Good morning everyone,  I'm looking for a subgroup of $Z(G)$ the center of the group $G$ that is not characteristic subgroup of $G$.  Did someone have that in his pocket ?  I wish you a very good day !
In the book I'm study is written:     A normal subgroup of a group need not be characteristic.   And as an exercise I'm supposed to find an example, it also said that is pretty hard to find one. After trying for two days I wasn't able to find one example.   So, I'm asking for an example of a group $G$ with a normal subgroup $H$ that is not a characteristic of $G$.  I will add some context because maybe it will clarify why the book said it is difficult to find an example: the main problem with the exercise is that until it the book had only covered Group Definition, Subgroups, Lagrange's Theorem and Homomorphisms. So I'm supposed to find a example with such lack of advanced tools.
How do I create an 8 sided die from one object? Simply put, is there a way to make a cube balance solely on one point/vertex?  I'm struggling. I do not want to have to join two objects, nor have to extrude an object. I just want to have a simple eight-sided die created with one mesh. I have enabled platform encryption at my DE and selected Case Subject, Description and Comments to be encrypted.  I have a created a sample case with these values:    And when I query from Developer Console it returns as follows:       How do I know, subject and description are encrypted at Rest? Is there any indicator which can confirm that this data is encrypted?   By the way, in case of File/attachment there is isEncrypted attribute is available which denotes the same as follows. Convex function as supremum of countably many affine functions Let $f:\mathbb{R}^n\to \mathbb{R}$ be a convex function. It is well-known that (see e.g.~) that there exists a set $A$ such that  $$ f(x)=\sup_{a\in A}A(a)x+b(a), \quad \forall x\in \mathbb{R}^n, $$ where $A(a)\in\mathbb{R}^{n\times n}$ and $b(a)\in \mathbb{R}^n$.  I was wondering, is there any reference saying we can choose $A$ as a countable set? Is $f(a)\!=\!0\!=\!f(b)\Rightarrow (x\!-\!a)(x\!-\!b)\mid f(x)\,$ true if $\,a=b?\ $ [Double Factor Theorem] What do the supercharges in extended supersymmetry do? What do the supercharges in  do?  In ${\cal N}=1$ supersymmetry there are a certain number of fermions and and equal number of bosons. You can transform all fermions to the bosons (and vice versa) in a 1 to 1 fashion using a single supercharge, $Q$.  So what happens when you have, for example, ${\cal N}=2$ supersymmetry with 8 supercharges? Since $Q$ is a generator of supersymmetry transformations, is it a linear combination of these supercharges that act on the particles? In which case could one particle be acted on by two separate linear combinations of $Q$? Or is it strictly one linear combination of $Q$ per fermion/boson?  Also, what does ${\cal N}$ mean physically? What difference does ${\cal N}=2$ have to ${\cal N}=1$ other than more supercharges? Or is that the only difference between the two theories? Creating Inky-looking characters in Blender I'm involved in a project to do with ink animation and experimentation. I found this incredible piece of animation:    I originally thought this was some sort of After Effects compositing (is it even possible in AE?!), but found out that this particular animation was created in 3DS Max with Krakatoa and FumeFX. I'm just wonder if a similar effect is doable with Blender's smoke simulator.  There are various YouTube videos that have pretty decent results of ink renders from Blender, but I can't seem to find anything of a similar sort of movement animation. Just blobs floating around! The closest thing I found was this animation of a         but again I can't seem to replicate this. The smoke just doesn't seem to follow my character's animation. Any help with this would be greatly appreciated. Thanks. copy files over ssh HTTP 401 twice before HTTP 200 when serving files What are the limits on how I can cast, change, and retract votes? Does an adjective or longer modifier apply to both nouns when they are joined with 'and' [or 'or']? How can I limit the headphone max volume on OS X? So I hurt my ears from excessively loud music last week at work. I'm well aware of the volume down key of course, but I tend to crank it up too much throughout the day without noticing it.    I need a limiter and it needs to be system-wide, since I listen to various other sound sources aside from iTunes. This isn't really an issue on my phone, since I can set a volume limit in the iOS settings.  Is there an equivalent built-in or third-party fix for OS X Lion? \dots versus \ldots - is there a difference? Is there a difference between \dots and \ldots, the ellipsis?  If yes: Which? When to use \dots and when to use \ldots?  If no: Why are there both at all? How to test randomness (case in point - Shuffling) First off, this question is ripped out from  question. I did it because I think this part is bigger than a sub-part of a longer question. If it offends, please pardon me.  Assume that you have a algorithm that generates randomness. Now how do you test it? Or to be more direct - Assume you have an algorithm that shuffles a deck of cards, how do you test that it's a perfectly random algorithm?  To add some theory to the problem - A deck of cards can be shuffled in 52! (52 factorial) different ways. Take a deck of cards, shuffle it by hand and write down the order of all cards. What is the probability that you would have gotten exactly that shuffle? Answer: 1 / 52!.  What is the chance that you, after shuffling, will get A, K, Q, J ... of each suit in a sequence? Answer 1 / 52!  So, just shuffling once and looking at the result will give you absolutely no information about your shuffling algorithms randomness. Twice and you have more information, Three even more...  How would you black box test a shuffling algorithm for randomness? Short story where time is accelerated and characters have to get back to ship? Looking for a short story where time is accelerated on a planet.  The characters are actually born at the beginning of the story and then grow up quickly and inadvertently learn that their forebears came from a rocket ship nearby.  After having aged to "teenagers" they try to reach the ship?  At some point there is a war between confused aging survivors.  I think it was in a compilation which included a story called "Cousins". Best WPF open source projects I have programmed with WPF for more than a year. When I started there weren't so many examples and guidelines to follow, now the situation is much more better.  I would like to ask you some help in listing the best WPF (open) source code examples or projects that you suggest to study in order to learn more about project structure, best practices, etc.  Thank you, Marco If I set multiple Alarms, can I tell them apart? I recall Asimov had a character who was The autoc of , clearly based on the word “autokrat” but I don’t recall the exact spelling.  This stumped Google, which would rather correct the spelling to some other word than find a unique word.  In the story, he had gone undercover and the reader did not know either that some other character was in fact the Autoc, until it had been revealed to the main character. I've been trying to wrap my head around the basic concepts of category theory, and I thought I would attempt to illustrate what I understand with the category of sets, probably the easiest example. Particularly, I've been trying to prove that $id_A$ (the identity morphism on $A$, for all $A \in Obj(\mathbf{Set})$) is $1_A \colon A \rightarrow A, x \mapsto x$.  This is a very intuitive and reasonable statement, and it's trivial to prove that $1_A$ is indeed an identity morphism on $A$, and I suppose uniqueness of $id_A$ can be demonstrated analogously to uniqueness of the identity element in a monoid (considering the subcategory which has $A$ as its only object, and endofunctions on $A$ as its only morphisms).  In this manner, it is not hard to prove that the proposition in the title is true, but this demonstration requires to make an assumption or guess as to what could $id_A$ be. Specifically, the scheme of the proof is: assume $id_A$ = $1_A$, see that it works with the definition of an identity morphism, show that the identity morphism is unique, and, in conclusion, $id_A$ can only be $1_A$. What I'm looking for, nonetheless, is a somehow more direct proof, that doesn't assume $id_A$ = $1_A$ at the start. I want to place myself in a state of little or no knowledge about sets and functions, and under this assumption, why would I assume $id_A$ = $1_A$ at first? Why not try with $id_\mathbb{Z}$ = $f \colon \mathbb{Z} \to \mathbb{Z}, x \mapsto x^2 + 1$, for example? It wouldn't work, but I don't have any reason to think that $1_\mathbb{Z}$ is a better guess for $id_\mathbb{Z}$.  I suppose that the proof for which I'm asking would work for categories of sets with additional structure, and probably for posets as well, although I'm not clear as to what modifications it would require to work.  Thanks. Login not redirecting back to page where login link was clicked if account created Standalone python app that uses qgis.core without installing qgis