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Is the above statement true? If I have already proven that the set is linearly independent, is it automatically given that the subset is also linearly independent? If so, how would I go about showing that as a proof. Web-mapping with editing capabilities? This is a special case of , which, I think, might have drawn little attention because it was too general. In this question, I would like to consider a specific case.  Take a classical system given by the action $$ S[x] = \int dt \left[ \frac{\alpha}{2} \ddot{x}^2 + \frac{\beta}{2} \dot{x}^2 + \frac{\gamma}{2} x^2 \right]. $$  This action is Lagrangian, but it is not what we are usually dealing with in physics, because the Lagrangian contains second time derivatives.  The equations of motion are:  $$ \alpha \ddddot{x} - \beta \ddot{x} + \gamma x = 0. $$  This can be solved as usual by employing Fourier transform:  $$ x(t) = a_1 e^{i \omega_1 t} + a_1^{*} e^{-i \omega_1 t} + a_2 e^{i \omega_2 t} + a_2^{*} e^{-i \omega_2 t}, $$  where $\omega_1^2$ and $\omega_2^2$ are the two roots of $\alpha \omega^4 + \beta \omega^2 + \gamma$, given by  $$ \omega_{1,2}^2 = \frac{- \beta \pm \sqrt{\beta^2 - 4 \alpha \gamma}}{2 \alpha} .$$  Now by definition the phase space is the space of solutions of equations of motion. In this case it is parametrized by two complex co-ordinates $a_{1,2}$, meaning that it is 4-dimensional.  I would like to know if there's a natural way to associate the symplectic structure (Poisson bracket) on this phase space. I.e.,  $$ \{a_1, a_1^{*}\} = ? $$ $$ \{a_1, a_2^{*}\} = ? $$ $$ \{a_2, a_1^{*}\} = ? $$ $$ \{a_2, a_2^{*}\} = ? $$ $$ \{a_1, a_2\} = ? $$ $$ \{a_1^{*}, a_2^{*}\} = ? $$ I would be happy to get some hints on the following integral: $$ \int_0^1\frac{x^{19}-1}{\ln x}\,dx $$ Why we don't use gamma rays, x-rays or ultraviolet to transmit data? Why "a user" instead of "an user"? I am getting confused while using an article an. Because an is used while referring to vowels (a,e,i,o,u).  But today I read an article where for user they used a user.  My question is, why was an not used?  Could someone explain this with an example? DataView.Sort - more than just asc/desc (need custom sort) Running macOS Sierra(Version 10.12.2 16C67).  I installed  version wine-1.8.6 (through a .pkg file). I opened up a game and it worked fine.  The next day when I downloaded Origin installer, it wouldn't open. When I double clicked on it, the Wine icon showed up in the dock and said it was open, but no windows showed up. When I clicked on another program(or on the desktop), Wine closed. Now when I try to start that same game I opened when I first installed Wine, the same thing happens that happened when I started Origin installer.  When I go to the terminal and type wine {path of .exe} I got this:   I installed wine tricks using brew install winetricks and it installed fine, yet it didn't change my situation.  winecfg works properly, and so does sh winetricks. Could the missing antimatter lie outside the observable universe? I recently got the book "selected problems in real analysis", and I'm stuck solving the very first problem  $(u_n)$ is a binary sequence iff it only contains $0$ and $1$ in the sequence  Let $A$ be the set of all binary sequences  I have to prove that $A$ and ${2}^{\mathbb{N}}$ have the same cardinality, that is to say there exists a 1-1 function from one set to another  I've thought about maybe considering integers as base-2 numbers  Thanks for your help Book featuring a child learning from a crowdsourced AI book I need help with STATS homework about normal distribution The distribution of sale price for new MacBook Air’s is approximately normally distributed with a mean of \$800 and a standard deviation of \$70. (a) Mr. Bob plans to spend between $700 \text{ and } $850 on a new Apple MacBook Air. What proportion of the available MacBook Airs can Mr. Bob afford? (b) What is the 90th percentile of computers of this type? (c) Show that a MacBook Air priced at \$1000 would be considered an outlier using the 1.5 × IQR rule. For a, I tried doing 850-700=50 and $\frac{50}{800} = \frac{1}{40}$, for b, I tried $m+(0.5 \times R)/y$ for c, I tried $1.5 \times 700 - 1.5 \times 850$ How do I determine whether a drive is an HDD or SSD in Windows 8? I've recently received an assembled PC. I don't know if the disk is an SSD or old HDD (whether SATA or other technologies).  I can't find a way to tell if the disk is SSD or not, apart from the size of the disk. However, now that the sizes of SSDs are getting close to that of HDDs, checking the size is not a good and robust way to distinguish drive type. Is there some other way to determine if the disk is an SSD? Proving inequality $(a+\frac{1}{a})^2 + (b+\frac{1}{b})^2 \geq \frac{25}{2}$ for $a+b=1$ Why does everything spin? The origin of spin is some what a puzzle to me, everything spin from galaxies to planets to weather to electrons.  Where has all the angular momentum come from? Why is it so natural?  I was also thinking do photons spin? we always think of the wave as a standard 2d sin wave but could this rotate in 3d? What implications would this have?   And what about spacetime how does all the spinning effect?  This has always been avoided in all lectures and classes I ever went to. How and when is old-releases.ubuntu.com updated? The old-releases.ubuntu.com mirror hosts the repositories for EOL Ubuntu releases. The  currently lists up to 11.10 (or 12.04.3 if you count point upgrades), but the  lists up to 13.04. Why the discrepancy? When will the main page be updated? When will the newest EOL release (13.10) be shifted here? Rigid body simulation not interacting with objects that have an array modifier Can I use my iPhone as a "remote" for my iPad? Let $p, q \in \mathbb R$ be positive reals for which $p + q = 1$. How to prove that for any two natural numbers $n, m$ the following inequality is true?  $(1 - p^n)^m + (1 - q^m)^n \ge 1$  I don't have a big knowledge about solving inequalities, so I tried to use cauchy-schwarz inequality, binomial theorem and some other baisc techniques, but it lead me nowhere. I've been thinking about it for a long time and now I'm completly stuck. How to format a USB drive? Possible Duplicate:             How to format a pen drive in Ubuntu? I have this pen drive which I made bootable using some other system (OS). But I need to format it from my Ubuntu 12.04. Please guide me.