## Friday, 8 December 2017

### Some recent maths activity

Yesterday I received the question from my engineering friend:

Let $f$ be a real function so that for all $x,y\in \mathbb{R}$, $f(x+y) = f(x)+f(y)+xy(x+y)$ and $\lim _{x \to 0} f(x)/x = 1$ hold. Find $f$.

It does not like a casual question to ordinary university students, not even for maths students...but anyway one may notice that $xy(x+y) = (x+y)^3 - x^3 - y^3$ if you know symmetric polynomials well, and the free linear term makes up the limit we have $f(x) = x^3/3 + x$.

Well, it is easy to show that the function is continuous by exploiting the equality $f(2x) = 2f(x) + 2x^3$, but even stronger we can prove differentiability. Rearranging gives

$\frac{f(x+y)-f(y)}{x} = \frac{f(x) + xy(x+y)}{x}$

Taking limit $x\to 0$ yields $f'(y) = 1 + y^2$ - not only that the derivative exists, we also get a complete DE with an initial value $f(0) = 0$. That easily solves to $f(x) = x^3/3 + x$.

What if the limit condition is changed? Say, $\lim _{x\to 1} f(x)/(x-1) = 1$? We can rewrite the expression as the following:

$f(x+y-1) = f(x-1)+f(y)+(x-1)y(x+y-1)$

Dividing both sides by $(x-1)$ reduces the question to the original case which gives the same solution.

Let is consider the functional equation at a much generalized form: $f(x+y) = f(x)+f(y)+g(x,y)$. According to the above argument if we managed to show that
$\lim _{x\to 0} (f(x)+g(x,y))/x$ exists then we can easily reduce it back to a DE where existance or uniqueness is clear. However this is hard to work around if we do not assume the limit condition because we know pretty much nothing about $f$. It does not work by assuming continuity of $f$ or $g$, since we may come across to some very nasty functions like the Weierstrass function which makes no sense in these questions. We leave a few observations here without solving it (or even getting close):

1. $g$ must by symmetric. This is obvious by observing the rest of the term. In particular, if $g$ is a polynomial then it is in the ring generated by $\sigma _1 = x+y$ and $\sigma _2 = xy$.

2. If $g(x,y) = O(xy)$ for small $x,y$ then it is possible to recover $\lim _{x\to 0}f(x)/x = 1$ using estimates like $f(x) = 2^n f(2^{-n}x) + O(x^2)$ or $f(x) = nf(x/n) + \log n O(x^2)$.

3. If $g$ is Lipschitz we know immediately that it's differentiable a.e. but that says we could have uncountably many non-differentiable points that we not want to deal with...

But that's it. I do not want to spend more than 60 minutes on this useless (for me) problem :d

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The 1st Simon-Marais (aka the Pacific Putnam) was held on October 2017 and the statistics are finally out (compare the efficiency against IMO marking team...). It's very surprising that only 1~2% of the students got problem A4 (and A3). I expected the top rankers to be close to 42 (aka 6 correct answers) but it turned out that not many olympiad players participated the event as can be judged from the award list. I expected the event to be much harder next year.
\$\

## Wednesday, 6 December 2017

### 6/12/17

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2017年10月30日

「想不到你剛來一天就能把這支弄到手真是厲害呢。」

「我剛好相熟的朋友在銀座經營酒舖，要弄一支來還是可以的。在外面難找倒是真的呢。」他徒手撕下一塊烤肉放進口，順手把威士忌倒進兩人的杯裡：「這個我剛讓廚房做出來的，趕快趁熱試試，我覺得烤得蠻不錯的。」

「也不能說是難搶，只是價錢問題而已。網上教人去唐吉訶德或者西武超市搶的攻略變多才是看起來難搶的原因；你不差那一萬幾千塊的話amazon也可以訂到呢。可惜我可沒你那麼有錢，上次搶到兩支帶回去望了好久最後還是決定轉手賣掉。」隨手打開amazon鍵入響21，サントリー官方售價是零售價的一倍以上，媽的。

「轉賣賺多少了？這麼說來你應該還沒試過它吧，不試的話你肯定會後悔的。」

「賺了大概五六單FGO吧，我沒玩就是了。最近都忘著玩火紋手遊給任天堂賺錢呢。」

「微課不如無課，課個五六單還不如把錢留著去下面的游泳池玩呢。說起來你這個燻雞粉應該也去朝聖過了吧？」

「我又不是朝聖狂熱者，再說不帶個伴進去看上去就太寒酸了呢……咦不對，上次你明明說過明明不喜歡希爾頓的怎麼又來了？」

「還不是女友想來……結果一抵就逛了一整天，晚上又趕去高田馬場那邊喝酒了。聽說今晚有音遊界DJ樣子，下次讓她給你介紹一個好了。」

……

「說實話我實在不太想接手你家的公司……股東成份有點複雜，我又沒有多餘的票灌下去。在這個節點惹到別的勢力的話可是不太妙呢。」

「票也不用太多每季10張就好了啦，反正中價區價格彈性比較大，要調節也比較不會被投訴。」

「我就真的沒有餘裕能把票分出來了……按照我的模型來看如果我家公司的票再減三張左右的話高價循環就會崩潰，能套錢的地方少了一個的話會十分不方便。對面也盯票流盯得很緊，我換過來應該很快就會被發現了。」

「嗯……反正我這邊有的是人，來交易一下如何？我把你加進換票網絡裡，這樣對手會比較難鎖定你的動向；相應地我希望你能放5-8張票在我們這邊公司，這方面同樣可以透過網絡進行。」要比喻的話換票網絡大概就像洋蔥網絡(TOR)一樣，將點對點的連結遮蔽起來。

「這個可以。不過我在明處不需要太多小動作，放20票進網絡就夠了。你公司需要的票數我會從我手下分出來。」

「真夠精打細算呢。你的目標定好了嗎？」

「其實我沒有，不過手下們可是有一大串目想炸的人呢。有一位想搶新的碧藍艦娘經理不但被守下來，對方還真接把公司炸爛，每次提到那公司他都氣得牙癢癢的呢。反正現在是第一個賽季，多收攏人心才是最重要的。」

「多收下線準備下季再大幹一場嗎？」

「老了肝不起來了啦。有互助圈子才能確保大家不會餓死也不會輕易被攻擊，玩下去就當多認識朋友好了……總之你幫我將票吃掉，我幫你經營集火，這樣沒問題吧。」

「沒問題，那就拜托你囉。」

「合作愉快。」

「Cheers」

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