Linespacing editor
TeX editor with line-spacing preferences
Jensen's Inequality implies convexity? Suppose $\phi$ is a real function on $\mathbb{R}$ such that $$\phi \Bigl(\int_0^1f(x)\,dx\Bigr)\leq \int_0^1 \phi(f)\,dx$$ for every real bounded measurable function $f$. Prove that $\phi$ is then convex.    This one is in Rudin's Real and Complex Analysis, page 74, and, to be quite honest, I am unsure what my first step to solve this problem should even be. To help think of a general methodology, I attempted this similar question, from page 71 of the same book:      Assume $\phi$ is a continuous real valued function on $(a,b)$ such that $$\phi\Bigl(\frac{x+y}{2}\Bigr)\leq\frac{\phi(x)+\phi(y)}{2}$$ for all $x, y\in (a,b).$ Prove that $\phi$ is then convex (the conclusion does not follow if continuity is omitted from the hypothesis).   I take it that the latter question is just a special case of the former (assuming $\phi$ is bounded on $(a,b)$), so I attempted to solve the latter problem first. My attempt was to note that \begin{align*} \phi\big(\lambda x + (1-\lambda)y\big) &amp;\leq \phi\Bigl(\frac{2\lambda x + 2(1-\lambda)y}{2}\Bigr) \\ &amp;\leq \frac{1}{2}\Bigl(\phi\bigl(2\lambda x\bigr)+\phi \bigl(2(1-\lambda)y\bigr)\Bigr) \\ &amp; \leq \phi\bigl(\lambda x\bigr)+\phi\bigl((1-\lambda)y\bigr) \end{align*} with the latter inequality justified simply by applying the hypothesis again twice to the previous expression.   From here, however, I am stuck; I cannot justify "popping out" the $\lambda$ and the $(1-\lambda)$ in the last inequality via some bound afforded by the hypothesis and/or the continuity of $\phi$. Analogous results to Euler's theorem Fermat's little theorem and its generalization, Euler's theorem holds for pair of relatively prime integers. Are there any analogous results for not relatively prime integers ? Thanks in advance. How can one set the executable bit on files using umask? According to documentation, a umask of 000 should work, but in practice this only works on directories.  Also open to other methods of automatically setting +rwx on a particular user's created files if you have any. I'm trying to setup OpenVPN using NetworkManager. The GUI seems buggy and unresponsive. Are there any other tools for setting up a VPN client? Does the artificer's Enhanced Weapon stack on top of hexblade's Improved Pact Blade? Perhaps this is trivial: how does one prove that the work done by a homogeneous gravitational field on a system of particles is equal to the work done on a point mass with the total mass of the system, located at the center of mass of the system. How do I earn SC2's "Hurry up, it's raid night" achievement? The "Hurry up, it's raid night" achievement in Starcraft 2 involves "beating Wings of Liberty, on normal, in 8 hours or less."  Is there a recommended mission path to take? Which mission should I take for the branching paths? Does it count time spent on the Hyperion in between missions, or is it exclusively time elapsed on missions. Does replaying a mission affect the time? General tips / hints from those who've unlocked the achievement?  Edit: Just got this achievement tonight, finally! The answer was perfect. Double decimal formatting in Java Many people (at least in Germany) think that Italians add olive oil to the boiling water for spaghetti. However, Italians tend to tell the opposite (from my experience).  So, can you please state objective reasons for or against adding olive oil to the boiling water for spaghetti? What kind of beings are shown by the Marauder's Map? The map definitely shows people (humans) inside the castle But I remember reading Harry watching Peeves bouncing around in the trophy room and Peeves is not human And I don't remember if I had exactly read it or not, but  says that it shows the ghosts as well So what kind or creatures or beings come under the map's radar? Any living creature? Or humanoids? Or creatures with a significant size or intelligence? Would it show the house elves in the kitchen, centaurs in the forests and the merpeople in the lake as well? What about the giant squid and the thestrals or unicorns? Would it have shown the troll from the Philosopher's stone if they had the map then? What about Quirell/Voldemort? Would he have shown up on the map? And the basilisk from the chamber of secrets (the book)? May be the chamber itself was unplottable but if the map could show the basilisk he would have been visible when he was slithering around in the pipes. i want to write a device driver but not able to find the header file can someone please help me find them?Also if someone can point out some important site links that would be really appreciated Several years ago I heard of something called dummy subjects in high school. It was then stated that, for example, it is a dummy subject when it starts many instances of sentences, e.g.        It is rainy today.   It is hard to imagine how hard is to live alone.   It is not my university degree that matters here.     My question is if we can use it with plural subjects? For example in       If some think that in Mahmoud Dowlatabadi’s works it is only social themes that matter and the writer is ignoring fictional arrays and artistic techniques, at least, endings of his stories show wrongness of this idea.     My question deals with the italicized part. If this usage is OK, can we also write they are social themes that matter? I mean can they also play the role of a dummy subject? What is Active Directory Domain Services and how does it work? This is a  about Active Directory Domain Services (AD DS).         I find myself explaining some of what I assume is common knowledge about it almost daily. This question will, hopefully, serve as a canonical question and answer for most basic Active Directory questions. If you feel that you can improve the answer to this question, please edit away. So I was thinking... If heat I feel is just lots of particles going wild and transferring their energy to other bodies, why am I not burned by the wind?  When I thought about it more I figured out that wind usually carries some humidity, and since particles of liquid are moving same speed as the wind, they are basically static relative to each other, so no energy is transferred between them (wind and water particles). And if that water sticks to my skin and wind blows, it'll evaporate thus taking energy from my skin and make me feel cold.  Thing is, I don't think that's really the case but even if it is, if I somehow dry out the wind, will it burn me if it's strong enough? And winds can reach some pretty high velocities (though I must admit I'm not sure if they are comparable to movement of atoms in warm bodies etc...).  So. Bottom line. Can I be burned by wind in some perfect scenario? Java reading a file into an ArrayList? Pgfplots: How to fill area under a curve in a 3d plot similar to \closedcycle in 2d How can I fill the area under a curve in a 3d plot, similar to the \closedcycle command in ordinary 2d plots? Applying this command does a filling which is not rotated.  Minimal working example:  \documentclass{article}  \usepackage{tikz} \usepackage{pgfplots} \begin{document}   \begin{tikzpicture}      \begin{axis}[     xmin=0,xmax=3,     zmin=0,zmax=2     ]         \addplot3[red,domain=0:1,fill=blue,opacity=0.5,samples y=0] (2,x,x^2) \closedcycle ;      \end{axis}   \end{tikzpicture} \end{document}   In 2D it looks like this  \documentclass{article} \usepackage{tikz} \usepackage{pgfplots} \begin{document}   \begin{tikzpicture}   \begin{axis}[domain=0:1     ]         \addplot[red,domain=0:1,fill=blue,opacity=0.5,samples y=0] {x^2} \closedcycle ;   \end{axis}   \end{tikzpicture} \end{document}   How can I easily apply this to a 3D plot? Help with completing the proof: Any unbounded sequence is either infinitely large or has a convergent subsequential limit Given $\{x_n\}$ is an unbounded sequence, prove:   $$ \lim_{n\to\infty}x_n = \infty\ \  \text{or}\ \ \lim_{n_k \to \infty} = A $$   In other words any unbounded sequence either infinitely large or has a convergent subsequence.     Everything before udpate section is wrong.  Part 1. Proving $|x_n| &gt; M \implies \lim_{n\to\infty} x_n = \infty$  $\Box$ Start with the definition of an unbounded sequence: $$ \forall M &gt; 0 \in \Bbb R \ \exists N \in \Bbb N: \forall n &gt; N\implies |x_n| &gt; M $$  Suppose that: $$ \lim_{n\to\infty}x_n \ne \infty $$ That means: $$ \forall n\in\Bbb N\ \exists C &gt;0 \in\Bbb R : |x_n| &lt; C $$  Now taking $M^\prime = \max\{M, C\}$ we have that: $$ \begin{cases} \begin{align} \exists N \in \Bbb N : &amp;\forall n&gt;N \implies &amp;|x_n| &gt; M^\prime \\ &amp;\forall n \in \Bbb N: &amp;|x_n| &lt; M^\prime \end{align} \end{cases} $$  So we have arrived to a contradiction which means the assumption is wrong and: $$ \lim_{n\to\infty}x_n = \infty $$ $\Box$     Part 2. It's not clear to me how to prove that if unboundedness does not imply infinity limit then it must imply the existence of a convergent subsequence.   Two questions is my mind:   Is part 1 correct?  How do I proceed with the second part?     I'm using the following definition. $$ \lim_{n\to\infty}x_n = +\infty \stackrel{\text{def}}{\iff} \forall \epsilon &gt; 0 \exists N \in \Bbb N :\forall n &gt; N \implies x_n &gt; \epsilon \\ \lim_{n\to\infty}x_n = -\infty \stackrel{\text{def}}{\iff} \forall \epsilon &gt; 0 \exists N \in \Bbb N :\forall n &gt; N \implies x_n &lt; -\epsilon \\ $$ A sequence is considered infinitely large when: $$ \lim_{n\to\infty}x_n = \infty\ \ \text{when}\ \ \lim_{n\to\infty}x_n = + \infty \ \ \text{or} \ \ \lim_{n\to\infty}x_n = - \infty $$    Update.   Looks like i've messed up a lot of things. I will try once again. As shown in comments and answers the statement holds only in case: $$ \lim_{n\to\infty} |x_n| = \infty \iff \forall \epsilon &gt; 0 \exists N \in \Bbb N: \forall n &gt; N \implies |x_n| &gt; \epsilon $$  Consider the following: $$ \lnot P = \lnot\left(\lim_{n\to\infty}|x_n| = \infty \right) \iff \exists \epsilon &gt; 0\ \forall N_1 \in \Bbb N : \exists n &gt; N_1 \land |x_n| &lt; \epsilon $$  But on the other hand it is given that $x_n$ is unbounded: $$ Q = \forall M &gt; 0\ \exists N_2 \in \Bbb N : |x_{N_2}| &gt; M $$  Construct a negative expression for boundedness: $$ \lnot P = \exists M &gt; 0\ \forall N_2 \in \Bbb N : |x_{N_2}| &lt; M $$  If $S = \lnot P \implies \lnot Q$ then $S = P \lor \lnot Q$  Let $\epsilon = M$, choose $N = \max\{N_1, N_2\}$ then both statements are true and: $$ \exists \epsilon &gt; 0\ \forall N = \max\{N_1, N_2\}: \exists n &gt; N \land |x_n| &lt; \epsilon $$  Now either $P$ is true, which would mean $\lim |x_n| = \infty$, or $\lnot Q$ is true which would mean the sequence is bounded and hence contains a convergent subsequence. Is there a rising trend in Number Series I am trying to build a stock screening utility.   What  I am trying is to find if there is a rising trend in a time series of profit margins of a company. I know there can be dips in some years but I want to find if mathematically that overall there is a rising trend as well as towards the recent years there is higher margins as compared to previous years.   Is there any way we can find it mathematically?          Profit Return Year,Company A,Company B,Company C 2005,38.97,20.44,22.01 2006,22.4,20.59,20.63 2007,22.4,20.59,20.63 2008,21.56,22.01,23.87 2009,25,20,23.52 2010,25,22.38,18 2011,30,24.13,24.85 2012,32,25.43,27.47 2013,30,20,22 2014,35,27,20 Image of the tensor product of strict maps of Banach spaces Word or phrase that can mean something and its opposite