This is my first question, so I’m sorry if I made mistakes.    $\left(\begin{array}{ccccc}1&amp;1&amp;1&amp;\cdots&amp;1\\2&amp;2^2&amp;2^3&amp;\cdots&amp;2^n\\3&amp;3^2&amp;3^3&amp;\cdots&amp;3^n\\\vdots&amp;\vdots&amp;\vdots&amp;\ddots&amp;\vdots\\n&amp;n^2&amp;n^3&amp;\cdots&amp;n^n\end{array}\right)$    In order to solve the above inverse matrix,  I tried to solve the below liner simultaneous equations in n-th unknowns.    $kx_1+k^2x_2+\cdots+k^nx_n=1+2^{n-1}+\cdots+k^{n-1}$ $k=1,2,\cdots,n$  Inductively, I know the solution of the liner simultaneous equations.  $x_k=\frac{B_{n-k}}{n-k} {n-1 \choose n-k-1} (1≦k&lt;n-1)$ $x_{n-1}=\frac{1}{2}$ $x_n=\frac{1}{n}$  But even though I solved the solutions of the equations, I have no ideas how to use it.    I thought Vandermonde’s déterminant may be effective, but I have no ideas how to use it.
I'm working through Exercise 40 from section 1.2.3 of Knuth's The Art of Computer Programming volume 1, but am finding myself unable to produce a rigorous proof, and the one  is suspect and not quite clear enough (for me, at least) in some of the steps; in particular, how to "[identify] the $k$th order coefficient in [the] two polynomials."  The problem is this: given a Vandermonde matrix $[x_j^i]_n$, show that the inverse is given by  $$ [b_{ij}]_n = \left[ \frac{    \sum_{\substack{1 \leq k_1 &lt; \dotsc &lt; k_{n-j} \leq n\\k_1,\dotsc,k_{n-j} \neq i}} (-1)^{j-1} x_{k_1} \dotsc x_{k_{n-j}} }{    x_i \prod_{\substack{1 \leq k \leq n\\k \neq i}} (x_k - x_i) } \right]_n\text{.} $$  The author gives a hint by stating that the sum in the above numerator is just the coefficient of $x^{j-1}$ in the polynomial $(x_1-x)\dotsc(x_n-x)/(x_i-x)$; and he gives an intermediate result showing the explicit multiplication of the matrix and its inverse as  $$    \sum_{1 \leq t \leq n}b_{it}x_j^t    =    \frac{       x_j \prod_{\substack{1 \leq k \leq n\\k \neq i}} (x_k - x_j)    }{       x_i \prod_{\substack{1 \leq k \leq n\\k \neq i}} (x_k - x_i)    }    =    \delta_{ij}\text{.} $$  The only other hint as to the type of solution he was expecting is a reference to A. de Moivre's The Doctrine of Chances, 2nd edition, pp. 197-199, which deals with polynomial recurrence relations and difference products (available ).  At the least, I was just hoping someone could either verify the proof is correct at proofwiki and possibly fill in exactly how one identifies the $k$th order coefficient in the proof; or perhaps explain a proof strategy as to what steps to take where the intermediate result is obtained at some point before the final result.  Thanks so much for any help.
Implement numbering scheme like A,B,C... AA,AB,... AAA..., similar to converting a number to radix26 I want to implement numbering scheme like Microsoft Word uses for numbering.  first one gets = A,next is  B, next is C, .... then AA, then AB,....and so on.  as shown below   A        B  C  .   .  AA  AB  AC  .  .  AAA  AAB  ....   '=>' here means converted to.  some examples:       1 =&gt; A 26 =&gt; Z 27 =&gt; AA 52 =&gt; AZ 53 =&gt; BA   and heres the code for it:     var convertToNumberingScheme = function(n){         var x = n-1,             r26 = x.toString(26),             baseCharCode = "A".charCodeAt(0);          var arr = r26.split(''),             len = arr.length;          var newArr =arr.map(function(val,i){             val = parseInt(val,26);              if( (i === 0) &amp;&amp; ( len &gt; 1)){                 val = val-1;             }              return String.fromCharCode(baseCharCode + val);         });         return newArr.join('');     }   It seems to work fine, but any ideas if there are some potential bugs or ways to  optimize this. How to overload __init__ method based on argument type? Lecturer with anxiety. How to survive in academia? I've been a long time member of academia.SE but I created this alt account because the question is a bit embarrassing. I finished my PhD 4 years ago and after a 2 year postdoc at a different university secured a contractual lecturership. During my PhD I never realised I had fear of public speaking, because I always felt secure presenting in front of my supervisor. Even at conferences, his presence in the audience was enough to keep me relaxed.  Fast forward to my postdoc, I was mostly working and publishing by myself. So, even if I had to present something it was to my coauthors, with which I am okay. But as a lecturer doing research part of my time, I am involved in projects and seminars where I am expected to present my latest work. Being on my own has brought out my inner anxiety, so much so that I choked during a presentation, had to drink water to calm my nerves and apologise for my breathlessness and slurred speech. That was the most embarrasing thing in my life. Please note that the anxiety is not the result of me not publishing or doing research. I have published more than most in my department. Knowing that hasn't helped with the anxiety though. I am constantly making excuses to get out of project presentations and seminar talks. Fortunately, the anxiety is less intense in teaching and I am able to manage.   How do I survive as an academic? Now even conference presentations seem daunting. I can publish exclusively in journals, but how do I get out of giving project and departmental presentations? How can I politely refuse to present without revealing my situation? Or better yet, how do I fix the whole thing? That one bad presentation has really messed me up. Now that I have Google Chrome all set up, I got myself a new laptop. Does anyone have any idea how to transfer my settings, cookies, favorites, etc?   I checked documents and settings\me\application data, and found nothing. How to Identify Overfitting in Convolutional Neural network? I understand that dropout is used to reduce over fitting in the network. This is a generalization technique. In convolutional neural network how can I identify overfitting? One situation that I can think of is when I get training accuracy too high compared to testing or validation accuracy. In that case the model tries to overfit to the training samples and performs poorly to the testing samples. Is this the only way that indicates whether to apply dropout or should dropout be blindly added to the model hoping that it increases testing or validation accuracy Calculate total sum of all numbers in c:forEach loop Compiling with g++ using multiple cores Determining a tricky limit What is the limit of   $$\lim_{n \to \infty} \frac{\log n}{n^\delta}$$  where $\delta &gt; 0$? Setting CSS pseudo-class rules from JavaScript I want to ask how to get id of element which was clicked by right clicking in open menu. I must know ID of this element to show only that history the element what was clicked fom open menu: Can somebody please check if my proof is okay?  Prove that $||z_1|-|z_2|| \leq |z_1-z_2|$.  Attempt: This proof seems similar to the triangle inequality proof.  $$\begin{split} |z_1-z_2|^2 &amp;= (z_1-z_2) \overline{(z_1-z_2)} \\ &amp;= (z_1-z_2)(\bar z_1 - \bar z_2) \\ &amp;= z_1\bar z_1 - z_1 \bar z_2 - \bar z_1 z_2 + z_2 \bar z_2\\ &amp;=|z_1|^2 - 2Re(z_1 \bar z_2) + |z_2|^2\\ &amp; \geq |z_1|^2 -2|z_1 \bar z_2| + |z_2|^2 \\ &amp;= |z_1|^2 - 2|z_1||z_2| + |z_2|^2 \\ &amp;= (|z_1| - |z_2|)^2. \end{split}$$  Taking the square root of both sides gives $|z_1|-|z_2| \leq |z_1-z_2|$  Can I just take the norm of both sides so that $||z_1|-|z_2|| \leq ||z_1-z_2|| = |z_1-z_2|$?  Thank you. The space $c_0$ of sequences that converge to 0 is not reflexive Thunar files view, date format "today, yesterday" Why is gksu no longer installed by default? I have always recommended pressing ALT+F2 and typing for example gksu nautilus or gksudo nautilus both are equivalent; to run graphical programs with root privileges.  However, I have just installed raring on a test system and this no longer works.  gksu and gksudo are not installed as part of a default raring install; though they are in the repositories. In 12.04 at least gksudo was just a symbolic link to gksu  The reasons I have always done this are because while it is often safe to use sudo under some circumstances it can cause problems with graphical applications because gksu has you running the program as root while with sudo you are running as yourself but with elevated privileges.  This can mean you inadvertently change the owner of critical system files.  This can cause problems.  This reasoning is explained better here:         Why was command removed from a fresh install and what should we be using instead? How to install a "pdf printer"? Is there a way to install a printer that prints to pdf?  There's a lot of discussions about this in the web foruns, but none seens to work. Defaulting to the sign-up form? Recently I've noticed more and more sign-in / sign-up workflows defaulting to the sign-up option first and providing a minimal "Sign-In / Login" action somewhere else on the page. What's the reason behind this practice?  Examples where I've seen this recently: Derivation of Maxwell's equations from field tensor lagrangian I've started reading Peskin and Schroeder on my own time, and I'm a bit confused about how to obtain Maxwell's equations from the (source-free) lagrangian density $L = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$ (where $F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu A^\mu$ is the field tensor).  Substituting in for the definition of the field tensor yields $L = -\frac{1}{2}[(\partial_\mu A_\nu)(\partial^\mu A^\nu) - (\partial_\mu A_\nu)(\partial^\nu A^\mu)]$.  I know I should be using $A^\mu$ as the dynamical variable in the Euler-Lagrange equations, which become $\frac{\partial L}{\partial A_\mu} - \partial_\mu\frac{\partial L}{\partial(\partial_\mu A_\nu)} = - \partial_\mu\frac{\partial L}{\partial(\partial_\mu A_\nu)}$, but I'm confused about how to proceed from here.  I know I should end up with $\partial_\mu F^{\mu\nu} = 0$, but I don't quite see why.  Since $\mu$ and $\nu$ are dummy indices, I should be able to change them: how do the indices in the lagrangian relate to the indices in the derivatives in the Euler-Lagrange equations? My daughter bought a ticket for a flight with a layover in Emirates (from Dallas, Texas to India). She had a stop of 48 hours in Dubai. When she was checking in in Dubai for Delhi, she was informed that her visa for India has expired in 2012. Did Emirates not have an obligation to check the visa for India before selling a ticket? How can we unlock an iPhone backup file if we don't know the password? Limit problem arctg (1 ^ infinty) Can anyone help me with this limit problem without L'Hopital rule and Taylor series? $$\lim_{x\rightarrow\ -\infty}\left(-2\frac{\arctan(x)}{\pi}\right)^x$$