does anyone know where I can read up on how Cramer's rule allows us to determine which linear systems are inconsistent and which systems are indeterminate?
wikipedia's article mentions the results, but no explanation
I have some algebra question. I want to show that if $F[x,y]$ is a polynomial ring with two variables over a field $F$ and $(x^5,y^6,xy)$ is an ideal of that ring, then it cannot be generated by two elements i.e. there is no $f,g\in F[x,y]$ such that $(f,g) = (x^5,y^6,xy)$.
Is there some easy way to show this without using very high concept of algebra?
Hi @robjohn ! How are you? I hope you are fine! I got 2 answers to this question. I upvoted both. If you have time, can you at least give a little comment under the question? I have answers. But I want to get teacher approval. Can you help me?
I found this question that caught my attention at MSE.
Something is going wrong with my "solution", but I can't find it. I doubt the solution is that simple. There could be a terrible mistake in the solution.
Original problem says:
Prove that for any real values of $c$, the equation $x(x^2-1)...
Imagine the top-right quadrant of the plane is an infinite sheet of paper, and define the "origami area" of a point to be the area of the quadrilateral obtained by picking up the corner and folding it onto the point, as shown.
Then the lemniscate of Bernoulli (this nice figure eight shape), rotated 45 degrees, has the nice property that the origami area of any of its points is constant, and in fact equals the area of one lobe of the lemniscate.
This, I believe, is related to the fact that the lemniscate of Bernoulli is the circle inversion of the hyperbola y=1/x.
Yeah I'm pretty sure if you define the "rectangular area" (I don't know a good name!) of a point $(x,y)$ to be $xy$ (the area of the rectangle to the left and below it), the "origami area" I defined is the rectangular area of the point's inversion
That is, the inversion of $(x,y)$ is $(r^2/x,r^2/y)$ where $r^2=x^2+y^2$, and the origami area of $(x,y)$ is $r^4/xy=(r^2/x)(r^2/y)$
@Ted how do I understand sections of the dual of the tautological bundle? it's possible to do those by calculating with cocyles in charts, but I'm hoping for a more geometric perspective to exist
yeah, homogenous polynomials of degree 1 are all linear functionals on $\mathbb{C}^{n+1}$ and the section just comes from restricting a fixed functional to the span of $[x]$ for any $[x]\in\mathbb{P}^n(\mathbb{C})$, so that makes sense, but how do I see these are all possible sections?
@Thor: Usually, when you think of this in terms of a sheaf, you start with the presheaf of homogeneous functions of degree $k$ on the preimage of the open set.
I think the easiest way to nail it is just to use transition functions. If you know something about sheaf cohomology, I can offer an inductive argument.
If $H\subset\Bbb P^n$ is any hyperplane ($\cong \Bbb P^{n-1}$), then you have a short exact sequence $$0\to \mathscr O(-1)\mathscr O \to\mathscr O_H\to 0,$$ and can tensor this with $\mathscr O(1)$ and look at the beginning of the long exact sequence on cohomology.
I don't see anything wrong with my argument as a heuristic and transition functions as a rigorous proof. But, in general, sheaf cohomology is all about asking (among other things) when does a global section come from restricting an ambient global section, etc.
i would hopefully attempt to make contact with two mathematicians who could answer a quick question:
my book says $h$ is a map with domain $\mathbb{R}^2$, and explicitly sets its basis to $\{ (2 \ \ 0), ( 1 \ \ 4) \}$. Am I to understand that this isn't actually $\mathbb{R}^2$, but rather a subspace of it?
it appears to be regarding R^2 as a vector space with that chosen ordered basis. i agree that depending on where this is in a book, it could be confusing to see that, because proving that that list is a basis would be a common exercise.
and not just something you say in the setup to something else.
yeah, i just think that "consider R^n with the basis [list]" contains an implicit assertion (but no argument) that the list is a basis. which to me might be fine in chapter 20 but not in chapter 2.
so.. we could use a representation using this basis, or just the standard basis. it's confusing because the author introduces this basis in order to explain representation in another basis
the author introduced $h$ to begin explaining changes of basis. but I asked the above question because that change of basis looked totally superfluous and arbitrary
Arnold was supposed to be at a conference I went to in Lyon (in honor of Elie Cartan's 100th birthday, I think), but they didn't allow him out of Russia.
@shintuku In my books I use the following sorts of examples. Suppose you want to find the standard matrix of, say, reflection across a line in the plane. With respect to an obviously-chosen convenient basis this matrix will be $\left[\begin{matrix} 1 & 0 \\ 0 & -1\end{matrix}\right]$. Now the change of basis formula gives you the matrix you want. Even better in higher dimensions ...
I love his stuff! His Diff.Eq book and his Mathematical Methods of Classical mechanics are two of my all-time favorites. I have scoured the internet for talks and found a few but I wish there were more in english.
@Quin Indeed. And Phillip Griffiths was too, still alive. Robert Bryant was a grad student the same time I was and we became friends, but he is just an astounding super-star.
@TedShifrin I have watched Robert Bryant talk a few times (on YouTube) and understanding some of his work is a long term goal of mine. I really appreciate his style. (Similar to Arnold, I gather, making hard stuff simple instead of a weird trend of making not so hard stuff hard!)
self gratification was the euphemism. as a youngster i read a lot of my mum's stuff, and took me many years to understand why there was so much discussion about said topic. my mum was usually at pains to explain was uncharacteristically silent.
Consider a function $f: R \to C$ lying in $L^2 (R)$, so $\int |f(x)|^2 dx < \infty$. Suppose $f$ has an analytic continuation to complex $z=x+iy$ for $y$ at least in some interval $I$. Are there any general statements or theorems regarding the convergence of $\int |f(x+iy)|^2 dx$ for all $y \in I$? Have you seen something similar yet, maybe examples or so?
if analyticity is part of the assumption, a useful search term might be 'hardy class'
it doesn't answer your question so much as assume the hypotheses of your question, but people have generalized them a lot
so maybe there is stuff known about the gap if any between hardy class functions and functions that just happen to analytically extend to square integrable functions as you describe.
the standard hardy classes would be for the upper half plane. i.e. $I = [0, \epsilon)$ with nothing below the real axis. i haven't thought about this in long enough to know if that makes a difference.
Right, I know about Hardy classes, should have said that. I would need something concerning a symmetric strip around the real axis, not upper half plane or unit disk
on at least a closed interval or an interval you don't mind shrinking a bit on the x-axis, those extensions seem like they ought to have finite integral no matter what, and it's just a question of controlling the growth at endpoints of your x-range or your y-range.
which you see even in just the usual definition of H^2, which i think assumes a uniform bound on the L^2 norms of the slices. if that wasn't automatic they probably wouldn't put that in the definition.
I'm asking because the function I have here is quite complicated (involving several integral transforms and other shenanigans), so proving brute-force that the analytic continuation is still in L^2 is hardly possible. But I figured maybe there are some general results I couldn't find in the literature yet... I'm not an analyst by the way
i've definitely seen generalizations of classical hardy where they don't require uniform boundedness on the whole upper half plane, only some kind of niceness close to the x-axis and maybe some growth condition or not even having square integrable slices past some point. but this vagueness is useless to you.
several integral transforms. i like the sound of that. i would have suggested looking at papers specific to those transforms but if you're combining them, they may not talk to each other in any useful way. they meaning the references.
as long as this isn't part of some plan to prove the riemann hypothesis, i wish you luck.
yeah, consulting papers about these integral transforms was my first attempt, of course. But to no avail. So I thought, maybe there are general results on that matter
nah, no Riemann hypothesis. Just some physically motivated problem.
do you have any suggestions where I could look for those generalizations of classical hardy classes you were mentioning?
a good start might be a google scholar search for papers with 'hardy spaces' in the title which would likely to push beyond the classical theory and not recapitulate books with the phrase 'hardy space' in the title (although if you can find those, they might be helpful)
the book might have been 'generalized analytic continuation' by ross and shapiro, if not it is probably cited in that book.
i had a couple of books by bill ross and coauthors. his papers might be worth a glance although i think he was generally focused elsehwere. but people do work on this stuff.
What can we say about a riemannian manifold that has minimal submanifolds everywhere and in any "direction"? Is it, for example, of constant sectional curvature?
By minimal submanifolds "everywhere" and "any direction" I mean that for any given point $p\in M$ and any subspace $E\subset T_pM$, we have a minimal submanifold $\Sigma$ with $p\in \Sigma$ and $T_p\Sigma = E$ (therefore $\dim \Sigma = \dim E$).
I know that if we consider "totally geodesic" instead of "minimal", the ambient manifold is of constant sectional curvature.
@Ders This is an interesting question, and my original thought was that I had absolutely no idea. But here's one example. Consider $\Bbb CP^2$ with its usual Kähler metric of constant holomorphic sectional curvature $4$. Any smooth complex curve is a minimal $2$-dimensional submanifold. And there are minimal $3$-spheres anywhere you want. See this, for example. I expect you can play similar games with $\Bbb CP^n$ with higher $n$.
@robjohn or anyone else who knows a solution to the number of regions in a konnected graph inscribed in a circle problem: is splitting the graph into overlapping planar graphs likely tto be a profitable move to explore? Or would i best be served by following a brute force geometric approach of counting intersections?
from a post on the cross-validated stack exchange website to confirm what the MLE is and to find the MOM estimator of $\theta$. I found it to be $\bar{x}$. However, the confirmed answer on the post says that the estimator is $x_{1}$. Is this jsut because $\theta > x$?
@TedShifrin Pretty out of nowhere. I was attending to a course and the lecturer commented the result for the totally geodesic hypothesis and then it came to me "what if minimal instead?"
Well, kudos to you for asking a non-obvious question. I don't know the ultimate answer, but I'll ask one of my friends. Have you worked the exercise to prove the totally geodesic result? I assigned that last time I taught graduate differential geometry. ...
My friend just answered. He thinks it's true if you allow your minimal submanifolds to be suitably small with boundary, but surely false if you require closed submanifolds. He thinks it's false even for minimal surfaces (with arbitrary tangent plane) in a general $3$-dimensional manifold.
Oh, you definitely need to learn multivariable analysis before you do serious manifolds ... get to work on my other book! :) Then read Guillemin & Pollack Differential Topology before going to abstract manifolds.
ted can correct me but i think a lot of the delta between single variable calc and undergrad level differential geometry is getting comfortable with doing calculus-like things with more than one variable at the same time.
My lectures are for the multivariable analysis/linear algebra course. Sadly, the crew of undergrads got pooped and didn't want to record the diff geo lectures my last semester.
All calculus books steal unabashedly. Having said that, I stole a few of my favorites from Apostol and put them in Spivak's Calculus. I have some unique exercises in my multivariable math book, too. But most appear somewhere or other.
This is why I think that exercises determine whether a book is really good or not, although a horridly written text won't be saved by exhilarating exercises.
@Jack: You should ask Leslie. He's bragging that he's becoming a geometer.
For algebraic geometry, you need a lot of commutative algebra (beyond standard grad algebra), and it doesn't hurt so have learned some Riemann surfaces/algebraic curves first. Just as it doesn't hurt to learn undergrad diff geo before you get to the abstract graduate stuff.
A course on Dynamical systems really tied together a lot of the geometry for me. In particular, Hamiltonian systems. I'm still not sure if it would have been better to do dynamics before the diff. geo / riemann surfaces or not but something to think about.
Not really. UGA taught undergrad number theory just assuming you knew modular arithmetic and basics of rings. You don't need Galois theory at the undergraduate introductory level.
Well, if you're going on in mathematics, you definitely need to learn multivariable analysis, derivative as a linear map, inverse and implicit function theorems, etc.
not sure what you mean, the jet bundle doesn't depend on coordinates
I actually like the jet bundle perspective a lot, but my issue at the moment is that I can't figure out the relationship between the jet bundles and taking differentials on the iterated tangent bundles
I found this question that caught my attention at MSE.
Something is going wrong with my "solution", but I can't find it. I doubt the solution is that simple. There could be a terrible mistake in the solution.
Original problem says:
Prove that for any real values of $c$, the equation $x(x^2-1)...
