I have a set of maps I am creating with Data Driven Pages. Each map has a scale and rotation set by the feature class that is driving the map creation.  I would like to create a polygon that is the extent of each page that is created - not the polygon from the DDP index feature class. Is this possible?  I am playing around with the Fishnet, as my page size is the same, but not sure I can put the rotation into that.
I need to label the four corners of a dataframe as shown in the image. Can this be done with dynamic text scripts?
Was Harry Potter a direct descendant of Godric Gryffindor? Proving that $A_n$ is the only proper nontrivial normal subgroup of $S_n$, $n\geq 5$ There is a famous Theorem telling that:  For $n≥5$, $A_n$ is the only proper nontrivial normal subgroup of $S_n$.  For the proof, we firstly start with assuming a subgroup of $S_n$ which $1≠N⊲S_n$. We proceed until at the last part of proof's body, we assume $N∩A_n=\{1\}$. This assumption should be meet a contradiction with normality of $N$ in $S_n$. There; we get $N=\{1,\pi $} in which $\pi$ is an odd permutation of order $2$. Now for meeting desire inconsistency, I have two approaches:  (a) Since every normal subgroup, having two elements, lies in the center of $G$ so, our $N⊆ Z(S_n)=\{1\}$ for $n≥5$ and then $N=\{1\}$.  (b) Clearly, $1≠N$ acting on set $\Omega=\{1,2,...,n\}$ is intransitive wherein $|\Omega|≥5$ and according to the following Proposition $S_n$ would be imprimitive.  Proposition 7.1: If the transitive group $G$ contains an intransitive normal subgroup different from $1$, then $G$ is imprimitive (Finite Permutation Groups by H.Wielandt).  May I ask if the second approach is valid? I am fond of knowing new approach if exists. Thanks. Number of preimages of a group element under a homomorphism. I want the entries in the TOC for the indices to align with the Chapter entries. I can do this with \addcontents. \printindex also puts an entry in the TOC but it is aligned with the section entries.   So I want to:    change how printindex makes its entry into the TOC or  to eliminate printindex entry.   MWE:  \documentclass{book} \usepackage{multind}        % create the indices \makeindex{names}            \makeindex{subject} \newcommand{\Ndx}[1] { \index{names}{#1} }     % define the type of entry \newcommand{\Sdx}[1] { \index{subject}{#1} }   % to the indices \usepackage[headings]{fullpage}         % 8-1/2, 11 page with 1" margins \usepackage{fancyhdr} \pagestyle{fancy} \begin{document} \fancyhead{} \fancyfoot{} \renewcommand{\sectionmark}[1]{\markright{\thesection.\ #1}} \fancyhead[LE,RO]{\rightmark} \fancyhead[RE,LO]{\thepage} \tableofcontents        % put table of contents here  \chapter{\bf Logic, Plausibility, and Probability} \label{Ch:Logic} \section{Logical Arguments and Reasoning } It is obvious that logical arguments are essential to the judicial system. The logic must be able to treat syllogisms\Sdx{syllogism} that range from deductive\Sdx{syllogism!deductive} to abductive\Sdx{syllogism!abductive}, that is from strong to weak syllogisms \Sdx{syllogism!weak}\Sdx{syllogism!strong} as illustrated in the table taken from Author \Ndx{Author} \Ndx{Cohen} \Sdx{Cohen on probability} \clearpage \addcontentsline{toc}{chapter}{Subject Index} \printindex{subject}{Subject Index} \addcontentsline{toc}{chapter}{Name Index} \printindex{names}{Name Index} \end{document} How to control spacing in a TikZ tree? How to address a professor in person at US universities? At US universities, how should one address a professor while talking to him or her? In my home country, we usually address the professors as either "sir" or "madam" while talking about our queries in person.   What's the norm at US universities? Centre of $GL_n(\mathbb R)$ How to prove that center of $GL_n(\mathbb R)$ is scalar matrices?  Again what can you say about the center of set of all upper triangular matrices and center of set of all diagonal matrices. My office is going to buy a bulk number of desktop PCs (for developmental purposes).  I encouraged them to buy PCs without a pre-installed OS and then install Linux. My colleagues agree with me but they are not regular users of any Linux-based OS, so they want to choose a Linux OS which is best for our purposes and user-friendly and I suggested Ubuntu.  Now I want some data to support my position, what's so good about Linux and specifically Ubuntu? Best File System for Sharing Between OS X and Windows How can I '\input' a file without getting the implicit newline? Send data from Lightning Web Component (LWC) to a Visualforce page I am adding an LWC component in visualforce and need to add a handler for an event fired in the LWC component on the visualforce page. When googling all I can find is examples of how to do this with aura component using application events. LWC controller: handleSearchClick() {              const searchinfo = {          shipper: this.shipper,         system: this.system,         product: this.product,         status: this.template.querySelector('c-lwc-multi-select').value(),         period: this.period       };     this.dispatchEvent(new CustomEvent(         'dosearch',          {             detail: { data:  searchinfo},             bubbles: true,             composed: true,         }     ));     console.log({ 'lwcNominationSearch searchinfo': searchinfo }); }  AURA app: &lt;aura:application extends=&quot;ltng:outApp&quot; access=&quot;GLOBAL&quot;&gt; &lt;aura:dependency resource=&quot;c:lwcNominationSearch&quot; /&gt; &lt;aura:dependency resource=&quot;markup://force:*&quot; type=&quot;EVENT&quot;/&gt; &lt;/aura:application&gt;  Visualforce: &lt;div id=&quot;rcontainer&quot;&gt; &lt;/div&gt;  &lt;script&gt;     $Lightning.use(&quot;c:NotificationSearch_ltngoutApp&quot;, function()     {         $Lightning.createComponent(         &quot;c:lwcNominationSearch&quot;,         {},         &quot;rcontainer&quot;,         function(cmp)         {             console.log('component created');              //$A.eventService.addHandler({ &quot;event&quot;: &quot;c:myEvent&quot;, &quot;handler&quot; : visualForceFunction})               $A.eventService.addHandler({                 event: 'dosearch',                 handler: function (event) {                          console.log(&quot;vf event success&quot;);                 }             });          });     }); &lt;/script&gt; Isomorphism between two models of Hilbert's Axioms with different notions of congruence Cast ray to select block in voxel game Using the properties of real numbers, verify commutativity on complex numbers. I have just started self-studying Linear Algebra after a 3 year hiatus from mathematics (in an extended neuroscience foray), and I want to ensure that I am sufficiently verifying commutativity of complex numbers by using the properties of real numbers.  My attempts...  (1) Commutativity of Addition:   Verify that $w + z = z + w$ for all $w, z \in \mathbb{C}$.  $$w + z = (a + bi) + (c + di) \hspace{10mm}\text{by definition of }\mathbb{C}$$ $$\hspace{13mm} = a + (bi + c) + di \hspace{10mm}\text{by associativity in }\mathbb{R}$$ $$\hspace{17mm} = a + (c + bi) + di \hspace{10mm}\text{by commutativity in }\mathbb{R}$$ $$\hspace{16mm} = (a + c) + (bi + di) \hspace{10mm}\text{by associativity in }\mathbb{R}$$ $$\hspace{19mm} = (c + a) + (di + bi) \hspace{10mm}\text{by commutativity in }\mathbb{R}$$ $$\hspace{13mm} = c + (a + di) + bi \hspace{10mm}\text{by associativity in }\mathbb{R}$$ $$\hspace{17mm} = c + (di + a) + bi \hspace{10mm}\text{by commutativity in }\mathbb{R}$$ $$\hspace{17mm} = (c + di) + (a + bi) \hspace{11mm}\text{by associativity in }\mathbb{R}$$ $$\hspace{13mm} = z + w \hspace{34mm}\text{by definition of }\mathbb{C}$$  $$\blacksquare$$  This seems a bit verbose to me - is it acceptable to combine steps? For example, rather than:  $$w + z = (a + bi) + (c + di) \hspace{10mm}\text{by definition of }\mathbb{C}$$ $$\hspace{13mm} = a + (bi + c) + di \hspace{10mm}\text{by associativity in }\mathbb{R}$$ $$\hspace{17mm} = a + (c + bi) + di \hspace{10mm}\text{by commutativity in }\mathbb{R}$$ can I instead write: $$w + z = (a + bi) + (c + di) \hspace{10mm}\text{by definition of }\mathbb{C}$$ $$\hspace{48mm} = a + (c + bi) + di \hspace{10mm}\text{by associativity and commutativity in }\mathbb{R}$$?  (2) Commutativity of Multiplication:  Verify that $wz = zw$ for all $w, z \in \mathbb{C}$.  $$\hspace{-14mm}wz = (a + bi)(c + di) \hspace{24mm}\text{by definition of }\mathbb{C}$$ $$\hspace{4mm} = a \cdot c + a \cdot di + bi \cdot c + bi \cdot di \hspace{10mm}\text{by distribution in }\mathbb{R}$$ $$\hspace{9mm} = a \cdot c + bi \cdot c + a \cdot di + bi \cdot di \hspace{10mm}\text{by commutativity in }\mathbb{R}$$ $$\hspace{35mm} = c \cdot a + c \cdot bi + di \cdot a + di \cdot bi \hspace{10mm}\text{by multiplicative commutativity in }\mathbb{R}$$ $$\hspace{1mm} = c(a + bi) + di(a + bi) \hspace{20mm}\text{by distribution in }\mathbb{R}$$ $$\hspace{5mm} = (c + di)(a + bi)\hspace{34mm}\text{by distribution in }\mathbb{R}$$ $$\hspace{4mm} = zw \hspace{55mm}\text{by distribution in }\mathbb{R}$$ $$\blacksquare$$  This also seems a bit wonky to me. Any and all suggestions on clearing up my verification, reasoning, etc. would be greatly appreciated! Taylor Expansion $\log(\frac{1+z}{1-z})$ Thunderbird - remove unused dictionaries from menu Every Group is a Fundamental Group If the big bang was the birth of everything, and the big bang was an event in the sense that it had a location and a time (time 0), wouldn't that mean that our universe has a center?  Where was the big bang? What is there now? Are we moving away from the center still? Are these even valid questions? Uninstalling QGIS that was installed using OSGeo I installed QGIS using OSGEO4W, and would love to know how to uninstall it, because when I go to "Control Panel > Programs > Uninstall" the program does not appear in the list of installed programs.   How can I do this? What's the difference between  var A = function () {     this.x = function () {         //do something     }; };   and  var A = function () { }; A.prototype.x = function () {     //do something };