mm, presumably you mean a level set of this function. expressing it locally as a graph, in informal terms, means writing one 'variable' (subject to the constraint F = 0, or whatever condition you are using to define the cylinder) as a function of other 'variables.'
in particular examples you can do this with algebra, the theorems show conditions where you can do it even if nice algebra is not available.
i don't think anyone would need general facts of functions defining manifolds to analyze a cylinder, although i understand you took that as an example.
Also the idea of being expressed as a "function" versus not being function. For this I refer to the expression $x^2 + y ^2 = 1$. So around the points $\pm 1$ we can't write this expression as a "function" of the other variable unless we restrict the domain. and in this example taking it to higher dimensions the same comes about to an extent. So since this isn't a "function" on the whole domain what do we call it? Aren't all hypersurfaces a function of their variables?
Thanks for your question.
A subspace is made-up with a set of vectors (called bases) that have to be independent from each other, otherwise, they will be multiple from each other.
Once you have a set of vectors making up a subspace, they should be independent from each other, that is if you take ...
That is what I remember from my linear algebra class :/
it introduces some stuff that is superfluous. that the things defining a subspace be linearly independent (not required, although the vectors in this problem are), and the dot product (always present in R^n but not always required for analysis).
Yes that is what I do understand about the manifold definition @leslietownes . So if we are not able to express this level set as a graph, how do we "express" the level set?
I understand what is needed to check to determine if it is or isn't a manifold. I'm just trying to wrap my head around the concept in relation to the hypersurface, whatever it may be. Because the hypersurface is described in terms of a function
Let me see if describing the picture in my head/ looking at the illustration from Ted's book helps: I got the hypersurface, then some tangent plane to that surface at the point $a$ where we are examining things.....In this neighbourhood I can express this "special set of points (i.e manifold)" as a $C^{1}$ function.
What's going on outside of this neighbourhood? Say I can't express any set of points outside of the neighbourhood as $C^{1}$ functions. How would I be able to describe these parts of my hypersurface?
I think I've articulated it the way I'm seeing the gaps in my understanding...
I'm gonna read the wiki article and see what happens
unit-length closed interval is an interval of length 1?
determines the smallest set of unit-length closed intervals that contains all of the given points!
By sorting points in nonincreasing order, we can take smallest point and build unit interval [xi, xi+1] (unit interval). Following this, we can cover all points!
I don't know what does the previous even is trying to say
The solution as you see is straightforward, so what is the point behind it please?
How this is the smallest set that covers all points if we are basically taking UNIT intervals around each smallest point we pick from the set points? If nothing makes sense to you, I will exit the room :/
dc3 i think it's fairly common for the functions you get from the implicit function theorem not to be globally defined and not to exhaust the points on a manifold. there are probably topological characteristics that distinguish graphs of functions of n variables from general n-manifolds
with most of the exciting stuff probably happening in how various patches of a thing fit together
yesterday my son returned :-(, my wife got in an accident on the way back (she's ok, car not so) :-( & my friend's son returned from is 1.7k mile tout of ca.
when i first moved back here, my wife bought a car, and someone ran a red light and totaled it before i could even see the thing.
daughter came home and said "i told miss fuentes [a teacher] all about the dead heron." this dead bird she saw at the duck pond. she's been obsessed with it. "miss fuentes said it was sick." great. "am i sick?" not so great.
my daughter is something of a celebrity at day care. kids mob her when we drop her off, and when we pick her up. i don't know why.
the teachers use her as an example, i think because they don't want anyone else breaking a limb there. "let's all walk slowly so we don't break our legs."
with regards to the question you asked about $\epsilon(i) \to 0$ as $i \to \infty$
you use the fact that if a fraction has a limit at infinity, it implies that given a side of the fraction there's an error bound on the other side at all $n$
@Koro $\lim \limits_{n \to \infty} \sum_{i=1}^n b_i + \lim \limits_{n \to \infty} \sum_{i=1}^n \epsilon(i) = \frac{L_a}{L_1}$. suppose either limits diverge. then we get a contradiction
we still prove they exist the same way: suppose they don't. then it equals $\frac{L_a}{L_1}$ or $L_a$, depending on at what point you begin contradiction. But this is a contradiction: neither can diverge if they equal a constant
We have $\frac {a_i}{\epsilon(i)+L_1}=b_i$ and since $\epsilon (n)\to 0$, there exists an M such that for all $i\ge M$ we have $|\epsilon (i)|\lt \frac {L_1}2$
Using this we have $\sum_{i=M}^n \frac {a_i}{\epsilon (i)+L_1}\ge \frac 2{3L_1} \sum_{i=M}^n b_i$
So by limit comparison, the result follows. But I think you wanted to avoid limit comparison.
@shintuku no, that's not correct. $\sum b_i \epsilon (i)\ne (\sum b_i)(\sum \epsilon (i)$
I found an article which says it's possible in hyperbollic and elliptic geometry for certain radiuses and angles of square, and the construction of square and circle don't depend on each other
@robjohn I have been looking at the Dominated Convergence Theorem (DCT), as given here for example. If it is applicable, then for sufficiently small $h(n)$ (i.e. large $n$), the remainder, viewed as a sequence of functions $g_{h(n)}(u)$ could be replaced by the converging function. Since it is $o(h^m)$ (presumably uniformly in $u$), the converging function seems to be $g(u)=0$.
Although this is a constant, for the purpose of moving the remainder outside the integral it would maybe be more sensible to end up with a function only depending on $h$, if that would make any difference. Maybe you were alluding to the DCT.
Hey guys! I just wanted to ask a quick question as a sanity check.
I have been asked to expand $e^{-x^2}$ in the neighborhood of x = 0 to three terms plus remainder (writing the remainder in Lagrange's form). A simple enough question, as you just have to calculate a bunch of derivatives.
From calculating the derivatives, it's clear the odd powers of x will vanish in the Taylor expansion. Thus, in order to get a Taylor polynomial with three terms we need to calculate up till the 4th derivative. However, since the 4th degree and 5 degree Taylor polynomial are equal, it made sense to me to c…
However, the textbook claims that the error should be: $$\frac{x^6}{3!}e^{-\theta^2 x^2}$$
I am 98% sure the textbook is wrong on this one, but just in case I'm being a numpty I wanted to get an outsider's opinion
Some more food for thought, @robjohn. If $\frac{g_{h(n)}(u)}{h^m}$ converges uniformly to $0$ (by the assumption that the Taylor expansion occurs uniformly in $u$), then so does the remainder $g_{h(n)}(u)$. Since by the definition of uniform convergence, given an $\epsilon$, there is an $N(\epsilon)$ that does not depend on $u$; that is, there is a function $y(h)$ such that $$\left |\frac{g_{h(n)}(u)}{h^m}\right|<y(h)<\epsilon.$$ $y(h)$ exists in order to solve $y(h)<\epsilon$ for $n$ in $h(n)$ to obtain $N(\epsilon)$. Multiplying the inequalities above by $h^m$ and attaching $<\epsilon$ on…
Thus $g_{h(n)}(u)$ is bounded above by some function only depending on $h$. Since $g_{h(n)}(u)$ is unknown, I'm unsure though if it is bounded above by some function only depending on $u$ (denoted $g(x)$ in the link above). This is a requirement for the application of the DCT.
It is maybe clearer to avoid it, but I have been somewhat inconsistent with writing out that $h$ is a function of $n$, i.e. $h=h(n)$.
consider page 172-173 in Topology and Geometry by Glen Bredon. In particular, Lemma 3.1.
The result is: If $f,g$ are paths in $X$ such that $f(1)=g(0)$ then the $1-chain$ $f\cdot g - f - g$ is a boundary. Here, $f\cdot g$ is the concatentation of the paths $f$ and $g$.
He proves the result as fol...
11 Days till Halloween. I'll be waiting in the pumpkin patch like Linus, with my blanky, waiting for the Great (albeit Mean) Pumpkin! But I'm okay with a Mean Scare Crow, or a Mean Gourd. Hint... HINT... @robjohn ??
@leslietownes Wow. I don't know if @robjohn will want to appear here as a mean strawberry. As you can imagine, the Mean pumpkin/Gourd is a natural fit with his orange attire. I'll search for it. I'm quite sure I saved the image file of his Mean Pumkin costume.
Aren't dragons kind of like unicorns, i.e., imaginary critters? I wonder if reincarnation of dinosaurs over the history led the human imagination to recall/imagine their existence, in the form of dragons and unicorns??
i'm not sure she could wear a pokemon costume to her day care. they have a ban on commercial characters. it's part of some kind of anti-capitalist indoctrination.
@leslietownes There is a school in Seattle that cancelled its Pumpkin Parade and the costume day because they said it was rascist. It was claimed that "As a school with foundational beliefs around equity for our students and families, we are moving away from our traditional 'Pumpkin Parade' event and requesting that students do not come to school in costumes."
@robjohn I do not mean to tell you what to do... I just have enjoyed very much, as have a lot of others, your periodic "holiday spirit." Perhaps I should leave you to your own creative whims. Maybe it stops being fun fun for you, when requests are made. In any case, on this site, it is woke to have fun.
@leslietownes It's sad that the our local Human Society needs to send out warnings about keeping one's black cat safe and indoors, during this time of year.
@robjohn Actually, the average life span of more or less strictly indoor cats is almost 8 times the average lifespan of indoor cats that can come and go. Smart move, really, for kitties sakes'
we have a heated bed, a bed shaped like a black cat head (you crawl into the mouth to sleep in it), a bed shaped like a cupcake, various items of upholstered furniture, and our daughter's bed. some days she sleeps in all of them, one after the other,
@leslietownes Tell me about it. I spend so much time at my desk, that Shai owns most of the upholstered furniture. One he reserves for our interactive play. He catches toys, and bats them, very well. The furniture, when he's being "spunky" is his playground. Else his beds, except for his beds by the heat registers.
But anyone in CA doesn't need a heated anything. ;P Perhaps in Northern CA. But then again, look at Texas, last winter.
@PM2Ring got a SageMathCell question for you: do you have any idea how to make it do insert instead of overtype when you click into the script? i keep getting little things wrong and have to delete the entire line
(actually, maybe a more pertinent question: where tf is Insert on my laptop keyboard)
but what is b, as a data structure? does it have entries and if so how many? do you want \ell to feature in the sum somehow? at the moment it's just (u-1) b_{u+1} because all of the terms being added are the same
sum ell=1..u-1 b_{ell+1} would be a sum of the u-1 terms b_2 + b_3 + ... + b_u; if you wanna write this as a dot product it may help to know what b is, so we know what we can or cannot dot it with
if we don't know where b lives it's hard to be sure what we can dot it with. maybe you're asking for a dot product of whatever b represents with a sequence of u-1 copies of 1 in it and zeros elsewhere?
i can think of reasons why doing this might be desirable in programming, if you have primitives for dot product type operations and you don't want to reinvent the wheel, but if ell and u are changing it might not be faster than just computing the sum some other way
it has nothing to do with programming, I have to convert the summation into dot product. I don't understand how to do it since the index of the summation mentions upper bound instead of the lower bound
the thing we're wrestling with is that the question seems to presuppose a thing called b that has entries, but we don't know how many entries, so we don't know where it lives
in a world where (1,0) dot (1,0) and (1,0,0) dot (1,0,0) are different dot products (even if they result in the same thing), this is a problem
the simplest way i can think of (aside from something correct but silly) is to define two vectors. first one is B=(b,b,b,...,b), second is U=(1,1,1,...,1)
and then B.U is just the sum of the elements in B
it's a bit silly here, though, b/c you can just write B=bU
