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!

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!)

banditry!

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.

[ from "Black Hole: Photographs of Paint Flung by Centrifugal Force" by Fabian Oefner ]

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 ]

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 ...

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 ..?

... "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" ..!

[sighs]
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.

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

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" ..! (-: