As far as I know, nowadays most of the mathematical literature is written and published in English and mathematicians communicate with each other in English. Although there are certain number of books written in other languages (like EGA), but at the same time their counterparts also appear in English (like Stacks Project). However, many graduate programs still require their students to pass a language translation test in French, German or Russian (which a paper dictionary, not a dictionary app in cellphones, which seems even more ridiculous to me...). I wonder what makes it still necessary to have foreign language requirement as of 2010s.  I believe my question has different focus than this one     Where the questioner specifically asked for advice for the most useful language among French, German and Russian:  I personally have no preference on which to learn, but I was wondering if there were other reasons that would make one language more advantageous over the others in terms of a general mathematical career.  while I am asking why we ever need a second language for mathematical study in 2010s:
I will soon be attending a Ph.D program in mathematics. I do not know how common this practice is, but the department has a requirement that graduate students must learn a foreign language and be able to read and write mathematics in that language. The three choices are French, German, or Russian. I personally have no preference on which to learn, but I was wondering if there were other reasons that would make one language more advantageous over the others in terms of a general mathematical career. Thanks in advance in any response.
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I read about the Quine-Duhem holistic thesis, but I do not know if is the same as underdetermination. Determine the Galois group of $\mathbb{Q}(\sqrt{a+b\sqrt{d}})$ Suppose $L=\mathbb{Q}(\sqrt{a+b\sqrt{d}})$,($d$ and $a+b\sqrt{d}$ are square free algebraic integers) when is $L/\mathbb{Q}$ a normal extension? When does $Aut(L/\mathbb{Q})=\mathbb{Z}/4\mathbb{Z}$ or $Aut(L/\mathbb{Q})=\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$? Horizontal spacing issue in \listoftheorems using amsthm + thmtools How to reference sections, equations, and figures in a separate document (e.g., supplemental information)? I have a main document (manuscript) and a supplemental document.   I wonder if there is a way that I can "link" the two (make them recognize each other) so that section, equation, and figure references across documents are recognized? Remove preinstalled apps on Big Sur MacBook Pro M1 Sorry for my English, I'm French and  I have  tried a lot (even with SIP disabled) but nothing works. I'm kind very desperate since I’m not sure how I would set up screen time to limit my children’s usage of the Pre-installed apps. The reason why I need to do so, is that my kid waste time in stead of studying. Is there a guide how to set this up? I was wondering if there is a viable place on Stack Exchange to ask economics-related questions.  Questions I'd have in particular could be equation-based or theoretical.  I suppose the equation-based questions could possibly fit into mathematics but I haven't checked their FAQ. In the simple linear regression model, $\hat{\beta}$ is the sum of independent normally distributed random variables.  Is it false because in linear regression there is $\beta$ and not $\hat{\beta}$? Traveling without paying lodging fees in America I want to travel around the East Coast without paying for lodging.  I'd be willing to sleep in my car or even pitch a tent somewhere. Is this a problem?  Legal?  I really want to get to the beach this season but I’m afraid that lodging will be out of my price range.  Specifically Maryland, Delaware, maybe New Jersey or Virginia. I am interested in the situation where you have a very interesting result. For instance, you have solved a very important open problem. However, you are not known in the field and do not have any remarkable publications. Your supervisor thinks the work is good and you submit the work to a high profile journal, but you get rejected.  The thing is that the contribution is very strong. It breaks what most people believe or what they have already proven: e.g., you solve the P vs. NP problem or any other well known open problem.   The reviewers strongly reject your work with no justification and they do not state why the result is wrong. Examples of reviewer comments include:   "The proof must be wrong."  "You cannot achieve such a result."  "You do not understand well the notion of ..."   My question is what to do in this situation? Where to go? If your advisor accepts the work, but the reviewers from the top journal reject the work without even explaining the mistakes,  what should you do? How to preserve detail when downscaling scanned photos? I'm mixing analog with digital photography — I tend to take better photos when I can't look at a small digital screen, unconsciously forcing me to take the picture before I shoot instead of checking afterwards if it succeeded. I also like the grain film gives me.  However, most shots aren't worth spending much time in the darkroom, and scanning them is a lot faster. I'm wondering how best to convert the film to digital.  Scanning at lower than maximum (non-interpolated) resolution of my image scanner tends to exaggerate the contrast of the grain and screw up the mid-tones because of that.1 On the other hand, the resulting size of the scans are major overkill in most cases (I don't print them at A0 size), so I want to downscale the pictures. However, the grain confuses most methods and results in loss of detail and/or loss of the characteristic grain. For example, bicubic sharper tends to result in a "bigger" grain than the original. Now, the best compromise I've found so far is simply using bilinear interpolation, but I doubt that this is the best method to preserve both overall details and the look of the grain.  In short: when processing the scanned negative, how do I "best" preserve both image quality and the characteristic look of the grain?   As Stan Rogers pointed out below, there's a difference between truly scanning the grain of the film, and scanning the "grainy" character of film at lower resolutions. This question is about the latter; Stan Rogers explains how to do the first.  I'm looking for alternatives to the standard Photoshop options to try out, and reasons for each. Is the fact that every film has a different grain characteristic a problem? Are customized advanced noise reduction methods perhaps an option?  1. I'm using a Minolta DiMAGE Scan Elite 5400, using the software provided by Minolta.  As requested, here is a scanned picture with different resize methods. Just scanned this, scanning a B&amp;W negative to get better tonality range. I turned off grain dissolver, automatic dust and scratch removal, etc, for obvious reasons. On the right the pictures are upscaled again with Nearest Neighbor. I set the 8 bit scan to 16 bits before downscaling, assuming it reduces risks of banding if I choose to mess with curves, channel mixers, etc. No other modifications (usually I first retouch dust &amp; scratches, and in the end channel mix the picture to B&amp;W, then switch to 8 bits again). So far I've been using public void run() {} methods to execute my code in Java. When/why might one want to use main() or init() instead of run()? What does "I have straight A's." mean? This is a follow-on from my  about samples from a distribution.  Suppose $X_1 \ldots X_{n-1}, X_n$ are random variables all following some fixed distribution $D$. How do I prove that $P(X_n &gt; X_1 \wedge \ldots \wedge X_n &gt; X_{n-1}) = \frac{1}{n}$ (this is true intuitively and experimentally). \documentclass[]{phstyle} \usepackage{latexsym,amsfonts,amssymb, undertilde} \usepackage[fleqn]{amsmath} \usepackage{makeidx}  %Preamble  \numberwithin{equation}{chapter} %\renewcommand{\theequation}{\thesection.\arabic{equation}}  \begin{document}  %\include{chapI}   these 2 lines are because the chapters are different tex files  %\include{chap1}  \renewcommand{\thechapter}{\Roman{chapter}}  %\setcounter{chapter}{0} %\renewcommand{\theequation}{\thechapter.\arabic{equation}}  \chapter{Introduction}  \section{Chap I Section1}  \begin{equation} this is equation I.1 \end{equation}  \begin{equation} this is equation I.2 \end{equation}  \section{Chap I Section2}  \begin{equation} this is equation I.3 \end{equation}  \chapter{Chapter 1}  \section{Chap 1 Section 1}  \begin{equation} this is equation 1.1.1 \end{equation}  \begin{equation} this is equation 1.1.2 \end{equation}  \end{document}     I have been typing Chapters 1-5 that contain sections and subsections and equations. The equations were numbered globally, i.e. if I had 3 equations in section 1 of Chap 1, they were numbered 1.1.1, 1.1.2, 1.1.3 and for Section 2, the equations were 1.2.4, 1.2.5, etc.  Recently I had to change Chap 1 and everything in it to Chapter I (roman number) and Chap 2 were renumbered to Chap 1, etc.  My problem is that LaTeX started numbering the equations in Chap I (the other chapters did not have this problem) according to the sections and not globally anymore. That is, the 3 equations in Chap I were I.1, I.2, I.3, and the equations in the second section of Chap I became I.1, I.2, etc instead of I.4, I.5, etc  I needed the prefix to be the chapter and not the sections. I don't understand why there is a change. I have never used \numberwithin in my book  Here are some of my codes  \documentclass[]{phstyle} \usepackage{latexsym,amsfonts,amssymb, undertilde} \usepackage[fleqn]{amsmath}   for the equation numbering in Chap I  \renewcommand{\thechapter}{\Roman{chapter}} \renewcommand{\theequation}{\thechapter.\arabic{equation}} **   Previously when Chap I was Chap 1, ** was  \renewcommand{\theequation}{\thesection.\arabic{equation}}   I have also tried the line below but it didn't work  \renewcommand{\theequation}{I.\arabic{equation}}   Thank you in advance Consider the atomic spectrum (absorption) of hydrogen.     The Bohr's model postulates that there are only certain fixed orbits allowed in the atom. An atom will only be excited to a higher orbit, if it is supplied with light that precisely matches the difference in energies between the two orbits.   But how precise does 'precisely' mean. Of course, if we need energy $E$ to excite the electron to a higher energy level, and I supply a photon with just $E/2$ I would expect nothing to happen (since the electron cannot occupy an orbit between the allowed ones). But what if I supplied a photon with energy $0.99E$, or $1.0001E$ or some such number. What will happen then?  I think that the electron should still undergo excitation precisely because the lines we observe in the line spectrum have some thickness. Which means that for a given transition, the atom absorbs frequencies in a certain range.   Is my reasoning correct? If not, why? How does Bohr's model explain this? How about modern theory?   If I'm right, what is the range of values that an atom can 'accept' for a given transition?