## Sunday, 13 December 2009

### Duals Duals Duals

Is there a connection between all these "duals" in math?
There seem to be two main classes of duals.
One from linear algebra / functional analysis, and one from optimization theory. (Lagrangian/convex duals)

Well actually there is.

Recall that dual space of a vector space V is simply the set of all linear functionals of V. To put it another way, dual space is simply all the row vectors if the original space is finite dimensional.

Dual norm comes up in convex optimization. Dual norm of z is defined as sup_x { | ||x||<=1}. This is actually the same as the operator norm of the vector z when you think of z as an element of the dual space (row vector). In great generality, we can think of dual norm as INDUCED NORMS ON THE DUAL SPACE.

This really plays a role in Lp duality.  Lp dualtity says that the dual of Lp space is isometric to Lq. Here we are talking about infinite-dimensional spaces. Also the dual norm of an element of the dual space of Lp will be equal to the Lq norm of that linear functional when mapped into Lq. (See any real analysis text)

You can think about the dual cone as one way of generating a INDUCED CONE IN THE DUAL SPACE. (Again, we think of row vectors as elements in the dual space)

I don't see the relation between the Fenchel dual (Legendre transform), Lagrangian dual, and the dual space though.

## Wednesday, 9 December 2009

### Syndicated from another blog.

“男人有 老婆又有情人是男人的本事。请放心，情人不会成为老婆，两个概念不一样。男人的大部分心思还是在你身上，因为你是他老婆，是孩子的妈妈，名正言顺，任何时 候都改变不了。而情人只不是婚姻的一种补充，是成功男人一种精神上的追求和情感上的需要，其实也是挺正常的事情，也不必太认真。”——这是昨天中午我在一 个案宗中看到的，记录着一个成功男人的一段非常有代表性的“经典话语”，这就是他们奉行“家中红旗不倒，外面彩旗飘飘”，今朝有酒今朝醉，自嘲“自己喜新 不厌旧，风流不下流”。

人生 短暂，即使你通过努力拼搏成功了也不要为自己找许多漂亮的借口，而去三番五次的寻找情人当作自己的“标志”。虽然在你的脸上贴上美丽好看的“情人标志”， 但那都啻是你一厢情愿的虚荣心在作怪罢了，人们一定会对你深恶痛绝而讥笑藐视你的，同样你也将会是臭名远洋、遗臭万年的。之所以，你还是做一个洁身自好的 成功男人为好，这样人们就会对你另眼相看而铭记你的忠贞气节，却给以你流芳百世。

现在 和谐的社会需要更多的是老实敦厚、温文尔雅的成功男人，同样也需要更多的是安分守己、贤惠善良的温柔女人。只有男人永远不再做纸醉金迷、拈花惹草的男人， 女人不再做水性扬花、放任自流的女人，那么整个世界将会是更加充满人情味的，而幸福的婚姻家庭就会是永远的牢固不破了。