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How do airlines determine ticket prices? I have come across these three definitions of the limit superior (or upper limit) and the limit inferior (or lower limit) of a sequence of real numbers and I wonder how to establish the equivalence of these.    Walter Rudin: PRINCIPLES OF MATHEMATICAL ANALYSIS, 3rd edition:  Definition 3.16: Given a sequence $\{s_n\}$ of real numbers, let $E$ be the set of numbers $x$ (in the extended real number system) such that $s_{n_k} \to x$ for some subsequence $\{s_{n_k}\}$. This set contains all subsequential limits ..., plus possibly the numbers $+\infty$, $-\infty$. We now ... put  $$s^{*} = \sup E,$$ $$s_{*} = \inf E.$$ The numbers $s^{*}$, $s_{*}$ are called the upper and lower limits of $\{s_n\}$; we use the notation  $$ \lim_{n \to \infty} \sup s_n = s^{*}, \, \, \, \lim_{n \to \infty} \inf s_n = s_{*}.$$ Tom M. Apostol: MATHEMATICAL ANALYSIS, 2nd edition:  Sec. 8.3: Definition 8.2: Let $\{a_n\}$ be a sequence of real numbers. Suppose there is a real number $U$ satisfying the following conditions:    i} For every $\epsilon &gt; 0$ there exists an integer $N$ such that $n &gt; N$ implies $$a_n &lt; U + \epsilon.$$ ii) Given $\epsilon &gt; 0$ and given $m &gt; 0$, there exists an integer $n&gt; m$ such that $$a_n &gt; U - \epsilon.$$ Then $U$ is called the limit superior (or upper limit) of $\{a_n\}$, and we write $$U = \lim_{n \to \infty} \sup a_n.$$ Statement (i) implies that the set $\{a_1, a_2, a_3, \ldots \}$ is bounded above. If this set is not bounded above, we define $$\lim_{n\to\infty}\sup a_n = +\infty.$$ If the set is bounded above but not bounded below and if $\{a_n\}$ has no finite limit superior, then we say $\lim \sup_{n\to\infty} a_n = -\infty$. The limit inferior (or lower limit) of $\{a_n\}$ is defined as follows: $$\lim_{n\to\infty}\inf a_n = -\lim_{n\to\infty}\sup b_n,$$ where $b_n = -a_n$ for $n= 1, 2, 3, \ldots$.    Robert G. Bartle and Donald R. Sherbert: INTRODUCTION TO REAL ANALYSIS, 3rd edition: Exercises for Section 3.3: Problem 10: Let $(x_n)$ be a bounded sequence of real numbers, and for each $n\in \mathbb{N}$ let $s_n \colon= \sup \{x_k \colon k \geq n\}$ and let $t_n \colon= \inf \{x_k \colon k \geq n\}$. Prove that $(s_n)$ and $(t_n)$ are monotone and convergent. Also prove that if $\lim (s_n) = \lim (t_n)$, then $(x_n)$ is convergent. [One calls $\lim (s_n)$ the limit superior of $(x_n)$ and $\lim (t_n)$ the limit inferior of $(x_n)$.]   Now how can one show that the above three definitions are equivalent (i.e. these three definitions are of the same pair of numbers)? I recently obtained a Seagate Expansion 3TB external harddrive. I'm trying to copy over all of my stuff to prepare for reformatting and installing a new OS. A handful of my mp3s did not transfer when I was doing my music folder. I checked them after the fact and the ORIGINAL files became corrupted somehow. I am certain of at least a few that were not corrupted in anyway before doing this. Every now and then during my transfer I got  There was an error copying the file into /media/***/Seagate Expansion Drive. Error splicing file: Input/output error   Each of these files that I check after are corrupted at some point in the duration and will not play. Why did this happen and what can I do to fix it? Alternatives to Shapefiles as open-source, cross platform dataset types I read a book that answers one of my questions, and I would like to quote it as an answer. Is that illegal?    Thanks. I think this was more of a fantasy series, though it was more subtle than The Lord of the Rings IIRC. Set in modern times - though there was some alternate realm thing that was less modern I think.   I read it about ten years ago, I think, and there were a number of books at that time. My brother and I stopped reading it shortly after the main character summoned a small demon somehow who was a bad influence... I think.  It involved a boy who somehow was at some kind of training school/academy. Unsure where it was located, but it needed a classic portal to get to it I think. I clearly remember some scene involving a car driving at specific speeds to get past some barriers. There may have been something about how the school was (slowly) moving or the wards were shrinking... unsure. I don't think it was only magic based, but that was some element of it. The boy had some kind of strange power that made him an incredible fighter, but he wasn't sure how (at least from what I remember). The recordings of him training just showed a white flash I think. I remember some scene in an elevator going down to the training floor where he went to a level below some adult with him.  There was something else about him joining a group that had twelve members, one for each month, whose powers grew and lessened based on how close the date was to "their" month.  Oh, and there was something about him looking for his mother I think? Perhaps in another "realm"?   Any help is appreciated! When typing either of the commands nautilus . or xdg-open . into terminal, they execute correctly, but with a warning:  Nautilus-Share-Message: Called "net usershare info" but it failed: Failed to execute child process “net” (No such file or directory)   As I say, it opens the file manager correctly and I can detect no further problems. There are several instances of users experiencing this message in other questions , but each of the questions either addresses a side-issue I'm not experiencing, or simply goes unanswered.    What does this message mean, and can I do anything to prevent it? Can I have my iPhone play sound for MacBook? Say I am playing a video on my MacBook Pro, and I don't want to distract others with the sound. The problem is, I don't have a 3.5 mm Headphone or AirPods at my hand, but I have an iPhone 8 Plus and its EarPods with Lightning Connector. Can I play the sound from the video playing on macOS with the iPhone, preferably without connecting the iPhone to the Mac so that I can plug in the EarPods to the iPhone's socket?  This is basically the reverse of Extend tourist visa stay in USA due to coronavirus and waive I-539 fee My mom has come to US on a tourist visa and her I-94 expires on the first week of May.   Due to the coronavirus, travel restrictions have been imposed in our home country (India). Also we want to avoid the risk of traveling in flights at this time.  Is there an easy way to to extend her stay in the US by one or two months?  I am aware that we should fill out an  to extend our stay. But in-order to submit the online application it requires a fee of $370.    Is filing I-539 the easiest way to extend her stay at this time? If so, is it possible to waive the fee due to the coronavirus situation?  After filing I-539 should she wait for an approval to continue her stay? Or is the receipt good enough?   Can you please help me with these? Thanks!  [Edit 1]:  With the current situation of the virus, traveling anywhere sounds like a very risky operation, especially for elders.   That is the main reason behind this question. If the US government will allow visitors to overstay until Covid-19 settles down! Choosing between $z$-test and $t$-test Background: I'm giving a presentation to colleagues at work on hypothesis testing, and understand most of it fine but there's one aspect that I'm tying myself up in knots trying to understand as well as explain it to others.  This is what I think I know (please correct if wrong!)   Statistics that would be normal if variance was known, follow a $t$-distribution if the variance is unknown CLT (Central Limit Theorem): The sampling distribution of the sample mean is approximately normal for sufficiently large $n$ (could be $30$, could be up to $300$ for highly skewed distributions) The $t$-distribution can be considered Normal for degrees of freedom $&gt; 30$   You use the $z$-test if:   Population normal and variance known (for any sample size) Population normal, variance unknown and $n&gt;30$ (due to CLT) Population binomial, $np&gt;10$, $nq&gt;10$   You use the $t$-test if:   Population normal, variance unknown and $n&lt;30$ No knowledge about population or variance and $n&lt;30$, but sample data looks normal / passes tests etc so population can be assumed normal   So I'm left with:    For samples $&gt;30$ and $&lt;\approx 300$(?), no knowledge about population and variance known / unknown.   So my questions are:   At what sample size can you assume (where no knowledge about population distribution or variance) that the sampling distribution of the mean is normal (i.e. CLT has kicked in) when the sampling distribution looks non-normal? I know that some distributions need $n&gt;300$, but some resources seem to say use the $z$-test whenever $n&gt;30$... For the cases I'm unsure about, I presume I look at the data for normality. Now, if the sample data does looks normal do I use the $z$-test (since assume population normal, and since $n&gt;30$)? What about where the sample data for cases I'm uncertain about don't look normal? Are there any circumstances where you'd still use a $t$-test or $z$-test or do you always look to transform / use non-parametric tests? I know that, due to CLT, at some value of $n$ the sampling distribution of the mean will approximate to normal but the sample data won't tell me what that value of $n$ is; the sample data could be non-normal whilst the sample mean follows a normal / $t$.  Are there cases where you'd be transforming / using a non-parametric test when in fact the sampling distribution of the mean was normal / $t$ but you couldn't tell? I know that to better balance paragon and epic level players against similar level monsters, WotC changed the math used for monsters for MM3, and I believe that they changed some of the types of powers they gave monsters.   If I want to update monsters from MM1 or MM2 to be in line with MM3 era monsters, what changes should I make? Which function ($f$) is continuous nowhere but $|f(x)|$ is continuous everywhere?   I found this question , the question seems much interesting but for obvious reason it is closed there, I was wondering how to derive such a function? I have a Lubuntu (now 14.04) installation. Recently, I logged in to find that the Wi-Fi icon had disappeared, and there was some sort of a keyboard icon in it's place. I have restarted multiple times, but it doesn't fix it. I can't find the same applet in the panel settings, and resetting the panel didn't help. When I try to manually start nm-applet, it starts, but prints a bunch of warnings to the console:  nm-applet-Message: using fallback from indicator to GtkStatusIcon   It then starts, using ugly icons for the Wi-Fi connectivity symbol.  What do I need to do to make it start correctly? Once it starts, why isn't it using the right icons? I unplugged my USB stick from Ubuntu 16.04 without the save remove option, but exfat should be fine doing so. No read write was happening, the partition does not use journaling. What I end up with is an unmountable USB stick with checksum error.   What tools are there to recover the partition table or at least the data on the USB stick? What I am doing now is scanning via parted. Has anyone experienced the same problem with exfat and recovery options? Unfortunately parted probably won't detect the partition because .  I also tried to use Partition Wizard on Windows but it failed to detect anything, just showing unallocated space. How do I proceed? are there helpful tools under Windows or Ubuntu?    I also tried to use gpart -v /dev/sdb but it failed to detect anything. On OS X there is a tool called fsck_exfat, is there any on Windows?  update The disk is preinstalled with Parted Magic. After quick search no partition shows up. Blank white desktop icons on new Windows 7 install I am looking for an Ubuntu version that consumes less Memory and CPU. I have read both Lubuntu and Xubuntu (The homepages, wikipedia, phoronix and other sites comparing both). But from experience, which one uses less memory and is less CPU intensive. I need to install them in very old hardware and want to persuade the owner of the hardware of the benefits of Ubuntu.  in this case I want to install 11.10 or 12.04 when it comes out. How are each behaving in those versions?  The 2 PCs I will be installing either Xubuntu or Lubuntu are:  Granpa PC:  CPU - Pentium 2 450Mhz RAM - 64MB DIMM Video - 16MB Used for - Documents and Internet. No listening to music, no looking at videos. Just using it for document writing.  The other old meat:  CPU - Pentium 3 550Mhz RAM - 128MB DIMM Video - 16MB Used for - Documents and Internet also but they want.. or maybe they are wishing for it to use it to see movies and listen to music. This one has internet. The other one does not.  UPDATE  Wanted to update this question with the test results for over 8 months worth of using Lubuntu and Xubuntu. When it comes to very old hardware, am talking about 64 MB, 128 MB or even 256 MB of RAM computers, Lubuntu is better than Xubuntu when comparing Memory usage. When it comes to CPU usage, both are the same.  This will not improve I/O Disk performance which will be the bottleneck for old computers. Also the difference in performance between Xubuntu and Lubuntu is about 10% to 20% percent in memory usage. Again, CPU usage is the same.  Now, when it comes to how responsive the Desktop is, LXDE (Lubuntu) has a much better response time than XFCE (Xubuntu). I would say about 50% or just simply noticeable to the eye. Lubuntu is similar to how Windows XP looks while Xubuntu is more like how Gnome 2 looked.  The computers (The old ones mentioned here) and 2 more are still working in excellent conditions thanks to the Ubuntu family. I also want to add I am using 13.10 for both distros. I'm using  to go through JavaScript, and it's returning many suggestions to replace == (two equals signs) with === (three equals signs) when doing things like comparing idSele_UNVEHtype.value.length == 0 inside of an if statement.  Is there a performance benefit to replacing == with ===?   Any performance improvement would be welcomed as many comparison operators exist.  If no type conversion takes place, would there be a performance gain over ==? Solve $f(x)f(2x^2) = f(2x^3+x)$ I built a simple bridge rectifier.         &ndash; Schematic created using   I connected the output of the function generator with CH1 of my oscilloscope and  with the input of the rectifier (using a T-Piece). The output goes into CH2.   The original wave looks like this:    After connecting CH2 the picture looks like this:   While CH2 looks good,  CH1 now shows the wave I would expect without capacitator but not the expected original wave.   Could somebody explain why? I'm using a Gwinstek GDS-1072A-U Oscilloscope and a Gwinstek SFG-1003 function generator. Prove the equality: $\det\left[\begin{smallmatrix} -2a &a+b &a+c \\ b+a&-2b &b+c \\ c+a&c+b &-2c \end{smallmatrix}\right] = 4(a+b)(b+c)(c+a)$ i have to prove:   $$\begin{vmatrix} -2a &amp;a+b  &amp;a+c \\   b+a&amp;-2b  &amp;b+c \\   c+a&amp;c+b  &amp;-2c  \end{vmatrix} = 4(a+b)(b+c)(c+a)$$   I have tried many calculations between the rows and columns of the determinant to get to the answer i want to solve the exercise however none of them gave me something right. Can someone help?