Friday, March 29, 2013

under "house arrest" .. so to speak.

some european traditions still strike me strangely. despite a general agnostic sentiment here in finland, the nation still celebrates the christian holidays as official ones.

so today, being good friday and three days before easter, is one of those holidays. next monday is also off, too. as a result ..
the university buildings are locked,
i forgot the numerical entry code for mine,
most shops and cafes are closed,
and so i'm "stuck" working at home.
at this point one wonders if i should just stop working for once [1] and just enjoy the (forced) time off.

thinking about it, it feels like ages since i've just .. not gotten up in the morning, immediately thought about the research problem or task at hand [2], and barring other immediate promises and responsibilities, dwelt on it for the rest of the day.

who knows? maybe a change of scene will do me some good. there's still most of the day left, here in helsinki. it's been ages since i've been to the cinema, for one thing. it's not hard to take a ferry to estonia and see the old town there ..

mathematics tends to be an obsessive kind of habit;
the key, i suppose, is the right distraction ..

[1] it could be just the colleagues that i know, but there's a common folklore among us that academics work all the time. (compare this with the general populace, who think that university faculty are lazy and only work "8 months of the year" ..!)

related to this point, i don't think that i work all the time; for instance, after returning from the office, i would never work at home in the evenings .. well, until recently anyway. i get the impression, though, that other people think i work all the time. two fellow postdocs pointedly asked me about my plans for the weekend, for instance ..

[2] on a related note: somehow over the years, i became a morning person. figure that out!

Tuesday, March 26, 2013

switching gears, drawing pictures.

this morning i felt like a change, so i decided to think about something else for a while ..

.. something, in particular, that i could draw.

(maybe the abstraction of baηach spaces is starting to get to me; besides, i promised a co-author or two that i'd prove this one particular lemma .. let's hope it's still true!)

Monday, March 25, 2013


well, it took me more than 18 months .. but today i finally got around to getting a university library card. as a result, immediately afterwards i sped to the analysis shelves and made out like a bandit!

if you're curious, here's my loot:

my only regret is that measure theory and fine properties of functions by evans and gariepy wasn't available .. which isn't surprising. for one thing, it's almost impossible to find that book for sale.

(i'm starting to think that to find a reasonably priced copy, i'll have to inherit it from someone's estate!)

on an unrelated note, often i try to cite the most original sources as possible .. which often leads me to wild goose chases. figuring out, for example, who was first to characterise the dual of $L^\infty(X,\mu)$ --- say, for σ-finite measures $\mu$ --- is taking longer than i thought.
so far, i've traced it as far back as a transactions paper of Hildebrandt from 1934 .. in the case of $X$ being an interval on the real line, anyway.

it's never that simple, of course. in the same year there is a competing paper from studia mathematica by Fichtenholz and Kantorovich, who treat the same setting.
at this point, i wouldn't be surprised if the result can be found in lebesgue's thesis .. or, for that matter, on a bit of scroll from the days of archimedes!

MoAR: bare-bones, this week.

this week's roundup is a bit sparse, without too much deep commentary. i guess i've been distracted, if not busy, with writing and editing a manuscript or two, and havne't had too much spare brainpower to ponder meaningful opinions to these things i've read.

MOOCs: yes's and no's, from the faculty.

i should stop posting article excerpts about this topic. the phenomenon is in full swing, but it's too early to conclude anything about it. for instance, there is no first generation of MOOC college "graduates" yet, so we don't know how stable the framework is.

it's also not clear to me if the objectives for MOOCs have been fully made clear and compatible to the public. as for the faculty, however ..
"John Owens was drawn to MOOCs because of their reach. He also did not want to be left behind... It does not take a programming expert to decrypt the writing on the wall: No matter where you teach, online education is coming. "I would rather understand this at the front end," said Mr. Owens, "than be forced into it on the back end."
As far as awarding formal credit is concerned, most professors do not think their MOOCs are ready for prime time. Asked if students who succeed in their MOOCs deserve to get course credit from their home institutions, 72 percent said no.

~ from "The Professors Who Make the MOOCs" @the_chronicle

beyond public key encryption.

well, if number theory can encrypt data, then i suppose that abstract algebra can improve things further ..
"The numbers in the file remain encrypted at all times, so Bob cannot learn anything about them. Nevertheless, he can run computer programs on the encrypted data, performing operations such as summation. The output of the programs is also encrypted; Bob can’t read it. But when he gives the results back to Alice, she can extract the answer with her decryption key.

The technique that makes this magic trick possible is called fully homomorphic encryption, or FHE. It’s not exactly a new idea, but for many years it was viewed as a fantasy that would never come true.

~ from "Alice and Bob in Cipherspace" @americanscientist

beyond the tenure track.

first, the excerpt ..
"Enter “Beyond Academia,” the first career conference at the University of California, Berkeley, organized solely by Ph.D. students and postdoctoral fellows, an unlikely group for a non-academic job fair. The sold-out event — to be held in Berkeley this Friday, March 22 — is a quiet revolution if one considers the investment of time and money that goes into grooming a grad student for a tenure-track position.

“There are Ph.D. students who feel they can’t come out and say they want to leave academia, they’re too afraid,” said Els van der Helm, a fifth-year Ph.D. student in psychology and lead organizer of the conference. “This will give them a chance to explore other options. We have to start a conversation about this because academia is not for everyone.”

~ from "Ph.D. students rethink the tenure track, scope out non-academic jobs" @newscenter.berkeley
and now, a few comments: academia is self-selective. the faculty that one meets during one's ph.d. are in fact the survivors, and those in their cohort who "didn't make it" are unrepresented and probably un-measured.

so despite the motives for this kind of conference, it kind of makes sense. it's not like one's ph.d. supervisor would have a lot of information about non-academic jobs, because (s)he's spent an entire career gearing for the opposite.

hacking creativity.

part of me hopes that the creative process will always remain a mystery, with a level of randomness and experience that makes it fundamentally human. on the other hand, if someone could give me a recipe for being able to prove the theorems i want .. then yes: i'm sold! (-:

at any rate, the human brain is a rather interesting entity, and these excerpts are about its creative impulses.
"Time is the raw material of creation. Wipe away the magic and myth of creating and all that remains is work: the work of becoming expert through study and practice, the work of finding solutions to problems and problems with those solutions, the work of trial and error, the work of thinking and perfecting, the work of creating. Creating consumes. It is all day, every day. It knows neither weekends nor vacations. It is not when we feel like it. It is habit, compulsion, obsession, vocation. The common thread that links creators is how they spend their time. No matter what you read, no matter what they claim, nearly all creators spend nearly all their time on the work of creation. There are few overnight successes and many up-all-night successes."

~ from "Creative People Say No" @medium

once again, a few more thousand words.

this time around: one is a work of art, the other an infographic.

it is possible to have rows and rows of windmills, occupying surface area .. but at this scale, i think length is slightly more meaningful.

[ from the land art generator archive, with self explanatory title ]

Friday, March 22, 2013

in-medias-res: tired

some days ago, i told my friends that i won't work this weekend .. and, of course, i meant it at the time. (the last two weeks have been a lot of scratchwork on paper and a lot of $\LaTeX$, and i may have said that i need a vacation.)

i'm not so sure anymore.

sure, i still feel like i need a vacation .. but if tomorrow morning i wake and i want to attack one particular research problem or polish the manuscript .. then, why not?

i've re-read the previous passage and it sounds like bravado or machismo.
that's not my intention, though.

the thing is that i never fully plan out a schedule for the days of the weekend; if i do, then usually it's in the evening when the only viable options are to meet friends or to go to bed early.

an unhindered day like saturday often becomes a very attractive time, say several multi-hour blocks, to work something out and even to type up the associated ideas (which, if you're ambitious, might one day form the basis for another research article). while at home, nobody at the department ever contacts me, and the situation lends itself to complete vegetation. despite not being obligated to work, it is that selfsame flexibility for work (or its absence) that drives it progressively ...

Wednesday, March 20, 2013

argh .. hmph!

for a while this afternoon, i kept getting compiler errors with my $\LaTeX$ and i couldn't figure out what was wrong ..

wtf? what's wrong with my \begin{itemise} command?

it took a lot longer to figure out than i care to admit,
and maybe i've been away from the u.s. for too long ..

.. but apparently, it's spelled itemize ..! 7-:.

on a related note, i am aware that brackets of the form \[ and \] can be use in place of dollar signs $\$$, in order to render $\LaTeX$, and it may even be the best practice, these days ..

.. but the truth is that i can't stand them.
there: i said it! for me, it's always dollar signs!

as a result, i always work with with u.s. layout on a finnish keyboard .. which causes no end of typing troubles, as the keys say one thing but pressing them gives another.

the trouble really comes when, at some point, my fingers stop moving on their own and i have to think about which one is which ..

// added: 10:49EST, same day

i'm running out of bracket symbols: argh!
curly brackets are typically used for sets ..
square brackets are already used for equivalence classes,
i'm already using open paren's for pairs of objects,
and double square brackets represent the induced current of the associated object.
now i have to write down the quotient norm of an equivalence class of a pair of finitely-additive measures, which so far looks ugly: $\| [ (\mu,\nu)] \|$.

ugly, ugly notation .. but it's a technical lemma, so maybe the referee will forgive me ..?

Tuesday, March 19, 2013

... "i sat in abject horror, my mathematical blood ran cold" ...

I will show you something different from either
        Your shadow at morning striding behind you
                Or your shadow at evening rising to meet you;
                        I will show you fear in a handful of dust. [0]

you know, i had always been dismissive of those constructivists who, among other things, refuse to accept the axiom of choice as part of their proof-writing toolbox: a bunch of mealy-mouthed naysayers and contrarians, i thought!

it had always seemed to me a handy, albeit strange, tool .. but it gets the job done, right?
one of its consequences, the hahη-baηach separation theorem, is incredibly useful .. if not highly magical and never leading to any concrete example. if i can't build something by hand, then usually i use hb.

similarly, for me baηach-alaοglu is like crack: i think i have some mental addiction to weak-star convergent subsequences .. or, if the situation calls for it, nεts. [1]
the last few days have been conceptually difficult .. to the point where i thought i stumbled onto either a paradox or a counterexample to one of my own results.

i felt like my imagination was being stretched to its (very limited) capacity and that i was teetering over the edge of conventional sanity. this afternoon i attained some kind of resolution, though. at the same time,
i sat in abject horror,
my mathematical blood ran cold,
and i was ready to throw down some printed pages to the ground,
step on them repeatedly,
and immediately afterwards, run away, screaming.
fortunately (for my officemate, anyway) and as creepy as this feeling was, i restrained myself and sat calmly at my desk, trying to look at the bright side:

well, at least the theorem's not wrong.

for the record, i've been thinking (too much and too often) about the dual of the βanach space of functions $L^\infty(\mathbb{R}^n)$, which consists of bounded, fιnitely-additive sιgned measures on $\mathbb{R}^n$ that vanish on sets of leþesgue measure zero.

if you have never been curious about these objects, then don't start now. seriously.

although they have a not-unnatural role in functional analysis, my obsession with them has gotten to the point
where i think i have become a generally worse person and perhaps less human.

i learned of the following results from an old paper of hewιtt and yοsida, called finitely additive measures from 1952. in that sense, it reads like a gothic novel .. all quiet and calm at first, and then the monsters come.
theorem 3.3: fubini's theorem fails for finitely additive measures in $[L^\infty(\mathbb{R})]^*$.

theorem 3.4: there exists a nonzero finitely-additive measure $\zeta$ on $\mathbb{R}$ so that $$ \int_{-\infty}^\infty c(t) \, d\zeta(t) \;=\; 0 $$ for all bounded continuous functions $c$ ... and any such measure $\zeta$ must be purely finitely additive.

roughly speaking .. by "purely finitely additive" here, they mean that the object cannot have any nonzero part that behaves like a usual (countably additive) measure. in other words, it's a distinctly exotic object.

theorem 3.6. for any real number $a$, there exists $\zeta_a$ in $[L^\infty(\mathbb{R})]^*$ so that $$ \int_{-\infty}^\infty x(t+u) \, d\zeta_a(u) \;=\; x(t+a) $$ for all essentially bounded functions $x \in L^\infty(\mathbb{R})$ and a.e. $t \in \mathbb{R}$ [2].

keep in mind that $\zeta_a$ is not a point-mass at $a$. in particular, it vanishes on all βorel sets of leþesgue measure zero! (this, by the way, was something close to the paradox i had in mind, believing that it was impossible ..)
thinking about it, the results aren't that much more surprising than the banach-tarsκi paradοx. then again, it's been an obsessive week or two.

i think i need a vacation.

[0] re-reading this line by eliot, notions like "cantοr dust" and singular measures come to mind.
[1] not everything in life is metrisable, you know. for some reason, my work has taken me to these exotic locales, lately.
[2] the original statement was over-simplified. thanks to L for pointing this out.

initiating self-destruct countdown: ... 6, 5, 4 ...

.. a week or two ago, i had an idea for a new theorem;
yesterday i was about to put the polish on the proof ..

.. and this morning, i almost constructed a counterexample for it,
with an emphasis on the word "almost" ..!

i guess it's another session at the library today ..

the more i learn about duality in Banach spaces, the subtler it seems to become.

Monday, March 18, 2013

MoAR: so that you can get some work done, today ..

.. i've narrowed down this week's roundup to only a few shared posts, this week. enjoy!

i can't tell if it's the same graph as before ..

.. but there's bad news for STEM ph.d's out there. when they said that they want more graduates working in science and engineering, maybe they meant only undergraduate degrees?
"Jordan Weissmann, an editor at The Atlantic, analyzed the latest NSF figures. Upon graduation, he says, "Ph.D.s in general have a less than 50 percent chance of having a full-time job, and that percentage has been decreasing for about 20 years."

Worse yet, as of 2011, approximately one-third of people graduating with a doctoral degree in science, technology, math or engineering had no job or post-doctoral offer of any kind.

~ from "Are There Too Many Ph.D.s And Not Enough Jobs?" @npr

a controversial issue: gender in maths, worldwide.

well, 1.5 million data points sound like a lot. so provided they accounted for the usual national, societal, and social factors (e.g. percentage of girls that have access to primary education), i'd say that the result is rather striking.
"We did not find a sex difference in mathematics among the lowest performing students, but this is where the sex difference in reading was largest. In contrast, the sex difference in mathematics was largest among the higher performing students, and this is where the sex difference in reading was smallest. The implication is that if policy makers decide that changes in these sex differences are desired, different approaches will be needed to achieve this for reading and mathematics. Interventions that focus on high-achieving girls in mathematics and on low achieving boys in reading are likely to yield the strongest educational benefits."

~ from "Sex Differences in Mathematics and Reading Achievement Are Inversely Related .." @plos-1

two different kinds of advice.

the first bit of advice is about .. well, advising.  (it reminds me of the mind-set of grant writing, actually.)
"A project that is going to take eight years of construction work before it produces any scientific results cannot and should not be built by a PhD student. On the other hand, a project that dries up in two years is equally bad. In other words, no matter what idea I come up with, I need to be able to say that all the candidates I hire should find enough material to write a thesis and graduate—no matter what the experimental outcome.

This means that any big idea I come up with also needs to be partitioned into chunks of the right size. If it can't, then it doesn't work in an academic institution. Since all experimental results need to be thesis-worthy, the questions I want to answer should be open enough to accommodate failure. For instance, my ideas are often based on a single experiment: if we conduct experiment "a," we could measure property "b," and that would be so cool! But, what if "a" doesn't work? Does the student go home?

~ from "From idea to science: Knowing when you’ve got a good idea" @arstechnica
the next bit of advice is about writing .. or more precisely, rules about storytelling. here are two:
"8. Finish your story, let go even if it’s not perfect. In an ideal world you have both, but move on. Do better next time.

9. When you’re stuck, make a list of what WOULDN’T happen next. Lots of times the material to get you unstuck will show up.

~ from "Pixar’s 22 Rules of Storytelling" @aerogrammestudio
i like to think of #8 as "just submit the damned paper!" and #9 as 'it never hurts to try building counter-examples' .. (-:

lastly, another thousand words.

~ from "Inside Wonderland" by Jaume Plensa

Thursday, March 14, 2013

in which i feel like a fake .. (UPDATED)

sometimes i really question what i'm doing .. if i've really turned to the dark side of the force ..
  1. on a set $X$, complement and union, strictly speaking, form .. well, a field on the collection $P(X)$ of all subsets of $X$. it happens that i'm in the setting where there is another set $X_1$ for which there is an .. er, field isomorphism from $P(X)$ onto $P(X_1)$ ..

    .. and that's not all; this isomorphism gives rise to an isometric isomorphism from finitely-additive measures on $X$ to finite Borel regular measures on $X_1$.

    ye gods: with words like field and isomorphism (and there's that hidden word functor around, somewhere) .. should i just hand in my resignation and rename this blog to the "frustrated pseudo-analyst" ..?

  2. earlier today, i wrote that
    "as usual, a measure here refers to a non-negative signed measure."

    re-reading that sentence later, i felt slightly disgusted with myself .. and wondered how i came to this sorry state in my life.
// updated: 15 march 2013 @11:04EEST

the last few posts have been both strange and technical. i blame this on having been fully immersed in solving a recent problem .. to the point where i probably don't make much sense to people, including friends.

in fact, all this week it's taken quite the effort to switch from "maths mode" back to "human mode." i think i'm better now ..

.. well, depending on your definition of "better" ..! (-:

Tuesday, March 12, 2013

the screen sucks away my soul.

earlier, staring at the $\LaTeX$ code on my computer screen, i despaired of it never being finished ..

it wasn't until i compiled it again and looked at a printout that i felt better, even when i started marking typos, errors, and fixes on it. $\LaTeX$ really is a wonderful markup language, making neat and elegant these symbols and scrawls ..

.. which still need work, of course. the lemmas and theorems are taking shape, though, and i can see more clearly a clean, rigorous end to it.

Monday, March 11, 2013

MoAR #10: not five articles, but five sections.

maybe i should spend less time on the internet .. because it's getting harder to keep my word. as you may recall from the first roundup, i promised that i'd only share the top 5 articles in any given week.

well, this week there are considerably more than that .. but if this were more like a newspaper, then they do fit neatly into five different sections, each with at least three articles.

oh well;
today being a monday, you didn't expect to get any work done, right? (-:

the Technology pages:
collective search for research, and how computers see.

odd: it seems that the best implementations of machine vision do not involve making the machine able to see well .. but instead, making it good at remembering what it (or its predecessors) have seen before.
"The latest work on the Kinect uses the same sort of machine-learning approach to distinguish between an open hand and a clenched fist. Although there are no details, its general method was to use a large number of images of people's hands and supervised training to distinguish between open and closed hands. The learning algorithm is based on a forests of decision trees, which is the same general method used to implement the skeleton tracking."

~ from "Kinect Can Detect Clenched Fist" @i-programmer
so in some ways, it's not really "vision" .. but computers taking advantage of what things should look like (instead of what they "really are" .. whatever that means).

going another direction, computers and networks do have a knack for putting things together at a scale of complexity that is otherwise hard for mere humans to fathom, much less detect.
"Enter search engines. Much like Google Flu Trends reveals influenza outbreaks by tracking flu-related search terms, search queries about drug combinations and possible side effects—say, "paroxetine," "pravastatin," and "hyperglycemia"—might enable researchers to identify unanticipated downsides to medications, says bioinformatics researcher Nigam Shah of Stanford University in Palo Alto, California. "If a lot of people are concerned about a symptom, that in itself is valuable information."

Although many bad reactions to drugs never get reported to doctors, people talk about what's bothering them all the time on a casual basis to their friends or online, notes computational biologist Nicholas Tatonetti of Columbia University, who was also involved with the study. "They don't really know," he says. "They're just reporting on their symptoms, which is just a normal thing that humans love to do.""

~ from "Should You Mix Those Two Drugs? Ask Dr. Google" @sciencenow
lastly, part of why i support the open source movement is its independence.  it is a not a corporation dictating the terms of service to you (though that happens often in other situations).  it is about people building software and solutions for other people.  the profit incentive does drive innovation .. but being ultimately economical in nature, it has its limitations: if it is not cost-effective and/or profitable, then corporations won't touch it ..

.. even if society has a need for it.  enter, then, the hackers: corporations may be efficient in what they do, but what they do does cover everything a society needs.

this isn't their fault, either: it's an issue of their self-preservation;
sometimes the people must take the impetus to help themselves.
"... the pharmaceutical industry isn't interested in paying for compute time—or any sort of research at all—on many rare and not-so-rare diseases, simply because the potential financial payoff for finding a drug is so low. While researchers working for universities, disease advocacy groups, and other nonprofit organizations have found thousands of target proteins for rare and "neglected" or "orphaned" diseases such as malaria, they have not had the resources to use software such as TerraDiscoveries' to look for good drug candidates.

That's where the Quantum Cures effort comes in. TerraDiscoveries is providing its software for free to the organization and is repackaging it for use on individual Windows, Mac OS, and Linux PCs as "screen saver" software. The software, which will be available starting in June for download, installs with user-level permissions and will allow individuals to set how much of their compute time is made available. Husick said that a version for Android and Apple's iOS will follow later in the year, allowing individuals to donate time on their mobile devices."

~ from "Crowdsourcing the cloud to find cures for rare and “orphaned” diseases" @arstechnica

the Tabloid pages:
a scandal, an accusation, and a ~100year old op-ed

my mind is split about this first article. on one hand, students should expect to be treated fairly.  on the other hand, if you're going to be taught by a world-class researcher, then you can't possibly expect to be given your professor's full 100% attention.  these are men and women who are not formally trained in pedagogy, but through training and experience have among the sharpest minds in science and letters.

there are plenty of small, liberal arts colleges full of faculty who are concerned only with education and the well-being of students.  if a student really wanted this kind of experience, then why wouldn't they go there, instead?

at any rate, the tone of this writer suggests that harvard students will do enough, but not more.  in that sense, they are clever and efficient young people.  i would have thought, however, that they would be suspect that they are more intellectually driven that that and more than mere mercenaries .. but not being present on their campus, the educational climate remains a mystery to me. [1]
"“The modal Harvard student takes their courses as seriously as they think the instructor is taking the course,” says Lewis, now Director of Undergraduate Studies in Computer Science. “Situations in which you get multiple cases of breach of academic integrity [are] where there is a general consensus or a general feeling in the student body that the instructor is somehow behaving dishonorably, or unfairly, or bizarrely in some way that is contrary to Harvard students’ general expectation that if they’re going to have to work hard, the professor should damn well seem to be working hard too.”"

~ from "The Fall of Academics at Harvard" @thecrimson
taking a different turn, one point that i think many people overlook is that philosophy isn't supposed to be science.  historically, science is the successor to the so-called "natural philosophy" of the 18th and 19th centuries, made systematic through quantitative and repeatable methods.
"If so, then we are duped and bound to be disappointed, says Wittgenstein. For these are mere pseudo-problems, the misbegotten products of linguistic illusion and muddled thinking. So it should be entirely unsurprising that the “philosophy” aiming to solve them has been marked by perennial controversy and lack of decisive progress — by an embarrassing failure, after over 2000 years, to settle any of its central issues. Therefore traditional philosophical theorizing must give way to a painstaking identification of its tempting but misguided presuppositions and an understanding of how we ever came to regard them as legitimate."

~ from "Was Wittgenstein Right?" @nyt:opinionator
philosophy is different.  its generality is astounding in that it encompasses science, mathematics, art, and the social sciences.  by its nature it is therefore messy.  you have to deal not only with atoms and/or mitochondria .. but with humans and not only their observable and measurable behavior, but their wants and desires and inclinations .. however irrational.

the critics of philosophy are also ignorant of another point: time scales and evolution.

the body of knowledge that is relevant to philosophy has only increased with time. in the late 19th century, for example, the very human discipline of sociology was born. in the last 15-20 years, there has also been a systematic study of the psychology of motivation; referring only to what little populist books and articles i've read, motivation is a highly non-intuitive phenomenon that hardly fits perfectly with, say, first-order symbolic logic.

think of it this way: if you can't trust economics to make consistent, unequivocal predictions .. then why are you insisting that something of a more general framework be even more precise, especially when it becomes more general with time? we already have a well-established discipline called science, so why are you trying to insist that philosophy narrow down to one of its subcategories?

lastly, here's an excerpt from chesterton:
"To begin with, of course, there is no such thing as Success. Or, if you like to put it so, there is nothing that is not successful. That a thing is successful merely means that it is; a millionaire is successful in being a millionaire and a donkey in being a donkey. Any live man has succeeded in living; any dead man may have succeeded in committing suicide. But, passing over the bad logic and bad philosophy in the phrase, we may take it, as these writers do, in the ordinary sense of success in obtaining money or worldly position. These writers profess to tell the ordinary man how he may succeed in his trade or speculation—how, if he is a builder, he may succeed as a builder; how, if he is a stockbroker, he may succeed as a stockbroker."

~ from "The Fallacy of Success" by g.k.chesterton.
the tone of this almost sound fatalist, but he makes a good point.  as the quote goes: "the world is all that is the case."  to figure out causality is quite hard .. assuming that there is even a clear causality in a given situation.  how would one discount the possibility that a colleague's success was not, say, due purely to luck?

highlight on Education: on- or off-campus?

simply put, perhaps education should not refer to what, but how.
"Institutions of higher learning must move, as the historian Walter Russell Mead puts it, from a model of "time served" to a model of "stuff learned." Because increasingly the world does not care what you know. Everything is on Google. The world only cares, and will only pay for, what you can do with what you know. And therefore it will not pay for a C+ in chemistry, just because your state college considers that a passing grade and was willing to give you a diploma that says so. We're moving to a more competency-based world where there will be less interest in how you acquired the competency -- in an online course, at a four-year-college or in a company-administered class -- and more demand to prove that you mastered the competency."

~ from "The Professor's Big Stage" @nyt
as for the actual logistics of education: i'm all for the e-book revolution, as long as it means that students have more access to resources.  there's a catch, though: what if you need the internet to do your homework, but you have no internet?
"The e-textbooks used in the project, run by the Fairfax County Public Schools, worked only when students were online—and some features required fast connections. But it turns out that even in such a well-heeled region, many students did not have broadband access at home and were unable to do their homework, sparking complaints from parents that led the school system to approve the purchase of $2-million in printed textbooks for those who preferred a hard copy."

~ from "'Bandwidth Divide' Could Bar Some People From Online Learning" @the_chronicle
lastly, having spent more than a year in finland and spent an extended work trip to spain, the economic advantages to staying with family and attending the local university are becoming more apparent to me.  i'm all for independence and self-realisation .. but not every parent is the helicopter sort.  as long as the university is good enough, why spend additional precious resources just to move away?
"That approach might also help address one of the most serious potential objections to the idea of killing off our current aid system: that it could accidentally make school more expensive for some of the poorest families. The reality is that tuition is not the biggest expense for most full-time undergrads at public colleges: It's cost of living. After all the other aid low-income students receive, Pell Grants and tax breaks often end up paying for their meals and rent. But if colleges continued charging at least some tuition for wealthier students, the money could be cycled back into living expense grants for the neediest"

~ from "How Washington Could Make College Tuition Free (Without Spending a Penny More on Education)" @the_atlantic
in fact, i wonder if the family environment would make a good, stabilising effect.  college is fundamentally different from high school, with all kinds of new aspects of life and learning.  why not keep intact something comfortable from your previous stage of life, then?

the Feature pages:
celebrating int'l women's day

i can't tell whether (a) there have always been women in science, but just unreported due to the societal mores of the time, or (b) women were that effectively blocked from those areas of study and work. (it's probably both.)

little by little, i think mathematics is paying homage to the great female mathematicians, after centuries of acknowledging men only; of those whom come to mind, i remember  kovalevskaya, ladyzhenskaya and uraltseva, noether, uhlenbeck, and daubechies. my hope is that the next generation achieves even more greatness.

as for some lesser known female researchers ..
"Cartwright herself was always somewhat diffident when asked to assess the lasting importance of her war work. She and Littlewood had provided a scientific explanation for some peculiar features of the behaviour of radio waves, but they did not in the end supply the answer in time. They simply succeeded in directing the engineers' attention away from faulty equipment towards practical ways of compensating for the electrical "noise" - or erratic fluctuations - being produced."

~ from "A Point of View: Mary, queen of maths" @bbcnews
"Up until this point Hopper had worked on punch card programming, but with the move she now began to program in C-10, which required her to learn octal, the base-8 number system. But it wasn't a great solution and she wanted to simplify the programming system.To this end, in 1952 she invented the first compiler, A-0, which translated mathematical symbols into machine code, and updated the system with A-1 and A-2 the following year. "Nobody believed that," she said. "I had a running compiler and nobody would touch it. They told me computers could only do arithmetic; they could not do programs.""

~ from "On International Woman's Day we remember Grace Hopper" @theregister
"Lamarr realized that by transmitting radio signals along rapidly changing, or "hopping," frequencies, American radio-guided weapons would be far more resilient to detection and jamming. The sequence of frequencies would be known by both the transmitter and receiver ahead of time, but to the German detectors their message would seem like gibberish. "No jammer could detect it, no German code-breaker could decipher a completely random code" .."

~ from "Hedy Lamarr: Not just a pretty face" @sciam
to explain, lamarr was in fact a pretty face: she was a hollywood movie star who decided, in her spare time, to work on signal processing.  (having seen a few of her films, i can testify that she is a terrific actress.)

man: beauty and brains!

the Science section
progress and its discontents.

what i want to know is: if there are three bodies in the system, how many parameters are there, and why would they combine into 16?
"Specific repeating solutions have been hard to come by, however. The famed mathematicians Joseph-Louis Lagrange and Leonhard Euler had come up with some in the 18th century, but it wasn't until the 1970s, with a little help from modern computing, that U.S. mathematician Roger Broucke and French astronomer Michel Hénon discovered more. Until now, specific solutions could be sorted into just three families: the Lagrange-Euler family, the Broucke-Hénon family, and the figure-eight family, the last of which was discovered in 1993 by physicist Cristopher Moore at the Santa Fe Institute.

The discovery of 13 new families, made by physicists Milovan Šuvakov and Veljko Dmitrašinović at the University of Belgrade, brings the new total to 16. "The results are beautiful, and beautifully presented," says Richard Montgomery, a mathematician at the University of California, Santa Cruz, who was not involved with the discovery."

~ from "Physicists Discover a Whopping 13 New Solutions to Three-Body Problem" @sciencenow
let me get this straight: as a community, you achieve the predicted goals from a generation ago .. and you're still not happy?  ye gods: what more could you possibly want?
"In the (unlikely) event that the particle doesn’t have a spin of zero, physics would be turned on its head! It would be a revolution — the universe that we thought we knew and understood is weirder than our weirdest dreams! In case you haven’t already guessed, this scenario excites physicists, a lot.

The current scenario (that’s looking most likely) would prove the existence of a particle theorized in the 1960s, thereby tying up the Standard Model of physics in a pretty, neat, red quantum bow. This is decidedly boring in comparison. What’s worse, of all the possible theorized Higgs species, what if the most “vanilla” option is proven to be correct?
And there you have a strange juxtaposition — a profound discovery that’s also an anticlimax."

~ from "Higgs-Hatin': 'Vanilla' Boson May Be Real" @discovery
you know what's even more inaccessible than physics?  maths!
"The language of physics is mathematics, and it cannot be done honestly without mathematics. That makes it inaccessible. The language of literature is English or Chinese or whatever, and that makes it accessible. And literature is about the human condition. Physics is about the nonhuman condition. It's not a taste that all human beings have."

~ from "Can Physics Experiments Catch Up?" (an Interview) @slate

[1] rereading the earlier version of this paragraph, it occurred to me that the article didn't ever indicate the nature of the course and its organisation .. or lack thereof. i still maintain that a good student (and imagine that harvard should be full of them) would rise to the challenge of accepting their own responsibility for their education, by picking up the organisational slack.

on the other hand, this reminds me of my early years in a ph.d. program, with weekly problem sets due on various days of the week. there is a difference, though: the faculty encouraged collaboration, as long as you wrote up your own solutions and, in some cases, referenced with which students you worked. in point of fact, i was once "cited" in a problem set from a course that i wasn't even taking!

Thursday, March 07, 2013

in medias res: a three-star rating .. (UPDATED).

// initially posted: 2013-03-07 @ 13:34EEST

today i took the double dual $(V^*)^{**}$ of a certain dual Banach space $V^*$.
it was rather confusing, but seemed like a good idea at the time ..

// updated: 2013-03-08 @ 13:30EEST

i think i've either jumped off the deep end or gotten nostalgic for topics i learned during my student days. in the last two weeks i've gone to the library four times to read the books on functiοnal analysιs by rudin and by dunford-&-schwartz.

more and more i am impressed by the latter volume, but that's more an artifact of my current mathematical necessities; lately, you see, i have become wont of more gory details regarding some seeming familiar Banach spaces. this sounds like pure folly ..

.. but i want to understand better the space $L^\infty(\mathbb{R}^n)$ and its Banach dual! [1]
there is already a handy characterisation, but it uses words like "finitely additive" and "set function" .. which, admittedly, worry me. the awesome thing about D&S is that they have an entire chapter written in terms of these things. apparently, most of the usual measure and integration theory runs analogously through.

more to the point, i don't have to re-prove everything myself; these guys are lifesavers!
on a related note, these two passages from D&S caught my eye [2].
  1. Theorem III.5.13 & 14 (Алексaндров). [If $\mu$ is] a bounded, regular, complex-valued, [finitely-]additive set function defined on a field $\Sigma$ of subsets of a compact topological space $S$, [t]hen $\mu$ is countably additive ... [Moreover] there is a unique regular, countably additive extension to the $\sigma$-field determined by $\Sigma$.

    i'm probably too naive, but it's hard for me to appreciate the difference between finitely additive measures and the usual (countably additive) ones. in light of this theorem, though, there are two points that come to mind:

    (A) once the set function is assumed regular --- i.e. that it has "good limits" for sets that fit the topology of the space --- then on compacta, there is no difference between finitely- and countably- additive. on the other hand, set functions that are strictly outside of $L^1(\mathbb{R}^n)$ must therefore lack these nice limit-properties, which means that they will be hard to work with!

    (B) this seems to be one of the rare criteria for checking whether a bounded linear functional of $L^\infty(\mathbb{R}^n)$ is actually a Lebesgue integrable function .. or, more precisely, part of a criterion. the theorem guarantees that we can work with measures as usual, so the Radon-Nikodym theorem would apply in this setting.

  2. some parts of the book provide suggestions and instructions for how to proceed. for example, on pg. 122, III.3.6 it reads:

    The reader will more easily perceive the significance of the somewhat complicated conditions (ii) and (iii) in the following [Theorem 6], if [s]he reads the statement and proof of Theorem 7 after the statement of Theorem 6 but before its proof.

    wait .. so why didn't they just switch the order of theorems 6 and 7? (-:

// updated: 2013-03-08 @ 17:30EEST

the further i read on, the more uneasy i feel. some of these constructions are just .. creepy: i can't think of another way to put it.

one result is outright unnerving:
Theorem IV.6.18-19: If $S$ is a compact Hausdorff space, then there exists a(nother) totally disconnected, compact Hausdorff space $S_1$ and an embedding $i: S \hookrightarrow S_1$ so that $i(S)$ is dense in $S_1$ and that induces an isometric isomorphism $$ i_*: L^\infty(S,\mu) \,\to\, C(S_1). $$
to get a sense of what i mean, one can take as nice of a space $S$ as possible and extend it to some $S_1$ so that $S$ is dense, yet incredibly scattered within $S_1$ .. to the extent that characteristic functions $\chi_A$ of subsets $A$ in $S$ extend to continuous functions in $S$.

the real kicker is that the extension is also a characteristic function of some subset in $S_1$ as well (IV.9.10). roughly speaking, then, the continuous extension is not done through interpolating between the values $0$ and $1$, but by sufficiently separating the preimage sets in $S_1$ between these values .. and none of the subsets $A$ are chosen in advance!

in short, $S_1$ must be a weird, messed-up space.

.. i think i'm going to stop reading for today.

[1] .. and in case you're keeping count, no: $[L^1_\mu(\mathbb{R}^n)]^{**} \cong [L^\infty_\mu(\mathbb{R}^n)]^*$ is only two stars. (i might get to the third one in a later update.)

[2] the square brackets []'s indicate my reformatting.

Rιemannian metrιcs for .. wait: what?

it's been a few days since i last checked the arXiv, so this title/abstract from a few days ago caught me by surprise:

Riemannιan metrιcs for neural netwοrks

We describe four algorithms for neural network training, each adapted to different scalability constraints. These algorithms are mathematically principled and invariant under a number of transformations in data and network representation, from which performance is thus independent. These algorithms are obtained from the setting of differential geometry, and are based on either the natural gradient using the Fisher information matrix, or on Hessian methods, scaled down in a specific way to allow for scalability while keeping some of their key mathematical properties.
well, if fisher information and shannοn entrοpy are involved, then the word "Riemanniaη" makes a little more sense, if only because of connections to log-Sobolev inequalities on manifolds ..

Tuesday, March 05, 2013

i have glimpsed the future. (it looks pretty cool.)

// initially posted: 14:46EEST
so today i gave an expository talk [1] ..
.. neither on a chalkboard nor a whiteboard, nor with beamer .. well, maybe 2 slides ..

.. but with a SMΛRT Board!

i must say that the interface is pretty cool. at some point i could copy-&-paste an entire equation (though, at the time, i was trying to take just one side of the equation).

as a downside, though, my handwriting is slightly worse than on white (marker) boards. it resembles that of a 6-year old:

oh well: at least this is a sign that the future i want is slowly coming to pass .. (-:

(on a related note, i didn't import a PDF and try to write on it. i wonder how flexible that interface would be ..)

// added: 15:22EEST
ok, one last comment:

it was really cool that i never really had to "erase" anything other than if i wrote the wrong symbol. there was always the option of appending another "page" to the end of the SMΛRTboard document, and so i had the benefit of scrolling back to something i had written before.

[1] lab scientists would call this a "journal club" talk, i think.

Monday, March 04, 2013

MoAR: cool, hopefully cool, weird-but-cool, uncool, and weird.

this past week was .. busy, i suppose. more than that, it just seemed full of distractions when all i wanted to do was rewrite some manuscripts.

anyway .. the roundup:

1. sensing the invisible.

i think it's generally accepted that there are more than five human senses, but imagine having even more .. and new ones, at that:
"In a study published last week, for instance, Nicolelis’s group at Duke used brain implants to allow mice to sense infrared light, something mammals can’t normally perceive. They did it by wiring a head-mounted infrared sensor to electrodes implanted into a part of the brain called the somatosensory cortex.

Similarly, Nicolelis thinks in the future humans with brain implants might be able to sense x-rays, operate distant machines, or navigate in virtual space with their thoughts, since the brain will accommodate foreign objects including computers as part of itself.

~ from "The Brain Is Not Computable" @techreview
.. "virtual space," you say? how virtual?

admittedly, after reading this i pondered how possible it could be to experience four dimensional space .. or at least some representation of a space with four independent parameters, where geometry can be converted to something observable.

2. on open access (or: sweet!)

these days i don't know mathematics journals so well, mostly because i usually find out new and interesting results from arXiv preprint server first. (this can lead to trouble sometimes, because often i have no idea where to submit an article, when it's ready.)

despite my own habits, i think this is a good step forward for maths and science.
"The White House has moved to make the results of federally funded research available to the public for free within a year, bowing to public pressure for unfettered access to scholarly articles and other materials produced at taxpayers' expense.

"Americans should have easy access to the results of research they help support," John Holdren, the director of the White House Office of Science and Technology Policy, wrote on the White House website.

~ from "White House directs open access for government research" @reuters
however, publishing companies are probably not eager to see their profits drop .. so i predict that they will start charging authors in order to meet the new mandate.

i don't think that it will get to the point where the company will refuse to publish the article, especially if it's been accepted for publication under peer review. on the other hand, never underestimate the power of greed.

3. familiar yet unexpected

i don't know about you, but i see a familiar shape here.

This and similar images show the stability of the hexagon even 20+ years after Voyager. Movies of Saturn's North Pole show the cloud structure maintaining its hexagonal structure while rotating. Unlike individual clouds appearing like a hexagon on Earth, the Saturn cloud pattern appears to have six well defined sides of nearly equal length. Four Earths could fit inside the  hexagon.

~ from "Saturn's Hexagon and Rings "

4. different assumptions, flawed data.

what i'd like to know is what the average "split" is, in north america.
" The test that Henrich introduced to the Machiguenga was called the ultimatum game. The rules are simple: in each game there are two players who remain anonymous to each other. The first player is given an amount of money, say $100, and told that he has to offer some of the cash, in an amount of his choosing, to the other subject. The second player can accept or refuse the split. But there’s a hitch: players know that if the recipient refuses the offer, both leave empty-handed. North Americans, who are the most common subjects for such experiments, usually offer a 50-50 split when on the giving end. When on the receiving end, they show an eagerness to punish the other player for uneven splits at their own expense.
When he began to run the game it became immediately clear that Machiguengan behavior was dramatically different from that of the average North American. To begin with, the offers from the first player were much lower. In addition, when on the receiving end of the game, the Machiguenga rarely refused even the lowest possible amount. “It just seemed ridiculous to the Machiguenga that you would reject an offer of free money,” says Henrich. “They just didn’t understand why anyone would sacrifice money to punish someone who had the good luck of getting to play the other role in the game.”

~ from "We Aren't the World" @psmag
there's more.
"But the most chilling potential problem is that the data we use to guide ourselves can be incomplete or overly reductionist. Many crimes go unreported, which could fool predictive policing software into thinking a neighborhood is safe. Cops on the beat, however, might be able to tell when things don’t seem quite right there and keep an eye out. ­Morozov fears a future in which such “intuitive knowledge” about how to deploy resources is overruled by algorithms that can work only with hard data and can’t, of course, account for the data they don’t have."

~ from "The Problem with Our Data Obsession" @mit_techreview

5. an interesting analogy..

i take it that the scifi writer, s. lem, wasn't aware of "pair of pants" decompositions from hyperbolic geometry. (-:
"Let us imagine a mad tailor who makes all sort of clothes. He does not know anything about people, birds, or plants. He is not interested in the world; he does not examine it. He makes clothes but does not know for whom. He does not think about it. The tailor is only concerned about one thing: he wants to be consistent. […] He takes the finished clothes to a massive warehouse. If we could enter it, we would discover that some of the clothes fit an octopus, others fit trees, butterflies, or people. We would find clothes for a centaur and for a unicorn as well as for creatures they have not even been imagined yet. The great majority of his clothes would not find any applications.

Mathematics works in the same way. It builds structures but it is not clear of what. These are perfect models (i.e., perfectly accurate), but a mathematician does not know what they are models of. He is not interested. He does what he does because such an action has turned out to be possible.

~ from S. Lem's "Summan Technologiae" @facebook

Saturday, March 02, 2013

haunted, mathematically.

i feel exhausted. aside from attending seminars and a few bureaucratic things, i've been writing and re-writing all week long [1] .. which is no end of frustation.

there's been a research question, however, constantly sitting in the back of my mind. i really want to know the answer ..

.. but again: i feel exhausted and i can't summon the energy to attack it. i haven't the heart right now to try a dozen ideas and have half of them fail easily, and then trod through the remaining ones and see which of them survive their implementation.


i'm still wondering what the answer is, though.

the problem comes so easily to mind. yesterday it was hard to talk to people sometimes, because suddenly i'd think about it and lose track of what friends would be saying. at certain points i just guessed what they were talking about, responded to something related to my guess (in order to suggest my interest), and they would go back to talking.

sometimes maths is like a strange, cerebral purgatory.

[1] i've been taking evenings off but worked through the weekend, last week. every morning i've been waking up tired.