Let's say I have a 90mm and an 135mm lens, and Nikon's 24mp full frame and 24mp APS-C DSLR. Assume they are the same optically.  Now, based on what I understood, doing a DX-crop on the image taken by 90mm lens on full frame camera will give you exactly same view of an image taken by 135mm on full frame, with less pixel. But, 24mp when cropped, will still provide enough pixel for viewing on computers. So, in terms of sharpness and details (I don't know their difference), will it be the same with an image taken with 135mm lens? How about when viewing on 100% size?  I find this confusing because, when you shoot further away, you will lose some tiny details, vs if you just shoot it with a longer lens. But that will mean APS-C inherently takes in less details than full frame, which doesn't sounds right. Or does using APS-C gives more details than doing a DX-crop on a full frame body? Is it because it gives more pixel? I read that pixel got nothing to do with sharpness, but in this case?  Sorry for the messy questions, this is so confusing that I am unable to tidy it up in my mind. Thanks!
Take an FF body and a crop body, with similar MP rating - and the FF body will give a sharper image of the same object. At least, that seems to be the case - as here:    I assume a low ISO was used in both images.  Why does the FF body, with the same lens and almost the same MP, give an image that is so much sharper?   My best guess is that the diffraction / airy still play a role, and the larger pixels reduce effective overlap - thus increasing contrast/sharpness by displaying less of the neighboring diffraction, as below?
How to test with Math.random in JavaScript? What is "umask" and how does it work? I believe that umask is something that controls , but do not fully understand it.  After running umask 0644 in a , I cannot read the files I create with the command-line text editor nano. I noticed that the permissions of that file are set to 0022 instead of the default 0755.  How does umask work? I thought I could just remove the each digit in the umask from 0777, 7 - 6 = 1 and 7 - 4 = 3, so I expect the permissions to be 0133, but apparently, this is not the case.   What is umask exactly? Explain it to me like I was a "Linux noob" How do I calculate with umask? What are use cases for umask? Provide two arguments to one option using getopts In below code when I give option r then getopts requires one arguments:  while getopts ":hr::l:" opt; do     case $opt in         r ) echo "Run Numbers - argument = $OPTARG " ;;         l ) echo "Latency range - argument = $OPTARG" ;;         h ) helptext             graceful_exit ;;         * ) usage             clean_up             exit 1     esac done   But I need to pass two arguments after -r option, instead of one. Is there an easy way to do this? Why (in relatively non-technical terms) are Calabi-Yau manifolds favored for compactified dimensions in string theory? Is there a "win64" identifier in Qmake project files?  documentation does not mention other than unix / macx / win32.  So far I've tried using:  win32:message("using win32") win64:message("using win64") amd64:message("using amd64")   The result is always "using win32".  Must I use a separate project-file for x32 and x64 projects, so they would compile against correct libraries? Is there any other way to identify between 32-bit and 64-bit environments? Is it possible to use multiple here-docs in bash? Can one use multiple here-docs to provide input to a command in bash?  $ cat &lt;&lt;&lt;foo &lt;&lt;&lt;bar bar $ cat &lt;&lt;EOF1 &lt;&lt;EOF2 &gt; foo &gt; EOF1 &gt; bar &gt; EOF2 bar   Obviously, in both cases, the second here-doc is used as stdin, and replaces the first reference. Is the solution to use echos instead?  $ cat &lt;(echo -n foo) &lt;(echo bar) foobar   Also, for some reason, using a combination didn't work for me. Why would that be?  $ cat &lt;&lt;&lt;foo &lt;(echo bar) bar $ cat &lt;(echo -n foo) &lt;&lt;&lt;bar foo Show that the following series of function defines a continuous differentiable function function in $\mathbb R$. $$\sum_{n=1}^{\infty}\frac{\sin (nx^2)}{1+n^3}.$$  We have , $|f_n(x)|=\left|\frac{\sin (nx^2)}{1+n^3}\right|\le \frac{1}{1+n^3}\le \frac{1}{n^3}=M_n \text{ (say) }$  As, $\sum M_n$ is convergent so, the given series is uniformly convergent. Also , each $f_n(x)$ is a continuous function in $\mathbb R$. So, the given series converges to a continuous function , say $f(x)$.  But how we can show that $f(x)$ is differentiable function ?  Please help... How to solve $\frac 1{1.2.3}+\frac 1{2.3.4}+\frac 1{3.4.5}+...$ As my question says,   $$\frac 1{1.2.3}+\frac 1{2.3.4}+\frac 1{3.4.5}+...$$  As far as $\frac 1{1.2}+\frac 1{2.3}+\frac 1{3.4}+...$ is concerned, we can write it as $\frac 1{n-1} - \frac 1{n}$ What can we do here? Topological proof on discrete topology where $X$ is infinite How must I prove this problem? Let $X$ be an infinite set and let $T$ a topology in $X$ in which all infinite subsets of $X$ are open.   Prove: $T$ is a discrete topology in $X$. Getting a collection of index values using a LINQ query Is there a better way to do this?  string[] s = {"zero", "one", "two", "three", "four", "five"};  var x =  s .Select((a,i) =&gt; new {Value = a, Index = i}) .Where(b =&gt; b.Value.StartsWith("t")) .Select(c =&gt; c.Index);   i.e. I'm looking for a more efficient or more elegant way to get the positions of the items matching the criteria. How to handle a deliberate duplicate question I got  question in review. First sentence is:     I know there is a  question, but no answers.   And indeed the two questions are virtually identical. I started to flag as duplicate - but the original question had no answer at the time, so it wouldn't be correct and apparently . (I didn't go so far as to try to submit the flag).  What is the proper action here? How do we want such a question to be handled? Should both (or more) questions stay open indefinitely until either is answered and somebody gets the good idea to do the duplicate linking?   As the new question is essentially a bump to the front page (perhaps with re-tagging), especially in the case where the OP knows that a previous question exist, perhaps something should be done to enable bumping re-emphasizing without writing the new question.  As I wrote this,  actually got an answer. So the old one should now be closed as a duplicate of the new - but my question remains for the case where all known/found duplicate questions have no answer?  Ironically, just before asking this, the "Questions that may already have your answer" gave me ... (Although the suggestion has no answer). So it has been asked before - the comment there even suggest the ability to re-emphasize. Stone-Weierstrass theorem with $p(1/x)$ I am trying to prove that for a continuous $f\colon[1,\infty)\rightarrow\mathbb R$ and $f(x)\to a$ as $x\to\infty$ it could be approximated by $g(x)=p(1/x)$ where $p$ is a polynomial. The projection of an ellipse is still an ellipse While studying the problem of determining the orbit of a binary star system, my astronomy notes say: "the projection of an ellipse onto a plane is still an ellipse (except the special case in which it's a segment)". How could I prove this?  EDIT: As suggested i'm adding what I know about ellipses. I know the basics of ellipses in the plane, the equation of an ellipse in a plane in cartesia coordinates, the equation of an ellipse in polar coordinates, the fact that ellipses can be obtained by the intersection of planes with a cone. In my geometry course we defined ellipses as the the points of the plane such that the ratio between the distance from a point and a line is constant and less than 1. I also know that the sum of the distances of the points of the ellipse from the foci is constant. Did all Headmasters get a portrait in Hogwarts? Throughout the Harry Potter series, we see that deceased Headmasters of the school get a semi-sentient portrait of themselves in the Headmaster's Office, at the service of the current Headmaster. Do all Headmasters get a portrait? What are the conditions of getting one? For instance, there are a couple of times when Dumbledore is temporarily deposed as Headmaster: Dolores Umbridge takes over the job in Order of the Phoenix, and Minerva McGonagall takes the job in Chamber of Secrets and Half Blood Prince. Will Umbridge and McGonagall have portraits in the Headmaster's Office when they expire? Is it possible to run the Windows 10 Command prompt on Windows 7? Suppose Alice chooses a random Prime $p$ and a random private Key $a \in \mathbb{Z}^*_p$. By accident, she also chooses a random number $g \in \mathbb{Z}^*_p$, which is not a generator of $\mathbb{Z}^*_p$ and therefore  $$\langle g\rangle \subset \mathbb{Z}^*_p$$  as opposed to $\langle g\rangle = \mathbb{Z}^*_p$, which would yield a valid key. Alice then computes $A \equiv g^a \pmod{p} $ and publishes the Tuple $(p,g,A)$ as her public key.  Bob now encrypts a message $M$ using Alices public key by computing  $$C_1 = g^b \pmod{p}$$ $$C_2 = M \cdot A^b \pmod{p}$$  Is it possible for an attacker (Eve) to distinguish a “real” ciphertext $(C_1,C_2)$ from a random ciphertext $(Z_1,Z_2) \stackrel{$}{\longleftarrow} \mathbb{Z}^*_p$ with significant advantage (say $Adv \geq \frac12$)?    Edit: I guess it should be enough to show, that the ciphertext cannot be random, if  $$\langle g \rangle \neq \langle C_1 \rangle \quad(\text{or $Z_1$, respectively})$$  But how can Eve check for this property efficiently? Can someone point me in the right direction here?    Edit: A real-world situation where this flaw occurred 3 more votes from other users are needed to close this question.   This is the message I got hovering the mouse over close voting tab.   I wonder can we begin any sentence with a numeric value? Is it good practice? Spacing issue for newenvironment wrapper of equation environment If I create a custom environment based on the equation environment, I get a spacing issue: the next line of text after my environment closes doesn't start on the left margin.  Example code:  \documentclass{article} \begin{document} \newenvironment{myenv}{\begin{equation}}{\end{equation}} \noindent $| \leftarrow$ this is the left margin. \begin{myenv} e ^{i \pi} + 1 = 0 \end{myenv} $| \leftarrow$ this isn't the left margin. \end{document}   Output:    Removing the newline after \end{myenv} solves the spacing problem, but the resulting code is not readable.  Of course, in this example I could just use the equation environment itself. My real issue involves a more complex custom environment that suffers from the same problem, and which also includes an equation environment within it. I would like to know how to put a time delay in a Python script. How to prove: if $a,b \in \mathbb N$, then $a^{1/b}$ is an integer or an irrational number?