Is it physically possible for gravity to accelerate an object with mass to the speed of light?
Will an object always fall at an infinite speed in a black hole?
How do you prove that $\sum\limits_{n=0}^{\infty}{2n \choose n}x^n=(1-4x)^{-1/2}$? I tried to identify the sum as a binomial series, but the $4$ and the $-1/2$ puzzle me. (This series arises in studying the first passage time of a simple random walk.) I just read this very insightful post , where the author stated that the variance of $\hat\beta$ is:  $$\text{var}(\hat\beta) = \sigma^2(\textbf{X}^\prime \textbf{X})^{-1}.$$  I couldn't figure out why it is like this. Can anyone elaborate a bit? How to create and design hoops net and attach ot to a hoop? Distribution of the maximum of two correlated normal variables Are two norms equivalent if they induce the same topology on a vector space? Blender render crash when using subsurf modifiers. Essentially, my problem is that i am trying to render a scene, however, blender crashes on me early during the render. I only have about 14 total objects in the scene. (It's not that complex) I tried to render a different blender file which was bigger than the one that i am having problems with, and the scene renders just fine. There is no difference in the render presets for both of these files. However, i still cannot render the other scene. It gets to preparing scene data, but then it crashes.  Thank you. How to perfectly align UV coordinates automatically? I have this setup:   The mesh is only the visible top side, with a mirror modifier for the bottom part. Now I want to unwrap it like in the picture. I want the inner circle (red) to go to the top of the UV map...  The problem is that no matter what automatic unwrapping I use, it never works as intended.  The closest I've gotten was by using "follow quads", but that skewed the UV map a lot:   I used the result from the second image and manually adjusted the vertices with a lot of scaling and "align auto", but its very far from perfect as you can see in the first image.  Here's what I want exactly:   The uv vertices should be distributed evenly (horizontally) The middle line (yellow) should be on a "natural" y height (so there's no stretching)   I hope it's clear what I want to achieve. I'm sure there must be some automatic way to perfectly align and distribute the UVs in this way.  By the way: If you look closely you can see that I already placed a uv seam (aligned with the green y axis in the screenshot) but that didn't really help. I've always wondered why the number of protons in the Universe exactly matches the number of electrons. They are such different particles with totally different cross sections. So, first of all, is this true? And, secondly, how could it be that they have been produced in the same amount after the Big-Bang? Is the .minecraft folder hidden on a Mac? I have a friend who is playing with me on a server we've created. Unfortunately, he is on a  Mac whereas I am on a Windows machine. I told him that I should be able to compile and send a .minecraft folder to him so that we can both play. Unfortunately, he can't find his .minecraft folder where it is supposed to be. Is it hidden and how can I unhide it? $V$ be a vector space , $T:V \to V$ be a linear operator , then is $\ker (T) \cap R(T) \cong R(T)/R(T^2) $? Let $V$ be a vector space , $T:V \to V$ be a linear operator , then is it true that   $\ker (T) \cap R(T) \cong R(T)/R(T^2) $ ( where $R(T)$ denotes the range of $T$ ) ?    I know that the statement is true when $V$ is finite dimensional as I can show that if $V$ is finite   dimensional , then $\dim (\ker (T) \cap R(T))=rank (T) - rank (T^2)=\dim R(T)/R(T^2)$ ,   but I don't know what happens in general . Please help . Thanks in advance How to open a PDF form file that says Please wait...? I'm looking for a good, consistent and easy way to do a report of sudoers permissions for every user and group on the system. For users, I want the report to include permissions granted via group membership. How to create 2D shadows I'd like to create lights and shadows in a 2D environment, like in the image below:    I did several searches but got no results. Do you have any hint? Do Indian passport holders with UK Tier 2 need to get a tourist visa for Mexico? I'm an Indian passport holder currently resident in the UK under a Tier 2 (General) visa and planning on going to Mexico on a holiday. So I was looking into the rules on the Mexican embassy in the UK and :     According to the new regulations who entered into effect on 18th, May   2016 (article 26 of the Guidelines and Requirements for Migratory   Services) regarding foreign nationals visiting Mexico, holders of a   valid UK visa, regardless their nationality, are not required to apply   for a Mexican visa as a tourist, business (Non-Lucrative Activities)   or transit visitors for a stay of up to 6 months. This regulation does   not apply to British Travel Document holders.   However, :     PERMANENT RESIDENT IN THE UK (ILR / ILE / PERMANENT RESIDENTS)        According to Mexican regulations which came into force in June 2009,   Permanent residents in the United Kingdom wishing to travel to Mexico   do not require a visa to enter the country as tourists or business   visitors for up to 180 days and as visitors in transit for up to 30   days, regardless their nationality.   So which one is correct? Are Tier 2 holders with an Indian passport allowed visa-free / visa on-arrival entry to Mexico? What's the difference between executing a script like this:  ./test.sh  and executing a script like this:  . test.sh?  I tried a simple, two-line script to see if I could find if there was a difference:  #!/bin/bash ls   But both . test.sh and ./test.sh returned the same information. This is Exercise 26 from Chapter V Algebraic Extensions in Lang's Algebra.  $F$ is finite normal extension over $k$ and $f(x)$ is irreducible in $k[x]$. If $f(x)=g(x)h(x)k(x) \in F[x]$ where $g(x),h(x)$ are monic irreducible factors in $F[x]$, I need to show that there exists an automorphism of $F$ which takes $g(x)$ to $h(x)$.  Any hints would be appreciated.  Sending one root to another is okay but how to do that for a collection of roots? Finding package name of installed software Since I have not been able to open Dash Home in 12.04.4 see , I am hoping for a work-around. Is there another convenient way to open apps/programs without having to search through folders? Is this question clear enough? For example, I just installed Kile (publishing program) from the Software Center, but I can't find a link/tile/icon/shortcut to open it. Thanks, to whomever can help. Best Place to put Feature Requests? The only place I am aware of is the . Wedge product and change of variables Where is the right place to register/enqueue scripts & styles I am using WordPress 3.1.4 by now. I am confused with where (which hook) do I use:   to register and/or enqueue  scripts and styles  on front- and back-ends?   Questions:   Which are right hooks to use? All front end register/enqueue scripts/styles in init? Why is there no admin_print_styles-{xxx}?