I have to use a Divide-and-Conquer procedure for the computation of the i-th greatest element at a row of integers. I have tried the following:http://pastebin.com/rPnc74gd The time complexity will be $T(n)=\Theta(n)+T(q-1)+T(n-q)$, right? But how can we compute the latter without knowing the value of q? @cirpis @robjohn
Steps to build Huffman Tree
Input is array of unique characters along with their frequency of occurrences and output is Huffman Tree.
Create a leaf node for each unique character and build a min heap of all leaf nodes (Min Heap is used as a priority queue. The value of frequency field is used t...
eh, doesn't strike me as terribly surprising. once you find one convergent family, you can add a lot of different things to it without spoiling the convergence
Let $p \in M$ be a a point in a manifold and let $\varphi^X_t$ and $\varphi^Y_t$ be the local flows of the vector fields $X$ and $Y$ respectively. Define the commutator of flows: $\alpha(t)= \varphi^Y_{-t} \varphi^X_{-t}\varphi^Y_t\varphi^X_t$. I'm trying to prove:
$$\left .\frac{d}{dt} \right|_...
one is then interested in taking certain sums of a certain function evaluated at the roots of that characteristic polynomial. that can be written as a contour integral, and one can pull out the leading behavior (terms of order $N$ and $N^{0}$) pretty easily
woops, denominator in $p_N$ should've had $2N+1$ not $2n+1$
now, what's intriguing is the following limit
suppose i take $N$ to be very large. then from numerical results i know i can approximate that function as $N^{-1} f(N(\lambda-1))$ for some smooth function $f$
the question is how that limiting behavior arises, and in particular if one can find a closed-form for it
You can study some of it from the chapters at the back of Dummit-Foote. I went through those a long time ago, just the basics.
Now what's amazing is that I recently found it mentioned on Hatcher that group homology of a finite group $G$ is actually just the usual homology of $K(G, 1)$.
but, if i've got a space with first homotopy group being $G$, then isn't the first homology group the abelianization of $G$? (that's the hurewicz (spelling?) lemma, i think)
so if the group homology of $G$ is actually just the homology of $K(G,1)$, i'd think that means that the first group homology of $G$ is the first homology of $K(G,1)$
after reading this very interesting fact from hatcher, it seems to me that there should be some kind of connection between (etale?) (co)homology of Spec(Q) and galois cohomology of Gal(\bar Q/Q), even though i don't know any of the etale stuff, neither what Spec(Q) means.
@Semiclassical ah. well, homology with nonabelian coefficients wouldn't serve the fundamental property for which they are studied in the topological context -- the group won't be abelian.
So I want to find the interaction between two spheres. My intuition is to calculate the interaction between one point of the sphere and the whole of the other sphere and then integrate over the first sphere.
but I don't know how to define the other sphere geometrically relative to an arbitrary point on the first sphere
i may have gotten on the wrong footing there, since while searching for some physics instance of it i got onto the topic of "group cohomology" which has made some impact in condensed matter physics
e.g. the last bit of this section of Wikipedia's page on group cohomology
They are given an area $T$, restricted by $x^2/9 + y^2/4 = 1$, $z = x^2+y^2$ and the $xy-$plane. The problem is to calculate $$ \iint_{S_1} \hspace{-0.65cm} O \qquad F \cdot \hat{n}\,\mathrm{d}S $$ where $S_1$ is the cylindrical part of $T$ (eg. exclude the top and the bottom of the closed surface).
Ugh.... Was anyone taught this property in basic calculus or earlier? $$\frac{a}{(bx^2+c)^2}=\frac{a}{2c}\left(\frac{1}{bx^2+c}-\frac{bx^2-c}{(bx^2+c)^2}\right),\; a,\,b,\,c,\,d\in\mathbb{R}$$
Wait, partial fraction decomposition leads nowhere.... For example, solving $\frac{3}{(4x^2+5)^2}=\frac{Ax+B}{4x^2+5}+\frac{Cx+D}{(4x^2+5)^2}$ yields $A=B=C=0$ and $D=3$...
@ᴇʏᴇs I put on a "front" in social settings, so as to avoid ostracism. It's a lose-lose scenario; pretending to have a high opinion of myself lowers it even further, but the same happens if I perpetuate the low opinions of myself
@teadawg: I don't wish to meddle, but my advice would be to just enjoy math and your friends as much as you can without thinking about what other people think ... obviously, don't go out of your way to be an asshole :) I've enjoyed interacting with you here.
BTW, @teadawg, when Owatch was doing partial fractions the other day, he was clearing denominators and saying equality of polynomials necessitates equality of corresponding coefficients. This gives a system of linear equations. This is the standard way we teach the material in Calc II. You wanted to be sneaky and substitute values of $x$ to make it easier; that works most of the time (and is sort of what we do with residues in complex analysis), but it doesn't work all of the time.
@Mike: The perils of the electronic era. I just noticed I had set the due date for the WeBWork assignment to be April 28, rather than March 28. I just changed it :P Interestingly, none of the students called my attention to that. I suspect they all knew :)
@TedShifrin Hopefully you can help me clear up something :p It should be easy for you as it is multicalc. Trying to calculate the flux out of an area bounded by $z = x^2 + y^2$, $x^2/9 + y^2/4=1$ and thhe $xy-plane$. The question asks just for the flux out of the side, eg the cylindrical part
@TedShifrin The method I was using was induction. We've shown it for the case n=1. Then $\displaystyle\tilde{\frac{d^{k+1}x}{dt^{k+1}}} = \tilde{\frac{d}{dt}(\frac{d^kx}{dt^k})} = \frac{d}{dt}((-i\omega)^k\tilde{x})$
@TedShifrin Yeah, but not here. It is just the side. Later one is to prove the flux out of the top by using the flux out of the side, and the divergence theorem
People who do serious PDE need to use multivariable analysis/vector calculus stuff.
At the undergraduate level, you use some vector calculus in complex for sure, and multidimensional real analysis (the stuff in my course) is very important. But a first real analysis course is just single-variable.
@TedShifrin I did something like $\int_0^{2\pi} \int_0^1 \int_0^{(3 r \cos \theta)^2 + (2 r \cos \theta)^2} 6r \,\mathrm{d}r,\mathrm{d}\theta$ after using the jacobian and etc.
Let $R \supset \Bbb{Z}$. Now consider $S = \{ \alpha \in R: \alpha = \sum_{n=1}^{\infty} \dfrac{a_n}{n^x}$ for some $x \in \Bbb{Z}, a_n \in \Bbb{Z}\}$. It is a ring because of Dirichlet convolution arguments which are easy. What can be done with this ring?
Between the usual metric on R and the absolute value function on R , is one of those more fundamental than the other ( is there an intrinsec hierarchy ) ?
Or are both notions equally as fundamental ?
By one notion A being more fundamental than the other notion B i mean that its more natural to consider notion B in terms of notion A , than the opposite./
For instance, we can see either of those in terms of the other
d(x,y) = |x-y|
and |x| = d(x,0)
But is any of those hierarchies more fundamental than the other in the sense that we should naturally choose that hierarchy in organizing the absolute value function on R and the usual metric on R ?
@Chris'ssis Well, it fluctuates due to various factors. I have made some plans to solve my problems in the next few months, but I am scared there might be obstacles hard to overcome.
@Chris'ssis Actually, when I look at all the suffering in the world, it's really scary. So many people living with physical illness, mental illness, poverty, etc.
@JasperLoy Well, as I told you many times, you can focus on the positive things, not on the negative ones. It depends on you how you wanna look at the world.
Let F be a compact subset of a closed interval [a,b]⊂R. Let Γ⊂R^2 be the graph of the function δF(x)=dist(x,F). I am trying to find a formula for the lenght of Γ. For the case that F does include a and b then the lenght is sqrt(2)*m(F^c)+m(F). However I cant find a formula for the lenght in any other case. I need a formula only including m(F^c) and m(F), a and b, where m is the lebesque measure. Can ayone help?
I was trying to figure out the intersection of $z = x^2 + y^2$ and $x^2/a+y^2/b = 1$
$$ r(u,v) = \begin{cases} x = a u \cos v \\ y = b u \sin v \\ z = (a u \cos v)^2 + (b u \sin v)^2 \end{cases} \ , \quad u \in [0,1] \ v \in [0,2\pi] $$
Steps to build Huffman Tree
Input is array of unique characters along with their frequency of occurrences and output is Huffman Tree.
Create a leaf node for each unique character and build a min heap of all leaf nodes (Min Heap is used as a priority queue. The value of frequency field is used t...
Let $R \supset \Bbb{Z}$. Now consider $S = \{ \alpha \in R: \alpha = \sum_{n=1}^{\infty} \dfrac{a_n}{n^x}$ for some $x \in \Bbb{Z}, a_n \in \Bbb{Z}\}$. It is a ring because of Dirichlet convolution arguments which are easy. What can be done with this ring?
