Monday 28 June 2010

Osu! 第22話

『沒想到在這裡會遇到Friday的一員,好險剛沒發生什麼事。』
『疑?請問你是?』
一同下車的另一個男人居然主動靠過來說話,這個人我看過,他是在火車經過節奏之都時上車的。全身長滿肌肉、只穿了一條三角褲在車廂內走動的身影很難讓人不注意到他。
『俺叫做[][][][][][][][][],阿!不對~少數一個,俺叫做[][][][][][][][][][][],不對不對!是[][][][][][][][][][]才對。』
這個人是腦殘阿?疑?仔細一看他頭上居然真的有個洞!對不起我誤會你了……我當然沒把這話說出口,好吧!遇到那麼多奇怪的事我腦袋也怪怪的了。
『沒想到Friday也加入Aakiha的行列了,真是糟糕哪!你們也有急事去機城吧!俺剛好也要去,要不要一起同行啊?俺們邊跑邊聊。』
『恩,好。』他看起來不像壞人,而且好像也是這件事的關係人,或許能問出更多東西。
我們用全速在機城的近郊草原奔跑,以我們的腳程十分鐘就能到達,意料之外的是[]男跟得上我們的步調,不過他居然是用腳上的推進器低空飛行的……仔細一看他頭上那極不自然的洞在飛行時還發出了機械般的微微藍光…
『你是闇天使吧!』比起其他重要問題我忍不住先問了這個…
『俺不知道闇天使是什麼,俺只知道我身體是鋼鐵般的肌肉構成的。不過確實是有人叫俺闇天使過。』
不管了…看起來不像黑米的手下…..
『剛才Observer說叛變的城主是Aakiha…我記得他應該就是秋楓城的城主,所以是秋楓城攻打機城的囉!』
『沒錯!而我剛提到的Friday是個由13個神秘高手組成的集團,每個成員都擁有各自的拿手絕活,並擁有自己的編號和代稱,據說身為首領的十三號成員Commander至今都還沒有人見過其真實面貌。』
『那Friday是秋楓城底下的組織嗎?』
『不是,他們是只為達到集團目的而行動的一群人,他們現在幫助Aakiha肯定是他們的目標一致了。』
『那可真是糟糕阿~剛那個Observer看起來蠻強的。』
『放心吧!如果只有他的話沒有你強的,但是實力那麼強的人有十三個真的是大問題。』
『你是什麼人?為什麼會知道這些事?』
『俺不知道…俺沒有記憶…』
『疑?』
『俺不知道自己是誰…很久很久以前的某一天,當俺一張開眼睛,發現自己坐在一個隕石坑的正中間,俺不知道自己之前發生了什麼事,唯一的記憶是一個博士以及一個機器少女的故事…』
『是嗎…原來你失去記憶了呀。』我語氣變得有點感傷,不過為了釐清所有事情,我決定繼續問下去。『那你是怎麼知道Friday跟Aakiha的事情的?』
『俺為了找回自己的記憶,於是用繪本把博士跟少女的故事畫出來,在世界各地流浪,只要遇到人就拿畫本跟他講故事,希望能找到一些線索…不過不知道為什麼大家都不願意聽俺講故事,至今只有兩個人願意聽俺把故事講完,今天在節奏之都遇到的一個少女是第二個。』
看到你這幅德性誰還敢接近你啊?我這句話當然也沒說出來,奇怪的是我今天怎麼那麼愛吐槽別人,好在在我想到下一句話下之前他又繼續說了下去。
『隨著到處流浪,俺漸漸了解這個世界的局勢,知道秋楓之城的城主Aakiha長期用機城暗中壓榨秋楓城資源的名義煽動人民攻打機城,也在多次進出機城後發現那並不是事實。不久之後當俺經過港都附近的遺蹟時,俺偶然遇見了Friday的三號成員Hacker,他跟你一樣用闇天使這奇怪的名詞稱呼俺,並且對俺很有興趣的,他聽完故事之後對俺說他是隸屬於Friday這個組織,希望俺說出金Q拉的情報,當俺說不知道的時候Hacker突然把俺壓到牆上逼問,最後是一個叫做黑紗的女子出現幫助攻擊他俺才得救,他們兩個人的戰鬥誇張到把整個遺蹟都毀掉了,所幸最後她險勝擊退了Hacker。』
『黑莎是什麼人?為什麼會出現在那種地方?』
『黑莎說她的全名叫做波頓安娜.黑紗,她是為了消滅Friday的成員才尾隨Hacker到那個地方的,當俺被抓住時她發現突襲機會因而現身,不過最後還是讓Hacker給跑了~干!後來黑紗跟俺說明了Friday是什麼樣的組織,不過也僅止於俺剛跟你講的那些情報。』
『恩,我大致上了解了,那你去機城的目的是要阻止秋楓軍跟Friday發動最終協奏曲嗎?』
『不,俺沒有這種能力。』
『那你為何還要跑來這麼危險的地方?』
『因為黑莎說她會來到這個城鎮辦些事情,叫俺今天到古堡跟她會合,沒想到Aakiha居然在今天發生戰爭、古堡又燒了起來,這些事情肯定都有關聯。俺原本想趁亂潛入城中,不過既然遇到你,或許俺們可以直接殺進去。』
『我們必須先去搭救我們的同伴,不過我也不知道他在哪裡,只知道一位可以信賴的人他住在半山腰,從正中央的大道攻進去也剛好會經過古堡。我們就一起合作吧!』
『OK!俺沒問題!』
不久就抵達機城的入口了,果真堆滿了秋楓軍的軍隊。
『好~要上了!』
我抽出觸控筆帶頭跳了進去,在裡面轉了一圈,周圍的數十名敵人立即被震飛,對複雜魚來幾個都沒問題!
yoki與Darri則默契有佳,Darri在後方用Edit技為製造掩護的障壁等輔助工作、yoki則善用這些輔助發揮了比平常更強的戰力。
而[]男…他的戰力居然跟肌肉成反比,頻頻陷入危機之中,不久陷入跪在地上,不好了…我得先幫助他!
『俺生氣啦!!!繪本之怒!』
沒想到他頭頂上的洞居然發出了一條巨大光束把周圍的敵人都掃光,喂!基本戰鬥力跟絕招威力未免相差太懸殊了吧!好吧…看來不用太擔心他了(吧?)
大道的末端就是古堡,古堡後面的巨大高山的半山腰就是團長家的位置,眼前有將近個1000敵軍,沒問題的!我們上!
四個人往人海裡面衝了進去…

Thursday 24 June 2010

Osu! 第20.21話

間章
獵寶隊家中。
nagi細心地替jac洗好身上的傷口時,皎潔的月亮已經高高地掛左天上了。累了一整天的飛馬被拴左後園吃草,而被打昏了的Rider則被緊緊地綁在地上。
nagi問道:「jac,你覺得這半塊玉珮和羊皮卷有甚麼意見?」
「啊,上面寫的文字我都看不明白呢……」
「羊皮卷已經褪色了,我想賣給osu學院做研究應該不錯;至於這玉……」
她還沒說完,jac已經搖頭道:「都褪色成這樣了,跟普通的羊皮紙差不多,這樣賣不了錢啊。我想小雨之前也是在找這些東西吧,等他回來問問也許會有新的線索吧!」
這時,門那邊傳來了三長三短的暗號。nagi暗忖:「是小雨!」
nagi小聲道:「小雨大概不會在考試期間回來吧……?」到最後,她還是開了門。
門外沒有任何動靜。當nagi暗自奇怪著關門時,一陣寒風吹了進來,她被吹得跌坐在地上。
街上一個行人也沒有,再加上這陣奇怪的寒風……nagi想到這裡有點害怕。
背後突然傳來了一把年輕女性的聲音:「抱歉打擾了……」幾乎把身經百戰的兩人嚇得跳起來,「我是小雨的朋友,他告訴我你們兩個在替他尋找他需要的東西對嗎?」
只見聲音的主人穿著一件長長的黑色的斗篷,除了那蒼白而優美的臉蛋外,從頭到腳都是黑色;她的斗篷胸前有一個令人不寒而慄的骷髏頭,教人不敢直視。
「嗯,我們找到一些物品,但是我不知道是不是小雨所要的。」說罷她把玉珮和羊皮拿出來。
「對,就是這兩樣了。作為證明,我可以把羊皮卷復原一部分給你們看。」說罷,黑衣人的手上出現了一支筆,她在羊皮卷下畫了個魔法陣,一部分褪了色的字發出了淡藍色的光芒,nagi把頭伸過來想要看清楚:
「……此乃伊格瑙˙莉絲之日記……有一天,我發現了一種與別不同的共嗚,是由長時間的準確共嗚而產生,所產生的能量比一般還要高出2倍……然而,我找不到純正的新力量……」
復原出來到這裡的字到這裡就完了,但是nagi好像完全不懂當中的意思。
「也許你們以後就會明白當中的意思……我先走了,你們保重。」她對兩人微笑了一下就走了。

haruhiteamsos

『我們一知道消息就趕過去了,說不定還來得及。』雖然嘴巴這麼說,但是動腦想一想既然假黑米能如此游刃有餘地跟我們講這件事,那黑米肯定早就開始行動了。
已經晚上了,我椅著頭看著窗外等待,車窗上是自己清晰的倒影,在倒影上呼嘯而過的則是一片漆黑的田野,田野後方的山脈更加得黑暗了,不過在深藍的天空映襯下反而清晰可見,只要火車出了那座山就能看到機城了。
『以工業發達聞名的機城座落在巨大的高山下,從山腳下的古堡為中心輻射出去的是一座座工廠,在外面一圈連接草原的則是民眾居住的現代化都市。被包圍在最裡面的古堡顯得相當突兀,不過它突兀的地方不只在於建築的年紀,而是它的高大雄偉與至今都無法再現的建築設計—那是由好幾位古代最偉大的Mapper所創建出來的,牆上的每個孔洞都經過精密的構思,裡面的建築設計也完全善用材料的音響性質,只要配合季風的風向,古堡會不間段地發出悅耳的歌聲,而且不同的季節會發出不同的音樂,這些偉大的工藝設計也深遠地影響到現在機城的工業發展…. 』
為了舒緩一下緊張的情緒,darri像導遊般為大家講解機城的歷史。
不久後,漸漸有建築物從山脈邊緣的輪闊露出來,古堡的一角也露了出來
『看,那就是機城最偉大的……』darri沒繼續說下去,原本吵雜的車廂也頓時安靜下來,遠方我們看到的只有一座陷入火海的巨大城堡,仔細一看,剛剛看到的城市燈光居然都是火光。
『機城到底發生了什麼事!?』
最後火車決定緊急折返,只有我們三個以及另外兩個人下了車。
『科科科….已經開始了呀!』一個笑聲詭異的男子看著火海中的機城自言自語,他的樣子居然給人一種在欣賞風景的感覺,不過我想他或許知道些什麼。
『你知道機城到底發生了什麼事嗎?』
『恩?』男子帶著狐疑的眼神看向我『那還用說嗎?看就知道機城已經被攻陷了呀~!』
『為什麼?是誰這麼做的?毀掉整個城市得不到任何好處阿!機城的價值不就在於它發達的工業科技嗎?』darri聽到這裡生氣得插嘴。
『小哥別機動嘛…那還不是機城那邊的城主太死板了,不願意聽從我們城主大人的發動”最終協奏曲”,如此窩囊的態度,大人覺得機城的城主是不配擁有聖地的。』
『你們城主是什麼人?我們這個大陸之所以是唯一沒有統治者土地就是因為如果有外敵入侵城邦間就會聯合對抗、內部有人挑起爭端,所有城主都會討伐反叛者,這是大戰後fish哥不希望世界被絕對的權力支配才訂下的城邦協定呀! 最終協奏曲還有聖地是什麼?無論有什麼理由,犧牲那麼多人以及文化遺產都是不值得的呀!』yoki也忍不住插嘴了。
『科科科…你們真是煩人。算了,反正消息很快就會傳出去了,我就告訴你們吧!我們的城主是偉大的Aakiha大人。至於最終協奏曲嘛…你們待會就知道了!科科科科科….』那詭異的男人身影漸漸發光,是傳送到附近某定點的節奏技發動了!
『喂!等等!你到底是誰?』
『恩?我是七號Observer,科科科…你是鼎鼎大名的節奏之都高手頻果吧!或許我們待會會再見面喔!科科科科…』當Observer全身被光壟罩後,光芒突然凝聚成一點,然後就消失了,留下來的是更深的黑暗。

Monday 21 June 2010

Osu! 第19話

在下山的途中,傷痕累累的大家以聊天支撐著自己的意志……
我說道:「對不起團長,我們害那間屋子毀了啊……」
「不要緊,房子毀了可以再起,你們沒事就好了。」
「對了團長,你認識那家伙嗎?他自稱是你老朋友呢。」
「我當然認得他……」團長一邊說,一邊把思緒帶到了好一段日子以前。……
我從這裡看到的,是倒塌的城市。
所看到的是絕望。所看到的是死,所看到的是無。
無任何一物,無任何動靜,無任何生氣。
傳入我耳邊的只有連線不絕的雨聲……
我吃力地爬起來,心裡咒罵著那個該死的空賊團。
fish哥才出去一會,他們就吃豹子膽嗎?
他們來襲之前我正好在地底的研究室工作,剛收到有空襲的消息,我頭頂大概被轟得亂七八糟,而我的研究室也劇烈地搖晃起來。我被一個從天而降的文件夾打中,就昏倒了好一會兒。
在我眼前的還是同樣摸著腫起的頭起來的教官abalee,他為人跟教學一樣溫柔……嗯,現在好像不是說這些的時候吧!

我拉把他道:「外面的學生不知道怎樣了,我們快出去看看吧!」
外面的景象把我們倆嚇壞了。眼前已經沒有驚慌走避白學生--要麼都走光了,要麼被打中趴在地上--同時,一隊空賊竟敢衝進來!
我跟abalee互相點一點頭,便抽出滑鼠、鍵盤和板子逕自衝往空賊團去。
起初的嘍囉簡直不堪一擊,任何一個節奏都可以輕易地把他們轟出老遠;漸漸地,我們發現了有不對勁的地方。困著我們的人似乎越來越強,完全不像那些紀律散渙的空賊。難道…?
我稍一分神,abalee被被對手沖到遠處,把我倆分開來。我下意識地伸出手,手卻被一條冰冷的觸手纏住,把我拖到另一個地方。
眼前是一位長得跟我一模一樣的人!唯一不同的是他身上的冰冷氣息,我看一看剛剛被纏住的手,竟然在短時間內被凍傷了。
「你……到底是誰?」
他笑著,我身邊的空氣、景物都凝固了。「我是誰,那不重要啊。我跟那幫雜碎不同,我才不會來搶東西呢。我只是幫人拿一點東西罷了。」
我正想伸出手擋住他時,一陣冰涼觸感在我胸前滑過,然後我就失去知覺了。
後來我聽說我被鎖在某個儲物格中,而abalee則靠自己的力量撐到了fish哥回來……
原來團長也有一段這樣的過去啊……
此時學姊的腳步卻硬生生地停了下來。
沒有玻璃呢……是絆倒了?不是--黑米站在不遠處笑著迎接我們……

關卡: OSTER project - Frozen rain [Normal]
http://osu.ppy.sh/p/beatmap?b=5434&m=0

失敗路線
「學姊你怎麼了?」
「我我我……忽然喘不過氣來……」說完就咕咚一聲倒在地上。
居然嚇成這樣了……

成功路線
「靠,又是他嗎……」
「?」學姊沒有見過他。
「他駛了我10金幣啦……」
「好,我來幫你出氣!」
笨學姊啊……

Tuesday 15 June 2010

15-6-10

Why? You'll never win this cursed war...
~dirge of swans~

始終考試不是一個好選擇呢。
筆記的話大概就差Chem一科了~
大家加油。

Sunday 13 June 2010

Economics -- Price elasticity and Government intervention (rearranged)

The price elasticity of demand/supply (Ed/Es) refers to the responsiveness of the Qd/Qs to a price change.

Ed = %ΔQd/%ΔP and Es = %ΔQs/%ΔP where %ΔX = (X2-X1))/[(X1+X2)/2]*100%.

Elasticity only refers to the magnitude of Ed/Es only. Ed<0 since P and Qd is negatively related. Note that the elasticity refers to % change rather than absolute difference.

Types of elasticity:
  1. Perfectly inelastic (Ed / Es =0) if %ΔQd / Qs=0, i.e., remains unchanged for a change in price. Its curve is vertical.
  2. Inelastic (0<Ed / Es <1)%ΔQd / Qs<%ΔP, its slope it somehow straight.
  3. Unitarily elastic: (Ed / Es =1) %ΔQd / Qs=%ΔP, it forms a rectangular hyperbola
  4. Elastic: (1<Ed / Es <∞) %ΔQ / Qs>%ΔP, the slope is somehow flat.
  5. Perfectly elastic: (Ed / Es =∞), where %ΔP0 causing an infinitely large change in Qd/Qs.. Its slope is horizontal.
In a free market, TR of firm = TE of buyers = Total exchange value = PQd.

When Ed>1, PTR, PTR when Ed<1 and TR remains unchanged neglecting the chance of P if Ed=1.

Note that when the curve is linear, the mid-point between the intersection with x-axis and y-axis will have Ed=1, the part above that point will have Ed>1 and the part below the point will have Ed<1. We can proof this one by considering TR maximized at the midpoint.

Factors affecting the price elasticity of demand:
  1. Closer substitutes available→more elastic
  2. Demand of necessities→more inelastic, demand for luxuries→more elastic
  3. Consumption habits: harder to change the habit→more inelastic
  4. Proportion of expenditure to income↑ → more elastic
  5. Price range↑ / More useful / More durable → more elastic
Factors affecting the price elasticity of supply:
  1. Nature of goods, e.g. land, artworks of deceased painters will be perfectly inelastic.
  2. Flexibility of production: ease of adjusting production (e.g. faster training to unskilled workers)↑→Es↑.
  3. Factor mobility ↑→Es↑
  4. Reserve capacity (Max. amount of resources that can be put into production)↑→Es↑
  5. Time allowed to adjust production↑→Es↑
  6. Lower entrance to the industry→Es↑
Market intervention
  1. Price ceiling / Max price control: Under this condition, producers/sellers are not allowed to charge more than the max price. (e.g. rent control of residential units before 1999 in HK). Effective price ceiling P’<Pe, otherwise the equilibrium point is still at the interception point of the demand curve and supply curve. In this case a shortage appeared where the quantity transected refers to the intersection of supple curve and the price ceiling. In this case P’<P, Q’<Q, new TR = P’Q’<PQ=original TR. In this case price failed to allocate resources, so non-price competition arises such as queuing, drawing lots, by needs, etc. Black market will also arise. Graphically the price of black market will be the vertical projection of the new quantity transected to the demand curve.
  2. Price floor / Min price control: Under this condition the consumers/buyers are not allowed to pay less than the minimum price. (e.g.: min wage to the foreign domestic helpers) The price floor is effective only if P’>P0. In this case the new quantity transected will be the intersection between demand curve and the price floor. As the new point is on the demand cure, P’>Pe but Q’<Q0. We check the change of TR by the price elasticity of (original) demand: if Ed<1, TR↑, if Ed>1, TR↓, Ed=1, TR unchanged. In this case surplus appeared and sellers compete to sell products. Since they can’t cut the price, they competes by non-price competition such as free gifts and improve their skills. Some even cut their price illegally (to the vertical projection of the new quantity transected to the supply curve) to eliminate excess supply.
  3. Quantity control / quota: It is the maximum amount of output. (e.g. # of taxis and minibus in HK) Effective quota implies that the Q’<Qe. Graphically the supply curve was bent into a vertical line at the quota. The new quantity transected will be the intersection between the new supply curve and the demand curve. In this case P’>Pe and Q’<Qe so we check the TR by considering the price elasticity of the demand just same as the price floor. Since the sellers can’t sell more in quantity, they improve the quality which gives them a higher TR.
  4. Unit tax: same amount of tax imposed to every unit of product (different from valorem tax which the tax refers to a % of the price) (e.g., unit tax on petrol) In this case the supply curve shifts upwards by the amount of tax. In this case P’>Pe and Q’>Qe, and the part of the TR (Tax*Q’) will be the tax given out. For the consumers the burden will be (P’-Pe)*Q’ while the producer’s burden will be the remaining part (Tax-P’+Pe)*Q’. After the tax is imposed the TR is reduced by (Tax*Q’+(Qe-Q’)*Pe. The second of TR loss was due to the shift on demand curve. The TE of consumers refers to the elasticity of the demand curve like the previous intervention. The distribution of tax burden can also be given from the formula:
Consumer’s tax burden/producer’s tax burden = Es/Ed.
  1. Unit subsidy: A same amount of subsidy is granted to each unit of output.
(e.g., government schools). In this case the supply curve shifts downwards by the amount of unit subsidy. The new equilibrium will be the intersection between the new supply curve and demand curve. In this case P’<Pe and Q’>Qe. We find the change in TE by considering the price elasticity of demand curve like the previous cases. The total subsidy will be Tax*Q’. In this case the consumers’ share on the unit subsidy will be Q’*(Pe-P’) while the producers get the remaining part of subsidy, that is, Q’(Tax-Pe+P’). Therefore the new TR of the producers will be Q’(2P’+Tax-Pe). The distribution of unit subsidy can be given by:

Consumer’s share of subsidy / producer’s share subsidy = Es/Ed.

Economics -- Basic Demand and Supply

Demand = Want + purchasing power = willingness + ability to buy
An individual demand of a good refers to the quantities that a buyer is able (purchasing power) and willing to buy at different prices over a period of time, ceteris paribus. We draw the demand curve that P (price) on the y-axis and Q (units/time) on the x-axis to draw a demand curve. Quantity demanded (Qd) is the quantity that a buyer is willing and able to buy at a particular price over a period of time. Graphically demand is the whole curve and Qd refers to one single point on the curve. Note that Qd is the quantity that is planed to buy, but not the actual unit bought (transected). Demand schedule is a table to show the Qd at different prices.
Individual demand curve: shows P VS Qd of a good to an individual buyer.
Market demand curve: shows P VS Qd of a good for all individual buyers.
The market demand curve is the horizontal summation of all individual demand curves.
Law of demand: the demand curve is ↓slopping, ↑P↓Qd, we can also say that the P and Qd is inversely related, ceteris paribus.
Supply = Willingness + ability to sell
Individual supply, supply curve shares the similar explanation with the terms about demand.
Law of supply: supply curve is ↑slopping, ↑P↑Qs or positively related, ceteris paribus.
Note that quantity transected = min{Qs,Qd}.
Equilibrium (market-clearing) price: At Pe, Qs=Qd and we denote the equilibrium quantity as Qe. In this case the price will have no tendency to change, ceteris paribus. At that price quantity is equal to the equilibrium quantity. Note that when the two curves have no intersection the equilibrium price will be zero (free goods). The equilibrium quantity will be zero too since there will be no need for transaction.
When PPe, excess goods/surplus appears and there’re unsold goods. In this case sellers may raise the price to make consumers buy more until the price reaches the equilibrium.
The automatic change of price is called the price system or market mechanism.
Money price: price expressed in terms of money, relative price: price expressed in terms of another good. Note that the relative price also obeys the law of demand, given the price of the good for comparison remains unchanged.
Consider the relative price of X is k units of good Y where k>1. If the price of X,Y raised the same amount, then the new relative price of X will be k’ units of Y, where k’beef and leather) Qs of X↑→ supply of Y↑
2) Competitive supply which requires similar factors of production, Qs of X↑→ supply of Y↑
3) Prices of inputs (price of factors of production)↑ → Supply↓
4) State of Technology, # of producers and other factors

Ceteris paribus: everything being kept constant.

Note that the affect of new equilibrium price change will not be shown here since it is a table.

Friday 11 June 2010

Mechanics IV

Circular motion:
Radian is a measurement of angle. 180˚ = π rad.
Physical quantities about circular motion:
1) Angular displacement: θ (in radian, rad) =arc length/radius=s/r
2) T: period: time to complete one revolution, SI unit is s.
3) Angular speed (velocity but we don’t care about the direction): ω=2π/T, with unit rad s-1.
4) Linear velocity v = s/t = rθ/t = rω = 2πr/T, the unit is m s-1.
5) (Centripetal) acceleration: Consider the object moves Δθ in a short interval Δt. Then the limit of Δv is equal to the arc of the circular path. That is, Δv=vΔθ. By a = dv/dt = vdθ/dt = vω. Therefore a = vω = v2/r = rω2. The acceleration is pointing towards the center of the circular path.
6) Centripetal force: F = ma = mvω = mv2/r = mrω2 is also pointing towards the center of the path. Moreover it isn’t a new type of force, but it can be provided through different sources like tension and friction.
Under uniform circular motion (v and ω are constant), when F> mv2/r, then the excess F will accelerate more seriously towards the center until F= mv2/r’ where r’ is the new radius smaller than the original one, therefore it will spiral inwards. Oppositely if F < mv2/r, r will increase so that the object will spiral outwards.
Complex case on circular motion:
1) Conical pendulum: A light and inextensible string of length l, making angle θ with the vertical line. In this case the centripetal force is given by the tension of the string T. Now consider the vertical motion (is zero), by ΣF=ma, we have the vertical component is equal to the weight of the pendulum. i.e., Tcosθ = mg. In the horizontal view of the circular path we have the horizontal component of tension as the centripetal force. i.e., Tsinθ=mv2/r. Combining the two equations will give tanθ=v2/rg. With the assistance of v = rω we can find the remaining information. Note that since F = Tsinθ = mrω2 = mlsinθω2, we will have T = mlω2.
2) In the flight of aircraft is actually a conical pendulum model, while the lifting force gives the centripetal force instead of tension. The inclined angle of the aircraft is also equal to the angle made with the vertical line.
3) Cycling in a level road: when the bicycle turns, its c.g. have to be closer to the center of the circular path. Otherwise the friction produced when the bicycle is turning around is unbalanced. If it is suitably stanted, the moment produced by the couple mg and R is offset by the friction. Let the horizontal and vertical distance between c.g. and the wheel be a and h respectively. Now F = mv2/r, R=mg and Fh=Ra by taking moment on G. Combining the equations together gives tanθ=v2/rg and the resultant forces acting on the wheel is given by R+F.
4) Cycling in a banked road of θ: In this case we assume the bicycle is perpendicular to the road. The centripetal force is given by the resultant of R and mg. by Rsinθ = mv2/r and Rcosθ = mg, we will have tanθ=v2/rg.

Mechanics III

The affect of collision depends on the mass and velocity of the object, so a physical quantities is created as momentum, which is the product of mass and velocity. Mathematically p=mv, which is a vector with unit kg ms-1.
By F=ma=m(Δv/t)=( Δmv)/t, we can conclude that F = change in momentum / time of action. In other works, the net force acting on the object is equal to the rate of change of momentum of it.
Impulse is also another physical quantities that Ft=Δ(mv), which is a vector with unit Ns.
Object deforms during collision, and the force that make it deforms is called impact force while that t is called the time of impact. The longer time of impact results in smaller impact force.
We can draw a force-time graph (F-t)graph to represent the collision, while ∫F dt=Δmv which is the impulse delivered to the object, while it is also the change of momentum of the object.
The law of conservation of momentum states that the total of momentum of a system is conserved if no net external force is acting on it. Mathematically mAuA+mBuB=mAvA+mBvB.
The collision can be classified as following:
1) Elastic collision: Total K.E. conserved. i.e., mAuA2/2+mBuB2/2=mAvA2/2+mBvB2/2.
2) Inelastic collision: Some K.E. lost : i.e., mAuA2/2+mBuB2/2>mAvA2/2+mBvB2/2
3) Perfectly inelastic collision: vA=vB, then K.E. lost is maximized.
Note that total K.E. won’t increase during collision if there’s no net external force.
Explosion: In this case chemical energy was transferred into K.E. The momentum can be given by mAvA=-mBvB.
Apparent loss of momentum: consider the whole system (Earth), or the external force(friction).
Proof on law of conservation of momentum:
During collision there’s a force acting on A by B and a force with equal magnitude but opposite direction acting on B by A by Newton’s third law of motion.
FA=-FB, (mAuA - mAvA)/t = (mBvB - mBuB)/t gives the required equation.
Oblique collision in a plane: Assume B is resting and A crashes into B with velocity uA (We assume that A is moving horizontally since we can “turn” the plane.).
After collision, A moves with a velocity vA which makes angle θ with the horizontal, while B move with velocity vB which makes an angle φ with the horizontal. Now let θ be positive (above the horizontal).
Now we decompose the velocity and consider the two direction:
Along x-axis: mAuA = mAvAcosθ + mBvBcosφ
Along y-axis: 0= mAvAsinθ - mBvBsinφ, i.e., mAvAsinθ = mBvBsinφ
Another method is that by mAuA =mAvA+mBvB and they are all vectors, draw a triangle that the angle between mAuA and mAvA is θ and the angle between mAuA and mBvB is φ. By sine law, their relationship is given by mAuA/sin(θ+φ) = mAuA/sin φ = mBuB/sin θ.
Special case: when the two masses are the same and the collision is elastic, the angle θ+φ must be equal to 90 degrees.
Proof: By mAuA+mBuB=mAvA+mBvB , uB=0 and mA=mB, we get uA=vA+vB. (1)
Again since it is an elastic collision we have mAuA2/2+mBuB2/2=mAvA2/2+mBvB2/2 and so that uA2=vA2+vB2. (2)
By (1) we can draw a triangle that adding vA and vB by tip-to-tail method and uA as the sum. Then by (2) and Pythagoras theorem we will get the angle between vA and vB must be 90 degrees.
We call a motion as projectile when the object is given a initial velocity and move under gravitational acceleration (or air resistance as well).
‘When a initial velocity u is given, making angle θ with the horizontal, we will have:
Horizontal: ux=ucosθ, uy=usinθ.
Note that when the vertical velocity is changing due to g, the horizontal velocity WILL NOT CHANGE. (This refers to the Monkey and Hunter experiment.)
Then at an instant t, vx=uxt and vy=uy-gt.
The displacement is given by x=ux and y=uyt-gt2/2.
Now consider some important physical quantities about projectile motion:
1) Max. height: It reaches the highest point when vy=0, i.e., t=uy/g. Substitute t=uy/g into y=uyt-gt2/2 we will have H=ymax=uy2/2g=u2sin2θ/2g.
2) Time of flight: Reaching the highest point and back to the ground takes the same time when we consider a complete projectile motion. Therefore T=2uy/g=2usinθ/g,
3) Range of flight: Obviously R=uxT=2usinθ(ucosθ)/g=u2(2sinθcosθ)/g=u2sin(2θ)/g. That explains that why flying in θ and (90˚-θ) gives the same range.
4) Angle with the horizontal is given by θ’=tan-1(vy/vx)=tan-1(uy-gt/ux).

Thursday 10 June 2010

Mechanics notes II

*This set of notes act as a remainder rather than the actual application on different problems. Readers should try themselves in different problems as well.
I will complete the DSE syb. of mechanics except gravitation this year. This include:
I: Position and moment, Newton's Law of Motion
II: Fotce in a plane and moment, work, energy and pwoer
III: Momentum, projectile and circular (this one is much more easier than AL)

Methods to add vectors up:
Tip-to-tail method: Consider two forces F1 and F2. Firstly draw the vector F1 starting from the origin, and draw F2 from the tail of F1. The tail of F2 will be the resultant force.
Parallelogram of forces: Draw the two forces at the origin, project them to another’s tail. The intersection point will be the tail of resultant force.
Consider F is a force with magnitude F and angle between the force and the x-axis is θ. Then it can be resolved into Fx=Fcosθ and Fy=Fsinθ. Mathematically F==F(cosθI+sinθj).
When we apply Newton’s Law of Motion on a plane, we always resolve the force along the plane and perpendicular to the plane. Then each of the forces will never be offset by each other. Also note that when the object is at rest or in uniform motion then the net force is zero.
Moment or a torque at a point by a force is defined as Fd, which is the force times he perpendicular distance from the point. Note that if the fixed point, point giving force and the direction of force is parallel then its moment will be zero since it can’t rotate the object. Alternatively if the force makes an angle θ with the object then the perpendicular distance will be dsinθ. Comparing gives the moment = Fdsinθ. Moments can have 2 direction: clockwise or anti-clockwise.
The principle of moments states that the object is balanced if the total clockwise moments is equal to that of anti-clockwise moments.
When two forces with equal magnitudes acting in opposite directions on an object, they will form a couple, which produce a moment of force(one force, not the sum of the two forces) times perpendicular distance between the two forces, no matter which point is fixed to calculate the moment.
Equilibrium is defined as net force acting on it is zero and net moment at any point of it is zero.
Centre of gravity (c.g.) is the point through which the weight of the whole object acts. If any object is hanged, then it will move until the c.g. is vertically below the fixed point. Otherwise a couple is produced to rotate that object to that stable case.
There will be three types of equilibrium: Stable equilibrium implies that the c.g. have to move upward in order to move it. Unstable equilibrium implies that the c.g. will fall as to make the whole object moves. Neutral equilibrium implies that the vertical position of c.g. won’t change even the position of the whole object changes.
Doing work means the process of energy transfer.
Work done by a force = component of force in the direction of displacement times displacements.
If the forces makes an angle θ with the displacement, then the component will be Fcosθ. Mathematically W=Fscosθ. It is a scalar with unit Joule (J).
Note that W=0 if (1) s=0 or (2) cosθ=0, then the force is perpendicular to the displacement, which the component of force with the same direction with the displacement is zero.
Work done by a force is also the energy transferred by the force. A positive work done means that the object gains mechanical energy, and a negative work done means that it losses mechanical energy (but internal energy can be gained).
Now consider the object moving in a straight line (therefore consider K.E. only):
W=Fs=(ma)(v2-u2/2a)=mv2/2-mu2/2=ΔK.E., therefore we can identify that K.E. of an object is mv2/2 where v is its velocity.
Potential energy can be divided into elastic potential energy (e.p.e.) and gravitational potential energy (g.p.e.). e.p.e. is stored in an elastic object when it is stretched, compressed or bent. When it is released the energy will be released too.
For g.p.e. we have ΔP.E.=mgΔh, we can’t determine the g.p.e. of an object since the “height” is a relative concept, but Δh can be accurately measured.
The conservation of mechanical energy states that initial K.E.+ initial P.E.=final K.E.+ final P.E.
That is, the mechanical energy of an object is conserved, provided that no work done is on the object. This implies that:
1) If mg is the only force acting on it, then ΔKE+ΔPE=0.
2) If the resultant force is mg or component of mg, then ΔKE+ΔPE=0. This mainly happens when the object is on a inclined plane, where part of the mg offsets with R.
3) In other case, ΔKE+ΔPE = work done by the forces other than the (component) of gravitational forces.
In a simple pendulum, v is maximized when it is in the lowest position where we can assume it’s PE is relatively zero.
The law of conservation of energy states that energy can’t be destroyed or created. But it can transfer from one to another.
Power is the rate of energy transfer or work is done. Mathematically P=E/t=W/t. It is a scalar with unit Watt (W). Also, P=W/t=Fs/t=F(s/t)=Fv. Therefore a power P it required to apply a force to an object such that it is moving in constant velocity v. (Note that if it is moving in constant velocity, there’ll be another force opposite to this force.)

Wednesday 9 June 2010

Pilot Ch 2.2

我們所生活的世界並非如我們所看到的那麼單純。
一件叫「裡世界」的事把子翼生活徹底改變了。
放學後,校園一角。
「真是的,今天早上又無故怪笑,現在又叫我過來……要知道學期初很忙的啊~」子翼抱怨道。
「嗯,這我也能理觸。可是你剛才不是說要完成委託嗎?我在幫助你耶!」羽恆雖然這樣道,子翼當然不知道她帶子他過來的用意。
眼前的建築是位於頂樓的課室,門牌說上了"Mathematics ICT Lab",也就是電腦實驗室,通常只有教師才有機會用到--這裡的系統是Linux而不是常用的Windows,加上長年在運行研究用的程式,加上要長年開啟冷氣防止機件過熱,根本沒人會有興趣進來。
羽恆打開了實驗室的門,氣溫立刻下降了好幾度。
子翼一進來第一句話就是:「冷氣真凍,難怪沒人用了呢。」
羽恆沒有理會子翼,隨手按開了其中一部的螢幕,上面顯示著這部電腦仍然運行著某個計算程式。
子翼道:「這部電腦看起來有別人在用了吧。」
然而,子翼眼都沒眨就把那程式終止了。
「你、你把程式關掉了?那別人的功夫就白費了啊!」
「沒關係呢。這個程式會透過內聯網接收要做的工作。這部電腦上失去了的資料,總可以很快補上去的。」說罷就把電腦關掉。
關機這動作使子翼更驚訝,他問:「你不是要電腦才關掉別人的程式嗎?怎麼現在又關機?」
羽恆卻賣了個關子沒有回答他,只是帶他離開了實驗室向下一個目的地出發。

同時間,教員室。
雪晴呆呆的站在教員室外等著。
憑著她息樣貌只要站著不動的話,大概就會惹來不少男孩走過來。可是雪晴沒空去理會他們。
她正在等一個老師,但是雪晴沒有進去自動找他。
三分鐘後,那個老師自動跑了出來。
「老師。」雪晴帶著勝利的笑容把老師截了下來。
看到這種笑容,一般人都都會著迷而呆上一陣子,然而老師仍是氣急敗壞地說:「啊啊,實驗室的電腦好像出了故障。有課業問題的話一會再說吧?」
「老師,那不是故障唷~」雪晴掏出了一張紙,上面寫滿了毫無意義的英文和數學符號。她指著最後一行:「Error code 22呢。」
老師嘆了一口氣道:「……好吧,大家也知道彼此的立場,所以我就答你一個問題吧。」
(你在研究的,是2V project吧?」
「真聰明呢,丫頭。還是說我低估了你吧?」
「在前輩面前我可沒有甚麼花樣好玩的,我只是做我應該做和能做的事罷了。」
「所以你還是要跟我對立囉?」
「也許是吧。老師不會扣我平時分吧?」
「那當然不會啊~好了,我上去把程式修一下,你也保重。」老師說完便轉身離去,忽然又回過頭來補充一句:「對了,聽說你們也帶了個拍檔來吧?這個大概對你們有用。」他把一個塑料袋拋到雪晴手上就往實驗室跑去了。

從實驗室下來時,已經快六點了。大部分學生都已經離去,只剩幾個學生依然在打籃球。
「我們就等一下雪晴吧。」羽恆把書包放在旁邊的桌子後便打暱罐裝綠荼自己喝著。
「你們兩個,本來就是裡世界的人吧?」
「嗯。情報販這一個行業似乎天生就註定了靠裡世界混飯吃呢。」
「我這個新人要進來的話,有甚麼事情值得你們幫手呢?」
「裡世界不是隨便可以進的呢。那些不惜一切進入的話,就有可能變成我們所看到的黑社會組織了吧。」
這時,雪晴也過來了。
「好吧,我帶你去基地看一下好了。」羽晴補充道:「所謂的基地,就是裡世界不同人聚集的地方,同時也能通向不同地會社本部呢。」
既然是裡世界的核心,地點也理所當然地左旺角。在銀行中心進入地庫,跟上次一樣走過狹窄的通道來到了一道黑色的門面前。這扇門看起來會自動上鎖,沒有適常的鑰匙睪絕對開不了的。
雪晴把塑料袋拿出來,交給子翼道:「嗯,這就是進入基地的鑰匙,就交給你好了。」
「啊,你們不要嗎?」
裡世界的潛規則之一就是情報商不會主動找人。所以我們倆個也很少進來。另一方面,這個鑰匙可不是普通的鑰匙呢。」
子翼打開塑料袋,裡面的是一條銀匙和一個奇怪的匙扣。
「你把手指按在匙扣上吧。」
子翼依言按了下去,此時奇怪的事發生了:匙扣對著木門發出了一陣光芒,然後匙扣上便出現了"identification completed"的字樣。
「認證完畢了呢。這條銀匙大概只是個幌子吧?以後你帶著這個匙扣,這扇門會自動認得了~」
眼前的通往裡世界的門已經開啟……

Tuesday 8 June 2010

Mechanics notes I

Caution:
1) In order to avoid the usage of equation editor, reader should be careful about whether the physical quantities are a vector or scalar.
2) The concept of calculus is used.
3) Graph is not given in the sense that it is driven from a set of datum. However the concept will be discussed. Readers may try to draw the graph themselves.
4)Sorry that I have no time to make the power/subscribed words/bolded set/...

Time (t): SI unit second (s), reaction time of human = 0.2s
Length: SI unit metre (m).
Scalar: A physical quantity describes the magnitude only. Mathematically, x∈R.
Vector: A physical quantity that gives size and direction. Mathematically x∈R2, Usually we will add an arrow on the physical quantities to mark that it is a vector. Another representation will be V=(a,b), a,b∈R. Note that |v|=(a2+b2)0.5, and it’s direction will be tan-1(a/b). Also (a,b)+(c,d)=(a+c,b+d).
Distance (s) (scalar) is defined as the total length of the path, with SI unit m.
Displacement (s) (vector) measures the change in position.
Displacement on a 1D case: Definition of positive direction, the displacement will be a scalar, where the +ve/-ve sign denotes the direction.
Displacement on a 2D case: Define every straight line travelled as s1=(x1,y1) to sn(xn,yn), then distance travelled = Σ(|sn|) while displacement = |(Σsn)|
Average speed (v) = (distance) s/t which is a scalar with SI unit m s-1.
Average velocity (v) = (displacement) s/t which is a vector with SI unit m s-1.
Considering the velocity on 1D/2D is similar to the displacement.
Instantaneous speed: average speed in a very short time. Mathematically it is Δs (distance) /t
Instantaneous velocity: average velocity in a very short time. v=ds (displacement) /dt
Uniform motion: when v(t) is a constant we call the motion of the object as uniform.
We define final velocity as v while initial velocity is u. Then the average acceleration over a time period is defined as a=(v-u)/t. The instantaneous acceleration will be a=dv/dt.
Uniformly acceleration: if a(t) is a constant, then we say it is uniformly accelerated. By F=ma if it is uniformly accelerated then the force is constantly given.
Displacement-time graph (s-t graph): y=s(t). y’=v(t).
Velocity-time graph (v-t graph): y=v(t), y’=a(t), ∫y dt=s(t).
Acceleration-time graph (a-t graph): y=a(t), ∫y dt=v(t).
Equations on uniform motion: (Under vertical motion, a is fixed as a=g=10ms-1.
1) s=(u+v)t/2, which can be proven by ∫y dt=s(t).
2) v=u+at, which can be proven by ∫y dt=v(t)
3) s=ut+at2/2 since s=(u+v)t/2=(2u+at)t/2=ut+at2/2.
4) v2=u2+2as since v-u=at, so (v+u)(v-u)=at(v+u)=2as.
Note: In vertical motion, neglecting air resistance, the s-t graph is symmetric and is a parabola.
Newton’s First Law of Motion: An object will remain at rest or constant velocity as long as the net force acting on it is zero.
Note that net force = ΣF which is the sum of all force acting on it.
Net force can be zero if the several force is balanced, or no force acting on it.
Inertia: the tendency of an object to remain at rest or in uniform motion, measured in mass and the unit is kg. It is harder to move an object with larger inertia.
Force (vector) (F): can be measured by spring balance/force sensors, with SI unit Newton (N).
Force can be divided into contact force or non-contact force. Some common force will be the weight (W=mg), normal reaction (R), friction (Ff) and tension (T).
Resultant force = ΣF which is the sum of given forces.
Calculation on force refers to the calculations of vectors.
Notes on free-body diagram:
1) Weight: vertically downwards from the center of mass
2) Normal reaction: pointing up from the contact surface, perpendicular to the surface. If the surface is an inclined plane, then the starting point should be the projection of center of mass vertically downwards to the contact surface.
3) Friction: On the contact surface.
Newton’s Second Law of Motion: a is proportional to ΣF (same direction) and inversely proportional to m. Considering the SI unit, then F=ma.
Application:
1) Weight: W=mg
2) Apparent weight during acceleration: The apparent weight will be R (not W); by ΣF=W-R=ma, we have R=W-ma=m(g-a) which implies that when the acceleration has the same direction with the gravity, our apparent weight will be reduced.
3) Friction (Max Ff=μR, where μ is the friction constant) appears when two surfaces slide or tends to slide over each other. When ΣF ≧ Max Ff it will start to move. Note that the max. (static) friction is a bit larger than the kinetic friction.
4) Fluid (liquid/gas) resistance increases as the difference between velocity and terminal velocity increase. The velocity of free-fall object will tends to the terminal velocity.
Newton’s Third Law of Motion: For every action there exist an equal but opposite reaction.
The action-reaction pair will be the force acting on A by B and the force acting on B by A.

Saturday 5 June 2010

Osu! 第18話

三對三的戰鬥中
黑衣男子阻擋了我對黑米的攻擊,不過他也漸漸被我逼到退路,在最後一擊的時候把他抓起來丟往跟yoki戰鬥中的女子,太好了!一次解決兩個敵人,剩下黑米被我們三個包圍了。
「哼哼~還是在預料之中,不過你以為這樣就能阻止我嗎?」黑米被我們包圍後仍保持一貫的從容。這是怎麼回事?難道他自暴自棄了嗎?我不理會趁這機會把筆刺了過去。
「Normal-hitfinish!」
鏘!
被強力的鑼聲震穿一個洞的黑米倒在地上,不過他被打穿的肚子中露出的居然全是機械,他不理會自己受的傷朝笑著我們: 「哈哈哈!我只不過是一個被命令來搶回[黑米理論]的闇天使罷了,[黑米理論]已經被你們打下去的兩個人帶走了,黑米大人早就預料到你們會把注意力放在跟他相貌一樣的我身上,所以我只要把理論交給他們兩個,然後故意在大樓上跟你對戰,再讓他們兩個很自然得掉下去就能成功取回去了,哈哈哈哈!」
「什麼?居然會這樣…」我沮喪得跪在地上。
「對了~」偽黑米又補充一句「真正的黑米大人正朝向帶有創造之力的戒指過去喔!哈哈哈哈..哈...哈...」說完假黑米就停止動作了
我心頭一促,立刻站起來,糟了!我必須先趕回去!三人趕緊往機城方向出發。

wmfchris

「呼哈、呼哈,終於趕上了……」眼前是久違了的osu學院。
我看看手錶,是十一點五十八分。還好,我沒有遲到……這時,頭上傳來一把熟悉而洪亮的聲音。
fish哥正向我們致辭:「大家好,歡迎大家來到我們osu學院一年一度的考試(peppy's advanced level exmination for beaters and mappers),咳唔……照慣例我們先說說這個考試的背景……
「在很多年前,peppy把我們這個世界造了出來。但是當時的波動很混亂,於是peppy就把beater和mapper創造了出來,利用他們控制節奏的技巧,調節大地的波動。但是peppy創造出來的beater和mapper畢竟還是人類,終有一天會死去。peppy想了一個方法,就是讓他創造出來的beater和mapper把知識傳給下一代,並設立不同的測試來驗證他們的實力。當他們通過測試時就會得到更大的beater和mapper力量;時至今日,我等還是負著教育的責任,把beater和mapper的技巧傳下去!」
台下的人早已入睡。
「嗯……好!我們來說一說規則吧。Beater的話與以往一樣,我們的考核期是一個月,共三部分;頭十天是論文寫作,測試大家對osu世界的見解。值得一提的是這部分並沒有固定答案,大家可以盡情抒發喔!
中間的十天是測試大家的實力,在這十天裡有一張ranking chart,寫著是次考試的15首曲目,電腦Auto的測試結果是15首歌的最高難度合共1億分,大家只要達到70%,也就是七千萬分就算是合格了。大家可以在這十天任意練習,但只有最高分的一次才被計算成考生的分數。
最後十天是測試大家的對戰能力,考生每天都會被分到一個對戰鬥房裡。房裡會有八位考生,大家可以輪流選擇曲目共嗚向大家對戰。這一部分對總成績的影響只加不減,所以就算是輸了也沒關係。」
「現在請用你們熱烈的掌聲歡迎今次的主考官!」
一陣熱烈的掌聲過後,fish歌請出了兩個人。
「第一個人是abalee,他是我校的榮譽教官,專業是osu史和mapping理論。他會負責你們今次的論文審核!」
「另一個人是思寒,他是我校絕頂高手之一,專業是戰鬥理論。他會負責你們對戰的事務!」
「忘了補充一句,第二部分的審核100%由電腦完成,同時大家也可以看到即時TOP40和總成數的排名。另外,在這一個月期間學院為你們安排了特別房間和徽章,讓大家可以自由使用學院的設施。
現在我要宣佈這次的論文題目--論spinner和節奏感的關係。好啦,廢話小說,詳情你們可以問你們的助理。解散!」
啊,終於講完了。我的助理名叫……xipan來著?
(得到[考試徽章](限期: 1個月)x1!)
「請問你叫……小雨嗎?」身後突然有人拍我的肩。
「我是啊~」
「我是xipan,是你這次考核的專屬助理。」他聳聳肩,繼續道:「原本是數人分到一個助理的,可是上頭只示我只看你一個。聽說,是你哥的主意吧?」
「那笨老哥最好消失啦……」我還沒說完,一把磁性聲音從側面傳來:「我消失了,你不就考試都不用來吧?」
眼前是我那又帥氣又強的笨老哥,雖然笨,但卻是世上一流的beater和mapper,例如其名作Justice to believe,作品一出,其共嗚就傳遍了天下。不知道多少人為了征服此曲,虛渡了多少時日、奉上了多少鮮血……拿到的也只是一個pass而已。
「梆」一聲,我的頭上立刻起了個腫包。「哼,我不是說了不要認識陌生人的嗎?看來你又忘記了。至於你那邊的事我從校長那邊聽說了。我會過去做我該做的事。而xipan,如果我妹敢離開osu學院一步的話,直接敲昏她拖回房就對了。」
我能做的就只有狠狠地瞪著他遠去。
晚上,睡房。
「可惡啊!spinner明明跟節奏沒有關係!還是我自己試一次好了……」
關卡:Loituma - Ievan Polkka (unranked)
http://osu.ppy.sh/s/2463

Wednesday 2 June 2010

SIMC report (5) Challenge 3iii

7)Long term consideration
In model 1 (Challenge 1), our locating strategy is about to miximize the market share with the least number of new shops opened. That is, only consider the marginal influence of the shops.
In model 2 (Challenge 2), our locating strategy is about to maximize the market share within the given amount of budget, proportional to the k-junction. That is, the cost of opening cost is the only new factor.
Now let's do some math. Assume the cost for the n-th month is C(k). Generally C(k) is increasing due to inflation. Extend C(k) so that the domain the real number. (negative k implies that cost projected in the past), also C(k)≦C([k])≦C(k+1)  for any integer k. Therefore it is monotonically increasing. Integrate once on C(k) from zero to infinity, which is diverging. Therefore the cost of running a shop under unlimited time requires infinity cost. Comparing with the cost of opening a new shop, which is a finite cost, the ratio of run cost is far (infinity-ly) more important than the set-up cost. So in case we should take the maximization of profit as the second aim after the maximization of market share. Indeed, what we have to do is build up a model to consider the lattice that supplies. To extend this model, the vector field of supplier and consumer can be built, which will be the ultimate model of this problem.
"We can't open the shop under an unlimited time. Why we have to consider this case?"
This assumption helps us to realize that the running cost is more important than the set-up cost in running a shop, which will be more important than the locating strategy to win the competition.
"If the maximization of profit is concerned, using multiple shops to block the Starbucks will be useless. Then all previous models will be demolished."
Yes this is the main problem of the simplified models.
8)Debug
So far we have finished the analysis of this problem. You may ask that why we finish this problem with just some simple maths, but that is the truth. In fact, our proof about the locating strategy is incomplete, but in the sense of considering every simple case, it works. Combining together, it will works too.
Some advanced tools like matrix and programmiung is  highly appreciated in the competition but we tried to finish the problem by simple mathematics, we will be proud of this.
The full solution from different team will be released in 1 month and I'm looking for this and a further analysis.

End of SIMC report: The Challenge

Tuesday 1 June 2010

SIMC Report (4) Challenge 3ii

ii)A complicated model
When we built up the previous model, we assume that the population only flow within the two closest shop. In fact the population flows everywhere. That how can we express the average population density as a average of a whole day? Putting them in equal proportion is obviously not fair. Our solution will be:
Assume that for a particular street, the closeset distance from the street to the Starbucks shops willbe a1,a2,..., am units while the distance to the Coffee Bean will be b1,b2,...,bn units. Then our market share will be: [Σ(1/bi)/Σ(1/bi)+Σ(1/ai)]. The idea is that we still have to show that the closer distance will have a higher chance of providing service to a (averagely) closer customers.  It can also shows that a very close shop does have a very strong influence on that street (which is the real case).
"Distance tends to zero will results in an infinity-ly large influence to the streer?"
This problem makes no sense. The distance is assumes to be an integer, and the indfluence is measured in terms of the whole street. Even though the man living besides the shop goes there everyday, the shop can't influence the whole street at all.
Now note that consider the model when there's only 1 Starbucks and 1 Coffee Bean shop. The market share will be (1/b)/(1/a+1/b)=a/(a+b), which is just same ax the model of (i)! Then we are success in considering the extended model.
in this model, the locating strategy is nearly the same as (i) so we need not to discuss again. Just a point to note that the locating of n-junction will be much more important.
One important function of this model is that the share of a shop to a particular segment of a street is not as exact as 1 or 0 or 0.5, but they're equalizing to half-half. They are competing. That's why we need some strategy to win the competition.
(Note that the share originally is 1, falled to less than one, higher than 0.5, if the share is half-half it's still the same, and if it is zero it will raise a bit.)
6)Population denstiy
In the above model we are considering the flow of population. However one of the assumption still exist -- the population density is even. If it is not even, the proportion will be different.
In the real society, Coffee Shop starts their business in the business zone where people used to negotiate there. A rural area will have a higher potential population density than a urban area. It will be very complicated for us to compute those values. In order to show that the model is efficient to compute the real model.
consider the above map (2*4) again. Now assume that the segment (1,2) to (3,2) is a trade zone where people likes coffee so much there (they don't put a coffee machine at office right?), hence a double share to the coffee shop. The new share will be 5.5/16, where the rate of increasing will be 14% which compansated the deficit of locating in a bad location which fewer customers.
Combining thw two models, the new share will be about 7.2/16=0.45 which is nearly half! We see that the effect of trade zone is even larger than the density.

we will take care of the long-run considerationg in the next discussion.