Prove that $(a,b)^2=(a^2,b^2,ab)$
Expanding GCDs Reasoning: $\gcd(a,c) \cdot \gcd(b,c) = \gcd(ab,bc,ac,cc)$
What is it called when a non-verb is used as a verb? This is in very common use on the internet. I just read this:     "[The] hotel employee walks in, I say "uh, puppy" and she just NOPEd the [heck] out of the room."   This is often done by adding 'd to the word. using tar showing, "No Such file or directory" But when I try to run it using  tar zxvf jre-8u151-linux-i586   I get tar (child): jre-8u151-linux-i586: Cannot open: No such file or directory tar (child): Error is not recoverable: exiting now tar: Child returned status 2  What is wrong? The name and place matches prefectly. Proving that without the axiom of choice there is a set with an accumulation point that isn'tthe limit of a convergeant sequence I've been reading the simple parts of Thomas Jech's book on the Axiom of Choice and came across the following proof on page 141. The proof assumes a mathematical model without the axiom of choice.     Lemma. $X$ is D-finite if and only if $X$ does not have a countable subset.       Theorem. There is a set $S$ of real numbers and a real number $a \not \in S$ such that a is in the closure of $S$ but there is no sequence $(x_n)^{\infty}_{n=0} \subset S$ such that $(x_n)^{\infty}_{n=0} \longrightarrow a$.       Proof. Let $D$ be a D-finite infinite set of reals. We claim that the set $D$ must have an accumulation point. To prove this claim suppose that all $d \in D$ are not accumulation points, then let $\{I_n:n=0,1,...\}$ be a fixed enumeration of open intervals with rational endpoints. Assign the least $n$ such that $I_n \cap D={d}$ to each $d \in D$. This makes $D$ countable and by the above lemma gives us our contradiction. So $D$ has an accumulation point $a$. Let $S=D \setminus\{a\}$ and since $S$ is D-finite, every convergent sequence in $S$ is eventually constant.   Could somebody explain this proof? Why is the last sentence true/ Why does this mean every convergent sequence is eventually constant?  I think the following theorem is important     Theorem. There is a model of ZF which has an infinite set of real numbers without a countable subset How to distribute proprietary Mathematica code resizable serial console window? A sine integral $\int_0^{\infty} \left(\frac{\sin x }{x }\right)^n\,\mathrm{d}x$ The following question comes from   post $$\int_0^{\infty} \left(\frac{\sin x }{x }\right)^n\,\mathrm{d}x$$ but now I'd be curious to know how to deal with it by methods of complex analysis. Some suggestions, hints? Thanks!!!  Sis. Featured On Meta section doesn't list MSE posts on meta sites Footnotes layout in multicols environment I am using the package multicol to typeset a one-column (spanning the entire text width) heading and a two-column main text block. In order to do so, I begin the multicols environment along with the start of the text block. The one-column heading spans a small fraction of the text height in the front page and I need to set a footnote there, before starting the two-column environment. I'd like to have that footnote typeset under the first column in the two-column running text. How could I possibly do this?   Moreover, I'd like to have the footnotes typeset in a single column under one of the text columns -- I don't even mind having them as two-column footnotes, but what I do not want is a footnotes layout spanning the entire text width. I have tried with the command \twocolumnfootnotes from the class memoir, with the packages ledmac, dblfnote, and even with ftnright (the latter complains about its being used in a single-column document, due to that one-column heading). None has proved useful in this setting. As a less than MWE, just to convey a more concrete depiction of my scenario, let us consider:  \documentclass{memoir}  \usepackage{multicol}  \begin{document}  A single-column heading\footnote{Here I have my footnote.}  \begin{multicols}{2} Here I am in the two-column environment, and I want my footnotes being set in accordance with the document's ``columnwidth''.\footnote{A footnote in the two-column environment.} \end{multicols}   \end{document} Underfull \hbox in bibliography I am getting lots of "Underfull \hbox" warnings that point to the *.bbl file. It looks like these messages appear only when URLs are very long, like in the below example:  @techreport{nistguidesec,   author    = {Wayne Jansen and Timothy Grance},   title     = {Guidelines on Security and Privacy in Public Cloud Computing},   month    = {January},   note     = {Draft Special Publication 800-144. Available at \url{http://csrc.nist.gov/publications/drafts/800-144/Draft-SP-800-144_cloud-computing.pdf}}, }   In the produced PDF text is correctly hyphened and links are correctly split over several lines (I am using hyperref package)  How can I avoid annoying warning messages about "underfull \hbox" pointing to bibliography? During graduate studies online application submission period, is there any preference given for an applicant if he/she submits his application at the beginning of that period compared with someone who submits his/her application couple of hours before the deadline?  Or as long as the applications are complete there is absolutely no difference! I'm trying to calculate the amount of settings the enigma machine has. I have found several sites regarding this topic, but it seems like there are two answers to my question. The first answer is 158,962,555,217,826,360,000 and the second is 107,458,687,327,250,619,360,000. Where the difference is that in the second answer, it has been taken into account that the notched rings on the enigma could be set as well, giving 26 * 26 = 676 more settings than the first answer.   But which answer is the right one, and why? How many possible different Enigma machine settings are there and how can one determine this number? I want to prove that: $\ln(x+1)&lt; x$.  My idea is to define:  $f(x) = \ln(x+1) - x$, so:  $f'(x) = \dfrac1{1+x} - 1  = \dfrac{-x}{1+x} &lt; 0, \text{ for }x &gt;0$.  Which leads to $f(x)&lt;f(0)$, so $\ln (x+1)-x&lt;0$.  Is that a valid proof? Any other ideas?  Thanks. In Python, how do I get a function name as a string, without calling the function?  def my_function():     pass  print get_function_name_as_string(my_function) # my_function is not in quotes   should output "my_function".  Is such function available in Python? If not, any ideas on how to implement get_function_name_as_string, in Python? Is there any adjustment required when computing the correlation coefficient between:   continuous and discrete variable  two discrete variables ?    Thanks ! Deep learning model (LSTM) with temporal and non temporal attributes .gitignore for Visual Studio Projects and Solutions Work done in adiabatic process I have an alias that I would like to use both as a regular user and as root, via sudo. Specifically, it is this:  alias rm=trash   This works fine as a normal user, and it works fine when I use sudo -i to get a root shell prompt, but if I use sudo rm, the alias does not apply. So where do I need to put my alias so that it works in one-off sudo commands? How can I design a tree graph with node shapes as this photo? $\delta$ differential notation Various textbooks that I am currently consulting (including Spacecraft Dynamics and Control An Introduction - Anton H.J. De Ruiter | Christopher J. Damaren | James R. Forbes Section 1.4, page 32) use $\delta$, not $d$ or $\partial$ to express an infinitesimal quantity. In the context of the reference text above, the symbol is used as part of a general definition for the derivative of a vector. Specifically, given   $$\mathbf{\vec{r}}=\mathcal{\vec{F^{T}_{1}}}\mathbf{\vec{r_{1}}}$$   where   $$\mathcal{\vec{F^{T}_{1}}}$$   is a vectrix   $$[\mathbf{\vec{x_1}} \phantom{s} \mathbf{\vec{y_1}} \phantom{s} \mathbf{\vec{z_1}}].$$  The time derivative of the vector is defined as   $$\dot{\mathbf{\vec{r}}}\triangleq \lim_{\delta t\to 0}\frac{\delta\mathbf{\vec{r}}}{\delta t}.$$  In this context, what is the difference between $\frac{\delta\mathbf{\vec{r}}}{\delta t}$ and $\frac{d \mathbf{\vec{r}}}{d t}$ ?  Note about thermodynamic use of $\delta$: My understanding is that $\delta$ is used in thermodynamic equations to express path dependence of a scalar quantity such as heat or work. What does it mean in the context of an abstract physical vector?