.. among other things, "imagine ELIZA passing calc ii?" (updated)

there has been a lot of discussion about the phenomenon of MOOCs (i.e. massive open online courses) but i've come across very few discussions by academic mathematicians.

there's a new opinion by k. devlin, though. in true mathematical fashion, he avoids grand-standing predictions in favor of a few valuable insights. one of them is:

At the level of the individual student, MOOCs are, quite frankly, not that great, and not at all as good as a traditional university education. This is reflected (in part) in those huge dropout rates and the low level of performance of the majority that stick it out. But in every MOOC, a relatively small percentage of students manage to make the course work to their advantage, and do well. And when that initial letter M refers not to tens of thousands but to "millions," those successes become a lot of talented individuals.

One crucial talent in particular that successful MOOC students possess is being highly self-motivated and persistent. Right now, innate talent, self-motivation, and persistence are not enough to guarantee an individual success, if she or he does not live in the right part of the wor[l]d or have access to the right resources. But with MOOCs, anyone with access to a broadband connection gets an entry ticket. The playing field may still not be level, but it's suddenly a whole lot more level than before. Level enough, in fact. And as with Google search, in education, "level enough" is level enough.

it's a good point.

one could make the argument that successful MOOC participants exhibit more persistence than on-campus students, because they wouldn't have the benefit of a community of peers to signal that they should study more or allocate time to do the work. unless they have actively looked for study partners in her/his local area, MOOC participants are on their own and their success is truly theirs ..

.. provided that the MOOC participants actually do their own work, of course.

so maybe the article title "the darwinisation of higher education" is apt;
nature also rewards those species that try their hand at camouflage.

i may be cynical, but then again, i'm also a mathematician who appreciates certainty and tries to pay attention to nonexamples.

---

my experience with web homework, though possibly unrelated, comes to mind. here was the setup:

• since the questions were multiple-choice, the grading was automatic;
• since they were allowed several chances to get each problem right,
  often without time limits and with possible access to wikis and online examples,
  most students had very good scores. many even had close-to-perfect ones.

there was always a lot of variation in their in-class quiz scores, though.

even if all of my former students were entirely ethical and did everything by themselves (in that they asked no other humans for help) there still remained a difference between what they did on a computer and what they did on their own.
..
" [thinks]
'on their own' ..
..

that betrays my biases. there's nothing wrong with looking everything up and putting it together, i suppose, especially if one always has access to the internet.

on the same token, an information engine like wοlfram α can probably do simpler but similar tasks and more efficiently, too.

many fellow educators may agree with me that most test problems in entry-level maths courses are essentially algorithmic; in fact, "good" exam problems are hard to write and i think it takes a nontrivial amount of work to write a challenging problem that requires little of either axiomatic proof or purely algorithmic computation. this suggests (but does not prove) that successors of these programs could soon perform that much better than our students on the same tasks; if you even go so far as to believe in the singularιty, then it shouldn't be that far in the future.

so as the pundits like to say, if we need new workers for an "information economy," then why should they be human workers, if machines can do a systematically better job? on a related note, imagine an eliza program passing calculus ii? that would be both very cool and very scary.

to be continued later:
some ideas on how to make money from this,
and what might go wrong.

---

added: evening, 8 jan 2013:
never mind. as i expected, my idea's not original and they already thought of it.

Coursera recently announced another route to help students earn credit for its courses — and produce revenue. The company has arranged for the American Council on Education, the umbrella group of higher education, to have subject experts assess whether several courses are worthy of transfer credits. If the experts say they are, students who successfully complete those courses could take an identity-verified proctored exam, pay a fee and get an ACE Credit transcript, a certification that 2,000 universities already accept for credit.

~ from "Students Rush to Web Classes, but Profits May Be Much Later" @nyt

indeed, if they think that students are going to (be tempted to) cheat, then they may as well make money on it ..

so i got my wish .. sort of.

oddly enough, it happens that everybody in my family is feeling ill with fever except me [1]. nobody else has any energy to do anything, except take prolonged naps.

not to be too ruthless about this turn of events, but if i'm on my own .. then it makes a perfectly good day for mathematics .. (-:



[1] first of all, no: i didn't cause this purposely. i may be itching to get some work done, but i'm not that much of a mercenary! equally odd, though, is that i've been the one who has been wearing thin sweaters and light coats and no hat while outdoors .. and running outside in 40*C^oF weather in the rain. i'm starting to believe that infectiousness [2] is either a wholly random process, or that i'm some sort of immune carrier of this particular flu .. which is still odd, since i haven't been sick lately.

[2] this is probably a made-up word. on the other hand, "infection" doesn't seem like the right word, especially as it suggests something bacterial in origin.

mildly relevant: creativity, DIY.

this is probably one of the recurring themes in my posts, such as the last one about mornings ..

.. but, if only to become more effective in one's goals, there's a lot to be said for hacking one's own habits and inclinations, conscious or otherwise. there's a difference between working effectively at a fixed task (where considerations of time can be measured quantitatively) versus being prolific at a creative task, for example.

the more i think about it, the more they seem like different modes of thinking. it's therefore appropriate to consider ways of making these experiences more efficient and separately so ..!

---

Creativity in the Wild: Improving Creative Reasoning through Immersion in Natural Settings

Ruth Ann Atchley¹, David L. Strayer^2*, Paul Atchley¹

1 Department of Psychology, University of Kansas, Lawrence, Kansas, United States of America, 2 Department of Psychology, University of Utah, Salt Lake City, Utah, United States of America

... However, the impact of nature on higher-level tasks such as creative problem solving has not been explored. Here we show that four days of immersion in nature, and the corresponding disconnection from multi-media and technology, increases performance on a creativity, problem-solving task by a full 50% in a group of naive hikers. Our results demonstrate that there is a cognitive advantage to be realized if we spend time immersed in a natural setting. .

---

Creativity Happens When You Least Expect It

By Sian Beilock | Dec 07, 2012

... But not all tasks require working memory for success. In fact, sometimes people’s ability to think about information in new and unusual ways can actually be hampered when they wield too much brainpower. This means that what we think of as our optimal time of day, may not be optimal for everything.

Recent research confirms this idea. In a paper published last year in the journal Thinking and Reasoning, psychologist Mareike Wieth and her colleagues found that when people have to solve “insight problems” that require a high degree of creativity, solvers are much more successful when they tackle these problems at the time of day in which they are least alert.

---

i must say: the more i learn about the human brain, the more amazed i am at how it works ..!

a quick one, while he's away .. (updated)

// started: 24 dec 2012, mid-morning.

i haven't worked in a coffeehouse in a while:
comforting but slightly unfamiliar.

the background christmas music is distracting, though.

// added: 25 dec 2012, early morning.

due to oceanic jet lag, i woke up dreadfully early, well before sunrise on EST .. and managed to improve a technical lemma in one of my manuscripts.

so if all goes well, then this will make an extant proof even clearer,
and will cut away ~2 pages from the previous writeup.

not bad: a little thinking done, even before any of the family's woken up. you see, the older i get, the harder it seems to carry a normal conversation, first thing in the morning.

put otherwise, i feel more encouraged to work out ideas, right away upon waking up ..

.. as if it were a perfect time to hack my unconscious,
see what strange intuitions have occurred to me while i slept, the night before ..

.. so it happens fairly often that i can't wait to get up and start the day, even if i'm not fully awake. ultimately, i'm a more effective (and less irritable) person when left alone in the mornings .. at least long enough to sort out an idea or two.

// added: 25 dec 2012, late afternoon.

i was about to check the arXiv .. but then realised:

if i do look for preprints, then it's likely that i'll browse through some of them and start thinking about ideas. once that happens, then the whole week will turn into something degenerate.

it won't really be a holiday, nor will it be a proper workweek. i'll probably judge it as one or the other; in either case, i won't be satisfied by what will happen.

no good can come out of this.

so i left the arXiv alone, for now. it's one thing to work in the early mornings when everyone else is asleep .. but if i've made a point to visit family during the holidays, then shouldn't i be consistent and set aside the maths?

[sighs]

mildly relevant: campus life is relatively rare.

this is an excerpt from an article called "the end of the university," as found on the american interest.

more appropriately, it should be called "the end of the american university" because there is a strong focus on tuition, student debt, and the 'business' of running a university.

to wit: quite a few places in europe still have very affordable [1] university education for all citizens of the e.u. without a lot of tuition fees .. though the tide seems to be turning, especially in the u.k.

it's funny .. in the united states, they're talking about tuition-free higher education, whereas much of europe is discussing the exact opposite!

anyway, about that article ..

-✂--
..
To borrow an analogy from the music industry, universities have previously sold education in an “album” package—the four-year bachelor’s degree in a certain major, usually coupled with a core curriculum. The trend for the future will be more compact, targeted educational certificates and credits, which students will be able to pick and choose from to create their own academic portfolios. Take a math class from MIT, an engineering class from Purdue, perhaps with a course in environmental law from Yale..
..
-✂--
..
It’s worth noting that while the four-year residential experience is what many of us picture when we think of “college”, the residential college experience has already become an experience only a minority of the nation’s students enjoy. Adult returning students now make up a large mass of those attending university. Non-traditional students make up 40 percent of all college students. Together with commuting students, or others taking classes online, they show that the traditional residential college experience is something many students either can’t afford or don't want. The for-profit colleges, which often cater to working adult students with a combination of night and weekend classes and online coursework, have tapped into the massive demand for practical and customized education. It’s a sign of what is to come.
..
-✂--

[1] well, compared to private universities in the u.s., at least.  from what little i've read, the enrollment cost to a university student per semester is less than 1000€ per year in spain and france, and i think that finnish universities don't charge tuition at all ..?

it's not the fault of the holidays .. it's just me.

this and last week i've lacked focus. every day i get up, i try to work, but little if anything happens. there are some ideas on a few sheets of paper from the last few mornings ..

.. what's not made it to the recycling bin, anyway ..

.. and then there are a few manuscripts to rewrite, papers that i told myself i'd actually read [1], co-authors to whom i should really write back ..

the only thing that's not on my to-do list is preparing another talk, which is relieving. it's not that i don't like giving talks; i could use a break from the road, though.

---

i'd like to think that i'm still tired from recent travels .. which i probably am, but that's not entirely it [2]. it really is a lack of focus, or more accurately: commitment.

lately i've found it difficult to choose what to do. i have too much on my plate these days ..

barely relevant, but ..

after reading this mentalfloss article,  now i'm tempted to use the character/symbol ..

(literally, "that" .. a combination of thorn and t.)

.. as a replacement for "so/such that" ..! (-:

mildly relevant: disruption in education.

an interesting call to action,
as passed to me in november by a colleague.

(FYI: here "disruptive" is meant in the sense of 'disruptive innovation.')

But you know what? Those classes weren’t like jazz compositions. They didn’t create genuine intellectual community. They didn’t even create ersatz intellectual community. They were just great lectures: we showed up, we listened, we took notes, and we left, ready to discuss what we’d heard in smaller sections.
-- ✂ --
Outside the elite institutions, though, the other 75% of students—over 13 million of them—are enrolled in the four thousand institutions you haven’t heard of: Abraham Baldwin Agricultural College. Bridgerland Applied Technology College. The Laboratory Institute of Merchandising. When we talk about college education in the US, these institutions are usually left out of the conversation, but Clayton State educates as many undergraduates as Harvard. Saint Leo educates twice as many. City College of San Francisco enrolls as many as the entire Ivy League combined. These are where most students are, and their experience is what college education is mostly like.
-- ✂ --
We have several advantages over the recording industry, of course. We are decentralized and mostly non-profit. We employ lots of smart people. We have previous examples to learn from, and our core competence is learning from the past. And armed with these advantages, we’re probably going to screw this up as badly as the music people did..

(continued in "napster, udacity, and the academy" @clayshirky)

don't worry: i still remember .. and i understand completely.

less than half of december remains. i guess it means the end of the year is truly upon us .. and for many of you, the season of job applications is partially over [1].

of course, the pessimist in me would point out the many deadlines yet to come .. and how there is really no rest for temporarily-hired persons in maths.

what a life!

after grading batches and batches of final exams,
doling out grades for the course,
answering upset student emails [2],
maybe a free afternoon to spend revisiting an idea,
getting back to co-authors ..

.. after that, it then becomes time to finish another 30-50 job applications [3]!

---

so to those of you on the market, this is to say: i'm sorry. yes, it's not my fault, so it's not really an apology .. but i know your pain, have felt it, and you know how i feel about it too.

i remember how it feels to worry so much about the future that you cannot even concentrate on what really matters to a mathematician ..

.. that matter, being mathematics, the freedom of calm and contemplation ..

.. so maybe this is my way of dealing with a mathematical version of survivor guilt. at any rate,

mildly relevant: a web-based, collaborative latex editor.

interesting: writelatex is a latex version of etherpad, a collaborative writing and editing tool.

i would imaging that coupling dropbox with your favorite $\LaTeX$ program would accomplish something similar but without the option of editing the same document simultaneously.

(to me, though, the only instance where this could be really useful would be if several authors are composing the introduction of a paper together.)

for mathematicians (like me), an open problem is like a revealing mystery.

it's for title/abstracts like these that i constantly check cνgmt for updates. (sure, some researchers do post survey articles and lecture notes on the arχiv, but not as often.)

L. Ambrοsio - M. Colοmbo - S. Di Marinο

Sobοlev spaces in metric measure spaces: reflexιvity and lower semicοntinuity of slοpe

Submitted Paper (2012)
Keywords: Sobolev spaces, Metrιc measure sρaces, Weak Gradιents

Abstract. In this paper we make a survey of some recent developments of the theory of Sοbolev spaces $W^{1,q}(X,d,m)$, $1 < q < \infty$ in metric measure spaces $(X,d,m)$. In the final part of the paper we provide a new proof of the reflexivity of the Sobolev space based on $\Gamma$-convergence; this result extends Cheegεr's work because no Poincaré inequality is needed and the measure-theoretic doubling property is weakened to the metric doubling property of the support of $m$. We also discuss the lower semicοntinuity of the slope of Lipschitζ functions and some open problems.

one cool thing about these kinds of expositions is that open problems of the field are explicitly stated, just put out there. it's not that i expect to solve them, but there's something .. enchanting? alluring, i suppose, about encountering something that nobody knows how to solve (yet).

I love rumors! Facts can be so misleading, where rumors, true or false, are often revealing.
~ col. hans landa

it drives one's ideas, sharpens one's focus to some good end;
also, open problems suggest ..

.. though mathematicians vary by talent, inclination, and drive in very large degrees ..

.. that we are all equal in a few ways, at least until someone solves the problem at hand. then again, there are always problems and unknowns, just like there are always books i've never read in any public library.

their existence is somehow very comforting to me. (-:

on math ed: the (possible) danger of breaking down problems ..

whoa -- from "u.s. math achievement: how bad is it" @psychologytoday

When the researchers broke problem-solving activities down into procedural activities and conceptual activities, they expected to find that the higher performing countries engaged in more conceptual problem solving. They found no such difference. But then they took a second step. They coded the data based on whether the teachers made the conceptual problems easier by converting them, for the students, in to procedural problems.

Looked at this way, it became clear that the US was an outlier (as was Australia, the only other low-performing nation in the study). Teachers in the US almost always converted challenging conceptual problems into procedural problems. In doing so, they did exactly the wrong thing. According to a seminal study by Hiebert and Grouws (2007) the two features of instruction that predict good math outcomes are

1. Being explicit about the conceptual structure, and interconnectedness, of mathematics
2. Allowing students to struggle to understand mathematical concepts.

By converting conceptual struggle into procedural learning, US math teaches were unintentionally depriving their students of two crucial elements of effective learning.

the unnerving thing is that, to me, "breaking down a problem" seems like a natural thing to do. it's how research goes, all the time.

the point, however, seems not for the teacher to do all of the reduction .. as (s)he would already know how .. but for the students to learn how to do it.

.. there and back.

today and yesterday i felt like everyone treated me a little like a carton of milk, on which the expiration date was hard to read.

they: "janus, hi! how long are you in town?"
me: "hi! er, well .."

.. it was a short visit this time: a little more than 48 hours. i felt like i was constantly saying hello and sorry and measurablε dιfferentiable structure and goodbye to many colleagues, both new and familiar.

it happened that my colleague/friend/host asked me if i've visited his (new) university before. i nodded.

thrice before, in the last 10 years:
one of the first conferences that i ever attended, too.

speaking of a next time, another colleague asked me if i'd be free to visit next spring .. which means, i hope, late spring. honestly, i need a break from traveling [1].

besides that, there's something about the finnish winter that makes me want to retreat indoors, and not come out for anything but trail running, indoor rock climbing, a long session of sauna, or a pub crawl.



.. on an unrelated note, there's something very soothing about traveling on trains.

---

the more i think about it, the more collaborations feel unnatural to me. i find them stressful, honestly .. though it's often a good type of stress that leads to some productive end.

one problem is that i'm just not that quick .. not right now, anyway;

if i believe it and if i act effectively towards it,
then who knows: maybe i can actually become smarter.

on the other hand, maybe i should just let myself be "stupid" and throw out ideas, instead of sitting quietly, contemplating, working them out into a polished form by myself.

to a certain extent, it doesn't matter if a given idea is a good one or a bad one. what matters instead is that you eventually get a few good ideas.

so the point is to generate good-enough ideas and resolve them effectively, and more processing power (in terms of number of minds thinking about it) generally leads to swifter resolutions.

still, i hate sounding stupid and i can't stand not knowing what to do next .. which is a frustratingly impasse, i tell you. maybe i just have to get over it.





[1] in 2012 i've been to lappeenranta, madrid, new york, tampa, ann arbor, lansing/cadillac, turku, amsterdam & brussels & brugge & lille & reims & strasbourg & heidelberg & frankfurt, krakow, jyvaskyla, new york again, pittsburgh, paris, san francisco, london, madrid again, segovia, and jyvaskyla again .. which totals 88 days (or ~3 months) i spent out of town ..

[realises]

.. ye gods! now that i think about it, i criss-crossed the atlantic four times this year, and gave fourteen (14) talks & lectures. is something wrong with me? i must be a masochist..!

mildly relevant: story-telling, neurologically speaking ..

interesting.. if the brain really works out this way, then this could prove useful for more effective teaching or perhaps giving better talks:

Evolution has wired our brains for storytelling—how to make use of it

Now all this is interesting. We know that we can activate our brains better if we listen to stories. The still unanswered question is: Why is that? Why does the format of a story, where events unfold one after the other, have such a profound impact on our learning?

The simple answer is this: We are wired that way. A story, if broken down into the simplest form, is a connection of cause and effect And that is exactly how we think. We think in narratives all day long, no matter if it is about buying groceries, whether we think about work or our spouse at home. We make up (short) stories in our heads for every action and conversation. In fact, Jeremy Hsu found [that] "personal stories and gossip make up 65% of our conversations."

~ from "the science of storytelling" @lifehacker.

lessons in (not) traveling.

[written earlier on the train]

i don't know if i'm tired from work or tired from traveling.

as for this week, i'm taking a short trip and spending a few days at a nearby university: a colleague (and friend) moved there for a position and i've been invited to speak in his new seminar, maybe work together on a problem or two.

---

it's been a week since i've returned from a 2-week stay in spain. i don't know where those seven days went, but they didn't amount to too much. then again, it's hard to compare the daily grind to, say, working for hours with collaborators every day for a while; the latter case certainly sets a rather high standard, of course.

i know that i showed up to the office, committed to a list of tasks, and completed some of them .. yet it feels like i've not accomplished anything since coming back. part of the problem is that i've been too preoccupied with this next trip, for no other reason than i have to plan for it and stick to the plan.

it is a curse to have a one-track mind, incapable of multi-tasking.

at any rate, a week is too short: i should have arranged more space between these trips .. at least two weeks, in order to rebuild a routine.

habit and routine have become my main tools to develop any kind of work efficiency.
it sounds boring, but it really works.

i suspect that this is something that everyone just knows ..
.. well, except me, who had to actually learn it.

the holidays are coming. taking time off sounds very good right now, but there are so many things to do. the year is ending and all i can think about are the things i've not yet done but wanted to do.

those kinds of thoughts are exhausting ones, and they only lead to a vicious cycle of unproductivity and self-recrimination ..

.. that is, i think about my shortcomings,
feeling bad about them, i sulk and do nothing;

having done nothing, i develop new shortcomings,
then think about them again .. 7-:

i don't know if i'm really tired, but that's how i feel.



[written later, having arrived]

epilogue: it's always worth visiting friends. (-:

mildly relevant: the fault lies not in the stars ..?

it may seem like students are getting worse every year .. which is what an educator would find very convenient to believe. it would imply that we're doing a good job but everyone else isn't.

i find that answer a little too convenient.

it is true that school standards change with time .. but i remember suggesting once to colleagues that the reality may be purely relative to our perspective. instead, what if we are simply getting more familiar with teaching the same topics?

But if you've never been to a place before, you need more than a description of a place; you need an exact definition, or a precise formula for finding it. The curse of knowledge is the reason why, when I had to search for a friend's tent in a field, their advice of "it's the blue one" seemed perfectly sensible to them and was completely useless for me, as I stood there staring blankly at hundreds of blue tents.

This same quirk is why teaching is so difficult to do well. Once you are familiar with a topic it is very hard to understand what someone who isn't familiar with it needs to know. The curse of knowledge isn't a surprising flaw in our mental machinery – really it is just a side effect of our basic alienation from each other. We all have different thoughts and beliefs, and we have no special access to each other's minds..

~ from "why we are so bad with directions" @bbc

life wasn't simpler, back then.

over the weekend i learned about blogger's label feature ..

for an example, click here; also .. [0]

.. so i ended up reading quite a few posts i wrote when i was in the middle of my ph.d. and shortly after i started doing actual research. it makes for more interesting reading than my life now, but .. ye gods:

I. it was a dark, uncertain point in my life. i forgot how often i worried about whether everything would work out or even whether i was cut out to be a mathematician [1], how it took so long to ever learn anything .. much less work with it .. and then there were 2-3 rounds of job searches: one just before the financial crisis hit with full force, the other(s) after it.

in comparison, my life now feels relatively secure and even routine.

that's not to say that i've stopped worrying. rather, i think i worry just as much as i used to [2] .. just that i've gotten used to not getting any resolution out of it, and gotten better at pushing it out of mind.

II. i wrote a lot more often about technical details, and i don't exactly know why. maybe it was because all this research stuff was new to me and i found it hard to separate it from the rest of my life.

maybe it's because by writing them, i could partially convince myself that i had some idea of how they worked .. which i didn't, of course.

maybe it's because i've encountered enough non-experts, relative to my field, and have given enough unsuccessful explanations that i've grown tired of writing or thinking about them, during my non-work hours.

contrary to popular expectation, not all of us researchers work all the time;
these days i would consider myself a non-example.

who knows? maybe it's progress .. that i see a bigger picture now and that the details, though still essential and important, aren't the main focus anymore.

related to this, my research problems have been asking why a bit more often than how, these days.

III. travel wasn't routine to me, yet. to be fair, i didn't particularly like traveling back then, either, but i was more excited by the trips i took.

the first flight to finland (2003) was an adventure to me, for example.

quotable: not a 'can say,' but "can do" attitude.

from "the second test" @randsinrepose:

I am not suggesting that the hardworking people in these other disciplines don’t have amazingly complex and difficult jobs, but I do think they should be able to clearly describe the work they do and the value they provide… to anyone. They need to pass the Second Test, and that means being able to fully and clearly explain your job to the rest of your team not with words, but with action.

Most folks believe that if they can describe a job that they can do it. Most folks are wrong. I’ve been spun and burned by too many fast-talking, charismatic experts in my career to trust anything but results. The Second Test is not the exclusive domain of engineers. In most groups of people, there is a means by which you earn your stripes. The difference with engineers is a combination of their low tolerance for spin and their deep desire for measurability.

.. and in a similar vein, from "surely you're joking, mr. feynman!"

All the time you're saying to yourself, 'I could do that, but I won't,'--which is just another way of saying that you can't.

class dismissed: as for the rest ..

overall it's been a good visit, but intense and tiring. i've tried to do a good job with the lectures, but not at the expense of research.

.. which would explain the suboptimal planning of my lectures!

as of now, my colleagues and i have gotten headway into a well-defined project, maybe two.  there are still many details to consider, but we've thought long enough and hard enough on a few problems that i believe something can be done with them.

one part makes me cringe, though: i might have to revisit some of the basics of some slightly arcane topics in functional analysis ..

.. such as the weak-star operator topology ..! [1] 7-:

---

to be fair, i think that it's been easier on me than on them.  often i do most of the listening and little of the talking.  it went the other way around, this time.

over the last two weeks i felt like i explained a lot about a theory that nobody seems to study very much .. which is a shame, since i've consistently found it a rather useful theory. by sheer exposure and effort, i think i've also convinced my collaborators of the same.

so i did end up being useful, in some fashion .. which is relieving.  in spain these days, research funds aren't easy to come by.  imagine if they had invited me and gotten little out of the deal! (-:

for my own part, i learned quite a bit about fractals .. and in some sense, fubini's theorem, too.



[1] to be precise, the wiki directs to the weak operator topology, not the weak-star operator topology .. but on the other hand, what i would call the weak-star operator topology differs from what the corresponding wiki would suggest; i claim that one doesn't actually need the underlying source and target spaces to be the same .. much less hilbert spaces .. \-: