My book is Connections, Curvature, and Characteristic Classes by Loring W. Tu (I'll call this Volume 3), a sequel to both Differential Forms in Algebraic Topology by Loring W. Tu and Raoul Bott (Volume 2) and An Introduction to Manifolds by Loring W. Tu (Volume 1).  I refer to .  Let $M$ be a set, possibly empty. Let $G$ be a group, possibly a singleton. Let $G$ act right on $M$ by the action $\mu: M \times G \to M$. For each $x \in M$, let $\text{Stab}(x):=\{g \in G | \mu(x,g) = x\}$ denote a stabilizer subgroup of $G$. Let $1_G$ be the identity of $G$.  I understand definitions of $\mu$ to be free as follows:   : $\mu$ is free if for all $g \in G$, if there exists $x \in M$ such that $\mu(x,g)=x$, then we have that $g=1_G$.  in : (same as 's, given above) : $\mu$ is free if for all $x \in M$, $\text{Stab}(x) = \{1_G\}$   Question 1: For Wikipedia's and jgon's definitions, there is no explicit reference to stabilizers. For Tu's definition, how do I understand $\text{Stab}(x)$ for $M$ empty and $G$ not a singleton?  Question 2: Similarly, for the definition of faithful as     $$\bigcap_{x \in M} \text{Stab}(x) = \{1_G\} \tag{2a}$$   How do I understand $\mu$ as never faithful for $M$ empty and $G$ not a singleton?  My attempt to understand:   For Question 2, I think I can apply , by $M$ empty assumption to say $\bigcap_{x \in M} \text{Stab}(x) = G$. Then I apply $G$ not a singleton assumption to get $\bigcap_{x \in M} \text{Stab}(x) \ne \{1_G\}$. For Question 1, I think we somehow say $\text{Stab}(x) = G$ for all $x \in M = \emptyset$ by some vacuousness argument. I'm not really sure.
Let $G$ be a group, possibly a singleton. Let $M$ be a set, possibly empty. Let $\mu: M \times G \to M$ be a right group action. Let $1_G$ be the identity of $G$.  I understand 's second definition of faithful as follows:     $\mu$ is faithful if for each $g \in G$ such that $g \ne 1_G$, there exists $x \in M$ such that $\mu(x,g) \ne x$. $\tag{1}$   Later  says if $\mu$ is free and $M$ is a non-empty set, then $\mu$ is faithful.  Question: Given the idea that we can have $M$ as an empty set, I would like to clarify: Are these correct?   If $G$ is a singleton, then every action $\mu$ is faithful, whether or not $M$ is empty. If $G$ is not a singleton, but $M$ is empty, then every action $\mu$ is not faithful.
Please help me  to evaluate combinatorially the following sum: $$\sum_{k=0}^n \binom{n}{k}$$   Thank you. Enabling CORS in GeoServer (jetty)? I'm trying to prove : Given a simple graph $G$ with $n$ vertices, where $n$ is even, prove that if every vertex has degree $\dfrac n2 + 1$, then $G$ must contain a (simple) $3$-cycle. A (simple) $3$-cycle is a set of $3$ (distinct) vertices, $a, b, c$ such that $ab$ is an edge, $bc$ is an edge and $ac$ is an edge.  I know there are similar ones already on here, but I can't comprehend any of the answers. This is what I have so far:  Base Case: Since the graph has to have at least $3$ vertices to have a simple $3$-cycle, the lowest possible even number of vertices is $n = 4$.   $\dfrac42 + 1 = 3$ degree (edges attached to node). This means each vertex connects to each of the other $3$ vertices since $G$ is a simple graph. Thus, any combination of $3$ vertices will be a simple $3$-cycle.  Inductive Hypothesis: Assume a simple graph with $k$ vertices, where $k$ is even, contains a simple $3$-cycle if each vertex has degree $\dfrac k2 + 1$.  Inductive Step: Since only concerned about when the graph has an even number of vertices, going to show a simple graph with $k + 2$ vertices, where $k$ is even, contains a simple $3$-cycle if every vertex has degree $\dfrac{k+2}{2} + 1$   This is as far as I get and don't know how to finish the Inductive Step (Also please feel free to let me know if I messed anything up, up to this point) What is the purpose of the word 'self'? Entire function bounded by a polynomial is a polynomial Suppose that an entire function $f(z)$ satisfies $\left|f(z)\right|\leq k\left|z\right|^n$ for sufficiently large $\left|z\right|$, where $n\in\mathbb{Z^+}$ and $k&gt;0$ is constant. Show that $f$ is a polynomial of degree at most $n$. Unexpected behavior of rule matching a pattern Now that tags have this erm...  which appears when you move your mouse over them (try it! ), one question remains: What in the billion blue blazes does the 'subscribe' button do?   It's not RSS, since a traditional RSS feed link appears beside it, and clicking on it seems to increment the number of subscriber by one, but other than that scouring over the Stack Overflow UI does not appear to reveal anything. Surely this feature would have more use than serving as a popularity contest for the tags? How do I prove that then $f(x) = 1$ for some $x ∈ [0, 1]$? We are asked the following:      let $f(x)$ be a continuous function on $[0, 1]$ such that   $$\int _ {0} ^ {1} \ f(y) dy = 1$$   prove that $f(x) = 1$ for some $x ∈ [0, 1]$   The question asks to take into account Rolle's Theorem and Mean Value Theorem.  I'm aware of the definitions, but I am really unsure how to answer this question and my tutors are not really responding till after easter.    Could someone please help? Why is there a "y" involved?  Thanks you UK coastline shapefile? Book for proof writing Deployed web parts don't show up in Web Part Gallery I am trying to balance underbraces in the equation below, how can I ensure that both underbraces at the same depth?  \begin{equation} y = \underbrace{ \left [ x^2 + r^2 (a^2+\frac{a^{0.3}}{b}) \right]}_\text{growth rate} \times  \underbrace{ C_0}_\text{initial} \end{equation} Showing that ${\rm E}[X]=\sum_{k=0}^\infty P(X>k)$ for a discrete random variable Let $X$ be a discrete random variable whose range is $0,1,2,3,\ldots$. Prove that $$ {\rm E}[X]=\sum_{k=0}^\infty P(X&gt;k). $$  How to prove this? I tried a bit but unable to post due to formatting issue. Can anyone help? I have a 24/7 always-on Debian Jessie based headless home server that has a large 1TB SSD for the OS and all of my frequently accessed files. This same system has 4 larger hard disk drives in a SnapRAID array. These are mainly for archiving infrequently accessed Blu-rays and want those drives to remain spun down in standby unless I actually read or write to them. They are all formatted as ext4 and mounted with noatime and nodiratime enabled.  So even though no process or program should be regularly accessing those drives in any direct way, the hard drives constantly get spun up from standby. It seems to be related to graphical programs that provide a gui file browser, even something like Chromium. If I don't even browse into those drives, I'm thinking that these processes by simply getting a list of available drives spins up the hard disks. Much like blkid does. The problem is, it's hard to determine the root cause of this since none of these processes are actually reading or writing the filesystem on those drives, so no files are actually changing or being touched. Is there some sort of cache that I can populate or a buffer to prevent these programs from spinning up the hard drive simply by getting a list of available disks? This is honestly driving me insane, since I can't find a reliably way to keep these disks spun-down even though there is no direct access of the filesystem.  UPDATE: Thanks to Stephen's answer, I was able to trace the disk activity to gvfs and udisks. It's a real shame that these processes insist on waking up disks in standby when they aren't actually being accessed to do any real I/O with the filesystem. So far I just uninstalled them, knowing that it will remove some functionality from PCManFM and the like. bordermatrix with blocks I need to write a complicated matrix, that has a structure as in the attached picture. For about 2 days I'm trying to configure Ubuntu so that I can send an email, but when I type in:  echo "Subject: sendmail test" | sendmail -v my@email.com   I'll get:  postdrop: warning: unable to look up public/pickup: No such file or directory Mail Delivery Status Report will be mailed to &lt;andreigbr&gt;.   Even if I change my@email with mine, the inbox will be empty. Can somebody help me, please? I watched about 5 tutorial about how to send an email fom the command line but still doesn't work. Particularly, does proc always list the VmRSS value in kB? I can't find a solid answer on the , although it appears that it sticks to one unit. MySQL convert date string to Unix timestamp It is generally taught that  function as adjectives modifying a noun (or pronoun).  In a sentence such as:     A major accident occurred on our bus's route to school, making us late for class.   or     The gas tank exploded, sending shrapnel through the air.   are the participle phrases modifying their preceding clauses? (If not, what are they modifying?) In doing instanton problems or when connecting quantum field theory to statistical mechanics, I often see people trying the Wick rotation trick by defining an imaginary time $\tau\equiv it$. So, in this case, we should make sure that $\tau$ is imaginary, right? However, I often see places where people seem to handle $\tau$ as a real number, for example setting $\tau$ to $\pm\infty$. Moreover, I remember in one of my quantum mechanics classes, the professor said the Euclidean time $\tau$ was real. I am so confused and need some good explanations.