Three Haiku for Steve Jobs

nineteen eighty-four.
the mouse: I thought, "the future"
just nineteen myself

for the rest of us!
typography and quickdraw!
a machine with depth

now victorious:
your pinnacle of design
in every pocket

The Gramophone (A Dream)

My gaze fell upon my laptop. I told Erin, “I miss using my laptop. Sometimes I wish I could log in to it once in a while.”

“Maybe we can run the generator tonight. Then you could use it for a bit.” That was typically kind of her; gasoline for the generator took a lot of effort to obtain, and firing up the laptop qualified as a frivolous use of the power. Besides, the Internet was gone, with so many dead from the plague. What did I really want to see? I could write some code, but there weren’t any problems that I needed a computer to solve any more.

After thirty minutes’ walk through the empty, silent streets, I walked right in to the doctor’s office; there weren’t any other patients. I had never met him, and there didn’t seem to be much point in small talk. “I wonder if you would give me some Prozac, please,” I asked.

“Do you have a prescription?” the doctor asked me. “I can't find it,” I replied, and I wasn’t sure that I had ever really had one.

“Then I will have to examine you, so that I can write a prescription for it, if, in fact, it is indicated.” The doctor’s leather swivel chair squeaked as he leaned back.

“Look, it’s not like I’m asking you for anything dangerous. What could possibly go wrong? Who would care at this point?”

“It’s fortunate that you’re not looking for anything dangerous,” the doctor replied, “for everything that had even the slightest potential to get anyone high was looted from the pharmacy a long time ago. But it doesn’t matter whether what you want is dangerous or not. I am a doctor, and doctors have rules. If you want me to prescribe medication for you, I must examine you first.”

But the doctor did not mean to examine me personally. He escorted me to a little room where there was a stool set before a video screen. A computer-generated image of a nurse appeared on the screen and began to ask me questions, building up my medical history from scratch. I had trouble remembering the answers to the questions. It had been so long since I had thought about any of this. I wondered where the doctor got the power to operate the device.

Night was falling as I walked home. I saw that while I was away, Erin had somehow managed to find a gramophone, and a stack of old gramophone records to go with it. It was powered by a spring wound by a crank, and played the music through a metal horn, so we could listen to music without running the generator. She hadn’t heard me come in. I watched her look at the gramophone records one by one, trying to find something that she thought we might enjoy hearing, I supposed, though the records were very old; the most recent was from the thirties. My heart swelled with admiration for her. She is making the best of a bad situation, I thought.

it’s his country

As if I had not made enough spectacle of myself already, I took this Mr. Tallman by the sleeve and told him to look over the side, explaining that the sea had turned yellow. I am afraid Mr. Tallman turned white himself instead, and turned something else too—his back—looking as though he would have struck me if he dared. It was comic enough, I suppose—I heard some of the other passengers chuckling about it afterward—but I don't believe I have seen such hatred in a human face before. Just then the captain came strolling up, and I—considerably deflated but not flattened yet, and thinking that he had not overheard Mr. Tallman and me—mentioned for the final time that the that the water had turned yellow. “I know,” the captain said. “It’s his country” (here he jerked his head in the direction of the pitiful Mr. Tallman), “bleeding to death.”

— Gene Wolfe, Seven American Nights

Scientific Terminology Spikes

It turns out to be interesting to give obscure mathematical terms to Google's Ngram service, which measures the prevalence of words in historical published works. Take haversine, a function from trigonometry that is not taught in high school today but is frequently used in navigation:


Looking at the spikes, I conjecture that during the World Wars, a lot of people were studying navigation!

On a somewhat melancholy note, look at tensor and differential equation:

I interpret the peaks here as the Kennedy Moon Race, and the push for increased science education that went with it.

A Poem of Casie's

Several months ago among Casie's effects I found issue XXVII of The Quatrain, Saint Anselm College's literary journal, published her senior year. I wanted to share a poem she published in that issue with you on the occasion of her birthday.

Untitled


Now is the time for sleep,
    A time to slip away,
        into the peaceful darkness.

Suspended until after I have caressed my infant's skin,
    heard my friend's troubles,
    dwelled in the hearts of my family,
    become the salted tears of my husband,
    and smiled my final smile.

I am not alone
    I am with myself.
I AM.
Carry my soul not in your minds, but within your hearts


C. Marie Fillio '92

A Dream

I am on my hands and knees, peering furtively out of the bay window in the front of my home in Africa. I am looking for a break in the irregular stream of cars on the street across the lawn. While there are gaps, I can always see the next car in the distance approaching. This makes it impossible to achieve my goal: to twist the clear plastic rod that will close the slats of the blinds without anyone seeing the slow change of the slats' angle, from horizontal to vertical. If any of the occupants of the cars sees any motion in my home, they will know that someone remains within.

"C_____ was right," I think to myself, as I wait for my opportunity to quickly render the front window opaque to passersby. "I have waited too long to get out." Where is out? Wherever the cars on the street are going. They are all nice cars, big, dusty silver sedans. They have beautiful grillwork of the type you don't see any more: too elaborate to be a Benz, too understated to be a Rolls. They are driving at a measured pace, but it is clear they are leaving without expecting to return. Their single direction of motion portends that the frontier of security is collapsing toward me, slowly but irreversibly by now, soon to leave me on the other side.

Later, in my darkened house, more tastefully decorated than my real one, cozy, with an unforced antique charm, I am looking at the blinds on the door connecting the back of the kitchen to the backyard. These are of similar manufacture to those in the front, which I must have closed because I am no longer concerned about them. I don't even intend to close these blinds to the back garden. But I am looking at the slats which are made of rare African wood: cut so thin they are translucent, almost like the hull of a vanilla bean. I glumly contemplate how expensive they were.

π day in classical Wiener space

It's π day so I thought I might do a math post.

The other day I was doing laundry and I thought to myself, "if I finish this whole pile, my laundry will really only be done for a moment. It won't stay done even past bedtime."

This caused me to flash back to Physics I class in high school. The teacher gave the definition of an object being "at rest" whenever the first derivative of position was zero, i.e., dx/dt = 0. But this meant that a rock, thrown perfectly vertically from the surface of the earth, is "at rest" at the top of its flight. Right at the apex of the parabola you would get on an x vs. t graph.

This upset my classmate Jeff, since the rest state was only achieved instantaneously. He argued that there was a fundamental difference between a rock at rest because it was lying on the ground, and one at "rest" because it had reached the apex of its vertical flight. At the teacher's provocation he made his own definition of rest precise. I'm not sure I remember it but I think it was more or less the idea that an object was at rest at time t if there was an open interval (t−ε,t+ε) such that dx/dt was zero on the entire interval.

But this made me think of the interesting function f (x)=e^-1/x. This function has the property that f(0) = 0 and f'(0)=0 but f(x) ≠ 0 when x ≠ 0. Therefore it might be said to be at rest in my physics teacher's definition but not in Jeff's. But on the other hand, this f occupies a kind of middle ground: the flatness it experiences at 0 is much more profound that the flatness of the parabola traced by the thrown rock at its apex. The function is infinitely differentiable, but the derivatives of all orders are zero at x=0: f ⁽ⁿ⁾(0) = 0 ∀n. Indeed, this function is not analytic at zero, i.e., its Taylor series expansion has zero radius of convergence there (and is therefore useless). This is the classic example of a non-analytic smooth function. The zero function z certainly satisifes the trait of having z ⁽ⁿ⁾(x) = 0 ∀n, and indeed this holds on an open interval containing any x; so we see that f is kind of at the boundary of functions absolutely at rest and those instantaneously at rest. If the Taylor series of a function f is identically 0 at a point x, it is very hard to argue that we are dealing with motion rather than rest.

All of this brought to mind an interesting digression in Miles Reid's Undergraduate Algebraic Geometry (which I confess that I have only skimmed), in which he notes:

[O]ne can reasonably consider the following rings of functions on U [an open interval]

• C⁰(U) = all continuous functions f: U→ℝ;
• C^∞(U) = all smooth functions (that is, differentiable to any order);
• C^ω(U) = all analytic functions (that is, convergent power series);
• ℝ[x] = the polynomial ring, viewed as polynomial functions on U.

There are of course inclusions ℝ[x] ⊂ C^ω(U) ⊂ C^∞(U) ⊂ C⁰(U).

These rings of functions correspond to some of the important categories of geometry: C⁰(U) to the topological category, C^∞(U) to the differentiable category (differentiable manifolds), C^ω(U) to real analytic geometry, and ℝ[x] to algebraic geometry. The point I want to make here is that each of these inclusions represents an absolutely huge gap, and that this leads to the main characteristics of geometry in the different categories. Although it's not stressed very much in school and first year university calculus, any reasonable way of measuring C⁰(U) will reveal that the differentiable functions have measure 0 in the continuous functions (so if you pick a continuous function at random then with probability 1 it will be nowhere differentiable, like Brownian motion).

What a fascinating thought! Constructing a function which is continuous on a whole interval but nowhere differentiable on that same interval requires a fair bit of effort. The effort required creates the false sense that such functions must be rare. But Reid asserts that they are insanely common in the wild landscape of function space. I wondered whether Wikipedia had enough information to supply some information about this and I was somewhat pleasantly surprised that it did. The answer is found in classical Wiener space (why didn't I think to look there?)

Right in the the article I just linked there's a section on the density of nowhere-differentiable functions. It turns out, though, that measure in function space is more complicated than on ℝⁿ; I was surprised to learn that "there is no infinite-dimensional Lebesgue measure," and that in fact classical Wiener space was invented to create a notion of measure that worked for spaces of continuous functions. A great deal of the previous, I wish to stress, is way over my head. I can't understand the definition of Wiener space at all; I checked out of real analysis too early. But I do find it fascinating when exotic things seem to dominate familiar things in number the way Brownian functions seem to do with continuous functions.