convert some more fixes

\documentclass[12pt]{article}

\usepackage{amsmath}

\begin{document}

\begin{align*}

 &{\bf F}_{j}(t) = {\bf F}^C_j(t)-\int \limits_{0}^{t} \Gamma_j(t-s) {\bf v}_j(s)~\text{d}s + {\bf F}^R_j(t) \\

 &\Gamma_j(t-s) = \sum \limits_{k=1}^{N_k} \frac{c_k}{\tau_k} e^{-(t-s)/\tau_k} \\ 

 &\langle{\bf F}^R_j(t),{\bf F}^R_j(s)\rangle = \text{k$_\text{B}$T} ~\Gamma_j(t-s)

\end{align*}

\end{document}

\documentclass[preview]{standalone}

\usepackage{varwidth}

\usepackage[utf8x]{inputenc}

\usepackage{amsmath,amssymb,amsthm,bm,tikz}

\usetikzlibrary{automata,arrows,shapes,snakes}

\begin{document}

\begin{varwidth}{50in}

\begin{tikzpicture}

%Global

\node (v1) at (0,6.0) [draw,thick,minimum width=0.2cm,minimum height=0.2cm] { $\bm{v} \leftarrow \bm{v}+L_v.\Delta t/2$ };

\node (s1) at (0,4.5) [draw,thick,minimum width=0.2cm,minimum height=0.2cm] { $\bm{s} \leftarrow \bm{s}+L_s.\Delta t/2$ };

\node (r)  at (0,3.0) [draw,thick,minimum width=0.2cm,minimum height=0.2cm] { $\bm{r} \leftarrow \bm{r}+L_r.\Delta t$ };

\node (s2) at (0,1.5) [draw,thick,minimum width=0.2cm,minimum height=0.2cm] { $\bm{s} \leftarrow \bm{s}+L_s.\Delta t/2$ };

\node (v2) at (0,0.0) [draw,thick,minimum width=0.2cm,minimum height=0.2cm] { $\bm{v} \leftarrow \bm{v}+L_v.\Delta t/2$ };

\draw[line width=2pt, ->] (v1) -- (s1);

\draw[line width=2pt, ->] (s1) -- (r);

\draw[line width=2pt, ->] (r) -- (s2);

\draw[line width=2pt, ->] (s2) -- (v2);

%Spin

\node (s01) at (6,6.0) [draw,thick,minimum width=0.2cm,minimum height=0.2cm] {$\bm{s}_0 \leftarrow \bm{s}_0+L_{s_0}.\Delta t/4$ };

\node (sN1) at (6,4.5) [draw,thick,minimum width=0.2cm,minimum height=0.2cm] {$\bm{s}_{\rm N-1}\leftarrow\bm{s}_{\rm N-1}+L_{s_{\rm N-1}}.\Delta t/4$};

\node (sN)  at (6,3.0) [draw,thick,minimum width=0.2cm,minimum height=0.2cm] {$\bm{s}_{\rm N} \leftarrow \bm{s}_{\rm N}+L_{s_{\rm N}}.\Delta t/2$ };

\node (sN2) at (6,1.5) [draw,thick,minimum width=0.2cm,minimum height=0.2cm] {$\bm{s}_{\rm N-1}\leftarrow\bm{s}_{\rm N-1}+L_{s_{\rm N-1}}.\Delta t/4$};

\node (s02) at (6,0.0) [draw,thick,minimum width=0.2cm,minimum height=0.2cm] {$\bm{s}_0 \leftarrow \bm{s}_0+L_{s_0}.\Delta t/4$ };

\draw[line width=2pt,dashed, ->] (s01) -- (sN1);

\draw[line width=2pt, ->] (sN1) -- (sN);

\draw[line width=2pt, ->] (sN) -- (sN2);

\draw[line width=2pt,dashed, ->] (sN2) -- (s02);

%from Global to Spin

\draw[line width=2pt, dashed, ->] (s1) -- (s01.west);

\draw[line width=2pt, dashed, ->] (s1) -- (s02.west);

\end{tikzpicture}

\end{varwidth}

\end{document}

\documentclass[12pt]{article}

\begin{document}

$$

  v(t+\frac{\Delta t}{2}) = v(t) + \frac{\Delta t}{2}\cdot a(t),

$$

$$

  r(t+\Delta t) = r(t) + \Delta t\cdot v(t+\frac{\Delta t}{2}),

$$

$$

  a(t+\Delta t) = \frac{1}{m}\cdot F\left[ r(t+\Delta t), v(t) +\lambda \cdot \Delta t\cdot a(t)\right],

$$

$$

  v(t+\Delta t) = v(t+\frac{\Delta t}{2}) + \frac{\Delta t}{2}\cdot a(t+\Delta t)

$$

\end{document}