Irreducibility of polynomial $x^k-x^{k-1}-\ldots-x-1$
Irreducibility of an infinite sequence of polynomials
How does ACF & PACF identify the order of MA and AR terms? It's been more than 2 years that I am working on different time series. I have read on many articles that ACF is used to identify order of MA term, and PACF for AR. There is a thumb rule that for MA, the lag where ACF shuts off suddenly is the order of MA and similarly for PACF and AR.   Here is  I followed from PennState Eberly College of Science.   My Question is why is it so? For me even ACF can give AR term. I need explanation of thumb rule mentioned above. I am not able to understand thumb rule intuitively/mathematically that why -  Identification of an AR model is often best done with the PACF. Identification of an MA model is often best done with the ACF rather than the PACF  Please note:- I don't need how but "WHY". :) Why are swap partitions discouraged on SSD drives, are they harmful? I often read that one should not place swap partitions on a SSD drive, as this may harm the device. Is this true? Can you please explain the reason to me?  Because I otherwise would have thought that placing swap on an SSD is the best choice, as it's much faster than HDDs and therefore swapping RAM contents to the SSD is not as slow as it would be with the HDD... Mail on Big Sur does not reliably find dollar values Missing abstract with amsart class I'm not very good at LaTeX but I've been using it to write my dissertation. Have just come to the end and trying to format everything correctly. Whenever I try to put my Abstract paragraph into the form \begin{abstract} (writing here) \end{abstract} and click play the whole abstract section just disappears. What am I doing wrong?  \documentclass[12pt]{amsart}  \begin{document}  \begin{abstract} (my abstract here) \end{abstract} \end{document} When would you use a voltage regulator vs a resistor voltage divider?  Are there any uses for which a resistor divider is particularly bad? What do I if I used "Reset All" or "Erase All Content and Settings" on a jailbroken iOS Device? I recently received an iOS device and it was jailbroken. When I wanted to unjailbreak it, I used "Erase All Content and Settings." The loading screen came on and it has not changed. After rebooting the device, it is now stuck at the Apple logo. What should I do? Probability Interval for Gamma Distribution There is something that I am doing wrong in the exercise below and I would appreciate some help figuring it out.  Let $X_1, X_2, \ldots, X_5$ be a random sample from a $\Gamma \left(3,3 \right)$ distribution . Define $W=\sum_{i=1}^5 X_i $.    Obtain the distribution of $W$ and of $2W/3$ Find $c_1$ and $c_2$ such that $P \left( c_1 &lt; W &lt;c_2 \right)=0.95 $     Since $W$ is the sum of iid gamma variables, $W \sim \Gamma \left(15,3 \right)$ and now a straightforward transformation argument shows that $Y=\frac{2}{3} W \sim \chi^2 \left(30 \right) $. This completes question 1.  Now for question 2, my thoughts were to transform these probabilities using the result from question 1, namely that $\frac{2}{3} W \sim \chi^2 \left(30 \right) $. Hence, unless I have made a horrible mistake here:  $$P \left( \frac{2}{3} c_1 &lt; Y &lt; \frac{2}{3} c_2 \right)=0.95 $$  We can find the constants from a chisquare table for $30$ degrees of freedom. In my case, they are $16.791$ for the lower bound and $46.979$ for the upper one.   Then if we equate $\frac{2}{3}c_1=16.791$ and solve for $c_1$ we should be able to find the constant for the above gamma confidence interval, right?   The problem is that according to $R$, $P \left( W&lt;c_1 \right) =1 $, instead of  the desired $0.025$ which means I have gone wrong somewhere. I do not think it is in question though $1$ as I have verified the result many times.   Thank you. Suppose that $S_0$ and A are positive numbers, let $$S_{n+1} = \frac{1}{2}\left(S_n +\frac{A}{S_n}\right)$$ with $n \geq 0 $.      (a)Show that $S_{n+1} \geq \sqrt{A} $ if $n \geq 0$       (b)Show that $S_{n+1} \leq S_n $ , if $n \geq 1$      (c) Show that $s= \lim\limits_{n \rightarrow \infty} S_n$ exists      (d) find s   (a) Show that $S_{n+1} \geq \sqrt{A} $ if $n \geq 0$  Given $$P_n:  S_{n+1} = \frac{1}{2}\left(S_n +\frac{A}{S_n}\right) \geq \sqrt{A}$$  $$P_0: S_{1} = \frac{1}{2}\left(S_0 +\frac{A}{S_0}\right) \geq \sqrt{A}  $$  We assume that $P_n$ is true  $$P_{n+1}: S_{n+2}= \frac{1}{2}\left(S_{n+1} +\frac{A}{S_{n+1}}\right)$$ by assumption $$S_{n+2}= \frac{1}{2}\left(S_{n+1}\left(1 +\frac{A}{(S_{n+1})^2}\right)\right) \geq \frac{1}{2}\left(\sqrt{A}\left(1 +\frac{A}{(\sqrt{A})^2}\right)\right)$$ $$ S_{n+2}= \frac{1}{2}\left(S_{n+1} +\frac{A}{S_{n+1}}\right) \geq \sqrt{A}  $$  It follows that $S_{n+1} \geq \sqrt{A} $    (b) Show that $S_{n+1} \leq S_n $ , if $n \geq 1$  $$S_{n+1} \leq S_n$$ $$\frac{1}{2}\left(S_n +\frac{A}{S_n}\right) \leq S_n $$ Dividing by $S_n$ $$\frac{1}{2}\left(1 +\frac{A}{S_n^2}\right) \leq 1 $$ $$\frac{A}{2S_n^2} \leq \frac{1}{2}$$ $$A \leq S_n^2$$ $$S_n \geq \sqrt{A}$$ As  $S_{n+1} \leq S_n$ yields a true statement, it follows $S_{n+1} \leq S_n$ is true.    (c) Show that $s= \lim\limits_{n \rightarrow \infty} S_n$ exists  Since $S_{n+1} \leq S_n$, the sequence is non-increasing,   using the non-increasing theorem stating that      if $\{S_n\}$ is non-increasing then $$\lim\limits_{n \rightarrow &gt; \infty} S_n = \inf\{S_n\} $$     (d) find s    Is the argumentation in (a) and (b) appropriate? Also, I have to admit I m getting less confident in my argumentation (c) and (d). How to proceed in (c) and (d)?  Much appreciated for your input or help. Rounded corners in TikZ trees. How do I enable the "Universe" repository from the command line? I have been experiencing an issue that once I have a material created and looking the way I want it, I cannot seem to export it correctly with the textures mixed.  I don't know the correct words to use, but in blender I have the material created using the nodes editor, and I have placed two textures inside of there (one for the rock texture and one for moss, my object is a rock). When I export this model and import to Unity (the game engine I am using), I find a materials folder but the material inside of it does not have the textures assigned, it's just a plain white material?  How can I export my material from blender in a way that it will allow me to import it WITH TEXTURES into Unity? Financial proof to get a Schengen visa to France I am planning to apply for a Schengen visa to visit France. I have an attestation d'acceuil, but the visa requires I submit bank statement for the past 3 months. I can get this statement, but the problem is that my company has been paying my salary in cash for the past months, and I place the money into the account by myself.  Can I submit the following?   my accounting book (current account)  letter from the bank stating that I have money in it payslips for the past 3 months   Will that satisfy the financial requirements for the visa? Which damage dice exactly does the Great Weapon Fighting fighting style allow you to reroll? OSX Server: should I use Open Directory? Find all rings $R$ satisfying $\mathbb Z \subset R \subset \mathbb Z[\frac {1}{2}]$ Checking out  and  (and some papers) was a bit difficult for understanding Group Action. The closest to understanding was this part:    I understand the definition of homomorphism (a structure-preserving map between two algebraic structures of the same type). And from what I understand the symmetric group is a group of functions (bijective functions) mapping elements of a set $X$ onto itself.  However I get confused at this line:     ...an action of $G$ on $X$ may be formally defined as a group homomorphism $\phi$ from $G$ to the symmetric group of $X$.   That seems to be the definition of group action, but I am having trouble understanding how to apply that.  What that says to me is it is a homomorphism (function) $f \colon X \rightarrow S$ where $S$ is a set of bijective functions on $X$. That itself seems weird, I thought a group action would result in a number or something related to $\{1, 2, 3\}$ (defined next).  Continuing, if we have a set $X = \{1, 2, 3\}$ and a group $(*,\{4, 5, 6\})$, then the symmetric group of $X$ would be something like $S = \{1 \rightarrow 2, 2 \rightarrow 3, \dots \}$, and $\phi$ would be something like $f : \{4, 5, 6\} \rightarrow S$. That doesn't make sense to me and I feel like I'm interpreting something wrong. I don't see how mapping to a set of functions $S$ will result in an action on the set. What is the principle of underdetermination? While studying I read about the principle of underdetermination of scientific theories. I made some researches online but I am more confused than before. I read about the Quine-Duhem holistic thesis, but I do not know if is the same as underdetermination. There does not exist an entire function which satisfies $f({1\over n})={1\over 2^n}$? There does not exist an entire function which satisfies $f({1\over n})={1\over 2^n}$, what I tried is if possible then define $g(z)=f(z)-{1\over 2^{1\over z}}$ Then $g({1\over n})=0$ and so $g(z)$ is entire and its $0$ set has limit point in it and so $f(z)={1\over 2^{1\over z}}$ which is not analytic at $0$? Please help!  Edit: OOps! the way I defined $g(z)$ that is not entire! could any one give me hint? Custom action to set emblem in thunar Open Mapping Theorem: counterexample