I have some trouble understanding the concept of a covariance matrix. For instance, I'm going over this question that says: assume that we have U1, U2 and U3 as independent zero-mean, unit-variance Gaussian random variables. Furthermore, we have X = U1, Y = U1 + U2, and Z = U1 + U2 + U3. We're asked to find the covariance matrix for (X,Y,Z).  I just require some guidance/hints on how to solve this sort of a problem. And I'd really appreciate any help on this matter.
I try to understand and visualize myself covariance matrix. Supposing I have a matrix A = [ 2 3 4; 5 5 6 ], how do I calculate its covariance matrix, and what is its practical meaning? (All I was able to understand by now is that on the diagonal of the covariance matrix, variances for particular variables are placed, and on the upper and lower fields correlations between those variables.)
Gravity as a gauge theory Currently, (classical) gravity (General Relativity) is NOT a gauge theory (at least in the sense of a Yang-Mills theory).   Why should "classical" gravity be some (non-trivial or "special" or extended) gauge theory? Should quantum gravity be a gauge theory?  Remark: There are some contradictory claims in the literature to this issue. To what extent gravity is "a gauge" theory?Obviously, GR is not a YM theory. So, why do some people say that gravity "IS" a gauge theory? I found this question relevant, e.g., then we handle GR in the Einstein-Cartan theory or any other theory beyond GR, like teleparallel theories or higher-derivative gravitational theories. So I think it could be useful to discuss the "gauge flavor" of gravity here. Installing Numpy on 64bit Windows 7 with Python 2.7.3 Using (or not) the word "to" in a sentence ending with "be able" Which of these is grammatically correct?   As educators, we are always looking for ways to reach all the learners in our classroom as effectively as we are able.   As educators, we are always looking for ways to reach all the learners in our classroom as effectively as we are able to. 1) Up until now, during practical work sessions, I always used these formulas for uncertainty propagation:  if $C = A+B$  or  $C = A-B$ $$\Delta C = \Delta A + \Delta B$$ if $C = AB$ or $C = \frac{A}{B}$ $$\frac{\Delta C}{C} = \frac{\Delta A}{A}  + \frac{\Delta B}{B}$$ if $C = A^m$ $$\frac{\Delta C}{C} = |m|\frac{\Delta A}{A}$$   These formulas are derived from the expression for the differential of a function of multiple variables:  if $C = f(A,B)$ $$ dC = \frac{\partial C}{\partial A}dA + \frac{\partial C}{\partial B}dB \Rightarrow \Delta C = \frac{\partial C}{\partial A}\Delta A + \frac{\partial C}{\partial B}\Delta B$$  and I think that makes sense because the value C is just a function of two variables that happen to be measurements.  2) But this morning, I was told that this is wrong and I should actually use these instead:  if $C = A+B$  or  $C = A-B$ $$(\Delta C)^2 = (\Delta A)^2 + (\Delta B)^2$$ if $C = AB$ or $C = \frac{A}{B}$ $$\left(\frac{\Delta C}{C}\right)^2 = \left(\frac{\Delta A}{A}\right)^2  + \left(\frac{\Delta B}{B}\right)^2$$  which are apparently derived from a general formula:  $$(\Delta C)^2  = \left(\frac{\partial C}{\partial x_1}\Delta x_1\right)^2 + \left(\frac{\partial C}{\partial x_2}\Delta x_2\right)^2 +   \hspace{0.3cm}...$$  or   $$\Delta C  = \sqrt{\left(\frac{\partial C}{\partial x_1}\Delta x_1\right)^2 + \left(\frac{\partial C}{\partial x_2}\Delta x_2\right)^2 +   \hspace{0.3cm}...}$$  3) So:   Why is this formula better?  Where does it come from? What does it actually represent? (Do I recognize the shape of a norm in that last formula?)  What is wrong with the formula of the differential?   I'm asking a lot, but the propagation of uncertainty always confused the ship out of me. What are the answers for these basic quetions? When to use 'by' instead of 'at' or 'on' to specify time/date? Let $A$ be a non-empty set and $n$ be the number of elements in $A$, i.e. $n:=|A|$.  I know that the number of elements of the power set of $A$ is $2^n$, i.e.  $|\mathcal{P}(A)|=2^n$.  I came across the fact that exactly half of the elements of $\mathcal{P}(A)$ contain an odd number of elements, and half of them an even number of elements.  Can someone prove this? Or hint at a proof? I accidentaly forbid SSH connection to a remote server... What's next? Let's say it again, we all make mistakes, and I have just made one.  A brief history: I was doing some stuff on a VPS (Debian) I'm renting, when I noticed some strange behaviour. Using the netstat command I saw an non-authorized connection through SSH. I didn't know what to do, so I decided to close his connection using iptables:  iptables -A INPUT -p tcp --dport ssh -s IP -j DROP   But I am tired, and I wrote  iptables -A INPUT -p tcp --dport ssh -j DROP   and I kicked myself (and everyone else) out...  How do I fix this? Is there an alternative to DVD-Shrink, that can compress a DVD image to a smaller size to fit on a writeable DVD? Is there a linux alternative to the Windows application DVD-Shrink, that can compress a DVD image to a smaller size to fit on a writeable DVD?  In backing up some DVD's the ripped ISO file size is over 7000MB is there an alternative linux software, similar to Windows "DVD Shrink" that can compress this to fit on a writeable DVD which is about 4000MB LWC to display salesforce objects as a picklist onclick I have a requirement where I need to display list of salesforce objects as a picklist when a radio button is clicked.  Being new to LWC development, I could not figure out what is the right approach to do so. I have come across  from Salesforce articles where we can display radio buttons but as per my scenario, i need to display list of all salesforce objects as picklist when a radio button option is selected.  Can anyone please suggest the right approach to do so. GRUB Help and Reinstalling Ubuntu Recently, I installed Ubuntu 10 Netbook edition (on my netbook, yes, obvious). My battery died literally, while I was using Ubuntu.   Now, when I restart the computer, it seems that GRUB has duplicated all of the entries in the MBR that pertain to the partition that houses my Ubuntu install. For example:  Ubuntu ... Ubuntu ... (Recovery Mode) Ubuntu ... Ubuntu ... (Recovery Mode) Memtest ... Memtest ... Windows (Vista loader) Windows (Windows 7 starter loader)    When I try to access either of the two Ubuntu instances in the boot loader it presents me with an error akin to this:   ...numbers... Kernel panic ... VFS unable to mount...   What I would like to do is:  Wipe out grub, and Ubuntu, and then reinstall Ubuntu (I really only use it for emacs and ess for R so there's nothing in the home directory that I need to backup)  How would I go about "resetting" my system? Or fixing the issue that I have.   Thanks in advance!   BEB How can I use Bengali script in an English document? In the context of metric (and in general first-countable) topologies, it's reasonably clear what a closed set is: a set $F$ is closed if and only if every convergent sequence of points in $F$ converges to a point also in $F$. This naturally generalises to the definition of a closed set in an arbitrary topological space using the concept of limit points... but limit points are defined in terms of open sets, which are, to me, somewhat more mysterious than closed sets.  I was once told that ultrafilters axiomatise the concept of (sets of) big sets. I'm hoping here to find a similar conceptual picture of topologies defined by systems of open sets, preferably without reference to closed sets. One such explanation I've seen is that open set axiomatise the concept of nearness, which, I suppose, is fair enough at least for metric topologies. Indeed, if something is "happening" at points near $x \in X$, then it's usually the case that every open neighbourhood $U \subseteq X$ of $x$ contains an open subset $V \subseteq U$, $x \in V$, such that the something "happens" at all points in $V$. But what about non-metrisable topologies, particularly the coarse ones where there are no "small" open sets?  Consider, for example, the Zariski topology on affine $n$-space $\mathbb{A}^n$. $\mathbb{A}^n$ is irreducible, so every non-empty open set is dense. It seems reasonable to interpret this to mean that every non-empty open set is "large". Indeed, if we work over the complex numbers, then in the usual metric geometry, Zariski-open sets are unbounded and have full measure, so are very large indeed, so I don't think it's fair to say that open sets are capturing the notion of nearness here.  I'm also curious about the history of point set topology. When were the axioms first written down? What were the first "non-geometric" examples of topological spaces — "non-geometric" here referring to either non-metrisable topologies or topologies on sets other than sets of points of some intuitively geometric object — and were they part of the motivations for creating point set topology? Proving the snake lemma without a diagram chase How does "virus been detected" on android pop up on chrome when visiting some sites? What is the cause of this chrome bug on android? The last site I went to was www.destructoid.com and all of a sudden it gets redirected to the following url. This happened previously awhile back, but I didn't document everything as I did today. I'm also on android chrome canary if that matters.   I took some screenshots before closing out of that tab. But my question is:   What chrome bug or malware is causing these random redirects? How can one detect if they have this vulnerability?   The url is: $\sum x_n^2$ and $\sum y_n^2$ converge, what does $\sum (x_n + y_n)^2$ do? So I know that $\sum x_n^2$ converges, and $\sum y_n^2$ converges. ${x_n}$ and ${y_n}$ are both sequences of real numbers. How can I prove that $$\sum(x_n+y_n)^2$$ converges? So far I have $$\sum(x_n+y_n)^2=\sum(x_n)^2 +\sum (y_n)^2 +\sum(2x_ny_n)$$ so I need to prove that $\sum 2x_ny_n$ converges. Is this the right way to approach it? If it is, I am completely lost as to the next step.  I also need to prove that $$\sqrt{\sum(x_n+y_n)^2}\leq \sqrt{\sum(x_n)^2}+\sqrt{\sum(y_n)^2}$$  I assume this will follow pretty easily from a result in the first proof, but I'm stuck on that now, and perhaps it is easier to start with the second one. I have a lot of work I've done on the first problem, but none of it seemed to go anywhere. Any hints?  Thanks! Why can't photons have a mass? Could you explain this to me in a short and mathematical way? Twin Paradox, calculating spacetime intervals from both perspectives I've very recently started to try to understand special relativity. I'm want to get a decent understanding of the twin paradox. I'll post what I've done so far and highlight what's gone wrong for me.  The situation is that Alice and Bob both begin at point $x_1$ in Alice's coordinate system $(x,t)$ (we have orientated the axis so that y and z will not matter). Bob then instantaneously moves off with velocity $v$ in the positive $x$ direction. Bob's coordinate system is now $(x',t')=(\gamma (x-vt), \gamma (t-v\frac{x}{c^2}))$. In Alice's coordinates, Bob reaches point $x_2$ after having moved distance $d$, then instantly turns around and travels back to Alice at $-v$. I want to show that Alice's space time interval $\Delta s_A^2$ is greater than Bob's $\Delta s_B^2$, as each path's space time interval is proportional to the proper time passed along that path.    According to Alice: $\Delta s_A^2=-c^2\Delta t^2 + \Delta x_A^2$ where $\Delta t=\dfrac{2d}{v}$ and $\Delta x_A=0$ as she didn't move, but the time elapsed is the time it took bob to travel the distance and then come back. Thus $\Delta s_A^2=-\dfrac{4c^2d^2}{v^2}$  Still according to Alice, Bob's $\Delta s_B^2=-c^2\Delta t^2+\Delta x_B^2$, where the change in time is the same, but now bob has moved distance $d$ twice, thus $\Delta x_B=2d$ ($\Delta x$ refers to total distance traveled rather than displacement which is 0 in this case). Now we have $\Delta s_B^2=-\dfrac{4c^2d^2}{v^2}+4d^2$.     The |size| of Bob's spacetime interval now definitely smaller than Alice's, and this would be all OK, except that when I do the calculations in Bob's frame, they don't agree. This is contradictory to the fact that $\Delta s^2$ is conserved under Lorentz transformations.    According to Bob: $\Delta {s'}_A^2=-c^2\Delta t'^2 + \Delta {x'}_A^2$ where $\Delta t'=\dfrac{2d'}{v}$. I'm not sure if I'm right in saying that $v$, the relative velocity of the two frames, is the only velocity upon which both Alice and Bob will agree on, other than the speed of light. Anyway, $d=\gamma d'$ as lengths contract by $\gamma$ i.e. $d'$ is smaller than $d$ by factor $\gamma$. Also, $\Delta {x'}_A=2d'$, so we have $\Delta {s'}_A^2=-\dfrac{4c^2d'^2}{v^2}+4d'^2=-\dfrac{4c^2d^2}{\gamma ^2 v^2}+\dfrac{4d^2}{\gamma ^2}=4d^2\dfrac{v^2-c^2}{\gamma^2 v^2}\neq \Delta s_A^2$  Although I haven't included them here, my calculations for $\Delta {s'}_B^2$ and $\Delta s_B^2$ agree. Sorry for the ultra-long post, but any help would we well appreciated! How do I find a command if I don't know its name? I'm pretty sure the answer to this should be obvious - even to me ...  It's been a while I've been reading and attempting to construct simple electronic circuits. The supply almost always is either through a battery/cell, or AC duly rectified to DC.   I have the impression electronic circuits rarely ever work off an AC supply.    Is this impression correct? If yes, why is an electronic circuit rarely (if ever) powered/biased using straight AC?