This Tuesday, I went to teach in the morning as usual. I demonstrated the power of Pappus's Theorem, where we use the arclength and centroid to find the surface area of a surface of revolution. Then I came home and had leftovers for lunch, watched some TV and did a lot of cleaning the apartment simultaneously. In the late afternoon, I went to the gym since I didn't go yesterday (I was on a stroll, and Dad said that was enough exercise). I'm trying to retain my drive to exercise roughly every other day (thrice weekly) by not giving myself too many excuses. I've realized that it's always a slippery slope—and I'd rather not start. In the evening, I went to prayer meeting as usual. Olympics: Gold medal match between USA and Brazil women's indoor volleyball. I have some thoughts, maybe I'll post them later. Too tired now. Good night.
[Non Sequitur: Helvetica, my hero]
|Stroll and Chat||0 comments|
This Monday, I continued writing my third paper in the morning. I wrote a series of four theorems, all relating to each other. I still need to fill in the proofs, but I'm pretty sure they are all in my head. To conjure them up, I went out for a stroll in the afternoon. I ended up talking to Dad for most of that walk—but it was very very good. We came up with a lot of good ideas (not about my research)—ideas that neither he nor I had on our own—this shows the power of discussion and positive synergy. After I got home, I took a nice hot bath (not shower) and figured out most of the details of the proofs. I'll probably write them down on Wednesday, unless I have strength to do that after teaching tomorrow. In the evening, I talked to Caleb for some time, then had a late dinner, and prepared my lesson for tomorrow.
Oh, on a side note, when I was cooking rice for lunch today, I accidentally made "egg rice." Indeed, I usually steam eggs while I steam rice by putting a divider in the rice cooker. Today it tipped over so the eggs got mixed in with the rice. It turned out quite nicely though. Pleasant surprises in life. =)
[Non Sequitur: FlyeBaby hammock]
|[TT] Four People||0 comments|
This Sunday, I woke up at 6 again, but stayed in bed until 7 since I was pretty tired. I then went to church early and got ready. I did the usual: translated, played guitar, and did offering. I finally did not have a meeting during lunch! Of course, by the time I was done with offering, everyone's done with lunch. But at least I got to talk to a bunch of people while eating, since they were not gone yet (as opposed to when I come out of the meetings in the previous weeks). It was nice to catch up a bit with Anne, Jennifer, and Mari. And no, I was not trying to talk to sisters. I was simply sitting alone on a bench and they came over to talk to me. Afterwards, I went to Caltech to play table tennis with Ran, Matt, and his friend Howard. It's been a while since we've had four people, which is so much better than just three. I really miss playing table tennis—this has been the sentiment each time I played. I wish I would play more regularly so I stop feeling this way. Since I twisted my ankle during a game (I still won it), I decided to take off early and come home. As such, I was able to have dinner at home. I ate the Har Lam Kee leftovers that I inherited from the Ohio people yesterday. I did some light work and reading. Another week.
[Non Sequitur: Oh the places you'll go...]
This Saturday, I got up at 6 again, after four days of waking up at 7, which was preceded by 8 days of waking up at 6, after waking up at 7 after the retreat. I then went to New/Valley for a rice stick for breakfast, and went to church early to set-up. Today I twiddled with the AC settings, so I think I wouldn't have to come early any more—it's set to turn on and off automatically now. After Bible study, the Ohio people (including Jonah) and I got lunch from the Taiwanese lunch box place and brought it back to the church building to eat. We stayed around for a bit while Steve was finishing some basketball or football fantasy thing. It was good to spend some time with these people—I'm starting to get more comfortable with them. I then called Charley and talked for 1.5 hours or so while driving to and shopping at 99 Ranch Market, and continued all the way until I got home. It's nice to catch-up with him—I haven't talked to him extensively since I left Seattle about a month ago. In the evening, I went to the gym, but it was closed. Alas. I decided to come home, do some work, and then relax.
[Non Sequitur: Fonts and trust]
This Friday, I couldn't fall asleep last night until past 2 AM, but still woke up at 7 AM naturally. I then did some math reading, Bible reading, chores, and errands. In the afternoon, I talked with Caleb while walking to the gym (like with Steph on Wednesday), exercised, and then walked back while talking to him again. As for Olympics, I watched both May/Walsh dominate Italy in women's beach volleyball quarter final match, and also the women's gymnastic team final. This is the first time I watched gymnastics, and I have no idea what's going on.
[Non Sequitur: Descriptive camera]
|No TV||0 comments|
This Thursday, I woke up early to prepare for teaching. I found something worth-while to show them, but I didn't end up having time. I was not trying to push it in too much, since I'll have Tuesday discussion again before there is any lecture. Indeed, usually there is a lecture on Monday, but since it is a holiday, there are no lectures between my two discussion sections. As such, I didn't really feel like using up all the interesting material. Today I did a calculation to show them that the surface area of a sphere between two parallel planes is the same as that of the circumscribed cylinder with axis orthogonal to the planes. It is very cool actually.
In the afternoon, I held office hours, and then came home. I was fairly hungry, but it wasn't dinner time yet, so I decided to go out for a drive and hunt for interesting food. I ended up eating at Ramen Yamadaya. It was pretty good—thanks to Tammy for the recommendation.
I feel like I've been watching too much TV lately, so I decided to go (at least) one day without watching TV. I did it fairly easily actually. Now that I know I'm not addicted, I might go back to it with more peace tomorrow.
This Wednesday, I did some work, prepared for teaching tomorrow, and took care of some errands. I cooked a bit too much for lunch, but ate most of it anyway. Oops. In the afternoon, I talked to Steph for a while—we haven't chatted for a long time. It's good to catch up for a bit. And then I went jogging again—this time slightly faster (6.5mph) for the same amount of time, ending with a bit longer stretch (2.5min) of 8mph. I'll slowly build my tolerance back up. In the evening, Caleb called me and talked for a while. I continued watching the Olympics: finished watching women's volleyball quarter-final USA sweep DOM in 3.
[Non Sequitur: Edison's cradle]
|Late Lunch||0 comments|
This Tuesday, I went to school to teach in the morning. There was this girl who wanted to meet with me privately to talk about the midterm, so I waited for her during my lunch hour. She was very late, and got here as I was about to leave. I wish I had left a few minutes earlier so I wouldn't have bumped into her. Oh well. After helping her, I eventually got home close to 2 PM, and I hadn't eaten lunch! I was soooo hungry. I talked to Steph on the way but then hung up immediately as I got home since I really wanted to eat food. (I did have a granola bar earlier, but still, this is three hours past my usual lunch time, and seven hours past my breakfast—good thing I tried to eat dinner a bit late today and not right when I woke up.) I was so hungry and tired that basically I spent the rest of the day recuperating—doing chores around the house, reading Bible, some simple programming, and talking to Caleb. In the evening, I went to prayer meeting as usual, and stayed pretty late talking (partly about Proof without Words, the fantastic book that Barmore bought for me—more about it another day). It's a bit late, but I need to stay up a bit more to calm down before I can sleep.
[Non Sequitur: FreeKey key ring: The easiest way to fasten your keys]
This Monday, I did some math in the morning (what else is new?), read Bible (not this), and cleaned my apartment (and not this either). I continue doing the same stuff in the afternoon. Then as a change of pace, I went groceries shopping (how do you like this as a change?). In the evening, I went to jog in the grad student gym (KREC) since I don't have access to the Wooden Center during the summer. My leg injury is healing nicely so I'd like to get back to jogging. I always jog a bit longer when I jog in the gym, so I like to do that even though there is the overhead of getting there (and back). Today I started really slow, ran at 6mph for a while, and then increased to 8mph at the end. The whole thing was over in 15 minutes. I figured I would start small and build up again—given that I haven't ran pretty much since the beginning of summer when my leg started to hurt a whole lot. At night, I spent an hour preparing for teaching tomorrow (I actually don't remember any of this stuff, and really need to look at it before hand), and then called mother. She talked animatedly about her Bible reading on Kings. I told her that we also started reading Kings in our Bible study, partially due to my suggestion. She was very amused.
[Non Sequitur: Art made out of candy]
This Sunday, I was not feeling well, so I skipped the meeting. I had a HSG planning meeting to go to, and also wanted to attend the farewell meeting for JR, so I kept waiting to see if I feel better so I could at least come in the afternoon. Alas, it didn't happen. I cooked simple tofu, broccoli, eggs, and pasta for lunch, and continue resting. I eventually felt better in the afternoon, and was able to go do some tutoring near Hurry Curry. Then I bought Hurry Curry to-go and ate at home. When I was eating, Caleb called me, so I let him talk while I ate. In the end, we talked for about two hours.
[Non Sequitur: How to get paid to watch Netflix]
|Olympics (After)math||0 comments|
This Saturday, I went to Bible study, had lunch with the gang at Cafe Spot, got a haircut, and then came home. Here are some thoughts about the Olympics.
Round Robin Play
Some of the sports feature multiple groups of round robin preliminary round. Take, for example, women's volleyball. There are 12 countries partitioned into 2 sets of 6 countries each. Let us focus on only one of the set for now. The round robin means that each country plays the other 5 countries precisely once. It is kindergarten mathematics to realize that there are a total of 6*5/2=15 games. (Here "game" is the "match" in volleyball; just to avoid confusion when I use "matching" later in the graph theoretic sense.) Without having a team play twice a day, each team needs to play five different days. Here comes the interesting part: these games can be played in only five days. Of course, if it can be done, then every team needs to play one game per day. An event scheduler would need to specify a match-up (matching) between three teams with three other teams for each day, such that each pair of teams plays precisely once during the five days.
A graph is a collection of vertices and edges. We will denote each country as a vertex, and an edge represents a game between the two. Since every pair of countries need to play in the round robin, every pair of vertices are joined by an edge. This is known as the complete graph (on 6 vertices). There are 3 games in the first day, which is represented by three edges. Notice that these edges are disjoint, and cover all the vertices. This is known as a perfect matching of the given graph. If we proceed through all 5 days, we have 5 perfect matchings. Furthermore, they are all pairwise disjoint, and they cover all the edges. This is called a 1-factorization of the graph. (A perfect matching is also called a 1-factor.) The specific claim at the end of the paragraph is that the complete graph on 6 vertices admits a 1-factorization.
In fact, any complete graph on (necessarily) an even number of vertices admit a 1-factorization. Suppose we have 2n vertices. Draw a circle and put one of the vertices at the center of the circle. Put the other 2n-1 vertices on the circle evenly spread out. Now pick two adjacent vertices on the circle and pair them together, pair their neighbours together, and their neighbours' neighbours, and so on, until only one vertex remains on the circle. Pair this one with the vertex in the center. We just paired n pairs of vertices together. This gives a 1-factor. To have a 1-factorization, we would need 2n-1 such 1-factors, and the center vertex will have to be paired with a different vertex each time. Simply rotate the previous picture so that this happens (with all the other pairings adjusted accordingly), and we are done.
This is all very elementary, and perhaps not worth writing a note about. What I actually wanted to mention is that for a complete graph on six vertices, the 1-factorization is unique in some sense.
Imagine if we have a 1-factorization (namely, the schedule of how the 6 countries should play with each other). If we call the first 1-factor the third and call the third the first, nothing much has changed. That is, we should consider if we swap the team match-ups for day 1 and day 3, we essentially have the same set-up. (I know psychologically, it is very different: For example, in men's event, since USA beat SRB and GER 3-0, we were more confident going in to play BRA and RUS—which definitely helped us. But mathematically, let's consider that luck of the draw.) Also, let us consider that the order of the three games each day to be inconsequential. That is to say, a 1-factorization is an unordered set of 1-factors, each being an unordered set of edges.
Let me illustrate this concretely. Consider women's group A and group B.
Day 1: A: ALG-JPN, GBR-RUS, ITA-DOM. B: CHN-SRB, USA-KOR, BRA-TUR
Day 2: A: DOM-RUS, ITA-JPN, GBR-ALG. B: CHN-TUR, SRB-KOR, USA-BRA.
Day 3: A: DOM-JPN, ALG-RUS, GBR-ITA. B: SRB-TUR, USA-CHN, BRA-KOR.
Day 4: A: JPN-RUS, GBR-DOM, ALG-ITA. B: BRA-CHN, TUR-KOR, USA-SRB.
Day 5: A: ALG-DOM, GBR-JPN, ITA-RUS. B: CHN-KOR, USA-TUR, BRA-SRB.
Notice that if you substitute
CHN for ALG,
SRB for JPN,
TUR for GBR,
BRA for RUS,
KOR for ITA, and
USA for DOM
in the group A schedule for days 1 and 2, we get the group B schedule (albeit with different order of matches—but we said we didn't care about that). If we do the same substitution for Day 3 in group A, we get group B Day 4, whereas days 4 and 5 give days 5 and 3, respectively. That is to say, if we don't care about the days either (which we said, for the purpose of this discussion, we don't), then we found a 1-factorization preserving map between group A and group B.
What is the same then is the structure of the match-ups. Indeed, you can say that "When JPN/SRB play against RUS/BRA (day 4/5, respectively), GBR/TUR play against DOM/USA, and the remaining two teams play against each other." This means that once you put any pair of teams together, you know exactly how to match up the other four teams during that day! It turns out that this can always be done for 1-factorizations of complete graphs up to 6 vertices. Indeed, for instance, for 4 vertices: player 1 must play with players 2, 3, and 4 each. After we paired that up, the other two must play with each other. Uniqueness can also be proved like this (with a bit more care) for a complete graph with 6 vertices. I would hazard to say that it is a neat little exercise that you should consider trying. On the other hand, don't try to prove uniqueness for 8 vertices, because it is not true. There are 6 ways to do it. There are 396 non-isomorphic 1-factorizations for 10 vertices, and the next couple of numbers are 526915620 and 1132835421602062347.
Finally, it would not be right if I did not at least briefly mention that there is a huge theory behind this. The 1-factorization of the complete graph with 6 vertices is also called a 2-(6,2,1) design, and combinatorial design theory study these objects that are simply named "designs." One of the experts in this field is my undergrad advisor Professor Rick Wilson.
Suggested reading: Baranyai's theorem, http://oeis.org/A000474, A Course in Combinatorics by van Lint and Wilson.
[Non Sequitur: Patrick is a new dad, granddad, and great-granddad after three babies arrive in three months]
This Friday, I woke up and continued grading. I finished grading the last problem in 1.5 hours, and then spent another hour post-processing—sorting, adding up all the scores, and keying them in to the online grade book. I wanted to finish all this today since it is Drop Day and I think it would be nice for the kids to know their midterm scores before the day's over. Then I went to pick up Giang and drive to Ocean Star in Monterey Park for dim sum with our officemate Davide, who is leaving LA soon. He is going to Chicago for postdoc, and we will miss him. After lunch, Giang and I went to the 99 Ranch Market across the street to do some light groceries shopping, and then came home.
[Photo: Farewell lunch for Davide with Giang, officemates for three and two years, respectively.]
[Non Sequitur: 11 works of art made from coins]
This Thursday, I slept for more than 8.5 hours for the first time in a while—I went to sleep before 10 PM and woke up after 6:30 AM. I went to school to teach in the morning, had lunch, and then held a shortened office hours. Then I came home and spent the afternoon and evening grading. So far I've graded 3 out of 4 problems, taking a total of 3.5 hours so far. The 4th problem has a lot of parts, so might take a long time. We'll see how things go tomorrow.
[Non Sequitur: Robot learns to recognise itself in the mirror]
This Wednesday, there is a midterm for the class I am TAing. The instructor sent me the midterm via email while I was sleeping, so I went to school and printed it out. Then I came home and did the exam, finding several problems with the exam. I gave him suggestions of what to change, and then I did some work in the morning and essentially finished a different version of my paper. I went to school again to meet with Jackie to discuss GSO matters over lunch, and then prepped for teaching there. Then I made minor adjustments to my paper, printed it out, and came home. In the evening, I relaxed and talked to Caleb for two hours. (I watched the tennis men's doubles final in the Olympics, Bryan brothers beat Tsonga and Llodra, and we were discussing a lot of details thereof.) This is definitely longer than some of the days when he was physically here.
[Non Sequitur: The walking table makes moving easy]
|Free Combinatorics Lesson||0 comments|
This Tuesday, I gave review for the midterm in the morning, and then went to Mitsuwa for lunch again. Today is my final session teaching this client, after three years and hundreds of hours with trig, calculus, and physics. I decided to throw in a free session of combinatorics as parting gift. I used the material I made for my Los Angeles Math Circle lecture and taught him about Catalan numbers. I was pretty excited and this was one of our longest sessions. I'll miss tutoring him—he was one of my best students. Best of luck at Harvard, kid.
Afterwards while driving home, traffic was extra bad and it took me perhaps double the normal time to get home. I rested for a bit and then went to prayer meeting.
[Non Sequitur: Why waiting in line is torture]
|Extra Office Hours||0 comments|
This Monday, I forced myself to do some work in the morning to get back into the grove of things. I then went to school to run some errands and also host extra office hours. I was very popular today (as there is a midterm this week) and the extra hours were due to student request. Thankfully the colloquium room was empty so I used that instead. Afterwards, I went to Mitsuwa for dinner followed by, you guessed it, tutoring once more. I purposefully scheduled it at night so I can have eat dinner and let my client eat up (pun intended) my evening so I need not fret over having nothing to do and wasting too much time watching TV or such.
[Non Sequitur: Projection instructions for Gone with the Wind]
|More Meetings||0 comments|
This Sunday, I woke up naturally at 7 and got ready for church, getting there right before the meeting started (which is late in my book). I translated for breaking of bread, did offering, and then had a meeting with all the Sunday school class coordinators. It's always very reassuring talking to them. Today Jennifer joined us. Yay! She is taking over Steph's Upper Elementary class. After the long meeting, Tammy was already turning off lights, and most people had gone. I will have one more planning meeting next week to round off this four straight weeks of meetings. Actually from May 20 till next week, there are 15 Sundays. I was not here 6 Sundays (Utah 1, Taiwan 3, Seattle 2), which leaves 9 Sundays. There were 2 coordinators' meetings, 4 HSG planning meeting, and 2 retreat meetings. This means there was only one Sunday, namely, June 3, where I had no meetings. Thus for 12 weeks, I was not here half of the weeks and had meetings the other half. I'm definitely going to enjoy having my lunch back come September, and perhaps having time again to sing.
[Non Sequitur: Women lazier than men]
This Saturday, I woke up naturally around the usual time. Most people were already up! I led morning devotion for the guys (Beryl did for the girls) and then we had some pastries for breakfast. Originally we were planning to go on a hike, but since it was really hot, we decided to stay inside. We had some free time, and then had our second meeting. This was mostly free sharing by the kids, and closed up by Beryl's testimony.
Then Kevin cooked some burgers for us as lunch. Then we played some more Taboo (we've been playing this during most of the free times) and also Scattergories. Rosy with her experience (in life) and also trickery wiped us pretty clearly—though most of us also came up with some interesting answers once in a while. Afterwards, we played Mafia, one of my least favourite games—but since Jireh kept on asking for it, I suggested that we play it for a bit.
The rest of Matthew Wu's family arrived in the afternoon, and David shared the closing message. Then we played some more Taboo before dinner, which consisted of Costco pizza that the Wu's brought and leftover salads from lunch. We then packed up and left.
It's been a tiring two days. I'm glad to be back and sleeping in my own bed.
[Non Sequitur: Lenovo CEO gives part of his bonus to employees]
This Friday, I took Caleb to the Flyaway stop—his visit is over! Then I drove directly to the east side. Since I was a bit early, I went and bought some pastries at 99 Ranch Market before going to Beryl's house. I left my car in their garage and went with Beryl and Jireh to the retreat. This is the ACF high school and middle school retreat, hosted by Rosy and Benjamin. Besides Beryl, Kevin and I are the older ones going as chaperone/musician. I mostly just sing really loudly, and Kevin plays the guitar.
After arriving, we made simple sandwiches for lunch, and played some Taboo. Steven Lee shared the opening message on the theme: Seek First His Kingdom and His Righteousness.
Then we played some charades where we gave the other team the choice to act out Melchizedek (chosen) and Zebedee, while we acted out Matthew the Tax Collector. We realized our suggestions were too hard while theirs were too easy, so we, in good faith, tried to compensate. We then gave them Abel and Hannah (chosen), while they gave us Hezekiah.
After lasagne for dinner, we had an activity where the kids tried to put water, rocks, and sand in a small jar. After a little bit of sharing from Beryl, we had a bon fire where Uncle Benjamin shared, followed with some singing and mash-mellows.
[Non Sequitur: In Absentia: optical illusions series by Regina Silveira]
|Fire Alarm||0 comments|
This Thursday, Caleb drove me to school and we went to get $1 Jamba together. Then he left for a day at Caltech, while I went to teach. The first hour was interrupted by a fire alarm, but I finished teaching most of the stuff I wanted by then. The second hour went a lot smoother. Then I had a quick lunch and then held two hours of office hours. A lot of students showed up at my office—it was fairly packed. Then I came home and relaxed a bit, did laundry, had dinner, and packed for tomorrow's mini retreat.
[Non Sequitur: 3D display frame]