Chinese Remainder for x= 7 (mod 8), x= 2 (mod 9), x= -1 (mod 5)
Solving multiple congruences by CRT = Chinese Remainder Theorem
What Happens When A Gravitational Wave Interacts With Another One? If two gravitational waves came in contact with each other what would happen? In another question entirely, what happens when a higher gravitational field interacts with a weaker one. Why is the decay of a neutral rho meson into two neutral pions forbidden? Create Task in loop but all tasks take the same parameter I'm doing some exercises on Apostol's calculus, on the floor function. Now, he doesn't give an explicit definition of $[x]$, so I'm going with this one:     DEFINITION Given  $x\in \Bbb R$, the integer part of $x$ is the unique $z\in \Bbb Z$ such that $$z\leq x &lt; z+1$$ and we denote it by $[x]$.   Now he asks to prove some basic things about it, such as: if $n\in \Bbb Z$, then $[x+n]=[x]+n$  So I proved it like this: Let $z=[x+n]$ and $z'=[x]$. Then we have that  $$z\leq x+n&lt;z+1$$  $$z'\leq x&lt;z'+1$$  Then $$z'+n\leq x+n&lt;z'+n+1$$  But since $z'$ is an integer, so is $z'+n$. Since $z$ is unique, it must be that $z'+n=z$.  However, this doesn't seem to get me anywhere to prove that  $$\left[ {2x} \right] = \left[ x \right] + \left[ {x + \frac{1}{2}} \right]$$  in and in general that   $$\left[ {nx} \right] = \sum\limits_{k = 0}^{n - 1} {\left[ {x + \frac{k}{n}} \right]} $$  Obviously one could do an informal proof thinking about "the carries", but that's not the idea, let alone how tedious it would be. Maybe there is some easier or clearer characterization of $[x]$ in terms of $x$ to work this out.  Another property is $$[-x]=\begin{cases}-[x]\text{ ; if }x\in \Bbb Z \cr-[x]-1 \text{ ; otherwise}\end{cases}$$  I argue: if $x\in\Bbb Z$, it is clear $[x]=x$. Then $-[x]=-x$, and $-[x]\in \Bbb Z$ so $[-[x]]=-[x]=[-x]$. For the other, I guess one could say:  $$n \leqslant x &lt; n + 1 \Rightarrow  - n - 1 &lt; x \leqslant -n$$  and since $x$ is not an integer, this should be the same as $$ - n - 1 \leqslant -x &lt; -n$$  $$ - n - 1 \leqslant -x &lt; (-n-1)+1$$  So $[-x]=-[x]-1$ How do servlets work? Instantiation, sessions, shared variables and multithreading Mean Value Theorem: $\frac{2}{\pi}<\frac{\sin x}{x}<1$ I need to show that $\dfrac{2}{\pi}&lt;\dfrac{\sin(x)}{x}&lt;1$ for $0&lt;x&lt;\dfrac{\pi}{2}$.  I know I need to use the mean value theorem, would I just say that since $f$ is continuous in the interval (call it I).  We know $\exists c\in I$ such that $f'(c) = \dfrac{f\left(\dfrac{\pi}{2}\right)-f(0)}{\dfrac{\pi}{2}}$. I don't know where this is going though.  Also, would I need to show $\lim_{x \to 0}\dfrac{\sin(x)}{x} = 1$? Is there a way to do this without L'H Rule? How to run a specific program as root without a password prompt? I have seen several answers and proofs here at MSE that a self-adjoint operator $T$ on a Hilbert space $H$ where $T^2=T$, implies $T$ being an orthogonal projection operator.   The thought struck me; what would happen if the condition $T^2=T$ is replaced by $T^3=T^2$ or instead by $T^4=T^3$ or why not more generally by $T^n=T^{n+1}$ for some $n\in\mathbb{N}$? Would this still imply $T$ to be an orthogonal projection operator?  I guess someone must have thought of this before me? In that case perhaps there is a reference in the literature for this or perhaps the proof is way simpler than I might expect? The pronunciation of words which begins 'con' and 'com' Where do I use delegates? What are some real world places that call for delegates?  I'm curious what situations or patterns are present where this method is the best solution.  No code required. Biblatex 3.3 name formatting I would like to create a custom format for citing authors' names in the text, let's call it "newformat". One option I can think of is creating a new bibmacro for names where the parameters are changed to my liking, such as   \documentclass{article}    \usepackage{filecontents}   \usepackage[backend=biber]{biblatex}[2016/01/01] %the latest version of biblatex?   % biber is version 2.4; the latest?  \begin{filecontents}{\jobname.bib} @misc{Doe12,   author = {Doe, John},   year = {2012},   title = {A macro for formatting names}, } \end{filecontents}  \addbibresource{\jobname.bib} \nocite{*}  \newbibmacro{name:newformat}{\printnames{authors}} \DeclareNameFormat{newformat}{%   \usebibmacro{name:newformat}{\textbf{#1}}{\textbf{#3}}{#5}{#7}%   \usebibmacro{name:andothers}% }  \begin{document}    \section*{Testing the format here}    I am citing   \citename{Doe12}[newformat]{author}    \printbibliography % to make sure that the .bib file is properly formated.  \end{document}   Here, I just used \textbf for the sake of the example, but I ultimately want to use other functions, say upper case or palindrome!  However, I get this weird error message:  ! Illegal parameter number in definition of \blx@defformat@d. &lt;to be read again&gt;                     3 l.20   \usebibmacro{name:last}{#1}{#3                                      }{#5}{#7}%   which seems to indicate that the second option {#3} is not expected. I am a bit at a lost here; I checked the biblatex package and \usebibmacro{name:last} must be given 4 parameters, isn't it? Please advise. Let $A,B$ be any two sets. I really think that the statement $|A|\leq|B|$ or $|B|\leq|A|$ is true. Formally:     $$\forall A\forall B[\,|A|\leq|B| \lor\ |B|\leq|A|\,]$$   If this statement is true, what is the proof ? Prove that the sum to $n$ terms of a sequence:  $$\frac{1^2}{1\times 3}+\frac{2^2}{3\times 5}+\frac{3^2}{5\times 7}+\cdots $$ is  $$ \frac{n(n+1)}{2(2n+1)} $$ Can you be sued for using publicly used images on a blog without copyright permission? How do you reopen a closed question? To reopen a closed question,  must agree that the question is suitable for the site and cast votes to reopen the question. But:  How does one actually vote to reopen? Is there a reputation level that you must have before you can see this function? Or do I just leave a comment along the lines of, &quot;I vote to reopen&quot;? What happens if someone with less than 3000 reputation leaves a comment that they wish to vote to reopen? How can one draw attention to their closed questions? Is it done via a posting new question? Or do you edit the closed question?   Related:  For more information, see &quot;&quot; in the . How to detect Windows 64-bit platform with .NET? In a  2.0 C# application I use the following code to detect the operating system platform:  string os_platform = System.Environment.OSVersion.Platform.ToString();   This returns "Win32NT". The problem is that it returns "Win32NT" even when running on Windows Vista 64-bit.  Is there any other method to know the correct platform (32&nbsp;or 64&nbsp;bit)?  Note that it should also detect 64&nbsp;bit when run as a 32&nbsp;bit application on Windows 64&nbsp;bit. How to make a domain receiving email behind cloudflare? Progressive Draw 2 and Draw 4 Uno When playing with progressive Draw 2 and Draw 4 and (say that there are 2 players playing) and    Player 1 goes out with a Draw 2  Player 2 puts down a Draw 2    Does player 1 have to pick up 4 cards? How to upload a file of size 2GB from Apex? Is there any sobject that I can make use of? How should a family with young children apply for UK visas? We are sponsoring a family to visit us in UK as friends from Madagascar for two weeks. They have a 6 year old and a 16 month old.   Do the kids need their own visas, and are they the standard visitor visa or a special child visa?   Also, when I try and apply on the website I can only apply for one person so how do I apply for all the individuals in the family?   I have looked on the Immigration website and can't find the answers I need.