Carl Brannen
  • Member for 10 years, 10 months
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Why is $\pi$ = 3.14... instead of 6.28...?
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46 votes

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 ...

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What is a real world application of polynomial factoring?
12 votes

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, ...

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Proof $1+2+3+4+\cdots+n = \frac{n\times(n+1)}2$
10 votes

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)...

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Is there any information on power series whose coefficients are only 0 and 1?
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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 $\...

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Calculate sum of an infinite series
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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.

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a question on definite integral $\int\limits_0^t e^{\alpha t}\sin(\omega t)\,\mathrm dt$
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5 votes

This is the imaginary part of a similar integral that is easy to compute.

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An inequality on a convex function
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 $n=12$...

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Hilbert's 19th problem: Why do we care?
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 ...

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Theorem on behaviour of real continuous functions on the integers
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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 ...

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Floer theory or Floer homology, an introduction for physicists needed
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 ...

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What applications of the Residue Theorem to real integration have had the biggest impact outside of pure math?
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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 ...

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densities and isosceles triangles
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 $4\pi$.

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Birth and Death Process Question (Queuing)
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 ...

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Counting eleven digit integers with the sum of the digits 2
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3 votes

Per the OP's edit, to remove this from the unanswered question list, the 11th entry is 20,000,000,000.

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Simple 4-cycle permutation
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 ...

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A line which bisects two sides of a triangle is parallel to the third.
2 votes

Use "like" triangles, i.e. side angle side, in proportion.

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Linear subspaces of rotation matrices
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 ...

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Confusion about joint distribution between two independent random variables
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 $x>y | x>1/2$.

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What is a good language to develop in for simple, yet customizable math programs?
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 ...

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Finding best response function with probabilities (BR) given a normal-matrix representation of the game
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.

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How to find that two circles form a knot?
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 \...

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Equilateral polygon in a plane
2 votes

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 $\pi/n$ ...

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How to graph equation
2 votes

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?

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Given two basis sets for a finite Hilbert space, does an unbiased vector exist?
1 votes

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 ...

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Find the equation of the plane passing through a point and a vector orthogonal
1 votes

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

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Does solution of $ x=\sum_{n=0}^\infty e^{-A_n/x}$ exist?
1 votes

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 $A_n$...

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A Logic for Digital Circuits
1 votes

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 ...

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Ratio Language Help
1 votes

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 ...

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Expected value of the product of the sum of a specific distribution
1 votes

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: $(K\;E(a))^L$

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Books and Papers that have treatment of properties like Idempotence and related operations
1 votes

I work on elementary particle theory. Calculations using "Feynman diagrams" are done using "propagators" (which are Green's functions for wave functions, i.e. solutions to the Klein-Gordon or ...

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