I recently watched Tenet. And I have lots of questions and one of them is that. If we see from the Protagonist's perspective, he was going to disarm/collect the bomb at staircase in the opera house, and is stopped by some armed man. Then we see a hole in the wall/seat getting filled up by concrete remains, and then we see Neil firing a shot. I understand that it was inverted Neil who actually fired a shot (if seen from his perspective) but when we see it from the viewer/Protagonist's perspective, we see bullet going in the gun that Neil is holding. So strictly from the viewer's/Protagonist's perspective, who put the bullet in the wall/hole at the first place. Imagine if I was one of the persons sitting there at the opera, I must have not seen any hole at the wall because it was not there before shooting started, then at the time Neil fired that shot, that bullet hole must be there for it to work. So how did it work from the viewer/Protagonist's perspective? Sorry if I sound confusing and for my bad English as it's not my first language.
In the first half of Tenet, there is a scene where The Protagonist and Neil break into Freeport and find a glass wall with bullet holes (could only find a good photo with Christopher Nolan in it):  In the next scene, we find out that these bullet holes were caused by an inverted armed man entering the room (exiting the room from their perspective) and reverse-shooting the glass, which removes the bullet holes from the future-facing perspective. This all makes sense, but raises a question: how long were the bullet holes there in the first place? Were they in place as soon as the glass was installed, possibly years ago? Or even back when the glass was manufactured?
Is it acceptable to change colors on a logo to make it fit a website's theme? I have a few payment brand images that need to go in the footer of a website, the footer is very dark grey, to the point where making the logos grey-scale wouldn't be much of a good idea, especially as we still need the logos to stand out.  For example I took the following logo:   And changed it to fit with our very dominant red theme:    Is this good in terms of the user's perspective? Is this legal or misrepresenting the brand? 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The exercise asks me to consider that $g(t) = f(x_0+ht,y_0+kt)$ is a parabola, but I couldn't understand how it helps. Maybe since it's a parabola, I know that this derivative has to be $0$ in some point, and also that this function $g(t)$ has a global extrema point, but I couldn't derive anything from this. I also tried to find the derivatives, equal them to $0$ and solve the system. I got:  $$x_0 = -\frac{2 b d - c e}{4 a b - c^2}, y_0 = -\frac{   c d - 2 a e}{-4 a b + c^2}$$  But when I try to find   $$f\left(-\frac{2 b d - c e}{4 a b - c^2},-\frac{   c d - 2 a e}{-4 a b + c^2}\right)$$ so I can try to prove:  $$f(x,y)-f(x_0,y_0)$$  is either greater or less $0$ depending on the case (so I can do $f(x,y) \le f(x_0,y_0)$ and $f(x,y) \ge f(x_0,y_0)$)  things get hairy. Speed of light travel Is there a way to get the raw SQL string executed when calling PDOStatement::execute() on a prepared statement? For debugging purposes this would be extremely useful. I have a lot of filesystems:   / on a PCI-E SSD swap on a SATA SSD /media/ned is a RAID5 array /tmp lives on RAM Parts of /home/oli are symlinked and bind-mounted in from ned I also have /media/jeff/ which is RAID1 that I store backups and bind-mount ~/.wine in off.   It's actually a lot more complicated than that. There are all sorts of extra symlinks that I need to untangle in the long run but for now I just need a nice way of taking a path and finding out where it physically lives. Prove $\frac{a+b+c}{abc} \leq \frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}$. So I have to prove $$ \frac{a+b+c}{abc} \leq \frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}.$$  I rearranged it   $$ a^2bc + ab^2c + abc^2 \leq b^2c^2 + a^2c^2 + a^2b^2 .$$  My idea from there is somehow using the AM-GM inequality. Not sure how though. Any ideas?  Thanks ARMA/GARCH estimation in sequence I have a time series that shows a nonstationary seasonal autoregressive component as well as known heteroshedasticity. In order to model the series, I have fit a seasonal ARIMA model for the mean with the auto.arima model in the forecast package in R and a GARCH model on the residuals of the ARIMA model.   Is the procedure of sequentially estimating ARIMA and GARCH model correct or would it have been better to jointly model the mean and the variance of the series? If this were correct, is there a (possibly R) function to do it? Is it possible to have an object ignore a certain light source but still be affected by others? I want to create an effect on a character where their skin and clothing are effected by two different light sources. The clothing should only reflect a bright bounce light, but have the skin should not be affected by this light source.    In the example image I have two objects a cube, a sphere and three emission  lights . I want the two white lights to hit both objects, and I need the purple light to hit and reflect off of the sphere, but ignore the cube entirely (no reflections).  Someone suggested to composite the layers together, it works... but this is very impractical and cumbersome because it forces the background to be transparent and it will be a pain to try and layer together a more complex scene, especially if its animated which is what I want this effect for.  I know you can have objects ignore shadows and all kinds of other things so im hoping someone knows how to get an object to ignore light with a few settings. Discovering new features Order a MySQL table by two columns How do I sort a MySQL table by two columns? What I want are articles sorted by highest ratings first, then most recent date.  As an example, this would be a sample output (left # is the rating, then the article title, then the article date)  +================+=============================+==============+ | article_rating | article                     | article_time | +================+=============================+==============+ | 50             | This article rocks          | Feb 4, 2009  | +----------------+-----------------------------+--------------+ | 35             | This article is pretty good | Feb 1, 2009  | +----------------+-----------------------------+--------------+ | 5              | This Article isn't so hot   | Jan 25, 2009 | +================+=============================+==============+  The relevant SQL I'm using is: ORDER BY article_rating, article_time DESC  I can sort by one or the other, but not both. I recently saw a question titled "" (10k+ users only). I thought, "surely someone has asked this question before", and began looking for a duplicate.  is a question with the exact same title, just with a little different punctuation and casing. It was "Jquery vs Javascript" and "jQuery vs. Javascript?". The only difference being the ., ?, and the casing.  Since we now , I propose that when checking for questions with duplicate names, certain punctuation be striped out to prevent the duplicate from occurring in the first place.  As pointed out in the comments, if isn't already, white-space and non-printable characters may want to be removed for duplicate detection as well. Show $||Tx||_2 \leq ||(v_n)||_2 \cdot ||x||_{\infty}$ Let $(v_n)_{n \geq 1} \in \ell ^2$ be a fixed bounded sequence of real numbers. Define a mapping $T$ on $\ell ^{\infty}$ using the formula $$T(a_1, a_2,...)=(v_1a_1,v_2a_2,...), \, \, \, \, \, x=(a_1,a_2,...) \in \ell^{\infty}$$  We can use the fact that, $Tx \in \ell^2 $ for every $x \in \ell^{\infty}$ and $T: (\ell^{\infty},||\cdot ||_{\infty}) \rightarrow (\ell^{2},||\cdot ||_{2})$ is a linear operator.  Show $||Tx||_2 \leq ||(v_n)||_2 \cdot ||x||_{\infty}$ for all $x \in \ell^{\infty}$    Here is what I tried:  $||Tx||_2 = \bigg( \sum v_k^2 a_k^2 \bigg)^{1/2} \leq (||x||_{\infty}^2 \sum v_k^2 )^{1/2}= (||x||_{\infty}^2 ||(v_k)||_2^2 )^{1/2} = ||(v_k)||_2 \cdot ||x||_{\infty}$  Is this OK? I am new to compact sets, and I had some hard time trying to solve this:  Let $X$ be an infinite set; $T$ the discrete topology on $X$ .Prove that $(X,T)$ is not compact.  I know that I may consider the open cover $\{\{x\},x \in X\}$ and prove that it doesn't have a finite subcover,but I don't know how to write it exactly.  Moreover, I seriously can't understant why $X$ itself cannot be considered as a finite subcover?  Any help is appreciated. I can't understand how the "cryptographic functions" are to be selected RSA + OAEP which are used in OAEP. How to choose these "cryptographic functions"(G and H)? Where is the denoise option in 2.90? describes how to get the changelog for a given package. However, it's the whole changelog.  What I'd like is a way to see what will be changed in whatever I'm about to upgrade. For example, I was imagining something like:  $ sudo apt-get upgrade --show-changelogs Reading package lists... Done Building dependency tree        Reading state information... Done Calculating upgrade... Done The following packages will be upgraded:   foo     -- adds the bar feature for better snazziness     -- removes the deprecated baz feature  1 upgraded, 0 newly installed, 0 to remove and 0 not upgraded. Need to get 8,864 kB of archives. After this operation, 285 kB disk space will be freed. Do you want to continue [Y/n]?    Unfortunately, I don't see anything like that in the man pages for apt-get. Does this or something similar exist?  UPDATE: I've written a  talking about apt-listchanges and Debian changelogs in a little more depth that incorporates some of the feedback in this question. How to reformulate differential equation problem as OrnsteinUhlenbeckProcess