If time is the 4th dimension, I can think of 2 ways to describe the flow of time. Either matter is moving forwards along the t axis, or space is. In the former, each particle has its own time and &quot;exists&quot; only once, while in the latter, everything Big Bang to Heat Death already exists, and space is just going through it, like frames in a movie. Are  any of the 2 correct? Do we even know? And, if neither are, how does it actually work?
This is an attempt to gather together the various questions about time that have been asked on this site and provide a single set of hopefully authoritative answers. Specifically we attempt to address issues such as:   What do physicists mean by time? How does time flow? Why is there an arrow of time?
Prove that $∂A$ is closed given $∂A = \text{Cl}(A) − \text{Int}(A)$ Similar questions have been asked, but none with the given information. My textbook doesn't give me the fact that  $∂A = \text{Cl}(A) − \text{Int}(A)$. If that were the case, I could just state that definition and note that it's the intersection of closed sets.   I have very little knowledge of set theory and proofs, so I'm not sure how else to prove this. As always, I appreciate any help.  The question comes from “Introduction to Topology: Pure and Applied” by Colin Adams and Robert Franzosa.   "Let $A$ be a subset of a topological space $X$. Prove that $∂A$ is closed given  $∂A = \text{Cl}(A) − \text{Int}(A)$" Where is the Layers panel in Blender 2.8? I am starting working on Blender 2.8, but I can't find the layer panel. Where is it? Can I install Ubuntu inside Windows? Ubuntu 12.04 won't come with the Windows Installer (Wubi) pre-added. But is there any way to install 12.04 using Wubi? How do you style an input type="file" button? Can you say "are not we all?" instead of "aren't we all?" Because "aren't" translates to "are not" I pose the question, can you use both interchangeably (in the context of "aren't we all?")? "Are not" sounds very grammatically incorrect in this situation. Can you say "you need not go there" as opposed to "you needn't go there"?   Basically, I thought you can say "you need not" instead of "you needn't" but that you can't say "are not" instead of "aren't" (in the context I've provided) because I've never heard the second.  My friend believes that if you can use one example interchangeably, you can use the other as well.  So,   Can you use example '1' interchangeably? Can you use example '2' interchangeably? How do I theme a webform? LCM. What am I missing? Problem statement:  Let $S$ be the set of all positive integers $n$ such that $n^2$ is a multiple of both $24$ and $108$. Which of the following integers are divisors of every integer $n$ in $S$? The choices are: $12, 24, 36, 72$. The solution is $12$ and $36$. My solution was $S = \{36, 36* 6^3, 36*6^5, ...\}$  I can't figure out this problem. I started by finding the $\text{lcm}(24,108)$ and solved for $n$. I got some of the numbers in the set $S$, but not all of them, and I don't see what I'm missing. Update: added the choices and what my results Look and you shall find (a correlation) I have several hundred measurements. Now, I am considering utilizing some kind of software to correlate every measure with every measure. This means that there are thousands of correlations. Among these there should (statistically) be a high correlation, even if the data is completely random (each measure has only about 100 datapoints).  When I find a correlation, how do I include the information about how hard I looked for a correlation, into it?   I am not at a high level in statistics, so please bear with me. Geometric proof for irrationality of $\pi$ Is there a geometric proof for irrationality of $\pi$? That would be neat. mysql select distinct rows into a comma delimited list column The Wifi is working fine but no internet connection in Windows 8 RSA leak bits to factor N How do I find out which process is eating up my bandwidth? I think I'm being the victim of a bug here. Sometimes while I'm working (I still don't know why), my network traffic goes up to 200 KB/s and stays that way, even tough I'm not doing anything internet-related.   This sometimes happens to me with the CPU usage. When it does, I just run a top command to find out which process is responsible and then kill it. Problem is: I have no way of knowing which process is responsible for my high network usage. Both the resource monitor and the top command only tell me my total network usage, neither of them tells me process specific network info.  I've found questions here about monitoring total bandwidth usage, but, as I mentioned, that's not what I need. Is there another command I can use to find out which process is getting out of hand?  The command iftop gives results that disagree entirely with the information reported by System Monitor. While the latter claims there's high network traffic, the former claims there's barely 1 KB/s.  I've already tried killing all the obvious ones (Firefox, update-manager, Pidgin, etc) with no luck. So far, restarting the machine is the only way I found of getting rid of the issue. I'm using soul to letterspace headings in a Hungarian document and compiling it with lualatex.  For some reason soul drops some of the accented letters at the end of a word, but not all. In the example below, only ő and ű are affected:  \documentclass{minimal} \usepackage{fontspec} \setmainfont[Ligatures = TeX]{TeX Gyre Pagella} \usepackage{soul} \begin{document} \noindent \so{áll llá ÁLL LLÁ}\\ \so{íll llí ÍLL LLÍ}\\ \so{űll llű ŰLL LLŰ}\\ \so{őll llő ŐLL LLŐ}\\ \so{üll llü ÜLL LLÜ}\\ \so{öll llö ÖLL LLÖ}\\ \so{úll llú ÚLL LLÚ}\\ \so{óll lló ÓLL LLÓ}\\ \so{éll llé ÉLL LLÉ} \end{document} Show that:  If $U$ is a principal ultrafilter, then the canonical inmersion $j$ is an isomorphism between $\mathcal A$ and $Ult_{U}\mathcal A$ When should I publish in interdisciplinary versus subject-specific scientific journals? Most science research is published in subject-specific journals, of which there are many. However, there are a relatively small number of interdisciplinary journals, some of which are the most highly cited of all journals (Nature, Science); others have impact factors comparable to subject-specific journals (e.g., PLOS ONE, J. R. Soc. Interface, etc.).  If we discount arguments around open science and similar business model arguments (which could favor journals like PLOS ONE), and assume that the purpose of publishing is simply to get your research read (and cited) by the largest "relevant audience", when should one publish in an interdisciplinary journal versus a subject-specific journal (assuming they have comparable impact factors)? (Are there any papers out there that compare the fate of publications [or their authors!] in the two different types of journal?)  Thanks in advance. When you edit a message and repost the contents of  its first revision, the repost is eaten up by chat.  Steps to reproduce:   Join any chat room Post something. Let's say you post abcd Edit that to something else. For example, --- Try to post abcd again. The green (pending) message will show up, but when the response is received, it disappears (the server spits out {"id":null,"time":null} as the response)   This happens more often than you think. Many a time I post something, only to realize that I want to use the magic separator (---) to separate it from a previous block. So I up-arrow, Ctrl-X, and type ---, and then I repost the cut text. Which gets eaten up by the server -- making a small change to the repost fixes things. Are repository lists secure? Is there an HTTPS version? Are repository updates secure?  As a bear of little brain from the developer side, I cannot understand why the repository list is http://security.ubuntu.com and the other http(unsecured) sites listed in /etc/apt/sources.list.  Without a certificate chain match this appears as "ask any responder for a list of packages to update" instead of "ask the ubuntu.com site..."  Can any network choose to spoof the update sites, and is this a common practice to provide a locally cached and vetted copy? When am I entitled to participate in a stock split? — record date, split date, ex date Stock split announcements usually mention a few dates — record date (&quot;date of record&quot;), split date (&quot;distribution date&quot;, &quot;pay date&quot;, or &quot;payment date&quot;), ex date (&quot;ex-split date&quot;). Examples:  :  Each Apple shareholder of record at the close of business on August 24, 2020 will receive three additional shares for every share held on the record date, and trading will begin on a split-adjusted basis on August 31, 2020.   :  Each stockholder of record on August 21, 2020 will receive a dividend of four additional shares of common stock for each then-held share, to be distributed after close of trading on August 28, 2020. Trading will begin on a stock split-adjusted basis on August 31, 2020.    In the case of the Tesla split, the record date is August 21, the split date is August 28 (Friday), and the ex date is August 31 (Monday). Apple's  provides definitions of these dates:  What is the effective date of the split? There are several key dates. The Record Date – August 24, 2020 - determines which shareholders are entitled to receive additional shares due to the split. The Split Date – August 28, 2020 - shareholders are due split shares after the close of business on this date. The Ex Date – August 31, 2020 - the date determined by Nasdaq when Apple common shares will trade at the new split-adjusted price.  However, further down in the FAQ, there is an explanation that seems to contradict the definition of the record date given above:  What happens if I buy or sell shares on or after the Record Date and before the Ex Date? [...] If you buy shares on or after the Record Date but before the Ex Date, you will purchase the shares at the pre-split price and will receive (or your brokerage account will be credited with) the shares purchased. Following the split, you will receive (or your brokerage account will be credited with) the additional shares resulting from the stock split.  Judging by the Apple FAQ, the way the record date works for stock splits is completely different from the way the record date works for cash dividends. With cash dividends, people who become shareholders of record after the record date are not entitled to the dividends. With stock splits, however, people who purchase shares after the record date (but before the ex date) are entitled to the stock split. Furthermore, for stock splits, the record date comes before the ex date, whereas for cash dividends, the record date comes after the ex date. It appears that I am not the only one confused about this. A quick internet search reveals years and years of confusion regarding the record date for stock splits. An extremely common point of confusion is the apparently incorrect assumption that the record date for stock splits works in the same way as the record date for cash dividends. A few questions:  Am I right that the record date for stock splits works differently from the record date of cash dividends? The only relevant dates for a stock split seem to be the split date (on which I will receive new shares after the market closes), and the ex date (on which I can sell the new shares on stock exchanges at a post-split price). It seems that the record date is completely irrelevant. What is the record date for? From my understanding, the price of the stock will be reduced at the instant the new shares are distributed (on the split date). Is this correct? I am currently studying  answer on the Moore-Penrose pseudoinverse and the Euclidean norm by the user &quot;Etienne dM&quot;. The first point of their answer proceeds as follows:  Let $x$ be $A^+y$.  Let me begin by the second point. For all $z$, we have: \begin{align} \lVert Az-b \rVert_2^2 &amp;= \lVert Ax-b \rVert_2^2 + \lVert A(z-x) \rVert_2^2 + 2 (z-x)^TA^T(Ax-b)\\ &amp; \geq  \lVert Ax-b \rVert_2^2 + 2 (z-x)^TA^T(Ax-b) \end{align} Moreover, because $(AA^+)^T = AA^+$, $$ A^T(Ax-b) = ((AA^+)A)^Tb - A^Tb = 0$$ Thus, we prove that for all $z$, $\rVert Az-b \lVert_2^2 \geq\rVert Ax-b \lVert_2^2$, that is to say $A^+b$ is as close as possible to $y$ in term of the Euclidian norm $\lVert Ax-b\rVert_2$.   I realise that $x = A^+ y$, but I don't understand how any of this implies that &quot;$A^+ b$ is as close as possible to $y$ in terms of the Euclidean norm $\lVert Ax-b\rVert_2$&quot;. I would greatly appreciate it if people would please take the time to explain this to me.