How many unique n-tuples add to integer m?
How to use stars and bars for solving number of integral solutions to an equality?
How to read a csv file and use the values as x and y points in blender? Probability of sampling with and without replacement In sampling without replacement the probability of any fixed element in the population to be included in a random sample of size $r$ is $\frac{r}{n}$. In sampling with replacement the corresponding probability is $\left[1- \left(\frac{1}{1-n}\right)^r\right]$.  Please help me show how this is proved. How does one find which files are taking up 80% of the space on a Linux webserver? The drive is constantly filling up. You've hunted down all the spare and random files you can. grep'd for coredump files and even removed some of the un-needed backups...  What would be your next move.  The actual server in question has 10GB of website files and the OS shouldn't take any more than 10GB so how do you track down what's filling a 50GB (virtual) drive? Should a math paper use non-alphabetical ordering of authors' names for unequal contributions? Using alphabetical ordering for authors' names is a long established tradition in mathematics. In the situations of extremely unequal contributions, is it reasonable to break from this tradition?  Has there been any examples?  If it is common, what is the threshold for breaking from alphabetical ordering? Difference between declaring variables before or in loop? getting error when I compile the Java code using package in commandline? How many triples $(x,y,n)$ are there such that $x^n - y^n = 2^{100}$ How do I reinstall Unity? I tested out Unity 5, and of course, it broke things, so I tried downgrading by    sudo apt-get install ppa-purge sudo ppa-purge ppa:unity-team/staging   It removed Unity completely.    apt-get install unity   seemed to work, but the login screen now offers only Gnome.  How do I get back to Unity 4? Is the "live" in "He only lived a few days after the accident" intrasitive or transitive? Showing that the set of fixed points is a smooth manifold Let $M$ be a smooth manifold and Let $G$ be a finite group acting on $M$ by diffeomorphisms. Show that the set of fixed point $F=\{m \in M: g.m=m \}$ is a smooth manifold. I am unable to deal with the set of fixed points. It doesn't seem to be an open set. It would have been trivial if it was open. Any hints? Nested Mathematical Functions This question has bugged me for a while. I want to write a nested function, f(g(x)). If I naively enter it:  \begin{align}     f(g(x))=y \end{align}     The problem is that each and every parenthesis is the same size as all of the others. Shouldn't the outer parentheses be larger? But if I do this:  \begin{align}     f\left(g(x)\right)=y \end{align}     Now it looks even less like the functions are nested. What's the proper way to do this? Here's a module with castellated pads attached to an eval board:  (source: ) Looking closely at the wires interfacing with the castellated pads, they're clearly soldered to the eval board. However, they don't look like they're soldered to the module, but rather they're being pressed against the pads with the tension in the wire. It also looks like the free end of the loops are protruding down into a channel cut in the PCB. Do these kind of connectors/metal loops have a name? Clarifying question on meaning of derivative of a linear transformation In my analysis class, my professor (and the textbook) reference the fact that $\mathbf{D}T(\bar{x}) = T$ for any $\bar{x} \in \mathbb{R}^n$ where $T \in \mathcal{L}(\mathbb{R}^n,\mathbb{R}^n) $. I'm wondering about the precise meaning of this statement. Is it that the differential of any linear transformation is also linear? Or is that the differential of any linear transformation is itself (and therefore also linear)? The second seems confusing to me because the derivative of a function is often not itself. For example, $f(x) = 2x$ is a linear transformation but $f'(x) = 2 \ne 2x$ for all $x \in \mathbb{R}$. Between you and ("me" or "I")? Is it correct to say "between you and me" or "between you and I"?  I am not a native English speaker, so please bear with me. Prove that if $V$ and $W$ are three-dimensional subspaces of $\Bbb R^{5}$, then $V$ and $W$ must have a non-zero vector in common.  So far I got $V = \{v_{1}, v_{2}, v_{3}\}$ and $W = \{ w_{1}, w_{2}, w_{3}\}$ because they are 3-dimensional subspaces. Then $V \cup W = \{v_{1}, v_{2}, v_{3}, w_{1}, w_{2}, w_{3}\}$  My prof said that now I need to prove that they are linearly independent so this is a contradiction and the statement is wrong because we are in $\Bbb R^{5}$ and we have 6 elements on it.  I don't know how to express them as a linearly independent proof, so I hope anyone can give me a good explanation on it. Proving that cardinality of the reals = cardinality of $[0,1]$ Radiation– white vs black house, hot or cool? How do I repeat a theorem number? Often in my papers I want to mention theorems in the introduction that are introduced in more detail and proved in a later section.  I'd like to include the statement of these theorems twice over, each time numbered as the theorem would naturally appear in its later section.  My current solution (using amsthm) is, e.g.  %(in the introduction) \newtheorem*{thm:associativity}{Theorem \ref{thm:associativity}} \begin{thm:associativity} Lorem ipsum ... \end{thm:associativity}  %(in a later section) \begin{thm} \label{thm:associativity} Lorem ipsum ... \end{thm}      Can anyone think of a cleaner solution? $\{\emptyset\}$ is a set containing the empty set. Is $\{\emptyset\}$ a subset of $\{\{\emptyset\}\}$?  My hypothesis is yes by looking at the form of "the superset $\{\{\emptyset\}\}$" which contains "the subset $\{\emptyset\}$". Why do I have to solve for $x$ in this proof, if I choose not to find a counterexample?