Manipulate - Plot - Function with Parameters
How are parameters evaluated for a Plot in Manipulate
Let $A$ be a real square matrix en let $I$ be the identity matrix with the same size. Let $n&gt;0$ be the smallest natural number for which there exist  $a_0,a_1,...a_{n-1} \in {R}$ such that $A^n+a_{n-1}A^{n-1}+....+a_1A+a_0I=O$.  Show that A is invertible if and only if $a_0\neq0 $.  So i need to proof    $A$ invertible $\rightarrow$$a_0 \neq 0$ $a_0 \neq0 \rightarrow$ $A$ invertible    I don't really know where to start with 1, and with 2 I tried to prove it with a contradiction but I don't think it's right either. What was the point of testing carbonite-freezing technology on Han Solo? Darth Vader tested carbonite-freezing technology on Han Solo because he didn't want to kill Luke who was precious to the emperor. But, why was it necessary? Wasn't it already a tested and proven technology? Even Darth Vader (when he was Anakin) himself used it along with others decades ago (Star Wars: The Clone Wars S03E18). Diagonalisation and characteristic polynomial Let $A\in M_3(\Bbb R)$ which is not a diagonal matrix. Let $P$ be a polynomial (in one variable), with real coefficients and of degree 3 such that $P(A) = 0$. Pick out the true statements:  a. $P = cP_A$ where $c \in\Bbb R$ and $P_A$ is the characteristic polynomial of $A$;  b. if $P$ has a complex root (i.e. a root with non-zero imaginary part), then $P = cP_A$, with $c$ and $P_A$ as above;  c. if $P$ has a complex root, then $A$ is diagonalizable over $\Bbb C$.  I guess b and c are true although I am unable to disprove a. Because A could satisfy $(X-A)$ but this is not a polynomial. Now if $P$ has a complex root, then its conjugate too is its root as the polynomial has  real coefficients. As we all know An odd degree polynomial has always a real root, what we have is three distinct roots. Since $A$ satisfies the polynomial the only difference between the characteristic polynomial and $P$ has to be a constant. Proving $|P(A\cap B)-P(A)P(B)|\leq \frac{1}4$ Let $A$ and $B$ be two events of a probability space. Prove that  $\displaystyle|P(A\cap B)-P(A)P(B)|\leq \frac{1}4$   I think it's a very challenging problem, and I've made no progress so far ...  Can someone give me a hint ? Where are temporary files in Chrome stored? Is there an easy way to TeX geometric pictures? Best way to find all factors of a given number IPv4 vs IPv6 priority in Windows 7 I have IPv6 connectivity through Hurricane Electric tunnel. Since IPv6 day this year, many services (google.com, facebook.com, etc.) enabled IPv6 on their main domains. On my Windows machine, IPv6 is preferred over IPv4. This means that whenever I visit Google, all traffic goes through my tunnel to Hurricane Electric, which raises the latency by more than 100%:  C:\&gt; ping www.google.com  Pinging www.l.google.com [2001:4860:8005::68] with 32 bytes of data: Reply from 2001:4860:8005::68: time=85ms Reply from 2001:4860:8005::68: time=84ms Reply from 2001:4860:8005::68: time=112ms Reply from 2001:4860:8005::68: time=86ms  Ping statistics for 2001:4860:8005::68:     Packets: Sent = 4, Received = 4, Lost = 0 (0% loss), Approximate round trip times in milli-seconds:     Minimum = 84ms, Maximum = 112ms, Average = 91ms   C:\&gt; ping -4 www.google.com  Pinging www.l.google.com [173.194.79.103] with 32 bytes of data: Reply from 173.194.79.103: bytes=32 time=28ms TTL=48 Reply from 173.194.79.103: bytes=32 time=28ms TTL=48 Reply from 173.194.79.103: bytes=32 time=55ms TTL=46 Reply from 173.194.79.103: bytes=32 time=29ms TTL=46  Ping statistics for 173.194.79.103:     Packets: Sent = 4, Received = 4, Lost = 0 (0% loss), Approximate round trip times in milli-seconds:     Minimum = 28ms, Maximum = 55ms, Average = 35ms   Question: How can I make Windows 7 to always prefer IPv4, when both IPv4 and IPv6 records are available for a specific domain name? How can I write SQL for a table that shares the same name as a protected keyword in MySql? How do I instantiate a string array object into a Spring.Net container? Alternatively, why does the force created by blowing out air feel so much stronger than the force created by sucking in air?  Ok, so forget the human factor involved in blowing out candles. Consider a vacuum cleaner with a suction end and a blower end. Anyone who has tried it out notices the blower end creates a much stronger force than the suction end, despite the discharge being (more or less) equal at both ends.  Why does this happen? How can a file be copied? How do I copy a file in Python?  I couldn't find anything under . how to stop firing oncomplete if save action has errors My requirement is to stop oncomplete call if save action has errors. I know I am doing some silly mistake.   Here I am unable to stop calling oncomplete function.  Requirement is, from VFP save method, data will get saved and then perform callout from oncomplete function. If there are any errors during save then stop making callouts.  Visualforce  &lt;apex:page id="hasErrorTestPage" controller="ErrorMessageFinder"  showHeader="false"&gt;     &lt;apex:pageMessages id="msgId"/&gt;     &lt;script&gt;         function performCallOutMethod()         {             var msg = document.getElementById('{!$Component.MSEForm.errorMsgInd}').value;             alert(msg);              if(msg == 'false' || typeof(msg) == 'undefined')             {                 alert('entering callout')                 performCallOut();                 return true;             }             else             {                 return false;             }         }     &lt;/script&gt;     &lt;apex:form id="MSEForm"&gt;         &lt;apex:actionFunction name="performCallOut" action="{!submit}" reRender="panelId,msgId"/&gt;          &lt;apex:outputPanel id="panelId"&gt;             &lt;apex:outputText value="{!hasError}" id="errorMsgInd"/&gt;         &lt;/apex:outputPanel&gt;         &lt;apex:inputText id="customerNm" value="{!CustomerName}"/&gt;&lt;br/&gt;         &lt;apex:commandButton value="Save" action="{!save}"                       oncomplete="return performCallOutMethod();" reRender="panelId,msgId"/&gt;     &lt;/apex:form&gt; &lt;/apex:page&gt;   Controller  public class ErrorMessageFinder {     public String CustomerName {get;set;}      public Boolean hasError {          get {              return ApexPages.hasMessages();          }            }      public PageReference save()     {            if(String.isBlank(CustomerName))         {             ApexPages.Message msg = new ApexPages.Message(ApexPages.Severity.Error, 'Name cannot be null');             ApexPages.addMessage(msg);             return null;         }         return null;      }       public PageReference submit()     {         try         {   //navigate to view page             PageReference pg = new PageReference('/'+ '0019000001UwnVS');             pg.setRedirect(true);             return pg;         }         catch (Exception ex)         {             System.debug('Error in submit ' + ex);             ApexPages.Message msg = new ApexPages.Message(ApexPages.Severity.Error, ex.getMessage());             ApexPages.addMessage(msg);              return null;         }     } }   Based on this code, if I hit save button without entering CustomerName then it is not showing error after save. It is showing error during oncomplete call and then navigating to view page, which I do not want.  Any help is appreciated. I have put a compiled code so that anyone can try with this code. sample code for Dynamically populating dropdown in JSP How do I remove "Ubuntu" in the bios boot menu? (UEFI) Why do we say that irreducible representation of Poincare group represents the one-particle state? Only because   Rep is unitary, so saves positive-definite norm (for possibility density), Casimir operators of the group have eigenvalues $m^{2}$ and $m^2s(s + 1)$, so characterizes mass and spin, and It is the representation of the global group of relativistic symmetry,    yes? How to use Cubic to create a custom Ubuntu live CD image? Can I concatenate multiple MySQL rows into one field? Why, oh why, don't the Weasleys "conjure" up some diamonds/gold/etc, sell 'em to Muggles, and get rich? They could buy a mansion (like the Malfoys), and live there happily ever after.  I know there's  that prohibits one from conjuring food (which doesn't make much sense to me, either). Even if it applied to gold, what about all the other elements/compounds/materials? Could the law really be 'don't let a wizard conjure something if it'll make them richer'? That's totally nonsensical.  Why does Wizard society still have poor people, especially if they're talented? Random plot based on a specific random seed