Htam::Diary.Theme(math)

This Tuesday, I woke up and continued where I left off last night making figures for my paper.  Then suddenly inspiration hit, and I started having all these ideas to tackle the problem I've been trying to solve for the last two months.  I was so excited that I kept walking around in my room in circles while I think about the details.  I was originally going to eat at 11 but since this discovery I wanted to use the momentum and figure out some details sufficiently so I could present it to Igor.  I ended up not having time to eat lunch.  I rushed to school and met with Igor.  I told him this might be premature, but I was really excited.  He told me he thinks I should be, and is quite pleased with this result as well.  I have a lot of details to work out.  One major hole right now is bridged by my limited understanding of logic—I feel like it is a true statement due to general theories in logic.  So I spend the afternoon talking to a logician.  I have some more ideas how to approach it now.  Basically, I will have quite a bit of work to do in the next couple of weeks.  This is perfect timing, too, since the previous paper is wrapping up as we speak.

I should mention that besides proving a small result that was not publishable but worthy of including in my dissertation, I had not really solved any open problems since solving the first one in late January and second one in early February.  I've waited for this day to come for four months!  I've been working very hard for the last few months, writing the paper but also trying to solve new open problems.  It is funny that today I realized I can use the aforementioned small result, that was thought to be not publishable, as part of solving this bigger problem.  Everything is fitting together nicely!

I stayed at school until 7 PM.  Then I called Steph, and she said she would call me back in a bit.  So I drove to prayer meeting directly from school.  In the middle she called me and we talked.  I told her about this wonderful news, and then we talked a bit more after I got to the parking lot.  Then I wished her well for her one-week trip, and went into prayer meeting.  I was there a lot earlier than usual, but since I had not had dinner (I ate lunch after my meeting with Igor, at about 2:30), it was perfect.  I actually ate dinner for once!  Dan even noticed and asked me what's going on.  Haha.

This Sunday, I was the 7th person to get to church, as opposed to the 5th.  I did recording today since Ernest was not here.  In the afternoon, I went to the Ruddock reception as part of Caltech's alumni weekend.  I was planning on going to play table tennis in the gym afterwards, but it was used for a fencing tournament.  So I came back to church and did some work, including chordbook stuff and actual math.

Today is a momentous day—I started a file called dissertation.tex.  Recently I proved a new result that is not big enough to be published by itself, but important enough to be written down.  So Igor suggested that I write it down briefly somewhere, and I could include it into my dissertation ("not just a union of papers").  And I thought, what better place than to just put it in a file called that name?

In the evening, people came back to church, and we had a potluck dinner for Chris Tom, who is leaving on Tuesday to Maryland.  Afterwards we had some sharing and prayer time for him.  We will definitely miss him.  May the Lord be with him each day.

[Photo: Ruddock House crest cake at the alumni reception.]

This Wednesday, I woke up early and went through my proof again.  Then I started doing work, and came up with a brilliant idea to make the one-dimensional Wang tile emulation from yesterday two-dimensional.  Now I need to flesh out the details and see if it can actually be made to work like how I imagined it in my head.  I stopped and looked at interesting tiles on a building while walking to school, probably not inspiring my research, but still cool.  After class/lunch/class I had a meeting with Igor.  Today I presented the entire proof in detail—he agrees with the correctness.  Yay!  At the very least, I am confirmed to have solved one open problem!  I stayed in the department a little for tea, and then came home and had dinner.  Now I'm very tired, and want to go sleep.  I'll try to stay awake until 8, since that's probably a good idea.

This Sunday, I went to church in my rental car—this allowed me to be there early like I am used to.  I set up everything, and then had some quiet time.  It's been a while since I did translation for breaking of bread.  It went sufficiently well, I think.  Today was sharing meeting, after a few sharing, I went outside to eat a snack since I was hungry.  Steph asked me for help on something, so since I plan to listen to the message later anyway, I went ahead and helped her.  Today I also went around telling some people that I got an advisor.  And afterwards I sat down with Steve and I explained my current problem to him.  Steph wanted to see my rental car, and consequently got way too excited and even went in and sat in it.  So silly.  Her dad also got excited and wanted to look under the hood.  That's a bit more normal I guess.  Afterwards I drove Anne home, and we had a nice chat, when I tried to explain to her in very simple terms what my current research topic is about.

At home, I started working again, and to my surprise, I think I solved the problem!!  I am trying to not get too excited just yet, since there are many details to check, but I think the heuristic that I came up with can be translated into a rigorous proof in a few days (hopefully).  Here comes the description.

The setup is that we are given a finite set of tiles.  Without rotating, we want to many copies of these tiles to perfectly tile some region.  Obviously given any region and any set of tiles, we can ask, "is the region tileable?"  This decision problem is NP-complete (a.k.a. "hard") in general.  Now suppose we simplify it, that instead of having arbitrary tile sets, we have a fixed tile set.  Since this decision problem is easier (it is trivially inside the previous problem), it is actually more difficult to prove that is still NP-complete.  Moore and Robson did just this, when they embedded a (variant of the) k-SAT problem (which they also proved to be NP-complete) into tileability of a region with a set of fixed tiles.  However, their region is not simply connected.  An alternate simplification is tileability of arbitrary tile sets on simply connected regions, which is also NP-complete.  Igor thus asked me to consider the even simpler problem, where we have fixed tile set and simply connected region.  (Again, let me remind you that the simpler the decision problem, the harder it is to prove that it is NP-complete.)  I decided to follow Moore and Robson's approach and embed the same variant of k-SAT they used, namely, Cubic Monotone 1-in-3 SAT, which is NP-complete.

Now here is the fun part.  Remember yesterday I made a set of 7 tiles that would be able to tile an arbitrary rectangle given one border?  Well, I am going to use that idea!  I will introduce a parity peg uniformly on all tiles in order to make the tiles conduct a boolean value horizontally (parallel to the border).  By extending the bottom row to a double-row that acts as an explicit crossover gadget, I should be able to embed the NP-complete problem mentioned above.  A rough estimate is that I need a variable tile in 2 variants, a clause tile in 3 variants, 7 tiles in 2 variants, with another tile in 4 variants (this enables crossover).  This total to, wow, 23 tiles!!!  Cool thing.

Okay, so here's the plan.  This proof relies on a few things.
1) NP-completeness of Cubic Monotone 1-in-3 SAT.
2) My rectangle construction from yesterday.
3) Overall logic in glueing the two.

(1) is a known result, but I should make sure that I understood it correctly.  (2) is difficult to check thoroughly, so I need to let it rest a little and check again.  I checked (3) again today after I came up with it.  It should be fairly sound.

Anyway, tomorrow I will probably try to check (2) again and read up on (1).  Again, there might be something wrong with this proof—if I discover it, that's not the end of the world, since I can then try to fix the hole in the proof (pun intended!).  But in any case, shouldn't celebrate yet.  I emailed Igor and asked for a Wednesday meeting.  Let's see if I can't get this worked out by then.

Oh man, I am saying I shouldn't be too excited yet, but I am.

This Saturday, I woke up relatively early and did some math in the morning before going to Bible study.  I started reading a paper I printed yesterday, and I couldn't finish reading because ideas kept coming to me that I had to try out.  So I did.  Anyway, the question is trying to come up with some crazy region that is still simply connected.  The paper I am reading uses regions with a lot of holes.  I am trying to plug up the holes.  So today I came up with 7 tiles that can make a rectangle.  I can control the length and width of the rectangle by encoding it as one boundary of the region.  This is a tiny step, but I am really excited and motivated at the moment.  Hopefully from rectangles can come more general regions, and then regions beyond regions, and finally plug up all the holes.  I know this is incomprehensible.  This entry is probably mostly for myself, just in case I actually solve this problem and become famous and needs to write a memoir one day. :-D  Just kidding.

Anyway, this was what I did in the morning and in the afternoon.  But between it, I went to Bible study, had lunch, and then went groceries shopping in Arcadia.  How do I manage all that?  Thanks to my car rental.  Thanks Steph for convincing me.

This Friday, I went to school fairly early and had breakfast there.  In the afternoon, after Igor's class, I went to talk to him.  We made an appointment first day of the term and had postponed it 5 times and skipped 2 options.  Finally, at the end of the third week, we were able to meet.  We talked for over an hour regarding various things.  And at the end, I proposed, and he accepted.  I now have an advisor.  Yay!!  He already gave me two problems to think about.  Afterwards I went to the library to borrow references and to the reading room to print out papers to read regarding these topics.

Then I rent a car from Enterprise (they came to pick me up).  It's some kind of "bug" or something from Volkswagen.  I picked the Corolla class but they were out, so they gave me "free upgrade."  Uggh, I'd rather just drive a Corolla, and didn't really want to pay for a Camry (maybe I should have).

Instead of ending with a quote from The Little Prince, I'll post what Tammy wrote to me in an email:

"I, Jed, take you, Igor, to be my academically-wedded advisor.
I promise to cherish you in sickness and in health,
Till PhD completion do we part."

Thanks Tammy.

This Tuesday, I spent most of the day trying to understand the bijective proof of the hook length formula as written in Sagan.  This proof is due to Igor and his coauthors.  He told me I don't need to read it, since he feels bad making me read his result.  But I am, of course, going to read it.  It is quite involved, and this is the first time in a while that I had to do a 2-pass on some proof, reading once purposefully skimming it to get the bigger picture, and then re-reading it a second time to get the details.  [This is similar to 2-pass video encoding.]  The proof spans several pages, so it was a major endeavour.

Besides this, there are a few interesting things today.
1) I was thinking about math as I was jogging, and I completely missed a turn I had intended to make.
2) I went to Whole Foods to get breakfast possibly for the first time on a school day, this is due to the fact that I forgot to defrost bread last night.
3) We had a power outage today, but everything is restored.
4) I finished watching two movies today.  I did not watch movies at all for the first half of the year; but since June, I've watched 12.  Half of them were watched on planes or with friends, and a quarter were watched in this past week.
5) I got a delivery just as I opened my door.  Perfect timing!

This Monday, I printed out Igor's paper titled The Nature of Partition Bijections I. Involutions which he sent to me during the holiday.  I started reading it on Friday but was having trouble reading it on the screen since I need to refer to so many different parts back and forth, and flipping PDF file is infinitely more awkward than spreading out papers on my gigantic U-shaped "whoosh" desk.  So I stopped then and decided to wait until I can print it out.

It was a fun read today.  I spent quite a bit of time thinking about it.  This paper outlines some idea which is on one hand exciting, but yet on the other hand disappointing.  The exciting thing is that we now have a natural and uniform way of extracting involutions out of bijections, appealing to every mathematician's desire to generalise and bestow order on the chaos.  On the other hand, it takes out the magic and romantic nature of these seemingly clever involutions, which would have appealed to the kid inside of every combinatorialist.  Indeed, the reason Igor sent this to me was because I emailed him about Andrews's involutive proof of the Gauss identity.  There is a fairly trivial way to prove a more general identity using a bijection due to Sylvester.  And now by projecting Vahlen's involution (an involutive proof of the "trivial identity" $(1-t^i)/(1-t^i)=1$) along this bijection, we recover Andrews's clever involution.  Now the magic is gone.

This Wednesday, I started reading the third chapter of Sagan's The Symmetric Group.  The chapters are

1) regarding group representations in general,
2) representations of the symmetric group specifically,
3) combinatorial algorithms thereof,
4) symmetric functions, and finally
5) applications and generalisations.

The third chapter is on combinatorial algorithms that I already learned from reading chapter 7 of Stanley regarding symmetric functions.  As one see, symmetric functions, representations of the symmetric group, Young tableaux, and combinatorial algorithms thereof are intimately related.  However, Stanley's approach is with symmetric functions, whereas Sagan's is obviously of representation.  So this should give a different light to these algorithms.  In fact, since symmetric functions is developed in chapter 4 of Sagan, these combinatorial algorithms are developed and proved without using symmetric function machinery.  That is, they are proved directly by combinatorial arguments—as combinatorial results ought to be proved (if possible).  This means I am going to have some reading that would remind me of the old days when I simply delight in the cleverness and tenacity of combinatorialists.  In any case, I read 4 out of 11 sections today.  It looks like I will be able to comfortably finish this before the quarter is over.

This Monday, I woke up at 05:59:44.  Last night I slept a bit later, since I ate too much for dinner (though it was partly due to eating too little for lunch, so I compensated; but since I usually eat much more for lunch than dinner, it felt really extra much—despite the fact that this morning I weigh myself and I actually lost a bit of weight), and therefore had more trouble waking up today.  In the morning I reviewed whatever I read from Stanley so I could meet with Igor in the afternoon.  I did not want to repeat last time, where I basically did not remember anything to be able to answer his questions intelligently, even though I did read carefully and thought about every little thing—I simply did not retain enough information to hold a conversation.  So this time I spent two days reviewing.  And I think I had a more successful meeting with him, despite the fact that I still did not know enough, of course.

After that, I came home and Jonah suggested visiting me, so I went to pick him up from school, went to Whole Foods together to get some groceries, and then came home.  I cooked and we ate.  I had four dishes today (one was leftover from yesterday).  Then we talked and sang hymns.

This Friday, I woke up at 4 something, decided to "sleep in" until 5, and got up.  I did so much work in the morning that I felt okay to respond to a few Facebook posts before lunch.  Then I bought a taco and ate two, courtesy of "one free taco" coupon from Rubio's.  Since I printed the coupon out, I might do this a few times before it expires.  Then I went to Murphy Hall to take care of something.  While there, I figured I might as well pick up my diploma.  I now have physical proof that I have Mastered the Art of mathematics.  Pretty cool, eh?  Today's class was about Dyson's rank.  It was pretty fun.  After class, my officemate asked me lots of questions and so we discussed for an hour again—I actually quite like doing this.  Perhaps due to the light lunch, I was really hungry, so I ate dinner at 4 something, and relaxed a bit.  Now I'm ready to sleep, so I can wake up way too early again tomorrow.  Yay!

This Wednesday, I did math in the morning, went to have lunch with Athipat, and then went to class.  After class, Igor told me he thinks I am right to have been suspicious about a problem he gave as homework a while ago.  He indeed confused it with something else, and there is no easy way to do it—as I suspected.  This is one of the two problems I was unable to do.  The other one has a solution using heavier machinery that we have not learned yet at that time.  So I guess I did pretty well overall.  Then I stayed in the classroom a while longer to think.  He gave three applications to the Sylvester bijection (between partitions with odd parts and distinct parts) to enumerate distinct parts by largest part, odd parts by distinct sizes, and odd parts by number of parts.  I was able to figure out the first and third applications in the room within a few seconds of thought.  But the middle one took me a long time.  I moved to the grad lounge and continued working.  After about half an hour of thinking, I finally figured it out.  It's actually quite clever and non-obvious.  It involves looking at outer corners of the Young diagram and interlacing them across a slanted diagonal.

At night I went to Mitsuwa for dinner on the way to tutoring.  Going early and eating in that area helps avoid traffic.  Next week I will go even earlier, since today we finished around 8 and I was so tired I was falling asleep driving back.

This Monday, I started reading again since the great homework was over.  I am enjoying this reading a lot.  I just alternate between reading and doing random stuff on the computer, with the ratio heavily titled towards reading.  Like, read a long time, come and do a simple trivial task, go back and read a long time, and such.  It's a good mix.  I also try to read on my couch, so I am forced to get up from my desk-chair and make the arduous journey across my room over to the couch.

This Tuesday, I got up at 5 something and went out to jog before the sunrise.  It feels really good.  Then I posted on my Facebook status "counting pretty trees (each label is greater or less than all its neighbours), perky increasing trees (each non-root vertex has even number of sons), and petty increasing trees (each vertex at odd distance from the root has at most one son)."  I basically set my goal to enumerate three classes of trees!  Yesterday I already started working on enumerating pretty trees, which I completed today.  Perky trees took me a long time, since there is a parity problem.  I started reading Stanley, and I came across the composition formula for exponential generating functions.  I paused for a beat, and then went crazy.  This is exactly what I need!  So I whipped up the hyperbolic cosine and solved the problem.  Armed with this confidence and (slight) experience, I quickly enumerated the petty trees.  I was really happy, since I actually set out to do quite a bit, and I was able to do them.  I am starting to see the extreme beauty (and power) of exponential generating functions.  Cool!!  (I'm going to sleep now, and skipping prayer meeting.  Once my sleep time stabilises I'll consider going again.  For now, I plan to sleep around 8 and wake up around 5.)

This Monday, I spent the whole day working on math as usual.  I tried to prove that the number of alternating trees on $n$ edges is the same as the regions in the complement of a hyperplane arrangement known as the Linial arrangement.  On the surface, it looks very similar to the Shi arrangement that we learned in class.  But we drop the conditions that $x_i\neq x_j$.  As such, they can now make complicated collision patterns.  I thought about this long and hard, starting last week, and I finally came up with this brilliant way to count the number of vectors not on the hyperplane arrangement reduced to a finite field.  Even after getting an answer, now I have to enumerate alternating trees, which is a whole other problem.  I'll continue to tackle this tomorrow.  For now, I'm going to try to sleep early(ish) and take advantage of the end of Daylight Saving Time.

This Friday, besides eating lunch at school, and going to class ("the usual" for Friday), I spent the rest of the day reading.  Reading what?  Stanley's Enumerative Combinatorics Chapter 7 on symmetric functions.  I am doing a reading course with Igor and this is what I ought to have been doing all along.  I have of course been doing it, but with his homework taking up most of my time, I had not read much.

This Thursday, besides jogging, eating lunch at home, and going to school for seminar ("the usual" for Thursday), I spent the rest of the day reading.  Reading what?  Stanley's Enumerative Combinatorics Chapter 7 on symmetric functions.  I am doing a reading course with Igor and this is what I ought to have been doing all along.  I have of course been doing it, but with his homework taking up most of my time, I had not read much.

This Tuesday, I went to the dentist.  Apparently I am getting cavities on the fillings, despite brushing AND flossing.  I am now on a fluoride program.  Sigh.  Today is my first day of rest, besides going to the dentist, I also vacuumed my room (way over due), opened up a new fan (I'll write about this tomorrow when I get a picture), and bought some necessary items.  In the evening, Michael Woods called me and we talked for 1.5 hours... about math!  Actually, it was about differentiable manifolds, diffeomorphisms, and the Implicit Function Theorem.  It was a great way to have spent the evening.  For two days in a row now, I've only watched one block (20 minutes) of TV each.

This Monday, I did the finishing touches to my proofs, and submitted the homework set.  I have about 50 pages of scratch work for these 10 problems.  I guess it's not that impressive, since some problems were easy.  But there were some problems for which I wrote more than 10 pages.  After class, I explained some problems to a fellow student for about an hour.  When I got home, I was really tired, so I rested for a little bit.  Then I went to tutoring, followed by driving to Torrance.  I had dinner with Steph, and we talked.  It was quite good—we haven't had dinner alone since I went to Taiwan during summer break.  Well, save once when we shared a (super-expensive) pizza at the airport when we flew up to the Bay together, but that was hardly quality time.  Anyway, it's been a while, so it's good to catch up.  I am extremely tired, so for once I am going to go to bed "early."  (Indeed, it's been a while since I slept before midnight that going to bed now is actually sort of considered early, despite the fact that it is still rather late.)

This Sunday, I went to church as usual, but left early.  I gave Anne a ride back to USC (for the first time), since she wanted to go early too.  This definitely contributed to me leaving basically right after lunch, but I also wanted to work.  I revised my proof for a problem that I just wrote up (since I solved it pretty late in the game), and also proofread all other proofs.  I am fairly satisfied with my work.  I even found a modest simplification of one of the proofs during this process, so I re-wrote that as well.  This set is due tomorrow.  I'd have to say I probably worked for at least 4 hours on this pretty much every day since it was assigned about 12 days ago, and some days I worked even more than 10 hours.  I spent a lot of time, yet I was unable to solve 2 subproblems.  I think this is probably good enough, since Igor openly stated that he does not expect us to solve all the problems.

[Photo: My cleaning products.]

This Thursday, I found the number of spanning trees containing a given forest with known component sizes.  I also wrote up some proofs, including a combinatorial proof of the Cayley—Hamilton theorem.  I worked till quite late at night, and my proof is very ill-organised.  This is because I had only had the idea in my head, without writing any scratch work.  And since this is a bookkeeping-heavy problem, I typeset a lot of formulae, including an alternating sum of a product summed with a double sum of another product.  My proof ran about a full page with 13 paragraphs or so.  Only after I finished I realised I probably only need 1/3 of this.  I will reorganise and trim this later.  Now I am too tired and should go to rest.

This Wednesday, I went to tutoring in the afternoon, and then was going to head to Torrance to meet up with Steph.  But she contacted me last minute saying she can no longer meet with me.  So I came home.  I immediately started working, and wrote up many proofs, since I was a bit tired.  I like to write up proofs when I am tired so I could feel productive while not over-working.

This Monday, I spent most of the day trying to solve two problems.  One is the eqidistribution of parking functions by its sum with labelled trees by inversions.  The other is centrally symmetric non-intersecting chord arrangements on the circle.  I worked quite hard on both, and eventually solved the first one.  I decided to then go to sleep.  This was about 23:45.  As I was lying in bed, I started running through the proof of the problem.  Suddenly I had a flash of insight for the other problem!  So I had to get up.  This was about 00:15.  I flushed out the idea, noticing that it was flawed.  However, I had a sense that this idea could be salvaged.  If it was right, I would remember it after I wake up; but since it was wrong, I need to attempt to salvage the heuristic right now, otherwise I would surely lose the feeling.  So I worked on it, and I solved it in about 20 minutes!  So interesting that a problem I worked on for many hours during the day, I would solve in a short period from a flash of insight while falling asleep.  This is further proof of Wilson's claim that I mentioned yesterday.

This Friday, I worked at home in the morning, went to school for lunch, and went to class.  Then I discussed some combinatorics with my officemate, and then came home.  Since then, besides taking short breaks, I've been working till now.  My work is inundating my life once more.  I made some good progress today, as opposed to yesterday, where I worked very long hours but to no avail.

This Thursday, I worked for probably more than 10 hours.  So tired.  Besides the usual, I went to an unrelated colloquium talk today.  It was sandwiched between combinatorics seminar and departmental tea, so I figured I would go.  I brought my own work just in case I can't understand anything.  But it was really interesting.  The topic was on classifying knot invariants.  So in some sense we are looking for invariants of invariants.  He looked at skein algebra with a parameter A.  If we take A to be the multiplicative identity or the additive inverse thereof, the algebra becomes commutative, and using the skein relations twice essentially allows the knot to pass through itself.  I noticed this and asked after the talk if this is why in those cases we reduce to the fundamental group of a surface, since the knots become homotopy classes.  He was delighted and said he should have mentioned this.  That means I understood the talk!  Yay, so happy.

This Saturday, I attended my first American Mathematical Society meeting.  This time the sectionals is held at and hosted by UCLA.  So it is super convenient to attend.  I went to 9 combinatorics talks from 9 AM to 5:30 PM.

Afterwards, I came home and organised my stuff since I am too tired to work.  I pulled out my scratch work from my algebra and algebraic geometry classes, and placed them in my scratch work "recycling" bin.  To call it a recycling bin is misleading, since I don't throw it out.  (I still have my box of four years worth of scratch paper from Caltech.)  To make it easier to navigate, I put coloured paper between the different classes and labelled them properly.  Okay, I guess I am OCD.  Then I reorganised the more "important" papers on my slots and also labelled them with post-its.  I should have done this long ago—and wanted to do this for quite some time now.  Having done it today, I am so happy!  This is what Saturdays are for.

[Photo: My paper stacks with post-it labels.]

This Thursday, I ate leftovers at home, and only went to school at 2.  Usually I'll have to go earlier, but today there is a Combinatorics Colloquium so the usual seminar is cancelled.  The colloquium is from 3 to 4.  The talk was about the static and stochastic Ising models.  It was quite a good talk, in my opinion: very easy to understand at first, and then it turns engaging, and towards the end, indecipherable.  This is necessary so everyone gets something—those with less background in combinatorics than I, people like myself, and then those who know way more than I.  Afterwards, I stayed at school and worked.  Why, you may ask, when I could also work at home?  Well, I am nearing completion of this homework set, and I like to print it out and proof-read it on paper, and then revise, and repeat.  This requires that I have access to a printer, and hence I stayed at school.  Since we had tea served between the two colloquia (I did not attend the second one, but went to work), I was able to last until later.  At 8, Josh Zahl and I walked out to eat dinner at Enzo's Pizza.  It is good to catch up with him again.  I told him the beautiful injection, and he was able to appreciate it.  Afterwards, I kept working in the math department until around 10, and I finally went home at 10:30.  It's a long day, but it is a good day.

This Monday, I did work in the morning as usual, and then went to school for lunch at 11:30.  Afterwards, I kept working, and besides having an hour of class and also getting some snacks, I worked from noon till about 8 PM.  At around 7, I was getting tired, plus I was not making progress on the one problem I've spent ~10 hours on.  So I was about to come home.  But then I saw a flash, and voila, the problem was solved!  I was so happy I was jumping up and down and shouting.  There were still some people in the math department; they probably thought I was crazy.  Anyway, then I came home and had dinner around 9 PM.  It's a long day—total working over 8 hours.  But it is worth it.

This Sunday, I worked 8 or 9 hours!  I went to church in the morning, translating and doing treasury.  Afterwards, they decided to play football, which is great since then I could just come home and work.  I focused on work quite well—only taking small breaks to eat leftovers and such.  I got a lot of work done.  It feels really good.

At the end of the day, I had a lot of windows open.  2 firefox, 2 mplayer, 1 thunderbird, 1 xdvik (my homework set), 9 xpdf (lots of paper references), 21 mrxvt (my console of choice, of course).  A total of 36 windows on 12 desktops means average of 3 per desktop (pretty good, eh?).  I actually have 3 empty desktops and then on the 9 used, 6 of them are fixed with the same things all the time.  So 2 of them were very crowded with all the work windows.  I originally had it on one desktop, but it got too much to handle, so I split them into two.  This is how much my work is going on.  I had 33 pts open, and 19 Firefox tabs (3 non-work related, the rest were work tabs).

I don't usually work this hard (or at all) on Sundays.  I am happy to have once again found work.  I think I haven't worked this hard since studying for my quals half a year ago.  It's been quite some time.

I should also mention that in the last 6 days, I've gone to the groceries store 5 times—only Friday (the day I came home from school at 9:30 PM) was without shopping.

This Thursday, I only have one hour of seminar.  I also have leftovers from yesterday so I did not have to cook.  Hence I spent almost the entire day working.  It feels good to be working hard again.  At the end of the day, I had accessed seven books, now scattered about my desk.  I decided that it is so messy that it merits a picture.  So here it is.  I also have a tagged version [DEPRECATED] on Facebook (you'll have to login to see the tags).

[Photo: My messy desk after a day of studying.  It is about to become pristinely clean since I am obsessive compulsive.]

This Monday, I went to talk to Igor after his class.  After discussing several possibilities, we've decided that I should learn more about symmetric functions.  So I will be reading Chapter 7 from Stanley's Enumerative Combinatorics (volume 2) which I am buying from Amazon.  It should be here this week.  In the mean time, I am to read the first few sections of Macdonald's Symmetric Functions and Hall Polynomials that I borrowed from the library.  Reading is going to be fun.

This Wednesday, I went to sleep at 1 and woke up at 10.  I think I'm slowly getting back to a more normal sleep schedule.  Yay!  Today I tutored again, and also stopped by a (different) groceries store on the way home.  I got an email from Igor about the course he is teaching (which I plan to take).  I looked at the books list and I have one of the four books.  So I got it out and started reading it.  I read Stanley's Enumerative Combinatorics (vol. 1) late into the night.  It's a good feeling.

This Sunday, my family went to the reunion of an extended family.  No, we did not go to an extended family reunion, but rather, to the reunion of an extended family.  My father's father's younger brother shall be referred to as Om.  The descendants of Om had a reunion.  Om had four children, referred to as O1m, O2m, O3f, and O4f (the m and f denote sex).

Of the first family, O1m, his wife, their son, and their daughter.  The son has a pregnant wife, and a daughter.  The daughter has a dog and a boy friend.

Of the second family, O2m is deceased.  His widowed wife is the hostess.  She has three daughters, O21f, O22f, and O23f.  O21f has a husband (who was the first to greet us), his dad (who was my father's English teacher from high school), their son, and a baby about a week old.  O22f has a husband, a son, and is also pregnant.  O23f has a boy friend, who was the one who grilled the meat; they just returned from England a few days ago.

Of the third family, O3f is present.  Her husband was not present.  They have two children, a son and a daughter.  The son has a wife, and they have a child.  The daughter is by herself.

Of the fourth family, O4f is present.  She has two daughters.  Her husband and her younger daughter are in Hsinchu, and thus not present.  Her older daughter is present.  She is in my school year, and is my junior high classmate Judy Chang's elementary school classmate.

Moreover, the husband of the eldest daughter O21f of the second child O2m of my father's father's younger brother Om, is also the son of the older brother (my father's aforementioned English teacher) of the husband (not present) of the third child O3f of my father's father's younger brother Om.  That is, O21f married her father's sister's husband's brother's son.  They are not blood related, though their fathers are brother-in-laws.

I shall formalize my naming scheme:
- O denotes the root (in this case, my father's father's younger brother).
- Number N denotes the Nth child of the previous identifier.
- A letter is either a modifier or a relational letter.
- O, numbers, and letters are considered characters.
- Each segment is a string of numbers followed by precisely one modifier letter, and then appended with any nonnegative number of relational letters.
- Only the first segment may contain no numbers.
- Each name is of the format O followed by any positive number of segments.
- When parsing a character, we consider the substring from O to the current character as the current identifier, unless the current character is a modifier letter, in which case we exclude it.
- The previous identifier is the current identifier without its last character.
- m, f, k, u are modifier letters denoting male, female, kid of unknown sex, or unborn baby of the previous identifier.
- h, w, b denote husband, wife, and boyfriend of the current identifier (complementary sex pairing is understood).
- d denotes dad (alliteration!) of the previous identifier.

Then a concise list of those present beside my family can be presented: O1m, O1w, O11m, O11w, O111f, O112u, O12f, O12b, O2w, O21f, O21h, O21hd, O211m, O212k, O22f, O22h, O221m, O222u, O23f, O23b, O3f, O31m, O31w, O311k, O32f, O4f, O41f.

Those with two numbers after the root, except for the one with a "d" relational letter, are in my generation.  The ones with only one number are in my father's generation.  Those with three numbers are then, of course, a generation below me.

Anyways, to anyone but myself, this is probably a very pointlessly dry read.  I know not why anyone would bother reading this.  But it is nice to keep a record, and I might as well keep it here.  I might have gone a bit over with the formalization of the naming scheme.  This ought to appeal to those with a mathematical mind. :-)  This is also my longest post ever.  With a close second [DEPRECATED] containing 3863 characters.  This has 4167.  Well, they were close before I provide proof that this post was the longest, when it only had 3871 characters, differing by 8 characters.  This post also took me 80 minutes to compose.

American Mathematics Competition 12.
The individual results were known long ago, I scored 146.5 out of 150 possible.
I just received the Summary of High School Results and Awards in the mail.
I am regional first place (as noted a few posts ago, now I know what that was for).
There are 10 regions of USA and an additional region for Canada.
I am in region 8 and it consists of the following (12) states:
Alaska, Arizona, Colorado, Guam, Hawaii, Idaho, Montana, Nevada, New Mexico, Oregon, Utah, Washington, and Wyoming.
Additionally, I beat the top scorer of regions 3 through 5:
Alabama, Florida, Georgia, Puerto Rico, and Virgin Islands; Indiana, Michigan, and Ohio; Arkansas, Iowa, Kansas, Minnesota, Nebraska, North Dakota, Oklahoma, South Dakota, and Wisconsin.
As noted before, I am part of the three-way tie and places 13th in the nation.
Praise the Lord.

Try to solve the second-order non-homogeneous ordinary linear differential equation with the method of variation of parameters

and you'll see that this equation is so humorous!!

I just received a plaque from USPS today, which reads:
  ANNUAL HIGH SCHOOL
    MATH CONTEST
        2004
Highest Regional Score
      JED YANG
but I don't really know what this is all about...
I better call the mailer and confirm what I did to deserve this.

This decomposes each y in Span {u_1, ..., u_p} into the sum of p
projections onto one-dimensional subspaces that are mutually orthogonal.

This is why orthogonal basis is so cool!

To see if the infinite series (n^2+1)/(n^4+1) converges,
I used the limit comparison.
The ratio of that previous thing and 1/n^2 goes to 1 as n goes to oo.
So since 1/n^2 converges, the previous thing also converge.

Correct? Yes.