"Writing is inhibiting. Sighing, I sit, scribbling in ink this pidgin script. I sing with nihilistic witticism, disciplining signs with trifling gimmicks --impish hijinks which highlight stick sigils. Isn't it glib? Isn't it chic? I fit childish insights within rigids limits, writing schtick which might instill priggish misgivings in critics blind with hindsight. I dismiss nit-picking criticism which flirts with philistinism."
Let $f:[0,1]\to\Bbb R$ be a function with $f(0)=0$ and $0<f(y)/y\leq f(x)/x$ for any $0<x<y\leq 1$. Then is it true that $\int_0^1f(x)dx<\frac{1}{2}f(1)$?
So far for polynomial $f$, I think I can prove it because $f(0) =0$ implies $f(x) = xg(x)$ for some $g$. I'm not sure I'm able to write for just function.
Let $f:[0,1]\to\Bbb R$ be a continuous function with $f(0)=0$ and $0<f(y)/y< f(x)/x$ for any $0<x<y\leq 1$. Then is it true that $\int_0^1f(x)dx>\frac{1}{2}f(1)$?
For polynomial $f$, $f(0) = 0$ implies $f(x) = xg(x)$ for some polynomial $g$. But for continuous $f$, I'm not sure I'm able to write in that way.
such a g might not be continuous at 0, or even bounded near 0. e.g. f(x) = sqrt(x)
thinking geometrically it looks like the hypothesis implies that f is strictly 'concave down' and hence that its graph ought to lie strictly above any secant line to the graph. if that's true, then f(1)/2, being the trapezoidal rule approximation of the integral corresponding to partition [0,1], ought to be an underestimate of the integral
Currently I am studying singular value decompositions. I learnt something about its use in transferring signals over long distances but don't remember that now.
So I'll soon see what SVD is geometrically or what its applications are.
No Ted, I mean I don't know yet. Axler does SVD a bit differently (different than Strang's). There should be similarity between the two but I have not yet tried to see the similarity). If T is an operator on V then there exist orthonormal basis $e_i$'s and $f_i$'s such that $Tv= s_1\langle v,e_1\rangle f_1+...s_n\langle v,e_n\rangle f_n$ for every $v\in V$ and $s_i$'s are singular values of $T$.
i think it's to complicate the project of attacking your server, on the assumption that most people are right handed and they have closed the distance. i'm left-handed, though.
Okay so this is where it is mentioned, i do not understand what is happening here. I am reading the proof of a the "Cauchy Riemann differential equations" theorem. let me type it:
$ a,h \in \mathbb{C}$ with the identification $ \mathbb{C} \simeq \mathbb{R^2}$ the complex multiplication with $ a $ represents a real linear function $ A :\mathbb{R^2} \rightarrow \mathbb{R^2}$
But $ a * h = a_1h_1 - a_2h_2 + i ( h_1a_2+h_2a_1) $ how is this real?
Hi folks! We can't obtain the closed form solution for the maximum likelihood estimator of the second parameter of the Gamma ditribution, because of the digamma term in there. However we know that gamma distribution is a member of the expon ential family and we know the MLE for it, however from it we would only get the estimation of theta, right? I'm trying to fugure out if it's somehow possible to decompe calculate the MLE for gamma using that fact
It has been a while since i did Linear algebra. so i am out of touch. How can a two dimensional vector space have one spanning vector as basis vector? For me it is not clear why we use the describtion "real vector space" even though the element i is not in the real numbers... Surely $ R^2 $ is real, however atleast the basis vectors consists of elements that are in the real numbers..
@MadSpaces A vector space V over field F is called a real vector space if $F=R$ and complex vector space if F=C, the set of all complex numbers.
If you take $V=\mathbb C$ and $F=\mathbb R$ then vector space definition is satisfied for V with the usual "complex" addition and usual scalar multiplication where the scalars come from $\mathbb R$.
That is if $z_1,z_2\in \mathbb C$, then for any scalars $a,b\in \mathbb R$ the vector $az_1+bz_2\in \mathbb C$ etc.
Let $\phi:\mathbb C\to \mathbb R^2$ be the map $z\mapsto (Re(z), Im(z))$
Then this map is an linear isomorphism
The preimage of the canonical basis (1,0), (0,1) of $\mathbb R^2$ is $1, i$, which is a basis of $\mathbb C$ as a real vector space
the action of multiplication by $a\in \mathbb C$ through $\phi$ is precisely $\phi(a\phi^{-1}(h))$ where $h=(h_1, h_2)$. You can compute that $\phi(a\phi^{-1}(h)) = (a_1h_1 - a_2h_2 , h_1a_2+h_2a_1)$
where $a = a_1 + ia_2$
Now this formalism is very heavy, and to avoid it we often identify $z$ and $\phi(z)$, which is what robjohn did when he said that $i=(0,1)$
since $\sigma(\sqrt[3](2))^3= 2$ and so $\sigma(\sqrt[3]{2})=\sqrt[3]{2},\sqrt[3]{2}\rho, \sqrt[3]{2}\rho^2$, but the last two are not in the the field extension
@mathsresearcher Because it is faster to say "Find the structure" than it is to say "Find an isomorphism to a well-known group". And because mathematicians often use language in a specialized manner, and learning that specialized language is a part of learning mathematics.
(Of course, every profession has specialized language, and learning a profession includes learning that language.)
Oh, jeebus. I have a student who just sent me an email explaining to me how I should teach the class next time around. He is frustrated that I gave them too much homework in which I asked them to compute volumes using cross-sections and cylindrical shells.
Completely ignoring the fact that I have repeatedly indicated that homework is optional, and that I will give credit to any effort which is turned in (because the exams are where you are going to succeed or fail).
But, hey, 19 year old student, I am sure that you have much more pedagogical experience than I, and would run the class perfectly. :/
Hey everyone, I believe I may have solved a question that I posted. Is it better to edit the original post, or reply to my own question with an "answer"? My solution was inspired by someone who initially answered my question but then deleted their post so I can't give credit :(
@TedShifrin i had an analysis quiz last week, and i've successfully proved that $\lim_{(x,y)\to(0,0)} \sin(xy)(x+y)=0$ with the simple fact that if $d((x,y)(0,0))<\delta$, then $x<\delta$ and $y<\delta$.
remember you taught me that fact!
@ChrisLangfield i believe an answered question is better than one without.
you can mention how you solved it and your inspiration.
oh, how nice. how is it going, do you enjoy it? :)
@LeakyNun i now remembered something else. i think i've started to be interested in proof assistants because of you. i remember you sent a formal proof of a simple proposition, here.
If $G$ is any open set and $\gamma : [a,b] \to G$ is any rectifiable curve, why does it follow that $dist(\text{Im } \gamma, \partial G) > 0$? I know it has something to do with compactness, but I can't see the proof.
rectifiable curves are continuous by definition, no? so (as continuous image of compact set is compact, and compact = closed and bounded) im(gamma) is a closed subset of R^2. if that distance is 0, then, there's actually a point in im(gamma) on the boundary of G.
Yes, they are continuous, so their image is compact. That's the part I'm having trouble seeing. Why does having zero distance mean they have a common point.
user: for each n you can pick a point p_n on im(gamma) with d(p_n, partial G) < 1/n. by compactness p_n has a subsequence convergent to some p in the image of gamma. i think you can show d(p, partial G) = 0. but this subsequence is also a convergent sequence in the closed set G union partial G = the closure of G, so it's in the closure of G. p can't be in G (ruled out by d(p, partial G) = 0), so p is in partial G.
i may be circling around the same idea in one or two different guises there.
Thanks again for the guidance the other night, Ted. Your course notes were really helpful in understanding what was going on in the Euler-Lagrange. Everywhere else I was looking was missing just that little bit
@Under It's like asking whether the fundamental theorem of calculus $\int_a^b f'(x)\,dx = ...$ applies if you integrate something other than a derivative.
The more I read through math literature, the more I feel like I was never introduced to some key foundational concepts and their notations, and now there's so much behind the points I'm at that I don't know where to even go back and review.
It's bad enough that the book I was reading on Diff Geo years had me contemplating a religious war with the author over whether or not 0 was a natural number.
That's something I thought was weird when I started my proofs class. I screwed up a few problems because I assumed $0 \in \mathbb N$, until my professor clarified.
One professor was very much in the $0 \in \mathbb N$ camp that he told the whole class "This class will not be any easier if you start working with fewer numbers. 0 is natural and if you disagree, tough ****."
I lived down there during a drought for like half a year. It got so hot that it melted my shoes to the pavement. I was supposed to borrow a friend's car while I was down there, and the heatwave melted the engine belt apart.
The core problem I had with Houston was that it just felt like people living there were either doing very well for themselves or struggling to just get by. Very high income disparity and it showed in how the city was laid out.
I don't like the idea of a number being natural-fluid. Makes definitions and theorems situational and subjective instead of objective claims and conclusions.
I vaguely remember that someone here was working on something-like-a-field in which division by the zero element was well-defined. Did anything come from that?
