When matter is converted to energy by means of $E=mc^2$, it produces quite a lot of "energy".  What I am having trouble understanding is exactly how the matter is transformed to energy at the atomic level. Do the atoms gain something or lose something in their internal structure? Do they just vibrate at different frequencies when the conversion occurs? Edit: This is a duplicate of the question, pardon me. Sorry.
I know that there's a difference between relativistic and rest mass. Relativistic mass is "acquired" when an object is moving at speeds comparable to the speed of light. Rest mass is the inherent mass that something has regardless of the speed its moving at.  When fusion happens, a certain percentage of the rest mass of two hydrogen atoms is converted into energy. I understand that $E = m_{rest}c^2$.   How does the rest mass of an object turn into energy? Does this involve particle physics? Doesn't this suggest that rest mass can be viewed as some sort of potential energy? It seems to me that it's a sort of locked energy that has to undergo a certain process to expel its energy to surrounding particles.  Also, Why does only a small percentage of rest mass turn into energy? Why not all of it. What dictates how much rest mass gets converted into energy?
I've shown the existence of unique square roots of all positive rational numbers, so now I want to prove that the square root is monotone:   $0&lt;a&lt;b$ if and only if $\sqrt{a} &lt; \sqrt{b}$ AMD-V is not enabled in VirtualBox on AMD APU I am running on a dual core AMD E450 CPU. When I tried to run a 64-bit OS that requires hardware virtualization using VirtualBox, it showed me an error message that "AMD-V is not enabled".    My AMD processor is supposed provide AMD-V support, but I can find no option for enabling AMD-V in BIOS.    How can I solve this problem? How can I enable AMD-V for my CPU?  Thanks in advance  lscpu :-  Architecture:          x86_64 CPU op-mode(s):        32-bit, 64-bit Byte Order:            Little Endian CPU(s):                2 On-line CPU(s) list:   0,1 Thread(s) per core:    1 Core(s) per socket:    2 Socket(s):             1 NUMA node(s):          1 Vendor ID:             AuthenticAMD CPU family:            20 Model:                 2 Stepping:              0 CPU MHz:               1650.000 BogoMIPS:              3291.72 Virtualization:        AMD-V L1d cache:             32K L1i cache:             32K L2 cache:              512K NUMA node0 CPU(s):     0,1   EDITED:-  Error message from VirtualBox:-  Failed to open a session for the virtual machine XXX.  AMD-V is disabled in the BIOS. (VERR_SVM_DISABLED).  Result Code: NS_ERROR_FAILURE (0x80004005) Component: Console Interface: IConsole {1968b7d3-e3bf-4ceb-99e0-cb7c913317bb} n-dimensional vector space isomorphic to the n-th product of a field as abelian groups Let $V$ be a finite dimensional vector space (of dimension n) over a field $F$. I need to show that $V$ is isomorphic to $F^n$ as abelian groups. However, I don't really understand what does "isomorphic as abelian groups" mean, my (poor) attempt to solve this was:  Let $f:V\to F^n$, $f((v_1,v_2,\ldots,v_n)):=v_1+\ldots+v_n$, clearly $f$ is a group homomorphism, but it is not biyective since, for example, in the case where $dim(V)=2$ and $F=\mathbb{R}$ we have $f((2,-2))=0=f((3,-3))$, I've also tried it by defining $f$ to be the product of a vector's entries but obviously it didn't work so I'm stuck!   I think this should be an easy problem but since I don't quite understand it, it's giving me problems. Thank you for your help. Is there any quality difference between PNG vs BMP image formats? I have just learned that PNG is a lossless image format while BMP can be both uncompressed or lossless format. Should there be a quality difference between these two? If not, does it mean that other lossless formats like TIFF, JPEG2000 and PNG offer the same quality as BMP? Recently I came across a debate about whether it's fair to call RAW 'unprocessed' or 'unaltered' sensor data of a DSLR. As far as I know, the analog sensor data is processed into a digital RAW file using the ISO setting, a process during which the Bayer-sensor data is demosaiced as well. However, those are just the necessary steps for saving a digital file while maintaining maximum quality.  Are those assumptions correct? If so, are there any additional steps that need to be performed between the sensor receiving light and the saving of the RAW file? And can those be reasonably be described as an 'alteration' of the sensor data, or is it fair to say that a RAW file holds the unaltered sensor data? Proving $\int_{0}^{\infty} \mathrm{e}^{-x^2} dx = \frac{\sqrt \pi}{2}$ Is $x^3+y^3+z^3-1$ irreducible over a field $k$ of characteristic $\neq 3$? Is $x^3+y^3+z^3-1$ irreducible over a field $k$ of characteristic $\neq 3$?   If I can show that $y^3+z^3-1$ is irreducible, I believe I can use Eisenstein's criterion. But I don't see how to show even this.  The other approach on my mind is to show that $k[x,y,z]/(x^3+y^3+z^3-1)$ is an integral domain, but this doesn't seem particularly promising either. What is the location of lock screen images on Windows 10? Where, or what format, are the images used by Windows 10 when logged out or in lock-screen?  I searched the entire drive for *.jpg, *.bmp, *.png, and *.tif but the one I was looking for did not appear.  At first glance, , although not exactly a duplicate, appeared to have the path I wanted.  But when I went there, the pre-login image was there, but the lock-screen image was not.  (And contrary to the claim in that answer, the files there did have extensions.)  UPDATE: I just did a lockscreen, and got an image that IS in that directory, but earlier today, it was consistently an image of an interesting footbridge that is not in that directory. How come gas molecules don't settle down? If the earth's gravity exerts a net downward gravitational force on all air molecules, how come the molecules don't eventually lose their momentum and all settle down? How is the atmosphere is still miles thick after billions of years? Understanding the length extension attack I have been trying to understand exactly how a length extension attack works on SHA-1. I'll detail below what I've understood so far, so that I can convey my understanding of the same and hopefully get advice on where I'm going wrong.  Lets assume that for a message $m$, a secret $s$ is appended to it and the SHA-1 hash is calculated. If $\textrm{len}(m) + \textrm{len}(s)$ is not a multiple of the block size, a certain amount of padding is added to it to make it a multiple, and the hash is generated using what is essentially $s || m || \textrm{padding}$.  Now SHA-1 internally uses 5 registers — it takes in an block, generates new register values, turns the crank, takes in another block, generates new register values, turns the crank ... and so on till the end. When all blocks are exhausted, the register values are concatenated and spit out.  Assume that an attacker knows a certain value of $m$, for which he also knows the hashed key. In such a case, he knows the state of the 5 registers when $(s || m || \textrm{padding}$ was processed. Lets also assume that the attacker knows the length of $s$, so that he can predict how much padding was added.  Here is where I get confused. From what I've read so far, the attack involves taking the known hash, adding a certain amount of padding and then the value that we want. However, isn't the padding already used up in creating the hash? Why are we using padding in our attack? According to specification, to create data that will be signed with server private key for CertificateVerify message, we need to concatenate 64 space characters, the string &quot;TLS 1.3, server CertificateVerify&quot;, null and the hash of all handshake messages to this point. During the handshake process, we choosing the cipher suite and signature scheme... So, if the cipher suite is, for example, TLS_AES_256_GCM_SHA384 and the signature scheme is rsa_pss_rsae_sha256 (different hash algorithms), hash of which algorithm should be used as handshake hash in data to sign, SHA-384 or SHA-256? Is there a such thing as an exchange program in Harry Potter? In the Harry Potter universe, is there any mention of an exchange student at Hogwarts or one of the Hogwarts students going on exchange at another school?  This happens all the time in the "muggle" school systems I'm sure it would happen in the wizarding schools as well. Spurious whitespace with algpseudocode and noend I'm currently typesetting some pseudocode and was up to now pretty happy with the algorithmicx package and the algpseudocode command set. However, I would now like to use the noend option to omit end lines. That works, but unfortunately introduces additional whitespace (see lines 6 and 7 in the example below). I fiddled around with negative \vspaces but could not come up with a consistent value that works in all cases.    \documentclass{article} \usepackage[noend]{algpseudocode}  \begin{document}  \begin{algorithmic}[1]   \Function{lsolve}{$L$, $b$}     \State $x \gets b$     \For{$j \gets 0$ to $n - 1$}       \State $x_j \gets x_j / l_{jj}$       \ForAll{$i &gt; j$ where $l_{ij} \neq 0$}         \State $x_i \gets x_i - l_{ij} x_j$       \EndFor     \EndFor     \State \Return $x$   \EndFunction \end{algorithmic}  \end{document}   Is there any way to patch algorithmicx or do I have to finetune the pseudocode on a case by case basis? Pagination broken when merging search results with additional WP_Query On the search results page (search.php) I would like to perform an additional WP_Query to keep a specific order of search result types e.g. search results from authors should always be shown before the other results -  The first few pages should hold the author results, the other results should be shown on all the following pages only (not mixed on each page).  global $wp_query; //holds the results of the initial search query  $authorposts = $wp_query-&gt;posts;     $query = new WP_Query( 'year=2012&amp;monthnum=12&amp;day=12' ); $otherposts = $query-&gt;posts;  $wp_query-&gt;posts = array_merge( $authorposts, $otherposts  ); $wp_query-&gt;post_count = count( $wp_query-&gt;posts );  if ( have_posts() ) : while ( have_posts() ) : the_post(); ?&gt;   //show results loop...   The problems appear with the behaviour of pagination - while the $authorposts seem to paginate correctly i.e. correct page numbers for author posts and pagination works as only those where existent. But the $otherposts on the other hand are shown at the bottom of each page, right under the author posts. And they are the same on every page! Also the page count reflects the count of the first query only.  I have also tried to merge with the plus operator:  $wp_query-&gt;posts = $authorposts + $otherposts;   but in this scenario the $otherposts don't show up at all and the page count stays the same (page count for author posts).  All the examples with similar goals using two separate queries I've seen looked quite the same -  So I'm wondering if the my way of extending the original search query results with that additional WP_Query is sufficient or if I'm missing some $wp_query fields to be populated (correctly)? What's the correct way doing this?  Update: It should be noted that I'm using the WP-PageNavi plugin to paginate but this should not be relevant in the first place as the first page is already rendered wrong. And it's also initialized with the query object: wp_pagenavi( array( 'query' =&gt; $wp_query )); Recent fluid-mechanical experiments by the groups of Couder in Paris and Bush at MIT, mimic a surprisingly wide range of quantum effects. The essential ingredient of these fluid-mechanical systems is a background or pilot-wave that guides the droplets.  Now, surprisingly, a simple analysis of a Bell-type experiment shows that, in the presence of a background field, one of the premises of the Bell inequality, namely measurement independence (MI), is violated. See the paper  (available ). Therefore such classical droplet experiments could violate a Bell inequality. More importantly, if this analysis is correct, background-based hidden-variable theories are admissible, even if they are local (in the sense of ‘involving only (sub)luminal interactions’) and even if they are compatible with free will.   My question: to me the analysis seems fully sound, but maybe there is still an unphysical hypothesis that slipped in ? Given matrices $A$ and $B$, where that $AB = A + B$, prove $AB = BA$. I keep coming up with AB = AB. It seems like basic algebra, but for the life of me, I'm getting nowhere :/. Someone help please? Why are C character literals ints instead of chars? In C++, sizeof('a') == sizeof(char) == 1. This makes intuitive sense, since 'a' is a character literal, and sizeof(char) == 1 as defined by the standard.  In C however, sizeof('a') == sizeof(int). That is, it appears that C character literals are actually integers. Does anyone know why? I can find plenty of mentions of this C quirk but no explanation for why it exists. What compiler warning level do you recommend for different C/C++ compilers?  gcc and g++ will let you get away with a lot on the default level.  I find the best warning level for me is '-Wall'.  And I always try to remove fix the code for the warnings it generates.  (Even the silly ones about using parenthesis for logical precedence rules or to say I really mean 'if (x = y)')  What are your favorite levels for the different compilers, such as Sun CC, aCC (HPUX ?), Visual Studio, intel?  Edit:  I just wanted to point out that I don't use "-Werror" (but I do understand it's utility) on gcc/g++ because, I use:   #warning "this is a note to myself"   in a few places in my code.  Do all the compilers understand the #warning macro? Standard concise way to copy a file in Java? Why do I see a double variable initialized to some value like 21.4 as 21.399999618530273?