Defining a homomorphism from S_n to Z_2 and showing that A_n is its kernel.
Show that $A_n$ is the kernel of a group homomorphism of $S_n \rightarrow \{−1,1\}$.
Casting a Somatic, non-Material spell while holding a focus In  a side issue came up of whether you can cast a spell with Somatic components but no Material components while holding a focus in the hand used for Somatic components.  My reading of the rules sees "A spellcaster must have a hand free to access a spell's material components--or to hold a spellcasting focus--but it can be the same hand that he or she uses to perform somatic components." - it doesn't say you have to be holding the focus for the purpose of avoiding materials.  However, @Adeptus and @enkryptor argue that - since that rule is in the section on Material components - you don't consult it in for spells without Material, and fall back to the Somatic rule "the caster must have free use of at least one hand".  (I also appeal to the logic of fluff - why should it be harder to cast a spell with fewer components?)  Is there an official RAW or Word Of God on this matter? Can you cast a material-less, somatic spell while holding a focus in your only free hand? What's the best way to parse an XML dateTime in Java? Primary decomposition of a monomial ideal Can anyone give me an idea about the primary decomposition of the ideal $I=(x^3y,xy^4)$ of the ring $R=k[x,y]$?    I am trying to connect the primary decomposition with the set Ass(R/I) which i have find before.. I've got a bunch of files named with a the pattern 99 - DescriptiveName.txt and I'd like to remove the number from the front so I just have DescriptiveName.txt.  How can I do this? Can I do it from the command line or is there a utility that can do this? How to remove session entries from LightDM? Remove Background in Inkscape I'm trying to find an alternative to the standard tombstone qed symbol for more informal papers. I initially thought of an empty square with Pub written inside, but what would actually be better would be a small beer mug symbol. (much like the one found in this set of icons )  Unfortunately there doesn't seem to such a symbol in the comprehensive latex symbol list (although many funny symbols can be found there). Has anyone thought about this? Is there any quick-and-dirty solution? If $p,q$ are primes $(pq,a)=1$ and $p-1|q-1$ then prove $a^{q-1}\equiv 1\pmod{pq}$ $p,q$  primes, $(pq,a)=1$ and $p-1|q-1$ then prove $a^{q-1}\equiv 1\pmod{pq}$   Got this in an exam today and had no idea how to do it. What Is the Real Name of the #? My wife is an Indian national with an Indian passport. She holds an indefinite leave to remain in the UK (permanent residence card). Does she need to apply for a Schengen visa before travelling to France or any other Schengen countries? Is this metric space incomplete? Array versus List: When to use which? MyClass[] array; List&lt;MyClass&gt; list;   What are the scenarios when one is preferable over the other? And why? LWC: Difference between "window" and "this" I am trying to understand what exactly is the difference between window and this in the context of a LWC component.  According to this  the Locker intercepts calls to the Window and uses Secured Window instead which acts as:     Secure wrapper for the window object, which represents a window containing a DOM document.   As for this what I  was:     To add an event listener to an element that a template doesn’t own,   call addEventListener directly: this.addEventListener()   What does salesforce mean by an element that a template doesn't own? Isn't window an element that the template doesn't own? Value of $\frac{x^n -a^n}{ x-a}$? Does the potential energy of a single object exist? I know that potential energy exists for a system of objects. But why? Is there no potential energy for a single object system? I assume that there is.  Consider the earth-object system. Suppose an external force lifts the object above the earth. Then the work done by it gets stored in the earth-object system as potential energy.Right?  Now, my question is that is the potential energy always zero for a single object system.   MY REASONING  If we consider the object only as the system and lift it above the Earth, we are doing positive work on it and filling it with energy and the gravitational field of the Earth is doing negative work on it and draining it of energy. Thus whenever the object is moving it has kinetic energy which is fine. But when it is not moving,i.e. when its weight is equal to the force applied by us, the net work done is zero and so the object should not have any kinetic energy. Does it have potential energy? A generic list of anonymous class I think that all the zeros of the Riemann-Zeta function ${\zeta}( z ) = \frac{1}{1-2^{1-z}} \sum_{n = 0}^{\infty} \frac{1}{2^{n+1}} \sum_{k = 0}^{n} (-1)^k \binom{n}{k} (k+1)^{-z}$  have real part smaller than 1, but I don't know how to prove it. If Re(z) > 1 than  ${\zeta}( z ) = \sum_{n = 1}^{\infty} \left( \frac{1}{n} \right)^z$ Troubleshooting "Windows cannot be installed to this disk. The selected disk has an MBR partition table" Where were the cannons in final fight in Wakanda in Infinity War? In Black Panther, Ulysses Klaue describes his arm cannon as a leaf blower in comparison to the cannons/weapons that Wakanda has.  Infinity War comes after the events of Black Panther, so the weapons should exist. During the final fight with the children of Thanos and Space Dogs (as Rocket refers to them), Wakandans could have used these powerful cannons, pointing them at the opened section of barrier. Instead they chose a fist fight.  Even M'Baku says that it would be the end of Wakanda, so the magnitude is huge and risks are very high. Why were these powerful weapons not used?    Edit 1: I watched the movie again and planes are there in the fight. But they were only used to destroy the spaceships that crashed. Does Overhand Chop override the damage bonus from wielding a two handed weapon? I was showing a GM friend of mine a build I was making that uses the Two-Handed Fighter Archetype and we are both unsure of how it interacts with the damage bonus you get from two-handed weapons (which is 1-1/2 times your STR.) It reads:     Overhand Chop (Ex): At 3rd level, when a two-handed fighter makes a single attack (with the attack action or a charge) with a two-handed weapon, he adds double his Strength bonus on damage rolls. This ability replaces armor training 1.   So, does Overhand Chop turn the 1-1/2 times multiplier from two-handed weapons into a 2 times multiplier, or do the two add up? Or, do they interact in some other way?   This assumes a Medium sized character with 20 STR and a greatsword with 3 levels.