Htam::Diary

This Saturday, I finished reading the short story The Speckled Band in The Adventures of Sherlock Holmes.  I am doing leisure reading away from scientific magazines now!  I also borrowed The Hound of the Baskervilles from the Pasadena Public Library.  This is the third of four novels.  Today there was a small Bible study, but it was quite good, most people shared.

This Friday, I have understood the counter-example of a strong k-tournament of order n where n = k + 1, which is not Hamiltonian (does not admit a Hamiltonian cycle).  I tried to adapt the construction for n = k + 2 to disprove vertex-pancyclicity, but did not succeed.  It is also noted that now I just passed the half-way mark of my SURF: I am returning to Taiwan in 31 days and SURF started 32 days ago.

This Thursday, I verily enjoyed table tennis.  I first played with Ayman and won.  Then I played doubles with John, Miguel, and Mehdi (sp?).  We played with all three team configurations, with Mehdi losing thrice.  I scored some pretty good shots there.  Later, I played doubles with Olga, Paul, and Miguel.  I was with Olga, and we won.  Again, I performed well.  Today was very fun, and playing doubles is quite more social and with less pressure.

This Wednesday, I went to the SURF talk and it was impressively boring.  Afterwards, I went on the free tour of Huntington Library.  After walking indoors for a while, we went to the desert section of the gardens.  We saw a lot of cacti.  With my little tour group was Caleb Ng and Paul M (SURF student).  Walking for so long is very tiring.  We went to the Boone Garden of Sculpture afterwards to have some desserts.  I talked with Tony Wang a lot about our respective projects; he is a math SURF student from Hong Kong.  We even exchanged emails and we send each other a copy of our SURF progress reports.

At night, CCF council meeting was over (extremely late) dinner near Tim Kwa's house.  Jon Gardner drove me and Kevin Chen, picked up Nathan Lau on the way, and met with Tim and Scott Hsieh at a Chinese seafood restaurant.  We then went to Tim's house to finish the concil meeting.  Because we got lost a lot on the way back, tensing started half an hour late.

This Tuesday, I did all the usual things I do on Tuesdays: I have a meeting with my SURF mentor and presented him with a proof of the improvement I made last time, and telling him of the improvement I made yesterday.  I also played table tennis today with Roger, Freddy, and Wilson.  At night, I talked with Charley on the phone for 1.5 hours!  This is quite a long time, probably because we haven't talked on the phone since summer started.  Wow, so much catching up to do.

This Monday, I made more general improvment!  Until now, I have been working with the d = 1 case, so I have not even introduced the parameter d in any of my posts.  The general problem is, instead of asking for 1 cycle of each length, we want d disjoint cycles of an arbitrary length, through an arbitrary vertex.  To achieve this, the proposed characterization is d-edge-connectedness, where 1-edge-connectedness is simply strong-connectivity (or simply, strong) as we have been discussing.  For k = 3 case, I previously showed that for d = 1, vertex-pancyclicity and strong connectedness are equivalent for n at least 15.  Now for general d, I showed today that n at least 14_d_ + 1 works.  Thus what I got earlier is simply a special case of what I got today.  Along the same lines, and doing some preliminary investigation, I am conjecturing that for general k at least 8 and arbitrary d, n at least k + 2_d_ + 1 works.  This has not been verified.  Also, the photo shows what math research might look like. ;-)

This Sunday, I am feeling better.  But I still did not got to church.  I sang hymns by myself, and listened to a message online.  The rest of the day is still primarily spent by resting.

This Saturday, I am slightly sick.  Yesterday I was unable to James's house because of this.  Today I rested and did not go to church.  I went to sleep at 8pm (which is highly unusual) until 10pm.  I woke up, sent a few prayer requests by email to some brothers, and then went back to sleep.

This Friday I had a short meeting with my SURF mentor.  Since we only met Wednesday (due to July fourth), I did not have much time to work on things: the only result I had was for k > 3 (which is a lot, actually; but it was easy to present the result).  My mentor is pleased with my results.  He wants me to write up (typing, of course) the proofs for our next meeting.

Usually we meet on Tuesdays and Fridays, giving me equal time to work if I consider Sunday off (or Sunday 70% off and Saturday 30% off, or more generally Sunday p off and Saturday 1 - p off, where p is between 0 and 1).

This Thursday, I made some big progress on my research.  For k = 4, n needs only be at least 11.  For k between 5 and 7, n needs be at least k + 4.  For k > 7, n needs only be at least k + 3.  Gutin and Yeo demonstrated non-Hamiltonicity of k-hypertournaments of order n for n = k + 1, and since non-Hamiltonicity implies non-vertex-pancyclicity, I now only need to decide (find a proof or a counter-example) the case of n = k + 2.  Then I would have a few (finite) cases left in classifying the vertex-pancyclic hypertournaments.

At night Ning Bao came over to my apartment and played Karaoke Revolution for a while.  He brought volume 2 and 3 over.  I do not really know those songs.

This Wednesday, my SURF mentor professor Wilson approved my first progress report, and I turned it in, in order to get my paycheck.  I also presented the proof for the case of k = 3 reducing the requirement of n at least 29 to n at least 14.  We both agree this is a much more major improvement than the one bringing n  at least 32 to 29.  I plan on applying this same argument for k > 3 in the next few days.

This Tuesday is Independence Day.  I was quasi-sick.  I played puzzle games and did computer-related things.  I suspected that the table tennis club might not be playing today, so I did not go.  I finished up my progress report for SURF as well.

This Monday, many people have a day off.  To me, it doesn't really differ: I can work on math any day, and I can take any day "off" by not working.  However, contray to my prediction, somehow knowing that most people are not working made me not work today.  I thought about my problem a bit, looked at my progress report for a few minutes, but nothing substantial was done today.  I spend most of the day playing GUI puzzle games.

This Sunday, after missing about three (3) weeks of Caltech table tennis club, I finall was able to make it this week.  I first played with Mary, and then played with three other Chinese guys (Yongning He, postdoc, 2 grad) in doubles.  We played best out of three, and utilized all three different team configurations.  Che-Fung won all three configurations.

This Saturday, I went to Bible study at night.  It's been a while since I prepared before hand and shared at the meeting.  Today I prepared before hand, and shared at the meeting.  I really think preparing before hand and sharing at the meeting is hihgly beneficial.  Now eighty percent of the sentences in this post includes the phrases (or variation of) "prepare before hand" and "share at the meeting."

This Friday, the ABSK students went to Ray's house for dinner and games again.  We had pasta and garlic bread.  I really liked the bread.  Then we played Blokus [DEPRECATED] for four players.  This game utilizes the free n-polyominoshttp://mathworld.wolfram.com/Polyomino.html% for n from 1 to 5.  Each player has 21 pieces summing to 89 squares (1 monomino, 1 domino, 2 triomino, 5 tetromino, and 12 pentomino).  A n-polyomino is a generalization of a domino (2-polyomino), where a piece has n squares.  A fixed polyomino can be rotated, and a free polyomino can be flipped (and rotated).  Since we are playing with physical pieces that can be rotated and flipped, we use free polyominos.  Enough math—now the game play.  We play on a 20x20 board, each player start at a corner and we take turns placing one of our pieces on the board.  A player's piece must always touch his other pieces at a corner, but never on an edge (nor overlaps).  The goal is to put as many squares (not pieces) on the board as possible.  I played with Ray, Will (from MIT), and Jay Yang.  I won the first game decisively with only 13 squares left, whereas the others all had more than 25 left (I think).  The second game Ray won with 10 squares left.  I had 17 left.  Summing the squares left for both games, I won.  Bing Huo won even more spectacularly with four squares and zero squares!  He played with Tim Dong, Jian Yu Fung, and T C Neo.  It was an enjoyable evening.

This Thursday, our internet connectivity is finally restored.  I spend more time going over my progress report, editing proofs and clarifying statements.  At night, I ordered a new IBM T60 ThinkPad online.  After clicking on the confirmation, it reads:

Your request may take a moment to process, so please take this opportunity to...do nothing at all! Just relax!

I thought the message was atypical of IBM behaviour and quite amusing.

This Wednesday, still without internet, I went to the library to get the requirements of the first SURF progress report.  I started answering weird questions that does not really apply to math research.  I also started writing out the proof of the result I got for relaxing the condition from n at least 32 to at least 29.

This Tuesday, I woke up and discovered that my internet connectivity is gone.  This is quite annoying and inconvenient.  Of course this also led me to realize how much I rely on a constant connection to the internet.  I ended up spending more time in the library, using public computers.

This Monday, I made progress on my SURF project.  I am working on the problem posed by Gutin and Yeo in 1997, to characterize vertex-pancyclic hypertournaments.  Petrovic and Thomassen in 2006 proved that for a k-tournament with n vertices that is large compared to k, vertex-pancyclicity and strongly connectedness are equivalent.  Specifically, for k = 3, they required n to be at least 32.  I lowered this to 29.