I'm using this program desmos.com/calculator to draw graphs. I need to draw a sin function. However I want that the range is between $-1/2$ and $1/2$ and not between $-1$ and $1$
Hi. Let $M$ be an $N\times N$ correlation matrix (symmetric, psd, ones on the diagonal). Also let $w_i$ for $i=1,\dots,N$ be nonnegative weights summing up to 2, and $R_i$ denote the sum of the $i$-th row of $M$. Then I wish to maximize $|\sum_i (N w_i-1)R_i|$, and perhaps it's optimal to take $w_1=2$ and $w_i=0$ for the rest. However, the problem is choosing valid $R_i$ (i.e., maintaining psd assumption). Any ideas? Or is there perhaps a better approach?
@Semiclassical, true. I only fix the assumptions on $M$ and $w_i$. So basically I'm looking for the most extreme $M$ along with the weights, in the sense of what I want to maximize
@ÍgjøgnumMeg Muss es unbedingt auf deutsch sein? es gibt das auch auf Englisch: amazon.co.uk/dp/0486623424 (Der Inhalt ist derselbe, auch wenn der Titel irgendwie anders ist)
Is it true that a sequence of functions which converges uniformly on countably many sets also converges uniformly on the union? Or is this only true for finitely many sets?
It's not true. Every monotonous sequence of continuous functions which converges to a continuous function converges uniformly on compact subsets (by Dini's theorem), but the real line may be written as a countable union of compact subsets, so it suffices to give a monotonous sequence which converges pointwise but not uniformly to a continuous function. Consider $f_n(x)=\frac{|x|}{n}$ (defined on $\Bbb R$)
this converges pointwise and monotonously to the zero function, but not uniformly
any bibtex experts around? I'm a bit out of my depth
I'm trying to reproduce a certain bibliography item that has multiple papers (all by the same authors) in one citation.
@Julius Right, that's what I thought. I can't comment much atm, but I'd point out that the row sums can be conveniently expressed as $R=Mu$ where $u$ is a vector of ones.
One point of confusion, as well: What is $Nw_i-1$ supposed to be? $w_i$ is evidently a scalar, but then $Nw_i$ is a matrix and so $Nw_i-1$ doesn't make much sense. (It would if $1$ is really the identity matrix, but I'd rather not presume.)
Suppose that $f : X \to Y$ is a continuous function, and $\{x_n\} \subseteq X$ is a sequence such that $\{f(x_n)\}$ converges. Does this imply $x_n$ converges? I think the answer is no: take $f : (0,\infty) \to (0,\infty)$ defined by $f(x) = 1/x$, and $x_n = n$. Then $f(x_n) = 1/n \to 0$ yet $n \to \infty$.
@Semiclassical, $N$ is also a scalar as the matrix $M$ is $N\times N$. And thanks for $R=Mu$, perhaps it will be useful. Now I'm thinking that maybe $M=uu'$ will be optimal as $|\sum_i (N w_i-1)R_i|$ is not only about asymmetry (i.e., one big row with $w_1=2$ and other rows as small as possible) but also about magnitude (which gets hurt if I have both positive and negative entries and want to preserve the psd property)
Continuous injective is not enough. Consider the function $f:[0,1) \to S^1$ $f(x)=e^{2\pi i x}$ take the sequence $1-\frac{1}{n}$, the image of this converges, but the sequence itself doesn't
Let $n \in \Bbb N_{>0}$ and let $(a_i)_{i=1}^n$ be a sequence of positive integers.
Then, must $\displaystyle \frac {\left( \sum_{i=1}^n a_i \right)!} {n! \prod_{i=1}^n a_i!}$ be an integer?
I know that without $n!$ it is just the multinomial coefficient, so it must be an integer. I also know...
@MatheinBoulomenos I have a union $X = X_1 \cup X_2$ and know that $f_n|_{X_1} \to f|_{X_1}$ and $f_n|_{X_2} \to f|_{X_2}$ uniformly. Then $f_n \to f$ uniformly.
Hey guys, with really strong letters of rec and a GPA > 3.7 as a community college student, do I stand any chance at being accepted to somewhere like Berkeley or UCLA? And if not, what kind of caliber of schools am I looking at?
@Dattier there are less than $500$ primes in $[1,3000]$, (430 to be exact) and only one value divisible by no prime, so for some prime $p$, $p$ divides both $a_i$ and $a_j$ for $a_i \neq a_j$
Sure! Haha. If $K\supset \mathbb{Q}$ is an algebraic extension, then apparently there is some $\alpha\in K\setminus \mathbb{Z}$ which is integral over $\mathbb{Z}$. Having that every element is algebraic isn't enough, sadly. When we try to get a common denominator to make the given equation in $\mathbb{Z}$, we affect the leading coefficient.
If $K \supset \Bbb Q$ is a proper algebraic extension, then we can choose $a\in K \setminus \Bbb Q$. $a$ is algebraic over $\Bbb Q$, so we have a relation $a^n+q_{n-1}a^{n-1}+\dots+q_0=0$, where the $q_i$ are rational numbers. Choose $k\neq 0$ in $\Bbb Z$ such that $k\cdot q_i \in \Bbb Z$ (take a common denominator), then we multiply the whole thing by $k^n$. We get $(ka)^n+kq_{n-1}(ka)^{n-1}+\dots+k^nq_0=0$, thus $ka$ is integral over $\Bbb Z$
$g_n=\begin{cases}f_k(x) & \text{if }n=0\\ \dfrac{c^n}{k(k+1)\cdots(k+n-1)}f_{k+n}(x) & \text{otherwise.}\end{cases}$ , where $c$ is a natural number.
Claim 2: For each $x\in \mathbb R$, $\lim_{k\to+\infty}f_k(x)=1.$
*Now, it follows from claim 2 that each $g_n$ is greater than $0$ if $n$ is large enough and that the sequence of all $g_n$'s converges to $0$.*
How do we get those conclusions that are written in italic?
for the first conclusion: c is a natural number,so $c^n\gt 0$, I can say as n gets bigger $f_{k+n}(x)\rightarrow 1 $. But how about ${k(k+1)\cdots(k+n-1)}$ ? Since k is a rational number it can be a negative number too
@CookieToast Great job, Cookie. BTW, taking $y=0$ is quite a bit easier than taking $y=2$ :P Cool technique, eh? If you look through my various problem sets, you'll see that problem appears as a challenge problem a few weeks into the second course.
@EricSilva: The expert I consulted (short of Robert Bryant) doesn't know about the result we were discussing. He thinks it might follow from Cartan-Kähler (which assumes real analyticity, not smoothness). I might have to ask Bryant. Ugh.
My book says "Let $C(a,b)$ be the set of continuous, complex-valued functions defined on the compact interval $[a,b]$ and put $$\langle f,g \rangle = \int_{a}^{b} f(x) \overline{g(x)} dx$$"
Then it tells me to orthogonalize the functions $e^{-x}, xe^{-x}, x^2e^{-x}$ in $C(0, \infty)$
I was chatting with the nurse before my oral surgery today. She said she could never survive any place but San Diego. Too cold/foggy in SF, too hot/humid in FL (where she has lots of relatives).
So I reminded her we only had to worry about drought and fire.
Let $n \in \Bbb N_{>0}$ and let $(a_i)_{i=1}^n$ be a sequence of positive integers.
Then, must $\displaystyle \frac {\left( \sum_{i=1}^n a_i \right)!} {n! \prod_{i=1}^n a_i!}$ be an integer?
I know that without $n!$ it is just the multinomial coefficient, so it must be an integer. I also know...
