How to evaluate $\lim_{x\to0} \frac{1-\cos(x)}{x^2}$ in terms of $\alpha = \lim_{x\to0} \frac{\sin(x)}{x}$?
Finding the limit of $(1-\cos x)/x^2$
Lille Metro schedule... is it 24/7? The M1 and M2 lines in Lille, France have a confusing . It's 7 days per week, but it looks like it's "running" basically 24h hours per day (minus some ten-minute delay). I never took the metro later than 11 PM when I was in Lille last year, but people told me that it stops just before midnight. But... it looks like it starts right back up again immediately, after about 10 minutes.  What are the actual hours of the Lille M1 and M2 lines? I have a hard time believing that they're 24/7... "For security reasons, framing is not allowed; click OK to remove the frames." on Imgur upload Ok first, there's that lugubrious use of the semicolon; not necessary.  Second, I didn't open a frame!  You did!  Alls I tried to do is upload a 5MB animated .gif file via Imgur.    Looking further into this, Imgur actually posts an error when you upload an animated .gif larger than 2MB.    Go figure! We have no mention of Voldemort in Harry's sixth year. So, I was wondering- what could Voldemort have been doing? conflict between hyperref and enumitem packages (and \refstepcounter) The solution outlined for each of the following two posts appears to conflict with the hyperref package.         Here is a MWE.  The desired output for the example is  1. Outer 2.1. This is line 2.1, in block 2.  2.2. Inner again 3. Outer again 4. Let’s end here   This output is indeed produced by the LaTeX source below, but if I uncomment the \usepackage{hyperref} line in the header, LaTeX gives me an error just after the \skipitem command, as described in the inline comment below:  \documentclass{article}  % \usepackage{hyperref} % if we uncomment the above line, we get an error as described below  \usepackage{enumitem}  \newlist{steps}{enumerate}{9} \setlist[steps]{itemindent=0pt} \setlist[steps,1]{label=\arabic*.,ref=\arabic*} \setlist[steps,2]{label=\arabic{stepsi}.\arabic*.,ref=\arabic{stepsi}.\arabic*}  \makeatletter \newcommand{\skipitem}{\refstepcounter{\@enumctr}} \makeatother  \begin{document} \begin{steps} \item Outer   \skipitem   % If we uncomment the \usepackage{hyperref} line in the header,   % we get the following error after the \skipitem above   %    % ./tmp.tex:20: Missing number, treated as zero.   % &lt;to be read again&gt;    %   \c@0.0ptxcii    % l.20   \skipitem   % \item[] \label{blockA}   \label{blockA}\item[]   \begin{steps}   \item\label{lineB} This is line~\ref{lineB}, in block~\ref{blockA}.   \item Inner again   \end{steps} \item Outer again \item Let's end here \end{steps}  \end{document}   Can someone help me work around this conflict, so I can use hyperref with this use of enumitems and refstepcounter?  Thanks.  (Note: smaller working examples are possible, but they give different error messages.  I'm hoping to get the current example working.) Finding the index of an item in a list This next year, I am helping an effort to clean several SQL Server environments.    We have about 10,000 stored procedures and estimate that only about 1000 of them are used on a regular basis, and another 200 or so are used on a rare occasion, meaning we have a lot of work to do.    Since we have multiple departments and teams that can access these databases and procedures, we are not always the ones calling the procedures, meaning that we must determine what procedures are being called.  On top of that, we want to determine this over a few months, not in a few days (which eliminates some possibilities).  One approach to this is to use the SQL Server Profiler and track what procedures are being called and compare them to the list of what procedures we have, while marking whether the procedures are used or not.  From then, we could move the procedures to a different schema in case a department comes screaming.  Is using the Profiler the most effective approach here?  And/Or have any of you done something similar and found another way/better way to do this? Let $R$ be a sub-ring of a ring $S$. Let $f,g$ be non-zero polynomials in $R[x]$ and assume that the leading coefficient of $f$ is a unit in $R$. If $f$ divides $g$ in $S[x]$, show that $f$ divides $g$ in $R[x]$.   Thoughts: If $f$ divides $g$ in $S[x]$, then  $g(x) = f(x)q(x)$ for some $q(x)$ in $S[x]$. From here I'm not sure what the question is asking. Do I have to show that $q(x)$ is an element of $R[x]$ ? Insights appreciated. I need help running python in ubuntu  I can run python without a problem in my terminal but when I create a .py file I can't get it to run. Why does a light beam in a moving light clock follows a V shaped path? If I am standing on a platform moving along the X-axis and I throw a basketball vertically upward along the z-axis, the ball also travels along the x-axis. Does the same apply to a photon? According to a special relativity postulate  speed of light is not affected by the velocity of the inertial frame or the velocity of the emitter. If this postulate is true, shouldn't a photon fired vertically as above should travel only along the z-axis but not along the x-axis? Interpretation of log transformed predictor and/or response Hi! I'm a beginner learner and I'd like to know how to snap these three objects. I'd like to snap the two cylinders with the different faces of the cube. The short one on the left and the highest cylinder on the top of the cube. I'd like to have the center of the cylinder aligned with the center of the cube (the base of the cylinder inscribed into the face of the cube)  I'm sorry for bad explanation I hope someone can help me  Thanks!   I add a screenshoot in which you can see what I have to do   Thanks again if someone can help me with this! Non-normality in residuals When you boot from live CD, or doing the first boot after installing Ubuntu, splash screen looks awesome. But as soon as you enable nvidia-current driver, installed with apt-get, splash screen goes all crazy.  With crazy I mean that resolution is very low, font(I assume) is very weird and it all looks like it's been broken.  This happened with 10.04 and now again with 10.10.  How do I fix splash screen after enabling nvidia drivers? Small caps font warning in beamer using biblatex Consider the following MWE  \documentclass{beamer} \usepackage{silence} \WarningFilter{biblatex}{Patching footnotes failed}  \usepackage{lmodern} \usepackage[T1]{fontenc}  \usepackage[style=authoryear-comp,backend=biber]{biblatex} \addbibresource{biblatex-examples.bib}  \begin{document} \begin{frame}   \fullcite{sarfraz} \end{frame} \end{document}   This generates the warning:  LaTeX Font Warning: Font shape `T1/lmss/m/sc' in size &lt;10.95&gt; not available (Font)              Font shape `T1/lmr/m/sc' tried instead on input line 28.   Is there a way to avoid the warning without using the silence package? Is it considered normal to publish job offers inviting candidates to apply based on their gender and/or race in academia? I was browsing PhD position offers in Europe on different websites and I was quite surprised to find that several offers contained sentences such as     University of X value equality and diversity. We strongly encourage women and BAME applicants for this position.    where BAME stands for Black, Asian and Minority Ethnicities; or     The research group aims to increase the number of women in scientific positions. Female candidates are therefore encouraged to apply.   Is it considered normal in the academic environment to discriminate between  candidates according to their race and/or gender for "egalitarian" reasons?   How is the choice of a candidate based on their gender and/or race considered "egalitarian"? C-3PO frequently speaks of “the Maker.” Is this a true droid religion, or is this a religious reverence by droids for their specific creators (in this case, Anakin, even though C-3PO has forgotten who he is)? Is this even a droid thing or is it just C-3PO? What else do we know about this “Maker”-worship? How do I see that $$f_{n+1}f_{n-1} = f_n^2 + (-1)^n$$ for $n \ge 2$, where $f_1 = 1$, $f_2 = 1$, and $f_{n+2} = f_{n+1} + f_n$ for $n \in \mathbb{N}$? 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. Differences between hash- and encryption algorithms? There is a lot of confusion between hash- and encryption algorithms; I would like to understand:   When do we use a hash algorithm and when do we use an encryption algorithm (a cipher)? How difficult it is to reverse hash- and encryption algorithms? From the theoretical/mathematical perspective, what differentiates a hash- and encryption algorithm? With Unix script, I'd like to add values in column 1 if text in column 2 is the same and print the sum.  I have a file that is:    7 GeneA 2 GeneB 6 GeneC 2 GeneB 9 GeneA   I want the output to look like:   16 GeneA 4  GeneB 6  GeneC   Thanks