Let $f(x)$ be a monic polynomial of odd degree. Prove that $\exists A\in \mathbb{R}$ s.t. $f(A)&lt;0$ and there exists $B \in \mathbb{R}$ such that $f(B)&gt;0$.  Deduce that every polynomial of odd degree has a real root.    There are questions that answer the final part, but they do not do so by proving the first part. I am fairly sure that this involves the intermediate value theorem, but not sure how to implement it in this case.
I want to prove that every real polynomial of odd degree has at least one real root, using  the intermediate value theorem.  Let $P(x) = x^{2n+1} + a_n x^{2n} + . . . + a_0$ for each $a_i \in \mathbb{R}$ and $n \in \mathbb{N}$.  By the fundamental theorem of algebra I know that $P(x)$ has exactly $2n+1$ complex roots, so  $P(x) = (x+r_1)(x+r_2) . . . (x+r_{2n+1})$ for each $r_i \in \mathbb{C}$  I do not know how to complete this but I do know that, at some point, I probably have to show that each root with imaginary part non zero has to come in conjugate pairs, and since $2n+1$ is odd there is at least $1$ root that is imaginary part $0$ and thus real.
I've heard that you need to get a Mac if you want to develop iPhone apps. Is this true?   Is it possible to develop iPhone apps using Linux? If yes, what do I need and where do I download the necessary tools? I recently found out about Ubuntu for Android Multi-Core Smartphones and was digging around on the Ubuntu website as well as the Canonical website but have yet to find any information on if it available to the public yet ?  I am very interested to try it our on my Android and if it is available for the public am wondering where and how I can get a copy of "Ubuntu for Android" ?  If it is not available then perhaps someone can shed some lite on what sort of timeframe it will be until it is available to the general public? Does "no larger than" imply shapability? Acknowledgements outside of document environment How do I add an acknowledgements section (maybe not specifically \section) outside of (before) my document environment?  in my .cls, there is  \def\@acknowledge{} ... \def\acknowledge#1{\booltrue{@ack} \gdef\@acknowledge{#1}} ... \def\ackpage{ \chapter*{Acknowledgments} \@acknowledge \newpage }   but it seems that I find problems when I put newlines in the  \acknowledge{folks in here}  (see  ) I'm very familiar with how symbols display differently in inline mode (with $ ... $) vs. how display math shows them (with \[ ... \] or $$ ... $$).  Two examples would be with limits  $\lim_{n\rightarrow \infty}f(x)$   vs.  \[ \lim_{n\rightarrow \infty}f(x) \]   and with sums  $\sum_{n=1}^{x} n^2$   vs.  \[ \sum_{n=1}^{x} n^2 \]   My Question is:     How do you display these things in inline mode (with all the nice inline formatting) as if they were in display mode ?   The inverse -- how to display things in display mode as if they were in inline mode -- is also of interest. Homology Groups of $S^n$ Intuition behind $k$-algebra, $k$-algebra morphisms? How to install VirtualBox from command line? I am a newbie. For some reasons, I need to install Virtualbox. I see that I can easily install it using the Software Center, but I trust most the apt-get install way, because I find it safer, less buggy, and it helps me accommodate more with the Terminal.  So, my question is the following: is there a way to install VirtualBox using the command line? I would prefer something easy, like apt-get install not unzipping, archiving, etc, because I still don't know how to do these in the Terminal. Looking for a book I read in the 90's (not sure if it was published in the 90's or not).  The story as far as I can remember goes like this. Human-looking aliens evolve on a very hostile planet. They encounter some big bad threat and they cast about for help. They encounter a rich industrialist on Earth and they send a woman to recruit him. She teaches the guy to use magic and how to transform his body. They build a ship and travel to her planet but just before they arrive, she vanishes and he crash lands on this hostile planet.  He goes through many trials and tribulations in order to get to her and eventually he has enough and uses the ability she teaches him on how to transform his body. He transforms himself into a half-man, half-dragon-like creature which allows him to travel to the other side of the planet to get to the girl. On the way he manages to neurally link into a large flying creature that flies very fast, only to discover that it is intelligent and they soon become friends. When he gets to the girl she doesn't recognise him so he gets mad and starts attacking only to realise that it's all been a test set up by these people to allow him to reach higher levels of magic.   There were supposed to be sequels but I never got around to reading them.  I remember the cover art depicted a large dragon like creature with a torso of a man and a woman standing in front of it looking back at him.  Any idea what this book may be? note: this question has been updated to reflect that we are currently using MySQL, having done so, I would like to see a how much easier it would be if we switched to a CTE-supporting database.  I have a self-referencing table with a primary key, id and a foreign key parent_id.  +------------+--------------+------+-----+---------+----------------+ | Field      | Type         | Null | Key | Default | Extra          | +------------+--------------+------+-----+---------+----------------+ | id         | int(11)      | NO   | PRI | NULL    | auto_increment |  | parent_id  | int(11)      | YES  |     | NULL    |                |  | name       | varchar(255) | YES  |     | NULL    |                |  | notes      | text         | YES  |     | NULL    |                |  +------------+--------------+------+-----+---------+----------------+   Given a name, how can I query the top-level parent?  Given a name, how can I query all of the id's associated with a record of name = 'foo'?  context: I am not a dba, but am planning to ask a dba to implement this type of hierarchical structure and would like to test some queries. The motivation for doing so is described by .    Here is an example of the relationships among ids in the table:    -- ----------------------------------------------------- -- Create a new database called 'testdb' -- ----------------------------------------------------- SET @OLD_UNIQUE_CHECKS=@@UNIQUE_CHECKS, UNIQUE_CHECKS=0; SET @OLD_FOREIGN_KEY_CHECKS=@@FOREIGN_KEY_CHECKS, FOREIGN_KEY_CHECKS=0; SET @OLD_SQL_MODE=@@SQL_MODE, SQL_MODE='TRADITIONAL';  CREATE SCHEMA IF NOT EXISTS `testdb` DEFAULT CHARACTER SET latin1 COLLATE latin1_swedish_ci ; USE `testdb` ;  -- ----------------------------------------------------- -- Table `testdb`.`observations` -- ----------------------------------------------------- CREATE  TABLE IF NOT EXISTS `testdb`.`observations` (   `id` INT NOT NULL ,   `parent_id` INT NULL ,   `name` VARCHAR(45) NULL ,   PRIMARY KEY (`id`) ) ENGINE = InnoDB;  SET SQL_MODE=@OLD_SQL_MODE; SET FOREIGN_KEY_CHECKS=@OLD_FOREIGN_KEY_CHECKS; SET UNIQUE_CHECKS=@OLD_UNIQUE_CHECKS;  -- ----------------------------------------------------- -- Add Example Data Set -- -----------------------------------------------------   INSERT INTO observations VALUES (1,3), (2,5), (3,NULL), (4,10),     (5,NULL), (6,1), (7,5), (8,10), (9,10), (10,3); Are GUID collisions possible? I'm working on a database in SQL Server 2000 that uses a GUID for each user that uses the app it's tied to. Somehow, two users ended up with the same GUID. I know that microsoft uses an algorithm to generate a random GUID that has an extremely low chance of causing collisons, but is a collision still possible? You can think this of the following as a 3d QFT where we try to calculate the self-energy of two fields. $I$ is a this external self-energy and let us assume it does not depend on the loop momenta which I denote a $\vec{x}=(x,y)$. Consider the following propagator  $$ P(\vec{r}) = \frac{x-ia}{2i^2[\,(x-ia)^2+(y-ib)^2+(z-ic)^2]\,} \,\, \gamma_i. $$ Then we want to calculate $$ I(x) = \int \frac{\,dx \,dy\, dz}{(2\pi)^3} \text{Tr} \left( \frac{x-ia}{2i^2[\,(x-ia)^2+(y-ib)^2+(z-ic)^2]\,}\gamma_x \gamma_x  \right), $$ where Tr$(\gamma_i \gamma_j)=2\delta_{ij}$. Therefore  I am trying to resolve the following integral where $x,y,z \in \mathbb{R}$ and $a,b,c$ are constants (internal self-energies if you wish) in $\mathbb{R}$.  Then, after simplifying the propagator $P(x,y)$ we end up with the following expression for the $x$ component:  $$ I = \frac{1}{2i^2} \int \frac{dxdy}{(2\pi)^2} (x-ia) \int \frac{dz}{2\pi} \frac{1}{[\,(x-ia)^2+(y-ib)^2+(z-ic)^2]\,} $$ but it can be shown that  $$ \int \frac{dq}{(q+iv)^2+w^2} = \frac{\pi}{|w|}\theta(|w|-|v|).  $$ (e.g. see here ) Thus we get that for $|(y-ib)^2+(x-ia)^2|&gt;|a|$, $\theta=1$ (for this is what we are interested about) and then we are left with $$ I = \frac{1}{4i^2} \int \frac{dxdy}{(2\pi)^2} \frac{(x-ia)}{|(x-ia)^2+(y-ib)^2|}.   $$ The question is wether I can make the transformation $$ (x-ia) = q \cos \theta  $$ $$ (y-ib) = q \sin \theta $$ in polar coordinates. Then $dx dy = qdqd\theta$ and the previous integral becomes, assuming $|q|&gt;0$, $$ I = \int_0^{\infty} dq \int_0^{2\pi} d\theta \frac{q \cos \theta}{|q|} =0. $$ Is this correct? I am confused since one can find complex poles in the $I$. If not, what would the correct way to resolve this be? I have a suspicion that this integral should indeed give zero but how can we find this? How do I define global variables in CoffeeScript? What's the purpose (if any) of "javascript:" in event handler tags? I've been making a concerted effort to improve my javascript skills lately by reading as much javascript code as I can. In doing this I've sometimes seen the javascript: prefix appended to the front of event handler attributes in HTML element tags. What's the purpose of this prefix? Basically, is there any appreciable difference between:  onchange="javascript: myFunction(this)"   and  onchange="myFunction(this)"   ? I understand intuitively that this is true, but I'm embarrassed to say I'm having a hard time constructing a rigorous proof that $|a+b| \leq |a|+|b|$. Any help would be appreciated :) Lufthansa airline offers the &quot;Economy Flex&quot; fare. The listed advantages of this fare are:  Rebooking possible plus fare difference Refundable except 70 EUR plus fare difference  What does the &quot;plus fare difference&quot; mean? Imagine I buy a ticket for 200 €. Later I decide to cancel. They keep 70 Euro, so I get 130€ back. What's with the fare difference? Fare difference to what? As a comparison, the &quot;Economy classic&quot; states:  Rebooking: 70 EUR plus fare difference Refund not possible Make Javascript do List Comprehension What year is the series "Chilling Adventures of Sabrina" supposedly set in? Almost everything in Greendale seems to be vintage 1960's (cars, movies, dress). I assume it is to reflect the original comic book, but I swear I saw Ambrose using a laptop computer in the first episode. Was this just to make it more topical? Or is it some kind of alternate reality? threeparttablex table notes for tabu tabulars with less than \textwidth Existence of area-preserving diffeomorphism between two sets in $\mathbb{R}^2$ Suppose that we have open sets $A, B\subset \mathbb{R}^2$ such that $A$ and $B$ are simply connected and have smooth boundaries. Furthermore, let us assume that $m(A) = m(B) &gt; 0$, where $m$ is the Lebesgue measure on $\mathbb{R}^2$. My question is as follows: when does there exist an area-preserving diffeomorphism between $A$ and $B$? In this context, an area-preserving diffeomorphism between $A$ and $B$ is a is a $C^{\infty}$-diffeomorphism $\phi : A\to B$ such that $m(S) = m(\phi(S))$ for all measurable $S\subseteq A$. If an area-preserving diffeomorphism doesn't always exist between sets $A$ and $B$ as above, are there any extra conditions we can impose on $A$ and $B$ such that there does exist an area-preserving diffeomorphism?  My thoughts thus far are that we always have a $C^{\infty}$-diffeomorphism between $A$ and $B$: this is a consequence of the . It's also obvious to me that if we can deform $A$ linearly to get $B$, then that deformation will be area-preserving. Furthermore, it seems that aside from this case, there's no reason why an area-preserving diffeomorphism should exist. However, I'm having a hard time proving that there necessarily isn't an area-preserving diffeomorphism between $A$ and $B$ if $B$ is not a linear deformation of $A$. Does anybody have any ideas?