We know that:       $a_1,...,a_n\in \mathbb R , \forall a_i\neq0 \\ b_1,...,b_n\in \mathbb R : b_j\neq b_k : \forall j\neq k$      Prove that there are $n-1$ roots in $(0,\infty)$:       $$a_1x^{b_1}+a_2x^{b_2}+...+a_nx^{b_n}=0  $$   Using induction, for n=1 it's obvious that there are 0 roots.  Suppose the statement is true for n-2 and prove for n-1:  From here on I'm not sure how to continue, I noticed that $x^{b_1}(a_1 + \ldots + a_n x^{b_n - b_1})=0$ but at what point can it come into the induction process ? Also, how does the IVT is applied here ?
I have two related problems that are causing me big trouble:  Let $\;a_1,...,a_n\in\Bbb R\;,\;\;a_i\neq 0\;\;\forall\;i\;$ , and let $\;b_1,...,b_n\in\Bbb R\;,\;\;b_i\neq b_j\;\;\forall\,i\neq j\;$ .  (1) Prove that the equation  $$a_1x^{b_1}+\ldots+a_nx^{b_n}=0$$  has at most $\;n-1\;$ different zeros in $\;(0,\infty)\;$ (Hint: Use induction)  (2) Prove that the equation  $$a_1e^{b_1x}+\ldots+a_ne^{b_nx}=0$$  has at most $\;n-1\;$ different zeros in $\;\Bbb R\;$ .  Now, the following is what I've done so far: for a differentiable function $\;f\;$ ,  I can prove that if $\;f'(x)=0\;$ has $\;n-1\;$ different zeros, then $\;f(x)\;$ has at most $\;n\;$ different zeros, using the mean value theorem, say.  But I can't see how to use induction in this case: if I differentiate in the first problem, I get $$0=f'(x)=a_1b_1x^{b_1-1}+\ldots +a_nb_nx^{b_n-1}=\frac1x\sum_{i=1}^na_ib_ix^{b_i}$$  ...and now?! How does induction kick in here? I though perhap something like: if we suppose WLOG $\;b_1&lt;b_2&lt;...&lt;b_n\;$ , then perhaps inducting on $\;\lfloor b_n\rfloor\;$, but it gets messy and blurry if $\;b_n&lt;0\;$...  Any help, hints are very much appreciated.
I'd like to get an advanced frame in Firefall. I've been told that I can use "Pilot Tokens" to unlock other frames without having to pay money.  How soon would I be able to unlock an advanced frame? Do I have to level a base frame all the way to level 40? More than once? How much space does Ubuntu take up after it's already installed? I know how much space is required to install Ubuntu.  How much space does a Ubuntu installation take up on the disk? How to get the length of longest string in an array I'm trying to create an annotated bibliography. Here's my .tex document:  \documentclass[12pt]{article} \title{Annotated Bibliography}  \begin{document}  \nocite{*}  \bibliography{annotatedbib}  \end{document}   And here is the associated .bib document:  @Misc{hacker,     AUTHOR = `{Andrew Hacker'},     TITLE = {Is Algebra Necessary?},     HOWPUBLISHED = {New York Times},     MONTH = {July 31},     YEAR = {2012},     ANNOTE = {This article was published in the New York Times.}, }   No matter into which format I compile my .tex document, the annotation will not show up! Is there something that I'm missing? Why are some faces in my mesh darker? Can I have dynamic custom variables in Word? I'm basically trying to create a variable in a document, let's say the name of a person, so that I can change the value in a single place and it updates through the whole document.  I've been reading about Fields and about a DOCVARIABLE, but I can't find where to edit the value (right clic didn't show an "edit" option). I'm using Word 2011 for Mac (that's what we have at work right now) so I'm wondering whether this version of Word is just not intuitive, or I cannot have variables like that in Word and I'm misinterpreting the documentation.  This is what I intend to do:  Hello, my name is PERSON_NAME, I have PERSON_AGE years old.   You get the idea.  PS: We actually want to modify the variables' values from our app, rather than from Word itself, kind of like injecting values to a .*docx file. I'm currently reading Marcus' "Number Fields," and I'm having difficulty proving the following result:  Corollary 2.2: Let $m$ be a squarefree integer. The set of algebraic integers in the quadratic field $\mathbb{Q}(\sqrt{m})$ is $$ \left\{ a + b \sqrt{m} : a,b \in \mathbb{Z} \right\} $$ if $m \equiv 2,3 \pmod{4}$; and$$ \left\{ \frac{a+b\sqrt{m}}{2} : a,b \in \mathbb{Z} ~,~ a \equiv b \pmod{2} \right\}$$ when $m \equiv 1 \pmod{4}$. $\square$  It is easy enough to see that the claimed sets do, indeed, consist of algebraic integers. The other half of the proof proceeds something like this: suppose that $\alpha = r + s \sqrt{m}$ is an element of $\mathbb{Q}(\sqrt{m})$. Then $\alpha$ is a root of the polynomial $$ x^2 - 2rx + r^2 - s^2m $$ In particular, this means that $\alpha$ is an algebraic integer if and only if both $2r$ and $r^2 - s^2 m$ are integers. At this point, the proof stops, and says that this implies the result. And therein lies the problem: I don't see how to make the connection. I do know, for example, that if one of $r,s$ is an integer, then the other one must be, too, i.e., either both are integers, or both are not. However, I cannot get any further than that. Can anyone give me a gentle nudge in the right direction? Thank you very much! Node Package Manager got corrupted in some way, now it cannot be installed I ran into a problem today when I decided to use npm update -g in a nodejs app directory. This was a bad decision, because it caused me to run into a .   The solution is to reinstall nodejs and npm. Unfortunately when I go to install npm using sudo apt-get install npm it doesn't work.  The full error in the terminal is here:  Some packages could not be installed. This may mean that you have requested an impossible situation or if you are using the unstable distribution that  some required packages have not yet been created or been moved out of Incoming. The following information may help to resolve the situation: The following packages have unmet dependencies:  npm : Depends: nodejs but it is not going to be installed    Depends: node-abbrev (&gt;= 1.0.4) but it is not going to be installed    Depends: node-ansi but it is not going to be installed    Depends: node-archy but it is not going to be installed    Depends: node-block-stream but it is not going to be installed    Depends: node-fstream (&gt;= 0.1.22) but it is not going to be installed    Depends: node-fstream-ignore but it is not going to be installed    Depends: node-github-url-from-git but it is not going to be installed    Depends: node-glob (&gt;= 3.1.21) but it is not going to be installed    Depends: node-graceful-fs (&gt;= 2.0.0) but it is not going to be installed    Depends: node-inherits but it is not going to be installed    Depends: node-ini (&gt;= 1.1.0) but it is not going to be installed    Depends: node-lockfile but it is not going to be installed    Depends: node-lru-cache (&gt;= 2.3.0) but it is not going to be installed    Depends: node-minimatch (&gt;= 0.2.11) but it is not going to be installed    Depends: node-mkdirp (&gt;= 0.3.3) but it is not going to be installed    Depends: node-gyp (&gt;= 0.10.9) but it is not going to be installed    Depends: node-nopt (&gt;= 2.1.1) but it is not going to be installed    Depends: node-npmlog but it is not going to be installed    Depends: node-once but it is not going to be installed    Depends: node-osenv but it is not going to be installed    Depends: node-read but it is not going to be installed    Depends: node-read-package-json (&gt;= 1.1.0) but it is not going to be installed    Depends: node-request (&gt;= 2.25.0) but it is not going to be installed    Depends: node-retry but it is not going to be installed    Depends: node-rimraf (&gt;= 2.2.2) but it is not going to be installed    Depends: node-semver (&gt;= 2.1.0) but it is not going to be installed    Depends: node-sha but it is not going to be installed    Depends: node-slide but it is not going to be installed    Depends: node-tar (&gt;= 0.1.18) but it is not going to be installed    Depends: node-which but it is not going to be installed E: Unable to correct problems, you have held broken packages.   What can I do to get npm properly installed and running again?  Thanks. What physical properties of materials trigger the capacitive touch screen? I have watched some youtube videos about capacitive touch screen      The videos show that a conductor(finger), in which charges can be induced, can be used to trigger the capacitive touch screen. However, I tried to wear a plastic glove to hold a metal rod to touch the screen and no detection is triggered.  I really want to know the physics theory about the capacitive sensing.  Can someone tell me what physical properties or shape trigger the capacitive touch screen?  Is there any examples that something WITHOUT conducting to human body can trigger the screen?  (P.S. I have read other posts about capacitive touch screen here and do not find a good answer so this is not a duplicate) Difference between var = {} and var = []? $a^m=b^m$ and $a^n=b^n$ imply $a=b$ Let $D$ be an integral domain and let $a^m=b^m$ and $a^n=b^n$ where $m$ and $n$ are relatively prime integers, $a,b \in D$.   How do I show $a=b$? A thought experiment: Imagine the Sun is suddenly removed. We wouldn't notice a difference for 8 minutes, because that's how long light takes to get from the Sun's surface to Earth.   However, what about the Sun's gravitational effect? If gravity propagates at the speed of light, for 8 minutes the Earth will continue to follow an orbit around nothing. If however, gravity is due to a distortion of spacetime, this distortion will cease to exist as soon as the mass is removed, thus the Earth will leave through the orbit tangent, so we could observe the Sun's disappearance more quickly.  What is the state of the research around such a thought experiment? Can this be inferred from observation? heredoc for Windows batch? Why is the gap between the translation and the last line of the gloss so wide in my interlinear gloss? I notice if I take away some letters, the translation line goes to where it should be, but this big block of white space is pretty untenable. Ideas?    \documentclass[11pt,a4paper]{book}  \setcounter{secnumdepth}{4} \usepackage{gb4e} \usepackage{tipa} \usepackage{setspace} \usepackage[T1]{fontenc} \begin{document} \begin{exe} \ex \glll Jichax    Qamaqix lij parki,  aynacha,    amsta,  parki, q'awanaks,   khuyt'asis  sarawjchix. \\ \{jicha-x(a)    Qamaqi-x(a) lij(u)  parki   aynacha amsta   parki q'awa-nak(a)-_{c}$\oslash$-s(a)   khuy(u)-t'a-s(i)-s(a)   sara-wj(a)-ch(i)-i-x(a)\}    \\ now-\textsc{top}    Fox-\textsc{top}    all acclivity   declivity   ascent  acclivity ravine-\textsc{pl-acc-ad} whistle-\textsc{m}-\textsc{refl}-\textsc{subr} go-\textsc{bfr-cnj-3sim-top}\\ \glt `And so, Fox must've gone whistling on all the ascents, slopes, ascents, mountainsides, and even in the ravines.' \end{exe} \end{document} What is the best way to execute a script when entering into a directory? When I move into a new directory I would like bash to execute the projectSettings.bash script much like RVM does. I have an existing trigger which converts leads into peron accounts if the company name is blank.  Trigger AutoConvert on Lead (after insert) {      LeadStatus convertStatus = [           select MasterLabel           from LeadStatus           where IsConverted = true           limit 1      ];      List&lt;Database.LeadConvert&gt; leadConverts = new List&lt;Database.LeadConvert&gt;();       for (Lead lead: Trigger.new) {           if (!lead.isConverted &amp;&amp; lead.Company == null) {                Database.LeadConvert lc = new Database.LeadConvert();                String oppName = lead.Name;                 lc.setLeadId(lead.Id);                lc.setDoNotCreateOpportunity(true);                lc.setConvertedStatus(convertStatus.MasterLabel);                 leadConverts.add(lc);           }      }       if (!leadConverts.isEmpty()) {           List&lt;Database.LeadConvertResult&gt; lcr = Database.convertLead(leadConverts);      } }   I need to also write a trigger that stops records from coming into Salesforce if that email already exisits in a contact or an account (we use person accounts).  The issue is how do I get it working with the existing trigger (I need it to run before the trigger that converts it into a person account is run).   Is there a way to combine the two or add the functionality to create the record only if the email address doesn't already exist as a contact in the org? Standard purpose of favorites I am wondering if my behavior regarding favorites is typical.  More importantly, if it isn't typical, does my behavior conflict with an internal metric designed to reward favorite'd questions?  I mark a question as a favorite for a few reasons:  A -- I find the question/answer(s) remarkable enough that coming back to them time and time again is of benefit to me.  B -- I am waiting to see if a sub-par quality question is going to be revised enough for me to come back and answer it.  Either way, I eventually remove my favorite.  C -- I have answered a question (of any quality) and am wanting to be notified of any competing answers.    Situations B and C are the most common for me and they typically mean I will go back and remove my favorite star later; I don't remove it because the quality of the question degraded, I merely don't want to be notified any longer.  Even stranger, I am often favoriting questions that are rather low-quality (reason B) making me wonder if low-quality posts are being rewarded via some invisible metric because I (at least temporarily) have given it what looks like a stamp of approval akin to an up-vote.    I suppose the feature being called "Favorite" makes me confused because most of the questions I select are not my favorite.  Am I accidentally messing with an invisible metric that is drawing attention to posts when that isn't desired behavior? What can I do when getting “This question body does not meet our quality standards”? While trying to ask a question, I got the error message     This question body does not meet our quality standards. Please make sure that it completely describes your problem - including what you have already tried - and is written using proper grammar.    Why am I getting this message? What can I do to get the system to accept my question? Can you be more specific? What are good resources on how to ask high-quality questions?      For more information, see  in the . Computing irreducible components of algebraic set There's a department at my university where the head of department is routinely put as author on papers he didn't contribute to, and where researchers from overseas who had no input (but who provide funding to the department) are put on papers.    As a pragmatic matter, how should a PhD student respond when they are told to put three names on a paper, when those people didn't contribute?  Assume that refusing to put the co-authors on is not an option, because:   There will be bad consequences if they do not, and  This practice is routinely done at a high level, and there is nobody at the university to complain to about it.   This is not a duplicate of , because its answer assumes that it is practical for the student to refuse.