sh script name -sh: /bin/sh^M: bad interpreter: No such file or directory
shebang line not working with cr-lf
Add CSS or JavaScript files to layout head from views or partial views No notification was received for an edit to a post TSA broke a lock for which they have a master key. How can I claim compensation? Prove that the equation: $c_0+c_1x+\ldots+c_nx^n=0$ has a real solution between 0 and 1. Let $c_0,c_1,c_2,\ldots ,c_n$ be constants such that :  $$c_0+\frac{c_1}{2}+\ldots+\frac{c_{n-1}}{n}+\frac{c_n}{n+1}=0$$  I have to prove that the equation: $$c_0+c_1x+\ldots+c_nx^n=0$$  Has a real solution between 0 and 1.  Didn't know how to start...I thought that maybe I could use something about derivative...But I'm lost... Any help,much appreciated!!! I found an example in the  for Dynamic LINQ that allows you to use a sql-like string (e.g. OrderBy("Name, Age DESC")) for ordering. Unfortunately, the method included only works on IQueryable&lt;T&gt;. Is there any way to get this functionality on IEnumerable&lt;T&gt;? How can I clip an image via a bezier-path? Can you have a class in a struct? Why aren't logarithms defined for negative $x$? Given a logarithm is true, if and only if, $y = \log_b{x}$ and $b^y = x$ (and $x$ and $b$ are positive, and $b$ is not equal to $1$), are true, why aren't logarithms defined for negative numbers?  Why can't $b$ be negative? Take $(-2)^3 = -8$ for example. Turning that into a logarithm, we get $3 = \log_{(-2)}{(-8)}$ which is an invalid equation as $b$ and $x$ are not positive! Why is this? "Lunch" vs. "dinner" vs. "supper" — times and meanings? I've seen cases where a noon-time meal is referred to as dinner, and the evening meal is called supper. There's also lunch around noon followed by dinner in the evening. Is there a particular difference between dinner and supper, or a circumstance where lunch becomes dinner? Is my solution correct for solving $\cos(z)=2$, $z\in \mathbb{C}$ Joel Spolsky asked what rhymes with orange. The official answer is, "Nothing," although a creative poet can get close by using half words, just the -nge part or resorting to place names and foreign words.  Does orange somehow violate the basic phonotactics of English? (And hence we wouldn't expect other English words to rhyme with it?) Subdivision surface modifier creates pinching at corners I've only been 3D modelling for a few days now, so this may just be a newbie mistake:  However, because this model is for use in a game I am trying to stay away from proximity loops for performance reasons. Instead, I have used edge creases to maintain the structure of the model. It has led to certain vertices not being affected by the subsurf modifier, causing undesirable lighting at high subsurf levels.  Manually moving each point in a bit seems kind of sloppy but I'm not too sure what the best way to go about this is.  If any more information is needed I'll give what I can, thanks in advance :) This is the problem I am currently having trouble with:  If $\lambda_1$ and $\lambda_2$ are eigenvalues of matrix  $$         \begin{bmatrix}         a &amp; b\\         c &amp; d\\         \end{bmatrix}$$prove that $\lambda_1^2$, $\lambda_1\lambda_2$ and $\lambda_2^2$ are eigenvalues of matrix $A$.     $$A=\begin{bmatrix}     a^2 &amp; ab &amp; b^2\\     2ac &amp; ad+bc &amp;2bd\\     c^2 &amp; cd &amp;d^2\\     \end{bmatrix}$$  What have I considered so far:   I tried to find characteristic polynomial of first matrix and then find the roots which would be $\lambda_1$ and $\lambda_2$. I was hoping I can make a connection with characteristic polynomial of matrix $A$, but it got too complicated and did not seem like the right idea. Since there are two eigenvalues for first matrix, there are two eigenvectors. I thought about somehow using $A\vec v_i=\lambda_i\vec v_i$, where $i=1,2$  Also, I thought about finding a connection between these two matrices. What I see is that the sum of elements of first and second column, of first matrix, are $(a+c)$ and $(b+d)$, respectively.  Sum of first, second and third column of matrix $A$ are $(a+c)^2, (a+c)(b+d)$ and $(b+d)^2$. Now, if I could assign $\lambda_1$ and $\lambda_2$ to $(a+c)$ and $(b+d)$, then I would get that sum of first, second and third column of matrix $A$ are $\lambda_1^2$, $\lambda_1\lambda_2$ and $\lambda_2^2$, but what would be next?   I thought it would be good to combine 2. and 3rd idea, but I am not making any progress.  I appreciate any help. Symmetric Inequality in $\mathbb{R}$ Error with \newcommand using \multicolumn in table I want to define a new command \tableheadline that (1) formats the headline text in some way and (2) takes an optional argument for spanning multiple columns while centering the headline.  I tried the following implementation:  \documentclass{minimal}  \newcommand{\tableheadline}[2][1]{\multicolumn{#1}{c}{\textit{#2}}} % \newcommand{\tableheadline}[2][1]{\multicolumn{1}{c}{\textit{#2}}}  \begin{document}   \begin{tabular}[\textwidth]{cc}      \tableheadline[2]{C}     \tableheadline{A} &amp; \tableheadline{B} \\     0 &amp; 1 \\     2 &amp; 3 \\   \end{tabular} \end{document}   This, however, gives me an error:  ! Misplaced \omit. \multispan -&gt;\omit                    \@multispan l.8 \tableheadline[2]{C}   I also get the error when I do not even use the optional argument (see commented line in the MWE). Any ideas what I am doing wrong? I am looking for a closed form (ideally expressed as elementary functions) of the function $\exp_n(x) = \sum_{k=0}^n x^k / k!$.  I am already aware of expressing it in terms of the gamma function. Background / Motivation When counting combinations of objects with generating functions, it is useful to be able to express the partial sum $1 + x + \cdots + x^n$ as $\frac{1-x^{n+1}}{1-x}$. For example, to count the number of ways to pick 5 marbles from a bag of blue, red, and green marbles where we pick at most 3 blue marbles and at most 2 red marbles, we can consider the generating function $f(x) = (1+x+x^2+x^3)(1+x+x^2)(1+x+x^2+\cdots)$. By using the partial sum identity, we can express it as $f(x) = \left(\frac{1-x^4}{1-x}\right)\left(\frac{1-x^3}{1-x}\right)\left(\frac{1}{1-x}\right)$.  Simplify, express as simpler product of series, and find the coefficient of the $x^5$ term. I want to be able to do the same for a generating function in the form $g(x) = \exp_{n_1}(x)^{p_1} \exp_{n_2}(x)^{p_2} \cdots \exp_{n_j}(x)^{p_j}$ The easiest way to extract the coefficient of a given term $x^p / p!$ would be to use a similar closed form expression for $\exp_n(x)$ and a similar technique to $f$. Attempted Solutions Differential equation Recall that the way to prove the identity $1+x+x^2+\cdots+x^n = \frac{1-x^{n+1}}{1-x}$ is to define $S = 1 + x + x^2 + \cdots + x^n$ and notice that: $S - Sx = 1 - x^{n+1}$. Likewise, notice that $y(x) = \exp_n(x)$ satisfies $y - y' = x^n/n!$.  Via SAGE, the solution is $y(x) = \frac{c+\Gamma(n+1,x)}{n!}e^x$.  Our initial condition $y(0) = 0$ so $c=0$.  By , $\Gamma(n+1,x) = n! e^{-x} \exp_n(x)$ so we have $y(x) = \exp_n(x)$. Recurrence Relation Notice that $\exp_n(x) = \exp_{n-1}(x) + x^n/n!$. Using the unilateral Z-Transform and related properties, we find that $\mathcal{Z}[\exp_n(x)] = (z e^{x/z})/(z-1)$. Therefore, $\exp_n(x) = \mathcal{Z}^{-1}\left[(z e^{x/z})/(z-1)\right] = \frac{1}{2 \pi i} \oint_C z^n e^{x/z}/(z-1)\;dz$. $(z^n e^{x/z})/(z-1)$ has two singularities: $z = 1$ and $z = 0$.  The point $z = 1$ is a pole of order one with residue $e^x$.  To find the residue at $z = 0$ consider the product $z^n e^{x/z} (-1/(1-z)) = -z^n \left( \sum_{m=0}^\infty x^m z^{-m} / m! \right) \left( \sum_{j=0}^\infty x^j \right)$.  The coefficient of the $z^{-1}$ term is given when $n - m + j = -1$. The residue of the point $z=0$ is then $-\sum_{m,j} x^m / m! = -\sum_{m=n+1}^\infty x^m / m!$. Let $C$ by the positively oriented unit circle centered at the origin.  By Cauchy's Residue Theorem, $\frac{1}{2 \pi i} \oint_C z^n e^{x/z}/(z-1)\;dz = \frac{1}{2 \pi i} 2 \pi i \left(e^x - \sum_{m=n+1}^\infty x^m / m!\right) = \exp_n(x)$. Finite Calculus I've tried to evaluate the sum using finite calculus, but can't seem to make much progress. How do I parse command line arguments in Java? My chain broke and in a moment of silliness (I was cold wet and slightly annoyed)  I threw the broken one away before thinking about measuring the new one.  Is there a guide to getting the chain length/tension correct ?  It's an 8 speed road bike (Shimano Sora back, double ring front) How to reduce the number of authors above >3 to 'First Author et al., (year)'? How did the phenomenon of doubling words come about? I am referring to phrases such as: "Do you like her, or do you like like her." Can someone provide an explanation of this? There are many more examples but none come to mind at the moment.