my daughter eats red meat with about the same frequency that i do, maybe one meal per month. but freely eats fish, poultry, and the aforementioned cheese sticks.
you should put screech on instagram. cat instagram is a huge thing.
an enormous number of my cat's followers are in japan. i don't know why this is. #2 nation is turkey.
black cat instagram may be different from general housecat instagram.
i'm reminiscing with a now-former coworker who left. she claims that on her first day i spent the whole day in her office talking about inane distracting BS instead of letting her work.
i disagree, it was at most half of the day. i have the parking lot records to prove it.
I'm going through that phase right now. The last one I read was Halmos'. I did a bit of a deep dive on Ramanujan and Hardy for a school assignment, so I didn't end up reading the actual biography.
@copper.hat Norbert Weiner? I'm not sure if I've heard of him.
one of my advisor's prized possessions was a book, authored by NW, and signed by NW after some event somewhere. the pure joy he had in having it was palpable.
his proof of the PNT is the only one that makes sense to me.
there are a lot of proofs out there. a lot of what wiener did with the fourier transform has just so become part of what people use the fourier transform to do, that it's second nature now.
Interesting, well I know that $\displaystyle \pi(x)\underset{+\infty}{\sim}\frac{x}{\log x}\iff p_{n}\underset{+\infty}{\sim}n \log(n)$. I read a bit about this a while back in a number theory book. However, my memory is not good.
It is an interesting problem about NT.
The series $\displaystyle \sum_{n=1}^{+\infty} \frac{\mu(n)}{n^{s}}$ converges for all $s>\frac{1}{2}$. Show that this statement implies the Riemann Hypothesis.
what is the integrating factor for: $\frac{\mathrm{d}y}{\mathrm{d}y}= \frac{2xy^2+y}{x-2y^3}$?
I can solve this ODE and here's my solution: the simplification leads to $2xy^2\mathrm{d}x+y^2 \mathrm{d}(\frac xy)=0$ and integrating both sides gives an implicit solution.
I think that the given ODE can't be converted to an exact differential equation.
@Koro I suppose there is a typo and the correct problem is $dy/dx=\frac{2xy^{2}+y}{x-2y^{3}}$ that is not exact equation as you said. But the integrate factor is $\mu(y)=\frac{1}{y^{2}}$ then we have a exact equation. Finally we get the solution: $x^{2}+y^{2}+\frac{x}{y}=C$ with $C$ a constant.
$p(D)$ is a polynomial operator with constant coefficients. For example if we have $y''+ay'+by=g(x)$ then $p(\frac {\mathrm{d}}{\mathrm{d}x})y=g(x)$, where $p(z)=z^2+az+b$. :)
what if $p(b)=0$? Then, it can be shown that $y_p= \frac x{p'(b)} e^{bx}$, if $p'(b)\ne 0$
Good night, copper!
And if $p'(b)$ is also zero, then $y_p=\frac{x^2}{p''(b)}e^{bx}$, if $fp''(b)\ne 0$.
Usually, we find particular solution using variation of parameters or indetermined coefficients method. But operator method gives a very short solution. :)
Suppose that the sequence $(x_n)$ does not converge to L. This means that there is a $d>0$ and a subsequence $(x_{n_k})$ such that $|x_{n_k}-L|\ge d$ for every $k\in \mathbb N$.
Now, note that either $\{n_k:k\in \mathbb N\}\cap A$ or $\{n_k:k\in \mathbb N\}\cap B$ is an infinite set.
WLOG. let the former be an infinite set. So we can make a subsequence of $(x_{n_k})$ out of those indices. Such subsequence is also a subsequence of $(x_a), a\in A$. :)
Hence the subsequence must converge to $L$, which contradicts out assumption for some $k$.
@Koro other demo I made if you are interested :P For $\epsilon > 0$ by hyphotesis there are $m_1,m_2 \in \Bbb N$ such that $|x_a - L| <\epsilon$ for all $n\ge m_1$ and $|x_b - L| <\epsilon$ for all $n \ge m_2$. Now, every element of $\{x_n\}$ is either an element of $\{x_a\}$ or $\{x_b\}$. So, given $n\ge \max(m_1,m_2)$ we have $|x_n-L|<\epsilon$
I wonder if the assumption $A\cap B = \emptyset$ is needed.
I think it suffices that $A\cap B = C$ where $C$ is finite.
Consider 2 eigenvectors v1 and v2 of a matrix M. v1 generates the eigenspace A and v2 generates the eigenspace B. Are v1 and v2 orthogonal? Or just if A is symmetrical?
@Odestheory12 The assumption that $A\cap B=\emptyset$ is not needed. Such is the beauty of taking limits, ‘eventuality’ or should I say ‘ultimate behaviour of the sequence’ matters and you can ignore first 10 billion or 20 trillion etc. terms of a sequence and still the limit of the sequence will remain the same.
The assumption that $A\cap B$ is a finite set works just fine.
@Koro I don't see why $A\cap B$ is of any consequence. It seems to me that as long as $A\cup B=\mathbb{N}$, and that $\lim\limits_{j\in A}x_j=\lim\limits_{j\in B}x_j=L$, we have $\lim\limits_{j\in\mathbb{N}}x_j=L$
Hello everyone. I want to know from specialists in polynomials - is there any progress in solving a system of equations consisting of Vieta formulas? For instance:
Solution for x1, x2 and x3.
A very interesting topic, but as far as I understand, there is not much progress in this area yet?
It's appendix material for a math phys / integrability project but it's got a lot of basic calculus in it like cesaro summation or generalisations of partial fractions
dtn: that seems equivalent to finding the three roots of a cubic with coefficients given by d/a, c/a, b/a, 1. lots of work done on polynomial root finding in general. maybe some of it is useful, depending on your desired interpretation of what it means to 'solve' a polynomial.
my daughter's very confused right now. our phones blew up with a tsunami warning, i guess because of that thing in the south pacific. no evacuations are ordered, so it's basically a warning for people who might literally be on the beach to move their sand castles.
our daughter heard some mild discussion of this and saw a video clip of what was going on in the south pacific and now thinks that it's going to get very wet here.
it started raining, which didn't help. "see? i told you"
i agree that as stated, it is confusing. what is AA?
this seems to be setting up some kind of thing where only the cardinalities of A and B are relevant, but the desired conclusion about h is confusing as stated.
i was expecting the conclusion to be that there is a bijection between A and B. may involve the axiom of choice if A and B are infinite sets.
oh, well that's a general thing. things named with different letters should not be assumed to be different (or required to be different) unless explicitly stated.
different naming usually just reflects potentially different roles or properties in an argument, but not actual distinction.
sometimes textbooks do mess this up, saying stuff like "let a, b, and c be integers" and from context it's clear that they intend a, b, and c to be distinct integers. but without context and in generality, labels are just labels.
VLC: this gets back to my suggestion that the exercise is broken. if f and h aren't required to be different, soupless's suggestion works. take h = f.
i don't know why an exercise would ask "if A and B, then A." that led me down the path of guessing what the exercise might have been intended to do.
VLC: it does. are A and B finite? in the infinite setting this will involve some version of the axiom of choice, so if that is unfamiliar to you, it may be a signal that something else was intended.
i think. all of this has the air of something where i post a comment on main and then asaf karagila responds with three comments curb-stomping my comment into the ground.
soupless: I think that the set of all such functions is defined to have cardinality $1$. I think that's done so that cardinal arithmetic $|B|^{|A|}$ holds with the definition that $0^0=1$.
wait, what is that cardinal arithmetic? i can't keep up. never mind, asking questions only that i can understand for now. thanks, Koro and leslie townes
my daughter cracked open a package of old masks that we don't use anymore because they aren't the N95/KN95/etc standard. she applied four masks to her stuffed animal and two masks to me.
There is a No Right Turn sign going into a parking structure at UCLA, positioned so that you see it right in front of you as you are walking into the structure. Unfortunately, you need to turn right to walk into the structure. When I had my son with me, I would make 3 left turns to walk in.
I just noticed, since the set of rationals is a field, and it is a subset of the set of reals, this forces the set of reals to be a field. Is this correct?
if $y=x\cos x$ is a solution of the ODE $y_n+a_1y_{n-1}+...+a_n y=0$ where $a_i$'s are real constants and $y_j$ denotes jth derivative of $y$ w.r.t. x. Then what is the minimum possible value of $n$?
We solve such ODEs by substituting $y=e^{rx}$. This substitution gives us a polynomial equation in $r$. Repetition of roots is same as multiplication to $e^{rx}$ by $x$.
depending on your ODE background theorems i wonder if you can get only a candidate for a minimum value, and then maybe have to check by hand that the smaller ones don't work.
