If $A = \begin{bmatrix} x & 1\\ y & 0\end{bmatrix}, B = \begin{bmatrix} z & 1\\ w & 0\end{bmatrix}$, for $x,y,z,w \in \Bbb{R}$.
I have observed by considering many random values of $x,y,z,w$ that:
If all the eigenvalues of $A^2B$ and $AB^2$ are less than one in absolute value $\implies$ $\det(AB...
Can anyone please explain why in a T1 space X, X being countably compact implies X is limit point compact?
Given any infinite subset of X, one can get a sequence {a_n} out of it. Noting that X=$\cup_i (X-\{a_i\})$, we have finitely many indices $i_1, i_2,…,i_n$ such that $X= \cup_{k=1}^{k=n}(X-\{a_{i_k}\})$. So one of the sets on RHS ( say$ X-\{a_{i_j}\}$ contains infinitely many terms of the sequence. I want to now prove that $a_{i_j}$ is a limit point of the sequence.
The K-distribution is a special case of the variance gamma distribution, which in turn is a special case of the generalised hyperbolic distribution. The generalised hyperbolic distribution is a superclass containing the student T's distribution, Laplace distribution, hyperbolic distribution, norm...
@AkivaWeinberger If this holds, then for every smooth function $f$ and every smooth vector field $V$, we have $0=[X,fV]=(Xf)V+f[X,V]=(Xf)V$. This implies that, for every smooth function $f$, we have $Xf=0$, thus $X=0$.
If $A = \begin{bmatrix} x & 1\\ y & 0\end{bmatrix}, B = \begin{bmatrix} z & 1\\ w & 0\end{bmatrix}$, for $x,y,z,w \in \Bbb{R}$.
I have observed by considering many examples of $x,y,z,w$ that:
If all the eigenvalues of $A^2B$ and $AB^2$ are less than one in absolute value $\implies$ $\det(AB+A+I)<...
@robjohn Indeed. :) Sage can save static plots as SVG (using matplotlib), but the resulting code is fairly huge, especially if it contains text, since matplotlib embeds the fonts into the SVG file. I do use that for a lot of stuff, but for basic things I prefer to write directly in SVG, or generate the SVG by my own Python code, which is what I did for the stuff I've posted here recently.
@XanderHenderson It's not a GIF! And I intentionally gave it a long period (15 seconds) so that it wouldn't be too disturbing for you. I suspect that the version with a 1 second period would probably be nauseating for you.
@robjohn They aren't permitted. There's an obscure bug that allows animated GIF avatars, but they tend to get removed fairly quickly, and I think the devs are working on fixing that bug.
If there's a direct proof and a proof by contradiction that are of similar complexity, it's generally preferable to give the direct proof. Unless you're specifically asked for a proof by contradiction. ;)
Sometimes, people get too focused on looking for a proof by contradiction and they overlook a more straightforward direct proof.
In the "real world", I suspect that most mathematicians attempt to prove things by first building some toy examples, and trying to figure out what parts of those examples generalize. But you don't even know if a result is true or not until you get a proof (one way or the other).
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@NotTfue Mathematics beyond high school is not just "plugging and chugging". It requires creativity. You build up a lot of tools, and you bring them to bear on problems. It is almost never obvious which tools are going to work until you try something.
In this case I wonder why do exam even test proving theorem. Most of the theorem I guess requires longer than an hour to come up. Verifying proof is already time consuming I can't think about coming up with one with fixed time interval.
@NotTfue The history of mathematics is a distinct discipline from mathematics itself. It is actually a pretty difficult area to study, as one needs to be both a skilled historian and a skilled mathematician to do it well.
Hence most mathematicians don't know a ton of history (except, maybe, in their own area).
And you need to deal with old notation (or no notation) and obsolete ideas. Eg, historical treatments of the theory of quadratic equations are tedious because they don't use negative numbers, so you have a bunch of separate cases.
@Thor @leslie This is something I've never seen before. I don't see anything obvious (since there isn't a projection from a Banach space to a finite-dimensional closed subspace, is there?).
@NotTfue What do you mean by "test proving theorem"? If you are referring to memorizing proofs of important theorems in your textbook, then they are basically important examples of some proving techniques, and you train by efforts of understanding these proofs.
@NotTfue Of course in real mathematics courses we're going to include proofs on exams. In some courses, all the questions will be proofs, although I always tested examples/counterexamples, as well. Some proofs may be regurgitations of proofs from the book/class. But some will be new things that are similar to what you've seen/done for homework. You learn only by practicing LOTS.
is it correct to write the following? Suppose f(x,y) depends on x,y. I want to integrate its PARTIAL derivative w.r.t x, where the integration variable is x. Can I write the following? Here is screen shot.
say $X$ is a Banach space and $A$ a finite-dimensional subspace. pick a basis $v_1,\dotsc,v_n$ of $A$. then, the maps $A\rightarrow\mathbb{R}$ mapping an element of $A$ to its $v_i$-coordinate are linear and trivially bounded, so they admit extensions $F_i\colon X\rightarrow\mathbb{R}$ by Hahn-Banach. then, $F(x)=\sum_{i=1}^nF_i(x)v_i$ is a bounded projection $X\rightarrow A$, I think.
but I never actually looked at a proof of Hahn-Banach, so take this with multiple grains of salt
Yes, I agree with that, and base fields are injective objects in the category of locally convex Hausdorff TVS's, thus so are their products (not necessarily finite).
Oh, maybe I misunderstood what you said. I thought that you construct splits of injections $V'\to V$ where $V'$ is of finite dimension via constructing splits of projections $V\to V/V'$.
@Yai0Phah No, I was referring to the linked question. One needs to project the domain onto the finite-dimensional kernel and split the finite-dimensional cokernel back to the codomain.
Fun fact: People who follow the Zoroastrianism have fire temples. In those fire temples they have 16 types of flames, they perform 1128 rituals on every flame.
Oh I see. I miss some of the foods I used to take almost daily before I came to college for Masters. One of them is this https://en.wikipedia.org/wiki/Dabeli apart from the one whose image I shared above.
My friend and I used to have this after coming from office.
It has been 20 years since I took any anthropology, but my recollection is that anthropologists typically define a religion to be a set of practices and beliefs pertaining to the supernatural.
@Thorgott Do you happen to know a book that treats number rings, as opposed to just considering rings of integers? (number ring being a subring of a number field)
@copper Those skills I actually do have to a reasonable extent. But my love of cooking probably would have worn out having to do it, rather than enjoying doing it for friends.
It's sort of like musicians. Very few make it as soloists, chamber musicians, or good orchestral players. But the super-talented ones tend to do well (albeit work their butts off).
