Abelian group (Commutative group)
Proof of property of abelian group
What is the lowest level of mathematics required in order to understand how encryption algorithms work? How to make files protected? How to make some file that contains crucial data protected? For instance, suppose I have some files that are important, is there any method or strategy to make these kind of file more protected on Ubuntu? I even want to make some files ask for a password whenever someone tries to open it. Does it affect the version of Ubuntu in terms of security policy? May I use any encryption algorithm externally? Regards. Homology of $S^1 \times (S^1 \vee S^1)$ I'm trying to solve question 2.2.9(b) in Hatcher's Algebraic Topology.   Question: Calculate the homology groups of $X=S^1 \times (S^1 \vee S^1)$.  My attempt: I try to use the Mayer-Vietoris sequence.   Let $A=S^1 \times (S^1 \vee \text{small bit}) =S^1\times S^1$ and $B=S^1 \times ( \text{small bit} \vee S^1)=S^1 \times S^1$. (Unsure how to express this, but hopefully this is clear enough.)   Then $A\cap B=S^1\times \{\text{point}\}=S^1$ and $A\cup B=X$.  The Mayer-Vietoris sequence then gives us an exact sequence in reduced homology as follows:  $$0 \to \mathbb{Z}^2 \to \tilde{H}_2(X) \to \mathbb{Z} \to \mathbb{Z}^2 \to \tilde{H}_1(X) \to 0.$$  Is it possible from this exact sequence to determine the homology groups, in some algebraic fashion? Or do I need to determine what the maps are? How do I determine the maps? Is my ESTA valid for a second trip? Find the limit if it exists of $S_{n+1} = \frac{1}{2}(S_n +\frac{A}{S_n})$ Say that Player X is 2nd in the initiative. 1st is the Black Court vampire that is in a rage and taking a swing at Player X. Player X gets a built in defense [Athletics/Fists + 4dF] but can Player X instead use his action this turn to cast his Rote Evocation Shield?  It is not his turn in the initiative order. A Block is an Action that a Player can take on his turn.   Thematically though, it seems to make sense that the char could cast a Rote Evocation Shield in response to an attack ahead of him in the initiative order. At least thematically in the books. Also, the example in YS252, shows him reacting to an attack by Agatha Hagglethorn by bring up his shield.  I am hoping there is some as yet undiscovered dark corner where this concept is covered or squashed. Why is there one P in "hoping" and two P's in "hopping"? Hoping is the present participle of hope. hopping is the present participle of hop. Hoping has only one p while hopping has two. What difference does it make? Why is hoping not hopping?  Edit: I changed mop to hop because of a comment by Kat.  Kat: Why would you compare hope to mop instead of hope to hop (in which case I think your answer will become obvious)? How do I reset a sequence in Oracle? In , I can do something like this:  ALTER SEQUENCE serial RESTART WITH 0;   Is there an Oracle equivalent? What is the easiest way to handle huge numbers in C? I need to store values in the Area 1000^900, or in more human readable form 10^2700. Does anybody know of an easy way to do that? Any help would really be appreciated! How can I keep recent files from appearing in Unity? I sometimes browse erotic media files but I would like to avoid having them appear in files &amp; folders / recent. How can I do that?  The old "making .recently-used.xbel a folder" trick from 10.10 and older doesn't work any more.   Plus I'd prefer to turn off tracking only temporarily.  I also prefer not to use a separate user for this, as switching users sometimes isn't fast enough. Is it possible to construct a multi-colored double-bladed lightsaber? I was just watching  video of some SW dueling at the 2015 World Fencing Championship, and wondered if it's actually possible in the SW universe to construct a multi-colored double-bladed lightsaber?   Are there any accounts of such a thing? If not, would it at least be theoretically possible to make one, or, are there obvious limitations? Superluminal neutrinos I was quite surprised to read this all over the news today:     Elusive, nearly massive subatomic particles called neutrinos appear to travel just faster than light, a team of physicists in Europe reports. If so, the observation would wreck Einstein's theory of special relativity, which demands that nothing can travel faster than light.   &mdash;  Apparently a CERN/Gran Sasso team measured a faster-than-light speed for neutrinos.   Is this even remotely possible? If so, would it be a real violation of Lorentz invariance or an "" effect?   The ; a webcast is/was planned .  News conference video If $G$ is isomorphic to $H$, then ${\rm Aut}(G)$ is isomorphic to ${\rm Aut}(H)$ If a group $G$ is isomorphic to $H$, prove that ${\rm Aut}(G)$ is isomorphic to ${\rm Aut}(H)$.   Can someone provide a step by step solution? Explaining along the way our strategy in proving this. Migrate database from Postgres to MySQL In the simple linear regression model, $\hat{\beta}$ is the sum of independent normally distributed random variables.  Is it false because in linear regression there is $\beta$ and not $\hat{\beta}$? How can I embed a view inside a content of a node in Drupal 8?   I found something similar, for example in the question about "" or the  module, but it is all for Drupal 7.  I tried with an Entity reference field pointing to a View but it display only a label and from the Manage Display tab I can choose only from Label and Entity ID. I need something like the "Rendered entity" option that is present on Entity reference fields pointing to other nodes. I have 2 partitions (Windows and Ubuntu 13.04) and I am asked which one to boot each time I turn on my computer.  I want to install Ubuntu 13.10 to the entire disk.  I do not mind if all previous Ubuntu/Win data are lost.  How can I do that? How to toggle background image on button click? This is a  about understanding and remediating the Heartbleed security issue.   What exactly is CVE-2014-0160 AKA "Heartbleed"? What is the cause, what OSs and versions of OpenSSL are vulnerable, what are the symptoms, are there any methods to detect a successful exploit?   How can I check to see if my system is affected? How can this vulnerability be mitigated? Should I be concerned that my keys or other private data have been compromised? What other side effects should I be concerned about? How to add a vertical line to a plot