So in my notes it says that when you write $g(t)= (t-1)u(t-1)-(t-1)u(t-2)$, the notes say that this makes the function turn (t-1) ON at t=1 and turns (t-1) OFF at t=2. The problem I have with wrapping my head around this is that, why does it turn off at t=2, because at t=2 only the second term $(t-1)u(t-2)$ gets turned off, the first term remains, so how does the whole thing turn off?
it comes from starcraft 2, where you call an expansion (a new base with minerals to gather) which has been filled with "enough" workers "saturaded". @AkivaWeinberger
i guess i'm just mad, that my High School degree doesn't give me a shit in studiing math. you can easily pass thourgh High School without any creativity.
@AkivaWeinberger but is an excercise of the same variety simply exchanging variables or slightly changing what's asked for? (i.e. speed instead of distance)
Another formulation: what do you think is a question of the same variety? Suppose someone asks: "i can bake 1 cake in 2 hours, how many cakes will i bake in 4 hours?"
and how do i give someone the tools to do it by himself without spoiling the result? in case of "i can bake 1 cake in 2 hours, how many cakes will i bake in 4 hours?" telling them about the rule of three?
Suppose that Y is subspace of normed space X, then if we have $T \in B(Y,l_{\infty})$ then there exists $S \in B(X,l_{\infty})$ such that $S|_Y = T$ and $||S|| = ||T||$
we just consider the sequence $T_i : Y \rightarrow K$ and use hahn banch theorem seperately on each i.
cool stuff happen in infinite dimension it is actually much more rich than finite dimension stuff.
I know that recursive equation of this algo is
$T\left ( n \right )=2T\left ( \frac{n}{2} \right )+2 $
Given that $T\left ( 1 \right )=0,T\left ( 2 \right )=1 $
and its solution is also given here,ijust want to clear my doubt where i am stuck at.
I solved this eqtn as--:
$\Rightarrow T\left...
Hey everyone :-) I've a quick question. Are the parametric coordinates for any parabola given by $(at^2, 2at)$ ? ($a$ is one quarter of the length of the latus rectum and $t$ is the parameter)
https://www.kickstarter.com/projects/2086970389/whisk-wiper-whisking-made-easy It feels like there's a certain maths principle being exploited here, however i am not sure what it is
I know that recursive equation of this algo is
$T\left ( n \right )=2T\left ( \frac{n}{2} \right )+2 $
Given that $T\left ( 1 \right )=0,T\left ( 2 \right )=1 $
and its solution is also given here,ijust want to clear my doubt where i am stuck at.
I solved this eqtn as--:
$\Rightarrow T\lef...
@Secret I don't see why there would be. It is just one of those simple ideas that are obvious in hindsight but requires someone to be the first to come up with them
I don't know, I might be distracted by that nice flower pattern formed when the whisk slide out of it and somehow it exploit the geometry of something of the whisk
We have that a particle is moving with the effect of the gravity $F=-mgj$ and it has the initial position $r(t=0)=10j$ and initial velocity $r'(t=0)=10i$.
We assume that $m=g=1$.
How can we calculate the velocity when it falls on the ground ($y=0$) ?
Do we use the formula $F=ma=m\frac{du}{d...
When we have that the motion of a particle in polar coordinates is givwn by the realtions $r=\cosh (t) \ \ \theta=t$, do we have that the acceleration is equal to $r''=\cosh (t)$ ?
I know that recursive equation of this algo is
$T\left ( n \right )=2T\left ( \frac{n}{2} \right )+2 $
Given that $T\left ( 1 \right )=0,T\left ( 2 \right )=1 $
and its solution is also given here,ijust want to clear my doubt where i am stuck at.
I solved this eqtn as--:
$\Rightarrow T\lef...
@TedShifrin Bonjour Ted, j'ai regardé livre de Stephen Smale, il est bien fait. Cependant, je n'ai pas trouvé d'exercices où Cauchy-Lipschitz; théorème d'explosion en temps fini; Inversion locale soient mélangés. Aurais-tu, par hasard, des noms de Théorème (ou exercices) utilisant ce mélange ? Par exemple, il y a le théorème d'Hadamard qui stipule que si f est de classe C^2 et que l'image réciproque de tout compact est compact alors f est un difféomorphisme. Merci d'avance
Note if we impose the axiom aa=a and aq=e, then ae=e can be derived. After that, when all possible words were written out, it is clear (assuming the algebra is non abelian but associative) that defining the word qa will give most of the entries in the table
Caption: Therefore one example is setting the axiom qa and then ee, and you lock all the entries in the cayley table. There are other choices, but note the correlations introduced when a axiom is set
I remember from high school that one can construct the image of the inversion at a circle by drawing chord and tangents (or the other way around, depending on whether we start inside or outside of the circle). I want to prove that this construction indeed yields inversion at a circle, and I see how I can do this with this theorem (a^2 = pc) of which I don't know the English name.
no, I want to prove MP' = r^2 / MP. I know it follows directly from a^2 = pc (given in the first picture), but I want to avoid anything related to Pythagoras
@BalarkaSen: take P inside the circle. construct the chord perpendicular to MP, intersect with the circle. construct tangents. their intersection is P'. theorem: MP' / r = r / MP
(and the same should work if P is outside the circle)
@Huy The quadrilateral you have in your picture is symmetric about MP', not? Because the two sides are radius, and the other two are tangents from the same point, which have the same lengths.
So the central angle at M is divided into two equal angles by MP'. That should imply similarity of the appropriate triangles.
Well, the important part is that the angle at P is divided into two equal parts by MP', but w/e.
I'm reading about disc/vector bundles. If $p \colon E \to B$ is a disc bundle with $B \subseteq E$ and $p$ is the identity on $B$, then why is the inclusion of $B$ into $E$ the zero section of $p$?
@abenthy Eh, the zero section is just the canonical way the base space is included in the total space of the disk/vector bundle. There may be other inclusions if eg $E$ is the trivial bundle.
How is a child supposed to make a scale replica of Egypt one week and then write a book report on Martin Luther King the next and STILL have time to make a working baking soda volcano?
I've seen $f^n$ for iterated functions, but it's usually stated what it means the first time the symbol appears since it could be interpreted in different ways
@saturatedexpo Balarka's given notation is a good choice as it's not ambiguous, but $f^2$ is fine as long as $f$ isn't something like $\sin$ where the notation $\sin^2 x$ means $(\sin x)^2$.
while looking it up I found out that there's a university in France which is also called UGA @Ted, now I only need to find out where Georgia is because my knowledge of the US is pretty terrible
I've also seen $f^n$ for the $n$-th derivative, but that's usually written $f^{(n)}$, just use the symbol you prefer but write explicitely what it means the first time you do
I know that recursive equation of this algo is
$T\left ( n \right )=2T\left ( \frac{n}{2} \right )+2 $
Given that $T\left ( 1 \right )=0,T\left ( 2 \right )=1 $
and its solution is also given here,ijust want to clear my doubt where i am stuck at.
I solved this eqtn as--:
$\Rightarrow T\left...
We have that a particle is moving with the effect of the gravity $F=-mgj$ and it has the initial position $r(t=0)=10j$ and initial velocity $r'(t=0)=10i$.
We assume that $m=g=1$.
How can we calculate the velocity when it falls on the ground ($y=0$) ?
Do we use the formula $F=ma=m\frac{du}{d...
The area of triangle formed by the lines
$$a_1x + b_1y + c_1=0$$
$$a_2x + b_2y + c_2=0$$
$$a_3x + b_3y + c_3=0$$
Is $$\frac {\Delta^2}{2\lambda_1 \lambda_2 \lambda_3}$$
Where
$$\Delta =
\begin{vmatrix}
a_1 & b_1 & c_1 \\
a_2 & a_2 & c_2 \\
a_3 & b_3 & c_3 \\...
