mildly mathematical: a webcomic.

it's been a while since i visited smbc comics.  i forgot how math-geeky they can be. for instance:

show's over, folks: it's time for a new act.

i feel like i've gone on a kind of speaking tour, across various cities in finland:

jyväskylä and espoo in october,
lappeenranta and helsinki in january,
espoo again in february,
helsinki again, this month; also turku. [1]

it's a fine thing to travel within finland, because the train becomes a viable option.  on a related note, i'm starting to hate airplanes.

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it was a good april, which i spent not traveling.  i worked steadily, went back to reading, and became frustrated with new ideas.  strange as it sounds, i think that's fine .. maybe even the way it should be, for a mathematician. [2]

having gone back "on the road" this and last week, i've realised something: it's not just the traveling.  there was and is an additional, wearying aspect that's interwoven into these trips.

it's not been so many times, but i've nonetheless lost count of how many times i've given this particular talk about measurablε differentιable structures.  it's in my head now and i can deliver it well .. maybe a little too well.

today i realised that it would take another half-hour to fully explain some details, so on the fly i came up with a new, more succinct sketch of the ideas involved.

in particular, i drew many diagrams and pointed at them a lot.

nobody complained,
but it's not clear to me how much of the audience i lost.

there was a period in my life when i attempted to give every talk only once .. foolish, i know .. but i like to think that my intentions were honest and of the good:

it's similar to my current rule, that i have something new to talk about, every semester: the point is to stay active in research, to keep moving, to keep creating.

There will be time to murder and create,
And time for all the works and days of hands
That lift and drop a question on your plate;
Time for you and time for me,
And time yet for a hundred indecisions,
And for a hundred visions and revisions,
Before the taking of a toast and tea.
~ t.s. eliot

so i think i'm getting complacent.

i still like the ideas in the paper i wrote, but giving the related talk poses no real challenge anymore.  i don't get as excited about presenting the details again, and that isn't very fair to the audience.

if i can't get as excited as before, as i would like,
then how can i expect an audience to be interested in what i have to say?

it doesn't seem honest, not anymore .. which is why i'm shelving this talk from now on.  so unless a colleague invites me and explicitly requests this talk, i plan to never give it again.

it was a good, long run and i think my audiences enjoyed it well enough.  like a good comedian, though, it's only right that i come up with new material .. (-:

Deferential, glad to be of use,
Politic, cautious, and meticulous;
Full of high sentence, but a bit obtuse;
At times, indeed, almost ridiculous—
Almost, at times, the Fool.

[1] 10 talks in 8 months, accounting for madrid and the united states, too. everyone tells me that i'm young and that i should travel a lot when i can .. but ye gods: it's exhausting.

[2] frustrated is probably the wrong word, here.  what i mean is: just as a writer is one who writes, edits, and rewrites, a mathematician should ponder, innovate, polish, and write.  

mathematical lurking; also, an interesting title/abstract.

i like lurking on cνgmt, the research page for GMΤ and Calculus of Varιations at Pisa (and elsewhere).  it's a great example of a solid on-line presence for a mathematical community.

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when i feel tempted to travel, sometimes i look through their "events" page.

when i was looking for jobs recently, sometimes i'd browse through their "open positions" page.  (most of the time these adverts, along with other european positions, never make it to mathjοbs.org.)

twice a week i check the arXιv for new and interesting preprints; the same goes for cνgmt.  some of the content is exclusive to that website (though the trend has been changing, it seems).

sometimes i even look through their seminars list, just to see what people are thinking about.  this kind of news is as fresh as it gets:

it can easily take a year to see a paper appear in print,
it takes weeks or months to draft out a preprint,
but it takes a few hours, maybe days, to prepare a good talk from recent ideas.

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so today i stumbled upon the following title/abstract:

Rademacher's theorem for Euclidean measures
Andrεa Marchesε

Abstract. For every Euclιdean Radon measure $\mu$ we state an adapted version of Rademachεr's theorem, which is, in a certain sense, the best possible for the measure μ. We define a sort of fιbre bundle -- actually a map $S$ that at each point $x \in \mathbb{R}^n$ associates a vector subspace $S(x)$ of $T_x\mathbb{R}^n$, possibly with non-constant dimension $k(x)$ -- such that every Lipschιtz function $f : \mathbb{R}^n \to \mathbb{R}$ is differentiable at $x$, along $S(x)$, for $\mu$-a.e. $x$.

We prove that $S$ is maximal in the following sense: there exists a Lιpschitz function $g : \mathbb{R}^n \to \mathbb{R}$ which doesn't admit derivative at $\mu$-a.e. $x$, along any direction not belonging to $S(x)$. Joint work with Gιovanni Albertι.

intriguing!

i wonder if a "Euclιdean Radοn measure" means more than the name suggests, i.e. a Radοn measure on a Euclιdean space $\mathbb{R}^n$. [1] in that case, i imagine that this fibre bundle $x \mapsto S(x)$ can potentially be everywhere $0$-dimensional.  (think of a cantοr set.)

moreover, this result is of a poιntwise nature.  what i want to know is: does the operator $$D_S : {\rm Lip}(\mathbb{R}^n) \to L^\infty( \mathbb{R} ^n,\mu)$$defined by the formula $$D_Sf(x) \,:=\, \langle \nabla f(x), S(x) \rangle$$has any good contιnuity properties?  For instance, given a sequence of (uniformly) Lipschitz functions $\{f_j\}$ converging poιntwise to $f$, does $D_Sf_j \to D_Sf$ in any reasonable sense?

now i want to know when the preprint will be ready.  i wonder if they explicitly construct the fibre bundle $S$, or if it is adapted from the covering theorems of albertι, csörnyeι, and preιss ..?

maybe i should pay a visit to pisa.
(i've never been to italy, for that matter.)


[1] generic terms for mathematical objects are always dangerous.  for example, what kind of curreηt is a "nοrmal curreηt" or a "cartesian curreηt" ..?

article(s) post: on scientific rigour.

since a colleague of mine passed along this article to me on g+, i thought i'd share it too.

-- ✂ -- --
The government has drafted in the Wikipedia founder Jimmy Wales to help make all taxpayer-funded academic research in Britain available online to anyone who wants to read or use it.
..
..
A government source said that, in the longer term, Wales would help to set up the next generation of open-access platforms for British researchers. "He's also going to be advising us on the format in which academic papers should be published and data standards.One of the big opportunities is, right now, a journal article might be published but the underlying data isn'tand we want to move into a world where the data is published alongside an article in an open format, available free of charge."
-- ✂ -- --

from "Wikipedia founder to help in government's research scheme"
@ guardian.co.uk (by alοk jha)

interestingly enough, the arχiv already has a bare-bones dataset support policy.  being a mathematician and a theorist, though, i've never made use of it.  besides, it seems subject to cancellation at any time .. \-:

one things comes to mind, though:

if all taxpayer-funded research became publicly available in the united states .. and in particular, to politicians and interest groups .. then i have a hard time believing that it will be left alone, and not used in some political shenanigans to shut down some perfectly good veins of academic inquiry.

imagine if a PI in a biology lab has to enter a political fight simply because some legislators are creationists (or their constituents are), think that the laboratory's agenda only furthers the theory of evolution, masterfully arrange control of the appropriate u.s. senate committee, and threaten to cut off the PI's funding.

likely such a scenario wouldn't come to pass, but still: i imagine [1] ..

i'm not saying that public access is a bad thing.  it might, however, open up new problems in place of the partially-solved ones.  it is clear that transparency is a different thing than surveillance, but both are practices to ensure that those under watch are on their best behavior.

besides, when has a change in society not caused more work for those involved? [2] (-:

nonetheless, this would be the sort of step that fosters greater transparency in research.  judging on their various movements that .. to put it mildly, are holding scientists to their word, maybe transparency is a partial solution for the alternative:

".. someone is going to check your work. A group of researchers have already begun what they’ve dubbed the Reproducibility Project which aims to replicate every study from those three journals for that one year. The project is part of Open Science Framework, a group interested in scientific values, and its stated mission is to “estimate the reproducibility of a sample of studies from the scientific literature.” This is a more polite way of saying “We want to see how much of what gets published turns out to be bunk.”

~ from " Is Psychology About to Come Undone?"
@ the chronicle (by tοm bartlεtt)

sometimes i'm glad that i'm a theorist, and needn't suffer the gory mess that are experiments.  on the other hand, imagine if computer proof verification became streamlined and feasible for every mathematician:

barring small, easily-fixed gaps, i wonder how many proofs out there would actually satisfy an automated check?  i have a feeling that most working mathematicians take a lot of logical assumptions for granted.

after all, the proof of  1 + 1 = 2 supposedly takes 379 pages ..!

[1] suffice it to say that i am not a creationist: just not my thing, sorry.
[2] i had in mind the "broader impact" section of an NSF research proposal ..

mildly mathematical: it takes more than one formula to kill wall st ..

i'm running a kind of gauntlet again: over a span of 7 days, i'll be giving two talks.

the good thing is that the subjects aren't too different from talks i've given before, so the preparations won't be a problem.  (i even have slides for one of them, depending on how lazy i feel ..)

maybe it's some kind of spring fever or something [0],
maybe it's the jolly feeling of vappu, a national holiday in finland ..

.. but i want to write today.
on the other hand, i just can't get myself to do it.

*sighs*
so yes, procrastination ensues.

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having seen a few headlines about it already, i expect a few more math blogs will link to the bbc news article about the black-schοles equation, or more notoriously, "the fοrmula that kιlled wall street."  so rather than discussing that in detail, i figured that i'd re-post an article from wired that i read and enjoyed, a few years ago.

it's about gaussιan cοpula functions, which gave rise to yet another "fοrmula that kιlled wall street" [1].  at the heart of the article is probability and correlation, which can be explained quite intuitively.

here's an excerpt:

-- ✂ -- -- -- -- -- -- -- --
The reason that ratings agencies and investors felt so safe with the triple-A tranches was that they believed there was no way hundreds of homeowners would all default on their loans at the same time. One person might lose his job, another might fall ill. But those are individual calamities that don't affect the mortgage pool much as a whole: Everybody else is still making their payments on time.

But not all calamities are individual, and tranching still hadn't solved all the problems of mortgage-pool risk. Some things, like falling house prices, affect a large number of people at once. If home values in your neighborhood decline and you lose some of your equity, there's a good chance your neighbors will lose theirs as well. If, as a result, you default on your mortgage, there's a higher probability they will default, too. That's called correlation—the degree to which one variable moves in line with another—and measuring it is an important part of determining how risky mortgage bonds are.

Investors like risk, as long as they can price it. What they hate is uncertainty—not knowing how big the risk is. As a result, bond investors and mortgage lenders desperately want to be able to measure, model, and price correlation.
-- ✂ -- -- -- -- -- -- -- --

as for the gory details, here's the diagram that is completely lifted from the article from wired [2]:

Here's what killed your 401(k)   David X. Lι's Gaussian copula function as first published in 2000. Investors exploited it as a quick—and fatally flawed—way to assess risk. A shorter version appears on this month's cover of Wired. 

Probability Specifically, this is a joint default probability—the likelihood that any two members of the pool (A and B) will both default. It's what investors are looking for, and the rest of the formula provides the answer.	Survival times The amount of time between now and when A and B can be expected to default. Lι took the idea from a concept in actuarial science that charts what happens to someone's life expectancy when their spouse dies.	Equality A dangerously precise concept, since it leaves no room for error. Clean equations help both quants and their managers forget that the real world contains a surprising amount of uncertainty, fuzziness, and precariousness.
Copula This couples (hence the Latinate term copula) the individual probabilities associated with A and B to come up with a single number. Errors here massively increase the risk of the whole equation blowing up.	Distribution functions The probabilities of how long A and B are likely to survive. Since these are not certainties, they can be dangerous: Small miscalculations may leave you facing much more risk than the formula indicates.	Gamma The all-powerful correlation parameter, which reduces correlation to a single constant—something that should be highly improbable, if not impossible. This is the magic number that made Lι's copula function irresistible.

[0] in my previous postdoc, the semester would typically end around this time of year; i'd even have grades submitted by now.  coincidentally enough, the same calendar also fits the universities of my ph.d and my bachelor's degree.


so after 12 years of conditioning, i think my body expects to be tired by 1st of May .. \-:


[1] it's funny how the media will always refer to a "wall street killer" to sell a story.  meanwhile, not too recently ago protesters were trying to occupy wall street .. which suggests that wall st has been alive and well for most of this time.

it just goes to show you, as a general rule, that people want to make money. any mechanism for doing so -- whether technical or social -- will survive as long as that desire survives.  i'd even go as far as to say that if some version of the apocalypse comes -- say skynet does launch those nukes and we have to fight those damned T-800s -- and if we somehow pull through, one of the first things we'll re-institute would probably be some sort of stock market.


[2] that said, owners of wired.com: please don't sue.

i think it's clear to everyone that it's your succinct explanation and not mine.  besides, i would have rendered the actual formula with LaTeX and mathjax.

TGIM!

admittedly, i like mondays. 

i think it's because i can go to the office and back to work.  that either means that i really like my job .. or that i really hate my life.

i like to think that it's the first case that holds. (-:

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back when i lived in the states, i used to have no qualms about going to the office on weekends .. especially where i was a student.  on the other hand, that was a college town in which, to get from one end to the other, it would probably take at most 30-40 minutes by bicycle.

here in finland, however, i somehow intuit it as "culturally wrong" to stop by the office between friday evening and monday morning.  that doesn't mean that my colleagues don't work on weekends .. but those who do, they tell me that they work at home.

as for me, i do a little work on weekends .. but having all sorts of alternatives around me,

• like a stack of (leisure) books to read, 
• a journal to write in, 
• a trail system near my home to explore,
• friends to call and meet up ..

it's harder to keep to task, you know?  i must be getting lazier with age.

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i could go to a cafe, of course.  they seem different, though, here in finland and other parts of europe.

maybe as an american, the notion of a cafe is foreign to me and the closest approximations are indie coffeehouses that are most easily found in college towns or highly-urbanised cities, or those ubiquitous starbucks chains.  in both cases, it's perfectly common to see people drinking coffee and "working" on laptops.

perhaps i go to the wrong cafes or perhaps i visit them at the wrong hours, but i rarely see that sort of thing in helsinki.  it could be argued, of course, that this is a sign of civilised life. 

i mean, imagine it:
cafes as places where people actually chat and enjoy themselves ..?

i should get my head examined! q-:

mildly mathematical: bezier curves.

on how bézier curves are constructed:

these are hasty screenshots of animations, as found on jason davies's webpage.

i recommend going and having a look;
they're really cool and much more intuitive in real time.

sometimes the derivative is an oracle, at least one-dimensionally .. (link to a preprint)

just saw this on the arxiv today, off a preprint of n. katzourakis. it's about infinite-harmonic maps from the plane into space, but it was this result that piqued my curiosity [1]:

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Theorem (κatzοurakis). Suppose $\Omega \subseteq \mathbb{R}^n$ is open and contractible and $u : \Omega \to \mathbb{R}^N$ is in $C^2(\Omega)^N$.  Then the following are equivalent:

1. $u$ is a Rank-One map, that is ${\rm rank}(Du) \leq 1$ on $\Omega$, or equivalently there exist $\xi: \Omega \to \mathbb{R}^N$ and $w : \Omega \to \mathbb{R}^n$ such that $Du = \xi \otimes w$;
2. there exists $f \in C^2(\Omega)$, a partition $\{B_i\}_{i=1}^\infty$ of $\Omega$ of Borel sets, where each $B_i$ equals a connected open set with a boundary portion and Lipschitz curves $\{\nu^i\}_{i=1}^\infty$ in $W^{1,\infty}_{\rm loc}(\mathbb{R})^N$ such that on each $B_i$, $u$ equals the composition of the curve $\nu^i$ with the scalar function $f$:
   $$u \;=\; \nu^i \circ f, \hspace{.5in} \text{ on } B_i \subseteq \Omega$$Moreover, $|\dot{\nu}^i| \equiv 1$ on $f(B_i)$, $\dot{\nu}^i \equiv 0$ on $\mathbb{R} \setminus f(B_i)$, and there exist $({\nu}^i)''$ on $f(B_i)$, interpreted as $1$-sides on $\partial f(B_i)$, if any.  Also,
   $$ Du \;=\; (\dot{\nu}^i \circ f) \otimes Df, \hspace{.5in} \text{ on } B_i \subseteq \Omega$$and the image $u(\Omega)$ is a $1$-rectifiable subset of $\mathbb{R}^N$.

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interesting:

roughly speaking, if the derivative tells you that a smooth mapping has 1-dimensional behavior, then you can actually cut it up into a single function that mimicks the mapping's behavior through curves.

(i haven't read the proof, but my guess is that the hard work is somehow done through the partitioning.  i wonder if there is a Rank-$M$ version of this result, for $M \in \mathbb{N}$ ..)

there's also a version purely for maps with components in $W^{1,\infty}(\Omega)^N$, in the same paper, but with an $L^\infty$-approximation condition via smooth maps.



[1] the blue text was added today (1 May 2012). i could swear that it was there before, but it seemed to have disappeared after reloading the blog page.  also, i decided to indent some of the last few paragraphs to highlight my guess on the subject.

an article excerpt from the guardian is below.
(it seems that the movement is gaining steam.)

Harvard University says it can't afford journal publishers' prices

Exasperated by rising subscription costs charged by academic publishers, Harvard University has encouraged its faculty members to make their research freely available through open access journals and to resign from publications that keep articles behind paywalls.

A memo from Harvard Library to the university's 2,100 teaching and research staff called for action after warning it could no longer afford the price hikes imposed by many large journal publishers, which bill the library around $3.5m a year.
..
..
Robert Darnton, director of Harvard Library told the Guardian: "I hope that other universities will take similar action. We all face the same paradox. We faculty do the research, write the papers, referee papers by other researchers, serve on editorial boards, all of it for free … and then we buy back the results of our labour at outrageous prices."

[the article continues here]

a personal gripe ..

*argh*

sometimes i really, really hate working with weaκ-star limits;
it's like working with black magic.

it's hard to extract any geometric information at all, from them .. at least, i've never been particularly good at it.

*sighs*