My camera became all distorted and if I add a different camera, it does the same thing.
Please let me know how to change the size of the camera view. The red dotted line of the square is too small for me.
Statistical Entropy and Information theory How to troubleshoot unwanted white spaces smartly? Can one introduce magnetic monopoles without Dirac strings? To introduce magnetic monopoles in Maxwell equations, Dirac uses special strings, that are singularities in space, allowing potentials to be gauge potentials. A consequence of this is the quantization of charge.  Okay, it looks great. But is this the only way to introduce magnetic monopoles? How to find relevant and new open research topics? How can one find relevant and new open research topics?  When working on my PhD thesis, I find it hard to find open questions in my very specific research area. Which approach do you take to find new and relevant topics? Reading output of a command into an array in Bash I am a software engineer but try to keep up with maths as I really enjoyed the subject at school. I just saw a great TED talk:   The talk states that the current largest known prime is $2^{57885161}-1$  This got me thinking - how does one actually prove whether a number this large is prime or not?  Is it just done by brute force via a computer algorithm or are there any well known mathematical techniques for doing so?  I would be really grateful for any pointers as I am eager to learn about these techniques (if they exist). I have very little background in physics, so I apologize if this question is painfully naive.  Consider the following thought experiment: an observer is in a closed room whose walls, floor, and ceiling are made entirely of mirrors, with a single light source in the middle of the room. When the light is on, the observer can see many copies of his reflection all over the place.  Suddenly, the light source turns itself off. Intuitively, I would expect the observer to "instantly" see darkness. However, I can't figure out why that is the case under the "particle" interpretation of light. There are obviously lots of photons already in the room from before. Furthermore, we know that they don't get "consumed" when they hit a wall, because otherwise the observer wouldn't see so many reflections of himself. Basically, when the light goes off, what happens to the photons already in the room?  I suspect the answer goes something like this: the photons in the room lose a little bit of energy every time they bounce off a mirror, but it's so minuscule that we can still see more reflections than our eye can resolve anyway. When the light goes off, however, it takes them a very small fraction of a second to bounce around the room enough times to diffuse completely, which our eye cannot detect.  Is that about right? If we had a theoretical "perfect reflector", would the light remain trapped in the room forever? If we had instruments that could measure such things very finely, would it take (slightly) longer for the light to go out in a room made of mirrors as opposed to a room made of, say, black cloth? Polygon label repeated for each tile KKT points and nonlinear optimality (theory) I have some questions about Karush Kuhn Tucker conditions. I am not clear about the theory. There are plenty of theorems and I am confused with it. Let us say we have the following problem: We have a general problem, not a convex problem. $$\min f(x)\\g_{j}(x)\le 0,\forall j\in J\\h_{k}(x)=0,\forall k \in K. $$ I will solve it.....Lagrange function, verifying primal feasibility, dual feasibility and complementary slackness and I will get some KKT points. Now what is the conclusion?  If I get no KKT point, can I say that the problem has no solution or what is the conclusion? If I get only one KKT point, is it a local minimum or even a global minimum? If I get more KKT points, what is the conclusion? In which case I will verify SOSC condition? If I get more KKT points or even for one point is it necessary? And the conclusion is what? The KKT points which satisfies the SOSC condition are strick local minimum and the poins which does not satisfy it are not the solution of the problem?  I am so sorry for so many questions, but any help would be very appreciated. Thank you very much in advance. I have a directory with numerous files. Part of the files have the 755 permissions and the other part have 644 permissions. I'd like to convert the files with 755 permissions to 644. I have tried the following line by running it from the directory itself:  find . -perm 755 -exec chmod 644 {}\;   However as a result, the permission changed only for the directory itself and after changing it back I found out that the files permissions remained unchanged. Do I miss something? Choosing research ideas to include in a statement of purpose For fixed $z_i$s inside the unit disc, can we always choose $a_i$s such that $\left|\sum_{i=1}^n a_iz_i\right|<\sqrt3$? Let $z_1,z_2,\ldots,z_n$ be complex number such that $|z_i|&lt;1$ for all $i=1,2,\ldots,n$. Show that we can choose $a_i \in\{-1,1\}$, $i=1,2,\ldots,n$ such that $$\left|\sum_{i=1}^n a_iz_i\right|&lt;\sqrt3.$$ Proof of A ⊆ C and B ⊆ C ⇔ A ∪ B ⊆ C I know that this is valid because by thinking about it, you can see it's true, but I'm just not sure how I would go about actually proving it mathematically. I've kind of tried proof by contradiction, since that's what I use most of the time, but it doesn't seem to work.    In my mind, A ∪ B ⊆ C kind of just implies that A ⊆ C and B ⊆ C. Subspaces of separable normed spaces Let $X$ be a separable normed space. Is it true that every subspace is separable? If it was Hilbert space I would take the dense set and then their projections.  It sounds trivial but I cannot prove or disprove it...  Thank you. As the title asks, why didn't the basilisk bite received in the Chamber of Secrets destroy the Horcrux inside the bitten person?   Yes, I know Fawkes quite quickly healed the bite wound with his phoenix tears, but when Harry used the basilisk fang to destroy Tom Riddle's diary the destruction of that Horcrux was immediate. When Ron hit the locket with the Sword of Gryffindor, it was destroyed immediately. Although the destruction of the ring occurred off-page, Dumbledore used the Sword of Gryffindor to destroy it, so presumably, like the locket, it was gone right away. Nagini died straightaway when Neville decapitated her. The destruction of the Hufflepuff cup happens off-page. The diadem seems to have been destroyed by Fiendfyre immediately as well, as when Harry examined it outside the Room of Requirement it was already leaking a blood-like substance, and then it cracked in half. When Voldemort hit Harry with Avada Kedavra in the forest, Harry woke up in limbo with the last fragment of Voldemort's soul - that awful flayed baby-like being - separated from him immediately.   Is there a logical/canon-based explanation for why the fragment of Voldemort's soul residing in Harry wasn't destroyed upon Harry being bitten by the basilisk?  NOTE: I've read the HP Wiki's entry on Horcruxes; however, I find the HP Wiki to be inconsistent in its accuracy, so I'm hesitant to accept the Wiki's explanation on the matter. I'd prefer an explanation constructed from information in the books or a statement by JKR, or the like. Principal component analysis (PCA) is usually explained via an eigen-decomposition of the covariance matrix. However, it can also be performed via singular value decomposition (SVD) of the data matrix $\mathbf X$. How does it work? What is the connection between these two approaches? What is the relationship between SVD and PCA?  Or in other words, how to use SVD of the data matrix to perform dimensionality reduction? Sarrus Rule with LaTeX Why are there no line numbers in the code listings? Would it make discussing code easier if you could simply refer to a particular line or range, or am I wrong? $n\ge 8\,\Rightarrow\, n = 3j + 5k\ $ for $\,j,k,n\in \Bbb N$ I'm new to Math Induction, I don't know how it goes from $𝑘 + 1 = 3𝑥 + 5𝑦 + 1$ into $5𝑦 = 5𝑦 − 5 + 5.$  I'm trying to prove that:   Any whole number of at least 8 cents can be obtained using coins of 3 and 5 cents  Assume: There is a number $𝑘$ for which there are $𝑥$ and $𝑦$ such that $𝑘 = 3𝑥 + 5𝑦$  Here's my working:  $𝑘 = 3𝑥 + 5𝑦$  Therefore $𝑘 + 1 = 3𝑥 + 5𝑦 + 1$  $5𝑦 = 5𝑦 − 5 + 5$  $5𝑦 = 5 𝑦 − 1 + 5$  $𝑘 + 1 = 3𝑥 + 5𝑦 + 1$  $𝑘 + 1 = 3𝑥 + 5(𝑦 − 1) + 5 + 1$  $𝑘 + 1 = 3𝑥 + 5(𝑦 − 1) + 6$  $𝑘 + 1 = 3(𝑥 + 2) + 5(𝑦 − 1)$  $𝒌 + 𝟏 = 𝟑𝒙' + 𝟓𝒚'$ Can't boot XP after Ubuntu Installation, how to fix? I installed Ubuntu and after the installation, i can't boot to XP. When i choose XP from the Grub menu, I got black screen for like 30 sec, and then it returns to the Grub menu.  Thanks!