Show that any number of partitions of 2r + k into r + k parts is the same for any k using ferrers diagram.  Already tried to do by checking transpose etc. but cannot come up with a solution.
Show that any number of partitions of $2r + k$ into $r + k$ parts is the same for any $k$.   I've tried this, but I haven't come up with anything; hence why I have nothing written here. But in any case, any sort of help is appreciated.
Is an increasing union of subspaces a subspace? How do I choose between "as to", "on", "of", and "about"? CUDA Driver API vs. CUDA runtime Importing 3D DXF file into QGIS with z information as attribute value? A site (or scraper) is copying content from Stack Exchange. What should I do? Roots in a polynomial ring About the relation between $\int_{0}^{\infty}\frac{\cosh((\alpha-\beta)x)}{(\cosh(x))^{\alpha+\beta }} \,dx$ and $B(\alpha,\beta)$ How do I get from   $$B(\alpha,\beta)=\int_{0}^{1}{ t^{ \alpha -1}  (1-t)^{ \beta -1} \,dt}$$ to:   $$2^{2- \alpha - \beta } \int_{0}^{ \infty}{ \frac{ \cosh (( \alpha - \beta )x)}{( \cosh(x))^{ \alpha + \beta }} \,dx}$$  ? I have tried different change of variables such as $t=1- \frac{1}{ \cosh (x)}$ and got $\int_{0}^{ \infty}{ \frac{ \sinh (x)  ( \cosh (x)-1)^{ \alpha -1} }{( \cosh(x))^{ \alpha + \beta }} \,dx}$ but without success. how to add a custom lock for the recovery and/or bootloader? Unlocking the bootloader and/or the recovery is very popular discussion across many devices. But doing so exposes a very obvious and inevitable issue with security!  An unlocked recovery means that no matter how secure you make your android lock screen, someone can easily reset phone to restore the phone to a fully functional (and lockless/insecure) state and the worst part, they don't have to erase your data(meaning they can access all your personal files and folders)!  It is almost hilarious! In fact if the phone supports insertion of external SD cards, you can even flash your own software into the phone and maybe bypass the lock screen without wiping any data.  For example: One possible loophole is using the 'Tasker' app which has a feature that can temporarily disable the keyguard. And the best part is that Tasker can be flashed using the recovery!  So is there any method by which I could add a security measure that prevents complete access to my recovery and/or boot loader that will make my phone less insecure?!  Expected answer: Method by which a custom pin/password must be entered before entering the bootloader/recovery. So that my bootloader/recovery is still technically unlocked but simply requires authentication to use (this also ensures that conventional unlocking methods won't work as the phone is not really locked!). This also means that one does not have to lose root access to ensure security! I'm trying to use Python to place a camera. I know the camera's location, and the forward and up vectors for the camera (i.e. the direction it's pointing and its orientation). How do I do this?  I know that it's possible to calculate a raw world matrix from the above information, which I can then apply to the camera, but I was rather hoping not to have to. Is there an easier way? The Ball From A Pass What exactly is a distribution? I know very little of Probability and Statistics, and am wishing to learn.  I see the word "distribution" used all over the place in different contexts.  For example, a discrete random variable has a "probability distribution."  I know what this is. A continuous random variable has a probability density function, then for $x\in\mathbb{R}$, the integral from $-\infty$ to $x$ of the probability density function is the cumulative distribution function evaluated at $x$.   And apparently just "distribution function" is synonymous with "cumulative distribution function," at least when talking about continuous random variables (question: are they always synonyms?).  Then there are many famous distributions.  $\Gamma$ distribution $\chi^2$ distribution, etc.  But what exactly is a $\Gamma$ distribution?  Is it the cumulative distribution function of a $\Gamma$ random variable?  Or the probability density function of a $\Gamma$ random variable?  But then a frequency distribution of a finite data set appears to be a histogram.  Long story short: in Probability and Statistics, what is the definition of the word "distribution"?    I know the definition of distribution in Mathematics (an element of the dual space of the collection of test functions equipped with the inductive limit topology), but not Probability and Statistics. How do I enable error reporting in PHP? Are there any universal rules in appropriating “_an,” “_sh,” “_es,” “_ch” for the demonyms of countries? Suffixes indicating people and language of country vary by country:  _an: American, German, Italian, Belgian, Australian, Russian, Ukrainian, Korean,  Mexican, Brazilian, Chilean, Argentinean, Indonesian, Malaysian, Iranian, Egyptian, Somalian, Palestinian  _sh: English, British, Polish, Danish, Swedish, Turkish,    _ch: French, Dutch  _ese: Chinese, Taiwanese, Japanese, Vietnamese, Portuguese  _no: Filipino, Latino  _i: Israeli  _: Thai  Among the above, _an seems to outnumber other suffixes.  Are there any simple rule to govern appropriation of suffix turning a country name into adjective, indicating people and language of countries in the world in terms of say, region, race, or cultural background? Or is it just discretional? Derivative of inverse function at a point. Let $f\left(x\right)=x^{13}+3x^{9}+2x^{5}+x^{2}+x+2$ find $\left(f^{-1}\right)'\left(1\right)$. Can somebody show me the method? I am a little confused. Where did Magister Illyrio Mopatis get the dragon eggs from? Evaluating $\frac{1}{2\pi}\int_{0}^{2\pi}\frac{1}{1-2t\cos\theta +t^2}d\theta$ I need solve this integral, and I tried various methods of solving and did not get it. The integral is:   $$\frac{1}{2\pi}\int_{0}^{2\pi}\frac{1}{1-2t\cos\theta +t^2}d\theta,$$  where $t$ is a positive integer. What do ellipsis [...] mean in a list? If a function $f : \mathbb Z\times \mathbb Z \rightarrow \mathbb{R}^{+} $ satisfies the following condition  $$\forall x, y \in \mathbb{Z}, f(x,y) = \dfrac{f(x + 1, y)+f(x, y + 1) + f(x - 1, y) +f(x, y - 1)}{4}$$  then is $f$ constant function? How could I detect (preferably with Python) duplicate MP3 files that can be encoded with different bitrates (but they are the same song) and ID3 tags that can be incorrect?  I know I can do an  checksum of the files content but that won't work for different bitrates. And I don't know if ID3 tags have influence in generating the MD5 checksum. Should I re-encode MP3 files that have a different bitrate and then I can do the checksum? What do you recommend? Why doesn't \makeatletter work inside \newcommand? After hours of banging my brain against my skull, I narrowed down an issue I'm having to the fact that \makeatletter doesn't work inside \newcommand. Commands that I call after \makeatletter are registered as undefined, as if \makeatletter didn't do anything.  Why is this the case?  In the example below, I define two commands. Calling one of them from the document body causes the undefined error, while the other doesn't.  \documentclass[11pt]{book} \usepackage{kvsetkeys}  \newcommand{\commandThatFails}{     \makeatletter \comma@parse{}{} \makeatother }  \makeatletter     \newcommand{\commandThatWorks}{         \comma@parse{}{}     } \makeatother  \begin{document}  Hi there.  \commandThatFails \commandThatWorks  \end{document}