When and why do we take into account the curvature of the curve at infinitesimal scale in calculus?
Why is area of a surface of revolution integral $2\pi y~ds$? not '$dx$'?
I am looking for a exact derivation of a relation between redshift $z$ and distance $d$.  What I know is the definition $$z=\frac{\lambda_{\text{observed}}}{\lambda_{\text{unshifted}}}-1=\sqrt{\frac{1+\frac{v}{c}}{1-\frac{v}{c}}}-1$$ and that the Hubble constant $H$ as a function of $z$ is: $$H^2=H_0^2\left(\Omega_m\left(1+z\right)^3+\Omega_{\Lambda}\right)$$  How can I use this to derive the distance? To my (British) ear, it never sounds correct to say you have a doubt about something.  I expect that doubt to be pluralised, even if there's only one specific thing that I'm doubtful of, in one specific way, so I would always say "I have doubts about something".  Answers to  say that using doubt as a synonym for question is "Indian English". That seems to be the case in  which starts with "I have a doubt on my construction", because the OP really is saying "I have a question" (which he goes on to actually ask). Okay, it should have been a question/doubt about something, not on, but I'm not concerned with that detail here.  But in  the OP says "...there is a doubt that can I use [some construction]". In that case it's not easy to see how the word doubt can be replaced with question.  I don't have a problem with, for example, , because there I interpret it as doubt=uncertainty (a state of mind, syntactically equivalent to words like happiness, agitation, sorrow, etc.). But you certainly can't "have a happiness about something". So my question is...     Is it acceptable today to use singular noun doubt in any contexts where it means something other than state of mental uncertainty? If so, what are those contexts? File changed listener in Java I'd like to be notified when a file has been changed in the file system. I have found nothing but a thread that polls the lastModified File property and clearly this solution is not optimal. Resistor circuit that isn't parallel or series How does one attack a two-time pad (i.e. one time pad with key reuse)? My question might appear the same as the question , but actually I did read all the answers and none of them helped me with the details I need.  I am new to cryptography and my problem is with two time pad attacks on OTP. The problem I had in my course was that I have 10 ciphertexts encrypted with the same key $K$. I am then given another ciphertext that I should decrypt.  I know that XOR-ing two ciphers gives me the XOR of their original messages.  My question is what is the correct thing to do after that?  I tried to take 3 ciphertexts $C_1, C_2$ and $C_3$.  Then get $S_1 = C_1 \oplus C_2  \oplus $' ', also get $S_2 = C_1 \oplus C_3 \oplus$ ' '.  After that I compared all corresponding characters in $S_1$ and $S_2$,  and if $S_1[i] = S_2[i]$ then I calculate $S_1[i] \oplus C_2[i]$ to get $K[i]$.  I tried this on paper before coding and it worked, but I might be missing something.  Is this the right approach? Why does it work? Ubuntu boots on terminal only, how do I recover the GUI desktop? So I'm using an app that stores images heavily in the DB. What's your outlook on this? I'm more of a type to store the location in the filesystem, than store it directly in the DB.  What do you think are the pros/cons? Empty intersection and empty union Strange Pop-up appldnld.apple.com:80 I received a strange Pop Up message. The unsettling part is password will be sent in clear text.  Is my phone affected with any malware or app virus ? I am using Ubuntu 16.04 LTS . I have python3 installed. There are  two versions installed, python 3.4.3 and python 3.6 . Whenever I use python3 command, it takes python 3.4.3 by default. I want to use python 3.6 with python3. python3 --version shows version 3.4.3 I am installing ansible which supports version &gt; 3.5 . So, whenever, I type ansible in the terminal, it throws error because of python 3.4 sudo update-alternatives --config python3 update-alternatives: error: no alternatives for python3 While traveling in the Berlin Tegel airport (TXL), I wanted to get on Wi-Fi. I found three unsecured networks in addition to a number of password-protected networks.   _Free_TXL_WiFi airberlin Berlin Tegel Airport WiFi   I had no idea which of these were an official Wi-Fi network provided by the airport and which were provided by other people, potentially for malicious purposes like capturing my network traffic. Presumably the official one uses a captive portal to allow internet access.  Is there any way to determine which Wi-Fi network is the official one provided by the airport? Would airport staff such as at the ticketing booths know? Are there official lists available somewhere?  I’d like an answer that is applicable to airports generally, rather than specific to the Berlin Airport. Also, please don’t just give a blanket “don’t use any public Wi-Fi network” answer; I know there are risks associated with it, but sometimes it is a necessity and so I want to know how to identify the airport-provided one.   suggests there are no official Wi-Fi networks, but the answers are six years old, so it’s not clear if things have changed or if none of the networks I saw were official. What exactly happened with Dr. Crusher's husband and Captain Picard? Hydrostatic pressure? What what I understand, hydrostatic pressure is the "weight" of the water pushing against objects. But if this is true, why is hydrostatic pressure perpendicular to the surface it acts on instead of always going down?  For example, if you placed a book on a desk, the book's weight would push against the desk, but gravity is pulling it "down". But if you put another book beside this book the first book wouldn't apply any force on it. The weight exerted from the book always goes down.  Similarly, why doesn't hydrostatic pressure always go down? I understand that if you put a plate on the seabed, the weight of the water would push down on it, but if you just had a vertical plate standing on the bed, why would force push it from the sides?  To sum up, why does hydrostatic pressure act perpendicular to the object instead of always down? gedit "cannot open display" , on remote computer , using ssh - Ubuntu 12.04 Urgent Sydney transit visa question How to tell which row number is clicked in a table? What is a referee in the manuscript submission process? Please submit, with the manuscript, the names, addresses (including   countries), and e-mail addresses of five potential referees who are   outside the authors' institutions and must not have conflict of   interest with the authors.   Do you think I should name 5 researchers from a similar area that are not affiliated with my university and co-authors? Is this what referees are? A slick proof that a field which is finitely generated as a ring is finite It is a known fact that if $k$ is a field that is finitely generated as a ring, which is the same as having a surjective ring homomorphism $f:\mathbb{Z}[x_1,\dots,x_n]\to k$ for some $n\in \mathbb{N}$, then $k$ must be finite. Since finite generation as a ring implies finite generation over the prime field ($\mathbb Q$ or $\mathbb F_p$), by Noether normalization it follows that $k$ must be a finite extension of its prime field. In positive characteristic this finishes the job and in zero characteristic, should lead quickly to a contradiction, though I don't see immediately how. Is there an elementary/slick proof of this fact? Getting javax.crypto.IllegalBlockSizeException: Input length must be multiple of 16 when decrypting with padded cipher? Equivalence between least squares and MLE in Gaussian model I am new to Machine Learning, and am trying to learn it on my own. Recently I was reading through  and had a basic question.   Slide 13 says that "Least Square Estimate is same as Maximum Likelihood Estimate under a Gaussian model". It seems like it is something simple, but I am unable to see this. Can someone please explain what is going on here? I am interested in seeing the Math.   I will later try to see the probabilistic viewpoint of Ridge and Lasso regression also, so if there are any suggestions that will help me, that will be much appreciated also.