I'm currently resolving a problem, and I came across the following matrix for which I need an expression of the determinant. $$     \begin{matrix}     a+1 &amp; 1 &amp; 1 &amp; 1 &amp; . &amp; . &amp; . &amp; 1 &amp; 1 \\     -a &amp; a+1 &amp; 1 &amp; 1 &amp; . &amp; . &amp; . &amp; 1 &amp; 1 \\     0 &amp; -a &amp; a+1 &amp; 1 &amp; . &amp; . &amp; . &amp; 1 &amp; 1 \\     0 &amp; 0 &amp; -a &amp; a+1 &amp; . &amp;  &amp;  &amp; . &amp; . \\     . &amp; . &amp; . &amp; . &amp; . &amp; . &amp;  &amp; . &amp; . \\     . &amp; . &amp;  &amp; . &amp; . &amp; . &amp; . &amp; . &amp; . \\     . &amp; . &amp;  &amp;  &amp; . &amp; . &amp; . &amp; 1 &amp; 1 \\     0 &amp; 0 &amp; . &amp; . &amp; . &amp; 0 &amp; -a &amp; a+1 &amp; 1 \\     0 &amp; 0 &amp; . &amp; . &amp; . &amp; . &amp; 0 &amp; -a &amp; a+1 \\     \end{matrix} $$ where $a$ is a real number. This matrix can also be described like this:  $$ m_{i, j} =  \begin{cases} 1,  &amp; \text{if $i&lt;j$} \\ a+1, &amp; \text{if $i=j$} \\ -a, &amp; \text{if $i=j+1$} \\ 0, &amp; \text{if $i&gt;j+1$} \end{cases} $$  The size of the matrix is not known, potentially high (more than 1000).  I do not know how to solve this problem, nor can it be solved.  Edit:  Following achille hui advice, the problem is equivalent to compute an expression of the determinant of the next matrix:  $$     \begin{matrix}     2a+1 &amp; -a &amp; 0 &amp; . &amp; . &amp; . &amp; 0 &amp; 0 &amp; 0 \\     -a &amp; 2a+1 &amp; -a &amp; . &amp; . &amp; . &amp; 0 &amp; 0 &amp; 0 \\     0 &amp; -a &amp; 2a+1 &amp; . &amp; . &amp; . &amp; 0 &amp; 0 &amp; 0 \\     . &amp; . &amp; . &amp; . &amp; . &amp;  &amp;  &amp; . &amp; . \\     . &amp; . &amp; . &amp; . &amp; . &amp; . &amp;  &amp; . &amp; . \\     . &amp; . &amp;  &amp; . &amp; . &amp; . &amp; . &amp; . &amp; . \\     0 &amp; 0 &amp; 0 &amp;  &amp; . &amp; . &amp; 2a+1 &amp; -a &amp; 0 \\     0 &amp; 0 &amp; 0 &amp; . &amp; . &amp; . &amp; -a &amp; 2a+1 &amp; -a \\     0 &amp; 0 &amp; 0 &amp; . &amp; . &amp; . &amp; 0 &amp; -a &amp; a+1 \\     \end{matrix} $$ Be aware that the last coefficient is  $a+1$ and not $2a+1$.  Edit 2:  By developping the last row, we get ride of the last coefficient and we deal with tridiagonal matrices with constant diagonals.  Using the answer from , we get the following expression: $$ det_n = \frac{1}{\sqrt{4a+1}} \biggl[ \frac{1+\sqrt{4a+1}}{2} \biggl(\frac{2a+1+\sqrt{4a+1}}{2}\biggl)^n - \frac{1-\sqrt{4a+1}}{2} \biggl(\frac{2a+1-\sqrt{4a+1}}{2}\biggl)^n \biggl] $$
How to show that the determinant of the following $(n\times n)$ matrix  $$\begin{pmatrix} 5 &amp; 2 &amp; 0 &amp; 0 &amp; 0 &amp; \cdots &amp; 0 \\ 2 &amp; 5 &amp; 2 &amp; 0 &amp; 0 &amp; \cdots &amp; 0 \\ 0 &amp; 2 &amp; 5 &amp; 2 &amp; 0 &amp; \cdots &amp; 0 \\ \vdots &amp; \vdots&amp; \vdots&amp; \vdots &amp; \vdots &amp; \vdots &amp; \vdots \\ 0 &amp; \cdots &amp; \cdots &amp; 0 &amp; 2 &amp; 5 &amp; 2 \\ 0 &amp; \cdots &amp; \cdots &amp; \cdots &amp; \cdots &amp; 2 &amp; 5 \end{pmatrix}$$  is equal to $\frac13(4^{n+1}-1)$?    More generally:     How does one compute the determinant of the following  (where the three diagonals are constant)?      $$M_n(a,b,c) = \begin{pmatrix} a &amp; b &amp; 0 &amp; 0 &amp; 0 &amp; \cdots &amp; 0 \\ c &amp; a &amp; b &amp; 0 &amp; 0 &amp; \cdots &amp; 0 \\ 0 &amp; c &amp; a &amp; b &amp; 0 &amp; \cdots &amp; 0 \\ \vdots &amp; \vdots&amp; \vdots&amp; \vdots &amp; \vdots&amp; \vdots &amp; \vdots \\ 0 &amp; \cdots &amp; \cdots &amp; 0 &amp; c &amp; a &amp; b \\ 0 &amp; \cdots &amp; \cdots &amp; \cdots &amp; \cdots &amp; c &amp; a \end{pmatrix}$$      Here $a,b,c$ can be taken to be real numbers, or complex numbers.   In other words, the matrix $M_n(a,b,c) = (m_{ij})_{1 \le i,j \le n}$ is such that $$m_{ij} = \begin{cases} a &amp; i = j, \\ b &amp; i = j - 1, \\ c &amp; i = j + 1, \\ 0 &amp; \text{otherwise.} \end{cases}$$  There does not seem to be an easy pattern to use induction: the matrix is not a diagonal  of the type $M = \bigl(\begin{smallmatrix} A &amp; C \\ 0 &amp; B \end{smallmatrix}\bigr)$ (where we could use $\det(M) = \det(A) \det(B)$ for the induction step), and there are no lines or columns with only one nonzero entry, so  gets complicated quickly.  Is there a general pattern that one could use? Or is the answer only known on a case-by-case basis? It's possible to compute the determinant by hand for small $n$:  $$\begin{align} \det(M_1(a,b,c)) &amp; = \begin{vmatrix} a \end{vmatrix} = a  \\ \det(M_2(a,b,c)) &amp; = \begin{vmatrix} a &amp; b \\ c &amp; a \end{vmatrix} = a^2 - bc \\ \det(M_3(a,b,c)) &amp; = \begin{vmatrix} a &amp; b &amp; 0 \\ c &amp; a &amp; b \\ 0 &amp; c &amp; a \end{vmatrix} = a^3 - 2abc \end{align}$$  But there is no readily apparent pattern and the computation becomes very difficult when $n$ gets large.
Strictly increasing sequence $\{ a_k \}$ of positive integers such that $\sum 1/{a_k}$ is finite Derivation of the general forms of partial fractions I'm learning about partial fractions, and I've been told of 3 types or "forms" that they can take  (1) If the denominator of the fraction has linear factors:  $${5 \over {(x - 2)(x + 3)}} \equiv {A \over {x - 2}} + {B \over {x + 3}}$$  (2) If the denominator of the fraction has quadratic factors that don't factorise:  $${{2x + 3} \over {(x - 1)({x^2} + 4)}} \equiv {A \over {x - 1}} + {{Bx + C} \over {{x^2} + 4}}$$  (3) If the denominator has a factor that repeats:  $${{5x + 3} \over {(x - 2){{(x + 3)}^2}}} \equiv {A \over {x - 2}} + {B \over {x + 3}} + {C \over {{{(x + 3)}^2}}}$$    I'd appreciate it if someone could explain to me how these equivalence relationships are derived, essentially how do they take the forms that they do? I sort of understand the first one, but the other two I don't.  Thank you This is a 2015 IMO problem.  It seems difficult to solve.     Find all triples of positive integers $(a, b, c)$ such that each of the numbers $ab-c$, $bc-a$, and $ca-b$ is a power of $2$.   Four such triples are $(a,b,c)=(2,2,2),(2,3,2),(3,5,7),(2,6,11)$. I am trying to pickle some mangoes the Indian way. I boiled the jar in hot water, dried it and then added the mixture of spices, salt and mangoes to the jars after the mixture had cooled. I made the mixture in a frying pan to try and remove as much water as I could. After it was cooled, I put it in the jar and heated some mustard oil till it just barely started smoking. I let that cool and poured it over the mango spice mixture.  For the first three days it was fine. On the fourth day, I opened it to test the taste and ended up adding more salt. Now, there seems to be gas forming in the jar. I've noticed it a few times now. When I open the jar, I hear a whooshing noise as air escapes. It's quite a lot, since I opened the container about 3 times in a 10 hour period and each time gas escaped. Has this pickle gone bad? Or is this something that's normal? tkz-graph: changing individual vertex type I'm using these settings for most vertices:  \GraphInit[vstyle=Classic] \tikzset{VertexStyle/.append style = {minimum size = 5pt, inner sep = 0pt}}   But I'd like to have one or two vertices of my graph be hollow with a thick outer edge. I'm very confused about what options can be used in \Vertex, which ones can go in \SetUpVertex, for which I have to use \tikzset, etc. What's the best option here? How do I set a user environment variable? (permanently, not session) This is irritating me. I seen several suggestions (all using different files and syntax) and none of them worked.  How do I set an environment variable for a specific user? I am on debian squeeze. What is the exact syntax I should put in the file to make ABC = "123"? Necessary and sufficient condition for the matrix $A = I - 2 x x^t$ to be orthogonal I would like to be able to execute a SOQL query that will allow me to see which users have access to a certain record type for a certain object.  For example, if I have a case record type called Phone Team, I would like to make a query that returns the names of all the users that have been assigned the Phone Team record type, as can be seen in the native user interface in a profile's Object Settings page.  So far I have managed to get a query that can get the names of all the users that have, for example, read access to the case object:  SELECT Assignee.Name FROM PermissionSetAssignment      WHERE PermissionSetId IN          (SELECT ParentId FROM ObjectPermissions WHERE SObjectType = 'Case'               AND PermissionsRead = true AND Parent.IsOwnedByProfile = true)   However, there doesn't seem to be any way to access record type assignment in the same way that you can object permissions.  I see from  that the info can be obtained via the metadata API, but I would much prefer to do it by SOQL if possible.  Thanks How can I copy a file from another directory to the current one? I have this problem as part of a school assignment:  NOTE: In case it's hard to read in the image above, here's the directory tree: $ tree . `-- sample_dir1     `-- sample_dir         |-- admin         |-- cambridge         |   |-- cafeteria         |   |-- library         |   `-- security         |       |-- annex         |       |-- building         |       `-- parking         |-- faculty         |-- history.exe         |-- markham         |   |-- annex         |   |-- building1         |   `-- parking         |-- oxford         |   |-- outline.doc         |   |-- programming         |   |   `-- report.pdf         |   `-- security         `-- stenton             |-- gen_ed             `-- lib_arts                 |-- english.txt                 `-- match.doc  15 directories, 11 files  I tried doing the command: $ cp ../cambridge/security/parking ./parking2  ...but it isn't working. The question in the image above states that the current directory is stenton, and that we need to make a copy of the file named parking (from the security directory), and to name the new file parking2, and place it in the current directory, stenton, using relative pathnames. What am I doing wrong? My cp command seems to be correct. I have a repeated composition of functions ${T_n}(z) = {\tau _0} \circ {\tau _1} \circ {\tau _2} \circ  \cdots  \circ {\tau _n}(z)$  By analogy with $\sum\limits_{i = 1}^n {} ,\prod\limits_{i = 1}^n {} ,\bigcup\limits_{i = 1}^n {} ,\bigcap\limits_{i = 1}^n {} ,$ I want to write ${T_n}(z) = \left( {\mathop  \circ \limits_{i = 0}^n {\tau _i}} \right)(z)$ or even ${T_n}(z) = {\mathop  \circ \limits_{i = 0}^n {\tau _i}} (z)$. Can I do this? Does moving variable declaration outside of a loop actually increase performance? After update to OS X 10.11 I started having these bugs:   Finder sidebar text flashing  Very high CPU load:      I have Macbook Pro Retina 13" Mid 2014  I reset NVRAM.  I deleted com.apple.finder.plist, it didn't help.  Also I did this: , no longer flashes. I've seen some of the other threads on using strings as x axis labels, but nothing that fits my situation.  I'm trying to create a bar chart from a CSV file. I can do it so long as everything in the data is numeric. However, in my actual data table, the first row is multi-word string descriptions. I need these descriptions to be the x axis labels for each bar.  I've tried symbolic x coords, but I get the "could not parse input as a floating point number" error. I've also tried adding an ID column and using that as the x values, and assigning xtick labels as the string descriptors. I get the same issue that way.  Here's an attempt at an MWE:  \documentclass{article}  \usepackage{lmodern} \usepackage{tikz} \usepackage{pgfplots} \usepackage{pgfplotstable}  \begin{document}  \pgfplotstabletypeset[columns/category/.style={string type},col sep=comma]{Book3.csv} \vspace{1cm}  \begin{tikzpicture} \begin{axis}[ ybar, xlabel=Xstuff, ylabel=Value, symbolic x coords={cat a,cat b,cat c,cat d,cat e,cat f,cat g}, %xtick labels={cat a,cat b,cat c,cat d,cat e,cat f,cat g},% this is from when i tried the xtick label method xtick=data, nodes near coords, nodes near coords align={vertical}] \addplot table[x=category,y=value]{Book3.csv}; %\addplot table[x=category,y=value]{Book3.csv};&amp; again, from trying the xtick label method  \end{axis}  \end{tikzpicture} \end{document}   Since I suspect this might get answered quickly by you chart gurus, I have a follow-up question:  I have to do a bunch of these charts, and I would like to automate the symbolic x coords process. Is there a way to populate the dictionary of accepted x coords by reading a column of the data? (something like symbolic x coords={columns/category/{Book3.csv}} )?  Book3.csv looks like this (ignore the ID column if using the xtick label method):  ID, category, value, value 2 1,  cat a,    1,     7 2,  cat b,    2,     6 3,  cat c,    3,     5 4,  cat d,    4,     4 5,  cat e,    5,     3 6,  cat f,    6,     2 7,  cat g,    7,     1 Set String via String.prototype function without return Can you use an external 301 redirect for a blog post hosted on wordpress.com? How do I synchronise two folders in real-time in Windows 7? I want to synchronise two folders in real time under Windows 7.  Basically, I want to monitor a folder and synchronise each change (new files, changed files, deleted files) to another drive. It has to be in real time, so it gets synchronised instantly when a change happens. A one-direction synchronisation is enough.  I tried Microsoft's , but it does only syncing by hand or scheduled (thus not satisfying the "real-time" requirement).  Can this be achieved with Windows 7 itself? If not, are there other methods of doing this? How to repair Ubuntu installation after interrupted dist-upgrade without losing any custom settings? During my attempt to upgrade my Xubuntu 14.04 installation with  sudo apt-get dist-upgrade   I made the mistake of interrupting the process by closing my laptop, which caused it to suspend to disk. After re-opening, the monitor remained black, so I was forced to reboot the system. I have since discovered many failures of drivers (wlan0 missing, USB memory stick cannot be mounted, USB mouse not recognised, and maybe more undiscovered problems with hardware).   My beginner's attempts to manually repair the single problems could have made it even worse. During the installation of the current distribution I chose to encrypt my whole hard disk. How do I format a Microsoft JSON date? I'm taking my first crack at  with jQuery. I'm getting my data onto my page, but I'm having some trouble with the JSON data that is returned for Date data types. Basically, I'm getting a string back that looks like this:  /Date(1224043200000)/   From someone totally new to JSON - How do I format this to a short date format? Should this be handled somewhere in the jQuery code? I've tried the jQuery.UI.datepicker plugin using $.datepicker.formatDate() without any success.  FYI: Here's the solution I came up with using a combination of the answers here:  function getMismatch(id) {   $.getJSON("Main.aspx?Callback=GetMismatch",     { MismatchId: id },      function (result) {       $("#AuthMerchId").text(result.AuthorizationMerchantId);       $("#SttlMerchId").text(result.SettlementMerchantId);       $("#CreateDate").text(formatJSONDate(Date(result.AppendDts)));       $("#ExpireDate").text(formatJSONDate(Date(result.ExpiresDts)));       $("#LastUpdate").text(formatJSONDate(Date(result.LastUpdateDts)));       $("#LastUpdatedBy").text(result.LastUpdateNt);       $("#ProcessIn").text(result.ProcessIn);     }   );    return false; }  function formatJSONDate(jsonDate) {   var newDate = dateFormat(jsonDate, "mm/dd/yyyy");   return newDate; }   This solution got my object from the callback method and displayed the dates on the page properly using the date format library. How to remove URI encoding from file names? I realise this might be quite a simple question, but I'm still quite new to the command line and only have a grasp of basic commands.  I've downloaded some lecture presentations (~25 or so) from my University however on doing so they've been named things like...  L2%20Development%20of%20immune%20system.pptx L4%20Molecular%20Recognition.pdf   As you can see they've downloaded with the URL encoding %20 instead of a space.  My question is how to batch rename all of these files so that the %20 is removed and replaced with a space instead? How to capture a series of screenshots in Web Driver using Java