What is the trick for evaluating the determinant of this matrix?  $$\begin{bmatrix} 2 &amp; -1 \\ -1 &amp; 2 &amp; -1 \\ &amp; -1 &amp; 2 &amp; -1 \\ &amp;&amp; -1 &amp; 2 &amp; -1 \\ &amp;&amp;&amp; -1 &amp; 2 &amp; -1 \\ &amp;&amp;&amp;&amp; -1 &amp; 2 \end{bmatrix}$$  $$%Commenting out original picture ![enter image description here][1]$$
Let $A_n$ denote an $n \times n$ tridiagonal matrix.  $$A_n=\begin{pmatrix}2 &amp; -1 &amp; &amp; &amp; 0 \\ -1 &amp; 2 &amp; -1 &amp; &amp; \\ &amp; \ddots &amp; \ddots &amp; \ddots &amp; \\ &amp; &amp; -1 &amp; 2 &amp; -1 \\ 0 &amp; &amp; &amp; -1 &amp; 2 \end{pmatrix} \quad\text{for }n \ge 2$$  Set $D_n = \det(A_n)$  Prove that $D_n = 2D_{n-1} - D_{n-2}$ for $n \ge 4$.
What are those zigzagged folded paper things you see in supernatural anime? In a lot of supernatural anime, like Shin Sekai Yori, or Tokyo Ravens there often appear these large ropes with folded paper things hanging from them:    As far as I can tell, they serve to seal or mark places that are sacred or in some way supernatural, like shinboku, or holy trees:    What are these folded paper things, and what purpose do they really serve? Overriding the final destination of the OAuth2.0 flow from a custom OAuth2.0 Auth provider Background:  I have created a custom Auth. Provider in Salesforce to reach out to an authorization server that isn't on the default list of Provider Types. I used  as a good starting point to extend the Auth.AuthProviderPluginClass. Once appropriately added to my org, it then became an option on the list of Provider Types. I was able to create an Auth. Provider using that new provider type and it works to initiate the OAuth2.0 flow. I am appropriately able to get the access token and refresh token from the authorization server.  The Problem:  As it currently stands, we will have users click on a button that will send them to the provided "OAuth-Only Initialization URL" link that was generated when I created the actual Auth. Provider record. The trick is that when the OAuth2.0 flow is completed, the user is redirected back to the Salesforce Setup Home page, and I don't see a good way in the documentation to override the final destination to be something else (like a success page or something).  Question:  Does anyone know of a way to override the final destination of the OAuth2.0 flow from a custom OAuth2.0 Auth provider?    UPDATE: I tried using the startUrl (and also tried retUrl for good measure) as query params, but doing so didn't change the behavior after the OAuth2.0 flow finished.  Meaning, both of the following URLs gave me the same final destination as not having any query params at all:  https://&lt;scratch org random url&gt;.my.salesforce.com/services/auth/oauth/MyCustomAuthProvider?retUrl=lightning%2Fpage%2Fhome   https://&lt;scratch org random url&gt;.my.salesforce.com/services/auth/oauth/MyCustomAuthProvider?startUrl=lightning%2Fpage%2Fhome     Another Update: This WORKED:  https:/&lt;scratch org random url&gt;.my.salesforce.com/services/auth/oauth/MyCustomAuthProvider?startURL=%2Flightning%2Fn%2FCustomPage   Apparently, it was important to add the leading /, (of course, url encoded to be %2F) and %2Flightning%2Fpage%2Fhome gets interpreted as the setup home and sends you there.    (Parallel posted to Salesforce Dev Forum: ) In LWC am trying to iterate over a list and show the details from the list, this is working correctly , but as a header want to show the record number in a displayed order, below is desired outcome and simplified sample code.  Desired OutCome:   Employee Record 1: A  Employee Record 2: B  Employee Record 3: C   JavaScript:  Var empNames = ["A" , "B" , "C"];   HTML:  &lt;template for:each={empNames} for:item="employee" for:index="index"&gt; Employee Record{index} {employee} &lt;/template&gt;    Actual outcome  Employee Record 0: A  Employee Record 1: B  Employee Record 2: C   I understand as i am using {index} which indicated the position of element in an array which starts from 0 my actual outcome also starts like Employee Record 0 instead of Employee Record 1, but is there a simple way without writing much logic on JS to handle this in html and to make my headers starts from 1 instead of 0? Sum of columns for rows containing a specific term How this java code produces deadlock? I want to animate a character in a realistic but easy way. I was thinking that I could use videos of me performing the actions I want to have on my characters, however I can't find a way of placing the videos and make the character to follow my movements in an automated way, like motion tracking. How can I animate automatically using this technique? Are there any Plug-ins to work like this? Is this mehod actually more useful than standard animation techniques? limit: $u_n=\frac{1+2^2+3^3+...+n^n}{n^n}$. Let $u_n=\dfrac{1+2^2+3^3+...+n^n}{n^n}$. Prove that : $$\lim u_n=1$$ I tried to prove that $u_n&lt; something$ have limit  is 1 ; but I can't List of lists changes reflected across sublists unexpectedly Why does Simplify ignore an assumption? In a standing wave, how does energy travel past a node?  It should just get reflected. Assume the case of first overtone and you strike the string at a place. How will energy distribute itself?  If it distributes over the whole string, then how is it the wave traveled past a node?  Basically the problem is a standing wave has been set up in the string . Now you strike the string at a place, and this creates a new pulse. Now this pulse will travel past the next node in the string (not the boundary) or will it reflect off even that node and remain confined to a particular region of the string? Why are there holes in the icing? Bash - Integer expression expected Salesforce query on profile not working on user language change I have a query on profile like  profilename = [Select name from profile where id=:userinfo.getUserprofileid ].name  Now I have a check in  VF page  like  profilename  == 'System Administrator' . this works fine if the logged in user language is english. If the user language is different then profile name will be  'System admin' in that language . So how to prevent this beahviour?   Also  is there any particular field in profile which I can use  to identify the profile is system admin? so that I can have check on that field instead of checking for name which is causing the problem, for example I think modifyall filed is only checked for system admin. I'm looking for a Drupal Commerce Shipping in-cart estimate Does anyone know the major differences between the Java and .Net garbage collectors? A web search has not revealed much, and it was a question that came up in a test. How come $32.5 = 31.5$? (The "Missing Square" puzzle.) What is a good introduction to statistics for a mathematician who is already well-versed in probability?  I have two distinct motivations for asking, which may well lead to different suggestions:   I'd like to better understand the statistics motivation behind many problems considered by probabilists. I'd like to know how to better interpret the results of Monte Carlo simulations which I sometimes do to form mathematical conjectures.   I'm open to the possibility that the best way to go is not to look for something like "Statistics for Probabilists" and just go to a more introductory source. Parallelize Bash script with maximum number of processes Lets say I have a loop in Bash:  for foo in `some-command` do    do-something $foo done   do-something is cpu bound and I have a nice shiny 4 core processor. I'd like to be able to run up to 4 do-something's at once.  The naive approach seems to be:  for foo in `some-command` do    do-something $foo &amp; done   This will run all do-somethings at once, but there are a couple downsides, mainly that do-something may also have some significant I/O which performing all at once might slow down a bit. The other problem is that this code block returns immediately, so no way to do other work when all the do-somethings are finished.  How would you write this loop so there are always X do-somethings running at once? Is there any known relationship between $\sum_{i=1}^{n} f(i)$ and $\sum_{i=1}^{n} \dfrac {1}{f(i)}$ If we knew $\sum_{i=1}^{n} f(i)=S$  ($n$ can be $\infty$), is there anything we could say about $\sum_{i=1}^{n} \dfrac {1}{f(i)}$ in terms of $S$?  The only thing I was able to find on the web is . If there is no known relationship, is it possible for there to exist one that we just haven't discovered? That's really embarrassing, however I need to ask it. I could not prove that the normed space    $\langle \mathscr{C} [0, 1], \| \cdot \|_1 \rangle $ is complete (as a metric space), where $\| f\|_1 = \int_0^1 |f(x)|dx$. I don't really know how to explicit a continuous functions as a limit of a Cauchy sequence. I tried to prove that the pointwise limit (of the Cauchy sequence) have just a finite number of discontinuities (I don't know if it's true) and then approximate it by a continuous function.  Thanks in advance.