How to prevent my temporary horse from disappearing after Hosea stables mission?
The speed of light is always locally constant i.e. every observer measures the speed of light at their location to be $c$. Therefore there cannot be an observer who sees light to be stationary. There can be no observer who moves at the speed of light.
In fact it was when Einstein realised this that he discovered special relativity.
First, remove all the nested figure, adjustbox and minipage stuff. Then scale down the larger plot using the xscale and yscale options of tikz (or scale up the smaller plot, if you prefer). For the x-axis, you can compute the factor since the domain is given; it is 5/5.5=0.90909.... For the y-axis, I chose the practical approach and just modified it until it looked the same. So we have to add
\begin{tikzpicture}[...,xscale=0.90909,yscale=0.75]
in the second plot.
The advantage of this kind of scaling over \adjustbox is that just the coordinate system is scaled but not fonts and line widths.
Regarding your second question of how to place two figures side by side, may I refer you to Two figures side by side, LaTeX figures side by side and many similar posts.
\documentclass[border=2mm]{standalone}
\usepackage{tikz}
\usetikzlibrary{calc}
\begin{document}
\begin{tikzpicture}[my plot/.style={thick, smooth, samples=100, domain=0.1:5}, my grid/.style={densely dotted,opacity=0.5, every node/.style={black,opacity=1}}, my axis/.style={latex-latex}]
\draw[my plot, color=black] (0,0) plot (\x,{ln(\x)});
\coordinate (start plot) at (0.1,{ln(0.1)});
\coordinate (end plot) at (5,{ln(5)});
\draw[my axis] ([shift={(-0.5cm,0.5cm)}]start plot |- end plot) node[above] {$u(\cdot)$} |- node[coordinate](origin){} ([shift={(0.5cm,-0.5cm)}]start plot -| end plot) node[right] {$\cdot$};
\def\x{0.5}\def\y{4}\def\p{0.55}
\coordinate (Ux) at (\x,{ln(\x)});
\coordinate (Uy) at (\y,{ln(\y)});
\coordinate (Up) at ({\p*\x+(1-\p)*\y},{ln(\p*\x+(1-\p)*\y)});
\draw (Ux) -- coordinate[pos=1-\p] (Up-mid) (Uy);
\path let \p1=(Up-mid), \n1={pow(e,\y1*0.03514)} in (28.4576*\n1,\y1) coordinate (Up-mid2);
\draw[my grid] (Ux) |- node[below,font=\scriptsize]{$x$} (origin) |- node[left,font=\scriptsize]{$u(x)$} cycle;
\draw[my grid] (Uy) |- node[below,font=\scriptsize]{$y$} (origin) |- node[left,font=\scriptsize]{$u(y)$} cycle;
\draw[my grid] (Up) |- node[below, yshift=2.25pt, font=\scriptsize]{$px+(1-p)y$} (origin) |- node[left,font=\scriptsize]{$u(px+(1-p)y)$} cycle;
\draw[my grid] (Up-mid) |- (origin) |- node[left,font=\scriptsize]{$pu(x)+(1-p)u(y)$} cycle;
\draw[my grid] (Up-mid) -- (Up-mid2);
\end{tikzpicture}
\quad
\begin{tikzpicture}[my plot/.style={thick, smooth, samples=100, domain=0.1:5.5}, my grid/.style={densely dotted,opacity=0.5, every node/.style={black,opacity=1}}, my axis/.style={latex-latex},xscale=0.90909,yscale=0.75]
\draw[my plot, color=black] (0,0) plot (\x,{(\x)});
\coordinate (start plot) at (0,{(0)});
\coordinate (end plot) at (5.5,{(5.5)});
\draw[my axis] ([shift={(-0.5cm,0.5cm)}]start plot |- end plot) node[above] {$u(\cdot)$} |- node[coordinate](origin){} ([shift={(0.5cm,-0.5cm)}]start plot -| end plot) node[right] {$\cdot$};
\def\x{0.5}\def\y{4}\def\p{0.55}
\coordinate (Ux) at (\x,{(\x)});
\coordinate (Uy) at (\y,{(\y)});
\coordinate (Up) at ({\p*\x+(1-\p)*\y},{(\p*\x+(1-\p)*\y)});
\draw (Ux) -- coordinate[pos=1-\p] (Up-mid) (Uy);
\path let \p1=(Up-mid), \n1={pow(e,\y1*0.03514)} in (28.4576*\n1,\y1) coordinate (Up-mid2);
\draw[my grid] (Ux) |- node[below,font=\scriptsize]{$x$} (origin) |- node[left,font=\scriptsize]{$u(x)$} cycle;
\draw[my grid] (Uy) |- node[below,font=\scriptsize]{$y$} (origin) |- node[left,font=\scriptsize]{$u(y)$} cycle;
\draw[my grid] (Up) |- node[below, yshift=2.25pt, font=\scriptsize]{$px+(1-p)y$} (origin) |- node[align=right,font=\scriptsize,xshift=-40pt]{$pu(x)+(1-p)u(y)=$\\$=u(px+(1-p)y)$} cycle;
\end{tikzpicture}
\end{document}
Edit: Here is the wrapper to fit the plots, including captions, on an article page. I additionally use the trick to remove the width of the long labels using \llap{...} such that the labels stick into the margin and the other figure, respectively.
\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{calc}
\usepackage{blindtext}
\begin{document}
\blindtext
\begin{figure}
\newcommand\commonscalefactor{0.64}%
\begin{minipage}[b]{\dimexpr0.5\textwidth-1em}
\centering
\begin{tikzpicture}[my plot/.style={thick, smooth, samples=100, domain=0.1:5}, my grid/.style={densely dotted,opacity=0.5, every node/.style={black,opacity=1}}, my axis/.style={latex-latex},scale=\commonscalefactor]
\draw[my plot, color=black] (0,0) plot (\x,{ln(\x)});
\coordinate (start plot) at (0.1,{ln(0.1)});
\coordinate (end plot) at (5,{ln(5)});
\draw[my axis] ([shift={(-0.5cm,0.5cm)}]start plot |- end plot) node[above] {$u(\cdot)$} |- node[coordinate](origin){} ([shift={(0.5cm,-0.5cm)}]start plot -| end plot) node[right] {$\cdot$};
\def\x{0.5}\def\y{4}\def\p{0.55}
\coordinate (Ux) at (\x,{ln(\x)});
\coordinate (Uy) at (\y,{ln(\y)});
\coordinate (Up) at ({\p*\x+(1-\p)*\y},{ln(\p*\x+(1-\p)*\y)});
\draw (Ux) -- coordinate[pos=1-\p] (Up-mid) (Uy);
\path let \p1=(Up-mid), \n1={pow(e,\y1*0.03514)} in (28.4576*\n1,\y1) coordinate (Up-mid2);
\draw[my grid] (Ux) |- node[below,font=\scriptsize]{$x$} (origin) |- node[left,font=\scriptsize]{$u(x)$} cycle;
\draw[my grid] (Uy) |- node[below,font=\scriptsize]{$y$} (origin) |- node[left,font=\scriptsize]{$u(y)$} cycle;
\draw[my grid] (Up) |- node[below, yshift=2.25pt, font=\scriptsize]{$px+(1-p)y$} (origin) |- node[left,font=\scriptsize]
{\llap{$u(px+(1-p)y)$}} cycle;
\draw[my grid] (Up-mid) |- (origin) |- node[left,font=\scriptsize]{\llap{$pu(x)+(1-p)u(y)$}} cycle;
\draw[my grid] (Up-mid) -- (Up-mid2);
\end{tikzpicture}\\
\caption{Risk Aversion}\label{RA}
\end{minipage}%
\hspace{2em}%
\begin{minipage}[b]{\dimexpr0.5\textwidth-1em}
\centering
\begin{tikzpicture}[my plot/.style={thick, smooth, samples=100, domain=0.1:5.5}, my grid/.style={densely dotted,opacity=0.5, every node/.style={black,opacity=1}}, my axis/.style={latex-latex},xscale=0.90909*\commonscalefactor,yscale=0.75*\commonscalefactor]
\draw[my plot, color=black] (0,0) plot (\x,{(\x)});
\coordinate (start plot) at (0,{(0)});
\coordinate (end plot) at (5.5,{(5.5)});
\draw[my axis] ([shift={(-0.5cm,0.5cm)}]start plot |- end plot) node[above] {$u(\cdot)$} |- node[coordinate](origin){} ([shift={(0.5cm,-0.5cm)}]start plot -| end plot) node[right] {$\cdot$};
\def\x{0.5}\def\y{4}\def\p{0.55}
\coordinate (Ux) at (\x,{(\x)});
\coordinate (Uy) at (\y,{(\y)});
\coordinate (Up) at ({\p*\x+(1-\p)*\y},{(\p*\x+(1-\p)*\y)});
\draw (Ux) -- coordinate[pos=1-\p] (Up-mid) (Uy);
\path let \p1=(Up-mid), \n1={pow(e,\y1*0.03514)} in (28.4576*\n1,\y1) coordinate (Up-mid2);
\draw[my grid] (Ux) |- node[below,font=\scriptsize]{$x$} (origin) |- node[left,font=\scriptsize]{$u(x)$} cycle;
\draw[my grid] (Uy) |- node[below,font=\scriptsize]{$y$} (origin) |- node[left,font=\scriptsize]{$u(y)$} cycle;
\draw[my grid] (Up) |- node[below, yshift=2.25pt, font=\scriptsize]{$px+(1-p)y$} (origin) |- node[align=right,font=\scriptsize,xshift=-0.5em]{\llap{$pu(x)+(1-p)u(y)=$}\\\llap{$=u(px+(1-p)y)$}} cycle;
\end{tikzpicture}\\
\caption{Risk Neutrality}\label{RN}
\end{minipage}%
\end{figure}
\end{document}
On Ubuntu you can use purge-old-kernels to purge old kernels , to install it:
For ubuntu 16.04
sudo apt-key adv --keyserver keyserver.ubuntu.com --recv-keys F430BBA5
sudo add-apt-repository ppa:byobu/ppa
sudo apt-get update
sudo apt-get install byobu
Also you can install it without adding the ppa: sudo apt-get install byobu
For Ubuntu versions lower than 16.04
sudo add-apt-repository ppa:bikeshed/ppa
sudo apt-get update
sudo apt-get install bikeshed
Run the following command to keeping the latest 2 kernel :
sudo purge-old-kernels
You can specify the number n of kernel to keep:
sudo purge-old-kernels --keep n
n=1:
sudo purge-old-kernels --keep 1
On centOS install yum-utils package :
yum install yum-utils
To keep the latest n kernels , run:
package-cleanup --oldkernels --count=n
n=1:
package-cleanup --oldkernels --count=1