Difference between space-like and time-like singularities
What is the definition of a timelike and spacelike singularity?
Code is supposed to be exact and "as-is". Sometimes, showing two spaces is important. In a comment, it is not possible to show code with  multiple   spaces. This became apparent in  (see comments).  EDIT: apparently this bug is larger then I thought: multiple    spaces   here are neither preserved in normal question text. Pity (though I couldn't find this reported before, I'd assume it is).  Workaround: use Alt+0160 (windows), or U+00A0 (Unicode NBSP), as I did in the example above. A few years ago, I applied for a UK visa. The visa was initially refused and I had no rights of appeal. However, the refusal letter had some glaring mistakes and I sent an email to the Embassy. The refusal was "overturned" by an ECM.  I am now making an application for a visa which has the question "Have you ever been refused a visa?". How should I answer it? How can I backup my photos while travelling? I'm going travelling around the world for a year in September and am already panicking about how best to send my photos home while in various countries.  Requirements:   I only need to backup JPEG files (I don't shoot RAW but that's another thread ;) ) I need to be able to upload in a web-based manner (I will be using Internet cafes and so can't assume I can use FTP etc) I need plenty of space, preferably unlimited (I racked up about 200GB last year) I need to store the full image (no lossy compression is acceptable!)   Options I can think of:   Burning and posting DVDs (can I trust these in the post and not to get corrupt? It may be better than waiting for upload speed though...) Flickr (I have a pro account but not sure how hard it'll be to get the photos back down again. Will I even be able to get the original snaps?) Setting up a server at a friend's house and hosting some kind of file server with a web-app front end   What have other people done? Or does anyone else have a better idea? Can I use Google Maps as a layer in ArcMap?  Is it possible? Calculate the limit of two interrelated sequences? Search across Stack Exchange sites in Android app launch sms application with an intent jQuery ajax load() java script not executing? When should you use a class vs a struct in C++? How do I install drivers for Realtek RTL8812BU? So I've been experimenting with Ubuntu on and off for a while, but I honestly have no idea what the hell I'm doing. So I have a realtek RTL8812bu WiFi dongle that I'm hoping to get working. I did some research, and came across  that a couple people across two threads said worked for them:  At the bottom it said there is a new version . So somewhere it said it was confirmed to work on kernels up to 4.16, so I checked and I'm on 4.15.0. But the problem I'm having, is what am I supposed to do? There is some installation guide at the bottom that is about as clear as mud to me since I don't know much. Claim: If $(x_\alpha)_{\alpha\in A}$ is a collection of real numbers $x_\alpha\in [0,\infty]$ such that $\sum_{\alpha\in A}x_\alpha&lt;\infty$, then $x_\alpha=0$ for all but at most countably many $\alpha\in A$ ($A$ need not be countable).  Proof: Let $\sum_{\alpha\in A}x_\alpha=M&lt;\infty$. Consider $S_n=\{\alpha\in A \mid x_\alpha&gt;1/n\}$.  Then $M\geq\sum_{\alpha\in S_n}x_\alpha&gt;\sum_{\alpha\in S_n}1/n=\frac{N}{n}$, where $N\in\mathbb{N}\cup\{\infty\}$ is the number of elements in $S_n$.  Thus $S_n$ has at most $Mn$ elements.   Hence $\{\alpha\in A \mid x_\alpha&gt;0\}=\bigcup_{n\in\mathbb{N}}S_n$ is countable as the countable union of finite sets. $\square$  First, is my proof correct? Second, are there more concise/elegant proofs? How to revert to tabbed overview in Chrome? I recently updated my Chrome for Android (currently on v68) and noticed that tapping on the box icon (next to address bar) gives a tab list instead of SystemUI type overview that Chrome used to give. See the two images below for comparison.  I know that I can get that same view by swiping downwards from the address bar but I'm habitual of getting it from that box icon.   So what can I do to get back that view from that box without reverting to a previous version of Chrome and without switching to an another browser (unless it is Chrome based)? My question is really ''what is the definition of a conservative vector field''?  I've consulted 3 textbooks that all say a vector field $\vec{F}$ is conservative by definition if there exists a scalar potential $\phi$ such that $\nabla \phi = \vec{F}$. Then, they go on to talk about connected domains, path independence and the equality of mixed partials and how they are all related.   In particular, they emphasize that e.g. in $\mathbb{R}^2$ given $\vec{F} = \bigl&lt;F_1,\,F_2\bigr&gt;$, if $\frac{\partial F_1}{\partial y} = \frac{\partial F_2}{\partial x}$ on a simply-connected domain, then $\vec{F}$ is conservative on that domain.  However, without fail, all of them then offer the example of $\vec{F} = \bigl&lt; \frac{-y}{x^2+y^2},\, \frac{x}{x^2+y^2} \bigr&gt;$, pointing out:   it's line integral is not path independent, even though $\frac{\partial F_1}{\partial y} = \frac{\partial F_2}{\partial x}$   and this is explained by pointing out that the domain is not simply connected (if the path contains the origin).  That much makes sense to me: we cannot conclude $\vec{F}$ is conservative based on the partial derivatives, because the domain is not simply connected - totally consistent with what has been presented.  What none of them address is why $\vec{F} = \bigl&lt; \frac{-y}{x^2+y^2},\, \frac{x}{x^2+y^2} \bigr&gt;$ is not conservative when there exists a potential $\phi = \arctan(y/x)$ such that $\nabla \phi = \vec{F}$. None of the texts mention any necessary conditions on the scalar potential. So is the existence of a scalar potential the definition of a vector field being conservative or not? How often is the Golden Ratio actually used in modern design? The  is the stuff of legend () and is . My question is two-fold, really: Is the Golden Ratio really a good tool used in modern design (defined here as 20th century on forward), and if so, how often is it used outside of the aforementioned known uses?  EDIT: To @e100's first comment, examples of modern use of the Golden Ratio would be a great thing to add here if there any documented or arguably conclusive uses of it. PHP PDOException: "SQLSTATE[HY093]: Invalid parameter number" Present Perfect or simple past? When is the Laplace transform of a function continuous? Are there any general conditions on a function so that its Laplace transform is continuous? e.g. what conditions on $f:\mathbb {R} \rightarrow \mathbb {R}$ guarantee that its Laplace transform  $$F\left(s\right)=\int_{0}^{\infty}f\left(t\right)e^{-st}dt$$ is continuous? It seems to me that if $f$ has compact support and is bounded, then it ought to be. The idea I have is that we get  $$\left|F\left(s'\right)-F\left(s\right)\right|\leq\int_{0}^{y}\left|f\left(t\right)\right|\left|e^{-st}\right|dt$$ by making $s'$ and $s$ close enough, it seems we can make the RHS arbitrarily small. I haven't been able to complete a proof. Additionally, can we change the assumptions? Are there more natural assumptions to get continuity? SHA-256 "almost unique"? I have seen numerous references on the internet of people describing SHA-256 as generating an "almost unique" hash.  . there are more.  Is there some mathematical basis to the almost uniqueness? I would have thought that as SHA-256 is an evenly-distibuted (?) hash algorithm which can accept any message (infinitely many) as an input and produces a fixed length (finite) hash - then there will in fact be an infinite number of each possible output - which I would think hardly qualifies as "almost unique".  Is there a basis for this statement? 1) How do I prove the following:   Let A = {6a + 4b ∈ Z : a, b ∈ Z} and B = {2a ∈ Z : a ∈ Z}. Show that A = B.  Thank you all for the help! Is it normal for BibTeX to replace similar author names with "------"? In my .bib file, I have two entries with exactly the same author names:  @article{Seshadrinathan2010A-Subjective-St,     Author = {K. Seshadrinathan and R. Soundararajan and A. C. Bovik and L. K. Cormack}, ....  @article{Seshadrinathan2009Study-of-Subjec,     Author = {K. Seshadrinathan and R. Soundararajan and A. C. Bovik and L. K. Cormack}, ....   When cited after each other, and typeset with the IEEEtran bibliography style, the .bbl file reads:  \bibitem{Seshadrinathan2010A-Subjective-St} K.~Seshadrinathan, R.~Soundararajan, A.~C. Bovik, and L.~K. Cormack, ``A   subjective study to evaluate video quality assessment algorithms,''   \emph{SPIE Proceedings Human Vision and Electronic Imaging}, 2010.  \bibitem{Seshadrinathan2009Study-of-Subjec} ------, ``Study of subjective and objective quality assessment of video,''   \emph{IEEE Transactions on Image Processing}, 2009.   As you can see, the author names have been replaced with ------. It looks like this in the final PDF:    Is this normal behavior?