3d rendering: transform to be applied to a screen Rect to match the perspective projection done by a arbitrary camera in an arbitrary 3d world.
Finding the Transform matrix from 4 projected points (with Javascript)
The age of the universe Many times I have read statements like, "the age of the universe is 14 billion years" . For example this wikipedia page .   Now, my question is, which observers' are these time intervals? According to whom 14 billion years? Prove that the projection $p_i$ from the product space $X_I S_i \tau$ into $S_i \tau_i$ is open for each $i \in I$ Full question: A function $f$ from a space $X, \tau$ to a space $Y, \tau'$ is said to be open if $f(V)$ is open in $Y$ whenever $V$ is an open subset of $X$. Prove that the projection $p_i$ from the product space $X_I S_i \tau$ into $S_i \tau_i$ is open for each $i \in I$.  Attempt: Let $S = \Pi_{i \in I} S_i$ be the product space. Then, let $p_i: S, \tau \to S_i, \tau_i$ be the projection $p_i(s) = s_i$ for each $s \in S$. Then, $p_i$ is continuous for all $i \in I$.  Then, if $V_i$ is open in $S_i$, then $p_i^{-1}(V_i) = S$ is one of the open sets in the basis, namely $\{\Pi_I V_i : V_i \text{ open in } S_i \text{ and } V_i = S_i \text{ for all but at most finitely many i}\}$, and we see clearly that $V_i$ fits the criteria for all $i \in I$ because we defined $V_i$ as being open in $S_i$ and $I$ is a finitely countable set, hence there will be at most finitely many $i$ for which $V_i = S_i$.  Thus, the projection $p_i$ is open for each $i \in I$.  I'm not sure if I tackled this correctly and wanted to make sure I was on the right track or got it right. Best practices for storing UI settings? What will I miss out on if I play Diablo 3 without Reaper of Souls? I own the base game of Diablo 3.  I didn't really like the "post-Inferno" content of the game, so I stopped playing shortly after I curb-stomped the final boss for the n-th time.    I understand now that a lot of things have changed in the base game.  There's also an expansion.  I don't plan to purchase the expansion, but some of these changes sound like they could make the game more fun for me.  I can guess that probably the new character and act are expansion-only content.  I know I'm probably , but if I go back and play now, what else will I be missing out on if I don't have the expansion? Using wilcox.test() and t.test() in R yielding different p-values I bought a Magic the Gathering display box today with cards from the new edition Innistrad. Unfortunately one of the cards in a booster pack was damaged. It is not just a simple scratch - one corner is bent and a small part of the card is missing. According to different trading platforms this card would be worth about 6 Dollars minimum in mint condition. So it is not really expensive but also not totally worthless.  What can I do about this? Can I obtain a replacement card from the producer (WotC) or from the store I bought it from? How can I prove that the card was damaged when I opened the booster pack? I am sure I am not the first person who got a damaged card from a booster pack. How are such issues usually handled? Can I get SSD rw performance while keeing data security if I combine SSD & HDD in btrfs RAID1 I have 2 storage devices; classical slow HDD (750GB, /dev/sda) and faster SSD (128GB, /dev/sdb). Currently I have installed Ubuntu &amp; Mint on same btrfs partition on SSD (/dev/sdb5). My btrfs pool consists of /dev/sdb5.  What I want to achieve is data replication on HDD while keeping SSD performance: Something like this:  For read case it would be ideal for procedure to go like this:   read from SSD check checksum if it verifies.. awesome, give me the data if not, fetch data from HDD and perform error correction through duplicate data   I'd manually run (some tool I'm not sure which one... btrfs check --repair perhaps?) every month or so to check HDD checksums and correct them via SSD data against bitrot.  For write case:   write to SSD sync to HDD when write requirements are low, thus not slowing down the system.   Is this possible and how would I do it?  Bottom line is, I like btrfs and would like it to duplicate data across HDD &amp; SSD drive automatically while keeping SSD excellent performance.   Using SSD as write cache is not an option since it's missing data duplication and security. Number of homomorphisms from $(\mathbb Q^+ , \,\cdot\,)$ to $(\mathbb Z_n , +)$ How many homomorphism are there from $(\mathbb Q^+ , \,\cdot\,)$ to $(\mathbb Z_n , +)$?   Could anyone give a hint how to proceed? I couldn't even start Assume $T \in \mathbb{R}^{d\times d}$. Let $\rho(T)$ be the spectral radius of $T$ and $\rho \ge 0$. Prove that  $\rho(T) &lt; \rho$ if and only if there exists a $P \succ 0$ satisfying $T^TPT - \rho^2 P \prec 0$.  I can prove the Necessity: If $P = I$, for any $x \in \mathbb{R}^d$,   \begin{equation} x^T T^T P T x - \rho^2 x^T P x = \| Tx \|^2 - \rho^2 \| x\|^2 &lt; 0 \end{equation} so $T^T PT - \rho^2 P \prec 0$  I get stuck in proving the Sufficiency. I am trying to solve this exercise from Jech's book on set theory:  Ex. 3.6: There is a well ordering of the class of all finite sequences of ordinals such that for each $\alpha$, the set of all finite sequences in $\omega_{\alpha}$ is an initial segment and its order-type is $\omega_{\alpha}$. Definition: If $W$ is a well ordered set and $u \in W$, then the set $\{ x \in W \mid  x &lt; u \}$ is an initial segment in W  Any help? Cycles not rendering image or movie clip used as background in the viewport Question about $\lim _{q \rightarrow \infty}\|f\|_{q}=\|f\|_{\infty}$ Let $(X,B,\mu)$ be a complete measure space,Show that $$\lim _{q \rightarrow \infty}\|f\|_{q}=\|f\|_{\infty}, \quad \forall f \in \bigcup_{p} \bigcap_{p \leqslant q&lt;\infty} L^{q}$$ So,$\lim _{q \rightarrow \infty}\|f\|_{q}$ , $\|f\|_{\infty}$ are equal-norm with space $ L^{\infty} \cap\left(\bigcup_{p} \bigcap_{p \leqslant q} L^{q}\right)$. Case 1: $m(X)&lt;\infty $.It's easy to prove that. Case 2: $m(X)=\infty $. I have no idea about it,And I started to doubt the correctness of this conclusion. Can somebody give me a hint for this problem or just give an example to prove that this is a wrong conclusion when $m(X)=\infty $. Thanks in advance. Bank Deposit Scam I was on Instagram and saw someone's post about making extra cash.  I inboxed her for details and she said her manager will deposit funds into my account, then I would send them $2000 to split between the both of them.  She stated that I would need to overnight my bankcard to her so her manager can process the deposit.  She didn't ask for anything (neither SSN nor routing or account number) but the bank card only.  I assume that it is a scam.  Can anyone elaborate on this type of scam?  She claim it's not a scam. How do comment @replies work? Why are lamps and mesh lights not illuminating anything? There are several techniques to do a 3 point lighting system.   How do you properly light a scene while using mesh planes as emission light sources?   I'm following , however despite setting up planes to act a lighting sources, the scene is still very dark.      Actual Blender file here: What's the tax on a hobby orchard? Comment flags are broken If I flag a comment now, it puts the dialog off the page:        It is only fixed by performing one of these actions:   Maximize window (if not already maximized) Restore window (if maximized)         I am running Windows and Firefox; also I tried Firefox in safe mode. I would be interested to see what testing was done on this beforehand - as this seems like a pretty glaring error. Please and thank you.    Inspired by the answer, the following CSS worked for me. It makes me sad to have to fix Stack Exchange CSS; I would figure this community over all others would get it right. Oh well.  aside {   transform: none !important; } I write some choice questions into beamer frame, after some research, here is my code:  \documentclass[aspectratio=169]{beamer} \usetheme{Berkeley} \usepackage{tasks,CJK} \begin{document} \begin{CJK*}{utf8}{bsmi} \begin{frame} \transwipe 1.Which one you want to choose? \begin{tasks}[counter-format=tsk[A].,debug=true](4) \task A \task B \task C \task D \end{tasks} \end{frame} \end{CJK*} \end{document}   output with pdflatex:  But I don't want the D choice goto the second line, I want them stay in one line, what should I do? Proving that $A+A \not\subset 2A $ using a particular definition for sum of sets Let $X$ de a vector space and $A\subset X$. If we define $A+B :=\{a+b|a\in A, b\in B\}$ I am trying to show that    $$ \begin{equation} 2A \subsetneq A+A, \end{equation} $$  i.e., it might happen that $2A \neq A+A$. I was able to show that $2A \subset A+A$, but I am not able to come up with a counterexample to show that the converse is not necessarily true. If I assume $A+A\subset 2A$ and let $x\in(A+A)$, then by its definition we have $x=a_1 + a_2$ for some $a_1,a_2\in A$ I guess I could naively say that $a_1+a_2 = 2(a'_1+a'_2)$ where $a_i=a_i/2$ for $i=1,2$ and I guess $(a'_1+a'_2)$ might not be an element of $A$ in some concrete case but I am not able to come up with anything and I've spent more time than I would like to admit. Any suggestions?  Thank you. I've never really understood why math induction is supposed to work.  You have these 3 steps:   Prove true for base case (n=0 or 1 or whatever) Assume true for n=k.  Call this the induction hypothesis. Prove true for n=k+1, somewhere using the induction hypothesis in your proof.   In my experience the proof is usually algebraic, and you just manipulate the problem until you get the induction hypothesis to appear.  If you can do that and it works out, then you say the proof holds.  Here's one I just worked out,  Show $\displaystyle\lim_{x\to\infty} \frac{(\ln x)^n}{x} = 0$  So you go:   Use L'Hospital's rule. $\displaystyle\lim_{x\to\infty} \frac{\ln x}{x} = 0$. Since that's $\displaystyle\lim_{x\to\infty} \frac{1}{x} = 0$. Assume true for $n=k$. $\displaystyle\lim_{x\to\infty} \frac{(\ln x)^k}{x} = 0$. Prove true for $n=k+1$.  You get $\displaystyle\lim_{x\to\infty} \frac{(\ln x)^{k+1}}{x} = 0.$   Use L'Hospital again: $\displaystyle\lim_{x\to\infty} \frac{(k+1)(\ln x)^{k}}{x} = 0$.  Then you see the induction hypothesis appear, and you can say this is equal to $0$.  What I'm not comfortable with is this idea that you can just assume something to be true ($n=k$), then based on that assumption, form a proof for $n=k+1$ case.  I don't see how you can use something you've assumed to be true to prove something else to be true.