Thickening the line in legend representation
Pgfplots: Linewidth when using \ref in custom legend
Let $a_k:=(-1)^k$ where $k\in\mathbb{N}$. $\mathbb{N}$ is the set of all non-negative integer.  And we define the partial sum $S_n:=\sum \limits_{k=0}^{n}a_k$. Notice that the sequence $\{S_k\}$ diverges which also implies that the infinite series $\sum \limits_{k=0}^{\infty}a_k$ cannot be defined.  If I consider the infinite sum $1-1+1-1+1-1+\cdots$, then this statement is equivalent that I just defined the infinite series $\sum \limits_{k=0}^{\infty}a_k$, which is contradiction.  But, suppose that the sum $1-1+1-1+1-1+\cdots$ exists and let the value of the sum be $S$. Then, we can easily observe that $S=1-S$, therefore $S=1/2$. The supposition of this proposition already proved as false, but if I ignore the definition of infinite series, it holds.  Which one is right? Or are these propositions depends on what we define? iOS 10 remember to "not trust this computer" keep duplicates out of $PATH on source I have the following code that's source-d by my .shellrc  PATH="${PATH}:${HOME}/perl5/bin" PATH="${PATH}:${HOME}/.bin" export PATH   but if I make changes to other code and then source this file, my path continues to get longer and longer with each source, each time appending these when they're already there. What can I do to prevent this? I am currently learning how to use TikZ/PGF to make some figures for a project report, and so I have a question:     Is there is a practical way of making a particular object into a node?   The object I have in mind is shown in the Figure below. I want the object to consist in the big circle, the two circular arcs, the label and anchors at north, south, east and west. The dotted lines are simply there to indicate that I am going to have paths going out of these points, which is why I need the anchors there.     I include the source, in the case that it is useful to anyone.  \begin{tikzpicture} \draw[help lines, gray, dotted] (-2,-2) grid (2,2);  \begin{scope}[shift={ (0,0) } ]     \begin{scope}[rotate=90]         \draw[thick] (-1,0) arc[start angle=-90, delta angle=90, %                               x radius=1, y radius=1] -- (0,1) %NW                      (0,-1) arc[start angle=180, delta angle=-90, %                                    x radius=1, y radius=1] -- (1,0); %SE         \draw[gray] (0,0) circle (1);     \end{scope} % Anchor replacements     \node(N1)       at (0,1)    {};     \node(S1)       at (0,-1)   {};     \node(E1)       at (1,0)    {};     \node(W1)       at (-1,0)   {}; % Label     \node(Crossing number) at (0,0) {1}; \end{scope}  \path[thick, gray, dotted] [ out=90,   in=270]     (N1.center) edge (0,2) [ out=180,  in=0]       (W1.center) edge (-2,0) [ out=270,  in=90]      (S1.center) edge (0,-2) [ out=360,  in=180]     (E1.center) edge (2,0); \end{tikzpicture}   PS. I have had a look at the Manual, but I could not find a description of how to do this. 32 bit or 64 bit OS on a 512 mb VPS? C# Mono - Low Level Keyboard Hook Setting the correct encoding when piping stdout in Python Unable to install Ubuntu on Lenovo Y500 Isomorphism between dual space and bilinear forms Studying from Roman's Advanced Linear Algebra, I want to prove that $$U^* \otimes V^* \cong (U \otimes V)^* \cong \hom(U,V;\mathbb{F})$$ The author proves that $U^* \otimes V^* \cong (U \otimes V)^*$ by showing that there is an unique linear transformation   $$\theta:U^* \otimes V^* \to (U \otimes V)^*$$  defined by $\theta(f \otimes g)=f \odot g$ where $(f \odot g)(u \otimes v)=f(u)g(v)$  I want to prove that $U^* \otimes V^* \cong \hom(U,V;\mathbb{F})$. I tried to use the universal property by fixing $f \in U^*, g\in V^*$ and defining $S: U^*\times V^* \to \hom(U,V;\mathbb{F})$ as $S(f,g)=F_{f,g}$, where $F_{f,g}(u,v)=f(u)g(v)$, but I got stuck. In low-cost mass-produced items I often run into black blobs of what looks like resin applied directly on top of something on the PCB. What are these things exactly? I suspect this is some kind of custom IC that is layed out directly on the PCB to save on the plastic  housing/connector pins. Is this correct? If so, what is this technique called?    This is a photograph of the inside of a cheap digital multimeter. The black blob is the only non-basic piece of circuitry present, along with an op-amp (top) and a single bipolar junction transistor. Compensation for missed connection due to delay with same airline but separate bookings Long story short, I had to fly from A (in Spain) to C (in the UK) and I booked two separate flights with the same budget airline, so the actual route was A (Spain) -> B (UK) -> C (UK).  The first flight was delayed by 90 minutes, which means that I was unable to catch the second one (the layover time was around an hour and a half) and had to buy another ticket for the following day (with another airline).  I googled a lot and found mixed responses. Is there any chance I can get a refund for my second flight? I will obviously ask them but I'm unsure what my best course of action is. Please recommend undergraduate-level textbooks on electrodynamics which emphasize practical applications and real life examples. Please describe the book's level and contents and its intended audience in as much detail as possible.   Please provide both applications from everyday life for curious students as well as engineering applications from a physics perspective. Is there any difference in equals and == for String variables? if ('text'.equals(str))   or   if ('text' == str)   Is there any difference? If yes, then what is it?   Added, after answered.  String q = 'teXt'; system.debug('teXt'.equals(q)); // true system.debug('teXt' == q);      // true system.debug('text'.equals(q)); // false system.debug('text' == q);      // true Use of "save" or "save for" in sentences Making data points different sizes based on data using QGIS? I am trying to plot sales data on a map using QGIS. I will add the disclaimer that I am a rookie at using the program. I added the different sales types by adding delimited text layers (utf16). The data included Longitude, Latitude, and Amount. I want to make the dots on the map scale with the value of the sale. Ive had no luck with trying to use Simple Marker->Data defined properties-> size and writing case functions. Some data points show up at different sizes while others show up at all data points. Here are my functions under different simple markers:   CASE WHEN Amount &lt;= 10000 THEN '.2' END CASE WHEN 10000 &lt; Amount &lt; 75000 THEN '.4' END CASE WHEN 75000 &lt; Amount &lt;= 250000 THEN '.6' END CASE WHEN Amount &gt;= 250000 THEN '1' END   The majority of my data set falls into the 10-75k range. However the .4 and .6 size circles show up at every data point on the map, while the .2 and 1 sizes only show up where the data specifies (along with the .4 and .6 sizes). At this point I am trying to figure out what is wrong with the equations, however I am stuck.   Is there a better way to go about this or am I just simply messing up the equations?   I wish I could share my whole map with you but it is looking great. I went with U/Joseph 's solution and here is an excerpt of the results for those interested. What's wrong with this? SSL certificate for a public IP address? I just tried buying a Comodo Positive SSL but it was rejected due to not supporting a public IP address, but instead they only support a domain name.  Does anyone know any SSL certificate provider that supports public IP address instead of a domain name?  My company has a dedicated server hosted with a web hosting company, which is used for running a bugtracker for multiple projects (for multiple clients). Since it's only used for a bugtracker, we don't need a domain name (our clients access it by typing the public IP in their browser). Prove that $Au\cdot v = u\cdot A^Tv$ Let A be an $n$ x $n$ matrix and let $u$,$v$ $\in$ $\mathbb{R^n}$. Prove that $$Au\cdot v = u\cdot A^Tv$$  I tried using the fact that $A^Tu=A\cdot u$. However, I cannot seem to get to this result. Could anyone please help me out? std::auto_ptr to std::unique_ptr SCP fails without error