Recent content by bham10246

Hi, this is a question to the members with some knowledge in algebraic geometry: 1. what are rational curves with self-intersection -2? How do they look like? 2. do you know why these correspond to the vertices of some of the Dynkin diagrams? 3. just something that's bothering me...

Do you have an example for this: "Some would say we study resolutions of singularities purely because of a classical hang up that smooth things are 'nicer' " ? Do you mean in terms of covering it locally with Euclidean open sets or when defining curves or geodesics on the surface?

For your example above: "A singularity, from a quotient by a group action, is a fixed point in the orginal space: generically the quotien X --> X/G is |G| to 1, but at some places it is not |G| to 1 - these give singularities. A simple example is R/{+/-1} (not in SL, but that's not the...

Thank you n_bourbaki! Your explanation makes sense. I found a paper from arXiv and it seemed interesting. The hardest part was that since it was a random paper, I had to look up definitions and basic concepts for every other word. I had to read through the paper a few times and think about...

Hi, I posted this under General Physics but I thought I should post this under the math section too in case there are mathematicians who do not read posts under General Physics. I read some papers online about Calabi-Yau surfaces but I have some basic questions about them if you can answer...

Hi Chris Hillman, I didn't mean it in a bad sort of way. I just thought you are one of the many people on this forum who leaves behind great stuff to think about, gives wise advices to students, and am able to combine different fields to help us to keep a bigger picture in mind. For...

Is this an excerpt from a book? Wow... So basic algebra is very important as we continue to study mathematics. By the way, your entry is pretty interesting...

Definition of Total Variation: Let f(t) be complex-valued function defined on the interval [0,1]. Let P: 0=t_0 < t_1 < ... < t_N = 1 be a partition of the unit interval. Then T_0^1(f) = \sup_{P} \:\sum_{i} |f(t_i)-f(t_i-1)| where we take the supremum over all partitions of the unit...

Does this work? By definition, ess \sup f(x) = \inf \{M : m\{x: f(x)> M\}=0 \} . So suppose such finite M does not exist. Then m\{x: f(x)> n\} >0 for all n. Then by Tchevbychev, \int_{\mathbb{R}} |f|^p \geq n m(E) >0 where E = \{x: f(x)> n \} . So as n \rightarrow \infty...

As for my own answer to Question 3, I think if f is in L^1 \cap L^\infty, then f\in L^p for every p\geq 1. So the converse of Problem 3 is certainly true! But I don't think this is true...

Homework Statement Let f_n, f: [0,1]\rightarrow \mathbb{R}, and f_n(x)\rightarrow f(x) for each x \in [0,1]. I need to show the following two things: a. T_0^1(f)\leq \lim\inf_{n\rightarrow \infty} T_0^1(f_n), and b. if each f_n is absolutely continuous and T_0^1(f_n)\leq 1 for each...

Hi ansrivas, you might be right, as long as the bounded variation is for a finite partition of the interval [a,b]. That is, \sum_{i=1,..., N} |f(x_i)-f(x_i-1)| \leq M for some M. It's because for your function f, the total variation of f is infinite, isn't it?

Thanks! I think I have seen your proposition before in some book!

Coming up with counterexamples is hard. So to prove or not to prove, that depends if there exists a counterexample. Question 1 has been ANSWERED!: If f has a bounded variation on [a,b] , then is it true that f is of Riemann integration on [a,b]? Question 2 has been ANSWERED!: Is it...

Thanks for all your input! I was joyful at first but I don't think f is Lipschitz continuous at x for almost all x implies that f is absolutely continuous. Maybe f is Lipschitz continuous at x for almost all x implies that f is almost absolutely continous?