Let $\langle X,\leq_X\rangle$ be a well-ordered set. I want to show that $X$ with the topology on $X$ relative to $\leq_X$ (the order topology) is not homeomorphic to any initial segment of $X$. And then I need to conclude that any two well-ordered sets $\langle X,\leq_X\rangle$ and $\langle Y,\leq_Y\rangle$ are homeomorphic iff they are isomorphic. I can feel both seem true, but while the first one I don't know how to proof, the second one I can prove, but not as a conclusion of the first. Any ideas on how can I show the second one as a conclusion of the first?  And on how can I prove the first?  Thanks!
I am trying to get a better feel for the topology of ordinals and just received a great answer to this  where the Cantor-Bendixson rank and degree turn out to be a complete homeomorphism invariant of countable compact spaces. Now I am curious about what is going on within homeomorphism classes of ordinals. In particular, I want to know how to characterize countable ordinals $\alpha$ which are minimal among all ordinals homeomorphic to $\alpha$.   Certainly finite ordinals and $\omega$ are easy. Next, if $\alpha\cong \omega+1$, then it seems like we would have to have $\omega+1\leq \alpha&lt;\omega +\omega$ since there can only be one limit point in $\alpha$. Things become a little fuzzier for me with ordinals that have infinitely many limit points.  For instance, suppose $\alpha$ is an ordinal homeomorphic to $\omega^{\omega}$. Is it necessarily true that $\alpha\geq \omega^{\omega}$? Is there a general rule for knowing when an ordinal is the smallest among all ordinals homeomorphic to it?
Why do universities have to spend money on journals? Obviously this is a question in the light of the recent Elsevier boycott. Currently we do have an arXiv, maintained by academia and where researchers regularly upload parts of their work. In such a case,   Why do universities spend lots of money to publish in third-party journals?   The question especially applies to journals that operate with a rigorous profit motive. The subscription is very high, so wouldn't publishing in such journals affect the paper's citation count and deter the spread of knowledge about the work within academic circles?   Why should not universities collaborate to create free, open access, peer-reviewed journals?   Moreover, given the need to conserve paper, why should journals spend on printing research papers? Wouldn't an online version suffice, as most people use only local computer printouts anyway? In other words, why can't we have a Wikipedia-like system of sharing research knowledge, having properly established standards for such journals? Export data as CSV in back end with proper HTTP headers I wrote a plugin that displays all products in a woocommerce store in admin settings option, now I want to add a link to download the products as CSV file.  The problem is, when I click the link I get a permission error saying I don't have permission to view this page.  Here's my code:  function extra_tablenav($which) {     if ($which == "top") {         echo '&lt;h3 style="display:inline"&gt;'         . __('These products are currently in the database:')         . '&lt;/h3&gt;' .         '&amp;nbsp;&amp;nbsp;&amp;nbsp;' .         '&lt;a href="' . admin_url('admin.php?page=download_csv.php') . '"&gt;' . __('Export to CSV') . '&lt;/a&gt;';     } }   How can I fix those permissions? Schengen visa. What qualifies as main destination? I'm planning a multi city trip soon. I'll be applying with the German embassy as I had planned on going to Berlin for 3 days before going to Prague, Vienna, and Budapest for 2-3 days each afterwards.   My question is, it is working out more conveniently for me to instead make Germany my last destination, rendering the first entry clause. And I can only apply with the German consulate as they are the only ones I can find an appointment for in time. Would staying in Germany for 4 days afterwards be enough to classify it as a main destination (as opposed to 2-3 days in the other cities)? I have two layers, one (A) has point features and the other (B) has polygons.  How do I produce a new table with some (the ID) or all the fields from the A table and some (the area code) or all from the B table where the B polygon contains the A point?  I am using QGIS. On  we try really hard to fix spelling, grammar, and capitalization mistakes in question titles in order to avoid the . But occasionally someone enters a question with an all-lowercase title ().  It would be useful if the system could nudge the user toward capitalizing titles property by showing a hint when the user appears to be capitalizing incorrectly. For instance:    As  and  note in comments, the feature should also try to detect and advise against Title Case and ALL CAPS. How is "Can anyone tell me how can I solve this" wrong? How to prove $x^{n}$ is not uniformly continuous How to prove $x^{n}$ is not uniformly continuous on the interval $[0, +\infty)$? Remove only the commas present within the double quotes In a text file, I want to remove , (commas) and also the " (quotes) (only if the double quotes  contains  numbers separated by commas).  56,72,"12,34,54",x,y,"foo,a,b,bar"   Expected ouput  56,72,123454,x,y,"foo,a,b,bar"   Note: I show the above line just as an example. My text file contains many lines like above and the numbers separated by commas present within the double quotes should vary. That is,  56,72,"12,34,54",x,y,"foo,a,b,bar" 56,92,"12,34",x,y,"foo,a,b,bar" 56,72,"12,34,54,78,76,54,67",x,y,"foo,a,b,bar" 56,72,x,y,"foo,a,b,bar","12,34,54" 56,72,x,y,"foo,a,b,bar","12,34,54","45,57,84,92","bar,foo"   Expected output:  56,72,123454,x,y,"foo,a,b,bar" 56,92,1234,x,y,"foo,a,b,bar" 56,72,12345478765467,x,y,"foo,a,b,bar" 56,72,x,y,"foo,a,b,bar",123454 56,72,x,y,"foo,a,b,bar",123454,45578492,"bar,foo"   There a n number of numbers present within the double quotes separated by commas. And also leave the double quotes which contains characters as it is.  I love sed text processing tool. I'm happy if you post any sed solution for this. Arrow keys, Home, End, tab-complete keys not working in shell I have installed ubuntu minimal(mini.iso) on my vm. I then used recovery mode to login as root and create an account with useradd -m admin and then set a password with passwd admin.  When I login on the new account, instead of the normal prompt I only see a $ sign. If I try to tab-complete a command or file name it prints a normal tab. If I try to use the arrow keys it prints ^[[A, ^[[B, ^[[C or ^[[D. Also, ls no longer adds colors.  None of these problems were in recovery mode. How can I fix this? What pressure should I run my Road Bike tyres at? There's a question for mountain bike tyres but I can't seem to find anything on here for road bikes. At the moment I'm running at around 90psi. Is that too much? too little?  Also, am I more likely to get punctures running at higher pressure?  I seem to be getting a lot lately. Reprojecting raster from 0 360 to -180 180 with cutting 180 meridian using gdalwarp I have a geotiff raster image that has a coordinate system with longitudes from 0 to 360. The horizontal center of the image is 180 longitude. See image below:    I want to transform it to EPSG:4326 SRS with -180 180 longtitude range.  And I want the center of the image to be at Greenwich meridian (0). I guess this srs is very widely used. I expect the result look like this:    So I use a gdalwarp command to reproject:  gdalwarp -s_srs '+proj=latlong +datum=WGS84 +pm=180dW' -t_srs EPSG:4326 test_col.tif test_4326.tif   But I only get a tiff with bigger dimensions (more pixels) and EPSG:4326 metadata. The image itself looks the same, as the initial one. But I expect it to swap the hemispheres.  How do I gdalwarp an image to be strictly -180 180 EPSG:4326 with the center in 0 longitude?  This is gdalinfo of my initial file:  Origin = (-0.102272598067084,89.946211604095552) Pixel Size = (0.204545196134167,-0.204423208191126) Metadata:   AREA_OR_POINT=Area Image Structure Metadata:   INTERLEAVE=BAND Corner Coordinates: Upper Left  (  -0.1022726,  89.9462116) (  0d 6' 8.18"W, 89d56'46.36"N) Lower Left  (  -0.1022726, -89.9462116) (  0d 6' 8.18"W, 89d56'46.36"S) Upper Right (     359.897,      89.946) (359d53'50.18"E, 89d56'46.36"N) Lower Right (     359.897,     -89.946) (359d53'50.18"E, 89d56'46.36"S) Center      ( 179.8975000,  -0.0000000) (179d53'51.00"E,  0d 0' 0.00"S)   This is gdalinfo after gdalwarp     Origin = (-180.102727401932952,89.946211604095552) Pixel Size = (0.091397622896436,-0.091420837939082) Metadata:   AREA_OR_POINT=Area Image Structure Metadata:   INTERLEAVE=PIXEL Corner Coordinates: Upper Left  (-180.1027274,  89.9462116) (180d 6' 9.82"W, 89d56'46.36"N) Lower Left  (-180.1027274, -89.9699975) (180d 6' 9.82"W, 89d58'11.99"S) Upper Right ( 179.8211116,  89.9462116) (179d49'16.00"E, 89d56'46.36"N) Lower Right ( 179.8211116, -89.9699975) (179d49'16.00"E, 89d58'11.99"S) Center      (  -0.1408079,  -0.0118929) (  0d 8'26.91"W,  0d 0'42.81"S) Also how do LEFT JOIN, RIGHT JOIN and FULL JOIN fit in? Why is inductance (L) proportional with turns-square (N²)? We start from the   $$ \mathbf{\nabla} \times \mathbf{B} = \mu \mathbf{J} + \overbrace{\mu \epsilon \dfrac{\partial \mathbf{E}}{\partial t}}^0. $$  We take surface integration of both sides, for the surface (\$s\$) inside the mean path (\$c\$) of the core.  $$ \int_s \left( \mathbf{\nabla} \times \mathbf{B} \right) \cdot d\mathbf{s} = \mu \int_s \mathbf{J} \cdot d\mathbf{s} $$  We use the  to rewrite the left hand side; where \$c\$ is in the same direction with the magnetic flux \$\Phi\$.  $$ \oint_c \mathbf{B} \cdot d \mathbf{\ell} = \mu N I $$  (The integral on the left hand side results \$NI\$, because there are \$N\$ different wires on the winding.)  The magnetic field density inside these kind of cores are considered to be uniform. So, we can write  $$ B \ell_c \overset\sim= \mu NI \implies B = \dfrac{\mu NI}{\ell_c}; $$  where \$\ell_c\$ is the mean path lenght of the core.  We can find magnetic flux from the magnetic flux density we have found by using the cross sectional area of the core \$A_c\$.  $$ \Phi = BA_c = \dfrac{\mu NIA_c}{\ell_c} $$  By definition, inductance is the amount of magnetic flux generated per applied current, that is  $$ L \overset\triangle= \dfrac{\Phi}{I}. $$  So, we find inductance of the system as  $$ \boxed{ L = \dfrac{\Phi}{I} = \dfrac{\dfrac{\mu NIA_c}{\ell_c}}{I} = \dfrac{\mu NA_c}{\ell_c} }. $$  But, all other sources () give inductance of an inductor like this as  $$ L = \dfrac{\mu N^2A_c}{\ell_c}. $$  What is the mistake I did in my derivation? Please explain in detail. I'm looking for the fastest way to determine if a long value is a perfect square (i.e. its square root is another integer):    I've done it the easy way, by using the built-in  function, but I'm wondering if there is a way to do it faster by restricting yourself to integer-only domain. Maintaining a lookup table is impractical (since there are about 231.5 integers whose square is less than 263).   Here is the very simple and straightforward way I'm doing it now:  public final static boolean isPerfectSquare(long n) {   if (n &lt; 0)     return false;    long tst = (long)(Math.sqrt(n) + 0.5);   return tst*tst == n; }   Note: I'm using this function in many  problems.  So no one else will ever have to maintain this code.  And this kind of micro-optimization could actually make a difference, since part of the challenge is to do every algorithm in less than a minute, and this function will need to be called millions of times in some problems.    I've tried the different solutions to the problem:   After exhaustive testing, I found that adding 0.5 to the result of Math.sqrt() is not necessary, at least not on my machine. The  was faster, but it gave incorrect results for n >= 410881.  However, as suggested by , we can use the FISR hack for n &lt; 410881. Newton's method was a good bit slower than Math.sqrt().  This is probably because Math.sqrt() uses something similar to Newton's Method, but implemented in the hardware so it's much faster than in Java.  Also, Newton's Method still required use of doubles. A modified Newton's method, which used a few tricks so that only integer math was involved, required some hacks to avoid overflow (I want this function to work with all positive 64-bit signed integers), and it was still slower than Math.sqrt(). Binary chop was even slower.  This makes sense because the binary chop will on average require 16 passes to find the square root of a 64-bit number. According to John's tests, using or statements is faster in C++ than using a switch, but in Java and C# there appears to be no difference between or and switch. I also tried making a lookup table (as a private static array of 64 boolean values).  Then instead of either switch or or statement, I would just say if(lookup[(int)(n&amp;0x3F)]) { test } else return false;.  To my surprise, this was (just slightly) slower. This is because . What documentation is needed for a tourist to drive a US citizen's car? How to determine the cause of MacBook battery not charging? I see that the question &quot;why is my MacBook not charging?&quot; gets asked a lot. The problem is, there are multiple reasons. Possible points of failure are: charger, cable, magsafe connector, DC input board, logic board, battery. (Have I missed any?). What is the best sequence for a user to follow, to diagnose the problem? By way of example, here are the first steps that I usually try:  Make sure the charger is working. Either use your charger on a different Mac, or test a spare charger with your Mac.  Check the connection. There might be gunk in your port that's stopping the connector from lying flat.  Try resetting the SMC and PRAM.  If the computer boots, run Coconut Battery to determine battery status.   This is where I get a bit stuck. For example, I have a machine here that boots fine when plugged in, but not from battery. Disconnecting the MagSafe causes the machine to switch off. Sounds like it's a dead battery, except that Coconut reports the battery as detected, in good condition, but not charging. What steps should I take next? Specifically, I'm asking for a generic algorithm to help determine the source of the fault (preferably without swapping out lots of components one by one)? Integration of $e^{ax}\cos bx$ and $e^{ax}\sin bx$ I want to see if my process makes a lot of context switches. I also want to see how manpulating task groups affects the number of context switches. Proof that the One-Dimensional Simple Harmonic Oscillator is Non-Degenerate? The standard treatment of the one-dimensional  (SHO) using the raising and lowering operators arrives at the countable basis of eigenstates $\{\vert n \rangle\}_{n = 0}^{\infty}$ each with corresponding eigenvalue $E_n = \omega \left(n + \frac{1}{2}\right)$. Refer to this construction as the abstract solution.  How does the abstract solution also prove uniqueness? Why is there only one unique sequence of countable eigenstates? In particular, can one prove the state $\vert 0\rangle$ is the unique ground state without resorting to coordinate representation? (It would then follow that the set $\{\vert n \rangle\}_{n = 0}^{\infty}$ is also unique.)  The uniqueness condition is obvious if one solves the problem in coordinate representation since then one works in the realm of differential equations where uniqueness theorems abound. Most textbooks ignore this detail (especially since they often solve the problem both in coordinate representation and abstractly), however I have found two exceptions:   Shankar appeals to a theorem which proves one-dimensional systems are non-degenerate, however this is unsatisfactory for two reasons:   Not every one-dimensional system is non-degenerate, however a general result can be proven for a large class of potentials (the SHO potential is in such a class). The proof requires a departure from the abstract solution since it classifies the potentials according to their functional properties.  Griffiths addresses this concern in a footnote stating that the equation $a \vert 0\rangle = 0$ uniquely determines the state $\vert 0\rangle$. Perhaps this follows from the abstract solution, however I do not see how. What keeps mass from turning into energy?