User q.q.j. - MathOverflow

The phenomena of eventual counterexamples

2010-02-16T13:00:56Z

Define an "eventual counterexample" to be

$P(a) = T $ for $a < n$
$P(n) = F$
$n$ is sufficiently large for $P(n) = T\ \ \forall n \in \mathbb{N}$ to be a 'reasonable' conjecture to make.

where 'reasonable' is open to interpretation, and similar statements for rational, real, or more abstractly ordered sets for $n$ to belong to are acceptable answers.

What are some examples of eventual counterexamples, famous or otherwise, and do different eventual counterexamples share any common features? Could we build an 'early warning system' set of heuristics for seemingly plausible theorems?

edit: The Polya conjecture is a good example of what I was trying to get at, but answers are not restricted to number theory or any one area.

Matrix representation for $F_4$

2010-02-26T04:07:34Z

Has anyone ever bothered to write down the 26-dimensional fundamental representation of $F_4$? I wouldn't mind looking at it. Is it in $\mathfrak{so}(26)$?

I'm familiar with the construction of the fundamental representation for $G_2$ where you can use use the fact that the groups is automorphism group of the octonions to put linear relations on $\mathfrak{so}(7)$. (Elements of ${\mathfrak{g}}_2$ are the derivations)

Answering the same question for $E_6$, $E_7$ or $E_8$ would also be welcome here.

The finite subgroups of SU(n)

2010-03-04T11:23:31Z

This question is inspired by the recent question "The finite subgroups of SL(2,C)". While reading the answers there I remembered reading once that identifying the finite subgroups of SU(3) is still an open problem. I have tried to check this and it seems it was at least still open in the Eighties.

Can anyone confirm or deny that the finite subgroups of SU(3) are not all known? And if this is true, then what is the source of the difficulty?

Secondly, what is known of the finite subgroups of SU(n) for n > 3?

UPDATE: Thanks to those below who have corrected my ignorance! It seems that I may have been tricked by some particularly sensationalised abstracts (or perhaps just misunderstood them.)

Answer by Q.Q.J. for Why is the half-torus rigid?

2011-10-11T01:35:31Z

For any surface patch with the first fundamental form $$g(u,v) = \left[\begin{array}{cc} (c+a\cos{v})^2 & 0\\ 0 & a^2\end{array}\right]$$ The Gauss and Codazzi Equations are

$$ \begin{align} ac\cos(v)+a^2\cos^2(v)-h_{11}h_{22}+h_{12}^2&=0\\ h_{11,v} - h_{12,u} + \frac{a\sin(v)}{c+a\cos{v}}h_{11}+\frac{\sin(v)(c+a\cos(v))}{a}h_{22}&=0\\ h_{22,u} - h_{12,v} + \frac{a\sin(v)}{c+a\cos{v}}h_{12}&=0 \end{align} $$ If we can show that the solution for the functions $h_{ij}$ is the same as that for the torus patch $(c+a\cos(v))\cos(u), (c+a\cos(v))\sin(u), a\sin(v))$, then we are done by uniqueness part of the Fundamental Theorem of Surfaces (patches with the same $g$ and $h$ differ only by a rigid motion).

Remark: If we make an overly strong assumption that $h$ is diagonal then this gives the result, but otherwise, as Deane comments, it is not immediately clear how/if we can prove the uniqueness of the $h_{ij}$ in the general case.

Update: Consider a particular local isometry of a patch on the torus that is small enough to not create any umbillic points. We can reparametrise in the neighbourhood of any non-umbillic point to a principal patch where both $g$ and $h$ are diagonal. The first fundamental form for the reparametrised isometric patch will have the form

$$g(u,v) = \left[\begin{array}{cc} \lambda(u,v)^2(c+a\cos{v})^2 & 0\\ 0 & \mu(u,v)^2a^2\end{array}\right]$$

for known $\lambda,\mu$ and then the Codazzi equations are now a linear system for $h_{11},h_{22}$: $$h_{11,v} = \frac12\partial_v(\lambda^2(c+a\cos{v})^2)(\frac{h_{11}}{\lambda^2(c+a\cos{v})^2} + \frac{h_{22}}{\mu^2a^2})$$

$$h_{22,u} = \frac12\partial_u(\mu^2a^2)(\frac{h_{11}}{\lambda^2(c+a\cos{v})^2} + \frac{h_{22}}{\mu^2a^2})$$

and the Gauss equation is $$h_{11}h_{22} = \lambda^2\mu^2a\cos(v)(c+a\cos(v)).$$

The quotient of a Lie group by the Levi factor of a parabolic subgroup

2011-02-04T11:13:26Z

I am interested in some references on the quotient spaces obtained by quotienting G, a simple Lie group, by L, the group generated by the Levi factor of a parabolic subalgebra.

Presumably the case where L is the maximal torus is understood?

I am mostly interested in the compact case.

Answer by Q.Q.J. for Is there an invariant theory explanation of the orbit structure of GL₂ acting on second-diagonal symmetric matrices by g∙X = gXJg^tJ ?

2010-09-04T16:28:31Z

I don't completely understand every part of your question but I will get the ball rolling and edit if necessary. The short answer to your question is yes (at least in the case of infinite fields which is all I know about).

The orbits can be characterised in a coordinate independent way by an orbit-type stratifaction of the orbit space. The orbit space is diffeomorphic to the semialgebraic variety defined on the space of invariant polynomial generators with inequalites given by the Procesi and Schwarz type approach. This space admits a primary stratification (in the sense of Whitney) and the strata in the two diffeomorphic spaces correspond.

In your case I seem to get that (for $F = \mathbb{C}$), the ring of invariant polynomials are generated by the real and imaginary parts of

$\gamma^2 +\alpha\beta$

and

$2|\gamma|^2 + |\alpha|^2 + |\beta|^2$

If you think of these as functions on $\mathbb{C}^3$ or $\mathbb{R}^6$ you can take gradients and then dot products to get a Grammian matrix. Then the equations defined by setting the various minors of the matrix equal to zero will identify the boundary of a semialgebraic region of $\mathbb{R}^3$. On the various pieces of the boundary of different dimensions the orbit type (equivalence class defined by conjugations of stabiliser groups) will be different.

The fact that the orbits can be characterised by a matrix of the form you give is then related to the fact that you can solve

$|\alpha|^2 + |\beta|^2 = C_1$

$\alpha\beta= C_2 + i C_3$

for arbitrary (C_1,C_2,C_3) in the semialgebraic variety.

If this is in anyway helpful let me know if I can add more details.

Answer by Q.Q.J. for Roadmap to learning about Ricci Flow?

2010-09-03T12:41:36Z

Another useful reference is Peter Topping's "Lectures on the Ricci Flow" which is freely available as a pdf at

Lectures on the Ricci Flow

and also links to buy the book therein.

Answer by Q.Q.J. for Mathematics of the Anthropic Principle

2010-06-20T17:49:56Z

The question could use some clarification, but perhaps you would like to read The Height of a Giraffe for an example of a fascinating calculation based on 'anthropic reasoning'. This was one of a number of similar papers to come out a couple of years ago and there was plenty of discussion about it on physics blogs at the time.

I suppose in some sense anthropic reasoning is the cousin of 'Fermi problem' type calculations. I think there probably are interesting mathematical/logical questions in there somewhere about the validity of such estimates but I don't know what the best way to frame them is either.

Answer by Q.Q.J. for Do names given to math concepts have a role in common mistakes by students?

2010-06-18T11:52:30Z

If you count "or" as a mathematical concept, the the fact that it is fundamentally inclusive in mathematics but often exclusive in most other uses of English can lead students to mistakes.

Good quality data/packages for statistical/structure analysis of words in the English language

2010-06-16T12:44:08Z

From time to time I find myself wishing to calculate basic statistics on words in the English language. For example, today I found myself wanting a graph of the number of English words vs. their length.

Admittedly, such queries usually arise for me in the context of conversational/recreational purposes, but with the obvious links to cryptography, computational game theory (Scrabble AI etc), and statistics, I think the following question easily falls within the purview of mathematical research.

What good quality resources exist for performing statistical/structural analysis on the set of English words?

One answer to this would be "get a decent word list and write an appropriate program", but, firstly, I don't know what the best word list is and where to get it , and, secondly, in a post-Wolfram Alpha world, I am compelled to search for something higher level, that I can consult from time to time with little set-up needed. For example Mathematica seems to have an elaborate "WordData" package, though I am somewhat unsure of how exhaustive the data set is, given the following excerpt from the Wolfram site:

"Total number of words and phrases in WordData: In[1]:= Length[WordData[All]] Out[1]= 149191"

If anyone has first-hand experience with this package, or even better (as a Maple user), a similar or better resource that is standalone, (or implemented in Maple), then it would be great to hear about it.

Answer by Q.Q.J. for Surveys on Navier Stokes Equations and its physical implications

2010-05-28T11:26:27Z

Terry Tao wrote a blog article a few years ago:

Why global regularity for Navier-Stokes is hard

and another which followed up on some aspects (and links through to an arxiv preprint too): A quantitative formulation of the global regularity problem for the periodic Navier-Stokes equation

Answer by Q.Q.J. for Is every G-invariant function on a Lie algebra a trace?

2010-05-21T03:50:17Z

If the representation is fixed as the fundamental representation, then in the case of $\mathfrak{so}(2n)$, you need Pfaffians as well as traces.

What is state of the art for the Shooting Method?

2010-04-13T20:26:18Z

I am interested in examples where the Shooting Method has been used to find solutions to systems of ordinary differential equations that are either

reasonably large systems, or
the search algorithm in the shooting parameters is somewhat prohibitive because of the nature of the solutions, or
both of the above.

Any references, descriptions, recent progress, folklore, in the ballpark would be of interest. Feel free to interpret "reasonably large" subjectively if necessary.

Answer by Q.Q.J. for Stable Tables on Fluctuating Floors

2010-03-11T15:36:00Z

"Mathematical table turning revisited" by Baritompa, L"owen, Polster, and Ross

I am no expert on what is or isn't possible but there are at least two different groups who have looked at this type of problem and this article contains a number of references that are probably relevant to you.

Answer by Q.Q.J. for What's the current state of Yang Mills Mass Gap question?

2010-03-06T05:47:42Z

There's some guidelines about open problems in the FAQ that you might want to read, but there was a good article by Faddeev last November that you should know about:

Mass in Quantum Yang-Mills Theory, by Faddeev

Answer by Q.Q.J. for Undergraduate Derivation of Fundamental Solution to Heat Equation

2010-03-05T08:06:36Z

I think the term fundamental solution (at least sometimes) conventionally includes the integral around your $K$. I will assume this. If I recall correctly then the following argument is from "Partial Differential Equations" by Strauss.

A particularly simple solution follows from the self-similarity principle, i.e.

If $u(x,t)$ is a solution then so is $u(cx, a c^2t)$

This suggests looking for a particular solution of the form $K(x,t) = g(p)$, where $p = \frac{x}{\sqrt{4at}}$

Substituting $g$ into the heat equation leads to the differential equation

$$g''+\frac{p}{2}g' = 0 $$

Then the fundamental solution as above follows from solving this.

The Angel Problem - was the bet paid?

2010-02-28T15:17:55Z

Did Conway pay the wager for either of the proofs to the The Angel Problem?

I'd check in on this every now and again when it was an unsolved problem and would like to know how the story ends. Anyone know more details?

Answer by Q.Q.J. for Resources on Invariant Theory

2010-02-27T12:21:13Z

Streklin has already provided a link to the best introductory reference, but two more worth noting are

some classical invariant theory material in an appendix (E or F?) at the end of Fulton and Harris.
"Lie groups: an approach through invariants and representations" by Procesi also contains a wealth of knowledge. I believe this is available on Springerlink if you have access.

Answer by Q.Q.J. for Are there any interesting connections between Game Theory and Algebraic Topology?

2010-02-27T11:59:20Z

Borel Determinacy and more modern improvements link game theory to topology, but more along the measure theory vein than algebraic topology

Answer by Q.Q.J. for Figure out the roots from the Dynkin diagram

2010-02-24T05:13:43Z

Just a related point for hashing out arguments quickly:

If you ever find yourself in the situation where you want to rule out some vector being a root in a hurry, you can check it against the other roots you do know to see if you get a Cartan integer out of their dot products.

Answer by Q.Q.J. for Value of "of course" in the mathematical literature

2010-02-24T03:52:22Z

If someone said "of course the equation is well-posed because it is elliptic, etc" it would teach the reader that the statement is not only true, but that there is a well known train of thought for proving the result.

In this sense there is no claim that the proof is short or contains only simple steps, but it does say that that among a certain class of people familiar with the area that it is in fact a well-walked path.

I briefly consulted a dictionary and found these two definitions for "of course" 1. "certainly; definitely" 2. "in the usual or natural order of things"

I am suggesting that "of course" is useful in the second instance as a pedagogical indicator.

I am not claiming that this is always the usage employed in mathematical writing but it is a good one if used in the right circumstances.

Answer by Q.Q.J. for The finite subgroups of SL(2,C)

2010-02-22T11:30:02Z

If you are interested in the compact real form then "On Non-Linear Realizations of the group $SU(2)$" by Mickelsson and Niederle lists the conjugacy classes of closed proper subgroups of SU(2) as a recap before going on the nonlinear cases. They are

i) The unitary subgroup $U(1)$

ii) The subgroup $N[U(1)]$ (normalizer of $U(1)$)

iii) $C_n$, the cyclic subgroups of order $n$

iv) The subgroups $\tilde{D_{2n}}$ where $\tilde{D_{2n}}/Z_2$ is isomorphic to the dihedral group $D_n$ of order $2n$.

v) The subgroup $\tilde{T}$, where $\tilde{T}/Z_2$ is isomorphic to the tetrahedral group T of order 12.

vi) The subgroup $\tilde{O}$, where $\tilde{O}/Z_2$ is isomorphic to the octahedral group O of order 24.

vii) The subgroup $\tilde{Y}$, where $\tilde{Y}/Z_2$ is isomorphic to the icosahedral group Y of order 60.

They attribute this result to a 'method of Murnaghan' whose book is "The Theory of group representations" and from memory it is in the back as an appendix.

They go on to say which of these lead to homogeneous spaces that are 3-manifolds. An interesting read and possibly of some relevance to your notes.

Answer by Q.Q.J. for Divide a square into 5 equal squares

2010-02-13T13:15:04Z

The Wallace-Bolyai-Gerwien Theorem theorem says:

Any two simple polygons of equal area are equidecomposable

(where simple means no self intersections and equidecomposable means finitely cut and glued).

For your problem you can take the first polygon to be a unit square and the second to be a sqrt(5) by 1/sqrt(5) rectangle and apply this theorem. Then perform the remaining four cuts.

Also, the generalisation of your question is the 2d analogue of Hilbert's 3rd Problem which asks whether given any two polyhedra with equal volume can one be finitely cut and glued into the other. The answer here, unlike in the 2d case, is "no" which was proved by Dehn using Dehn invariants in 1900.

Answer by Q.Q.J. for Geometric Interpretation of Trace

2010-02-13T13:25:59Z

An easy calculation that may help somehow:

Any square matrix $A$ can be written as

$A = \Sigma_{i,j} u_i v_j^t$

where $u_i,v_j$ are column matrices, and there are many different choices as to how to choose {$u_i$}, {$v_j$}. Then it follows that

$Tr(A) = \Sigma_{i,j} Tr(u_i v_j^t) = \Sigma_{i,j} u_i \cdot v_j$

and now that you have a sum of dot products you may be able to make various geometric interpetations.

Answer by Q.Q.J. for Alternative Undergraduate Analysis Texts

2010-02-02T13:48:40Z

Knapp's "Basic Real Analysis" covers a lot of material and takes care with some of the topics you mentioned. I'm not completely sure if I would have wanted it as my very first analysis book but it would have been good to have at hand and I think it would be a good text to work through.

Answer by Q.Q.J. for Which popular games are the most mathematical?

2010-02-02T12:28:19Z

Blokus is a fairly new game that's gaining popularity (though there are older games with a similar set-up). There are several versions, and the four-player version has some non-cooperative elements to the gameplay.

Each player takes turns to place polyominoes of size 1 squares through five (the monomino, domino, triominoes, tetrominoes, and pentominoes) so that they touch a previously played piece of their own colour, but only at the corners. The overall aim of the game is to try and cover as much area with your own pieces as possible. The countertactics to stop a player doing this involve placing your pieces in a way that will block them from making good moves.

I think this game would fit your criteria. It is relatively unstudied from a mathematical point of view as far as I know. I imagine some familiarity with some of the mathematical work on tessellations of polyominoes would have to give a player at least a marginal advantage in planning a long-term strategy. It probably fits the criteria in other ways too.

Name of upper triangular/lower triangular Lie Algebra decomposition

2010-01-29T09:30:52Z

What is the name of the Lie algebra decomposition where the positive root vectors are upper triangular and the negative root vectors are lower triangular?

Comment by Q.Q.J.

2011-10-11T03:48:13Z

I was mentally imagining switching out $\det{h}$ with $K\det{g}$ whenever it created trouble but that doesn't quite get the job done, does it. Just as an idea, perhaps $h_{12}$ can be assumed zero at a single point via a rigid motion, and then controlled via a Gronwall-type inequality regardless of the elliptic/hyperbolic nature of the system, hence leaving only a linear system in the diagonal variables.

Comment by Q.Q.J.

2011-10-11T01:59:49Z

Thanks George! So quadruple backslash is the trick.

Comment by Q.Q.J.

2011-10-11T01:37:44Z

Any tips on how to get the matrix to go over two lines?

Comment by Q.Q.J.

2011-05-01T23:58:54Z

Mariano, since I ask "Do different eventual counterexamples share any common features?", the possibility of different classes of general behaviour means that the plural seems more appropriate to me.

Comment by Q.Q.J.

2011-04-01T12:26:50Z

Gerry, I see what you mean but I think it is less presumptuous to suppose there is some possible plurality to the idea.

Comment by Q.Q.J.

2011-03-04T10:54:23Z

I vote for the theorem to be called "Chern-Gauss-Bonnet" instead of "Generalized Gauss-Bonnet" in the text of the question!

Comment by Q.Q.J.

2011-02-05T05:02:57Z

The Levi factor is the semisimple part of the parabolic... The imagined classification would ideally include a tabulation of the fundamental groups of all possible G/L for example.

Comment by Q.Q.J.

2011-02-05T01:00:19Z

The totality of the question is essentially "What is the status of the complete classification of all of the spaces of type G/L for simple G?" Perhaps you are telling me that a detailed answer might be too broad but I think that the question itself is not.

Comment by Q.Q.J.

2010-09-22T07:46:38Z

I take 'good' to mean able to solve, intepret, or extract qualitative behaviour due to the reduction/transformation. I think the question makes sense and that some vagueness correlates well with how ad hoc the study of PDEs is. Perhaps writing a simple PDE on the sphere and then using stereographic projections to put it on R^2 could help to reverse-engineer an illustrative example.

Comment by Q.Q.J.

2010-09-14T08:41:33Z

AS: Actually the principal definition from the Concise Oxford English dictionary is (adjective) "enabling a person to discover or learn something for themselves". Plenty much broad scope there to zero your aagghhument.

Comment by Q.Q.J.

2010-09-14T08:26:26Z

I'm not sure but I think this notion became popular in the context of renormalization, where it was tempting to interpret the somewhat ad hoc technique of 'subtracting the infinities' as correctly setting arbitrary constants to match the correct total energy content. There is a brief mention of the idea you refer to in the preface to the Feynman Lectures on Gravitation. Perhaps there are some other pointers there but I don't recall.

Comment by Q.Q.J.

2010-06-18T16:00:06Z

And it gets even more xorsting as the number of disjuncts increases!

Comment by Q.Q.J.

2010-06-17T20:18:44Z

I'll keep it in mind Michael!

Comment by Q.Q.J.

2010-06-17T20:14:04Z

Yes I was making some additional assumptions unthinkingly t3suji, thanks. Maybe I'll have another go if Malik tells us what kind of properties he/she is interested in.

Comment by Q.Q.J.

2010-06-17T15:53:39Z

"infinity minus 1 over infinity" is a great attempt and contains some meaning. Just throw in some compactification and rigorously establish infinitesemals as your daugter's mathematical sophistication grows.