You mean you don't have groupies?
184 karma · joined March 2, 2009
[1] Lipton, R. DNA solution of hard computational problems. Science 268 (1995), 542-545.
[2] Head, T. Aqueous Solutions of Algorithmic Problems: Emphasizing Knights on a 3 x 3. Lecture Notes In Computer Science; Vol. 2340
However you made a sligth mistake with the shifting part:
"The set part of the new object will be the union of the two sets with all the letters shifted as many places as specified by the integer part of the second TIAM object."
I think you are shifting a little bit too much.
On a more serious note: Other people have recommended very good books that will teach you problem solving so I would like to give you a feeling of what "real" math is. In high school and in most University degrees you only do applied math => how to use math to solve this or that problem. The more math you know, the more tools you have to solve your problem. For a given problem: could I solve it with a partial differential equation? with a Bayesian approach? etc. You can see how the more math you know, the wider your search for a solution can be.
However, most people never get to do "pure" math => to create your own new math. When you read a book or attend to a class you only get to see the finished product. When you see a finished building, they have removed all the scaffolding, the cranes, and all the other tools that were used to build it. If you spend some time creating your own new math you will get a very deep understanding of what math is and it will widen your learning abilities.
Of course I don't expect a high school student to create something truly new, this is just a learning exercise. I just want to provide a 'feeling', I won't be formal or rigorous.
Definition 1: An object of class TIAM is composed of an integer and a set. The set can be empty or contain letters from the English alphabet with no duplicates.
Definition 2: Operation "+" This operation will take two TIAM objects and produce a new TIAM object. The integer part of the new object will be the sum of the cardinalities of the sets. The set part of the new object will be the union of the two sets with all the letters shifted as many places as specified by the integer part of the second TIAM object.
Example 1: [1, {a,b,c}] + [2, {c, d}] = [cardinal({a,b,c}) + cardinal({c,d}), {a,b,c}U{c,d}>>2] = [3+2,{a,b,c,d}>>2] = [5, c,d,e,f]
Example 2: [0,{z}] + [1, {}] = [cardinal({z}) + cardinal({}), {z}U{}>>1] = [1+0,{z}>>1] = [1,{a}]
* TIAM objects are closed under +. (See http://en.wikipedia.org/wiki/Closure_%28mathematics%29). Do you understand why? Can you prove it?
* Is the operation + conmutative or non conmutative? (See http://en.wikipedia.org/wiki/Commutativity) Why? Can you prove that it is / it isn't?
* Can you find a neutral element? (See http://en.wikipedia.org/wiki/Identity_element) I mean, can you find a TIAM object (I will call it TIAM-ZERO) so that TIAM-A + TIAM-ZERO = TIAM-A for any given value of TIAM-A?
Etc.
This is just an example, try to build your own stuff and reason about it. Ask yourself questions about it and answer them. Once you get used to this kind of thinking, you will learn faster and deeper.
I STRONGLY disagree.
Let's address this with an example: "Two planes are either parallel or they intersect in a line" Someone who just finished high school should be able to "prove" it informally without even opening a text book. Of course high school education doesn't give you the tools to write a formal proof, but if you think about it for a while, then write an informal proof, when you finally open your book and read the formal proof it will be pretty obvious to you. Once you learn proper maths in University, you should be able to sketch a formal proof very quickly.
Obvioulsy you cannot reinvent all the math from the last 5000 years all by yourself, so you need massive amounts of effort and study to learn (a part of) it. In my view, learning the basics of your problem space, then thinking informally about it, then (informally) writing your own theorems and proofs is the best way of learning. You won't come up with everything, but when you go to your book and read that all-important theorem you will think: "Aaaahhh! of course! how didn't I think of it!?"
You have a circle with radius=1 centered in (0,0). Draw a line from the center of the circle to any point in the circle. Now you build a triangle using this line as your hypotenuse. Inmediately, you will see that the heigth and the width of the triangle will be in (-1, 1). That is why -1 <= sen(x) <= 1 and -1 <= cos(x) <= 1. You can see that your triangle is always a rigth triangle so you can apply Pitagoras. That is why sen^2(x) + cos^2(x) = 1. Keep thinking about it for a few weeks and you will realize that all the formulas you memorized in high school are just common sense, you can deduce all of them just by drawing a circle with a triangle in your mind.
Try to do the same with other stuff (set theory is a good second thing to look at). You will discover that math notation is just notation and that many proofs (at high school level) are just common sense written in a very formal language.
Think about what the maths are about, not about formulas or notation.
I'm a Spaniard, I have worked in five different countries in Europe and Latin America. I'm currently settled in the U.K. I have worked as a consultant, as a developer in a "one-product" company and now I´m working as an in-house developer in a non-tech company.
It simply does not make sense to relocate to USA for me. I have had a couple of offers to work in Boston or New York in the last few years and I turned them down. Weak dollar versus euro means that I won´t be able to pay my euro denominated mortgage while living in the U.S. I am free to work anywhere within the European Union, and I have always been very welcomed in Spanish speaking American countries while I would need some paperwork to work in the U.S.
The political climate doesn't help either. I have never been in the U.S. and I realize that T.V. is not necessarily correlated to real life, but every other day you get news about some guy who was traveling to the U.S. for some perfectly legitimate reason, with all their legal documents in good shape and gets detained at the border for 30 hours, their families are not notified and they are not allowed to drink, eat or sleep.
On the other hand, I personally know some people (about 15 persons) who travel frequently from Europe to the U.S.,for both professional and personal reasons. They have never had problems in the border. There is clearly a dissonance between my "real-life" experience and my "media-induced" experience. Nonetheless, if I had to travel there, I would be a bit scared... my mother would definitely be very scared.
I really hope that things change. My wife is eager to spend a few years living in the U.S. We won't be going for now... may be during Obama's second term (if it happens) it would be economically feasible (strong dollar) and professionally interesting for me. As anyone with a passion for our craft, I have read countless histories about Silicon Valley, and I think the odds would be better for my own (future) start-up in there rather than in Europe, where bureaucracy and lack of funding make things more difficult.