SKIPPED EQUATION: p_{x} ~WIKIEQUATION: {\displaystyle p_{i}}
SKIPPED EQUATION: \mu_{x} ~WIKIEQUATION: {\displaystyle \mu _{tc}}
SKIPPED EQUATION: \sigma ~WIKIEQUATION: {\displaystyle \sigma }
SKIPPED EQUATION: w_{x} ~WIKIEQUATION: {\displaystyle w_{+}}
SKIPPED EQUATION: w_{x} ~WIKIEQUATION: {\displaystyle w_{-}}
SKIPPED EQUATION: u_{x} ~WIKIEQUATION: {\displaystyle u_{1,2}}
SKIPPED EQUATION: (u) ~WIKIEQUATION: {\displaystyle (u)}
SKIPPED EQUATION: (v_{x}) ~WIKIEQUATION: {\displaystyle (v_{1,2})}
SKIPPED EQUATION: n ~WIKIEQUATION: {\displaystyle n}
SKIPPED EQUATION: m ~WIKIEQUATION: {\displaystyle m}
SKIPPED EQUATION: (x))-((x))|^{2}}{2*\sigma}}) ~WIKIEQUATION: {\displaystyle R_{\psi }(t)=\sum _{i=1}^{N}\exp \left(-{\frac {\left|(x_{i}(t),y_{i}(t))-(\mathrm {X} _{i},_{\psi },\mathrm {P} _{i},_{\psi })\right\vert ^{2}}{2\cdot \sigma }}\right)}
SKIPPED EQUATION: ((x)) ~WIKIEQUATION: {\displaystyle (x_{i},y_{i})}
SKIPPED EQUATION: ((x)) ~WIKIEQUATION: {\displaystyle (c_{i},r_{i})}
SKIPPED EQUATION: t ~WIKIEQUATION: {\displaystyle t}
SKIPPED EQUATION: \psi ~WIKIEQUATION: {\displaystyle \psi }
SKIPPED EQUATION: R ~WIKIEQUATION: {\displaystyle R}
SKIPPED EQUATION: N ~WIKIEQUATION: {\displaystyle N}
SKIPPED EQUATION: \sigma ~WIKIEQUATION: {\displaystyle \sigma }
SKIPPED EQUATION: R_{x} ~WIKIEQUATION: {\displaystyle R_{y}(t)}
SKIPPED EQUATION: {x} ~WIKIEQUATION: {\displaystyle _{\psi }}
SKIPPED EQUATION: t ~WIKIEQUATION: {\displaystyle t}
SKIPPED EQUATION: {{R}} ~WIKIEQUATION: {\displaystyle {\bar {R}}}
SKIPPED EQUATION: t ~WIKIEQUATION: {\displaystyle t}
SKIPPED EQUATION: n_{x} ~WIKIEQUATION: {\displaystyle n_{y}(t)}
SKIPPED EQUATION: r ~WIKIEQUATION: {\displaystyle r}
SKIPPED EQUATION: \tau ~WIKIEQUATION: {\displaystyle \tau }
SKIPPED EQUATION: p ~WIKIEQUATION: {\displaystyle p}
SKIPPED EQUATION: N ~WIKIEQUATION: {\displaystyle N}
SKIPPED EQUATION: N_{x}(\tau)^{2}-N_{x}(\tau)^{2} ~WIKIEQUATION: {\displaystyle \varepsilon _{\psi }(\tau )=N_{\psi }^{F}(\tau )^{2}-N_{\psi }^{B}(\tau )^{2}}
SKIPPED EQUATION: \varepsilon ~WIKIEQUATION: {\displaystyle \varepsilon }
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle x}
SKIPPED EQUATION: \theta_{x} ~WIKIEQUATION: {\displaystyle \theta _{p},}
SKIPPED EQUATION: v ~WIKIEQUATION: {\displaystyle v}
SKIPPED EQUATION: \theta ~WIKIEQUATION: {\displaystyle \theta }
SKIPPED EQUATION: H ~WIKIEQUATION: {\displaystyle H}
SKIPPED EQUATION: v_{x} ~WIKIEQUATION: {\displaystyle v_{1}<v<v_{2}}
SKIPPED EQUATION: v ~WIKIEQUATION: {\displaystyle v_{1}<v<v_{2}}
SKIPPED EQUATION: v_{x} ~WIKIEQUATION: {\displaystyle v_{1}<v<v_{2}}
UNPARSED EQUATION: (({\x{1}{2}})[1+\cos((x))]\)^{q} ~WIKIEQUATION: {\displaystyle b(\theta ,\theta _{p})=\left\{\left({\frac {1}{2}}\right)\left[\ 1+cos(\theta ,\theta _{p})\right]\ \right\}^{q}}
SKIPPED EQUATION: q ~WIKIEQUATION: {\displaystyle q}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle x}
SKIPPED EQUATION: p ~WIKIEQUATION: {\displaystyle p}
SKIPPED EQUATION: r ~WIKIEQUATION: {\displaystyle r}
SKIPPED EQUATION: i ~WIKIEQUATION: {\displaystyle i,j}
SKIPPED EQUATION: l ~WIKIEQUATION: {\displaystyle l}
SKIPPED EQUATION: x_{x} ~WIKIEQUATION: {\displaystyle x_{k}}
SKIPPED EQUATION: e^ ~WIKIEQUATION: {\displaystyle G(u)=e^{(u-u_{0})^{T}C(u-u_{0})}}
SKIPPED EQUATION: u_{x} ~WIKIEQUATION: {\displaystyle u_{0}}
SKIPPED EQUATION: C ~WIKIEQUATION: {\displaystyle C}
SKIPPED EQUATION: k ~WIKIEQUATION: {\displaystyle k}
SKIPPED EQUATION: l ~WIKIEQUATION: {\displaystyle l}
SKIPPED EQUATION: \tau ~WIKIEQUATION: {\displaystyle \tau }
SKIPPED EQUATION: G_{x} ~WIKIEQUATION: {\displaystyle G_{k}^{l}(t)}
SKIPPED EQUATION: H_{x} ~WIKIEQUATION: {\displaystyle H_{l}^{l}(t)}
SKIPPED EQUATION: P_{x} ~WIKIEQUATION: {\displaystyle P^{l}(t)}
SKIPPED EQUATION: l ~WIKIEQUATION: {\displaystyle l}
SKIPPED EQUATION: \tau_{x} ~WIKIEQUATION: {\displaystyle \tau _{s}}
SKIPPED EQUATION: H_{x} ~WIKIEQUATION: {\displaystyle H_{k}^{l}(t)}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle x}
SKIPPED EQUATION: y ~WIKIEQUATION: {\displaystyle y}
SKIPPED EQUATION: z ~WIKIEQUATION: {\displaystyle z}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle x}
SKIPPED EQUATION: y ~WIKIEQUATION: {\displaystyle y}
SKIPPED EQUATION: z ~WIKIEQUATION: {\displaystyle z}
SKIPPED EQUATION: f_{x} ~WIKIEQUATION: {\displaystyle f_{i}\ }
SKIPPED EQUATION: c_{x} ~WIKIEQUATION: {\displaystyle c_{j}\ }
SKIPPED EQUATION: f_{x}}x} ~WIKIEQUATION: {\displaystyle f_{p{\mbox{-}}int}\ }
SKIPPED EQUATION: 1 ~WIKIEQUATION: {\displaystyle p(c_{rational}|f_{p{\mbox{-}}int})=1\ }
SKIPPED EQUATION: 0 ~WIKIEQUATION: {\displaystyle p(c_{irrational}|f_{p{\mbox{-}}int})=0\ }
SKIPPED EQUATION: 05 ~WIKIEQUATION: {\displaystyle p(c_{even}|f_{p{\mbox{-}}int})=0.5\ }
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle r=I\{s_{1},s_{2},..,s_{n}\}}
SKIPPED EQUATION: R_{x} ~WIKIEQUATION: {\displaystyle R_{1}=R_{2}}
SKIPPED EQUATION: F_{x} ~WIKIEQUATION: {\displaystyle F_{1}>F_{2}}
SKIPPED EQUATION: F_{x} ~WIKIEQUATION: {\displaystyle F_{1}>F_{2}}
SKIPPED EQUATION: G_{x} ~WIKIEQUATION: {\displaystyle G_{1}<G_{2}}
SKIPPED EQUATION: G_{x} ~WIKIEQUATION: {\displaystyle G_{1}<G_{2}}
SKIPPED EQUATION: F_{x} ~WIKIEQUATION: {\displaystyle F_{2}}
SKIPPED EQUATION: F_{x} ~WIKIEQUATION: {\displaystyle F_{1}}
SKIPPED EQUATION: G_{x} ~WIKIEQUATION: {\displaystyle G_{2}}
SKIPPED EQUATION: F_{x} ~WIKIEQUATION: {\displaystyle F_{1}}
SKIPPED EQUATION: G_{x} ~WIKIEQUATION: {\displaystyle G_{1}}
SKIPPED EQUATION: G_{x} ~WIKIEQUATION: {\displaystyle G_{2}}
UNPARSED EQUATION: {\x{x ~WIKIEQUATION: {\displaystyle \quad \mathrm {IQ} ={\frac {\mathrm {mental\;age} }{\mathrm {chronological\;age} }}\cdot 100}
SKIPPED EQUATION: x}{chronological ~WIKIEQUATION: {\displaystyle \quad \mathrm {IQ} ={\frac {\mathrm {mental\;age} }{\mathrm {chronological\;age} }}\cdot 100}
SKIPPED EQUATION: x}}*100 ~WIKIEQUATION: {\displaystyle \quad \mathrm {IQ} ={\frac {\mathrm {mental\;age} }{\mathrm {chronological\;age} }}\cdot 100}
SKIPPED EQUATION: i ~WIKIEQUATION: {\displaystyle i}
SKIPPED EQUATION: a ~WIKIEQUATION: {\displaystyle a}
SKIPPED EQUATION: k ~WIKIEQUATION: {\displaystyle k}
SKIPPED EQUATION: n ~WIKIEQUATION: {\displaystyle n}
SKIPPED EQUATION: k ~WIKIEQUATION: {\displaystyle k}
SKIPPED EQUATION: a ~WIKIEQUATION: {\displaystyle a}
SKIPPED EQUATION: b ~WIKIEQUATION: {\displaystyle b}
SKIPPED EQUATION: p ~WIKIEQUATION: {\displaystyle p}
SKIPPED EQUATION: a ~WIKIEQUATION: {\displaystyle a}
SKIPPED EQUATION: b ~WIKIEQUATION: {\displaystyle b}
SKIPPED EQUATION: n ~WIKIEQUATION: {\displaystyle n}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle MRT}
SKIPPED EQUATION: K ~WIKIEQUATION: {\displaystyle K}
SKIPPED EQUATION: N ~WIKIEQUATION: {\displaystyle N}
SKIPPED EQUATION: v_{x}-v_{x} ~WIKIEQUATION: {\displaystyle \mathrm {log\;odds} (A\ {\text{beats}}\ B\mid v_{a},v_{b})=v_{a}-v_{b}}
SKIPPED EQUATION: \pi_{x} ~WIKIEQUATION: {\displaystyle p(H1)=\pi _{1}}
SKIPPED EQUATION: \pi_{x} ~WIKIEQUATION: {\displaystyle p(H2)=\pi _{2}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \Rightarrow {\frac {p(y|H2)}{p(y|H1)}}\geq {\frac {\pi _{1}}{\pi _{2}}}}
SKIPPED EQUATION: \tau_{x} ~WIKIEQUATION: {\displaystyle \tau _{MAP}}
SKIPPED EQUATION: \tau_{x} ~WIKIEQUATION: {\displaystyle L(y)\geq \tau _{MAP}}
SKIPPED EQUATION: U_{x} ~WIKIEQUATION: {\displaystyle U_{11}}
SKIPPED EQUATION: U_{x} ~WIKIEQUATION: {\displaystyle U_{12}}
SKIPPED EQUATION: U_{x} ~WIKIEQUATION: {\displaystyle U_{21}}
SKIPPED EQUATION: U_{x} ~WIKIEQUATION: {\displaystyle U_{22}}
SKIPPED EQUATION: U_{x}-U_{x} ~WIKIEQUATION: {\displaystyle U_{11}-U_{21}}
SKIPPED EQUATION: U_{x}-U_{x} ~WIKIEQUATION: {\displaystyle U_{22}-U_{12}}
SKIPPED EQUATION: P_{x} ~WIKIEQUATION: {\displaystyle P_{11}}
SKIPPED EQUATION: P_{x} ~WIKIEQUATION: {\displaystyle P_{12}}
SKIPPED EQUATION: P_{x}*(U_{x}-U_{x})-P_{x}*(U_{x}-U_{x}) ~WIKIEQUATION: {\displaystyle U'=P_{11}\cdot (U_{11}-U_{21})-P_{12}\cdot (U_{22}-U_{12})}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle P_{11}=\pi _{1}\cdot \int _{R_{1}}p(y|H1)\,dy}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle P_{12}=\pi _{2}\cdot \int _{R_{1}}p(y|H2)\,dy}
SKIPPED EQUATION: \pi_{x} ~WIKIEQUATION: {\displaystyle \pi _{1}}
SKIPPED EQUATION: \pi_{x} ~WIKIEQUATION: {\displaystyle \pi _{2}}
SKIPPED EQUATION: R_{x} ~WIKIEQUATION: {\displaystyle R_{1}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \Rightarrow U'=\int _{R_{1}}\left\{\pi _{1}\cdot (U_{11}-U_{21})\cdot p(y|H1)-\pi _{2}\cdot (U_{22}-U_{12})\cdot p(y|H2)\right\}\,dy}
SKIPPED EQUATION: U ~WIKIEQUATION: {\displaystyle U'}
SKIPPED EQUATION: U ~WIKIEQUATION: {\displaystyle U}
SKIPPED EQUATION: R_{x} ~WIKIEQUATION: {\displaystyle R_{1}}
SKIPPED EQUATION: \pi_{x}*(U_{x}-U_{x})*p(x)-\pi_{x}*(U_{x}-U_{x})*p(x) ~WIKIEQUATION: {\displaystyle \pi _{1}\cdot (U_{11}-U_{21})\cdot p(y|H1)-\pi _{2}\cdot (U_{22}-U_{12})\cdot p(y|H2)>0}
SKIPPED EQUATION: 0 ~WIKIEQUATION: {\displaystyle \pi _{1}\cdot (U_{11}-U_{21})\cdot p(y|H1)-\pi _{2}\cdot (U_{22}-U_{12})\cdot p(y|H2)>0}
SKIPPED EQUATION: \pi_{x}*(U_{x}-U_{x})*p(x) ~WIKIEQUATION: {\displaystyle \pi _{2}\cdot (U_{22}-U_{12})\cdot p(y|H2)\geq \pi _{1}\cdot (U_{11}-U_{21})\cdot p(y|H1)}
SKIPPED EQUATION: \pi_{x}*(U_{x}-U_{x})*p(x) ~WIKIEQUATION: {\displaystyle \pi _{2}\cdot (U_{22}-U_{12})\cdot p(y|H2)\geq \pi _{1}\cdot (U_{11}-U_{21})\cdot p(y|H1)}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \Rightarrow L(y)\equiv {\frac {p(y|H2)}{p(y|H1)}}\geq {\frac {\pi _{1}\cdot (U_{11}-U_{21})}{\pi _{2}\cdot (U_{22}-U_{12})}}\equiv \tau _{B}}
SKIPPED EQUATION: {\x{\pi_{x}*(U_{x}-U_{x})}{\pi_{x}*(U_{x}-U_{x})}} ~WIKIEQUATION: {\displaystyle \Rightarrow L(y)\equiv {\frac {p(y|H2)}{p(y|H1)}}\geq {\frac {\pi _{1}\cdot (U_{11}-U_{21})}{\pi _{2}\cdot (U_{22}-U_{12})}}\equiv \tau _{B}}
SKIPPED EQUATION: p ~WIKIEQUATION: {\displaystyle p=0.5}
SKIPPED EQUATION: p ~WIKIEQUATION: {\displaystyle p=0.5}
SKIPPED EQUATION: k ~WIKIEQUATION: {\displaystyle {\frac {\Delta I}{I}}=k,}
SKIPPED EQUATION: I ~WIKIEQUATION: {\displaystyle I\!}
SKIPPED EQUATION: \DeltaI ~WIKIEQUATION: {\displaystyle \Delta I\!}
SKIPPED EQUATION: S_{x} ~WIKIEQUATION: {\displaystyle S_{i}}
SKIPPED EQUATION: x_{x} ~WIKIEQUATION: {\displaystyle x_{ij}}
SKIPPED EQUATION: \sigma_{x} ~WIKIEQUATION: {\displaystyle \sigma _{i}}
SKIPPED EQUATION: R_{x} ~WIKIEQUATION: {\displaystyle R_{i}}
SKIPPED EQUATION: r_{x} ~WIKIEQUATION: {\displaystyle r_{ij}}
SKIPPED EQUATION: R_{x} ~WIKIEQUATION: {\displaystyle R_{i}}
SKIPPED EQUATION: S_{x} ~WIKIEQUATION: {\displaystyle S_{i}}
SKIPPED EQUATION: {\x{S_{x}-S_{x}}{\sigma}} ~WIKIEQUATION: {\displaystyle x_{ij}={\frac {S_{i}-S_{j}}{\sigma }}\,}
SKIPPED EQUATION: {S_{x}-S_{x}} ~WIKIEQUATION: {\displaystyle {S_{i}-S_{j}}}
SKIPPED EQUATION: x_{x} ~WIKIEQUATION: {\displaystyle x_{ij}}
SKIPPED EQUATION: \sigma ~WIKIEQUATION: {\displaystyle \sigma =1}
SKIPPED EQUATION: P_{x} ~WIKIEQUATION: {\displaystyle P_{ij}}
SKIPPED EQUATION: P_{x} ~WIKIEQUATION: {\displaystyle P_{ij}=0.84}
SKIPPED EQUATION: x_{x} ~WIKIEQUATION: {\displaystyle x_{ij}}
SKIPPED EQUATION: S_{x}-S_{x} ~WIKIEQUATION: {\displaystyle S_{i}-S_{j}\cong 1}
SKIPPED EQUATION: a ~WIKIEQUATION: {\displaystyle a}
SKIPPED EQUATION: a ~WIKIEQUATION: {\displaystyle a}
SKIPPED EQUATION: b ~WIKIEQUATION: {\displaystyle b}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle ab}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle ba}
SKIPPED EQUATION: x_{x} ~WIKIEQUATION: {\displaystyle x_{1}}
SKIPPED EQUATION: x_{x} ~WIKIEQUATION: {\displaystyle x_{2}}
SKIPPED EQUATION: a ~WIKIEQUATION: {\displaystyle a}
SKIPPED EQUATION: b ~WIKIEQUATION: {\displaystyle b}
SKIPPED EQUATION: a ~WIKIEQUATION: {\displaystyle a}
SKIPPED EQUATION: b ~WIKIEQUATION: {\displaystyle b}
SKIPPED EQUATION: b ~WIKIEQUATION: {\displaystyle b}
SKIPPED EQUATION: a ~WIKIEQUATION: {\displaystyle a}
SKIPPED EQUATION: (x)) ~WIKIEQUATION: {\displaystyle p(e)=p\left({\tilde {\chi }}_{{\boldsymbol {w}},{\boldsymbol {k}},{\boldsymbol {\lambda }},0,0}^{2}\right)<0}
SKIPPED EQUATION: 0 ~WIKIEQUATION: {\displaystyle p(e)=p\left({\tilde {\chi }}_{{\boldsymbol {w}},{\boldsymbol {k}},{\boldsymbol {\lambda }},0,0}^{2}\right)<0}
SKIPPED EQUATION: {\x{\mu_{x}-\mu_{x}}{\sigma_{x}^{2}-\sigma_{x}^{2}}} ~WIKIEQUATION: {\displaystyle {\boldsymbol {w}}={\begin{bmatrix}\sigma _{a}^{2}&-\sigma _{b}^{2}\end{bmatrix}},\;{\boldsymbol {k}}={\begin{bmatrix}1&1\end{bmatrix}},\;{\boldsymbol {\lambda }}={\frac {\mu _{a}-\mu _{b}}{\sigma _{a}^{2}-\sigma _{b}^{2}}}{\begin{bmatrix}\sigma _{a}^{2}&\sigma _{b}^{2}\end{bmatrix}}.}
SKIPPED EQUATION: A ~WIKIEQUATION: {\displaystyle A}
SKIPPED EQUATION: (\lambda) ~WIKIEQUATION: {\displaystyle \lambda }
SKIPPED EQUATION: (x) ~WIKIEQUATION: {\displaystyle {\begin{aligned}dy_{\text{1}}\ =\ (-ky_{\text{1}}-wy_{\text{2}}+I_{\text{1}})dt\ +\ cdW_{\text{1}}\\dy_{\text{2}}\ =\ (-ky_{\text{2}}-wy_{\text{1}}+I_{\text{2}})dt\ +\ cdW_{\text{2}}\end{aligned}},\quad y_{\text{1}}(0)\ =\ y_{\text{2}}(0)=0}
SKIPPED EQUATION: k ~WIKIEQUATION: {\displaystyle k}
SKIPPED EQUATION: w ~WIKIEQUATION: {\displaystyle w}
SKIPPED EQUATION: (x) ~WIKIEQUATION: {\displaystyle {\begin{aligned}dy_{\text{1}}\ =\ I_{\text{1}}dt\ +\ cdW_{\text{1}}\ -\ u(I_{\text{2}}dt\ +\ cdW_{\text{2}})\\dy_{\text{2}}\ =\ I_{\text{2}}dt\ +\ cdW_{\text{2}}\ -\ u(I_{\text{1}}dt\ +\ cdW_{\text{1}})\end{aligned}},\quad y_{\text{1}}(0)\ =\ y_{\text{2}}(0)=0}
SKIPPED EQUATION: u ~WIKIEQUATION: {\displaystyle u}
SKIPPED EQUATION: k_{x} ~WIKIEQUATION: {\displaystyle k_{\text{inh}}}
SKIPPED EQUATION: w ~WIKIEQUATION: {\displaystyle w'}
SKIPPED EQUATION: S ~WIKIEQUATION: {\displaystyle S}
SKIPPED EQUATION: K ~WIKIEQUATION: {\displaystyle K}
SKIPPED EQUATION: k ~WIKIEQUATION: {\displaystyle k}
SKIPPED EQUATION: S ~WIKIEQUATION: {\displaystyle S}
SKIPPED EQUATION: C ~WIKIEQUATION: {\displaystyle C}
SKIPPED EQUATION: C ~WIKIEQUATION: {\displaystyle C}
SKIPPED EQUATION: S_{x} ~WIKIEQUATION: {\displaystyle S_{0}}
SKIPPED EQUATION: p ~WIKIEQUATION: {\displaystyle p=0}
SKIPPED EQUATION: S_{x} ~WIKIEQUATION: {\displaystyle S=S_{0}}
SKIPPED EQUATION: C ~WIKIEQUATION: {\displaystyle C}
UNPARSED EQUATION: 0463^{-0072} ~WIKIEQUATION: {\displaystyle \Delta I/I=0.463{(I/I_{0})}^{-0.072}}
SKIPPED EQUATION: B ~WIKIEQUATION: {\displaystyle B}
SKIPPED EQUATION: \DeltaB ~WIKIEQUATION: {\displaystyle \Delta B}
SKIPPED EQUATION: C ~WIKIEQUATION: {\displaystyle C}
SKIPPED EQUATION: B ~WIKIEQUATION: {\displaystyle B}
SKIPPED EQUATION: \DeltaB ~WIKIEQUATION: {\displaystyle \Delta B}
SKIPPED EQUATION: B ~WIKIEQUATION: {\displaystyle B}
SKIPPED EQUATION: \DeltaB ~WIKIEQUATION: {\displaystyle \Delta B}
SKIPPED EQUATION: B ~WIKIEQUATION: {\displaystyle B}
SKIPPED EQUATION: \Delta ~WIKIEQUATION: {\displaystyle \Delta }
SKIPPED EQUATION: \Delta ~WIKIEQUATION: {\displaystyle \Delta }
SKIPPED EQUATION: \Delta ~WIKIEQUATION: {\displaystyle \Delta }
SKIPPED EQUATION: \Delta ~WIKIEQUATION: {\displaystyle \Delta }
SKIPPED EQUATION: \Delta ~WIKIEQUATION: {\displaystyle \Delta }
SKIPPED EQUATION: \Delta ~WIKIEQUATION: {\displaystyle \Delta }
SKIPPED EQUATION: \Delta ~WIKIEQUATION: {\displaystyle \Delta }
UNPARSED EQUATION: {\x{07^{8}*03^{4}}{07^{8}*03^{4}+03^{8}*07^{4}}} ~WIKIEQUATION: {\displaystyle {\frac {0.7^{8}\times 0.3^{4}}{0.7^{8}\times 0.3^{4}+0.3^{8}\times 0.7^{4}}}}
SKIPPED EQUATION: \alpha_{x}}(V)*(1-m)-\beta_{x}}(V)*m ~WIKIEQUATION: {\displaystyle {\frac {dm(t,V)}{dt}}={\frac {m_{\infty }(V)-m(t,V)}{\tau _{\mathrm {m} }(V)}}=\alpha _{\mathrm {m} }(V)\cdot (1-m)-\beta _{\mathrm {m} }(V)\cdot m}
SKIPPED EQUATION: I_{x}}\[t_{x}}-R_{x}}C_{x}}\log(1-{\x{V_{x}}}{x_{x}}}})]^{-1} ~WIKIEQUATION: {\displaystyle f(I)={\begin{cases}0,&I\leq I_{\mathrm {th} }\\\left[t_{\mathrm {ref} }-R_{\mathrm {m} }C_{\mathrm {m} }\log \left(1-{\tfrac {V_{\mathrm {th} }}{IR_{\mathrm {m} }}}\right)\right]^{-1},&I>I_{\mathrm {th} }\end{cases}}}
SKIPPED EQUATION: \tau_{x} ~WIKIEQUATION: {\displaystyle \tau _{m}}
SKIPPED EQUATION: \tau_{x} ~WIKIEQUATION: {\displaystyle \tau _{k}}
SKIPPED EQUATION: C_{x}}{\x{d^{\alpha}V_{x}}(t)}{d^{\alpha}t}} ~WIKIEQUATION: {\displaystyle I(t)-{\frac {V_{\mathrm {m} }(t)}{R_{\mathrm {m} }}}=C_{\mathrm {m} }{\frac {d^{\alpha }V_{\mathrm {m} }(t)}{d^{\alpha }t}}}
SKIPPED EQUATION: V ~WIKIEQUATION: {\displaystyle V}
SKIPPED EQUATION: V_{x} ~WIKIEQUATION: {\displaystyle V_{T}}
SKIPPED EQUATION: \tau_{x} ~WIKIEQUATION: {\displaystyle \tau _{m}}
SKIPPED EQUATION: E_{x} ~WIKIEQUATION: {\displaystyle E_{m}}
SKIPPED EQUATION: \Delta_{x} ~WIKIEQUATION: {\displaystyle \Delta _{T}}
SKIPPED EQUATION: V_{x} ~WIKIEQUATION: {\displaystyle V_{T}}
SKIPPED EQUATION: V_{x} ~WIKIEQUATION: {\displaystyle V_{T}}
SKIPPED EQUATION: \tau ~WIKIEQUATION: {\displaystyle \tau }
SKIPPED EQUATION: V_{x} ~WIKIEQUATION: {\displaystyle V_{T}}
SKIPPED EQUATION: \tau ~WIKIEQUATION: {\displaystyle \tau }
SKIPPED EQUATION: \tau_{x} ~WIKIEQUATION: {\displaystyle \tau _{m}}
SKIPPED EQUATION: \sigma ~WIKIEQUATION: {\displaystyle \sigma }
SKIPPED EQUATION: I_{x} ~WIKIEQUATION: {\displaystyle I(t)=I_{0}}
SKIPPED EQUATION: I_{x} ~WIKIEQUATION: {\displaystyle I_{0}}
SKIPPED EQUATION: V_{x} ~WIKIEQUATION: {\displaystyle V_{th}}
SKIPPED EQUATION: V_{x} ~WIKIEQUATION: {\displaystyle V_{th}}
SKIPPED EQUATION: f ~WIKIEQUATION: {\displaystyle f}
SKIPPED EQUATION: \tau_{x} ~WIKIEQUATION: {\displaystyle \tau _{0}}
SKIPPED EQUATION: \beta ~WIKIEQUATION: {\displaystyle \beta }
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \beta \to \infty }
SKIPPED EQUATION: t_{x} ~WIKIEQUATION: {\displaystyle t_{n}}
SKIPPED EQUATION: V_{x} ~WIKIEQUATION: {\displaystyle V_{th}}
UNPARSED EQUATION: 05[1+\tanh(\x\max)] ~WIKIEQUATION: {\displaystyle F(x)=0.5[1+\tanh(\gamma x)]}
SKIPPED EQUATION: \gamma ~WIKIEQUATION: {\displaystyle \gamma }
SKIPPED EQUATION: f ~WIKIEQUATION: {\displaystyle f}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle V(t)=\sum _{f}\eta (t-t^{f})+\int _{0}^{\infty }\kappa (s)I(t-s)\,ds+V_{\mathrm {rest} }}
SKIPPED EQUATION: \tau_{x} ~WIKIEQUATION: {\displaystyle \tau _{0}}
SKIPPED EQUATION: \beta ~WIKIEQUATION: {\displaystyle \beta }
SKIPPED EQUATION: \vartheta_{x} ~WIKIEQUATION: {\displaystyle \vartheta _{0}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \beta \to \infty }
SKIPPED EQUATION: w_{x} ~WIKIEQUATION: {\displaystyle w_{ij}}
SKIPPED EQUATION: \varepsilon_{x} ~WIKIEQUATION: {\displaystyle \varepsilon _{ij}(s)}
SKIPPED EQUATION: {{t}} ~WIKIEQUATION: {\displaystyle {\hat {t}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle V(t)=\eta (t-{\hat {t}})+\int _{0}^{\infty }\kappa (s)I(t-s)\,ds+V_{\mathrm {rest} }}
SKIPPED EQUATION: \eta_{x}(t-{{t}}_{x})+\Sum(x)w_{x}\Sum(x)\varepsilon_{x}((x))+V_{x}} ~WIKIEQUATION: {\displaystyle V_{i}(t\mid {\hat {t}}_{i})=\eta _{i}(t-{\hat {t}}_{i})+\sum _{j}w_{ij}\sum _{f}\varepsilon _{ij}(t-{\hat {t}}_{i},t-t^{f})+V_{\mathrm {rest} }}
SKIPPED EQUATION: {{t}}_{x} ~WIKIEQUATION: {\displaystyle {\hat {t}}_{i}}
SKIPPED EQUATION: \varepsilon_{x} ~WIKIEQUATION: {\displaystyle \varepsilon _{ij}}
SKIPPED EQUATION: V_{x} ~WIKIEQUATION: {\displaystyle V_{th}}
SKIPPED EQUATION: \eta ~WIKIEQUATION: {\displaystyle \eta }
SKIPPED EQUATION: \eta ~WIKIEQUATION: {\displaystyle \eta }
SKIPPED EQUATION: i ~WIKIEQUATION: {\displaystyle i}
SKIPPED EQUATION: t ~WIKIEQUATION: {\displaystyle t}
SKIPPED EQUATION: W_{x} ~WIKIEQUATION: {\displaystyle W_{j\rightarrow i}}
SKIPPED EQUATION: j ~WIKIEQUATION: {\displaystyle j}
SKIPPED EQUATION: i ~WIKIEQUATION: {\displaystyle i}
SKIPPED EQUATION: g_{x} ~WIKIEQUATION: {\displaystyle g_{j}}
SKIPPED EQUATION: i ~WIKIEQUATION: {\displaystyle i}
SKIPPED EQUATION: t ~WIKIEQUATION: {\displaystyle t}
SKIPPED EQUATION: \x(s ~WIKIEQUATION: {\displaystyle L_{t}^{i}=\sup\{s<t:X_{s}(i)=1\}.}
SKIPPED EQUATION: 1) ~WIKIEQUATION: {\displaystyle L_{t}^{i}=\sup\{s<t:X_{s}(i)=1\}.}
SKIPPED EQUATION: g_{x} ~WIKIEQUATION: {\displaystyle g_{j}}
SKIPPED EQUATION: W_{x} ~WIKIEQUATION: {\displaystyle W_{j\to i}}
SKIPPED EQUATION: {{g}}_{x}}m_{x}*(V-V_{x}})+{{g}}_{x}}w*(V-V_{x}})+{{g}}_{x}}*(V-V_{x}}) ~WIKIEQUATION: {\displaystyle I_{\mathrm {ion} }(V,w)={\bar {g}}_{\mathrm {Ca} }m_{\infty }\cdot (V-V_{\mathrm {Ca} })+{\bar {g}}_{\mathrm {K} }w\cdot (V-V_{\mathrm {K} })+{\bar {g}}_{\mathrm {L} }\cdot (V-V_{\mathrm {L} })}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle P_{\text{spike}}(t\in [t',t'+\Delta _{t}])=\Delta _{t}\cdot g[s(t)]}
UNPARSED EQUATION: {\x{P_{x}(t ~WIKIEQUATION: {\displaystyle R_{\text{fire}}(t)={\frac {P_{\text{spike}}(t;\Delta _{t})}{\Delta _{t}}}=[y(t)+R_{0}]\cdot P_{0}(t)}
SKIPPED EQUATION: g_{x}*((x) ~WIKIEQUATION: {\displaystyle y(t)\simeq g_{\text{gain}}\cdot \langle s^{2}(t)\rangle ,}
SKIPPED EQUATION: {{g}}_{x}}*[O]*(V(t)-E_{x}}) ~WIKIEQUATION: {\displaystyle I_{\mathrm {AMPA} }(t,V)={\bar {g}}_{\mathrm {AMPA} }\cdot [O]\cdot (V(t)-E_{\mathrm {AMPA} })}
SKIPPED EQUATION: {{g}}_{x}}*B(V)*[O]*(V(t)-E_{x}}) ~WIKIEQUATION: {\displaystyle I_{\mathrm {NMDA} }(t,V)={\bar {g}}_{\mathrm {NMDA} }\cdot B(V)\cdot [O]\cdot (V(t)-E_{\mathrm {NMDA} })}
SKIPPED EQUATION: {{g}}_{x}}}*([O_{x}]+[O_{x}])*(V(t)-E_{x}}) ~WIKIEQUATION: {\displaystyle I_{\mathrm {GABA_{A}} }(t,V)={\bar {g}}_{\mathrm {GABA_{A}} }\cdot ([O_{1}]+[O_{2}])\cdot (V(t)-E_{\mathrm {Cl} })}
SKIPPED EQUATION: {{g}}_{x}}}*{\x{[G]^{n}}{[G]^{n}+K_{x}}}}*(V(t)-E_{x}}) ~WIKIEQUATION: {\displaystyle I_{\mathrm {GABA_{B}} }(t,V)={\bar {g}}_{\mathrm {GABA_{B}} }\cdot {\tfrac {[G]^{n}}{[G]^{n}+K_{\mathrm {d} }}}\cdot (V(t)-E_{\mathrm {K} })}
SKIPPED EQUATION: \varphi ~WIKIEQUATION: {\displaystyle \varphi }
SKIPPED EQUATION: \varphi ~WIKIEQUATION: {\displaystyle \varphi }
SKIPPED EQUATION: (\lambda) ~WIKIEQUATION: {\displaystyle \lambda }
SKIPPED EQUATION: r_{x} ~WIKIEQUATION: {\displaystyle r_{l}}
SKIPPED EQUATION: r_{x} ~WIKIEQUATION: {\displaystyle r_{m}}
SKIPPED EQUATION: {r_{x}}/{r_{x}} ~WIKIEQUATION: {\displaystyle \lambda ^{2}={r_{m}}/{r_{l}}}
SKIPPED EQUATION: (x) ~WIKIEQUATION: {\displaystyle G_{N}=G_{S}+\sum _{j=1}^{n}A_{D_{j}}F_{dga_{j}},}
UNPARSED EQUATION: {\x{B_{x}(x) ~WIKIEQUATION: {\displaystyle B_{\mathrm {out} ,i}={\frac {B_{\mathrm {in} ,i+1}(d_{i+1}/d_{i})^{3/2}}{\sqrt {R_{\mathrm {m} ,i+1}/R_{\mathrm {m} ,i}}}}}
UNPARSED EQUATION: {\x{B_{x}}(d_{x}}/d_{x}})^{3/2}}{\sqrt{R_{x}}/R_{x}}}}}+{\x{B_{x}}(d_{x}}/d_{x}})^{3/2}}{\sqrt{R_{x}}/R_{x}}}}}+\x ~WIKIEQUATION: {\displaystyle B_{\mathrm {out,par} }={\frac {B_{\mathrm {in,dau1} }(d_{\mathrm {dau1} }/d_{\mathrm {par} })^{3/2}}{\sqrt {R_{\mathrm {m,dau1} }/R_{\mathrm {m,par} }}}}+{\frac {B_{\mathrm {in,dau2} }(d_{\mathrm {dau2} }/d_{\mathrm {par} })^{3/2}}{\sqrt {R_{\mathrm {m,dau2} }/R_{\mathrm {m,par} }}}}+\ldots }
SKIPPED EQUATION: (x) ~WIKIEQUATION: {\displaystyle G_{N}={\frac {A_{\mathrm {soma} }}{R_{\mathrm {m,soma} }}}+\sum _{j}B_{\mathrm {in,stem} ,j}G_{\infty ,j}.}
SKIPPED EQUATION: f ~WIKIEQUATION: {\displaystyle f}
SKIPPED EQUATION: g ~WIKIEQUATION: {\displaystyle g}
SKIPPED EQUATION: {{x}} ~WIKIEQUATION: {\displaystyle {\dot {x}}}
SKIPPED EQUATION: {{u}} ~WIKIEQUATION: {\displaystyle {\dot {u}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \mu \ll 1}
SKIPPED EQUATION: x_{x}-x_{x}+c ~WIKIEQUATION: {\displaystyle {\begin{aligned}x_{n+1}=&f(x_{n},y_{n})=x_{n}^{2}\exp {(y_{n}-x_{n})}+k\\y_{n+1}=&g(x_{n},y_{n})=ay_{n}-bx_{n}+c\\\end{aligned}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle x}
SKIPPED EQUATION: y ~WIKIEQUATION: {\displaystyle y}
SKIPPED EQUATION: k ~WIKIEQUATION: {\displaystyle k}
SKIPPED EQUATION: a ~WIKIEQUATION: {\displaystyle a}
UNPARSED EQUATION: (a ~WIKIEQUATION: {\displaystyle (a<1)}
SKIPPED EQUATION: 1) ~WIKIEQUATION: {\displaystyle (a<1)}
SKIPPED EQUATION: b ~WIKIEQUATION: {\displaystyle b}
UNPARSED EQUATION: (b ~WIKIEQUATION: {\displaystyle (b<1)}
SKIPPED EQUATION: 1) ~WIKIEQUATION: {\displaystyle (b<1)}
SKIPPED EQUATION: c ~WIKIEQUATION: {\displaystyle c}
SKIPPED EQUATION: a ~WIKIEQUATION: {\displaystyle a=0.89}
SKIPPED EQUATION: c ~WIKIEQUATION: {\displaystyle c=0.28}
SKIPPED EQUATION: k ~WIKIEQUATION: {\displaystyle k=0.025}
SKIPPED EQUATION: 06 ~WIKIEQUATION: {\displaystyle 0.6}
SKIPPED EQUATION: 018 ~WIKIEQUATION: {\displaystyle 0.18}
SKIPPED EQUATION: k ~WIKIEQUATION: {\displaystyle k=0}
SKIPPED EQUATION: b ~WIKIEQUATION: {\displaystyle b<<a}
SKIPPED EQUATION: a ~WIKIEQUATION: {\displaystyle b<<a}
SKIPPED EQUATION: y_{x} ~WIKIEQUATION: {\displaystyle y_{f0}}
SKIPPED EQUATION: r ~WIKIEQUATION: {\displaystyle r}
SKIPPED EQUATION: N ~WIKIEQUATION: {\displaystyle N}
SKIPPED EQUATION: 0 ~WIKIEQUATION: {\displaystyle 0>d<1}
SKIPPED EQUATION: d ~WIKIEQUATION: {\displaystyle 0>d<1}
SKIPPED EQUATION: 1 ~WIKIEQUATION: {\displaystyle 0>d<1}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle x}
SKIPPED EQUATION: n ~WIKIEQUATION: {\displaystyle n}
SKIPPED EQUATION: i ~WIKIEQUATION: {\displaystyle i}
SKIPPED EQUATION: a ~WIKIEQUATION: {\displaystyle a=0.89}
SKIPPED EQUATION: b ~WIKIEQUATION: {\displaystyle b=0.6}
SKIPPED EQUATION: c ~WIKIEQUATION: {\displaystyle c=0.28}
SKIPPED EQUATION: k ~WIKIEQUATION: {\displaystyle k=0.02}
SKIPPED EQUATION: i ~WIKIEQUATION: {\displaystyle i}
SKIPPED EQUATION: j ~WIKIEQUATION: {\displaystyle j}
SKIPPED EQUATION: I ~WIKIEQUATION: {\displaystyle I}
SKIPPED EQUATION: J ~WIKIEQUATION: {\displaystyle J}
UNPARSED EQUATION: i*00033 ~WIKIEQUATION: {\displaystyle x^{ij}=i*0.0033}
UNPARSED EQUATION: y_{x}-(j*00066) ~WIKIEQUATION: {\displaystyle y^{ij}=y_{f}-(j*0.0066)}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle x}
SKIPPED EQUATION: 500*500 ~WIKIEQUATION: {\displaystyle 500\times 500}
SKIPPED EQUATION: a ~WIKIEQUATION: {\displaystyle a=0.89}
SKIPPED EQUATION: b ~WIKIEQUATION: {\displaystyle b=0.18}
SKIPPED EQUATION: c ~WIKIEQUATION: {\displaystyle c=0.28}
SKIPPED EQUATION: k ~WIKIEQUATION: {\displaystyle k=0.026}
SKIPPED EQUATION: p ~WIKIEQUATION: {\displaystyle p}
SKIPPED EQUATION: p ~WIKIEQUATION: {\displaystyle p=0.25}
SKIPPED EQUATION: b ~WIKIEQUATION: {\displaystyle b=0}
SKIPPED EQUATION: y ~WIKIEQUATION: {\displaystyle y}
SKIPPED EQUATION: b ~WIKIEQUATION: {\displaystyle b}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle x=1}
SKIPPED EQUATION: y ~WIKIEQUATION: {\displaystyle y=1}
SKIPPED EQUATION: k ~WIKIEQUATION: {\displaystyle k}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle x}
SKIPPED EQUATION: b ~WIKIEQUATION: {\displaystyle b}
SKIPPED EQUATION: a ~WIKIEQUATION: {\displaystyle a=0.89}
SKIPPED EQUATION: c ~WIKIEQUATION: {\displaystyle c=0.28}
SKIPPED EQUATION: k ~WIKIEQUATION: {\displaystyle k=0.026}
SKIPPED EQUATION: b ~WIKIEQUATION: {\displaystyle b}
SKIPPED EQUATION: 016 ~WIKIEQUATION: {\displaystyle 0.16}
SKIPPED EQUATION: 04 ~WIKIEQUATION: {\displaystyle 0.4}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle x}
SKIPPED EQUATION: b ~WIKIEQUATION: {\displaystyle b}
SKIPPED EQUATION: a ~WIKIEQUATION: {\displaystyle a=0.89}
SKIPPED EQUATION: c ~WIKIEQUATION: {\displaystyle c=0.28}
SKIPPED EQUATION: k ~WIKIEQUATION: {\displaystyle k=0.026}
SKIPPED EQUATION: b ~WIKIEQUATION: {\displaystyle b}
SKIPPED EQUATION: 016 ~WIKIEQUATION: {\displaystyle 0.16}
SKIPPED EQUATION: 04 ~WIKIEQUATION: {\displaystyle 0.4}
SKIPPED EQUATION: a_{x} ~WIKIEQUATION: {\displaystyle a_{i}}
SKIPPED EQUATION: L_{x} ~WIKIEQUATION: {\displaystyle L_{i}}
SKIPPED EQUATION: V_{x} ~WIKIEQUATION: {\displaystyle V_{i}}
SKIPPED EQUATION: c_{x} ~WIKIEQUATION: {\displaystyle c_{i}}
SKIPPED EQUATION: r_{x} ~WIKIEQUATION: {\displaystyle r_{Mi}}
SKIPPED EQUATION: r_{x} ~WIKIEQUATION: {\displaystyle r_{L}}
SKIPPED EQUATION: g_{x} ~WIKIEQUATION: {\displaystyle g_{1,2}}
SKIPPED EQUATION: g_{x} ~WIKIEQUATION: {\displaystyle g_{2,1}}
SKIPPED EQUATION: r_{x} ~WIKIEQUATION: {\displaystyle r_{1}=r_{2}\equiv r}
SKIPPED EQUATION: r_{x} ~WIKIEQUATION: {\displaystyle r_{M}=r_{M1}=r_{M2}}
SKIPPED EQUATION: ((x)) ~WIKIEQUATION: {\displaystyle ({\mathcal {G}},{\mathcal {M}},{\mathcal {P}})\ ,}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle g\in {\mathcal {G}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle m\in {\mathcal {M}}}
SKIPPED EQUATION: P ~WIKIEQUATION: {\displaystyle P}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle m_{\mathrm {temp} }\in {\mathcal {M}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle m\in {\mathcal {M}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle ({\mathcal {G}},\circ )}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \circ }
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle g\cdot m\in {\mathcal {M}},m\in {\mathcal {M}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle (g\circ g^{\prime })\cdot m=g\cdot (g^{\prime }\cdot m)\in {\mathcal {M}}.}
SKIPPED EQUATION: {{M}} ~WIKIEQUATION: {\displaystyle {\mathcal {M}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\mathcal {M}}\doteq \{m=g\cdot m_{\mathrm {temp} },g\in {\mathcal {G}}\}}
SKIPPED EQUATION: {{G}} ~WIKIEQUATION: {\displaystyle {\mathcal {G}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle m(u),u\in U\subset {\mathbb {R} }^{1}\rightarrow {\mathbb {R} }^{2}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle X\doteq \{x_{1},\dots ,x_{n}\}\subset {\mathbb {R} }^{3}\in {\mathcal {M}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle X\subset {\mathbb {R} }^{3}\in {\mathcal {M}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle m:U\subset {\mathbb {R} }^{1,2}\rightarrow {\mathbb {R} }^{3}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle m(u),u\in U}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle I\in {\mathcal {M}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle I(x),x\in X\subset {\mathbb {R} }^{1,2,3}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle I(x),x\in {\mathbb {R} }^{2}}
SKIPPED EQUATION: A ~WIKIEQUATION: {\displaystyle A}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle x\in {\mathbb {R} }^{n}}
SKIPPED EQUATION: n*1 ~WIKIEQUATION: {\displaystyle n\times 1}
SKIPPED EQUATION: n ~WIKIEQUATION: {\displaystyle n}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle y=A\cdot x\in {\mathbb {R} }^{n}}
SKIPPED EQUATION: (x) ~WIKIEQUATION: {\displaystyle \phi (\cdot )=(\phi _{1}(\cdot ),\phi _{2}(\cdot ),\phi _{3}(\cdot ))}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \phi \circ \phi ^{\prime }(\cdot )\doteq \phi (\phi ^{\prime }(\cdot ))}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \phi \circ \phi ^{-1}(\cdot )=\phi (\phi ^{-1}(\cdot ))=id}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle I(x),x\in {\mathbb {R} }^{3}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \phi \cdot I(x)=I\circ \phi ^{-1}(x),x\in {\mathbb {R} }^{3}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle X\subset {\mathbb {R} }^{3}\in {\mathcal {M}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle m(u),u\in U}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \phi \cdot m(u)\doteq \phi \circ m(u),u\in U}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \phi _{t},t\in [0,1]}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle x\in X}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle v_{t},t\in [0,1]}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\dot {\phi }}_{t}=v_{t}(\phi _{t}),\phi _{0}=id}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\frac {d}{dt}}\phi _{t}=v_{t}\circ \phi _{t},\ \phi _{0}=id\ ;}
SKIPPED EQUATION: \phi ~WIKIEQUATION: {\displaystyle \phi }
SKIPPED EQUATION: {{\phi}}_{x} ~WIKIEQUATION: {\displaystyle {\dot {\phi }}_{t}(x)}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle x}
SKIPPED EQUATION: t ~WIKIEQUATION: {\displaystyle t}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \phi _{t}^{-1},t\in [0,1]}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\frac {d}{dt}}\phi _{t}^{-1}=-(D\phi _{t}^{-1})v_{t},\ \phi _{0}^{-1}=id\ .}
SKIPPED EQUATION: ((x)) ~WIKIEQUATION: {\displaystyle (V,\|\cdot \|_{V})}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle v_{i}\in H_{0}^{3},i=1,2,3,}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle Diff_{V}\doteq \{\varphi =\phi _{1}:{\dot {\phi }}_{t}=v_{t}\circ \phi _{t},\phi _{0}=id,\int _{0}^{1}\|v_{t}\|_{V}dt<\infty \}\ ,}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \|v\|_{V}^{2}\doteq \int _{X}Av\cdot vdx,\ v\in V\ ,}
SKIPPED EQUATION: A ~WIKIEQUATION: {\displaystyle A}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle A:V\mapsto V^{*}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle Av\in V^{*}}
SKIPPED EQUATION: ((x)) ~WIKIEQUATION: {\displaystyle (V,\|\cdot \|_{V})}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \sigma (v)\doteq Av\in V^{*}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle (\sigma |w)\doteq \int _{{\mathbb {R} }^{3}}\sum _{i}w_{i}(x)\sigma _{i}(dx)}
SKIPPED EQUATION: ((x)) ~WIKIEQUATION: {\displaystyle (V,\|\cdot \|_{V})}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \langle v,w\rangle _{V}\doteq \int _{X}Av\cdot wdx,\ \|v\|_{V}^{2}\doteq \int _{X}Av\cdot vdx,\ v,w\in V\ .}
SKIPPED EQUATION: A ~WIKIEQUATION: {\displaystyle A}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle (Av|v)<\infty }
SKIPPED EQUATION: A ~WIKIEQUATION: {\displaystyle A}
SKIPPED EQUATION: ((x)) ~WIKIEQUATION: {\displaystyle (V,\|\cdot \|_{V})}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle K\sigma (x)_{i}\doteq \sum _{j}\int _{{\mathbb {R} }^{3}}k_{ij}(x,y)\sigma _{j}(dy)}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle (\sigma |v)\doteq \int v\cdot \mu dx}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \phi \cdot m\in {\mathcal {M}},\phi \in Diff_{V},m\in {\mathcal {M}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle d_{Diff_{V}}(\psi ,\varphi )=\inf _{v_{t}}\left(\int _{0}^{1}\int _{X}Av_{t}\cdot v_{t}dx\ dt:\phi _{0}=\psi ,\phi _{1}=\varphi ,{\dot {\phi }}_{t}=v_{t}\circ \phi _{t}\right)^{1/2}\ ;}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \phi \in Diff_{V}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle d_{Diff_{V}}(\psi ,\varphi )=d_{Diff_{V}}(\psi \circ \phi ,\varphi \circ \phi )}
SKIPPED EQUATION: d_{x}}:{{M}}*{{M}}x{R}^{+} ~WIKIEQUATION: {\displaystyle d_{\mathcal {M}}:{\mathcal {M}}\times {\mathcal {M}}\rightarrow \mathbb {R} ^{+}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle d_{\mathcal {M}}(m,n)=\inf _{\phi \in \operatorname {Diff} _{V}:\phi \cdot m=n}d_{\operatorname {Diff} _{V}}(id,\phi )\ ;}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle I\in {\mathcal {I}}}
SKIPPED EQUATION: d_{x}} ~WIKIEQUATION: {\displaystyle ,d_{\mathcal {I}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle v\doteq {\dot {\phi }}\circ \phi ^{-1}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle J(\phi )\doteq \int _{0}^{1}L(\phi _{t},{\dot {\phi }}_{t})dt\ ;}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle L(\phi _{t},{\dot {\phi }}_{t})\doteq {\frac {1}{2}}\int _{X}A({\dot {\phi }}_{t}\circ \phi _{t}^{-1})\cdot ({\dot {\phi }}_{t}\circ \phi _{t}^{-1})dx={\frac {1}{2}}\int _{X}Av_{t}\cdot v_{t}\ dx\ .}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle Av}
SKIPPED EQUATION: v ~WIKIEQUATION: {\displaystyle v}
SKIPPED EQUATION: A ~WIKIEQUATION: {\displaystyle A}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle ad_{v}:w\in V\mapsto V}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle ad_{v}[w]\doteq [v,w]\doteq (Dv)w-(Dw)v\in V}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle Av\in V^{*}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\frac {d}{dt}}Av_{t}+ad_{v_{t}}^{*}(Av_{t})=0\ ,\ t\in [0,1]\ ;}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle w\in V,}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \int _{X}\left({\frac {d}{dt}}Av_{t}+ad_{v_{t}}^{*}(Av_{t})\right)\cdot wdx=\int _{X}{\frac {d}{dt}}Av_{t}\cdot wdx+\int _{X}Av_{t}\cdot ((Dv_{t})w-(Dw)v_{t})dx=0.}
SKIPPED EQUATION: 2 ~WIKIEQUATION: {\displaystyle \leq 2}
SKIPPED EQUATION: (x) ~WIKIEQUATION: {\displaystyle (Av_{t}\mid v_{t})}
SKIPPED EQUATION: x_{x} ~WIKIEQUATION: {\displaystyle Av_{t}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle Av\in V^{*}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle A=identity}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle (Av_{t}\mid w)=\int _{X}\mu _{t}\cdot w\,dx}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\frac {d}{dt}}\mu _{t}+(Dv_{t})^{T}\mu _{t}+(D\mu _{t})v_{t}+(\nabla \cdot v)\mu _{t}=0\ ,t\in [0,1].}
SKIPPED EQUATION: v_{x} ~WIKIEQUATION: {\displaystyle v_{0}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle Exp_{\rm {id}}(\cdot ):V\to Diff_{V}}
SKIPPED EQUATION: \phi_{x} ~WIKIEQUATION: {\displaystyle Exp_{id}(v_{0})=\phi _{1}}
SKIPPED EQUATION: v_{x} ~WIKIEQUATION: {\displaystyle {\dot {\phi }}_{0}=v_{0}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\dot {\phi }}_{t}=v_{t}\circ \phi _{t},t\in [0,1]}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \int _{X}Av_{t}\cdot w\,dx}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle Av\in V}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \ \ \ {\frac {d}{dt}}Av_{t}+(Dv_{t})^{T}Av_{t}+(DAv_{t})v_{t}+(\nabla \cdot v)Av_{t}=0\ ;}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle Av\in V^{*}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle w\in V}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \ \ \ \int _{X}{\frac {d}{dt}}Av_{t}\cdot wdx+\int _{X}Av_{t}\cdot ((Dv_{t})w-(Dw)v_{t})dx=0.}
SKIPPED EQUATION: v_{x} ~WIKIEQUATION: {\displaystyle v_{0}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle Log_{\rm {id}}(\cdot ):Diff_{V}\to V}
SKIPPED EQUATION: \varphi ~WIKIEQUATION: {\displaystyle \varphi }
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle v_{0}\in V}
SKIPPED EQUATION: \varphi ~WIKIEQUATION: {\displaystyle Log_{id}(\varphi )=v_{0}\ {\text{initial condition of EL geodesic }}{\dot {\phi }}_{0}=v_{0},\phi _{0}=id,\phi _{1}=\varphi \ .}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \phi =Exp_{\varphi }(v_{0}\circ \varphi )\doteq Exp_{id}(v_{0})\circ \varphi }
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle Log_{\varphi }(\phi )\doteq Log_{id}(\phi \circ \varphi ^{-1})\circ \varphi }
SKIPPED EQUATION: v_{x}_{x} ~WIKIEQUATION: {\displaystyle \|v_{0}\|_{V}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle t\mapsto \phi _{t}\in \operatorname {Diff} _{V}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle t\mapsto v_{t}\in V}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\dot {\phi }}_{t}=v_{t}\cdot \phi _{t},\phi _{0}=id.}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle Av\in V^{*}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle p:{\dot {\phi }}\mapsto (p\mid {\dot {\phi }})}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\dot {\phi }}_{t}=v_{t}\circ \phi _{t}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle H(\phi _{t},p_{t},v_{t})=\int _{X}p_{t}\cdot (v_{t}\circ \phi _{t})dx-{\frac {1}{2}}\int _{X}Av_{t}\cdot v_{t}dx.}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\dot {\phi }}_{t}=v_{t}\circ \phi _{t},\phi _{0}=id,}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle dx}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle Av_{t}\in V^{*}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle t\in [0,1]}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle H(\phi _{t},p_{t})=H(\phi _{0},p_{0})={\frac {1}{2}}\int _{X}p_{0}\cdot v_{0}dx={\frac {1}{2}}\int _{X}Av_{0}\cdot v_{0}dx={\frac {1}{2}}\int _{X}Av_{t}\cdot v_{t}dx}
SKIPPED EQUATION: {{\phi}} ~WIKIEQUATION: {\displaystyle {\dot {\phi }}}
SKIPPED EQUATION: \phi ~WIKIEQUATION: {\displaystyle \phi }
SKIPPED EQUATION: p ~WIKIEQUATION: {\displaystyle p}
SKIPPED EQUATION: v_{x} ~WIKIEQUATION: {\displaystyle v_{0}}
SKIPPED EQUATION: x_{x} ~WIKIEQUATION: {\displaystyle Av_{0}}
SKIPPED EQUATION: t ~WIKIEQUATION: {\displaystyle t=0}
SKIPPED EQUATION: p_{x} ~WIKIEQUATION: {\displaystyle p_{0}}
SKIPPED EQUATION: t ~WIKIEQUATION: {\displaystyle t=0}
SKIPPED EQUATION: x_{x} ~WIKIEQUATION: {\displaystyle v_{0}=Kp_{0}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle Av}
SKIPPED EQUATION: v ~WIKIEQUATION: {\displaystyle v}
SKIPPED EQUATION: A ~WIKIEQUATION: {\displaystyle A}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\begin{matrix}{\text{Eulerian}}&\ \ \ \ {\frac {d}{dt}}\int _{X}Av_{t}\cdot ((D\phi _{t})w)\circ \phi _{t}^{-1})dx=0\ ,\ t\in [0,1].\\&\\{\text{Canonical}}&\ \ \ \ \ \ \ \ \ \ \ {\frac {d}{dt}}\int _{X}p_{t}\cdot ((D\phi _{t})w)dx=0\ ,\ t\in [0,1]\ {\text{ for all}}\ w\in V\ .\end{matrix}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle w_{t}=((D\phi _{t})w)\circ \phi _{t}^{-1}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\frac {d}{dt}}(Av_{t}|((D\phi _{t})w)\circ \phi _{t}^{-1})=({\frac {d}{dt}}Av_{t}|((D\phi _{t})w)\circ \phi _{t}^{-1})+(Av_{t}|{\frac {d}{dt}}((D\phi _{t})w)\circ \phi _{t}^{-1})=({\frac {d}{dt}}Av_{t}|w_{t})+(Av_{t}|(Dv_{t})w_{t}-(Dw_{t})v_{t})=0.}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle v_{0}\in V}
SKIPPED EQUATION: p_{x} ~WIKIEQUATION: {\displaystyle p_{0}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\frac {1}{2}}\int _{0}^{1}\int _{X}Av_{t}\cdot v_{t}dxdt}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle t\in [0,1)}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\text{min}}_{\phi :v={\dot {\phi }}\circ \phi ^{-1},\phi _{0}=id}C(\phi )\doteq {\frac {1}{2}}\int _{0}^{1}\int _{X}Av_{t}\cdot v_{t}dxdt+E(\phi _{1})}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\begin{cases}{\text{Euler Conservation}}\ \ \ \ \ \ \ \ &\ \ \ {\frac {d}{dt}}Av_{t}+ad_{v_{t}}^{*}(Av_{t})=0,\ t\in [0,1)\ ,\\{\text{Boundary Condition}}&\ \ \ \phi _{0}=id,Av_{1}=-{\frac {\partial E(\phi )}{\partial \phi }}|_{\phi =\phi _{1}}\ .\end{cases}}}
SKIPPED EQUATION: t ~WIKIEQUATION: {\displaystyle t=1}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle t\in [0,1)}
SKIPPED EQUATION: v_{x} ~WIKIEQUATION: {\displaystyle v_{0}}
SKIPPED EQUATION: x_{x} ~WIKIEQUATION: {\displaystyle v_{0}=Kp_{0}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \min _{v_{0}}C(v_{0})\doteq {\frac {1}{2}}\int _{X}Av_{0}\cdot v_{0}dx+E(\mathrm {Exp} _{\mathrm {id} }(v_{0})\cdot I_{0})\ ;}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \min _{p_{0}}C(p_{0})={\frac {1}{2}}\int _{X}p_{0}\cdot Kp_{0}dx+E(\mathrm {Exp} _{\text{id}}(Kp_{0})\cdot I_{0})}
SKIPPED EQUATION: v_{x} ~WIKIEQUATION: {\displaystyle v_{0}}
SKIPPED EQUATION: p_{x} ~WIKIEQUATION: {\displaystyle p_{0}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle q_{t}\doteq I\circ \phi _{t}^{-1},q_{0}=I}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle I(x),x\in X}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \phi \cdot I\doteq I\circ \phi ^{-1}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \phi _{t},t\in [0,1]}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle v_{t},t\in [0,1]}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\dot {\phi }}=v\circ \phi }
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle E(\phi _{1})\doteq \|I\circ \phi _{1}^{-1}-I^{\prime }\|^{2}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\begin{matrix}&\ \ \ \ \ \min _{v:{\dot {\phi }}=v\circ \phi }C(v)\doteq {\frac {1}{2}}\int _{0}^{1}\int _{X}Av_{t}\cdot v_{t}dxdt+{\frac {1}{2}}\int _{{\mathbb {R} }^{3}}|I\circ \phi _{1}^{-1}(x)-I^{\prime }(x)|^{2}dx\end{matrix}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\begin{cases}&{\text{Endpoint Condition:}}\ \ \ \ \ \ Av_{1}=\mu _{1}dx,\mu _{1}=(I\circ \phi _{1}^{-1}-I^{\prime })\nabla (I\circ \phi _{1}^{-1})\ ,\\&{\text{Conservation:}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Av_{t}=\mu _{t}\,dx,\ \mu _{t}=(D\phi _{t}^{-1})^{T}\mu _{0}\circ \phi _{t}^{-1}|D\phi _{t}^{-1}|\ .\\&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mu _{0}=(I-I^{\prime }\circ \phi _{1})\nabla I|D\phi _{1}|\ .\\\end{cases}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\begin{cases}&{\text{Endpoint Condition:}}\ \ \ \ \ \ Av_{1}=\mu _{1}dx,\mu _{1}=(I\circ \phi _{1}^{-1}-I^{\prime })\nabla (I\circ \phi _{1}^{-1})\ ,\\&{\text{Conservation:}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Av_{t}=\mu _{t}\,dx,\ \mu _{t}=(D\phi _{t}^{-1})^{T}\mu _{0}\circ \phi _{t}^{-1}|D\phi _{t}^{-1}|\ .\\&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mu _{0}=(I-I^{\prime }\circ \phi _{1})\nabla I|D\phi _{1}|\ .\\\end{cases}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle I(x),x\in X}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle q_{t}\doteq I\circ \phi _{t}^{-1}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\dot {q}}_{t}=-\nabla q_{t}\cdot v_{t}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\begin{matrix}&\ \ \ \ \ \min _{q:{\dot {q}}=v\circ q}C(v)\doteq {\frac {1}{2}}\int _{0}^{1}\int _{X}Av_{t}\cdot v_{t}dxdt+{\frac {1}{2}}\int _{{\mathbb {R} }^{3}}|q_{1}(x)-I^{\prime }(x)|^{2}dx\end{matrix}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle I(x),x\in {\mathbb {R} }^{2}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle M(x),x\in {\mathbb {R} }^{3}}
SKIPPED EQUATION: 3*3 ~WIKIEQUATION: {\displaystyle 3\times 3}
SKIPPED EQUATION: 3*3 ~WIKIEQUATION: {\displaystyle 3\times 3}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle M(x),x\in {\mathbb {R} }^{3}}
SKIPPED EQUATION: 123) ~WIKIEQUATION: {\displaystyle \{\lambda _{i}(x),e_{i}(x),i=1,2,3\}}
SKIPPED EQUATION: (\lambda)_{x} ~WIKIEQUATION: {\displaystyle \lambda _{1}\geq \lambda _{2}\geq \lambda _{3}}
SKIPPED EQUATION: (\lambda)_{x} ~WIKIEQUATION: {\displaystyle \lambda _{1}\geq \lambda _{2}\geq \lambda _{3}}
SKIPPED EQUATION: (\lambda)_{x} ~WIKIEQUATION: {\displaystyle \lambda _{1}\geq \lambda _{2}\geq \lambda _{3}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle I(x),x\in {\mathbb {R} }^{3}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \varphi \cdot M=(\lambda _{1}{\hat {e}}_{1}{\hat {e}}_{1}^{T}+\lambda _{2}{\hat {e}}_{2}{\hat {e}}_{2}^{T}+\lambda _{3}{\hat {e}}_{3}{\hat {e}}_{3}^{T})\circ \varphi ^{-1},}
SKIPPED EQUATION: {{e}}_{x}*{{e}}_{x} ~WIKIEQUATION: {\displaystyle {\begin{aligned}{\hat {e}}_{1}&={\frac {D\varphi e_{1}}{\|D\varphi e_{1}\|}}\ ,\ \ \ {\hat {e}}_{2}={\frac {D\varphi e_{2}-\langle {\hat {e}}_{1},D\varphi e_{2}\rangle {\hat {e}}_{1}}{\sqrt {\|D\varphi e_{2}\|^{2}-\langle {\hat {e}}_{1},D\varphi e_{2}\rangle ^{2}}}}\ ,\ \ \ {\hat {e}}_{3}={\hat {e}}_{1}\times {\hat {e}}_{2}\end{aligned}}}
SKIPPED EQUATION: n ~WIKIEQUATION: {\displaystyle n}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \int _{{\bf {s}}\in {\mathbb {S} }^{2}}\psi ^{2}({\bf {s}})d{\bf {s}}=1}
SKIPPED EQUATION: \phi_{x} ~WIKIEQUATION: {\displaystyle \phi _{t}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\dot {\phi }}_{t}=v_{t}(\phi _{t}),t\in [0,1],}
SKIPPED EQUATION: {x} ~WIKIEQUATION: {\displaystyle \phi _{0}={id}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\bf {s}}\in {{\mathbb {S} }^{2}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle x\in X}
SKIPPED EQUATION: \psi_{x}}((x)) ~WIKIEQUATION: {\displaystyle \psi _{\mathrm {targ} }({\bf {s}},x)}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\bf {s}}\in {{\mathbb {S} }^{2}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle x\in X}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \phi _{1}\cdot \psi (x)\doteq (D\phi _{1})\psi \circ \phi _{1}^{-1}(x),x\in X}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\begin{aligned}(D\phi _{1})\psi \circ \phi _{1}^{-1}(x)={\sqrt {\frac {\det {{\bigl (}D_{\phi _{1}^{-1}}\phi _{1}{\bigr )}^{-1}}}{\left\|{{\bigl (}D_{\phi _{1}^{-1}}\phi _{1}{\bigr )}^{-1}}{\bf {s}}\right\|^{3}}}}\quad \psi \left({\frac {(D_{\phi _{1}^{-1}}\phi _{1}{\bigr )}^{-1}{\bf {s}}}{\|(D_{\phi _{1}^{-1}}\phi _{1}{\bigr )}^{-1}{\bf {s}}\|}},\phi _{1}^{-1}(x)\right).\end{aligned}}}
SKIPPED EQUATION: \psi ~WIKIEQUATION: {\displaystyle \psi }
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\begin{aligned}C(v)=\inf _{v:{\dot {\phi }}_{t}=v_{t}\circ \phi _{t},\phi _{0}={id}}\int _{0}^{1}\int _{X}Av_{t}\cdot v_{t}dx\ dt+\lambda \int _{x\in \Omega }\|\log _{(D\phi _{1})\psi _{\mathrm {temp} }\circ \phi _{1}^{-1}(x)}(\psi _{\mathrm {targ} }(x))\|_{(D\phi _{1})\psi _{\mathrm {temp} }\circ \phi _{1}^{-1}(x)}^{2}dx\end{aligned}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \psi _{1},\psi _{2}\in \Psi }
SKIPPED EQUATION: (x) ~WIKIEQUATION: {\displaystyle {\begin{aligned}\|\log _{\psi _{1}}(\psi _{2})\|_{\psi _{1}}=\cos ^{-1}\langle \psi _{1},\psi _{2}\rangle =\cos ^{-1}\left(\int _{{\bf {s}}\in {\mathbb {S} }^{2}}\psi _{1}({\bf {s}})\psi _{2}({\bf {s}})d{\bf {s}}\right),\end{aligned}}}
SKIPPED EQUATION: ((x)) ~WIKIEQUATION: {\displaystyle \langle \cdot ,\cdot \rangle }
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle t\mapsto (\phi _{t},I_{t})}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \phi _{t}\cdot I_{t}\doteq I_{t}\circ \phi _{t}^{-1}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \min _{(v,I)}{\frac {1}{2}}\int _{0}^{1}\left(\int _{X}Av_{t}\cdot v_{t}dx+\|{\dot {I}}_{t}\circ \phi _{t}^{-1}\|^{2}/\sigma ^{2}\right)\,dt{\text{ subject to}}\ \phi _{0}=id,I_{0}={\text{fixed}},I_{1}={\text{fixed}}}
SKIPPED EQUATION: X ~WIKIEQUATION: {\displaystyle X}
SKIPPED EQUATION: 123 ~WIKIEQUATION: {\displaystyle d=1,2,3}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle m:u\in U\subset {\mathbb {R} }^{0,1,2,3}\rightarrow {\mathbb {R} }^{3}}
SKIPPED EQUATION: \varphi_{x} ~WIKIEQUATION: {\displaystyle \varphi _{t}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle X\doteq \{x_{1},\dots ,x_{n}\}\subset {\mathbb {R} }^{3}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\begin{matrix}&\ \ \ \min _{\phi :v={\dot {\phi }}\circ \phi ^{-1}}C(\phi )\doteq {\frac {1}{2}}\int (Av_{t}|v_{t})dt+{\frac {1}{2}}\sum _{i}\|\phi _{1}(x_{i})-x_{i}^{\prime }\|^{2}\end{matrix}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \displaystyle Av_{t}\in V^{*}\textstyle ,t\in [0,1]}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\begin{cases}&{\text{Endpoint Condition:}}\ \ \ \ \ Av_{1}=\sum _{i=1}^{n}p_{1}(i)\delta _{\phi _{1}(x_{i})},p_{1}(i)=(x_{i}^{\prime }-\phi _{1}(x_{i}))\ ,\\&{\text{Conservation:}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ Av_{t}=\sum _{i=1}^{n}p_{t}(i)\delta _{\phi _{t}(x_{i})},\ p_{t}(i)=(D\phi _{t1})_{|\phi _{t}(x_{i})}^{T}p_{1}(i)\ ,\ \phi _{t1}\doteq \phi _{1}\circ \phi _{t}^{-1}\ ,\\&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Av_{0}=\sum _{i}\delta _{x_{i}}(\cdot )p_{0}(i)\ {\text{with}}\ \ p_{0}(i)=(D\phi _{1})_{|x_{i}}^{T}(x_{i}^{\prime }-\phi _{1}(x_{i}))\end{cases}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \min _{\phi :v={\dot {\phi }}\circ \phi ^{-1}}C(\phi )\doteq {\frac {1}{2}}\int (Av_{t}\mid v_{t})\,dt+{\frac {1}{2}}\|\mu _{\phi _{1}\cdot m}-\mu _{m^{\prime }}\|_{\mathrm {mea} }^{2}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle m:u\in [0,1]\rightarrow {\mathbb {R} }^{3}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \phi \cdot m=\phi \circ m}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle m\circ \gamma }
SKIPPED EQUATION: \gamma ~WIKIEQUATION: {\displaystyle \gamma }
SKIPPED EQUATION: m ~WIKIEQUATION: {\displaystyle m}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \min _{\phi :v={\dot {\phi }}\circ \phi ^{-1}}C(\phi )\doteq {\frac {1}{2}}\int (Av_{t}\mid v_{t})\,dt+{\frac {1}{2}}\|{\mathcal {C}}_{\phi _{1}\cdot m}-{\mathcal {C}}_{m^{\prime }}\|_{\mathrm {cur} }^{2}}
SKIPPED EQUATION: {{C}}_{\phi_{x}*m}-{{C}}_{x}}_{x}}^{2}/2 ~WIKIEQUATION: {\displaystyle E(\phi _{1})=\|{\mathcal {C}}_{\phi _{1}\cdot m}-{\mathcal {C}}_{m^{\prime }}\|_{\mathrm {cur} }^{2}/2}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \|{\mathcal {C}}_{m}\|_{\mathrm {cur} }^{2}=\int _{0}^{1}\int _{0}^{1}K_{C}(m(u),m(v))\partial m(u)\cdot \partial m(v)\,du\,dv}
SKIPPED EQUATION: K_{x}} ~WIKIEQUATION: {\displaystyle K_{\mathcal {C}}}
SKIPPED EQUATION: m ~WIKIEQUATION: {\displaystyle m}
SKIPPED EQUATION: m ~WIKIEQUATION: {\displaystyle m'}
SKIPPED EQUATION: {{V}}_{\phi_{x}*m}-{{V}}_{x}}_{x}^{2}/2 ~WIKIEQUATION: {\displaystyle E(\phi _{1})=\|{\mathcal {V}}_{\phi _{1}\cdot m}-{\mathcal {V}}_{m^{\prime }}\|_{cur}^{2}/2}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \|{\mathcal {V}}_{m}\|_{var}^{2}=\int _{0}^{1}\int _{0}^{1}k_{\mathbb {R} ^{3}}(m(u),m(v))k_{\mathbf {Gr} }\left([\partial m(u)],[\partial m(v)]\right)\partial m(u){|}{|}\partial m(v){|}\,du\,dv}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle m:u\in U\subset {\mathbb {R} }^{2}\rightarrow {\mathbb {R} }^{3}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle m\circ \gamma }
SKIPPED EQUATION: \gamma ~WIKIEQUATION: {\displaystyle \gamma }
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \min _{\phi :v={\dot {\phi }}\circ \phi ^{-1}}C(\phi )\doteq {\frac {1}{2}}\int (Av_{t}\mid v_{t})\,dt+{\frac {1}{2}}\|{\mathcal {C}}_{\phi _{1}\cdot m}-{\mathcal {C}}_{m^{\prime }}\|_{\mathrm {cur} }^{2}}
SKIPPED EQUATION: {{C}}_{\phi_{x}*m}-{{C}}_{x}}_{x}}^{2}/2 ~WIKIEQUATION: {\displaystyle E(\phi _{1})=\|{\mathcal {C}}_{\phi _{1}\cdot m}-{\mathcal {C}}_{m^{\prime }}\|_{\mathrm {cur} }^{2}/2}
SKIPPED EQUATION: (x) ~WIKIEQUATION: {\displaystyle \|{\mathcal {C}}_{m}\|_{\mathrm {cur} }^{2}=\iint _{U\times U}K_{C}(m(u),m(v)){\vec {n}}(u)\cdot {\vec {n}}(v)\,du\,dv}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\vec {n}}=\partial _{u_{1}}m\wedge \partial _{u_{2}}m}
SKIPPED EQUATION: m ~WIKIEQUATION: {\displaystyle m}
SKIPPED EQUATION: m ~WIKIEQUATION: {\displaystyle m}
SKIPPED EQUATION: {{V}}_{x} ~WIKIEQUATION: {\displaystyle {\mathcal {V}}_{m}}
SKIPPED EQUATION: {{C}}_{x}_{x}}^{2} ~WIKIEQUATION: {\displaystyle \|{\mathcal {C}}_{m}\|_{\mathrm {cur} }^{2}}
SKIPPED EQUATION: (x) ~WIKIEQUATION: {\displaystyle \|{\mathcal {V}}_{m}\|_{\mathrm {var} }^{2}=\iint _{U\times U}k_{\mathbb {R} ^{3}}(m(u),m(v))k_{\mathbf {Gr} }\left([{\vec {n}}(u)],[{\vec {n}}(v)]\right){|}{\vec {n}}(u){|}{|}{\vec {n}}(v){|}\,du\,dv}
SKIPPED EQUATION: 0 ~WIKIEQUATION: {\displaystyle 0<t_{1}<\dots t_{K}=1}
SKIPPED EQUATION: t_{x} ~WIKIEQUATION: {\displaystyle 0<t_{1}<\dots t_{K}=1}
SKIPPED EQUATION: x_{x} ~WIKIEQUATION: {\displaystyle 0<t_{1}<\dots t_{K}=1}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle E(t_{k}),k=1,\dots ,K}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \min _{\phi :v={\dot {\phi }}\circ \phi ^{-1},\phi _{0}=id}C(\phi )\doteq {\frac {1}{2}}\int _{0}^{1}(Av_{t}\mid v_{t})\,dt+\sum _{k=1}^{K}E(\phi _{t_{k}})}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle v_{0}\in V}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle m_{0}\in {\mathcal {M}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle v_{0}\in V}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle n\doteq Exp_{id}(v_{0})\cdot m_{0}\in {\mathcal {M}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle I\in {\mathcal {I}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle I_{a}\in {\mathcal {I}}}
SKIPPED EQUATION: I ~WIKIEQUATION: {\displaystyle I}
SKIPPED EQUATION: x_{x} ~WIKIEQUATION: {\displaystyle Diff_{V}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle v\in V}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle I^{D}\in {\mathcal {I}}^{D}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle p(I^{D}|I_{a})=\int _{V}p(I^{D}|Exp_{id}(v)\cdot I_{a})\pi _{V}(dv)\ .}
SKIPPED EQUATION: v_{x} ~WIKIEQUATION: {\displaystyle v_{0}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle I\doteq \phi \cdot I_{\mathrm {temp} }\in {\mathcal {I}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle I^{D}\in {\mathcal {I}}^{\mathcal {D}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle I\in {\mathcal {I}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle I^{D}\in {\mathcal {I}}^{\mathcal {D}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle m\in {\mathcal {M}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle v_{0}\in V}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle Exp_{id}(v_{0})\cdot m\in {\mathcal {M}}}
SKIPPED EQUATION: x_{x} ~WIKIEQUATION: {\displaystyle Av_{0}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle w\in V}
SKIPPED EQUATION: {{g}}_{x}}m^{3}h ~WIKIEQUATION: {\displaystyle {g}_{Na^{+}}={\bar {g}}_{Na^{+}}m^{3}h,}
SKIPPED EQUATION: {{g}}_{x}} ~WIKIEQUATION: {\displaystyle {\bar {g}}_{Na^{+}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \varphi \in \operatorname {Diff} _{V}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\mathcal {I}}\doteq \{\varphi \cdot I\mid \varphi \in \operatorname {Diff} _{V}\}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle I\in {\mathcal {I}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\mathcal {M}}\doteq \{\varphi \cdot M\mid \varphi \in \operatorname {Diff} _{V}\}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle (\varphi ,I)\mapsto \varphi \cdot I}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\dot {\phi }}_{t},t\in [0,1],\phi _{t}\in \operatorname {Diff} _{V}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \varphi ,\psi \in \operatorname {Diff} _{V}}
SKIPPED EQUATION: \psi ~WIKIEQUATION: {\displaystyle \phi _{0}=\varphi ,\phi _{1}=\psi }
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \rho (\varphi ,\psi )=\inf _{\phi :\phi _{0}=\varphi ,\phi _{1}=\psi }\int _{0}^{1}\|{\dot {\phi }}_{t}\|_{\phi _{t}}\,dt}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \varphi \in \operatorname {Diff} _{V}}
UNPARSED EQUATION: *_{x} ~WIKIEQUATION: {\displaystyle \|\cdot \|_{\varphi }}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \varphi \in \operatorname {Diff} _{V}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \phi \in \operatorname {Diff} _{V}}
SKIPPED EQUATION: {{\phi}}_{x}_{\phi_{x}} ~WIKIEQUATION: {\displaystyle \|{\dot {\phi }}_{t}\|_{\phi _{t}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \varphi \cdot I\in {\mathcal {I}},\varphi \in \operatorname {Diff} _{V},M\in {\mathcal {M}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \varphi _{t},t\in [0,1]}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\frac {d}{dt}}\varphi _{t}=v_{t}\circ \varphi _{t},\ \varphi _{0}=\operatorname {id} ;}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle v_{t}={\dot {\varphi }}_{t}\circ \varphi _{t}^{-1},t\in [0,1]}
SKIPPED EQUATION: \x{x ~WIKIEQUATION: {\displaystyle {\frac {d}{dt}}\varphi _{t}^{-1}=-(D\varphi _{t}^{-1})v_{t},\ \varphi _{0}^{-1}=\operatorname {id} ,}
SKIPPED EQUATION: 3*3 ~WIKIEQUATION: {\displaystyle 3\times 3}
SKIPPED EQUATION: ((x)) ~WIKIEQUATION: {\displaystyle (V,\|\cdot \|_{V})}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle v_{i}\in H_{0}^{3},i=1,2,3,}
SKIPPED EQUATION: ((x)) ~WIKIEQUATION: {\displaystyle (V,\|\cdot \|_{V})}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \operatorname {Diff} _{V}\doteq \{\varphi =\varphi _{1}:{\dot {\varphi }}_{t}=v_{t}\circ \varphi _{t},\varphi _{0}=\operatorname {id} ,\int _{0}^{1}\|v_{t}\|_{V}\,dt<\infty \}\ .}
SKIPPED EQUATION: I_{x} ~WIKIEQUATION: {\displaystyle I_{temp}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle I\in {\mathcal {I}}\doteq \{I=I_{temp}\circ \varphi ,\varphi \in \operatorname {Diff} _{V}\}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\mathcal {M}}\doteq \{\varphi \cdot M_{temp}:\varphi \in \operatorname {Diff} _{V}\}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\mathcal {I}}\doteq \{\varphi \cdot I:\varphi \in \operatorname {Diff} _{V}\}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\mathcal {M}}\doteq \{\varphi \cdot M:\varphi \in \operatorname {Diff} _{V}\}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \varphi \in \operatorname {Diff} _{V}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \|{\dot {\varphi }}\|_{\varphi }\doteq \|{\dot {\varphi }}\circ \varphi ^{-1}\|_{V}=\|v\|_{V},}
SKIPPED EQUATION: ((x)) ~WIKIEQUATION: {\displaystyle (V,\|\cdot \|_{V})}
SKIPPED EQUATION: V ~WIKIEQUATION: {\displaystyle V}
SKIPPED EQUATION: V ~WIKIEQUATION: {\displaystyle V^{*}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \sigma \doteq Av\in V^{*}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle v,w\in V}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \langle v,w\rangle _{V}\doteq \int _{X}Av\cdot w\,dx,\ \|v\|_{V}^{2}\doteq \int _{X}Av\cdot v\,dx,\ v,w\in V\ .}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \int Av\cdot v\,dx\doteq \int \mu \cdot v\,dx=\sum _{i=1}^{3}\mu _{i}v_{i}\,dx.}
SKIPPED EQUATION: A ~WIKIEQUATION: {\displaystyle A}
SKIPPED EQUATION: ((x)) ~WIKIEQUATION: {\displaystyle (V,\|\cdot \|_{V})}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle KAv(x)_{i}\doteq \sum _{j}\int _{{\mathbb {R} }^{3}}k_{ij}(x,y)Av_{j}(y)\,dy\in V\ .}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle d_{\mathrm {Diff} _{V}}(\psi ,\varphi )=\inf _{v_{t}}\left(\int _{0}^{1}\int _{X}Av_{t}\cdot v_{t}\,dx\,dt:\phi _{0}=\psi ,\phi _{1}=\varphi ,{\dot {\phi }}_{t}=v_{t}\circ \phi _{t}\right)^{1/2}\ .}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \phi \in \operatorname {Diff} _{V}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle d_{\operatorname {Diff} _{V}}(\psi ,\varphi )=d_{\operatorname {Diff} _{V}}(\psi \circ \phi ,\varphi \circ \phi ).}
SKIPPED EQUATION: d_{x}}:{{I}}*{{I}}x{R}^{+} ~WIKIEQUATION: {\displaystyle d_{\mathcal {I}}:{\mathcal {I}}\times {\mathcal {I}}\rightarrow \mathbb {R} ^{+}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle d_{\mathcal {I}}(I,J)=\inf _{\phi \in \operatorname {Diff} _{V}:\phi \cdot I=J}d_{\operatorname {Diff} _{V}}(id,\phi )\ ;}
SKIPPED EQUATION: d_{x}}:{{M}}*{{M}}x{R}^{+} ~WIKIEQUATION: {\displaystyle d_{\mathcal {M}}:{\mathcal {M}}\times {\mathcal {M}}\rightarrow \mathbb {R} ^{+}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle d_{\mathcal {M}}(M,N)=\inf _{\phi \in \operatorname {Diff} _{V}:\phi \cdot M=N}d_{\mathrm {Diff} _{V}}(\operatorname {id} ,\phi )\ .}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle t\mapsto \phi _{t}\in \operatorname {Diff} _{V}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle t\mapsto v_{t}\in V}
SKIPPED EQUATION: \x{x ~WIKIEQUATION: {\displaystyle {\dot {\phi }}_{t}=v_{t}\cdot \phi _{t},\phi _{0}=\operatorname {id} .}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle Av\in V^{*}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle p:{\dot {\phi }}\mapsto (p\mid {\dot {\phi }})}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\dot {\phi }}_{t}=v_{t}\circ \phi _{t}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle H(\phi _{t},p_{t},v_{t})=\int _{X}p_{t}\cdot (v_{t}\circ \phi _{t})\,dx-{\frac {1}{2}}\int _{X}Av_{t}\cdot v_{t}\,dx.}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle H(\phi _{t},p_{t})=H(\operatorname {id} ,p_{0})={\frac {1}{2}}\int _{X}p_{0}\cdot v_{0}\,dx}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle x_{i},i=1,\dots ,n}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle p(i),i=1,\dots ,n}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle H(\phi _{t},p_{t})={\frac {1}{2}}\int _{U}\int _{U}p_{t}(u)\cdot K(\phi _{t}(m(u)),\phi _{t}(m(v)))p_{t}(v)\,du\,dv}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\begin{cases}v_{t}=\textstyle \int _{U}\displaystyle K(\cdot ,\phi _{t}(m(u)))p_{t}(u)\,du\ ,\\{\dot {p}}_{t}(u)=-(Dv_{t})_{|_{\phi _{t}(m(u))}}^{T}p_{t}(u),u\in U\end{cases}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle d^{2}=(p_{0}\mid v_{0})=\int _{U}p_{0}(u)\cdot \int _{U}K(m(u),m(u^{\prime }))p_{0}(u^{\prime })\,du\,du^{\prime }}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle H(\phi _{t},p_{t})={\frac {1}{2}}\int _{{\mathbb {R} }^{3}}\int _{{\mathbb {R} }^{3}}p_{t}(x)\cdot K(\phi _{t}(x),\phi _{t}(y))p_{t}(y)\,dx\,dy\displaystyle }
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\begin{cases}v_{t}=\textstyle \int _{X}\displaystyle K(\cdot ,\phi _{t}(x))p_{t}(x)\,dx\ ,\\{\dot {p}}_{t}(x)=-(Dv_{t})_{|_{\phi _{t}(x)}}^{T}p_{t}(x),x\in {\mathbb {R} }^{3}\end{cases}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \displaystyle d^{2}=(p_{0}\mid v_{0})=\int _{\mathbb {R} ^{3}}p_{0}(x)\cdot \int _{{\mathbb {R} }^{3}}K(x,y)p_{0}(y)\,dy\,dx.}
SKIPPED EQUATION: r ~WIKIEQUATION: {\displaystyle r}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle x}
SKIPPED EQUATION: y ~WIKIEQUATION: {\displaystyle y}
SKIPPED EQUATION: I_{x} ~WIKIEQUATION: {\displaystyle I_{temp}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle I\in {\mathcal {I}}\doteq \{I=I_{temp}\circ \varphi ,\varphi \in Diff_{V}\}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \phi _{t},t\in [0,1]}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\frac {d}{dt}}\phi _{t}=v_{t}\circ \phi _{t},\ \phi _{0}=id}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle v_{t},t\in [0,1]}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle v_{t}\in C^{1}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle v\in V}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \phi _{t}^{-1},t\in [0,1]}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\frac {d}{dt}}\phi _{t}^{-1}=-(D\phi _{t}^{-1})v_{t},\ \phi _{0}^{-1}=id\ .}
SKIPPED EQUATION: ((x)) ~WIKIEQUATION: {\displaystyle (V,\|\cdot \|_{V})}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle v_{i}\in H_{0}^{3},i=1,2,3,}
SKIPPED EQUATION: ((x)) ~WIKIEQUATION: {\displaystyle (V,\|\cdot \|_{V})}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle Diff_{V}\doteq \{\varphi =\phi _{1}:{\dot {\phi }}_{t}=v_{t}\circ \phi _{t},\phi _{0}=id,\int _{0}^{1}\|v_{t}\|_{V}dt<\infty \}\ .}
SKIPPED EQUATION: ((x)) ~WIKIEQUATION: {\displaystyle (V,\|\cdot \|_{V})}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \|v\|_{V}^{2}\doteq \int _{R^{3}}Av\cdot vdx,\ v\in V\ ,}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle Av}
SKIPPED EQUATION: V ~WIKIEQUATION: {\displaystyle V^{*}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle v={\dot {\phi }}\circ \phi ^{-1}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\textstyle \min _{v:{\dot {\phi }}=v\circ \phi ,\phi _{0}=id}C(v)\doteq {\frac {1}{2}}\int _{0}^{1}\int _{R^{3}}Av_{t}\cdot v_{t}dxdt+{\frac {1}{2}}\int _{R^{3}}|I\circ \phi _{1}^{-1}-J|^{2}dx}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \phi _{t1}\doteq \phi _{1}\circ \phi _{t}^{-1}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\begin{cases}&v_{t}^{new}(\cdot )=v_{t}^{old}(\cdot )-\epsilon (v_{t}^{old}-\int _{R^{3}}K(\cdot ,y)(I\circ \phi _{t}^{-1old}(y)-J\circ \phi _{t1}^{old}(y))\nabla (I\circ \phi _{t}^{-1old}(y))|D\phi _{t1}^{old}(y)|dy),t\in [0,1]\\&{\dot {\phi }}_{t}^{new}=v_{t}^{new}\circ \phi _{t}^{new},t\in [0,1]\end{cases}}}
SKIPPED EQUATION: t ~WIKIEQUATION: {\displaystyle t=0}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \mu _{0}^{*}=Av_{0}^{*}=(I-J\circ \phi _{1}^{*})\nabla I|D\phi _{1}^{*}|}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle Av_{t}^{*}=(D\phi _{t}^{*-1})^{T}Av_{0}^{*}\circ \phi _{t}^{*-1}|D\phi _{t}^{*-1}|}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \min _{v:{\dot {\phi }}_{t}=v_{t}\circ \phi _{t}}C(v)\doteq {\frac {1}{2}}\int _{0}^{1}\int _{R^{3}}Av_{t}\cdot v_{t}dxdt+{\frac {1}{2}}\sum _{i}(\phi _{1}(x_{i})-y_{i})\cdot (\phi _{1}(x_{i})-y_{i})}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle I\circ \phi _{t}^{-1}}
SKIPPED EQUATION: v_{x} ~WIKIEQUATION: {\displaystyle v_{t},}
SKIPPED EQUATION: \phi_{x} ~WIKIEQUATION: {\displaystyle \phi _{t}.}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \phi _{t1}\doteq \phi _{1}\circ \phi _{t}^{-1}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\begin{cases}&v_{t}^{new}(\cdot )=v_{t}^{old}(\cdot )-\epsilon (v_{t}^{old}+\sum _{i}K(\cdot ,\phi _{t}^{old}(x_{i}))(D\phi _{t1})^{oldT}|_{\phi _{t}^{old}(x_{i})}(y_{i}-\phi _{1}^{old}(x_{i})),t\in [0,1]\\&{\dot {\phi }}_{t}^{new}=v_{t}^{new}\circ \phi _{t}^{new},t\in [0,1]\end{cases}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle (\phi +\epsilon \delta \phi \circ \phi )\circ (\phi ^{-1}+\epsilon \delta \phi ^{-1}\circ \phi ^{-1})=id+o(\epsilon )}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \delta \phi ^{-1}\circ \phi ^{-1}=-(D\phi _{1}^{-1})\delta \phi }
SKIPPED EQUATION: \deltav ~WIKIEQUATION: {\displaystyle \delta v}
SKIPPED EQUATION: (x)_{x}\delta\phi_{x}+\deltav_{x} ~WIKIEQUATION: {\displaystyle {\frac {d}{dt}}\left(\delta \phi _{|\phi }\right)=(Dv)_{|\phi }\delta \phi _{|\phi }+\delta v_{|\phi }}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \delta \phi _{1}=(D\phi _{1})_{|\phi _{1}^{-1}}\int _{0}^{1}(D\phi _{t})_{|\phi _{1}^{-1}}^{-1}(\delta v_{t})_{\phi _{t}\circ \phi _{1}^{-1}}dt}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle E(\phi )=\int _{X}|I\circ \phi ^{-1}-J|^{2}dx}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\frac {d}{d\epsilon }}{\frac {1}{2}}\int _{X}|I\circ (\phi ^{-1}+\epsilon \delta \phi ^{-1}\circ \phi ^{-1})-J|^{2}dx|_{\epsilon =0}=\int _{X}(I\circ \phi ^{-1}-J)\nabla I|_{\phi ^{-1}}\delta \phi ^{-1}\circ \phi ^{-1}dx}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle =\int _{X}(I\circ \phi ^{-1}-J)\nabla I|_{\phi ^{-1}}(-D\phi _{1}^{-1})\delta \phi dx}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle =\int _{X}(I\circ \phi _{1}^{-1}-J)\nabla I|_{\phi _{1}^{-1}}(-D\phi _{1})_{|\phi _{1}^{-1}}^{-1}(D\phi _{1})_{|\phi _{1}^{-1}})\int _{0}^{1}(D\phi _{t})_{|\phi _{1}^{-1}}^{-1}(\delta v_{t})_{|{\phi _{t}\circ \phi _{1}^{-1}}}dtdx}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \phi _{t1}\doteq \phi _{1}\circ \phi _{t}^{-1}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\begin{aligned}{\frac {d}{d\epsilon }}C(v+\epsilon \delta v)|_{\epsilon =0}&=\int _{0}^{1}\int _{X}Av_{t}\cdot \delta v_{t}\ dx\ dt-\int _{0}^{1}\int _{X}(I\circ \phi _{1}^{-1}-J)\nabla I|_{\phi _{1}^{-1}}(D\phi _{t})_{|\phi _{1}^{-1}}^{-1}(\delta v_{t})_{|{\phi _{t}\circ \phi _{1}^{-1}}}\ dx\,dt\\&=\int _{0}^{1}\int _{X}\left(Av_{t}-(I\circ \phi _{t}^{-1}-J\circ \phi _{t1})\nabla I|_{\phi _{t}^{-1}}(D\phi _{t})_{|\phi _{t}^{-1}}^{-1}|D\phi _{t1}|\right)\cdot \delta v_{t}\ dx\,dt\\&=0\end{aligned}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \delta \phi \circ \phi }
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \sum _{i}(\phi _{1}(x_{i})-y_{i})\cdot D\phi _{1}|_{\phi _{1}^{-1}(\phi _{1}(x_{i}))}\int _{0}^{1}(D\phi _{t})_{|\phi _{1}^{-1}(\phi _{1}(x_{i}))}^{-1}\delta v_{t}|_{\phi _{t}\circ \phi _{1}^{-1}(\phi _{1}(x_{i}))}dt}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle =\int _{0}^{1}\int _{X}\sum _{i}\delta _{\phi _{t}(x_{i})}(x)(\phi _{1}(x_{i})-y_{i})\cdot (D\phi _{1})_{\phi _{t}^{-1}(x)}(D\phi _{t})_{\phi _{t}^{-1}(x)}^{-1}\delta v_{t}(x)dxdt=\int _{0}^{1}\int _{X}\sum _{i}\delta _{\phi _{t}(x_{i})}(y)(D\phi _{t1})_{\phi _{t}(x_{i})}^{T}(\phi _{1}(x_{i})-y_{i})\cdot \delta v_{t}(x)dxdt}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle I(x),x\in {\mathbb {R} }^{3}}
UNPARSED EQUATION: * ~WIKIEQUATION: {\displaystyle \|\cdot \|}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \varphi \cdot M=(\lambda _{1}{\hat {e}}_{1}{\hat {e}}_{1}^{T}+\lambda _{2}{\hat {e}}_{2}{\hat {e}}_{2}^{T}+\lambda _{3}{\hat {e}}_{3}{\hat {e}}_{3}^{T})\circ \varphi ^{-1},}
SKIPPED EQUATION: {{e}}_{x}*{{e}}_{x} ~WIKIEQUATION: {\displaystyle {\begin{aligned}{\hat {e}}_{1}&={\frac {D\varphi e_{1}}{\|D\varphi e_{1}\|}}\ ,\ \ \ {\hat {e}}_{2}={\frac {D\varphi e_{2}-\langle {\hat {e}}_{1},D\varphi e_{2}\rangle {\hat {e}}_{1}}{\sqrt {\|D\varphi e_{2}\|^{2}-\langle {\hat {e}}_{1},D\varphi e_{2}\rangle ^{2}}}}\ ,\ \ \ {\hat {e}}_{3}={\hat {e}}_{1}\times {\hat {e}}_{2}\end{aligned}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle I^{\prime }(x),x\in {\mathbb {R} }^{3}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle E(\phi _{1})\doteq \alpha \int _{{\mathbb {R} }^{3}}\|\phi _{1}\cdot I-I^{\prime }\|^{2}\,dx+\beta \int _{{\mathbb {R} }^{3}}(\|\phi _{1}\cdot I\|-\|I^{\prime }\|)^{2}\,dx).}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \min _{v:{\dot {\phi }}\circ \phi ^{-1}}{\frac {1}{2}}\int _{0}^{1}\int _{R^{3}}Av_{t}\cdot v_{t}dxdt+\alpha \int _{{\mathbb {R} }^{3}}\|\phi _{1}\cdot I-I^{\prime }\|^{2}\,dx+\beta \int _{{\mathbb {R} }^{3}}(\|\phi _{1}\cdot I\|-\|I^{\prime }\|)^{2}\,dx\ .}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle M^{\prime }(x),x\in {\mathbb {R} }^{3}}
UNPARSED EQUATION: *_{x} ~WIKIEQUATION: {\displaystyle \|\cdot \|_{F}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \min _{v:v={\dot {\phi }}\circ \phi ^{-1}}{\frac {1}{2}}\int _{0}^{1}\int _{R^{3}}Av_{t}\cdot v_{t}dxdt+\alpha \int _{{\mathbb {R} }^{3}}\|\phi _{1}\cdot M(x)-M^{\prime }(x)\|_{F}^{2}dx}
SKIPPED EQUATION: n ~WIKIEQUATION: {\displaystyle n}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \int _{{\bf {s}}\in {\mathbb {S} }^{2}}\psi ^{2}({\bf {s}})d{\bf {s}}=1}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \psi _{1},\psi _{2}\in \Psi }
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\begin{aligned}\rho (\psi _{1},\psi _{2})=\|\log _{\psi _{1}}(\psi _{2})\|_{\psi _{1}}=\cos ^{-1}\langle \psi _{1},\psi _{2}\rangle =\cos ^{-1}\left(\int _{{\bf {s}}\in {\mathbb {S} }^{2}}\psi _{1}({\bf {s}})\psi _{2}({\bf {s}})d{\bf {s}}\right),\end{aligned}}}
SKIPPED EQUATION: ((x)) ~WIKIEQUATION: {\displaystyle \langle \cdot ,\cdot \rangle }
SKIPPED EQUATION: \psi_{x}}((x)) ~WIKIEQUATION: {\displaystyle \psi _{\mathrm {temp} }({\bf {s}},x)}
SKIPPED EQUATION: \psi_{x}}((x)) ~WIKIEQUATION: {\displaystyle \psi _{\mathrm {targ} }({\bf {s}},x)}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\bf {s}}\in {{\mathbb {S} }^{2}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle x\in X}
SKIPPED EQUATION: \phi_{x} ~WIKIEQUATION: {\displaystyle \phi _{t}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\dot {\phi }}_{t}=v_{t}(\phi _{t}),t\in [0,1],\phi _{0}={id}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \phi _{1}\cdot \psi (x)\doteq (D\phi _{1})\psi \circ \phi _{1}^{-1}(x),x\in X}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\begin{aligned}(D\phi _{1})\psi \circ \phi _{1}^{-1}(x)={\sqrt {\frac {\det {{\bigl (}D_{\phi _{1}^{-1}}\phi _{1}{\bigr )}^{-1}}}{\left\|{{\bigl (}D_{\phi _{1}^{-1}}\phi _{1}{\bigr )}^{-1}}{\bf {s}}\right\|^{3}}}}\quad \psi \left({\frac {(D_{\phi _{1}^{-1}}\phi _{1}{\bigr )}^{-1}{\bf {s}}}{\|(D_{\phi _{1}^{-1}}\phi _{1}{\bigr )}^{-1}{\bf {s}}\|}},\phi _{1}^{-1}(x)\right).\end{aligned}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\begin{aligned}\min _{v:{\dot {\phi }}_{t}=v_{t}\circ \phi _{t},\phi _{0}={id}}\int _{0}^{1}\int _{R^{3}}Av_{t}\cdot v_{t}dx\ dt+\lambda \int _{R^{3}}\|\log _{(D\phi _{1})\psi _{\mathrm {temp} }\circ \phi _{1}^{-1}(x)}(\psi _{\mathrm {targ} }(x))\|_{(D\phi _{1})\psi _{\mathrm {temp} }\circ \phi _{1}^{-1}(x)}^{2}dx\end{aligned}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle q_{t}\doteq I\circ \phi _{t}^{-1},q_{0}=I}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle I(x),x\in X=R^{3}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\dot {q}}_{t}=-\nabla q_{t}\cdot v_{t}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\begin{matrix}&\ \ \ \ \ \min _{v:{\dot {q}}=v\circ q}C(v)\doteq {\frac {1}{2}}\int _{0}^{1}\int _{R^{3}}Av_{t}\cdot v_{t}dxdt+{\frac {1}{2}}\int _{{\mathbb {R} }^{3}}|q_{1}(x)-J(x)|^{2}dx\end{matrix}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\begin{cases}{\text{Hamiltonian Dynamics}}&\ \ \ \ \ \ \ \ \ \ {\dot {q}}_{t}=-\nabla q_{t}\cdot v_{t}\\&\ \ \ \ \ \ \ \ \ \ {\dot {p}}_{t}=-{\text{div}}(p_{t}v_{t}),\ \ \ \ t\in [0,1]\\&\ \ \ \ \ \ \ \ \ \ Av_{t}=\mu _{t}=-p_{t}\nabla q_{t}\\{\text{Endpoint Condition}}&\ \ \ \ \ \ \ \ \ p_{1}=-{\frac {\partial E}{\partial q_{1}}}(q_{1})=-(q_{1}-J)=-(I\circ \phi _{1}^{-1}-J)\\&\ \ \ \ \ \ \ \ \ \ Av_{1}=\mu _{1}=(I\circ \phi _{1}^{-1}-J)\nabla (I\circ \phi _{1}^{-1})\ \ t=1\ .\\{\text{Conserved Dynamics}}&\ \ \ \ \ \ \ \ \ \ p_{t}=-(I\circ \phi _{t}^{-1}-J\circ \phi _{t1})|D\phi _{t1}|\ ,\ \ t\in [0,1]\ .\\\end{cases}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle q_{t}=I\circ \phi _{t}^{-1}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\dot {q}}=-\nabla q\cdot v}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle Av=-p\nabla q}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle (-{\dot {p}}-\nabla \cdot (pv)|\delta q))=0}
SKIPPED EQUATION: (x) ~WIKIEQUATION: {\displaystyle (\delta p|{\dot {q}}+\nabla q\cdot v)=0}
SKIPPED EQUATION: {s} ~WIKIEQUATION: {\displaystyle \mathbf {s} }
SKIPPED EQUATION: {s} ~WIKIEQUATION: {\displaystyle \mathbf {s} }
SKIPPED EQUATION: D ~WIKIEQUATION: {\displaystyle D}
SKIPPED EQUATION: K ~WIKIEQUATION: {\displaystyle K}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle K\ll D}
SKIPPED EQUATION: {s} ~WIKIEQUATION: {\displaystyle \mathbf {s} }
SKIPPED EQUATION: K ~WIKIEQUATION: {\displaystyle K=1}
SKIPPED EQUATION: {v} ~WIKIEQUATION: {\displaystyle \mathbf {v} }
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle I(\mathbf {v} )=\int dxP_{\mathbf {v} }(x|spike)log2[P_{\mathbf {v} }(x|spike)/P_{\mathbf {v} }(x)]}
UNPARSED EQUATION: (\delta(x-{s}*{v})|spike)_{x}} ~WIKIEQUATION: {\displaystyle P_{\mathbf {v} }(x|spike)=\langle \delta (x-\mathbf {s} \cdot \mathbf {v} )|spike\rangle _{\mathbf {s} }}
SKIPPED EQUATION: {v} ~WIKIEQUATION: {\displaystyle \mathbf {v} }
SKIPPED EQUATION: K ~WIKIEQUATION: {\displaystyle K=1}
SKIPPED EQUATION: {v} ~WIKIEQUATION: {\displaystyle \mathbf {v} }
SKIPPED EQUATION: K ~WIKIEQUATION: {\displaystyle K>1}
SKIPPED EQUATION: 1 ~WIKIEQUATION: {\displaystyle K>1}
SKIPPED EQUATION: 1}^{K}\delta(x_{x}-{s}*{v}_{x})|spike)_{x}} ~WIKIEQUATION: {\displaystyle P_{\mathbf {v} ^{K}}(\mathbf {x} |spike)=\langle \prod _{i=1}^{K}\delta (x_{i}-\mathbf {s} \cdot \mathbf {v} _{i})|spike\rangle _{\mathbf {s} }}
SKIPPED EQUATION: 1}^{K}\delta(x_{x}-{s}*{v}_{x}))_{x}} ~WIKIEQUATION: {\displaystyle P_{\mathbf {v} ^{K}}(\mathbf {x} )=\langle \prod _{i=1}^{K}\delta (x_{i}-\mathbf {s} \cdot \mathbf {v} _{i})\rangle _{\mathbf {s} }}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\frac {d}{dt}}{\vec {X}}=-\alpha {\vec {X}}+{\frac {1}{\beta }}(I-\chi \Omega X)^{-1}\Omega {\vec {S}}}
SKIPPED EQUATION: \alpha ~WIKIEQUATION: {\displaystyle \alpha =0}
SKIPPED EQUATION: \alpha ~WIKIEQUATION: {\displaystyle \alpha }
SKIPPED EQUATION: {\x{R_{x}-R_{x}}{R_{x}}} ~WIKIEQUATION: {\displaystyle \chi ={\frac {R_{\text{off}}-R_{\text{on}}}{R_{\text{off}}}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \Omega }
SKIPPED EQUATION: \beta ~WIKIEQUATION: {\displaystyle \beta }
SKIPPED EQUATION: \alpha ~WIKIEQUATION: {\displaystyle \alpha }
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \langle |X(t+\Delta t)-X(t)|^{2}\rangle =2nDt}
SKIPPED EQUATION: \Xi ~WIKIEQUATION: {\displaystyle \Xi }
SKIPPED EQUATION: \Xi ~WIKIEQUATION: {\displaystyle m{\ddot {x}}+\Gamma {\dot {x}}-F(x,t)=\Xi ,}
SKIPPED EQUATION: \rho ~WIKIEQUATION: {\displaystyle \rho }
SKIPPED EQUATION: \Xi ~WIKIEQUATION: {\displaystyle \Xi }
SKIPPED EQUATION: \delta ~WIKIEQUATION: {\displaystyle \delta }
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle m{\frac {d^{2}x}{dt^{2}}}+\Gamma {\frac {dx}{dt}}+\nabla U(x)={\sqrt {2\varepsilon \gamma }}\,{\frac {d\eta }{dt}},}
SKIPPED EQUATION: k_{x} ~WIKIEQUATION: {\displaystyle k_{\text{B}}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \gamma \to \infty }
SKIPPED EQUATION: \delta ~WIKIEQUATION: {\displaystyle \delta }
SKIPPED EQUATION: {b}(X((x) ~WIKIEQUATION: {\displaystyle {\dot {X}}(t)={b}(X(t))+{\sqrt {2}}{B}_{e}(X(t)){\dot {w}}(t),\qquad \qquad (1)}
SKIPPED EQUATION: {B}_{x} ~WIKIEQUATION: {\displaystyle {B}_{e}}
SKIPPED EQUATION: {\x} ~WIKIEQUATION: {\textstyle X}
SKIPPED EQUATION: \gamma ~WIKIEQUATION: {\displaystyle \gamma }
SKIPPED EQUATION: x]}{\Deltat}} ~WIKIEQUATION: {\displaystyle a(x)=\lim _{\Delta t\rightarrow 0}{\frac {E[\Delta X(t)\mid X(t)=x]}{\Delta t}},}
SKIPPED EQUATION: x]}{2\Deltat}} ~WIKIEQUATION: {\displaystyle D(x)=\lim _{\Delta t\rightarrow 0}{\frac {E[\Delta X(t)^{T}\,\Delta X(t)\mid X(t)=x]}{2\,\Delta t}}.}
SKIPPED EQUATION: x] ~WIKIEQUATION: {\displaystyle E[\cdot \,|\,X(t)=x]}
SKIPPED EQUATION: \Deltat ~WIKIEQUATION: {\displaystyle \Delta t}
SKIPPED EQUATION: \Deltat ~WIKIEQUATION: {\displaystyle \Delta t}
SKIPPED EQUATION: x_{x} ~WIKIEQUATION: {\displaystyle x_{k}}
SKIPPED EQUATION: N_{x} ~WIKIEQUATION: {\displaystyle N_{t}}
SKIPPED EQUATION: (x) ~WIKIEQUATION: {\displaystyle \{x^{i}(t_{1}),\dots ,x^{i}(t_{N_{s}})\},}
SKIPPED EQUATION: t_{x} ~WIKIEQUATION: {\displaystyle t_{j}}
SKIPPED EQUATION: (x) ~WIKIEQUATION: {\displaystyle a(x_{k})=(a_{x}(x_{k}),a_{y}(x_{k}))}
SKIPPED EQUATION: x_{x} ~WIKIEQUATION: {\displaystyle x_{k}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle a_{x}(x_{k})\approx {\frac {1}{N_{k}}}\sum _{j=1}^{N_{t}}\sum _{i=0,{\tilde {x}}_{i}^{j}\in S(x_{k},r)}^{N_{s}-1}\left({\frac {x_{i+1}^{j}-x_{i}^{j}}{\Delta t}}\right)}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle a_{y}(x_{k})\approx {\frac {1}{N_{k}}}\sum _{j=1}^{N_{t}}\sum _{i=0,{\tilde {x}}_{i}^{j}\in S(x_{k},r)}^{N_{s}-1}\left({\frac {y_{i+1}^{j}-y_{i}^{j}}{\Delta t}}\right),}
SKIPPED EQUATION: N_{x} ~WIKIEQUATION: {\displaystyle N_{k}}
SKIPPED EQUATION: \varepsilon ~WIKIEQUATION: {\displaystyle \varepsilon }
SKIPPED EQUATION: \eta ~WIKIEQUATION: {\displaystyle \eta }
SKIPPED EQUATION: \varepsilon ~WIKIEQUATION: {\displaystyle \varepsilon }
SKIPPED EQUATION: \eta ~WIKIEQUATION: {\displaystyle \eta }
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle V(t)=\sum _{f}\eta (t-t^{f})+\int _{0}^{\infty }\kappa (s)I(t-s)\,ds+V_{\mathrm {rest} }}
SKIPPED EQUATION: \vartheta_{x} ~WIKIEQUATION: {\displaystyle \vartheta _{0}}
SKIPPED EQUATION: f ~WIKIEQUATION: {\displaystyle f}
SKIPPED EQUATION: \tau_{x} ~WIKIEQUATION: {\displaystyle \tau _{0}}
SKIPPED EQUATION: \beta ~WIKIEQUATION: {\displaystyle \beta }
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \beta \to \infty }
SKIPPED EQUATION: 1 ~WIKIEQUATION: {\displaystyle 1\leq i\leq N}
SKIPPED EQUATION: i ~WIKIEQUATION: {\displaystyle 1\leq i\leq N}
SKIPPED EQUATION: N ~WIKIEQUATION: {\displaystyle 1\leq i\leq N}
SKIPPED EQUATION: i ~WIKIEQUATION: {\displaystyle i}
SKIPPED EQUATION: w_{x} ~WIKIEQUATION: {\displaystyle w_{ij}}
SKIPPED EQUATION: \varepsilon_{x} ~WIKIEQUATION: {\displaystyle \varepsilon _{ij}(s)}
SKIPPED EQUATION: t_{x} ~WIKIEQUATION: {\displaystyle t_{n}}
SKIPPED EQUATION: \Deltat ~WIKIEQUATION: {\displaystyle \Delta t}
SKIPPED EQUATION: \vartheta ~WIKIEQUATION: {\displaystyle \vartheta }
UNPARSED EQUATION: 05[1+\tanh(\x\max)] ~WIKIEQUATION: {\displaystyle F(x)=0.5[1+\tanh(\gamma x)]}
SKIPPED EQUATION: \gamma ~WIKIEQUATION: {\displaystyle \gamma }
SKIPPED EQUATION: f ~WIKIEQUATION: {\displaystyle f}
SKIPPED EQUATION: V(t_{x})-\vartheta(t_{x}) ~WIKIEQUATION: {\displaystyle y_{n}=V(t_{n})-\vartheta (t_{n})}
SKIPPED EQUATION: t_{x} ~WIKIEQUATION: {\displaystyle t_{k}}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \{X_{j}(t_{m})\in \{0,1\};m=1,2,3,\dots \}}
SKIPPED EQUATION: \varepsilon_{x} ~WIKIEQUATION: {\displaystyle \varepsilon _{ij}(s)}
SKIPPED EQUATION: X_{x} ~WIKIEQUATION: {\displaystyle X_{j}}
SKIPPED EQUATION: \tau_{x} ~WIKIEQUATION: {\displaystyle \tau _{m}}
SKIPPED EQUATION: \tau_{x} ~WIKIEQUATION: {\displaystyle \tau _{m}}
SKIPPED EQUATION: 01 ~WIKIEQUATION: {\displaystyle u(t)={\begin{cases}0,&t<0\\1,&t\geq 0\end{cases}}}
SKIPPED EQUATION: 0) ~WIKIEQUATION: {\displaystyle V_{0}=V(t=0)}
SKIPPED EQUATION: t ~WIKIEQUATION: {\displaystyle t=0}
SKIPPED EQUATION: V_{x}e^{0} ~WIKIEQUATION: {\displaystyle V=V_{0}e^{0}}
SKIPPED EQUATION: V_{x} ~WIKIEQUATION: {\displaystyle V=V_{0}}
SKIPPED EQUATION: \tau ~WIKIEQUATION: {\displaystyle t=\tau }
SKIPPED EQUATION: 0} ~WIKIEQUATION: {\textstyle \lim _{t\to \infty }f(t)=0}
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle {\begin{aligned}V(t)&=V_{0}e^{-t/\tau }+{\frac {Ae^{-t/\tau }}{\tau }}\int _{0}^{t}\,dt'\ e^{t'/\tau }e^{j\omega t'}\\[1ex]&=V_{0}e^{-t/\tau }+{\frac {\frac {1}{\tau }}{j\omega +{\frac {1}{\tau }}}}A\left(e^{j\omega t}-e^{-t/\tau }\right).\end{aligned}}}
SKIPPED EQUATION: \tau ~WIKIEQUATION: {\displaystyle \tau }
SKIPPED EQUATION: (x) ~WIKIEQUATION: {\displaystyle {\frac {dV}{dt}}+{\frac {1}{\tau }}V=f(t)=Au(t),}
SKIPPED EQUATION: \tau ~WIKIEQUATION: {\displaystyle \tau }
SKIPPED EQUATION: \tau ~WIKIEQUATION: {\displaystyle \tau }
SKIPPED EQUATION: x ~WIKIEQUATION: {\displaystyle \tau =RC}
SKIPPED EQUATION: {t} ~WIKIEQUATION: {\displaystyle 5\tau ={\text{FO4}}}
SKIPPED EQUATION: x_{x}((x) ~WIKIEQUATION: {\displaystyle F=hA_{s}\left(T(t)-T_{a}\right),}
SKIPPED EQUATION: \tau ~WIKIEQUATION: {\displaystyle \tau }
SKIPPED EQUATION: V_{x}}(1-e^{-t/\tau}) ~WIKIEQUATION: {\displaystyle V(t)=V_{\textrm {max}}\left(1-e^{-t/\tau }\right)}
SKIPPED EQUATION: V_{x}}e^{-t/\tau} ~WIKIEQUATION: {\displaystyle V(t)=V_{\textrm {max}}e^{-t/\tau }}
SKIPPED EQUATION: \tau ~WIKIEQUATION: {\displaystyle \tau }
SKIPPED EQUATION: \tau ~WIKIEQUATION: {\displaystyle \tau }
SKIPPED EQUATION: \tau ~WIKIEQUATION: {\displaystyle \tau }
SKIPPED EQUATION: \tau ~WIKIEQUATION: {\displaystyle \tau }
