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Class option with arbitrary key value

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I'm working on a personal class where I absolutely need to get a seed number before loading anything in the document. I would like the seed in the class option, something like :

``````documentclass[seed=326]{my-class}
begin{document}

end{document}
``````

In my class, I want the seed number to be stored in a command called `mycommand` if the seed option is used, and I want `mycommand` to stay undefined if the option seed is not used.

Does anyone know if something like this is possible, and what should be written in `my-class`?

Exponentials of stochastic band matrices

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 $$E[X_{t + 1}] = X_t$$?

trouble with writing negative fractional exponential

Code:

``````begin{equation*}
v_o = (v_i - V_D) e^-frac{ t}{tau}
end{equation*}
``````

What I want:

What I am getting:

Erro in Ubuntu Server for service bind9

Error in service bind9 when I’m starting

If \$varphi\$ is a normal faithful semifinite weight, is \$eta_varphi(mathfrak{n}_varphicapmathfrak{n}_varphi^*)\$ dense in \$mathfrak{H}_varphi\$

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.

Can’t install any Nvidia driver

When i run:

``````sudo ubuntu-drivers devices
== /sys/devices/pci0000:00/0000:00:01.0/0000:01:00.0 ==
modalias : pci:v000010DEd00002520sv00001025sd00001547bc03sc00i00
vendor   : NVIDIA Corporation
model    : GA106M [GeForce RTX 3060 Mobile / Max-Q]
driver   : nvidia-driver-525-open - third-party non-free recommended
driver   : nvidia-driver-515-server - distro non-free
driver   : nvidia-driver-510 - distro non-free
driver   : nvidia-driver-470-server - distro non-free
driver   : nvidia-driver-515 - distro non-free
driver   : nvidia-driver-470 - distro non-free
driver   : nvidia-driver-515-open - distro non-free
driver   : nvidia-driver-525 - third-party non-free
driver   : nvidia-driver-525-server - distro non-free
driver   : xserver-xorg-video-nouveau - distro free builtin
``````

When i install the recommended driver (nvidia-driver-525-open), i get a black screen.

When i install nvidia-driver-515 (or any below version) i get stuck at login screen (keyboard, mouse, nothing is working… even ctrl+alt+f1 to open another tty).

This is the bug report log for the nvidia-driver-525:

Laptop: Acer Predator Helios 300 PH315-54-760S Gaming Laptop
GPU: NVIDIA GeForce RTX 3060
Kernel: 5.15.0-57-generic
OS: Ubuntu 22.04.2 LTS

How to determine if a subspace of \$ mathbb{R}^n \$ has an integer basis

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 $$W$$.

Centering a string in another

I have a string of length `N` which I want to center in another string.

Thus setting some string of length N

``````old_string="***"
``````

I want a new string (length 55) with the old string centered in it.

``````new_string="     ***     "
``````

How can I do this in bash ?

Although in actual fact the trailing spaces are redundant, so I can-simply have

``````new_string="     ***"
``````

What is the relationship between measurable or continuos cross-sections?

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.

Chromebook dying unexpectedly

My school provides me with a Chromebook.

Today, I closed the lid, then opened it up, expecting to see the lock screen, but instead, I saw where I last left off. I thought it might be because I had used the super zoom (ctrl+alt+brightness) because that usually happens, but instead of working again, it just turned off, and won’t turn back on.

It is charging but should show the low battery error, which it does not.

Find orthogonal vectors in relation to span

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.

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