Visiting and home

Today, even more than Friday, was a day of visiting. We had bread and soup, meat and cheese, coffee and cookies. I helped make bread in the morning -- something I haven't done often at home since my move, mostly because of the logistics of counter space management -- and was quite well pleased by my efforts. Basil, black pepper, and garlic are good bread seasonings. The soup was good, too: ham, beans, and potatoes was an old favorite when I was a kid, and it still is; and the veggie stew was nearly as good (both were good, but I'm biased). Then the aunts and uncles came, and Scott and Brittany, and we sat around and chatted. And then everyone dispersed -- except, of course, my parents and the cats and me.

In a way, I felt like I ought to be saying farewell, too. Home is an elusive word, but right now it's attached to an apartment in California as much to a house in the woods in Maryland. But I'm here for a little longer, and that's fine -- particularly since, now that the house is quiet, I can avoid the worst teasing about still being in graduate school. Of course, I'm sure I'll be just as hectored once I graduate, too: as it's said, if you're so smart, why ain't you rich? Well, I like comfort well enough -- particularly little comforts like warm socks and hot tea -- but I don't really think I want to be rich. Most of my family understands and approves of my aspirations well enough, though, so that's enough complaint from me.

One of my holiday books is Innovation and Its Discontents by Jaffe and Lerner. It's brief (about 200 pages with 20 additional pages of end-notes), informative, and well-written. The subtitle, How our Broken Patent System is Endangering Innovation and Progress, and What to Do About It, spells out the author's thesis pretty well: namely, that changes in the US patenting system since 1982 have caused patents to be granted too easily and patent rights to become too potent a weapon, so that the monopoly granted by a patent can hinder as much as it helps. As you might imagine, the anecdotes sometimes leave me grinding my teeth -- just as is the case with recent books on copyright law -- but I regard the information as not only interesting, but probably personally useful. It's like learning about export restrictions and government classifications of what things are sensitive: however boneheaded I think certain aspects of the current system might be, it's best to know enough to try to avoid running accidentally afoul of them.

On a tangential note: LAPACK, an enormous and widely-used library of freely available dense numerical linear algebra codes, is undergoing another revision, with which I'm tangentially involved (I was responsible for the last revision of CLAPACK, a C language translation of LAPACK; I hope that this LAPACK release will finally allow CLAPACK to slide into graceful oblivion, but the details of this hope are a topic for another day). Things are just getting under way, but one of the questions which came up in the very first discussions was how should the copyrights be managed? When the early versions of LAPACK came out, a bald statement along the lines of This is free; use it for commercial or non-commercial purposes as you see fit, but while the codes are as good as we know how to make them, we don't guarantee them and won't be held liable seemed like quite enough. But now it's not. The idea that lawyers must be involved in order to give software away rankles, but it seems that without the lawyers, some big companies will shy away. Fortunately, the universities involved do have legal departments who, I suppose, are capable of giving advice on such things.

On a ligher note, last night I picked up and re-read Susan Cooper's The Dark Is Rising, the second book in a sequence of the same title. It's billed as a children's book, and it has been on my shelf since I was much younger; but the writing is good (better than Rowlings', I think), and I like the story. Besides, the story is set around this time of year, and is a good deal less sappy than most such stories.

Children's literature is also a nice break in technical reading, which I've been pursuing in parallel with my leisure books. I brought with me Nick Trefethen's Spectral Methods in MATLAB, which I recommend for its clarity and brevity as well as for its usefulness (if you find yourself solving boundary value problems). Also, my parents gave me a copy of Luenberger's Optimization by Vector Space Methods, a book which presents a wide range of material -- much wider than might be suggested by the title -- in a very natural geometric setting. Luenberger uses the language of linear algebra and functional analysis (which is basically linear algebra in infinite-dimensional spaces) to describe and unify ideas that come up in all sorts of interesting areas: pure analysis, mechanics, statistics, control theory, finance, and numerical methods, among others. I wish I'd known about this book and read it as a companion to Royden when I was taking my graduate analysis course (or perhaps I wouldn't have wished for it when I took that sequence in 97-98; I've learned some things in the intervening years). In any case: it's a grand book, and I recommend it for those who are interested in -- and not terrified by -- the unifying language of functional analysis or the many applications of ideas of linear algebra, convexity, and optimization.

On a note somewhere between Susan Cooper and David Luenberger, let me note that the columns on the Mathematical Association of America web site are often both interesting and very widely accessible -- I'll recommend it even for my friends who didn't go much beyond high-school mathematics. And if you're a puzzle fan, read the December article in the Math Games column.

Did I say I planned to spend the break reading and entirely ignore this blog? Well, the best-laid plans fall awry, and perhaps it's just as well.

• Currently drinking: Mint tea

Piece of Pi

Today's stupid pi trick:

• Write (or find) a program to compute the binary digits of pi.
• Starting at the 1138694996 and 752552936 places after the binary point, expand a few digits.
• Group into five-bit chunks encoding characters.
• See what words are spelled out.

(Digits of pi courtesy David Bailey's web site.)

Letter from America

My family exchanged gifts today. One book, which I both gave and received -- very happily in both cases, I might add -- was Letter from America: 1946-2004, a collection of Alistair Cooke's radio series.

I enjoyed Cooke's America, and his Memories of the Great and the Good; but I think I like this collection better than either of those. The letters span six decades, and many more than six topics: everything from sketches of famous and not-so-famous people to the city of Washington, DC, the vagaries of dress styles to the origin of Golden Gate Park -- it's all there. And it's all done with beautiful style.

I recommend it highly.

From the Woods

At the moment, there are seven humans and four cats in the house: besides my parents, both my brothers and their respective significant others are here. And, of course, I'm here. And then there are the cats: Misty, who is getting up in years and mostly wants to be left alone; Pounce, who is a little mean and none too bright; and Thyme, who is convinced that she really could catch her tail if it would only stay still. Rick and Sarah's cat, Sake, is here for the moment, too.

I sometimes miss the dog, and the cats who have died.

It will be much quieter on Christmas Day, since my siblings are scattering to other surroundings. So we'll be moving our local traditions up a day, and after that the house population will be down to three humans, three cats, and a handful of errant moths. We humans will probably read for most of the day; and the cats will probably sleep for most of the day. The moths will do whatever it is that moths do.

With Scott and Brittany, I watched a tape of Scott's black belt testing. It was several years ago, but neither Brittany nor I had seen it before. I wrapped gifts -- books, of course, since they are easy to pack and since I'm not a terribly inspired gift-giver -- and I helped cut vegetables. I checked e-mail. And I sat and read, and listened to the rain.

I have work with me. I have one or two of the reference books which seemed most immediately relevant, and a few papers. I have my laptop, my pens, and my pad. I'm spending two weeks at home, in all, and I've no doubt that I'll be happy to have my work with me before I go back. But for now, I'm just happy to visit with people and cats, to sit and read, and to listen to the rain.

In the Details

Consider the following two recurrences:

• y_n+1 = y_n-1 - 3/2 y_n, y₁ = 1, y₂ = 1/2
• z_n+1 = z_m-1 - 8/3 z_n, z₁ = 1, z₂ = 1/3

What are y₁₀₀ and z₁₀₀? Try computing a few steps of each recurrence to see what happens.

Here's a little MATLAB script to compute y₁₀₀ and z₁₀₀. If you don't have MATLAB (which is expensive), it will also run in Octave (which is free). Try it, or write a program in your favorite language which does the same computation. Can you explain the results?

  y = [1, 1/2];
  z = [1, 1/3];

 
  for j = 3:100
    y(j) = y(j-2)-(3/2)*y(j-1);
    z(j) = z(j-2)-(8/3)*z(j-1);
  end
  
  disp( y(end) );
  disp( z(end) );

December

I returned yesterday from a trip to LA to visit Winnie's family, and I leave tomorrow to visit my family for two weeks. In other news, I have been reading, pondering, writing, and reading some more. All the writing has been technical, as has much of the reading. I have written no letters and no Christmas cards, and it seems unlikely that I'll do so. I've avoided getting sick, mostly, and I have made some effort to be sociable; but I've also been eating less, sleeping more, seeking work, and avoiding people more than I sometimes do. All this is utterly predictable: I'm this way most Decembers.

Posts will be sparse, if there are any at all, for the next couple weeks. It should be more regular come January. And if I owe you a letter, maybe that will come in January, too.

Tongues

If you speak three tongues, you're trilingual. Two, you're bilingual. One? American.

I can follow some of a conversation in German or in Spanish, if the topic is very simple or if the speakers are talking more slowly than any native speaker ever does. I can usually understand advertisements and instructions in Spanish, and on rare occasions have made my way through technical papers written in German. And I know enough Cantonese to identify a few types of tea and food, to count and maybe tell time, and to say things like thank you, I know, what is that?, where are you?, and chicken head.

I'm effectively monolingual, and I'm self-conscious about it.

Camels

Any article which includes the line

And we all know what humps mean! Humps mean local maxima! Or camels. But here they mean local maxima!

sounds good to me.

Haircuts again

Something like that.

Haircut and books

I had my hair cut this morning. I've thought for at least a month that a trim might be in order. But errands like haircuts (and grocery shopping and laundry) have seemed a good deal less interesting than other things, at least for a little while. There is at least a sense of consistency in wearing mismatched socks, piling books haphazardly on any available surface, preparing meals that are heavy on pickles and applesauce, and fighting a losing battle against my hair's creative impulses.

Winnie mentioned that I should not get my hair cut too short over the top. It makes you look even taller and thinner than you are, she said. I wasn't very clear in communicating this, though. When I say myself in the bathroom mirror later in the day, in one of those corner-of-the-eye moments that show so much more than a face-on examination, I realized that (in a hooded sweatshirt, at least), I looked like nothing so much as a stereotype of a monk. You know the image I mean: a sort of angular character, sharp around the nose, chin, eyebrows, and perhaps about ears -- the anti-Friar Tuck, if you will. I was amused by the thought; I'd be a terrible monk.

As an aside -- I'm not gaunt. I am bony around the joints -- nothing short of a sweater or thick jacket is going to make my shoulder a comfortable headrest; this has been empirically demonstrated -- but that's different.

I also did other errands, including a trip to get some books. I didn't find the biography of Euler that I wanted for myself, but I did find other books that I'd planned to get for family members. Reviews and recommendations will be forthcoming after the holidays, I'm sure.

The new SIAM Review was waiting in my mailbox when I went to the office. I'm very pleased by this issue. The Problems and Techniques suggestion includes an article on semi-separable matrices, and another on the analysis of Markov chain mixing times. The convergence of a Markov chain to it's stationary distribution is closely related to the gap between the dominant eigenvalue (one), and the second-largest eigenvalue, so building a fast-mixing Markov chains turns into a problem with a very interesting numerical linear algebra component. The authors are doing the standard Right Things with respect to the linear algebra -- including symmetry reductions -- but I'm curious whether they're doing anything that could benefit from some of my favorite algorithms.

There's also an article on the fast low-rank approximate solution of certain Lyapunov equations, too, and how the method involved (a variant of ADI, or alternating-direction-implicit methods) relates to a problem solved over a particular rational Krylov subspace. The author wrote a really cool thesis on model reduction a while ago -- it won the last Householder prize -- and the paper clarifies some connections that I didn't get when I read the thesis. And the survey article (A Survey of Public-Key Cryptosystems) also looks interesting, though I haven't yet even skimmed it. And, of course, the book reviews are always entertaining.

Anyhow, putting aside any technical jargon in the previous two paragraphs: I'm excited. The articles look cool, and I expect to learn some very useful things from them.

I've spent some of my time on leisure reading, too. I've been re-reading Ball's A Short Account of the History of Mathematics, and I've been going through another of Poincare's books. It's fascinating stuff. Ball is probably much more historically accurate than Bell, but I do sometimes find myself re-reading a section of Bell's book (Men of Mathematics) after reading a few pages of Ball's book. Bell is a lot less dry: did you know that Florence Nightingale was one of Sylvester's students; or that Poincare was notorious for his utter inability to make drawings that resembled anything in heaven and earth; or that Lobatchewsky spent a very active period as the curator of the University Museum at Kazan (in Russia), and continued to pitch in even after he became the university rector? On the other hand, Ball also mentions interesting anecdotes from time to time: I hadn't realized what a cad Cardan was, for example (a talented cad, but a cad naetheless). It's probably good for me to spend time with both books.

I wonder if there are any books out there exclusively on major mathematicians of the nineteenth and twentieth centuries? Maybe; but perhaps it's just as well they aren't in my current reading queue, or I'd spend less time on my other interesting reading on various methods for linear and nonlinear eigenvalue calculations (and applications, naturally).

• Currently drinking: Coffee

Symmetry and perturbation

Symmetry is an old topic of fascination for mathematicians. I know I've recommended Weyl's book, Symmetry, before; let me now do so again. It's a short book, written for a lay audience, and it describes types of symmetries in art, nature, and mechanics. While Weyl writes very clearly, the book does reflect a very deep sort of knowledge; I have another book by Weyl on my shelf on a more mathematical treatment of the classical groups, and his treatment of symmetry groups in quantum mechanics is considered classic.

The notion of a perturbation is similarly old, mostly because the real world tends to be full of problems that are Really Hard, and the only way anybody knows to tackle them is to pretend they are Tractable (or perhaps Trivial on a good day). This usually means dropping small terms that make the problem hard, and then analyzing the effect of the missing bit. Sometimes it's possible to correct the answer to account (at least partially) for the effect of the missing term; and sometimes the best one can hope for is to figure out roughly how bad the mistake was. The business of getting rid of the hard parts of a problem by estimating or bounding them is at the heart of mathematical analysis, together with the notion of a limiting process (which sometimes allows estimates to be parleyed back into equalities).

Elementary courses on differential equations tend to emphasize a small set of equations which can be solved by hand. While this seems sensible to me -- after all, we choose our models rather than accept them as gifts from on high -- it does have the unfortunate side effect that many otherwise well-educated people fail to realize how fundamentally hard it is to get exact analytical solutions. Differential equations with solutions in terms of elementary functions are exceedingly rare; and equations for which such a solution can be found and understood by a reasonably educated human are rarer still. Nevertheless, a colleague of mine, an engineer who should have known better, was once inspired to ask why I didn't just solve a particular equation analytically; and when I explained to him that the integration was provably intractable, he snorted in apparent disbelief, shrugged, and observed that at least computers make it trivial to compute numerical solutions. I'm not sure whether I disabused him of this notion in our subsequent conversation, but I surely tried.

Those equations which can be analyzed at all are usually analyzed by exploiting symmetries, which deliver interesting qualitative information even in the cases when they don't lead to a full solution. Fourier analysis depends on translational symmetry; separation of variables depends on a certain symmetry in the shape of the domain where an equation lives; and familiar basic conservation laws (conservation of energy, momentum, etc) are closely linked to other symmetries (a fact proved by Emmy Noether). Dimensional analysis (or the study of dynamic similarities) is another type of symmetry reduction, though most people who know what dimensional analysis is probably have never heard of Sophus Lie or Emmy Noether; the matter is only confused by the fact that dimensionless parameters are often called dimensionless groups, a name which bewildered me for years.

Equations which are almost symmetric are immensely interesting. Symmetric systems show all sorts of behaviors that don't usually occur if there's no symmetry -- such behaviors are nongeneric -- and a perturbation which changes the symmetry therefore often alters the solution enormously. At the same time, an lot of both the natural world and the engineered world is almost -- but not quite -- symmetric; and so beams buckle, atoms bond to form molecules, shutters buzz in a strong breeze, whirlpools form when the sink drains, and dropped sheets of paper go flying all over the place when you drop them. Well, my papers fly every which way when I drop them; perhaps your papers drop directly to the floor, in which case I can only guess that you use really thick paper or that you live in a very rarified atmosphere indeed. Either way: huzzah for symmetry breaking! It makes the world a more interesting place.

Of course, to an unwary user who would like to simulate a physical system, symmetry breaking can herald interesting times indeed.

To solve a continuous problem on a computer, one discretizes the problem: in some way, we have to approximate an infinite-dimensional problem by something which is finite-dimensional (as the speaker at a recent talk observed, We do not need to go to infinity, which is good, because that is too big). One way to do this is by difference approximations: instead of computing smooth functions, we compute functions at a (large) number of discrete points; and when we need a derivative (tangent at a point), we replace it with a divided difference (a secant between successive points). This approximate system generally does not have all the same symmetries as the original system. For example, if the original problem remains the same if we move the coordinate system around, the best the discrete system can do is remain the same if we move the coordinate system around in a way that maps mesh points to other mesh points. Or suppose the differential equation preserves some invariant relationship involving a derivative; if we want a similar conservation law to hold for the difference equation, we have to ask which difference? For a nice function of a single variable, there is only one derivative at a point x; but there are two natural differences, one involving the point to the left of x, and one involving the point to the right of x. (There is a class of integrators for Hamiltonian systems which approximately conserve a differential relationship called a symplectic form; the analysis of these methods is complicated by precisely the issue indicated above, since the discrete system has two natural analogues of the symplectic form for the continuous system).

As another example, consider what happens if you want to know the few lowest resonant frequencies of a gong. A gong is highly symmetric: you can rotate it, flip it over, or reflect it across various planes, and after you've finished your mutilation, it will still look the same as when you started. Having great faith in the power of your computer, and being unwilling to go through the pain of hand analysis, you feed the problem to some standard finite element code, which is built to solve exactly such equations. The program runs a standard algorithm, and returns its estimate of the lowest few frequencies, and you discover to your dismay that the computation takes a lot longer than you thought, and misses some eigenvalues, too. Why? Because of the symmetries in the original problem (because O(2) is non-Abelian, if you like), many of the resonant frequencies of the gong correspond to multiple eigenvalues -- which is a very rare case for problems which lack such symmetries. The presence of these multiple eigenvalues (called a degeneracy) carries through exactly if the system is discretized carefully so that the discrete system has a symmetry that mimics the symmetry of the continuous system; if the discrete system does not have such a symmetry, the eigensolver might have less trouble, but you'll probably have to work harder (use more mesh points) in order to get a decent answer. There's no free lunch (or tanstaafl, if you read too much Heinlein in a mis-spent -- or maybe well-spent -- youth). Whether it's exactly preserved or only approximated, the presence of a degeneracy causes confusion for standard eigensolver algorithms.

Now, suppose you're a very clever blobby, and have figured out that to find the resonant frequencies of a gong, you can just restrict your attention to specific types of motions. If you start the standard algorithm (shift-invert Lanczos iteration with partial reorthogonalization) at a special starting vector which obeys a specific symmetry -- reflection, about some symmetry plane, say -- then all the subsequent iterations it looks at will also have the same symmetry. You've just managed to perform a symmetry reduction on your problem without changing the model at all! Of course, if you've had a course in numerical linear algebra which was sufficiently competently executed that you learned about Lanczos's iteration, you probably know what will go wrong. Slight differences in rounding errors act as a perturbation, causing the iterations to drift a little, so that your iteration no longer stays strictly symmetric -- and suddenly you're faced with a symmetry-breaking behavior again, and it will make your life... interesting.

Now, if you're a reasonably clever blobby who has spent too much time thinking in detail about the behavior of floating point arithmetic, you might realize that there are situations in which the symmetry indicated will be preserved exactly, even in floating point arithmetic. But it will only work that way in some situations; and though the situations aren't that hard to figure out, most people have fuzzy mental models of what actually goes on inside of the floating point unit on their machine, and will either be completely oblivious to the behavior described above, or will be immensely spooked by the fact that the program can be broken by changing the parentheses in a program so that one expression is computed rather than an algebraically equivalent alternative. Of course, if you had enough knowledge of all the different pieces to realize such a subtle way to factor out a symmetry, you probably know enough to understand how to factor out the symmetry explicitly in a pre-processing step, and get rid of all the subtleties and potential sources of instability.

And this is the real art in numerical analysis: recognizing what symmetries and problem structures can be reasonably conserved under discretization or under the action of transformations used in a numerical method, and what symmetries can just be approximated; and then parleying that knowledge into algorithms which are simultaneously fast and accurate.

What, never?

According to my mother, if you were ever able to quit thinking about problems on command, it was not within the times of my experience with you.

If I had one ability that I could magically gain, I don't think the ability to quit thinking about problems would be it. I still think it would be pretty cool to be able to digest cellulose, though.

Day of the Sock Monster

It has been about a week now since Elena finished moving into the flat with us. With her came a variety of things: there's a wicker magazine holder now, a little dish of sea shells and drift glass in the bathroom, and a toaster oven that doesn't even look disreputable, much less like a potential fire hazard. She also brought a Siamese cat, Niko. So far, the cat and I have treated each other with a wary sort of respect. I came in the first evening she was her, and there she sat in the doorway. We looked at each other for a minute or two; I offered her a hand, which she deigned to sniff; and then I went to prepare a meal, and she disappeared.

I've felt like she was sitting there, silently watching, while I ate many of my other dinners this week. Perhaps she found them entertaining. I've come home several times this week in a distracted mood, usually with visions of symmetry groups dancing in my head, or sometimes -- when hunger, fatigue, and personal frustrations overcome my technical fascinations -- less pleasant thoughts involving Donald Trump morphing, a la Terminator 2, into a giant animated Santa Claus. There are more prosaic fears and frustrations, too, but the seasonal homicidal reality-television robot sounds much more impressive as a symbol of terror than the fine gentleman who charged through a red to nearly run me over, right? Whatever the case, most days this week I've stepped in the door, taken off my shoes, and realized that I've eaten little or nothing since my morning cereal, and that I probably ought to make food. So the cat watched on, and I foraged, tripping over boxes and discovering that the bananas have been -- correctly -- reclassified as inedible and disposed of accordingly; that the stale bread that I thought to use for onion soup has been similarly removed; and that, though they could not possibly have gone bad, the egg noodles I'd intended to use as the base for my dinner were missing, possibly because a UN humanitarian mission mistakenly evacuated them, but more probably because I ate them and then forgot about it.

So I've gulped down salty pickled vegetables and followed them with apple sauce (a word to the curious: the salt from the pickle brine sets off the apple sauce taste nicely; and Dad, I apologize for ever having made fun of the Wheat Thins + ice cream combination, which is pretty good after all). I've munched down meals of apples, cheese, and crackers, and then -- having taken care of the most urgent edge of hunger -- become distracted and wandered off to spend most of the rest of the evening studying numerical dispersion relations. I've scrambled a couple of eggs to have with an apple; I've mixed yogurt, honey, cinnamon, and barley cereal to accompany an apple; and in one case I accompanied my apple with -- uh -- another apple. And the cat watches on, in that inscrutable way that cats manage so well. Except, of course, when the cat has decided to sleep in the corner; then we change roles, and I watch the cat while I munch.

The cat is much better than I am at gazing inscrutably.

Fortunately, I do have human friends who sometimes eat with me as well. I had a good meal and a cup of tea and a ramblingly disorganized conversation on Tuesday with a friend, and we each read drafts of the paper the other was working on. And on Wednesday, I had dinner with Winnie. I fear I was not terribly good company, as I kept getting distracted by visions of symmetry groups dancing in my head. Yes, I know that's a trite way to put it -- but dancing symmetry groups are far less insipid than dancing sugar plums, and far less frightening than the dancing homocidal Trump-Santa-T2 robots, so I'll adapt boldly a cliche which so many have adapted before. Whether my head is full of dancing algebraic abstractions or differential operators doing their thing (their thing is beat poetry -- or at least you can pretend it is, if you don't know what a differential operator is), I do sometimes chew on problems to the point that my interactions with the world become less graceful. I always hope that this amuses my fellows and myself, rather than exasperating my friends and causing me to wander toward a path of peril -- but since my friends (usually) forgive me, and since I regain the lost weight easily enough, I try not to worry about such little character flaws. Besides, you don't have to know me well to recognize the distracted air and the chaotic hairstyle (math hair, as a couple friends call it) that signals that I'm thinking about a problem; so those acquaintances who find me truly aggravating when I'm in such a mode can easily figure out when I'm best avoided.

I tried to ask the cat her opinion about this. She graciously has not proferred her opinion of my character flaws. She also won't play with her string-on-a-stick cat toy when I'm wielding it, but I think that's more from shyness than from any real disapproval. Besides, you can exhibit some truly fascinating nonlinear wave phenomena by waving the stick back and forth and varying rates. Jim came home one evening while I was illustrating this to myself (and to the cat, but since the cat was hiding somewhere, I was the only obvious audience). I think Jim was far more amused by my fascination with the cat toy than he would have been if the cat was involved.

It's possible that I was once able to quit thinking about problems on command. If so, that time has long past. Sometimes I can put aside a problem over the course of a long walk -- this weekend I took a wonderful ramble from campus up the hillside, along Euclid and Grizzly Peak to Tilden, then back along Spruce, down Marin, down Indian Rock path, and then along Colusa and Portland into El Cerrito -- but even that is sometimes only a temporary measure. Fortunately, with a little effort, I can usually switch between problems. For instance, my shoes seem to rub around my ankles far more now than when I first got them; and while I was bandaging the blood blister induced from the weekends ramblings, I pondered a variety of possible ways that I might be able to patch my shoes in order to prevent further rubbing. My ponderings ultimately led me to conclude only that an extra pair of socks might keep my feet warmer, and that the sides of my heels are perfectly capable of developing the additional callusses needed to handle any remaining confrontations with my shoes that the socks don't quell. But during the time I was pondering my shoes, I didn't think about mathematical problems at all.

Anyhow, I've been distracted; my main point was not how good salty applesauce tastes, nor how distracted I can get by technical problems, nor the new collection of Cooke's Letters from America: 1946-2004 which I saw when I was visiting Barnes and Noble with Winnie. The point is to describe the cat, and perhaps to set the stage for the startling revelation about the cat that occurred to me when I folded my laundry this evening.

Niko, though as innocent-looking as any creature with an unnervingly intense stare and an expressed desire to pounce on things can possibly be, has a dark secret. She is, in fact, a Sock Monster.

My mother used to tell me that lost socks went to a colony on the moon, but it took only a little maturity and reflection to find the problem with the sock colony theory. While it's true that socks are gregarious, and tend to be found in packs, they don't seem to collaborate effectively in groups of size any greater than two; and while rocket science is perhaps not as complicated as the old cliche might suggest, I believe effective rocket engineering is probably beyond the intellectual capabilities of even the most prodigious pair of socks. Boots, maybe; socks, never. Besides, where is a pair of socks going to get the materials to build a booster rocket, let alone enough rocket fuel to reach escape velocity? No, the lunar sock colony theory cannot hold; and so there is only one explanation for sock disappearance

No, it's not the theory of spontaneous combustion -- that's been thoroughly discredited at this point. The reason that socks disappear is that they are eaten by Sock Monsters. Further, anyone who has seen a cat batting at a sock, or trying to climb into a basket of clean laundry, might reasonably begin to infer the true identity of the Sock Monsters. But Niko is a truly remarkable sock monster, as she somehow conspired to make my socks disappear before I even arrived here. My sock supply has dwindled alarmingly over the course of this semester, faster than can be accounted for by the flow of socks from my drawer to the bag of cleaning rags (the BOSSCHEDAWOWET or Bag of Socks So Clearly Holey Even DAvid WOn't WEar Them). So I laid it out plainly for the cat while I folded my laundry: Cat, I said, you may freely prey on any of the socks in the bosschedawowet; or, for that matter, any of the socks in my drawer with holes large enough for me to fit both thumbs through, since soon enough I'll relent and turn those into cleaning rags, too. I give you this freely, as tribute; take it, and please leave my other socks in peace! I looked over my shoulder to see if the cat was taking it in -- I needed to tuck my chin to hold the shirt I was folding in place anyhow -- but the cat was concentrating on her forepaw, which she was cleaning with a sort of cross-eyed look of concentration for which I felt immediate sympathy. I left her to contemplate her task in peace while I finished my folding; then I turned, crouched down, and looked at her intently to see if she had any reaction to my peace plan.

She just stared back at me inscrutably. Cats are so cool.