I now work for a multinational with a large presence in the US. As a consequence I am obliged to learn how to respond in the event of an active shooter event
One runs, if one cant run, one hides, if there is nowhere to hide, one must accept that their life is in immenent danger and do what they can to fight back
The video was actually chilling in how matter of fact it was
In linear algebra, the Frobenius normal form or rational canonical form of a square matrix A with entries in a field F is a canonical form for matrices obtained by conjugation by invertible matrices over F. The form reflects a minimal decomposition of the vector space into subspaces that are cyclic for A (i.e., spanned by some vector and its repeated images under A). Since only one normal form can be reached from a given matrix (whence the "canonical"), a matrix B is similar to A if and only if it has the same rational canonical form as A. Since this form can be found without any operations that...
it never occurred to me that i would see the plaintext passwords. obvious of course, but that was not why i was doin git
i wanted to connect a single board computer to the mainframe so i could do some data logging for experiments i was running. this was before such things were done.
i was studying elec eng, so had an idea of digital electronics, but the basic was self taughtt
when you are trying to figure stuff out even little hints are very helpful. so i would see stuff used in byte and try i out. things like peek & poke, that sort of stuff
when we found out that the screen was memory mapped that was a big deal
It got pretty late here last time.. about $:30 in the morning...had to catch up on some sleep
Last time I guess things were really unclear for mostly my fault. I will try to sort of clear things up.
Let T:\R^n \rightarrow \R^n be a function(non-linear). ONe thing for sure is I dont know whether lim _{\theta \tends \infty} T(\theta) is bounded or not.
Given this I want to assume \lim_sup_{\theta}\inf_{x \in S}\|Ax - T(\theta)\|_2 where A is \R^{n \times d} and S is comapact subset of \R^d is finite
To support my assumption I can think of two choices
Either \|T(\theta)\| = \exp(-theta) so that lim\|T(\theta)\| remains bounded. Or as I was trying yesterday assume, \sup_\theta\inf{x \in \R^d}\|Ax - T(\theta)\|_2 is bounded. And then claim that \sup_\theta\inf{x \in S}\|Ax - T(\theta)\|_2 is bounded for any compact set S
@rostader I mean, I would like to get some random proof-writing question generator
I am looking forward to write the PRMO and hence I need to prep myself in number theory so as to write the proof questions/numerical answer questions that may come up in later stages.
If you have no idea regarding such a thing, no problems, sir :)
It was out of a longing to get a hand in number theory that I asked you so.
@rostader I bet you don't have ChatJax installed (from the half-latex in your comments). The link is in the info for the room (at the upper right of the page in the desktop version or in the page info for the mobile browser).
I have seen some "Anti-Pi Rant" videos where they say that Pi being a boring number. There argument is that "Sure pi is an irrational number , but so is root 2 , root 3 , the golden ratio phi etc".My confusion is that , isn't pi also a transcendental number too ? I heard that to this day , very few numbers were proven to be transcendental.
proof verification: given $H: X \times I \to Y$ continuous, where $I = [0,1]$, every $\gamma_{t_0} := (x \mapsto H(x, t_0))$ is continuous
proof: given $U_Y \in \tau_Y: \gamma_{t_0}^{-1}(U_Y) = \{x \in X: H(x, t_0) \in U_Y\} = \pi_X ( H^{-1}(U_Y)\cap \pi_I^{-1}(t_0) ) = \pi_X(H^{-1}(U_Y)) \cap \underbrace{\pi_X(\pi_I^{-1}(t_0))}_{= X} = \pi_X(H^{-1}(U_Y))$ which is open since projections are open mappings.
but in this specific case it should be easier than invoking this general result: obviously [0,1] embeds into R, and R embeds into [0,1], and for standard Borel spaces this is enough to have an isomorphism
”Cantor-Schroeder-Bernstein holds for standard Borel spaces”
Ok this is pretty much the same as proving the big result I mentioned earlier though so I’m not sure what’s my point
anyway all of this can be found in Kechris Classical Descriptive Set Theory if you want to see the details
XP ah, but that makes sense, I really just needed a measurable function whose inverse is measurable which turns out to be the obvious one (I was just hung up on the fact that [0,1] was compact while R was not)
This is a (very) partial answer: Suppose the polynomial is real and of odd degree. Then this polynomial is the characteristic polynomial of some matrix acting in an odd number of dimensions. Restricting this matrix to the sphere and then projecting onto the tangent space of the sphere defines a v...
once you had that expresison, you can actually evaluate some nasty integrals. Multiply both side of the expression with sin(nx) and integrate from -pi to pi. You will see all terms die off in the right side except one, and left side you have a really ugly integral
If $(0, 1)$ is a solution for $y = \pm \sqrt{e^{-x^2}}$, the author infers it implies $y = \sqrt{e^{-x^2}}$, i.e., without $\pm$. Does anyone have a clue why? Of course, it is obviously true if $x=0$ given our initial supposition, but how do we know this holds for other values of $x$?
I have a quick group theory question. If a group G acts on a set S, I understand that an orbit is a minimal set of elements in S that G sends into itself. Is there a name for a maximal set that G always sends outside of S?
I'm not sure. The complement of an orbit contains other orbits. Thus, that complement has some elements that G does not send outside of the complement.
Here's what I'm thinking. Suppose we have C2 group acting on a set of 4 points $\{1,2,3,4\}$ with the orbits $\{1,3\},\{2,4\}$ what I want is a name for sets like $\{1,2\},\{3,4\}$. or $\{1,4\},\{2,3\}$.
@LucasHenrique the nature of mathematics is that many things go from from "don't know what to do" to "obvious" (clearly a bad characterisation) with little transition.
@TedShifrin Not sure really, but its related to an exercise about kernel density estimation. $k(u)$ is a kernel density estimator. There are more conditions on $k(u)$ that I left out.
@schn Let $k_\alpha(x)=\frac{[0\le x\le \alpha]}\alpha$, then $k_\alpha^2(x)=\frac1\alpha k_\alpha(x)$, so you can adjust the ratio of the two integrals to be anything you wish.
@copper.hat That is true. When I see people ask what someone has done on something when they say they have no idea what to do, you know that even a small nudge in the right direction will help. However, if you give a hint, people complain that hints are not good.
@TedShifrin You've never used Approach0? it is a good, TeX-friendly search engine.
I was talking to a professor. I need to find a research area I like... someday, sometime...
topics in algebraic topology are really interesting and I liked algebra. I thought of something related to homological algebra or something but I'm really not sure
@Ted @Thorgott Speaking of Chern classes, I realized a good way to think about them is as intersection of homotopies (homotopes between homotopies, etc) of sections of a given bundle with the zero section.
But then I found out Danny Calegari already knows this
what you do during undergrad probably has less impact on what comes after than you're imagining
@BalarkaSen hmm, intersection of generic section and zero section is dual to the Euler class, which is the top Chern class, and you're saying something inductively based on the inductive construction of the Chern class or something?
That sounds like the right idea to me. Here is the penultimate Chern class for a complex vector bundle $E \to B$; take any section $s : B \to E$, multiply it with the fiberwise $S^1$-action in virtue of the complex structure to get a family of sections $B \times S^1 \to E$ which can be filled to a disk of sections $B \times D^2 \to E$ because the space of sections is contractible.
Intersect this with the zero section $E_0 \subset E$ and this is the representative for $c_{n-1}$, $n = \mathrm{rk}(E)$.
It's relatively easy to check by naturality; I believe the inductive Gysin construction is related but have not checked. For Stiefel-Whitney classes you can do the same but with $S^0$-action. Which screams Wu's formula, because you're saying SW classes = homotopy-coherent self-intersection number of 0-section. Remember cup product does not commute, it commutes upto homotopy, and the failure is recorded by Steenrod squares.