光与暗

以前一直很顺，阳光只是一种天性和惯性
但如果经历过挫折之后，还能阳光，那才是真正的阳光
这两种阳光的质量是不一样的
后者更持久
更经得起考验

以前一直很不顺，黑暗只是一种天性和惯性
但如果经历过快乐之后，还能黑暗，那才是真正的黑暗
这两种黑暗的质量是不一样的
后者更持久
更经得起考验

Back in Singapore

After a very long travel (20 hours or more), I am back in Singapore. I will write some afterthought during this travel.

1) There exists food that tastes the same with SAF combat ration (the old version in the 90s). The food was served as lunch by the flight. The thing that made it worse is that I will be taking the same flight when returning to Canada...

2) If anyone of you have seen the SAF advertisment, you will see the same phenomenon at the taxi area at Changi airport terminal. Basically seven taxis will come in and park. Afterwhich, the attendent will direct the people to the respective taxis. Once all taxis are filled, the attendent will stop the passenger flow and allow the taxi to go off, at the same time, ushering more taxis to come in and the process repeat. The interesting thing is also when moving out, you will see an army of taxi out on the expressway.

3) SQ 800++ series has a lot space between the seats, which is pretty nice. Although the SQ flight will only take me to Hong Kong. The airline bringing me from Hong Kong to Canada does not have that much spaces...looking forward to 15 hours of sufferings.

4) Before I was in Canada, I used to find air-con area sometimes too cold for comfort. Now I find them too warm for comfort.

5) Canada hottest day in summer is not even as hot as an average day in Singapore.

6) Canada rainy days are not as heavy as a normal rainy day in Singapore.

7) You can walk a distance of three bus-stops or more in Canada without sweating (even in summer!) but you will sweat profusely in Singapore even by walking to the nearby coffe-shop.

8) Jet lag is as bad as what people describe. Either that or I have high resistance to it.

9) For a split of a second, I forgot that Singapore operates using left-hand drive, resulting in a short wave of panic in the taxi.

10) The Guess x3 show broadcasted in Singapore lagged behind Taiwan by 5 weeks or more.

三国演义的另类男人

Found this in a forum. Thought that it's pretty amusing, so have it copied on my blog.

最廉价的杀手--关羽
替袁绍斩了华雄，酬劳是一杯不太热的酒；替曹操杀了两大名将颜良、文丑，曹操给的酬劳倒是不少，可是他最后挂印封金而去，一点儿都没要（免费）。所以说便宜啊！真便宜！！

最顾家的男人--吕布
曹操重兵围城，生死攸关之即，人家吕布照样按心得在家里陪老婆孩子，正是现代模范丈夫、好男人的典范啊！

最不顾家的男人--刘备
“兄弟如手足，妻子如衣服”绝对一个不顾家的男人，有危险了自己先开溜，老婆孩子全丢给别人，最可恨的是为了笼络人心，居然把自己亲生儿子往地上摔，绝对是妇女儿童保护协会声讨的反面典型。

最小气的人--曹操
人家杨修只不过偷吃他一口酥，他就怀恨在心，后来因为一块鸡肋把人家给杀了。

最大方的人--陶谦
这么大一块地盘，说送人就送人，人家不要还几次三番的硬塞给别人，大方，真大方！

最幸运的人--刘备
没兵器的时候有人送兵器（送双股剑）；没地盘的时候有人送地盘（三让徐州）；没大将的时候有人送大将（赵云来投）；没军师的时候有人送军师（走马荐诸葛）；没老婆的时候有人送老婆（过江招亲）。

最命苦的人--也是刘备
好不容易做个官（县令），让督由给搅了；好不容易有块地盘（徐州），让吕布给夺了；好不容易有个军师（徐庶），让曹操给抢了；好不容易娶小老婆（孙尚香），让孙权给扣了；好不容易有点儿兵（75万），让陆逊给烧了；好不容易有个儿子（阿斗），让自己给摔傻了。

最老不死的人--廖化
很早就出场，跟着关羽混，关羽死了以后跟着刘备混；刘备死了以后跟着诸葛亮混；诸葛亮死了以后跟着姜维混；等姜维都死了他还没死，真是个老不死的，生命力太强了！

最诚实的人--刘禅
连一句瞎话也不会说，真够诚实的。

最具演艺精神的人--司马懿
装病，扮病人，一扮就是十年。

最贪吃的人--夏侯敦
连自己的眼睛也要吃，幸好因为口味不佳，没有吃第二口。（这要是吃了第二口的话，就要尊称他为“盲侠”了）

最贪酒的人--关羽
为喝一杯酒，宁愿拿着大刀和大将华雄去拼命；为喝一顿酒，宁愿冒着生命危险过江去单刀赴会。

最不讲文明的人--弥衡
出场就骂人，见谁骂谁，一直骂到死为止。

最偷懒的人--徐庶
“终生不献一策”很典型的光吃饭不干活。

最乱用职权的人--鲁肃
国家领土（荆州）你说借就借了？（一定拿了人家红包）

最不讲工作效率的人--曹操
带着83万大军，可还是让赵云跑了。

最讲工作效率的人--甘宁
应该三千人干的工作，他带一百人就干好了。（百骑劫营）

最恐怖的人--诸葛亮
活着的时候一个人坐在城头上吓跑15万人，就连死了以后他的木雕遗像也能吓跑一群人。

最透明的人--司马昭
因为：“司马昭之心，路人皆知。”

最一目了然的人--夏侯敦
他当然只能“一目”了然喽！

最爱抢东西的人--赵云
在长板坡这么危险情况下，还顺手抢了一把宝剑（青杠剑）三条朔（夺朔三条）和一个小孩（阿斗）。

最能偷东西的人--胡车儿
虽不能在百万军中去上将首级，但至少能在百万军中偷上将兵器。

最会炒作自己的人--庞德
懂得用棺材对自己进行负面炒作。

最说到做到的人--孙坚
曾说过：“我要是拿了玉玺，就死于刀箭之下......”后来就真的被人用箭射死了。（所以没事儿不要乱发誓）

最不团结的家庭--诸葛家
家里一共才三个兄弟，死也不肯呆在一起，分别效力于魏蜀吴三国，成天是各为其主你挣我斗，太不注意团结了。（人心散，队伍不好带）

走路最慢的人--曹值
一首诗都念完了，还没走完7步路

Hijacked!!!

Today is a bad day for me. My laptop got hijacked by some spywares yesterday. I can't seem to remove it even with spybot SD. Just when I was thinking of enduring and living with it, it causes my window explorer to collapse.

Thus the only solution I can think of is my recovery disk. Now the problem is solved but my files are all gone in the recovery process. AHHHHHHHHH!!! SIANZ AH!!!

Some Galois Cohomology

I will give an overview of Galois cohomology in this post. Again you are bewarned.

Before we can define galois cohomology groups, we need to understand what a profinite group is. A profinite group is, by definition, an inverse limit of finite groups. An equivalent formulation of a profinite group is that it is a compact, Hausdorff and totally disconnected topological group.
It can be shown that such a group has a basis of neighbourhoods (consisting of open normal subgroups) around the identity element. By compactness, it follows that these open normal subgroup are of finite index. (In fact, generally every open subgroup of a compact group has finite index) Examples of profinite groups are Galois groups of (not necessarily finite) Galois extensions.

Now let G be a profinite group. We say that an abelian group M is a discrete G-module if M has a continous G-action (here we endow M with the discrete topology). For each n>0, we denote the abelian group of continous maps from G^{n} (with the usual product topology) to M by C^{n} (notations here are not standard!). In the case of n=0, we take the group to be M. There is then the differential maps (which I will not define) that go from C^{n} to C^{n+1}. Hence we have constructed a cochain complex and taking cohomology, we obtain our so-called galois cohomology groups. (There is also the non-abelian Galois cohomology where M is not necessarily an abelian group, but I shall not go into that. Also if G is a finite group, then the Galois cohomology groups will coincide with the usual cohomology groups.)

We will give a sketch of how Galois cohomology can be used in local class field theory. Let F be a nonarchimedean local field. Equivalently to say, F is a finite extension of either the field of p-adic rationals or the field of quotient of the power series ring in one variable over a finite field of order p. (Here p is a prime number in the usual sense.) Fix a separable algebraic closure, say K.
Let G be its associated Galois group. Then one has an isomorphism (denoted by inv) between the second cohomology group of G with coefficients in the multiplicative group of K and the additive group Q/Z. For a finite separable extension L over F of degree n, the inv map induces an isomorphism between the second cohomology group of Gal(L/F) with coefficients in the multiplicative group of L and (1/n Z)/Z. Under this isomorphism, the element of the cohomology group corresponding to 1/n mod Z is called the fundamental class of the extension L/F. The establishment of the inv map is the first step in the proof of the existence of the local Artin map. (We remark that the development of class field theory was classically done by cohomological methods. Today, there are other ways to develop the theory of class field theory which are also important in their own rights.)