I was hoping to get a little help on bayesian inference, since my TA is being unresponsive. I got this HW problem mathb.in/154743, description and my thought process is in the link.
I was just thinking about unprovability. I just wanted to know if it is possible to make a concrete boundary between provable problems and unprovable problems in a certain axiomatic system.
We know that there is a statement that is true yet unprovable. Then is it possible that a statement is tr...
Hmm. It turns out that the deadline for the next grant I want to apply for is Octiber 4th rather than sometime in April as I thought. Plus, there is also one further one with deadline October 1st. I hope I can recycle a large part of my proposal.
It seems like I can use essentially the same project description for one of these new ones, but the one on October 1st has a 2 page limit which is bonkers.
Hi. I am interested in the limit $L=\lim_{K\to\infty}\sum_{k=1}^Kk^{-\alpha}P_K(k)$ for some $\alpha>0$, and where $P_K(k)$ for $k=1,\dots,K$ are nonzero probabilities summing up to 1. Also, for each $k=1,2,\dots$ I have that $P_K(k)\to 0$ as $K\to\infty$. Am I right that since $L\leq 1$, I can use the dominated convergence theorem and in this way $L=0$?
i want to know, in the third line, how did they write (ab)(cd) = (acb)(acd)? and they wrote "Every element of An is a product of terms of the form (ab)(cd) or (ab)(ac), where a,b,c,d are distinct elements of { 1 ,2, . . . , n }", so where is the element (abc)=(ac)(ab)
In this paper1 on the history of functional analysis, the author mentions the following example of an infinite system of linear equations in an infinite number of variables $c_i = A_{ij} x_j$:
\begin{align*}
\begin{array}{ccccccccc}
1 & = & x_1 & + & x_2 & + & x_3 & + & \dots \\
1 & = & & &...
Every element of Sn can be generated by the (n-1) transpositions (12)(13)(14)...(1n). I want to ask what was the intuition behind this? Or was it just observation that (1i)(1j)(1i)=(ij). I hope there was some intuition, because most proofs rely on some tricky non-trivial identity, that i cannot "observe".
@MATHASKER If it helps you should draw a graph to see what is going on. You can see then that as x gets closer and closer to 1/3, y gets closer and closer to -1, so the required limit is -1. It's not important what y is when x is 1/3 itself in this case.
because I have in mind that the bottom is the output, and that permutations act from left to right. so the leftmost transposition would be the last one performed
@MATHASKER Once you understand something, everything will follow. The problem with many students now is that they practise endelessly without understanding, wasting all their time.
@Jasper thats partly a teaching issue, I think: a good teacher designs their course so as to make it hard for students to blindly pattern-match
There is pattern matching in math, of course, but of a different sort: a novice tried to match problems to patterns of symbols, whereas an expert matches problems to patterns of reasoning
Reminds me of a distinction I made in an undergrad paper which I was always proud of: the difference between being literal, literate, and literary
What if someone asks what the power is in this expression $5^3$ and they say it's 3. Would that be incorrect as the entire expression is actually the power?
@NV-US: What you'll want to understand at some point is what happens when you do what's called conjugation. When you do $\sigma\tau\sigma^{-1}$ it's like a change of coordinates. So if $\tau = (12)$ and $\sigma$ sends $1 to i$ and $2 to j$, then $\sigma\tau\sigma^{-1} = (ij)$.
@CausingUnderflowsEverywhere: Imprecise language, again. $3$ is the exponent.
Okay I'll wait for the day I'll work with someone from the US and they will be pale when I say to the exponent and it doesn't make sense. Im sure it's not a big deal I'll just have a striking accent to do with how I pronounce (exponent vs power)
Have you proved in class that if $a$ has order $n$, then it follows that if $a^k=e$, it MUST be that $n|k$? ... That's where the contradiction proof goes.
Well, if you've proved that fact once and for all, you do not need to reprove it. By definition (?), the lcm is the smallest positive number that is divisible by both. So there's nothing more to do.
@TedShifrin our prof said that spivak would be his preffered text but he didn't want us to have to buy two textbooks so only assigns readings from James Stewart :(
My current reading list is: Calculus of One Variable by Keith E. Hirst. Calculus With Applications by Peter D. Lax and Maria Shea Terrell. Calculus with Vectors Jay S. Treiman. Spivak's Calculus James Stewart Calc
@Semiclassical Dang it, I tried to apply your Euler-Maclaurin approach to a seemingly well behaved integral, that is,$$\int_0^{2\pi} \ln\left(1+\frac{\sin(2x)}{2\sin(x)} +\frac{\sin(3x)}{3\sin(3)} \right)~\mathrm dx$$Though it doesn't seem to work. I assume its not analytic lol, which would suck.
In this paper1 on the history of functional analysis, the author mentions the following example of an infinite system of linear equations in an infinite number of variables $c_i = A_{ij} x_j$:
\begin{align*}
\begin{array}{ccccccccc}
1 & = & x_1 & + & x_2 & + & x_3 & + & \dots \\
1 & = & & &...
