I have been confused by different interpretations of polynomial and monomial. Is $\sqrt[3]{x^3}$ really a monomial?
Are $(x^3)^{1/3} $ and $\sqrt{x^2}$ monomials?    In mathematics, a  is, roughly speaking, a polynomial which has only one term. Two different definitions of a monomial may be encountered:  For the first definition, a monomial, also called power product, is a product of powers of variables with nonnegative integer exponents, or, in other words, a product of variables, possibly with repetitions.   For the second definition, a monomial is a monomial in the first sense multiplied by a nonzero constant, called the coefficient of the monomial.
Does "staff" take a plural verb? Which one of these two statements is correct?     Our staff do ...      Our staff does ...   And is staffs ever correct? Do I have an SSD and a hard drive on my MacBook Pro? I have two small questions that I can't solve. So, there are 8 axioms for vector spaces. One of them is that $1 \cdot a = a$ for all $a \in V$ and of course $1$ is multiplicative identity in $\mathbb{F}.$ How can I be sure this axiom cannot be derived from other 7 axioms? Also, why is there no definition of $\alpha a= a\alpha, \forall \alpha \in \mathbb{F}, \forall a \in V$? Why is multiplication of scalars and vectors not defined in commutative way? JavaScript equivalent to printf/String.Format I'm looking for a good JavaScript equivalent of the C/PHP printf() or for C#/Java programmers, String.Format() (IFormatProvider for .NET).  My basic requirement is a thousand separator format for numbers for now, but something that handles lots of combinations (including dates) would be good.  I realize Microsoft's  library provides a version of String.Format(), but we don't want the entire overhead of that framework. It is me who am to blame.    It is I who am to blame.      Which one is correct here? I am bewildered about these two sentences. How do I Parameterize a null string with DBNull.Value clearly and quickly Bernoulli numbers, taylor series expansion of tan x I found the following formula here: .  Taylor series of $\tan x$:  $$\tan x = \sum_{n\,=\,1}^\infty \frac {(-1)^{n-1}2^{2n} (2^{2n}-1) B_{2n}} {(2n)!} x^{2n - 1} $$.  I do not understand what the Bernoulli number means in this formula. My problems are the following ones:  I do not know anything about Bernoulli numbers. I do not know how to expand the Bernoulli number in the formula you see further up. What expansion will be the final result? How did you arrive there? Why did you arrive there? How did you know that you should use the specific Bernoulli-related terms that you used in your answer? What is the purpose of the Bernoulli number in this formula?  A great deal of thanks. Set horizontal width on custom listings environment I want to create a custom listings environment and want to set its width as something like 90% of textwidth, centred.  Per an  for algorithms, the first thing I tried was to embed the listing in a minipage environment. However, . I found a way to still make standard environment work, but it requires to use the escape command every time before ending the environment.  The second thing I tried was this latter command, but as far as I can see, it doesn't offer a feature to set the width of the listing or centre it.  \documentclass{article} \usepackage{listings} \usepackage{lipsum}   \lstnewenvironment{queryl} %% APPROACH 1  {\lstset{frame=shadowbox,escapechar=`}}  {}  \newenvironment{query} %% APPROACH 2  {\begin{minipage}{4cm}\centering\begin{queryl}}  {\end{queryl}\end{minipage}}   \begin{document} %% APPROACH 1 Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Ut purus elit, vestibulum ut, placerat ac, adipiscing vitae, felis. \begin{queryl} begin { do nothing } end ; \end{queryl} Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Ut purus elit, vestibulum ut, placerat ac, adipiscing vitae, felis.  %% APPROACH 2 Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Ut purus elit, vestibulum ut, placerat ac, adipiscing vitae, felis.  \begin{query} begin { do nothing } end ;`%&lt;- needs escape-to-LaTeX character to work \end{query}  Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Ut purus elit, vestibulum ut, placerat ac, adipiscing vitae, felis. \end{document}     One can note that for the first environment, the frame extends beyond the width of the text. Also, puzzlingly the minipage environment doesn't seem to want to centre and swallows space (though I guess I could perhaps at least get that to work with tinkering).  In any case, does anyone have any ideas on how to set a width for a custom listings environment and centre it without requiring use of the escape character each time to end the environment? Clicking community promotion ad renders an instance of SO within the frame of that ad Today I came across an ad on this question on SO:  (I don't expect it to stay very long).   There was a community promotion ad on the right hand side of the site, I think for one of our English sites. It had a sentence that I only partially remember:     Idiom ... poor or even poorer   I found it interesting, so I clicked that ad. Which caused a new instance of SO to open within the ad space. Unfortunately I failed to take a screenshot. I've been refreshing the page like crazy, but can't get the ad to appear again.  Edit: I managed to get a new community add, for the photography site, and captured a screenshot:   Edit2: This bug still persists, I saw it today with an OOP add from CodeReview, and an area 51 add. I met a guy on a sugar daddy dating site and we talked, I told him my situation and he said he would help.   He asked for my bank account information and I told him you can use cash app or PayPal to go through. He said his account manager deals with his finances. So I set up an account with a bank that I don’t use so that my primary account isn’t used.  Eventually he was in the process of sending me money and we had agreed on a amount that wasn't crazy. But then he asked for my address and I asked why and he said just wondering. Then said he wanted to send me a surprise for being his baby. Which I didn’t give.  Then later on he sent the money and said he sent me a bonus and that I was supposed to use that to pay his mothers caregiver, which he didn’t mention until after.  I kept questioning him and he was saying that he works in construction and he’s trusting me to carry out the errand. Then he said don’t let him down and I asked him if he was threatening me and he kept saying I need this done but didn’t really answer my question.  Am I being scammed? I’m about to change my password but is this a scam? Why are there so many places to put a binary in Linux? There are at least these five:   /bin/ /sbin/ /usr/bin/ /usr/local/bin/ /usr/local/sbin/   And on my office box, I do not have write permissions to some of these.  What type of binary goes into which of these bins? I have 3 accounts that I am deleting - however, several hours after the 24 hour (+ grace period - end about 4-5 hours ago), the countdown completed, but disappeared and the accounts are still there.   I would rather not go through another 24 hours, and would like to have these accounts deleted (and yes, I have thought about it and this decision is not up for discussion).  Currently, the accounts are still there.  Could this please be seen to? I need help!  I need to prove that for any $2 \le n,k$ positive integers $(n - 1)^2 \mid n^k -1$ if and only if $(n - 1) \mid k$  Thanks! Story with VR worlds for the wealthy Windows 10 doesn't shut down properly I exported a map and everything looks fine when viewing the PDF. But when printing it on my plotter, all the text disappears. The scale bar would bring but the text associated to the scale bar wouldn't. The title block also disappears. I've tried messing around with the settings. I've checked/unchecked &quot;Embed All Document Fonts&quot; and &quot;Convert Marker Symbols to Polygons&quot;. I'm not sure what I'm doing wrong here. What's weird is I exported this map a few weeks ago and it printed just fine. Today, I needed to make a change to the map and exported it out, and now I am experiencing this issue. I must have changed one of the export settings in the past few weeks but I'm not sure what it is. Has anyone experienced this? where instance variables get initialized How can I make a macOS Mojave bootable USB installer in Windows 10? Natural Deduction - How to define literals? I'm currently working on the following question:  "At least one of Plato and Democritus believed in the theory of forms. Plato believed in the theory of forms only if he was not an atomist, and Democritus was an atomist only if he did not believe in the theory of forms. Democritus was an atomist. Prove, using Natural Deduction, that Plato was not an atomist"  I have been working for quite some time to take the information and turn it into a set of literals and statements I can work with.  So far, I have got $P \lor D$,  $P \Rightarrow \lnot a$,  $D \Rightarrow A$,  $D = A$  However, I cannot seem to make these prove the statement using natural deduction, any hints? Reprojecting LiDAR data with libLAS, error: "latitude or longitude exceeded limits"? I have a lidar file (file.las) with the following spatial reference:  Spatial Reference:            PROJCS["NAD83 / UTM zone 11N",     GEOGCS["NAD83",         DATUM["North_American_Datum_1983",             SPHEROID["GRS 1980",6378137,298.2572221010002,                 AUTHORITY["EPSG","7019"]],             AUTHORITY["EPSG","6269"]],         PRIMEM["Greenwich",0],         UNIT["degree",0.0174532925199433],         AUTHORITY["EPSG","4269"]],     PROJECTION["Transverse_Mercator"],     PARAMETER["latitude_of_origin",0],     PARAMETER["central_meridian",-117],     PARAMETER["scale_factor",0.9996],     PARAMETER["false_easting",500000],     PARAMETER["false_northing",0],     UNIT["metre",1,         AUTHORITY["EPSG","9001"]],     AUTHORITY["EPSG","26911"]]   Then I reproject it to WGS84 with las2las:  las2las --a_srs EPSG:26911 --t_srs EPSG:4326 -i file1.las -o output.las   And I get:   Spatial Reference:                GEOGCS["WGS 84",         DATUM["WGS_1984",             SPHEROID["WGS 84",6378137,298.257223563,                 AUTHORITY["EPSG","7030"]],             AUTHORITY["EPSG","6326"]],         PRIMEM["Greenwich",0],         UNIT["degree",0.0174532925199433],         AUTHORITY["EPSG","4326"]]   But when I try to revert the transformation: from WGS84 to NAD83/UTM 11N, I get an error:  las2las --a_srs EPSG:4326 --t_srs EPSG:26911 -i output.las -o wgs2utm.las ERROR 1: latitude or longitude exceeded limits error: Could not project point for ReprojectionTransform::latitude or longitude exceeded limits0.   Why I got this error, if I am just reversing the transformation (originally from NAD83 to WGS84, and then from WGS84 to NAD83)?.