Let $f(x)$ be a polynomial of degree n with real coefficient and such that $f(x)\geq0$ for every real number $x$ prove that $ f(x)+f'(x)+\dots+f^{(n)}(x)\geq0 $ for all real number $x$. I thought about the following. Since $f(x)$ is positive for all real number, then $f(x)$ must be even function, with a positive leading coefficient. Any hints will be appreciated
Let $p(x)$ be a polynomial of degree $n$ satisfying $p(x)\geq 0$ for all $x$. That is, for all $x$, $p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \geq 0$, $a_n\neq 0$.  Show that $p(x)+p&#39;(x)+p&#39;&#39;(x)+\cdots+p^{(n)}(x)\geq 0$ for all $x$ where $p^{(i)}(x)$ is the $i^\text{th}$  derivative.  My interest: I know that, we can rewrite the LDE as follows  $$ p(x)+p&#39;(x)+p&#39;&#39;(x)+\cdots+p^{(n)}(x) = Lp(x) $$  where $L := I + D + D^2  + \cdots + D^{(n)}$. Can we say anything about a linear operator of this kind so that it does not change the sign of the input it takes? I can try to solve the question by writing for all the derivatives and factoring them using $p(x)$, but I think there should be a clever way of showing this by the properties of $L$. Can I figure out a solution by just looking at $L$ and the sign of $p(x)$ as in the question? Where should I look for that?
What's a better way to define individual link colours for a url in a macro? I've got a document where I'd like two colours of links - cyan for links in headers that since it's brighter, and navy blue for links in the main body which is more subtle/less distracting. I already tried using the xcolor package and changing the text colour with textcolor, like \textcolor{cyan}{\href{url.com}{link name}}, but it doesn't override whatever link or url colour I have defined in hypersetup.   So this is what I've done:   I've defined the colour navy:   \definecolor{navy}{HTML}{2F729C}    And also imported the hyperref package and defined commands to set the global link/url colours to cyan / navy.   \usepackage{hyperref}                         \hypersetup{colorlinks=true,linkcolor=navy,urlcolor=navy}                        \newcommand{\navylinks}{\hypersetup{linkcolor=navy,urlcolor=navy}}               \newcommand{\bluelinks}{\hypersetup{linkcolor=cyan,urlcolor=cyan}}     Then I've defined another command specifically for the header links, where I turn on cyan links globally, include the header text and link, and then turn on navy links again:   \newcommand{\headerlink}[1]{\bluelinks\href{#2}{#1}\navylinks}   My issue with this is that I feel like there could be a better way to do it than to set the global link colour every time I use a link macro; is there a way to define the link colour just for one link, like one would do with \textcolor{...}, that would be able to override hypersetup? Does an ability or spell grant me the bonus action with which to use it? The  description of casting times says that spells with a casting time of &quot;bonus action&quot; require a bonus action to cast (p. 79, emphasis mine):  Most spells require a single action to cast, but some spells require a bonus action, a reaction, or much more time to cast. Bonus Action A spell cast with a bonus action is especially swift. You must use a bonus action on your turn to cast the spell, provided that you haven't already taken a bonus action this turn. You can't cast another spell during the same turn, except for a cantrip with a casting time of 1 action.  It's pretty clear that bonus actions aren't automatic (p. 69):  You can take a bonus action only when a special ability, spell, or other feature of the game states that you can do something as a bonus action. You otherwise don't have a bonus action to take.  So to cast a spell like Healing Word (p. 93), do I  need to be granted a bonus action that I can use, or do I have a bonus action I can use?  I guess I'm unclear on whether I need to have a bonus action before I can spend it on something, or whether having an ability, spell, or whatnot that can be done as a bonus action grants me the bonus action I need to use it. Insert a paragraph into a list that uses the whole text width Let $R$ be a Artinian commutative ring with $1 \neq 0$. If $I$ is prime, then $I$ is maximal. Prove: Let $R$ a Artinian commutative ring with $1 \neq 0$. If $I$ is prime, then $I$ is maximal.     I got stuck on this. I understand that every ideal is generated by finitely many elements. Here is my approach:  It's enough to prove the following implication: $$ R/I \ \text{is a domain} \quad \Rightarrow \quad R/I \ \text{is a field}$$ Then only finding an inverse is left. So let $x \in R$ in arbitrary. We have to find an inverse element of $x+I \in R/I$, that as an element y so that $x \cdot y \in I+1$. We denote $I = (x_1, x_2, \cdots, x_n)$.  Now I have to find y so that $xy -1 = r_1 x_1+ r_2 x_2 + \cdots + r_n x_n$. This is where I no more knew what to do.   Your advise will be appreciated. Recently I decided to use only Ubuntu and there is no Windows in my laptop. When I turn on my laptop the GRUB menu is showing but I don't need to choose OS anymore.   How can I remove or hide this? And after removing this how could I access to recovery mode? find exec '{}' not available after > Exec allows us to either pass all arguments at once with {} + or to pass them one by one with {} \;  Now let's say I want to rename all jpeg, no problem doing this:  find . \( -name '*.jpg' -o -name '*.jpeg' \) -exec mv '{}' '{}'.new \;   But if I need to redirect output, '{}' isn't accessible after redirection.  find . \( -name '*.jpg' -o -name '*.jpeg' \) -exec cjpeg -quality 80 '{}' &gt; optimized_'{}' \;   This wouldn't work. I'd have to use a for loop, storing find's output into a variable before using it. Let's admit it, it's cumbersome.  for f in `find . \( -name '*.jpg' -o -name '*.jpeg' \)`; do cjpeg -quality 80 $f &gt; optimized_$f; done;   So is there a better way? How can I change pdf page size? I would like to split pages in pdf so that each page would be divided into 2 pages ( e.g. form 100 pages I would go into 200 but smaller - exactly half size).  I need this to read pdfs on my ebook reader.  Anybody knows how can this be done ? Is there a 'biblatex in a nutshell' guide out there? I'd like if someone explained to me the essentials on how to use biblatex (what lines I have to write in my document, which files I have to have, how many times and what I have to compile), so then I can go to  to customise it further. Refreshing Formula Fields Dynamically (without save) I have an "invoice" custom object which has a couple of summarising formula fields in. I also have a complicated and dynamic visualforce page for building the invoice objects.  As the user is completing this form, different areas are dynamically updated to show "latest" values. This is all fine where I am copying data about or performing bespoke lookups and maths, but - in what is really the simplest case - a formula field, I can find no way to get the value to update, without performing some weird cyclic save + reload of the whole object.  This is inappropriate for a couple of reasons (the object should not be saved until the invoice is complete + accepted) and the other option I have found online is to spoof the value by duplicating my formula field maths into the custom controller and re-executing them there - also unacceptable from a code/logic location/duplication perspective.  Is there anyway to "force" an object/page to recalculate it's formula fields, either through standard or extended controllers? p:commandbutton action doesn't work inside p:dialog Is it (and if yes how?) possible to define an an binary operation $*$ for an arbitrary set $M$ such that $(M,*)$ is a group?  If $M$ is finite or countable infinite this is trivial, but is it also possible for any other sets? And if yes how can you construct $*$? If no can you provide a counterexample? What is the minimum $ \sigma$-algebra that contains open intervals with rational endpoints What are the minimum $\sigma$-ring and $\sigma$-algebra on $\mathbb R$ which contain the open intervals with rational endpoints?  Is there a relation between this $\sigma$-algebra and Borels? How to delete a file with a weird name? I wrote a program that uses POSIX memory-mapping function (mmap)  The program takes a file (a.dat) and memory-maps it for reading/writing.  Due to errors in the program, every time I run the program a file with some weird names (e.g., ?d?P?^z??d?P?^z?) is created. The error is resolved but I am not able to delete the files.  I am not able to delete it either using command line or by select/deleting from window manager.  So how should I delete it? I'm using Ubuntu 11.04. I am getting FATAL_ERROR|System.QueryException: List has no rows for assignment to SObject. I know that it is because it is not returning any rows in the SOQL query but what I want is that if it returns any row, it should update ALO field on contact object, if it is null then ALO field should be set to null.  Here is my code:  ziptoLOD = [SELECT Id, Name, LOD__c             FROM Zip_3_to_LOD__c             WHERE Name =: contact.Extract_Zipcode__c ];  Zip_3_to_LOD__c ziptoLOD2 = (ziptoLOD != null) ? ziptoLOD : null;  if (ziptoLOD2 == null) {     contact.ALO__c = ' ';    } else {     System.debug(System.LoggingLevel.ERROR, 'Zip to LOD' + ziptoLOD2);     contact.ALO__c = ziptoLOD2.LOD__c; }   My code gets stuck at the SOQL query and gives me the list has no rows error. Looking for a scifi novel with advanced pheromone technology I'm trying to solve the well known Coupon Collector's Problem by explicitly finding the probability distribution (so far all the methods I read involve using some sort of trick). However, I'm not having much luck getting anywhere as combinatorics is not something I'm particularly good at.   The Coupon Collector's Problem is stated as:  There are $m$ different kinds of coupons to be collected from boxes. Assuming each type of coupon is equally likely to be found per box, what's the expected amount of boxes one has to buy to collect all types of coupons?  What I'm attempting:  Let $N$ be the random variable associated with the number of boxes one has to buy to find all coupons. Then $P(N=n)=\frac{|A_n|}{|\Omega _n|}$, where $A_n$ is the set of all outcomes such that all types of coupons are observed in $n$ buys, and $\Omega _n$ is the set of all the possible outcomes in $n$ buys. I think $|\Omega _n| = m^n$, but I'm not even sure about that anymore, as all my attempts so far led to garbage probabilities that either diverged or didn't sum up to 1. Space of probability measures "complete"? (In the other sense) What will be a smoother transition into GNOME for Ubuntu 18.04-- Ubuntu GNOME or Vanilla Ubuntu? ^^^^ Will Ubuntu GNOME upgrade to Ubuntu 18.04, or will support discontinue when 18.04 is released?? Just want to know what's better to run right now so I can upgrade and not have to reinstall later(I intend to see this transition through)-- GNOME installed correctly on vanilla Ubuntu, or Ubuntu GNOME? Is it possible to save settings for Microsoft Paint? After closing paint the last used settings went back to default. I want them to be saved. Like the selected Brush-Tool, Color, Line-strength, etc.  Either after every usage or I define one I use most. Maybe through the registry or by start parameters? How to prove that derivatives have the Intermediate Value Property