harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
I_3^N \leq C(\bar{R}_N+\bar{R}_N^\prime)+\frac18\frac1N \Big\|\sum_{i=1}^NX_iv_i\Big\|_{L^2}^2+\frac{1}{4}\sum_{i=1}^N\| v_i\|_{H^1}^2 &+C\Big(\sum_{i=1}^N\|v_i\|_{L^2}^2\Big)\Big[R_N^Z+1+\frac{1}{N}\sum_{i=1}^N\Big(\|X_i\|_{L^4}^{4/(1-2s)}+\|\Lambda^sX_i\|_{L^4}^4\Big)\Big],
\bigcup_{i = 1}^{n}(G'_{2i-1}-G'_{2i}) = \left(\bigcup_{i = 1}^{n}(G_{2i-1}-G_{2i}) \right)\cap K_{2n}\subseteq \left(\bigcup_{i = 1}^n (K_{2i-1}- K_{2i}) \right)\cap K_{2n}=\emptyset
\Gamma=(\begin{matrix}\frac{\partial A}{\partial x}&\frac{\partial A}{\partial y}\\ \frac{\partial A}{\partial y}&-\frac{\partial A}{\partial x}\end{matrix})+\frac{1}{2\pi\nu}e^{\lambda x}K_{o}(\lambda r)(\begin{matrix}1&0\\ 0&1\end{matrix}),\Pi=\frac{\rho}{2\pi}\nabla(lnr)
\ddot{\phi}({\bf k}_{n}, t ) + \omega_{n}^{2}\phi ({\bf k}_{n}, t ) + \sum_{n^{\prime}}\Gamma_{n n^{\prime}}^{D}{\dot{\phi}}({\bf k}_{n^{\prime}}, t ) + \omega_{n}\sum_{n^{\prime}}\Gamma_{n n^{\prime}}^{C}\phi ({\bf k}_{n^{\prime}}, t ) + \frac{\delta V}{\delta \phi ( -{\bf k}_{n}, t )}= \eta ({\bf k}_{n}, t ) ,
\psi^{\prime}(x):=\psi^{\prime}(\lambda_1(x))v_1+\psi^{\prime}(\lambda_2(x))v_2+\psi^{\prime}(\lambda_4(x))v_4 &\psi^{\prime\prime}(x):=\psi^{\prime\prime}(\lambda_1(x))v_1+\psi^{\prime\prime}(\lambda_2(x))v_2+\psi^{\prime\prime}(\lambda_4(x))v_4
C_{\ref{eq:c_D_var}}= \left( \frac{\log^2(2 D_{\min})}{1/2 - D_{\min}}\vee 2 \log^2 2 \right)+ \left( \frac{\log^2(2 - 2 D_{\max})}{D_{\max}- 1/2}\vee 2 \log^2 2 \right) &+ \left( \frac{\log(e/(2 D_{\min}))}{2 D_{\min}^2}+ \frac{\log(e/(2 - 2D_{\max}))}{(2 - 2 D_{\max})^2}\right),
H[\Psi,\xi] \!\!:=\!\! \frac{i}{4}\int[\bar{\Psi}(t,\vec{\bf r})\vec{\gamma}\cdot\vec{\nabla}\Psi(t,\vec{\bf r})-\vec{\nabla}\bar{\Psi}(t,\vec{\bf r})\cdot\vec{\gamma}\Psi(t,\vec{\bf r})]dr^3
a^{j + 1}_j \|u^{j+1}\|^2 \leq \sum^{j}_{s = 1}(a^{j + 1}_s - a^{j + 1}_{s-1})\|u^s\|^2+ a^{j + 1}_0 \Big(\|u^0 \|^2 + \frac{T^{\alpha}\Gamma(1 - \alpha)}{2c\ln 2}\|f^{j + \sigma}\|^2 \Big).
\displaystyle{1+\sqrt{\frac{2\pi}{g^2_{nc}\theta}}\sum_{l=1}^{+\infty}\frac{\exp(-\frac{\pi l}{g^2_{nc}\theta})}{\sqrt{l}}\left(1-\exp\left[i\frac{Lx}{\theta}+\frac{1}{l}\frac{g^2_{nc}L^2x^2}{16\pi^3\theta}\right]\right)}.
\begin{array}{l}{w_{2 i + 1}= x_{i}^{- 1}x_{i + 1}, i = 0 , 1 , 2 , \cdots , N - 1 ,}\\{w_{2 j}= y_{j}, j = 1 , 2 , \cdots , N - 1 ,}\\{w_{2 N}= \prod_{l = 1}^{N - 1}y_{l}^{- 1}.}\\ \end{array}
\dfrac{S^{(n+3)}(0)}{(n+3)!}=-(n+2)(n-1)\left( n^{2}+13n+6\right)+15(n+3)n\left( n^{2}+9n+2\right) -45(n+3)(n+2)n\left( n+5\right) +35(n+3)(n+2)(n+1)(n+2)+ \dfrac{1}{2}(n+7)(n+6)(n+5)(n+4)(n-1)= \dfrac{1}{2}\left( n-1\right) \left( n+4\right) p(n)
\begin{aligned}(X_{t_{i-1}+t})_{t\in[0,t_i-t_{i-1}]}=F(X_{t_{i-1}},V), (\widetilde X_{t_{i-1}+t})_{t\in[0,t_i-t_{i-1}]}=F(\widetilde X_{t_{i-1}},\widetilde V).\end{aligned}
\frac 38 I = 3M(z-m) +3z(M-m) + [ 2(z-m)^2 -2 (z-m)(M-z) -(M-z)^2 ] = 3M(1-M-2m) + 3(1-M-m)(M-m) + 2( 1-M-2m )^2 -2 (1-M-2m)(2M+m-1) - (2M+m-1)^2 = -4M^2 + 3 + 8mM -15m+14m^2 = -4M^2 + 3 + (-m)(15-8M )+14m^2 > 7\epsilon,
d^*(\mathfrak{D}f_1,\mathfrak{D}f_2) =sup \{|\mathfrak{D}f_1-\mathfrak{D}f_2|~u,v\in U\}&=\frac{25}{36}\leq \frac{2}{3}\times \frac{11}{6}&=\lambda d^*(f_1,f_2).
\sup_{K\geq 1}\,\sum_{j=0}^{{1\over \delta_K}-1}h_K G\left(\overline{(i_K-j)\delta_K}\,\log K\right)e^{L\,\log K\, \rho(j \delta_K,i_K \delta_K)}=\ :\ \overline G(\alpha)<+\infty.
\theta(q,a) =(q^2;q^2){}_\infty(a;q){}_{\infty}{}_{2}\phi_1\bigg[\genfrac{}{}{0pt}{}{-a/q,-a}{a^2};q^2,q^2\bigg], \theta(q,b) =(q^2;q^2){}_\infty(b;q){}_{\infty}{}_{2}\phi_1\bigg[\genfrac{}{}{0pt}{}{-b/q,-b}{b^2};q^2,q^2\bigg],
m_\gamma\leq A^{1-\frac{\gamma}{n}}\left(n^nA^{n+1}\right)^{\frac\gamma{n}}=\gamma^\gamma\big(\tfrac{n}{\gamma}\big)^\gamma A^{\gamma+1}\leq\gamma^\gamma\left(\tfrac32A\right)^{\gamma+1}A\geq2\|\Phi_2\|_2,
M =\left(\begin{array}{ccc|ccc|c}0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 \hline 0 0 0 0 0 0 0 0 0 0 0 1 -\frac{1}{a}m_{5,7}0 0 0 0 0 0 0 \hline0 0 0 0 0 0 0\end{array}\right),
\begin{aligned}\widetilde{H}^\textsc{cs}_\pm = -\frac{1}{2}\sum_{j=1}^N \partial_{x_j}^2 + \beta \sum_{i < j}^N \frac{\beta\mp P_{ij}}{4 \sin^2[(x_i - x_j)/2]}& = \frac{1}{2}\sum_{j=1}^N \, (z_j \, \partial_{z_j})^2 + \beta \sum_{i < j}^N \frac{z_i \, z_j}{z_{ij}\, z_{ji}}\, (\beta\mp P_{ij})\, .\end{aligned}
\frac{d}{ds}\frac{s}{h(s)}=\frac{d}{ds}\frac{1}{\int_\mathbb{R}\frac{1}{s^2+u^2}d\mu(u)}=2s\cdot\frac{\int_\mathbb{R}\frac{1}{(s^2+u^2)^2}d\mu(u)}{\left(\int_\mathbb{R}\frac{1}{s^2+u^2}d\mu(u)\right)^2}.
\lambda_{a|b|c|c}^{c|c|a|a}+ \lambda_{a|b|c|c}^{c|c|a|b}\lambda_{a|b|e|e}^{e|e|a|a}= \lambda_{a|b|e|e}^{e|e|a|a}+ \lambda_{a|b|e|e}^{e|e|a|b}\lambda_{a|b|c|c}^{c|c|a|a},
{\rm r.h.s.}= \left(\frac{A'_{i-1,j-1}}{A'_{i,j}}\prod_{k=i-a-1}^{b-j-1}\lambda_k\right)\left( \frac{A_{i-1,j-1}}{A_{i,j}}\prod_{k=i-a}^{b-j}\lambda_k\right)= p^*\big(L_{i,j}'L_{i,j}\big).
\begin{aligned}G(z) = A(z) \beta(z^{-1}) = A(z)\bigg( 1+ z \sum_n \frac{a_n \bar b_n t_n \nu_n}{z- t_n}\bigg) & =A(z) \bigg( 1+ \sum_n a_n \bar b_n t^2_n \nu_n \Big(\frac{1}{z- t_n}+\frac{1}{t_n}\Big) \bigg).\end{aligned}
C = 2\gamma-2\sum_{p}\frac{\log p}{p^2-p+1}\ \ + \frac{\zeta(2)\zeta(3)}{\zeta(6)}\sum_{k=1}^{\infty}\frac{\mu(k)}{k^2}\left(-2\sum_{p|k}\frac{(p-1)p\log p}{p^2-p+1}+\log k\right)\prod\limits_{p|k}\left(1+\frac{p-1}{p^2-p+1}\right).
\frac{i}{\eta_{22}}\left(\Gamma_{2s}^{1}\eta_{12}+\Gamma_{2i}^{2}\eta_{22}\right) =\frac{i}{2}\frac{\eta_{22,s}}{\eta_{22}}-\frac{i}{\eta_{12}}\left(\Gamma_{1s}^{1}\eta_{12}+\Gamma_{1s}^{2}\eta_{22}\right) =-\frac{i}{2}\frac{\eta_{12,s}}{\eta_{12}}.
[0,1]^{|V(F)|-||S_0(F)||}\ni(z_{v_1},\cdots,z_{v_p})\in\Delta_F\setminus S_0(F) :\Longleftrightarrow \left(\{v_1,\cdots,v_p\}=V(F)\setminus\{v'\in s_v|s_v \in S_0(F)\}~\wedge~(v_i\preceq v_j \Leftrightarrow z_{v_j}\leq z_{v_i})\right).
\begin{cases}\frac{1}{s_{i}}+\frac{\alpha_{i}}{n-1}>0, \beta_{i}\geq0,\end{cases}\quad\mathrm{or}\begin{cases}\frac{1}{s_{i}}+\frac{\alpha_{i}}{n-1}>0, \beta_{i}<0, \frac{1}{s_{i}}+\frac{\alpha_{i}+\beta_{i}}{n}>0,\end{cases}i=1,2,3,
R_{\kappa,i}(x;q)=\sum_{n\geq0}\frac{x^{\kappa n}q^{\kappa n+\kappa n^2+n-in}(1-x^iq^{2ni+i}) (-1)^nq^{\frac{n(n-1)}{2}}}{(q)_n(xq^{n+1})_{\infty}}.
\rho(x)n =\rho(s(x_{1})x_{1}\hdots s(x_{l-2})x_{l-2}) \rho(s(x_{l-1})x_{l-1}s(x_{l})x_{l})n &=\rho(s(x_{1})x_{1}\hdots(x_{l-2})x_{l-2}) s(x_{l-1})\rho(x_{l-1}s(x_{l})x_{l})n &=\rho(s(x_{1})x_{1}\hdots s(x_{l-2})x_{l-2})s(x_{l-1})x_{l-1}s(x_{l})x_{l}n &=\rho(s(x_{1})x_{1}\hdots s(x_{l-2})x_{l-2})n'
\sigma_{\alpha+\phi(\lambda)}(\vartheta)(\mathrm{W}) =a(\lambda)b(\lambda,\alpha)\sigma_\alpha(\vartheta)(\mathrm{W}+c^t(\lambda)), &=a(\lambda)b(\lambda,\alpha)T^*_{c^t(\lambda)}\circ\sigma_\alpha(\vartheta)(\mathrm{W}).
f(x) = \lim_{n}D_{\varphi \circ \pi}^{(g_{n})}(z)(f)\leq \lim_{n}\| f \|_{L^{\infty}(X,D_{\varphi \circ \pi}(z))}- \delta D_{\varphi}^{(g_{n})}(z)(A) &= \| f \|_{L^{\infty}(X,D_{\varphi \circ \pi}(z))}- \delta \delta_{\pi(x)}(A) = \| f \|_{L^{\infty}(X,D_{\varphi \circ \pi}(z))}- \delta
\frac{d \varepsilon}{d t}= \frac{q^{2}c^{3}}{2 A^{2}m{\frac{d \tilde{\tau}}{d t}}}\left[ \frac{d | \vec{E}|^{2}}{d t}+ 2 v^{j}E_{i}\partial_{i}E_{j}\right] \,{,}
f_{j}(z) = \prod_{k \neq j, k = 1}^{N+n}\frac{z-\varrho_{k}}{1 - z - \varrho_{k}}\frac{1 - \varrho_{j}- \varrho_{k}}{\varrho_{j}- \varrho_{k}}, j = 1, \ldots, N+n.
|\lambda_i^j(P)-\nu_{i-1}^j(P)|<\delta_{i-1}~\textrm{for all}~\lambda_i^j\in \mathfrak{M}_{\sigma}(M_i^j)~\textrm{and}~P\in \bigcup_{t\leq t_{i-1}}Bl_t(M_{i-1})(j=1, \cdots, k).
\frac12\frac{d}{dt}\|{\Delta}_j b\|^2_{L^2}\leq \|[{\Delta}_j,\,u\cdot \nabla]b\|_{L^2}\|{\Delta}_j b\|_{L^2}+\|{\Delta}_j(b\cdot \nabla u)\|_{L^2}\|{\Delta}_j b\|_{L^2}\|{\Delta}_j b\|_{L^2}.
u=[\begin{matrix}e^{i2t}s+e^{-i2t}\overline{s}\\ \frac{1-i}{2}e^{i2t}s+\frac{1+i}{2}e^{-i2t}\overline{s}\\ -\frac{i}{2}e^{i2t}s+\frac{i}{2}e^{-i2t}\overline{s}\end{matrix}]+O(\alpha+|s|^{2})
\Pr(Y \geq X) =\sum_{k = \lfloor np - b_p \rfloor}^{\lceil np + b_p \rceil}\sum_{\ell = \lfloor nq - b_q\rfloor}^{\lceil nq + b_q \rceil}1_{\{k \leq \ell\}}\Pr(X = k) \Pr(Y = \ell) +o(n^{-1})
\langle hw_\eta^q\rangle_{x,\eta}=\langle \langle h(\eta)|\eta \cdot w(x)|^q\rangle_\eta\rangle_x & \geq \langle \langle h(\eta)|\eta \cdot w(x)|^q\mathbf{1}_{A_{t,x}}(\eta)\rangle_\eta\rangle_x & \geq \langle c^q |w(x)|^q \langle h(\eta)\mathbf{1}_{A_{t,x}}(\eta)\rangle_\eta\rangle_x =C\langle |w|^q\rangle_x, C>0.
\iint_{\Omega}\mathcal{A}\nabla u\cdot \nabla (u-u_j)\frac{dtdx}{|t|^{n-d-1}}=\iint_{\Omega}\mathcal{A}^j\nabla u_j\cdot \nabla (u-u_j)\frac{dtdx}{|t|^{n-d-1}}=0.
R_{\sf t t}= - \frac{d - 1}{a}\left( \frac{d^{2}a}{d{\sf t}^{2}}\right) \; , R_{i j}= \left[ a \left( \frac{d^{2}a}{d{\sf t}^{2}}\right) + ( d - 2 ) \left( \frac{d a}{d \sf t}\right)^{2}+ ( d - 2 ) k \right] \gamma_{i j}\; .
\begin{aligned}I_{Vf}(x,t):=I_{3}-I_{4}=\int_{0}^{t}\int_{\partial\Omega}\bigg( \frac{\partial \varepsilon_{n}\big(x-\xi,b(t,\tau)\big)}{\partial\nu(\xi)}a(\tau)(Vf)(\xi,\tau) &-\varepsilon_{n}\big(x-\xi,b(t,\tau)\big)a(\tau)\frac{\partial (Vf)(\xi,\tau)}{\partial\nu(\xi)}\bigg) dS_{\xi}d\tau=0.\end{aligned}
B_{k,0}= D_{k-1}B_{k-1,0}=D_{k-1}D_{k-2}\cdots D_2 B_{k,i}= D_{k-1}B_{k-1,i}+B^2_{k-1,i-1}B_{k,k-1}= B_{k-1,k-2}=\cdots=B_{2,1}=1.
\tilde{z}_1+\tilde{z}_2=3^{3b+1}\alpha^3,\ \tilde{z}_1^2-\tilde{z}_1\tilde{z}_2+\tilde{z}_2^2=3\beta^3,\ \alpha\beta=\tilde{z}_4,\ 3\nmid \alpha\beta,\ \gcd(\alpha,\beta)=1.
| B , p = 0 \rangle = \int D x ( \sigma ) \exp \left({\frac{i}{8 \pi}}F_{\mu \nu}{\int_{0}}^{2 \pi}d \sigma ~ x^{\mu}( \sigma ) \cdot \partial_{\sigma}x^{\nu}- i{\int_{0}}^{2 \pi}d \sigma P_{\mu}\cdot x^{\mu}\right) | B \rangle_{- 1}
\Vert g \Vert_{W^{\alpha,2}}= \left(\int_0^T \vert g(t)\vert^2 \mathrm{d}t + \int_0^T \int_0^T \frac{\vert g(r)-g(s)\vert^2}{\vert r - s \vert^{2\alpha+1}}\mathrm{d}r\mathrm{d}s\right)^\frac{1}{2}.
\begin{array}{c}{{\overline{{{A}}}^{\prime 1}{}_{\nu}\longrightarrow W_{\nu}^{-},}}\\{{\overline{{{A}}}^{\prime 2}{}_{\nu}\longrightarrow W_{\nu}^{+},}}\\{{\overline{{{A}}}^{\prime 3}{}_{\nu}\longrightarrow Z_{\nu},}}\\{{\overline{{{A}}}^{\prime 4}{}_{\nu}\longrightarrow A_{\nu}.}}\\ \end{array}
\vec{B_0}(\mathbf{0}) = (\tilde{g}_u^{ij}-g^{ij})\partial_j\psi \cdot\partial_i|_{\mathbf{0}}; \ B_1(\mathbf{0}) = \frac12\tilde{g}_u^{ij}\partial_i(\log \det[\tilde{g}_u])\partial_j\psi(\mathbf{0})\,,
\sum_{T_t \in \mathcal{C}(p_1,\ldots, 2q_N)}1 \ \ =\ \ \binom{r}{p_1,\ldots,2q_N}< \frac{r!}{(p_1)!\cdots (p_n)!}\frac{1}{(p/2)!}\binom{p/2}{q_{n+1},\ldots,q_N}.
\varepsilon_{BOLD}=\frac{1}{T}\sum_{t=1}^{T}\frac{|BOLD_{simulation}(t)-BOLD_{real}(t)|^2}{|BOLD_{real}(t)|^2}. \varepsilon_{h}=\frac{1}{T}\sum_{t=1}^{T}\frac{|h_{simulation}(t)-h_{real}(t)|^2}{|h_{real}(t)|^2}.
\dfrac{z+\psi_{q_{\zeta}W}\left( z\right)}{1+\varphi_{W}\left( z\right)}=\dfrac{\dfrac{1}{\zeta}\left( \left( \zeta+a_{1}\right) \left(m_{W}(z)-m_{W}(\zeta)\right) +\zeta^{2}-z^{2}\right)}{1-\dfrac{z}{\zeta}}& =\dfrac{\left( \zeta+a_{1}\right) \left( m_{W}(z)-m_{W}(\zeta)\right)}{\zeta-z}+\zeta+z
\left[{\cal D}^{\mu}\cdot \Gamma \right]_{\mathrm{f l a t}}= x^{\nu}\left[{\Theta}_{\nu}^{\mu}\left( x \right) \cdot \Gamma \right]_{\mathrm{f l a t}}- x^{\nu}\left[ w_{\nu}^{\mu}\left( x \right) \cdot \Gamma \right]_{\mathrm{f l a t}}- \left[ \Lambda^{\mu}\left( x \right) \cdot \Gamma \right]_{\mathrm{f l a t}},
\dim\overline{N}=\binom{c+a-e+3}{3}+\binom{c+b-e+3}{3}+\binom{c-a+3}{3}&+\binom{c-b+3}{3}-\binom{a+b-e+3}{3}-\binom{b-a+3}{3}-\binom{2a-e+3}{3}&-\binom{2b-e+3}{3}-3-t(e,a,b),
\frac{F l^{4}}{E^{2}( x - y )^{4}}\left[ \frac{1}{G}( \frac{\mathrm{d}x}{\mathrm{d}\tau})^{2}+ \frac{1}{F}( \frac{\mathrm{d}y}{\mathrm{d}\tau})^{2}\right] = \frac{E^{2}- V^{2}}{E^{2}}
\alpha_{i,\ell}:=w_{i,\ell}\frac{\vert \lambda_{\ell+1}-\lambda_{j(\ell+1)}\vert}{\vert \lambda_{\ell}-\lambda_{j(\ell)}\vert}\ \ \ \ \ \mathrm{for\all\}\i\ge 1\ \mathrm{and\}\ell\ge 2,
\begin{aligned}\Vert \nabla \partial_Y w_2 \Vert_{L^2}\lesssim 1+\Vert w_1\Vert_{H^1}+\sup_Y\left|\frac{h^0_e}{u^0_e}\right|\Vert \nabla \partial_Y w_1\Vert_{L^2}.\end{aligned}
u(t) = \frac{1}{4 \cdot 0!}- \frac{1}{4}+ \left( \frac{1}{4 \cdot 1!}+1 \right) t + \frac{1}{4 \cdot 2!}t^2 + \frac{1}{4 \cdot 3!}t^3 + \frac{1}{4 \cdot 4!}t^4 + \dots + \frac{1}{4 \cdot k!}t^k + \dots = t - \frac{1}{4}+ \frac{1}{4}\sum\limits_{k=0}^{\infty}\frac{1}{k!}t^k,
\|f\|_{C^\tau}:=\displaystyle\sum_{|\alpha|\le [\tau]}\|\partial^\alpha_{x}f\|_{L^\infty(\mathbb{R}^n)}+\displaystyle\sum_{|\alpha|= [\tau]}\sup_{x\neq y}\frac{|\partial^\alpha_{x}f(x)-\partial^\alpha_{x}f(y)|}{|x-y|^{\tau-[\tau]}}<\infty.
- \int d^{1 0}\sigma \sqrt{- \mathrm{d e t}\, \big ( \eta_{\mu \nu}+ F_{\mu \nu}- 2 \bar{\lambda}\Gamma_{\mu}\partial_{\nu}\lambda + \bar{\lambda}\Gamma^{\rho}\partial_{\mu}\lambda \bar{\lambda}\Gamma_{\rho}\partial_{\nu}\lambda \big )}.
\theta_{ab}=\left(\begin{array}{ccccc}0 \theta_1 0 0 0 -\theta_1 0 0 0 0 0 0 0 \theta_2 0 0 0 -\theta_2 0 0 0 0 0 0 \theta_{ij}\end{array}\right) ,\i, j=5,\dots, 2p \ .
\begin{aligned}\dot{z}= z- (i a_{20}- b_{20}) z^2 -\frac{1}{4}( r_{11}+ 2 i r_{20}) z^3 +\frac{1}{4}(r_{11}-2 i r_{20}) z w^2, \dot{w}= -w +\frac{1}{2}( i b_{11}-2 i a_{20}) z^2 - \frac{1}{2}( i b_{11}+ 2 b_{20}) w^2-\frac{1}{4}(r_{11}+ 2 i r_{20}) z^2 w + \frac{1}{4}(r_{11}- 2 i r_{20}) w^3.\end{aligned}
d_{\Sigma,1}=\max_{(d^{[{\sf d}]}_1,\cdots,d_{N_1}^{[{\sf d}]},d^{[{\sf u}]}_1,\cdots,d_{N_2}^{[{\sf u}]})\in\mathcal{D}}\left\{\sum_{i=1}^{N_1}d_i^{[{\sf d}]}+\sum_{j=1}^{N_2}d_j^{[{\sf u}]}\right\},
x_1^2 y_1^2 (x_1 y_2 + x_2 y_1) + x_1 y_1(x_1^2 y_2^2 + x_2^2 y_1^2 + a x_1 x_2 y_1 y_2) + + b(x_1^3 y_2^3 + x_2^3 y_1^3) + c x_1 x_2 y_1 y_2 (x_1 y_2 + x_2 y_1) + + x_2 y_2 (d(x_1^2 y_2^2 + x_2^2 y_1^2) + ex_1 x_2 y_1 y_2) + f x_2^2 y_2^2 (x_1 y_2 + x_2 y_1) =0 \, .
\frac{d \phi^2}{d \phi^1}=-\frac{\partial W^1}{\partial \phi^1}\left( \frac{\partial W^1}{\partial \phi^2}\right)^{-1}\hspace{0.5cm}\equiv \hspace{0.5cm}\frac{\partial W^1}{\partial\phi^1}d\phi^1+\frac{\partial W^1}{\partial \phi^2}d\phi^2 =dW^1=0
\P_x(\tau_{\Gamma}>t) = \P_{t^{-1/\alpha}x}(\tau_{\Gamma}>1) \leq c \left(1+\left(t^{-1/\alpha}|x|\right)^{\alpha-\beta}\right) M_{\Gamma}(t^{-1/\alpha}x) &= c\left(t^{-\beta/\alpha}+t^{-1}|x|^{\alpha-\beta}\right)M_{\Gamma}(x).
h^\mu_{~\nu}= \left [\begin{array}{ccccc}h_1 h_3 0 0 0 -d^2 h_3 /f h_2 0 0 0 0 0 h^\chi_{~\chi}h^\chi_{~\theta}h^\chi_{~\phi}0 0 h^\theta_{~\chi}h^\theta_{~\theta}h^\theta_{~\phi}0 0 h^\phi_{~\chi}h^\phi_{~\theta}h^\phi_{~\phi}\end{array}\right ]
\iota_{\xi}\, E^{A}\, = \, \xi^{A}\ \Rightarrow \ \iota_{\xi}\, \Omega_{p}\, = \, \frac{1}{( p - 1 ) !}E^{A_{1}}. . . E^{A_{p - 1}}\xi^{A_{p}}\, \Omega_{A_{p}. . . A_{1}}.
s = \sqrt{3},B_1 =\left(\begin{tabular}{rcr}$1+s$ $3+s$ $-1+s$ $3-s$\end{tabular}\right),\; \textrm{and}B_2=\left(\begin{tabular}{rcr}$3-s$ $1-s$ $-3-s$ $1+s$\end{tabular}\right).
\hat{E}(z) = \frac{1}{1-C + C \sqrt{1-z}}= \frac{[1-C] - C \sqrt{1-z}}{(1-2C) + C^2 z}= \frac{[1- C] - C \sqrt{1-z}}{2C -1}\sum_{k=0}^{\infty}\left(\frac{C^2}{2C -1}\right)^k z^k.
D_{2k;w_1,w_2}^{e^{2\Upsilon}g}\bigl( u \otimes v \bigr) = e^{(w_1+w_2-2k)\Upsilon}D_{2k;w_1,w_2}^{g}\bigl( e^{-w_1\Upsilon}u \otimes e^{-w_2\Upsilon}v \bigr)
\psi : \mathcal{J}_{K}^{\mathfrak{m}}\longrightarrow D\ell_{K}^{\mathfrak{m}}\alpha=(\alpha_{\mathfrak{p}}) \longmapsto \psi(\alpha)=\prod_{\textrm{place finie de $K$}}\mathfrak{p}^{\widetilde{v_{\mathfrak{p}}}(\alpha_{\mathfrak{p}})}
t r K ( t ) = ( 4 \pi t )^{- 3 / 2}\sum_{k = 0 , 1 / 2 , 1 , . . .}^{\infty}( \int_{B^{3}}d v a_{k}+ \int_{S^{2}}d s c_{k}) \exp ( - t V^{\prime \prime}( \phi ) ) t^{k},
e^{i}= g(\pi_{L}(\cdot), e_{i}), \eta^{*}= g(\xi, \pi_{\xi}(\cdot)) = \eta (\pi_{\xi}(\cdot)) = \eta - \eta \circ \pi_{L}, e^{i}= f^{i}\circ \phi, f^{i}= -e^{i}\circ \phi.
Z_{c}= \lim_{L \rightarrow \infty}\int \exp \{i \int_{- L}^{L}d t \sum_{\alpha}- ( \frac{i}{2}\frac{\partial \bar{c}^{\alpha}}{\partial t}+ m_{\alpha}\bar{c}^{\alpha}) c^{\alpha}+ h . c . + \chi_{\alpha}( \bar{c}^{\alpha}+ c^{\alpha}) \}d \bar{c}^{\alpha}d c^{\alpha}
\int\limits_{\mathbb{R}^{N}}\left\vert v\right\vert^{2(q-1)}\frac{dx}{\left\vert x\right\vert^{s}}=d^{q}\int\limits_{\mathbb{R}^{N}}\frac{\left\vert u\right\vert^{2(q-1)}}{\left\vert x\right\vert^{N+sd-Nd}}dx
{\mathbb{W}_{\kappa}}:=\begin{cases}\widetilde{\mathbb{W}}_{\kappa}\oplus \left(\mathbb{W}^{\ell,\left(\frac{-}{\ell}\right)}\otimes_{\Lambda}\mathcal{R}_{\mathcal{P}_{\kappa}}\right)[f_\kappa] \textrm{if $r=1$ and $\psi_\ell$ is the trivial character,}\widetilde{\mathbb{W}}_{\kappa}\textrm{otherwise,}\end{cases}
\Phi^{T^*Q}_g(\alpha_q) =T^*_{g\cdot q}\Phi_{g^{-1}}(\alpha_q), \Phi^{(T^1_k)^*Q}_g(\alpha^1_{q},\dots,\alpha^k_{q}) =(T^*_{g\cdot q}\Phi_{g^{-1}}(\alpha^1_q),\dots,T^*_{g\cdot q}\Phi_{g^{-1}}(\alpha^k_q)).
V_{n m}^{<}( B ) = \frac{- 1}{2 \pi^{2}l^{3}}\int_{0}^{1}d \eta \int_{0}^{1}d u \frac{u^{2}\left[ f ( u ; r ) \right]^{n - 2 m}\left[ g ( u ; r ) \right]^{m}}{\left[ ( r u + \frac{\pi}{2})^{2}+ \bar{c}^{2}\right]^{n}}.
P_{r h}^{-}={\frac{g^{2}L}{4 \pi^{2}}}\left[ \sum_{N \neq 0}{\frac{C_{N}^{3}C_{- N}^{3}}{N^{2}}}+ \sum_{N}\left[{\frac{C_{N}^{+}C_{- N}^{-}}{( Z_{R}+ N )^{2}}}+{\frac{C_{N}^{-}C_{- N}^{+}}{( Z_{R}- N )^{2}}}\right] + \Pi_{R}^{2}\right]
(\frac{1}{2}iax^{2}+iJx)=\frac{1}{2}ia(x^{2}+\frac{2Jx}{a}+(\frac{J}{a})^{2}-(\frac{J}{a})^{2})=-\frac{1}{2}\frac{a}{i}(x+\frac{J}{a})^{2}-\frac{iJ^{2}}{2a}.
Q_0^1=\frac{1}{2}(A_6x_1+B_6x_2+C_6)+\frac{1}{4}(A_1Jx_1+B_1Jx_2+C_1J), &P_2^1= -\frac{1}{2}(A_1A_6x_1^2+(A_1B_6+B_1A_6)x_1x_2+B_1B_6x_2^2+(C_1A_6+A_1C_6)x_1 &\quad+(C_1B_6+B_1C_6)x_2+C_1C_6+N),
[\begin{matrix}J_{x}\\J_{y}\\J_{z}\end{matrix}]=[\begin{matrix}\sigma_{xx}&\sigma_{xy}&\sigma_{xz}\\ \sigma_{yx}&\sigma_{yy}&\sigma_{yz}\\ \sigma_{zx}&\sigma_{zy}&\sigma_{zz}\end{matrix}][\begin{matrix}E_{x}\\E_{y}\\E_{z}\end{matrix}]
\begin{aligned}p(P)= \frac{1}{\pi R^{2}\sin^{2}\theta}\int_{0}^{R\sin\theta}\int_{0}^{2\pi}p(r',\theta')r'dr'd\theta' &+\frac{2}{\pi R^{2}\sin^{2}\theta}\int_{0}^{R\sin\theta}\int_{0}^{r'}\int_{0}^{2\pi}\frac{(\hat{u}-u)v_{\theta'}}{\rho}r'd\theta'd\rho dr'\doteq I_{1}'+I_{2}'.\end{aligned}
\bar{z}_{tt}= \partial_t \bar{z}_t=\partial_t \Big(F(z(\alpha,t),t)-\sum_{j=1}^N \frac{\lambda_j i}{2\pi(z(\alpha,t)-z_j(t))}\Big) = F_z(z(\alpha,t),t) z_t+F_t+\sum_{j=1}^N \frac{\lambda_j i(z_t(\alpha,t)-\dot{z}_j(t))}{2\pi (z(\alpha,t)-z_j(t))^2}
L_{\mu\nu}[\xi|s] \longrightarrow L_{\mu\nu}[\xi|s] + \sqrt{6}(4\pi \bar{N}\dot{\xi}(s)^2)^{-1}\epsilon_{\mu\nu\rho\sigma}{\cal D}^\sigma(s) \{\Phi_\xi^{-1}(s_+,0) \tilde{\Lambda}^\rho[\xi|s] \Phi_\xi(s_+,0) \}.
\frac{R_{(1,-1)}(\frac{m-1}{2}-1, 1;q)\cdot R_{(2,-1)}(\frac{m-1}{4},\frac{m-1}{4};q)}{R_{(2,-1)}(\frac{m-1}{4}+1,\frac{m-1}{4}-1;q)}= \frac{(q^m;q^m)_\infty}{(q)_\infty}\cdot \theta(q^{\frac{m-1}{2}};q^m) \cdot \frac{1}{\theta(q^\frac{m-1}{2};q^m)}&= \frac{(q^m;q^m)_\infty}{(q)_\infty}&= \prod_{t=1}^\infty \frac{(1-q^{mt})}{(1-q^t)}.
\lfloor r_i u \rfloor = \lfloor r_i x_0 \rfloor\xleftarrow{z_1 \gamma_1^\lor}\cdots \xleftarrow{z_{p-1}\gamma_{p-1}^\lor}\lfloor r_i x_{p-1}\rfloor = x_p\xleftarrow{\gamma_{p+1}^\lor}\cdots \xleftarrow{\gamma_r^\lor}x_r =v.
\int \exp \{a^{4}\sum_{i = 1}^{2}\sum_{x}\bar{\psi}_{i}( x ) ( \hat{D}+ m ) \psi_{i}( x ) \}d \bar{\psi}d \psi = \lim_{L \rightarrow \infty , b \rightarrow 0}\int \exp \{- S \}d \phi_{n}^{*}d \phi_{n}d \chi^{*}d \chi
I_{l_1,l_2,k,n}< cq^{\frac{1}{4}(n-2k-2l_2)^2+(k-2l_1)(n-2k-2l_2)+2l_1(n-2k-2l_2)+2l_2(k-2l_1)+4l_1l_2+2l_1^2+2l_1(k-2l_1)+(k-2l_1)^2}&= cq^{\frac{1}{4}n^2-l_2(n-2k-l_2)-2l_1(k-l_1)}.
\psi ( x ) = \widetilde \phi ( x ) - \phi ( x ) = t_{1}\left({K^{- \frac{1}{{R^{2}}}}- 1}\right) \cos \left({\frac{x}{R}}\right) + t_{2}\left({K^{- \frac{4}{{R^{2}}}}- 1}\right) \cos \left({\frac{{2 x}}{R}}\right) + \cdots ,
\theta^{5}=\frac{\xi^{2}}{[1+\frac{\xi^{2}}{3}]^{3/2}}+\frac{3\xi^{2}}{9[1+\frac{\xi^{2}}{3}]^{5/2}}=\frac{9}{9[1+\frac{\xi^{2}}{3}]^{5/2}}
D_{\; \; i j}^{k l}= \frac 1 2 \left( \delta_{\; \; i}^{\left[ k \right.}\delta_{\; \; j}^{\left. l \right]}- \frac 1{\triangle}\delta_{\; \; m}^{\left[ l \right.}\partial_{}^{\left. k \right]}\delta_{\; \; \left[ j \right.}^{m}\partial_{\left. i \right]}^{}\right) .
R[ U_{1}] / R[ V_{1}] \cong ( R [ U_{1}] + R [ V_{2}] ) / ( R [ V_{1}] + R[V_{2}] ) &= ( R [ U_{2}] + R [ V_{1}] ) / ( R [ V_{1}] + R[V_{2}] ) & \cong R [ U_{2}] / R [ V_{2}].
\Z=H^{-1}\left(\begin{array}{ccc}(1-z)(i\cal E \rm +1)-{\rm \Phi}^2 \hspace{5mm}i\cal E\it+z i\cal E\it+z \hspace{5mm}(1+z)(i\cal E \rm -1)+{\rm \Phi}^2 \end{array}\right),
U(a,b;z)=\frac{\pi}{\sin(\pi b)}\left(\frac{{}_{1}F_{1}(a,b;z)}{\Gamma(1+a-b)\Gamma(b)}-\frac{x^{1-b}{}_{1}F_{1}(1+a-b,2-b;z)}{\Gamma(a)\Gamma(2-b)}\right).
\eta_{ij}^{\pm}+\eta_{ji}^{\pm}=1 + \delta_{x_i,x_j}, (\eta_{ij}^{\pm})^{2}= \eta_{ij}^{\pm}, \eta_{ij}^{\pm}\eta_{jk}^{\pm}= \eta_{ik}^{\pm}\eta_{ij}^{\pm}+\eta_{jk}^{\pm}\eta_{ik}^{\pm}- \eta_{ik}^{\pm},
\gamma^{\beta}({- \triangle}) = \int_{0}^{\infty}{\frac{{\mathrm{d}}s}{s}}\left[ \theta_{3}\Big ( 0 ,{\mathrm{e}}^{- \frac{\beta^{2}}{4 s}}\Big ) - 1 \right]{\mathrm e}^{- s m^{2}}f ( - s \triangle ) ,
\|\nabla \bar{\rho}\|^2 = - \langle P_N \bar{\rho}, \Delta P_N \bar{\rho}\rangle - \langle Q_N \bar{\rho}, \Delta Q_N \bar{\rho}\rangle= \|\nabla P_N\bar{\rho}\|^2 + \|\nabla Q_N\bar{\rho}\|^2 \geq\|\nabla Q_N\bar{\rho}\|^2 \geq \lambda_N \|Q_N\bar{\rho}\|^2 \,,
T_{i+1}> N_{i+1}=N_i\geq T_i+\eta_i l_{i,1}\geq \, l_{i,1}+\dots+l_{i,M+1}+(L_1+\dots+L_{M}-1)l_{i,1}= (L_1 l_{i,1}+l_{i,2})+\dots+(L_{M}l_{i,1}+l_{i,M+1}) = \, l_{i+1,1}+\dots+l_{i+1,M}=T_{i+1}
N(Z,W) =\frac{1}{2}\Big{(}M(Z,W)+M(W,Z) \Big{)}&= \frac{1}{2}\Big{(}M(Z,W)+ J_Z J_W(\nabla_WJ)_Z \Big{)}&=\frac{1}{2}\Big{(}M(Z,W)- ( -J_WJ_Z-2 \langle Z,W \rangle ) (\nabla_ZJ)_W \Big{)}&=M(Z,W)+ \langle Z,W \rangle (\nabla_ZJ)_W.
t_k(n)\sim \begin{cases}\dfrac{1}{2^{2k}E_{2k}}\cdot\sigma_{\chi_{-4};2k}(2n+k+1), k\equiv 0\pmod{2}, -\dfrac{1}{E_{2k}}\cdot \sigma_{\chi_{-4};2k}(2n+k+1), k\equiv 1\pmod{2},\end{cases}
(\sigma_{\rm HHJ}-\sigma,\tau)= \sum_{K\in\mathcal{T}}\int_{\partial K}M_{\nu t}(\tau)\frac{\partial(u_{\rm HHJ}-\Pi_{\rm D}u)}{\partial t}ds = -\sum_{K\in\mathcal{T}}\int_{\partial K}M_{\nu\nu}(\tau)\frac{\partial(u_{\rm HHJ}-\Pi_{\rm D}u)}{\partial\nu}ds.
(b^{*_\sigma})^{*_\sigma}=\big(\overline{\sigma(g,g^{-1})}b^*\big)^{*_\sigma}=\overline{\sigma(g^{-1},g)}\big(\overline{\sigma(g,g^{-1})}b^*\big)^* &=\overline{\sigma(g^{-1},g)}\sigma(g,g^{-1})(b^*)^*=b.
(1+\delta)\|T^\ast \omega\|_{\varphi_1}^2 + \|S\omega\|_{\varphi_3}^2 \ge \int_{\Omega_\varepsilon}\sum_{j,k=1}^n \left(\frac{\partial^2\varphi}{\partial z_j\partial \bar{z}_k}+\frac12 \frac{\partial^2\kappa\circ s}{\partial z_j\partial \bar{z}_k}\right)\omega_j\bar{\omega}_k e^{-\varphi'}.
< \alpha ( x , 0 ) > \equiv{\frac{< \psi | \alpha ( x , 0 ) | \psi >}{< \psi | \psi >}}={\frac{2}{\hbar L}}\sum_{n = 1}^{\infty}\mathrm{R e}[ n ~ C_{n}~ e^{2 \pi ~ i ~ n ~ x / L}]
\frac{\hat{r}u_{0}\eta_{0}}{\omega D_{0}}=\frac{\hat{r}}{4\omega D_{0}}(\hat{u}_{0}^{*}\hat{\eta}_{0}+\hat{u}_{0}\hat{\eta}_{0}^{*})+\frac{\hat{r}}{4\omega D_{0}}(\hat{u}_{0}\hat{\eta}_{0}e^{-2it}+\hat{u}_{0}^{*}\hat{\eta}_{0}^{*}e^{2it})
\mathcal{S}_{coord}:= \bigcap \{\mathcal{S}\subseteq \mathbb{S}^n : \mbox{$\mathcal{S}$ is admissible and a coordinate subspace}\}, \mathcal{S}_{part}:= \bigcap \{\mathcal{S}\subseteq \mathbb{S}^n : \mbox{$\mathcal{S}$ is admissible and a partition subspace}\}.
\phi \left(\left[\begin{array}{cc}\tilde{A}0 0 0\end{array}\right]\right) = \left[ \begin{array}{cc}Q 0 0 1\end{array}\right]\left[\begin{array}{cc}D\tilde{A}D^{-1}0 0 0\end{array}\right]\left[ \begin{array}{cc}Q^T 0 0 1\end{array}\right]\textrm{for all $\tilde{A}\in M_2$,}
I_{E}= - \frac{\pi}{4}\int_{\sigma = 0}^{\sigma_{\partial M}}\left[ ( f(R) - R f'(R)) b^{3}+ 6 \left (f'(R) b \dot{b}^{2}+ f''(R) \dot{R}b^{2}\dot{b}+ b f'(R) \right )\right] d \sigma
Q^T L_5^\prime(\lambda) Q = \left[\begin{array}{ccc|cc}\lambda P_5+P_4 P_3 P_2 -I_n 0 P_3 -\lambda P_3+P_2 -\lambda P_2+P_1 \lambda I_n -I_n P_2 -\lambda P_2+P_1 -\lambda P_1+P_0 0 \lambda I_n \hline-I_n \lambda I_n 0 0 0 0 -I_n \lambda I_n 0 0\end{array}\right],
\kappa (X, K_X+\Delta) =\kappa (X', K_{X'}+\Delta') &\geq \kappa (X', a(K_{X'}+\Delta')-af'^*K_{Y'}+f'^*H) &\geq \kappa (X'_y, K_{X'_y}+\Delta'|_{X'_y})+\dim Y' &=\kappa (X_y, K_{X_y}+\Delta|_{X_y})+\dim Y.