Show that if $u$ is harmonic in $\mathbb{R}^n$ and $u = o(|x|)$, then $u$ is constant.  I am trying to prove this result by the mean value property. But it doesn't work. Please help me. Thanks!  PS: I just realized that one can prove it easily by the estimates on the derivatives of harmonic functions, which implies a stronger result:   If $u$ is harmonic in $\mathbb{R}^n$ and $u \le C(1+|x|)^N$, then $u$ is a polynomial of degree at most $N$.   But I am still curious about how to prove it directly by the mean value property, which is the "Hint" for this problem.
Show that if $u$ is harmonic in $\mathbb R^n$ and $u=o(|x|)$, then $u$ is a constant.(Hint: use the solid version of the mean value property $u(x)=\frac{1}{\omega R^n} \int _{B_{R(x)}} u(y)dy$, and estimate $|u(x)-u(x')|$.)  Can anyone tell me how to use this hint?
Detach (move) subdirectory into separate Git repository I have a  repository which contains a number of subdirectories. Now I have found that one of the subdirectories is unrelated to the other and should be detached to a separate repository.  How can I do this while keeping the history of the files within the subdirectory?  I guess I could make a clone and remove the unwanted parts of each clone, but I suppose this would give me the complete tree when checking out an older revision etc. This might be acceptable, but I would prefer to be able to pretend that the two repositories doesn't have a shared history.  Just to make it clear, I have the following structure:  XYZ/     .git/     XY1/     ABC/     XY2/   But I would like this instead:  XYZ/     .git/     XY1/     XY2/ ABC/     .git/     ABC/ How to use unnumbered chapters with KOMA-script? I am having some issues with the hyperref package. I would like to create a Introduction chapter without a number so I use \chapter*{Introduction} and \addcontentsline{toc}{chapter}{Introduction} because I want it in the table of contents (see example).  But then the link in my PDF viewer gets it wrong and also the automatically created header is no longer correct (see picture). The table of contents shows the right page number (p.1) but the link is also wrong.   What is my mistake here?  \documentclass{scrbook}  \usepackage{hyperref} \usepackage{blindtext} \usepackage{glossaries}   \makeglossaries  \newacronym{eg}{e.g.}{% for example }   \begin{document}   \frontmatter  % abstract before the table of contents \chapter*{Abstract} \addcontentsline{toc}{chapter}{Abstract} \blindtext  \tableofcontents  % chapter with acronyms \cleardoublepage \phantomsection  \printglossary[type=\acronymtype] \addcontentsline{toc}{chapter}{Glossary}  \cleardoublepage \mainmatter   \phantomsection % \chapter{Introduction} % with this everything is correct but I don't want the number \chapter*{Introduction} \addcontentsline{toc}{chapter}{Introduction}  \blindtext  \pagebreak  \blindtext  % first chapter \chapter{Chapter1}  \gls{eg} \blindtext  \pagebreak  \blindtext   \end{document} How specific does "beast you have seen" have to be for Wild Shape? Starting at 2nd level, you can use your action to magically assume the   shape of a beast that you have seen before.   This doesn't say anything about the gender or age of the beast, so can a druid change into a female bear if he has only seen male bears? Can he change into a bear cub if he has only seen adult bears? What about hens/roosters/chickens, which look quite a bit different? Or does the gender of the animal shape just match the gender of druid? Surface area inside cylinder Find the surface area of the part $\sigma$: $x^2+y^2+z^2=4$ that lies inside the cylinder $x^2+y^2=2y$  So, the surface is a sphere of $R=2$. It looks there should be double integral to calculate the surface, but how, which way? Show/Hide hidden files without restarting finder? I came to understand that the following is the way to toggle the visibility of hidden files:  defaults write com.apple.finder AppleShowAllFiles YES # replace YES with NO to hide hidden files killall -HUP Finder /System/Library/CoreServices/Finder.app   Is there a way to show/hide hidden files without killing the Finder? How to transfer between International T2 to Domestic 1B terminals in Mumbai? I will be arriving in Mumbai Terminal T2 on a Turkish Airlines international flight. I have a separately-purchased IndiGo ticket 5 hours later from Terminal 1B (Domestic). In the past I heard of a free bus to transfer.    Is that still the case and if not how do I get to Terminal 1B from Terminal 2? I just installed Ubuntu 14.04 Server and cannot get wifi configured correctly to work with WPA2 personal and could use some help.    There was a simple wizard during install where I selected my SSID from a list and entered my passphrase and that worked great.  Now that the install is done I am having trouble configuring wifi.  My Access Point is setup to WPA2 Personal TKIP or AES. Any advice would be greatly appreciated.  I have been messing around with WPA supplicant ant my /etc/network/interfaces file with no luck.  Thanks I read the definition of  as  $$W ~=~ \vec{F} \cdot \vec{d}$$  $$\text{ Work  =  (Force) $\cdot$ (Distance)}.$$ If a book is there on the table, no work is done as no distance is covered. If I hold up a book in my hand and my arm is stretched, if no work is being done, where is my energy going? Strange character when using listings package I'm seeing an odd character like a very flat U, where there should be a space, with the listings package. See example below. Does anyone have any idea why this is happening, or how to get rid or work around it?  \documentclass{standalone} \usepackage{underscore} \usepackage{listings} \usepackage[usenames,dvipsnames, rgb, svgnames, x11names]{xcolor} % Text layout \topmargin 0.0cm \oddsidemargin 0.5cm \evensidemargin 0.5cm \textwidth 16cm \textheight 21cm  \lstnewenvironment{cmd}[1][]{%   \lstset{language=sh, backgroundcolor=\color{LightSteelBlue1}, #1}} {}  \begin{document} \begin{cmd}[caption=Configuration file section including filters] [affy6] [...]     [[hapmap]]     phenofiles = "data/Hapmap6.0/Pheno/hapmappheno_909.csv",     genofile = "/data/snppy/Hapmap6.0/Geno/birdseed-v2.calls.txt"     anno_filter = "chromosome = 5"     idlink_filter = "studyid = 'CEU'"     some_other_stuff = "foo = bar" \end{cmd} \end{document} I'm having serious problems installing the Broadcom drivers for Ubuntu. It worked perfectly on my previous version, but now, it is impossible.   What are the steps to install Broadcom wireless drivers for a BCM43xx card?  I'm a user with no advance knowledge in Linux, so I would need clear explanations on how to make, compile, etc.    lspci -vnn | grep Network showed:   Broadcom Corporation BCM4322 802.11a/b/g/n Wireless LAN Controller [14e4:432b]      iwconfig showed:  lo        no wireless extensions. eth0      no wireless extensions.   NOTE: Answer below is updated every time new information is added and confirmed working. How can I save the command and its execution output to file at the same time Suppose I have a command java -cp src/ nlp.lm.BigramModel wsj/ 0.1, I want to save the command itself and the command execution output to the file at the same time.  One way to do this is to   echo "$ java -cp src/ nlp.lm.BigramModel atis/ 0.1" &gt; trace/bigram-trace.txt     java -cp src/ nlp.lm.BigramModel atis/ 0.1 &gt;&gt; trace/bigram-trace.txt   Is there a better way to do this? How can I find more music similar to the ones I like? WITH ROLLUP WHERE x IS NULL Does $k+\aleph_0=\mathfrak{c}$ imply $k=\mathfrak{c}$ without the Axiom of Choice? I'm currently reading a little deeper into the Axiom of Choice, and I'm pleasantly surprised to find it makes the arithmetic of infinite cardinals seem easy. With AC follows the Absorption Law of Cardinal Arithmetic, which states that for $\kappa$ and $\lambda$ cardinal numbers, the larger infinite and the smaller nonzero, then $\kappa+\lambda=\kappa\cdot\lambda=\max(\kappa,\lambda)$.  I was playing around with the equation $k+\aleph_0=\mathfrak{c}$ for some cardinal $k$. From the above, it follows that $\mathfrak{c}=k+\aleph_0=\max(k,\aleph_0)$, which implies $k=\mathfrak{c}$.  I'm curious, can we still show $k=\mathfrak{c}$ without the Axiom of Choice? Is it maybe possible to bound $\mathfrak{c}-\aleph_0$ above and below by $\mathfrak{c}$? But then I'm not quite sure such algebraic manipulations even mean anything, or work like that here. Certainly normal arithmetic does not! Thanks. Exercise $1.8$ of chapter one in Hartshorne. In exercise 1.8 of chap I in Hartshorne algebraic geometry,       Let $Y$ be an affine variety of dimension $r$ in $\mathbf A^n$. Let $H$ be a hypersurface in $\mathbf A^n$, and assume that $Y \nsubseteq H$. Then every irreducible component of $Y \cap H$ has dimension $r-1$.     I refered to a . In this solution, why $f$ is not a unit in $B$? Setting constraints causes the location of the constrained objects/bodies not be accessible I set up 2 bodies manually &amp; assigned Copy Location constraint to one with respect to another body. Then I set up the keyframe for the guiding body &amp; then I started to view the change in the location of the constrained body. What I noticed in the 3D viewport was the constrained object moves, but when I look into its real position the values are stagnant. Why is that? I was unable to find my answer in previously asked questions by other people !! How to adjust the gap between figure and caption within the same subfloat My question is:  How to adjust the gap between figure and caption within the same subfloat  Not the gap between two subfloat.  For the following example:  \begin{figure}     \centering     \subfloat[A]{\includegraphics[ scale=0.5]{images/a.jpg}}     \vspace{-10pt}     \subfloat[B]{\includegraphics[scale=0.5]{images/b.jpg}}     \vspace{-10pt}     \caption{}\label{fig:online_test} \end{figure}   I got the following figure:    The sub-caption of (a) is too close to the sub-figure B. I hope the caption (a) to stay closer to sub-figure A. How can I adjust the space between the caption and figure within a subfloat?  Thank you all for helping me and I would prefer a solution for subfloat, instead of using other packages. But I would also appreciate a solution of using other packages.   I know my question looks similar to . But please make it clear. I'm asking for how to adjust the gap between figure and caption within a subfloat, not the gap between two subfloat. That is, the gap between the caption "a" and the figure "A"  Thank you. \includegraphics optional argument not resolved if macro Convergence of $\int\limits_1^{\infty}x^\alpha\sin(x^\beta)dx$ I'm trying to use Linq to return a list of ids given a list of objects where the id is a property. I'd like to be able to do this without looping through each object and pulling out the unique ids that I find.  I have a list of objects of type MyClass and one of the properties of this class is an ID.  public class MyClass {   public int ID { get; set; } }   What I want to do is write a Linq query to return me a list of those Ids  How do I do that given an IList&lt;MyClass&gt; such that it returns an IEnumerable&lt;int&gt; of the ids?  I'm sure it must be possible to do it in one or two lines using Linq rather than looping through each item in the MyClass list and adding the unique values into a list.  Any help in creating an elegant solution would be much appreciated!