I am having trouble at aligning the following plots (see code below). I have removed the x values to better use the available space. But as you can see in the picutre the horizontal distance is not equal because of the x-labels in the last row. Additionally the labels on the right side are off. I could solve the issue with the y-values by introducing phantom characters but there should be a better way I think.  Is there a way to solve these problems?    \begin{figure} \centering \begin{tikzpicture} % longitudinal speed without disturbance \begin{axis}[...,      name=plot1,      clip mode=individual,...]      ... \end{axis} \begin{axis}[...,      name=plot2,      clip mode=individual,      at=(plot1.right of south east), anchor=left of south west,...]      ... \end{axis} \begin{axis}...\end{axis} \begin{axis}...\end{axis} \begin{axis}...\end{axis} \begin{axis}...\end{axis} \end{tikzpicture} \end{figure}
I once already asked a very similar question; . But this is some what different.  I have the following code;  \documentclass{standalone} \usepackage{graphics} \usepackage{tikz,pgfplots} \pgfplotsset{compat=1.6} \begin{document} \begin{tikzpicture}  \begin{axis}[% width=3cm, height=3cm, scale only axis, every x tick label/.append style={font=\scriptsize\color{gray!80!black}}, xmin=0, xmax=20000, xlabel={$t$}, xmajorgrids, xminorgrids, every y tick label/.append style={font=\scriptsize\color{gray!80!black}}, ymin=0.0004, ymax=0.0012, ylabel={$\phi_{\mathrm{i}}(t)$}, ymajorgrids, yminorgrids, name=plot1]  \end{axis}  \begin{axis}[% width=3cm, height=3cm, scale only axis, every x tick label/.append style={font=\scriptsize\color{gray!80!black}}, xmin=0, xmax=20000, xlabel={$t$}, xmajorgrids, xminorgrids, every y tick label/.append style={font=\scriptsize\color{gray!80!black}}, ymin=0.2, ymax=1.2, ylabel={$h(t)$}, ymajorgrids, yminorgrids, name=plot2, at=(plot1.right of north east), anchor=left of north west]  \end{axis}  \begin{axis}[% width=3cm, height=3cm, scale only axis, every x tick label/.append style={font=\scriptsize\color{gray!80!black}}, xmin=0, xmax=20000, xlabel={$t$}, xmajorgrids, xminorgrids, every y tick label/.append style={font=\scriptsize\color{gray!80!black}}, ymin=0.0004, ymax=0.0012, ylabel={$\phi_{\mathrm{o}}(t)$}, ymajorgrids, yminorgrids, name=plot4, at=(plot1.below south west), anchor=above north west]  \end{axis}  \begin{axis}[% width=3cm, height=3cm, scale only axis, every x tick label/.append style={font=\scriptsize\color{gray!80!black}}, xmin=0, xmax=20000, xlabel={$t$}, xmajorgrids, xminorgrids, every y tick label/.append style={font=\scriptsize\color{gray!80!black}}, ymin=2000, ymax=12000, ylabel={$p_{\mathrm{i}}(t)$}, ymajorgrids, yminorgrids, name=plot5, at=(plot4.right of north east), anchor=left of north west]  \end{axis}  \end{tikzpicture} \end{document}   This creates the following plot:    I think there is to few vertical/horizontal spacing between the plots. I want to increase this. Currently I use x- yshift, this gives the following result:  \documentclass{standalone} \usepackage{graphics} \usepackage{tikz,pgfplots} \pgfplotsset{compat=1.6} \begin{document} \begin{tikzpicture}  \begin{axis}[% width=3cm, height=3cm, scale only axis, every x tick label/.append style={font=\scriptsize\color{gray!80!black}}, xmin=0, xmax=20000, xlabel={$t$}, xmajorgrids, xminorgrids, every y tick label/.append style={font=\scriptsize\color{gray!80!black}}, ymin=0.0004, ymax=0.0012, ylabel={$\phi_{\mathrm{i}}(t)$}, ymajorgrids, yminorgrids, name=plot1]  \end{axis}  \begin{axis}[% xshift=1cm, width=3cm, height=3cm, scale only axis, every x tick label/.append style={font=\scriptsize\color{gray!80!black}}, xmin=0, xmax=20000, xlabel={$t$}, xmajorgrids, xminorgrids, every y tick label/.append style={font=\scriptsize\color{gray!80!black}}, ymin=0.2, ymax=1.2, ylabel={$h(t)$}, ymajorgrids, yminorgrids, name=plot2, at=(plot1.right of north east), anchor=left of north west]  \end{axis}  \begin{axis}[% yshift=-1cm, width=3cm, height=3cm, scale only axis, every x tick label/.append style={font=\scriptsize\color{gray!80!black}}, xmin=0, xmax=20000, xlabel={$t$}, xmajorgrids, xminorgrids, every y tick label/.append style={font=\scriptsize\color{gray!80!black}}, ymin=0.0004, ymax=0.0012, ylabel={$\phi_{\mathrm{o}}(t)$}, ymajorgrids, yminorgrids, name=plot4, at=(plot1.below south west), anchor=above north west]  \end{axis}  \begin{axis}[% xshift=1cm, width=3cm, height=3cm, scale only axis, every x tick label/.append style={font=\scriptsize\color{gray!80!black}}, xmin=0, xmax=20000, xlabel={$t$}, xmajorgrids, xminorgrids, every y tick label/.append style={font=\scriptsize\color{gray!80!black}}, ymin=2000, ymax=12000, ylabel={$p_{\mathrm{i}}(t)$}, ymajorgrids, yminorgrids, name=plot5, at=(plot4.right of north east), anchor=left of north west]  \end{axis}  \end{tikzpicture} \end{document}     I want to know if there is a more consistent manner to do this.
Let $D \subset \mathbb{C}$ be an open, connected set and let $\{ u_n \}$ be a sequence of harmonic functions with $u_n: D \longrightarrow (0, \infty)$. Show that if $u_n(z_0) \rightarrow 0$ for some $z_0 \in D$, then $u_n \rightarrow 0$ uniformly on compact subsets of $D$.  If you could offer a hint or a helpful question to get me started I'd appreciate it.   Progress: I now see why showing it for the unit disk suffices (any compact subset in $D$ can be covered by finitely many disks each which is contained in $D$). Windows Subsystem for Linux (WSL): what can't I do with the Ubuntu application for Microsoft Windows? Spectral radii and norms of similar elements in a C*-algebra: $\|bab^{-1}\|<1$ if $b=(\sum_{n=0}^\infty (a^*)^n a^n)^{1/2}$ Let $A$ be a unital $C^*$-algebra and $a\in A$ such that $r(a) &lt; 1$. Define b  = $(\sum_{n=0}^\infty (a^*)^n a^n)^{1/2}$. We can prove that $b\geq e$ and that $b$ is invertible. I want to show $\| b a b^{-1} \| &lt; 1$.   From the definition of $b$ we see that $a^* b^2 a = b^2-e$ and we know $r(bab^{-1}) = r(a) &lt;1$.   So it suffices to prove $r(b a b^{-1}) = \| b a b^{-1} \|$. It can follow from the fact $c = ba b^{-1}$ is a normal element... I don't know how to prove it (I have tried to compare $c^* c$ and $c c^*$...).    Context: The question appears in Murphy's book, page 74.   I have managed to prove the first part. The second part of the question is to prove $$r(a)= \inf_{c\in Inv(A)}\{\|cac^{-1} \| \}$$ It's easy to see $$r(a) \leq \inf_{c\in Inv(A)}\{\|cac^{-1} \| \}$$ But I can't prove $$r(a) \geq \inf_{c\in Inv(A)}\{\|cac^{-1} \| \}$$ If we have had $r(a) = \|b a b^{-1} \|$ then it was obvious... But this is not true, so how we can prove this inequality? How can I filter items with a certain enchantement? Map a network drive to be used by a service Suppose some Windows service uses code that wants mapped network drives and no UNC paths. 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For our purposes, this is a regular convex polygon (triangle, square, pentagon, hexagon and so on).  It can be assumed that the centre of the regular polygon is at the origin $(0,0)$, and the radius is $1$ unit.  If there's no such equation, can the non-existence be proven? If there are equations, but only for certain polygons (for example, only for $n &lt; 7$ or something), can those equations be provided? Proving that a Hölder space is a Banach space I am trying to show that the Hölder space $C^{k,\gamma}(\bar{U})$ is a Banach space. To do this, I successfully proved that the mapping $\| \quad \| : C^{k,\gamma}(\bar{U}) \to [0,\infty)$ is a norm, by proving its properties.   But how do I show that the sequence $\{u_k\}_{k=1}^\infty \subset C^{k,\gamma}(\bar{U})$ converges to $u \in C^{k,\gamma}(\bar{U})$, that is, how do I show that $$\lim_{k \to \infty}\|u_k-u\|=0,$$ which would mean the normed linear space is complete, and hence a Banach space?  Here are the following taken from PDE Evans, 2nd edition, page 255:     Definition. The Hölder space $$C^{k,\gamma}(\bar{U})$$ consists of all functions $u \in C^k(\bar{U})$ for which the norm $$\|u\|_{C^{k,y}(\bar{U})}:= \sum_{|\alpha|\le k} \|D^\alpha u \|_{C(\bar{U})}+\sum_{|\alpha|=k} [D^\alpha u]_{C^{0,\gamma}(\bar{U})}$$ is finite.   Also from page 254,     Definitions. (i) If $u : U \to \mathbb{R}$ is bounded and continuous, we write $$\|u\|_{C(\bar{U})}:=\sup_{x\in U}|u(x)|.$$      (ii) The $\gamma$th-Hölder seminorm of $u : U \to \mathbb{R}$ is $$[u]_{C^{0,\gamma}(\bar{U})}:=\sup_{\substack{x,y\in U \\ x \neq y}} \left\{\frac{|u(x)-u(y)|}{|x-y|^\gamma} \right\},$$ and the $\gamma$th-Hölder norm is $$\|u\|_{C^{0,\gamma}(\bar{U})}:=\|u\|_{C(\bar{U})}+[u]_{C^{0,\gamma}(\bar{U})}.$$   This is all I got so far: \begin{align} \|u_k-u\|_{C^{k,\gamma}(\bar{U})}&amp;=\sum_{|\alpha|\le k} \|D^\alpha u \|_{C(\bar{U})}+\sum_{|\alpha|=k} [D^\alpha u]_{C^{0,\gamma}(\bar{U})} \\ &amp;= \sum_{|\alpha|\le k} \sup_{x\in U} |u_k(x)-u(x)|+ \sum_{|\alpha|=k} \sup_{\substack{x,y\in U \\ x \neq y}} \left\{\frac{|[u_k(x)-u(x)]-[u_k(y)-u(y)]|}{|x-y|^\gamma} \right\}. \end{align} Now, where can I go from here, to show that $\lim_{k \to \infty}\|u_k-u\|_{C^{k,\gamma}(\bar{U})}=0$? The sequence is Cauchy, and I have to use that fact somehow. We are supposed to prove that the zero vector is unique with the additive identity $x+O=x$ and then by creating another additive identity with $O'$, then $x+O'=x$.  We set them equal $x+O=x+O'$ and use the vector cancelling theorem to get $O=O'$, but how does this prove that the zero vector is unique? The default is that clicking on the folder in the unity dock opens the home directory.   Since there are shortcuts for all the sub-folders of the home directory in the sidebar already, I'd prefer the default directory to move to a different location (~/Documents).  How can I achieve this? Is momentum conserved in this situation? Suppose we have object A , which has momentum mv. Then it elastically collides with a wall, so it's momentum now -mv, So how is momentum conserved here ? Can't transfer purchases from iPhone to iTunes in iOS 9 After upgrading to iOS 9, I can't transfer purchases from my iPhone to iTunes. 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