I think that this might be a stupid question, but I never figured this out. I think this might be a stupid question. If I get rid of the that, is it a run on? I see it a lot, so I think that it is right, but I can't figure out what it's called or the exact rules of it since &quot;I think&quot; and &quot;This might be a stupid question&quot; both seem like independent clauses, so why can they go together? I remember also seeing the following sentence in a book too. I want to tell her I would've known. Thank you! This is my first time using Stack Exchange, so please tell me if I did something wrong.
In general, when can I omit that from a sentence?   Can I omit that (emphasized) in sentence (2) below?        We say that such algorithms handle concept drift and can learn from time-changing data streams.   When the statistical properties of the target variable change, we say that concept drift has occurred.   She told me that some of her classmates failed.      To me, sentences (1) and (3) sound better without that. If I omit that from (2), it just doesn't sound right when I read it out loud. (I should point out I'm not a native speaker.)  How do you decide when to omit it? Is it just a matter of taste? (I know that is sometimes mandatory.)  Can you think of sentences that "sound better" with that? Can you think of sentences that "sound better" without that? Can you think of a few sentences where it is wrong to omit that?
$100$-th derivative of the function $f(x)=e^{x}\cos(x)$ I've got this task I'm not able to solve. So i need to find the 100-th derivative of $$f(x)=e^{x}\cos(x)$$ where $x=\pi$.  I've tried using Leibniz's formula but it got me nowhere, induction doesn't seem to help either, so if you could just give me a hint, I'd be very grateful.  Many thanks! Came across this little practice exercise, and I couldn't properly convince myself of this relation:  Let $X,Y \subset \mathbb{R}^n$ and $g:X\times Y \rightarrow \mathbb{R}$. Show that $$\sup_{y \in Y} \inf_{x \in X}g(x,y) \leq \inf_{x \in X} \sup_{y \in Y} g(x,y).$$  My thinking was starting with $\inf_{x \in X}g(x,\overline{y}) \leq\sup_{y \in Y} g(\overline{x},y)$, but does that even hold for all $\overline{x} \in X$ and $\overline{y} \in Y$? I have an upcoming trip where I will be flying to and from Copenhagen, through Toronto.  I notice that my departure and return gate in the Philadelphia airport is not in the international terminal.  Where will I clear customs? Possible bug in sidebar "question asked:" date I am working on a racing game in Unity3d and I have to show the rank as the winners touch the winning point.  As the winners reach at the winning point the image of the winner number should pop up for 1-2 seconds that is it.  How do I do it? Can anyone sort out this problem? Any help would be appreciated.  Thanks in advance :) How to hook into the CPT's title placeholder? Objective: To separate text column from figures column in every page I don't know if this has been asked before. But so far nothing has come up with my search.  In some books, the figures are placed at the outer side of the page like in the picture below.    Is this achievable in TeX/LaTeX? That is how does one set up the page so that all text will appear in a 'text column' and the figures appear in a 'figures column' in every page? \textcite macro produces "Author (year)" correctly but colours first paren with hyperref I am using a custom \textcite macro that I found somewhere. It works nicely except that it makes the opening paren of "Author (year)" part of the link (also colouring it blue) which is not desired.  This is likely just a minor change to the definition of the macro, but the solution eludes me (I don't get where the opening paren is added at all).    Opening paranthesis should not be part of link and should not be coloured.  (M)WE:  \documentclass{scrartcl}  \usepackage{csquotes}  \usepackage[backend=biber,             bibencoding=utf8,             hyperref=true,             natbib=true,             minnames=1,     %% prevent truncation to year-only             backref=true, %% TODO REMOVE before final             bibstyle=apa,             citestyle=authoryear,             ]{biblatex} \DeclareLanguageMapping{english}{english-apa} \addbibresource{bib.bib}   \DeclareCiteCommand{\textcite}   {\boolfalse{cbx:parens}}   {\usebibmacro{citeindex}%    \printtext[bibhyperref]{\usebibmacro{textcite}}}   {\ifbool{cbx:parens}      {\bibcloseparen\global\boolfalse{cbx:parens}}      {}%    \multicitedelim}   {\usebibmacro{textcite:postnote}}   \usepackage[table,svgnames]{xcolor}  \definecolor{solblue}{HTML}{268BD2}  \usepackage[pdftex]{hyperref}  \hypersetup{    pdfhighlight = /P,    colorlinks=true,    urlcolor=solblue,    filecolor=solblue,    linkcolor=solblue,    citecolor=solblue,    raiselinks=true,    breaklinks,    unicode }   \begin{document}  foobar \textcite{thesis_pockrandt} foobar  \end{document}   bib.bib:  @phdthesis{thesis_pockrandt,     author = {Christopher Maximilian Pockrandt},     title = {{Approximate String Matching: Improving Data Structures and Algorithms}},     year = {2019} } $P$ is a prime ideal of $R$ iff $R/P$ is an integral domain. : $P≠R$ Here is the theorem and proof in my textbook.     Thm.  Let $P$ be an ideal of a commutative ring $R$ with identity $1$.  Then $P$ is a prime ideal of $R$ if and only if $R/P$ is an integral domain.   Proof.   Suppose that $P$ is a prime ideal of a commutative ring $R$ with $1$.  Then $P$≠$R$ implies $1+P≠0+P$. Hence $R/P$ is a commutative ring $R$ with identity.  Assume that $(a+P)(b+P)=0+P$.  Then $ab+P=0+P$ and $ab∈P$.  By the definition of a prime ideal P we get $a∈P$ or $b∈P$.  That is, $a+P=0+P$ or $b+P=0+P$.  Thus $R/P$ is an integral domain.   Conversely, if $R/P$ is an integral domain, then $1+P≠0+P$ and $R/P$ is a commutative ring $R$ which has no zero divisors.  Hence $P≠R$.  Assume $ab∈P$.  Then $ab+P=0+P$ and $(a+P)(b+P)=0+P$.  Since $R/P$ is an integral domain, we get $a+P=0+P$ or $b+P=0+P$.  So $a∈P$ or $b∈P$.  Thus $P$ is a prime ideal.    I am having problems understanding proof with "Conversely, if $R/P$ is an integral domain, then $1+P≠0+P$.".  I know that $R/P$ has an identity. (cause it is ID)  But, why doesn't "$P$" have an identity?  Isn't there a case $P=R$?  Please help me understanding. I think $P≠R$ should be with condition of Thm..  Thank you in advance. How do I tell which point-and-shoot cameras take good low light photos? I am writing a journal which has two column. I am using {figure} as well as {figure*} to place the figures in one column as well as stretch across two columns.   My problem is the order of the figures are not correct. e.g. their order of appearance is:                  Figure 1.                 Figure 2.                 Figure 5.                 Figure 3.                 Figure 4.   What should I do to make them appear in order? Itemize: Identation is different between first and second line How to change keyboard layout via command line cmd.exe on Windows XP/7? Is is possible to change the keyboard layout via command line on Windows XP/7? Is "from where" grammatically equivalent to "from which"? What's the preferred method of writing AJAX-enabled plugins? save as a video as .mp4 So I made an intro using animation and it saves it as a .blend and I want it so save as a video like .mp4 or something like that. I've tried changing the codec because I tried opening the intro with windows media instead of blend. I didn't try all the codec's but I tried some of them. How many ways can the average of n dice be a? How do I pass JavaScript variables to PHP? For which $n$ does $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a+b+c}$ imply $\frac{1}{a^n}+\frac{1}{b^n}+\frac{1}{c^n}=\frac{1}{a^n+b^n+c^n}$ I'm having trouble finishing a problem on an old national competition.  As the title states, the question says asks:  Given $a,b,c \neq 0,a+b=c$ such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a+b+c}$,  Find all integers $n$ such that $\frac{1}{a^n}+\frac{1}{b^n}+\frac{1}{c^n}=\frac{1}{a^n+b^n+c^n}$.  I know that if it is true for $n$, it is also true for $-n$ since $(a^n+b^n+c^n)(\frac{1}{a^n+b^n+c^n})=1$ and $(a^n+b^n+c^n)(\frac{1}{a^n}+\frac{1}{b^n}+\frac{1}{c^n})=1$ and the statment follows for $-n$ given that we can make the substitution for $n$. Thus I found out that $n=-1$ satisfies the equality.  I also know that it fails for all even $n$. If $n$ were even, $(a^n,b^n,c^n)$ would all be positive, and Titu's Lemma asserts that $\frac{1}{a^n}+\frac{1}{b^n}+\frac{1}{c^n}\geq \frac{9}{a^n+b^n+c^n}$. Thus $n$ must be odd.  I also observed that $(2,-1,1)$ was a solution, and from here noted that this particular solution satisfies all odd $n$.  I have been trying to prove that all other solutions satisfy all odd $n$ with no success.  Any help would be greatly appreciated. Thanks! After taking a statistics course and then trying to help fellow students, I noticed one subject that inspires much head-desk banging is interpreting the results of statistical hypothesis tests.  It seems that students easily learn how to perform the calculations required by a given test but get hung up on interpreting the results.  Many computerized tools report test results in terms of "p values" or "t values".  How would you explain the following points to college students taking their first course in statistics:   What does a "p-value" mean in relation to the hypothesis being tested?  Are there cases when one should be looking for a high p-value or a low p-value? What is the relationship between a p-value and a t-value?