Archive for the ‘mathematics’ Category

Other useful courses

January 28, 2014

Continuing on the theme of Chalk : an introduction, here are some hypothetical useful courses that nobody organizes for M.Sc. students in mathematics.

  • The Greek alphabet: how to tell \xi and \varsigma apart on the blackboard
  • How to pronounce foreign names: the language l’Hospital
  • How to read upside-down text: the art of checking your student’s answers in a hurry
  • How to use the copier: How duplexes get done, and why you need to be VERY careful with transparencies
  • The Laptop and the Data Projector: One Thousand Years of Anger and Sorrow
  • Seven exercises in following a flood of information transparency after transparency and slide after slide and oh god why’s she wiping that text away nooo I didn’t write it down yet—
  • One exercise in trying to listen while you’re making notes too; it’s like those sleep tapes in that it doesn’t work
  • What is the sound of an unasked question? The nine types of silence in the classroom
  • If every student seems stupid, it’s probably you that is
  • Practical examples for everything, A — Acad
  • Acupressure by chalk
  • How much caffeine is too much caffeine? (This is a trick question)
  • Acupuncture by chalk: an introduction to the Omerta of the classroom

New Year’s Night: Mathematics is impossible!

January 5, 2014

So, the first night of the year, one thirty AM. Me and my brother are at our parents’ place, watching TV. (Apparent cause: Hey, home together, let’s spend every possible moment together. Real cause: Brother sent me to buy snacks; I panicked and overperformed and now there’s a shopping bag full of salty stuff and we’re scattering come morning.)

We’re getting ready to fight over what to watch (Me: Hobbit extras! He: America’s Funniest Home Videos reskinned for Finland!), when I see mathematics on the random channel open and say “Hey! Holy MS Word equation editor, Batman!”

(My brother’s not Batman; he’s a physicist. But we had watched the 1966 Adam West Batman film a few days ago before that, and it had been one of the funniest films I’ve ever seen.)

But this show: it is one of those nighttime live call-in shows: it shows a puzzle or a question, and if you call in and answer it correctly, you win a small sum of money. (Well, if you call in you’re fed into a series of multiple choice questions, and if you answer them correctly, and are the quickest of the round — one round’s a minute or two of callers, maybe? then you get on air and are allowed to guess.)

DSC_0146(Click for bigger.)

(Translation: “Laske kaikki mahdolliset luvut yhteen!” is “Add up all the numbers you can!” — more literally but less fluently “Count all the possible numbers together!”)

(“Nopeita kierroksia” isn’t “Thus I suffer for my sins” but “Quick rounds!”)

Me and my brother watched, with increasing puzzlement, hatred and incredulity, this program for an hour and a half, while smarter members of the family slept.

There were dozens and dozens of callers, most of them not audibly inebriated, and equally many different answers, all of them wrong. The spread was in thousands. The MC or people-goader — a nice, a little awkward young guy doing his first night of this, at the lowest end of the TV-personality totem pole — grew more and more anxious with every “No! Sorry, that wasn’t the right answer…” until his turn ended and he was replaced by an equally shocked, though more experienced, woman.

By now you think this is an illustration of the stupidity of the sort of people who watch TV on a fine New Year’s night; but no. Me and my brother both have Ph.D.’s, though admittedly only mine is in mathematics; his is in the soft and almost humanistically unrigorous subject of physics. We came up with a dozen ways to interpret the problem to explain why the right answer hadn’t been called in yet; but eventually someone always called in with our most likely guesses, and proved them wrong.

(We didn’t call, because then we would have been rubes, not amateur ethnologists.)

For example: The equation’s not that difficult. But it’s been called already, so that’s not it.

Hang on a minute, that 14 is really badly aligned. Is it a trick, a 1^4? Do the people behind this show know what exponentiation is?

Hey, wait, that 5+5. That’s not a plus sign, that’s division! You need to get really close to a passably big screen to notice that…

Hey, wait, the problem is “Count all the possible numbers together!” Not “solve the equation for question mark” — oh, how our education misleads us.

Oh, so it is addition. Disregard the multiplication, the minus signs, all that. (I hold forth for a few minutes on “The numbers in 5-7 are 5 and 7, not 5 and -7, unless you remember, as one does, that subtraction is defined as the addition of the sum-reciprocal number of the second operand,* in which case that’s 5 plus -7, but what kind of mathematical knowledge can we assume of this program and its audience — you tell me, you have the degree in a soft science, physicist.”)

(* = This could be accurate.)

Wait, “all possible numbers”? Does that mean… all natural numbers? All real numbers, all complex numbers, and… fuck, that’s a lot. And that’s either undefined or zero.

No, you physicist, I’m not going to call in with “undefined or zero”, I’m sure that’s not how they mean it. It’s not my fault mathematics makes you read things like a paranoia patient. And we call that rigor, thank you very much.

Oh, those numbers on the left-hand side of the screen, one to ten. Oh you clever bastards.

Hang on a minute. The sevens on the first and last row look different. Maybe the first one isn’t a number… look, it’s the same as those definitely-not-numbers squiggles around the equation!

Look at that 10. That’s not the same 1 as in the 19 on the next row… more fake numbers! Subtract ten! And that 5 on the last row is just a squiggle!

Wait, does this mean the numbers in the “Hyvää uutta vuotta!” (“Good New Year!”) rectangle, or does it include the phone number too? The reward money number? The 18 in the K-18 age limit? (Is that minus eighteen?) Nobody’s called in with a number over seventy million, but the spread is astonishing — here’s a partial record of about an hour — I’ve inserted comments where our best guesses were shot down:

-2030

625

169

103

418

393

620

448

73

114

403

390

634

533

913

-175

175

195

1543

398

122

78

613

131

410

154

373

390

64

91

128

88

162

178

70

1478

172

129

177 (fuck)

1185

83

188

24

175

160 (double fuck)

169

172

Please note the -2030 and the 1543. That’s worse spread than with first-year non-math-majors on Introduction to Small Integers!

The second MC, the woman, eventually grew really desperate with the hints: add up the numbers, add them up, listen to what I’m saying, don’t solve the equation, add up the numbers, all the numbers, all the numbers, all the numbers, in this rectangle to the left of me, oh God, how can we be doing the third hour of this, usually this isn’t more than an hour — that didn’t help us, or the callers.

DSC_0148

And that scroll at the very bottom of the screen? I hope it was a general rules-scroller, because it advised one to consider “all Arabic, Roman or written-out numbers while solving the problem”.

Oh, the reward? Began at 200 before we started watching; eventually crept up to 750 euros when we stopped (3 AM), and to 950 by 4 AM when the show stopped. (Parents have a provider that offers TV with a two-week recall.)

The next morning I checked the scheduled early-morning continuation of the show, but it had been replaced with an SMS forum — you text them, and a slow scroll of the received texts shows on the screen — there were, as usually there are, racists and xenophobes and some that were both actually, but no answers.

The good people of eurojahti.com, I hope you are happy with destroying my faith in mathematics and the Finnish people just two hours into 2014.

Ritual dialogues of mathematicians having coffee

December 3, 2013

(To be read in a dull monotone by a set of two people for the amusement of the complement.)

*

I could go for a cup more.

You could always go for a cup more.

Better then that I do not start at all; for by induction I would never stop.

Nonsense, for there is a boundary condition in the worldwide availability of coffee beans.

But are not coffee beans, with respect to time, a renewable resource beyond the rapidity I can consume them, even in a liquid concentrate?

Ah, I see you are right; you should not even have had the first cup.

Indeed. And I shall not have another.

*

Was that good?

That was better than yesterday.

But was it good?

It was the best I’ve ever had.

But was it good?

That I’d rather not say.

Rats.

*

In this cup, coffee frozen to brown snow. In this cup, a boil under the lid. Let us call the temperatures zero and one hundred.

From cup to cup you pour this, back and forth, portion and portion.

So the temperatures change, but do they converge?

They do; I have proven this.

Where do they converge? We may assume the cups to be identical, and containing an identical amount of coffee.

At fifty do they converge.

Let me sketch this. You graduate student over there, stop eating the chalk and give me one. Scribble scribble. Oh, yes, right, they converge. Hooray.

It is proven, then?

It is proven, with reasonable assumptions on “pouring”, “back and forth”, and “portion”, for all measurable cups of finite Lebesgue measure in any fixed dimension. Results for Hausdorff cups of non-integer dimension to be investigated next.

Yes. I will get more coffee.

Coffee is life.

No.

Correction: Lack of coffee implies lack of life, by the Erdös definition of life; “life: doing mathematics”. This is the standard definition of life.

Correction accepted.

Correction acceptance accepted.

*

Topologically speaking…

Never speak topologically when I’m here!

I speak topologically; you vanish.

I do.

Topologically speaking, this coffee cup is the same as this donut.

How so? I only know function theory.

Both could, assuming they are malleable, be deformed to the other.

But your coffee cup isn’t malleable!

Not at this temperature, no. But that is hardly the point.

What would you do with a ceramic donut anyway?

Interdisciplinary research.

How?

I would give it to my son.

I am puzzled.

So I would solve the longstanding open problem in theology, “Which of you, if your son asks for bread, will give him a stone instead?”

But what of the donut-matter coffee cup?

That I would eat.

How would you drink coffee, then?

Oh.

See, your plan fails like the commutativity of addition and the square root operation, and you resemble a mathematics student in such a person’s first year by claiming the negation of this statement!

Ha ha. I am amused.

Yes you are!

I am amused by your wit. Ha ha.

Math and visa

June 6, 2013

What math is

Mathematics is not a science. It’s not a natural science: it does not study nature. It’s not a human science: it doesn’t study humans.

Mathematics is an art. It’s not a human art; it’s not useful for elucidating human feelings and passions. It’s a natural art: it is very, very useful for elucidating nature’s laws and actions.

(Well, this does not seem immediately fallacious on a cursory one-eyed glance, so it’s up to my usual self-review standards of my own personal philosophies…)

*

Visa

So I’m going to China to a mathematics conference for next week; this meant getting a visa. I filled up the papers, sent them in, and waited. A week later the departmental secretary brought me my passport back; I sat down to wait for the other papers.

A week passed. I grew anxious.

I went to the secretary and asked, “Hey, where’s my visa at?”

She said, “Da fuq, graduate student? It’s in da passport.”

I said, “Wait, what’s a visa look like anyhow?”

She said, “Oh em gee, graduate student” — I might be using false tones and idioms here.

I went, looked inside the passport; hidden on a random page halfway through it there was an official-looking sticker which, apparently, is what a visa looks like.

So now I know that.

The evolution of a solution

March 25, 2013

As a TA, I sometimes TA for a course whose coursework has no ready solutions. Then the following happens.

*

Iteration 0: No solution. These problems are impossible. The lecturer is a sadist. These are his research problems.

Iteration 1: Death is the only solution.

Iteration 2: Hey, if we know “A” this problem is easily solved. Hooray!

Iteration 3: Oh, “A” follows from this problem. Dang.

Iteration 4: If we assume “B” is known, this is both easy and elegant!

Iteration 5: If we assume “B”, we’re assuming something not known or proven on this course!

Iteration 6: It’s one page if I hand-wave the hard part!

Iteration 7: It’s three pages and no hand-waving!

Iteration 8: It’s three pages, no hand-waving, and assuming 1 < 0 holds!

Iteration 9: Hang on a minute, this is not a problem about “X”. That’s why no “X”-literature had a peep of it.

Iteration 10: This is about “Y”! And it’s an easy “Y”-problem!

Iteration 11: Three lines, easy peasy… aw crap, that inequality’s not strict.

Iteration 12: Three lines, plus eleven special cases, can this really be— (phone rings)

Iteration 13: “Misprint, Mr. Lecturer? The one in Problem 3, right? Right right. What? I meant the inequality… oh, that’s a different misprint?”

Iteration 14: It’s not an “Y”-problem. It’s an “X”-problem, and the zero was clever misdirection for infinity.

Iteration 15: I have… a solution? A skeleton anyway; let’s throw some meat on this pony!

Iteration 16: It’s a elephant. I can’t give this solution to the little ones. First thing, my wrist would break at the blackboard.

Iteration 17: I could use transparencies… Wait, no, I’d better try simplifying this. Get some jumping jacks, elephant solution!

Iteration 18: Right, I don’t need the special case where r>1 and r<-1; silly me.

Iteration 19, the homework meeting: “Mmh, yeah. You can prove it that way too.” (crushes paper, cries a silent tear, moves to the next problem)

*

Iteration 19 can be averted by having a handout. (“Yeah, I guess you could use the obvious, elegant solution Mr. Poopypants put on the blackboard. If on the other hand you want a solution with pizazz and loxodromic Möbius transformations… well, one out of two ain’t too bad… here’s a handout… Aw, come on people, don’t you have saunas to set fire to or something?”)

The mathematical life, part aleph

March 11, 2013

A harmonic function, for the purposes of this discussion, is a function f for which \Delta f = 0. A superharmonic function is one for which \Delta f \leq 0, and a subharmonic one one for which \Delta f \geq 0. Consequently, a function is harmonic if and only if it is both superharmonic and subharmonic.

Thus, a superharmonic function is (generally speaking) not harmonic.

When I explained this to the teaching-of-mathematics studying fellow the next desk over, his comment was: “Are you telling me that Superman isn’t even a man?”

(To which I should have said, “He’s from Krypton, isn’t he? I don’t even know if we should call him a he! What the hell, he might have tentacles or nothing at all down there — wait, let me check, there must be fan fiction about this. Let me google for ‘superman duck penis’.”)

To which I answered, “Ja, but if ve take der Super-Man und der Sub-Man, they together make a Man!”

*

A question, from the same discussion: As is well known, a topologist is a person who doesn’t know the difference between a donut and a coffee cup. This being so because in the topological sense they’re the same thing: if they were made of clay, you could morph one into the other without destroying or introducing any holes. (A donut has one, in the middle; a coffee cup has one, in the handle.)

The question now rises, how many holes should a donut have to be topologically equivalent to a human being?

Probably more than one, as the digestive pathway, mouth to fartmaker, is not the only one. But this quickly becomes a quest into the insides of the human being; it is not clear to me if even the male and the female of the species are topologically equivalent. (Either “Physiological gender as a topological concept” or “The topological equivalence of the sexes: Towards a mathematical feminism”, forthcoming once I get the funding.)

Research into this is on hiatus because the damn biologists, who surely have the requisite expertise, are far away across the frozen waste in a different building.

My daily life: communication and solutions

September 27, 2012

A day at the math department. Midday. I go to the toilet to drop excrement and read Twitter. That done, I reach into the toilet paper conch.

It’s empty.

Well, reaching deep within I can feel the cardboard tube, but that’s no good for wiping.

For I moment I just sit there, dull surprise on my face.

Then I read some more Twitter and FMyLife, resisting a slight urge to comment on one of those about my position.

Then, when there are no particular sounds of footsteps from the corridor, I crack the cubicle door open and reach into the antechamber, the likewise closet-sized pre-toilet with a handwashing basin and a single male-peeing bowl. (I realize my terminology is weird; but you rarely read or talk about toilets.) There on the wall, two paces away, next to the corridor/toilet door, is a dispenser for hand towels.

Paper hand towels. And not the sandpapery kind either, but the nice ones. (If it had been the sandpaper towels, I might have resorted to some real commando methods.)

I calculate — one human being, two hands; one needed for ripping out a stack of towels, one needed for keeping the door closed, one needed to keep my underpants awkwardly halfway up, covering the worm and the dumplings.

Left for towels, right for pants; mercifully nobody picks these four seconds for a time to come in.

I wipe, the toilet eats the towels without too much burping; and as I stand up I really notice something I had glanced at coming in: a wadded, unused paper towel, like the last of a bunch held by a sweaty hand, in the nook between the seat and the wall.

Apparently, it seems, I was not the first to resort to these methods.

And as I walk out, a physics assistant rushes past me, into the toilet. For a split second I try to find a polite way to tell him ERMAHGERD THERE’S NO TOILET PAPER; but a split second isn’t enough.

Besides, it’s a problem with a proven solution; and as a mathematician, I’m happy with that.

Where I disprove physics

August 21, 2012

Author’s note — Coming to work this Tuesday, I found the following in my mailbox from Monday. The work is mine; the content, I fear, is the result of a Monday rashly started, with no heed given to proper plus-minus stretching before trying to engage serious mathematical issues.

*

AN NOVEL APPLICATION OF A TIME-COUNTING THE KARTOFFEL-CAUCHY-BUNYAKOVSKY MEASURE

(submitted to Utt. Bull.)

I have heard that this kind of a device is used to measure seconds:

“One potato, two potato, three potato” (etc. etc.),

the logic being that the addition of the word “potato” pads the number-repetition to approximately one second in duration, like this:

(t=0 s) “One potato” (t=1 s) “two potato” (t=2 s) “three potato” (t=3 s; etc. etc.).

I have heard this is an easy way to count time, and that any fool (i.e. a “standard person”) can do it.

However.

The word “one” is clearly shorter when said (i.e. its common saying-out-loud measure is less) than the word (and measure of) “two thousand seven hundred and ninety-eight”. For the Potato Method (PM) to work as described above, the enunciation of the word “potato” needs to be commensurably briefer.

However, it is well know that for any length of time, we can find a number whose enunciation takes, for a standard person, approximately that length of time. (To an error of less than epsilon for the average over a set of sufficient size, obviously; also see note one.)

Thus we can assume a number n_1 such that it cannot (by any standard person) be said in an amount of time less than one second. When that number is reached using PM, the standard person needs to pronounce the word “potato” in the remaining time, which is, zero point zero, zero repeating, seconds or less.

Worse still, we can take a number n_2 such that its pronunciation takes two seconds (to an epsilon); and our standard person is then required to pronounce “potato” in minus one second, i.e. to cause the reversal of time’s arrow solely by the enunciation of three common syllables.

Since I am assured by good practical sources, who never expected to be involved in the creation of a mathematical paper, that the PM is usable by all standard people, I thencefore have proven the existence of time travel.

Applications to follow in a later article. Referees, get cracking like you got this yesterday! Potato potato potato!

yrs obdnt srvnt,

LORD OF TIME
potato potato potato

*

Note one: The existence of numbers of arbitrarily long enunciation

Due to the fact that the enunciation of numbers uses the decimal system, this follows from the well-known nine-filling principle, i.e. that numbers of the form 10^n - 1 require n instances of the number 9, and thence the word “nine”, to be expressed.

Ex. Let n=3. Then 10^3 = 1000, and 10^3-1 = 999, pronounced “nine hundred ninety-nine“, emphasis mine. Note the three repetitions of the word “nine”.

If the enunciation of the word “nine” takes t_9 seconds, then we achieve a number of enunciation-length t_e seconds if n t_9 > t_e, that is, no later than at the number

\displaystyle{10^{\lceil t_e/t_9 \rceil}-1}.

qed

(the same)

Mathematicians solve real-life problems

May 15, 2012

Out of toilet paper.

What is toilet paper? A tissue that through repeated contact transfers fluids and solids from the nether area to itself, after which the tissue plus fluids and solids can be disposed of.

The most common substitutes ignore this last part: using the person’s own hand, for example, is a highly suboptimal solution since the hand cannot be disposed of in any easy fashion. This is because the hand is attached to the body.

Likewise, the use of underpants, t-shirts, and overcoats is troublesome because of the disposal problem. Leaving overcoats with fluid and solid remnants at one’s local waste disposal point can result in loss of social prestige.

The optimal solution to running out of toilet paper are small birds. Their own motor effects serve to enhance the transfer of fluids and solids, and once released at a window, they dispose of themselves.

Exercise. (a) Compare the effectiveness of utilizing pet birds vs. wild birds, as regards costs of care vs. immediate availability. (b) How to catch birds when not wearing clothing on the waist-ankles interval. (c) Are specific toilet rooms necessary? Design a portable intra-rectal toilet with pneumatic compression.

Opening doors.

What is a locked door? A rotational system attached to a doorframe at n points, that can be orthogonalized from the frame if at most n-1 of those points are fixed, enabling access through the doorframe.

If, however, all n of those points are fixed (i.e. the key is lost), the usual layman solution is to “bust” the door, that is, to reduce the structural integrity of the door rectangle until (a) one of the points gives, or (b) a sufficient subset of the door rectangle orthogonalizes.

Such brute force methods are not elegant and cannot be recommended. The local order authority may react to them negatively and possibly induce unforeseen fatalities.

Exercise. (a) Formulate the door analogue in an arbitrary dimension. (b) Build a 5-dimensional door. Do not open it for any reason whatsoever. (c) Study the theory of doorknobs, e.g. T. Setting’s Mathematics for the Knobhead. (d) Write a search algorithm for arbitrary keys, then ask the author for a test key.

This problem of opening doors without keys is as of yet unsolved. Those interested in collaboration on the matter should contact Prof. Holzbein, Königstrasse 2a (the porch), Stuttgart.

Vocation seeping into the hobby

April 12, 2012

…was great for Tolkien; he did languages for his academic living, and as a result could do Elf languages not only enthusiastically, but expertly.

Me? I’m a mathematician.

That cannot carry over very well.

* * *

The Adventure of Ruprecht Generick
in the Magical Land of Ba Nach

(a derivative tale)

TABLE OF CONTENTS

  1. Falling into fantasy: Life is complex when the real and the imaginary meet
  2. More dimensions than these three: an N-chanted tunnel in the sky!
  3. The Set of All Sets, or is this Egyptian a god of an alien realm?
  4. To the limit of the sum of your fears: the dread Integral Riders arrive!
  5. Oh no! There’s a conjecture about a savior, but where’s the proof?
  6. A lacunary series of meetings in the shadow of the discontinuity called Death
  7. One, two, too many Integral Riders: the Great Battle for Ba Nach, commences!
  8. Cold equations and the calculus of defeat: You win if and only if you run away!
  9. I have found a fixed point in my life! The heroic transformation of Möbius!
  10. An exponent of peace, a minimizer of strife: the solution comes from the left-hand side
  11. Farewells, squares and ghost references

* * *

Oy, it would be the sad tale of Polly Nomial (which would be a nice spoken-word piece in an interdisciplinary gathering, and then the mathematicians would be forever alone) crossed with Terry Brooks.

Also, the titles might show how much anime I’ve been watching lately.

But seriously, I think that as a result of my academic education, I would probably get lost in a sea of details if I tried to write heroic fantasy. (Wait, that’s not “if I tried”, but “when I will try”. One day!) And then there would be a footnote, saying: “For more detail on the Elvish system of integration that Magister Luchaliber alluded to, see Appendix Theta.”

Which, you know, starts from the halfway point of the book, just after Appendix Rho, “The High Elven Concept of Infinity: The Legend of Rumikol”, and Appendix Sigma, “The Intuitive Analogues of the Standard (\epsilon,\delta)-Proof in Wood Elf Spoken-Word Pseudo-Real Analysis: The High Cost of Low Rigor”.

With the first line being “Portions of this appendix were published as ‘A novel but non-pedagogical approach to visualizing limits’, by M. Ascaras and E. Riz, in Comm. Soc. Chic. Ken., 148 (2012), 34–40.”

Which, you know, would be fun to write; but books are often written to be read, too.

(But hey, “mathematically rigorous worldbuilding” sounds nice, doesn’t it? I wonder if there are books that investigate the history of mathematical ideas and conceptions among non-scholars. Presumably it can’t have been one sheep two sheep many sheep all the time; nowadays even non-mathies have intuitions about zero, infinity, and the like.)