Please help me understand the following concept. Let A be a matrix of constants and Y is a random vector. Var(AY) = AVar(Y)A', where A' is A transpose. I understand that AA' = A^2. What I don't understand is why A' goes in the end, instead of AA'Var(Y). Is there some kind of property in linear algebra that would help me understand the correct order? Thank you!
$\newcommand{\Var}{\operatorname{Var}}$In this  is claimed that if the equation of errors in OLS is given by: $$u=y - X\beta$$ Then in the presence of heteroscedasticity the variance of $u$, will not be constant, $\sigma^2 \times I$, where $I$ is an identity matrix, but: $$\Var(u\mid X)=\sigma^2\Omega$$ In order to account for the heteroskedasticity, we can estimate the transform system, such that $P$ is a transformation matrix. $$Py=PX\beta-Pu$$ Where "the variance of a constant matrix $P$ times a random vector $u$" is: $$\Var(Pu\mid X)=P\Var(u\mid X)P'=P(\sigma^2\Omega)P'$$ Can somebody explain me the proof for that?
Not getting Rep Notification for new up votes So just a little while ago i was flipping out because i was 2 rep away from my 10k rep goal    but then all of a sudden it was then 10,008 rep because someone upvoted me. however i didn't get any notification about this +10  [    now as you can see in the screenshots above, over the past 2 days i got 2 upvotes to a question i posted, however the notifications only shows 1 +10 notification. and they system knows i got the upvote because i knew that i now have access to mod tools.  I am on a Windows 10 Laptop using Chrome 78.0.3904.97 (Official Build) (64-bit) If $\sum_na_n=\infty$ and $a_n\downarrow 0$ then $\sum\limits_n\min(a_{n},\frac{1}{n})=\infty$ If $\sum_na_n=\infty$ and $a_n\downarrow 0$ then $\sum\limits_n\min(a_{n},\frac{1}{n})=\infty$     My  idea:  $$b_{n}=\min(a_{n},\tfrac{1}{n})\Longrightarrow b_{n}\le a_{n},b_{n}\le\tfrac{1}{n}$$ then I don't know how to go on. Confusion between Voltage gain & Voltage gain in decibels (dB) The definition of voltage gain is \$V_{\text{out}}/V_{\text{in}}\$.  However, I read some articles about the gain in decibels, and I have a confusion now.  Here is an article about it:   Here, I understand the definition of Power gain in decibels, which is $$ \text{Gain} = 10 \log \left( {P_{\text{out}} \over P_{\text{in}}} \right)\text{ dB} $$  However, I can't understand why Voltage gain in decibels is $$ 20 \log \left( {V_{\text{out}} \over V_{\text{in}}} \right)\text{ dB} $$  If \$ 20 \log \left( {V_{\text{out}} \over V_{\text{in}}} \right)\text{ dB} \$ is derived from $$ 10 \log {\left( {V_{\text{out}}^2 \over R_{\text{out}}}\right) \over \left({V_{\text{in}}^2 \over R_{\text{in}}}\right) }\text{ dB} $$ then this is the power gain, not the voltage gain, isn't it? However, the Wikipedia says it is a formula for the Voltage gain in decibels. I thought the voltage gain in decibels would be \$ 10 \log \left( {V_{\text{out}} \over V_{\text{in}}} \right)\text{ dB} \$. Actually, the example section in that linked page uses voltage gain \$ V_{\text{out}} \over V_{\text{in}} \$.  Why did \$V^2/R\$ suddenly come out from the voltage gain in decibels? Does upvoting a duplicate comment do anything? When a question is initially flagged as a duplicate, there is a comment. If the duplicate flag gets enough votes, the question is closed as a duplicate, and the comment is deleted.  Does upvoting the duplicate comment count as a close vote? Does it do anything at all? Stripboard/veroboard/matrix board design software I'm prototyping a simple layout for a small project on matrix board, but figured there must be a better way than drawing it all out on paper. Are there any aimed at matrix board?  I've found the following software for stripboard, but nothing explicitly for matrix board:     Stripboard Magic (seems to be abandonware, but there's page about it )  Stripboard Editor Limited free version, designed to integrate well with     Any more I've missed? Any recommendations? The Frobenius Coin Problem I am asked to prove that:       For integers $n, x,y &gt; 0$, where $x,y$ are relatively prime,        every $n \ge (x-1) (y-1)$ can be expressed as $xa + yb$, with nonnegative integers $a,b \ge0$.   How should I approach this? I have very limited knowledge in number theory. $X$ and $Y$ are independent $U(0,1)$ random variables.   I have to find $E[X|X&gt;Y]$.   I found it out by integrating $\int_0^1 \int_y^1 x dxdy$ to get $\frac{1}{3}$ as the answer but this is wrong according to the answer key. What is the mistake that I have made? Why can a 1 element set be a member of another set but not a subset of it? I have came across this in a textbook:     $\{2\}\nsubseteq\{\{2\}\}$ but $\{2\}\in\{\{2\}\}$   I understand that $\{2\}$ is an element (member) of the other set but considering $\{2\}$ is a set itself, why is it not a subset? Can I put \fbox along the diagonal so that they touch each other? Definition of autocorrelation time (for effective sample size) I've found two definitions in the literature for the autocorrelation time of a weakly stationary time series:  $$ \tau_a = 1+2\sum_{k=1}^\infty \rho_k \quad \text{versus} \quad \tau_b = 1+2\sum_{k=1}^\infty \left|\rho_k\right| $$  where $\rho_k = \frac{\text{Cov}[X_t,X_{t+h}]}{\text{Var}[X_t]}$ is the autocorrelation at lag $k$.    One application of the autocorrelation time is to find the "effective sample size": if you have $n$ observations of a time series, and you know its autocorrelation time $\tau$, then you can pretend that you have  $$ n_\text{eff} = \frac{n}{\tau} $$  independent samples instead of $n$ correlated ones for the purposes of finding the mean.  Estimating $\tau$ from data is non-trivial, but there are a few ways of doing it (see ).  The definition without absolute values, $\tau_a$, seems more common in the literature; but it admits the possibility of $\tau_a&lt;1$.  Using R and the "coda" package:  require(coda) ts.uncorr &lt;- arima.sim(model=list(),n=10000)         # white noise  ts.corr &lt;- arima.sim(model=list(ar=-0.5),n=10000)    # AR(1) effectiveSize(ts.uncorr)                             # Sanity check     # result should be close to 10000 effectiveSize(ts.corr)     # result is in the neighborhood of 30000... ???   The "effectiveSize" function in "coda" uses a definition of the autocorrelation time equivalent to $\tau_a$, above.  There are some other R packages out there that compute effective sample size or autocorrelation time, and all the ones I've tried give results consistent with this:  that an AR(1) process with a negative AR coefficient has more effective samples than the correlated time series.  This seems strange.    Obviously, this can never happen in the $\tau_b$ definition of autocorrelation time.  What is the correct definition of autocorrelation time?  Is there something wrong with my understanding of effective sample sizes?  The $n_\text{eff} &gt; n$ result shown above seems like it must be wrong... what's going on? Integral $\int_{\frac{1}{2}}^1 \frac{\ln x}{\sqrt{x^2+1}}dx$ Create an encrypted ZIP file in Python I faced this issue when I want to use sudo command :  sudo: /etc/sudoers is owned by uid 1000, should be 0 sudo: no valid sudoers sources found, quitting sudo: unable to initialize policy plugin   then I tried a solution from a similar question in order to change the owner back to the root:   $ pkexec visudo   also I tried this one   $ pkexec chown root:root /etc/sudoers /etc/sudoers.d -R   but I have faced another issue with these commands:   Error getting authority: Error initializing authority: Could not connect: No such file or directory   So is there a solution to this issue?    Full shell issue snippets:   ubuntu@LAPTOP-D4T16P7J:~$ sudo -i sudo: /etc/sudoers is owned by uid 1000, should be 0 sudo: no valid sudoers sources found, quitting sudo: unable to initialize policy plugin ubuntu@LAPTOP-D4T16P7J:~$ pkexec visudo Error getting authority: Error initializing authority: Could not connect: No such file or directory ubuntu@LAPTOP-D4T16P7J:~$ pkexec chown root:root /etc/sudoers /etc/sudoers.d -R Error getting authority: Error initializing authority: Could not connect: No such file or directory     This is my /etc/sudoers file:   ubuntu@LAPTOP-D4T16P7J:~$ cat /etc/sudoers # # This file MUST be edited with the 'visudo' command as root. # # Please consider adding local content in /etc/sudoers.d/ instead of # directly modifying this file. # # See the man page for details on how to write a sudoers file. # Defaults        env_reset Defaults        mail_badpass Defaults        secure_path="/usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/sbin:/bin:/snap/bin"  # Host alias specification  # User alias specification  # Cmnd alias specification  # User privilege specification root    ALL=(ALL:ALL) ALL  # Members of the admin group may gain root privileges %admin ALL=(ALL) ALL  # Allow members of group sudo to execute any command %sudo   ALL=(ALL:ALL) ALL  # See sudoers(5) for more information on "#include" directives:  #includedir /etc/sudoers.d     OS info:    Ubuntu WSL2. Does the Divergence Theorem Work on a Surface? The divergence theorem in $\mathbb{R}^3$ says that the integral of the divergence of a vector field over a solid $\Omega$ in $\mathbb{R}^3$ equals the flux through the surface of $\Omega$ denoted by $\partial \Omega$.   My question is that can you use the divergence theorem in $\mathbb{R}^3$ where you're integrating the divergence of a vector field over a surface $S$ in $\mathbb{R}^3$ to get the flux through the edges ($\partial S$) of the surface? So if the surface $S$ is closed in $\mathbb{R}^3$ such as a sphere, then clearly $\partial S = 0$ so the integral of the divergence over this particular $S$ is $0$? Fixed points under action of conjugate group elements Lecture notes:  If $g$ and $h$ are conjugate then $fix(g)=fix(h)$. To see this, say $g=k^{-1}hk$  If $g.s=s \iff k^{-1}hk.s=s \iff h.(k.s)=(k.s)$  Me: I understand the last line- I just fail to see how it implies $fix(g)=fix(h)$  EDIT: As Ben S says I think there was a typo in the notes and they forgot modulus signs so- New Aim: show $|fix(g)|=|fix(h)|$ What is the best way to remove texture from a scanned textured photo paper? I have a bunch of scanned old family photos where the photo paper has a texture. Unfortunately, the texture of the photo scans quite well. What is the best way to remove the texture?  (Photoshop CS5) How is the passage of time handled in Pokemon? I stopped following the Pokemon anime a while ago, but I know the series has had Ash go through many regions, including retreading at least one of them.  When Ash was about to get his eight badge, Misty mentioned that they'd been on the road for a year at that point.  I saw some recent episodes, and Ash looks nearly identical to when he first started as a trainer.  Normally I'd attribute that to cartoon sitcom logic, where the characters never age; but in a series with continuity and references to earlier seasons, this makes no sense.  What's the official word on a timeline for the Pokemon anime? Infinite product with sign changes. For infinite products I know the following theorems:  Theorem I. If $a_n \geq 0$ for all $n$ then $\prod (1+a_n)$ converges if and only if $\sum a_n$ converges.  Theorem II. If $-1 &lt; a_n \leq 0$ for all $n$ then $\prod (1+a_n)$ converges if and only if $\sum a_n$ converges.  My question is what are necessary and sufficient conditions (possibly including more restrictions than in I and II) for $\prod (1+a_n)$ convergence/divergence as related to  $\sum a_n$ convergence/divergence  if the sequence of terms $a_n$ can switch signs repeatedly but always stay greater than $-1$.  I already know the trivial equivalence of $\prod(1+a_n)$ and $\sum\ln(1+a_n)$. I have an armature set up, but the bones aren't moving the mesh properly  The mesh and armature:    The vertex group of the bone I am going to move:    As you can see, the circled region doesn't belong to the vertex group. But, when I move the bone look what happens:    That part of the mesh has moved with the bone.. Why? How to plot a surface from a set of data? Disclaimer: I already read questions like  and  but these did not solve my problem.  I have this set of data:  x       y       z 2.17    0.001   0.82044815 2.17    0.002   0.82345825 2.17    0.004   0.82679255 2.17    0.008   0.83334715 2.17    0.016   0.84395915 2.17    0.032   0.8584953 2.21    0.001   0.77582165 2.21    0.003   0.78520505 2.21    0.009   0.80205985 2.21    0.027   0.83085105 2.24    0.001   0.7227885 2.24    0.002   0.73391615 2.24    0.005   0.7543979 2.24    0.015   0.78798745 2.24    0.003   0.74176635 2.24    0.009   0.77064805 2.24    0.027   0.81042375 2.26    0.001   0.66545585 2.26    0.003   0.7012046 2.26    0.005   0.721067 2.26    0.009   0.7447984 2.26    0.015   0.76715245 2.26    0.027   0.794177 2.27    0.001   0.62916195 2.27    0.003   0.6774642 2.27    0.009   0.72961785 2.27    0.027   0.7861086 2.28    0.001   0.5750828 2.28    0.003   0.65059675 2.28    0.005   0.6802631 2.28    0.009   0.7145367 2.28    0.015   0.74447695 2.28    0.027   0.7774403 2.29    0.001   0.51357255 2.29    0.002   0.581053 2.29    0.003   0.6173075 2.29    0.009   0.6972096 2.29    0.027   0.76793225 2.31    0.001   0.36997965 2.31    0.002   0.474415 2.31    0.003   0.53649295 2.31    0.009   0.6587164 2.31    0.016   0.70870255 2.31    0.027   0.7482423 2.31    0.05    0.7912395 2.34    0.001   0.2204104 2.34    0.002   0.316308 2.34    0.003   0.39256745 2.34    0.004   0.45240835 2.34    0.009   0.5883453 2.34    0.016   0.6590771 2.34    0.027   0.71444205 2.34    0.05    0.7690014 2.38    0.001   0.13286995 2.38    0.002   0.1828288 2.38    0.004   0.2980268 2.38    0.008   0.4507145 2.38    0.016   0.58417075 2.38    0.032   0.6833616   And if I plot it in Mathematica (with ListPlot3D[data]) it looks, without further tweaking and fiddling around, like:    But if I try this pgfplots code  \begin{tikzpicture}   \begin{axis}     \addplot3 [surf] table {data.txt};   \end{axis} \end{tikzpicture}   I only get this:    What do I have to do to get a surface like in Mathematica (color, smoothness, style, etc. does not matter, just a closed surface)?