you'll be about halfway down the stairs, they're split up into two flights, and suddenly a black flash out of nowhere will bite you in the leg and run away.
i went to a restaurant the other day for my anniversary and the fiercest looking pit bull you have ever seen ran up to me and licked my hand. i have a soft spot for pits.
we are undoing all of this with my daughter, she is very open about everything.
my grandmother said something like, nobody is alive forever, and you can't be bothered to feel anything about anything that happens on earth because none of it matters. and then some catholic stuff i forgot.
i remember some of it at christmas and easter.
my daughter's very open about everything and she yells at me if i don't reciprocate.
she will be the emotionally healthiest member of the family
$\pi(x) = \pi(\sqrt{x}) - 1 + \sum_{k = 0}^{\pi(\sqrt{x})} \sum_{\ell = 0}^{\pi(\sqrt{x})}(-1)^k{\pi(\sqrt{x}) \choose \ell }{\ell \choose k } \tag{5}$ Can we do something with those binomial coefficients in particular?
Here's a problem that I'm trying: **Problem**: Suppose $f$ is continuous and periodic on the reals with period 1. Prove that if $\alpha\in[0,1]$ is an irrational number, then
Hint: First consider $f(t) = \exp(2\pi ikt)$ where k is an integer.
--- **Attempt**: I have verified the equality for $\exp(2\pi ikt)$. I know (Theorem 4.25, Big Rudin) that for any continuous function $f$ and any $\epsilon > 0$, we can find a trigonometric polynomial $P$ such that $|f(t) - P(t)| < \epsilon$ for all $t\in\mathbb R$, whe…
The fact that this question: math.stackexchange.com/questions/482599/… was closed got me thinking: Is there a need for a "mathematical proofs" stackexchange? I think the stackexchange format would lend itself nicely to a proof collection in contrast to proofwiki.org/wiki/Main_Page you would have a natural "statement" (the question) and "proofs" the answers, and people can comment on proofs for clarifications, corrections, etc.
is it obvious that this P will satisfy the above inequality? any tips for going about showing it I dont remember strategies for showing this matrix inequalities
@robjohn Thanks. It is a bit ambiguous which variable is meant with $o$, since $cx$ is also a variable for a constant $c$. My notation here first assumed $cx$ was the variable and then $x$, which is confusing. My short proof here assumed $cx$ was the variable, but then the limit was taken with respect to both $cx$ and $x$, which is not very consistent.
@leslietownes: I've finally found a reference that uses the CORRECT™ definition of strictly decreasing: Terrance Tao's Analysis I. I've updated my answer.
@anakhro: I was being half-facetious when I said that. It transpires that a number of authors (including Spivak and Rudin) only define "monotonocity" on intervals. By that definition, whether the function $f(x)=1/x$ is strictly decreasing is undefined. You can read my answer for more details
@anakhro: I think it's because some of the main theorems in this topic are about functions which are monotonic on an interval. Still, that's no reason to make the definition itself so restrictive
@anakhro: I guess I see people often being pushed towards the "classics" like Spivak or Rudin, but it's good to know that Abbott's being given some recognition
Heh, doesn't stuff like Stewart's calculus have $\varepsilon-\delta$ arguments for limits and stuff for the sake of including the formal stuff in introductions?
@anakhro: Yeah it does. The course I first did at school had nothing of the sort, which is quite sad really. It was a highly calculation-based course. That made no sense—if you wanted to become a mathematician, then it didn't have anywhere near enough rigour; if you wanted to become an engineer, then it overlooked the fact that engineers often don't do calculations by hand—there's this thing called Wolfram Alpha for that
@anakhro: On second thoughts, I agree with you that rigorous calculus shouldn't be taught at school. But there's not enough emphasis on building understanding or intuition either
I agree there. So then the question is how do you build understanding/intuition without forcing students to take a course they don't want/need, meanwhile being engaging and accessible. Should be a Millennium prize problem.
he let the better students teach the class, which was actually fairly effective, but maybe not the best use of taxpayer dollars.
we also had a german language instructor whose main qualification was a german surname. she had probably taken it in high school. hopeless.
sometimes the lack of resources was good. the honors fourth year spanish class was the freshman spanish for spanish speakers class (our high school had a lot of resources for new arrivals from mexico). much more effective than textbook based instruction.
The way people drive around here, I would believe that the requirement for a driver's ed teacher be that they know the difference between a car and a dog.
there was something else they cut that used to be a mainstay. shop class. woodworking and the like. a vocational high school opened and they just moved it over there.
some of the kids in that class were really talented. professional level stuff. it was probably the only thing about school that interested them.
apparently my high school has a computer science class now. i wonder what that's like.
certainly nobody was qualified to teach that in my day. they had one IT guy and he 'outsourced' most of his work to students. i spent several days installing ethernet cards in a bunch of PCs on the day that the internet came. i should have been compensated.
each department had its own set of computers, moving the equipment was such a pain in the ass. they should have just had one computer lab. nobody did anything on computers then.
