How to install Sentellic fspc package?
How do I resolve unmet dependencies after adding a PPA?
How can I integrate $\ln \left( x+\sqrt{1+x^2} \right) $? Order of a homomorphism of groups Let $f: G \to H$ be a homomorphism of groups. Assume that $a\in G$ and $\operatorname{ord}(a)=n$. Prove that the order of $f(a)$ is a divisor of $n$. I know that if $H$ is a subgroup of $G$, then $$|G|=|H|\cdot|\text{distinct cosets of $H$}|$$ so $|H|$ must be divisor of $|G|$, but I don't understand how to apply that logic to homomorphisms (or if that will even work). Any help is appreciated! How precise must the energies match for absorption of photons? According to Quantum Mechanics, in order for an atom to absorb a photon the energy of the photon must be precisely that of a "jump" between energy states of the atom.  How precise must it be?   If I create a photon with an energy within an error of 0.0001% of that of an energy state, will it be absorbed by my atom? Epsilon delta proof for $\lim_\limits{x\to a} \sqrt{x} = \sqrt{a}$ where $a>0$ $f(x) = \sqrt{x}$. For right limit, $x-a &gt; 0$. Let $x-a &lt; \delta$ then $(\sqrt{x}-\sqrt{a}) \lt \frac{\delta}{\sqrt{x}+\sqrt{a}}$. Then chose we must choose $\epsilon $ so that $\epsilon &gt; \frac{\delta}{\sqrt{x}+\sqrt{a}}$. Is this correct? How would I conclude the proof?  Edit - Follow up  From answers, I try a proof for the same limit but this time Left side:  For left side limit, $x-a &lt; 0$ so $a-x &gt; 0$. Let $a-x &lt; \delta$ then $\sqrt{a}-\sqrt{x} \lt \frac{\delta}{\sqrt{a}+\sqrt{x}} \lt \frac{\delta}{\sqrt{a}}$.   So let me begin the proof by saying to choose $\delta = \epsilon \sqrt a$ then $a-x \lt \delta $ so $(\sqrt a - \sqrt x )(\sqrt a + \sqrt x) \lt \epsilon \sqrt a$ so $\sqrt a - \sqrt x \lt \epsilon$ I am executing my ./shell.sh which contains   eval "$(pyenv init -)" pyenv shell 3.8.2   But nothing happens. If i run this code in bash manually it works fine and shell version of python switches from 2.7.10 to 3.8.2.  I am in the right directory and I gave the .sh file execution permissions. Why my differential equations become True? What is the result of measuring $\sigma_x$ on the state $|01\rangle+|10\rangle$? I confused about how to calculate the probabilities and getting a certain result of measuring Bell's states with Pauli matrices as the operator. When you measure something, the state involved would be projected onto an eigenstate of the observable. given $|\psi\rangle = \frac{1}{\sqrt2} (|01⟩ + |10⟩)$ as the state and $\sigma_x = \left[\begin{matrix}0&amp;1\\1&amp;0\\\end{matrix}\right]$ as the observable. if the probability is 1/2, how to calculate them actually? What is the state after measurement? Is there ever a time to create a user for a company? I'm wondering if there's ever a reason or time when a company should have its own user (like say, Stack Overflow), instead of users that make it up (like, Jeff Atwood).   I figure this is kind of a bad idea as someone with access to multiple accounts could upvote himself over and over.... but is there any point this is okay and/or a good idea? Statistics module doesn't show the "Enable access log" checkbox I dont know what to do to show "Enable access log" option.  I activated logging all messages and checked all statistics permissions.  Thanks. SemanticImport vs Import differ on same CSV file? During my attempt to upgrade my Xubuntu 14.04 installation with  sudo apt-get dist-upgrade   I made the mistake of interrupting the process by closing my laptop, which caused it to suspend to disk. After re-opening, the monitor remained black, so I was forced to reboot the system. I have since discovered many failures of drivers (wlan0 missing, USB memory stick cannot be mounted, USB mouse not recognised, and maybe more undiscovered problems with hardware).   My beginner's attempts to manually repair the single problems could have made it even worse. During the installation of the current distribution I chose to encrypt my whole hard disk. As I show below, typical alien depictions include an elongated, pale face, two dark elliptical eyes, and a small mouth.  Who first used this design for an alien?  Where does this design come from? Passive aggressive people will sometimes veil insulting, critical, derogatory or generally aggressive comments with humor. The patina of humor makes the comment seem like a joke, not to be taken seriously, all in good fun, and it safeguards the aggressive person. If someone gets offended, they are being too serious, can't take a joke, and so they cannot voice their upset without seeming to be the one escalating the situation, when in fact it was the aggressor who did that.   This behavior seems to lie on a spectrum. On one side are comments which are simultaneously aggressive and genuinely funny (though of course both, especially the latter, are fairly subjective). On the other side would be lame attempts at the behavior, something like "you are fat!... just joking!" in which nothing in the tone or delivery signals a register of humor. In the middle of the spectrum, I imagine, are aggressive statements genuinely delivered as jokes. Their register signals that they are jokes, but they are aggressive rather than funny. They are aggressive comments masquerading as jokes.      A friend once called this "joking on the square", but I have never heard or seen this phrase. This type of comment is both pervasive and psychologically subtle and in my experience there is often an exact word describing such things. If not a word, is there an astute phrase which cuts to the heart of this? Changing order of features in shapefile I have a shapefile and I want to change the order of the features. Is that possible? I want to sort the features in the shapefile because I'm embedding this shapefile in a mobile app. The app will display a list of features, and although I could do the sorting at load time, I'd prefer to have them sorted. I was surprised to only recently notice that      An object of any density can be large enough to fall within its own   Schwarzschild radius.   Of course! It turns out that supermassive black holes at galactic centers can have an average density of less than water's.  Somehow I always operated under the assumption that black holes of any size had to be superdense objects by everyday standards.  Compare the Earth to collapsing into a mere 9mm marble retaining the same mass, in order for the escape velocity at the surface to finally reach that of light.  Or Mt. Everest packed into one nanometer.  Reading on about this , it increases proportionally with total mass.  Assuming matter is accumulated at a steady density into a spherical volume, the volume's radius will only "grow" at a cube root of the total volume and be quickly outpaced by its own gravitational radius.  Question: For an object the , what would have to be its diameter for it to qualify as a black hole (from an external point of view)?  Would this not imply by definition that:   The Earth, Solar system and Milky Way are conceivably inside this black hole? Black holes can be nested/be contained within larger ones? Whether something is a black hole or not is actually a matter of perspective/where the observer is, inside or outside? "To your left" versus "on your left" [See  for more on this topic, as well as better illustrations of the final product.]  I have a document with a lot of short excerpts from another book scattered through it.  Putting frames around them looks a little too 'clean'. How can I get a 'torn page' type effect, to make it look as if the excerpt has been physically torn out of a book?  (I've occasionally seen this done in printed books, although I can't find an example right now.)  I'm not wedded to a very specific look here --  gives plenty of examples. But just to give this some frame of reference, here are some of the nicer ones:          (The strong colouration in this second example is a probably a little too distracting for actual use, although it might be interesting as an option. It's more the edge effect I'm interested in.)    Edit (again): I tried typesetting my text according to the updated versions of the answers below. Here are the results:    Or with interpolated shading: I have an unused copy of Mass Effect 2 Collector's Edition bought via Amazon.com in the US, including a special code for Inferno Armour (also unused).  Will it work fine if I install and start playing the game in the UK? Does geographical location matter?  Thanks! Can I smoothly upgrade from one LTS to next LTS release? When there are several cottages each with several tariffs (depending on the season), should the plural be "cottage tariffs" or "cottages tariffs"?  For context, this is to be a section title on a website renting out the cottages.  An explanation of the grammatical rules would be helpful.