User jose brox - MathOverflow

Infinite dimensional central simple algebras

2011-09-15T09:21:28Z

When constructing the Brauer group of a field, only the finite-dimensional central simple algebras are considered (because of Artin-Wedderburn's characterization).

But what happens to the infinite-dimensional ones? (I.e., to simple algebras which are infinite-dimensional over their centers).

Is there an analogue for the Brauer group?
Is there some structure theorem like Artin-Wedderburn's?
What can be said about cross products of these algebras?

Which mathematicians have influenced you the most?

2009-11-14T13:28:17Z

There are mathematicians whose creativity, insight and taste have the power of driving anyone into a world of beautiful ideas, which can inspire the desire, even the need for doing mathematics, or can make one to confront some kind of problems, dedicate his life to a branch of math, or choose an specific research topic.

I think that this kind of force must not be underestimated; on the contrary, we have the duty to take advantage of it in order to improve the mathematical education of those who may come after us, using the work of those gifted mathematicians (and even their own words) to inspire them as they inspired ourselves.

So, I'm interested on knowing who (the mathematician), when (in which moment of your career), where (which specific work) and why this person had an impact on your way of looking at math. Like this, we will have an statistic about which mathematicians are more akin to appeal to our students at any moment of their development. Please, keep one mathematician for post, so that votes are really representative.

Answer by Jose Brox for Why were matrix determinants once such a big deal?

2010-12-23T16:46:05Z

The simple fact that the invertible square matrices of order $n$ are precisely those which have nonzero determinant has a real bunch of theoretical applications (in ring theory, topology, differential geometry, etc).

For example, if you consider the general linear group $GL(n, \mathbb{R})$ as a Lie group, to see that its associated Lie algebra is (isomorphic to) $gl(n, \mathbb{R})$, an essential step is to get the equivalence of tangent planes $T_e(GL(n,\mathbb{R})) \cong T_e(gl(n,\mathbb{R}))$ (where $e$ is the neutral element). But this is trivial if we have in mind that $GL(n,\mathbb{R}) = \{A\in gl(n,\mathbb{R}) : det(A) \neq 0 \}$ and that $det$ is continuous, because then $GL(n,\mathbb{R})$ is automatically an open set of $gl(n,\mathbb{R})$.

Answer by Jose Brox for Asymptotics of Product of consecutive primes

2010-11-02T11:24:53Z

You can also prove that $\displaystyle \ \lim_n \ \ \sqrt[p_n]{\prod_1^n p_i} = e$

(where $p_i$ is the $i$-eth prime number and $e$ is Euler's exponential number)

Answer by Jose Brox for Why are polynomials easier to handle with than integers?

2010-10-29T08:33:28Z

Following the commentary by Amri, I think that his idea can be explained in terms of graduations: The natural graduation of $F[X]$ over $F$, generated by the degree, allows us to compute simultaneously a lot of $F$-sums without mixing information. Maybe the rest of facts can be also described by graduation properties? (I'm not taking into account trivial graduations for $Z$).

How to locate the paper that established Robinson Arithmetic?

2010-07-05T18:10:58Z

If I'm not mistaken, it was in his seminal paper “An Essentially Undecidable Axiom System”, published in

Proceedings of the International Congress of Mathematics (1950), 729–730,

where R.M. Robinson proved that Gödel Incompleteness Theorem still applies to Peano Axioms if we drop the induction schema (hence showing that infinite axiomatization is not necessary for essential undecidability), in what we now call Robinson Arithmetic.

I would like to know:

Is actually this paper what I should be looking for?
Can it be found anywhere on the net? (I already tried on MathSciNet, SpringerLink, JSTOR and Google Scholar, without success)
Can anyone pinpoint to closely related, or at least similar, accessible papers?

(Note: I already have the book "Undecidable theories", which he published in collaboration with Tarski, but I'd prefer to locate papers about 'Robinson theory', specifically).

Answer by Jose Brox for Widely accepted mathematical results that were later shown wrong?

2010-08-15T20:18:01Z

In the 1960s, John Horton Conway verified the 1920s efforts of Alexander and Seifert in Knot Theory to tabulate all the knots of at most 10 crossings (1). He found several omissions and one duplication, but somewhat famously failed to discover another one. This error would propagate when Dale Rolfsen added a knot table (as Appendix C) in his influential book Knots and Links in 1976, based on Conway's work. This addition happened in spite that Kenneth Perko had noticed, in 1974, the other pair of entries in classical knot tables that actually represent the same knot (2). In Rolfsen's knot table, this supposed pair of distinct knots is labeled $10_{161}$ and $10_{162}$. Now this pair is called the Perko pair, for obvious reasons :)

(1) An enumeration of knots and links, and some of their algebraic properties, Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967), Pergamon, pp. 329–358

(2) On the classification of knots. Proc. Amer. Math. Soc. 45 (1974), 262--266

(The information of this post is quoted from the following Wikipedia articles:

)

Finding the largest integer describable with a string of symbols of predefined length

2010-07-22T03:55:16Z

(This question is motivated by the reading of the article Large numbers and unprovable theorems by Joel Spencer, which can be found at http://mathdl.maa.org/images/upload_library/22/Ford/Spencer669-675.pdf and that I recommend).

We all know of the game where a card of a predefined size, say 3x5 cm, is given to every contender, and whoever writes the biggest (positive) integer on his, wins. Naive answers are easily defeated by iteration of fast-growing functions; those are defeated by induction, and these by transfinite induction. However, if a system of axioms is prefixed, then we cannot pursue this strategy forever: for example, if we are only willing to accept Peano's Axioms (PA), then $f_{\alpha}(n)$ (where $\alpha$ is an ordinal number and $f$ is defined à la Ackermann) is computable (in the sense that the axioms ensure that a program to compute it exists and will terminate in finite time) when $\alpha < \epsilon_0$, but not if $\alpha = \epsilon_0$.

This problem suggests two related questions to me:

It is known that the Ackermann function is well-defined inside AP, and that other functions which grow much faster, like the one in the strong version of the finite Ramsey Theorem of Paris-Harrington, or Goodstein's function whenever it grows fast (I think), or $f_{\epsilon_0}(9)$, cannot be defined everywhere just by application of AP, because they "grow too fast for AP". Is there a rigorous definition of what it means for a function to grow too fast for AP (or any other arithmetical axiom system)? Can we establish in any sense a "limit" for this process? For example, can we find a "threshold function" F, depending on the axioms, such that if f dominates F then f is not computable and if F dominates f then it will be? (I'm thinking about something among the lines of the convergence of the p-series for p>0 whenever p>1 and its divergence whenever p<=1).
Building in the exposition above Spencer observes that, between experts, this game is not funny and reduces to claims of legitimacy (over the validity of the axioms they are supposed to use), since if we allow just a fixed amount of characters for describing our number, and our axioms system is prefixed also, then THERE in fact IS a largest number computable on that system (and thus competitors would come to a draw). However, what happens if we consider the following metagame? Instead of fixing the axiom system beforehand, we allow every contender to (secretly) choose his own system of axioms for arithmetic, in the hope that his will allow faster-growing computable functions than those of the others. Doing this, the contender takes the risk, while trying to get more and more power from the axioms, of actually getting an inconsistent system! Whoever gets the biggest (computable) number in a consistent axiom system wins. Is this game interesting, or is it "flawed" too? In adittion, inconsistency may be proved within the axiom system, but its consistency would have to be proven in a more powerful framework. Which one would you select and why? What about the metametagame of letting those frameworks to the election of the players? Is that still interesting?

Answer by Jose Brox for Is there an image for you that epitomizes mathematics?

2010-07-07T00:43:42Z

For category theory, from Abstract and Concrete Categories: The joy of cats, by Adámek, Herrlich and Strecker, page 12:

(http://katmat.math.uni-bremen.de/acc/acc.pdf)

Categorists have developed a symbolism that allows one quickly to visualize quite complicated facts by means of diagrams.

For me, this represents the fact that most, if not all of mathematics, is about structures and relations: even the simplest of them, when combined and interrelated, can give birth to fairly complex behaviour.

Answer by Jose Brox for Applications of classical field theory

2010-07-07T16:53:30Z

Well, I'm not really sure about whether you wish to refer just to relativistic classical field theories or you are interested on non-relativistic ones as well.

Either way, you have:

Classical thermodynamics, where you study the internal energy, entropy, temperature, pressure and volume fields of a classical sytem. This theory serves to:

Understand and apply at wish the mechanisms of heat conduction (freezers, air-conditioners, determining the hour of death of a corpse...).

Make rough cosmological assesments (heat death of the universe and the like).

Understand, measure and determine the restrictions on life at a thermodynamical level (need for feeding, maximal safe temperature, maximal attainable speed, construction of thermomethers).

Understand how to construct, manage and study combustion-powered engines (Carnot efficiency, automotive industry).

Fluid dynamics, where you study the velocity, temperature, density and pressure fields of a (liquid or gaseous) fluid. This theory serves to:

Design fluid-efficient machines, whether they are flying machines - planes and spaceships -, running machines - Formula 1 cars - or navigating machines - ships and submarines -.

Understand and implement interesting flows through pipes, nozzles and turbines (we need them, for example, to conduct or transport liquids, gasses and colloids and to know how to expel them properly).

Get insight on some transport phenomena that occur in Biology (since water is an important component of most living creatures and of their environments).

Predict the weather on a given (not too big) zone, in a given time (not too far in the future).

Get knowledge on the important ocean currents and on how to predict tsunamis.

Have the correct tool for (fluid) acoustical engineering, since sound is no more than disturbances of the average pressure of the fluid.

Magnetohydrodynamics, which studies the dynamics of electrically conducting fluids (like plasmas, liquid metals or salt water). Its equations are the mixture of Navier-Stokes' and Maxwell's. This theory serves to:

Model the core of the Earth as a liquid-metal dynamo (generation of its global magnetic field, Seismology).

Accurately describe the internal dynamics of stars and another space objects made mostly from plasma (predict the number, size and movement of the Sunspots).

Know how to cool high temperature systems by means of liquid metals.

Research a totally new generation of engines.

Electrohydrodynamics, which studies the dynamics of ionised particles on the sine of a fluid, subject to electric fields. This theory serves to:

In general, to convert electrical energy into kinetic energy (and vice versa).

Know how to cool systems by means of ionized liquids.

Construct liquid electrical generators.

Build propulsion devices without moving parts (EHD thrusters).

Add some more pseudoscience to our already-too-magical world: Designing air ionizers and claiming that they have all kind of health benefits.

Answer by Jose Brox for Is there an image for you that epitomizes mathematics?

2010-07-07T00:31:01Z

Keizo Ushio's topological "impossible" sculptures on stone

Answer by Jose Brox for eBook readers for mathematics

2010-07-05T14:47:43Z

Another page for taking a fair look at the eReaders market is

http://www.ereaderguide.info/

They do extensive reviews of the machines and rate them by several criteria. If you go into the page of a particular eReader, you will see which file formats it supports (e.g., you can see that, in fact, the EPocketBook 301 supports DJVU!) and also a long, detailed analysis about how good it is for reading.

(The page looks like a little biased towards Amazon and their Kindles, though).

I recommend you navigate this site for a while before making your mind up!

Answer by Jose Brox for Famous mathematical quotes

2010-02-06T01:53:29Z

Not famous yet, maybe from now on!

At a purely formal level, one could call probability theory the study of measure spaces with total measure one, but that would be like calling number theory the study of strings of digits which terminate.

Terence Tao

Answer by Jose Brox for Number of valid topologies on a finite set of n elements

2009-12-15T15:26:34Z

The best result I know is found in the article The number of finite topologies, by D. Kleitman and B. Rothschild, where they state that the base-2 logarithm of the number of distinct topologies on a set of $n$ elements is asymptotic to $n^2/4$.

Answer by Jose Brox for Justifying a theory by a seemingly unrelated example

2009-12-13T12:14:48Z

The Laplace transform to systematically solve homogeneous ODE's with constant coefficients by transforming them in polynomial equations (and then transforming the solution back).

Solving a differential equation by the Laplace transform

Answer by Jose Brox for Justifying a theory by a seemingly unrelated example

2009-12-13T12:09:42Z

Root systems and Dynkin diagrams for classification matters (see both Root system and ADE classification at Wikipedia).

Answer by Jose Brox for Favorite popular math book

2009-12-12T00:05:07Z

Title: Innumeracy: Mathematical Illiteracy and its Consequences

Author: John Allen Paulos, mathematician and well-known skeptic (in the good, modern sense of the word).

Short description: Paulos explains for the general public (and he does it fairly well) why it should understand a little more mathematics and know how to do Fermi calculations and educated probabilistic guesses. The dangers coming from pseudosciences are also highlighted, something very needed nowadays in my opinion.

Math level: +. Some chapters are not that easy to read without a mathematical background, more because of the great number of calculations and logic steps involved in the explanations than because of the math level of those per se.

Price: 9.89$ at Amazon.

Answer by Jose Brox for Number theory textbook with an algebraic perspective

2009-12-07T12:33:00Z

If I understand it well now, what you want are books about basic number theory with a good algebraic foundation. I can recommend the following:

Elementary methods in number theory (2000), by Nathanson. Graduate Texts in Mathematics. It starts low, but it reaches quite high.
Elementary Number Theory with Applications (2007) by Koshy. Elsevier. Truly basic. Not very very algebraic, but a really nice textbook.
Algebra and number theory by Andrew Baker. Online notes. Fairly basic.
A computational introduction to number theory and algebra (2005) by Shoup. Cambridge University Press / Online Free Version. Despite the title, I think it satisfies the conditions you are looking for.

Answer by Jose Brox for Number theory textbook with an algebraic perspective

2009-12-07T10:49:44Z

Well, it depends on the actual subject you want to approach and the "decent command of modern algebra" already assumed; without knowing more, I would recommend:

Number fields (1995) by Marcus. Universitext. Just as the title says, a (great!) introduction to number fields.
Algebraic Number Theory (1986) by Cassels and Frölich. Academic Press. It explains the basics (class field theory, zeta functions) to understand the Langlands Program.
A course on arithmetic (1996) by Serre. Graduate Texts in Mathematics. P-adic fields, quadratic forms, zeta functions and modular forms.

Answer by Jose Brox for Some equivalent statements about primitive algebras

2009-12-07T10:29:27Z

Lam (A first course in noncommutative rings, 2ed) does it for (unital) rings $R$ in Lemma 11.28 (page 186):

If such a $B$ exists, we may assume (after an application of Zorn's Lemma) that it is a maximal left ideal. The annihilator of the simple left $R$-module $R/B$ is an ideal in $B$, and so it must be zero. This shows that $R$ is left primitive. Conversely, if $R$ is left primitive, there exists a faithful simple left $R$-module, which we may take to be $R/B$ for some (maximal) left ideal $B \subsetneq R$. A nonzero ideal $C$ cannot lie in $B$ (for otherwise $C$ annihilates $R/B$) and so must be comaximal with $B$.

Here, the statement equivalent to the Axiom of Choice is Zorn's Lemma; it says that:

Every partially ordered set in which every chain (i.e. totally ordered subset) has an upper bound contains at least one maximal element.

In this proof we get to see one of its most common uses: it assures that any unital ring has a maximal ideal (see the Wikipedia page for more information).

Answer by Jose Brox for books well-motivated with explicit examples

2009-12-06T16:50:27Z

Visual Complex Analysis, by Tristan Needham.

Really nice to get a thorough geometrical understanding of (one) complex variable.

Topological Langlands?

2009-11-30T15:32:20Z

In a workshop about the geometry of $\mathbb{F}_1$ I attended recently, it came up a question related to a mysterious but "not-so-secret-anymore" seminar about... an hypothetical Topological Langlands Correspondence!

I had never heard about this program; I have found this page via Google:

http://www.math.jhu.edu/~asalch/toplang/

I only know a bit about the Number-Theoretic Langlands Program, and I still have a hard time trying to understand what is happening in the Geometrical one, so I cannot even start to draw a global picture out of the information dispersed in that site.

So, the questions are: What do you know (or what can you infere from the web) about the Topological Langlands Correspondence? Which is the global picture? What are its analogies with the (original) Langlands Program? Is it doable, or just a "little game" for now? What has been proved until now? What implications would it have?

(Note: It is somewhat difficult to tag this one, feel free to retag it if you have a better understanding of the subject than I have!)

Answer by Jose Brox for Best mathematics conference?

2009-11-26T03:18:58Z

The 1900's Paris ICM... just because of Hilbert's talk, in which he modelled the shape of a rather big amount of the mathematics that were to come; and not just by becoming fancy, but by identifying really difficult and interesting problems and, even more importantly, by pointing to logic and rigor so that others could see the necessity for a thorough cleaning of their mathematical building.

(Sorry, I could not locate any link to the proceedings ;P)

Answer by Jose Brox for Can infinity shorten proofs a lot?

2009-11-18T20:17:15Z

While browsing Wikipedia, I came across this statement:

"The introduction of projective techniques made many theorems in algebraic geometry simpler and sharper: For example, Bézout's theorem on the number of intersection points between two varieties can be stated in its sharpest form only in projective space".

I think this fact can be stated in simple terms without any explicit reference to algebraic geometry and projective space, just talking about intersections of graphics of polynomials, the product of their degrees, and "combing space to get the point of infinity" (or some better and longer explanation).

What are tame and wild hereditary algebras?

2009-11-18T01:43:26Z

What are tame and wild hereditary algebras?
Are they related to hereditary rings? (Those are rings for which every left (resp. right) ideal is projective, equivalently, for which every left (resp. right) submodule of a projective module is again projective).
Googling them I can see they seem related to the "tame representation type", but this concept is also new to me.
I would also like to know what is their relation to path algebras, since sometimes they appear mentioned together.

Do you know any good (newbie) references for all this? (Or can you elaborate in any of the questions?)

Answer by Jose Brox for Is there an agreed name for partial ordering based on Pareto Dominance relation?

2009-11-18T02:04:33Z

Depending on if you allow "greater" to mean "greater or equal" or to mean "strictly greater" we have two answers coming from Order Theory, respectively:

The product order: http://en.wikipedia.org/wiki/Product_order
The reflexive closure of the direct product of two strict total orders

Those two, along with the lexicographic order (which I searched to find this answer), are the three best-known choices among the possible orders on the Cartesian product of two totally ordered sets, as is stated in

http://en.wikipedia.org/wiki/Total_order#Orders_on_the_Cartesian_product_of_totally_ordered_sets

Answer by Jose Brox for Can infinity shorten proofs a lot?

2009-11-17T15:09:45Z

Maybe you could use some complex transformation related to Möebius functions (of the kind "circles and lines are the same") to prove several 2D-geometric statements at once.

Answer by Jose Brox for Where can I find questions motivating important ideas in math?

2009-11-16T17:51:46Z

I suggest you take a look at the Tricki:

http://www.tricki.org

Answer by Jose Brox for ubiquity, importance of path algebras

2009-11-16T17:42:12Z

You also have the rather new field of Leavitt Path Algebras (in which I happen to be working right now), where you take a field $K$ and a directed graph $E$, generate its extended graph $E'$ (add to $E$ its own edges reversed, denoted as $e^*$ for every edge $e$), and compute the Leavitt path algebra of $E$, $L(E)$, as the path algebra $KE'$ modulo some relations called the Cuntz-Krieger relations, inherited from the $C^*$-algebras setting, concretely:

(CK1) $e^* f=\delta_{ef}$ for any two edges $e,f$ of $E'$.

(CK2) $\sum_{e\in s^{-1}(v)}ee^* = v$, for $v$ a vertex which emits a nonzero finite number of edges, and $s^{-1}(v)$ the set of those edges.

(One can look at (CK1) and (CK2) as an abstract generalization of the product of matrix units).

These associative algebras provide us simultaneously with a purely algebraic analog of $C^*$-algebras of graph and a generalization of the Leavitt algebras (some associative algebras which do not satisfy the IBN property).

The full matrix rings over $K$ of order $n$ then arise as the Leavitt path algebras of the graphs with $n$ (consecutive) vertices and $n-1$ arrows, one between every pair of consecutive vertices.

Another simple example of Leavitt path algebra is the ring of Laurent polynomials over $K$, $K[x,x^{-1}]$, which appears associated to the graph with one vertex and a single loop.

The theory of LPAs is useful, and even beautiful, because:

They provide simple, visually attractive representations of well-known algebras.
They allow us to look at their algebraic properties by means of the combinatorial properties of their associated graphs. This happens to equip us with some rather powerful tools.
Conversely, they also enable "algebraic engineering", since they give us a straightforward, visual way to construct new algebras, customized with any algebraic or ring-theoretic properties we may desire. For example, we can show an algebra generated by five elements such that it is exchange but not purely innitely simple, by constructing a particular (small) graph with some (easy) graph-theoretic features.

Some references:

G. Abrams, G. Aranda Pino. "The Leavitt path algebra of a graph", J. Algebra 293 (2), 319-334 (2005). (Available at http://agt.cie.uma.es/~gonzalo/papers/AA1_Web.pdf).
P. Ara, M.A. Moreno, E. Pardo. "Nonstable K-Theory for graph algebras", Algebra Repr. Th. DOI 10.1007/s10468-006-9044-z (electronic). (Available at http://www.springerlink.com/content/pu701474q5300m63/).
G. Abrams, G. Aranda Pino, F. Perera, M. Siles Molina. "Chain conditions for Leavitt path algebras". (Available at http://agt.cie.uma.es/~gonzalo/papers/AAPS1_Web.pdf).
K.R. Goodearl. "Leavitt path algebras and direct limits", Contemp. Math. 480 (2009), 165-187.

Answer by Jose Brox for Which mathematicians have influenced you the most?

2009-11-14T13:46:24Z

Who: Leonhard Euler.

When: As a highschool student.

Where: On the book "Euler: the master of us all" by William Dunham.

Why: The amount of creativity and genius dispersed among the so-different works of Euler continues to amaze me just now, so it only could have a devastating effect on me 10 years ago. He not only addressed a lot of distinct topics, he layed the foundations of many branches of mathematics and solved with ease many problems that were interesting me at that moment of my life. I learned a lot from him: he really deserves the title of "master of us all".

Comment by Jose Brox

2011-04-22T11:06:10Z

Well, the poster said: "Many mathematical areas have a notion of "dimension", either rigorously or naively, [...]", and thus I felt the p-notion to fit the question.

Comment by Jose Brox

2011-03-29T03:00:25Z

This is what is usually called 'the homological study' of ring theory

Comment by Jose Brox

2011-01-28T10:57:38Z

A really nice answer to my question. Thanks!

Comment by Jose Brox

2011-01-17T07:44:37Z

Please, one mathematician per post!

Comment by Jose Brox

2010-12-14T12:58:17Z

It is a set with a geometrical depiction of great beauty and intricacy, but... how is it a structure?

Comment by Jose Brox

2010-08-17T02:05:39Z

I really loved this one

Comment by Jose Brox

2010-08-15T19:48:10Z

I somewhat disagree with C. Schultz. Are there any other kinds of theorems I am not aware of?

Comment by Jose Brox

2010-08-11T12:02:04Z

I totally agree. +1

Comment by Jose Brox

2010-08-11T11:50:13Z

Can you give some references for derivatives of polynomials over general rings? (The stranger, the better). Thanks!

Comment by Jose Brox

2010-08-04T02:38:41Z

Oops! I slipped deep and hard here! Obviously, the group (R,+) should be the only group with three elements, but it is not; if you fix that, then the multiplication exposed above does not satisfy the distributivity property by the left. Hence I'm deleting my post...

Comment by Jose Brox

2010-07-27T10:43:05Z

Yes, I have knowledge of it, and I also think it is indeed lovely and a very-well written exposition! Sadly, it does not (if I remember correctly) address directly the issue I was thinking about (but I highly recommend it for introducing students to these "more-complex-than-it-seems-at-first-sight theme!)

Comment by Jose Brox

2010-07-22T13:03:10Z

Moreover, who is really capable of judging if "none of the answers is good" when talking about a soft-question (and be confident that he could convince the rest of the people about him being right)? I learned some things from these answers, and I am pretty sure that there probably is at least one person that found good any particular answer - namely, its poser!

Comment by Jose Brox

2010-07-08T22:48:46Z

Thanks for the reference to that article, it may be useful for me! (btw, I think you should accept your own answer!)

Comment by Jose Brox

2010-07-08T10:44:20Z

How does the standard proof for $R[X]$ go? We may be able to extend that proof, since $R[X,X^{-1}]$ is a localization of $R[X]$.

Comment by Jose Brox

2010-07-08T09:58:20Z

Sorry, you are right! The sketch I quickly made in my miind was clearly wrong. In consequence, I will delete this answer! (and give it some more thought). Thank you!