Recent content by klackity

Haha, stop this. Listen: Do not worry about grades except inasmuch as they show you a lack in your ability. If you get a B+, that almost surely means you did not master the material as well as you could have. Getting a B+ just means you now need to take the time to try to learn what you don't...

You couldn't have a "line which smoothly develops into a surface". Suppose for the sake of contradiction such a manifold, M, existed. Then about some point p, there would be a 1 dimensional coordinate chart, and about some point q, a 2 dimensional coordinate chart. Now, by "smoothly develops...

Here's the simplest way to see it: 1. We know that two vectors a and b are orthogonal if and only if <a,b> = 0. 2. We can see that <e1, a2 - <a2,e1>e1> = <e1,a2> - <e1,<a2,e1>e1> = <e1,a2> - <a2,e1><e1,e1> = <e1,a2> - <a2,e1> = <e1,a2> - <e1,a2> = 0 3. Therefore, e1 and a2 - <a2,e1>e1...

It's 20 credits. Basic Probability, Complex Variables II, Differential Geometry II, and Functional Analysis are 3 credits each. Advanced Logic and Philosophy of Mind are 4 credits each. It seems strange that the harder courses are 3 credits, but they are grad courses so they meet for less...

Let's see: Basic Probability: B Functional Analysis: B Complex Variables II: A- Philosophy of Mind: A- Differential Geometry II: A Advanced Logic: A And, that is probably the most difficult semester I will ever take.

Here's my recommendation: 1) Don't, under any circumstances, finish high school early. It will put you at a severe disadvantage. 2) Take as many advanced classes as you can at your high school, but don't worry about rushing it. 3) Take classes at a community college or local university over...

Well, compactness is a topological notion, not an algebraic one. Here's the formal definition: A topological space is compact if every open cover has a finite subcover. In order to understand this definition, you need to know the definitions of "open", "cover", and "finite subcover". I'll...

You all are way too pessimistic. The people who fail aren't the people who mess up. The people who fail are the ones who don't go head-first into what they want to do. You get confused, because all these great mathematicians seem to have perfect records. You think that their greatness...

RichardParker: You will absolutely perish in the future if you do that often. If you don't understand the proof, chances are you don't really understand the theorem that well either.

Most proofs aren't actually so complicated, once you get past the details and see the intuition. When reading a proof, what you should be looking for most is the intuition behind it. After you read through all the details of a proof, you should try to summarize the major steps of the proof in...

Not really, because the 3.7 guy had a 3.9 in major. A 3.7 in major would not nearly be good enough.

Yeah, but "mathgrad31455" went to an ivy (and had a 3.9 in major, mind you), while "mrb" went to a "Big state school, not a great math reputation" The point is that even a 3.97 an was not enough to gain "mrb" admission into 9/10 of the (admittedly top-tier) places he applied to, even with a...

You'll see a few people with lower GPAs being accepted to top places, but they are applying to an applied math or statistics program, or are female. To have a realistic shot at a top 10 pure math program, it looks like GPA of 3.9+ is essentially required, even if you have taken grad courses...

Laser BattleAxe Eye Surgery

Yes. This is a result of the Heine-Borel theorem: http://en.wikipedia.org/wiki/Heine%E2%80%93Borel_theorem