## Saturday, 31 March 2012

### Basics of Integration. Part I. Technical basis.

Unlike the passages I've written before, I hope this one serves as a basic theory on integration. We may discuss more technical issue in a different approach than the textbook.

I assume some basic knowledge on differentiation and basic algebra, while some good basic knowledge on real analysis like technical refining on Riemann sum, MVT (mean value theorem) or upper/lower limit of Riemann sum, are not assumed but maybe discussed a bit here. The issue discussed will be less vigour then undergrad. course.

We start with FToC (fundamental theorem of calculus), anti-derivatives, then some technical work on substituion and by part; some real application on area and volume. That should be enough to cover most A-level techniques. The application goes deeper for differential equations, some simple contour integration, integration on polar coordinates and multiple integral. To end with we will focus on the art of integration, some special and beautiful techniques which kills many ugly integrands. Questions will be from A-level materials and the series "Today's calculation of Integral" (TCoI).

## Tuesday, 27 March 2012

### 27-3-2012

Quite long from my last time typing some progress/diary here...

Well, I'm not a secondary school student anymore so don't I expect I produce notes like in the past (unless some specified topics are required), and I'm still thinking what should I put here in the future.

There's no doubt that I'll continue writing stuffs and making osu! stuffs here, and some generalization or summary on the game vs reality theory -- it's really an interesting topic for me, there are too many subjects related there, like economics (game development, in-game vs market), mathematics, history (The projection of past into game), animation & music, psychology (self-projection on anime), etc.

The nature of RPG is different statistics, a simplified life model. Attack, defence value is determined by series of events, damage done is calculated by a certain formula, evasion is determined by fake-random variable. Everything is expectable, and we know the final result by mathematics. It is not that meaningful to raise a number in such a way instead of writing 1+1=2 on the paper --- (that reminds me George Orwell's 1984 --- freedom is the freedom to say 2+2=4, and if the above condition is satified, everything is not a problem"), so we should appreciate other elements of the game. They often success because of its pictures and stories --- great achievements when completing a story, many doujin CG and musical pieces -- they are the attractive points for ordinary players. We shall analysis the game deeply, the nature of the story, setting of the story, some comparative study with the past or other games -- and that's what I'm going to do.

And yeah. My major in undergrad is mathematics, so I'll still deal with maths here. The first targeted notes will be about integration, starting from FToC (Fundamental Theorem of Calculus) we will go through some typical skills, then some beautiful tricks. As a training of myself I'll try to make some complex integral here and some extreme real integral may need the assistance of complex number here. The material could be mainly from A-level questions, as well as TCoI (Today's calculation of Integral). Integration as a part of real analysis, own its beauty and we shall learn them seriously.

Today is Saturio's birthday, happy birthday!

wmfchris, 27 Mar 2012

## Wednesday, 21 March 2012

### 4 years of Osu! -

Joined: Mar 21, 2008
Last Visit: Mar 21, 2012
Happy birthday to my osu! account, and I am going to make a review here even though I make osu!-related stuffs here rarely.
Some "first of - " in my osu!
First solo play - 21/3/2008 - Wizards of Winter
First rank achieved (from zero rank) - #3173 (out of 7401 player)
First avatar - Sirius Black doujin
First achievement - 21/10/2008 - 500 combo
First taiko play - unknown, 2008
First taiko #1 - unknown, 2008
First ctb play - unknown, 2009
First ctb #1 - unknown, 2009
First solo #1 - 17/2/2009 http://osu.ppy.sh/s/3640
First map attempt - 24/2/2009 http://osu.ppy.sh/s/5462
First mod attempt - 6/4/2009 http://osu.ppy.sh/s/8474
First taiko overall #1 (with screenshot) - 31/8/2009 http://puu.sh/lzYj
First ranked map - 9/4/2010 http://osu.ppy.sh/s/14561
First taiko GD ranked - 8/5/2010 http://osu.ppy.sh/s/11909
First self-made solo diff ranked - 8/6/2010 http://osu.ppy.sh/s/7280
First map with long-time trial - http://osu.ppy.sh/b/3990
First map with 100+ retry at the same time - http://osu.ppy.sh/b/57627

Now let's reivew some of the important history in my Osu! life...
1) Mar 08' - Yes I resistered!
2) Oct 08' - My first play with screenshot, Beethoven virus [Hard] FC with and B. #90. Many of the new players could never imagine "an B = #90" because in harder maps that have few 100 and 1 miss while the easy maps all players just being defeated with 4mod SS...

3) end of 08 - My 500 combo achievement is from DDR megamix with halftime enabled. (LOL 32 min and half time...) but by the end of 2008, the multiplier of halftime changed from 0.6x to 0.3x, all HT scores are halved. My ranked score fall from #21 to #45...

4) Dec 08' - CTB score reset. I think I started to play CtB since that time, when I tried to make bunch of #1 in Winter... and I have already forgotten when did I start to play Taiko mode, but that must be far eariler than the time I start to play CTB.

5) Feb 09' - Mario is my favourite series both in BGM and the games, and there are some epic mario maps at that time like Underground mozart http://osu.ppy.sh/s/2857, Super Mario Bros. (TECHNO remix) http://osu.ppy.sh/s/628, Hyadain - Super Mario http://osu.ppy.sh/s/2442  and we played so much. Take an example, I takes me 4 year to FC 3 diffs in Super Mario...
[Easy] - 2008 http://osu.ppy.sh/ss/259835
[Normal] - 2010 http://osu.ppy.sh/ss/259830
[Hard] - 2012 http://osu.ppy.sh/ss/259833
And I #1-ed by my first map playing with FL on a [Normal] diff after like 40 play counts. You can still see me #12 there...

6) Jun 09' - Taiko slider hitpoint added. I don't think many of you know that in the past we can freely spam slider... and since a very few proportion of the score is made before the taiko hitpoint is added, a bunch of taiko score is quite funny. My #1 (in the past) - http://osu.ppy.sh/s/1369 Lullaby for you in TWEWY (The world ends with you, I liked that game too much) - I have had 1.2 million point on that map, with a C only. I remembered that many people reported my score, like the spinner bug in very early osu! history -- and that reminds me another unranked epic map, spinning Ievan Polka by loginer. Also, hats off to the mapper of Lullaby for you -- he's the one of the earilest HK player (UID 72!) and earilest HK famous player and mapper. You can still see replays from him, in 2012, 5 years after he play....

(now, here's another early HK mapper yuching, she mapped http://osu.ppy.sh/s/3027 M2U - Space of Soul)
7) Aug 09' - #1 in one of my favourite song http://osu.ppy.sh/s/7238 but only for 1 minutes after it ranks. Since that time I started to realize what's touhou and play touhou games and music. Now I can play Reach for the moon, Immortal smoke by piano.
8) 31 Aug, 09' - By the very end of summer holiday, I went to the top of the world, for 20 hours, defeating rokodo and march9th. At the time, bancho's announce's still in #osu, and their reaction was quite great.
9) 6 Sept, 09' - Finally, the day has come. Taiko scores are reseted. Then in the coming several day the taiko ranking is totally a mess. I remembered solo players like Lesjuh, WhiteWolf has been Taiko overall #1 for... 5 minutes? Well that's remarkable too. After like 1 month, there're several players with significantly more ranked score, namely, suinagi, and wmfchris. That it's still suinagi's era but I has been a strong challenger. #3 was youngdow, which is known as OFC_youngdow, or 3UNI now. #4 ~ 10 contain come familiar ID, like KJS (not kds!), NatsumeRin (!!), march9th, lovless,...

10)
http://puu.sh/f4V2 a very short term #1 that time.
http://puu.sh/lIBM And I start my #1 run on Oct 09. Interestingly, I need 111 million score to "advance to next rank" and I'm #1... This bug has a several variation. In my first time #1, it's "undefined" score, this time "111million = my ranked score", the third time "= - my ranked score" because #0 = 0 points, and the fourth time it's ------- because we don't need anymore points to go next rank ; )

11) Dec 09' - I start drilling CTB again. A bunch of no mod SS and HD SS allows me to stay in top 6 and overall #10. I simply get the 2000000 fruit achievement and leave CTB.

12) Jan 10' - First and last tournament joined as player, Saturio and me play as a team. We walked through mafiamaster team due to their absence, we are defeated by strager. Clearly I'm not enough to FC Border of Death easily that time......

13) I played with Saturio mainly at that time, she taught me how to play solo, and in return I taught her to play taiko and ctb... She has some remarkable results that is still in TOP 1/2 : )
However, I don't really have time to deal with solo+taiko+ctb rank-fighting mode (I hv to spend like 5 hours everyday for this, more on weekend), my solo rank slide slightly...
http://puu.sh/cEVy - secret screenshot! >w<
14) 1 Apr, 10' TOUHOUSU! Marisa invaded osu! http://puu.sh/cEWE (I didn't have osu! direct at that time... I wonder what's on another side of the butterfly-shotgun) and the forum as well. It's a pity that I've forgotten fools' day 09 and I didn't screenshot the forum.
15) 9 Apr, 10' Holy Moon - My first ranked map, and it takes 6 days only, very short as a new mapper that time already. It's mainly Saturio's effort to make that epic diff -- I can now SS that on my android by one finger! This map is from YuFu, and is used in a Taiko flash game, in which the website is closed now. This is the referred website which has nothing there now. However if someone knows that where the flash website go, please tell me.
16) Apr, 10' V3 - second map ranked. There are two notable points for the map:
Firstly, this is the first victim under "3I rule" started that time. Just 20 minutes after the rule is set, 111 checked by map and popped... [Vivace] is remapped into [Climbb's vivace], losing half of my style. btw I've already lost the original map orz Also, abalee helpped me to pubble and bubble this map lol Not many of you know that MAT at the very first has no actual power (but some managerial power like silence...faq? All ppl are free to summon faq that time. We do !multi, !where, !faq colour, and all chinese player know that kiou is a girl, moe: kioukiou, kioukiou = SS queen and kioukiou is powerful!!!!) Then it becomes MAT pubble + 2 BAT bubble/rank, it's trashed very soon. abalee was a MAT when he pubbled my map, then becomes BAT when he bubble... lol
Secondly, my ranked map has a bunch of V3, including Beethoven Virus and V3 (piano ver.)... here's two quoting:
i) 111(he checked all 3 my V3): "wtf V3 again!?!"
wmf: "this sounds better = d"
111: "well yes..."
ii) (ignorethis's V3 ranked), wmf: "Glass mod my V3?"
Glass: "ign's V3 is just ranked..."
wmf: "it sounds slightly different..."
If you listen the intro part carefully, you can find some difference, like the difference Al-azif and ignorethis' Eternal fullmoon, ign's version has more echo.

17) May 10' - The official HK player community is established. HKOSP -- HK osu! SandPackage -- is my idea. Sandbag is a name for players slightly stronger than newbies... and package sounds like a group of sandbag...

18) May 10' - Mar 11' What happened? I don't remember anything notable remembered. I started to leave taiko ranked play at Jan 11'. My time in Osu! is simply put in the mapping and modding stuff, as well as community effort.
19) Apr 11' - Nightcorization! http://puu.sh/lIIm This is the 2011 fools' day, songs automatically DT without tuning, all frequency goes into 1.5 speed and it becomes true nightcoreized song! Some song sounds better in nightcorized ver, but some does not. Nightcorized song does not sounds good under double nightcorization.
20) Apr 11' - Taiko World Cup 2011. In the public perspective, as a staff or referee, it's a huge success; in a private perspective, as a supporter of HK team, it's... ummm... a shame.
TWC is a sign for the rise of status of taiko as a mode in Osu!, and it also lead to a larger demand on taiko mapping, modding, playing, as well as the channel. As a ref. I can also feel the passion of the players, some dare to map taiko maps after TWC and aim for being selected, and more of them simply do more mapping on that one. Besides, the match is really, really interesting. I would describe the final as "historic" where TW wins 5:4 in a heroic way. The streaming can be referred to:
As a supporter, or as a tactic resolver, HK lost their game, to JP, under a bunch of problems that should be controlled, like not choosing AAA, didn't practice Etude 10-4... till now when I play Etude 10-4 in piano I'm still a bit sad
21) May 11' - I lost my Taiko overall #1... and 1 year after, I'm still in #3 w

22) May 11' - In very rare case I played once Demetori's fierce map, and I luckily passed lol http://osu.ppy.sh/ss/89059, I like sanae's theme - Faith is for the Transient people so much, and this year's TWC you can see my Sanae remix map.
23) Oct 11' - In order to achieve this one http://puu.sh/6N4E I started to play solo and mod again. It's #999 / lv 99 / 999kd, but that should be my last time solo within 3-digit ranking...
24) Jan 12' - Joined MAT. I don't really need to explain this w
25) Feb 12' - now TWC#2 going on, being vice-manager I'd really exciting about this one :3
That's most of the notable moment in my osu! life. Now here's some notable players that plays with me, or assisted me in my osu! life, and some of them may not be famous enough, but they are actually good friend to me...

Solo gameplay ---
Victorfox - mp buddies in 2008, play Rainspark [Insane] a lot.
manna - HK good players

Taiko gameplay ---
Rokodo
March9th
kanopu
KDS, KJS, 3UNI
NatsumeRin --- he's taiko pro in the past :3
Mapping ---
Al-azif - legendary
James
These two guys simply dominates the map when I play solo and taiko, the secret seven will be legend in the past, now and forever.
Community ---
old HK players like yanki, manna, beat4you, antony11, diela, and ALL members in HKOSP, you know I won't forget you : )
Some TW players like Alapang, f-cat, proseyu, jerrysr, QQsister, Vibty...

I tried to list some older players here only, for those who are new, my friend list is a better indicator :3 ~

and of course, you know that one I didn't mention in the above.

Saturio.

……

## Monday, 12 March 2012

### Visualization. 2

「這裡……是那裡？」少女皺了一下眉頭問道，這裡荒涼得可怕，絕對不是她平日出沒的地方；她只記得她中了迷藥，然後……然後就被綁在地下室了。

「這裡大概是大華城郊外某個地方吧，我只是跟著買回來的情報走過來而已。只要回到城裡面就甚麼事都過去了～反正那兩個笨蛋還真的沒有下手啊，換了是我綁了妳回來一定先吃掉妳再讓你妳心服口服地服從我啊對不對？」他笑吟吟地調戲著少女。

## Friday, 9 March 2012

### Osu! Taiko TOP 10 Chinese Players 2011

Simply proud of the community. However we now have a lot to do with rules discussion, map quality raising blablabla other than game playing and performance art.

## Friday, 2 March 2012

### M2: Further application of vectors on Euclidean Geometry II

Dot product is a bridge between vector and segment length. When ordinary length calculation does not work, we may rely on position vector to deal with the problems. More importantly we can freely switch between the calculation of dot product of vector and length, while both of them obeys most arithmetic law so that the calculation is convinent.
Besides length, dot product can be used to calculate angle between two vectors, and in the case when we lack information we can still show that two lines are perpendicular.
Example 4. In a convex quadrilateral ABCD where $|AB|^2+|CD|^2=|AD|^2+|BC|^2$, show that AC is perpendicular to BD.

In Euclidean geometry method, we can deal it with simple sine law across the angle between intersection of AC and BD.
In vector method, all we have to do is to show that $\vec{AC}\cdot\vec{BD}=0$
Set origin at A. Let the position vector to B,C,D be B,C,D with length b,c,d respectively.
$|\vec{AB}|^2+|\vec{CD}|^2=|\vec{AD}|^2+|\vec{BC}|^2$
$\vec{b}\cdot\vec{b}+(\vec{c}-\vec{d})\cdot(\vec{c}-\vec{d})=\vec{d}\cdot\vec{d}+(\vec{b}-\vec{c})\cdot(\vec{b}-\vec{c})$
$\vec{c}\cdot\vec{d}=\vec{b}\cdot\vec{c}$
$\vec{c}\cdot(\vec{b}-\vec{d})=0$
Therefore AC is perpendicular to BD.