How can I get a line of text spread across the entire line width? As shown; with letter spacing increased (not only word spacing).   from:    to:    Lastly, how could I set a line-to-line spacing that would most approximate to that shown in the previous illustrations?  PS. Here's the code for the first image output:  % methamatics \documentclass{article} \usepackage{unicode-math} \usepackage{fontspec,xltxtra} \usepackage{lipsum}  \emergencystretch 3em % ------------------- \tolerance=1 \emergencystretch=\maxdimen \hyphenpenalty=1000 \hbadness=1000 % ------------------- \renewcommand{\normalsize}{\fontsize{12}{7}\selectfont} \setmathfont[]{STIX Two Math} \setmainfont[Ligatures = {Historic}, Contextuals = Alternate, Kerning = On]{Hoefler Text Pro}  % \setmathrm % [ %   Style = Historic, %   Ligatures = Historic, % ] % {Hoefler Text Pro}  % \setmathsf % [ %   Style = Historic, %   Ligatures = Historic, % ] % {Hoefler Text Pro}  % \setboldmathrm % [ %   Style = Historic, %   Ligatures = Historic, % ] % {Hoefler Text Pro Bold}  %-CMDS \newcommand{\tib}[1]{{\fontspec     [         Contextuals = LineFinal,         Kerning = On,     ]     {Hoefler Text Pro Engraved One}     \fontsize{24}{12}     \selectfont #1 \normalfont \normalsize}}  \newcommand{\tisc}[1]{{\fontspec     [         Letters = SmallCaps,         Kerning = On,     ]     {Hoefler Text Pro}     \fontsize{20}{12}     \selectfont #1 \normalfont \normalsize}}  %-DCMNT \begin{document}     \begin{center}         \tib{VECTOR ANALYSIS}\\[\baselineskip]         \tisc{an introduction\\         to \\         vector methods} \\          $$  $$     \end{center} \end{document}
Is there a "sentence stretch" that acts horizontally in a similar way as \setstretch acts vertically?
Really stuck with this, \begin{cases}  2tx'(t)-1=x^2t \\  x(t=1)=1 \\  \end{cases} I don't understand the variables. What does it mean that $x$ differentiated have $t$ as a function value, like this: $x'(t)$? Don't get it :( Transparency/Masking : See through an Outer Sphere and see Inner Sphere in true colours Summary  I want to be able to look through an outer sphere - BlueSphere, and see another sphere inside - RedSphere  This is the target goal I'm trying to get to:    Issues  Here is what I have tried, and the problems I'm having:  1)  If I set the spheres transparency alpha values to 0.75    - then the inner RedSphere does not take on its "true" colour and shows as blue instead of red.    2)  If I set the spheres transparency alpha to lower values of 0.5   the spheres lose their crispness which I don't want and the inner RedSphere still does not take on its "true" colour   { can't paste a 3rd sample - need more reputation points }   Background  I am animating two spheres, one inside the other, superimposed about a common origin.   At the start:   the outer BlueSphere is at full radius the inner RedSphere is at zero radius  As time progresses:   the outer BlueSphere shrinks the inner RedSphere expands    So what happens is the expanding inner RedSphere "bursts through" the contracting outer BlueSphere.  So the effect I am looking for is to see the RedSphere appearing as Red as it expands inside the BlueSphere.  Thinking/Logic  So I figure that transparency is not the way to go.  The transparency settings are behaving correctly - as one would expect, in a "real world" situation - the inner sphere (Red) takes on the colour of the outer sphere (Blue) seeing as we are looking through a (transparent) Blue material.  Any ideas on what approach to take ?  It's almost as if I should be "removing" the part of the BlueSphere that the RedSphere "occupies" so that we see the RedSphere directly without any BlueSphere between us and the RedSphere.  Some sort of masking ??  Advice, comments, suggestions welcome. Thanks.  [requested edit]  Using internal render.  [ My End Solution - Thanks for the guidance ]  Right, so here is the way I got to my final result:  1) I created python drivers for each sphere   I used the frame_current scene property for input Passed that to a scripted expression which used sin() as the scaling factor Used some if / elif conditionals against frame_current to make the spheres visible only when I needed them   2) I created 3 layers   On Layer-1 - background   I put RedSphere AND BlueSphere with drivers driver_Outer_RedSphere() &amp; driver_Outer_BlueSphere() limiting the display of each sphere to show only where each radius > halfway  On Layer-2 - foreground-1   I put RedSphere ONLY with driver driver_Inner_RedSphere() limiting the display of RedSphere to show only where its radius &lt;= halfway  On Layer-3 - foreground-2   I put BlueSphere ONLY with driver driver_Inner_BlueSphere() limiting the display of BlueSphere to show only where its radius &lt;= halfway    3) I rendered each layer to .PNG sequence [RGBA] (alpha channel for transparency)   Render Layer-1 - background   background : [sky]  Render Layer-2 - foreground-1   background : [transparent]  Render Layer-3 - foreground-2   background : [transparent]    4) I brought all 3 sequences into the Video Sequence Editor   Strip-1 - background   blend [cross] opacity [1.00]  Strip-2 - foreground-1   blend [alpha-over] opacity [0.50]  Strip-3 - foreground-2   blend [alpha-over] opacity [0.50]    5) Rendered the 3 layer final sequence to H.264  So the "aha moment" struck me right at the end when I was in the Video Sequence Editor:   "Don't try to make the Outer Spheres transparent, and see through to the Inner Spheres" "Instead overlay the Inner Spheres over the Outer Spheres, and limit the opacity of the Inner Sphere overlay"     So now I see my Inner Spheres in their true colours, but the colours are muted due to the .75 opacity - and hence it gives the visual impression that the Inner Spheres are in fact inside the Outer Spheres.  Thanks all for the input - it got me going in the right direction. How to change boot splash screen in 18.04 I have done this in the past in 16.04 and 17.10 but in 18.04 I can not get plymouth manager to work.  Downloaded it from the software center but it doesn't work.  Is there a guide available that will let me do it in terminal? Why and how does C# allow accessing private variables outside the class itself when it's within the same containing class? Circular References Cause Memory Leak? I'm trying to run down a memory leak in a windows forms application.  I'm looking now at a form which contains several embedded forms.  What worries me is that the child forms, in their constructor, take a reference to the parent form, and keep it in a private member field.  So it seems to me that come garbage-collection time:  Parent has a reference to the child form, via the controls collection (child form is embedded there).  Child form is not GC'd.  Child form has a reference to the parent form, via the private member field.  Parent form is not GC'd.   Is this an accurate understanding of how the garbage collector will evaluate the situation?  Any way to 'prove' it for testing purposes? Questions in titles need to be repeated, why?  To me this pattern seems ridiculous. But I frequently encounter the "not a real question" situation, when I put my question in the title, then the context for that specific question in the 'question body'.     So:  I am to take it that the title is almost completely irrelevant for establishing subsequent question body is asking? I choose my titles to expose the actual question so it need not be repeated below. If questions need to be repeated in the body, could we simply title it "A question, click here and read below" ?   Tagged bug, as I have found this "problem" to be quite reproducible.  Steps to reproduce:   Ask a question with the clear, simple question in the title, and the evidence/context/example in the question body. Await close votes. MySQL autoincrement column jumps by 10- why? I have a couple tables in which I created an object ID as either an Int or Bigint, and in both cases, they seem to autoincrement by 10 (ie, the first insert is object ID 1, the second is object ID 11, the third is object ID 21, etc).  Two questions:   Why does it do that? Is that a problem? Given two input layers  The clip layer - an arbitrary layer (though in my case it's a float 1 band geotiff) The mask layer - a paletted raster in which every pixel is either NO DATA or a single &quot;color&quot; value, same for every pixel.  I want to create an output raster whose overall extent matches the extent of the mask layer and each pixel is either NO DATA if the corresponding pixel of the mask layer is NO DATA or the value of the corresponding pixel from the clip layer if the corresponding pixel of the mask layer is not NO DATA. I know I can mask the clip layer with a polygon, but the mask layer is a raster. Polygonizing the raster layer is possible in principle, but would result in a polygon with a lot of very short edges. Is there a direct way to do this in QGIS? I suppose there's probably a path that involves creating a new layer and using the raster calculator, but I'm hoping there's something more direct. Further complicating the matter is that the two rasters might not have the same resolution and origin. I believe this is equivalent to  in ArcMap. Why do we need to dummy code categorical variables How to select (highlight) text in Skype chat without using a mouse? Recently I started learning to use keyboard shortcuts almost for everything I'm doing on the laptop, but soon enough after I started, I bumped into the problem of being unable to highlight (select) simple text in Skype chats.  I did some research and I couldn't find any solution to my problem. Have I missed an already answered question that is the same like mine or if not what solution can you suggest in order to achieve the results I want without touching mouse or touchpad or any cursor moving device? Proof : $2^{n-1}\mid n!$ if and only if $n$ is a power of $2$. I want to prove that:  $2^{n-1}\mid n!$ if and only if $n$ is a power of $2$. What are the proper uses of:   static_cast dynamic_cast const_cast reinterpret_cast C-style cast (type)value Function-style cast type(value)   How does one decide which to use in which specific cases? When you're logged in to a Stack Exchange account, it's possible to add and remove logins (associated OpenID/OAuth accounts) without performing any form of reauthentication.  This is not a full vulnerability in itself, but it provides an extremely easy way for an attacker who's found a temporary XSS vulnerability to turn that into a permanent account hijack. An XSS exploit on its own may only provide a minute of indirect access to a user's account. But with this login configuration, the attacker could use that minute to automatically:   Add a new attacker-controlled login to the account. For example:   The XSS JavaScript would generate a random identifier, and use that to construct a login URL of the form http://hacker.example/openid/1234567. The hacker.example server would automatically authenticate the first OpenID login request made with a given identifier, allowing it to be added to the account without user interaction. All subsequent login attempts would be automatically rejected for the user. The hacker.example server would notify the operator with the account name and the generated identifier, and would allow them to login to it.  Remove all existing logins from the account. Log the user out of their account, with no way to get back in.   If this exploit were used to hijack moderator and staff accounts it could makes it more difficult to quickly respond to an attack, creating a greater opportunity for extended abuse.  This is a broader instance of the same fundamental vulnerability which was previously-reported as . That instance is a little more straightforward to fix: just ask for the password again when changing it. However, safe reauthentication with OpenID/OAuth logins is more complicated. Most users' OpenID providers will be configured to automatically re-authenticate upon request, so there would be minimal security gain from requesting that.  Instead or as well, my suggestion is that for users that have a verified email address, require an emailed confirmation link to be clicked before committing any login changes.    I'm not a real security expert, but I've just found that fellow Stack Overflow user , who is, previously described this vulnerability in his 2012 blog post :     If an attacker can add additional login accounts then they can create a persistent mechanism to get back into that Stack Overflow account at any time without needing to hijack the session. Now of course the victim can always go into their account settings and identify the presence of the new login, but it’s a question of how much damage is done before then.      How would an attacker mitigate the risk of having their newly added account removed? Just remove the victim’s original account(s) instead.      [...]      Finally, this is also a lesson for those considering using third party authentication providers. Yes, it’s great that you can farm off this work to someone who has a solid implementation, but you still have responsibility for securing the authenticated session on your site. If you choose not to, be aware of the risks because certainly we’ve seen them exploited before. When do I have to use 'will' instead of 'going to'? Does going to only express an intention and will some kind of prediction that doesn't necessarily happen?  EDIT: Thanks for all your answers. I asked this question because I always fail to complete exercises like this from an ESL book:   We  will fly to Venice in June. (Correct: But, how can I be sure that I'm not sick in June?) Philipp will be 15 next Wednesday (Correct: Since his birthday is a fixed day, this will happen). They will get a new computer. (Wrong: Why?, If I need a new computer, I simply get a new one) In 2020 people are going to buy more hybrid cars.  (Wrong, Why? Who can tell for sure). "strlen(s1) - strlen(s2)" is never less than zero How do I split a forest tree into two trees with proper vertical alignment? \documentclass[oneside,12pt]{article} \usepackage{geometry} \usepackage{microtype} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage{lmodern}  \usepackage{enumitem} \setlist[enumerate,1]{label=\bfseries\Alph*,align=left,leftmargin=*,   labelsep=1.5em} \setlist[enumerate,2]{label=\arabic*.,labelindent=1em,labelsep=1.5em,   leftmargin=*}  \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsthm} \theoremstyle{definition} \newtheorem*{sol}{Solution}  \usepackage{forest} \forestset{   ass/.append style={     before computing xy={l=\baselineskip},     no edge   },   prooftree/.style={     for tree={       child anchor=north,       parent anchor=south     }   } }  \begin{document} \subsection*{Exercises 29} \begin{enumerate} \item Use QL trees to evaluate the entailment claims (1) to (10) in Exercises   28A.   \begin{enumerate}   \item $\forall x(Fx\supset Gx)\vDash\forall x(Gx\supset Fx)$      \begin{forest}       prooftree       [$\forall x(Fx\supset Gx)$       [$\neg\forall x(Gx\supset Fx)\checkmark$,ass       [$\exists x\neg(Gx\supset Fx)\checkmark$,ass       [$\neg(Ga\supset Fa)\checkmark$,ass       [$Ga$,ass       [$\neg Fa$,ass       [$(Fa\supset Ga)\checkmark$,ass       [$\neg Fa$] [$Ga$]]]]]]]]     \end{forest}      $\neg Fa$ and $Ga$ make the premisses and the negation of conclusion true.     Thus, the entailment is invalid.   \item $\forall x(Fx\supset Gx)\vDash\forall x(\neg Gx\supset\neg Fx)$      \begin{forest}       prooftree       [$\forall x(Fx\supset Gx)$       [$\neg\forall x(\neg Gx\supset\neg Fx)\checkmark$,ass       [$\exists x\neg(\neg Gx\supset\neg Fx)\checkmark$,ass       [$\neg(\neg Ga\supset\neg Fa)\checkmark$,ass       [$\neg Ga$,ass       [$\neg\neg Fa$,ass       [$(Fa\supset Ga)\checkmark$,ass       [$\neg Fa$ [*,ass]] [$Ga$ [*,ass]]]]]]]]]     \end{forest}      Since the tree closes, the entailment is valid.   \item $\forall x\exists yLxy\vDash\forall y\exists xLyx$      \begin{forest}       prooftree       [$\forall x\exists yLxy$       [$\neg\forall y\exists xLyx\checkmark$,ass       [$\exists y\neg\exists xLyx\checkmark$,ass       [$\neg\exists xLax\checkmark$,ass       [$\forall x\neg Lax$,ass       [$\exists yLay\checkmark$,ass       [$Lab$,ass]]]]]]]     \end{forest}      \begin{forest}       prooftree       [$\neg Lab$,ass [*,ass]]     \end{forest}   \item $\forall x((Fx\wedge Gx)\supset Hx)\vDash\forall     x(Fx\supset(Gx\supset Hx))$   \item $(\forall xFx\vee\forall xGx)\vDash\forall(Fx\vee Gx)$   \item $\forall x(Fx\supset Gx),\forall x(\neg Gx\supset Hx)\vDash     \forall x(Fx\supset\neg Hx)$   \item $\exists x(Fx\wedge Gx),\forall x(\neg Hx\supset\neg Gx)\vDash     \exists x(Fx\wedge Hx)$   \item $\forall x\exists y(Fy\supset Gx)\vDash\forall y\exists x(Gx\supset     Fy)$   \item $\forall x\forall y(Lxy\supset Lyx)\vDash\forall xLxx$   \item $\forall x(\exists yLxy\supset\forall zLzx)\vDash\forall x\forall y     (Lxy\supset Lyx)$   \end{enumerate} \end{enumerate} \end{document}   Usually, I have to manually split a forest tree into two to emulate page breaking in a forest tree. However, the second tree doesn't vertically align with the first as in the picture below. How can I vertically align them together? @cfr When the automorphism group is trivial...... If $G$ is a trivial group , obviously $Aut(G)$  is  trivial . Does the converse  hold $?$ .  When  we  are  given  that a  group $G$  has  trivial  automorphism  group , can we  conclude  that the group  will  be  trivial $?$ .  I  guess  not . For  one  thing , we know  that  $$G/Z(G)\cong Inn(G)$$. So , $Aut(G)$ being  trivial  would  imply  that $$G\cong Z(G)$$  i.e.  $G$  is  commutative . But I  see  nothing about $G'$s being trivial .   The problem  is  that  while I cannot  prove  it , I  could  not  find  a  counter example  either , from  among  the  groups  I  am  familiar  with .   What  is  the non-trivial  commutative  group  that  has  trivial  automorphism  group  $?$ gnome maps program never runs I have this problem while running GNOME Maps ~$ gnome-maps  (org.gnome.Maps:10414): Gjs-WARNING **: 19:43:34.553: Some code called array.toString() on a Uint8Array instance. Previously this would have interpreted the bytes of the array as a string, but that is nonstandard. In the future this will return the bytes as comma-separated digits. For the time being, the old behavior has been preserved, but please fix your code anyway to explicitly call ByteArray.toString(array). (Note that array.toString() may have been called implicitly.) 0 &lt;TOP LEVEL&gt; [&quot;resource:///org/gnome/Maps/js/osmTypes.js&quot;:32] 1 &lt;TOP LEVEL&gt; [&quot;resource:///org/gnome/Maps/js/osmEditDialog.js&quot;:35] 2 &lt;TOP LEVEL&gt; [&quot;resource:///org/gnome/Maps/js/osmEdit.js&quot;:25] 3 &lt;TOP LEVEL&gt; [&quot;resource:///org/gnome/Maps/js/contextMenu.js&quot;:33] 4 &lt;TOP LEVEL&gt; [&quot;resource:///org/gnome/Maps/js/mainWindow.js&quot;:33] 5 &lt;TOP LEVEL&gt; [&quot;resource:///org/gnome/Maps/js/application.js&quot;:35] 6 &lt;TOP LEVEL&gt; [&quot;resource:///org/gnome/Maps/js/main.js&quot;:43] 7 start() [&quot;resource:///org/gnome/gjs/modules/package.js&quot;:209] 8 &lt;TOP LEVEL&gt; [&quot;/usr/bin/gnome-maps&quot;:2]   (org.gnome.Maps:10414): Gdk-WARNING **: 19:43:36.525: Native Windows taller than 65535 pixels are not supported Gdk-Message: 19:43:36.574: Error 71 (Protocol error) dispatching to Wayland display.  19:43:36.574 (This message has changed many times) What's this? $ not defined using jQuery in WordPress I know that jQuery is loaded, because I can switch out the $ for 'jQuery' and everything behaves as expected, but this will be a messy script if I can't fix this  This script:  jQuery(document).ready(function(){     $("ul.vimeo_desc_feed li a").click(function(){         alert($(this).attr('href'));         return false;     }) });   Produces the error $ is not a function  This script:  jQuery(document).ready(function(){     jQuery("ul.vimeo_desc_feed li a").click(function(){         alert(jQuery(this).attr('href'));         return false;     }) });   works fine. How do I enumerate the properties of a JavaScript object?  I actually want to list all the defined variables and their values, but I've learned that defining a variable actually creates a property of the window object.