Aproximation (Proof)
On finding polynomials that approximate a function and its derivative (extensions of Stone-Weierstrass?)
first post between 12/26 and 1/1   Does that mean your very first post as a user should be made in those days?  Or that you should be the first user to post any post on a site starting 12/26?  Or something else?  The first option seems somewhat more likely but I'm not sure I'm parsing the description right. UnboundLocalError on local variable when reassigned after first use When to use "respectively"? I have been wondering what it means when people use "respectively" in, before, and after sentences. For example:     We are looking for a babysitter to pick up and supervise our kids ages   6 and 3, respectively, on Monday and Friday.   I don't know if this person used it correctly, but if they did what does it mean? If they didn't use it properly what would be the proper usage?  Thanks in advance. I am confused about the usage of the words like and unlike in sentences. Like and unlike make me confused because I see them being used everywhere almost interchangeably, and to make matters worse I even saw commas making difference in meaning.  Which of the following sentences is most correct, and least ambiguous?      Unlike Xbox, Playstation can't play Halo.     Playstation can't play Halo unlike Xbox can.     Playstation can't play Halo, unlike Xbox can.     Like Xbox, Playstation can't play Halo.     Playstation can't play Halo like Xbox can.     Playstation can't play Halo, like Xbox can.     Playstation can't play Halo unlike Xbox.     Playstation can't play Halo, unlike Xbox.     Playstation can't play Halo like Xbox.     Playstation can't play Halo, like Xbox.    In case you don't know what I actually wanted to say in those sentences: I'm trying to say that Playstation can't run the Halo video game and Xbox can run that game. I probably should have tried sentences with no brands but I can't imagine any meaningful sentences at this moment which contains like/unlike. Disable storage of invisible files "._" on my cfs or smb network storage Using fflush(stdin) How to get the insert ID in JDBC? Due to my ignorance, I find that most of the references for mathematical analysis (real analysis or advanced calculus) I have read do not talk much about the "multivariate calculus". After dealing with the single variable calculus theoretically, it usually directly goes to the topic of measure theory.    After reading the wiki article "", I'd like to find the rigorous proof for this test. The first book comes to my mind is Courant's Introduction to Calculus and Analysis which includes the multivariate case in the second volume.   Motivated by this, I'd like to put the question here:     What are the usual references for the   theoretical treatment for   ? Too many of your edits were rejected, try again in 7 days First time I've got the following message on SO:     Too many of your edits were rejected,   try again in 7 days.   Although, I have made ~50 accepted edits this day.  I'm sure that should outweigh the ~5 rejected edits, right?    My suggestion:  bool allowEdit(USER u) {     if(u.rejected * 5 &gt; u.accepted)         return(false);     else         return(true); }   Think of it this way: if I do a 1000 accepted edits, and mess up on 5 [rejected], does that mean I should get banned for 7 days?    Hmmm... There seems to be a bug; the message only appears when clicking edit from review.  But otherwise, I can't edit , even when going to it from the homepage. Dollar signs breaks formatting of 'Hot Network Questions' Related to  and .  This :    looks like this in the 'Hot Network Questions' section of e.g. :    I think Latex formatting of question titles should be disabled if the question comes from a site that don't use Latex formatting. If $\sum a_n$ converges and $b_n=\sum\limits_{k=n}^{\infty}a_k $, prove that $\sum \frac{a_n}{b_n}$ diverges Let $\displaystyle \sum a_n$ be  convergent series of positive terms and set  $\displaystyle b_n=\sum_{k=n}^{\infty}a_k$ , then prove that $\displaystyle\sum \frac{a_n}{b_n}$ diverges.   I could see that $\{b_n\}$ is monotonically decreasing sequence converging to $0$ and I can write $\displaystyle\sum \frac{a_n}{b_n}=\sum\frac{b_n-b_{n+1}}{b_n}$, how shall I proceed further? Volume form on $(n-1)$-sphere $S^{n-1}$ Let $\omega$ the (n-1) form on $\mathbb{R}^n$ $$\omega=\sum_{j=1}^{n}(-1)^{j-1}x_{j}dx_{1}\wedge\cdots\wedge \hat{dx_{j}}\wedge\cdots dx_{n}$$ Show that the restriction of $\omega$ to $S^{n-1}$ in precisely the volume for this sphere.  What I did was: $\omega$ never vanish on the sphere, because, defining $\sigma\in \Omega^{n-1}(S)$ for $$\sigma_{p}(v_{1},...,v_{n-1})=det(p,v_{1},...,v_{n-1})$$ and $i:S^{n-1}\rightarrow \mathbb{R}^{n}$ the inclusion function, then $\omega=i^{\ast}(\sigma)$ then $\omega\not=0$ and therfore is a volume form.   How proof that $\omega$ is the volume form?   The first thing that comes to mind is show that $\int_{S^{n-1}}\omega=Vol(S^{n-1})$ but I have serious problems with the definition, I think that is to much.   How see that $\omega$ is invariant on $\mathbb{R}^{n}$ under action of $O(n)$ Is it hard to transfer contacts from one iPhone to another iPhone? I just bought a new iPhone 6s, and want to move all the contacts from the old one to the new one. Can iTunes or iCloud help me do this, or are there any other ways? Someone asked me this question and I don't think I gave him an adequate answer (I was trying to think of the extreme case - that of neutron stars) sort() and reverse() functions do not work Can't learn the third Word of Power from Wulfgar Why do I get "unresolved external symbol" errors when using templates? When I write C++ code for a class using templates and split the code between a source (CPP) file and a header (H) file, I get a whole lot of "unresolved external symbol" errors when it comes to linking the final executible, despite the object file being correctly built and included in the linking.  What's happening here, and how can I fix it? Auto.arima with daily data: how to capture seasonality/periodicity? I am fitting an ARIMA model on a daily time series. Data are collected daily from 02-01-2010 to 30-07-2011 and are about newspaper sales. Since a weekly pattern in sales can be found (the daily average amount of copies sold is usually the same from Monday to Friday, then increases on Saturday and Sunday), I am trying to capture this "seasonality". Given the sales data "data", I create the time series as follows:  salests&lt;-ts(data,start=c(2010,1),frequency=365)   and then I use the auto.arima(.) function to select the best ARIMA model via AIC criterion. The result is always a non-seasonal ARIMA model, but if I try some SARIMAs model with the following syntax as example:  sarima1&lt;-arima(salests, order = c(2,1,2), seasonal = list(order = c(1, 0, 1), period = 7))   I can obtain better results. Is there anything wrongs in the ts command / arima specification? The weekly pattern is very strong so I would not expect so many difficulties in capturing it.  Any help would be very useful. Thank you, Giulia Deppieri  Update:  I have already changed some arguments. More precisely, the procedure selects ARIMA(4,1,3) as the best model when I set D=7, but AIC and the others good of fit indexes and forecasts as well) do not improve at all. I guess there's some mistakes due to confusion between seasonality and periodicity..?!   Auto.arima call used and output obtained:  modArima&lt;-auto.arima(salests,D=7,max.P = 5, max.Q = 5)     ARIMA(2,1,2) with drift         : 1e+20  ARIMA(0,1,0) with drift         : 5265.543  ARIMA(1,1,0) with drift         : 5182.772  ARIMA(0,1,1) with drift         : 1e+20  ARIMA(2,1,0) with drift         : 5137.279  ARIMA(2,1,1) with drift         : 1e+20  ARIMA(3,1,1) with drift         : 1e+20  ARIMA(2,1,0)                    : 5135.382  ARIMA(1,1,0)                    : 5180.817  ARIMA(3,1,0)                    : 5117.714  ARIMA(3,1,1)                    : 1e+20  ARIMA(4,1,1)                    : 5045.236  ARIMA(4,1,1) with drift         : 5040.53  ARIMA(5,1,1) with drift         : 1e+20  ARIMA(4,1,0) with drift         : 5112.614  ARIMA(4,1,2) with drift         : 4953.417  ARIMA(5,1,3) with drift         : 1e+20  ARIMA(4,1,2)                    : 4960.516  ARIMA(3,1,2) with drift         : 1e+20  ARIMA(5,1,2) with drift         : 1e+20  ARIMA(4,1,3) with drift         : 4868.669  ARIMA(5,1,4) with drift         : 1e+20  ARIMA(4,1,3)                    : 4870.92  ARIMA(3,1,3) with drift         : 1e+20  ARIMA(4,1,4) with drift         : 4874.095   Best model: ARIMA(4,1,3) with drift           So I assume the arima function should be used as:  bestOrder &lt;- cbind(modArima$arma[1],modArima$arma[5],modArima$arma[2]) sarima1&lt;-arima(salests, order = c(4,1,3))   with no seasonal component parameters and period specifications. Data and exploratory analysis show that the same weekly pattern can be approximatively considered for each week, with the only exception of August 2010 (when a consistent increase in sales is registered). Unfortunately I have no expertise in timeseries modeling at all, in fact I am trying this approach in order to find an alternative solution to other parametric e non-parametric models I have tried to fit for these problematic data. I have also many dependent numeric variables but they have shown low power in explaining the response variable: undoubtedly, the most difficult part to model is the time component. Moreover, the construction of dummy variables to represent months and weekdays turned out not to be a robust solution. Why didn't the Hobbits get frostbite? Let $\mathbb{S} = \{ (X,Y,Z) \in \mathbb{R}^3 :X^2 + Y^2+Z^2=1\}$ denote the Riemann sphere. The following problem is taken from Gamelin "Complex Analysis" chapter one:     What is the image of the spherical cap $ C := \{(X,Y,Z) \in \mathbb{S} : A \leq X \leq 1\}$, where $A$ is some fixed real number between $-1$ and $1$, under the stereographic projection $ \pi(X,Y,Z) = \frac{X+iY}{1-Z}$?   Here is my work so far:  ( The case $A=1$ is trivial, so assume $A &lt;1$)  We want to find $\pi(C) = \{z \in \mathbb{C} \mid \exists(X,Y,Z) \in C: \pi(X,Y,Z) = z \}= \{z \in \mathbb{C} \mid \pi^{-1}(z) \in C\}$, where the last equality holds by $\pi$ being a bijection. Write $z=x+iy$.  Here, $ \pi^{-1}(z) \in C\iff A\leq \frac{2x}{1+x^2+y^2} \leq 1 \iff (\star) :A \leq \frac{2x}{1+x^2+y^2}$, since the inequality $2x \leq 1+x^2+y^2$ holds for all $x,y \in \mathbb{R}$.  $A=0$: $( \star)$ reduces to $ 0 \leq x$, so in this case the image is $\{z \in \mathbb{C} \mid \Re(z) \geq 0 \}$.   $A &gt;0$: Since $A \neq 0$ in $(\star)$ we have,  $ \frac{2x}{1+x^2+y^2} \geq A \iff Ax^2-2x+Ay^2+A \leq 0 \iff (x-\frac{1}{A})^2 +y^2 \leq \frac{1}{A^2}-1$. Thus the image is the interior and boundary of the circle centered at $\frac{1}{A}$ with radius $\sqrt{\frac{1}{A^2}-1}$. Note $\frac{1}{A^2}-1 \geq 0$ as $A \in (0,1)$.  $A &lt;0$ : By symmetry and the above, here the image is $ \mathbb{C} / \{ $ interior and boundary of circle centered at  $  \frac{1}{A}$ with radius $\sqrt {\frac{1}{A^2}-1} \}$.  Is my working correct? Many Thanks!