Find a set A of rational numbers that is isomorphic to sup{$\omega$, $\omega^\omega$, $\omega^{\omega^\omega}$, $\cdots$}
Question of an isomorphism of $\epsilon_ 0$ and a subset of the rationals.
Short story about a woman that travels to other planets. Read online recently. She just manages to live long enough to embrace new technology How to log source file name and line number in Python Image data URI and SEO Do you think the use of image data URI is SEO friendly?   On the one hand saves you a HTTP request. On the other hand it really messes up your code. Especially for image files larger than 20KB or moving GIF image files of up to 200KB, the quantity of code inserted is enormous. As Googlebot reads usually from top to bottom, all data after the image data URI encoding will be harder to rank well in Google.   Consider that image data URI is not in the CSS files as they should normally be and for the time being I can use only TinyMCE editor which provides me with just two options, either insert image with the image data URI method or the traditional image link.   I would like to know which is more search engine friendly. How to adapt the color of a label using pgfplots I created this style that shifts the labels and put them on the bottom of the ybars.  However, I have a problem with the bars that have too short length (for example, see the 8.7 and 5.1 in the example below). I want to check if the values are less than some threshold, or the length of the bar is shorter than the label (number) then change its color.  How can I overlay another label on top of the one in white when the bar is shorter? or maybe change the color of the label to a gradient color? or just overlay the part that is outside of the bar?    \documentclass{standalone}  \usepackage{tikz} \usepackage{pgfplots} \usepackage{pgfplotstable} \usetikzlibrary{calc}  \begin{document} \makeatletter \begin{tikzpicture} \begin{axis}[ ybar, every node near coord/.style={   check for zero/.code={ % If meta=0, make the node a coordinate (which doesn't have text)     \pgfmathfloatifflags{\pgfplotspointmeta}{0}{         \pgfkeys{/tikz/coordinate}     }{}   },   check for zero, color=white, text opacity=1, font=\footnotesize, inner ysep=0.5pt, },% % Bottom nodes  calculate full offset/.code={     \pgfkeys{/pgf/fpu=true,/pgf/fpu/output format=fixed}     \pgfmathsetmacro\testmacro{((\pgfplotspointmeta-\pgfkeysvalueof{/pgfplots/ymin})*10^\pgfplots@data@scale@trafo@EXPONENT@y)*\pgfplots@y@veclength}     \pgfkeys{/pgf/fpu=false} },% nodes near coords bottom/.style={     every node near coord/.append style={         /pgfplots/calculate full offset,         yshift=-\testmacro,         rotate=90, anchor=west,%I need the rotate text here and not in the          %general style, as the ybar redefines the style     } },% ymin=0, nodes near coords={\pgfmathprintnumber[fixed zerofill,precision=1]{\pgfplotspointmeta}}, nodes near coords bottom, ] \pgfplotstableread{ % set real imag mag 1 8.7 2 5.1 3 91.8367  }\mydata \addplot table[x index=0,y index=1,header=false] {\mydata}; \end{axis} \end{tikzpicture} \makeatother \end{document}   * EDIT *  I need to create some special gradient on each label that is below the bar height. For example, I need to adapt something like  to each label that is not showing. However, I cannot find a way to put the macro in the style section of the codes. Or is there any equivalent solution? I need to add a specific column if it does not exist. I have something like the following, but it always returns false:  IF EXISTS(SELECT *           FROM   INFORMATION_SCHEMA.COLUMNS           WHERE  TABLE_NAME = 'myTableName'                  AND COLUMN_NAME = 'myColumnName')    How can I check if a column exists in a table of the SQL Server database? I know that there exists at least one solution, because $ax \equiv b \mod m \implies ax - b = qm$  Rearranging terms gives $ax - qm = b$, and since $d | a$ and $d | m$, so $d | b$ it can be said that $b \equiv 0 \mod d$.  I implemented Euclid's extended algorithm and have been able to experimentally verify that the claim holds, but I do not see a pattern or how I can begin to show there are exactly $d$ solutions.  def euclid_extended(x, y, i=0):   if y == 0:     print("iteration {0} returns ({1}, {2}, {3}) - base case hit with ({4}, {5})".format(i, x, 1, 0, x, y))     return (x, 1, 0)   else:     (d, a, b) = euclid_extended(y, x % y, i + 1)     print("iteration {0}. ({4}, {5}) =&gt; ({1}, {2}, {3})".format(i, d, b, a - b * (x//y), x, y))     return (d, b, a - b * (x//y)) These questions are really bothering me. Any help would be much appreciated.  Let $p$ be prime  a) If $(G, \oplus)$ has order $2p$, prove that every proper subgroup of $(G, \oplus)$ is cyclic  b) If $(G, \oplus)$ has order $p^2$, prove that $(G, \oplus)$ has a subgroup of order $p$ Enabling CORS in GeoServer (jetty)? Why are airline passengers asked to lift up window shades during takeoff and landing? I have always wondered why the windows shades have to be raised during take off and landing. I can't think of any safety issues by having a closed window. $11$ out of $36$? I got this by writing down the number of possible outcomes ($36$) and then counting how many of the pairs had a $6$ in them: $(1,6)$, $(2,6)$, $(3,6)$, $(4,6)$, $(5,6)$, $(6,6)$, $(6,5)$, $(6,4)$, $(6,3)$, $(6,2)$, $(6,1)$. Is this correct? This is problem 31b) from Dummit and Foote chapter 14.2. I am looking for a hint on how to attack the problem, since I have been thinking about it for a couple of hours but I don't even know where to start. The problem says:  Let $K$ be a finite extension of $F$ of degree $n$. Let $\alpha$ be an element of $K$. Prove that the minimal polynomial for $\alpha$ over $F$ is the same as the minimal polynomial for the linear transformation $T_{\alpha}$. In this problem, $T_{\alpha}$ is an $F$-linear transformation of $K$ that arises from $\alpha$ acting by left multiplication on $K$.  I appreciate any helpful suggestions on how to start attacking the problem or any possible hint. Thanks! I've read that flash memories can "only" be reprogrammed 100000 to 1000000 times, until the memory storage "deteriorates"  Why exactly does this happen with flash and not other memory types, and what does "deteriorate" refer to, internally?  EDIT: Since it's not only flash that this happens, I'd like to generalize a bit and inquire about the memories that have this problem. Also, does the wearing out between these memory types occur due to the same phenomenon? The Monty Hall Problem - where does our intuition fail us? How to combine array items with string substitution? Merge duplicate records into 1 field php execute a background process I need to execute a directory copy upon a user action, but the directories are quite large, so I would like to be able to perform such an action without the user being aware of the time it takes for the copy to complete.  Any suggestions would be much appreciated. What's the difference between stochastic and random? What is the term for taking it out on someone even though you will be harmed by this too? When you are so mad at someone that you make a bad decision, even though you will be harmed by it as well, in order to harm that person; is there a good term for this?   For example burning $100 because he owns $20 of it. Do tag edits count for Archaeologist? How does a cell phone charger work?