math « Yet Another Mathblog

The enlightenment of Professor Bigglesnot

An enthusiastic “Yippee!” echoed down the corridor. so loud it woke several faculty members in
nearby offices. Some even got up out of their chairs and looked up and down the hallway before
returning to grading or research or freecell before falling asleep again. But Bigglesnot was excited.
After all, computing automorphism semigroups of quantum hyperalgebras was his life’s passion, ever since he was a graduate student. In front of him, was the latest issue of the Quantum Hyperalgebra Journal, newly released from its plastic shrink-wrap. It was opened to the article which was the focus of Bigglesnot’s attention – the esteemed Ziggotwat’s discussion of a new algorithm to compute automorphism semigroups of quantum hyperalgebras. Bigglesnot could see immediately from the tables of new data presented that Ziggotwat’s implementation was faster, better and more general than his own program. Whispering “Awesome! Awesome! So, awesome! …” under his breath, he shot off an email to Ziggotwat asking for more information, and, if at all possible, further details on the implementation. Could he please post or email the code, for others to look at? That would be awesome!

Days and weeks went by, but with no reply. One morning B found an email from Z: “…the code needs to be cleaned up first …”, “… so sorry for the late reply …”, but “.. thanks for your interest!”, was the gist. As luck would have it, B spotted Z a few weeks later at the annual meeting of the Society of Quantum Hyperalgebraists. Undaunted, one night after the talks of the meeting were finished, B bombarded Z with free beer and flattery peppered with questions about his program. “In fact”, Z finally confessed, “all the work was done by my former student Pipperpop, who has graduated and does not reply to my emails. I can send you what I have – but no promises!” Drunk, but now estatic, Bigglesnot managed to say “Awesome!” before he fell off his barstool.

In the months that followed, B pored through the incomplete, undocumented code. It was provided as a sequence of files, each one seeming dependent on another. They wouldn’t compile, no matter how B tried. Each day for a month, after attending to his classes, B would try to modify one of the files, hoping that a small change would allow him to compile the files into a functioning program. Each day, he would draft an email to Z (or to P, or to the editor of QHJ) asking what kind of LSD was he tripping on to make him high enough to think this mess of code would ever compile?
And each day, he wisely deleted the draft. Bigglesnot, usually filled to overflowing with self-confidence, was defeated.

Finally, B realized the solution! Not the solution of to how to compile Pipperpop’s poop, but the solution to the general problem. Software computations submitted to scientific journals must be “open” – if scientific data obtained as a result of a software computation is part of research paper submitted for publication then the source code for that software must also be made public and verified before publication. Enlightened, Bigglesnot was optimistic once again.

This postwas inspired by the excellent paper: Pederson, Empiricism is Not a Matter of Faith, Computational Linguistics, Volume 34, Number 3, pp. 465-470, September 2008.
http://www.d.umn.edu/~tpederse/full-pubs.html

August 18, 2009 Posted by wdjoyner | math, software | open source, sage, software | No Comments Yet

Hadamard’s maximal determinant problem

This blog entry is to remind or introduce to people the fascinating problem called the Hadamard maximal determinant problem: What is the maximal possible determinant of a matrix M whose entries are of absolute value at most 1? Hadamard proved that if M is a complex matrix of order n, whose entries are bounded by $|M_{ij}| \leq 1$ , for each i, j between 1 and n, then
$|det(M)| \leq n^{n/2}$ (equality is attained, so this is best possible for such matrices).

If instead the entries of the matrix are +1 or -1 and the size of the matrix is nxn where n is a multiple of 4, then the problem of the maximal determinant presumably boils down to the well-known search for Hadamard matrices. This is discussed in many books, papers and website but in particular, I refer to

http://en.wikipedia.org/wiki/Hadamard_matrix
http://www.research.att.com/~njas/hadamard/
Hadamard matrices and their applications, by K. J. Horadam
http://www.uow.edu.au/~jennie/hadamard.html

What I think is fascinating is the entries of the matrix are only assumed to be real and not of size 4kx4k. In this case, the maximal value of the determinant is less clear. The results are complicated and depend in a fascinating way on the congruence class of n mod 4. Please see the excellent webpages (maintained by Will Orrick and Bruce Solomon)

http://www.indiana.edu/~maxdet/, and in particular,
http://www.indiana.edu/~maxdet/bounds.html

In particular, the case of an nxn matrix with n=4k+3 seems to be open.

June 27, 2009 Posted by wdjoyner | math | Hadamard matrix problem, math | No Comments Yet

Sage at the NSF-CDI+ECCAD conferences

This is a very quick view of some highlights of the conference http://www4.ncsu.edu/~kaltofen/NSF_WS_ECCAD09_Itinerary.html. I think further details of the talks will appear on the conference webpage later. This is very incomplete – just a few thought I was able to
jot down correctly.

Panel discussion:
Q1: What are the grand challenges of symbolic computing?
Is the term “symbolic computation” to broad? (Hybrid symbolic/numerical, algebraic geometric computation, algebraic combinatorial/group-theoretical, computer proofs, tensor calculations, differential, mathematical knowledge/database research, user interfaces, …)

General ans: No. Hoon Hong points out that user interfaces are lower level but below to the same group.

Q2: How can the latest algorithmic advances be made readily available: google analog of problem formulation? (Idea: suppose someone has a clever idea for a good algorithm but not enough discipline to implement it …)

One answer: Sage can put software together – is this the right way? Analog of stackoverflow.com?

Q3: What is the next killer app for symbolic computation? (Oil app of Groebner bases, cel phone app, robotics, …)

Q4: Can academically produced software such as LAPACK, LINBOX, SAGE compete with commercial software?

Hoon Hong answer: Yes but why? Why not cooperate. Support Sage very much but more research on interfaces and integration of different systems could lead to cooperation of the commercial systems with Sage.

Another panel:
Q: What are the spectacular successes and failures of computer algebra?

Failures:
(a) Small number of researchers.
(b) Sage could fail from lack of lies with the symbolic/numerical community (as Maxima/Axiom did). Matlab may fail due to uphill battle to integrate Mupad into good symbolic toolbox. (Many voiced view that Matlab is strong because of its many toolboxes, on the panel and privately.)
(c) Education at the High School level using CA.
(d) Presenting output of CA intelligently and in a standard format.
(e) Failure to teach people how to properly use a CA system.

Successes:
(a) Sage – interesting new effort (with caveat above)
(b) Groebner bases, LLL.

My talk on Sage raised a lot of questions. My There is both strong support for Sage and some questions on its design philisophy. My page 6 at http://sage.math.washington.edu/home/wdj/expository/nsf-eccad2009/
was a source of lots of questions.

At ECCAD http://www.cs.uri.edu/eccad2009/Welcome.html, Sage was mentioned a few times in talks as well as in some posters. The “main” Sage event was a laptop software demo Which Karl-Dieter Crisman set up for Sage.

Overall, a good experience for Sage.

May 7, 2009 Posted by wdjoyner | math, sage | computer algebra, NSF, sage | No Comments Yet

Sage at AMS-MAA 2009 Washington DC

I think Sage gained a lot of publicity this year both by being at
the booth but also having an MAA panel discussion and an AMS session.
The panel discussion was good to be able to meet others in the
teaching community. I think this is related to Sage development because
projects like the educational open source software webworks has a
funding model which seems to be successful. I think Karl Crisman said he
would try to follow up on that.

A few people I met at the booth said they were interested in Sage development
but more stopped by saying that either they or their students could not afford
Maple or Mma and was looking into a cheaper quality alternative. The collaroration
possibilities of the Sage server was a strong “selling point” for smaller schools
which could load sage on a webserver.

There were some really good talks at the Sage session. For example,
Marshall’s talk had amazing graphics and Robert Miller’s talk was very well attended
(with maybe twice as many people in the audience as some of the others).
I thought the quality overall was great, but I’m very partial to such topics of course.

The general message I got from many was that more written material
on Sage in use would be welcomed, especially books. I was touched by one guy
who explained to me that his students were very poor (waitresses, for example)
who cannot afford calculus texts and commercial math programs. The point
he was implicitly making was that by offering software and documentation for
free we are actually improving the quality of such peoples’ lives in a real way.

January 8, 2009 Posted by wdjoyner | math, sage | sage, sage math | 2 Comments

Future coding theory in Sage projects

Here are a few ideas for future Sage projects in error-correcting codes.

Needed in Sage is

a rewrite in Cython, for speed,
special decoding algorithms, also in Cython,
for special decoders, it seems to me that one either needs

more classes (eg, a CyclicCodes class), each of which will have a decode method, or
another attribute, such as a name (string) which can be used to determine which decoder method to use.

codes over Rings (Cesar Agustin Garcia Vazquez is working on this)
codes defined from finite groups fings – for example split group codes,
a fast minimum distance function (followed by a fast weight distribution function), valid for all characteristics.

It seems more “Pythonic” to add more classes for decoders, but I am not sure.

December 11, 2008 Posted by wdjoyner | math, sage | coding theory, sage | 2 Comments

Steiner systems and codes

A t-(v,k,λ)-design D=(P,B) is a pair consisting of a set P of points and a collection B of k-element subsets of P, called blocks, such that the number r of blocks that contain any point p in P is independent of p, and the number λ of blocks that contain any given t-element subset T of P is independent of the choice of T. The numbers v (the number of elements of P), b (the number of blocks), k, r, λ, and t are the parameters of the design. The parameters must satisfy several combinatorial identities, for example:

$\lambda _i = \lambda \left(\begin{array}{c} v-i\\ t-i\end{array}\right)/\left(\begin{array}{c} k-i\\ t-i\end{array}\right)$

where $\lambda _i$ is the number of blocks that contain any i-element set of points.

A Steiner system S(t,k,v) is a t-(v,k,λ) design with λ=1. There are no Steiner systems known with t>5. The ones known (to me anyway) for t=5 are as follows:

S(5,6,12), S(5,6,24), S(5,8,24), S(5,7,28), S(5,6,48), S(5,6,72), and S(5,6,84).

Question: Are there others?

A couple of these are well-known to arise as the support of codewords of a constant weight in a linear code C (as in the Assmus-Mattson theorem, discussed in another post) in the case when C is a Golay code (S(5,6,12) and S(5,8,24)). See also the wikipedia entry for Steiner system.

Question: Do any of these others arise “naturally from coding theory” like these two do? I.e., do they all arise as the support of codewords of a constant weight in a linear code C via Assmus-Mattson?

Here is a Sage example to illustrate the case of S(5,8,24):

sage: C = ExtendedBinaryGolayCode()
sage: C.assmus_mattson_designs(5)
['weights from C: ',
[8, 12, 16, 24],
‘designs from C: ‘,
[[5, (24, 8, 1)], [5, (24, 12, 48)], [5, (24, 16, 78)], [5, (24, 24, 1)]],
‘weights from C*: ‘,
[8, 12, 16],
‘designs from C*: ‘,
[[5, (24, 8, 1)], [5, (24, 12, 48)], [5, (24, 16, 78)]]]
sage: C.assmus_mattson_designs(6)
0
sage: blocks = [c.support() for c in C if hamming_weight(c)==8]; len(blocks)
759

October 2, 2008 Posted by wdjoyner | math, sage | "block designs", "steiner systems", coding theory, combinatorics, math, sage | No Comments Yet

new Rubik’s cube bound of Tom Rokicki

For some time, Tom Rokicki has been working on lowing the upper bound for the face-turn metric of the Rubik’s cube. His main work (finished in Feb or March 2008) is described in http://tomas.rokicki.com/rubik25.pdf. More recently, John Welborn came up out of the blue and offered Tom CPU time on the “farm” of computers which Sony Imageworks uses to render animated movies. As a result of this generous offer, Tom was able to extend the same method used in the 25-move upper bound to show: in the face turn metric, every position can be solved in <=N moves, where N=20,21,22 (which one, we don’t know yet).

Recently, New Scientist ran a short article on Tom’s result. Here is the link, though the full article requires a subscription: http://www.newscientist.com/channel/fundamentals/mg19926681.800-cracking-the-hardest-mystery-of-the-rubiks-cube.html .

Please see Herbert Kociemba’s cube
performance page for more info.

September 10, 2008 Posted by wdjoyner | math | math, Rubik's cube | No Comments Yet

The Assmus-Mattson Theorem, Golay codes, and Mathieu groups

A block design is a pair $(X,B)$ , where $X$ is a non-empty finite set of $v>0$ elements called points, and $B$ is a non-empty finite multiset of size $b$ whose elements are called blocks, such that each block is a non-empty finite multiset of $k$ points. A design without repeated blocks is called a simple block design. If every subset of points of size $t$ is contained in exactly $\lambda$ blocks the the block design is called a $t$ – $(v,k,\lambda)$ design (or simply a $t$ -design when the parameters are not specfied). When $\lambda = 1$ then the block design is called a $S(t,k,v)$ Steiner system.

Let $C$ be an [ $n,k,d$ ] code and let $C_i = \{ c \in C\ |\ wt(c) = i\}$ denote the weight $i$ subset of codewords of weight $i$ . For each codeword $c\in C$ , let $supp(c)=\{i\ |\ c_i\not= 0\}$ denote the support of the codeword.

The first example below means that the binary [ $24,12,8$ ]-code $C$ has the property that the (support of the) codewords of weight 8 (resp, 12, 16) form a 5-design.

Example: Let $C$ denote the extended binary Golay code of length $24$ . This is a self-dual [ $24,12,8$ ]-code. The set $X_8 = \{supp(c)\ |\ c \in C_8\}$ is a $5-(24, 8, 1)$ design; $X_{12} = \{supp(c)\ |\ c \in C_{12}\}$ is a $5-(24, 12, 48)$ design;and, $X_{16} = \{supp(c)\ |\ c \in C_{16}\}$ is a $5-(24, 16, 78)$ design.

This is a consequence of the following theorem of Assmus and Mattson.

Assmus and Mattson Theorem (section 8.4, page 303 of [HP]):

Let $A_0, A_1, ..., A_n$ be the weight distribution of the codewords in a binary linear [ $n , k, d$ ] code $C$ , and let $A_0^\perp, A_1^\perp, ..., A_n^\perp$ be the weight distribution of the codewords in its dual [ $n,n-k, d^\perp$ ] code $C^\perp$ . Fix a $t$ , $0<t<d$ , and let $s = |\{ i\ |\ A_i^\perp \not= 0, 0<i\leq n-t\, \}|$ .
Assume $s\leq d-t$ .

If $A_i\not= 0$ and $d\leq i\leq n$ then $C_i = \{ c \in C\ |\ wt(c) = i\}$ holds a simple $t$ -design.
If $A_i^\perp\not= 0$ and $d^\perp\leq i\leq n-t$ then $C_i^\perp = \{ c \in C^\perp \ |\ wt(c) = i\}$ holds a simple $t$ –design.
If $A_i^\perp\not= 0$ and $d^\perp\leq i\leq n-t$ then $C_i^\perp = \{ c \in C^\perp \ |\ wt(c) = i\}$ holds a simple $t$ –design.

In the Assmus and Mattson Theorem, $X$ is the set $\{1,2,...,n\}$ of coordinate locations and $B = \{supp(c)\ |\ c \in C_i\}$ is the set of supports of the codewords of $C$ of weight $i$ . Therefore, the parameters of the $t$ -design for $C_i$ are

$t$ = given,
$v = n$ ,
$k = i$ , (this k is not to be confused with dim(C)!),
$b = A_i$ ,
$\lambda = b*binomial(k,t)/binomial(v,t)$

(by Theorem 8.1.6, p. 294, in \cite{HP}). Here is a SAGE example.

sage: C = ExtendedBinaryGolayCode() sage: C.assmus_mattson_designs(5) ['weights from C: ', [8, 12, 16, 24], 'designs from C: ', [[5, (24, 8, 1)], [5, (24, 12, 48)], [5, (24, 16, 78)], [5, (24, 24, 1)]], 'weights from C*: ', [8, 12, 16], 'designs from C*: ', [[5, (24, 8, 1)], [5, (24, 12, 48)], [5, (24, 16, 78)]]] sage: C.assmus_mattson_designs(6) 0

The automorphism group of the extended binary Golay code is the Mathieu group $M_{24}$ . Moreover, the code is spanned by the codewords of weight 8.

References:
[HP] W. C. Huffman, V. Pless, Fundamentals of error-correcting codes, Cambridge Univ. Press, 2003.
[CvL] P. Cameron, J. van Lint, Graphs, codes and designs, Cambridge Univ. Press, 1980.

May 25, 2008 Posted by wdjoyner | math, sage | coding theory, math, sage | No Comments Yet

Some favorite quotes on math, science, learning

There are some things which cannot be learned quickly,
and time, which is all we have,
must be paid heavily for their acquiring.
They are the very simplest things,
and because it takes a man’s life to know them
the little new that each man gets from life
is very costly and the only heritage he has to leave.
Ernest Hemingway
(From A. E. Hotchner, Papa Hemingway, Random House, NY, 1966)

I believe that a scientist looking at nonscientific problems is just as dumb as the next guy.
Richard Feynman

The best thing for being sad is to learn something. That is the only thing that never fails. You may grow old and trembling in your anatomies, you may lie awake at night listening to the disorder of your veins, you may miss your only love, you may see the world about you devastated by evil lunatics, or know your honor trampled in the sewers of baser minds. There is only one thing for it then to learn. Learn why the world wags and what wags it. That is the only thing which the mind can never exhaust, never alienate, never be tortured by, never fear or distrust, and never dream of regretting.
T. H. White in The Once and Future King

Truth, like gold, is to be obtained not by its growth, but by washing away from it all that is not gold. Leo Tolstoy

Education is what survives when what has been learnt has been forgotten.
B. F. Skinner

The advantage is that mathematics is a field in which one’s blunders tend to show very clearly and can be corrected or erased with a stroke of the pencil. It is a field which has often been compared with chess, but differs from the latter in that it is only one’s best moments that count and not one’s worst. A single inattention may lose a chess game, whereas a single successful approach to a problem, among many which have been relegated to the wastebasket, will make a mathematician’s reputation.
Norbert Wiener, in Ex-Prodigy: My Childhood and Youth

Science is a differential equation. Religion is a boundary condition.
Alan Turing

Everything is vague to a degree you do not realize till you have tried to make it precise.
Bertrand Russell

For every complicated problem there is a solution that is simple, direct, understandable, and wrong.
H. L. Mencken

If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.
John Louis von Neumann

To be what we are, and to become what we are capable of becoming, is the only end in life.
Baruch Spinoza

April 21, 2008 Posted by wdjoyner | math | learning, math, quotations, science | No Comments Yet

Math blogs, SAGE, and latex

I only created this wordpress site thinking that I could not type LaTeX on my blogspot page (wdjoyner.blogspot.com). Since I figured out how, following http://wolverinex02.googlepages.com/emoticonsforblogger2, maybe this blog will stay dormant for now.

A latex test:

$\frac{\pi}{4}=\int_0^1 \frac{dx}{1+x^2}$ is smallest