Partial fraction decomposition algorithm
General rule for partial fraction decomposition
(a) $S_k=\{A\subset\mathbb{N}: |A|=k\}$ for $k\in\mathbb{N}$  (b) $S = \bigcup_{k=1}^\infty S_k$   Work: For (a), I am not too sure about what approach I should use. I think finding a bijective function between the $S_k$ and $\mathbb{N}$ is hard to do . I also thought about using the Schroeder-Bernstein theorem and find injective functions $f:\mathbb{N}\rightarrow S_k$and $g:S_k\rightarrow\mathbb{N}$ in order to prove that $S_k$ and $\mathbb{N}$ are numerically equivalent. However, I have a hard time finding and proving these injective functions. I haven't really looked at (b) as I have yet to complete (a). Here's the context       "Her heart leaping into her throat, she tiptoed into the salon. Such was her astonishment at seeing Mike chatting with her boss that she leaned against the wall, incapable of making another move."   Is this a phrase or what? Could it be used in other forms as well? Compute $\int\limits_{-\infty}^{\infty}\frac{1}{(1+x^2)^{n+1}}dx$ via residue calculus. Let $\Gamma_R$ be the semicircle of radius $R$ in the upper half plane. Then,  \begin{align} \int\limits_{-\infty}^{\infty}\frac{1}{(1+x^2)^{n+1}}dx  &amp;= \lim_{R\to \infty}\int_{\Gamma_R}\frac{1}{(1+z^2)^{n+1}}dz \\ &amp;= 2\pi i \operatorname{Res}\left(\frac{1}{(1+z^2)^{n+1}},i\right) \end{align} The pole of the function at $i$ is of order $n+1$, so the residue is computed by \begin{align} \operatorname{Res}\left(\frac{1}{(1+z^2)^{n+1}},i\right) &amp;= \frac{1}{n!}\lim_{z\to i}\frac{d^n}{dz^n}\left(\frac{1}{(z+i)^{n+1}}\right) \\ &amp;= \frac{1}{n!}\lim_{z\to i}(-1)^n\frac{(n+1)(n+2)\cdots(2n+1)}{(z+i)^{2n+1}} \\ &amp;=\frac{(2n+1)!}{i2^{2n+1}(n!)^2} \end{align} Hence, $$\int_{-\infty}^{\infty}\frac{1}{(1+x^2)^{n+1}}dx = \pi\frac{(2n+1)!}{2^{2n}(n!)^2}$$ The answer provided is $\frac{1\cdot 3\cdot 5\cdots(2n-1)}{2\cdot 4\cdot 6\cdots (2n)}\pi$. How do I manipulate my answer to obtain this answer? How do you return to the default view (the one you see when blender starts)? Why are floating point numbers inaccurate? php loop through json array Post Link for Tag Info Wiki's display text should be better than "unknown" The displayed text for a [Post Link] where the underlying Id corresponds to a Tag/Info Wiki (where the link redirects successfully) is just &quot;unknown&quot;. For example, if you've ever edited a tag wiki: . It would be nice if this could be better identified. Attacks of the MAC construction $\mathcal{H}(m\mathbin\|k)$ for common hashes $\mathcal{H}$? Consider a common practically-collision-resistant hash function $\mathcal{H}$ (e.g. , , , ), perhaps based on the  as are the first three. We define a  $\mathcal{C}$ $$(k,m) \mapsto \mathcal{C}(k,m)=\mathcal{H}(m\mathbin\|k)$$ where $\mathbin\|$ denotes concatenation, $k$ is a secret key (constant, or at least of fixed size), and $m$ is a message (possibly of variable length). Assume that an adversary can (iteratively) submit queries with $m_j$ and obtain $C(k,m_j)$, and wants to obtain $k$ or otherwise compute $C(k,m)$ for some $m\ne m_j$.  That MAC $\mathcal{C}$ is not trivially bad. In particular, if $\mathcal{H}$ was indistinguishable from a random function in the , $\mathcal{C}$ would be secure. And even though $\mathcal{H}$ may have the , it does not turn into a devastating attack on $\mathcal{C}$.  The less impractical generic attack that I see is that if a collision was known for $\mathcal{H}$ with the colliding messages of moderate identical length, it could be deduced countless collisions for $\mathcal{C}$. Hence security is demonstrably not better than collision-resistance of $\mathcal{H}$ (for identical-length messages). We could assume that $k$ is half the size of the result of $\mathcal{H}$, and hope that the security is about  or  or , 280, 2128, 2256 hash rounds for SHA-1, RIPEMD-160, SHA-256, SHA-512.   What are the known attacks against $\mathcal{C}$ (better than the above), and their cost, for each of these common hashes?  Is there hope for an argument that an attack against $\mathcal{C}$ would turn into an attack of similar cost against $\mathcal{H}$, or hint of the contrary?  Update: this  is of interest, but I fail to find that it really answers the present question.  Update 2: I am aware that the construction considered is weaker than , and in particular is vulnerable to collision on $\mathcal H$; I stated that, and that it is thus pointless to have the key wider hopeless to target security against some attacks better than half the hash's size. I'm asking exactly what cryptanalytic attack better than finding a collision on $\mathcal H$ there are. There is room for such an attack only by exploiting a weakness in the structure or/and the round function of a concrete $\mathcal H$. How to use different domains for multilingual sites and keep users logged in? Each time I changed and saved .config/user-dirs.dirs, and after restarted the system, the XDG_DOWNLOAD_DIR are reverted:  diff --git a/.config/user-dirs.dirs b/.config/user-dirs.dirs index 8f6e2db..dcff34e 100644 --- a/.config/user-dirs.dirs +++ b/.config/user-dirs.dirs @@ -7,11 +7,9 @@  #   XDG_PUBLICSHARE_DIR="$HOME/public"  XDG_TEMPLATES_DIR="$HOME/public/template" -  XDG_DESKTOP_DIR="$HOME/private/desktop"  XDG_DOCUMENTS_DIR="$HOME/private/doc"  XDG_PICTURES_DIR="$HOME/private/picture"  XDG_MUSIC_DIR="$HOME/private/music"  XDG_VIDEOS_DIR="$HOME/private/video" - -XDG_DOWNLOAD_DIR="$HOME/var/downloads" +XDG_DOWNLOAD_DIR="$HOME/"   How to make my changes permanent? Why don't we consider centrifugal force on a mass placed on earth? How can I do a full shutdown of windows 10 without logging in? I have a computer with both Windows 10 and Ubuntu 17.04 installed on it. I can't log in to Windows, and think it's a problem with the registry. Luckily, there is a way to restore the registry from backup from Ubuntu, but you have to be able to modify files on the Windows partition. Right now, I can't write to the NTFS partition because Windows didn't do a full shutdown.  How can I force windows 10 to do a full shutdown without being able to log in? Possible Duplicate:          How to download a Youtube video from command line?  Mac OS X and/or Ubuntu Linux. Optionally, i'd like to save it in .avi or .mkv format. I am a bit financially strapped and received a Gateway GT5056 tower. The tower has an AMD dual core 64-bit processor in it. Will Ubuntu 12.04 run on this machine? I will not have any other application other than LibreOffice installed. Word or phrase for mere coincidence that brings happiness Let $n$ is a natural number. Prove that $$\left\lfloor\frac{n}{1}\right\rfloor+\left\lfloor\frac{n}{2}\right\rfloor+....+\left\lfloor\frac{n}{n}\right\rfloor+\left\lfloor{\sqrt{n}}\right\rfloor$$   is even.   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