@XanderHenderson When I reread what I wrote, I saw that I considered the PDE to be $F(u)=0$ while giving the defn of a linear PDE by saying that the PDE $F(u)=0$ is linear if $F$ is linear.
But I now feel that this might not be complete.
To be more precise, I must say that in general for a PDE $F(u)=c$, we say that it's a linear one if $F$ is linear.
With this minor modification I think the defn of linearity of a PDE becomes more accurate. I hope you'll agree, right?
@XanderHenderson Ok, so then my understanding for this is: A PDE is quasi linear if the PDE is non linear but the coefficients infront of the highest-ordered partial derivative terms are all functions of the independent variables or lower order derivative terms.
Is the above interpretations correct?
I think with your help I have gained a fairly precise amount of understandings of all these definitions. As always, thank you so much for your time! Your references and comments really enriched my understanding.
Assume we have a cubic polynomial
$ x^3 +bx^2+xc+d=0 $, with $b,c,d$ real numbers.
Let $x_1, x_2, x_3 $ be the roots, either real or complex.
What is the relation of the coefficients $b,c$ and $d$ in order to have the roots inside the unit sphere, that means
$ |x_i| < 1$ for $i=1,2,3$ ?
baymax, if p(z) is a polynomial of degree 3 without 0 as a root then q(z) := z^3 p(1/z) is also a polynomial of degee 3 without 0 as a root, and a nonzero number w is a root of q if and only if 1/w is a root of p. so such a p will have all roots with norm more than 1 iff q has all roots with norm less than 1. the coefficients of q relate to those of p in a simple way, so conditions from answers to the question you posted can be used to get conditions of the kind you are looking for
$\mathbf {question 1}$
$E$ $=$ $\bigcup_{\alpha \in I} {\alpha} × U_{\alpha} × \Bbb R^{n}$/${equavalance- relation=:\rho}$.
(The cup is disjoint union.)
For Hausdorffness, take two elements first. Then, by the Hausdorffness of $M$, if $\pi(x) ≠ \pi(x')$, they are contained in each neighborhoods ...
one of those was posted only an hour ago? i don't think the site moves so quickly that i would expect any votes in that time. i dunno about the other one. not everybody knows differential topology. people who don't understand the question (which has only 2 upvotes since 2015) are probably not going to read, let alone upvote, any answers.
i used to know a tiny amount of differential topology, and don't see how your answer to the question re hirsch is responsive to the question. part of this might be due to the fact that the original post is not really one question, but more like two questions (when site guidelines would suggest one question per post). or maybe it's a request for a single reference and not for two answers to the two questions. either way, i don't understand it.
i'm not saying that i doubt you, i just don't have the understanding that i would need to have in my head to upvote the answer. this is pretty high level stuff for MSE and i would not expect tons of upvotes generally.
when i answer a really old question, it is not uncommon for me to get at max 1 or 2 upvotes, and they usually don't arrive right away. just to put the site activity in perspective. a lot of the really busy stuff and the high activity is on lower level questions.
for finite vectors spanning a vector space, we can argue that there exists a linearly independent set inside this that spans the same, by starting with one element $\alpha$, and asking algorithmically, does this span the set? if not, there is an element outside the span($\alpha$), so we append this outside element and ask the question again, and so on... Now, is thre a way to extend this argument to arbitrary sets of vectors?, how do I generalise this algorithm for infinite steps
I am getting hung up on a sentence in my textbook:
> Since the indicator function of an open rectangle (i.e. $(a_1,b_1)\times (a_2,b_2)\times\cdots\times (a_n,b_n)$) is the increasing limit of a sequence of continuous functions with compact support, the probability measure $\mu$ is also determined by the values of $\int \varphi\,\mu(\mathrm{d}x)$ when $\varphi$ varies in the space $C_c(\mathbb R^n)$ of all continuous functions with compact support from $\mathbb R^n$ into $\mathbb R$.
1. Why is an indicator function on an open rectangle an increasing limit of a sequence of continuous functions with compact support? 2. I don't see how $\mu$ is determined by the values $\int \varphi\,\mu(\mathrm{d}x)$. Is this clear to someone?
This was a quite useful answer to 1. and 2., i.e. that all the nonnegative bounded continuous functions, when integrated against the measure $\mu$, determine $\mu$. Although I believe the continuous functions with compact support are a subset of the bounded nonnegative continuous functions, so the statement in the quoted passage seems to be weaker.
@Thorgott Consider a fibration $M = \Sigma_2\times S^1\xrightarrow{f} S^1$ where $\Sigma_2$ is a genus 2 closed oriented surface. If there is an embedded closed oriented surface $\Sigma$ in $M$ which is transversal to a fiber $S = \Sigma_2$ and not homotopic (or isotopic) to $S$. Then $f(\Sigma) = S^1$?
I wonder in this case if it's possible that $\Sigma$ is embedded in $\Sigma_2\times (-1,1)$ once I view $S^1 = [-1,1]/\sim$ and $S = \Sigma_2\times\{0\}$.
@psie It depends on $U$. If the closure of $U$ is compact, then $f_n$ have compact support. In my case, I was considering $U$ to be an open rectangle, so indeed, they have compact support.
@nickbros123 eiter transfinite recursion or take a maximal set by Zorn's lemma
If $V$ is your vector space ten well-order $V$ and proceed wit your aloritm by transfinite recursion (wen a coice is required, pick least element wrt your well-orderin)
Assume we have a cubic polynomial
$ x^3 +bx^2+xc+d=0 $, with $b,c,d$ real numbers.
Let $x_1, x_2, x_3 $ be the roots, either real or complex.
What is the relation of the coefficients $b,c$ and $d$ in order to have the roots inside the unit sphere, that means
$ |x_i| < 1$ for $i=1,2,3$ ?
