Is there a preferred format for numbers on Stack Exchange?
Should the network use thousands separators in numbers?
When is it appropriate to end a question without a question mark? Is there any way to do n-level nested loops in Java? In other words, can I do something like  for() {     for {        for {        }     } }   Except N times? In other words, when the method creating the loops is called, it is given some parameter N, and the method would then create N of these loops nested one in another?  Of course, the idea is that there should be an "easy" or "the usual" way of doing it. I already have an idea for a very complicated one. Current  enables one to upgrade to macOS Mojave from a stable release, but I want to try out a fresh installation instead of upgrading with previous data, so that I can assess it better.  I've downloaded an application named "macOS Mojave Beta" from the App Store, and running it would presumably result in an upgrade. Now I wonder what to do next. Would  and booting from it be a good idea, when I'm working with a Beta version? The question is if you have a situation where one of the singular values is equal to 0 in a singular value decomposition of a matrix, how to do you procede to find the column vector of U corresponding to this singular value?   Usually I found the columns of U with this relation, so let's say that we refer to $\vec{u_2}$ as the U column vector corresponding to singular value equal to 0:  $\vec{u_2} = (1/s_2)Av_2$  Now the as we can see the 0 is in an illegal place, since you can't divide by zero. But if you move it to the left side, I still can't quite see how to find the column of U corresponding to it. Any suggestions?  Here is an example of a matrix that provided this specific challenge  $ A = \left(\begin{array}{rrr} -3 &amp; 1 \\ 6 &amp; -2 \\ 6 &amp; -2 \end{array}\right). $  $ \lambda{_1} = 90$,  $\lambda{_2} = 0$,  $s_1 = 3 \sqrt{10}$,  $s_2 = 0$,  How can one find $\vec{v_2}$ when $\lambda{_2} = 0$? In this case, you know that the eigenvectors of $A^{t} A$ are orthogonal, solve for $\lambda{_2}$ and find the eigenvector if it's orthogonal to the eigenvector corresponding to the first eigenvalue you have your solution.  So we end up with the following normalized column vectors for V, which we can see are orthogonal:  $ \vec{v_1} = \left(\begin{array}{rrr} -3/\sqrt{10} \\ 1/\sqrt{10} \\ \end{array}\right). $  $ \vec{v_2} = \left(\begin{array}{rrr} 1/\sqrt{10} \\ 3/\sqrt{10} \\ \end{array}\right). $ Correct spelling of Polish surname Dziubenofska? My Polish grandmothers last name was Dziubenofska, but I know that's not how it is spelt! I'm looking for the proper spelling.   Can you help find it for me?  She came over to Ellis Island and they changed it on her to dziuk. This is only a prefix to the full name. How long does the new dwellers queue last? When clicking the radio room icon that attracts a new dweller to your vault, how long will they stay queued out in front of the vault for before giving up?  Or will they?  I'm at max population and would rather not kill existing dwellers to accommodate them in my vault. In an Integral Domain is it true that $\gcd(ac,ab) = a\gcd(c,b)$? Entries I can safely exclude doing backups To be honest, I have no idea how to even start this problem. I'm sorry I don't have any work to show, but I'm just at a blank. Help?  Chapter 2: Problem 27:  University B, once boasted $17$ tenured professors of mathematics. Tradition prescribed that at their weekly luncheon meeting, faithfully attended by all $17$, any members who had discovered an error in their published work should make an announcement of this fact, and promptly resign. Such an announcement had never actually been made, because no professor was aware of any errors in her or his work. This is not to say that no errors existed, however. In fact, over the years, in the work of every member of the department at least one error had been found, by some other member of the department. This error had been mentioned to all other members of the department, but the actual author of the error had been kept ignorant of the fact, to forestall any resignations. One fateful year, the department was augmented by a visitor from another university, one Prof. X, who had come with hopes of being offered a permanent position at the end of the academic year. Naturally, he was apprised, by various members of the department, of the published errors which had been discovered. When the hoped-for appointment failed to materialize, Prof. X obtained his revenge at the last luncheon of the year. &quot;I have enjoyed my visit here very much&quot;, he said, &quot;but I feel that there is one thing that I have to tell you. At least one of you has published an incorrect result, which has been discovered by others in the department.&quot; What happened the next year?&quot;   Chapter 2: Problem 28:  After figuring out, or looking up, the answer to Problem 27, consider the following: Each member of the department already knew that Prof.X asserted, so how could his saying it change anything? Are all n-ary operators simply compositions of binary operators? Take for example $A \times B \cdot C$ = $(A \times B) \cdot C$ where $A, B, C$ are 3-component real vectors.  We can define a 3-nary operator $\times - \cdot$ that is a composition of the two common binary operators $\times$ and $\cdot$.  The same thing happens with most functions (operators) - the way we calculate them is by doing smaller binary problems and adding together.  Every time I try to come up with an $(n &gt; 2)$-ary operator my mind automatically looks for binary operators.  So, the question is, do there exist operators (of some weird kind in some branch of math) that cannot be decomposed into 2-ary and 1-ary operators?  Thanks. I have this weird object (vascular tree) which is empty inside, but closed from the top and the bottom:  However, it is empty inside, which makes me unable to use boolean on a bigger object (I want to make a hole in an object through which this object will go through). So I thought I should fill it and then try to boolean. But, how do I fill it?  Thank you sincerely. Is there anything like GF(6)? What *is* an Artificial Neural Network? In other words, why do we need to connect the battery positive to the negative to get electron flow? As far as I know, voltage difference is what drives current flow.  From what I understand, there's a surplus of negative charge (electrons) in the positive end of a battery (weird I know, but I guess they do it for mathematical reasons). Between the positive and negative ends of the battery is some kind of wall that prevents the electrons from diffusing, so they have to go the long way (through a wire to the other end of the battery) to diffuse and reach the receiver atoms/molecules.  So why not just have the surplus of electrons on one end go to ground to equalize (e.g. 5 volts to 0 volts should produce current)? Why does it have to be the other end of the battery? I'm sure there's atoms/molecules in the ground that could accept the extra electrons (receivers atoms/molecules for the donors in the battery). The following quote from  assumes "the URL" comes from somewhere and uses it to restrict the Host header. Where does "the URL" come from? Is it from the first line of a HTTP request (e.g. after GET), or from the input to a web client (e.g. an argument to curl, or the input in the address bar of a web browser)? Thanks.     The Host header specifies the Internet host and port number for the   resource being requested, as obtained from the original URL:  Host = "Host" ":" host [ ":" port ]       In particular:      • If the Host header does not contain a port, the default port for the   scheme is assumed.      • If the URL contains an IP address, the Host header should contain   the same address.      • If the URL contains a hostname, the Host header must contain the   same name.      • If the URL contains a hostname, the Host header should not contain   the IP address equivalent to the URL’s hostname, because this will   break virtually hosted servers, which layer multiple virtual sites   over a single IP address.      • If the URL contains a hostname, the Host header should not contain   another alias for this hostname, because this also will break   virtually hosted servers. I installed Windows 7, which ate Ubuntu's boot file. When starting up the computer, it now goes straight to Windows, without giving me the option of booting Ubuntu.   How can I get Ubuntu back? Windows installer for Ubuntu 14.04 LTS onwards Disable cache for some images When I add the picture, the word and pdf is the full page and the Latex moves to the right of the page. How can I solve this problem?  this is my work. this is my work and pdf link: https://we.tl/t-cgDSwDuu8H  I couldn't stop the pictures from sliding right. What should I do to make the image look like the original PDF file? If I reduce the scale of the picture is centered but this time it becomes very small.  Is a situation caused by page margins?  \documentclass [12pt, a4paper]{article} \usepackage{color} %\usepackage[turkish]{babel} \usepackage{graphicx} \usepackage{tabularx} \usepackage{multirow} \usepackage{graphicx} %\usepackage[pass,showframe]{geometry} %\usepackage[T1]{fontenc}  \begin{document}  \definecolor{renkbir}{RGB}{0,32,96}  \begin{tabular}{ m{8,5cm} m{6cm} } \Large{\textbf{MsdasdasdN }} \large{\textbf{\textcolor{renkbir}{Firma Adı }}} &amp;\small {Uçak Mühendisi \newline T.C No: asd \newline asd \newline asd \newline asd} \end{tabular} \ %Uçak mühendisi Tc No kısımını sağa yaslamak istedim fakat yapamadım.  \begin{center} \large{\textbf{\textcolor{renkbir}{KİŞİSEL BİLGİLER}}} \end{center}  \begin{center} \begin{tabular}{ l l r } Ad &amp; Mehm &amp; \multirow{9}{*}{\includegraphics[scale=0.15]{2.pdf}} \ Soyad &amp; asdasd \ Doğum Tarihi &amp; sanane (23 Yaşında) \ Adres &amp; asd \ Telefon &amp; asd \ Mail &amp; asd \ Sigara Kullanımı &amp; Hayır \ Ehliyet &amp; B \ Askerlik &amp; Tecilli (31.12.2020) \ \end{tabular} \end{center}  %\vspace{1cm} Burası boşluk bırakma komutu.  \begin{center} \begin{tabular}{ m{4cm} m{6cm} m{5cm} } \multicolumn{3}{c}{\large{\textbf{\textcolor{renkbir}{EĞİTİMLER}}}} \ \hline \multirow{2}{}{Üniversite(YL)} &amp; asd \newline Makine Mühendisliği \newline Özel Öğrenci / Türkçe \newline Durumu : Devam ediyor &amp; \multirow{2}{}{Eylül 2018 / } \ \hline \multirow{2}{}{Üniversite} &amp; asd \newline Uçak ve Uzay Bilimleri Fakültesi \newline Uçak asd \newline Örgün Öğretim / İngilizce \newline Durumu : Mezun &amp; \multirow{1}{}{Ekim 2013 / Haziran 2018} \ \hline \multirow{1}{}{Lise} &amp; asd \newline Anadolu Lisesi (100 / 90,02) &amp; \multirow{1}{}{Ekim 2013 / Haziran 2018} \ \end{tabular} \end{center}  \begin{center} \begin{tabular}{ m{3cm} m{4cm} m{4cm} m{4cm} } \multicolumn{4}{c}{\large{\textbf{\textcolor{renkbir}{YABANCI DİLLER}}}} \ \hline \textbf{İngilizce} &amp;\textbf{Okuma:} İyi &amp;\textbf{Yazma:} İyi &amp;\textbf{Konuşma:} Orta \ \end{tabular} \end{center}  \begin{center} \begin{tabular}{ m{5cm} m{5cm} m{5cm} } \multicolumn{3}{c}{\large{\textbf{\textcolor{renkbir}{BİLGİSAYAR BİLGİLERİ}}}} \ \hline \multicolumn{3}{l}{Donanım - Ofis Programları} % donann ofis koyu yazmak istedim yapamadım. \ Ms Excel &amp; Çok İyi &amp; +5 Yıl \ MS Word &amp; Çok İyi &amp; +5 Yıl \ Ms Powerpoint &amp; Çok İyi &amp; +5 Yıl \ \hline \multicolumn{3}{l}{Yazılım Geliştirme} % donann ofis koyu yazmak istedim yapamadım. \ Office &amp; Orta &amp; 2-3 Yıl \ Matlab &amp; Orta &amp; 2-3 Yıl \ Phyton &amp; Orta &amp; 0-1 Yıl \ Fortran &amp; Başlangıç &amp; 0-1 Yıl \ C++ &amp; Başlangıç &amp; 0-1 Yıl \ Ansys &amp; Başlangıç &amp; 0-1 Yıl \ Miktex &amp; Başlangıç &amp; 0-1 Yıl \ \hline \multicolumn{3}{l}{Bilgisayar Destekli Tasarım} % donann ofis koyu yazmak istedim yapamadım. \ SolidWorks &amp; Orta &amp; 2-3 Yıl \ AutoCAD &amp; Orta &amp; 0-1 Yıl \ \end{tabular} \end{center}  \begin{figure} \includegraphics[width=\linewidth]{2.pdf} \end{figure}  \begin{figure} \includegraphics[scale=1]{2.pdf} \end{figure}  \end{document}   Latex SS is here:  PDF SS is here: Let $M,N$ be $n \times n$ square matrices over an algebraically closed field with the properties that the trace of both matrices coincides along with all powers of the matrix. More specifically, suppose that $\mathrm{Tr}(M^k) = \mathrm{Tr}(N^k)$ for all $1\leq k \leq n$. The following questions about eigenvalues is then natural and I was thinking it would be an application of Cayley-Hamilton but I am having trouble writing out a proof.  How do we show that $M$ and $N$ have the same eigenvalues?  Added (because this question is now target of many duplicates, it should state its hypotheses properly). Assume that all the mentioned values of $k$ are nonzero in the field considered; in other words either the field is of characteristic $0$, or else its prime characteristic $p$ satisfies $p&gt;n$.