samples/texts/202179/page_7.md

To perform fits to data, it is most useful to consider yet another mathematically equivalent representation to Eqs. (25) and (26), a Lagrangian ($L$) representation in a path-integral context [6]. (This reference contains many references to other published works.) The long-time probability distribution $P$ at time $t_f = u\Delta t + t_0$, evolving from time $t_0$, in terms of the discrete index $s$, is given by

$$P=\int \text{\hspace{0.17em}⁣}\cdots \int \underset{‾}{DM}\exp (-\sum _{s=0}^u\mathrm{\Delta }t{L}_s),P\; =\; \backslash int\; \backslash dots\; \backslash int\; \backslash underline\{DM\}\; \backslash exp(-\backslash sum\_\{s=0\}^\{u\}\; \backslash Delta\; t\; L\_s),$$

$$\underset{‾}{DM}=g_{0+}^{1\mathrm{/}2}(2\pi \mathrm{\Delta }t{)}^{-1\mathrm{/}2}\prod _{s=1}^u{g}_{s+}^{1\mathrm{/}2}\prod _{G=1}^{\mathrm{\Theta }}(2\pi \mathrm{\Delta }t{)}^{-1\mathrm{/}2}d{M}_s^G,\backslash underline\{DM\}\; =\; g\_\{0+\}^\{1/2\}\; (2\backslash pi\backslash Delta\; t)^\{-1/2\}\; \backslash prod\_\{s=1\}^\{u\}\; g\_\{s+\}^\{1/2\}\; \backslash prod\_\{G=1\}^\{\backslash Theta\}\; (2\backslash pi\backslash Delta\; t)^\{-1/2\}\; dM\_s^G,$$

$$L=\frac{1}{2}({\dot{M}}^G-h^G)g_{G{G}^{\mathrm{\prime }}}({\dot{M}}^{G^{\mathrm{\prime }}}-h^{G^{\mathrm{\prime }}})+\frac{1}{2}{h}^G{}_{;G}+R\mathrm{/}6-V,L\; =\; \backslash frac\{1\}\{2\}(\backslash dot\{M\}^G\; -\; h^G)g\_\{GG\text{'}\}(\backslash dot\{M\}^\{G\text{'}\}\; -\; h^\{G\text{'}\})\; +\; \backslash frac\{1\}\{2\}h^G\{\}\_\{;G\}\; +\; R/6\; -\; V,$$

$$[\cdots {]}_{,G}=\frac{\mathrm{\partial }[\cdots ]}{\mathrm{\partial }{M}^G},[\backslash cdots]\_\{,G\}\; =\; \backslash frac\{\backslash partial[\backslash cdots]\}\{\backslash partial\; M^G\},$$

$$h^G=g^G-\frac{1}{2}{g}^{-1\mathrm{/}2}(g^{1\mathrm{/}2}{g}^{G{G}^{\mathrm{\prime }}}{)}_{,G},h^G\; =\; g^G\; -\; \backslash frac\{1\}\{2\}g^\{-1/2\}(g^\{1/2\}g^\{GG\text{'}\})\_\{,G\},$$

$$g_{G{G}^{\mathrm{\prime }}}=(g^{G{G}^{\mathrm{\prime }}}{)}^{-1},g\_\{GG\text{'}\}\; =\; (g^\{GG\text{'}\})^\{-1\},$$

$$g_s[M^G({\bar{t}}_s),{\bar{t}}_s]=\det (g_{G{G}^{\mathrm{\prime }}}{)}_s,g_{s+}=g_s[M_{s+1}^G,{\bar{t}}_s],g\_s[M^G(\backslash bar\{t\}\_s),\; \backslash bar\{t\}\_s]\; =\; \backslash det(g\_\{GG\text{'}\})\_s,\; \backslash quad\; g\_\{s+\}\; =\; g\_s[M\_\{s+1\}^G,\; \backslash bar\{t\}\_s],$$

$$M^G({\bar{t}}_s)=\frac{1}{2}(M_{s+1}^G+M_s^G),{\dot{M}}^G({\bar{t}}_s)=(M_{s+1}^G-M_s^G)\mathrm{/}\mathrm{\Delta }t,M^G(\backslash bar\{t\}\_s)\; =\; \backslash frac\{1\}\{2\}(M\_\{s+1\}^G\; +\; M\_s^G),\; \backslash quad\; \backslash dot\{M\}^G(\backslash bar\{t\}\_s)\; =\; (M\_\{s+1\}^G\; -\; M\_s^G)/\backslash Delta\; t,$$

$${\bar{t}}_s=t_s+\mathrm{\Delta }t\mathrm{/}2,\backslash bar\{t\}\_s\; =\; t\_s\; +\; \backslash Delta\; t/2,$$

$$h^G{}_{;G}=h^G{}_{,G}+{\mathrm{\Gamma }}_{GF}^F{h}^G=g^{-1\mathrm{/}2}(g^{1\mathrm{/}2}{h}^G{)}_{,G},h^G\{\}\_\{;G\}\; =\; h^G\{\}\_\{,G\}\; +\; \backslash Gamma^F\_\{GF\}\; h^G\; =\; g^\{-1/2\}(g^\{1/2\}h^G)\_\{,G\},$$

$${\mathrm{\Gamma }}_{JK}^F\equiv g^{LF}[JK,L]=g^{LF}(g_{JL,K}+g_{KL,J}-g_{JK,L}),\backslash Gamma^F\_\{JK\}\; \backslash equiv\; g^\{LF\}[JK,L]\; =\; g^\{LF\}(g\_\{JL,K\}\; +\; g\_\{KL,J\}\; -\; g\_\{JK,L\}),$$

$$R=g^{JL}{R}_{JL}=g^{JL}{g}^{JK}{R}_{FJK},R\; =\; g^\{JL\}\; R\_\{JL\}\; =\; g^\{JL\}\; g^\{JK\}\; R\_\{FJK\},$$

$$R_{FJKL}=\frac{1}{2}(g_{FK,JL}-g_{JK,FL}-g_{FL,JK}+g_{JL,FK})+g_{MN}({\mathrm{\Gamma }}_{FK}^M{\mathrm{\Gamma }}_{JL}^N-{\mathrm{\Gamma }}_{FL}^M{\mathrm{\Gamma }}_{JK}^N),(28)R\_\{FJKL\}\; =\; \backslash frac\{1\}\{2\}(g\_\{FK,JL\}\; -\; g\_\{JK,FL\}\; -\; g\_\{FL,JK\}\; +\; g\_\{JL,FK\})\; \backslash \backslash \; +g\_\{MN\}(\backslash Gamma^M\_\{FK\}\backslash Gamma^N\_\{JL\}\; -\; \backslash Gamma^M\_\{FL\}\backslash Gamma^N\_\{JK\}),\; \backslash qquad\; (28)$$

Such a path-integral representation may be derived directly for many systems, ranging from nuclear physics [11], to neuroscience [12-15]. (These references for both systems contain many references to other published papers.) In nuclear physics, the parameters include coupling constants and masses of mesons. In neuroscience, the parameters include chemical and electrical synaptic parameters, obtained by averaging over millions of synapses within minicolumnar structures of hundreds of neurons.

In the combat models, even relatively simple functional drifts and diffusions give rise to Lagrangians nonlinear in their underlying parameters. Even extremely simple Lagrangians can present subtle nonlinearities [16]. In the nuclear physics and neuroscience systems, the drifts and diffusions are quite nonlinear in their underlying variables ($M^G$), as well as being nonlinear in their underlying parameters. (For the nuclear physics quantum-mechanical problem, cost-