Confusion in understanding why empty set is a subset of every set
Is "The empty set is a subset of any set" a convention?
Avoiding if statement inside a for loop? How do I speed up my design work? I've been a designer for 6 months, having previously had a 1st class honours degree in Graphic Communication. My issue is that I am a Very good designer, but I work too slow. It's beginning to drive a rift between myself and the creative director who desperately wants me to work quicker. I've always been a workaholic even in university, I put in 110% all the time, really work my ass off, but I'm still not fast enough to complete projects quickly, my work is good, but it's not efficient.  Do you guys have any tips to help me speed up? Was the singularity at Big Bang perfectly uniform and if so, why did the universe lose its uniformity? Right click not working on Ubuntu 18.04 I have installed Ubuntu 18.04 LTS on my HP Pavilion series laptop. After installing this version of Ubuntu, right click is not working of my touchpad. How can I solve this?  BTW:  does not have any solution which which is working for me. PS:- Problem remains after selection area in gnome-tweak-tool The following time-frequency correspondence illustration is in .    Also a gif animation version (if the animation doesn't show, please open the following image in a new window):    This picture has so much explanatory power, and I would like to replicate it in TikZ for future use.  Here is what I came up with:  \documentclass{minimal} \usepackage{tikz}  \begin{document} \begin{tikzpicture} [x={(1cm,0.5cm)},z={(0cm,0.5cm)},y={(1cm,-0.2cm)}] \draw[-&gt;,thick,blue!90] (0,6.5,0) -- (6.2,6.5,0) node[right] {Frequency}; \draw[-&gt;,thick,red!90] (0,0,0) -- (0,6.5,0) node[below] {Time}; \draw[-&gt;,thick] (0,0,0) -- (0,0,2) node[above] {Magnitude}; \foreach \x in {0.5,1.5,2.5,3.5,4.5,5.5}{   \draw[blue!50] (\x,0,0)   \foreach \y in {0,0.02,...,6.28}{     -- ({\x},{\y},{sin(\x*\y*(157))/sqrt(2*\x)})   };   \draw[blue!90, thick] (\x,6.5,0) -- (\x,6.5,1/\x); }  \end{tikzpicture} \end{document}   This is the output so far.    I have two questions:   How to produce that red superposed sine wave of all the blue sine waves? I don't know if there is a sum function or I have to use loop yet again? How to make the camera projection in TikZ more similar to the perspective in that wikipedia illustration?   Any suggestion and tweaking of the parameters I used in the sample drawing are welcome as well.  Thanks in advance!    Update 1:  Here is a new version using tikz, more readable than the first one. Yet the superposition of the sine waves are done manually...I still don't know how to use foreach to produce a sum.  \begin{tikzpicture}[x={(1cm,0.5cm)},z={(0cm,0.5cm)},y={(1cm,-0.2cm)}] \draw[-&gt;,thick,black!70] (0,6.5,0) -- (6.2,6.5,0) node[right] {Frequency}; \draw[-&gt;,thick,black!70] (0,0,0) -- (0,6.5,0) node[below right] {Time}; \draw[-&gt;,thick] (0,0,0) -- (0,0,2) node[above] {Magnitude}; \foreach \y in {0.5,1.5,...,5.5}{ \draw [cyan!50, domain=0:2*pi,samples=200,smooth]   plot (\y,\x, {sin(4*\y*\x r)/\y }); \draw[blue, ultra thick] (\y,6.5,0) -- (\y,6.5,1/\y); } \draw [red, thick, domain=0:2*pi,samples=200,smooth]  plot (0,\x, {sin(4*0.5*\x r)/0.5 + sin(4*1.5*\x r)/1.5 + sin(4*2.5*\x r)/2.5 + sin(4*3.5*\x r)/3.5 + sin(4*4.5*\x r)/4.5 + sin(4*5.5*\x r)/5.5} ); \end{tikzpicture}   The result is as follows: Predict probabilities from Firth logistic regression in R I'm using the logistf package in R to perform Firth logistic regression on an unbalanced dataset. I have a logistf object:  fit = logistf(a~b)   Is there a predict() function like on that's used in the lm class to predict probabilities for future data points? Or do I have to manually input the estimated parameters from the Firth regression. As a student, I make mistakes all the time, and stealing my idea will not face heavy retaliation. I don't mind if someone comes up with concurrent works, but I do mind if my key idea is leaked. Many graduate students in my field therefore only use pre-print services if their papers are either publicly presented or conditionally accepted. I am looking for a website that can do most of what ArXiv does, with an additional private option or limited visibility, for example an option to make the paper unindexed by search engines. This way, my paper cannot be seen for at least a few months. By doing so, I can at least claim that I am one of the first few researchers who got the idea right. And most importantly, I can privately correct my mistakes, if any. Publishing mistakes is embarrassing and sometimes harms one's career. Many mistakes will be revealed during the peer-review processes and I have a chance to correct them before publicizing. Please kindly think twice before giving ethically and &quot;politically correct&quot; comments like no one will scoop me, everyone will cite me, and the scoopers will be punished. At least in my field, scooping and idea-stealing is commonly observable, and big-names don't care about publicizing pre-prints because they have resources to punish the scoopers. I do understand that keeping my work private might harm the community as the information becomes less transparent. However, it could also be possible that my un-reviewed flawed works misguide the community, and a paper reviewed will benefit the society more. I personally wasted six months because of someone else's mistake in a published work.  I heard that in molecular biology there is a widely-used service to upload your research data, like protein structure and sequencing, to get a timestamp. The uploaded materials can be set to be private for a period, and fully disclosed later. A lot of people do use this function. This is not exactly what I am looking for but shares some similar ideas. If I remember correctly, there is an interesting paper studying the scooping patterns with the aid of this database.  Solutions found!  and some other university-run pre-print services do have an embargo function. Moreover,  is a widely used repository, that, unlike arxiv, only displays the newest version to the general public, although still keeps a tag on the first submitted date! Why is my membership length only "Six months"? I was viewing the  and I noticed that I've apparently only had my account for six months, which is news to me... I opened my first site account here in February of 2014.  My account is much older than this, what's causing this bug?  To add some detail... On the per-site list, it does line up the &quot;member for X&quot; text with my actual membership time on that site. For example, when I look at the Meta Stack Exchange leaderboard, it shows me as a member for 2 years, 10 months. This is expected.  Update. With some work, today, I've managed to exceed my Interpersonal Skills hat count of with my Meta Stack Exchange hat count. This has changed my account age to match my MSE account age...  Still not reflective of my actual age... but better. Do "twice as high xxxxx as" , "twice as high" and "twice higher" mean the same thing? I have an elliptic curve $E$ over $\mathbb{F}_{11}$ defined by $y^2=x^3+4x$ with the point at infinity $\mathcal{O}$  I have a divisor of $E$, defined by $$D=\left[(0,0)\right]+\left[(2,4)\right]+\left[(4,5)\right]+\left[(6,3)\right]-4\left[\mathcal{O}\right]$$  I know that $\text{deg}(D)=1+1+1+1-4=0$ and it is clear that $\text{sum}(D)=\infty$  Therefore $D$ is the divisor of a function - we want to find this function  I am given \begin{align}\text{div}(y-2x)&amp;=\left[(0,0)\right]+2\left[(2,4)\right]-3\left[\mathcal{O}\right]\\ \text{div}(x-2)&amp;=\left[(2,4)\right]+\left[(2,-4)\right]-2\left[\mathcal{O}\right] \end{align}  I am then told that we can express $D$ as follows:  $$D = \left[(2,4)\right]+\text{div}\left(\frac{y-2x}{x-2}\right)+\left[(4,5)\right]+\left[(6,3)\right]-3\left[\mathcal{O}\right]$$  However, when I expand this back out, I get the following:  \begin{align}D &amp;= \left[(2,4)\right]+\text{div}\left(\frac{y-2x}{x-2}\right)+\left[(4,5)\right]+\left[(6,3)\right]-3\left[\mathcal{O}\right]\\ &amp;= \left[(2,4)\right]+\text{div}(y-2x)-\text{div}(x-2)+\left[\left(4,5\right)\right]+\left[(6,3)\right]-3\left[\mathcal{O}\right]\\ &amp;= \left[(2,4)\right]+\left(\left[(0,0)\right]+2\left[(2,4)\right]-3\left[\mathcal{O}\right]\right)-\left(\left[(2,4)\right]+\left[(2,-4)\right]-2\left[\mathcal{O}\right]\right)+\left[\left(4,5\right)\right]+\left[(6,3)\right]-3\left[\mathcal{O}\right]\\ &amp;= \left[(0,0)\right]+2\left[(2,4)\right]+\left[(4,5)\right]+\left[(6,3)\right]+2\left[\mathcal{O}\right]-\left[(2,-4)\right] \end{align}  which clearly isn't equivalent to the original $D$ unless $$\left[(2,4)\right]+2\left[\mathcal{O}\right]-\left[(2,-4)\right]=-4\left[\mathcal{O}\right]$$  Can anyone either explain what I've done wrong or show me how I can obtain the above equation please    This question comes in three parts:   Step 1 - This one Step 2 -  Step 3 - I am trying to connect to a Linode (running Ubuntu 12.04 LTS) from my local machine (also running Ubuntu 12.04 LTS)  I have created a private and public key on my local machine and copied my public key to my Linode's authorized_keys file.  However, whenever I try to ssh to my Linode I get the error message Permission denied (publickey).    It's not a problem with how ssh is set up on my Linode because I can ssh to it from my Windows machine using key authentication.    In my .ssh directory on my local Ubuntu machine, I have my id_rsa and id_rsa.pub files.  Do I need to create an authorized_keys file on my local machine?    EDIT:  This is what I get when I run ssh -vvv -i id_rsa [youruser]@[yourLinode]:  debug3: authmethod_lookup publickey debug3: remaining preferred: keyboard-interactive,password debug3: authmethod_is_enabled publickey debug1: Next authentication method: publickey debug1: Offering RSA public key: id_rsa debug3: send_pubkey_test debug2: we sent a publickey packet, wait for reply debug1: Authentications that can continue: publickey debug2: we did not send a packet, disable method debug1: No more authentication methods to try. Permission denied (publickey). How can I navigate my scene FPS style similar to how Unity does? I am aware of ShiftF fly, but it's extremely cumbersome having to use the scroll wheel to change the velocity of flight and moving the mouse from the center to spin.  Is there a way to move the viewport camera (not the scene camera) like Unity lets you right click and fly around with WASD and moving the mouse? 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. Styling an input type="file" button The  tag doesn't appear to be very useful.  It's used in a variety of ways (for reading, writing, showing, transforming, compiling, translating, obfuscating, obtaining, ... source code) on 1747 questions right now, and doesn't appear to be adding any classification value to the site.  Can we get rid of it? Installing QGIS on Ubuntu 14.04? Following the QGIS installation guide I have tried both UbuntuGIS and QGIS stable, but both fail with the error message:  The following packages have unmet dependencies:  python-qgis : Depends: python-qgis-common (= 2.6.1.1+trusty1) but it is not going to be installed             Depends: libqgispython2.6.1 but it is not going to be installed            Depends: libqgis-analysis2.6.1 but it is not going to be installed            Depends: libqgis-core2.6.1 but it is not going to be installed            Depends: libqgis-gui2.6.1 but it is not going to be installed            Depends: libqgis-networkanalysis2.6.1 but it is not going to be installed   qgis : Depends: libgdal1h (>= 1.8.0) but it is not going to be installed      Depends: libqgis-analysis2.6.1 but it is not going to be installed     Depends: libqgis-core2.6.1 but it is not going to be installed     Depends: libqgis-gui2.6.1 but it is not going to be installed     Depends: libqgis-networkanalysis2.6.1 but it is not going to be installed     Depends: qgis-providers (= 2.6.1.1+trusty1) but it is not going to be installed     Recommends: qgis-plugin-grass but it is not going to be installed     Recommends: qgis-plugin-globe but it is not going to be installed   E: Unable to correct problems, you have held broken packages.    I had QGISv.2.4 running on my machine with Ubuntu 12.04. After upgrading to Ubuntu 14.04 I was getting error message that pyqgis could not be loaded.  So I uninstalled QGIS using the Ubuntu Software Center and also run  sudo apt-get purge qgis   After that I tried to install the latest version of QGIS Debian Stable and UbuntuGIS unstable but getting the error messages described above.  The weird thing is that neither Ubuntu Software Center nor apt-get list have QGIS listed as installed, but if I run a search for qgis on my file system qgis and all its dependencies are still there. That's why I getting told to have broken packages. parsing a date string from FTPClient.getModificationTime() Styling an input type="file" button How do you style an input type="file" button? finding the index of a max value in R Show that the set of all finite subsets of $\mathbb{N}$ is countable.