If I do not phrase this question right, please forgive me in advance because I am a layman on the subject of physics, but a software engineer, nonetheless. I do understand many technical subjects. That said, I currently am writing a science fiction novel and an important thought occurred to me while describing the fiction's technology- could a theoretical  actually allow for faster than light travel?  I ask this because in my reading on the subject I learned that both gravity and gravitational waves propagate at the speed of light and no faster. If this is true, then how can a ship ride a wave of distorted space-time that moves faster than light? I know that in a local frame of reference the ship technically would not move at all, but if you consider the "warp bubble" the ship is inside as moving at some velocity relative to an outside observer, then it seems pretty clear that the warp bubble could not surpass the speed of light.  Of course, this is a science fiction I am writing, so I have ample room for "bending the rules" with the theoretical, but I have great respect for science and literature, and want to get things right enough as to seem plausible.
I'm talking about this nonsense:   Now, I'm aware that there are problems with the practicalities (or possibilities) of methods involving exotic matter with negative mass, and that kind of thing has been addressed here:   But in this case my question is different and more specific. Sweeping all the other problems of constructing such a beast under the rug, wouldn't this violate causality?  SR says that the simultaneity and relative ordering of events is dependent on the relative motion of the observer, but that no matter how it works out, if event A causes event B, no observer will see B first. One of the many problems with FTL travel (or signals) is that if any information is sent faster than light, there will exist an inertial frame in which the signal arrives before it is sent, a blatant violation of causality. (I think we can all agree, can we not, that the threshold should be extremely high before breaking assumptions that we're in a causal universe?)  So how is any scheme to travel faster than light, even if it may locally satisfy the equations of GR, not a blatant violation causality according to SR?  Is my understanding of GR incomplete and there's something in it that obviates the causality implications of SR? Or am I correct that FTL claims should be dismissed with extreme prejudice on this basis alone? (Pending a serious replicable experiment showing causality to be breakable.)
How to pass anonymous types as parameters? $\mbox{rank}(A)=1$ implies $\det(A+I)=\mbox{trace}(A)+1$ This problem comes from exercise from my lecturer  Let $A$ be a square matrix such that $\mbox{rank}(A)=1$. Prove that $\det(A+I)=\mbox{trace}(A)+1$.  ($I$ is the identy matrix)  I usually haven't any idea in solving linear algebra problem. Why every polynomial over the algebraic numbers $F$ splits over $F$? postfix discard or change Message-ID header based on another header line I want to change the Message-ID header or better yet, discard it. I know everyone is going to ask "Why would you want to do that?" because I know it sounds weird to even write it.  Well I am developing a system that saves an incoming e-mail in EML format, changes it and sends it to the user, at some point the user can request to have the original EML sent to him/her. The issue is that when I re-send the EML it has the same Message-ID header which causes exchange servers to drop the re-send.  Of course I am hoping there is some option through postfix to do this, otherwise I will need to parse and change the EML but I rather not do that.  I tried to do this with "header_checks" and it works BUT I need to be able to do this conditionally, meaning only on a specific occasion, which I can identify by looking at the other headers, I want to remove the Message-ID. Let A and B be sets. Show that A is a subset of B if and only if for any set C, one has A union C is a subset of B union C. Can you verify my proof if it is right?  Let $A$ and $B$ be sets.  (a) Show that $A$ is a subset of $B$ if and only if for any set $C$, one has $A$ union $C$ is a subset of $B$ union $C$. (b) Show that $A$ is a subset of $B$ if and only if for any set $C$, one has $A$ intersect $C$ is a subset of $B$ intersect $C$.  Edited Solution :  (a)  (=>) Given A is a subset of B, this implies A U B = B. Given a set C, A U B U C = B U C. Since A is a subset of B, this implies A U C is a subset of B U C.      Solution 1 :(&lt;=) Given A U C is a subset of B U C , For A U C , x is a member of A U C . For B U C, x is a member of B U C. Suppose x is a member of A but not a member of B, meaning A is not a subset of B. Therefore , A U C is not a subset of B U C. Hence, A must be a subset of B.      Solution 2 :(&lt;=) Given A U C is a subset of B U C, For A U C, x is a member of A U C and for B U C, x is a member of B U C. Suppose C is a disjoint set. Since A U C is a subset of B U C, it follows that A is a subset of B.  (b) (=>) Since A is a subset of B , it follows that A U B = B. [A intersect C] is a subset of A and  [B intersect C ] is a subset of B by definition of intersection. Let x be member of A and member of C. Since [ A intersect C ] is a subset of A, x must be in A. Let x be a member of B and member of C. Since [ B intersect C ] is a subset of B, x must be in B. Therefore, A intersect C is a subset of B intersect C.  (&lt;=) Since A intersect C is a subset of B intersect C, suppose we let Set C be a subset of B and a subset of A as well. C is a subset of A implies A intersect C is equals to C. C is a subset of B implies B intersect C is equals to C as well. Since A intersect C is equals to B intersect C, A is a subset of B. How to fade one material into the other? Hello blender community.  Could someone please let me know how to fix this issue? I have one material for the whole face and a different one for the lips.  there's a noticeable edge that I want to get rid of. I just want it to have an ease fade instead of a sharp edge.   what's the best way to go about this?  Thank you in advance!    Finished Update  I wanted to give an update on how it was fixed with @hexbob6 and @Lemon 's help.    I needed two textures. One is the actual colored version and the 2nd is the black and white masked version. Hexbob6 described very well on the process here:   A: The node process on how to do it.    The B/W masked texture is going to be connected to the Fac of the mixed shader along with its UV coordinates. Make sure the image is on clip instead of repeat. the 1st shader is going to be the black part of the B/W masked texture. An important thing to note is if you have a texture already for the object, be sure to connect the other UV coordinates! the 2nd shader will be the white part of the B/W masked texture.   B: This section is the UV layout called "mainUV" where all the color goes.  C: This section is the B/W masked texture to focus on the lips.  D: Just the render view  E: Best way to texture something is to be able to see it in 3D view and texture paint.  F: Just showing that there are two textures to keep track of. One is the main UV and the 2nd is to focus on the lip.   That was it. Once again thanks to those who replied and helped me figure this problem. In the process I learned other things that I didn't know about blender even though I've been using it for years. For my MAC laptop, I only have 250GB and out of this 250GB MAC is taking up 129GB in the other section. I looked everywhere to the best of my knowledge. Please note my home directories is only 45GB. Can you somehow please help me? Colors and legend in groupplots/barplot In order to compare three items I use the groupplot command. For each item I have n different criteria to compare.    Is it possible to set the color for each criteria itself? Meaning the first bar in every group goes gray, the second pink etc.. The legend than should contain the n compared criteria.  And... how do I manage to let the whole tikz-picture have a width of 0.9\textwith  \documentclass{standalone} \usepackage{pgfplots,pgfplotstable} \usetikzlibrary{pgfplots.groupplots}  \pgfplotstableread{ nr  fii faa foo a   10  15  12 b   12  18  14 c   13  19  15 d   9   13  10 e   7   11  8 f   10  15  12 }\datatable  \begin{document} \begin{tikzpicture} \begin{groupplot}     [group style={columns=3},         ybar,         ymin=0,         ymax=20,         xtick=\empty,         tickpos=both]  \nextgroupplot[xlabel=fii] \addplot [fill=blue] table [x expr=\coordindex, y=fii] {\datatable}; \nextgroupplot[xlabel=faa] \addplot [fill=green] table [x expr=\coordindex, y=faa] {\datatable}; \nextgroupplot[xlabel=foo] \addplot [fill=red] table [x expr=\coordindex, y=foo] {\datatable};  \end{groupplot} \end{tikzpicture} \end{document} Let's have breakfast    Let's have a good breakfast      I've learned that indefinite articles aren't used before uncountable nouns, so why is a used before the adjective "good" here?    Is there an exception to the rules or is it wrong? Show that any prime divisor of $x^4+x^3+x^2+x+1$, with $x\in\mathbb{N}$, is $5$ or $1$ mod $5$ We can write the "polynomial" as follows: $$x^4+x^3+x^2+x+1=\frac{x^5-1}{x-1}.$$ For even $x=2y$, we have that $x^5-1=(2y)^5-1=32y^5-1\equiv1$ mod $5$.  For odd $x=2y+1$, we have that $(2y+1)^5-1\equiv_532y^5\equiv2$ mod $5$.  I find it hard to believe my computations above. I guess this makes me rather clueless. I was hoping there was some way out with Legendre symbols. Any help is appreciated. Is it possible to test apex that relies on field history tracking? I have some code that relies on field history tracking by querying the History for a custom object. This works fine, but it appears that the history records aren't created during the test so none of my tests will pass. Why is the Descriptor not getting called when defined as instance attribute? Standard graded algebra Change .jpg quality of terminal screenshots Polynomials vanishing on an infinite set I'd like some help making this argument complete and rigorous (if it's correct - if not, help with that would be nice).  Here $k$ is a field.     Let $A_1,\ldots,A_n \subseteq k$ be infinite subsets. Then any polynomial in $k[x_1,\ldots,x_n]$ that vanishes on $A_1\times\cdots\times A_n\subseteq k^n$ must be $0$ (as a polynomial).   This is what I have ...  For the case $n=1$, a non-constant polynomial can only have as many roots as its degree, and in particular, it must have a finite number of roots. The only polynomial in one variable that has an infinite number of roots is $0$, so if a polynomial in $k[x_1,\ldots,x_n]$ vanishes on an infinite subset then it must be $0$.  For the inductive step, suppose the proposition is true for less than $n$ subsets and variables. Let $p\in k[x_1,\ldots,x_n]$ vanish on $A_1\times\cdots\times A_n$. Fix $x_n$ as some $a\in A_n$, and we have a polynomial in $n-1$ variables that vanishes on the set $A_1\times\cdots\times A_{n-1}$, so by the inductive hypothesis it must be identically $0$.  (Now it gets sketchy). Since this is true for any of the infinite values in $A_n$, and , $p$ must be $0$. How to reset Kali linux root password What is a substitute for red or white wine in a recipe? What intuitive notion is formalized by "natural transformation" in category theory? I was trying to install minted package in TeXLive. I installed it and then run sudo tlmgr update --all. It was running slow so I stopped it using Ctrl+Z in an ubuntu terminal and then I run sudo tlmgr update --list. After that I installed texlive-latex-extra and python-pygments. Later on I run sudo tlmgr update --all again and let it finish.  After that I am trying to run a simple code using listings and greek but I get an error     Bad Character code (256)   A MWE that illustrated the problem is the following  \documentclass{book}  \usepackage[english,greek]{babel} \usepackage[iso-8859-7]{inputenc} % %\usepackage[utf8]{inputenc} \usepackage{kerkis} \usepackage{pifont} % %  \newcommand{\sw}{\selectlanguage{english}} \newcommand{\sq}{\selectlanguage{greek}} \newcommand{\eng}[1]{\latintext#1\greektext} \newcommand{\gre}[1]{\greektext#1\latintext}  \usepackage{xcolor} \usepackage{listings} \usepackage{textcomp}  \definecolor{listinggray}{gray}{0.9} \definecolor{lbcolor}{rgb}{0.9,0.9,0.9}  \lstdefinestyle{JavaStyle}{         backgroundcolor=\color{lbcolor},     tabsize=4, %   rulecolor=,     language=Java,      % choose the language of the code         basicstyle=\scriptsize,     % the size of the fonts that are used for the code         upquote=true,         aboveskip={1.5\baselineskip},         columns=fixed,         showstringspaces=false,         extendedchars=false,         breaklines=true,         prebreak = \raisebox{0ex}[0ex][0ex]{\ensuremath{\hookleftarrow}},         frame=single,         numbers=left,         showtabs=false,         showspaces=false,         showstringspaces=false,         identifierstyle=\ttfamily,         keywordstyle=\color[rgb]{0,0,1},         %commentstyle=\color[rgb]{0.026,0.112,0.095},         commentstyle=\itshape\color{green!40!black},         stringstyle=\itshape\color{red!90!black},         numberstyle=\itshape\color{yellow!50!black} }  \lstdefinestyle{CppStyle}{         backgroundcolor=\color{lbcolor},     tabsize=4, %   rulecolor=,     language=C++,       % choose the language of the code         basicstyle=\scriptsize,     % the size of the fonts that are used for the code         upquote=true,         aboveskip={1.5\baselineskip},         columns=fixed,         showstringspaces=false,         extendedchars=false,         breaklines=true,         prebreak = \raisebox{0ex}[0ex][0ex]{\ensuremath{\hookleftarrow}},         frame=single,         numbers=left,         showtabs=false,         showspaces=false,         showstringspaces=false,         identifierstyle=\ttfamily,         keywordstyle=\color[rgb]{0,0,1},         %commentstyle=\color[rgb]{0.026,0.112,0.095},         commentstyle=\itshape\color{green!40!black},         stringstyle=\itshape\color{red!90!black},         numberstyle=\itshape\color{yellow!50!black} } \begin{document}  This is a dummy text. kjhbki \end{document}   The .aux file says  \relax  \select@language{greek} \@writefile{toc}{\select@language{greek}} \@writefile{lof}{\select@language{greek}} \@writefile{lot}{\select@language{greek}}   while the .log file can be found   Any idea why is this happening and how to fix this? Note that if I remove the listings preamble, delete all produced files and build the document, everything works fine. Is $0\,\mathrm{m}$ dimensionless? Is $0 \, \mathrm m = 0 \, \mathrm s = 0 \,\mathrm {kg} = 0$? How do we define $[0 \, \mathrm m]$?  I once was given an assignment where I was asked to deduce and write down some physical quantity. It turned out that this quantity was $0$ in one unit or another, so I decided to drop the physical units because I figured that it didn't matter anyhow.  But I got to thinking:  If $0 \, \mathrm m = 0$, then we could add units of different dimensions (an operation not normally defined) like so: $5 + 0 \,\mathrm m = 5$.   Does $[ \cdot ]$ map $0 \, \mathrm m$ to both $0$ and $L$? Is there an established regime of thinking about physical quantities as an abstract algebra? Perhaps as a vector space where formal sums such as $5 \, \mathrm m + 10 \, \mathrm s$ are allowed? Would it be improper to drop the units of measurement of $0 \,\mathrm m$ like this in an academic paper?