harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\delta \bar{f}_{h_k,2i}= \bar{f}_{h_k,2i}- \bar{f}_{h_{k-1},i}, \delta \bar{f}_{h_k,2i+1}= \bar{f}_{h_k,2i+1}- \bar{f}_{h_{k-1},i}, i=0,1,\ldots,N_{h_{k-1}}.
S_{R^{2}, b u l k}^{( 1 )}= \frac{1}{2 \kappa^{2}}\int d^{D}x \, \sqrt{- G}\frac{\alpha^{\prime}}{4}e^{\gamma \Phi}( R^{M N P Q}R_{M N P Q}- 4 R^{M N}R_{M N}+ R^{2}) ~ .
\sin_q (z) = \frac{\theta_1(z\mid \tau')}{\theta_1\left( \frac{\pi}{2}\bigm| \tau' \right)}\cos_q (z) = \frac{\theta_1\left( z+\frac{\pi}{2}\bigm| \tau' \right)}{\theta_1 \left( \frac{\pi}{2}\bigm| \tau' \right)}(\tau' = \frac{-1}{\tau})
\frac{1}{2\pi i}\int_{d-i\infty}^{d+i \infty}\frac{x^s}{s(s+1) \cdots (s+k)}\,ds = \begin{dcases}0 0 < x < 1, \frac{1}{k!}\left(1 - \frac{1}{x}\right)^k x \geq 1. \end{dcases}
[I,\bar{I}]^{A}{}_{B}= \, -(I^{A}{}_{R}\bar{I}^{R}{}_{B}-\bar{I}^{A}{}_{R}I^{R}{}_{B}),{\big(I\odot\bar{I}\big)_o}^{A}{}_{B}= \, \big(I^{A}{}_{R}\bar{I}^{R}{}_{B}+\bar{I}^{A}{}_{R}I^{R}{}_{B}\big)_o, \langle I,\bar{I}\rangle= \, I^{R}{}_{S}\bar{I}^{S}{}_{R}.
\begin{multlined}[t][12.5cm]\left|\widetilde{\Pi}^{P}_{\nu_{G},6c+\nu_{1}+3\nu_{2}}\left((z_{0},z_{1},z_{2}),(z_{0},z_{1},z_{2})\right)\right|\sim\frac{9\sqrt{3}c}{2\pi^3}\left(\frac{1}{3}\right)^{\nu_{1}+\nu_{2}}\left(\frac{1}{3}\right)^{\nu_{2}}.\end{multlined}
\Vert u(t)\Vert_{\infty}\leq\begin{cases}K \, t^{-\frac{N}{2q_0+N(m_2-1)}}\left( t + \| u_0 \|_{q_0}^{1-m_2}\right)^{-\frac{2q_0}{(m_2-1)[2q_0 + N(m_2-1)]}}\forall t \in \left( 0, 1 \right) , K \, t^{-\frac{1}{m_1-1}}\forall t > \| u_0 \|^{\frac{2q_0}{N}}_{q_0^{\phantom{a}}}\, ,\end{cases}
\begin{array}{c}(V_{n-2}|_S-N|_S)^2=(C_{2,Q}-V_{n-1}|_S)^2= C_{2,Q}^2-2C_{2,Q}\cdot V_{n-1}|_S + (V_{n-1}|_S)^2 =-1+(V_{n-1}|_S)^2\geq 0\end{array}
c x_{0}^{2}x_{1}^{2}+ c x_{0}^{2}x_{3}^{2}+ d x_{1}^{2}x_{3}^{2}+ x_{0}^{4}+ x_{0}x_{2}^{3}- x_{1}^{3}x_{3}+ x_{1}x_{3}^{3}= 0,
\sum_{j = 1}^{n}q^{\rho_{j}- 2 ( n - j )}\Lambda_{j}^{1 / 2}\mu_{j}^{- 1 / 2}{\cal X}^{j}{\cal X}^{- j}+ \frac{q^{- 2 n + 1}}{1 + q}{\cal X}^{0}{\cal X}^{0}= - \frac{1}{a^{2}}~ .
\begin{gathered}\iota(s)=s,\iota(t)=t,\iota(y)=y,I_1=\iota(x_s) = u\, x_s,\iota(y_t)=k, J_{0,1}=\iota(u_t)=\frac{u_t}{u},I_2=\iota(x_{ss})=u_s\, x_s + u\, x_{ss},\iota(y_{tt})=0, J_{0,2}=\iota(u_{tt})=\frac{u_{tt}}{u},J_{1,1}=\iota(u_{st})=\frac{u\, u_{st}-u_t\, u_s}{u^2}.\end{gathered}
\sum \frac{f ( z_{n})}{s^{\prime}( z_{n})}= w + \int_{0}^{\infty}{f ( x ) d x}+ \int_{0}^{\infty}{\left[ \frac{f ( i x )}{e^{- 2 \pi i s ( i x )}- 1}- \frac{f ( - i x )}{e^{2 \pi i s ( - i x )}- 1}\right] d x},
\big\|{\rm\bf Def}_\mathcal{C}(\mathbf{V})\big|\mathbb{L}_p(\mathcal{C})\big\|:=\left[\sum_{j,k=1}^n \big\|\mathfrak{D}_{jk}(\mathbf{V})\big|\mathbb{L}_p(\mathcal{C})\big\|^p\right]^{1/p}, \mathbf{V}\in\mathbb{W}^1_p(\mathcal{C}).
f^{p^t-1}= x_1^{p^ta_1-a_1}\cdots x_d^{p^ta_d - a_d}= (x_1^{a_1-1}\cdots x_d^{a_d-1})^{p^t}\cdot x_1^{p^t-a_1}\cdots x_d^{p^t-a_d},
\left(\frac{p}{q}\right)^{L}= (-1)^{N}\prod_{k \neq j ,i}\frac{p + p z_k - p/q}{p +p z_k - z_k}&= (-1)^{N}\prod_{k \neq j ,i}\frac{(p/q) (q z_k - p)}{p - q z_k}&= (p/q)^{N-2}.
\left( \frac{1}{2|\beta|^2}- \frac{1}{2|\beta'|^2}\right)^2 + \left| \frac{\alpha}{\beta}- \frac{\alpha'}{\beta'}\right|^2 = \left( \frac{1}{2|\beta|^2}+ \frac{1}{2|\beta'|^2}\right)^2 + \frac{| \alpha \beta' - \alpha' \beta |^2 -1}{|\beta|^2 |\beta'|^2}&\ge \left( \frac{1}{2|\beta|^2}+ \frac{1}{2|\beta'|^2}\right)^2.
\left(\begin{array}{cc}R^{\alpha\beta}0 0 R^{\alpha\beta}\end{array}\right)- \frac{1}{2}\left(\begin{array}{cc}g^{\alpha\beta}0 0 g^{\alpha\beta}\end{array}\right)R = \left(\begin{array}{cc}T^{\alpha\beta}0 0 T^{\alpha\beta}\end{array}\right)
\overline{\delta_k}(a_ip_{\alpha_i,b_j}Y_j) = \ \overline{\sigma_k}(a_i)\overline{\delta_k}(p_{\alpha_i,b_j}Y_j) + \overline{\delta_k}(a_i)p_{\alpha_i,b_j}Y_j = \ \sigma_k(a_i)p_{\alpha_i, \delta_k(b_j)}Y_j + \delta_k(a_i)p_{\alpha_i,b_j}Y_j,
F [ f ] ( x ) - c_{2}^{\prime}\delta ( ( x^{0})^{2}- |{\bf x}|^{2}) = \sum_{k = 0}^{N}a_{k}\left( \sum_{\nu = 0}^{3}\eta^{\nu \nu}\left( \frac{\partial}{\partial x^{\nu}}\right)^{2}\right)^{k}\delta ( x ) .
\forall \phi\in\mathcal{C}_c^\infty (\mathbb{R}^n) \lim_{k\to\infty}\int_\Omega \phi^2 |\nabla u_k|^2 ~\mbox{d}x = \int_{\mathbb{R}^n}\phi^2 ~\mbox{d}\mu, \lim_{k\to\infty}\int_\Omega \phi^2 |D(u_k)|^2 ~\mbox{d}x = \int_{\mathbb{R}^n}\phi^2 ~\mbox{d}\nu.
I =( x_1x_2x_3, x_1x_3x_5, x_1x_4x_5, x_2x_3x_4, x_2x_4x_5, x_1x_2x_6, x_1x_4x_6, x_2x_5x_6, x_3x_4x_6, x_3x_5x_6 )
[\begin{matrix}e_{1}^{\prime}(t)\\e_{2}^{\prime}(t)\\e_{3}^{\prime}(t)\\ \end{matrix}]=\|\gamma^{\prime}(t)\|[\begin{matrix}0&\kappa(t)&0\\ -\kappa(t)&0&\tau(t)\\ 0&-\tau(t)&0\\ \end{matrix}][\begin{matrix}e_{1}(t)\\e_{2}(t)\\e_{3}(t)\\ \end{matrix}]
\sum_{n \geq 1}A(n, m) e\Bigl( \frac{n d}{c}\Bigr) \psi(n) = c \sum_{\pm}\sum_{n_1 \mid c m}\sum_{n_2 \geq 1}\frac{A(n_1, n_2)}{n_1 n_2}S(m \bar d, \pm n_2; m c n_1^{-1}) \Psi^{\pm}\Bigl( \frac{n_2 n_1^2}{c^3 m}\Bigr).
\frac{1}{2}\left[ a_{2m}\partial^{2m}+\partial^{2m}M_{a_{2m}}\right] =\partial^{m}M_{a_{2m}}\partial^{m}+\frac{1}{2}\left[ \partial^{m},\left[\partial^{m},M_{a_{2m}}\right] \right] & =\partial^{m}M_{a_{2m}}\partial^{m}+[\text{diff. operator of order at most}2m-2].
F_1(\Omega_n) = \int_{\Gamma_n}j_1(x,\nabla b_{\Omega_n}(x),H_{\Gamma_n}(p_n(y))) d\mu_{\Gamma_n}(x) = \frac{1}{2h}\int_{U_h(\Gamma_n)}j_1(p_n(y),\nabla b_{\Omega_n}(p_n(y)),H_{\Gamma_n}(p_n(y))) \, \det(d_{T_n^{-1}(y)}T_n) \, dy.
\zeta_1 = \frac{\alpha_0}{r}- \frac{3T+1}{2r}+ \frac{3t-x}{2r}, \zeta_2 = \frac{\beta_0}{r}- \frac{3T+1}{2r}+ \frac{t+x}{r},\zeta_3 = \zeta_2 - \frac{1}{r}, \zeta_4 = \frac{\alpha_0 - \beta_0}{r}- \frac{t+x}{2r},\zeta_5 = \zeta_4 + \frac{1}{r},
S_+(z) = S_-(z)\begin{pmatrix}1 0 z^{-\alpha}e^{-2n\xi(z)}1\end{pmatrix}, z\in \gamma_1 \cup \gamma_3, &S_+(z) = S_-(z)\begin{pmatrix}0 z^{\alpha}-z^{-\alpha}0\end{pmatrix}, z\in \gamma_2, &S_+(z) = S_-(z)\begin{pmatrix}1 z^\alpha e^{2n \xi(z)}0 1\end{pmatrix}, z\in (b,\infty).
\sum_{|\sigma_1|+\dots+|\sigma_s|=L}\ \prod_{i=1}^{s}Q(\sigma_i) = \sum_{j=1}^m\ \sum_{(\pi(\sigma_1),\dots,\pi(\sigma_s)) = T_j}\ \prod_{i=1}^{s}Q (\sigma_i) \leq \sum_{j=1}^m \ \prod_{k=1}^{s}\left( \sum_{\pi(\sigma_k) =T_{j,k}}Q(\sigma_k)\right).
F_i:=Y_{\frac i4}= \mathcal S\left(\frac 14\right) Y_{\frac{i-1}4}+\int_{\frac{i-1}4}^{\frac i4}\mathcal S\left(\frac i4-s\right)\alpha_s ds +\int_{\frac{i-1}4}^{\frac i4}\mathcal S\left(\frac i4-s\right)\sigma_s dW_s = \mathcal S\left(\frac 14\right) Y_{\frac{i-1}4}+\mu_i + \epsilon_i,
P\bigl(\mathcal{E}_n^c\bigr) \leq2p \bigl( \exp\bigl\{-c\varepsilon_{n1}^2 n^{\beta}\bigr\}+\exp\bigl\{-c\varepsilon_{n2}^2 n \bigr\}\bigr) = o\bigl(p^{-4}\bigr).
G_{2k}(-1/\tau) = \sum_{(m, n) \neq (0, 0)}\frac{1}{(-m/\tau + n)^{2k}}&= \sum_{(m, n) \neq (0, 0)}\tau^{2k}\frac{1}{(-m + n\tau)^{2k}}&= \tau^{2k}G_{2k}(\tau)
\left\Vert u\right\Vert_{\mathbb{G}_{\alpha}^{p,q}(\mathbb{T}^{d})}:= \left(\sum_{k\in \mathbb{Z}^{d}}\left(2\alpha \left\Vert k\right\Vert^{q}_{p}\right)\hat{u}_{k}\right)^{1/2}< \infty,
- D_{\mu}^{i j}[ A ] \langle j^{\mu , j}\rangle + D_{\mu}^{i j}[ A ] Y_{\nu}^{k}\langle \frac{\delta j^{\nu , k}}{\delta A_{\mu}^{j}}\rangle + g f^{i j k}( Y_{\mu}^{j}\frac{\delta W}{\delta Y_{\mu}^{k}}+ J_{\mu}^{j}\frac{\delta W}{\delta J_{\mu}^{k}}+ \xi^{j}\frac{\delta W}{\delta \xi^{k}}+ \overline{{{\xi}}}^{j}\frac{\delta W}{\delta \overline{{{\xi}}}^{k}}) = 0 .
\langle \varphi | \chi \rangle_{{\cal L}}^{l}= ( 2 l - 1 ) \int_{| z | < 1}\frac{d z d \bar{z}}{2 \pi i}( 1 - z \bar{z})^{2 l - 2}\overline{{{\varphi ( \bar{z})}}}\chi ( \bar{z})
A =(a_{1}-\frac{\chi_{1}k}{d_{3}}-|\Omega|(a_3)_-)(b_{2}-2\frac{\chi_{2}}{d_{3}}l-|b_{4}||\Omega|) b_0- (b_{2}-\frac{\chi_{2}}{d_{3}}l-|\Omega|(b_4)_-)(\frac{\chi_{2}k}{d_{3}}+|\Omega|(b_3)_+)a_0 -\left(\frac{\chi_{2}k}{d_{3}}-b_1+|\Omega|(b_3)_-\right)((b_{4})_+|\Omega|+\frac{\chi_2}{d_3}l)a_0,
L_{B}= -{\frac{1}{4}}H^{2}{}_{\mu \nu}( B ) +{\frac{m}{4 !}}\epsilon^{\mu \nu \rho \omega}\, B_{\mu}\, \partial_{\, [ \, \nu}A_{\rho \sigma \, ]}-{\frac{1}{2 \cdot 3 !}}F^{2}{}_{\mu \nu \rho}( A )
n^{-1}\left| \sum_{p=1}^n u^{p\bar p}u_{p\bar p \bar j}\right|^2- \sum_{p,q=1}^n u^{p\bar p}u^{q\bar q}|u_{p\bar q j}|^2 \leq n^{-1}\left| \sum_{p=1}^n u^{p\bar p}u_{p\bar p \bar j}\right|^2- \sum_{p=1}^n|u^{p\bar p}u_{p\bar p j}|^2\leq 0.
\Big(-\frac{d}{ds}b^{(i)}(s,\tau,\alpha^{(i)}(s,\tau,\hat y'))\Big)e^{-b^{(i)}}v_i^g(s,\tau,y')+e^{-b^{(i)}}v_{is}^g(s,\tau,y') = \frac{\partial\tilde w_i^g(s,\tau,\alpha^{(i)})}{\partial s}+\sum_{j=1}^{n-1}\frac{\partial \tilde w_i^g(s,\tau,\alpha^{(i)})}{\partial\hat y_j}\alpha_{js}^{(i)}(s,\tau,y').
2g_n-2 = -2p^n + \sum_{j=1}^n \varphi(p^j) ( 1+ d p^{j-1}) = -2p^n + p^n-1 + d \frac{p^{2n}-1}{p+1}= \frac{d}{p+1}p^{2n}- p^n - \frac{p+1+d}{p+1}.
B(0) - B_0 = \frac{1}{2}\sum_{\pm}Op(Ad( O_{\pm})_{<0})(0,x,D) Op(Ad(O^{-1}_{\pm})_{<0})(D,0,y)( B_0 \pm i|D|^{-1}B_1) - B_0 & = \big[\frac{1}{2}\sum_{\pm}Op(Ad( O_{\pm})_{<0})(0,x,D) Op(Ad(O^{-1}_{\pm})_{<0}(D,0,y))- I\big]( B_0 \pm i|D|^{-1}B_1)
N = O O^{*}= c^{2}t^{2}+ \frac{\hbar^{2}}{P_{x}^{2}}+ \frac{\hbar^{2}}{P_{y}^{2}}+ \frac{\hbar^{2}}{P_{z}^{2}}- x^{2}- y^{2}- z^{2}- \frac{c^{2}\hbar^{2}}{E^{2}},
\mathcal{C}_{net}= 2\pi\lambda_{b}\int_{0}^{\infty}x_{11}\mathrm{e}^{-\pi\lambda_{b}\max^{2}\left(x_{11},x_{11}^{\mu/2}\left(\frac{P}{N}\right)^{\frac{2-\mu}{2\alpha}}\left(\frac{1}{M}\right)^{\frac{1}{2\alpha}}\right)}\mathrm{e}^{\left(-\frac{sN}{P}\right)}\mathbb{\mathcal{L_{I}}}\left(s\right)\mathrm{d}x_{11},
\psi_{\pi\varphi\ldots}(\stackrel{\rightharpoonup}{x_1},\ldots, \stackrel{\rightharpoonup}{x_N})\rightarrow(-1)^{\#\pi-{\rm indices}}\psi_{\pi\varphi\ldots}(-\stackrel{\rightharpoonup}{x_1},\ldots, -\stackrel{\rightharpoonup}{x_N})
[C_D,B_D]= O+\frac{z_1\bar{z}_2}{4}\beta(l_j)(\coth(\beta(a))-\tanh(\beta(a)))\theta(\tau_{\beta}) & -\frac{\bar{z}_1z_2}{4}\beta(l_j)(\tanh(\beta(a))+\coth(\beta(a))\tau_{\beta}
\Psi_{s}( P ) = \frac{2^{- 1 / 4}\Gamma ( 1 + 2 i b P ) \Gamma \left( 1 + \frac{2 i P}{b}\right) \cos ( 2 \pi s P )}{- 2 i \pi P}( \pi \mu \gamma ( b^{2}) )^{- i P / b}.
\frac{2}{e^t+1}e^{xt}= \sum_{n=0}^\infty Ch_{n,\lambda}(x) \frac{1}{n!}\Big( e^{\frac{1}{\lambda}t}-1\Big)^n = \sum_{n=0}^\infty Ch_{n,\lambda}(x) \sum_{m=n}^\infty S_2(m,n) \lambda^{-m}\frac{t^m}{m!},
\begin{aligned}(\rho^{n+1},\phi^{n+1})= \arg \min_{(\rho, \phi) \in \mathcal{A},m}\left\{\frac{1}{2\tau}\sum_{i=1}^s \int_0^1\int_{\Omega}F(\rho_i, m_i)D^{-1}_i dx dt+E(\rho(\cdot, 1), \phi(\cdot, 1)) \right\}, \partial_t \rho_i + \nabla\cdot (m_i) =0,m_i\cdot \mathbf{n}= 0, x\in \partial \Omega, \; \rho(x,0)=\rho^n.\end{aligned}
\epsilon\Vert \hat{U}_1 \Vert_* \le \epsilon\Vert \hat{U}_1 \Vert_* + \alpha_{\hat{R}_1}(\hat{U}_1) \le (1 + C_{u}m)(\epsilon\Vert \hat{W}\Vert_* + \alpha_{\hat{R}_1}(\hat{W})) & \le (1 + C_{u}m)\epsilon\Vert W \Vert_* + C(\gamma+ \delta_4).
\mathcal{L}= \beta^{2}\sqrt{1 + \frac{m^{2}}{\beta^{2}}A_{\mu}A^{\mu}}- \frac{1}{4}B_{\mu \nu}B^{\mu \nu}- \frac{\chi \theta}{2}\varepsilon^{\mu \nu \rho \lambda}\partial_{\mu}B_{\nu \rho}A_{\lambda}\; .
\gamma_{t}= \phi \left( \frac{-y_{u}X + x_{u}Y}{\sqrt{x_{u}^{2}+y_{u}^{2}}}\right) + f \partial_{z}& = -\phi \frac{y_{u}}{\sqrt{x_{u}^{2}+y_{u}^{2}}}\partial_{x}+ \phi \frac{x_{u}}{\sqrt{x_{u}^{2}+y_{u}^{2}}}\partial_{y}+ \left( f - \phi y \frac{y_{u}}{\sqrt{x_{u}^{2}+y_{u}^{2}}}\right) \partial_{z}.
\vert T^{(n_1-1,n_2-1)}Q_L(f-Q_L(f;s_{\zeta},t_{\zeta});s_{\xi},t_{\xi})\vert &\le \sum_{i=0}^{n_1-1}\sum_{j=0}^{n_2-1}\vert D_s^iD_t^j\big(f-Q_L(f;s_{\zeta},t_{\zeta})\big)\vert_{(s_{\xi},t_{\xi})}(s-s_{\xi})^i (t-t_{\xi})^j=O((\mbox{diam}(\Omega_{R_{\zeta}}))^{k+1}).
|\mathbb{C}_N^{(f)}| = \sum_{\pi(N): N=\{N_1^{c_1};N_2^{c_2};\ldots N_f^{c_f}\}}\binom{|\mathbb{C}_{N_1}^{(1)}|+c_1-1}{c_1}\binom{|\mathbb{C}_{N_2}^{(1)}|+c_2-1}{c_2}\cdots\binom{|\mathbb{C}_{N_f}^{(1)}|+c_f-1}{c_f}.
\frac{i}{2}\int_{0}^{\infty}\frac{I_{i \omega}( k \xi_{1}) I_{- i \omega}( k \xi_{2}) - I_{- i \omega}( k \xi_{1}) I_{i \omega}( k \xi_{2}) \,}{I_{i \omega}( k \xi_{2}) K_{i \omega}( k \xi_{1}) - I_{i \omega}( k \xi_{1}) K_{i \omega}( k \xi_{2})}
f\circ(id_V\otimes \tilde{m}_B)\circ\tilde{a}_{V,B,B}=(\tilde{m}_A\otimes id_V)\circ\tilde{a}^{-1}_{A,A,V}\circ(id_A\otimes f)\circ \tilde{a}_{A,V,B}\circ(f\otimes id_B)
\lambda_{1}\geqslant \sup_{s \in (0, 1)}\left\{4s(1 - s) \frac{\pi^{2}}{d^{2}}+ sK \right\}= \begin{cases}0 \textrm{if}Kd^{2}< - 4 \pi^{2}, (\frac{\pi}{d}+ \frac{Kd}{4 \pi})^{2}\textrm{if}Kd^{2}\in [- 4 \pi^{2}, 4 \pi^{2}], K \textrm{if}Kd^{2}\in (4 \pi^{2}, (n - 1) \pi^{2}].\end{cases}
\begin{aligned}Length(\sigma,g^+) =\int_0^l|\dot{\sigma}(t)|dt=\int_0^l|\nabla r|\cdot|\dot{\sigma}(t)|dt \geq\int_0^l|g^+(\nabla r,\dot{\sigma})|dt=\int_0^l|(r\circ\sigma)'(t)|dt \geq|\int_0^l (r\circ\sigma)'(t)dt|=|r(q')-r(q)| \end{aligned}
\nabla_{e_0}e_{0}=0,\ \nabla_{e_0}e_{1}=-\mu_1e_1,\ \nabla_{e_0}e_{2}=-\mu_2e_2, \nabla_{e_1}e_{0}=-\mu_1e_1, nabla_{e_1}e_{1}=-\mu_1e_0,\ \nabla_{e_1}e_{2}=0, \nabla_{e_2}e_{0}=-\mu_2e_2,\ \nabla_{e_2}e_{1}=0, nabla_{e_2}e_{2}=-\mu_2e_0.
\begin{array}{c}{{e_{i}= A_{i}^{-}A_{i + 1}^{+}, 1 \le i \le n + 1}}\\{{f_{i}= A_{i}^{+}A_{i + 1}^{-}, 1 \le i \le n + 1}}\\{{k_{i}= q^{- N_{A_{i}}+ N_{A_{i + 1}}}, 1 \le i \le n + 1}}\\{{e_{0}= A_{n + 1}^{-}A_{1}^{+},}}\\{{f_{0}= A_{n + 1}^{+}A_{1}^{-},}}\\{{k_{0}= q^{N_{A_{n + 1}}+ N_{A_{1}}}}}\\ \end{array}
\gamma_F(\frac{1}{2}\psi \circ q_{13}) = \gamma_F(\frac{1}{2}\psi)^2 \cdot \gamma_F(-J_2, \frac{1}{2}\psi) \cdot (\frac{2 (a_1 - c_1 t)}{b_1 + d_1 t}, J_2)_F = \gamma_F(J_2, \frac{1}{2}\psi)^{-1}\cdot (\frac{2 (a_1 - c_1 t)}{b_1 + d_1 t}, J_2)_F.
\hat{H}_{f r e e}= \int \left\{\pi^{+}( \stackrel{\rightharpoonup}{x}) \pi ( x ) + ( \bigtriangledown \varphi^{+}( \stackrel{\rightharpoonup}{x}) ) \cdot ( \bigtriangledown \varphi ( \stackrel{\rightharpoonup}{x}) ) \right\}d^{3}\stackrel{\rightharpoonup}{x}
B(\hat y, u(\hat y), Du(\hat y))-B(\hat x, v(\hat x), Dv(\hat x)) = D_xB(\tilde x, u(\hat y), Du(\hat y)) (\hat y-\hat x) + D_zB(\hat x, \tilde z, Du(\hat y)) (u(\hat y)-v(\hat x)) + \delta D_pB(\hat x, v(\hat x), \tilde p) [Dd (\hat x)+ Dd (\hat y) - 2(\hat x-z)],
\forall i,j\in [n] \mbox{such that}i<j,\: f(i)>f(j) \Longleftrightarrow f'(i)>f'(j), f(i)=f(j) \Longleftrightarrow f'(i)=f'(j), f(i)<f(j) \Longleftrightarrow f'(i)<f'(j).
\lim_{\omega_{n+1}\rightarrow0}\omega_{n+1}^{1-2s}\Omega_{n+1}\omega_{n+1}^{\frac{2s-1}{2}}\overline{v}=\tilde{V}\omega_{n+1}^{\frac{2s-1}{2}}\overline{v}\quad\partial\mathcal{S}_{+}^{n}\times\mathbb{R}.
u_j(A_j(0,1)) = \frac{r_j^{n-2}G(X_0, q_j+r_j A_j(0,1))}{\omega(B(q_j,r_j))}= \frac{r_j^{n-2}G(X_0, A(q_j,r_j))}{\omega(B(q_j,r_j))}\sim 1.
\dim X' = 2n+1-\dim(Z+V) & = 2n+1-\dim Z-\dim V+\dim(Z\cap V) & =2n+1-(2a_2+1+d)-n+\dim(Z\cap V) & =n-2a_2-d+\dim(Z\cap V) & =a_1-a_2+\dim(Z\cap V).
\frac{1}{4\tan\frac{\pi lb}{p}\tan\frac{\pi l}{p}}=\frac{\cos\frac{\pi lb}{p}\cos\frac{\pi l}{p}}{4\sin \frac{\pi lb}{p}\sin\frac{\pi l}{p}}\geq \frac{\cos\frac{\pi}{4}\cos\frac{\pi}{4b}}{\frac{4\pi lb}{p}\frac{\pi l}{p}}=c_1(b)\frac{p^2}{l^2}.
\mathcal{R}_2\quad\equiv \begin{cases}f_1=y^1_{10}y^2_{01}-y^1_{01}y^2_{10}-1=0\,, f_2=y^1_{20}y^2_{01}+y^1_{10}y^2_{11}-y^1_{11}y^2_{10}-y^1_{01}y^2_{20}=0\,, f_3=y^1_{11}y^2_{01}+y^1_{10}y^2_{02}-y^1_{02}y^2_{10}-y^1_{01}y^2_{11}=0\,, f_4=y^1_{20}+ y^1_{02}=0\,, f_5=y^2_{20}+ y^2_{02}=0\,. \end{cases}
(D y)_{(i,k)}=\sum_{j=1}^m \sum_{\ell=1}^{k_j}D_{(i,k),(j,\ell)}y_{(j,\ell)}=\sum_{j=1}^m \left(\sum_{\ell=1}^{k_j}D_{(i,k),(j,\ell)}\right) x_j=\sum_{j=1}^m M_{i,j}x_j =\lambda\, x_i =\lambda\, y_{(i,k)}.
\displaystyle A \left( \begin{array}{c}u v\end{array}\right) = -\left(\begin{array}{cc}\displaystyle \partial_{xxx}a\partial_{xxx}\displaystyle \frac{ab}{c}\partial_{xxx}\frac{r}{c}\partial_{x}+\frac{1}{c}\partial_{xxx}\end{array}\right) \left(\begin{array}{c}u v\end{array}\right)
\aligned (\hat d_{2}^{i+1}\circ( \frak h_b^{i i+1}-\frak h_a^{i+1 i})+ (\frak h_b^{i+1 i}- \frak h_a^{i i+1})\circ \hat d_{1}^{i})_{\alpha'_2\alpha_1}&= (\frak n_b^{i+1}\circ \psi_1^{i+1 i}- \psi_2^{ii+1}\circ \frak n_b^{i}-\frak n_a^{i+1}\circ \psi_1^{i+1 i}+ \psi_2^{ii+1}\circ \frak n_a^{i})_{\alpha'_2\alpha_1}.\endaligned
M_{2}(\mathcal{NC}(D_{2m})) =\dfrac{(2m-2)(2m-3)^{3}}{2}+2\dfrac{(m-2)^{2}(m-3)^{2}+2m(m-2)(m-3)+m^{2}}{4}~~~~~~~~~~-3 \dfrac{(m-2)(m-3)+m}{2}(2m-3)^{2}+(2m-2-\dfrac{3}{2})((m-2)(m-3)^{2}+m) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-\dfrac{(m-2)(m-3)^{3}+m}{2}= \dfrac{1}{2}(8m^{4}-40m^{3}+64m^{2}-32m) =4m^{4}-20m^{3}+32m^{2}-16m .
\Gamma_k =\sum_{(t_i,t_{i+1})\subset[a,b]}g_{t_i}f_{t_i,t_{i+1}}, \mathcal{S}_k =\overline{Span}\left\{f_{t_l,t_{l+1}}:\{(t_l,t_{l+1})\not\subset [a,b]}\right\}.
\mathcal{A}_1 \subseteq \{y^*(Ax)\colon x\in S_X, \y^*\in S_{Y^*}, \y^*(Tx)=\|T\|\}& = \{y^*(Ax)\colon x\in M_T, \y^*\in S_{Y^*}, \y^*(Tx)=\|T\|\}=\mathcal{A}_2.
\Big(D_1\bar{\xi}^{A}_{x+1}(s)+D_2\bar{\xi}^{B}_{x+1}(s)\Big)-\Big(D_1\bar{\xi}^{A}_{x+1}(s^-)+D_2\bar{\xi}^{B}_{x+1}(s^-)\Big)=D_1-D_2\,.
\{x_{i}, x_{j}\}_{D B}= 0 , \hspace{1 . 0 c m}\{x_{i}, p_{j}\}_{D B}= \delta_{i j}- x_{i}x_{j}, \hspace{1 . 0 c m}\{p_{i}, p_{j}\}_{D B}= x_{j}p_{i}- x_{i}p_{j},
U(t) \geqslant \mathrm{e}^{-t}U(0)+\left(1-\mathrm{e}^{-t}\right)\left(U'(0)+U(0)\right) &=U(0)+\left(1-\mathrm{e}^{-t}\right)U'(0)\geqslant U(0)\geqslant 0.
r_{l}^{x,y,z}:=\min\big\{h_{l}((x,0))(2),h_{l}((y,0))(2),h_{l}((z,0))(2)\big\}, &s_{l}^{x,y,z}:=\max\big\{h_{l}((x,0))(2),h_{l}((y,0))(2),h_{l}((z,0))(2)\big\}.
a_j^n - \frac{1}{\sqrt{2\pi V n}}\exp\left(-\frac{X_{n,j}^2}{2}\right) - \sum_{\sigma=2}^s\frac{q_\sigma(X_{n,j})}{n^\frac{\sigma}{2}}\underset{n\rightarrow+\infty}=o\left(\frac{1}{n^\frac{s}{2}}\right)
\delta_{J}\langle0,x_{0}^{\prime}|0,x_{0}^{\prime\prime}\rangle_{J}=i\langle0,x_{0}^{\prime}|\int_{x_{0}^{\prime\prime}}^{x_{0}^{\prime}}dx_{0}\delta J(a^{\dagger}+a)|0,x_{0}^{\prime\prime}\rangle_{J}
\begin{array}{c c c}{{\Phi_{1}^{( 1 )}= P_{y}}}&{{{}}}&{{\Phi_{1}^{( 2 )}= P_{z}}}\\{{\Phi_{2}^{( 1 )}= P_{x}}}&{{{}}}&{{\Phi_{2}^{( 2 )}= z}}\\ \end{array}.
-\int_{X}\int_{X}\psi(\gamma(x,y))d\mu(x)d\mu(y)= b\int_{X}\int_{X}\gamma(x,y)d\mu(x)d\mu(y) &+ \int_{(0, \infty)}\left [\int_{X}\int_{X}e^{-r\gamma(x,y)}d\mu(x)d\mu(y)\right ] \frac{1}{r}d\lambda(r).
m_{|u-m^{\gamma}(A)|}^{\gamma}(A) \leq m_{|u- c| + |c-m^{\gamma}(A)|}^{\gamma}(A) = m_{|u-c|}^{\gamma}(A) + |c - m_u^{\gamma}(A)| &\leq m_{|u-c|}^{\gamma}(A) + m_{|u-c|}^{\gamma}(A),
{\cal H}_{F_{4}}={\frac{1}{2}}p^{2}-{\frac{g^{2}}{2}}\sum_{\alpha \in \Delta}x ( \alpha \cdot q ) x ( - \alpha \cdot q ) -{g_{s}^{2}}\sum_{\lambda \in \Delta_{s}}x ( \lambda \cdot q ) x ( - \lambda \cdot q ) .
W(\rho,\psi) =\lambda_{E/F}(\psi)\cdot W(\chi_E,\psi_E) =\lambda_{E/F}(\psi)\cdot \Delta_{E/F}(c)\det(\rho)(c)\psi(mc^{-1}) =R(\psi,c)\cdot\det(\rho)(c)\cdot\psi(mc^{-1}) =\det(\rho)(c)\cdot\psi(mc^{-1}),
\left. + \frac{1}{1 9 2}\left( 1 6{\sigma}^{2}- 4 0 \sigma + 9 \right) \left( \psi \left( \frac 3 2 + \sqrt{\sigma}\right) + \psi \left( \frac 3 2 - \sqrt{\sigma}\right) \right) \right] + O \left( ( \beta - \beta_{H})^{2}\right) .
\mathbf{v}_{k}(z):=\begin{cases}(\phi_{0}^{(k)},\ldots,\phi_{p-k-1}^{(k)},-\sum_{j=k}^{p}a_{0}^{(j)}\phi_{j-k}^{(k)},\ldots,-\sum_{j=1}^{p}a_{k-1}^{(j)}\phi_{j-1}^{(k)}), 0\leq k\leq p, begin{align*}0.5em](-a_{k-p}^{(p)}\phi_{0}^{(k)},-\sum_{j=p-1}^{p}a_{k-p+1}^{(j)}\phi_{j-p+1}^{(k)},\ldots,-\sum_{j=1}^{p}a_{k-1}^{(j)}\phi_{j-1}^{(k)}), k\geq p+1.\end{cases}
[(b_1-b_2)^2-2]R_{1212}+[(b_2-b_3)^2-2]R_{2323}+[(b_1-b_3)^2-2]R_{1313}&=\frac{5}{9}-\frac{16}{3}tr(A)+\frac{15}{2}tr(A)^2+\frac{3}{2}|A|^2-\frac{3}{2}|Ric|^2.
\varphi(\omega)=\varphi|_{\omega_{0}}+\frac{\partial\varphi}{\partial\omega}|_{\omega_{0}}(\omega-\omega_{0})+\frac{1}{2}\frac{\partial^{2}\varphi}{\partial\omega^{2}}|_{\omega_{0}}(\omega-\omega_{0})^{2}+...+\frac{1}{p!}\frac{\partial^{p}\varphi}{\partial\omega^{p}}|_{\omega_{0}}(\omega-\omega_{0})^{p}+...
\textnormal{Res}_{k=-\eta}T_{-\eta,1}=-\frac{3}{4\left(D_{0}^{-2}r^{-1}(\eta)-1\right)}\begin{pmatrix}\tilde{T}_{-\eta,1}^{(11)}\tilde{T}_{-\eta,1}^{(12)}\tilde{T}_{-\eta,1}^{(21)}\tilde{T}_{-\eta,1}^{(22)}\end{pmatrix}
\sum_{q > p}\lambda(E_p \cap E_q) = \sum_{q > p}\sum_{\substack{r \in T_p s \in T_q}}\lambda \left( E_{p,r}\cap E_{q,s}\right) \ll \sum_{q > p}\frac{\psi(q)}{q}\sum_{\substack{r \in T_p, \s \in T_q |sp-rq| < 3 q \psi(p)}}1.
P\bigl(| B(t) + \mu t| + x < y\bigr) ={}P\bigl(x-y-\mu t < B(t) <y-x-\mu t\bigr) ={}\int_{x-y-\mu t}^{y-x-\mu t}\frac{e^{- \frac{w^{2}}{4t}}}{\sqrt{4 \pi t}}\,\mathrm{d}w
\left|\nabla W(y,s) - \nabla \Phi_\alpha\left(\frac{y}{\sqrt s}\right)\right| = \left|\sqrt{T-t}\,\nabla U(x,t) + \frac{ye^{\Phi_\alpha\left(\frac{y}{\sqrt{|\ln (T-t)|}}\right)}}{(2 + 2\alpha) |\ln (T-t)|}\right| &\leq \frac{C(A, K_0)\ln |\ln (T-t)|}{|\ln (T-t)|^2}(|y|^3 + 1) + \frac{C(A, K_0) \ln |\ln (T-t)|}{|\ln (T-t)|^3}(|y|^4 + 1).
\omega_k(f\circ u,t)_p = \sup_{|h|\le t}\| \Delta_h^k f\circ u \|_{L_p(\Omega)}\le C(f) \sum_{\ell=1}^{k}\sum_{i=1}^\ell \omega_k(u^i,t)_p\| u \|_{L_\infty(\Omega)}^{\ell-i}.
\hat{\cal I}^{\alpha\beta}_{\mu\nu}={\it g_2}\bigg( \hat \theta_\rho\big( -i \tilde F_{\lambda\gamma,\sigma}+ F_{\lambda\gamma}\hat P_\sigma-{\sqrt 2}i \hat \theta_5 F_{\lambda\gamma,\sigma}\big) +\hat \theta_\lambda \big( -i \tilde F_{\rho\sigma,\gamma}+ F_{\rho\sigma}\hat P_\gamma-{\sqrt 2}i \hat \theta_5 F_{\rho\sigma,\gamma}\big) \bigg) \,,
H_{L C}= \int_{0}^{l}\mathcal{H}_{L C}= \int_{0}^{l}d \sigma \frac{G_{+ -}}{2 \pi_{-}}\left( G^{i j}\pi_{i}\pi_{j}+ \frac{1}{( 2 \pi \alpha^{\prime})^{2}}G_{i j}Z^{i \prime}Z^{j \prime}\right)
u_1\Theta_p^{v_p}u_1^*= T_1(\Phi_\lambda^{-1}(v_quv_p^*))T_1(\Phi_\lambda^{-1}(v_p\Theta v_p^*))T_1(\Phi_\lambda^{-1}(v_quv_p^*))^* = T_1[\Phi_\lambda^{-1}(v_quv_p^*v_p\Theta v_p^*(v_quv_p^*)^*)]=T_1[\Phi_\lambda^{-1}(v_q\Theta v_q^*)]=\Theta_q^{v_q}.
\exp ( - \mathrm{F i g . ~ 4 ( c )}) = - ( \alpha^{\prime}g )^{2}\ln ( \varepsilon ) \int_{0}^{2 \pi}d \theta ~ \mathrm{T r}~{\cal P}\left( U [ A ] ~ [ D_{a}F_{b c}, F^{b c}] \partial_{\theta}\overline{{{X}}}^{a}\right) ,
\mu_{n+1}\le \mu_0 + \frac{g}{2}\big(E_\mathrm{H}(\phi_0)\big)^2 + \frac{g}{2}\big(\tilde{\mathcal{E}}[n+1](\phi_{n+1})\big)^2 &+\frac{2gC}{c}\|\upsilon\|_{L^\infty}+\frac{gC^2}{Nc}\mathcal{E}_\mathrm{Hartree}(\phi_0),
{\cal L}_{D}[ A , H , V ] ={\cal L}_{G}[ A ] + \frac{1}{2 G}V^{\mu}V_{\mu}+ \frac{1}{2}\epsilon^{\mu_{1}. . . \mu_{D}}H_{\mu_{3}. . . \mu_{D}}F_{\mu_{1}\mu_{2}}[ A + V ] ,
\int_{\Omega_{\rho}^{+}( \eta_{k})}\frac{H_{\mu}^{( 1 )}( \lambda z )}{H_{\nu}^{( 1 )}( z )}F ( z ) d z + \int_{\Omega_{\rho}^{-}( \eta_{k})}\frac{H_{\mu}^{( 2 )}( \lambda z )}{H_{\nu}^{( 2 )}( z )}F ( z ) d z = \left( 2 - \delta_{0 \eta_{k}}\right) \pi i \,{\mathrm{R e s}}_{z = \eta_{k}}\frac{H_{\mu}^{( 1 )}( \lambda z )}{H_{\nu}^{( 1 )}( z )}F ( z ) ,
[X \xleftarrow p V \xrightarrow s Y; E] \circ_{\oplus}[ Y \xleftarrow q W \xrightarrow t Z; F] &: = [(X \xleftarrow p V \xrightarrow s Y) \circ (Y \xleftarrow q W \xrightarrow s Z);\widetilde{q}^*E \oplus \widetilde{s}^*F ],
r^2 - \epsilon^2 p_\epsilon^*\Phi^- = O( \epsilon^{2n}r^{2-2n}) = O( r_\epsilon^4 ), \nabla^2( r^2 - \epsilon^2 p_\epsilon^*\Phi^- ) = O( \epsilon^{2n}r^{-2n}) = O( r_\epsilon^2 ).
\mathbf{E\,}h_{n}^{-1}\,H\left( \frac{x-X}{h_{n}}\right) =h_{n}^{-1}\int_{\mathbf{R}}f(z)\,H\left( \frac{x-z}{h_{n}}\right) \,dz=\int_{\mathbf{R}}f(x-h_{n}y)\,H(y)\,dy
E_{n}(x) = \sqrt{2}\sum_{\ell=2}^{\infty}\frac{\left[ ( 2 + \sqrt{2})^{\ell -1}- (2 - \sqrt{2})^{\ell -1}\right]}{2^{2 \ell + 1}}E_{n}^{(2 \ell)}(4x + \ell - 2 ).
p^{(1)}\cdots p^{(n)}=\left(\left(\xi_{1,1}^{(1)}\cdots\xi_{1,l_1^{(1)}}^{(1)}\right)\cdots\left(\xi_{t,1}^{(1)}\cdots\xi_{t,l_t^{(1)}}^{(1)}\right)\right)\cdots\left(\left(\xi_{1,1}^{(n)}\cdots\xi_{1,l_1^{(n)}}^{(n)}\right)\cdots\left(\xi_{t,1}^{(n)}\cdots\xi_{t,l_t^{(n)}}^{(n)}\right)\right).
T r_{w}K_{H_{2}}={\frac{e^{-{\bar{s}/ 4}}}{( 4 \pi \bar{s})^{1 / 2}}}\int_{0}^{\infty}{\frac{d y \cosh y e^{-{y^{2}/ \bar{s}}}}{( \sinh^{2}y + \sin^{2}{\frac{w}{2}})}}~ ~ .
(1- x^{2}y-xy^{2}-y^{3})G_{T}(y) &=Q_{T,0}(x)+(Q_{T,1}(x)-x^{2}Q_{T,0}(x))y+(Q_{T,2}(x)-x^{2}Q_{T,1}(x)-xQ_{T,0}(x))y^{2}&\ \ +\sum_{n=3}^{\infty}(Q_{T,n}(x)-x^{2}Q_{T,n-1}(x)-xQ_{T,n-2}(x)-Q_{T,n-3}(x))y^{n}.
\langle P^\mathrm{t}\overline f,\varphi\rangle = \int_{\mathbb R^\times}\int_G \overline{f(x)}\,P_x\operatorname{Tr}[\pi_\lambda(x)\,\widehat \varphi(\lambda)]\,\mathrm d\mu(x)\mathrm d\lambda_\pi &= \int_{\mathbb R^\times}\langle P^\mathrm{t}\overline{f},\operatorname{Tr}[\pi_\lambda(\cdot)\,\widehat \varphi(\lambda)]\rangle\,\mathrm d\lambda_\pi,