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Capitalization of names that begin lowercased, at the beginning of a sentence How to use grep output as a path for cd? $$|a\sin x+b \cos x|\leq \sqrt{a^2+b^2}$$   I have tried: $$|a\sin x+b \cos x|\leq |a+b|\leq \sqrt{a^2+b^2}$$  enough to prove: $$|a+b|\leq \sqrt{a^2+b^2}$$  But I can find how to continue from here If $(a,b)=1$ then prove $(a+b, ab)=1$. Let $a$ and $b$ be two integers such that $\left(a,b\right) = 1$. Prove that $\left(a+b, ab\right) = 1$.   $(a,b)=1$ means $a$ and $b$ have no prime factors in common  $ab$ is simply the product of factors of $a$ and factors of $b$.  Let's say $k\mid a+b$ where $k$ is some factor of $a$.  Then $ka=a+b$ and $ka-a=b$ and $a(k-l)=b$.  So $a(k-l)=b, \  a\mid a(k-1)$ [$a$ divides the left hand side] therefore $a\mid b$ [the right hand side].  But $(a,b)=1$ so $a$ cannot divide $b$.  We have a similar argument for $b$.  So $a+b$ is not divisible by any factors of $ab$.  Therefore, $(a+b, ab)=1$.  Would this be correct? Am I missing anything? What's the standard "roadmap" to learning quantum physics? Why don't we read ''k'' here?  Know → /nəʊ/   Knotty → /ˈnɒt.i/   Knockout → /ˈnɒk.aʊt/ Are tile bonuses counted twice if 2 new words are formed? An example:    O  O?   There is a triple letter tile to the right of the bottom O, indicated with a ?.  If I place an X on it, making the words OX (vertical) and OX (horizontal), do I score 50 points (scoring the tile twice) or 34 (scoring it once)? How to convert a number to words in MS Excel Android app? Excel Macros are unsupported in Android and there is also no SpellNum Function in Android . I want a function which can convert a number to words. Is there any way we can achieve this ? eg -  40004  , should be converted to Forty Thousand Four Linux Command Not Functioning As Alias Package natbib Error: Bibliography not compatible with author-year citations I have a basic tex file and a bibliography which is exported from Mendeley  However, when I want to cite in the author-year style, I get the error:   Package natbib Error: Bibliography not compatible with author-year citations. (natbib) Press &lt;return&gt; to continue in numerical citation style.  Any clue what's wrong?  Minimum working example:  \documentclass[letterpaper, 10pt]{article}  \usepackage[authoryear]{natbib} \usepackage{lipsum}  \begin{document}  \lipsum[1-4] \nocite{*}  \bibliographystyle{apalike} \bibliography{articles}  \end{document}    A snippet of my .bib library:  @article{shortcode, author = {Lastname, Firstname}, file = {:/path/to/file.pdf:pdf}, journal = {Journal name}, number = {xx}, pages = {x--xx}, title = {Title}, volume = {xx}, year = {xxxx}     } So, I'm working with OpenGL (I'm not exactly sure of the version), and I want to do an old-style lighting setup by essentially drawing a black rectangle over the screen, and drawing white circles over the areas that should be lit. I already have the rectangle and the circles, but I can't figure out how to make it so that the white circles are transparent when applied to the black areas. Any ideas? I tried to use blending modes, but it still evades me...  EDIT: Here's the code that I'm using (Python).  view_buf = Buffer(GL_INT, 4) glGetIntegerv(GL_VIEWPORT, view_buf) view = view_buf.to_list() if hasattr(view_buf, "to_list") else view_buf.list  glDisable(GL_DEPTH_TEST)  glDisable(GL_LIGHTING)  glEnable(GL_BLEND)  if logic.lights['smooth']:     glEnable(GL_LINE_SMOOTH)  # # Make sure we're using smooth shading instead of flat # glShadeModel(GL_SMOOTH)  # Setup the matrices  glMatrixMode(GL_PROJECTION) glPushMatrix() glLoadIdentity() gluOrtho2D(0, view[2], 0, view[3]) glMatrixMode(GL_MODELVIEW) glPushMatrix() glLoadIdentity()  #glAlphaFunc(GL_GREATER, 0.5)  glColor4f(0.0, 0, 0, 0.75)  glBegin(GL_QUADS);                              # Draw Black screen glVertex2d(0.0, 0.0); glVertex2d(view[2], 0.0); glVertex2d(view[2], view[3]); glVertex2d(0.0, view[3]); glEnd()  glColor4f(1, 1, 1, 1)  degtorad = 3.14159/180.0   glBlendFunc(GL_ZERO, GL_SRC_ALPHA)  def DrawCircle(x, y, radius, fill = 1, step = 11):      glVertex2f(x, y)      if fill:         glBegin(GL_TRIANGLE_FAN)     else:         glBegin(GL_LINE_LOOP)      for a in range(0, 360, step):         angle = degtorad * a         glVertex2f(x + (sin(angle) * radius), y + (cos(angle) * radius))      glEnd()   #for layer in range(len(logic.lights['layers'])): #   size =  for light in logic.lights['points']:     DrawCircle(light[0][0] * view[2], (1 - light[0][1]) * view[3], light[1], step = light[2])     if logic.lights['smooth']:         DrawCircle(light[0][0] * view[2], (1 - light[0][1]) * view[3], light[1], 0, light[2])  # Reset the state glPopMatrix() glMatrixMode(GL_PROJECTION) glPopMatrix() Are there concerns with doing a generalized additive model with a smoothing term of multiple variables? In R, usually one performs a generalized additive model like so:  library(mgcv) x &lt;- gam(depvar ~ s(indvar1) + s(indvar2) + parametricvar)   Say I want to explore the non-parametric interaction of indvar1 and indvar2.  Is the following kosher?  x &lt;- gam(depvar ~ s(indvar,indvar2) + parametricvar) If a warlock's invocation states that they can use a spell without a slot does that mean they can cast the spell as often as they want? On page 111 of the 5E hand book the invocations Misty Visions, Mask of Many Faces, Master of Myriad Forms, and Otherworldly Leap all they that you can cast them without expending a spell slot. Does this mean the spells essentially become cantrips and I can cast them as often as I want? Why GCP Kubernetes Node host location for europe-north1 seems to be in US? If a commutative ring is semiprime and its prime ideals are maximal then it is von Neumann regular (absolutely flat). If a commutative ring is semiprime and its prime ideals are maximal then it is von Neumann regular (absolutely flat).  The converse, although not immediately apparent, can be proven quite easily. But I've been stuck on this one for awhile.  I found that Bourbaki has a question in his treatise on commutative algebra, namely: if all points in Spec(A) are closed (ie. prime ideals are maximal) then A/R (where R is the nilradical) is von Neumann regular (absolutely flat). If this can be proven, the above statement follows immediately since the nilradical of a semiprime ring 0?  Ideas?   Ps: this is my first time posting a question so apologies for any formatting issues. How do I identify predatory and low quality journals? With Beall's List gone, how can I tell if a journal is spam? . While the reasons for his doing this are not yet public, this is a real loss of a valuable service. Does anyone know of similar services available to the general public?  Edit:  is definitely relevant, but that approach is more appropriate for finding top journals rather than identifying bottom ones. I.e., following that method would probably exclude lots of valid, lower-tier journals. Are there any approaches to easily identifying a predatory publisher? With the listings package, I would like to color the integers out of strings and comments and out of keywords too. The question for coloring numbers     . Some answer handle the numbers occurring within comments, strings or barewords, but no solution given work when a keyword contain a number. Which is annoying because C and C++ have a lot of types with a number in them, like uint32_t.  Here is a small example.  \documentclass{standalone} \usepackage{xcolor} \usepackage{listings}  \definecolor{darkgreen}{rgb}{0.18,0.54,0.34} \definecolor{maroon}{rgb}{0.64,0.16,0.16} \definecolor{darkpink}{rgb}{0.75,0.25,0.5}  \lstdefinestyle{myc++}{     language=[ISO]C++,     keywordstyle=\color{darkgreen}\bfseries,     commentstyle=\color{blue}\textit,     stringstyle=\color{darkpink}\ttfamily, %     literate=*     {0}{{{\textcolor{red}0}}}1     {1}{{{\textcolor{red}1}}}1     {2}{{{\textcolor{red}2}}}1     {3}{{{\textcolor{red}3}}}1     {4}{{{\textcolor{red}4}}}1     {5}{{{\textcolor{red}5}}}1     {6}{{{\textcolor{red}6}}}1     {7}{{{\textcolor{red}7}}}1     {8}{{{\textcolor{red}8}}}1     {9}{{{\textcolor{red}9}}}1,     escapeinside={|}{|}, %     morekeywords={int32_t} }  \begin{document}  \begin{lstlisting}[style=myc++] // In comment: 42 int32_t in_keyword = 80085; char *in_identifier|2| = "And in string 1337\n"; \end{lstlisting}  \end{document}   And the result:    The star for literate avoid to coloring the digits inside comments or strings. The escapeinside avoid coloring the digits inside a bareword when specifically marked. But I have yet to find how to color int32_t as a keyword. PSTricks error "Undefined control sequence  \c@lor@to@ps" I have a problem with this file tex:  \begin{document} \begin{figure} \begin{center} \begin{pspicture}(0,0)(1,3.8) \psline{-&lt;}(0,0)(0,.2) \psline{-}(0,.2)(0,.4) \psline{*-&gt;}(0,.4)(0,.9) \psline{-*}(0,.9)(0,1.4) \psline{-&lt;}(0,1.4)(0,2.4) \psline{-}(0,2.4)(0,3.4) \psline{*-}(0,3.4)(0,3.6) \psline{&gt;-}(0,3.6)(0,3.8) \rput(1,.4){$y=0$}%testo \rput(1,1.4){$y=1$} \rput(1,3.4){$y=3$} \end{pspicture} \end{center} \caption{} \end{figure} \end{document}   but I don't understand where is the error. Remove border from IFrame How would I remove the border from an iframe embedded in my web app? An example of the iframe is:  &lt;iframe src="myURL" width="300" height="300"&gt;Browser not compatible.&lt;/iframe&gt;   I would like the transition from the content on my page to the contents of the iframe to be seamless, assuming the background colors are consistent. The target browser is IE6 only and unfortunately solutions for others will not help. I noticed the other day when serving the public that when asking for something, people were saying "Can I get an xyz, please". The previous time I had such a job it was "Could I have an xyz", or "May I have..." if they were being very polite.  Does anyone know when this trend started and what caused it?