Several earlier articles have explained the topological difference between a countably infinite bouquet of circles vs. a countable Infinite Earring of circles.  But my question is purely one of definitions. What has not been adequately answered in earlier articles (although frequently asked) is why the Infinite Earring does not fit the definition of a wedge sum. For example, on p49 Hatcher states &quot;one might confuse X [the Infinite Earring] with the wedge sum of an infinite number of circles.&quot;  Yet, using Hatcher's definition of wedge sums (p10), the Infinite Earring appears to be a wedge sum of an infinite number of circles. So is the Infinite Earring a type of wedge sum, or not?
The (countably infinite) wedge sum of circles is quotient of  disjoint countable union of circles $\amalg S_i$, with points $x_i\in S_i$ identified to a single point, while the infinite earring H is the topological space defined by the union of circles in the Euclidean plane $\mathbb{R}^2$ with center $(1/n, 0)$ and radius $1/n$ for $n = 1, 2, 3, ...$. In the definition of countably infinite wedge sum of circles, it is not specified the size of circles, the points which to be identified to a single point etc. So we can take disjoint union of circles of radius $1/n$ and identify a point from each to a common single point to get the infinite earring. I couldn't understood the difference between these two topological spaces. Can one explain more precisely the difference between these two spaces?
In Python 2.5, the following code raises a TypeError:  &gt;&gt;&gt; class X:       def a(self):         print "a"  &gt;&gt;&gt; class Y(X):       def a(self):         super(Y,self).a()         print "b"  &gt;&gt;&gt; c = Y() &gt;&gt;&gt; c.a() Traceback (most recent call last):   File "&lt;stdin&gt;", line 1, in &lt;module&gt;   File "&lt;stdin&gt;", line 3, in a TypeError: super() argument 1 must be type, not classobj   If I replace the class X with class X(object), it will work. What's the explanation for this? Chatroom owners can star their own messages I was able to star my own message by pinning it and unpinning it.  Pinning a message adds a star, even to your own messages, but when the pin is removed, the star remains. Solve this integral for free WiFi I saw this today, I checked in Mathematica and the integral comes out to $\pi$, but I have no idea how to solve it.     FREE Wi-Fi: The Wi-Fi password is the first $10$ digits of the answer.   $$\int_{-2}^2\left(x^3\cos\frac x2+\frac12\right)\sqrt{4-x^2}\ dx$$ Ok guys, continuing my passage through edwards... here is the question... thanks for hints/solutions in advance:  Suppose $f(x,y,z)=0$ can be solved for each of the three variables $x,y,z$ as a differentiable function of the other two. Then prove that  $\displaystyle \frac{\partial x}{\partial y} \frac{\partial y}{\partial z} \frac{\partial z}{\partial x} = -1$  Verify this is the case for the ideal gas equation $pv =RT$ where (where $p,v,T$ are the three variables and $R$ is the constant). How to mount read-write an ext4 partition on Windows? Ant-Man, shrinking, and weight Show Log of the Sql statements that QGIS generates to retrieve Data from Postgis Is there a ways to  - make QGis log its generated SQL Statement or to - force POSTGIS to log all interaction with QGIS?  There is something strange happening to the WFS layer I retrieve in QGIS, since it does  not select all data available in the respective database table. I need to find out, what criteria QGIS chooses in order to narrow down the select set. How do I alter a language item link? I am looking to modify the default output of Drupal 8 of links in a certain situation, the language switcher. Ideally I want to be able to do that in a twig template and not have to produce a  that overwwrites that whole link generation system and add a BIG if for the cases I need to overwrite.  I think the output layout should be controllable in the twig template similar to  (this answer outputs the same url for every languages)  I am using bootstrap as a base theme so i'd like to use the bootstrap markup for my links (btn btn-primary, dropdown)  Here is my code I am trying to make links--language-block.html.twig  {%- if links|length == 2 -%}     {# show only alternate language button #}     {%- for key, item in links -%}         {%- if not item.attributes['#options']['set_active_class'] -%} {# &lt;--- this is always true!? #}             &lt;a href="{{ item.link['#url'] }}" class="btn btn-primary"&gt;{{ item.link['#title'] }}&lt;/a&gt; {# #url is always the same what ever the language it is pointing to! #}         {%- endif -%}     {%- endfor -%} {%- elseif links|length &gt; 2  -%}     {# show selected language in button and other languages in drop down #}     &lt;button class="btn btn-primary dropdown-toggle" type="button" data-toggle="dropdown"&gt;{{ selectedLanguage }}&lt;span class="caret"&gt;&lt;/span&gt;&lt;/button&gt;     &lt;ul class="dropdown-menu"&gt;         {% for key, item in links %}             {%- if not item.attributes['#options']['set_active_class'] -%} {# &lt;--- this is always true!? #}             &lt;li&gt;&lt;a href="{{ item.link['#url'] }}"&gt;{{ item.link['#title'] }}&lt;/a&gt;&lt;/li&gt;             {% endif %}         {% endfor %}     &lt;/ul&gt; {%- endif -%}   Anyone has an idea on how to do this? What do repeating '^@' symbols in a logfile mean? A few times I have come across this in a log file:  Sep 17 09:35:57 00187d378926 ovpn-00187d378926[1701]: /sbin/ifconfig tun0 0.0.0.0 Sep 17 09:35:57 00187d378926 kernel: Kernel logging (proc) stopped. Sep 17 09:35:57 00187d378926 rsyslogd: [origin software="rsyslogd" swVersion="4.6.4" x-pid="1707" x-info="http://www.rsyslog.com"] exiting on signal 15. ^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@ ^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@ ^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@   What does it mean?  Sometimes it is a single line, other times it is lots of lines.  I usually see it happen on Debian 6 machines. How to sum a variable by group I can't seem to get these particular faces to take the UV Mapping. I selected the entire object, entered edit mode, hit U>Unwrap.   The rest of the mesh takes the uv wrapping, but not those faces for some reason.    It looks normal in solid mode.     Does anyone know what could be causing this? How do I check if an integer is even or odd? How do  methods compare in terms of model variance and bias?   My question is partly motivated by this thread: . The answer there suggests that models learned with leave-one-out cross-validation have higher variance than those learned with regular $K$-fold cross-validation, making leave-one-out CV a worse choice.  However, my intuition tells me that in leave-one-out CV one should see relatively lower variance between models than in the $K$-fold CV, since we are only shifting one data point across folds and therefore the training sets between folds overlap substantially.  Or going in the other direction, if $K$ is low in the $K$-fold CV, the training sets would be quite different across folds, and the resulting models are more likely to be different (hence higher variance).  If the above argument is right, why would models learned with leave-one-out CV have higher variance? I haven't been able to do this exercise:     Let $f: A \rightarrow B$ be any function. $f^{-1}(X)$ is the inverse   image of $X$. Demonstrate that if $f$ is surjective then $X = f(f^{-1}(X))$ where $X \subseteq B$.   Since $X \subseteq B$, all the elements in $X$ belong to the codomain of $f$.  Since $f$ is surjective, it means that all elements in the codomain $B$ have some preimage in $A$. Given that $X \subseteq B$, all elements in $X$ must also have a preimage in $A$.  Have $\triangle = f^{-1}(X)$, $\triangle$ is now a set containing the preimages of the elements in $X$. Because of this, $\triangle \subseteq A$.  If we evaluate $f(\triangle)$, we...... nope, I don't know what I'm doing now.    What do you think? On the  blog post, nearly all the images are broken. I assume this is because they are all hosted on stackoverflow.blog/wp-content/uploads/ rather than zgab33vy595fw5zq-zippykid.netdna-ssl.com/wp-content/uploads/, and they got destroyed in the removal of Blog Overflow. I looked around at a few other blog posts, and this post seems to be the only one having an issue. In ArcMap, when you export data to PDF it preserves the layering within the PDF so that you can turn on/off individual map layers (within the PDF itself).  When I export from QGIS to PDF, I don't appear to have the option for this.  Does QGIS support preserving layers when exporting to PDF?  Is there a special setting needed to enable this capability?  Or has anyone wrote a plugin for this feature? What to conclude from this lasso plot (glmnet) Following is the plot of glmnet with default alpha(1, hence lasso) using mtcars data set in R with mpg as the DV and others as the predictor variables.  glmnet(as.matrix(mtcars[-1]), mtcars[,1])     What can we conclude from this plot regarding different variables, especially am, cyl and wt (red, black and light blue lines)? How would we phrase the output in a report to be published?  I thought of following:   wt is the most important predictor of mpg. It is negatively affecting the mpg. cyl is a weak negative predictor of mpg. am may be a positive predictor of mpg. Other variables are not robust predictors of mpg.    Thanks for your thoughts on this.  (Note: cyl is the black line which does not reach 0 till very close to it.)  Edit: Following is plot(mod, xvar='lambda') which shows x-axis in order reverse of above plot: How do you add CSS rules (eg strong { color: red }) by use of Javascript? Mortarboard badges being awarded as a result of bounties after recalc I use Firefox 35.0 on my Macbook to access Stack Exchange. Recently whenever I access a message or Community site, the "spinning circle" indicating activity starts, and never stops. This doesn't seem to happen on other non-Stack Exchange sites.  Has there been any change to the SE code which may have caused this.    The Web Developer Console shows  The connection to ws://qa.sockets.stackexchange.com/ was interrupted while the page was loading. full.en.js:1 Use of getPreventDefault() is deprecated.  Use defaultPrevented instead. jquery.min.js:3