How is it possible to display the crosshairs and the spaceship correctly in a 3rd person space shooter game?

There are two quite different, but useful, theorems related to 3D rotation that are referred to as “Euler’s Rotation Theorem“.

The first definition is given as a quotation in the book by Kuipers

• Any two independent orthonormal coordinate frames can be related by a sequence of rotations (not more than three) about coordinate axes, where no two successive rotations may be about the same axis.

There is a similar, but more pithy definition, from Wolfram MathWorld

• An arbitrary rotation may be described by only three parameters.

This is related to the three-angle representation of an arbitrary rotation by Euler, roll-pitch-yaw, Tati-Bryan angles etc.

The second definition is from Wikipedia

• When a sphere is moved around its centre it is always possible to find a diameter whose direction in the displaced position is the same as in the initial position. (Also given in latin)

This related to the angle-axis representation of an arbitrary rotation. This version of the Rotation Theorem is found in many other places as well, including here on Math Overflow.

Both definitions are related to rotation, and the second at least has a reference to Euler 1776. The first is related to Euler axes and Euler angles, seems like the sort of thing Euler might have figured out, but Kuipers does not provide a reference.

Euler was notorious for not publishing much. Somebody, post Euler, has declared these theorems as Euler’s Rotation Theorem. Which one is it? Is there a more nuanced way to refer to them?

So i found a riddle on reddit which is a consequence of the axiom of choice.

There is a house with 100 rooms, and each room contains countably many boxes indexed with the natural numbers. Each box contains a random real number, which is the same over all the rooms (that is, box n contains the same real number in every room).

100 set theorists play a game. Each person will go into a unique room and open as many boxes as they like (perhaps countably many) as long as they leave at least one box in their room unopened. Then, each of them need to pick an unopened box in their room, and guess what real number is inside of it.

In order to win, 99 of them need to guess correctly.

The mathematicians can discuss a strategy beforehand, but after they go into their respective rooms, no more communication is allowed. What is a 100% winning strategy for this seemingly impossible task?

The solution is here: https://www.reddit.com/r/math/comments/rpbhos/predicting_random_real_numbers_with_the_axiom_of/?utm_source=share&utm_medium=ios_app&utm_name=iossmf

My question is why can the mathematicians agree on a specific representative for each equivalence class? I thought that the axiom of choice guaranteed a choice function existed, but didnt specify it. But for the solution to the riddle, they would need to know what the choice function is. Can someone explain?

This may be a duplicate, but I’ve done some searching and I can’t find exactly this problem setting, probably due to not knowing the right terminology for how to refer to the transition matrix.

I’m considering absorbing random walks ${X_t}$ which have symmetric transition probabilities inside a window of size $k$. For example, on ${0, 1, 2, dots, 100}$, with $k=3$ if $X_t = 5$
begin{align*}
P[X_{t+1} = 4] &= P[X_{t+1} = 6]\
P[X_{t+1} = 3] &= P[X_{t+1} = 7]\
P[X_{t+1} = 2] &= P[X_{t+1} = 8]\
end{align*}

When the walk is within $k$ of the endpoint, the allowable transitions are truncated on both sides, e.g. if $X_t = 1$, $P[X_t = 3] = 0$.

I wanted to figure out the probability of the walk being absorbed at the upper endpoint, so I did some numerical experiments with walks on the integers from 0-100.
I calculated $T^n$ for the transition matrix $T$ and a large $n$. I then looked at $(T^n)_{i, 100}$ to see the probability of being absorbed at 100. It appears that the value is always equal to $i/100$, regardless of how I set the transition distribution or window-size.

The transition matrix for a walk on the integers in $[0, N]$ has the following properties:

1. $T$ is (row) stochastic
2. $T$ is a band matrix with upper and lower band size $k$
3. $T_{i, j} = 0$ if $|i – j| > max (i, N – i)$
4. $T_{i,i + r} = T_{i, i – r}$ (this is the main property I don’t know the name of)
5. $T_{0, 0} = T_{1, 1} = 1$

I would like a proof or counterexample of the following statement:
begin{equation*}
limlimits_{n to infty} (T^n)_{i, N} = i/N
end{equation*}

If the statement above is true, is it still true if the transition distributions are no longer left-right symmetric, but it’s still the case that $E[X_{t + 1}] = X_t$?

Let $M$ be a von Neumann algebra and $varphi: M_+to [0, infty]$ be a normal, faithful semifinite weight. Consider its associated semi-cyclic representation
$$pi_varphi: Mto B(mathfrak{H}_varphi)$$
(see Takesaki’s second book, chapter VII for details).

Consider the associated map
$$eta_varphi: mathfrak{n}_varphito mathfrak{H}_varphi.$$

Is it true that $eta_varphi(mathfrak{n}_varphicap mathfrak{n}_varphi^*)$ is norm-dense in $mathfrak{H}_varphi$?

Attempt: If $xi perp eta_varphi(mathfrak{n}_varphicap mathfrak{n}_varphi^*)$, then for $x,yin mathfrak{n}_varphi$
$$0 = langle xi, eta_varphi(x^*y) rangle = langle pi_varphi(x)xi, eta_varphi(y)rangle$$
so that $pi_varphi(mathfrak{n}_varphi)xi=0$. Maybe this is sufficient to conclude that $xi= 0$? I feel like non-degeneracy of $pi_varphi$ is relevant.