Yi-Wen
Liu's mathematics page Yi-Wen
Liu's mathematics page
about my personal interests in Mathematics....
Do we really know how to integrate?
In this essay I will present a problem that I asked and failed a Stanford Mathematics professor as of October, 1999. I will assume the readers already familiar with elementary calculus such as doing integration of cos(2x) from 0 to 2*pi, etc.
I was inspired by the existence of a non-Lebesgue integrable function over [0,1], of which the construction relies on the axiom of choice. This is to say, if you don't accept the axiom of choice, then any real function on [0,1] is Lebesgue-integrable. For people that just apply calculus without much ado, the best policy might hence be to deny the axiom of choice, and be satisfied that for all the functions we can ever deal with, we seemingly have a complete theory of integration, namely Lebesgue's. Nevertheless, as you read the axiom of choice, you would find it being so true that you really want to accept it. Even more frustrating is that in certain arenas such as abstract algebra, you have to accept the axiom of choice.
Now you have a dilemma: to be blind to the so-true axiom, or to admit that there is something that you don't know about integration.
I chose to admit how limited my knowledge was, and so did my professor. Accepting the axiom of choice in this context, I then raised the following question that failed the professor -- are there more Lebesgue integrable than non-integrable functions? I.e., is there a one-to-one correspondence between the collection of all the Lebesgue-integrable functions and the collection of else? Ever since I first thought about this question in 1995, I found it so profoundly hard either to construct such a one-to-one correspondence or to prove that it doesn't exist. So far I still don't know the answer to it, and my intuition told me that this problem leads to a very fundamental question in set theory and mathematical logic -- the consistency between the continuum hypothesis and the axiom of choice.
Oops, as a webpage I guess I've bored you enough. Anyway, please get back to me: jacobliu@stanford.edu if you are also interested or you know the answer to my question!
from Rubik Cube to Group Theory
I
spent 3 years when I was in high school, independently, to
develop my own solution to Rubik's cube. I did find a
solution and taught some of my school friends. They liked the
solution.
Using my solution, a novice will be able to learn in several days to unscramble a Rubik cube within about 3 minutes. I myself used to have an average speed of about 1 minute 40 seconds. However, speed is not what I would like to brag ---- after all, there are even quicker solutions that have helped people win in various competitions. Extremely skilled players can do it, somehow, within 30~40 seconds.
What I would indeed like to brag about is the beauty of my solution. Compared with other solutions, my solution doesn't require players to memorize long series of moves. You will know clearly what your are doing at each step. More over, when you learn my solution you actually learn many important concepts of the group theory. Among them are jargons such as coset, quotient groups, normal subgroups, isomorphism, commutability, orbits of group actions, fixed set of subgroups, etc.
Unlike most published methods that unscramble the cube in a layer-by-layer manner, my solution considers the 8 corners first. To convince you that my solution is thus pedagogical and beautiful, note that the subgroup of the actions that fixes the 8 corners turns out to be a normal subgroup, and the quotient group associated with this normal group turns out to be isomorphic to a 2*2*2 mini rubik cube. Here the subgroup being normal is really important, because it means that once fixing the 8 corners, you don't need to worry about them anymore, you will hence be able to fully concentrate on manipulating the 12 edges.
For a more thorough discussion on the group-theoretic perspectives of the Rubik's cube, readers are encouraged to take a look at professor W. D. Joyner's Homepage @ US Naval academy.
I recently learned that my "corner-first" solution was actually promoted by Minh Thai, the 1981 world champion whose record time was 26.04 seconds. A detail description of OUR method can be found in the webpage of this guy, Matthew Monroe. It is exciting to learn (via internet) 10 years after I formulated my solution that it is indeed in many senses recognized as the best solution.
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