harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathcal{L}^{\mathrm{optimal}}_{1}(\overline{\pmb{\theta}}, \overline{\pmb{\omega}}) = \max\{\mathcal{L}^{\mathrm{NCP}}_{1}(\overline{\pmb{\theta}}, \overline{\pmb{\omega}}), \mathcal{L}^{\mathrm{CLIP}}_{1}(\overline{\pmb{\theta}}, \overline{\pmb{\omega}}), \mathcal{L}^{\mathrm{KL}}_{1}(\overline{\pmb{\theta}}, \overline{\pmb{\omega}})\}.
\boldsymbol{\Sigma}_{\frac{n}{2}}:= \mathrm{diag}\left(\lambda_{3},\lambda_{5},\ldots,\lambda_{n+1}\right) + \lambda_{1}\mathbf{u}\mathbf{u}^{\top}\boldsymbol{\Gamma}_{\frac{n}{2}}:= \mathrm{diag}\left(\lambda_{2},\lambda_{4},\ldots,\lambda_{n}\right) + \lambda_{n+2}\mathbf{v}\mathbf{v}^{\top}
\mathcal{N}_{s+1}= \left\{j:j \in \mathcal{N}_{s}, j \neq \pi(s), \frac{\left|{\mathbf{f}_{i}^{[i,j]}}^{\mathsf{H}}\mathbf{b}_{s}^{[i]}\right|}{\| \mathbf{f}_{i}^{[i,j]}\| \|\mathbf{b}_{s}^{[i]}\|}<\alpha\right\}, s = s+1,
\frac{1}{2\pi}\int_0^{2\pi}\frac{1}{|1-re^{i\theta}|^p}d\theta = (1-r^2)^{1-p}{}_2F_1\left(1-\frac{p}{2}, 1-\frac{p}{2};1;r^2\right) \le \frac{\Gamma(p-1)}{\Gamma(p/2)^2}(1-r^2)^{1-p}.
S_{m m^{\prime}, n n^{\prime}}= \sqrt{\frac{2}{u^{2}( k + 2 )}}( - 1 )^{m^{\prime}( n + 1 ) + ( m + 1 ) n^{\prime}}e^{- i \pi ( k + 2 ) m^{\prime}n^{\prime}}\sin \left( \frac{\pi ( m + 1 ) ( n + 1 )}{k + 2}\right) .
\mathbb{E}^a_{Q,w,0}\left(\cosh\left((t+h)A_0\right)1_{\{0\longleftrightarrow \partial_{in}Q^a\}}\prod_{i\geq 1}\cosh(hA_i)\right) &=\sum_{\Gamma}\cosh\left(\left(t+h\right)|\Gamma|\right)\mathbb{P}^a_{Q,w,0}(\mathcal{C}_0=\Gamma)\mathbb{E}^a_{Q,w,0}\left(\prod_{i\geq 1}\cosh(hA_i)|\mathcal{C}_0=\Gamma\right),
\int_{|x|>1}u(\dot{x})|x|^{-d-\gamma}e^{-i(x,k)}dx=\sum_{0\neq z\in Z^d}e^{-i(z,k)}[\sum_{s=0}^{n-1}v_s(\dot{z})|z|^{-d-\gamma-s}+O(|z|^{-d-\gamma-n})]+g(k),
S = -{\frac{T}{4}}\int d^{p + 1}\xi \exp ( - \phi ) ( - g )^{1 / 4}( - \gamma )^{1 / 4}\left[ \gamma^{i j}( g_{i j}- g^{k l}{\cal F}_{i k}{\cal F}_{l j}) \varphi^{\frac{p - 3}{p + 1}}- ( p - 3 ) \varphi \right] .
W^0_{k+2}(p,p_1,\dots,p_{k+1})\ast W^0_{l+2}(p',p'_1,\dots,p'_{l+1}) =W^0_{m+2}(p,p_1,\dots,p_{m+1})=(\mathbf{1})\ast(\mathbf{1})\ast\dots\ast(\mathbf{1}), m=k+l
{\bf{x}}_{n, k}= [\frac{x[n] - x_k}{\|{\bf{v}}_n -{\bf{u}}_k\|_2}, \frac{y[n] - y_k}{\|{\bf{v}}_n -{\bf{u}}_k\|_2}, \frac{z[n]}{\|{\bf{v}}_n -{\bf{u}}_k\|_2}, \psi_n]^{}.
\sin_q z :=\frac{(q^2;q^2)_\infty}{(q;q^2)_\infty}(z)^{1/2}J^{(1)}_{1/2}(2 z;q^2) = \sum_{n=0}^{\infty}(-1)^n \frac{z^{2n+1}}{[2n+1]_q!},
\nabla_{P(p,q,r)}(z) = 1+ \frac{1}{4}\left( pq+qr+rp+1 \right)z^{2}, v_{3}(P(p,q,r)) = \frac{1}{64}\left( (p+q+r+1)(pq+qr+rp+1)+(p-1)(q-1)(r-1) \right)
\frac{\partial}{\partial r}\left(\log \left(\frac{F(p, r, \theta)}{F_k(r)}\right)\right)= \frac{\partial}{\partial r}\left(\log{F(p, r, \theta)}-\log{F_k(r)}\right) \le -\int_{0}^{r}\frac{s_{k}(t)^{2}}{s_{k}(r)^{2}}{\mathrm{Ric}}_{k}\left(\gamma_{p, \theta}^{\prime}(t)\right) d t.
\begin{aligned}\psi \omega^2 \leq \frac{c}{R^4}+\frac{c}{(\tau-t_0+T)^2}&\,\,\,\,\,\, +c\frac{\alpha^2}{R^2}+cK^2+c(a+(n-1)K)^2+\frac{ca^2g^2}{(\mu-g)^2}\end{aligned}
\sum_{m=0}^N\alpha^mc_1^{\frac{m}{2}}\|S_m^--S_m^0\|_{L^1}&\le\frac{1}{\beta h}(b\beta L^dU_{max}((\alpha+1)^2c_1+(\alpha+1)^4c_1^2))^2e^{b\beta L^dU_{max}((\alpha+1)^2c_1+(\alpha+1)^4c_1^2)}.
f_{\mathrm{e f f}}^{( \frac{3}{2}, 3 )}( t ) = - 4 . 9 3 t^{2}\frac{1 + 4 . 6 3 t^{2}+ 3 . 2 1 t^{4}- 9 . 4 8 t^{6}- 1 1 . 6 7 t^{8}}{( 1 + 1 . 1 6 t^{2}) ( 1 + 2 . 4 8 t^{2})^{2}},
\theta_{(3)}^{(1,0)}(\tau)=\sum_{(m_1,m_2)\in{\bf Z}^2}q^{\frac{1}{9}+m_1+3(m_1^2+m_2^2-m_1m_2)}=: q^{\frac{1}{9}}{\tilde \theta_{(3)}^{0}}(\tau),
\begin{array}{rcl}\dot{\tilde{V}}= -\dfrac{1}{V(x)(1 - \gamma^{-1}D(x))}\left[\dot{V}(x)\left(\gamma - D(x) \right) + V(x) \dot{D}(x) \right] &= -\dfrac{1}{V(x)(1- \gamma^{-1}D(x))}\left[\gamma \dot{V}(x) - \dot{V}(x) D(x) + V(x) \dot{D}(x) \right]\end{array}
\bar{V}_I \;\equiv\; 2(w^*_{I\,i}\, x^i) \;+\; 2 \left[(A+\lambda_I)^{-1}B w_I^* \right]^i \partial_i \;=\; 2\left[ \, w^*_{I\,i}\, x^i \;-\; (\Sigma w_I^*)^i \partial_i \, \right]\;\;,
\lambda_n\left(\left\{k\in\mathbb{N}:\,k\geq n\right\}\right) =\nu_n\left(\bigcup_{k\geq n}C_{k+1}\setminus C_{k}\right)=\nu_n\left(\bigcup_{k\geq n}C_{k+1}\setminus C_n\right) &\leq\nu_n\left(S\setminus C_{n}\right)<\varepsilon/2^{n+2}
s(\eta_x) = F(x) \star_0 \eta_{t_0(x)}\star_1 \cdots \star_{n-1}\eta_{t_{n-1}(x)}, begin{align*}0.5ex]t(\eta_x) = \eta_{s_{n-1}(x)}\star_{n-1}\cdots \star_1 \eta_{s_0(x)}\star_0 G(x),
\omega^{(g)}_{2,0}=\sum_{i+k=g}\frac{1}{24^{i}i!4^k(2k+1)!!}(D_{-x_1}^{3}+D_{-x_1}^3)^{i}(D_{-x_1}+D_{-x_2})^{k-1}(D_{-x_1}D_{-x_2})^{k+1}dz_1dz_2,
\mathbb{E}(\mathcal{Z}^{k}_{N}(\beta)) = \left(\frac{N}{2}\right)^{k}\int_{-1}^{1}\ldots \int_{-1}^{1}\mathbb{E}\left( \prod_{j=1}^{k}|p_{N}(x_{j})|^{2\beta}\right)\prod_{j=1}^{k}e^{-2\beta\mathbb{E}\log|p_{N}(x_{j})|}\rho^{q}(x_{j})\,dx_{j}
\bar{M}_{Z}^{2}= m_{1}^{2}\left( 1 + \left( \frac{\Gamma_{1}}{m_{1}}\right)^{2}\right)^{- 1}\quad \mathrm{a n d}\quad \bar{\Gamma}_{Z}^{2}= \Gamma_{1}^{2}\left( 1 + \left( \frac{\Gamma_{1}}{m_{1}}\right)^{2}\right)^{- 1},
\nabla f(\mathbf{x}^{\star}) + \frac{\rho(1-\eta)}{2}\mathbf{E}_{}^T\mathbf{E}_{}\mathbf{x}^{\star}+ \sqrt{\eta}\mathbf{E}_{}^T\boldsymbol{\alpha}^{\star}= \mathbf{0}\mathbf{E}_{}\mathbf{x}^{\star}= \mathbf{0}.
e^{ic_+(x)}\partial_{x}\left(\bar{u}_x(x) e^{ic_+(x)}\right) =-\frac{1}{\pi}\int_{\mathbb{R}}r_2(z) e^{2isx}\mathrm{d}z-\frac{1}{\pi}\int_{\mathbb{R}}r_2(z) e^{2 iz x}\left[M_{+,22}-1\right] dz &:=I_3(x)+I_4(x).
\Psi_{j}(x,t,y,s) = u^{\epsilon,\theta}(x,t) - u_{\theta}(y,s) - \frac{(x - y)^{2}}{2 \eta}- \frac{(t - s)^{2}}{2 \eta}- bt &- \sqrt{\theta}(T+1 - t)^{-1}+ g_{\eta,\nu}(-\nu y) - \alpha x^{2}
\sum_{k\geq 2}k^{1.5}\#_k=\sum_{j\geq 1}\sum_{k\in [2^j,2^{j+1}]}k^{1.5}\#_k\leq \sum_{j\geq 1}2^{1.5j+1.5}\delta^{-3}2^{-1.5j}\leq C' \delta^{-3}(-\log \delta).
\hat{\Gamma}_{(Y_+,D_{Y,+})}:=\bigoplus_{\vec d\in (\mathbb Z)^{|I_+|},f\in \mathbb K_{+}/\mathbb L}\frac{\prod_{i\in \bar{I}_+}\Gamma(1+\bar D_i-\langle D_i\cdot f\rangle)}{\prod_{i=1}^k\Gamma(1+v_i)}\prod_{i\in I_+,d_i<0}\frac{1}{\bar D_i-d_i}\textbf{1}_{f}[\textbf 1]_{(d_i)_{i\in I_+}}.
\psi^\mu_\mp |p,\pm\rangle_{RR}^0 = \psi^i_\pm |p,\pm\rangle_{RR}^0 = 0 \,, \qquad\psi^\alpha_{\pm}\equiv{1 \over \sqrt{2}}\left( \psi_0^{\alpha}\pm i \tilde{\psi}_0^{\alpha}\right) \,.
\omega_{Q_{XYU}}^{(\alpha)}(x,y|u):=\bar{\alpha}\Bigg(\log\frac{Q_{XY}(x,y)}{\pi_{XY}(x,y)}+\log\frac{Q_{XY|U}(x,y|u)}{Q_{X|U}(x|u)Q_{Y|U}(y|u)}\Bigg)+\alpha\log\frac{Q_{XY|U}(x,y|u)}{\pi_{XY}(x,y)}.
(1-\eta^{2})\frac{d^{2}S_{mn}(c,\eta)}{d\eta^{2}}-2\eta\frac{dS_{mn}(c,\eta)}{d\eta}+(\lambda_{mn}(c)-c^{2}\eta^{2}+\frac{m^{2}}{\eta^{2}-1})S_{mn}(c,\eta)=0
[\begin{matrix}1&3\\ 1&0\\ 1&2\end{matrix}]+[\begin{matrix}0&0\\ 7&5\\ 2&1\end{matrix}]=[\begin{matrix}1+0&3+0\\ 1+7&0+5\\ 1+2&2+1\end{matrix}]=[\begin{matrix}1&3\\ 8&5\\ 3&3\end{matrix}]
\Psi_{\mu_k}(w^{(k)}+ \lambda \triangle \tilde{w}^{(k)}) &\leq (1-\lambda) \Psi_{\mu_k}(w^{(k)})+ \varpi( \lambda\Psi_{\mu_k}(w^{(k)})) + \frac14 \lambda^2 C^2(1+\theta_1)^2\beta \Psi_{\mu_k}(w^{(k)})+ \lambda\sqrt2 \theta_1\Psi_{\mu_k}(w^{(k)}).
q_{1i}[\varphi, x_i]x_1 - x_1[\varphi ,x_i] = x_1 \sum_{M,N,b}\left( q_{1i}\beta (b,1) -1 \right)\alpha_{M,N,b}\rho_M \delta_N \sigma_b +\sum_{M,N,b}n_1 \alpha_{M,N,b}c_{M,N,b}\rho_M \delta_{N-e_1}\sigma_{b+e_1}.
\phi_{\xi'}(x_1,x_2)=e^{i[\chi_{\xi'}(x)-\chi_{\xi}(x)]}\phi_{\xi}(x_1,x_2)\ ,\ \ \ \ phi_C(x_1,x_2) = e^{i[\chi_C(x)-\chi_{\xi}(x)]}\phi_{\xi}(x_1,x_2)\ .
|\partial^\zeta_x(\nabla \phi^2 \cdot Q(v))-\nabla \phi^2 \cdot \partial^\zeta_x Q(v)|_2 \leq C\Big(|\nabla \phi|^2_\infty |\nabla^2 v |_2+|\phi|_\infty|\nabla^2 \phi|_6 |\nabla^2 v|_3 &+|\phi|_\infty |\nabla^3 \phi |_2|\nabla v|_\infty+|\nabla \phi|_\infty|\nabla^2 \phi|_2 |\nabla v|_\infty\Big) \leq Cc^3_3,
\begin{array}{l}\Omega^*=arg\min\limits_{\Omega \subseteq \mathbb{R}^n}\int\limits_{\Omega}d\mathbf{x}(k) s.t. \int\limits_{\Omega}dP(\mathbf{x}(k))= 1, \ \ \forall P\in \mathcal{P}_{\mathcal{X}_k}(\mathbf{X}(k)). \end{array}
Z_{+}^{*}( \tau^{+}) = - 2 \pi \int{\cal D}R^{+}{\cal D}\omega^{+}\delta [ \epsilon ( R^{\pm}- \dots ) ] e x p ( - \int_{X}\epsilon^{\mu \nu \rho \sigma}\tau^{+}R_{\mu \nu}^{+ a b}R_{\rho \sigma a b}^{+}) .
\Delta^{( 2 )}( x , x^{\prime}) = \frac{1}{2 \sqrt{\pi}}\int_{\Gamma_{l}+ \Gamma_{r}}f ( x , x^{\prime}, s ) e^{- \alpha}\alpha^{1 / 2}\left[ 2 + ( 4 / 3 ) \alpha \right] d s \quad , [ 0 . 3 c m ]
J(u_{t_{k}})=c_{0}(\sum_{i}\frac{1}{K_{i}^{\frac{n-2}{2}}})^{\frac{2}{n}}+o_{\varepsilon_{k}}(1)\geq c_{0}\frac{p^{\frac{2}{n}}}{(\max K)^{\frac{n-2}{n}}}+o_{\varepsilon_{k}}(1),
2n(n+ \alpha+\beta)(2n+\alpha+\beta-2)P_{n}^{(\alpha,\beta)}(x) &=(2n+\alpha+\beta-1)\left((2n+\alpha+\beta)(2n+\alpha+\beta-2)x+\alpha^2-\beta^2\right)P_{n-1}^{(\alpha,\beta)}(x) &\phantom{=}-2(n+\alpha-1)(n+\beta-1)(2n+\alpha+\beta)P_{n-2}^{(\alpha,\beta)}(x)
* \vee \left( *{\bf d}\omega - \frac 1 c \frac{\partial}{\partial t}{\cal I}_{*}\omega , \omega \right) - \vee \left({\cal I}_{*}\delta \omega ,{\cal I}_{*}\omega \right) = \frac{4 \pi}{c}* \vee \left({\cal I}_{*}{\cal J}, \omega \right) +
\bigg|\mathbb{E}_X\int fd\mu_X-\int fd\nu\bigg|= \bigg|\mathbb{E}_X\mathbb{E}_{\sigma}f\bigg(m^{-1/\alpha}\sum_{a\in\sigma}X_a\bigg)-\int fd\nu\bigg| \le \mathbb{E}_{\sigma}\bigg|\mathbb{E}_Xf\bigg(m^{-1/\alpha}\sum_{a\in\sigma}X_a\bigg)-\int fd\nu\bigg| \le \max_{\sigma}d_W(\rho_{\sigma},\nu).
| I_{2j}- \int_{K_j}\frac{e^{-itx}f_t (x) G(0)}{x- \lambda - i\delta}dx | = | \int_{K_j}\frac{e^{-itx}f_t (x) G(x)}{x-\lambda - i\delta}dx - \int_{K_j}\frac{e^{-itx}f_t (x) G(0)}{x- \lambda - i\delta}dx | \leq \sup_{K_j}\frac{|f_t (x)|}{|x- \lambda - i\delta|}\int_{K_j}| G(x) - G(0) | dx.
\int\limits_E h\circ\varphi_{i}\,\mathrm{d}\mu =\int\limits_{\varphi_{i}(E)}h \cdot f_{i}\,\mathrm{d}\lambda \quad\;\;\int\limits_E (h\cdot g_{i}) \circ \varphi_{i}\,\mathrm{d}\mu = \int\limits_{\varphi_{i}(E)}h \,\mathrm{d}\lambda .
\begin{aligned}\xi{(t,y)}= c_{1}+ c_{2}~{\sin{2t}}+c_{3}~{\cos{2t}}, \eta{(t,y)}={(c_{2}~{\cos{2t}}-c_{3}~{\sin{2t}})y}+c_{4}~{\sin{t}}+c_{5}~{\cos{t}}, B(t,y) = -{(c_{2}~{\sin{2t}}+c_{3}~{\cos{2t}}){y^{2}}}+{(c_{4}~{\cos{t}}-c_{5}~{\sin{t}})}{y}.\end{aligned}
H^{\textsc{e},\textsc{f}}(d;\Delta^{(1)},\dots,\Delta^{(\textsc{f})})=\sum\limits_{f\in C_{\Omega (z_1\dots,z_{\textsc{f}})}(d;\Delta^{(1)},\dots,\Delta^{(\textsc{f})})}\frac{1}{|\texttt{Aut}(f)|}\quad,
2=\left\langle r', r' \right\rangle = \left\langle a_1e_1+\cdots+a_ne_n, a_1e_1+\cdots+a_ne_n \right\rangle=\frac{\sum_{i=1}^nna_i^2-2\sum_{i<j}a_ia_j}{n+1}&=\frac{\sum_{i=1}^na_i^2+\sum_{i< j}(a_i-a_j)^2}{n+1}=\frac{(n-k)+k(n-k)}{n+1},
v^{(1,1)}_0 = v^{(2,1)}_0 = v^{(2,2)}_0 \I_2\times(0, T), h_{1,1}\partial_{x_2}v^{(1,1)}_0 = h_{2,1}\partial_{x_2}v^{(2,1)}_0 + h_{2,2}\partial_{x_2}v^{(2,2)}_0 \I_2\times(0, T),
k_x (x_{i+1/2}, y_{j}) u_{i+1, j}- \left( k_x (x_{i+1/2}, y_{j}) + k_x (x_{i-1/2}, y_{j}) \right) u_{i, j}+ & k_x (x_{i-1/2}, y_{j}) u_{i-1, j}+ k_y (x_{i}, y_{j+1/2}) u_{i, j+1}+ k_y (x_{i}, y_{j-1/2}) u_{i, j-1}- & \left( k_y (x_{i}, y_{j+1/2}) + k_y (x_{i}, y_{j-1/2}) \right) u_{i, j}= - h^2 f (x_{i}, y_{j}),
\int_{I}\cdot\cdot\cdot\int_{I}det[f_{j}(x_{k})]_{j,k=1}^{N}det[g_{j}(x_{k})]_{j,k=1}^{N}dx_{1}\cdot\cdot\cdot dx_{N}=N!det[\int_{I}f_{j}(x)g_{k}(x)dx]_{j,k=1}^{N}.
a_{i}(t-\tau,y_{\tau}(t),u_{\tau}(t))\ne 0,\forall t\in J_{\nu},i=1,\ldots,n-1 &\bar{a}_{\nu}<a_{\nu}-\varepsilon_{m,\nu}<b_{\nu}+\varepsilon_{m,\nu}<\bar{b}_{\nu},m=2,\ldots,n
\begin{array}{l}{t \ln t > ( t - 1 ) \ln ( t + 1 ) \qquad \mathrm{f o r}\quad t > 1 ,}\\{t \ln t < ( t - 1 ) \ln ( t + 1 ) \qquad \mathrm{f o r}\quad 0 < t < 1 ,}\\ \end{array}
\Lambda^{(m)}(t)=\frac{\delta}{m}\sum_{k=1}^{[mt]}Z_i=\frac{q\delta[mt]}{m}+\sqrt{\frac{q(1-q)\delta^2}{m}}\frac{1}{\sqrt{m}}\sum_{k=1}^{[mt]}\frac{Z_i-q}{\sqrt{q(1-q)}}.
\sum_{k=1}^{n}M_{h_0^k}x = \sum_{k=1}^{n}\sum_{m=0}^{3n}\alpha(k,m)x_m e_{m+k}= \sum_{i=1}^{4n}\Big(\sum_{m=0}^{i-1}\alpha(i-m,m)x_m \Big) e_{i}.
L^{\pi^{(x,y,i)}}_{r}(t) := L^{\pi^{(y,i)}}_{r}(t) 1_{\lbrace t>0 \rbrace}, R^{\pi^{(x,y,i)}}_{r}(t):= (y-x)1_{\lbrace t=0 \rbrace}+ R^{\pi^{(y,i)}}_{r}(t) 1_{\lbrace t>0 \rbrace}, t\geq 0.
\dot{z}_i =A_i^0z_i+f_i(x_i), \dot{\hat x}_{i}=S\hat x_i+dv+dK^{\top}(\hat x_i-\eta_i)+\bar g_i(\bar z_i,\,z_i,\,x_i) \dot{\eta}_i =S\eta_i+d v+ l_0c^{\top}\eta_{vi}\dot{\xi}_i =A_i^0\xi_i+f_i(x_i) \dot{w}=Sw+dv
\sqrt{\frac{1}{2 \pi \hbar \epsilon ( 1 + f ( q_{j}) \Delta q_{j})}}\; \exp \left[ \frac{( \Delta q_{j})^{2}}{2 \hbar \epsilon ( 1 + f ( q_{j}) \Delta q_{j})}\right] \, \approx \, \sqrt{\frac{1}{2 \pi \hbar \epsilon}}\; \exp \left\{\frac{\epsilon}{\hbar}\left[ \frac{1}{2}{\dot{q}}^{2}+ \frac{1}{2}\hbar f ( q ) \dot{q}\right] \right\}\, ,
h_{\frac{e}{2}}-h_{\frac{e}{2}-1}= \begin{cases}0 d_4\ge d_1+d_2+d_3 \frac{d_1+d_2+d_3-d_4}{2}-d_1+d_2+d_3\le d_4\le d_1+d_2+d_3 d_1 d_4\le -d_1+d_2+d_3 .\end{cases}
A^{(s,t)}_{k,l}=E{\bigg\{}\frac{\sum^{k}_{i=s}(-1)^{i-s}{{i-1}\choose i-s}{{m-i}\choose k-i}{S\choose i}}{{{m-s}\choose k-s}}\frac{\sum^{l}_{j=t}(-1)^{j-t}{{j-1}\choose j-t}{{n-j}\choose l-j}{T\choose j}}{{{n-t}\choose l-t}}{\bigg\}}.
\lim_{t\to\infty}\|e^t_k\| =\lim_{t\to\infty}L_k\|x^{t(k)}_k-x^{t(k)+1}_k\|+\left\|\nabla_kg\left(x^{t-1}\right)-\nabla_kg\left(x^{t(k)}\right)\right\|+\|x^{t}_k-x_k^{t-1}\| &=0+\|a_k^*-a_k^*\|+0=0,
\frac{1}{\prod\limits_{i<j}^n |t_i-t_j|^{2\lambda d(j-i)}}\sum\limits_{l\neq s}^{n}\frac{\partial}{\partial t_l}\Bigl[ (t_l-t_s)\prod\limits_{i<j}^n |t_i-t_j|^{2\lambda d(j-i)}\Bigr] = n-1 + 2\lambda\sum\limits_{l\neq s}^{n}\sum\limits_{j\neq l}^n d(|j-l|) \frac{t_l-t_s}{t_l-t_j}.
\frac{b_2^v}{1-c^2b_1^hb_2^v}=c^2b_1^h\hskip -0.3cm\sum_{j=m_1+1}^{m_1+m_2}\hskip -0.2cm\left(\frac{a_j^v\sqrt{A_j^v}}{\|u'_j\| (1-c^2b_1^hA_j^v)}\right){\!\!\!^{^{^{^{^{^{2}}}}}}}\hskip -0.1cm+\hskip -0.3cm \sum_{j=m_1+1}^{m_1+m_2}\hskip -0.2cm\left(\frac{(a_j^v)^2}{\|u'_j\|^2(1-c^2b_1^hA_j^v)}\right).
p_G^1(M) \stackrel{\eqref{equation:reg1}}{=}|G|^{n^*-g-1}\sum_{h_1,\dots,h_g \in G}\chi_{}(h_1^2\cdots h_g^2) &\stackrel{\eqref{equation:reg2}}{=}|G|^{n^*-g-1}\sum_{h_1,\dots,h_g \in G}\sum_{\rho \in \widehat{G}}n_{\rho}\chi_{\rho}(h_1^2\cdots h_g^2).
\hat{A}_{12}v_{p}'=-\frac{1}{\lambda_{p}}\hat{A}_{12}\hat{Q}^{-1}\mathbb{E}^{\tilde{\mathbb{P}}}\left[\pi\right]-\frac{1}{\lambda_{p}}\hat{A}_{12}\hat{Q}^{-1}\hat{A}_{12}^{\top}\mu'_{p}.
\Gamma_1 =r^{2n-1}\sum_{k=1}^n |w_1|^2 \eta_k\; \frac{d\psi}{2\pi}\wedge d\beta_{n-1}(w)\gamma_k(\eta)+ \textrm{vanishing terms}=r^{2n-1}\left(\sum_{k=1}^n\eta_k \gamma_k(\eta)\right) |w_1|^2\; \frac{d\psi}{2\pi}\wedge d\beta_{n-1}(w)+ \textrm{vanishing terms}\ .
u = \left[ q_s + 1, q_{s-1}, \ldots, q_1 \right] r_{s-1}+ \left[ q_{s-1}, \ldots, q_1 \right] r_s &= \left[ q_{s-1}, \ldots, q_1 \right] \left( r_{s-1}+ r_s \right) + \left[ q_{s}, \ldots, q_1 \right] r_{s-1}
\begin{pmatrix}0 \mu^2\lambda 1 0\end{pmatrix}\begin{pmatrix}1 1\end{pmatrix}=\begin{pmatrix}\mu^2\lambda 1\end{pmatrix},\begin{pmatrix}0 \mu^2\lambda 1 0\end{pmatrix}\begin{pmatrix}\mu 1\end{pmatrix}=\begin{pmatrix}\mu^2\lambda \mu\end{pmatrix},\begin{pmatrix}0 \mu^2\lambda 1 0\end{pmatrix}^2=\mu^2\lambda I_2.
a\#b \sim \sum_{k=0}^\infty \left(\frac{i}{2}\right)^k (a\# b)_k(a\# b)_k=\sum_{|\alpha|+|\beta|=k}\frac{1}{\alpha!\beta!}(-1)^{|\beta|}(\partial^\alpha_x\partial^\beta_\xi a)(\partial^\beta_x\partial^\alpha_\xi b)
- \frac 1{3C_2}(\partial_z u^z)(t,0) = \sum_{j\le M}\int F_3(\phi(t,r,z)) b_3(2^j(r,z)) dr dz & = \sum_{j\le M}\int F_3( 2^j \phi(t, 2^{-j}r, 2^{-j}z)) b_3(r,z) dr dz.
\sum_{n\geq 1}c(n)q^n | T_{p^2}= \sum_{n\geq 1}\left( c(p^2n) + \left(\frac{(-1)^kn}{p}\right)\chi(p)p^{k-1}c(n) + \chi(p)^2p^{2k-1}c(n/p^2)\right)q^n,
c_{p}=\frac{1}{4\pi^{2}p(2n-p)}\left( q_{p}+\sum_{k=1}^{p-1}\sum_{n_{1},n_{2},...,n_{k}}\frac{q_{n_{1}}q_{n_{2}}...q_{n_{k}}q_{p-n(k)}}{b(n,p,k)}\right) ,
\begin{aligned}\mathbb{E}^{u_1,u_2}[\tilde{H}_{1{v_1}}(t,x,y,z,z_1,z_2,u_1,u_2;q_1,k_1,k_{11},k_{21},p_1,Q_{11},Q_{21})|\mathcal{F}_t^1]=0, \mathbb{E}^{u_1,u_2}[\tilde{H}_{1{v_2}}(t,x,y,z,z_1,z_2,u_1,u_2;q_2,k_2,k_{12},k_{22},p_2,Q_{12},Q_{22})|\mathcal{F}_t^2]=0, \end{aligned}
2x^3+2xy^2-2x^2(u_1+u_2)-y^2(u_1+u_2)-2xy(v_1+v_2)+ x(a_1+a_2+ 2u_1^2+2u_2^2)+2y(u_1v_1+u_2v_2)-(a_1u_1+a_2u_2)=0 2y^3+2yx^2-2y^2(v_1+v_2)-x^2(v_1+v_2)-2xy(u_1+u_2)+ y(a_1+a_2+ 2v_1^2+2v_2^2)+2x(u_1v_1+u_2v_2)-(a_1v_1+a_2v_2)=0
Q_1=\frac{4\alpha \Phi}{(\lambda_1+\lambda_2)^3}\Big(\big((1+\alpha)\frac{\lambda_1}{\lambda_2}+(1-\alpha)\big)(\nabla_1h_{22})^2+\big((1-\alpha)+(1+\alpha)\frac{\lambda_2}{\lambda_1}\big)(\nabla_2h_{11})^2\Big).
S_{eff}=\frac{1}{2}E^aE^b\left(\delta^{ab}+\frac{22}{3}\frac{g^2}{(4\pi)^2}\ln\left(\frac{1}{\mu^2}\sqrt{2g^2E^aE^b)}\right)-\frac{11i}{48\pi}g^2\delta^{ab}\right)
S_m = \frac{1}{2}\sum_{i=1}^n \sum_{\substack{j=1 j\ne i}}^n \frac{z_i^m - z_j^m}{z_i-z_j}= \frac{1}{2}\sum_{i=1}^n \sum_{\substack{j=1 j\ne i}}^n \sum_{p=0}^{m-1}z_i^{m-1-p}z_j^p,
z_o^* + \sum\limits_{i=1}^n|z_i^*| = \lim (z_o+\sum\limits_{i=1}^n|z_i|) = \lim ( t+ \sum\limits_{i=1}^n|x_i| - \mu^2\sum\limits_{i=1}^n\cfrac{2}{t+|x_i|}& \leq \lim (t+\sum\limits_{i=1}^n|x_i|) = t^* + \sum\limits_{i=1}^n|x_i^*|=0.
\Gamma(s)G(z,s;B_{-b},A_{-c}) =\left( -2\pi i/k\right)^{s}k\left(A\left( z,s;B_{-b},\widehat{A}_{c^{-1}}\right) +e(s/2)A\left(z,s;B_{b},\widehat{A}_{-c^{-1}}\right) \right) & \quad+\Gamma(s)f^{\ast}(0)\left( L(s;A_{-c})+e(s/2)L(s;A_{c})\right).
\inf\left\{\beta\in(0,2):\sum_{k=1}^{\infty}\sum_{n=1}^{\infty}\frac{1}{k^{\beta}+n^{\beta}}\left(\frac{1}{k^{\alpha_1}+ n^{\alpha_2}}\right)^{2-1/\alpha_1-1/\alpha_2}<\infty\right\}&=\max\left\{3+\frac{\alpha_1}{\alpha_2}-2\alpha_1,2+\frac{2\alpha_2}{\alpha_1}-2\alpha_2\right\}.
\mathcal{E}_z(f,h) = -\int_{\Sigma_z}(\mathcal{L}_0 f) \, h\, d\mu_z\, = -\int_{\Sigma_z}f (\mathcal{L}_0 h) \, d\mu_z\, = \frac{1}{\beta}\int_{\Sigma_z}(a\Pi\nabla f) \cdot (\Pi \nabla h)\, d\mu_z\,,
\frac{\epsilon_1'}{1+\epsilon_1}-\frac{\epsilon_0'}{1+\epsilon_0}\ =\ \frac{\epsilon_1'}{1+\epsilon_1}- (1+\delta+\epsilon_1+\delta\epsilon_1)^{\dagger}&=\ \frac{\epsilon_1'}{1+\epsilon_1}- ((1+\delta)(1+\epsilon_1))^{\dagger}&=\ \frac{\epsilon_1'}{1+\epsilon_1}- \frac{\delta'}{1+\delta}- \frac{\epsilon_1'}{1+\epsilon_1}&=\ -\frac{\delta'}{1+\delta}
\frac{\partial S}{\partial\beta_{j}}=\frac{\partial S}{\partial\beta_{j}^{R}}\frac{\partial\beta_{j}^{R}}{\partial\beta_{j}}+\frac{\partial S}{\partial\beta_{j}^{I}}\frac{\partial\beta_{j}^{I}}{\partial\beta_{j}}=\frac{\partial S}{\partial\beta_{j}^{R}}-i\frac{\partial S}{\partial\beta_{j}^{I}}(j=1,2,3,...,n).
( f \star g ) ( p^{( 3 )}) = \frac{1}{( 2 \pi )^{1 3}}\int d p^{( 1 )}d p^{( 2 )}\delta^{2 6}( p^{( 1 )}+ p^{( 2 )}- p^{( 3 )}) f ( p^{( 1 )}) g ( p^{( 2 )})
v_N^{y,SO(5)}= ll^{\ast}\otimes 1 + 1 \otimes l + 1 \otimes l^{\ast}+ l \otimes l^{\ast}+ l^{\ast}\otimes l, v_N^{z,SO(5)}= ll^{\ast}\otimes 1 + 1 \otimes ll^{\ast}+ l \otimes 1 + l^{\ast}\otimes 1 + l \otimes l^{\ast}+ l^{\ast}\otimes l + ll^{\ast}\otimes l + ll^{\ast}\otimes l^{\ast}+ l^{\ast}\otimes l^2 + l \otimes (l^{\ast})^2,
f_{x_i}(n) \!=\! \frac{p}{N}\!+\! \left(1\!-\!p\right) \!\frac{3N\left(2n\!-\!1\right)\!-\!6n\left(n\!-\!1\right)\!-\!3}{N(N^2-1)}, n\!\leq\! N
E_{o p t}^{( p )}\simeq 0 . 4 2 0 7 9 8 7 - 1 1 . 9 8 \times 1 0^{- 5}/ p + 8 . 9 9 1 \times 1 0^{- 5}/ p^{2}+ \mathrm{O}( p^{- 3}) \, .
{\partial S_0\over\partial \Omega_{p+1}}= -{B\over 2\Omega_{p+1}\, (p+1)!}\left(\,\sigma_{\mu_1\dots\mu_{p+1}}-\sigma_{0)\mu_1\dots\mu_{p+1}}\right)^2-\frac{m_{p+1}}{2}\, p\, \Omega_{p+1}\,
\bar{I}^{(2l-2\varepsilon)}=\sum_{j=0}^{l}\left(\frac{m^2}{4m^2-q^2}\right)^{j-\varepsilon}\frac{\Gamma(2j-2\varepsilon-3)\Gamma(2-j+\varepsilon)}{i^{2j-2\varepsilon}\Gamma(j-1-\varepsilon)}+C^{l}_{\varepsilon},
A(\rho) =\tilde{\tau}^{-1}\rho(\tilde{\tau})\zeta_\phi^{p,\infty}(\rho) &=\tilde{\tau}_G^{-1}\zeta_\phi^{p,\infty}(\rho)\rho(\tilde{\tau}_G)\zeta_\phi^{p,\infty}(\rho)^{-1}\zeta_\phi^{p,\infty}(\rho) &=\tilde{\tau}_G^{-1}\zeta_\phi^{p,\infty}(\rho)\rho(\tilde{\tau}_G).
\inf_{\begin{subarray}{c}\mu_{N}\in \mathcal{M}(\mathbf{R}, 2(N+1)) \mathop{\mathrm{supp}}\mu_{N}= [-\alpha_{N}, \alpha_{N}] \end{subarray}}\left[\sup_{x \in \mathbf{R}}\left(V_{\mathcal{D}_{d}}^{\mu_{N}}(x) + \log w(x)\right)\right].
i!\,{\lvert \H\,}_{i}\left( f^{\ast}\right){\rvert}={\lvert \f_{\zeta^{(i)}}^{\ast}(\ldots f_{\zeta^{(j)}}^{\ast}(\ldots (f_{\zeta^{(1)}}^{\ast}({[V(\zeta^{(1)},\zeta^{(2)},\ldots ,\zeta^{(i)})]}^{2})\ldots )\ldots )\rvert ,}
\int_{0}^{1}g_{\mathrm{hyp},s}(z,\sigma_{q}w)du= \frac{4\pi y_{p}^{1-s}}{2s-1}\left(\delta_{p,q}v^{s}+ \alpha_{p,q}(s)v^{1-s}\right)+\frac{4\pi v^{1-s}}{2s-1}\sum_{n\not = 0}\alpha_{p,q}(n,s)W_{s}(n\sigma_{p}^{-1}z).
- \alpha \beta \frac{3 f^{2}}{6 4 \pi^{2}}\gamma_{5}/ \partial \delta^{4}( y ) = ( \alpha^{3}R + \beta^{3}L ) \hat{\Sigma}( y ) |_{\stackrel{{\scriptstyle \mathrm{a b n .}}}{{\scriptstyle \mathrm{p a r .}}}}.
[ Q , \bar{C}^{a}]_{P . B .}\stackrel{d e f .}{=}\tilde{K}_{a}= K_{a}+ f_{a b}^{c}\bar{C}_{c}C^{b}, [ \tilde{K}_{a}, \tilde{K}_{b}]_{P . B .}= f_{a b}^{c}\tilde{K}_{c}.
{\cal L}= - \frac{1}{4}F_{\mu \nu}F^{\mu \nu}+ \frac{\hbar}{2}\bar{\psi}\gamma^{\mu}\left( i \partial_{\mu}- e A_{\mu}\right) \psi - \frac{\hbar}{2}\left[ \left( i \partial_{\mu}+ e A_{\mu}\right) \bar{\psi}\right] \gamma^{\mu}\psi
m(f) \leq \prod_{i'=1}^df_{i'}^L + \sum_{i = 1}^d\Biggl( \prod_{i' \in \{\omega_1,\ldots,\omega_{i-1}\}}f^U_{i'}\Biggr) \cdot \Biggl(\prod_{i' \in \{\omega_{i+1},\ldots,\omega_{d}\}}f^L_{i'}\Biggr) \cdot (f_{\omega_i}- f_{\omega_i}^L),
\frac{1}{2}\mathbb{E}{[\|x_t-x^*\|^2]}=\mathbb{E}[G_{\Gamma t}] \le e^{-2\kappa\gamma\Gamma t}G_{0}+\frac{m\sigma^2 \Gamma}{4\kappa N}(1-e^{-2\kappa\gamma\Gamma t}).
\|A_{|I|}(\theta_{\ell_1})\|=\frac{1}{\prod_{\ell\in I}|\cos(\pi\theta_{\ell})|}\|F_{|I|}(\theta_{\ell_1})\|\leq C(\varepsilon) e^{\varepsilon (2q_n-|I|)}e^{|I| \ln 2}e^{|I| (\tilde{L}+\varepsilon)}\leq e^{3\varepsilon q_n}e^{L |I|}.
\mathfrak{p}_{s+1}= [\mathfrak{p}_s,\mathfrak{n}] + [J\mathfrak{p}_s,\mathfrak{n}] \subseteq [\mathfrak{d}_s,\mathfrak{n}] \subseteq \mathfrak{d}_{s+1}J\mathfrak{p}_{s+1}\subseteq J[\mathfrak{d}_s,\mathfrak{n}] \subseteq \mathfrak{d}_{s+1}
\begin{array}{c c c}{{D}}&{{:}}&{{\Omega^{p}\longrightarrow \Omega^{p + 1}\hspace{4 c m}}}\\{{D ( ( D F_{1}) . . . . ( D F_{p}) F )}}&{{\equiv}}&{{( - 1 )^{- 1}( D F_{1}) . . . ( D F_{p}) ( D F ) , \forall F , F_{1}, . . . , F_{p}\in \Omega^{p}({\cal A}) ,}}\\ \end{array}
\lim_{\lambda_1, \ldots, \lambda_d \rightarrow 1}\frac{\det(\varphi_i(\lambda_j))_{i, j = 1}^d}{\Delta(\lambda_1, \ldots, \lambda_d)}= \frac{\det(\varphi_i^{(j-1)}(1))_{i, j = 1}^d}{\prod_{j=1}^{d-1}j!}.
F(x+y)+F(x-y)-2 = F(x+y)-F(x)+F(x-y)-F(x) = \int_0^1 \langle\mathrm{D}F(x+t y)-\mathrm{D}F(x-t y), y\rangle\mathrm{d}t = 2\int_0^1\int_0^1 \langle \mathrm{D}^2 F(x-ty+2s y)y, y\rangle s\mathrm{d}s\mathrm{d}t \lesssim \| D^{2}F \|_{L^{\infty}(B_{3/2}\setminus B_{1/2})}F(y)^{2}.
{\cal R}_{ij}=\frac{1}{m}{\rm Tr}\left\{\left(\partial_{(i}{\cal E}\right){\cal P}^{-1}\left(\partial_{j)}{\cal E}^{\dagger}\right){\cal P}^{-1}\right\}=-\frac{1}{2m}{\rm Tr}\left\{\left(\partial_{(i}{\cal M}\right)\partial_{j)}{\cal M}^{-1}\right\}.
A_{1,\ell}=-\frac{16}{\lambda_{\ell}^3}\int_{C/\ell}^{\pi -C/\ell}\frac{\beta^2_{2,\ell}(\phi)}{(1-4 \alpha_2^2/\lambda_{\ell}^2)^{3/2}(1-4 \alpha_1^2/\lambda_{\ell}^2)^{1/2}}\sin \phi d \phi=-16 \int_{C/\ell}^{\pi -C/\ell}\frac{\beta^2_{2,\ell}(\phi)}{\ell^6}\sin \phi d \phi+O(\ell^{-2})=O(\ell^{-2}),
- \frac{1}{x_{1 2}^{4}\, x_{3 4}^{4}}\, ( 2 0 - 2 0 Y - \frac{1 6}{3}v + \frac{1 0}{3}Y^{2}+ \frac{8}{3}v Y + \frac{1}{3}v^{2}) \, \Phi ( v , Y )
V_{0}={\frac{\partial}{\partial \bar{z}}}+{\frac{1}{2}}\xi_{i}^{s}\lambda_{, \bar{z}}^{i}({\frac{\partial \sigma_{s}}{\partial p}}{\frac{\partial}{\partial q}}-{\frac{\partial \sigma_{s}}{\partial q}}{\frac{\partial}{\partial p}}) ,
\vec{q}_{\ell}={}^t\Biggl(\psi_{{1,r-1}}(t^nP_{\ell}(t)),\ldots, \psi_{{1,0}}(t^nP_{\ell}(t)),\ldots, \psi_{{m,r-1}}(t^nP_{\ell}(t)),\ldots, \psi_{{m,0}}(t^nP_{\ell}(t))\Biggr)\enspace.
\sup_{\boldsymbol{\alpha}\in\mathbb{A}_{\boldsymbol{r}}}\sup_{f_0\in\mathcal{B}^{\boldsymbol{\alpha}}_{\infty,\infty}(R)}\mathrm{E}_0\|\mathrm{E}(D^{\boldsymbol{r}}f|\boldsymbol{Y})-D^{\boldsymbol{r}}f_0\|_\infty\lesssim(n/\log{n})^{-\frac{\alpha^{*}\{1-\sum_{l=1}^d(r_l/\alpha_l)\}}{2\alpha^{*}+d}}.
{\frac{1}{2}}\sqrt{- \eta}\; \eta^{\mu \nu}={\frac{1}{1 - \lambda^{2}}}\; \left( \begin{array}{c c}{{{\lambda_{--}}}}&{{{{\frac{1 + \lambda^{2}}{2}}}}}\\{{{{\frac{1 + \lambda^{2}}{2}}}}}&{{{\lambda_{+ +}}}}\\ \end{array}\right)