Let $m$, $n$, and $p$ be real numbers such that $n\gt m\gt 0$ and $p\gt 0$. Prove that   $$\frac{m+p}{n+p}\gt\frac{m}{n}$$  My attempt:  $$\frac{m}{n}+\frac{n+p}{n+p}\gt\frac{m}{n}\implies\frac{m(n+p)+n(n+p)}{n(n+p)}\gt\frac{m}{n}\\\implies\frac{mn+mp+n^2+np}{n(n+p)}\gt\frac{m}{n}\implies\frac{n(m+p)+mp+n^2}{n(n+p)}\gt\frac{m}{n}$$  I can't seem to progress from here. I tried to find a way to force a cancellation with the $n$ in the denominator, but couldn't find a plausible approach. Any help?
Is there a general relation between $a/b$ and $(a+c)/(b+c)$ where $a,b&gt; 0$ and $c\geq 0$ ?   Is there a general proof for that relation ?
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} Stopping users voting multiple times on a website I'm planning to add some vote up/vote down buttons to a website I run. This seems easy enough but I want to stop people voting multiple times. One solution would be to make them register before allowing them to vote but I'd prefer not to have to force them to register.   Is there are a reasonably straightforward way of doing this? Checking the IP address doesn't seem like a good solution, since it's possible that multiple users may come from the same IP address.   Cookies might be the answer, but a savvy user could delete the cookie. Any better ideas? Say there are $m+n$ elements split into two groups ($m$ and $n$). The variance of the first group is $\sigma_m^2$ and the variance of the second group is $\sigma^2_n$. The elements themselves are assumed to be unknown but I know the means $\mu_m$ and $\mu_n$.  Is there a way to calculate the combined variance $\sigma^2_{(m+n)}$?   The variance doesn't have to be unbiased so denominator is $(m+n)$ and not $(m+n-1)$. How do I change the launcher commands? Isn't the uncertainty principle just non-fundamental limitations in our current technology that could be removed in a more advanced civilization? From what I understand, the  states that there is a fundamental natural limit to how accurately we can measure velocity and momentum at the same time. It's not a limit on equipment but just a natural phenomenon.  However, isn't this just an observational limit? There is a definite velocity and momentum, we just don't know it. As in, we can only know so much about the universe, but the universe still has definite characteristics.  Considering this, how do a wide range of quantum mechanical phenomena work? For example, quantum tunneling - its based on the fact that the position of the object is indefinite. But the position is definite, we just don't know it definitely. Or the famous light slot experiment? The creation of more light slots due to uncertainty of the photon's positions?  What I am basically asking is why is a limit on the observer, affecting the phenomenon he is observing? Isn't that equivalent to saying  because we haven't seen Star X, it doesn't exist? It's limiting the definition of the universe to the limits of our observation! Page number in title page I want to add page number from 531 to 536 in my latex IEEE conference template. But page number appearing from page 2 i.e. 532.  \documentclass[conference]{IEEEtran} %\IEEEoverridecommandlockouts  \makeatletter \let\old@ps@headings\ps@headings \let\old@ps@IEEEtitlepagestyle\ps@IEEEtitlepagestyle \def\confheader#1{%     % for all pages except the first     \def\ps@headings{%         \old@ps@headings%         \def\@oddhead{\strut\hfill#1\hfill\strut}%         \def\@evenhead{\strut\hfill#1\hfill\strut}%     }%     % for the first page     \def\ps@IEEEtitlepagestyle{%         \old@ps@IEEEtitlepagestyle%         \def\@oddhead{\strut\hfill#1\hfill\strut}%         \def\@evenhead{\strut\hfill#1\hfill\strut}%     }%     \ps@headings% } \makeatother  \confheader{ 2020 IEEE International Conference on Computation, Automation and Knowledge Management (ICCAKM) }   %%footer %\usepackage[pscoord]{eso-pic} %\newcommand{\placetextbox}[3]{ %   \setbox0=\hbox{#3} %   \AddToShipoutPictureFG*{ \put(\LenToUnit{#1\paperwidth},\LenToUnit{#2\paperheight}){\vtop{{\null}\makebox[0pt][c]{#3}}} %   } %} %\placetextbox{.23}{0.055}{\small{978-1-5386-8007-0/19/\$31.00~\copyright 2019 IEEE}}     \usepackage{cite} \usepackage{amsmath,amssymb,amsfonts} \usepackage{algorithmic} \usepackage{graphicx} \usepackage{textcomp} \usepackage{xcolor} \usepackage{float} \usepackage[export]{adjustbox} \usepackage{subcaption}     % %\usepackage[pscoord]{eso-pic} %\newcommand{\placetextbox}[3]{ %   \setbox0=\hbox{#3} %   \AddToShipoutPictureFG{ \put(\LenToUnit{#1\paperwidth},\LenToUnit{#2\paperheight}){\vtop{{\null}\makebox[0pt][c]{#3}}} %   } %} %\placetextbox{.2}{0.055}{978-1-4799-5522-0/15/\$31.00~\copyright 2015 IEEE}   \IEEEoverridecommandlockouts \IEEEpubid{\makebox[\columnwidth]{978-1-7281-0666-3/20/\$31.00~\copyright~2020         IEEE \hfill} \hspace{\columnsep}\makebox[\columnwidth]{ }}      \thispagestyle{plain}     \pagestyle{plain}     \setcounter{page}{531}   \begin{document}  \title{Performance Analysis of Physically     Unclonable Function Based on Voltage Divider Arrays} \author{\IEEEauthorblockN{Sumit Chaudhary$^{1}$}     \IEEEauthorblockA{\textit{Dept. of Electronics and Communication} \\         \textit{MNIT Jaipur, India}\\         2016pev5165@mnit.ac.in}     \and     \IEEEauthorblockN{Chitrakant Sahu$^{2}$}     \IEEEauthorblockA{\textit{Dept. of Electronics and Communication} \\         \textit{MNIT Jaipur, India}\\         chitrakant.ece@mnit.ac.in}}  \maketitle   \end{document} In MySQL, how do I get a list of all foreign key constraints pointing to a particular table? a particular column? This is the same thing as , but for MySQL. Is there a way to select the children of a parent object? I am carrying out research to look at differences in fish density and fish species richness when using two different underwater visual census methods. My data was originally count data but then typically this is changed to fish density but I have still decided to use a Poisson GLM, which I hope is right.   model1 &lt;- glm(g_den ~ method + site + depth, poisson)   My 3 predictor variables are method, site and depth which I ordered as factors when I input them.   My response variables are grouper species richness, grouper density and the same for other fish groups. I am aware that density is not an integer and it is numerical data e.g 1.34849. I am now however getting this error:      In dpois(y, mu, log = TRUE) : non-integer x = 0.037500   I've been reading up and many people suggest using an offset, is this the most advisable thing to do? Find conditional expectation from a 3-dimensional random vector? We are given two independent standard normal random variables $X$ and $Y$. We need to find out the M.G.F of $XY$.  I tried as follows : \begin{align} M_{XY}(t)&amp;=E\left(e^{(XY)t}\right)\\&amp;=\int_{- \infty}^{\infty}\int_{- \infty}^{\infty}e^{(XY)t}f_X(x)f_y(y)dxdy \\ &amp;=\dfrac{1}{2\pi}\int_{- \infty}^{\infty}\int_{- \infty}^{\infty}e^{(xy)t}e^{\frac{-x^2}{2}}e^{\frac{-y^2}{2}}dxdy \\ &amp;= \dfrac{1}{2\pi}\int_{- \infty}^{\infty}\int_{- \infty}^{\infty}e^{-\dfrac{(x-ty)^2}{2}}e^{\dfrac{-t^2y^2}{2}}e^{\frac{-y^2}{2}}dxdy \\ &amp;= \dfrac{1}{\sqrt{2\pi}}\int_{- \infty}^{\infty}e^{\dfrac{-t^2y^2}{2}}e^{\frac{-y^2}{2}}dy \\ &amp;= \dfrac{1}{\sqrt{2\pi}}\int_{- \infty}^{\infty}e^{-y^{2}(\frac{1}{2}+t^2)}dy \\ &amp;= \dfrac{1}{\sqrt{1+2t^2}} \end{align} Is this correct ? Why does an environment's label have to appear after the caption? \documentclass{article} \usepackage{algorithm}  \begin{document} \setcounter{section}{54} \section{A section with no label} \begin{algorithm} \label{myalg} \caption{An algorithm with the label `myalg'.} \end{algorithm} Reference to myalg: \ref{myalg}. \end{document}   With this code, the reference givens 55 (the section number); if I move the label after the caption command, I get the correct reference (1).  Why should this be the case? I find it confusing: when thinking about an algorithm as an environment, you expect to be able to label it right after you began it. At the very least I would expect some kind of error or "??" to appear, rather than the previous environment or reference-able entity being used. Processor: PIC18F66K22 Compiler:  C18 (XC8 is fine also)   I've just made (and succesfully tested) code to measure the voltage on AN4 analog pin.  Now I want to try something else, not quite sure if it's theoretically possible.  My test-board has a PIC processor, with VDD,AVVD connected to a supply (ranging from 0(if the battery is empty/disconnected) to 4,2(full battery).  Can I use ADC to measure the VDD(-in) of my board?  Why not?:   It's not an ANx pin, not sure how to make the ADC measure it? (Channel?) If the voltage on VDD(-in) drops, will my ADC reference voltage also drop? (So when my PIC only gets 3,3V the ADC will still say it's 100% (as the max Voltage on the chip, also the reference is 3,3V).   Why/what do you ask?:   Can I just measure Channel 30 (VDDCORE)? Could I set my Vref to Vdd and measure a precision voltage reference with the A/D converter. (Not sure how) Or I could add a wire from VDD-in to an ANx pin? Will Internal VREF+ (4.096V) stay 4+V when the VDD-in is around 3,4V? Is there some kind of transformator that makes it so? Internal VREF+ (4.096V) is a bit lower than the max of the battery (4,2V), does this mean I can damage the ADC? Or just that the result will be max when the battery is full. (4,2V out of 4,096V)     Please mind that my knowledge of Electronics is "low". I've learnt way more about the software part in my education, but I'd like to know more about (Embedded) Electronics as it's a crucial part in Embedded Engineering. (It's sad how my school doesn't see this.) What is economics like for ordinary people in the Star Trek universe? I'm trying to get the subcaptions and captions in sanserif font and Uppercased. According the manual of the caption package this should be possible in the following way.However, this doesnt seem to work. Am I defining something in the wrong way here?  \documentclass{article} \usepackage{floatrow} \usepackage[demo]{graphicx} \usepackage{subfig} \usepackage{caption} \floatsetup[figure]{subcapbesideposition = top}  \captionsetup{labelfont={sf,up}, labelsep=period, format = plain, singlelinecheck = false, textfont=rm, skip = 2pt} \captionsetup[subfloat]{labelfont = {up,sf},format = plain,textfont = rm,labelformat = simple, labelsep = period}  \begin{document}   \begin{figure}[h!]   \ffigbox[\textwidth]     {\begin{subfloatrow}         \sidesubfloat[]{\includegraphics[width = 0.4\textwidth]{A.pdf}}%         \sidesubfloat[]{\includegraphics[width = 0.4\textwidth]{B.pdf}}%       \end{subfloatrow}       \begin{subfloatrow}         \sidesubfloat[]{\includegraphics[width = 0.4\textwidth]{C.pdf}}%         \sidesubfloat[]{\includegraphics[width = 0.4\textwidth]{D.pdf}}%       \end{subfloatrow}}     {\caption{this is a caption}} \end{figure}  \end{document} What is Kirito's relationship with the girls of SAO? How to add sysadmin to user in SQL Server 2008 when no sysadmin accounts exist I have a SQL server 2008 instance running. Unfortunately during testing I deselected the sysadmin rights for my login and now can't readd it (because I don't have sysadmin rights).  There are no other sysadmin accounts for the instance except SA.  I was set up for just Windows Authentication so I hacked the LoginMode in the registry to 2 so I could login as SA using Sql Authentication. This does indeed set the login mode to Mixed, however the SA user is disabled by default and I can't reenable it because I am don't have sysadmin rights.  How do I enable the SA login so I can get in and reassign the sysadmin to my normal account? Is there a registry setting for this too or is it stored in the master database? How to get domain name from URL Galois group of irreducible cubic equation I've been trying a simple recipe I found on the internet, which is a simple recipe consisting mostly of gelatin.  The problem is that the end result is basically just a fruit-flavoured jelly (Jell-O?), rather than the more chewy gummy-bears (Haribo) style of jelly confectionery.  Can anyone suggest what I need to do to make them less springy and more chewy.  It's all difficult to articulate!  Here's what I used for the first test-batch:   12 tbsp sugar 12/3 cup fruit juice 8 tbsp Golden Syrup 8 tbsp gelatin   soften the gelatin in 1/2 cup of cold water. place sugar, syrup and the juice in a pan and gently heat until sugar dissolves. Stir in the gelatin and stir until dissolved. Pour into moulds and wait until set.    Thanks