Found this blog…

trying out some in wordpress.

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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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Nice

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*i*th 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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A list of notes, by date **Nov 8** – Densidad…

**Nov 15**

- Pere Menal ( 1992) Ver si un módulo artiniano tiene semilocal. Rosa Camp & W. Ricki Sí.
- Krull ’34 Módulos Artinianos satisfacen el Teorema de Krull-Schmidt? No es difícil ver que si es artiniano \footnotesize(se forma una cadena descendiente)\normalsize , indescomponibles. Es la única descomposición bajo isomorfía y reordenación de los sumandos? (Teorema de Krull-Schmidt). Facchini, H Levy, Vamos ’96 No.
- Un -módulo generado por elementos – es imágen homomórfica de

**Ene 13**

- Módulos proyectivos finitamente generados 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
- finitely generated finitely generated. Counterexample given by Geramnov, Sakhaev
- Fair-sized projective modules by Pavel ()
- The trace ideal is a bi-lateral ideal: … . Verify that
- Whitehead, as cited in Pavel’s article: a bi-lateral idempotent ideal of such that is finitely generated , with a projective ideal
- Hyman Bass 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 **

This is just some text under a subsection.

\subsubsection{Subsubsection}

This is just some text under a subsubsection.

\paragraph{Subsubsubsection}

This is just some text under a subsubsubsection.

\subparagraph{Subsubsubsubsection}

This is just some text under a subsubsubsubsection.

**3. Typesetting Commands **

Select a part of the text then click on the button Emphasize (H!),

or Bold (Fs), or Italic (Kt), or Slanted (Kt) to typeset

*Emphasize*, **Bold**, *Italics*,

*Slanted* texts.

You can also typeset \textrm{Roman}, \textsf{Sans Serif},

\textsc{Small Caps}, and \texttt{Typewriter} texts.

You can also apply the special, mathematics only commands

, ,

and . Note that blackboard bold and

calligraphic are correct only when applied to uppercase letters A

through Z.

You can apply the size tags — Format menu, Font size submenu —

{\tiny tiny}, {\scriptsize scriptsize}, {\footnotesize

footnotesize}, {\small small}, {\normalsize normalsize}, {\large

large}, {\Large Large}, {\LARGE LARGE}, {\huge huge} and {\Huge

Huge}.

You can use the \verb” etc. ” environment

for typesetting short quotations. Select the text then click on

Insert, Quotations, Short Quotations:

The buck stops here. *Harry Truman*

Ask not what your country can do for you; ask what you can do for

your country. *John F Kennedy*

I am not a crook. *Richard Nixon*

I did not have sexual relations with that woman, Miss Lewinsky.

*Bill Clinton*

The Quotation environment is used for quotations of more than one

paragraph. Following is the beginning of *The Jungle Books*

by Rudyard Kipling. (You should select the text first then click

on Insert, Quotations, Quotation):

It was seven o’clock of a very warm evening in the Seeonee Hills

when Father Wolf woke up from his day’s rest, scratched himself,

yawned and spread out his paws one after the other to get rid of

sleepy feeling in their tips. Mother Wolf lay with her big gray

nose dropped across her four tumbling, squealing cubs, and the

moon shone into the mouth of the cave where they all lived.

“*Augrh*” said Father Wolf, “it is time to hunt again.”

And he was going to spring down hill when a little shadow with a

bushy tail crossed the threshold and whined: “Good luck go with

you, O Chief of the Wolves; and good luck and strong white teeth

go with the noble children, that they may never forget the hungry

in this world.”

It was the jackal—Tabaqui the Dish-licker—and the wolves of

India despise Tabaqui because he runs about making mischief, and

telling tales, and eating rags and pieces of leather from the

village rubbish-heaps. But they are afraid of him too, because

Tabaqui, more than any one else in the jungle, is apt to go mad,

and then he forgets that he was afraid of anyone, and runs through

the forest biting everything in his way.

Use the Verbatim environment if you want \LaTeX to preserve

spacing, perhaps when including a fragment from a program such as:

#include // is used for standard libraries.

void main(void) // ”main” method always called first.

{

cout << ”This is a message.”;

// Send to output stream.

}

(After selecting the text click on Insert, Code Environments,

Code.)

**4. Mathematics and Text **

It holds \cite{KarelRektorys} the following

Theorem 1

(The Currant minimax principle.) Let be completely continuous

selfadjoint operator in a Hilbert space . Let be an

arbitrary integer and let be an arbitrary

system of linearly independent elements of . Denote

Then the -th eigenvalue of is equal to the minimum of these

maxima, when minimizing over all linearly independent systems

in ,

The above equations are automatically numbered as equation

(1) and (2).

**5. Lists Environments **

You can create numbered, bulleted, and description lists (Use the

Itemization or Enumeration buttons, or click on the Insert menu

then chose an item from the Enumeration submenu):

- List item 1
- List item 2
- A list item under a list item.
- Just another list item under a list item.
- Third level list item under a list item.
- Fourth and final level of list items allowed.

- Third level list item under a list item.

- Bullet item 1
- Bullet item 2
- Second level bullet item.
- Third level bullet item.
- Fourth (and final) level bullet item.

- Third level bullet item.

- Second level bullet item.
- Description List Each description list item has a term

followed by the description of that term. - Bunyip Mythical beast of Australian Aboriginal legends.
**6. Theorem-Like Environments**The following theorem-like environments (in alphabetical order) are available

in this style.This is an acknowledgement

This is an algorithm

This is an axiom

This is a case

**Claim 1**

This is a claim

This is a conclusion

This is a condition

This is a conjecture

**Corollary 2**

This is a corollary

This is a criterion

**Definition 3**

This is a definition

**Example 1**

This is an example

**Exercise 1**

This is an exercise

**Lemma 4**

This is a lemma

*Proof:*

This is the proof of the lemma.

This is notation

This is a problem

**Proposition 5**

This is a proposition

**Remark 1**

This is a remark

This is a solution

This is a summary

**Theorem 6**

This is a theorem

*Proof:*This is the proof.

\appendix

\chapter{The First Appendix}

The appendix fragment is used only once. Subsequent appendices can be created using the

Chapter command.\chapter{The Second Appendix}

\chapter{The Third Appendix}

Some text for the third Appendix.

This text is a sample for a short bibliography. You can cite a

book by making use of the command \verb”\cite{KarelRektorys}”:

\cite{KarelRektorys}. Papers can be cited similarly:

\cite{Bertoti97}. If you want multiple citations to appear in a

single set of square brackets you must type all of the citation

keys inside a single citation, separating each with a comma. Here

is an example: \cite{Bertoti97, Szeidl2001, Carlson67}.{9}

\bibitem {KarelRektorys}Rektorys, K.,*Variational methods in Mathematics,*, D. Reidel Publishing Company,

Science and Engineering

Dordrecht-Hollanf/Boston-U.S.A., 2th edition, 1975\bibitem {Bertoti97} \textsc{Bert\'{o}ti, E.}:

*On mixed variational formulation*, International Journal for Numerical Methods in

of linear elasticity using nonsymmetric stresses and

displacements

Engineering.,**42**, (1997), 561-578.\bibitem {Szeidl2001} \textsc{Szeidl, G.}:

*Boundary integral equations for*, Journal

plane problems in terms of stress functions of order one

of Computational and Applied Mechanics,**2**(2), (2001),

237-261.\bibitem {Carlson67} \textsc{Carlson D. E.}:

*On G\”{u}nther’s stress functions*, Quart. Appl. Math.,

for couple stresses**25**, (1967),

139-146.

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!

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

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

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

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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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Time to read a little bit about Grothendieck.

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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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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**3. Juegos con un Valor **

El juego se dice que tiene un valor si

y en este caso se llama *valor del juego* al número .

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

A estas relaciones (10) se hace referencia diciendo que es un *punto de silla* de la función , o que tiene un *punto de silla* en .

De (10) se sigue fácilmente que ; así pues (10) se puede escribir así

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

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

De las propiedades expuestas resulta que si hubiese dos puntos de silla, por ejemplo sucedería que

y

Sustituyendo en las primeras relaciones las variables por y en la segunda por resulta

de donde resulta . Además los cuatro puntos 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 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 se corresponde con el juego , donde 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.

Very interesting article on recent developments concerning solving open problems.

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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Machine Intelligence ?

Ideas, opinions and comments related to machine intelligence, cognitive robotics, computer vision, etc. For the blog in portuguese, see options below. By Joao Kögler.

Division by Zero

A blog about math, puzzles, teaching, and academic technology

UPROXX

News and Culture for the Digital Generation

The Math Less Traveled

Explorations in mathematical beauty

Picture this maths

maths is more than simply sums

The Dictionary of You

notes for my Costa Rican daughter on her country, its language, and motherhood far from home

These Football Times

The home of longform football writing

Yanis Varoufakis

thoughts for the post-2008 world

The Research Whisperer

Just like the Thesis Whisperer - but with more money

Zurdok Movimento

Espacio dedicado totalmente a Zurdok!

SantiCathorla's Blog

No fucking idea.

Scientific Clearing House

Carson C. Chow

What's new

Updates on my research and expository papers, discussion of open problems, and other maths-related topics. By Terence Tao

pwnetics

post-gradschool blog offerings to the index

mathbabe

Exploring and venting about quantitative issues

The Thesis Whisperer

Just like the horse whisperer - but with more pages

Indie Hero

Brian Marggraf, Author of Dream Brother: A Novel, Independent publishing advocate, New York City dweller

The Wandering Glitch

Simplex Sigillum Veri