harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\nabla^V_X \left(\nabla^V t\right) (Y)= \nabla^V_X \left(\nabla^V_Y t\right) -\nabla^V_{\nabla_X Y}t = \nabla^Q_{df(X)}\left(\nabla^Q_{df(Y)}t\right) -\nabla^Q_{\tilde{\nabla}_X Y-(\nabla_X df)(Y)}t = \nabla^Q_{df(X)}\left(\nabla^Q t\right) (df(Y))+\nabla^Q_{(\nabla_X df)(Y)}t = H_YK_X t-H_{(\nabla_X df)(Y)}s.
{\bf M}_2(y) = \left[\begin{array}{cccccc}y_{00}y_{10}y_{01}y_{11}y_{20}y_{02}y_{10}y_{20}y_{11}y_{21}y_{30}y_{12}y_{01}y_{11}y_{02}y_{12}y_{21}y_{03}y_{11}y_{21}y_{12}y_{22}y_{31}y_{13}y_{20}y_{30}y_{21}y_{31}y_{40}y_{22}y_{02}y_{12}y_{03}y_{13}y_{22}y_{04}\end{array}\right]
\left|\frac{1}{N}Tr\left(Q\left({\bf X}_N,Y{\bf Y}_N,{\bf Y}_N^{-1}\right)\right)-\right.\left.\frac{1}{N}Tr\left(Q\left({\bf X}_N,{\bf Y}_N,R({\bf Y}_N)\right)\right)\right|=\left|\frac{1}{N}Tr\left(P({\bf X}_N,{\bf Y}_N)\left({\bf Y}_N^{-1}-R({\bf Y}_N)\right)\right)\right|
\begin{cases}\partial_t W + (\Delta^{-1}\nabla^{\perp}W \cdot \nabla) W =0, W \Bigr|_{t=0}=W_0(x) = f^1(x) + f^2(x-x_W);\end{cases}&\begin{cases}\partial_t \omega^i + (\Delta^{-1}\nabla^{\perp}\omega^i\cdot\nabla) \omega^i=0, \omega^i \Bigr|_{t=0}=f^i(x),\end{cases}
ln(\begin{matrix}N\\x_{1},\cdot\cdot\cdot x_{n}\end{matrix})p_{1}^{x_{1}}\cdot\cdot\cdot p_{n}^{x_{n}}\approx-\frac{N}{2}\sum_{i}\frac{(\hat{p}_{i}-p_{i})^{2}}{p_{i}}=-\frac{1}{2}\sum_{i}\frac{(x_{i}-Np_{i})^{2}}{Np_{i}}
x y^{2}= \left[ x^{n_{c}}( x - \phi_{n_{c}}^{2}) \right]^{2}- 4 \Lambda^{4}x^{2 n_{c}}= x^{2 n_{c}}( x - \phi_{n_{c}}^{2}- 2 \Lambda^{2}) ( x - \phi_{n_{c}}^{2}+ 2 \Lambda^{2}) .
H_{m,n}\left(w_{1},w_{2}\right)=\begin{cases}\left(-1\right)^{n}n!w_{1}^{m-n}L_{n}^{\left(m-n\right)}\left(w_{1}w_{2}\right), m\ge n \left(-1\right)^{m}m!w_{2}^{n-m}L_{m}^{\left(n-m\right)}\left(w_{1}w_{2}\right), n\ge m\end{cases}
E_{C}^{( 2 ) r e g}= - \frac{M_{0}}{4 \sqrt{\alpha}}- \frac{M_{0}^{2}R}{8 \pi \alpha}\Gamma ( - 1 ) - \frac{M_{0}}{2 \pi \sqrt{\alpha}}\sum_{n = 1}^{\infty}n^{- 1}K_{1}\left( \frac{2 n M_{0}R}{\sqrt{\alpha}}\right) .
\begin{array}{rcll}R_{ijkl}+ R_{jikl}= 0 \ \ \ \ \ \ R_{ijkl}+ R_{ijlk}= 0 \ \ \ \ \ R_{ijkl}+R_{iklj}+R_{iljk}= 0 \ \ \ \ \ R_{ijkl}- R_{klij}= 0. \ \end{array}
\partial_t^+\xi^n =\partial_t^+u_{Ih}^n(u_{Ih}^n+u_{Ih}^{n-1})+\partial_tu_{Ih}^nu_h^{n-1}+\Delta t (\partial_t^+u_h^n)^2 &\quad+\partial_t^+u_h^n(u_{Ih}^n+2u_{Ih}^{n-1}-2e_h^{u^{n-1}}).
0 = \varepsilon \circ \psi(uvuv^{-1}[a_1,a_2]\cdots [a_{2m-1},a_{2m}])=\varepsilon \circ \psi(uvuv^{-1}) &= \varepsilon \bigl( w_u \ldotp g^{2k+1}w_v g^{-2k-1}\ldotp g^{2k+1+b}w_u g^{-2k-1-b}\ldotp\Delta_{4k+2}^2 \ldotp w_v^{-1}\bigr)= 2\varepsilon(w_u)+\varepsilon (\Delta_{4k+2}^2).
\left[\begin{array}{ccc}0 0 0 r_1 0 0 r_1 r_2 0\end{array}\right]\left[\begin{array}{c}x_1 x_2 x_3\end{array}\right]=\left[\begin{array}{c}r_1-1 r_2-1 r_3-1\end{array}\right] \, .
C_{2}(\vec{\varepsilon})=\left(1-(p-1)!!\varepsilon_{p}\sigma^{p}+\frac{\varepsilon_{q}^{2}}{2}(2q-1)!!\sigma^{2q}+\frac{\varepsilon_{p}^{2}}{2}(2p-1)!!\sigma^{2p}\right)^{-1}.
I(x) = \int_{0}^{\infty}dA(t)\int_{0}^{x+t}dF_n(u)\bar{G}(u) \left[B(x+t-u) - B(t-u) \right] = \int_{0}^{\infty}dA(t)\int_{0}^{x+t}dF_n(u)\bar{G}(u)B(x+t-u) - \int_{0}^{\infty}dA(t)\int_{0}^{x+t}dF_n(u)\bar{G}(u)B(t-u) =: I_1(x) + I_2(x).
\frac{I_{4}( \alpha , \beta , \gamma , \delta )}{\gamma \, \delta \, G ( 1 , \delta + 1 )}= \frac{G ( \alpha , \gamma + 1 )}{2 \mu - 3}S ( \mu - \alpha - 1 , \beta - 1 , \mu + \alpha - \delta - 2 , \delta - \beta ) + ( \alpha \leftrightarrow \beta ) \, ,
{\|D_{a^+}^{\alpha,\beta}y_{m}(x)-D_{a^+}^{\alpha,\beta}y(x)\|}_{C_{1-\gamma}[a,b]}={\|f(x,y_{m}(x))-f(x,y(x))\|}_{C_{1-\gamma}[a,b]}&\leq A{\|y_{m}(x)-y(x)\|}_{C_{1-\gamma}[a,b]}.
\textbf{w}_{1,k}= \frac{1}{\tau \lambda}\frac{\textbf{h}_{1,k}}{M \sigma_{\hat{h}_{1,k}}^2}\Big(\sum_{k'\neq k}^{K}\big(\textbf{p}_{1,k}\textbf{s}_{1,k'}^{H}\big)\textbf{h}_{1,k'}^H+\sum_{j\neq 1}^{L}\sum_{k'=1}^{K}\big(\textbf{p}_{1,k}\textbf{s}_{j,k'}^H\big)\textbf{h}_{j,k'}^H + \textbf{p}_{1,k}\textbf{N}_1^H \Big) \hat{\textbf{h}}_{1,k}.
t u=-1/2 (\eta^{\mu \alpha}\eta^{\nu\beta}+ \eta^{\nu\alpha}\eta^{\mu \beta}- \eta^{\mu \nu}\eta^{\alpha\beta})k^1_\mu k^2_\nu k^3_\alpha k^4_\beta \equiv-1/2K^{\mu \nu;\alpha\beta}k^1_\mu k^2_\nu k^3_\alpha k^4_\beta
( l + 1 ) N \gamma_{c}w^{c}f^{a_{1}. . . a_{l}}= \sum_{i = 1}^{l}w^{a_{i}}\left[ ( \gamma_{c}w^{c}- N ) \gamma_{b}f^{a_{1}. . .{\not a_{i}}. . . a_{l}b}\right] + d \left[ \left( \frac{l}{l + 2}\gamma_{c}w^{c}+ \frac{2}{l + 2}N \right) \gamma_{b}j^{a_{1}. . . a_{l}b}\right] ,
E_l(z) = \frac{f(z)}{l}\langle \gamma^Z(z), \nabla \phi(z/l)\rangle + \sum_{i = 1}^{m + n}\rho_{ii}^Z(z) \Big(\frac{2}{l}\partial_i f(z) \partial_i \phi(z/l) + \frac{1}{l^2}\partial_i^2 \phi(z/l) f(z)\Big)
\big(R+\frac{\alpha'}{4}R^2-r\big)\Delta f+\Delta\frac{\partial f}{\partial t}=(1+\frac{\alpha'}{2}R\big)\frac{\partial R}{\partial t}-\frac{\partial r}{\partial t}=(1+\frac{\alpha'}{2}R\big)\big(\Delta\Delta f+R\Delta f\big)-\frac{\partial r}{\partial t},
{\cal M}_{0}^{( r )}\sim{\textstyle{\frac{1}{2}}}( n_{1}+ n_{2}) \hat{Q}_{o}( 0 ) -{\textstyle{\frac{1}{2}}}\left[ 2 \cdot 2^{r / 2}\cdot k_{1}k_{2}+ 2 \right] ( m + \bar{m}) \hat{Q}_{v}( \zeta \tau ) .
\mathcal M_{n,\kappa,j}(p) := \begin{bmatrix}-1 -i (\lambda_n)^{(p-1)/2}2 e^{-L \sqrt{\lambda_n}}- \kappa (\lambda_n)^{p/2}e^{-2 L \sqrt{\lambda_n}}i (\lambda_n)^{(p-1)/2}e^{-2 L \sqrt{\lambda_n}}\delta_j e^{-2 L \sqrt{\lambda_n}}\kappa (\lambda_n)^{p/2}-i (\lambda_n)^{(p-1)/2}(1-\kappa) e^{-L \sqrt{\lambda_n}}\end{bmatrix},
\begin{aligned}G(u,v)=\left(1+\frac{R_{11}}{R_{13}}u+\frac{R_{12}}{R_{13}}v\right)^{i}\left(1+\frac{R_{21}}{R_{23}}u+\frac{R_{22}}{R_{23}}v\right)^{k}\left(1+\frac{R_{31}}{R_{33}}u+\frac{R_{32}}{R_{33}}v\right)^{N-i-k}&=\sum_{\substack{m,n=0 m+n\leqslant N}}^{N}\binom{N}{m,n}^{1/2}\,P_{m,n}(i,k;N)\,u^{m}v^{n},\end{aligned}
\begin{array}{lll}\left<\psi|O|\psi\right> = \left<h',j'\right|\overline{c}_{h'j'}\,O\,{c}_{hj}\left|h,j\right>={c}_{hj}\,\overline{c}_{h'j'}\left<h'|h\right>\left<j'\right|O\left|j\right> &={c}_{hj}\,\overline{c}_{hj'}\left<j'\right|O\left|j\right>=tr_{\cal F_{J^+}}\left\{{c}_{hj}\,\overline{c}_{hj'}\left|j\right>\left<j'\right|O\right\}\;.\end{array}
\mathfrak h=h_R-[h]_{R}v-\sum_{S\in \S\setminus \S_4}h_S +\sum_{S\in \S\setminus \S_4}\frac{[h]_S}{[v]_S}v_S-\sum_{S\in \S\cap \S_4}h_S +\sum_{S\in \overline{\S}_4}\frac{[h]_S}{[v]_S}v_S-\sum_{S\in{\S'_4}}\frac{[h]_S}{[v]_S}v_S
\int_{\sum_{i=1}^{3}\nu_{i}\alpha_{i}+2Kl>0}\prod_{i=1}^{3}\frac{d\alpha_{i}}{\sqrt{2K|q_{i}|}}\exp\frac{i\pi}{2Kq_{i}}\left[p_{i}\alpha_{i}^{2}-2\alpha_{i}(2K(n_{i}+q_{i}\lambda)+\mu_{i})+s_{i}(2Kn_{i}+\mu_{i})^{2}\right]
\mathbb{P}\left(\Psi(x,\gamma,Z)=y\right)= \mathbb{P}\left(( \mathbf{1}_{[0, \frac{3}{4}]}- \mathbf{1}_{(\frac{3}{4},1]})(Z)=1\right)=\mathbb{P}\left(Z \in \left[0,\frac{3}{4}\right]\right)=\frac{3}{4};
\frac{S_{k+1}}{S_k}= \frac{\frac{1+rS_k}{S_k}+2+\sqrt{\left(\frac{1+rS_k}{S_k}+2\right)^2-4}}{2},\frac{T_{k+1}}{T_k}= \frac{\frac{1}{T_k}+2+\sqrt{\left(\frac{1}{T_k}+2\right)^2-4}}{2},
\sum_{j=0}^\infty\sum_{i=0}^\infty\mathbb{P}(X=i)\left|\sum_{k=j+1}^\infty\left(Q_{i,k}-P_{i,k}\right)\right|=\left|\mathbb{E}\sum_{j=0}^\infty\sum_{k=j+1}^\infty\left(Q_{X,k}-P_{X,k}\right)\right|=\left|\mathbb{E}X-\mathbb{E}\sum_{j=0}^\infty\sum_{k=j+1}^\infty P_{X,k}\right|.
\frac{1}{k}\frac{B}{\phi(B)}\sum_{i=1}^k\frac{\phi(a_i)}{a_i}\#\mathcal{P}_{L_i,\mathcal{A}}(x)\ge (1+o(1))\frac{\Delta_K\#\mathcal{C}}{\phi(\Delta_K)\#G}\frac{x}{\log{x}},
F = \int d^{4}x \, \{- \frac{1}{4}B_{\mu \nu}B^{\mu \nu}- \frac{1}{4}\varepsilon_{a b}C_{\mu}^{a}C^{\mu b}- \frac{1}{1 2}\varepsilon_{a b}\varepsilon_{c d}C^{a c}C^{b d}\}
\mathbb{E}_\mu\mathbb{P}_{p_{W|U,V}^n}\Big(\big(\mathbf{U}(i),\mathbf{V}(i,j),\mathbf{W}\big)\notin\mathcal{A}_{\epsilon_1,\epsilon_2}\Big|\mathbf{U}=\mathbf{U}(i),\mathbf{V}=\mathbf{V}(i,j)\Big)\leq 2^{-n\beta^{(1)}_{\alpha,\epsilon_1}}+2^{-n\beta^{(2)}_{\alpha,\epsilon_2}},
s=-\sum_{i=1}^{N}\frac{\alpha_i}{\alpha}- \frac{Q}{\alpha}~, \quad ~\alpha_i=-\frac{Q}{2}+ \frac{1}{2}\sqrt{Q^2 + 4(2 - h_i)}, \quad ~\alpha=-\frac{Q}{2}+ \frac{1}{2}\sqrt{Q^2 + 8}
\nu \int_{0}^{\infty}f_{1}^{\prime \prime}( x ) f_{2}( x ) d x = - \nu f_{2}^{\prime}( 0 ) + \nu f_{1}^{\prime}( 0 ) + \nu \int_{0}^{\infty}f_{2}( y ) f_{1}^{\prime \prime}( y ) d y .
\begin{aligned}3t [ \partial_x u(t,x)]^2 + x [u(t,x)]^2 = 2\int_{-\infty}^x \partial_x u(t,x') Wu(t,x') ~\mathrm{d}x' + \int_{-\infty}^x [u(t,x')]^2 ~\mathrm{d}x' & \leq 2\|\partial_x u(t,)\|_{L^2}\|Wu(t,)\|_{L^2}+ \|u(t,)\|_{L^2}^2.\end{aligned}
\|(Y_1+h_1)^{\ast i_1+1}-Y_1^{\ast i_1+1}\|_{\mu}= \|Y_1\ast \left((Y_1+h_1)^{\ast i_1}-Y_1^{\ast i_1}\right)+h_1\ast (Y_1+h_1)^{\ast i_1}\|_{\mu}&\leq i_1\|Y_1\|_{\mu}\left(\|Y_1\|_{\mu}+\|h_1\|_{\mu}\right)^{i_1-1}\|h_1\|_{\mu}+\left(\|Y_1\|_{\mu}+\|h_1\|_{\mu}\right)^{i_1}\|h_1\|_{\mu},
\mathbf{\tilde{H}^*_n}= \mathbf{F_n^*}- \sqrt{n}\hat{\sigma}_n\mathbf{\bar{\Sigma}_n^{-1/2}}(Z_{n}^*-\bar{Z}_{n}^*)((\mathbf{E_*}L_n^*)^{-1}\Delta_n^*) +R^*_{5n}
\left|\begin{array}{lllllllll}\mu_0 \delta_1 0 0 0 \dots 0 0 0 \gamma_1 \mu_1 \delta_2 0 0 \dots 0 0 0 0 \gamma_2 \mu_2 \delta_3 0 \dots 0 0 0 0 0 \gamma_3 \mu_3 \delta_4 \dots 0 0 0 \vdots \vdots \vdots \ddots \ddots \ddots \vdots \vdots \vdots 0 0 0 0 0 \dots \gamma_{n-1}\mu_{n-1}\delta_{n}0 0 0 0 0 \dots 0 \gamma_n \mu_n \end{array}\right|=0,
\frac{dy}{dv}|_{u=u_{L}}=\frac{y_L}{2v_L}\pm \frac{\sqrt{y_L^2-4v_L}}{2v_L}, \frac{dR_1}{dv}|_{u=u_{L}}=\frac{y_L}{2v_L}-\frac{\sqrt{y_L^2-4v_L}}{2v_L}, \frac{dR_2}{dv}|_{u=u_{L}}=\frac{y_L}{2v_L}+\frac{\sqrt{y_L^2-4v_L}}{2v_L}.
\prod_{k=1}^{m}T_{k,j}^{\frac{\log s_{k}}{2\pi i}}= (4|\lambda_{j}|)^{-\beta_{j}}\prod_{\substack{k=1 k \neq j}}^{m}\widetilde{T}_{k,j}^{-\beta_{k}}, \mbox{where}\widetilde{T}_{k,j}= \frac{\left(\sqrt{|\lambda_{j}|}+\sqrt{|\lambda_{k}|}\right)^2}{\big|\lambda_j-\lambda_k\big|}.
S=\frac{M_{P}^{2}}{2}\int d^{4}xe(R+\frac{6}{L^{2}})-\frac{1}{2}\int d^{4}xe(\overline{\psi}_{\mu}\gamma^{\mu\nu\rho}D_{\nu}\psi_{\rho}+\frac{1}{L}\overline{\psi}_{\mu}\gamma^{\mu\nu}\psi_{\nu}),
K \sim \frac{\tau_3}{2}L^2 \frac{3}{2^{5/3}}\left(\frac{u^2{\alpha'}^2}{L^2}\right)^{4/3}= \frac{1}{8 \pi^2 g_{YM}^2}\frac{3}{2^{5/3}}\left(\frac{{\alpha'}^2}{L^2}\right)^{1/3}(u^2)^{4/3}\ ,
\sum\limits_{\sigma^{\prime}:\sigma \dagger \sigma^\prime}\sum\limits_{\substack{c\in\mathbb{Q}:c>0, \langle c\rangle=\hat{b}_{j}}}q^{d_{c,\sigma,j}}\frac{Rec(c)^{(\sigma^{\prime},b^{\prime})}_{(\sigma,b)}}{(-z+U_{j}(\sigma)/c)}[f_{(\sigma^{\prime},b^{\prime})}]_{z=U_{j}(\sigma)/c}.
D_I = \biggl\{\theta\in\{\pm 1\}^{\mathcal{D}}: \Bigl|\langle T b_{I}^{(\theta)}, b_{I}^{(\theta)}\rangle - \sum_{K\in\mathcal{B}_I}\langle Th_{K}, h_{K}\rangle \Bigr| > \eta_0 \biggr\}, I\in\mathcal{D}_{\leq n}.
\Gamma_q = \{\gamma : \gamma \textrm{is a}(q-1) \times (q-1) \textrm{symmetric positive semidefinite matrix}\}, \Pi = \{\pi : [0,1] \to \Gamma_q : \pi \textrm{is left continuous}, \pi(x) \leq \pi(x') \textrm{for}x \leq x' \},
\mathrm{m}_q = \frac{1}{2}c_q(n) \int_M \left(L_{(q)}+ (2q-1)! \left\langle \left[ T_{(2q-1)\alpha}, A_{\beta}\right] \cdot e_{\alpha}^{\top}, e_{\beta}^{\top}\right\rangle \right) \frac{1}{\sqrt{G}}dM & + \frac{1}{2}(2q-1)! c_q(n) \int_{\Sigma}H_{(2q-1)}d\Sigma,
R\cdot R(JX_1,JX_2,X_3,X_4;X,Y)=R\cdot R(X_1,X_2,JX_3,JX_4;X,Y) &=R\cdot R(X_1,X_2,X_3,X_4;JX,JY)=R\cdot R(X_1,X_2,X_3,X_4;X,Y),
(V(s)-V_*(s))-(V(s_1)-V_*(s_1)) =\int_{s_1}^{s}E(\tau,X(\tau))d\tau+\int_{s_1}^{s}V(\tau) \wedge B(\tau,X(\tau))d\tau &-\int_{s_1}^{s}E(\tau,X_*(\tau))d\tau-\int_{s_1}^{s}V_*(\tau) \wedge B(\tau,X_*(\tau))d\tau.
\, \frac{s_{\square}}{2\hbar}(\ln(z)-\ln(s_{\square}+m)+\ln(s_{\square})-\sum_{\square'\neq\square}\ln(s_{\square}-s_{\square'})+\sum_{\square'\neq\square}\ln(s_{\square'}-s_{\square}) &-\sum_{s=1}^3\sum_{\square'\neq\square}\ln(s_{\square'}-s_{\square}+\hbar_s)+\sum_{s=1}^3\sum_{\square'\neq\square}\ln(s_{\square}-s_{\square'}+\hbar_s)+o(\hbar)).
KS_{r_1,r_2}= \sup_{m \in I}\left|\sqrt{\frac{r_2}{r}}G^{r_1}_D(m) - \sqrt{\frac{r_1}{r}}G^{r_2}_E(m) + \sqrt{\frac{r_1 r_2}{r}}\mathrm E \left\{A_D-A_E \right\}(m) \right|,
\left|\sum_{k=1}^{n_1}\nu(k) (n_1n_2/k^2)-q_{n_1, n_2}\right|= \left|\sum_{k=1}^{n_1}\nu(k)\left(n_1n_2/k^2-\lfloor n_1/k\rfloor\lfloor n_2/k\rfloor\right)\right| \le (n_1+n_2)\sum_{k=1}^{n_1}(1/k)=o(n_1 n_2)\,.
i\hbar\frac{d\sigma_{j}}{dt}=[\sigma_{j},H]=[\sigma_{j},-\mu\sigma_{i}B_{i}]=-\mu(\sigma_{j}\sigma_{i}B_{i}-\sigma_{i}\sigma_{j}B_{i})=\mu[\sigma_{i},\sigma_{j}]B_{i}=2\mu i\epsilon_{ijk}\sigma_{k}B_{i}
|u|_{D^{3,q}}\leq \big(|\rho u_t+\rho v\cdot \nabla v|_{D^{1,q}}+|\nabla P|_{D^{1,q}}+|H\times H|_{D^{1,q}}+|u|_{D^{1,q}_0}\big) \leq C(c^6_1+c^2_1|u_t|_\infty+c^2_1|\nabla u_t|_q+c^3_1|v|_{D^{2,q}}).
\mathbf E(z;\mathbf a) = \mathbf E_0+ \sum_{k=1}^N \mathbf M_1(z;k) \cdot \mathbf E_0 + \sum_{k=1}^N \sum_{k_1 \neq k}^N \mathbf M_2(z; k, k_1) \cdot \mathbf E_0 + \cdots.
u(x,t) - e^{tL_{\ast}}[ u_0 + a(\cdot)^{-1}u_1] = - \int_{t/2}^t e^{(t-s)L_{\ast}}[ a(\cdot)^{-1}u_{tt}(\cdot, s) ] ds &-e^{\frac{t}{2}L_{\ast}}[ a(\cdot)^{-1}u_{t}(\cdot, t/2) ] &-\int_0^{t/2}L_{\ast}e^{(t-s)L_{\ast}}[ a(\cdot)^{-1}u_{t}(\cdot, s) ] ds,
F(z,v) := \sum_{n \ge 1}\sum_{m \ge 0}T_{n}\mathbb{P}\{L_{n}=m\}\frac{z^{n}}{n!}v^{m}= \sum_{n \ge 1}\sum_{m \ge 0}\mathbb{P}\{L_{n}=m\}\frac{z^{n}}{n}v^{m},
\omega = -\frac 12+\frac{q}{p},\kappa_n=\frac{n^2 \left(- n^4 - \frac{2s}{p^2}n^2 +\frac{4 r- s^2}{p^4}\right)}{2^2 (2 n - 1) (2 n + 1)},\quad\lambda_n=\frac{2 n^2 + 2 n + 1}{4}+ \frac{s}{2 p^2}.
G^{\left(N+1\right)}_{0;1,0, \dots,0,\dots, 0}(z)=G^{\left(N+1\right)}_{0;0,0, \dots,1,\dots, 0}(z)=\dots=G^{\left(N+1\right)}_{0;0,0, \dots,0,\dots, 1}(z)=0,
\Delta_{q}= ( q_{1}\q_{2}) \mathcal{M}_{0}^{- 1}\left( \begin{array}{c}{{q_{1}}}\\{{q_{2}}}\\ \end{array}\right) = e^{\phi_{0}}( q_{2}\chi_{0}- q_{1})^{2}+ e^{- \phi_{0}}q_{2}^{2},
0=\min_{0\le s\le t}\,(R^{\overline{X}}_s-\widetilde{\Lambda}_{s})=\overline{X}+\min_{0\le s\le t}\bigg(\int_0^s \frac{1}{R^{\overline{X}}_r}\,\mathrm{d}r+B_s-\widetilde{\Lambda}_{s}\bigg)\;\Longrightarrow\;\overline{X}\le \max_{0\le s\le t}\,(-B_s+\widetilde{\Lambda}_{s}), &0=\min_{0\le s\le t}\,(R^{\underline{X}}_s-\Lambda_{s}).
P_{m,2N}(i,j,q) = q^{\omega(m,i,j)}\genfrac{[}{]}{0pt}{}{N}{i,\j}_{q^6}(-q^3;q^3)_{N-i-j}, P_{m,2N+1}(i,j,q) = q^{\omega(m,i,j)}\frac{1-q^{3(N+1+i+(-1)^m j)}}{1-q^{6(N+1)}}\genfrac{[}{]}{0pt}{}{N+1}{i,\j}_{q^6}(-q^3;q^3)_{N+1-i-j},
\frac{\mu X_{j-1}(t^*)}{s X_j(t^*)}\leq \frac{\mu (1 + \delta)}{C_1 s}\cdot \frac{x_{j-1}(t^*)}{k_N^{-d_j}x_j(t^*)}= \frac{1 + \delta}{C_1 (1 - \delta)}\cdot \frac{j}{k_N^{-d_j}(e^{st^*}- 1)}\leq \frac{1 + \delta}{C_1 (1 - \delta)}\cdot \frac{k_N^2 k^*}{k_N^4 - 1}.
\overline{K}(a_{i,0},a_{i,1}, y_0, y_1)= (a_{i,0}a_{i,1}-t\sum_{\ell=0}^{2}d_{\ell-1,0}a_{i,0}^{\ell}a_{i,1}^{2-\ell})y_0y_1 + t(\sum_{\ell=0}^2 d_{\ell-1,-1}a_{i,0}^{\ell}a_{i,1}^{2-\ell})y_1^2.
\omega_{12}=-\frac{1}{6}*d\log(\kappa_2^2-\mu_2^2),\,\,\omega_{34}=2\omega_{12}+*d\log\kappa_1, \,\, \omega_{56}=\frac{\kappa_2\mu_2}{\kappa_2^2-\mu_2^2}*d\log\frac{\mu_2}{\kappa_2},
\pi(u) =\frac{2k-1}{4\pi}\int_{\tau=0}^{\infty}d\tau\; \sinh\tau \int_{0}^{2\pi}d\phi \left[ \cos\frac{\theta_{0}}{2}-i\sin\frac{\theta_{0}}{2}\hat{n}. \hat{r}\right]^{-2k}
\kappa= \begin{cases}\frac{N-2+p}{N-p}\hbox{if}1<p<2 \hbox{and}\mathcal G=0 \frac{N(p-1)}{N(p-1)-p}\hbox{if either}2\leq p < N\hbox{or}p =N \geq 3 2 \hbox{if}p=N=2. \end{cases}
\frac{(2\pi)^{D/2}}{G N \sqrt{2}}= \frac{\Lambda^{D-2}(1 - (eH/\Lambda^2)^{D/2 - 1})}{D/2 - 1}+\frac{(eH)^{(D-1)/2}}{\rho}\int_{\rho^2/eH}^{\infty}\frac{ds}{s^{1/2}}e^{-s}.
\nabla^{(2)}\cdot\frac{\partial f}{\partial(\nabla^{(2)}\rho)}=\sum_{\alpha,\beta=1}^{3}\frac{\partial^{2}}{\partial r_{\alpha}\partial r_{\beta}}\frac{\partial f}{\partial\rho_{\alpha\beta}}where\rho_{\alpha\beta}\equiv\frac{\partial^{2}\rho}{\partial r_{\alpha}\partial r_{\beta}}.
\mathcal{C}_{0}\xi =\int_{\Delta (A)}\widehat{A}_{0}(\xi +\partial \widehat{u}_{\xi})d\beta ,\ \ \mathcal{C}_{1}\xi =\int_{\Delta (A)}\left( \widehat{A}_{1}\ast \ast (\xi +\partial \widehat{u}_{\xi})\right) d\beta
\pi_{+}^{k}(\mathbf{A}(e_{\underline{i}})) = q^{k (k - 1) / 2}[k]! e_{\underline{i}}, \pi_{-}^{k}(\mathbf{A}(f_{\underline{i}})) = q^{k (k - 1) / 2}[k]! f_{\underline{i}}.
\pi_uf(x(\tau)) = \begin{pmatrix}0 x_1^2(\tau) 3 x_1^2 (\tau) \end{pmatrix}= \begin{pmatrix}0 E_{p}^{2}(-\tau^p)\sigma_1^2 3 E_{p}^{2}(-\tau^p)\sigma_1^2 \end{pmatrix}
- 4 b_{0}^{1}b_{0}^{0}{\frac 1{a^{2}}}+ 2 b_{0}^{0}b_{2}^{2}+ 3 b_{0}^{1}b_{7}^{2}= 0 , \, \, \, \, \, \, \, 4 \, b_{0}^{1}b_{0}^{0}{\frac 1{a^{2}}}+ 2 b_{0}^{0}b_{1}^{2}+ 3 b_{0}^{1}b_{8}^{2}= 0
t\in{\rm Stab}_{T}(\eta) \iff w_{0,J(w)}t w_{0,J(w)}\xi = \xi &\iff w_{0,J(w)}t w_{0,J(w)}\in{\rm Stab}_{T}(\xi) &\iff t \in w_{0,J(w)}{\rm Stab}_{T}(\xi)w_{0,J(w)}.
k_i^N(t,\nu^{N,1}_{\boldsymbol{x}},\ldots,\nu^{N,N}_{\boldsymbol{x}},\lambda^{N,1}_{\boldsymbol{\varphi}},\ldots,\lambda^{N,N}_{\boldsymbol{\varphi}}):=\rho_i^N\big(-R^{-1}B[P_t\nu^{N,1}_{\boldsymbol{x}}+\lambda^{N,1}_{\boldsymbol{\varphi}}],\ldots,-R^{-1}B[P_t\nu^{N,N}_{\boldsymbol{x}}+\lambda^{N,N}_{\boldsymbol{\varphi}}]\big).
\Big| \frac{1}{\varepsilon n^2}\sum_{x}\Phi_n(s, \tfrac{x}{n}) \sum_{r=1}^{\varepsilon n}\int_{\Omega}\overleftarrow{\eta}^{\ell}(x)[\eta(x+1)-\eta(x+1+r) ]f(\eta) d\nu_{b}\Big|.
(y(g),\gamma^\star,\delta^\star){z}= \sum_{E\subseteq\{1,\ldots,s\}}(\prod_{e\in E}y_e)(g^{(\tau_E)z},\gamma^\star,\delta^\star) &= \sum_{E'\subseteq\{1,\ldots,s\}}(\prod_{e\in E}y_e)((g^{z})^{\tau_{E'}},\gamma^\star,\delta^\star),
\sum_{k=E+1}^{E(\lambda)}k^{-\varrho}\|\partial_\nu\phi^{(1)}_k\|^2_{L^2(\partial B_R)}+ \sum_{k=E+1}^{E(\lambda)}k^{-\varrho}\|\partial_\nu\phi^{(1)}_k\|_{L^2(\partial B_R)}\|\partial_\nu(\Delta\phi^{(1)}_k)\|_{L^2(\partial B_R)}&\lesssim \sum_{k\geq 1}k^{-\varrho + 6/n}.
K_1 =K_1''+K_1'T(\phi_R) &= T(\phi)-T(\phi_0\cdots\phi_{R-1})T(\phi_R) &+\Big(T(\phi_0\cdots\phi_{R-1})-T(\phi_0)\cdots T(\phi_{R-1})\Big)T(\phi_R) &= T(\phi)-T(\phi_0)\cdots T(\phi_R)
c(u,s)=\frac{\varGamma\left(s+\displaystyle{\frac{b}{a}}\right)\varGamma\left(u+\displaystyle{\frac{b+1}{a}}\right)}{a\varGamma\left(s+\displaystyle{\frac{b+a+1}{a}}\right)\varGamma\left(u+\displaystyle{\frac{b}{a}}\right)}m
\begin{array}{l}{\partial_{i}x^{j}= \delta_{i}^{j}+ q \hat{\cal R}_{i l}^{j k}x^{l}\partial_{k},}\\{\tilde{\partial}_{k}x^{v}= \delta_{k}^{v}+ q^{- 1}\left. \hat{\cal R}^{- 1}\right._{k j}^{v i}x^{j}\tilde{\partial}_{i}.}\\ \end{array}
\aligned\Theta_{\tau}\ = \{\underset{\Xi}\sum}\ \Theta_{\Xi}\ = \{\underset{\Xi}\sum}\q^{(d-\frac12 )}\{\underset{\sigma \in{S_{\Xi}}}\sum}\Theta_{\sigma}&= \{\frac{(q-1)}{2}}\, q^{(d-\frac12 )}\{\underset{\rho(\sigma ) \, = \, d}\sum}\Theta_{\sigma}\ .\endaligned
\frac{1}{(x'(t)^2 + y'(t)^2)^{1/2}}\left( (m-1)\frac{y'(t)}{x(t)}- (n-1)\frac{x'(t)}{y(t)}+ \frac{x'(t)y''(t) - x''(t) y'(t)}{x'(t)^2 + y'(t)^2}\right) = \frac{1}{2}\frac{x'(t)y(t) - x(t) y'(t)}{(x'(t)^2 + y'(t)^2)^{1/2}}+ \lambda
R_{\mu \nu}-\frac{1}{2}g_{\mu \nu}R= 8\pi G \left(\frac{V}{\sqrt{1 + \nabla_\alpha T \nabla^\alpha T}}\,\,\nabla_\mu T \,\, \nabla_\nu T - g_{\mu \nu}V \,\sqrt{1 + \nabla_\alpha T \nabla^\alpha T}\right) \ ,
\left. H = t_{2 s}^{( 2 s )}( u )^{- 1}\frac{d}{d u}t_{2 s}^{( 2 s )}( u ) \right|_{u = - ( 2 s - 1 ) \lambda}= \prod_{j = 1}^{2 s}\sinh ( j \lambda )^{- 1}\left( \sum_{j = 1}^{N - 1}h_{j , j + 1}+ h_{N , 1}^{F}\right) ,
b_r :=- q^{-\frac t2 -\frac m2+\frac78 +\frac t{2t-1}r^2 +\frac t{2t-1}r}\frac1{(q)_\infty^3}\sum_{k=0}^{2t-1}(-1)^k q^{\frac12k^2+rk}\theta_{k-t,m}, C := (-1)^{t+m}q^{-\frac{(3t-m-2)(t+m-1)}{2(2t-1)}}.
\lim_{x\downarrow0}g'_0(x;a,b) =\lim_{x\downarrow0}g_1(x;a,b) +4a^3\int_{-1}^1 \int_{0}^sf'''(A)\,du\,s\,ds &=4af''(A)+\frac{8a^3}{3}f'''(A).
L_{eff}^{ii}= \frac{1}{2\pi}\left( \hat{\Phi}^i ~ \hat{\eta}^i \right)\left( \begin{array}{cc}A^{ii}C^{ii}/2 C^{ii}/2 B^{ii}\end{array}\right) ~ \left( \begin{array}{c}\hat{\Phi}^i \hat{\eta}^i\end{array}\right),
\left(V_{jn}^{\varepsilon}\right)_{t}-(A+\rho_{\varepsilon})V_{jn}^{\varepsilon}=e^{\rho_{\varepsilon}\left(t-T\right)}\left\langle F\left(\cdot,t;e^{\rho_{\varepsilon}\left(T-t\right)}v_{n}^{\varepsilon}\right),\phi_{j}\right\rangle +\left\langle \mathbf{P}_{\varepsilon}^{\beta}v_{n}^{\varepsilon},\phi_{j}\right\rangle ,
A_{\mu}= \mp{\frac{2 i}{g}}\sum_{I = 1}^{K}[ H_{I}{\frac{v_{\mu}^{I}}{\rho_{I}}}+ g \kappa E_{I K + 1}^{\pm}R_{\mu}^{I}\prod_{I = 1}^{K - 1}\delta ( R^{K}\cdot R^{I}) ] .
g^3 = (v_1^2v_2v_3v_4)(v_1^2v_2v_3v_4)(v_1^2v_2v_3v_4) &= (v_1^2v_2v_3\cdot v_4)(v_1^2v_2\cdot v_3v_4)(v_1\cdot v_1v_2v_3v_4) &= (v_1^2v_2v_3\cdot v_1^2v_2\cdot v_1)(v_4\cdot v_3v_4 \cdot v_1v_2v_3v_4) &= u(v_4\cdot v_3v_4 \cdot v_1v_2v_3v_4).
\lim_{t\to \infty}\frac{D_{1-x/t}-K_{d}}{A_{1}(b_{1}(t))}= \lim_{t\to \infty}\left[ \frac{{\bf{I}}(x,t)}{A_{1}(b_{1}(t))}- \frac{{\bf{II}}(x,t)}{A_{1}(b_{1}(t))}\right]= -\frac{c_{1}K_{d}}{\alpha \rho_{1}}(x^{-\rho_{1}/\alpha}-1).
p_\theta({\bf{x}})^{-1}= Z(\theta)^{-1}\exp \big[-h({\bf{x}}) - w(\theta)^\top f({\bf{x}})\big] p_\theta({\bf{x}})^{-1}= \exp \big[-h({\bf{x}})-Z'(\theta) -w(\theta)^\top f({\bf{x}})\big],
|a_{Q_1}-a_{Q}| =\frac{1}{|\widetilde{Q}_1| |\widetilde{Q}|}\left| |\widetilde{Q}| \int_{\widetilde{Q}_1}f(y) \,dy - |\widetilde{Q}_1| \int_{\widetilde{Q}}f(x) \,dx \right| &\leq\frac{1}{|\widetilde{Q}_1| |\widetilde{Q}|}\int_{\widetilde{Q}_1}\int_{\widetilde{Q}}|f(y)-f(x)| \,dy \, dx \,.
\frac{D\rho}{Dt}=\frac{1}{c^2}\frac{Dp}{Dt}- \frac{\rho \alpha_p T}{c_p}\frac{Ds}{Dt}, \frac{DT}{Dt}=\frac{T \alpha_p}{\rho c_p}\frac{Dp}{Dt}+ \frac{T}{c_p}\frac{Ds}{Dt}.
\lim_{\varepsilon\to 0^{+}}\frac{(q_{\frac{1}{\varepsilon}}\mu)(\mathbf{x}_{0}+\varepsilon \mathbf{x})}{\varepsilon^{\alpha-n}}= \ell|\mathbf{x}|^{\alpha-n}\ \ \ \mbox{weakly$^{\ast}$ in}\mathcal{K}_{M,r}'(\mathbb{R}^{n}) \ (\mbox{resp.}\mathcal{S}'_{r}(\mathbb{R}^{n})),
a_0 u_1^{0,N}(x) + a_1 u_1^{1,N}(x) + \dots a_{\lfloor \frac{N}{2}\rfloor}u_1^{\lfloor \frac{N}{2}\rfloor,N}(x) + b_1 u_2^{1,N}(x) + \dots b_{\lfloor \frac{N-1}{2}\rfloor}u_2^{\lfloor \frac{N-1}{2}\rfloor,N}(x) = 0,
\rho_0(n) = \rho_0(n-1) + q^n \rho_1(n) & = \rho_0(n-1) + q^n\left(\rho_1(n-1) + q^n\rho_2(n)\right) & = \rho_0(n-1) + q\left(\rho_0(n-1) - \rho_0(n-2)\right) +q^{2n}\left(\rho_0(n) - q^{-2}\rho_0(n-2)\right),
\text{LHS of (\ref{eqn:final})}= \sum_{m=0}^2 (-1)^m \binom{n-1 + 2-m}{n-1}\binom{n}{m}= \binom{n+1}{n-1}\binom{n}{0}- \binom{n}{n-1}\binom{n}{1}+ \binom{n-1}{n-1}\binom{n}{2}= \binom{n+1}{2}- \binom{n}{1}^2 + \binom{n}{2}= 0
\sum_{i=2}^{\nu}(i-1)k_i = k_2 + 2k_3 + \cdots + (\nu-1)k_\nu \geq \lambda_{\nu-1}+ 2\lambda_{\nu-2}+ \cdots + (\nu - 1)\lambda_1 = \sum_{i=1}^{\nu-1}(\nu-i)\lambda_i.
{\cal L}( \frac{3}{2}, \alpha - 1 ) = \bar{\Psi}[ \frac{2}{3}L_{\mu}( \partial_{\mu}+ i a_{\mu}+ i C_{\mu}) + M ] \Psi - \frac{i}{8 \pi ( \alpha - 1 )}\epsilon_{\mu \nu \lambda}a_{\mu}\partial_{\nu}a_{\lambda}\; ,
\begin{array}{l}\displaystyle{_cD_{0t}^{\alpha}\left(\int_0^x (x-t)^{\alpha-1}E_{\alpha, \alpha}[\lambda(x-t)^{\alpha}]f(t)dt\right) =}\displaystyle{\frac{1}{\Gamma(2-\alpha)}\int_0^x\frac{1}{(x-t)^{\alpha-1}}\frac{d^2}{dt^2}\left(\int_0^t (t-z)^{\alpha-1}E_{\alpha, \alpha}[\lambda(t-z)^{\alpha}]f(z)dz\right)dt.}\end{array}
c_L(\theta) =\hat\mu+\mathbf{r}_L(\theta)'\mathbf{R}_L^{-1}(\mathbf \Upsilon-\hat\mu \mathbf 1), \nabla_\theta c_L(\theta) =\hat\mu+\mathbf{Q}_L(\theta)\mathbf{R}_L^{-1}(\mathbf \Upsilon-\hat\mu \mathbf 1),
\mathcal{G}_\xi := \left\{z \in \mathbb{R}: \begin{array}{l}\lim_{r \rightarrow 0}\frac{\mu \left( B^T_{\xi, r}(z) \cap R_{(\omega_1 , \ldots , \omega_N )}\right)}{\mu \left( B^T_{\xi, r}(z) \right)}\mbox{exists}\mbox{for all}\omega_1 , \ldots , \omega_N \in \{0 , \ldots , \ell -1 \}\mbox{and}N \in \mathbb{N}\end{array}\right\}.
< \theta_{1}^{a}, \vec{Y}_{1}| e^{- l H}| \theta_{2}^{a}, \vec{Y}_{2}>_{B}= \frac{V ( B_{1}) V ( B_{2})}{\mathrm{V o l}( T_{6})}= \prod_{a}\frac{| \bar{n}_{a}^{( 1 )}\bar{n}_{a + 1}^{( 2 )}- \bar{n}_{a + 1}^{( 1 )}\bar{n}_{a}^{( 2 )}|}{\sin | \theta_{a}|}
- \frac{\pi^{2 \mu}a ( 1 ) a ( 2 \mu - 2 )}{2 ( \mu - 1 )}\left[ 3 ( \mu - 1 ) \Theta \Xi + \frac{3 \Xi}{( \mu - 1 )}- ( 3 \mu + 2 ) \Theta + \frac{2 \mu \Psi}{( \mu - 1 )}\right]
\sum_{b=0}^{a-r}(-1)^{b}\frac{1}{a+b}\binom{a+b}{2b+r}\binom{2b}{b}= \frac{(a-1)!(a-1)!}{(2a-1)!}\binom{2a-1}{r-1}\binom{2a-r-1}{a-r}= \frac{1}{2a-r}\binom{a-1}{a-r}.