Showing $ 2 $ matrices are similar
How do I tell if matrices are similar?
Angle notation for complex numbers in polar form Absolute Value Symbols What is the "best LaTeX practices" for writing absolute value symbols?  Are there any packages which provide good methods?  Some options include |x| and \mid x \mid, but I'm not sure which is best... As a Graduate Mathematics student, my interest lies in Number theory. I am curious to know if Number theory has any connections or applications to physics. I have never even heard of any applications of Number theory to physics. I have heard Applications of linear algebra and analysis to many branches of physics, but not number theory.  Waiting forward in receiving interesting answers! How can one show there is no holomorphic function $f$ on the open unit disc $\mathbb{D}$ such that it extends continuously to $\frac{1}{z}$ on $\partial\mathbb{D}$? I mean $f$ takes value $\frac{1}{z}$ when $|z|=1$ and continuous here.  Thank you. Why was Severus Snape placed in Slytherin? Why "hide" Luke from his father somewhere Anakin knows? In Episode II Anakin goes to the Lars homestead and meets his extended family. How would sending Luke there be "hidden" to Vader in any way? At least Leia had a random planet and family. all the LaTeX-tutorials tell me how LaTeX is an extention simplifying the building of TeX documents. Understandably, no one uses plain TeX anymore, right?  But I seem to learn better when I grasp the link between TeX and LaTeX. Does someone have a link to example files in plain TeX? I'd like to know which commands, macros etc. are LaTeX-specific. What is postselection in quantum computing? A quantum computer can efficiently solve problems lying in the complexity class . I have seen a claim the one can (potentially, because we don't know whether BQP is a proper subset or equal to PP) increase the efficiency of a quantum computer by applying postselection and that the class of efficiently solvable problems becomes now postBQP = .  What does postselection mean here? Unable to install ECW support on lubuntu 14.04 I have two integers, x and y so that x &lt; y. How many prime numbers are there between x and y (exclusive). Is there a formula or algorithm to compute? How do I stop my rim brakes making a squeaking/squealing sound? I bought a Schwinn Frontier Sport few months ago and started riding happily. After a few weeks my front rim brakes started creaking. It's so annoying in public places. My rear brake doesn't produce any sound but isn't as powerful as the front brake.  The retailer says cleaning would solve the front brake's issues and I got it cleaned, but nothing changed. Is there something I can do something myself to fix this? Equation System with 3 variables "Find all the possible solutions $(x,y,z) \in$  $\Bbb R$ that satisfies the following equation system".  My teacher gave me this problem and i got some clues in how to solve it. I evaluated $1$ and $-1$ and i got these solutions $(1,-1,1)$ and $(-1,-1,-1)$. I think these are the only ones but i can't prove it.  Am i wrong?, are there more solutions?.   Thanks  $$ \left\{  \begin{array}{c} x+y-z=-1 \\  x^2-y^2+z^2=1\\  -x^3+y^3+z^3=-1 \end{array} \right.  $$ I have been thinking about ways of teaching electronics and I'm wondering if the following is true...  For starters, when we talk about voltage as energy per unit charge, is this energy manifest simply as the kinetic energy of the electron?  So, all other things being equal, will the electrons coming out of a 4 volt battery will have twice the velocity of electrons coming out of a 1 volt battery because of ${1 \over 2}mv^2$?  This would seem to imply that the voltage drop across different parts of the circuit is essentially a comparison of the electron speeds at those points.  Furthermore doesn't this imply that the electrons are nearly halted by the time they reach the positive terminal of the battery? This makes intuitive sense to me, it seems pedagogically sound, and it provides an explanation for how electrons "know" to give up their energy across the circuit -- electrostatic repulsion communicates later resistances to earlier parts of the circuit.  How much of the above is correct? The one things that makes me suspicious is that, in this picture, electron density at the positive terminal of the battery is very high, and repulsive forces would prohibit this. On the other hand, if the Joules per Coulomb are not coming from kinetic energy, then what? The field? But wouldn't that just dissipate as light?  Something is wrong but I don't know what. "can't" versus "couldn't", what is the difference? That can't be Obama at the door, it's too early.  That couldn't be Obama at the door, it's too early.    What is the difference between them? Are they almost the same? I'm trying, but something is wrong with my code in LaTeX: \begin{asy} pair dynkinMark(real x, real y, string s = &quot;&quot;, pen p = defaultpen, pair B = dir(45)) { if (s != &quot;&quot;) { dot(&quot;$&quot;+s+&quot;$&quot;, (x,y), B, p); }     else { dot((x,y), p); }     return (x,y); }  dynkinMark(0,1, &quot;1&quot;, dir(135)); dynkinMark(0,2, &quot;2&quot;, dir(135)); dynkinMark(0,3, &quot;3&quot;, dir(135)); dynkinMark(0,4, &quot;4&quot;, dir(135)); dynkinMark(7,1, &quot;1'&quot;); dynkinMark(7,2, &quot;2'&quot;); dynkinMark(7,3, &quot;3'&quot;); dynkinMark(7,4, &quot;4'&quot;); path p1 = (0,1)..(2,2.0)..(5,3.5)..(7,4); path p2 = (0,2)..(2,1.2)..(5,2.5)..(7,3); path p3 = (0,3)..(2,2.8)..(5,1.5)..(7,1); path p4 = (0,4)..(2,3.8)..(5,3.5)..(7,2); draw(p1, blue); draw(p2, heavycyan); draw(p3, mediumcyan); draw(p4, lightblue); dot(&quot;$12$&quot;, IP(p1, p2), dir(90), heavygreen); dot(&quot;$13$&quot;, IP(p1, p3), dir(90), heavygreen); dot(&quot;$23$&quot;, IP(p2, p3), dir(90), heavygreen); dot(&quot;$14$&quot;, IP(p1, p4), dir(90), heavygreen); dot(&quot;$24$&quot;, IP(p2, p4), dir(90), heavygreen); real y = 0.8; label(&quot;$s_1$&quot;, (IP(p1,p2).x, y), heavygreen); label(&quot;$s_2$&quot;, (IP(p1,p3).x, y), heavygreen); label(&quot;$s_1$&quot;, (IP(p2,p3).x, y), heavygreen); label(&quot;$s_3$&quot;, (IP(p1,p4).x, y), heavygreen); label(&quot;$s_2$&quot;, (IP(p2,p4).x, y), heavygreen); \end{asy} Functional equation $f(r \cos \varphi)+f(r \sin \varphi)=f(r)$ Find all such monotonous function $f:[0,+\infty)\to \mathbb R$, such that for any real $r\ge0$ and $\varphi \in \left[\frac{\pi}6, \frac{\pi}4\right]$, $$f(r \cos \varphi)+f(r \sin \varphi)=f(r)\text.$$  My work so far:  If $\varphi \in \left[\frac{\pi}6, \frac{\pi}4\right]$ then $$\frac{\sqrt2}2\le\cos \varphi\le\frac{\sqrt3}2$$ $$\frac12\le\sin \varphi\le\frac{\sqrt2}2$$  If $r=0$ then $f(0)=0$  If $\varphi= \frac{\pi}{4}$ then $$f(r)=4f\left(\frac r2\right)$$  $f(x)=cx^2$ for $x\ge0$ and $c\in\mathbb R$ When true polymorphed creature dies, does it revert to its original form? The  description says:     Creature into Creature.   The transformation lasts for the duration, or until the target drops to 0 hit points or dies. If you concentrate on this spell for the full duration, the transformation lasts until it is dispelled.   Does the second sentence imply the creature won't revert to its normal form, even being killed? (providing the caster were concentrating on the spell for the full duration)  If so, isn't "until the target drops to 0 hit points" enough? Why the "...or dies" addition? A creature can't die having more that 0 hit points, can it? Which B2C option to choose? A stock Salesforce org is geared toward B2B companies, but I work in Higher Ed and we're B2C-focused.  There seem to be three main B2C options available:   Person Accounts Nonprofit Starter Pack (NPSP) 1-to-1 model NPSP bucket model   Why would I choose one option over the other? Uncommitted work pending in unit test with trigger and queueable callout I have an after insert trigger that invokes queueable Apex which makes a callout.  The trigger and callout work fine in the UI and I don't get any errors in normal operation.  However, my unit test keeps failing with the "uncommitted work pending" error despite all my attempts to separate the callout.  I've used a testVisible variable to disable the callout from the trigger so that I can invoke the queueable job explicitly in a startTest/stopTest block after the DML that fires the trigger.  I've pored over the callout class to make sure it's not doing any DML beforehand.  Anyway, if it was doing DML, I should get the uncommitted work pending errors in the UI, right?  I've tried API version 37 and version 41 but the unit test still fails.   I've had a case open with Salesforce premier support for 6 weeks now but haven't made any progress.  Any suggestions?  The unit test:  @isTest static void testCallout() {     // Disable the callout while setting up data.     CCx_EHR_CMCT_Services.isQueueSendEnabled = false;      // Set up the data and execute the DML.     / ...     insert act;      // Set up the http mock     // ...      // Now enable and execute the callout     Test.startTest();     CCx_EHR_CMCT_Services.isQueueSendEnabled = true;     Test.setMock(HttpCalloutMock.class, mock);     System.enqueueJob(new QueueSendToRedox(new List&lt;ActivityHL__c&gt;{act}, desk)));     Test.stopTest();  }   Trigger:  trigger HL_ActivityHL_c on ActivityHL__c (before insert, before update, after insert) {     if (Trigger.isAfter) {         if (Trigger.isInsert) {             CCx_ClientDataDomain.sendEHRClinicalComm(Trigger.new);         }     } }   Trigger handler class:  public without sharing class CCx_ClientDataDomain {     public static void sendEHRClinicalComm(List&lt;ActivityHL__c&gt; triggerNew) {     // do some processing here.     // ...     // then queue the callout job.     System.enqueueJob(new CCx_EHR_CMCT_Services.QueueSendToRedox(             deskActivities.get((Id) desk.getDeskId()), desk));     } }   Queueable job class with callout method:  public with sharing class CCx_EHR_CMCT_Services {     // isQueueSendEnabled is an attempted workaround for unit tests     @testVisible     private static Boolean isQueueSendEnabled = true;      public class QueueSendToRedox implements Queueable, Database.AllowsCallouts {         private List&lt;ActivityHL__c&gt; activities;         private CCx_DeskHierarchySelector.Desk desk;         private Boolean isEnabled;          public QueueSendToRedox(List&lt;ActivityHL__c&gt; activities, CCx_DeskHierarchySelector.Desk desk) {             this.activities = activities;             this.desk = desk;             this.isEnabled = isQueueSendEnabled;         }          public void execute (QueueableContext context) {             if(this.isEnabled) sendClinicalCommToRedox(activities, desk);         }     }      public static void sendClinicalCommToRedox(List&lt;ActivityHL__c&gt; activities, CCx_DeskHierarchySelector.Desk desk) {         // Set up data for the callout.  No DML actions are executed before the callout.         HttpRequest req = buildRequest();         // ...          // Callout         Http http = new Http();         HttpResponse resp = http.send(req);     } } How can I store and retrieve a custom list in a blend file? I am trying to create a script that will allow me to generate a list and store it in the blend file. The list starts out blank and then strings get added to the list, resulting in: [&quot;test1&quot;,&quot;test2&quot;,&quot;test3&quot;] I create a PropertyGroup Class for the enum, define a new scene property as a CollectionProperty type with access to that enum Class, before appending a new item into the enum: import bpy  class item_set(bpy.types.PropertyGroup):     myItems = []     enums = bpy.props.EnumProperty(name=&quot;My Items&quot;, items=myItems) bpy.utils.register_class(item_set)  bpy.types.Scene.myItemSet = bpy.props.CollectionProperty(type=item_set)      newItem = bpy.context.scene.myItemSet.add() newItem.myItems.append(&quot;test&quot;)    I need to be able to add items to the list, save the blend file, re-open the file, run the script again and retrieve the list from the scene. However, the  states:  One thing to keep in mind is that after saving and reloading your blend-file, your new property will have disappeared...    ...To make the values you assigned in a previous session available again, you need to redefine the property. So after reopening the blend-file you need to rerun: bpy.types.Object.myProperty = bpy.props.StringProperty()   I can do the appending into and retrieval of the property values absolutely fine upon an initial run, but as the documentation states, the property is unavailable when trying to view it's contents on a reload of the file (after having run the script, saved and re-opened the file), but I am unsure how to redefine it without clearing all it's values. If I redefined the scene property, it would presumably also need the enum Class to be available, which would currently set myItems back to [] (empty). Do I need to change the way I am initially setting the items in the class or am I on entirely the wrong track with all of this? All I need to do is store a list in the blend file, so if there is a better way please let me know. Thanks. (I would hope, eventually, to be able to store sub lists [[&quot;a&quot;,&quot;b&quot;],[&quot;c&quot;,&quot;d&quot;],[&quot;e&quot;,&quot;f&quot;]], I'm just trying to get normal lists to work first.)