@Jakobian @Koro I haven't read all the discussion while I was gone for hours and hours. But I propose the function $f(x) = \begin{cases} n, & x=1/n \\ x, &\text{otherwise}\end{cases}$. Did I misread the given hypothesis? It sure seems to me the limit in the hypothesis is $0$ but $f$ has no limit at $0$.
Hmm, I read the hypothesis as $\lim\limits_{x\to 0} f(x)\big(\frac1x-\left[\frac1x\right]\big) = 0$.
Any hint to find an upper bound for $\dfrac{x+y}{(y)(y^2-1)(\ln(x+y))}$ for $y \ge 2$ and $x \in \Bbb N$? What I have so far is that $y^2-1 > y$ so $\dfrac{1}{y^2-1}< \dfrac{1}{y}$ and hence $\dfrac{x+y}{(y)(y^2-1)(\ln(x+y))} < \dfrac{x+y}{y^2(\ln(x+y))}$. Also I've noticed that $\dfrac{x+y}{\ln(x+y)}>1$
I suspect an upper bound would be $\dfrac{1}{y^2}$ but $\dfrac{x+y}{\ln(x+y)}>1$ confuses me, unless its false.
It's a thing on the Art of Problem Solving website. They invite high schoolers and college students to work on open problems as a group. Anyone can join, so I thought it looked fun, but the message boards don't have much activity.
This year's problems are on "Factorizations in Additive Structures".
there's a lot of essentially BS awards and stuff given in the legal profession. "young attorneys to watch" etc. which sounds pervy to me.
i was asked by someone i knew if i was interested in being nominated for it and had to break the news to them that i am not, by their definition, a young attorney to watch.
i just seem young because of my up-to-the-minute cultural references.
tim robinson's series "detroiters" had a good scene where they film a local ad premised on the idea that tim, who is approximately my age, is popular with high schoolers. they go there.
im trying to solve these coupled differential equations - I know $\omega$ is in the answer, but when I calculate it it disappears when taking the determinant. Can anybody see what I’m doing wrong??
Does this sketch for the proof of Weierstrass' Approximation Proof make sense? BTW, the outline of the proof comes from my textbook: Let $f(x)$ be a continuous function in $[a, b]$; we want to show for every position $\epsilon$, there exists a polynomial $P(x)$ such that $|f(x) - P(x)| < \epsilon$ for all values of $x \in [a, b]$.
My textbook says to first apply the substitution $t = a + (b-a)x$ to transform the function $f(x)$ into a function $\phi(t)$, where $t \in [0, 1]$. Now, we approximate $\phi(t)$ using a piecewise linear approximation $\psi(t)$ to within $\epsilon/2$. We represent…
i'm trying to think about it. a is obviously the value at 0, no mystery about that.
you can form little hat functions ^ using functions of the given type, and then i guess you need to think about what happens when you add those together.
maybe there's some clear way of seeing that you get all piecewise linear functions from this procedure, immediately. it seems icky to me.
Any elegant way to show that $\dfrac{x+y}{y\ln(x+y)} <1$? I proved it with the mean value theorem but was looking for something without involving calculus.
my problem with $a + bt + \sum c_i |t-t_0|$ is that u can write it as $a + bt + |t-t_0|\sum c_i = a + bt + k|t-t_0|$, where $k = \sum c_i$, and i don't see how u can all the pieces of the linear approximation with this form
the general spirit of many proofs of the WAT is to establish it for some class of functions that you 'know' and then show that the properties of this class of functions extends to everything.
oh, is it supposed to be the same t_0 for all of them? that doesn't seem right.
david, the usual route if you do use korovkin is to show that the simple hypotheses of his result apply to the bernstein approximants. the bernstein approach is a good approach.
copper this is asking me for a credit card number. why are you hawking pornography on a math chat?
encyclopediaofmath.org/wiki/Korovkin_theorems provides background on the korovkin approach. roughly, certain approximation schemes have to converge uniformly for all continuous functions on a bounded interval if they converge uniformly for 1, x, and x^2.
it led to a bunch of fairly unimaginative ripoffs of the general theorem but is still surprising to most people.
because the bernstein scheme is within the realm of his theorem, you can prove WAT by proving the three bernstein approximants, of 1, x, and x^2, converge to what they're supposed to.
which is like two sum formulas. it's nothing.
i'd love to know what led korovkin to his result. it's fascinating.
not mind blown levels of fascinating, just, what led you there? he is gone now and we cannot ask. as far as i know he did not write about it in english.
so maybe the question is asking you to understand when the denominators of those terms in the 'rref' (which won't be if those things are zero) are zero
you sometimes get nonsense with a symbolic rref. the symbolic calculator has to choose what to divide by, and it can always do that if assumes the things it's dividing by are nonzero.
@CroCo i am afraid i do not understand the control conditions under which that happens. the conditions for gimbal lock are not present here, that's about my limit
@copper.hat I think it is a mathematical question here. The book says find the Jacobian matrix and determine when the rank doesn't reach its maximal value. From my calculation, it seems this occurs when sin(a2) != 0. But physically, there is another possibility when the robot is stretched.
it may be that other conditions arise that are not present in the symbolic form of the "rref" if the symbolic package used to define that knows how to simplify expressions, which most do.
i had a fun text exchange with one of my older friends today. we were making jokes premised on the human anatomy, just the stupidest stuff we could come up with. we met at age 15 which explains all of this.
croco the issue is that with symbolic matrices, sometimes the case analysis required to compute the rref is more complicated than finding the rank in some other way. i don't know that we're there here, but that is a background principle to keep in mind.
ted's suggestion makes sense to me, i just don't want to carry it to completion.
i'm lazy.
if one of s1 or c1 is nonzero you can turn one of the column 2 entries into a pivot by dividing by the relevant expression. then the column 3 entries become slightly more complicated expressions and you wonder if their numerators are zero.
you say simple. the configuration space of an arbitrary linkage, i think even with just one arm, is enough to generate all compact surfaces. so as simple as the classification of surfaces. which is not simple.
laughed out loud. reminded me of a time in grad school where we showed up at a BYOB party with 40s to learn that, uh, it wasn't the kind of party where you showed up with something in a brown wrapper.
we just committed to it. almost 20 years ago.
if they'd just said BYOB means bring someone else's drink, ok, we could have done that. i brought my own.
my wife actually knew me then. it's a miracle we're still together.
we didn't have google maps. i didn't know i was bringing malt liquor to a house with a nice deck.
it's funny how beer has different reputations in different areas. like stella here has some kind of cachet but it is a code word for lower class in other places.
and american budweiser is somehow not trash in some parts of the world.
i don't remember what i brought to the party but i do remember paying less than $2.50 for it.
the usual criticism of wine is, it's all the same, and that's not true at all. you can definitely tell the difference. i'm not as sure about beer. some of my favorites are very cheap.
i have trouble distinguishing whites. i can get oaked vs. non and sweet vs. dry but beyond that it is difficult. it doesn't help that it's served slightly under what we in southern CA would consider room temperature.
and a lot of white wines are just, ick, at room temp.
i don't want to tell them apart if they taste like that.
Can anyone suggest to me an informal definition of topology. I'm struggling to understand the topic. Also, any book recommendations for absolute beginners.
That is where the function $\frac{k^2+k}2$ comes in
so $k-\frac{r(k)^2+r(k)}2$ is $0$ at each $k$ that is in $\{0,0+1,0+1+2,0+1+2+3,\dots\}$
and increases by $1$ in between
So $k-\frac{r(k)^2+r(k)}2$ is $\overbrace{\ \quad0\ \quad}^{r(k)=0},\overbrace{\quad0,1\quad}^{r(k)=1},\overbrace{\ \ 0,1,2\ \ }^{r(k)=2},\overbrace{0,1,2,3}^{r(k)=3},\dots$
....Can anyone suggest to me an informal definition of topology. I'm struggling to understand the topic. Also, any book recommendations for absolute beginners....
@robjohn this without the zeroes does work as an example that I was looking for. I'm still trying to understand how to write its nth term. I'll get back for sure.
