@Nicholas The new distance and the initial distance are proportional to each other by some factor $k$. If the slope is constant, then that means the new point $(x_2^{'},\,y_2^{'})$ is also proportional to $(x_2,\,y_2)$ by the same factor $k$. Can you take it from there and derive an expression for the new point?
@DanielFischer If we want to compare two recurrence relations of the form $T(n)=aT\left( \frac{n}{b} \right)+f(n)$, do the two relations have to have the same b?
@DanielFischer Here at page 11 : http://www.scribd.com/doc/55552407/Master-s-Theorem#scribd I haven't undersood why since f(n) is the same in both recurrence relations , we try the first case.. Do you have an idea?
I'm reading Atiyah and Macdonald for commutative algebra, and Serre's "Linear Representations of Finite Groups" for representation theory. Hoping to make it through all of the first and the first 8 chapters of the second.
Haha, perhaps. I think I learn best by struggling through ideas a bit first. I read through Mendelson's "Topology" some time ago and did every exercise. Things were presented so thoroughly and clearly, I don't think I ever really internalized things for myself.
Maybe so. But I'd like to be well prepared going in, and really make the most of my graduate years. I've only seriously studied math for about two years now, and many know a great deal more than me. I'd like to bridge that gap.
It doesn't have to happen over night. But sooner rather than later would be good. :)
Thanks. :) Well, through Mendelson, I was exposed to the basic theory of topological spaces. Things like interior, closure, continuity from a topological point of view. Homeomorphism, weak and strong topologies. Basic notions of connectedness and compactness.
As far as Willard goes, just started. Learning some set theory, since I never did a lot of the basic problems, and would like to just have seen them before. Now onto metric spaces.
I have to describe the operation of the processure "RADIX SORT" at the following list of words:
COW, DOG, SEA, RUG, ROW, MOB, BOX, TAB, BAR, EAR, TAR, DIG, BIG, TEA, NOW, FOX
What does it mean to describe the operation of RADIXSORT?? To write how the list will look like at each step??
O...
@AlexW: Note that the movie turned into a standard Hollywood action movie the moment Donald started to help writing... and when Donald was no longer helping with the writing...
@Mike: that's what I figured. A kind of meta-satirization. Funny you should say that, since I was thinking "this is what the movie would look like if Donald wrote it..."
Still, it was so incongruous with Charlie's vision of the movie, which I was really enjoying. Disappointment, loneliness, self-loathing.
Well, two reasons. It's not really just a difference in vision, but also a difference in life philosophy; and with the exposure Charlie came out in the end to be a happier person and a better writer. And two: it would really be hard to write a movie where nothing truly happens, haha
The only good piece of comprehensible media I've aeen where nothing happens is Waiting for Godot.
Also yeah, I thought Fargo was hilarious. I was cracking up at the woodchipper scene, and I learned that people from Minnesota are inherently funny.
The whole sequence to the end is great. Like you said, the woodchipper scene is hilarious. And then when he's in the car with Marge, and she's saying she just doesn't understand... man.
William H. Macy. Agreed, just perfectly done.
Supposedly he insisted on it to the Coen brothers. I'm glad he did, he nailed it.
@MikeMiller OK, let's get this straight : I don't understand why Hatcher talks about linear simplices in the proof of excision at all. All what we need is a chain htpy equivalence between $C_n(X)$ and $C_n^\mathfrak{U}(X)$. The chain map $C_n^\mathfrak{U}(X) \to C_n(X)$ is clear. The other map $C_n(X) \to C_n^\mathfrak{U}(X)$ can be obtained by taking a singular simplex, subdividing like mad until I get every piece to be inside the cover by Lebesgue.
Don't see how linear chains interact.
[I'm actually a bit embarrased to ask this question : this is really basic stuff and I should have made this clear to myself long ago.]
@BalarkaSen: He has to do the calculations he did there at some point, whether it's in that section or later. The point of doing it in a convex open subset of Euclidean space is just notational convenience, I think.
@LeGrandDODOM http://math.stackexchange.com/questions/1045173/which-is-the-greatest-integer-value-of-a-for-which-a-is-asymptotically-fas/1104806?noredirect=1#comment2402348_1104806 So would you say that the greatest value of a is 48?
@DanielFischer What about this problem? Given a regular polyhedron such that two consecutive vertices have integer coordinates and a third vertice has integer coordinates, prove the polygon is a square
Oh, yes it does matter. $\sigma_\sharp$ is the pushforward $C_k(\Delta^n) \to C_k(X)$, sending $f$ to $\sigma \circ f$. The formulas he's using to show that things are chain maps or chain homotopies should depend on the calculations above - because $\Delta^n$ is homeomorphic to a convex subset of Euclidean space.
I think he's only even using the linear chain version on the identity map $\Delta^n \to \Delta^n$, so it remains that the generality was unnecessary; but it gives an idea of what's going on in the proof if he does it like he did there. I no longer object to it.
I am asked to describe the operation of the processure BUCKET SORT at the array $$A=\langle 0.75, 0.13, 0.16, 0.64, 0.39, 0.20, 0.89, 0.53, 0.71, 0.42, 0.19 \rangle $$
dividing the interval $[0, 1)$ in $10$ subintervals of equal length.
In my book I found the following algorithm:
BUCKETSORT...
@MikeMiller Oh, I didn't mean that. I meant that definition of Hausdorff spaces isn't hard, i.e., it doesn't take a lot of background to think about them.
But when you are struck by brilliant ideas, be sure to show your work. Otherwise, you'll be looking back and scratching your head about the result's validity (that's what's happened to me with this work on polylogs)
@DanielFischer Could I ask something. If you are given a function defined by $g(x,y): = \{ 4 \text{ if }x= 0\text{ or } y = 0 \text{ and } x^{2} + y^{2}~~\text{ otherwise} \}$ then does it follow that $g_{x}(0,0) = 0$ by simply noting that $g_{x}(x,y) = \{ 0 \text{ if }x = 0 \text{ or }y = 0 ~~\text{ and }2x ~~\text{ otherwise} \}$. I don't really see how we know that $g_{x}$ exists?
@DanielFischer A ok, we haven't said anything about it in class. We have the recurrence relation $T(n)=7T\left( \frac{n}{2}\right)+n^2$ and the solution is $T(n)=\Theta(n^{\log_{4}{49}})$. We also have the recurrence relation $T'(n)=\alpha T'\left( \frac{n}{4} \right)+n^2$.
$a=\alpha, b=4, f(n)=n^2$
$n^{\log_b a}=n^{\log_4 \alpha}$ For $\alpha>16$, $f(n)=O(n^{\log_b{\alpha}}- \epsilon), \epsilon>0$
So for $\alpha>16$: $T'(n)=\Theta(n^{\log_4{\alpha}})$
So that $A'$ is asymptotically faster than $A$, it has to hold $n^{\log_4{\alpha}}< n^{\log_4{49}} \Rightarrow \log_4{\alpha}<\log_4{49…
@JohnJack $g_x(0,y)$ exists only for $y = 0$ and $y^2 = 4$. You need to take the definition of the partial derivatives. Fix the $(x_0,y_0)$ you're interested in, and look at $$\frac{g(x,y_0) - g(x_0,y_0)}{x-x_0}.$$ If the limit exists, that's your partial derivative, otherwise $g_x$ doesn't exist at $(x_0,y_0)$.
@evinda If $\alpha$ is restricted to integer values, and "faster" means "strictly faster", it's correct. If $\alpha$ can be any (positive) real value, then there is no largest $\alpha$ such that $A'$ is asymptotically strictly faster. If "faster" is to be interpreted in the weak sense, then $\alpha = 49$ is the largest value.
I have to describe the operation of the processure "RADIX SORT" at the following list of words:
COW, DOG, SEA, RUG, ROW, MOB, BOX, TAB, BAR, EAR, TAR, DIG, BIG, TEA, NOW, FOX
What does it mean to describe the operation of RADIXSORT?? To write how the list will look like at each step??
O...
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