Is there a theorem in measure theory that discretization can approximate continuous functions when the discretization gets small enough? If this is the case, then we could map the set of integers to the continuous number line, so they would have the same measure. So I guess it is not the case. But in practice discretization works! So whatβs the relationship here?
There seems to be some research activity in the past years but I'm not sure if any of it is relevant outside of the field itself, which is not large at all. I know about Tao's recent advance in Sendov's conjecture, which truly seems intriguing and interdisciplinary. Also, I'm fairly convinced that the theory is useful because I've been taught quite some applications in my numeric analysis class when I was an undergraduate. The question is more like: is it an interesting field that not many people care about or is it simply dying?
I'm asking the question because I have the option to pursue approximation theory for my master's thesis. Kolmogorov-type inequalities, to be precise. This question may fit more into the career and education thread, however I think it is general enough to get its own thread since it is about a field of math and may (or may not) generate a healthy discussion.
I'm doing a summer project on approximation theory (focusing on Alice Roth's mathematical Swiss cheeses), primarily from a pure maths angle.
But as my eventual goal is to go into mathematical physics (probably something like quantum field theory or quantum gravity), it would be a shame if I missed any good opportunity to steer it in that sort of direction, or even nice links to observe for my own enjoyment.
So does anyone happen to be aware of any ways that approximation theory (ie how we can approximate complex functions by nicer functions on various sets) is used in mathematical physics, that an (ambitious) undergraduate might be able to grasp? Or any ideas in approximation theory that, while not necessarily connected, would complement other ideas in mathematical physics?
Can you look at set theory as an attempt to approximate stronger logics in first-order logic? The fact that functions are sets and you can quantify over them feels like emulating higher-order logic. Axiom of infinity gives you a way to do infinite conjuctions and disjunctions as in some kind of infinitary logic. Axiom of choice is basically kinda allowing you to have infinitely many quantifiers. And when we do mathematics informally we often use those properties not really caring about the underlying set theory and simply use them as if we really were in a stronger logic system.
So what I want to ask is: Am I speaking complete nonsense? Is there any formal way of treating this correspondence? Or at least some philosophical insights, can this be taken as a motivation behind constructing a set theory or does this follow from other motivations behind set theory? Are there any papers on the subject?
^(This questions got no answers in the simple questions thread so I post it as its own post. Hopefully it's okay, I think it could maybe generate some interesting discussion.)
A lot of ages throughout the series are unknown and everyone throws figures around, so I wanted to do my own rationalization and see what everyone else thinks.
As a point of clarity age, these estimate are at the point when the manga begins
i.e. If Naruto is 12 on the first page, then this person would be X years old.
> Known Ages
Sarutobi - 68
Jiraiya - 50
Mikoto Uchiha - 35
Kakashi - 26
Itachi - 17
Naruto - 12
There seem to be some constants here such as Sensei-Pupil, Mother-Child, etc so I want to estimate these constants into the rest.
Sensei-Pupil ages can vary a lot but Kishi generally seems to have them from 14-18 year difference (based off Jiraiya-Saru and Kakashi-Naruto, although they could have had many disciples before or after it seems like a rational age difference). I'd also guess that 18 yr old mothers wouldn't be much of a surprise considering Itachi-Mikoto, so Minato and Kushina being in the same class and probably age group, Naruto was probably born when Kushina was around 20.
> Unknown Ages Estimated
Hashirama - 96 This one is hard, since he's the grandfather of Tsunade I want to say his children were 25 when Tsunade was born, being in a new peaceful era. So if Tsunade is 51 at the beginning of the story and her parents were 25 and I'm guessing he was 96, then Hashi's was 30 at the time of Tsunade's parents being born.
Kakuzu - 80 Older than Sarutobi and younger than Hashi, probably tried to get an edge by attacking an older Hashi while Kakuzu was 18-20?, putting hashi at 34-36.
Nagato - 40 Most likely Jiraiya's first pupils, Jiraiya would only be 10 yrs older than Nagato. This seems appropriate due to Minato appearing younger when being trained by Jiraiya(who also appears older than the Hanzo battle J-man) Jiraiya would be 17ish when training Nagato.
Minato - 35 23 years older than Naruto, great time to be a dad and hokage after ending a war. I say ending a war because this would be nearly after the end of the 3rd shinobi world war(kakashi would be 13 when naruto was born) Jiraiya would be 21ish when training Minato.
Will add more from suggestions in the comments
I came across this math approximation today and was confused how it was derived. I guess it's related to normal approximation of binomial distribution, but I still can't figure it out. Highly appreciated for any help.
$1 = \sum_K (\begin{array}{c}N\\ K\end{array}) 2^{-N} \approx 2^{-N} (\begin{array}{c}N\\ N/2\end{array}) \sum_{r=-N/2}^{N/2} e^{-r^2 / {2\sigma^2}}$
Sorry for the long LaTeX equation. First time to ask a question here, not sure how to input a math formula.
I loved the Elocator prestige class in the 3.5 expanded psionics handbook. Basically this guy was a up close fighter who specialized in teleporting all around the battle-field.
He eventually got a spring-attack move with a teleport. So he could teleport, attack, teleport back in the same move.
I'm not asking for homebrew, I'm wondering what is the closest vanilla character that we can make that fulfills the flavor of this class?
Thanks!
https://math.stackexchange.com/questions/2560700/why-does-the-sup-norm-make-the-results-of-approximation-theory-independent-from
Wonderful Widgets Inc. has developed electronic devices which work properly with probability 0.95, independently of each other. The new devices are shipped out in boxes containing 400 each.
a) What percentage of boxes contain 390 or more working devices?
b) The company wants to guarantee that k or more devices per box work. What is the largest k such that at least 95% of the boxes meet the warranty?
Here's where I am so far:
Β΅ = 400 β 0.95 = 380
Ο = sqrt(380*.05) = 4.359
Then I'm stuck.
I was reading classical results on how shallow neural nets approximate functions and found the statement that seemed to claim that one can always normalize the gradient to be <=1 in a function class without changing the strength of oneβs approximation result. Is this true?
https://www.quora.com/unanswered/Does-normalizing-gradients-in-a-function-class-diminish-the-results-of-approximation-theory
http://math.stackexchange.com/questions/2158458/how-does-normalizing-gradients-in-a-function-class-affect-the-results-of-approxi
