[QUOTE=Shagnasty]
How do you fake it in math? What if you get bogged down in a field where there isn’t anything obvious to do at the moment? What if the remaining problems are so difficult that normal students can’t solve them?
[/QUOTE]
Then you don’t get a PhD. Or you choose a different field of Math.
How crappy? Crappy enough not to support any math grad students?
As far as I know, they do the same as anywhere else.
Here’s a slightly more technical question: I’ve been wondering about nowhere defined functions lately. As I was taught it, a function f in A -> B is a subset of A X B with (a, b) and (a, b’) in f iff b = b’. How does that work out when your domain is the null set?
What’s your Erdos number? 
[QUOTE=ultrafilter]
Here’s a slightly more technical question: I’ve been wondering about nowhere defined functions lately. As I was taught it, a function f in A -> B is a subset of A X B with (a, b) and (a, b’) in f iff b = b’. How does that work out when your domain is the null set?
[/QUOTE]
(I think instead of “iff” you meant “implies.”)
(Let 0 denote the empty set)
Right. A function is a certain set of ordered pairs. The empty function is the empty set of ordered pairs. What’s wrong with that? It just so happens that it is equal to the empty set.
Just as {x: x=1 and x=0}=0,
Also {(x,y): x is a member of the empty set, y is a member of Y, and blah}=0
Thus your use of the plural"functions" is also wrong. There is only one empty function, 0.
Unless, of course “nowhere defined function” means something else. It is not a term I have heard before.
[QUOTE=Thudlow Boink]
What’s your Erdos number? 
[/QUOTE]
It’s not finite. 
Ok I’ve asked this to numerous people…and I’ve probably had ultrafilter explain it to me too…but…
What is calculus?
(Not Ito Calculus. Calculus.)
I need this in mega idiot terms. I stopped understanding math in Algebra 2.
Anyone is free to answer - maybe if I have 10 people giving it to me, I can piece it together and understand it.
How exactly do you do research? Where do you find your problems, and how do you go about solving them?
Well, this might sound like a really stupid question, but since you’re here…
Why have I always been taught that as the size of an object increases, the ratio of surface area to volume decreases?
It’s demonstrably false, but everybody says it is true. I learned it in every math class, I learned it in high school physics, I’ve even seen it on this board.
But it’s not true.
Am I missing something, or do I have a Nobel Prize waiting for me?
[QUOTE=ZipperJJ]
Ok I’ve asked this to numerous people…and I’ve probably had ultrafilter explain it to me too…but..
What is calculus?
(Not Ito Calculus. Calculus.)
I need this in mega idiot terms. I stopped understanding math in Algebra 2.
Anyone is free to answer - maybe if I have 10 people giving it to me, I can piece it together and understand it.
[/QUOTE]
Lesson 1: Functions
Do you know what a function is?
A function is a rule (or an algorithm if you like) which has what is called a domain and a range. What a function does is for every thingy in the domain, it associates a thingy in the range. (in math notation f:X –> Y means f is a function with domain X and range Y)
Here’s an example: I’m going to make a function with domain being the set of cars, and range the set of colors. Then my function will be the rule which for every car gives you the color of that car. You can also think of it as a machine which you tell it something from the domain, and then it tells you something from the range, according to a fixed rule. That last part is important. If I tell my function “Jim’s car” then if it says “black”, then it will say “black” every time. (lets assume we can’t paint cars for now).
Here’s another example: f(x)=x+3 where the domain is the real numbers, and so is the range. If I say 4, my function will tell me 7. Or in symbols f(4)=7.
Good so far?
Calculus is the part of math that deals with differentation and integration.
differentation: Given the location of an object at each moment, calculate its speed.
integration: Given the speed of an object at each moment, calculate its location.
[QUOTE=Dijon Warlock]
Well, this might sound like a really stupid question, but since you’re here…
Why have I always been taught that as the size of an object increases, the ratio of surface area to volume decreases?
It’s demonstrably false, but everybody says it is true. I learned it in every math class, I learned it in high school physics, I’ve even seen it on this board.
But it’s not true.
Am I missing something, or do I have a Nobel Prize waiting for me?
[/QUOTE]
Not a stupid question. You are correct. What is the missing quantity, which is causing the confusion? Shape!
Look at a sphere:
SA=4pirr
V=(4/3)pirr*r
SA/V=3/r
This decreases as r gets bigger.
What about a rectangular solid which has length 1, width 1/x, and height x?
SA=2x+2+2/x
V=11/xx=1 (!)
SA/V=2x+2+2/x which approaches infinity as x gets really big.
So of course the answer is neither
[QUOTE=treis]
How exactly do you do research? Where do you find your problems, and how do you go about solving them?
[/QUOTE]
My advisor found me a problem. I talk about it with him when I get stuck. I do a lot of reading/studying to learn methods/results which might help me.
This may be way more narrow than you’re looking for, but you say you study set theory, so:
Are you one of those types who buys all the various large cardinal hypotheses as clearly, intuitively true, and, if so, could you perhaps shed some light on the source of that intuition?
[QUOTE=Indistinguishable]
This may be way more narrow than you’re looking for, but you say you study set theory, so:
Are you one of those types who buys all the various large cardinal hypotheses as clearly, intuitively true, and, if so, could you perhaps shed some light on the source of that intuition?
[/QUOTE]
I don’t think that they are intuitively true. I think the lack of counter proof (not for lack of trying!) and genuinely interesting theorems gives one a hint to just see what happens when we assume they are true. Look, if they are false, eventually we will get a contradiction. Enough weird shit is proved about them that somebody’s got to find a contradiction if it is there.
I do believe that 2^(aleph_0)=(aleph_2), though. It is a consequence of PFA.
I saw a presentation about I0-I3 (the top of the hierarchy) and it felt like we were standing on stilts. It was like, “that’'s all well and good, but I can’t wait to get back to solid ground.”
Do you know about the Reinhard cardinal, or the fact that most of the large cardinal axioms imply con(ZFC)?
[QUOTE=Jamaika a jamaikaiaké]
I do believe that 2^(aleph_0)=(aleph_2), though. It is a consequence of PFA.
[/QUOTE]
Interesting. I think I’ve heard that Goedel believed the same thing, though surely for some different reason. You take the Proper Forcing Axiom as clearly true, but not the large cardinal hypotheses? I must admit I’m fairly green to forcing (and all the crazy things set theorists do with it which are far removed from the concerns of “core” mathematicians), so perhaps with more experience, I’ll come to understand that perspective.
[QUOTE=Jamaika a jamaikaiaké]
Do you know about the Reinhard cardinal, or the fact that most of the large cardinal axioms imply con(ZFC)?
[/QUOTE]
Yeah on both counts, though only a tiny bit about the former. I imagine you bring it up because it’s the one large cardinal hypothesis so very strong as to be in conflict with Choice? I suppose most set theorists wouldn’t be happy to accept it as true, then, though when I can put myself in such moods as to think I have clear and distinct intuition into the set theoretical universe, it seems plausible enough to me as a description of … reality. But I don’t enter those moods often.
The latter I suppose would be because every large cardinal is >= a weakly inaccessible?; is that true, or are there large cardinals which don’t satisfy this property, but whose existence still either implies the existence of a weakly inaccessible or implies the consistency of ZFC in some other way?
[QUOTE=ultrafilter]
It’s a little tough to explain, but very roughly speaking, Ito calculus is a set of techniques for dealing with variables whose values change randomly over time. I’m mostly familiar with its use to model stock and bond prices, but I know that it also has applications in cell biology.
[/QUOTE]
Does it have any applications with respect to Gaussian Noise? Can it be used to process signals in a noisy environment?
Rats, Jaj already indicated that. I was wondering where one could find an example of Ito Calculus being used in signal processing.
In your estimation, who were/are the top 5 mathematicians?
Where did you go to college?
And thanks for availing yourself this way, pretty cool!
[QUOTE=user_hostile]
Does it have any applications with respect to Gaussian Noise? Can it be used to process signals in a noisy environment?
[/QUOTE]
Yep. For example, let’s say you have a signal at 1000 hz, and a noise at 60 hz, and want to filter out the noise with a simple RC filter. You use something called the Laplace to get a transfer function. With this transfer function you can calculate the magnitude of the attenuation of the signal, and the phase shift associated with the signal.
[QUOTE=Indistinguishable]
Yeah on both counts, though only a tiny bit about the former. I imagine you bring it up because it’s the one large cardinal hypothesis so very strong as to be in conflict with Choice? I suppose most set theorists wouldn’t be happy to accept it as true, then, though when I can put myself in such moods as to think I have clear and distinct intuition into the set theoretical universe, it seems plausible enough to me as a description of … reality. But I don’t enter those moods often.
The latter I suppose would be because every large cardinal is >= a weakly inaccessible?; is that true, or are there large cardinals which don’t satisfy this property, but whose existence still either implies the existence of a weakly inaccessible or implies the consistency of ZFC in some other way?
[/QUOTE]
That’s funny. I wouldn’t say “contradicts choice.” Instead I would say “inconsistent with ZFC,” meaning that I accept choice as a given.
I also mentioned Reinhardt because it was shown by Kunen to contradict ZFC within a year of its being defined. Thus my comment about lack of counter proof.
Honestly, I said most large cardinals imply con(ZFC) because I thought you needed strongly inaccessible. Whoops. But to be fair large cardinals aren’t my area of concentration.