Limit as n approaches infinity of (log(n!) / nlog(n))
Limit of $\frac{\log(n!)}{n\log(n)}$ as $n\to\infty$.
Coding for an ordered covariate I am performing ordinal regression, I have 5 response categories and several predictors both continuous and categorical. I would like to add a predictor which is categorical but ordered (1, 2, 3, 4). I don't think it would be appropriate to apply the usual dummy coding for unordered categorical predictors, but when I searched for how to code this I did not find much information.  In  "linear coding" or "assuming linearity of the predictor effect" is mentioned, but without further details.  Does it mean I just use my ordered values as they are, i.e. use them as a continuous variable? According to the , when a Shinigami dies, their remaining lifespan is given to the human they saved.     Shinigami who die are reduced to dust, and their remaining lifespan is given to the human they saved.   Since Rem killed L and Watari, she gained their remaining lifespans. Additionally she probably still had some lifespan left. On top of that Misa was already given a lot of lifespan from Gelus (from which she lost 75% again). So does this mean all of this lifespan is transferred to Misa Amane?!  Yet according to her , she dies in 2011 at the age of 26, only seven years after Rem died.  Does this mean that humans are also capable of reducing their own lifespans? Or would the author have made a mistake by letting Misa die as early as she has? I know she lost 75% of her remaining lifespan through the two eye deals, but you'd think that the cumulation of L, Watari and Rem's remaining lifespans would be greater than just one year, wouldn't you? According to the definition on , a multiple  is a root with  $n\geq 2$. For example, in the equation $(x-1)^2 = 0$, $1$ it the multiple (double) root.  Question: Why is $x = 1$ the multiple root of $(x-1)^2 = 0$?  What does multiplicity mean in this case? According to Wolfram multiplicity is the number of values for which a condition holds, or the number of times for which a polynomial equation has a root at a given point. In what way does the polynomial $x^2 - 2x + 1 = 0$ have multiple roots? How to show $\lim_{n \to \infty} a_n = \frac{ [x] + [2x] + [3x] + \dotsb + [nx] }{n^2} = x/2$? How do you forcefully remove apps in Windows 10? How can I remove apps that Windows doesn't seem to allow to be uninstalled, like Xbox and Groove Music? Computing the integral of $\log(\sin x)$ How to compute the following integral?  $$\int\log(\sin x)\,dx$$    Motivation: Since $\log(\sin x)'=\cot x$, the antiderivative $\int\log(\sin x)\,dx$ has the nice property $F''(x)=\cot x$. Can we find $F$ explicitly? Failing that, can we find the definite integral over one of intervals where $\log (\sin x)$ is defined? I want to programmatically create a new column in an MS Access table.  I've tried many permutations of ALTER TABLE MyTable Add  MyField DECIMAL (9,4) NULL; and got:      Syntax Error in Field Definition   I can easily create a number field that goes to a Double type, but I want decimal.  I would very strongly prefer to do this in a single ALTER TABLE statement and not have to create a field and then alter it.   I am using Access 2003. Divisibility Rule for 9 I'm working through an elementary number theory course right now and I think I've come up with a proof here but wanted some feedback on my logic.    Question:  If the sum of the digits in base 10 is divisible by 9, then the number itself is divisible by 9.    Proof: Suppose that $9|d_1+d_2+...+d_n$ then $d_1+d_2+...+d_n=0\mod9$  Now consider $d_1(10^{n-1})+d_2(10^{n-2})+...+d_{(n-1)}(10^1)+d_n(10^0)$ Each power of $10$ is equivalent to $1\mod9$   therefor  $d_1(10^{n-1})+d_2(10^{n-2})+...+d_{(n-1)}(10^1)+d_n(10^0)=(1\mod9)(d_1+d_2+...d_n)$  $9|(d_1+d_2+...d_n)$ by our assumption, thus $9|(1\mod9)(d_1+d_2+...d_n)$  Thus we have shown that if 9 divides the sum of the digits in base 10, 9 divides the number itself.    The only question I really have is whether I'm jumping the gun on my assumption concerning the powers of 10 being $1\mod 9$.  I think this is fair game here but not 100% confident.  Thanks. Shiny locked Legendaries in ORAS Serebii says that (probably) the only  in ORAS are Kyogre, Groudon, Rayquaza and Deoxys. This means that Reshiram and Zekrom, that were shiny locked in BW/B2W2, now can be found shiny.    Can anyone prove this? I'd like to know if it's true before waste hours in softresetting. I cheated on the Disciplined badge, how bad is it? I was looking for badges I didn't have yet, and I saw the Disciplined badge. So I deleted an answer of mine with 3+ votes, and waited. Five minutes later, sweet, a new badge! Then I went to my answer again, and undeleted it. No problem at all, and I keep the badge!   What will happen? Will my badge stay here or not? More generally, what happens when the conditions to get a badge are valid at one time, and then not valid?  Other badge in this case:   Unsung hero - "Zero score accepted answers: more than 10 and 25% of total"   What happens if answers get a vote after getting the badge? Are the badge frequently recalculated? Problem with \pspicture in a simple plain TeX code Just writing the following very simple program I had some problems. I use to write in plain TeX and in this case I tried to make a picture with PSTricks and first of all sized it with \pspicture. The program is:  \input pstricks \pspicture (0,0)(5,5) \endpspicture \bye   but when I compiled with TeX, I encountered the following error:  Command Line:   tex.exe --src --interaction=errorstopmode "figura.tex" Startup Folder: C:\Users\MT\Documents\TeX  This is TeXk, Version 3.14159265 (TeX Live 2015/W32TeX) (preloaded format=tex)  Source specials enabled. (./figura.tex (c:/texlive/texmf-dist/tex/generic/pstricks/pstricks.tex we are running tex and have to define some LaTeX commands ... (c:/texlive/texmf-dist/tex/generic/xkeyval/pst-xkey.tex 2005/11/25 v1.6 PSTricks specialization of xkeyval (HA) (c:/texlive/texmf-dist/tex/generic/xkeyval/xkeyval.tex (c:/texlive/texmf-dist/tex/generic/xkeyval/xkvutils.tex (c:/texlive/texmf-dist/tex/generic/xkeyval/keyval.tex)) 2014/12/03 v2.7a key=value parser (HA) (c:/texlive/texmf-dist/tex/generic/xkeyval/xkvtxhdr.tex 2005/02/22 v1.1 xkeyval TeX header (HA)))) (c:/texlive/texmf-dist/tex/generic/pstricks/pst-fp.tex `pst-fp' v0.05, 2010/01/17 (hv)) (c:/texlive/texmf-dist/tex/generic/pgf/utilities/pgfutil-common.tex (c:/texlive/texmf-dist/tex/generic/pgf/utilities/pgfutil-common-lists.tex)) (c:/texlive/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex (c:/texlive/texmf-dist/tex/generic/pgf/utilities/pgfkeysfiltered.code.tex)) (c:/texlive/texmf-dist/tex/generic/pgf/utilities/pgffor.code.tex (c:/texlive/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex (c:/texlive/texmf-dist/tex/generic/pgf/math/pgfmathcalc.code.tex (c:/texlive/texmf-dist/tex/generic/pgf/math/pgfmathutil.code.tex) (c:/texlive/texmf-dist/tex/generic/pgf/math/pgfmathparser.code.tex) (c:/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.code.tex (c:/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.basic.code.tex) (c:/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.trigonometric.code .tex) (c:/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.random.code.tex) (c:/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.comparison.code.te x) (c:/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.base.code.tex) (c:/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.round.code.tex) (c:/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.misc.code.tex) (c:/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.integerarithmetics .code.tex))) (c:/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfloat.code.tex) )) `PSTricks' v2.64b  &lt;2015/14/11&gt; (tvz) (c:/texlive/texmf-dist/tex/generic/pstricks/pstricks.con)) ! Undefined control sequence. \endpspicture ...z@ \box \pst@hbox \else \fboxsep                                                    =0pt \fbox {\rule [-\pst@s... l.5 \endpspicture  ? x No pages of output. Transcript written on figura.log.  _____________________________________________________________________   TeX Compilation Report (Pages: 0)   Errors: 1   Warnings: 0   Bad Boxes: 0 _____________________________________________________________________   What's the problem? Can an irrational number raised to an irrational power be rational?  If it can be rational, how can one prove it? I use my DSLR (Nikon D500) for both photography and video, and I prefer autofocus for photography and manual focus for video (to get rid of autofocus noise).  There is an option for changing between the two on both the body and the lens, and obviously it is very annoying to have to keep flicking 2 switches to do the same thing all of the time.  Do you only have to change one of them, and would doing this cause any damage to the camera and/or the lens?  (I am aware that a similar question has already been asked before, however the answers said more what is "best practice" and not so much if there are any consequences to only changing one of them)  Thanks! How do I calculate CRC32 of a string How do I calculate the CRC32 (Cyclic Redundancy Checksum) of a string in .NET? Conformal transformation/ Weyl scaling are they two different things? Confused! Which Ubuntu to use alongside Windows 7? In "Knots and Primes: An Introduction to Arithmetic Topology", the author uses the following proposition   Let $h: Y \to X$ be a covering. For any path $\gamma : [0,1] \to X$ and any $y \in h^{-1}(x) (x = \gamma(0))$, there exists a unique lift $\hat{\gamma} : [0,1] \to Y$ of $\gamma$ with $\hat{\gamma}(0) = y$. Furthermore, for any homotopy $\gamma_t (t \in [0,1])$ of $\gamma$ with $\gamma_t = \gamma(0)$ and $\gamma_t(1) = \gamma(1)$, there exists a unique lift of $\hat{\gamma_t}$ such that $\hat{\gamma_t}$ is the homotopy of $\hat{\gamma}$ with $\hat{\gamma_t}(0) = \hat{\gamma}(0)$ and $\hat{\gamma_t}(1) = \hat{\gamma}(1)$.  The author then follows with; "In the following, we assume that any covering space is connected. By the preceding proposition, the cardinality of the fiber $h^{-1}(x)$ is independent of $x \in X$."   I am not sure why this results is true. This is my attempt at explaining it to myself. Take two different $x_1, x_2$ and their fibers $h^{-1}(x_1), h^{-1}(x_2)$ such that $y_1 \in h^{-1}(x_1)$ and $y_2 \in h^{-1}(x_2)$. Take two paths $\gamma_1, \gamma_2$ such that $\gamma_1(0) = x_1$ and $\gamma_2(0) = x_2$. Then we get two lifts $\hat{\gamma_1}, \hat{\gamma_2}$ with $\hat{\gamma_1}(0) = y_1$ and $\hat{\gamma_2}(0) = y_2$. Now, since our covering space is connected we can continuously deform $\hat{\gamma_1}$ into $\hat{\gamma_2}$ and conclude that every elements in $h^{-1}(x_1)$ is also in $h^{-1}(x_2)$ and vice versa. I feel that this is wrong but cannot figure out the right way to see this. Any help would be appreciated. My wife and I didn't know we're pregnant and in a weekend fun with friends we smoked a bit of weed - we don't usually smoke weed or cigarettes.  This is not something we're regulary doing as I said, but how bad to the fetus can it be ? I am new to Ubuntu. I am Using Ubuntu 18.04 along with Windows10. I have gone through the below Q&amp;A and could able to change the name of Ubuntu,      But   if i edit the Ubuntu OS name to my choice, it is then adding "GNU/Linux" at the end. for ex. if i want "Ubuntu 18.04" it is then showing as "Ubuntu 18.04 GNU/Linux". how to avoid this text "GNU/Linux" how to edit the name to my choice which is shown as "Windows Boot Manager (on /dev/sda1)"        Update: I Confirm about the text "GNU/Linux" is solved by @Terrance comments. I still need to try about windows entry.      Final Update: Solved.    I thank @Terrance, @Oldfred, @WinEunuuchs2Unix. All the three methods are working great. i felt the small script is awesome and very very easy to use. which is written by WinEuuuchs2Unix. When signing in with Google (OAuth), it shows a list of accounts it already knows about. How can I remove some accounts from this list?    I know I can nuke all of them by deleting all cookies, but I'd prefer to remove one account that I don't want there, rather than all of them.  There used to be an ability to edit this list in the older version of this screen, but in 2018 redesign Google has removed [×] buttons next to the accounts.