?

∫∫[0->∞] e^(-(x+y)^2) dx dy = 1/2.

Is there a way to see this without doing the full integral?

∞

∫∫ e^(-x^2-xy-y^2) dx dy = 2π/√3.

-∞

How many paths are there from Start to Goal in n-dimensional space?

Rules:

(1) Paths are from Start [0, 0, 0, ... , 0] to Goal [a1, a2, a3, ... , an] where a1, a2, a3, ..., an, are positive integers.

(2) Only positive integer steps are allowed.

(e.g., For example in 5D space, we cannot pass through [0, 1, 2, 2, 4] after [0, 1, 2, 3, 4], because one of the coordinates is less.)

(3) 'Diagonal' steps are not allowed. (i.e., we cannot vary 2 or more coordinates in one move. e.g., we cannot go from [0, 1, 2, 3, 4] to [0, 1, 3, 4, 4] in one move.)

(4) We cannot pass through a point unless at least one of the coordinates is zero or the goal value (i.e., a1, a2, etc). (e.g., If we are in 3-dimension, this means we cannot go inside a rectangular box.)

m

∑ cos(2kΘ) = ( sin(mΘ) cos((m+1)Θ) ) / sin(Θ)

k = 1

for any positive integer m and any θ that is not an integer multiple of π.

I'm looking for a webpage where an audio clip for the pronunciation of "Hilfsgröße" is available.

I'm seeking a geometrical proof or interpretation of the above equation. Any ideas?

I'm also very interested in the derivation.

What is an antonym of "free" that can fill in the blank?

Suppose there's a piece of metal inside a closed container filled with air, and the whole set is on a scale. Let's say the scale read W kg.

(a) First we levitate the piece of metal with a magnet. How will the scale read compared with W?

(b) Then we drop it. When it's falling, how will the scale read compared with W?

My guess:

(a) less.

The weight of the piece of metal is suspended by the magnet.

(b) same.

I think of the air inside as water, guessing it doesn't make a difference. Then falling can be thought of as swimming in the water. And I can imagine that if I was swimming in a fish tank, the tank should weigh tank+water+me.

I'm not sure if they're correct, especially of (b). I'm seeking more convincing reasonings about these questions.

For example, the operational definition of the second uses the particular type of atomic clock. As Wikipedia says, it is supposedly the most "accurate" known. But how can one compare the "accuracy" of clocks?

I assume the procedure cannot (explicitly or implicitly) depend on the use of any type of clock to avoid being circular.

I don't understand the following part of the proof in this article: http://en.wikipedia.org/wiki/Spherical_law_of_cosi...

"..., whose direction is given by the component of v perpendicular to u. This means:

t_a = ( v - u (u ・ v) ) / | v - u (u ・ v) | = ( v - u cos(a) ) / sin(a)

where for the denominator we have used the Pythagorean identity sin^2(a) = 1 − cos^2(a). "

Could someone provide explanations and/or intermediate steps?

This is a particular part of a proof that K5 (complete graph of 5 vertices) is not planar. And I don't see how to go from the left side of '->' to the right side. Can anyone explain this?

I understand that if I remove 2 faces from an icosahedron, the resulted polytope cannot be convex because 6 vertices must meet. But I wonder if there's a way to prove the nonexistence combinatorially, perhaps using the Euler's formula.

The inside and outside of the wristband has the grid with 3 rows and 6 columns. (we call the direction of the hole of the wristband "column.") And we have 5 colors to color each box. How many distinct color patterns can we make?

Additional rules:

-The wristband can freely be rotated in any directions. And if 1 pattern looks the same as another when rotated, they are counted as 1 pattern.

-The wristband can be turned inside out. And if 1 pattern looks the same as another when turned inside out, they are counted as 1 pattern.

-The wristband can be rotated and then turned inside out, or vice versa. And if 2 patterns look the same after rotating and turning inside out, they are counted as 1 pattern.

Fit f(x) = Σ[k=1, n] a_k*g_k(x) to n points, where

①g_1, ..., g_n are linearly independent real functions given.

②all points given are in the domain of all g_1, ..., g_n.

③a_1, ..., a_n are real numbers you pick.

Intuitively, I can set up n simultaneous equations in n unknowns. But is it always the case?

Let's say you're on the 50th floor of a building, and you can see some buildings below. How do you estimate the height of the buildings below, as accurately as possible?

How about when you want to estimate the height of a skyscraper, which is taller than where you are? Can the same method be used?

(1) How do you tell if a function is absolutely symmetric/antisymmetric, locally symmetric/antisymmetric, or asymmetric?

(2) For a function f(x). How do you find the points of symmetry/antisymmetry if they exist?

(3) For a function f(x_0, x_1, x_2, ..., x_n). How do you find them if they exist?

Links for related materials are also appreciated.

Thank you.

For example,

http://www.wolframalpha.com/input/?i=INT%28%281-2x...

Can you tell if a definite integral is 0 just by looking at the integrand?

If you know some methods, I'd like you to explain why they work.

Links to websites where I can learn the mechanisms are also appreciated.

Thank you.

Why is ST so popular among physicist without experimental confirmations?

I wonder if there're people working on unification ideas other than ST.

If you're pro-ST. Please explain why.

If you're anti-ST. Please explain why.

Thanks.