Monday, August 29, 2011

so far, so-so.

odd: i haven't found an error yet in my "proof" of this one conjecture.

could it be .. that i have a correct, complete proof?

[winces, sighs]
as much it looks correct, experience tells me that it's probably not. likely i just haven't spotted the error yet.

on a wholly unrelated note ..

i was watching "stranger than fiction" starring will ferrell, and at some point it started bugging me: dustin hoffman's character is named professor hilbert and the main character was to die from a bus crash on the kronecker line ..

.. sure enough, on imdb:

The last names of all the characters (and the bus line and publishing firm names) are the names of mathematicians, scientists, engineers, artists, etc.

Friday, August 26, 2011

all work and no play .. something something something ..

wednesday was day 3 of a conference, here @ MSRI, and only one talk was scheduled. for once i had actually planned a fun, whimsical day of activity.

sure, i'd do some work in the morning, but ..
  • instead of dwelling in my hotel room, i'd pick a nearby cafe, try out their organic coffee blends .. and when stuck, watch the passerby through the windows;
  • then i'd take the train into san francisco, find a spot for lunch, and then wander around a few museums and well-suggested parts of town;
  • if luck would hold, then i'd acquire a crash pad from one of my contacts and then go outdoor climbing (bouldering) with a friend at indian rock park. otherwise, there is a 7-mile "fire trail" around the foothills that looked particularly interesting for a run;
  • then there would be a gourmet food-truck event, suggested by my sister, where i could sample local fine cuisine .. then a beer or two, and who knows?
well, i never got to half of it.

i had woken up and started LaTeXing in my room a little, if only to get a good pace started before heading to the cafe. that was before i found a gap in my proof ..

.. and before i knew it:
it was noon and the cleaning woman was knocking on my door.

that cast a strange color to the rest of the day: san francisco today would be out of the question. i went to the nearest lunch place, managed a sandwich, and suddenly remembered the climbing arrangements ..

.. wait: when was i supposed to meet ..?

.. but as these matters go, everyone else was happy to delay the climbing from 3pm to .. 5-6ish. there was plenty of time for a coffee and mull over the argument, so in the end i did go to a cafe ..

.. and find a cleaner, more elegant proof. by the time i met friends to go climbing, i didn't much worry that i was moving badly and (having taken a month's hiatus) lost all strength in my grip.

i've LaTeXing up the notes, and no more surprises have appeared. it's a good feeling.

Monday, August 22, 2011

perhaps it is time.

[i wrote this 4-5 days ago but was sidetracked by family matters]

yesterday i was math-walking .. which is a little like sleep-walking, except instead of being asleep i was distracted by mathematical thoughts.

i've had this germ of an idea. the first version, which came to me more than a week ago, clearly doesn't work, but aspects of it had merit ..

this morning i woke up at 7am [1], and i couldn't go back to sleep because i was too excited.

the idea can be modified,
it must be able to work ..

so i thought about it, then latexed it, and things seemed to "click" together .. which is fine, but not worth celebrating yet.


i've had too many occasions when things are looking really promising, only to discover a gap in the proof of a lemma leading to the result. (it's why i try to avoid saying that "i proved something" because initially, it's never fully clear whether there is enough rigor ..)

this time around, there is a difference. i think the idea is robust enough that i'm willing to discuss it with others;

maybe, with a few more pairs of eyes looking at it, the argument can be modified, clarified, optimised .. and i'll get the right proof.

[1] which is early-ish for me. these days i wake at 7:45am or so, if only because my father, who is retired, is up and about at 10am. that affords me two hours or so of work time, without interruption.

Wednesday, August 17, 2011

a few links, memory tricks.

i'm no number theοrist ..
unless it pertains to hausdοrff dimension [1], and even that's a stretch!

.. but i think that today's gοogle logo is pretty cool:

as long as i'm posting links, a friend of mine sent me this article from new scιentist:
Mental αbacus does away with words
by Ferrιs Jαbr (13:56 09 August 2011)

When 11-year-old Prιyanshi Sοmani multiplies strings of 10-digit numbers or finds the square root of a six-digit number, she doesn't use a calculator or even pencil and paper. Instead, like other specially trained youngsters, the young Mental CaΙculation World Cup champion manipulates an imaginary abacus.

Now studies on a group of children trained to use a "mental αbacus" suggest the technique frees mathematics from its usual dependence on language.
i wonder how geometric this approach is. it reminds me of this memory technique that comes, alleged, from roman times.
The method of loci is also commonly called the mental walk. In basic terms, it is a method of memory enhancement which uses visualization to organize and recall information. Many memory contest champions claim to use this technique in order to recall faces, digits, and lists of words. These champions’ successes have little to do with brain structure or intelligence, but more to do with their technique of using regions of their brain that have to do with spatial learning.

During the mental walk, people remember lists of words by mentally walking a familiar route and associating these objects with specific landmarks on their route. An example of this would be to remember your grocery shopping list in a mental walk from your bedroom to kitchen in your house. Let's say the first item on your list was bread; then mentally you can place a loaf of bread on your bed. As you continue mentally walking you can place the next item, assume it is eggs, on your dresser. The mental walk continues like this as you place consecutive items along a familiar route that you walk. So when you are at the grocery store, you can then think about this walk and “see” what you placed at each location. In your head you will remember bread being on your bed, and eggs being on the dresser. This can continue for as many items as you want to place on your path as long as the route continues. The more dramatic the images, the more vivid the memory. For instance: instead of "bread", try to visualize a giant loaf of bread; instead of "eggs", imagine broken eggs all over the place.

having tried neither technique, i have no idea how effective they can be. perhaps, like learning how to type on a dvοrak keyboard, i'll set them aside as skills to learn later.

[1] i learned of the following theorem from falcοner's book, some years ago. it's about rational approximations of real numbers and the hausdοrff dimension of the corresponding subsets:

Theorem. Take $a > 0$ and a sequence $\{n_i\}_{i=1}^\infty \subset \mathbb{N}$ so that $n_{j+1} \geq n_j^j$. Let $F$ be the set of real numbers $x$ so that

$$\textrm{dist}(n_jx, \mathbb{Z}) \,\leq\, n_j^a$$

for all $j \in \mathbb{N}$. Then the Hausdοrff dimension of $F$ is at most $1/(1+a)$.

Tuesday, August 16, 2011

thoughts between LaTeχing commands

i'm currently Teχing up a proof, which is good ..
.. but not only does it proceed by cases,
it (unfortunately) further proceeds by subcases.

[winces] .. [sighs] ..
sometimes life is necessarily complicated.

is it in good taste to write a short warning the reader,
before the \begin{proof} command ..?

on the brighter side of things, i think i'm unambiguously closer to proving a particular conjecture .. though by no means is it a full proof.

if there's anything i learned about the job, in the last few years, it's to appreciate any measure of progress and improvement. (-:

Thursday, August 11, 2011

i think i'm developing a dyslexic tic:
when writing out indices by hand, i keep writing '3' for 'ε' .. which ruins the estimates. up to symmetry, though, it's typographically the same
on a related note, i've decided to treat most of my stay in ny state as a 1/2-holiday. since one is supposed to relax and do things one wants to do, when on holiday ..

.. i'm going to work on another idea to attack this one conjecture.
wish me luck! (-:

Monday, August 08, 2011

when jargon creeps into colloquialism ..

it makes sense to think of maths as a language. it explains, for example, why students seem so able to write wholly nonsensical things on the pages of their calculus exam. [1]

on the other hand, the technical nature of mathematical jargon can be utterly confusing for the uninitiated. i mean something different from the ubiquity of "normal" (in the sense of normal subgroups, normal convergence, etc) and the loaded use of the word "trivial."

it's the smaller, subtler uses of technical words, like:

"modulo the republicans steering the debate to the right, congress will accomplish nothing this year."

"coke and pepsi are isomorphic, as far as i'm concerned."

last week for instance, i told my brother that 'if you condition on the fact that my last three relationships began when i was in a transition, then likely i'll never meet another woman again.'

when he asked me what the hell i meant by "condition," i explained to him what conditional probability is. (he subsequently thought it was a fine linguistic construct.)

tonight i told my friend that when it comes to real life, i don't feel embedded in it. when he asked me what i meant by embedded, i immediately said that sure, i feel immersed in things but that it wasn't quite a perfect fit.

after another odd look, i paused, thought a little, and told him that nowadays i feel a little less in touch with reality and society than i used to be .. which satisfied him.

[1] for instance, too often i've seen equal signs (=) treated as logical implication symbols (→). thinking about it, this is not so surprising: to most, mathematics is no more than computation, so an equals sign only confirms the logical progress of a computation.

Wednesday, August 03, 2011

the song remains the same (or: i found a cool preprint on the arχiv)

the more i think about it, the more it seems that the differentiabιlity property for functions seems to be a rather rigid property -- in terms of both the type of function and the geometry of the underlying (metrιc) space.

today i stumbled upon a further rigidity result on the arχiv:
Dιfferentiability, Pοrosity and Dοubling in Metrιc Measure Spaces
David Batε, Gareth Speιght [1]

We show if a metrιc measure space admits a dιfferentiable structure then pοrous sets have measure zero and hence the measure is pointwise dοubling. We then give a construction to show if we only require an approximate dιfferentiable structure the measure need no longer be pointwise dοubling.
a short-&-sweet abstract, an interesting result!

to give this result some context:
in functiοnal analysιs, one can make sense of derivatιves in terms of fréchet or g&ahat;teaux differentiabilιty. according to hearsay, radεmacher theorems in this context are quite hard ..

.. however, the dοubling condition implies that the underlying space must have a finite Hausdοrff dιmension. so in the context of measures [2], differentiabιlity (even in a generalized sense) must be a fιnite-dimensiοnal phenomenon!

[1] the names sound familiar; i think i met both of them before ..?

[2] strictly speaking, a(n outer) measure is not necessary in order to formulate a radεmacher-type property. it suffices instead to have a notion of what null sets are. according (again) to hearsay from my colleagues, there are quite a few ways to define notions of null sets in infinite-dimensional Baηach spaces ..

Tuesday, August 02, 2011

20 days and 20 nights?

currently i'm in limbo. i'm starting to think that moving in with my parents was a bad idea; maybe i should have inspected aιrbnb for a cheap one-month rental somewhere nice.
every so often i remember an acquaintance of mine named renzo, who spent a summer in costa rica, doing math, enjoying tropical weather, and entertaining guests who would visit him.

i think of the possibility of living in .. say, portland, OR, for a month .. working in cafes, hiking, drinking fine coffee and beer, wasting away in various powells bookstores.
sure, it's only for 3 weeks .. but i wonder if i'll get half of what i would like to get done.

on an unrelated note: i think geometers have moved in, across from my parents' house.