My grandma, aunt JoAnn are here to stay with us - hence the 11 cats. My sister and her dog are over, and my mom is off work at 8:30pm, and dad & sis went to get an x-mas tree
@PenAndPaperMathematics i have a reasonable merlot beside me, a grgich hills 2016 merlot, but will save it for guests, i will just have something light & white tonight unless i can persuade a friend to drink in which case it will be Amarone della Valpolicella
i find that beyond a certain investment, the relationship between price & enjoyment is somewhat tenuous. however, i have found the relationship between price & perspective next morning to be a lot less so.
I have one of those, we kept circling this block, missing our turn, then on the final one, I missed the green light slightly, they pulled us over at the liquor store. This is over 10 years ago, lol
I will have to check. When I answered, I think there was at most one. I did comment in my answer that there were plenty of other occurrences of the question. (A standard Frobenius Theorem question of the qualifying exam sort.)
It is now closed. I answered pretty soon after it was posted, so I don't recall any close votes right then. I guess I do treat graduate level stuff differently from low-level, but I never downvote an answerer. I guess that's our "friend"'s and others' new approach. I am really rather fed up.
@anonymous Oh, computerized proofs. AI in which I have marginal interest.
@TedShifrin We will go to my wife's cousins for the holiday, not exactly traditional, a blend of Indonesian & western. My daughter will spend the holiday in college in the UK, her first away from family.
https://chat.stackexchange.com/transcript/message/43019671#43019671 Unfortunately, so far kpop failed to produce anythingy that does not look like a homework question -_-
my daughter ate a bit of cheese/egg/shallot/dijon mustard on a baguette that we toasted in the toaster oven. she was planning on just PB&J and some frozen pineapple chunks but liked the way it looked.
koro: by 'same eigenvalues' they are probably assuming not just that the sets of eigenvalues are the same, but also (at a minimum) that the geometric multiplicities of the eigenvalues are the same for each operator.
it's not unheard of for a book to refer to the spectrum as not just the set of points but the set plus a multiplcity function. it's not common but i've seen it. if they were looking at something in a translation from another language maybe they missed this.
"same eigenvalues" to me does sound like just the point set.
our dishwasher appears to have broken. we don't use it a huge amount, but i'd specifically planned on hand washing a bunch of stuff while the dishwasher did the simple things and not having two sinks and four sets of hands was annoying this evening. merry christmas to me.
@Koro although for my statement you don't even need any assumptions on eigenvalues. If two matrices over some field commute such that both their characteristic polynomials split, then they share an eigenvector (but they don't necessarily correspond to the same eigenvalue)
basically just note that in the direct sum decomposition of the space with respect to one of the matrices, each generalized eigenspace is invariant with respect to the other matrix
I have this problem $$\begin{cases} y'=-x^2 y^2+\exp(-y^2)\\ y(0)=0\end{cases}$$ the question is: study the existennce and unicity of solution for this problem on $R=\{(x,y)\in\mathbb{R}^2, |x|\leq \frac12, |y|\leq 1\}$ using two methods.
can someone give me a hint on what is the two methods ?
the notes say it has a 'straightforward but messy ' proof
i think whats written after that doesn't make sense though. They say 'clearly $\psi_{\beta,\alpha}$ are diffeomorphisms on the intersection of their domains and codomains with the complement of $\mathbb{R}_{+}^{n}$', I think what they really mean to say here is with the complement of $\{0 \} \times \mathbb{R}^{n-1}$ unless im misunderstanding something
(and the complement of $\{0 \} \times \mathbb{R}^{n-1}$ here is with respect to $\mathbb{R}_{+}^{n}$)
@porridgemathematics It may also be worth noting that Schultz is a great guy, but probably should have retired several years ago. His approach to things is often a bit out of date (though I haven't looked at that document).
@BannedUser It has been a while since I thought about this, but a matrix is a way of representing a linear transformation from one vector space to another.
In the case of an $n\times n$ matrix, you are mapping $n$-dimensional vectors to other $n$-dimensional vectors.
So, consider the $n$ mutually orthogonal unit vectors in $\mathbb{R}^n$ (concretely, take $(1,0,0), (0,1,0), (0,0,10$ in $\mathbb{R}^3$).
These vectors define a cube of unit volume.
Now hit those three vectors with a $3\times 3$ matrix. The result will be a parallelepiped.
If I recall correctly, the volume of this parallelepiped is the determinant of the matrix.
So having a determinant of zero means that the image of the unit cube has zero volume. Which suggests that the cube has been smashed down to something which does not have volume (such as a portion of a plane, or a line segment, or a single point).
True. But when column vector is linearly dependent it makes sense that cube or parallelepiped is smashed down to plane or single point but when it is row it doesn't maketh sense. To me column vector represent where i j and k unit vector lies after transformation but the row doesn't makes sense to me in geometric point of view...
Okay, that isn't quite the standard definition of "linear independence". That is a definition of a linearly independent spanning set, or some such.
In any event, in your definition, linear independence describes a property of a set of vectors. So what does it mean for a single vector to be linearly dependent?
Suppose that I have an entire function $f$ and I want to check whether it extends to be analytic on the extended complex plane $\Bbb{C}_{\infty}$? How does one go about doing this? How does one check analyticity at $z = \infty$?
@user193319 Since $\infty$ must be an (isolated) singular point, the question only makes sense if you consider this as a map to the extended plane. You take coordinate charts at $\infty$ in both domain and range in the obvious way (let $u=1/x$ and $v=1/y$).
Theorem: Let $(f_n)$ be a sequence of real valued functions defined on metric space $E$. Suppose that $(f_n)$ converges uniformly on $E$ to the limit function $f$. Let $x$ be a limit point of $E$. Suppose that $ \lim_{t\to x}f_n(t)=:A_n$ exists for every $n\in \mathbb N$.
Then $\lim A_n$ exists a...
Please quit the use of unnecessary symbols (like $\land$) in a proof. Use words, or, better yet, say $m,n\ge N$. I quit reading at "taking $\lim_t\to x$ of both sides," because what you wrote is somewhat wrong.
Hello, can someone explain me this comment: You already found that $-\frac14\leq y'\leq 1$, so that at least $|y(x)|\leq |x|$.Thus there is no divergence in finite time
Pondering if I should write some sketchy notes about Steenrod squares while I'm reading it and also if it's going to be so sketchy that it's best done privately.
vrouvrou: if y(0) = 0 and |y'(t)| <= 1 for all t >= 0 then |y(x)| = |y(x) - y(0)| = |int 0...x y'(t) dt| <= int 0...x |y'(t)| dt <= int 0...x 1 dt = |x|
@Koro: This is similar to the non-proofs people give when they misapply the squeeze theorem and say $g(t)\le f(t)\le h(t) \implies \ell=\lim g(t)\le \lim f(t)\le h(t)=\ell \implies \lim f(t)=\ell$.
Leslie gave you the proof of the first statement, @Vrouvrou. "No divergence in finite time" means that the function cannot go to infinity in a finite interval $x\in [A,B]$.
@Semiclassic It presumes that $\lim f(t)$ exists, which is what you're trying to prove.
koro: i agree with ted, the issue is that last bit, "it follows by (1) that lim t->x |f_n(x) - f(t)| <= e." needs more reasoning as at this point it isn't clear that lim t->x f(t) exists, and you haven't made explicit why lim t->x |f_n(x) - f(t)| would exist for some other reason.
i really don't like boiling statements about limits down to symbols. at least for students. because a huge part of the conclusion, that "the limit exists and", is mentally insertable in front of "lim f(t) = L" but is not written there.
all of the "limit laws" have similar judo moves you can do, if limits of some of the inputs have exist, then another limit does too, and is given by the "law."
if one puts "lim f(t)" in quotes then i don't totally mind it as a shorthand way of saying "the limit of f(t), if it exists, lies between lim g(t) and lim h(t)"
I totally mind it, @Semiclassic. Students need to understand. Remember that the very definition of a limit is a squeeze. $0\le |f(t)-\ell|<\epsilon$ ...
most calculus books state the limit laws correctly and squeeze theorem correctly, but outside of the squeeze theorem, and maybe then only for those specific triginonmetric limits, i am not sure i have ever seen a book that emphasizes on the existence of the limit being part of the consequence of the 'law.'
it's formally printed in a theorem box, but the books never explain that it's part of the theorem. i am semi-OK with this but it does lead to multiple speed bumps in analysis.
"why am i suddenly worrying about laws that i already know"
In my calculus classes (including the Spivak one), I also made students write something like "$0/0$" at each step when they were doing L'Hôpital. And I always put a problem where it could trap them if they just did it mindlessly without checking.
koro: that's a common exercise in intro analysis books. you usually lose about half of the class right there. you can get them back, but you lose them for a minute.
pen: i'd expect anyone my age or older. more my dad's generation than mine.
koro with so much of this number and function convergence stuff, 99% of the work that goes into a rudin-style exposition vs. some other type of exposition is setting an argument up so all of the justifications arrive at once, right when needed, with as few sub-parts / side tours as possible, and as little case analysis as possible. not always the most illuminating way of learning about it in the first instance.
I'ma skip Rudin, and jump straight into Quantum Physics
Griffith's teaches it right, I think
I made it through measure theory using Papa Rudin, but never seemed to get passed the convexity part in the next chapter. That book is just too dense for my liking, though it is rigorous, etc. I need a more modern treatment
I like to jump around. E.g. Lang's Algebra has the best derivation of tensor product using UMP in category theory. He creates ad hoc a certain category just for discussion in the proof
Only my multivariable math book did I cover first page to last (with only a few skipped). In my other texts, I have never taught everything in one course.
Debugging Bootstrap Studio to Django template export code today for me :> C u every 1
I forgot, here's a group object definition: https://q.uiver.app/?q=WzAsOSxbMCwwLCJHXjMiXSxbMSwwLCJHXjIiXSxbMCwxLCJHXjIiXSxbMSwxLCJHIl0sWzIsMCwiRyJdLFsyLDEsIkdeMiJdLFswLDIsIkdeMiJdLFsxLDIsIkciXSxbMiwyLCJHXjIiXSxbMCwxLCJcXHRleHR7aWR9IFxcdGltZXMgbSJdLFsyLDMsIm0iLDFdLFsxLDMsIm0iLDFdLFs0LDEsIigxLFxcdGV4dHtpZH0pIiwyXSxbNCw1LCIoXFx0ZXh0e2lkfSwxKSJdLFs1LDMsIm0iLDFdLFs0LDMsIlxcdGV4dHtpZH0iLDFdLFs2LDIsIihpLFxcdGV4dHtpZH0pIl0sWzcsOCwiXFxEZWx0YSIsMl0sWzgsNSwiKFxcdGV4dHtpZH0sIGkpIiwyXSxbNyw2LCJcXERlbHRhIl0sWzAsMiwibVxcdGltZXNcXHRleHR7aWR9IiwyXSxbNywzLCIxIiwxXV0=
Everything commutes i.e.
I took the three diagrams involved in group objects from nLab and put them all into a single, connected diagram =)
Conclusion: it's a mess, athough each commuting square takes on a form $(f, \text{id})$ together with that pair permuted, sort of, since one of them has a diagonal map composed with it
the perspective never really occurred to me, but what's going on that a representing object for a functor, together with its universal element, is the same data as an initial object in the functors category of elements
i did ask her group members for their data for that lab (under the cover of "just wanting to check something") and confirmed that it matched what she submitted
Let $S_2$ act on products $\sigma (A\times B) = B \times A$. Similarly, let it act on tuples of maps $\sigma (f,g) = (g,f)$
Then $G$ is a group object with (insert associated maps) $\iff S_2 \{(1,\text{id}), m\times \text{id}, ((-)^{-1}, \text{id})\Delta\}$ when glued to $G^2 \xrightarrow{m} G\xleftarrow{m} G^2$ to form squares, commutes.
That's what I was looking at initially, but it doesn't seem to add any interesting viewpoint (yet)
