Show that $\text{trace}(A)\leq \sqrt{n}\sqrt{\text{trace}(AA^t)}$ where $A$ is in $M_{n}(\mathbb{R}).$  We know that $\text{trace}(A) =\sum_{i}\lambda_i$ where $\lambda_i$ are eigenvalues of $A.$ Then I was wondering if we could show that the eigenvalues of $AA^t$ are $\lambda_i^2$ then we could apply the AM-GM inequality to complete the proof. So we know that $A$ and $A^t$ have the same eigenvalues, but can we use this conclude that $AA^t$ have $\lambda_i^2$ as eigenvalues?
The following definition is needed for my actual question:  For $A, B \in \mathcal{M}_{n \times n}$ define $$  \langle A, B \rangle = \text{Trace}(B^{T}A) = \sum_{j=1}^{n}\sum_{i=1}^{n}b_{ij} \, a_{ij} $$  And here is the actual question:  For $B \in \mathcal{M}_{n \times n}$ show that $$ \text{Trace}(B)^{2} \:\leq\: n \,\text{Trace}(B^{T}B) $$  I know I have to use the Cauchy-Schwarz Inequality, but I wasn't exactly sure how to.  Here is my attempt:  Since I only have one matrix I decided to use the identity matrix $$ \begin{align} \text{Trace}(B)^{2} \:&amp;\leq\: n \,\text{Trace}(B^{T}B) \\ \text{Trace}(I^{T}B)^{2} \:&amp;\leq\: n \, \langle B, B \rangle \\ \langle I_{n}, B \rangle^{2} \:&amp;\leq\: n \, \Vert B \Vert^{2} \\ \langle I_{n}, B \rangle \:&amp;\leq\: \sqrt{n} \, \Vert B \Vert \end{align} $$  I feel like I either did it incorrect, or that I'm close but am missing something.
(semi)-Automatic pluralization around a citation 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. TeXstudio does a pretty good job with autocompletion. For example, when I type \eqref{, it already comes with suggestions, see the example below.    However, suppose that I don't want to use \eqref but a customized command using newcommand (I do this so that I can quickly switch to different styles, e.g., some for journals/conferences "Eq. (1)" is needed, whereas sometimes "(1)" suffices). In this case, TeXstudio does not do the auto-completion, see the example below.    Is it possible to configure TeXstudio such that it does the auto-completion for custom commands that require a reference?  Note that one option is to redefine \eqref using \renewcommand, but I'm looking for another way so that I can use if for many more commands, not just for \eqref.   Here is my MWE:  \documentclass{article} \usepackage{amsmath}  \newcommand{\alternative}[1]{Eq.~(\ref{#1})}  \begin{document}     \begin{equation}         \label{eq:emc2}         E = mc^2     \end{equation}     I can refer to the equation using \eqref{eq:emc2} and using \alternative{eq:emc2}. \end{document} Hadamard Product of $e^z-1$ Assuming the formulas $$\sin(z)=\frac{e^{iz}-e^{-iz}}{2i}\quad\text{and}\quad\frac{\sin(\pi z)}{\pi z}=\prod_{n=1}^\infty\left(1-\frac{z^2}{n^2}\right),$$ I want to find the Hadamard product of $e^z-1$. I think I've done the work correctly, but I end up having an extra factor of $2\pi i$ at the end (this is from Stein and Shakarchi's book and they list the answer for this particular problem). So, my work is \begin{align} e^z-1&amp;=e^{z/2}\left(e^{z/2}-e^{-z/2}\right)\\ \\ &amp;=e^{z/2}\left(e^{i(-iz/2)}-e^{-i(-iz/2)}\right)\\ \\ &amp;=2ie^{z/2}\left(\frac{e^{i(-iz/2)}-e^{-i(-iz/2)}}{2i}\right)\\ \\ &amp;=2ie^{z/2}\sin(-iz/2)\\ \\ &amp;=2\pi i ze^{z/2}\frac{\sin(-iz/2)}{\pi z}\\ \\ &amp;=2\pi i ze^{z/2}\frac{\sin\left(\pi(\frac{-iz}{2\pi})\right)}{\pi z}\tag{*}\\ \\ &amp;=2\pi i ze^{z/2}\prod_{n=1}^\infty\left(1-\frac{\left(\frac{-iz}{2\pi}\right)^2}{n^2}\right)\\ \\ &amp;=2\pi i ze^{z/2}\prod_{n=1}^\infty\left(1+\frac{z^2}{4n^2\pi^2}\right). \end{align}  The answer according to the book is $$e^z-1=e^{z/2}z\prod_{n=1}^\infty\left(1+\frac{z^2}{4n^2\pi^2}\right).$$Have I made a simple mistake in calculation, or are the two somehow compatible with each other?  Update: Thomas Andrews has pointed out a mistake in my infinite product formula. I've corrected that mistake.  $(*)$ The error occurs from this line to the next. Instead of multiplying and dividing by $z$, I should have multiplied and divided by $\frac{-iz}{2\pi}$. That would cause the appropriate cancellation. I want to run a command when there are only 10 entities left. I know you can use /execute unless entity to detect when all matching mobs have died, but it doesn't work if you want to run a command when there are for example 10 entities left.  A command like /execute if entity @e[limit=10] does not work, it also triggers if there is only one entity.  Such a counting system could also be used to determine once there is only one player left in an arena, meaning they have won a battle or similar. Testing equality of coefficients from two different regressions This seems to be a basic issue, but I just realized that I actually don't know how to test equality of coefficients from two different regressions. Can anyone shed some light on this?   More formally, suppose I ran the following two regressions:  $$ y_1 = X_1\beta_1 + \epsilon_1 $$ and  $$ y_2 = X_2\beta_2 + \epsilon_2 $$ where $X_i$ refers to the design matrix of regression $i$, and $\beta_i$ to the vector of coefficients in regression $i$. Note that $X_1$ and $X_2$ are potentially very different, with different dimensions etc. I am interested in for instance whether or not $\hat\beta_{11} \neq \hat\beta_{21}$.   If these came from the same regression, this would be trivial. But since they come from different ones, I am not quite sure how to do it. Does anyone have an idea or can give me some pointers?   My problem in detail: My first intuition was to look at the confidence intervals, and if they overlap, then I would say they are essentially the same. This procedure does not come with the correct size of the test, though (i.e. each individual confidence interval has $\alpha=0.05$, say, but looking at them jointly will not have the same probability). My "second" intuition was to conduct a normal t-test. That is, take   $$ \frac{\beta_{11}-\beta_{21}}{sd(\beta_{11})} $$  where $\beta_{21}$ is taken as the value of my null hypothesis. This does not take into account the estimation uncertainty of $\beta_{21}$, though, and the answer may depend on the order of the regressions (which one I call 1 and 2).   My third idea was to do it as in a standard test for equality of two coefficients from the same regression, that is take  $$ \frac{\beta_{11}-\beta_{21}}{sd(\beta_{11}-\beta_{21})} $$   The complication arises due to the fact that both come from different regressions. Note that   $$ Var(\beta_{11}-\beta_{21}) = Var(\beta_{11}) + Var(\beta_{21}) -2 Cov(\beta_{11},\beta_{21}) $$ but since they are from different regressions, how would I get $Cov(\beta_{11},\beta_{21})$?  This led me to ask this question here. This must be a standard procedure / standard test, but I cound not find anything that was sufficiently similar to this problem. So, if anyone can point me to the correct procedure, I would be very grateful! How can i integrate one SFDC org to another SFDC using Rest Api I am trying to integrate one Salesforce org to other org by using Oauth setup, I wanted to use  use consumer key and consumer secret key in my REST class to connect with the org, after connection part completion, I want to access Get(), Put(), Post(), etc. Including the ability to set Oauth and Authorization headers needed for Salesforce Rest API calls.  Actually I'm new in REST implementation so can you please explain me by using code samples. I have some commands to in my rc.local. I know that they are failing. How can I get log file with messages produced by executing rc.local? Where is it located?  I have checked the /var/log/boot.log. I know my messages are not there because I know already what is the reason of failure. But I still want to make sure from log file.  Note, I don't want to run script again, I could but I don't want. I would rather analyse wht happened during startup.  Thanks for any help.  Ubuntu 12.04 Desktop (if it matters) Journals selection - alternative criteria to impact factor While trying to select the journals I can submit my present work and future work, I realize the time between the date the authors send the article to the journal and when the revised article is submitted. Usually this is something between 6 months and more than 12 months.  Althought I am focusing on the Impact Factor (IF) to select journals, I think that 12 months to have a feedback that might include "add more variables analysis","this article will be more valuable if you test also model x,y and z", can have a huge impact on you work. It will make you stop what you are currently doing and possible spend up to 2 months making the changes. This might make a big mess on your work if you have deadlines to respect, however this happens to everyone in academia.  My question is, besides the IF journal based selection, how can we select a journal with enought quality with lower IF? What alternative criteria should one use to select the journal? Should we choose newly created Journals? Suppose Alice chooses a random Prime $p$ and a random private Key $a \in \mathbb{Z}^*_p$. By accident, she also chooses a random number $g \in \mathbb{Z}^*_p$, which is not a generator of $\mathbb{Z}^*_p$ and therefore  $$\langle g\rangle \subset \mathbb{Z}^*_p$$  as opposed to $\langle g\rangle = \mathbb{Z}^*_p$, which would yield a valid key. Alice then computes $A \equiv g^a \pmod{p} $ and publishes the Tuple $(p,g,A)$ as her public key.  Bob now encrypts a message $M$ using Alices public key by computing  $$C_1 = g^b \pmod{p}$$ $$C_2 = M \cdot A^b \pmod{p}$$  Is it possible for an attacker (Eve) to distinguish a “real” ciphertext $(C_1,C_2)$ from a random ciphertext $(Z_1,Z_2) \stackrel{$}{\longleftarrow} \mathbb{Z}^*_p$ with significant advantage (say $Adv \geq \frac12$)?    Edit: I guess it should be enough to show, that the ciphertext cannot be random, if  $$\langle g \rangle \neq \langle C_1 \rangle \quad(\text{or $Z_1$, respectively})$$  But how can Eve check for this property efficiently? Can someone point me in the right direction here?    Edit: A real-world situation where this flaw occurred Using the itemize environment inside a caption The following code does not compile. However, if I remove the itemize environment it does. I googled for this problem, and found suggestions that using the caption package would solve the problem, though I did not find any clear description of why the problem occurred in the first place. See for example    I'm using Debian squeeze with TeX Live 2009-11. The included version of caption has   \ProvidesPackage{caption}[2009/10/09 v3.1k Customizing captions (AR)]   I also tried downloading and using the caption package from     but this doesn't change anything. I put the sty files in my current directory. I assume that overrides the system files. This one corresponds to  \ProvidesPackage{caption}[2010/01/09 v3.1m Customizing captions (AR)]   Error message and example file follow.  *******************************************     ERROR: Argument of \@caption has an extra }.  --- TeX said --- &lt;inserted text&gt;                  \par  l.13     }  --- HELP --- From the .log file...  I've run across a `}' that doesn't seem to match anything. For example, `\def\a#1{...}' and `\a}' would produce this error. If you simply proceed now, the `\par' that I've just inserted will cause me to report a runaway argument that might be the root of the problem. But if your `}' was spurious, just type `2' and it will go away. ******************************************************  \documentclass{article} \usepackage{caption} \begin{document}  \begin{figure}     \captionsetup{singlelinecheck=off}     \caption     {       foo       \begin{itemize}       \item bar       \end{itemize}     } \end{figure} \end{document} Negative R squared contradicts ssa/sst? It seems that OTP encryption is unbreakable. I have read some articles (Wikipedia, Blogpost, etc.), but I could not find any information about how they are used.  If I wanted to create a messenger service, how would I distribute the OTP keys? Both the sender and the receiver need this key before the message is sent, right?   And how do I make sure no one else gets these keys? Hi everyone I'm attempting to install spotify, I've been running linux for less than 24 hours so please cut me some slack if this seems trivial. Here is a link to the webpage  and the example script.         Add this line to your list of repositories by editing your /etc/apt/sources.list    deb http://repository.spotify.com stable non-free    If you want to verify the downloaded packages, you will need to add our public key  sudo apt-key adv --keyserver hkp://keyserver.ubuntu.com:80 --recv-keys 94558F59    Run apt-get update  sudo apt-get update    Install spotify!      sudo apt-get install spotify-client      would someone be nice enough to walk me through this? I'm not even sure where to start. Is there a general formula for image size vs. print size? Do I have to buy a second wildcard certificate for a subdomain? We already have a wildcard certificate for *.mycompany.com. Our network has hosts that are only reachable internally. All of them belong to the internal.mycompany.com subdomain. There is a private server with the host name server.internal.mycompany.com on which I deployed our wildcard certificate.  When I visit the web server I get a host name mismatch error. Do I really have to get another wildcard certificate for *.internal.mycompany.com or is there another (free!?) way to use our wildcard certificate for all of our subdomains and its subdomains without getting an error in the browser? Why the name 'Rogue One'? So far, is there any suggestion, or speculation as to the reason for the name 'Rogue One' for the next Star Wars film?   To me, the name seems very close to the Rogue Squadron, most known from Hoth (their leader Luke Skywalker), especially the number following Rogue like a callsign.   However as far as I know, they were formed after the first Death Star destruction. So why Rogue One? "Coming soon" pages & SEO Why is the decay of a neutral rho meson into two neutral pions forbidden? Events triggered by dynamically generated element are not captured by event handler