格仔 Blog

Monday, August 24, 2015

【拍片】日出國 x 幻龍王

感覺剛養成的幻龍王隊潛力很大，
在大概 10:30 開始打了三場（11:00 結束的）日出國。

日出國的影片多數很好看，因為在尾 2 和最尾這兩關除了磨隊基本上沒有必勝法。
看看玩家怎樣爭扎到最後勝利也是一件樂事！

Sunday, August 23, 2015

雙極女神通關

今天又打了 10 石，最後覺得以下這隊平均較高機率通關（通常不到第 16 層都死不去），
所以不斷用這隊堅持嘗試：

全 297 和沒有任何潛覺。這一隊 6 隻寵物 5 隻都有技能 "加速" 的功能，
所以實際操作下等過 3, 4 回便可以再發動轉珠技。
如果潘的技能是加速便完美了！
4串暗的情況下必定單隻 100 萬。
暗 3 combo 及其中包含 1, 2 串 2way 的話月龍女的攻擊力會在 60~100 萬之間，
對一般平砍非常足夠了。

幻龍王隊回復力很缺，把心珠轉出來的寵是必不可少。

小過程. 以下是第 20 關的一些小截圖：

第 19 關是水赫拉，有幸不用開技便解決。
到第 20 關是蠻萬全的狀態。
全力第一擊是白虎 + 月龍，暗珠:火珠是 17:13，2way 表選最大火力，
因幻龍的技能關係順利地天降暗珠的 combo，不無小補。
第一擊把對方血量扣掉了 $\frac{2}{3}$（這傷害遠遠超過 2000 萬 XD）。
第二擊是潘 + 月龍（如圖所示），
4 暗 combo，3 個 2way 順利把她解決。

後記. 終於不用在這一關輪迴，通關的感覺很愉悅！（過程真的很痛苦
平日有新的想法的話會再進這關。
現在再挑戰這關的動機少了，石頭也可省點用。

Wednesday, August 19, 2015

Stochastic Integral w.r.t. continuous semimartigale

做個紀錄：

http://staff.ustc.edu.cn/~wangran/Course/

抛開嚴謹的證明，要對 local time process 有個大概的話：

http://streamdp.hhs.se/LinkedStaffDocs/download.aspx?dl=00037_015

這份 notes 不嚴謹的地份僅為推導致 definition 的部分，
在得到精確的 definition 後所有內容都是嚴謹生動。

生動的原因在於所有困難的 result（而且和 local time 關係不大）都會放到 appendix，
proof 也會 refer 其他書籍。
因此重要結果的推導非常明快，一目了然。

最後對 stochastic process 有個大概，想最深入了解整套 abstract theory，
那不得不推介 Olav Kallenberg 的 Foundations of Modern Probability。
我從中學習到很 abstract 的 martingale theory，
亦因此對自己所用充滿了信心。

Sunday, August 16, 2015

On a Question in Fourier Transform

Someone asked about the second last line below:

Where $A,\omega\in \R$. And I want to record the explanation:

Fact. For any $y\in \R$, we define $g(y):\R\to \R$ by \[
g(y) = \int_{-\infty}^\infty e^{-A(x+iy)^2}\,dx,
\] then $g'(y)=0$, and therefore \[
\int_{-\infty}^\infty e^{-A(x+iy)^2}\,dx = g(y) = g(0) = \int_{-\infty}^\infty e^{-Ax^2}\,dx ,
\] and the last one is well-known that can be converted to the standard normal density function.

Proof. Just note that $g(-y)=g(y)$ and therefore $g'(-y)(-1) = g'(y)$, next by the formula of $g'(y)$ we can show that $g'(-y)=g'(y)$, therefore \[
g'(y) = -g'(-y) = -g'(y) \implies g'(y)=0.\qed
\]

Friday, August 14, 2015

On Convex Functions

Today I came across a result for AC functions.

Theorem. If $f :I\to \R$ is absolutely continuous with $f'\in BV(I)$, then $f$ is representable as a difference of two convex functions.

We need this result for generalised Ito's Lemma which works on convex functions (with an indirect route: consideration of local time process). This result seems nonstandard, and I have paid a few hours finding standard results on convex functions, the following turns out to be what I want:

Lemma (Thm 14.14 of J. Yeh's Real analysis). Let $f$ be a real-valued function an open interval $I$ in $\R$. Suppose that

1) $f$ is AC on any closed subinterval of $I$; and
2) $f'$ is increasing on the subset, $A$, of $I$ on which $f'$ exists and $m(A)=m(I)$.

Then $f$ is a convex function on $I$.

Having the lemma we can prove the Theorem immediately.

Proof. Since $f$ is $AC$, fix $a\in I$ and for any $x\in I$ we have \[f(x)=f(a)+\int_a^xf'(s)\,ds.\] Now we can decompose $f'$ into a difference of two nondecreasing functions since $f'\in BV$, call them $H, K$, i.e., $f'=H-K$. As a result, \[
f(x) = f(a) +\int_a^x H\,ds - \int_a^x K\,ds,
\] finally we denote $h = \int_a^x H\,ds$ and $k = \int_a^x K\,ds$. Then $h,k$ are AC on any closed subinterval of $I$, moreover, $h' = H,k'=K$ a.e. and they are increasing, therefore we can apply the lemma to conclude that $h$ and $k$ are convex.$\qed$

There are many interesting and fundamental facts for convex functions that are not mentioned in UG curriculum of UST and I really suggesting reading them all, they are too standard to miss.

Still I want to record an important fact of convex functions: $f'_-(x)$ is increasing (of course) and always left continuous if $f$ is convex, this makes if possible to define Stieljes measure by using $f'_-$.