Moderators, to my knowledge, can't see votes and can't undo them. There's an automatic script that checks for serial voting every day (it should run in about three hours today); if he's downvoted more than one or two it should catch it and reverse the downvotes.
@PedroTamaroff It's not silly! I am trying to help the OP. BigM posted a whole load of rubbish. How is OP, presumably an undergrad, supposed to know that the answer is misleading?
When a plane contains the line $\overrightarrow{v}_1=\overrightarrow{a}+t\overrightarrow{u}$, does this mean that the vector $\overrightarrow{u}$ is parallel to the plane??
I might be mistaken, but I don't think there exists the notion of "a vector parallel to a plane", you could say the vector is embedded in the plane, yes; as well as the line.
I'm incorrect, the vector could be outside of the plane and, in that case, it would be parallel to it. It's just not as common as the notion or usefulness of perpendicular vectors in relation to planes.
I want to keep this as specific as possible while keeping the idea open to interpretation to different languages. Assume the class rectangle has the following items.
object Rectangle{
number x; (The center of the rectangle on the x axis.)
number y; (The center of the rectangle on the x axi...
for finite groups it's $\Bbb C[G]\cong\bigoplus_V{\rm End}_{\Bbb C}(V)$ as $V$ runs over irreducible representations. for compact the story is more complicated, and more complicated still for locally compact.
(if we're talking lie groups) it still forces the group to be finite dimensional, so it's not tooooo shocking. as far as i know there's nothing like it for infinite dimensional stuff.
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I think MSE shoud say "Thank you" for the contribution of the people, and ask their permissions for any rights.
Anyway, I don't wanna read such texts anymore because they annoy me far too much.
Some delete their accounts, but look ...
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let me clarify things further: i was in an interview and the interviewers presented me with a print out of a conic section and some basic geometric tools, and asked me to conclusively find out what it was. it could be anything except a circle
well if there's no numerical references, no equation, and only a section of the curve displayed, then i would look at how "bendy" the curve is-hyperbolas are "bendier"
you could i suppose guess at a focus, draw a directrix, and check distances-if it's a parabola, they would stay equal, if the drawing was at a scale where you could tell something-something you can also do to roughly guage the eccentricity
hey, i just thought of something: first i would construct the axis(that's easy), and then i would make two incident rays parallel to it. if they meet at a point on the axis, it's a parabola. isn't that right?
I am studying the same thing and intuitively... the statement used in Gödel's incompleteness theorem "This statement cannot be proved." Is an example of a statement that is verifiable yet unprovable. Now, if proof is equivalent to an algorithm (a supposition I am unsure of) then isn't this an instance of some language in NP but not in P?
Ya so take two points and find the difference between the set of points on the hyperbola with the two points and see that if the distance is constant@G-man
I said: "I dimly remember something about some guy named Oiler..."
which prompted his comment (after he smacked me).
someone had posed (just before) a query on proving the number of passes (saddle points) in relation to hills and valleys on a spherical planet. I dryly commented: "oh yeah, spheres have the two-number-thingie".
@Committingtoachallenge Not too good. Every day, I try to conquer my negative thoughts with positive ones. But it's hard to make progress. Even when progress is made, you can retrogress.
@Committingtoachallenge Well, sometimes people pronounce it a certain way because they heard from others, but it is not necessarily correct. So just pronounce it any way you want.
@DavidWheeler Yes, I am Jasper, the guy who keeps deleting accounts.
@DavidWheeler I usually have a blue square as my picture, so people recognise me, but there are also imposters, so beware.
I have been thinking about my entire past and my entire possible future lately. I am trying to be positive instead of negative about things. I need to believe that something extraordinary is possible.
If you wake up early in the morning, take a look at the rising sun-and try to forget about, all of the things you've done. There's another long night behind you, and another day's just begun. Let your eyes open wide, lay down all your fear. Give your heart to the sky, and all your sorrow will disappear.
How do u go about it....I understood that n and n+1 and all about that but after that it would be something like this no.... $n,n+1,n+1+1........../infinity$
Anyways thanks for directing me towards high order mathematics @BalarkaSen
Well I kept on studying differentiability and continuity yesterday but I don't get the difference between them......actually I had skipped the differentiability questions
If the right hand limit is equal to the left hand limit equal to the value of the function at a point a then it is a continuous function right @BalarkaSen
@robjohn @DanielFischer Hello!!! We have the function $$y(t)=\left\{\begin{matrix} 0 &, 0 \leq t \leq \frac{1}{2} \\ \frac{(t-\frac{1}{2})^2}{4} &, \frac{1}{2} \leq t \leq 1 \end{matrix}\right.$$
anyone know some good references on complex manifolds? i'd like to understand how the notion of a riemann surface generalizes to more than two complex variables, but my formal background isn't great
@MikeMiller: right now, i'm interested in the concepts and some of the terminology (mostly to be able to recognize the difference between terms like Kahler manifolds, K3 surfaces, etc.)
If there is a function which is converging when I keep on finding its derivatives like the first,second,......does that mean I can say the infinite derivative exist @BalarkaSen
the reason i have the complex case in mind is b/c in the Riemann surface context one is able to talk about Picard-Fuchs equations (which in that context i understand in terms of the cohomology of the surface)
presumably it's more general than that, but it's not something i know well
Let's start out with the auxiliary result
\begin{equation*}
\int_0^{\pi/2}\frac{x^2\log{(\sin(x))}}{\sin^2(x)}dx=\pi\ln{(2)}-\frac{\pi}{2}\ln^2(2)-\frac{\pi^3}{12}.
\end{equation*}
By the integration by parts all reduces to
$$\int_0^{\pi/2} \cot(x) (x^2 \cot(x)+2 x\log(\sin(x))) \ dx=\int_0^{\p...
@MikeMiller mostly i'm just trying to figure out how one takes the story i know in two complex variables (e.g. elliptic curves + picard-fuchs) and extends it to more complex variables
@evinda Compute the derivative for $x < 1/2$ and for $x > 1/2$, note that they have the same limit at $1/2$, and use a theorem that says if $f$ is continuous on $(a,b)$, differentiable on $(a,b)\setminus \{c\}$ and $\lim\limits_{x\to c} f'(x)$ exists, then $f$ is also differentiable at $c$, and $f'(c)$ is that limit.
right. and i suspect the new context would still have the fibers be algebraic varieties, but no longer something in 2 variables like $y^2=x(x-1)(x-\lambda)$
@EnjoysMath Open or not doesn't matter, for a finite union, you always have $$\overline{\bigcup A_i} = \bigcup \overline{A_i}.$$ A finite union of closed sets is closed, hence you have $\subset$, and each set on the right is a subset of the one on the left, hence $\supset$.
right. i still think that the same strategy that worked before is sensible, i.e. represent the sinc functions as fourier transforms and hope the integrals in $x,y,z$ are then simpler
and then try to evaluate the integrals over $x,y,z$. in the original case, those just simplify to delta functions, so you end up with a product of identical rectangle functions (if my quick mental review is right)
@SayanChattopadhyay if you had to call it part of something else, I'd say number theory. it also intersects with complex analysis and riemann surfaces. the more narrow topic of moonshine connects it with representation theory of sporadic simple groups too.
speaking of modular forms, i've heard people say modular forms can be visualized by looking at line bundles over modular surfaces. i've been dying to know what those are ever since i heard that.
I want to keep this as specific as possible while keeping the idea open to interpretation to different languages. Assume the class rectangle has the following items.
object Rectangle{
number x; (The center of the rectangle on the x axis.)
number y; (The center of the rectangle on the x axi...
Let's start out with the auxiliary result
\begin{equation*}
\int_0^{\pi/2}\frac{x^2\log{(\sin(x))}}{\sin^2(x)}dx=\pi\ln{(2)}-\frac{\pi}{2}\ln^2(2)-\frac{\pi^3}{12}.
\end{equation*}
By the integration by parts all reduces to
$$\int_0^{\pi/2} \cot(x) (x^2 \cot(x)+2 x\log(\sin(x))) \ dx=\int_0^{\p...
