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Modifying Latex of PDF in the PDF editor

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Is there any opensource PDF editor or Latex editor which allows us to edit the latex content from the PDF (created using tex) itself. It will be useful when modifying the Latex document. It should pop the Latex box of the PDF element when clicked.

Synctex highlights the Latex code in the latex editor, I want the same to be shown in a box beside PDF element in a editable box.

Is it possible in a normal Latex document? or Does it need the PDF to be Tagged completely?

Using Adobe software, I can only edit the PDF element and not the Latex of it.


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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

Please help me!

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


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

end{matrix}bigg]bigg)subseteq R^3,$$

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

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


I derived a formula for [x!]’ is it correct?


The starting point was that $ Gamma'(x+1)=Gamma(x+1)psi(x+1)$ where $psi(x+1)=-gamma+H_{x}$ . Hence $$ [x!]’ = x!biggl[-gamma+sum_{k=1}^{x}frac{1}{k}biggl]$$ For example $ [4!]’ = 24[-gamma+1+1/2+1/3+1/4]$ which gives 36.1462 that, put in the tangent equation, gives us exactly the tangent for x=4! Let me know if I made any mistakes in the derivation/generalization!


An inequality about the 2-Wasserstein distance


Let $W_2(mu,nu)$ denote the $2$-Wasserstein distance between two given probability measures $mu$ and $nu$ on $mathbb R^n$. For a probability measure $mu$ and $f:mathbb R^nto mathbb R^n$, let $f_{#}mu=mucirc f^{-1}$ denote the push-forward of $mu$ under $f$, i.e. $(f_{#}mu)(B)=mu(f^{-1}(B))$ for every Borel set $B$ in $mathbb R^n$. Why does the following inequality hold true?
$$W^2_2(f_{#}mu,g_{#}mu)leq int_{mathbb R^n}|f(x)-g(x)|^2,dmu(x) $$
for all $mu$-measurable functions $f,g:mathbb R^ntomathbb R^n$.

Some comment: the product measure $f_{#}muotimes g_{#}mu$ is a so-called transport plan and by definition of the Wasserstein distance
$$W^2_2(f_{#}mu,g_{#}mu)leq int_{mathbb R^ntimes mathbb R^n}|x-y|^2,d(f_{#}muotimes g_{#}mu)(x,y)=int_{mathbb R^ntimes mathbb R^n}|f(x)-g(y)|^2,dmu(x),dmu(y).$$