Is gravitational attraction increased by relativistic mass as well as rest mass?   If so what would happen if an observer travelled at relativistic speeds? Would it appear to him that the whole universe had acquired a greater mass as it was moving at relativistic speed towards or away from him and hence start accelerating towards him under greater gravitational attraction?   Similarly would it mean that the universe was suddenly hugely more energetic that it was before with much higher balancing gravitational potential?
I'm a big fan of the podcast Astronomy Cast and a while back I was listening to a Q&amp;A episode they did. A listener sent in a question that I found fascinating and have been wondering about ever since.   From the show :      Arunus Gidgowdusk from Lithuania asks: "If you took a one kilogram mass and accelerated it close to the speed of light would it form into a black hole? Would it stay a black hole if you then decreased the speed?"   Dr. Gay, an astrophysicist and one of the hosts, explained that she'd asked a number of her colleagues and that none of them could provide a satisfactory answer. I asked her more recently on Facebook if anyone had come forward with one and she said they had not.  So I thought maybe this would be a good place to ask.
When ever I save a gedit file in a directory, two copies of it get saved with identical content. Ex: If the gedit file is named as file_name and saved in Home directory then when u ls the Home directory you get file_name and file_name~ in the list. When the file command is run against them I get ASCII text, with very long lines for both of them when I view their contents using the less command they seem to largely contain identical content. The ~ file is a copy of the file when it was saved for the penultimate time. Can someone please help me in understanding as to why such a file (the file with a trailing ~ mark in its name ) is created? Prove that any continuous map on $\mathbb{R}$ which is the identity on $\mathbb{Q}$ is the identity map, hence Aut$(\mathbb{R}/\mathbb{Q})= 1.$  proof: Suppose $\sigma$ is continuous on $\mathbb{R}$ which is an identity on $\mathbb{Q}$. let $a \in R $. Then since $\sigma$ is continuous, then by definition there is $\delta_1 &gt; 0$ such that $|a - b| &lt; \delta_1$.  And assume $|\sigma(a) -\sigma(b)| &lt; \epsilon/2$ .  Let $\delta $= min{$\epsilon/2, \delta_1$}.  And now let $b \in \mathbb{Q}$ such that $|a - b| &lt; \epsilon$. Then  $|\sigma(a) - a| = |\sigma(a)  - b + b - a| \leq |\sigma(a) - \sigma(b) | + |b - a| &lt; \epsilon/2 + |b-a|$.  Can someone please help me? I am not sure . Can I say $|b - a| &lt; \epsilon/2 $? Add 'Page' above page numbers in Table of Contents Format timedelta to string I have no idea how prove that $$\{(z_0,\ldots,z_n)\in\mathbb{C}^{n+1} \quad| \quad z_0^d+z_1^2\ldots+z_n^2=0, \quad |z_0|^2+|z_1|^2\ldots+|z_n|^2=2\}$$ is a $(2n-1)$-compact manifold.  How give the charts. For $n=1$ and $d=2$ not distinguish which 1- manifdold is? My computer is loud: How do I tell which fan is the issue? On my new computer, my fan(s) is/are plain LOUD! Using SpeedFan, I find that my temps are all under 32 degrees C under my normal use (IDK how accurate that is but the CPU itself says 27 C).  The fan doesn't sound unbalanced and it isn't making sounds in rhythm (Not the usual rrrrrrrRRRRRR... rrrrrrrRRRRRR... rrrrrrrRRRRRR... rrrrrrrRRRRRR), just a constant fan. I am thinking about oiling the fans, but I can't decide if it is the PSU fan or a case fan or the CPU fan. If it's the CPU fan I will most likely just replace that with a quiet fan. How can I tell which fan is making the noise (if not multiple)?  One thing that I am thinking may be the problem is my CPU fan isn't PWM so would swapping that out help? There is no "linear voltage" or etc thing in BIOS so I think it may be running at full speed. Also, I could have too much airflow because it is also making a whistling noise that you can hear close to it that sounds like when I accedentially put part of the side cover over my floor vent to get it out of the way. Get all categories and posts in those categories I am looking for a solution that allows me to print the following:  Cat 1        Cat 2        Cat 3  Post 1       Post 1       Post 1  Post 2       Post 2       Post 2  Post 3                    Post 3                            Post 4   EDIT  I am looking for something that will only require one database query! So if you have a foreach in your code followed by a new WP_Query then that is not what I am looking for (I am planning to put this onto my website's homepage). How to derive the conjugate prior of an exponential family distribution I am trying to derive the conjugate prior of the univariate Gaussian distribution over both the mean and the precision. I know that the prior I'm looking for is the normal-gamma distribution, but the idea is to derive this result.  It seems that you can write down the conjugate prior of an exponential family distribution immediately if you represent it in the canonical exponential family form.  suggests a way to do just that. So far I have written the Gaussian probability function in this form and equated the following, but I don't see how the form of the conjugate prior is immediately obvious.  $$p(\mu, \lambda | D) \propto p(\mu,\lambda)p(D | \mu, \lambda)$$ how can I find sum of series $ \sum_{n=1}^{\infty} (n\cdot \ln \frac{2n+1}{2n-1} - 1) $? It is so weird for me because I put this to Mathematica and it tells me that sum does not converge...  Let consider sum no to infinity, but to n  $$ \sum_{k=1}^{n} (k\cdot \ln \frac{2k+1}{2k-1} - 1) =$$ $$ ln \frac{3}{1}\cdot \left(\frac{5}{3}\right)^2 \cdot...\cdot \left(\frac{2n+1}{2n-1}\right)^n - n = ln \frac{1}{1}\cdot \frac{1}{3}\cdot \frac{1}{5}\cdot ... \frac{1}{2n-1} - n $$ but $$ n = ln e^n $$ so it will be $$ln\frac{1}{e^n} \cdot \frac{1}{1}\cdot \frac{1}{3}\cdot \frac{1}{5}\cdot ... \frac{1}{2n-1}$$  So the limit of it is $-\infty$ Have I done this well or I missed sth? I would like to try a different video display driver, but I'm not sure which one I'm currently using.    What's the simple way to see what driver my system is using currently? What's the action of banging two beers together called? When beers in cans or bottles are banged together in rejoice of the company you're with, what is the action called? Our company would like to understand the effects:   If we have single table with Primary Key Identity column, int primary key IDENTITY(1,1),  and the Application is conducting multi-threading parallel processing, trying to conducts 1000s of simultaneous inserts on one table.   Would the identity primary key column fail, with a primary key violation in any way, for example: processors are trying to input the same identity number?  Or was SQL designed to be successful even in multithreading parallel insert?  Note: IDENTITY_INSERT is OFF Prove $\gcd(a,b) \gcd(a,c) \gcd(b,c) \,\text{lcm} (a,b,c)^2=$ $\text{lcm}(a,b)\,\text{lcm}(a,c) \,\text{lcm}(b,c) \gcd(a,b,c)^2$ I'm trying prove $$\gcd(a,b) \gcd(a,c) \gcd(b,c) \,\text{lcm} (a,b,c)^2=\text{lcm}(a,b)\,\text{lcm}(a,c) \,\text{lcm}(b,c) \gcd(a,b,c)^2$$  It is a part of solution of another problem I try a lot to prove it but I could not be succeed.  I tried using exponent manipulation in prime factorizations but it didn't help me. Windows vs Linux Local Time? Ever since I started using Linux in dualboot with Windows I noticed that when I rebooted the machine, the time on the other OS was wrong, e.g. it was 12:00 on Linux, then I rebooted and on Windows it was 18:00, and when I checked the BIOS, the RTC was set to 18:00. Since my timezone is -06:00 CST I assume Linux just sets the RTC’s time to UTC and re-calculates the time based on the timezone, whereas Windows just sets the RTC to the local time. Both use the same NTP server to sync the time.  The question is, which one is doing the right thing? On which one should I set the timezone setting to UTC in order to have the right time on both? Why do we need multiple approvers for suggested edits? I already have the "Edit Questions And Answers" privilege. Yet, if I approve someone else's edit, I'm told that more votes will be required before the edit is accepted.  This seems a bit peculiar to me, since I could just copy the edited version and paste it myself. Cardinal number subtraction We all know that $|\mathbb{N}| = \aleph_0$. Since $|\{-1\} \cup \mathbb{N}| = \aleph_0$ as well, I guess you could say that $\aleph_0 + 1 = \aleph_0$.  You can go on to derive that $\aleph_0 + \aleph_0 = \aleph_0$, and by induction $n \times \aleph_0 = \aleph_0$ (assuming that $n$ is finite, anyway).  Now, here's a thing: What is $\aleph_0 - \aleph_0$?  Well, $|\mathbb{P}| = \aleph_0$, and $|\mathbb{P} \backslash \mathbb{N}| = \aleph_0$, so perhaps $\aleph_0 - \aleph_0 = \aleph_0$?  No, wait. Consider the set $S = \{n \in \mathbb{N}: n &gt; 5\}$. Now we have $|S| = \aleph_0$, and yet $|\mathbb{N} \backslash S| = 5$. So maybe $\aleph_0 - \aleph_0 = 5$?  But that is absurd, since we can redefine $S$ to make the result any finite number we wish. Or we can define $S$ such that the result is countably infinite.  Does this mean that the notion of $\aleph_0 - \aleph_0$ simply has no definite answer? Or am I just being too simplistic here? Why is $\sqrt{-x}*\sqrt{-x}=-x?$ Q1 - Why is $\sqrt{-x}*\sqrt{-x}=-x?$  Q2 - I was thinking it would be:  $\sqrt{-x}*\sqrt{-x}=\sqrt{-x*-x}=\sqrt{x^2}$ but apparently not (why not?)  Q3 - What are the formal algebra rules to use? Can I calculate this without using i such as in: $\sqrt{-x}*\sqrt{-x}=i\sqrt{x}*i\sqrt{x}=-\sqrt{x^2}=-x$. Paucity of words for relationships Is it possible to install Windows 7 alongside Ubuntu and windows 10 dualboot? I installed windows 10 on my laptop in UEFI mode with HDD GPT disk format, then I installed Ubuntu 16.04LTS on another partition. So I have Ubuntu dualboot with Windows 10 now.  I also installed rEFInd (it's like GUI for you to choose between different OSs) because I don't wanna use Grub2 as the default bootloader. Now I want to have Windows 7 too, but when I boot from the USB stick, an error screen showed up saying something like  the windows boot manager is having problems due to a recent hardware change   My system is still working fine so I don't know if the problem came from my system or if the setup I want is impossible. I have learnt that the cardinality of the power set of the natural numbers is equal to the cardinality of the real numbers. What is the function that gives the one-to-one correspondence between these two sets?  I have also learnt that there exists no set whose cardinality is strictly between the natural numbers and the real numbers. Is there a proof of this or at least some intuitiveness behind it?