Like, if $x_n\in S^{\perp}$ and $x_n\to x$, then for $s\in S$, you have $\langle x,s\rangle = \lim_{n\to\infty} \langle x_n,s\rangle = 0$, so that's just continuity of the inner product
@0celo7 Oh! Well it's stated e.g. on the top of p.241 of Hatcher (not as a proposition of anything though..) / on Wikipedia - but in any case the first equality is a consequence of the definition of the cap product, and the second equality follows from $f_*[M] = deg(f) [f(M)]$ by definition of $deg(f)$.
@0celo7 oh. I don't know the proof off the top of my head - but I think you can find your answer in mathoverflow.net/questions/5001/… - think the upshot is to realise that the topological definition can be computed using local degrees using excision, and from there on it's not too hard (but I haven't thought about it very carefully)
@Mozzie I am on my way to bed, and my basic linear algebra is a bit rusty, but if the columns are linearly independent, then you should be able to do better than put it into some block form, and actually diagonalize it, no?
If x, y, z are distinct positive real numbers, the minimum value of {x^2 (y + z) + y^2 (x + z) + z^2 (x + y)} /xyz is ?
this is a classroom problem and the solution provided by my teacher is evaluate the expression @ x = y =z . I do not get it because the question states distinct positive real numbers .
So is my teacher right ? He is not able to satisfy my query.
Hi, I have the following problem: Given a finite size 2D lattice determined by the nearest neighbour links among sites, I need to generate all possible ways to divide the lattice in connected regions of at most C sites. Though I would like to be able to enumerate all the possible subdivisions, I would be content also with some programmatically way to generate them. Could someone point me to some article dealing with this problem? Thanks
Problem: Show that $\mathcal{C}[a,b]$, normed by the maximum norm, has a bounded sequence that fails to have any strongly convergent subsequence. Attempt: Consider $\{f_n\} \subseteq \mathcal{C}[0,1]$ defined by $f_n(x) = x^n$. Since $f_n'(x) = nx^{n-1} \ge 0$, $f_n$ is increasing and therefore $f_n(1)=1^n=1$ is the max. for every $n$. I.e., $||f_n||_{max} = 1$ for every $n$, so that it is a bounded sequence. Note that $||f_n-f||_{max} \to 0$ if and only if $f_n \to f$ uniformly in $\Bbb{R}$.
But any subsequence of $\{f_n\}$ converges pointwise the function $f$ that is $0$ on $[0,1)$ and $1$ at $x=1$, which is not continuous. Hence, no subsequence converges strongly in $\mathcal{C}[0,1]$...Does this sound right?
@MdAyquassar then so much the worse for them? this is a site for math Q&A, so to the extent that it's a question about writing in general (not mathematical writing in particular) it's simply off topic
@Semiclassical I don't think it's about writing in general. To draw an analogy, in maths I saw lots of folks avoiding the second comma and (depending on their style) even the period in "Therefore, x=[huge formula] (,) and y=[another huge formula] (.)", though the two clauses are independent and require a comma, and all English sentences require punctuation at the end.
How do you guys feel about a period after an expression thats at the end of a sentence, especially if it's its own line
Recall that $A\sim \pi^{n/2}r^n$.
It reminds me of that xkcd lemme find it
2004: Ian Stewart: Galois Theory (3rd ed.): "I have come to the conclusion that eliminating visual junk from the printed page is more important than punctuatory pedantry ... everything is much cleaner and less ambiguous without punctuation."
Lol "Whichever rule you follow, the journal you send it to will want the opposite."
There no "best way" for periods terminating sentences containing (block) formulas at the end. Without periods at the end of block formulas, you might not know where your sentences end, and with periods you might mis-take them for multiplication dots. That's why authors, publishers, and even whole states have their own styles regarding terminating periods. My question is about commas, and periods was an example of punctuation deviating from normal English.
@GFauxPas I wouldn't mind if anyone migrates. This should be someone with way, way higher reputation than me, since otherwise ac.se is likely to say it's not really about academia, but about maths and/or languages.
As of now, I'm only asking for getting the on-hold cleared.
If I have a linearly independent set of vectors say
\begin{bmatrix}0&0&9&0&0&0&0&0\\1&2&0&0&0&2&0&0\\0&0&0&0&20&0&13&0\\0&20&0&20&20&18&4&0\\0&4&0&8&5&0&0&19\\0&20&0&12&9&7&0&5\\0&4&0&0&0&0&0&0\\0&0&0&0&0&11&0&6\end{bmatrix}
is it possible to use row reduction to convert the matrix into two no...
but diagonalizing the matrix is not computationally sound for larger 1Mx1M matrices, therefore if the process could be lowered and only form sub blocks of size nxn a gradient decent algorithm could be used to solve sub blocks.
This ill conditioning makes it very hard to develop a robust numerical algorithm for the Jordan normal form, as the result depends critically on whether two eigenvalues are deemed to be equal. For this reason, the Jordan normal form is usually avoided in numerical analysis; the stable Schur decomposition[14] or pseudospectra[15] are better alternatives.
well calculating the Jordan normal form of the matrix shouldn't be that difficult, I would think? It's the change of basis matrices that are computationally intensive
I mean where the block are all orthogonal to each other
as to your jordan normal form comment, finding the det of matrices is a lot of work because first you must convert your matrix to reduced row form then multiply the diagonals
doing that along with solving $(c_1x^n + c_2x^(n-1) ...)$ is difficult
converting a matrix into increasing block sizes (1x1, 2x2, 3x3, 4x4, ....) might be easier, although that means that blocks would approach the size of the matrix
@GFauxPas No thanks; I've had enough of academia.se personally; I consider it even more toxic than here. If someone would like to ask there or to migrate, I wouldn't mind.
I'm reading Geramita's lectures on fat points, but I found a passage a bit confusing. (I'll post the reference as soon as I'll find the link)
I read that a fat point (of order $t$, defined by the ${p}^t$-primary ideal $p^t$) in $\mathbb{P}^n$ behaves like $\binom{n+t-1}{n}$ distinct points (in th...
Hello there, i want to get a comment on my approach to the extension of the problem in math.stackexchange.com/questions/2777056/… but as of now i could not had the time of the experts who have replied the question therein. Kindly help out!
Let $f:\mathbb{R} \rightarrow \mathbb{R}$.
Then consider the following delay equation :
$$f'(x) = f(x-1)$$
Let $S$ be the set of solution ot this equation. Then I would like to prove that :
$\forall f \in S -\{ x \mapsto 0\}$, $f$ is not bounded.
What I've done so far is that :
The set of...
Let $a,b,c$ and $d$ be positive real numbers , with $\frac{a}{b}<\frac{c}{d}$, then $\frac{a}{b}<\frac{a+c}{b+d}$. I shlould find the flaw in the following proof
If $\frac{a}{b}<\frac{a+c}{b+d}$, then $a(b+d)<(a+c)b$, so $ab+ad<ab+cb$, hence $\frac{a}{b}<\frac{c}{d}$
And the assertion follows.
Where is the flaw in this proof? I cant find any, all division are justified because everything is positive.
@BAYMAX actually i'm not sure it doesnt have a laplace transform now, Because maybe $f(x) = \displaystyle \sum_{n=0}^\infty \frac{e^{W_n(1)}}{e^{W_n(1)}+1}$ up to a constant scalar
well my thinking was that for the thing to converge, the term inside has to converge to $0$, so $\exp(W_n(1))$ also has to converge to $0$, so $\Re(W_n(1))$ has to diverge to $- \infty$, but then I have to admit I don't know by heart how that one behaves
for a function definded on $\mathbb{R}^2$ to $\mathbb{R}$ defined by $f((x,y))=x-9$
i want to find $f(A)$ where $$A=\{(x,y)\in\mathbb{R}^2, (x-1)^2+(y+1)^2<1\}$$
How to do ?
suppose you have $S=k[y_0,\ldots,y_n]$, and you consider $S_d$, i.e. the homogenous polynomials of degree $d$. Now we know that $S_d$ is isomorphic to $\mathbb{P}^N$, where $N=\binom{n+d}{d}-1$. How doeas this map associate a polynomial to a point in the projective space? I think I create a monomial basis of $S_d$, and then map the coefficient. Can anybbbody tell me if it is correct?
If x, y, z are distinct positive real numbers, the minimum value of {x^2 (y + z) + y^2 (x + z) + z^2 (x + y)} /xyz is ? this is a classroom problem and the solution provided by my teacher is evaluate the expression @ x = y =z . I do not get it because the question states distinct positive real numbers . So is my teacher right ? He is not able to satisfy my query.
Well, if $k, k^\prime \in \Bbb Z$ then $3k - k^\prime \in \Bbb Z$ because $3 \in \Bbb Z$ and $\Bbb Z$ is a ring, but I suppose that is unnecessarily pedantic
@LeakyNun if you want to construct a funtor from topological spaces to T1 spaces, here's what you can do: Let $X$ be a topological space, then define an equivalence relation by saying that $x \sim y$ iff for every continuous surjection $f:X \to Y$ where $Y$ is a T1 space, we have $f(x)=f(y)$
basically you write down all the axioms for a group as diagrams. You have morphisms for multiplication, inversion and unit element
then if you require that all these are actually morphisms in the category, that's called a group object
in the category of sets, these are just groups, in the category of topological spaces, topological groups, in the category of smooth manifolds Lie groups etc.
possibly pedantic: but I think it is slightly confusing to say $S^2$ is an affine variety (w.r.t. common terminology).. I think the term variety is always considered over an algebraically closed fields in the classical setting (of course you can define varieties w/ schemes, then we don't need to be working over alg. closed fields)
interesting - I didn't know that. Did he define varieties literally as zero set of polynomials in affine space over $k$, or did he implicitly also work with extensions though..
@loch Weil's foundations are now considered inelegant and there doesn't seem to be a reason to learn them except for understanding the history. So they're not treated in any modern literature afaik. But iirc, he worked embedded all fields involved into some large algebraically closed field
he did definitely work with extensions. The statement of the Weil conjectures involve the number of (closed) points on a variety over a finite field as you base change to larger finite fields
I was confused because I thought an affine variety surjecting to a projective thing (not a point) seemed weird (but I just realised there's nothing wrong with that lol TIL)
I know that every polynomial with complex coefficients has a solution in complex field. is it true that every 'multivariate polynomial' has solution in complex field?
If I have a linearly independent set of vectors say
\begin{bmatrix}0&0&9&0&0&0&0&0\\1&2&0&0&0&2&0&0\\0&0&0&0&20&0&13&0\\0&20&0&20&20&18&4&0\\0&4&0&8&5&0&0&19\\0&20&0&12&9&7&0&5\\0&4&0&0&0&0&0&0\\0&0&0&0&0&11&0&6\end{bmatrix}
is it possible to use row reduction to convert the matrix into two no...
Let $f:\mathbb{R} \rightarrow \mathbb{R}$.
Then consider the following delay equation :
$$f'(x) = f(x-1)$$
Let $S$ be the set of solution ot this equation. Then I would like to prove that :
$\forall f \in S -\{ x \mapsto 0\}$, $f$ is not bounded.
What I've done so far is that :
The set of...
for a function definded on $\mathbb{R}^2$ to $\mathbb{R}$ defined by $f((x,y))=x-9$
i want to find $f(A)$ where $$A=\{(x,y)\in\mathbb{R}^2, (x-1)^2+(y+1)^2<1\}$$
How to do ?
