Carl Brannen
• Member for 10 years, 10 months
• Last seen more than 1 year ago
• Pullman, WA

For mathematicians, $2\pi$ is a more natural number than $\pi$ because this is the circumference of the circle. The value $2\pi$ appears in things related to the circle such as Fourier transforms (as ...

You need polynomial factoring (or what's the same, root finding) for higher mathematics. For example, when you are looking for the eigenvalues of a matrix, they appear as the roots of a polynomial, ...

For example, $$X = 1+2+3+4+5+6$$ Then twice $X$ is $$2X = (1+2+3+4+5+6) + (1+2+3+4+5+6)$$ which we can rearrange as $$2X = (1+2+3+4+5+6) + (6+5+4+3+2+1)$$ and add term by term to get $$2X = (1+6)+(2+5)... View answer Accepted answer 7 votes If there's only a finite number of a_j=1, then, well, the series converges for all x. If there's an infinite number of a_j=1 then it will converge for -1<x<+1 due to comparison with \... View answer Accepted answer 7 votes Try considering what happens when you differentiate the following with respect to x: 1/(1+x) = 1-x+x^2-x^3+x^4... That should get you thinking in the right direction. View answer Accepted answer 5 votes This is the imaginary part of a similar integral that is easy to compute. View answer 5 votes Let \{a_j\} be a set of n numbers. Then the 4th power of the average of \{a_j\} is smaller than the average of the 4th powers of \{a_j\}:$$(\Sigma_ja_j/n)^4 \le \Sigma_j a_j^4/n.$$With n=12... View answer 4 votes Variational principles are incredibly important in physics. See the book "Variational Principles in Physics". For example, particle physics is typically done with a "Lagrangian". Newton began with ... View answer Accepted answer 4 votes \sin(\log(k^m)) = \sin(m\;\log(k)) is a subsequence of the form \sin(m\alpha) and may be easier to work with. The values of \alpha where the sequence \sin(m\alpha) is convergent have to be ... View answer 3 votes The reason I had asked this question is because a Floer theory proof was needed for a physics paper I was writing and I didn't want to include a proof I didn't understand. Eventually I gave up on the ... View answer Accepted answer 3 votes It's used all the time in physics, especially in quantum mechanics. The math physics text book "Mathematical Methods for Physicists" by Arfken & Weber has three applications in chapter 7.1 in the ... View answer 3 votes It might be useful to know the differential for area in spherical coordinates:$$d\Omega = \sin(\theta)\;d\theta\;d\phi$$And recall that the total surface area of a sphere is 4\pi. View answer 3 votes Re: "1) Evaluate the long-run average values for a) the number of arriving customers per hour who wait before they get served" For this to happen, someone has to arrive when there's 2 or 3 people ... View answer Accepted answer 3 votes Per the OP's edit, to remove this from the unanswered question list, the 11th entry is 20,000,000,000. View answer 3 votes I'm strictly shooting from the hip (i.e. this is just instinct), but it might help if you consider the following: (1) The 4th powers of 4-cycles are unity. I.e. (1234)^4 = e where e is the ... View answer 2 votes Use "like" triangles, i.e. side angle side, in proportion. View answer 2 votes Let's see if we can get 9 degrees of freedom together. First include the identity and the permutation matrices. This is a total of 6 matrices but they only include 5 degrees of freedom (as the matrix ... View answer 2 votes Try thinking about \int_0^1\int_0^1 h(x,y)\;\;dx\;\;dy where h(x,y) is a function that is true when x>y | x>1/2. View answer 2 votes This should be a community wiki. That said, I love Java because by the time I get it to compile, it runs due to the language being tight in terms of types. In addition, it has decent graphics and ... View answer 2 votes Why don't you compute the average payoff for player #1 for each of his three choices? Clearly he's going to choose the one that is largest. View answer 2 votes Gauss's linking integral will give the linking number:$$N = \frac{1}{4\pi}\int\int \frac{\vec{f}(\sigma)-\vec{g}(\tau)}{|\vec{f}(\sigma)-\vec{g}(\tau)|^3}\cdot \left(\frac{d\vec{f}}{d\sigma}\times \...

Maybe this is a start: Given: an n-step walk in the plane with each step of length 1 that begins with a step to (1,0) and ends at the origin, and all angles between steps being a multiple of $\pi/n$ ...

My instinct says you want to get rid of that $t$. Maybe you could find the value of $t$ that makes $f(x,y,t)$ maximum?

I've written a physics paper, "Unitary Mixing Matrix Theory" (for submission to Jour. Math. Phys.) that uses Sam Lisi's proof. I translated the proof into "physics language", and it's not unlikely ...

Try solving $(P-r)\cdot v=0$, where $r =(x,y,z)$.

Divide both sides by $x$ to get: $$1 = \Sigma\frac{e^{-A_n/x}}{x},$$ As $x\to \infty$, the left hand side is 1 while the right hand side goes to zero for each $n$. This might give a hint on what $A_n$...

Modern practice is to prefer designs where there are a relatively small number of "clock domains". A clock domain is a collection of sequential logic which is all clocked by the same clock net. Your ...

The ratio will be A/B = "worked" / "don't work". To see it my way, think about whether this ratio should be larger than 1. Assume that working pays better. So I want the larger number, "A" divided by ...

Since $E(XY) = E(X)E(Y)$ for random and independent variables as can be seen by: $$\int_x\int_y\;xy\;f(x)g(y)\;dx\;dy = \int_x xf(x)\;dx\int_y yf(y)\;dy$$ Didier Pau's answer is correct: $(K\;E(a))^L$