$GL(2,\mathbb{R})$ denote the group of two by two matrices with non-zero determinant. Let $A,B,C\in GL(2,\mathbb{R})$ and $A,B,C\neq aI$ where I is the identity of $GL(2,\mathbb{R})$ . Prove if $AB=BA $ and $BC=CB$ then $AC=CA$.     I try to proof this just use the properties of a group, but it seems not enough. There need to be some properties of matrices. More generally, does this property (if a commutes with b and b commutes with c,then a commutes with c).
$A,B,C$ are three $n\times n$ matrices such that $A$ has $n$ distinct eigenvalues, $AB=BA$, and $AC=CA$.   We need to show $BC=CB$.  Let $A$ be diagonalizable with eigenvectors say, $v_1,\dots,v_n$, so there exists $P$ such that $PAP^{-1}=D$. What more I can say from the information? Please help.
I have two large external drives with about 2.5 TB of data. About half the storage is images and videos that I want to put on a new NAS drive/device. I don’t want to carry over junk, duplicates and non photo/video files.   I am hoping Lightroom can help scour the mounds of folder structures on these drives and copy images/videos the new drive that I want to catalogue and keep clean as a working platform for all images and video. Once completed I plan to format the original drives and copy the cleaned-up content back to them as part of a backup strategy.  Can Lightroom help? If so, how? If not, is there some other application for Mac OS X that I can use for help with this? One problem I had with my original attempt is I wasn't sure how to move images and rename folders to get the new drive contents to make sense, so I formatted it and am looking to try again.  (steps for this operation would be helpful) Appending to the end of a file with NSMutableString Can't restart nginx, bind() failed (98: Address already in use) How do I skip specific tests when building an rpm? I am building an RPM of git version 2.13.0. The build has gone ok but one of the tests are failing:  Config variable ${LIB_HTTPD_SVN} is not defined Config variable ${LIB_HTTPD_SVN_PATH} is not defined   I cannot find any info about these variables so I was going to just skip the tests. How can I do that? Follow up: instance control in Java without enum How could I go about detecting (returning true/false) whether an ArrayList contains more than one of the same element in Java?  Many thanks, Terry  Edit Forgot to mention that I am not looking to compare "Blocks" with each other but their integer values. Each "block" has an int and this is what makes them different. I find the int of a particular Block by calling a method named "getNum" (e.g. table1[0][2].getNum(); How to take a screenshot of a whole desktop with app menu selection? help with some Hilbert style proofs in a propositional logic axiom system. I'm very new to classical propositional logic, and my lecturer is using a system with the following axioms:  A1. X→(Y→X)  A2. (X→(Y→Z))→((X→Y)→(X→Z))  A3. (¬Y→¬X)→(X→Y)  Use uniform substitution and Modes Ponens (MP) as rule of inference. Deduction Theorem also allowed.  I'd like to deduce the following:  (a) X, Y ⊢ ¬(X→(¬Y)) (conjunction, but expressed in terms of NOT and IMPLIES)  (b) X→((X→Y)→Y)  (c) (X→Y)→(((¬Y)→X)→Y)  (d) (((¬Y)→X)→Y)→(X→Y)  Remark: The last two are to show that the axiom (A3) can be replaced with a different axiom:  B3. (¬Y→¬X)→(((¬Y)→X)→Y).  Solution: Bram28 proves (b) and (c) below. I demonstrate (a) and (d).  (d) is alright; first, I claim that X, ((¬Y→X)→Y) ⊢ Y.  Axiom (A1) gives ⊢X→(¬Y→X) so that ((¬Y→X)→Y)⊢X→Y by Hypothetical Syllogism (HS), which Bram28 shows to be valid.  Now that we have ((¬Y→X)→Y)⊢X→Y just apply the deduction theorem to get what we want: ⊢((¬Y→X)→Y)→(X→Y).  (a) is slightly trickier. Axiom (A1) gives ⊢ Y→(X→Y) so that X, Y ⊢ (X→Y).  Next, by part (b), we have as a theorem that ⊢X→((X→¬Y)→¬Y).  Therefore X, Y ⊢(X→¬Y)→¬Y.  But ⊢(¬¬(X→¬Y))→(X→¬Y) by DN Elim, which Bram28 shows to be valid.  By HS, X, Y ⊢(¬¬(X→¬Y))→¬Y.  By Axiom (A3) and MP, X, Y ⊢(Y→¬(X→¬Y)).  Also, rather trivially X, Y ⊢Y.  Finally by MP we conclude X, Y ⊢¬(X→¬Y) as required. Hibernate: comparing current & previous record I want to compare the current value of an in-memory Hibernate entity with the value in the database:  HibernateSession sess = HibernateSessionFactory.getSession(); MyEntity newEntity = (MyEntity)sess.load(MyEntity.class, id); newEntity.setProperty("new value"); MyEntity oldEntity = (MyEntity)sess.load(MyEntity.class, id); // CODEBLOCK#1 evaluate differences between newEntity and oldEntity sess.update(newEntity);       In CODEBLOCK#1 I get that newEntity.getProperty()="new value" AND oldEntity.getProperty()="new value" (while I expected oldEntity.getProperty()="old value", of course). In fact the two objects are exactly the same in memory.  I messed around with HibernateSessionFactory.getSession().evict(newEntity) and attempted to set oldEntity=null to get rid of it (I need it only for the comparison):  HibernateSession sess = HibernateSessionFactory.getSession(); MyEntity newEntity = (MyEntity)sess.load(MyEntity.class, id); newEntity.setProperty("new value"); HibernateSessionFactory.getSession().evict(newEntity); MyEntity oldEntity = (MyEntity)sess.load(MyEntity.class, id); // CODEBLOCK#1 evaluate differences between newEntity and oldEntity oldEntity = null; sess.update(newEntity);   and now the two entities are distinct, but of course I get the dreaded org.hibernate.NonUniqueObjectException: a different object with the same identifier value was already associated with the session.  Any idea?  EDIT: I tried the double session strategy; I modified my HibernateSessionFactory to implement a map of session and then...  Session session1 = HibernateSessionFactory.getSession(SessionKeys.DEFAULT); Session session2 = HibernateSessionFactory.getSession(SessionKeys.ALTERNATE); Entity newEntity = (Entity)entity; newEntity.setNote("edited note"); Entity oldEntity = (Entity)session1.load(Entity.class, id);  System.out.println("NEW:" + newEntity.getNote()); System.out.println("OLD: " + oldEntity.getNote()); // HANGS HERE!!!  HibernateSessionFactory.closeSession(SessionKeys.ALTERNATE);   My unit test hangs while attempting to print the oldEntity note... :-( With services like Google Finance or Yahoo Finance it is very convenient to find the historical stock price metric of a listed company, but if the company is not listed, I can't find the data.  For example, I want to see the price metrics for Macromedia, which had been listed in 1993 on NASDAQ (MACR), then acquired by Adobe in 2005. How could I find its historical price metrics? How do I access previous promise results in a .then() chain? How fill face between irregular curves? Hello Blender Stack Exchange,  I am trying to fill a face between 4 surfaces. Two are straight edges and two are a curved line made up of short vertices that came from a SketchUp export.  The problem: I select the vertices on the top and bottom and fill the hole. It creates a strange face that ruins the curve. I would prefer it create some sort of complex curve. I want it to be smooth.  Images attached to better explain. Prove the inequality and limitation Suppose $\{a_k\}$ ,$\{b_k\}$ and $\{\xi_k\}$ are non negative, and for all $k\ge 0$ , we have $$a_{k+1}^2\le(a_k+b_k)^2-\xi_k^2$$   Prove   1.$$\sum_{i=1}^{k}\xi_i^2\le(a_1+\sum_{i=0}^kb_i)^2$$   2. If $\{b_k\}$ also satisfy $\sum_{k=0}^{\infty}b_k^2\lt+\infty $, then $$\lim_{k\to\infty}\frac{1}{k}\sum_{i=1}^k\xi_i^2=0$$   After doing this transformation:$$\xi_k^2\le(a_k+b_k)^2-a_{k+1}^2$$ $\Rightarrow$ $$\sum_{i=1}^{k}\xi_k^2\le(a_1+b_1-a_2)(a_1+b_1+a_2)+(a_2+b_2-a_3)(a_2+b_2+a_3)+\cdots(a_k+b_k-a_{k+1})(a_k+b_k+a_{k+1})$$ then I can't move forward any more. Identify a movie in which aliens believed a sci-fi TV show to be true documentary I remember watching a sci-fi movie in the early 00s. It was full of comedy. Here are the details I recalled:   There was a space opera TV show in the movie. It seemed a true copy of Star Trek (they used to pilot a ship and there were captain, first officer, science officer etc on bridge). The name of the TV show was possibly the name of the movie. Once the main crew of the TV show were celebrating some achievement, they were approached by humans having weird behavior (they were actually aliens in disguise; in reality, they looked like octopus). The alien race was innocent. They couldn't comprehend the concept of art, lies etc. So, they believed that the TV show (they got the EM waves) was true historical documentary of mankind. So, they came looking for help from the brave captain of the show. From the TV show details, they had built a starship which really worked. The aliens needed the TV show crew to pilot the ship to defeat another alien race which terrorized them. The TV show crew didn't have any problems piloting the ship as the controls were familiar, only these controls were real.   Can you please identify the movie? Email SMTP validator I need to send hundreds of newsletters, but would like to check first if email exists on server. It's called  validation, at least I think so, based on my research on Internet.    There's several libraries that can do that, and also a page with open-source code in  (), but I have hard time reading ASP Classic, and it seems that it uses some third-party library...    Is there some code for SMTP validation in C#, and/or general explanation of how it works? How to fix task manager memory column if it shows lies Parent Field in Lightning Datatable Are Parent Fields supported in lightning Datatable?  I'm trying to display Account Name with Contact information, but it seems parent fields are not supported in lightning datatable.  Component:  &lt;aura:component controller="PagingSortingController"&gt;     &lt;aura:attribute name="mydata" type="Object"/&gt;     &lt;aura:attribute name="mycolumns" type="List"/&gt;     &lt;aura:handler name="init" value="{! this }" action="{! c.init }"/&gt;     &lt;lightning:datatable data="{! v.mydata }"          columns="{! v.mycolumns }"          keyField="con.id"         onrowselection="{! c.getSelectedName }"/&gt; &lt;/aura:component&gt;   Controller:  ({     init: function (cmp, event, helper) {     cmp.set('v.mycolumns', [                 {label: 'Contact Name', fieldName: 'Name', type: 'text'},                 {label: 'Phone', fieldName: 'Phone', type: 'phone'},                 {label: 'Email', fieldName: 'Email', type: 'email'},                 {label: 'Account Name', fieldName: 'Account.Name', type: 'text'}             ]);         helper.getData(cmp);     }  })   Helper:  ({     getData : function(cmp) {         var action = cmp.get('c.getContacts');         action.setCallback(this, $A.getCallback(function (response) {             var state = response.getState();             if (state === "SUCCESS") {                 cmp.set('v.mydata', response.getReturnValue());             } else if (state === "ERROR") {                 var errors = response.getError();                 console.error(errors);             }         }));         $A.enqueueAction(action);     } })   Apex Controller:  @AuraEnabled     public static List&lt;Contact&gt; getContacts() {         List&lt;Contact&gt; contacts =                  [SELECT Id, Name, Phone, Email,Account.Name FROM Contact];         return contacts;     } Why does aptitude give me intel-microcode upgrades when I have an AMD processor? I have a Supermicro AS-2022G-URF (H8DGU motherboard) with 2 AMD Opteron 6272 16C/16T 2.1GHz and an LSI MegaRAID SAS 2108 RAID controller currently running Ubuntu Server 14.04 LTS (scheduled to be upgraded to 16.04 this summer). Several times over the last year, when I have done aptitude update/upgrade, the intel-microcode package has shown up as one of the packages that will be installed. Since this server has AMD processors, not Intel, Why does aptitude offer this package? Is it in any way necessary for other reasons? Any ill effects from going along with aptitude on this? Make a computer act as a virtual USB device for other equipments This question has troubled me for ages despite my several attempts of looking it up in dictionaries or usage books. Do we say, "Do you have any ideas" or "Do you have any idea"? I do see an example where "any" means "it doesn't matter who/which/what", therefore "You can borrow any book you like." Also, does it matter between using it in questions and negatives?