Neutral Drifts

Review of "Sage: Beginner's Guide" by Craig Finch

2011-07-29T08:33:00.000-07:00

I was asked by the publisher to review "Sage: Beginner's Guide" by Craig Finch. They sent me a free copy; I have no other conflicts of interest. I am generally biased towards Sage itself, as an avid user and minor developer.

On Amazon you can browse the table of contents, which gives a pretty good idea of the strengths of the book, namely basic computation and plotting, numerical calculations, and data analysis. The focus was an excellent choice considering what is already available. The current free Sage Tutorial is oriented much more towards pure mathematicians. There is a Numerical Computing With Sage as part of the standard documentation, but at the moment its quite short and nowhere near as helpful as Finch's book.

I liked the style of the book a lot. There are many code examples that illustrate how to accomplish concrete tasks, along with good explanations of what they are doing. Many of these are things that are unfortunately far from obvious to a beginner (or even intermediate) Sage user. Despite using Sage heavily for the last five years, I learned some new things. The book is particularly strong in showing how to use Numpy, Scipy, and Matplotlib. Sage wraps a lot of the functionality of these projects, but if you want to do something that isn't included in the standard interfaces it can be quite mystifying.

Chapter 9, "Learning Advanced Python Programming", might have been a little ambitious. There's nothing wrong with it, but its too short to provide enough. Fortunately there are a lot of good books, some of them free, that cover Python programming in much more depth. I would have preferred some of this space and effort to be devoted to using Cython and the @interact command, which are covered very briefly in Chapter 10.

I teach several classes using Sage and I will definitely advertise this text as a useful optional supplement (I consider it a little too expensive to add on as a mandatory second text). It would be nice if some institutions considered using Sage instead of its commercial competitors such as Maple, Matlab, and Mathematica - you could probably give every student a copy of this book for the money saved from license fees!

The only thing I disliked about the book was the quality of the illustrations. Sage output that was in LaTeX was not typeset, but instead looks as if a PNG was copied from a screenshot. Some of the examples would have benefited from being in color. The quality of the plots is also somewhat poor. This is not too big a deal if one is following along with Sage, since you can reproduce the figures. None of them are bad enough to obscure the content.

Overall this is a very impressive and useful introduction to Sage that should help any beginning user a great deal.

Top Ten Talk Titles at the International Congress of Industrial and Applied Mathematics

2011-07-19T12:58:00.001-07:00

10. The Most Likely Path to Systemic Failure

9. Exploding Rocks

8. Moving Mucus from the Outside In

7. Ducks in Array: Inferring Individual Rules from Collective Behaviour

6. The Nonlinear Dynamics of Jellyfish Swimming

5. The Neuromechanics of Insect Locomotion: How Cockroaches Run Fast and Stably Without Much Thought

4. Transformational Acoustics: Acoustic Cloaks, Carpets and Wormholes

3. A Semi-Implicit Blob Projection Method for Tiny Insect Flight

2. Mathematical Model for Contemplative Amoeboid Locomotion

1. Warping Peirce Quincuncial Panoramas

Ultrafilters

2011-05-16T12:36:00.001-07:00

One of the challenges in learning mathematics is the vocabulary. Its hard not just because of the sheer number of words. Some words make an old word strangely precise, like "continuous". But many others are naming concepts which have no common equivalent, and each of these requires wholly new paths of thought.

Recently I was reading a book on the positivity of multivariate polynomials, and I had to recall what ultrafilters are. Even having learned that before, its a bit of a struggle to internalize. It did inspire me to make the following "ad":

A family of Mobius transformations

2011-04-27T10:53:00.000-07:00

I was trying to sketch the behavior of a Mobius transformation in my complex analysis course today. Its hard to convey on the blackboard, so I tried making a video which shows a homotopy of the image of the unit disk, from the identity to (1+z)/(1-z) and back again.

Sci-fi history in a painting

2011-03-23T09:05:00.000-07:00

This is pretty fantastic - Ward Shelley's "The History of Science Fiction":

(The crop above is just the tip of the iceberg.)

Plotting the zeta function

2011-03-21T08:09:00.000-07:00

I lectured a tiny bit on the Riemann zeta function for the first time in my complex analysis course, which inspired me to make the following plot. Colors are the argument of the Riemann zeta function, brightness is proportional to magnitude. The default brightness map gives very low contrast, so I modified the magnitudes. To help with seeing the magnitudes a contour map with exponentially spaced contours is overlaid.


def xy_to_zeta_size(x,y):
    return abs(zeta(N(x+I*y)))
cvals = [e^i for i in srange(-7,1,.25)]
cp = contour_plot(xy_to_zeta_size,(-6,3),(-3,3),contours = cvals, fill=False, plot_points = 201)
rzeta(z) = zeta(z)/norm(zeta(z))^(.25)
rzf = fast_callable(rzeta,domain=CDF)
cparg = complex_plot(rzf,(-6,3),(-3,3))
show(cparg+cp,figsize=[18,12])

Youtube videos of n-body choreographies

2011-02-28T08:51:00.000-08:00

For a presentation I'm doing tomorrow, I made some animations of n-body choreographies (n equal masses which share a common trajectory). I put these up on youtube since that seems to make more sense than having them mangled by Blogger or hosting them myself:

Twenty-one bodies
Eight bodies
Three bodies on a figure eight

The videos are done using Sage, the Tachyon raytracer, and ffmpeg. The orbit data is from Carlos Simo.

Here's a 2D picture of the twenty-body choreography.

Are you listening, pig?

2010-09-29T06:59:00.000-07:00

I'd just like to share some recommendations that are all connected by the theme of visual and concrete computing. In the last few years I've become more and more convinced that (in the United States at least) our educational system is almost completely failing to provide adequate skills in computing - both the theory and practice of computer science. This is sad because I think there are many fun and interesting ways to learn about programming, logic, and algorithms. Here are some examples:

1. LOGO. When I was in elementary school I was exposed to Logo, a very visual platform for learning programming. If you have a child, make sure you check out an implementation and show your kid. No matter what computer you have, its almost certain that there are several free Logo programs for it.

2. Similar in spirit to Logo, but maybe more fun and less deep is the game Light-Bot at Armor Games. You can drag and drop visual program elements to write a program that instructs a robot to complete a variety of puzzles. Very fun, well done.

3. There are novels that glamorize hackers and computing to some extent, but nothing I've seen compares to the plot in Souls in the Great Machine by Sean McMullen. The plot is brilliant and complicated, but all you need to know is that they enslave all the mathematicians to create a giant human computer.

4. I was reminded of Eyes of the Calculor by this fantastic video on youtube. The creator has implemented a gigantic arithmetic unit within the virtual Minecraft environment. He mentions he was inspired by the book The Elements of Computing Systems: Building a Modern Computer from First Principles. I hadn't heard of that before but I will definitely take a look. If you watch the video you'll get the title of this post.

Torus knots in Sage with Tachyon and @parallel

2010-09-12T16:04:00.000-07:00

After working a bit more on a patch improving the Tachyon raytracer in Sage, I made a movie of a torus knot. The code below was parallelized to use 20 cores at a time on one of the Sage Foundation's lovely Sun Fire X4450s. Frames were animated with ffmpeg. Blogger might degrade the video, so I recommend downloading the original here.


pk = 3
qk = 23
res = 180
trad = .5
lab = 'tk_' + str(pk) + '_' + str(qk)
def tknot(theta):
    return [(2 + trad*cos(qk*theta/pk))*cos(theta), \\
(2 + trad*cos(qk*theta/pk))*sin(theta), trad*sin(qk*theta/pk)]
@parallel(ncpus = 20)
def tknotter(anum):
    th = anum*2.0*pi/res
    T = Tachyon(xres = 800, yres = 600, camera_center = (4,4,2.5), \\
raydepth=12, antialiasing=2)
    loc = tknot(th)
    nloc = [abs(x)/norm(vector(loc)) for x in loc]
    T.texture('s1',color=nloc,specular=1)
    T.texture('p1',color=(.5,.5,.5))
    T.sphere(loc,.05,'s1')
    ci = 0
    for sth in srange(0,th,2*pi/res):
        e1 = tknot(sth)
        e2 = tknot(sth+2*pi/res)
        ts = [abs(x)/norm(vector(e1)) for x in e1]
        T.texture('t'+str(ci), color=ts)
        T.fcylinder(e1, e2, .025,'t'+str(ci))
        ci += 1
    T.plane((0,0,-4),(0,0,1),'p1')
    T.light((6,6,11),.1,(1,1,1))
    T.light((5.9,5.9,11),.1,(1,1,1))
    T.save(DATA+lab+'%03d.png'%anum)
    return 1

qin = range(pk*res)
s = list(tknotter(qin))

for i in range(40):
    r2 = os.system('cp '+DATA+lab+'%03d.png '%(pk*res-1) + DATA+lab+'%03d.png'%(pk*res+i))

anum = qin[-1] + 41
th = anum*2*pi/res
T = Tachyon(xres = 800, yres = 600, camera_center = (5,5,3))
T.save(DATA+lab+'%03d.png'%anum)

The animation was then done with the following code; if you want to do something similar you would have to use the appropriate path to your ffmpeg executable.


import subprocess
subprocess.call('/home/mhampton/ffmpeg/ffmpeg -qmax 2 -i ' + \\
DATA + 'v%3d.png ./t2vest.mp4', shell=True, stdout=file('/dev/null','w'),stderr=file('/dev/null','w'))

Flight in Curved Spaces

2010-09-12T08:31:00.000-07:00

I was recently reminiscing about the movie Not Knot, and began thinking about recreating some of the hyperbolic scenes. But then today I stumbled upon a really nice application that does exactly what I was thinking about and more:

http://www.geometrygames.org/CurvedSpaces/index.html

Its a pretty small download, works on most platforms, very fun.

Tachyon in Sage: depth of field perspective

2010-09-03T20:48:00.000-07:00

Today I found a bug in the options to the tachyon raytracer in Sage. It should be possible to use alternate projections, including fisheye lens and perspective depth-of-field projections. I think I've fixed that, patch should get into sage-4.6 or so.

Here's a quick example:


T = Tachyon(xres=800,antialiasing=24, raydepth=12, projection = 'perspective_dof', focallength = '1.0', aperture = '.005')
T.light((0,5,7),1.0,(1,1,1))
T.texture('t1', opacity=1, specular = .3)
T.texture('t2', opacity=1, specular = .3, color = (0,0,1))
T.texture('t3', opacity = 1, specular = 1, color = (1,.8,1), diffuse=0.2)
T.plane((0,0,-1),(0,0,1),'t3')
ttlist = ['t1','t2']
tt = 't1'
T.cylinder((0,0,.1),(1,1/3,0),.05,'t3')
for q in srange(-3,100,.15):
    if tt == 't1':
        tt = 't2'
    else:
        tt = 't1'
    T.sphere((q,q/3+.3*sin(3*q),.1+.3*cos(3*q)), .1, tt)
T.show()

Glack

2010-07-25T08:39:00.001-07:00

Glack: n. a ravine, a defile; the fork of a tree, road, etc.; the angle between the thumb and forefinger; an opening in the wood where the wind blows briskly; a handful or small portion; as much grain as a reaper holds in his left hand; a slight repast.

Courtesy of Warrack's Scot Dialect Dictionary, 1911.

Book review: The Manga Guide to Molecular Biology

2010-05-09T09:53:00.001-07:00

After getting the Manga Guides to Statistics and Databases, I recently bought the Manga Guide to Molecular Biology. All of these guides are very basic, so if you already know a substantial amount you won't learn anything new, but I think they can be useful for teachers and for raw beginners.

The molecular biology guide does a good job explaining the classic, core ideas: DNA replication, transcription, translation, and some of the biochemical structures. To give an idea of the level of detail, Okazaki fragments, the structure of transfer RNA, and introns and exons are included, but not poly-adenylation. Some topics, like micro-RNA, are briefly described in sections that are in a more traditional textbook style.

Perhaps the best use of this book is as a gift to someone to encourage an interest in molecular biology - for example, a high school student relative.

A suspicious histone of Symbion pandora

2010-05-01T10:24:00.000-07:00

Recently on the blog Neatorama they highlighted a very strange creature, a parasite of the lobster that is pretty unrelated to other animals - Symbion pandora. Its got a rather complicated reproductive cycle, as described in the links below.

The articles mention that it was so unusual that a new phylum was proposed, the Cycliophora. I was curious about this so I looked up what sequences are available for Symbion pandora. There are some ESTs that are in the trace read archive, un-annotated as far as I can tell, and about 100 nucleotide records. Most of those are for RNA subunits, or cytochrome oxidase subunits, but there is one histone record. I've been reading about histones quite a bit recently so I BLASTed that record. It seems very suspicious to me that all the top hits were to lobsters (see alignment below, the top hit. Nephrops norvegicus is the Norway lobster). I strongly suspect that this was contaminated, and is not a correct Symbion pandora sequence.

The Neatorama link
More info is available from the New Scientist article that Neatorama links to.

Biological ontologies

2010-03-17T19:19:00.000-07:00

I've postponed really learning about ontologies in bioinformatics for a while now, but circumstances now dictate that I figure it out. It is a little bewildering. As just one example, the current Gene Ontology has the following processes as the top-level categories of "biological processes":

biological adhesion

biological regulation

carbohydrate utilization

carbon utilization

cell killing

cell wall organization or biogenesis

cellular component biogenesis

cellular component organization

cellular process

death

developmental process

establishment of localization

growth

immune system process

localization

locomotion

metabolic process

multi-organism process

multicellular organismal process

nitrogen utilization

phosphorus utilization

pigmentation

reproduction

reproductive process

response to stimulus

rhythmic process

signaling

signaling process

sugar utilization

sulfur utilization

viral reproduction

Most of these are reasonable, some seem odd. Pigmentation? A top-level process?

It reminds me of the fabulous first paragraph of Michel Foucault's "The Order of Things" (in my opinion his best book):

This book first arose out of a passage in Borges, out of the laughter that shattered, as I read the passage, all the familiar landmarks of my thought - our thought, the thought that bears the stamp of our age and our geography - breaking up all the ordered surfaces and all the planes with which we are accustomed to tame the wild profusion of existing things, and continuing long afterwards to disturb and threaten with collapse our age-old distinction between the Same and the Other. This passage quotes a ‘certain Chinese encyclopedia’ in which it is written that ‘animals are divided into: (a) belonging to the Emperor, (b) embalmed, (c) tame, (d) sucking pigs, (e) sirens, (f) fabulous, (g) stray dogs, (h) included in the present classification, (i) frenzied, (j) innumerable, (k) drawn with a very fine camelhair brush, (1) et cetera, (m) having just broken the water pitcher, (n) that from a long way off look like flies’. In the wonderment of this taxonomy, the thing we apprehend in one great leap, the thing that, by means of the fable, is demonstrated as the exotic charm of another system of thought, is the limitation of our own, the stark impossibility of thinking that.

weather maps

2010-03-04T08:32:00.000-08:00

Anyone who reads this blog, please chime in with your favorite weather maps or sites. I'm particularly interested in big, high-res satellite animations for the US. Best one I've found is here.

complex plot in sage

2010-02-27T19:37:00.001-08:00

I've been amusing myself all day with doing some complex plotting in Sage. The default color scheme has 0=black, infinity=white, and red=real. A lot of the time, this doesn't work too well, but if you plot f/abs(f) the color will just depend on the complex argument. For example, if f = (z^5+1)/(z^5-1) we get:

Or a more baroque example, f = z*log(z)*exp((z^5+1)/(z^5-1)^(1/2)):

Sometimes the default coloring is helpful, for example when showing the essential singularity at 0 of exp(1/z):

National Merit Corporation bullies blogger

2010-02-08T14:25:00.000-08:00

I just read an appalling story of a blogger being bullied by the National Merit Scholarship Corporation. She posted their cutoff PSAT scores for each state (I hadn't realized that they varied at all) and was threatened with legal action if she didn't take down the information.

Not surprisingly, after hearing about this someone else compiled the same information, and all the corporation has done is highlight their questionable judgement in a variety of ways.

GeoGebra and Sage

2009-11-12T07:57:00.000-08:00

Looks like there is some momentum building to integrate GeoGebra applets into the Sage notebook, which I think would be fantastic. I've been lazy about checking out GeoGebra until now, despite hearing good things about it. Dan Drake's demonstration of a nice applet made in less than an hour of experimenting inspired me to give it a try. In ten minutes I had downloaded GeoGebra, installed it, made a simple demo and moved it to my university account. I am very impressed, and hope to help a little to get this into Sage.

Plucker's Quartic Curve

2009-10-31T21:42:00.000-07:00

A quartic curve in the plane has a maximum of 28 real bitangents. Plucker came up with the following sort of example that attains that bound, which I illustrated in a particular case with phcpack and Sage:

More polytopal animations

2009-09-18T20:21:00.000-07:00

I was inspired by a still image from the Wikipedia entry on the 24-cell to make an animation of rotating and then stereographically projecting the 24-cell, showing the images of the original polytope edges. Math on Wikipedia is pretty spotty, but the polytope entries have really become top-notch in the last few years.
Here's a still from the animation:

and a medium and small resolution version of the animation:

small (300x300, 6 MB)
medium (512x512, 17 MB).

600-cell animations

2009-09-03T12:12:00.000-07:00

Over the last year or so, when I have time, I've been trying to fine tune ray-traced animations of polytopes (using Sage, Tachyon, ffmpeg, and cddlib). My latest effort was to do a loop of Schlegel projections of the beautiful 600-cell. The 600-cell is one of the six regular polytopes in 4 dimensions; its usually considered the 4D analog of the icosahedron. Here's a still from my animation:

Since Blogger's re-encoding butchers my movies, I will just post links. Here are four sizes:

small (320x200, 3.5 MB)
medium (640x400, 7 MB)
large (1280x800, 13 MB)
large, high-res (1280x800, 33 MB).

Twenty Four Cell, take 2

2009-07-24T12:36:00.000-07:00

Hmm...blogger re-encodes my video and for some reason their version looks like crap, so I won't even leave it up. I think my new attempt has much better lighting, and the faces have translucent panes now. But here is a still shot:

Three sizes of the new version:
small (320x200)
medium (640x400)
large (1280x800).

Witch of Maria Agnesi

2009-07-23T15:13:00.000-07:00

There's one on wikipedia like this, but for an upcoming class I wanted to do it in Sage:


xtreme = 4.1
myaxes = line([[-xtreme,0],[xtreme,0]],rgbcolor = (0,0,0))
myaxes = myaxes + line([[0,-1],[0,2.1]],rgbcolor = (0,0,0))
a = 1.0
t = var('t')
npi = RDF(pi)
def agnesi(theta):
    mac = circle((0,a),a,rgbcolor = (0,0,0))
    maL = line([[-xtreme,2*a],[xtreme,2*a]])
    maL2 = line([[0,0],[2*a*cot(theta),2*a]])
    p1 = [2*a*cot(theta),2*a*sin(theta)^2]
    p2 = [2*a*cot(theta)-cot(theta)*(2*a-2*a*sin(theta)^2),2*a*sin(theta)^2]
    maL3 = line([p2,p1,[2*a*cot(theta),2*a]], rgbcolor = (1,0,0))
    map1 = point(p1)
    map2 = point(p2)
    am = line([[-.05,a],[.05,a]], rgbcolor = (0,0,0))
    at = text('a',[-.1,a], rgbcolor = (0,0,0))
    yt = text('y',[0,2.2], rgbcolor = (0,0,0))
    xt = text('x',[xtreme + .1,-.1], rgbcolor = (0,0,0))
    #tt = text('t',[.15,.1], rgbcolor = (0,0,0))
    matext = at+yt+xt
    ma = mac+myaxes+maL+am+matext+maL2+map1+maL3+map2
    return ma

def witchy(theta):
    ma = agnesi(theta)
    agplot = parametric_plot([2*a*cot(t),2*a*sin(t)^2],[t,.001,theta], rgbcolor = (1,0,1))    
    return ma+agplot

a2 = animate([witchy(i) for i in srange(.1,npi-.1,npi/60)]+[witchy(i) for i in srange(npi-.1,.1,-npi/60)], xmin = -3, xmax = 3, ymin = 0, ymax = 2.3, figsize = [6,2.3], axes = False)

a2.show()

24 cell movie

2009-07-15T13:42:00.000-07:00

This is basically a test of my workflow for a more ambitious project, but its somewhat amusing in its own right. The original 48 MB version is here (much better quality).

Michael Corral's Calculus 3 book

2009-07-07T16:33:00.000-07:00

Last year I became aware of the fantastic free calc 3 book by Michael Corral. His website for it is here. I wanted to be able to compile the latex source files myself but couldn't get it to work on my mac or windows machines. I just gave it another try and finally succeeded on OS X.

Besides using his calc3book.sh script, I found it necessary to:

1) install pgf/tigz.
2) put picins.sty in my tex directory
3) Put the metapost files from m3Dplain in my tex directory.

Its a beautifully done book, so I am excited to learn some of the tricks Corral uses.

Hearing a trigonometric identity

2009-07-06T05:42:00.000-07:00

If you force a harmonic oscillator with natural frequency w_0 with a sinusoidal force of frequency w, the resulting steady-state is a linear combination of those two frequencies. If its something in the audible range, you hear either the two seperate frequencies or one frequency with a beat if they are close together. In other words, you hear either the right or left hand side of the identity shown below.

Sage source code available here. Ironically, because of my use of html I can't seem to show the source directly here.

Small frequency difference.

Smallish frequency difference (close to the Ma Bell tone).

Intermediate frequency difference.

Large frequency difference - you definitely hear it as a 2-note chord.

symmetry in chaos

2009-06-24T18:31:00.000-07:00

Recently SIAM had a deal to get some books from their catalog cheap if you bought the new edition of "Symmetry in Chaos" by Field and Golubitsky. So I did, and couldn't help but try to reproduce some of their figures. I think they might have a typo in their parameters for their figure 2.3 (or I am making some mistake, quite likely), but after exploring a bit with other parameters I got the following, with iterates of

Mixmead

2009-06-13T10:16:00.000-07:00

Recipe:

1.5 lbs Late Summer Wildflower Honey (Talking Oak Farm, WI)
3 lbs Northern Brewer Orange Blossom Honey (CA)
3 lbs Skalko's Honey Bee Farm Honey
3 lbs Duluth Whole Foods Coop (not the chain) Honey
1 cup Northern Brewer Gold Malt Extract
1 handful o' hops from Grant Anderson
Lalvin EC-1118 S.bayanus Champagne Yeast
...after 2 months added 1/2 cup of Adro Hungarian Forest Honey (great stuff)
...after 7 months added 1/4 cup of brown sugar and 2 Ceylon teabags
...bottled after 9 months with another 2 bags of Chai Black Tea.

All the tea is to offset the sweet taste - the champagne yeast should have gobbled it up but at the 7 month point it was still too cloying for me. Now it tastes pretty good...

Two recipes: beer and biscotti

2009-05-31T15:12:00.000-07:00

I tried two experiments this weekend, a beer and a variation on a biscotti recipe.

I won't know how good the beer is for another 2 months or so. Here is the ingredient list:

Wayward Monk Ale

6 lbs Northern Brewer Amber Malt Extract

3.15 lbs Northern Brewer Dark Malt Extract

1 1b Dark Belgian Candi Sugar

Wyeast Belgian Abbey II yeast (#1762)

1 oz Fuggle hops

1 oz Saaz hops

1 oz Hallertau Mittelfruh hops

1 gm of Myrica gale

1 pinch of Wormwood

1 lbs Belgian Biscuit Malt

1 lbs Franco-Belges Kiln Coffee Malt

The last two ingredients I forgot to order, and they will have to be added later. I'm not sure what effect that will have. I'm also not sure what this beer will be like - its kind of a winter-warmer/scotch ale/Belgian abbey hybrid. But that's why I homebrew - why make something you can buy in a store?

The biscotti is already a clear success. The recipe is essentially from Baking Illustrated, which I highly recommend.

2 cups of flour (I like to use some whole wheat in there)

1 tsp baking powder

1/2 tsp salt

3 tbsp unsalted butter

1 cup sugar (I like to use some portion of brown sugar)

2 eggs

1/2 tsp vanilla extract

2 oz sliced almonds, roasted briefly in a pan

1/2 cup of cocoa nibs (I used the organic Dagoba ones, which are the only thing I have ever seen in a supermarket)

You make the dough into two loaves, cook for about 25 minutes at 350 F. Take it out and slice into strips, then rebake at 325 for about 15 minutes. Its very easy but seems to impress people as much as much more difficult baked goods.

Polytopes in Sage, take 1

2009-05-25T12:13:00.000-07:00

Here's a first attempt at a short video intro to polytopes in Sage. Sage 4.0, which should be released soon, has some new functionality that was added during Sage Days 15 (thanks to David Perkinson for reviewing some of that, and Rob Beezer for helping David learn the review ropes).

The video is about 4 MB and can be found here. I am not sure how the format will work on other computers so if anyone takes a look and it doesn't work let me know. Also, I am interested in getting constructive criticism on the video.

boolean Adelaide

2009-05-12T11:06:00.000-07:00

Just a little follow-up post to my Adelaide Venn diagram post. I wanted to double-check its correctness, so I made this diagram showing whether each set was included or not. Color by numbers!

Protein of the Day #20: Sirtuin1

2009-05-04T09:20:00.001-07:00

This week there is an set of articles in Science on the protein Sirtuin1. The sirtuin family is very interesting, involved in ageing, epigenetic gene silencing, and other cellular regulation. The Science articles provide some new links of Sirtuin1 to circadian rhythms and matebolism.

Here's a static shot of the common 3D domain of the sirtuin superfamily:

The NAD+ dependence of Sirtuin1 seems to be an important link between the circadian system, the metabolic state of the cell, and genetic regulation. As always, I am wondering if there are connections here to mammalian hibernation...

Slouching towards Adelaide

2009-04-30T20:03:00.001-07:00

At some point when I was in grad school I became aware of some work on symmetric Venn diagrams. If you google this, you will find this link, which has been maintained but not changed too much since 1997. Other than that, there isn't a lot on the web and there is a particular lack of quantitative direction on how to construct the beautiful rotationally symmetric Venn diagrams such as Adelaide. This was named by Anthony Edwards after the city in which he discovered it. I have always wanted to go to Adelaide, it holds a strange attraction for me, so perhaps that is why that particular 7-set Venn diagram stuck in my head.

When I started working on my coloring book, I immediately thought of the Adelaide diagram but I didn't know how to construct it. After some mistakes today, I think I finally have it down. Here is a colored version:

Code (in Sage) for some version of this will be in the final coloring book.

Mixmead

2009-04-28T20:48:00.000-07:00

I've had a mead going for quite a while now - started on September 6th, 2008 - which I just racked for a second time. Its proving to be a troublesome brew, perhaps because I never had a clear vision of what it was supposed to be. I have been calling it the "Mixmead" since its a hodgepodge of all sorts of honey that I was able to get in 2008. Here's the exact recipe:

1.5 lbs Late Summer Wildflower Honey (Talking Oak Farm, WI)

3 lbs Northern Brewer Orange Blossom Honey (CA)

3 lbs Skalko's Honey Bee Farm Honey (Esko, MN)

3 lbs Duluth Whole Foods Coops Honey (God Knows Where, USA)

1 cup Northern Brewer's Gold Malt Extract (for protein, rather than that nasty chemical mead nutrient)

1 handful o'hops from Grant Anderson (thank you Grant!)

Lalvin EC-1118 Sabayanus Champagne yeast (Champagne yeast:beer yeast::SWAT team:suburban cop)

I racked this on November 16th, 2008, and added 1/2 cup of Hungarian Adro Forest Honey (fantastic stuff). Tasted too sweet then, despite the commando-like efforts of the champagne yeast.

This second racking, I added a tiny bit (3/16 cup) of brown sugar and 2 heavily steeped tea bags of some rather bitter Ceylon tea. I hope the tea will cut the excessive sweetness a bit (this is a pretty traditional thing to add to meads).

If all goes well, this will be bottled in a month.

Cayley Cubic

2009-04-21T16:37:00.000-07:00

To further my polynomial education I've been thinking some lately about the Cayley cubic, which in the affine form I was using is given by x^2 + y^2 + z^2 + z(y^2-x^2) -1 = 0.

I made a little movie of some of the real solution set of the Cayley cubic. Here's one of the stills:

I think the quicktime version works better in browsers, or you can download a slightly higher-quality mp4.

Protein of the Day #19: Fto (Fatso)

2009-04-20T09:03:00.000-07:00

Protein names are funny things. Often they are chosen before much of the functionality of the protein is characterized, which means they are usually misleading. I wonder how much progress in biology has been retarded over the years by that simple fact.

The abbreviation Fto comes from "fused toes" mutation, but was also initially called Fatso because it is a relatively large protein (about 500 amino acids, nothing close to the huge ones like titin). Ironically, this name is quite important since it seems that Fto is very important in energy homeostasis in mammals. It is highly expressed in the arcuate nucleus. There is some correlation between Fto mutations and susceptibility to diabetes. Because of all that, the preferred descriptive name now is "fat mass- and obesity-associated gene".

Protein of the Day #18: Cruciferina

2009-04-17T11:46:00.000-07:00

Cruciferina: sounds like an all-girl punk group or something. I couldn't find out too much about it, its a cupin-superfamily protein that is a desiccation-tolerant seed storage globulin. "Cupin" comes from the Latin "cupa" for small barrel; cupin proteins have a beta-barrel motif that apparently is useful for all sorts of things. Here's one representation of the barrel (not from Cruciferina in particular) from topsan.org:

Various cupin proteins seem to heavily expressed in the floral nectary tissues studied in my friend Clay Carter's lab.

Books I am reading right now

2009-04-15T16:10:00.000-07:00

Fiction:

0a. Anathem, by Neal Stephenson. I finished this one, but I am including it because I liked it so much. Not that Neal needs much advertising.

0b. The Algebraist, by Iain Banks. Fantastic cover, at least the US paperback. I finished this while in South Africa too, but it was one of my favorites this year.

1. The Sacred Book of the Werewolf, by Victor Pelevin. Trippy, odd, hard to describe - but good.

2. Wizard of the Crow, by Ngugi Wa'Thiong'O. Sort of like 100 Years of Solitude in Africa.

3. Victory of Eagles, by Naomi Novik. Good series.

4. Turn Coat, by Jim Butcher. I think I have read everything by him, he doesn't disappoint.

Non-fiction (I am leaving out a lot of math and bio books):

1. Imagining India, by Nandan M. Nilekani. Very good book about modern India, where its going and how it got there.

2. The Manga Guide To Statistics, by Shin Takahashi. Surprisingly good. The Manga Guide to Databases is also pretty good.

Curves of Pursuit

2009-04-15T13:26:00.000-07:00

When I visited Stellenbosch University in February, I noticed the math department had some nice posters on various mathematical topics that I thought could be implemented in Sage. The only one I did was "Curves of Pursuit", in which one has n points in a regular n-gon, and each point chases the next. The solutions are prettier when discrete time steps are taken. You can see the code at

http://wiki.sagemath.org/interact/graphics#CurvesofPursuit

Protein of the Day #17: mTOR

2009-04-14T09:11:00.001-07:00

mTOR (mammalian target of rapamycin) does a whole lot of things, as the attempted pathway image below shows (from Wikipedia, which I am not ashamed to link to for biochemistry). My interest in it comes from its role in energy balance, particularly in synthesizing information in the hypothalamus. Since I am primarily focused on mammalian hibernation, the fact that mTOR is a serine-threonine kinase is intriguing - its known that transcription and protein translation are shut down during mammalian torpor bouts so the control mechanisms are likely to involve post-translational protein modifications such as phosphorylation. Perhaps mTOR is involved in these mechanisms.

5-body woes

2009-04-13T11:53:00.000-07:00

I've been working for the last few weeks with Anders Jensen, Eduardo Leandro, Gareth Roberts, John Little, Manuele Santoprete, and Alain Albouy on some celestial mechanics problems. While I was in Goettingen, Anders and I tried hard to get some results on central configurations in the five-body problem. Unfortunately, we hit one very, very nasty special case in our finiteness proof, which has me musing over a couple of big polynomials that start out looking like this:

Ouch!

Back from CA and Goettingen

2009-04-12T08:57:00.000-07:00

Blog's not dead yet...just took a 2 month hiatus. I started feeling guilty about not posting, which doesn't make much sense since its my own blog and it has very few regular readers - even fewer now!

Since my wife now has a much nicer camera for her work, I can bring her small one with me on trips. Its too bad I didn't have it in South Africa, but at least I had it in Goettingen last week where I could take a few pictures of the famous goose girl:

Carl Witty does Sage on the G1

2009-02-19T13:03:00.000-08:00

"the entire compile took 20943 minutes of real time (about
14.5 days)"

This is just heroic, borderline legendary: compiling Sage on the Android G1. There is nothing to be done but read the post from sage-devel and see the picture.

I have compiled and run Sage 3.2.3 on my T-Mobile G1 cell phone, and
large portions of it actually work.

300 files had failing doctests; this means that all doctests passed in
864 files. A lot of the failing doctests are with pexpect (maxima,
gap, etc.); I don't know why these fail. When I try simple things
with gap and maxima, they do work. Many more tests fail due to
timeouts; the 300-second timeout is far too short for the G1.
(Doctesting any file with doctests takes >50 seconds; doctesting a
file with no doctests takes >4 seconds. The entire doctest run took
270724 seconds, or a little over 3 days.)

Unfortunately, it's far too slow to be useful for anything (just
starting Sage takes about a minute, if there's enough free memory; but
since Android likes to keep the memory full all the time, there never
is enough free memory, and it takes much longer to start). So I don't
plan to do anything further with the project.

Of course, the sensible way to do this is to find some nice fast
computer and set up a cross-compiler. I didn't do this the sensible
way; I actually did the build on the cell phone. And of course, the
build would stop every once in a while due to a bug, which I would
then patch around. So, adding up the 14 chunks of compilation
involved, the entire compile took 20943 minutes of real time (about
14.5 days) and 9438 minutes (about 6.5 days) of CPU time. You'll note
that the CPU time adds up to less than half of the real time; most of
the rest of the time was spent swapping. The G1 has about 100MB of
RAM; I set up a swap file on my micro-SD card, to allow the build to
proceed at all. On several files, the compiler uses more than 300MB.
While compiling those files, the CPU is typically less than 1% active;
I'm pretty sure that there were files that took more than a day to
compile. The phone was essentially unusable for smartphone activities
(web browsing, etc.) while the build was running. In fact, the whole
smartphone UI crashed and restarted on a fairly regular basis during
the build (I'm guessing it has some sort of watchdog timer and reboots
itself if it detects that some operation is taking an unreasonably
long time), but the build just continued anyway (running inside a
screen session).

The ATLAS tuning process took about 3.2 days (with almost as much CPU
time as real time; that didn't swap much). Skimming through the logs,
I see reports of numbers like 3 to 6 MFLOPs.

The build was performed inside a Debian armel testing chroot; it's
pretty nice being able to run real Debian on my cell phone.
(Everything works; I can ssh in and out, etc., although when it's on
the cell phone network, inbound connections are problematic because
it's behind a NAT.)

To make the whole project even more pointless, since the phone is
running real Debian (inside the chroot), I can probably just wait a
few weeks or months for Tim Abbott to get all the portability issues
fixed in the sagemath Debian package, and then install Sage on the
phone with apt-get.

Here's the processor:

cwitty@localhost:/tmp$ cat /proc/cpuinfo
Processor : ARMv6-compatible processor rev 2 (v6l)
BogoMIPS : 383.38
Features : swp half thumb fastmult edsp java
CPU implementer : 0x41
CPU architecture: 6TEJ
CPU variant : 0x1
CPU part : 0xb36
CPU revision : 2
Cache type : write-back
Cache clean : cp15 c7 ops
Cache lockdown : format C
Cache format : Harvard
I size : 32768
I assoc : 4
I line length : 32
I sets : 256
D size : 32768
D assoc : 4
D line length : 32
D sets : 256

Hardware : trout
Revision : 0080
Serial : 0000000000000000

Carl

Live from Cape Town

2009-01-25T04:55:00.000-08:00

I arrived at AIMS (the African Institute for Mathematical Sciences) yesterday. Its a great place, right on the Muizenberg beach near Cape Town - quite a thrill after being in Duluth this time of year.

I have volunteered to teach a short course on bioinformatics algorithms here, using biopython and Sage as the computational platform. It will be very interesting to see how it goes over. I am using the Plasmodium species which cause malaria as a source of examples and projects.

Below is a satellite picture of the cape peninsula and a view of Muizenberg that includes the main AIMS building.

Integral Apollonian Packings with Sage

2009-01-22T20:01:00.000-08:00

At the national math meetings this year I heard about some really interesting and fun work on integral Apollonian circle packings. The AMS has a nice introductory article about them. I couldn't resist trying to compute and draw some in Sage. Carl Witty greatly improved the speed of my first attempt, so what follows can be considered joint work of ours. (Sage can compute nicer PDF output of these, but blogger doesn't embed PDFs.)


def kfun(k1,k2,k3,k4):
    """
    The Descartes formula for the curvature of an inverted tangent circle.
    """
    return 2*k1+2*k2+2*k3-k4

def circfun(c1,c2,c3,c4):
    """
    Computes the inversion of circle 4 in the first three circles.
    """
    newk = kfun(c1[3],c2[3],c3[3],c4[3])
    newx = (2*c1[0]*c1[3]+2*c2[0]*c2[3]+2*c3[0]*c3[3]-c4[0]*c4[3])/newk
    newy = (2*c1[1]*c1[3]+2*c2[1]*c2[3]+2*c3[1]*c3[3]-c4[1]*c4[3])/newk
    if newk > 0:
        newr = 1/newk
    elif newk < 0:
        newr = -1/newk
    else:
        newr = Infinity
    return [newx, newy, newr, newk]

def mcircle(circdata, label = False, thick = 1/10):
    """
    Draws a circle from the data.  label = True
    """
    if label==True and circdata[3] > 0 and circdata[2] > 1/2000:
        lab = text(str(circdata[3]),(circdata[0],circdata[1]), fontsize = \
500*(circdata[2])^(.95), vertical_alignment = 'center', horizontal_alignment \
= 'center', rgbcolor = (0,0,0))
    else:
        lab = Graphics()
    circ = circle((circdata[0],circdata[1]), circdata[2], rgbcolor = (0,0,0), \
thickness = thick)
    return circ+lab

def add_circs(c1, c2, c3, c4, cutoff = 300):
    """
    Find the inversion of c4 through c1,c2,c3.  Add the result to circlist,
    then (if the result is big enough) recurse.
    """
    newcirc = circfun(c1, c2, c3, c4)
    if newcirc[3] < cutoff:
        circlist.append(newcirc)
        add_circs(newcirc, c1, c2, c3)
        add_circs(newcirc, c2, c3, c1)
        add_circs(newcirc, c3, c1, c2)

zst1 = [0,0,1/2,-2]
zst2 = [1/6,0,1/3,3]
zst3 = [-1/3,0,1/6,6]
zst4 = [-3/14,2/7,1/7,7]

circlist = [zst1,zst2,zst3,zst4]
add_circs(zst1,zst2,zst3,zst4)
add_circs(zst2,zst3,zst4,zst1)
add_circs(zst3,zst4,zst1,zst2)
add_circs(zst4,zst1,zst2,zst3)

sum([mcircle(q, label = True, thick = 1/2) for q in \
circlist]).save('./Apollonian3.png',axes = False, figsize = [12,12], xmin = \
-1/2, xmax = 1/2, ymin = -1/2, ymax = 1/2)

Protein of the Day #16: Maltase (plus a beer recipe)

2009-01-22T07:22:00.000-08:00

I just bottled a batch of my Chocolate Cake Stout, which is:

1 lb of Simpon's roasted barley
1 lb of crisp amber malt
6 lbs gold malt syrup (from Northern Brewer)
1 oz Yakima Magnum hops (13.5 % Alpha acid)
Safale S-04 yeast
8 oz Dutch process cocoa (from Penzey's Spices)
1 lb lactose sugar
1 whole vanilla bean (also from Penzey's).

Good stuff.

Maltase (alpha-D-glucosidase) is a crucial enzyme in beer making, and its regulation in response to other sugars is important to understand if you really want to grok brewing. Maltose is a disaccharide of two units of glucose; in most organisms it needs to be broken down to glucose before being used in other biochemical processes. The presence of glucose inhibits the production of maltase and maltose transporter proteins, so the glucose in a wort has to be mostly used up before the yeast will switch to consuming the maltose.

Sage at the joint math meetings

2009-01-06T19:13:00.000-08:00

Live from DC, we're halfway throught the joint mathematics meetings. This is the second year that Sage has had a booth in the exhibit hall.

Our location is not as good as last year's, but we are still getting a good amount of foot traffic - enough to occupy most of the people here most of the time. David Harvey, David Joyner, Robert Miller and I have been manning it most of time. Jason Grout helped out this afternoon quite a bit. I may be forgetting some folks because I'm a little frazzled.

The word of mouth about Sage is increasing, many people stopping by have tried it or at least heard about it. I think its very encouraging for the future. Several people expressed interest in how to become developers.

We had an MAA panel session today on open-source software in the undergraduate curriculum. Four speakers, two of which (David Joyner and Robert Miller) talked about Sage (others on R and WeBWork).

Overall, so far the booth has been another success in raising awareness and enthusiasm for Sage. Talking to different people has made me enthused about all sorts of development projects, enough to keep me busy for at least the next year...

Protein of the Day #15: Adropin

2008-12-18T20:12:00.001-08:00

Recently in Cell Metabolism there was a very interesting article on the newly characterized peptide Adropin. Adropin seems quite important in regulating the balance between sugar and lipid use as fuels; its one of several promising avenues to understanding and treating diabetes that have come out in the last year or two. The peptide is cleaved from the product of the gene Enho (energy homoestasis associated). Its extremely well conserved in mammals - for most species the amino acid sequence is 100% identical (a few ungulates have 1 amino acid difference). Here is a picture of the conservation from UCSC's genome browser:

Coloring book reject

2008-12-15T13:51:00.000-08:00

The image below is a Groebner fan that is just too complicated to put in the coloring book, buts I think its pretty impressive:

The ideal generating this is from what I call the "super three-vortex problem", the equations for the central configurations of a 1/r^2 potential:

I am mainly interested in the nonzero solutions of this system. To get at those, we can saturate the ideal - or in practical terms we can introduce a new variable, w, and add the equation w*s12*s13*s23 - 1 = 0 to the ideal. The 3D Groebner fan of the resulting system can be seen here.

A mathematical coloring book

2008-12-04T13:28:00.000-08:00

I've been working on a mathematical coloring book, with the pictures created using Sage. It still needs some work but I've put a preliminary version up at lulu.com. (I am not making any money on it, the cost is what lulu.com charges to print it.) I have also made the download freely available. I would appreciate feedback, especially from people with kids who try it out.

Protein of the Day #14: XP_001352106

2008-11-29T11:32:00.001-08:00

The genome of the malaria-causing Plasmodium falciparum is bizarre in a number of ways - the most striking feature being the extremely high (over 80%) A+T content. Looking on the protein level, there seem to be many proteins with long asparagine inserts. These asparagine inserts must serve some sort of purpose but it is unclear what it is. They do tend to confuse sequence-alignment and similarity searches, which is one reason so many of the proteins remain uncharacterized.
An extreme case of this is XP_001352106, which has a run of 83 asparagines in a row. It does have some similarity to a subunit of cyclin kinase but not enough to be very confident about its identity.

Symmetric Venn Diagram

2008-11-23T14:57:00.000-08:00

This was the start of a small industry of making symmetric Venn diagrams, which Branko Grunbaum found in 1975. I have been working on making a mathematical coloring book (first edition should be - needs to be - done by the holidays, so more details on that soon). I've been trying to making some symmetric Venn diagrams for it, this is a by-product of my first attempts:

Protein of the Day #13: ATP synthase, beta chain

2008-11-19T12:01:00.000-08:00

One of my students is doing a comparative genomics study of Plasmodium falciparum (with an ultimate goal of developing better alignment algorithms for organisms with extreme genomes), and I was curious about what the most conserved protein is relative to a model organism such as yeast. Turns out its ATP synthase; not too surprising but one might guess some other things too. The wikipedia entry is pretty good, although perhaps not as good as it could be. It does have a nice cartoon of the structure:

Protein of the Day #12: V1rf3

2008-11-17T14:03:00.000-08:00

V1rf3 vomeronasal 1 receptor, F3, as its called in mice, appears well-conserved in a variety of mammals, as shown below (sometimes under slightly different names). The vomeronasal system is distinct from the usual olfactory system for smelling, and can be sensitive to very different compounds. In mice the vomeronasal system is important for pheromones. It remains unclear whether humans have any sort of functional vomeronasal at all; my guess is that most of us do not, but perhaps a few people still do.

Permutohedron mirrors

2008-11-14T07:41:00.000-08:00

The image below is from a viewpoint in a mirror-faced 3D permutohedron (truncated cuboctahedron). It should link to a larger (1920x1200) version. This image was produced with Sage (using the Tachyon raytracer and some new code for polytopes that I've been working on).

Slices of the 600-cell

2008-11-13T10:54:00.000-08:00

Over the last 6 months or so I've been doing some work on visualizing polytopes, Groebner fans, and other geometric/algebraic objects. I gave a presentation last week about some of that, which forced me to finish up some projects. One of those was a movie of the 600-cell being sliced by 3-planes.

Protein of the Day #11: cystathionine gamma-lyase (CTH)

2008-10-27T11:51:00.000-07:00

In a recent article in Science, Yang et al found that cystathionine gamma-lyase can produce hydrogen sulfide gas in mice, and that this seems to help control blood pressure. Since I'm interested in mammalian hibernation, this made me wonder about connections with another Science article from 2005 which showed that a torpor-like state of lowered body temperature can be induced in mice by a particular level of H2S exposure. This protein is very well conserved; below is an alignment with the mouse version with some mammals and the chicken. Normally it is involved in cysteine metabolism and presumably does not create H2S; there are some splice variants and perhaps the variants are important in this respect.

Protein of the Day #10: TRPV1

2008-10-21T15:03:00.000-07:00

Mmmmm...spicy food. Wouldn't be as nice if we didn't have TRPV1, which responds to the capsaicin from hot peppers. Here is the predicted domain structure from the SMART database, with the transmembrane domains picked out (along with some ankyrin repeats):

Wordle Malaria

2008-10-13T20:35:00.000-07:00

Wordle is a fun little site; I fed the OMIM entry for malaria susceptibility into it and got:

Protein of the Day #9: Retrocyclin

2008-10-13T18:52:00.000-07:00

The defensins are an interesting protein family that is important in mammalian immune systems. It now seems that most mammals have some versions of the alpha- and beta-defensins, but only some primates have theta-defensins. In the human, there is a pseudo-gene for a theta-defensin that is post-translationally processed into a cyclic peptide called retrocyclin. It is possible that our loss of a functional retrocyclin contributes to our susceptibility to HIV and AIDS; its an interesting avenue for future gene therapy.

I can't find out too much about retrocyclin; since you pretty much have to use rhesus monkeys to study it, there isn't a lot out there yet. A good place to start is the OMIM entry for the alpha-defensins, and the paper by Cole et al.

Schlegel and the 600 cell

2008-10-09T18:18:00.000-07:00

I just wrote a patch to do Schlegel diagrams (a sort of projection) of 4D polytopes in Sage. The 4D regular polytopes are an awful lot of fun to think about; below is a picture of the 600 cell as rendered by my new code. Its much more fun to play with it interactively - check it out.

Protein of the Day #8: Hemoglobin

2008-10-02T16:29:00.000-07:00

So I should have called it "Protein of the Week". Ah well. Its the protein of the day, just not every day...

Hemoglobin: its a classic! Don't think that makes it boring. On the contrary, I think it remains a fascinating protein.

Its possible that it deserves the title of most-studied protein. Right now there are 4820 hemoglobin sequences at NCBI. It was discovered in 1851, and the structure solved in 1959 - I think that was the first protein structure found by x-ray crystallography. I could go on and on...

One of my interests in it at the moment is that hemoglobin is the food for the Plasmodium species that cause malaria. Amazingly, they synthesize their own heme groups. Hemoglobin is a funny food though, because of dealing with all those heme units, and Plasmodium has to accumulate hemozoin garbage.

Protein of the Day #7: Enolase

2008-09-25T05:41:00.000-07:00

Mammals have three enolases. A more descriptive name is "phosphopyruvate hydratase" - they catalyze the conversion of 2-phospho-D-glycerate to phosphoenolpyruvate.

In Plasmodium falciparum, there is still some controversy about their enolase. It appears that at least part of it comes from a migration from the apicoplast genome into the nuclear genome, but it may be a hybrid. Here's the telltale insertion that matches up with plants (in this case rice, but the apicoplast probably came from a red algae ancestor endosymbiont):

Movie of a Groebner fan

2008-09-18T08:21:00.000-07:00

This summer I spent some time thinking about animation and visualizing algebraic and geometric information. I have a longer to-do list than accomplishments but I have made some progress.

One of pilot project ideas was to take a 5-variable polynomial ideal and:
1) compute the Groebner fan using Sage and Gfan,
2) intersect it with a hyperplane (so now we're down to 4 dimensions)
3) slowly rotate the resulting polyhedral complex in 4 dimensions, rendering it using Tachyon/Sage
4) animate the resulting set of frames.

For step 4, I initially wanted to use Blender, but that was really overkill for what I needed and I didn't want to figure out how to get Sage and Blender using the same copy of Python (although someone should). In the end I used ffmpeg to get my movie.

Check out my current best effort.

My next goal in this direction is to do something with Sage's @interact command and JMol to highlight pieces of the fan, since the movie isn't really informative (more art than math I think).

Protein of the Day #6: CD36/Fatty acid translocase

2008-09-18T05:30:00.001-07:00

CD36 is a great example of the complexity of biology.

After some modification, it is the same thing as "platelet glycoprotein IV", an important protein in platelets and clotting - thrombospondin binds to it. Its also important in malaria, since Plasmodium infected erythrocytes can bind to CD36, and mutations in it can result in varying severity of malaria.

But its also "fatty acid translocase" and its a receptor for low density lipoprotein (LDL). Its been associated with a number of effects on the immune system, reaction to hyperglycemia, and oxidant stress.

Both these roles make it interesing in the context of mammalian hibernation, where the clotting reactions must be suppressed and metabolism switched to using ketone bodies derived from lipid stores.

Free Stanford!

2008-09-18T05:30:00.000-07:00

...just a little joke, no one is repressing it. But the Stanford Engineering school has done something extremely nice, namely put up entire course materials online for computer programming, AI, and some electrical engineering courses. Looks very well done. I really like the transcripts of the video lectures, since I like reading more than listening. I don't think MIT's OpenCourseWare does that, but that is another very nice open access project.

I am a little envious of today's self-motivated youth, it would be pretty easy to teach yourself almost anything these days. When I was a teenager I taught myself basic calculus from an old 1940s book (it was called something funny like Calculus for the Everyman, I can't remember exactly). It had nice line drawings but geez, being able to virtually sit in on MIT classes would have helped I think.

Protein of the Day #5: Aldehyde dehydrogenase 2

2008-09-12T12:29:00.000-07:00

Aldehyde dehydrogenase 2, or ALHD2, is the highlight of an article in Science this week showing it is related to mechanisms for protecting the heart from ischemia (lack of blood, which results in lack of oxygen (hypoxia)). There are cytosolic and mitochondrial versions; since this protein is important in metabolizing alcohol, having both versions seems to clear alcohol faster, although I can't find a definitive reference for that fact.

Protein of the Day #4: apical membrane antigen 1 (AMA1)

2008-09-11T05:45:00.000-07:00

In the Plasmodium species that cause malaria, the merozoite stage must invade red blood cells. It does this using the strange apicoplast organelle, which is a much-warped descendent of a chloroplast. Apicoplast:chloroplast as Gollum:Hobbit. Anyway, one of the proteins that helps this invasion is the apical membrane antigen 1, although like most Plasmodial proteins not that much is known about it.

Eloquent Javascript

2008-09-11T05:35:00.000-07:00

Because of my interest in Sage and eventually helping more with the notebook interface, I've been trying to learn some Javascript. Recently I found a fantastic online book that's a joy to use, partly because it has an interactive Javascript console and all code examples can be run within that framework: check out Eloquent Javascript.

Protein of the Day #3: Bone morphogenic protein 7

2008-08-21T07:13:00.001-07:00

In this week's Nature there is some exciting news about what makes brown adipose tissue, or "brown fat". It turns out that brown fat actually comes from muscle precursor cells, not fat cells, and bone morphogenic protein 7 (Bmp7) can induce the transformation.

Brown fat is very important in hibernation as it can generate heat by short-circuiting the mitochondrial proton pump. At very low temperatures, animals cannot shiver so they need other heat generating mechanisms. Its also important for infant mammals, including humans.

Bmp7 is part of the TGF-beta superfamily (transforming growth factors). In the human its on chromosome 20. 7 exons, very typical gene in that respect.

Brown fat has generated a lot of controversy over the years (since 1551!); this discovery seems like a big step forward.

Protein of the Day #2: Clathrin

2008-08-19T10:58:00.000-07:00

Clathrin is a very cool protein. It forms polyhedral lattices that help cells in endocytosis. Your synapses are using a lot of clathrin right now.

Protein of the Day #1: Alpha-2 Macroglobulin

2008-08-18T18:11:00.001-07:00

Long time no post. I've decided to try a "Protein of the Day" post mostly to help myself remember some of them.

For protein #1, I've picked alpha-2 macroglobulin, a serum protease inhibitor. NCBI's Online Mendelian Inheritance in Man (OMIM) is a nice curated resource for protein/gene information and its where I often turn first. Wikipedia has a nice entry too - the quality of wikipedia articles in biochemistry is usually excellent in my experience so far.

Its a glycoprotein, so there are some extra carbohydrates attached. There are four subunits held together by SS bonds. The gene structure is relatively complicated, with 36 exons (same number in human and mouse. It is thought to be evolutionarily related to the C3 and C4 proteins.

My interest in it mainly stems from the fact that it is important in mammalian hibernation. Among many other effects, it inhibits coagulation and fibrinolysis.

Anders Jensen's nonregular Groebner fan in 3D

2008-05-09T14:40:00.001-07:00

The only example of a non-regular Groebner fan that I am aware of is the following one from Anders Jensen's 2007 thesis, here plotted in 3D with Sage and gfan:

R4. = PolynomialRing(QQ,4)
idnp = R4.ideal([x*y*z+x^2*z-x*y,x*w^2-z,x*w^4+x*z])
gfnp = idnp.groebner_fan()
show(gfnp.render3d(), frame = False)

You have to follow the link for the plot since I am not sure how to include JMol applets on a blogger post.

OpenWetWare and Sage

2008-05-03T06:06:00.000-07:00

After reading a nice article by Julius Lucks on OpenWetWare, about python, biopython and SWIG, I suggested he check out Sage. He in turn suggested I write something up on OpenWetWare, and so I have. Hopefully this will lead to some more biology and bioinformatics interest in Sage.

Back from Lausanne

2008-04-27T10:43:00.000-07:00

Last week I went to a very nice conference at the Bernoulli Center, on real algebraic and tropical geometry. I gave a talk on some problems on finiteness and bounds on the real solution of polynomial systems coming from the n-body problem; Sage was featured in a variety of ways during my talk. Here's an animation of a 4-body central configuration (couldn't seem to upload it to the blog directly; maybe its too big).

making tracks

2008-04-17T10:35:00.000-07:00

I've finally added a feature to Sage that I've wanted for a long time: tracking the solution paths of polynomial systems through a homotopy continuation (using Jan Verschelde's phcpack). I am cleaning up my code for formal inclusion, but it seems to work pretty well. The picture below tracks 87 of 99 solutions of the Albouy-Chenciner equations for the three-body problem (in the complex plane). The initial solutions (small blue dots) are for masses m1 = 1, m2 = 2, and m3 = 3. The final solutions are for m1 = 1/100, m2 = 1/10, and m3 = 3. Some of the solutions are moving off to infinity: the mixed volume for the system with m1=m2=0 is only 18, so 81 solutions have to coalesce or move out to infinity. (Why only 87 of the 99? The other twelve are somewhat degenerate, and their solution paths are a little jumpy). Alex Jokela helped a lot with writing the parser for the phcpack path-tracking output.

color me gfan, now in rgb

2008-04-05T05:39:00.000-07:00

Various improvements to the gfan interface in sage are in the works; one of the minor things I've had fun doing is adding more flexible color functions to the render function. Here's the Groebner fan of the 3-vortex problem relative equilibria equations, where the color is determined by the polynomial in each reduced Groebner basis which has the highest degree in any one variable - the degrees of the polynomial are converted to RGB values.

Interactive coalescents

2008-03-26T21:14:00.000-07:00

Genetic coalescents are interesting statistically; the variance in the time to coalescence is large, which is the kind of quantity I think human intuition has trouble with. So it helps a bit to be able to play with them (code can be found here):

Gfan in 3D

2008-03-25T15:57:00.000-07:00

Having upgraded the Sage interface to gfan for version 0.3, I've been thinking about other ways to leverage Sage's capabilities in this respect. One thing I've been working on is a 3D Groebner fan representation. I have some working code for this now, which hopefully will end up in Sage-3.0 if I have time to polish it up. Below are a couple of screenshots of the 3D rendering of the Groebner fan of the ideal generated by (w^3-x, x^3-y, y^3-w, z^2-x-y-w):

restricted four-body problem

2008-03-21T19:25:00.000-07:00

The Albouy-Chenciner equations for the restricted four-body problem seem to generally have 191 solutions. For masses m1=17/20, m2 = 19/20, and m3 = 1, there are 160 complex solutions in 6 variables (the mutual distances between the particles). Plotting each sextuplet as a polygon gives the following plot, followed by the configurations formed by the positive real solutions. Computed with Sage and phcpack.

phcpack, sage, and interact

2008-03-21T12:33:00.000-07:00

I've been working on integrating Jan Verschelde's phcpack software with Sage. phcpack finds solutions to pretty nasty systems of multivariable polynomials by using polyhedral homotopy continuation. Sage can provide a nice frontend for this. Here's an interactive display of the complex solutions of the Albouy-Chenciner equations (from the paper "Le problème des n corps et les distances mutuelles", Inventiones Mathematicae 131 p.151-184, 1998) for the 4-vortex problem:

Color me Gfan

2008-03-17T05:08:00.000-07:00

The latest version of Gfan has some new capabilities that I am excited to use for testing whether ideals are zero-dimensional. But first I have to rewrite the Sage interface to Gfan. I thought that I should try to give some Sage-added-value while I was at it, so I am converting Gfan's xfig output to Sage graphics and adding some color. Here's one result so far: a map of all the reduced Groebner bases for the 3-vortex problem, colored by maximum degree:

Sage 2.10.3

2008-03-13T05:31:00.000-07:00

Sage 2.10.3 is out, with the first released version of the new interact command. As a spectator to the process, it looked like a tough fight between bugs and developers - which I think should be viewed as an entirely positive thing, since it is a consequence of improved QA practices.

Release notes are here, in case you need it the download page is here.

If I can stop playing with interact (which has already been useful to me in teaching and research after 1 day!!) I hope to contribute a wee bit to some upcoming releases. I am rewriting the gfan interface to make use of gfan 0.3. Also, a student and I are working on pretty graphics for path-tracking solutions from homotopy solvers of polynomial systems (phcpack). I'll have to revise the plan a bit to exploit interact.

Interact

2008-03-10T19:43:00.000-07:00

William Stein and Co. have delivered once again, with a new "interact" command that looks amazing even in its beta form. You can almost smell Sage-3.0; it should be out before the roses are blooming here in Duluth. Among other tests, I used it to explore the CpG content of the human mitochondrion averaged over a variable length window:

Music

2008-02-28T11:43:00.000-08:00

Okay, so its not about bioinformatics or Sage, but after a month of no posts I thought I should say something. In the near future I hope to put up some more technical stuff; right now I am working on updating the Gfan (Groebner basis fan software) component of Sage, which isn't very sexy.

So: some music picks. I have been very happy with Balkan Beat Box's eponymous first album. I highly recommend it. Also good recently was Dengue Fever's Escape From Dragon House.

coalescent

2008-01-26T19:12:00.001-08:00

Given the title of this blog, I've meant to do some coalescent stuff in sage for a while. Now that I am about to teach about it in a class, I've finally found the motivation:
(It's cooler animated, but I can't figure out how to post a gif animation here.)

Sage in 3D

2008-01-15T09:49:00.001-08:00

It appears that Jmol will be the backend for most 3D graphics in Sage. After some hard work by developers (that I had nothing to do with) some of that functionality was put in Sage in time for the joint math meetings in San Diego. William somehow managed to find a slew of 3D glasses so that folks could enjoy the stereoscopic view:

(photo by Robert Bradshaw)

It will be exciting to see what happens as the Jmol integration proceeds.

G+C content as a linked image map

2007-12-17T11:48:00.001-08:00

An image map of the human mitochondrion, linked to its GenBank entry by region, showing the G+C content in a small window. Created in Sage using the biopython optional module and this source code.

three body problem central configurations

2007-12-14T11:59:00.001-08:00

All 99 (complex) solutions of the Albouy-Chenciner equations for central configurations of the Newtonian three-body problem, with equal masses. There are three "distances" for each solution; each set of distances are drawn as a triangle. The unit circle is included for a sense of scale. The solutions of primary interest are real, but understanding the complex solutions is important for analyzing these equations.

This was computed with Sage, with the bulk of the work was done with Jan Verschelde's PHCpack. At the moment, PHCpack is in sage as an optional package, so you would have to install that to get the code to work: source code.

spheres

2007-12-13T14:26:00.000-08:00

Light reflecting in four tangent spheres....this relates to a math problem I abandoned a few years ago, maybe I'll work on it again someday. The sage code for the above:

t = Tachyon(camera_center=(0,-4,1), xres = 1200, yres = 800, raydepth = 12, aspectratio=.75, antialiasing = True)
t.light((0.02,0.012,0.001), 0.01, (1,0,0))
t.light((0,0,10), 0.01, (0,0,1))
t.texture('s', color = (.8,1,1), opacity = .9, specular = .95, diffuse = .3, ambient = 0.05)
t.texture('p', color = (0,0,1), opacity = 1, specular = .2)
t.sphere((-1,-.57735,-0.7071),1,'s')
t.sphere((1,-.57735,-0.7071),1,'s')
t.sphere((0,1.15465,-0.7071),1,'s')
t.sphere((0,0,0.9259),1,'s')
t.plane((0,0,-1.9259),(0,0,1),'p')
t.show()

Cobwebs

2007-12-13T14:14:00.001-08:00

This is mostly a test of how code looks in a post. The sage functions below are some initial attempts to write basic dynamical systems plotting functions for educational purposes.

def cobweb(a_function, start, mask = 0, iterations = 20, xmin = 0, xmax = 1):
'''
Returns a graphics object of a plot of the function and a cobweb trajectory starting from the value start.

INPUT:
a_function: a function of one variable
start: the starting value of the iteration
mask: (optional) the number of initial iterates to ignore
iterations: (optional) the number of iterations to draw, following the masked iterations
xmin: (optional) the lower end of the plotted interval
xmax: (optional) the upper end of the plotted interval

EXAMPLES:
sage: f = lambda x: 3.9*x*(1-x)
sage: show(cobweb(f,.01,iterations=200), xmin = 0, xmax = 1, ymin=0)

Note: This is very slow with symbolic functions.
'''
basic_plot = plot(a_function, xmin = xmin, xmax = xmax)
id_plot = plot(lambda x: x, xmin = xmin, xmax = xmax)
iter_list = []
current = start
for i in range(mask):
current = a_function(current)
for i in range(iterations):
iter_list.append([current,a_function(current)])
current = a_function(current)
iter_list.append([current,current])
cobweb = line(iter_list)
return basic_plot + id_plot + cobweb

def orbit_diagram(a_function,parameter_interval, domain=[0,1], mask = 50, iterations = 200, param_num = 500.0):
'''
Returns a plot of the iterations of a function as a function of a parameter value.

INPUT:
a_function: a function of one variable
parameter_interval: a two-element list of the lowest and highest parameters to plot.
domain: (optional) a two-element list of the lowest and highest input values to iterate
mask: (optional) the number of initial iterates to ignore
iterations: (optional) the number of iterations to draw, following the masked iterations

EXAMPLES:
sage: f = lambda x,m: m*x*(1-x)
sage: show(orbit_diagram(f,[3.4,4], mask = 100, iterations = 500), xmin=3.4, ymin=0)

NOTES:
This is pretty crude so far.
'''
point_list = []
plen = RDF(parameter_interval[1] - parameter_interval[0])
seed = random()*(domain[1]-domain[0])+domain[0]
for i in srange(parameter_interval[0],parameter_interval[1],plen/param_num):
for x in range(mask):
seed = a_function(seed,i)
for x in srange(iterations):
seed = a_function(seed,i)
point_list.append((i,seed))
return point(point_list,pointsize=1,rgbcolor=(0,0,0))

A blog by any other name...

2007-12-13T11:38:00.000-08:00

I've always hated the word 'blog', which I think is the only reason I haven't started one until now. But now with the sage projects new blog, I guess I'll give it a try.