

Division by Zero

While surfing the web the other day I read an article in which the author refers to a “topological map.” I think it is safe to say that he meant to write “topographic map.” This is an error I’ve seen many times before.

A topographic map is a map of a region that shows changes in elevation, usually with contour lines indicating different fixed elevations. This is a map that you would take on a hike.

A topological map is a continuous function between two topological spaces—not the same thing as a topographic map at all!

I thought for sure that there was no cartographic meaning for topological map. It turns out, however, that there is.

A topological map is a map that is only concerned with relative locations of features on the map, not on exact locations. A famous example is the graph that we use to…

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Cayley-Menger determinants

Nice

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:

tartaglia

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(\math{N}^k\subseteq\math{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

\chapter{Features of the Standard LaTeX Report Class}

1. Section

Use the \verb”

2. Section

” command for major sections, and
the \verb”

2.1. Subsection

” command for subsections, etc.

2.2. Subsection

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It holds \cite{KarelRektorys} the following

Theorem 1
(The Currant minimax principle.) Let {T} be completely continuous
selfadjoint operator in a Hilbert space {H}. Let {n} be an
arbitrary integer and let {u_1,\ldots,u_{n-1}} be an arbitrary
system of {n-1} linearly independent elements of {H}. Denote

\displaystyle \max_{\substack{v\in H, v\neq 0\\(v,u_1)=0,\ldots,(v,u_n)=0}}\frac{(Tv,v)}{(v,v)}=m(u_1,\ldots, u_{n-1}) \ \ \ \ \ (1)

 


Then the {n}-th eigenvalue of {T} is equal to the minimum of these
maxima, when minimizing over all linearly independent systems
{u_1,\ldots u_{n-1}} in {H},

\displaystyle \mu_n = \min_{\substack{u_1,\ldots, u_{n-1}\in H}} m(u_1,\ldots, u_{n-1}) \ \ \ \ \ (2)

 


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  • Bunyip Mythical beast of Australian Aboriginal legends.6. Theorem-Like Environments

    The following theorem-like environments (in alphabetical order) are available
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    This text is a sample for a short bibliography. You can cite a
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    , D. Reidel Publishing Company,
    Dordrecht-Hollanf/Boston-U.S.A., 2th edition, 1975

    \bibitem {Bertoti97} \textsc{Bert\'{o}ti, E.}: On mixed variational formulation
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    displacements
    , International Journal for Numerical Methods in
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    \bibitem {Szeidl2001} \textsc{Szeidl, G.}: Boundary integral equations for
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    , Journal
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    \bibitem {Carlson67} \textsc{Carlson D. E.}: On G\”{u}nther’s stress functions
    for couple stresses
    , Quart. Appl. Math., 25, (1967),
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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

Quartz

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

Grothendieck1975

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

Goedel1935

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)

y

\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.

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Very interesting article on recent developments concerning solving open problems.

Gödel's Lost Letter and P=NP

HamiltonClaypic

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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