Do spacelike singularities really exist in quantum gravity? Do spacelike singularities really exist in quantum gravity? If the memory of anything which falls into a black hole can't get out, is there any sense in which the interior of the black hole is real? Similarly, if we don't have any direct records of the spacelike singularity in the past, is there any sense in which it is real?
Do singularities have a "real" as opposed to mathematical or idealized existence? I was thinking of, for example a Schwarzchild metric at r=0, i.e. the gravitational singularity, a point of infinite density. I realise that there are different types of singularities--timelike, spacelike, co-ordinate singularities etc. In a short discussion with Lubos, I was a bit surprised when I assumed they are idealized and I believe he feels they exist. I am not a string theorist, so am not familiar with how singularities are dealt with in it.  In GR, I know the Penrose-Hawking singularity theorems, but I also know that Hawking has introduced his no-boundary, imaginary time model for the Big Bang, eliminating the need for that singularity. Are cosmic strings and other topological defects singularities or approximations of them (if they exist).  In what sense does a singularity exist in our universe? --as a real entity, as a mathematical or asymptotic idealization, as a pathology in equations to be renormalized or otherwise ignored, as not real as in LQG, or as real in Max Tegmark's over-the-top "all mathematical structures are real"?
"Above"/"below" before/after a noun I have seen sentences similar to the following:  (1) See the reference above. (2) See the reference below.  And,  (3) See the above reference.  But not,  (4) See the below reference.  Are all these forms acceptable? Which is/are preferred in formal writing? Citrix receiver error 1000119 I am trying to run Citrix receiver on the Xenial version of Ubuntu, but I am getting this error below  ERROR: Cannot connect to &quot;0.0.0.2 - Remote Desktop Connection&quot; Unknown error 1000119. Verify your connection settings and try again.  What should I do? Bring Debian to second Position in the GRUB Menu at startup Why does my document 'error' when using '%' I am designing gamecards in LaTeX  Everything is going pretty ok so far but when i try to insert the %-sign for a question about percentages my file keeps giving errors.  I tried putting the percentage between dollarsigns because i know just % will make it a 'remark' and that the dollarsign makes it 'mathematical data'.   Whats the reason for the error?   P.s. the error says: file ended while scanning use of \kaart. ...    Codesample:   {\kaart{Bekijk statistiek kaart F:   Schat het percentage mensen dat als lievelingseten spaghetti opgaven, je mag er slechts $ 2% $ langs zitten. } { $ 11% $}} Screen brightness control not working on Lenovo T530 What is the time complexity of conjugate gradient method? I have been trying to figure out the time complexity of the conjugate gradient method.  I have to solve a system of linear equations given by  $$Ax=b$$  where $A$ is a sparse, symmetric, positive definite matrix.  What would be the time complexity of the conjugate gradient method? Prevent malicious bots from posting spam I am visiting a friend in Switzerland for ten days. I have already got visitor's visa for the same on my Indian passport. Particulars mentioned on my visa are:  1. 'Etats Schengen' , 2. Single entry 3. Dates of visit. &amp; 4. Type of visa : C.  I understand that having already got ‘Etats Schengen' visa, I can travel to all Schengen countries.  I am also planning to visit Paris for a day during my stay in Switzerland. However, I never mentioned this in my visa application. I am planning to take a train from Geneva to Paris and book a hotel in Paris for one night. Can I possibly plan my Paris trip this way as I have Schengen visa, or I am not allowed to travel out of Switzerland as I never mentioned this in my visa application?   Also what does Single entry on my Schengen visa means: does it mean that I can enter and leave Switzerland only once or it means that I can enter and leave the Schengen region only once. In windows we have an option to view recently opened media files as my recent documents, I was  wondering does there exists something like that in Ubuntu?  via terminal or via GUI? Why won't OS X trust GitHub's SSL certificate? Backup and restore settings in Texstudio Does someone know, how to save all the settings of Texstudio and restore them in a simple way?  Thank you in advance! System Settings icons missing in 14.04 White cloudy areas on bottom of new stainless steel skillet I just bought my first few pot and pans "Calaphon" stainless steel trip-ply  I used my new 10" skillet tonight, and cooked pork chops. I heated the pan first, added olive oil, let it heat, then added chops. After I was done, cleaned it immediately under water, and noticed white cloud looking areas on the bottom of the skillet. They won’t wash or rub off  Do I need to return these? I did buy them for the look as well. When I create a UV sphere and add the subdivision surface modifier, the following wrinkle appears.    How can I avoid it?  I've read the article  but there seems to be no clear solution.  UPDATE 1  I've used the word 'apply' in the first sentence of the initial version of this question. It's misleading since I want to preserve the modifier without 'applying' it. Rather, what I meant is 'adding' the modifier.  UPDATE 2  I followed josh sanfelici's solution, but still there is an artifact at the pole.  Here is the screen: (A rendered view with a sun lamp)    Maybe it's the limitation of the general-purpose subdivision modifier.  BTW, when the cast modifier is added after the subdivision modifier, the sphere becomes perfect (even for my original wrinkled sphere).  But since I want to edit the resulting sphere (for example, scaling X, Y, etc) while modifier is added but not applied, it's not usable. In Harry Potter, what kind of home life do those born to a wizard and a Muggle have? Consider the curve $\frac{1}{x}$  where $x \geq 1$. Rotate this curve around the x-axis.   One Dimension - Clearly this structure is infinitely long.  Two Dimensions - Surface Area = $2\pi\int_∞^1\frac{1}{x}dx = 2\pi(\ln ∞ - \ln 1) = ∞$   Three Dimensions -  Volume = $\pi\int_∞^1{x}^{-2}dx = \pi(-\frac{1}{∞} + \frac{1}{1}) = \pi$   So this structure has infinite length and infinite surface area. However it has finite volume, which just does not make sense.  Even more interesting, the "walls" of this structure are infinitely thin. Since the volume is finite, we could fill this structure with a finite amount of paint. To fill the structure the paint would need to cover the complete surface area of the inside of this structure. Since the "walls" are infinitely thin, why would a finite amount of paint not be able to cover the outside of the "walls" too?  Please help me make sense of this whole thing. How do I know what symbols/characters are available in a font package? I usually use xelatex, but I'm exploring pdflatex now. When loading specific fonts in xelatex, I can always know what characters the font has available, since I can inspect the font file in any font viewing program, or I can find out the hard way by just entering the unicode and see if the character is displayed in the output. But how does this work with a font package?  Say, for instance, that I would like to use phonetic symbols in the  font. How do I know whether the font has phonetic symbols (I assume it does), and how do I know what command I need to type in order to make them appear? My only resource is , but this only tells me what characters/commands there are available in LaTeX2e by default (depending on the font encoding), and what symbols I can find in specific packages. Phonetic symbols, for instance, are said to be in the tipa package. Does that mean I am locked to tipa if I want to use phonetic symbols with pdflatex?  Here's a pointless MWE:  \documentclass{article} \usepackage{gentium} \begin{document} Please give me some phonetic symbols here? \end{document} Python __str__ and lists Example of a series! Give an example of a convergent series $$\sum_{n=1}^{\infty}a_n$$ such that the series $$\sum_{n=1}^{\infty}a_{3n}$$ is divergent..  I cannot find find any kind of series... I am also be very thankful if you find a divergent series (changing the term convergent)... Is there anyway to sync Outlook's calendar with Google's calendar? I've tried  before but it doesn't work properly. I'm using Windows 7 and Outlook 2010.  The thing is, it's really easy to check my Google Calendar on my phone since I have an Android terminal, but on my computer I prefer just opening Outlook instead of opening a web browser and logging into Gmail.