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. (-:

---

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.

---

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]:

---

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 $\mathbb{R}^N$.

---

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*

on acknowledgements ..

regarding papers and preprints ..

i've heard of authors giving thanks to cafes where they spent a lot of time, working out technical details of proofs and LaTeχ.

in one of my own papers, i've thanked a non-mathematician for helping improve the look of a diagram.  (in fact, he was a professional artist.)

i've never seen acknowledgements like the ones in this preprint, though. (-:

proofs, lost and found.

during the technical part of a seminar talk today, i withdrew into my own thoughts and came up with a wrong idea for a proof.  it was clearly wrong because it was too general in scope:

it didn't rely on the hypotheses i had in mind,
and it would have applied to a setting in which a counter-example already exists.

so yeah .. quite wrong! (-:

what i didn't know yet was why .. and that intrigued me;
that's the interesting part, you see.

so i thought and thought in circles, a bit lost .. and before i knew it, i was clapping with the rest of the audience.  exiting the seminar room, i strolled back to my office, still lost.

---

sometimes i love mathematics because of all the little twists and turns that can come up. a proof is like getting lost in an unfamiliar part of town, but then finding your way back:

sometimes you remember when you return,
sometimes you still get lost and deeply suspect that there must be a better way than this ..

on the other hand, a good, polished proof is like taking a shortcut or a scenic detour, the sort of path that you'd show your friends if they were tagging along.

many mathematicians give the analogy of a toolbox, and that every job requires the right tools.  whenever i'm working on a proof, though, i feel like i'm living out an analogy of an escape route of sorts:

there's got to be a way out of this.
what if we tried this door? no good. it's locked;
wait, is the the air vent wide enough ..?

it would probably explain why my proofs seem .. weird, to me.

they're probably not good ones .. not in the above sense, anyway.  i never think to go back and polish them up .. not unless i feel like i too easily get lost and need a better map ..

.. and i'm never surprised when, in the midst of explaining one, that my audience gives me strange looks.

article post: on good mathematical handwriting.

tips for writing out mathematical symbols, from john kerl. among them:

• Make a point come out of the top of the p, to distinguish it from a rho:
  , .
• Put a loop on the q, to avoid confusion with 9:
  , .

(more tips can be found here.)

uncommitted: no risk, no gain.

[written on friday]

maybe i should have waited until the end of today to submit the article. instead, i did so just before lunch.  it took only 5 minutes to upload and fill out the form.

otherwise, today was wholly unproductive.

Stephanie: What did you do today?
Leonard: Well, I'm a physicist, so I just thought about stuff.
Stephanie: That's it?
Leonard: I wrote some of it down.

[from the bιg bang theοry, the tv show]

---

when i think about it, most of my workdays are unproductive, in the sense that no stunning breakthrough occurs.  it's hard to say if i've ever had a moment that could really be called a "breakthrough."

i wouldn't call the process of research "continuous" either.  maybe it's like a (1-D) distribution function from probability .. but, of course, that's being rather optimistic:

who says that there is always progress in research? [1]
 

---

as for particulars, today i felt uncommitted to any one idea or problem.  my mind just started to wander and the rest of me was obliged to follow it.

i thought about curreηts,
then about metrιc spaces,
then returned to geοmetric measurε theοry;

all in all, in 5 pages i didn't prove anything .. maybe a lemma?

i wouldn't point to anything and call it a lemma, exactly.  these were five pages of .. observations, i suppose: the lazy kind of maths, where you ..

• trace through proofs and examples that you already know, 
• assume all the hypotheses you need to work through a special case, 
• trying, all the while, to identify the driving mechanism behind it all.

familiar terrain, you know?
no exploration of the unknown, no real struggle ..

.. and thus: nothing worthwhile earned.




[1] i've lost "theorems" before, due to sloppiness, which would make the graph decreasing on some sub-intervals.

in which i suggest an "N" instead of a "p" ..

while browsing the title/abstract of a preprint of cheη, pοnnusamy, and waηg, i read this excerpt ..

"In this paper, we investigate the properties of $p$-harmonic mappings in the unit disk $|z| \leq 1$. First, we discuss the convexity, the starlikeness and the region of variability of some classes of $p$-harmonic mappings."

.. and became excited:


new geometric results about $p$-harmonic mappings?
awesome! 

so i clicked on the PDF and read the first page ..

yeah, yeah, $p$-harmonic functions solve $\Delta^pf = 0$  ..
wait: why should $f$ be be $C^{2p}$-smooth?

and then i realised that i mistook a superscript for a subscript.  you see, these authors mean $p$ as an exponent for composition,
$$\Delta^pf \;=\; (\underbrace{\Delta \circ \cdots \circ \Delta}_\textrm{ $p$ times })f$$
whereas the $p$-Laplacian that i know and love from the literature is a nonlinear operator:
$$\Delta_pf \;:=\; \operatorname{div}[|\nabla f|^{p-2}\nabla f].$$

[sighs]
either i need more sleep or more coffee, today.

a mathematically historical whodunit? (fixed)

i'm used to historical notes where an established result in the west was known earlier to russian mathematicians .. such as the Cauchy-Schwarz-Буняковский inequality 
$$\vec{a} \cdot \vec{b} \;\leq\; \|\vec{a}\| \|\vec{b}\|$$ today, however, i learned that Егoров's theorem is supposed to be the Severini-Егoров theorem [1]!

if you believe the wiki, Severini published the result in 1911 in italian, whereas Егoров wrote his article in french in 1911, which was more widely circulated.

not being fluent in italian, i didn't bother to check the first reference.  it's also worth noting that Lebesgue wrote his ph.d. thesis "Intégrale, longueur, aire" in 1902 ..

.. so in that 8-year gap, could someone else have proven it first?

at any rate, enough speculation: i still have some LaTeχing to do .. (-:



[1] never mind the fact that Его́ров's theorem is usually listed as one of "Littlewood's three principles" .. so maybe the russians lost out again? \-:

an article link about fractions; also, my wistful memories of boyhood.

interesting!  to improve arithmetic skills, they're channeling another means of intuition for children:

Math teachers know that fractions can be hard for the average third-grader. Teachers at a public school in San Bruno, Calif., just south of San Francisco, are trying something new. They're teaching difficult [0] math concepts through music, and they're getting remarkable results.

~ from "fractions curriculum strikes right note in CA" by caitlin esch (NPR).

there's more to the story, of course:

In this class, more than half the students are English-language learners. Classroom teacher Gina Grites says Academic Music has especially helped her students, such as a little girl who volunteers to solve a problem, even though she barely speaks English.

"She knew that it was four counts and put it into a fraction," Grites says. "Just to have her get up and present in front of a class is a really big deal, and she raised her hand and wanted to, so I'm seeing a lot of these kids open up and want to try it, instead of hiding behind the desk and saying, 'Please don't call on me.' "

now that is awesome: learning lessons independently of language .. well, not completely, but it's a good plus to the program.

another point of interest is that there is a basic level of fun involved: making music.  it naturally incorporates a change of pace, breaks away the tedium of rows upon rows of numbers.

that reminds me of when i was a boy, and computers were new to schools.  i loved playing number munchers during Computer class [1].

also: those damnded troggles!
(i.e. the purple dudes on the top row.)

i always came in second or third; there was this one kid that i could never beat, though i could spell better than him .. not that spelling matters anymore.

oh well: one less thing to worry about, i guess.

---

i think i'm getting .. not old [2], but set in my ways. for example, it wouldn't occur to me that

"If you say, what's bigger one-eighth or one-fourth, they'll say one-eighth because eight is bigger than four," Courey says. It's all about visualization.

*blinks*
unreal: even as a kid, this was crucial stuff!

i always imagined a pizza, how many slices to cut it,
and how much of a share i would get! q-:



[0] the author's words, not mine. perhaps we shouldn't dismiss fractions so quickly, though: i know plenty of adults who are bad with them ..

.. come to think of it, the expression $$\frac{a}{b} + \frac{c}{d} \;:=\; \frac{ad+bc}{bd}$$ looks like a messed-up determinant!

[1] i remember a classroom containing three rows of apple IIe's, and being told to be very careful with the startup disk, because the computer wouldn't load without it.  the punishment was to get a zero for the day and sit with a blank screen, while everyone else got to play.  (needless to say, everybody was incredibly careful.)

on a related note, yes: in elementary school, we had a specific class to learn how to use computers. later in junior high, i took a required typing class, and it sounds exactly like what it is: you were graded for how well you could type.

[2] if i call myself old, then by the transitive property, some of my colleagues which are older than me, are also old.  being that they'd give me an earful about this, i can't possibly be "old" quite yet. q-:

growing pains, ill-fitting hats.

it's 16:30 on thursday [1], the building's pretty empty, but sunshine is still streaming from the windows in my office.  (good friday is to-morrow: a national holiday in finland.)

american or mathematician, some habits are hard for me to shake;
i still have some LaTeχing to do.

---

it's slowly sinking in: i have a job for next year.

it's job security for the next six years, for that matter .. maybe for the rest of my life, too.  i don't know yet.  it's conceptually relieving and simultaneously unnerving.

i've never had to "plan" for this long of a period, before.  grad school may have taken 5 years, but it was a well-structured program and i was following a standard template.  barring the creativity of research, it wasn't like i had to think about my life and what to do next.

the future's always been wide open, but it's more widely open now.

it's like having money in the bank; maybe i earned it, maybe i inherited it, but none-the-less the account's in my name.  with that many years to spend, i could easily be stupid, waste a lot of it, gain little, and accomplish nothing .. in so many different ways.

ye gods: i might have to grow up and be responsible, after all.

---

when i called my parents to tell them ..

me: .. so i found a position.
dad: another postdoc?
me: no, it's a tenure-track this time.

[brief pause]

dad: wait, you mean that you'll be a professor?
me: well .. technically, "assistant professor" ..

[distinct shouting in the background: "our son's going to be a professor!"]

*cringes*
*sighs*

i guess it's true.  i mean: it is true .. but it feels odd.

come to think of it, a lot of my colleagues and good friends are "professors" but i never really thought about it.  they're just my friends, you know?

the title of "professor" just seems loaded to me.  maybe i've adopted the european perspective too much.

it's not like i completed a habilitation or anything;
i just happen to have a new job, that's all.

my finnish colleagues would correctly say that i'm becoming a "lecturer" next fall .. which, of course, means something completely different in the states.

i could swear that i used to be just janus;
it was hard enough to be dr. geminus [2] ... but "prof. geminus" ..?

..
..

also, stay tuned: at some point i'll post about:

1. why i've been so vague about announcing the good news,
2. the complaints i have about having been on the market, this year and last.

[1] as a cultural difference from the states, business hours in finland tend to be from 8:00 to 16:00.

[2] i'm not the first "dr. geminus" in my immediate family.  suffice it to say that few of us know how to get a real job. q-:

mildly mathematical: from words to graphs ..

the geeky part of me thinks that this looks like a really cool graphic:

apparently, though, it is called a time worm.

it's an image collected from experimental data by Deb Roy from MIT, who collected 200 terabytes of data "to capture the emergence and refinement of specific words in [his] son’s vocabulary."

"In one 40-second clip, you can hear how “gaga” turned into “water” over the course of six months. In a video clip, below, you can hear and watch the evolution of "ball.""

[more @ MIT Scientist Captures 90,000 Hours of Video of His Son's First Words, Graphs It]

i'm not usually one for empirical science, but this is really cool.  the most interesting part about it (to me) is that it accounts for geometry .. and hence the relevance of time worms:

In a landscape-like image with peaks and valleys, you can see that the word “water” was uttered most often in the kitchen, while “bye” took place at the door.

The video was processed to show "time worms," below, charting the family's movement from room to room.

as for what the seed input was, the equipment consists of "fish-eye" lenses like this:

image courtesy of prof. deb roy and his research group
(and/or fastcompany.com)

the set-up is, however, a little creepy and bigbrother-ish:

From the day he and his wife brought their son home five years ago, the family's every movement and word was captured and tracked with a series of fisheye lenses in every room in their house. The purpose was to understand how we learn language, in context, through the words we hear.

i guess it's not really an invasion of privacy, since the experimenter volunteered his own family .. still, potentially being recorded for 5 years straight .. think about all the gossip you might have shared, or private opinions that you would prefer to keep private!


i think i'll stick to my abstract theories and notebooks, thank you! q-: