Class option with arbitrary key value

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.

Let $ W $ be a sub vector space of $ mathbb{R}^n $. How can we determine if $ W $ admits an integer basis?

This is equivalent to asking how to determine if $ W cap mathbb{Z}^n $ spans $ W $.

Let $G$ be a locally compact Polish (or compact) group acting continuously on a locally compact Polish (or compact) space $X$, and $mu$ a Borel measure on $X$. To be sure, continuity of the action means that the map $(g, x) in G times X mapsto g cdot x in X$ is continuous with respect to the product topology on $G times X$. Let $X/G$ denote the orbit space endowed with the quotient topology, and $pi : X rightarrow X/G$ denote the orbit map. A cross-section to the orbit map is a map $s: X/G rightarrow X$ satisfying $s circ pi = 1_{X}$. If $s$ and $t$ are measurable or continuous cross-sections to the orbit map, then it is known that their images $s(X/G)$ and $t(X/G)$ are measurable (closed in the case of a continuous cross-section with compact $G$ and $X$). What is the relationship between $mu(s(X/G)$ and $mu(t(X/G))$? Is it reasonable to expect $mu(s(X/G)) = mu(t(X/G))$?

PS: It is enough for me to consider the case of $X = G$, that is, $X$ is the underlying topological space of $G$, and the action of conjugation, and $mu$ Haar measure on $G$. Thanks.

Consider $R^3$ as an inner product space in relation to scalarmultiplication. Find all vectors in the subspace

$$Spanbigg(bigg[begin{matrix}1\2\1
end{matrix}bigg],bigg[begin{matrix}3\4\1
end{matrix}bigg]bigg)subseteq R^3,$$

which are orthogonal to the vector $bigg[begin{matrix}-1\1\1
end{matrix}bigg]$

Any help will be appreciated. I tried finding the random vector $v=bigg[begin{matrix}x_1\x_2\x_3
end{matrix}bigg]$ in the plane which i found to be ${x_3cdotbigg[begin{matrix}1\-1\1
end{matrix}bigg]bigg}$ but i dont know where to go from here.