harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
g_\alpha(x)=\, (x-1)\sum_{l=0}^{r-1}x^l\sum_{\substack{0\le i_0\le m-1\cr 0\le i_1\le m}}(-1)^{i_0+i_1}\binom{m-1}{i_0}\binom m{i_1}\cr \cdot\binom{i_0+i_1\tau+\alpha-\frac{\alpha+1}r+\frac lr}\alpha\frac{x^{r(i_0+i_1)}-1}{x^r-1}-1.
\tilde h_2 (x,u,\varphi) = \frac{u + x g_2(u, \varphi) + \cdots +x^{k-1}g_k (u,\varphi) + \frac{G_k (x,xu,\varphi)}{x}}{1 + x f_2 (u,\varphi) + \cdots + x^{k-1}f_k (u,\varphi) + \frac{F_k (x,xu,\varphi)}{x}}
{\cal A}_{\mathrm{N S}}= \frac{T}{2 \pi}{\frac{1}{\sqrt{8 \pi^{2}\alpha^{\prime}}}}\int_{0}^{\infty}{d t}\left( \frac{\pi}{t}\right)^{\frac{1}{2}}\, \mathrm{e}^{- b^{2}/ ( 2 \alpha^{\prime}t )}\, \left[ \left({\frac{f_{4}}{f_{2}}}\right)^{8}- \left({\frac{f_{3}}{f_{2}}}\right)^{8}\right] = -{\frac{T}{4 \pi \alpha^{\prime}}}| b | ~ .
B_{q,p}(\Omega)\leq\frac{diam(\Omega)^n}{n|\Omega|}\left(\frac{1-\delta}{{1}/{n}-\delta}\right)^{1-\delta}\omega_n^{1-\frac{1}{n}}|\Omega|^{\frac{1}{n}-\delta},\,\,\delta=\frac{1}{p}+\frac{1}{q}\geq 0.
B_\vee(1)= \;12+21, B_\vee(12+21)= \;\prescript{\uparrow}{}{1}\,\vert\,\prescript{\uparrow}{}{2}^{\uparrow}+\prescript{\uparrow}{}{2}^{\uparrow}\,\vert\,1^{\uparrow}= \;123+132+212+212+231+321.
\begin{cases}-\varphi''(x)-k^2\varphi(x)=\big(\widetilde a-a(x)-\tfrac12a_{c,2k}\big)\sin(kx),\quad[0,2\pi], \varphi(0)=\varphi(2\pi), \varphi'(0)=\varphi'(2\pi),\end{cases}
\sum_{q=1}^{N_1}\sum_{j=1}^{\Delta^n_{i\left(n,q\right)}N}|\Delta X^j_{i\left(n,q\right)}|{\bf 1}_{\{T_{q+1}-T_q < 2\Delta_n\}}\leq \sum_{i=1}^n \Delta_i^n N \sum_{j=1}^{\Delta_i^n N}|\Delta X_i^j| ({\bf 1}_{\{\Delta^n_i N \geq 2\}}+{\bf 1}_{\{\Delta^n_{i+1}N \geq 1\}})
\eta^{12}b_1 = (1+u(bE_4^2-aE_6))(\eta^{12},0)+u(bE_4^2-aE_6)(-uE_6,-6uE_4)+ u(bE_6-aE_4)(0,6\eta^{12})+u(bE_6-aE_4)(uE_4^2,6uE_6)
z_j''(t) = -\frac{2\bar{z}_j(t)z_j'^2(t)}{R^2-\vert z_j(t)\vert^2}+ \frac{4R}{\lambda_j(t)}\sum_{\underset{i\neq j}{i=0}}^{n-1}\frac{P_{j,i}(t)}{\Theta_{j,i}(t)^{3/2}}, 0\leq j\leq n-1,
n_{\alpha i}= \left( \begin{array}{c c c c c c}{1}&{1}&{1}&{1}&{1}&{1}\\{2}&{2}&{0}&{0}&{2}&{2}\\{3}&{3}&{0}&{0}&{3}&{5}\\{2}&{2}&{1}&{1}&{2}&{4}\\ \end{array}\right) .
\widetilde{X}^{(u)}_{t}\, =\, \mathbb E^{M}\Big[ \int^{t}_{0}\sum_{k=0}^{\infty}{\bf 1}_{\{M(t-s) \, =\, k\}}{\mathrm d}W_{s, k}\vert M(0) \, =\, 0\Big]\, ; t \ge 0 \, ,
\prod_{s'}\dfrac{\hbar'(p - l_{i}^{(s')}- 1) - (a_s - a_{s'})}{\hbar'(p - l_{i}^{(s')}) - (a_s - a_{s'})}= \prod_{(x',y') \in A_{\lambda,i}}\dfrac{\varepsilon_1 (x - x' + 1) + \varepsilon_2 (y - y' + 1)}{\varepsilon_1 (x - x') + \varepsilon_2 (y - y')}.
{\cal A}_0 \left(t,{\bf x}\right) = \int\limits_0^1{d\alpha\frac{1}{{1 + e^2 a\left({1 - 12\alpha^2 g^2 \partial^2}\right)}}}\frac{{x^i \partial_i^{\alpha x}\left({- J^0 \left({\alpha x}\right)}\right)}}{{\nabla_{\alpha x}^2 - \chi^2}}.
\max_{\tau_i \in \mathcal{S}^{i}}{J}_i(\tau_i)=\max_{\tau_i \in \mathcal{S}^{i}}\mathbb{E}\left\{\int_0^{\tau_i}e^{-\beta t}f(x_i(t))dt-e^{-\beta\tau_i}K\right\}.
t^{0 0}= \sum \hspace{- 1 . 4 e m}\int_{\lambda}E_{\lambda}\psi_{\lambda}^{*}( \textbf{x}) \, \psi_{\lambda}( \textbf{x}) - ( \xi - 1 / 4 ) \triangle \sum \hspace{- 1 . 4 e m}\int_{\lambda}E_{\lambda}^{- 1}\psi_{\lambda}^{*}( \textbf{x}) \, \psi_{\lambda}( \textbf{x}) ,
S(X_{(1)}) \otimes X_{(2)}\otimes S(Y_{(1)}) \otimes Y_{(2)}= K_{a}F_{a}\otimes 1 \otimes E_{a}K_{a}^{-1}\otimes K_{a}-K_{a}F_{a}\otimes 1 \otimes 1 \otimes E_{a}- K_{a}\otimes F_{a}\otimes E_{a}K_{a}^{-1}\otimes K_{a}+ K_{a}\otimes F_{a}\otimes 1 \otimes E_{a}.
\exp(-\beta \sum_{1 \leq i<j \leq N}\Phi(x_i,x_j)) = \prod\limits_{1 \leq i<j \leq N}\exp(-\beta \Phi(x_i,x_j)) &= \prod\limits_{1 \leq i<j \leq N}(1+f_{i,j}) &= \sum\limits_{g \in \mathcal{G}[N]}\prod\limits_{e \in E(g)}f_e,
0=(-1)^{p(a,c)}[[\tilde{g}(a),\tilde{g}(b)],\tilde{g}(c)]+(-1)^{p(b,a)}[[\tilde{g}(b),\tilde{g}(c)],\tilde{g}(a)]+(-1)^{p(c,b)}[[\tilde{g}(c),\tilde{g}(a)],\tilde{g}(b)],
\| \Gamma(Z_i, Z_j) \| \le \bigg\| \Gamma \bigg( \begin{bmatrix}Z_i 0 0 Z_j \end{bmatrix},\begin{bmatrix}Z_i 0 0 Z_j \end{bmatrix}\bigg) \bigg\| \le \| \Gamma(Z^{(0)}, Z^{(0)}) \| & = \| \Gamma(Z^{(0)}, Z^{(0)}) \| =\| \Gamma(Z^{(0)}, Z^{(0)})(I_{N_0}) \| & = \| K(Z^{(0)}, Z^{(0)})(I_{N_0}) \|
f_{J_{1}n_{1}\sigma_{1}; J_{2}n_{2}\sigma_{2}}^{J_{3}n_{3}\sigma_{3}}= ( - 1 )^{2 J_{1}( J_{2}- n_{2}- 1 )}{\tilde{f}}_{J_{1}n_{1}\sigma_{1}; J_{2}n_{2}\sigma_{2}}^{J_{3}n_{3}\sigma_{3}}.
H{\bf E}^{2}\equiv \frac{\Gamma r_{0}^{4}\{\bf E}^{2}}{4 R^{4}x^{3}}< \frac{1}{1 + \frac{\Delta \theta_{D}^{2}}{x_{*}^{2}}}\equiv \frac{{\bf l}^{2}}{R^{2}}\left( \frac{\Delta \theta_{D}^{2}}{x_{*}^{2}}\right)^{- 2}\ ,
\left( \begin{array}{c c c c | c c}{\times}&{\times}&{}&{}&{}&{}\\{\times}&{\times}&{}&{}&{}&{}\\{\times}&{\times}&{\times}&{\times}&{\times}&{\times}\\{\times}&{\times}&{\times}&{\times}&{\times}&{\times}\\ \hline{\times}&{\times}&{}&{}&{\times}&{}\\{\times}&{\times}&{}&{}&{\times}&{\times}\\ \end{array}\right)
\left\|(S(f,P,\xi_1)-S(f,P,\xi_2))\right\|=\left\|\sum\limits_{i=1}^{p}\dfrac{1}{N}z^{(i)}\right\|\geq \dfrac{1}{N}\left(\dfrac{p\beta}{2}-4\epsilon\right)\geq \dfrac{\alpha\beta}{4}.
\int_{B_{1/2}}|V_{k,r}|^2\,dx = \int_{B_{1/2}}\big |V+(1-\eta_r)(U_k-U_0) \big |^2\,dx &\le 2 \int_{B_{1/2}}\big ( |V|^2+|U_k|^2+|U_0|^2 \big )\,dx &\le \frac{1}{\mu}\Big ( \|v\|^2_{H^1(B_{1/2})}+ C(N,M)\Big ) = C(N,M,\mu) \Big ( \|v\|^2_{H^1(B_{1/2})}+1 \Big ).
\boldsymbol Q_l(i)=\boldsymbol\Lambda_l^{2i}\boldsymbol{\omega}_l(0)\big(\boldsymbol{\omega}_l^H(0)\boldsymbol\Lambda_l^{4i-2}\boldsymbol{\omega}_l(0)\big)^{-1}\boldsymbol\omega_l^H(0)\boldsymbol\Lambda_l^{2i-2}.
\int_{\Omega}ue^{\tau(\Phi-\overline \Phi)}dx=\sum_{\tilde y\in\mathcal H}e^{2i\tau\psi(\tilde y)}\frak F_{\tau,\tilde y}u+\frak J_{\tau}u+ o\left(\frac{1}{\tau}\right)\mbox{as}\,\,\tau\rightarrow +\infty.
\Psi={\sqrt{-1}}K_{i\bar j \alpha\bar\beta}{{v^i\bar v^j}\over G}dz^\alpha\wedge d\bar z^\beta,\ \omega_{FS}={{\sqrt{-1}}\over{2\pi}}{{\partial^2\log G}\over{\partial v^i\partial\bar v^j}}\delta v^i\wedge\delta\bar v^j,
\lim_{r \to 0}\mu(U_r)^{-1}\log \mu(\tau_r > t) = \lim_{r \to 0}\mu(U_r)^{-1}\log (1 - \mu(\tau_r \le t)) & = \lim_{r \to 0}\mu(U_r)^{-1}\log (1 - (t+1)\mu(U_r))= - (t+1) .
F_{d,\ell}(z;\tau) =\sum_{a\geq 0}\frac{(2\pi i\tau)^a}{a!}\sum_{n\geq 0}(\ell n+d)^{2a}\zeta^{\ell n+d}= \sum_{a\geq 0}\mathcal{D}_z^{2a}\left(\frac{\zeta^d}{1-\zeta^\ell}\right)\frac{(2\pi i\tau)^a}{a!}.
H^{2}= \left({\frac{\dot{a}}{a}}\right)^{2}={\frac{1}{3}}\rho -{\frac{k}{a^{2}}}={\frac{1}{3}}\left({\frac{1}{2}}\dot{\phi}^{2}+ V ( \phi ) + \rho_{\alpha}\right) -{\frac{k}{a^{2}}}\ .
\begin{aligned}u_3(x,t) =u_2(x,t)-H_{k+2}(x',x_n-P_{\Omega}(x',t),t) =u_1(x,t)-H_{k+1}(x',x_n-P_{\Omega}(x',t),t)-H_{k+2}(x',x_n-P_{\Omega}(x',t),t). \end{aligned}
\begin{aligned}\frac{v'(r)^{2}}{2}\leq \int_{0}^{r}t^{b}v^{q}(t)v'(t)f(w(t)) dt \leq r^{b}f(w(r))\int_{0}^{r}\left(\frac{v^{q+1}(t)}{q+1}\right)'dt \leq Cf(w(r))v^{q+1}(r)\quad\mbox{for all}\ 0<r<R,\end{aligned}
R_{(1)}^{\rm near}(r) = C_{\rm in}e^{-i \frac{\omega \ell^2}{2 r_{H}}\ln\left(\frac{r^2 - r_{H}^2}{r^2}\right)}+ C_{\rm out}e^{i \frac{\omega \ell^2}{2 r_{H}}\ln\left(\frac{r^2 - r_{H}^2}{r^2}\right)}.
\rho (\alpha (x))\circ \alpha_M=\alpha_M\circ \rho (x), \rho (\beta (x))\circ \beta_M=\beta_M\circ \rho (x), \rho ( [\beta (x), y ])\circ \beta_M=\rho (\alpha \beta(x))\circ \rho (y)- \rho (\beta (y))\circ \rho (\alpha (x)).
\mathbb K^{(\alpha,\frac{1}{2})}(x,y) = \frac{-1}{x-y}\begin{pmatrix}\vartheta_y^2 g(y) \vartheta_y g(y) g(y) \end{pmatrix}\begin{pmatrix}1 0 0 -2\alpha-\frac{1}{2}1 0 \alpha(\alpha+\frac{1}{2}) - 2\alpha-\frac{1}{2}1\end{pmatrix}\begin{pmatrix}f(x) - \vartheta_x f(x) \vartheta_x^2 f(x) \end{pmatrix}
\widetilde U_{ex}^{inst}(l) = - \frac{e^2}{2(2\pi)^3}\frac{(\overline u(p_1,\lambda_1) \gamma^\mu u(p'_1,\lambda'_1))}{\sqrt{2p_1^+}\sqrt{2p_1^{'+}}}\frac{(\overline v(p'_2,\lambda'_2)\gamma^\nu v(p_2,\lambda_2))}{\sqrt{2p_2^{'+}}\sqrt{2p_2^+}}\frac{\eta_\mu \eta_\nu}{q^{+2}}\,.
\begin{aligned}\int_0^T\int_0^{\tilde \lambda_1}\left(s\Theta a(v_x)^2 + s^3 \Theta^3\frac{(x-x_0)^2}{a}v^2\right)e^{2s\varphi}dxdt &\le C\left( \int_0^T \int_0^{\beta_1}h^2 e^{2s \Phi_1(t,-x)}dxdt+\int_0^T \int_\omega v^2dxdt\right)\end{aligned}
\ln\left[\begin{array}[c]{l}n k\end{array}\right] & \simeq-\ln\sqrt{2\pi}+\frac{1}{2}\ln\frac{n}{k\left( n-k\right)}+nS_{1}\left( \frac{k}{n},1-\frac{k}{n}\right) & =\ln\frac{1}{\sqrt{2\pi}}\sqrt{\frac{n}{k\left( n-k\right)}}+\left(-k\ln\frac{k}{n}-\left( n-k\right) \ln\left( 1-\frac{k}{n}\right) \right)
\hat{\chi}_{\mu \nu}= \Lambda_{\mu \nu}^{( 2 )}\, , \hspace{1 c m}\hat{\chi}_{\mu \underline{{{z}}}}={\textstyle \frac{1}{m}}\hat{\lambda}_{\mu}= \Lambda_{\mu}\, , \hspace{1 c m}\hat{\xi}^{\underline{{{z}}}}= \Lambda^{( 0 )}\, .
\hat{U}_m(F) ={\rm exp}\{(\hbar/ i) \hat{T}_m(F) \},\qquad\hat{T}_m(F) = \hbox{$\frac{1}{2}$}\epsilon_{ab}\{\bar{\Delta}_m^b, [ \bar{\Delta}_m^a, F ] \}+ (i/ \hbar)^2 m^2 F,
\displaystyle{\raisebox{0 . 9 e x}{\scriptsize{\diamond}}}\mathrm{\hspace{- 0 . 3 3 e x}}\rho_{\tau}( Q ) = \frac{\epsilon ( q_{0})}{\pi}\, I m \, \frac{1}{\displaystyle{\raisebox{1 . 1 e x}{\scriptsize{\diamond}}}\mathrm{\hspace{- 0 . 3 3 e x}}D_{\tau}( Q )}\, .
S_{A B}={\frac{i}{2}}e^{K_{V}/ 2}( \sigma_{x})_{A}{}^{C}\epsilon_{B C}P_{I}^{x}( q ) X^{I}( z ) ={\frac{i}{2}}( \sigma_{x})_{A}{}^{C}\epsilon_{B C}P_{I}^{x}( q ) L^{I}.
\| f_{0}-g_{n,0}\|_{\mathbb{W}^{1,p}(\Omega_0,\mathbb{R}^d)}=\|f_{0}-g_{n,0}\|_{C^0(\Omega_0)}+\left(\int_{\Omega_0}|Df_{0}-Dg_{n,0}|^p\,d\mu\right)^{\frac{1}{p}}\ .
\Big [{\bf a}_{p}( \vec{k}) ,{\bf a}_{p^{\prime}}^{\dagger}({\vec{k}}^{\prime}) \Big ] = 2 ( 2 \pi )^{n - 1}w_{p}( \vec{k}) \delta_{p p^{\prime}}\delta^{n - 1}( \vec{k}-{\vec{k}}^{\prime}) .
-{\cal L}= \frac{1}{2}\left[ 2 \sum_{i = 1}^{2}\partial_{+}\phi^{i}\partial_{-}\phi^{i}- \tanh^{2}\left({\frac{\phi^{1}- \phi^{2}}{2}}\right) \partial_{+}\phi^{3}\partial_{-}\phi^{3}\right] + \sum_{i = 1}^{2}\exp ( 2 \phi^{i})
\eta ( s , t ) = \frac{1}{B_{k_{1}k_{1}}( s^{2})}\mathrm{d e t}\left[ \frac{1}{s_{p 1}^{2}+ s_{q 2}^{2}}\left( 1 - \exp \left( - \frac{s_{p 1}^{2}+ s_{q 2}^{2}}{2 t}\right) \right) \right]_{p , q = 1 , \ldots , k_{1}}\ .
\pi ( d \alpha ) = \left( \begin{matrix}{- g^{\mu \nu}\partial_{\mu}a_{i 1}\partial_{\nu}b_{i 1}}&{- 2 \gamma^{\mu \nu}\gamma^{5}V_{\nu}a_{i 1}\partial_{\mu}b_{i 2}}\\{2 \gamma^{\mu \nu}\gamma^{5}V_{\nu}a_{i 2}\partial_{\mu}b_{i 1}}&{- g^{\mu \nu}\partial_{\mu}a_{i 2}\partial_{\nu}b_{i 2}}\\ \end{matrix}\right)
\lambda^{\mathrm{\footnotesize{G N}}}= \mu - 2 - \frac{2 ( \mu - 2 ) ( \mu - 1 ) \Gamma ( 2 \mu )}{( 2 \mu - 3 ) \Gamma ( 2 - \mu ) \Gamma^{2}( \mu ) \Gamma ( \mu + 1 ) N}
\begin{aligned}2 \delta \| \nabla f_{i}\|_{L^{2}(\Omega)}^{2}+ \| f_{i}\|_{L^{2}(\Omega)}^2 \leq \| \varphi_{i,0}\|_{L^{2}(\Omega)}^2, 2 \delta \| \Delta f_{i}\|_{L^{2}(\Omega)}^2 + \| \nabla f_{i}\|_{L^{2}(\Omega)}^2 \leq \| \nabla \varphi_{i,0}\|_{L^{2}(\Omega)}^2.\end{aligned}
t_{R}^{( n )}= s_{R}^{( n )}\left\{\theta ( s_{1}- t ){\frac{t}{s_{1}}}+ [ \theta ( s_{2}- t ) - \theta ( s_{1}- t ) ]{\frac{s_{2}- t}{s_{2}- s_{1}}}+ \cdot \cdot \cdot + \theta ( t_{n}- s ){\frac{t - s_{n}}{s - s_{n}}}\right\}.
{\widehat \gamma}^{1}= \left( \begin{array}{c c}{{{\widehat E}_{2}}}&{{0}}\\{{0}}&{{-{\widehat E}_{2}}}\\ \end{array}\right) ,{\widehat \gamma}^{2}= \left( \begin{array}{c c}{{0}}&{{\sigma_{x}}}\\{{- \sigma_{x}}}&{{0}}\\ \end{array}\right) ,
\hat C_{w_k}=\hat C_{y_k}-{\bf h}^{H}_{kB}{\bf\hat C}_{x_{B_k}}{\bf h}^{*}_{kB},\forall k\in\mathcal{C}, \hat C_{q_j}=\hat C_{z_j}-{\bf g}^{H}_{jl}{\bf\hat C}_{x_{l}}{\bf g}^{*}_{jl},\forall j\in\mathcal{D},
G_{A_{1}\, \dots \; A_{N}}= \left. \frac{( - i )^{N}}{Z^{( 0 )}\, ( 0 )}\; \frac{\delta^{N}Z^{( 0 )}}{\delta \, j_{A_{1}}\, \dots \; \delta \, j_{A_{N}}}\; \right|_{j = k = 0}
E[ 1_A \int_U \phi_t(x) \eta'(x) dx ] = \lim_{k \rightarrow \infty}E [ 1_A \int_U \phi_{n_k}(t,x) \eta'(x) dx ] =- \lim_{k \rightarrow \infty}E [ 1_A \int_U \partial_x \phi_{n_k}(t,x) \eta(x) dx ] & =- E [ 1_A \int_U \partial_x f_t(x) \eta(x) dx ].
\zeta(1-s,a)=\frac{\Gamma(s)}{(2\pi)^{s}}(e^{-\pi is/2}\sum_{n=1}^{\infty}\frac{e^{2\pi ina}}{n^{s}}+e^{\pi is/2}\sum_{n=1}^{\infty}\frac{e^{-2\pi ina}}{n^{s}}),
\int_{0}^{T}\left(\partial_t v^{\Delta t}_{h},P_h(u^{\Delta t}_{h}\phi)\right)\,dt =\int_{0}^{T}\left(\partial_t v^{\Delta t}_{h},u^{\Delta t}_{h}\phi\right)\,dt +\int_{0}^{T}\left(\partial_t v^{\Delta t}_{h},P_h(u^{\Delta t}_{h}\phi)-u^{\Delta t}_{h}\phi\right)\,dt=:I_1+I_2.
[ T_{m , n}, T_{m^{\prime}, n^{\prime}}] = 2 \sinh \left( \hbar ( m^{\prime}n - m n^{\prime}) \right) T_{m + m^{\prime}, n + n^{\prime}}+ \delta_{m , - m^{\prime}}\frac{\sinh ( \hbar m ( n + n^{\prime}) )}{\sinh ( \hbar ( n + n^{\prime}) )}C .
G_2(x)=x^2F_T(x)C(x)+\frac{x^3C(x)^2\big(F_T(x)-1\big)}{1-2x}+\frac{x^3C(x)^2}{1-2x}-\frac{x^3C(x)^2}{(1-x)^2}+\frac{x^3C(x)}{(1-x)\big(1-x-xC(x)\big)}\,.
kC_k^\mu(t) =2(k+\mu-1)tC_{k-1}^\mu(t)-(k+2\mu-2)C_{k-2}^\mu(t) (k+\mu)C_k^\mu(t) =\mu\big(C_k^{\mu+1}(t)-C_{k-2}^{\mu+1}(t)\big), 4\mu(\mu+k+1)(1-t^2)C_{k}^{\mu+1}=(k+2\mu)(k+2\mu+1)C_{k}^{\mu}-(k+1)(k+2)C_{k+2}^{\mu}.
\tilde{u}_{m, n+1}(t) = S(t-t_m)(u(\cdot, t_m),\partial_t u(\cdot, t_m)) + \int_{t_m}^t \frac{\sin ((t-\tau)\sqrt{-\Delta})}{\sqrt{-\Delta}}F(\tilde{u}_{m,n}(\tau)) d\tau.
W_{2p+1}=\frac{2p}{2p+1}\cdot\frac{2p-2}{2p-1}\cdot\cdot\cdot\frac{2}{3}W_{1}=\frac{(2p)!!}{(2p+1)!!}=\frac{2^{2p}(p!)^{2}}{(2p+1)!}.
0\le t(Y,f_*\O_X(((am-1)p^e+1)(K_{X/Y}+\Delta'')),H) = t(Y,{f^{(e)}}_*\L^{(e)}_{(X,\Delta)/Y}(amp^e(K_{X^e/Y^e}+\Delta)),H) \le t(Y,{F_Y^e}^*f_*\O_X(am(K_{X/Y}+\Delta)),H) = p^et(Y,f_*\O_X(am(K_{X/Y}+\Delta)),H),
\boldsymbol{K}_\alpha=\sum_{i=1}^n l_{i}^{(\alpha)}( \boldsymbol{q},\boldsymbol{p})\left(\frac{\partial}{\partial q_i}\otimes \mathrm{d}q_i + \frac{\partial}{\partial p_i}\otimes \mathrm{d}p_i\right ), \alpha=0,\ldots,n-1,
\left(\begin{array}[c]{ccc}l_{1}l_{2}l_{3}m_{1}m_{2}m_{3}\end{array}\right) \ =\ \Delta(l_{1},l_{2},l_{3})\S_{m_{1},m_{2},m_{3}}^{\ \l_{1},\l_{2},\l_{3}}\N_{m_{1},m_{2},m_{3}}^{\l_{1},\l_{2},\l_{3}}\ \ ,
y_{3}=y_{2}+hf\left( x_{2},y_{2}\right) +h\int\limits_{x_{0}}^{x_{3}}Kdt & =y_{2}+hf\left( x_{2},y_{2}\right) +\frac{h^{2}}{2}\left(K_{0}+K_{1}\right) +\frac{h^{2}}{2}\left( K_{1}+K_{2}\right) & =y_{2}+hf\left( x_{2},y_{2}\right) +\frac{h^{2}}{2}\left(K_{0}+2K_{1}+K_{2}\right)
\begin{aligned}\mu_{2,k}^{T}(u)\leq c(1+T)(\|u_0\|_{H^{s}}+\|\,(|x|^{r_1}+|y|^{r_2})u_0\|_{L^2_{xy}}) &+c(1+T)T^\gamma(\mu_{1,k}^{T}(u))^{k+1}+cT^\gamma(\mu_{1,k}^{T}(u))^{k}\mu_{2, k}^{T}(u).\end{aligned}
v(t)=v_{0}cos(\omega_{0}t)+\frac{v_{0}^{\prime}}{\omega_{0}}sin(\omega_{0}t)+\frac{\omega_{0}^{2}U\omega_{f}}{\omega_{f}^{2}-\omega_{0}^{2}}(\frac{1}{\omega_{0}}sin(\omega_{0}t)-\frac{1}{\omega_{f}}sin(\omega_{f}t)).
\int_0^{1/2}|g(x)|^Kdx= \sum_{j\leq K/2}D^j\binom{K}{j}\Gamma(K-j+1)(1+O(\exp(-C_7K)) &+\sum_{K/2<j\leq K}\binom{K}{j}\int_{\mathcal{A}}l(x)^KR(x,n)^jdx+O(\Gamma(K+1)\exp(-K))\:.
\sqrt{\frac{\theta_{3}}{\eta}}= q^{- 1 / 4 8}\prod_{n = 0}^{\infty}( 1 + q^{n + 1 / 2}) , \qquad \qquad \sqrt{\frac{\theta_{4}}{\eta}}= q^{- 1 / 4 8}\prod_{n = 0}^{\infty}( 1 - q^{n + 1 / 2})
p_d(N)=\mbox{dim}\, V^{\alpha}_{N+\frac{1}{2}{\alpha}^2}= \sum_{M=0}^{N}p_{d-2}(M)p_2(N-M) -p_{d-2}(N)+ \mbox{dim}\, T^{\alpha}_{N+\frac{1}{2}{\alpha}^2}\, .
\max_{\kappa,\lambda}\mathrm{R}\left(\frac{c_0}{\Phi_0(\nu;x)+\frac{\kappa_1}{x+\lambda_1+\frac{\kappa}{x+\lambda}}}-U(x)\frac{c_0}{\Phi_0(\nu;x+1)+\frac{\kappa_1}{x+1+\lambda_1+\frac{\kappa}{x+1+\lambda}}}-V(x)\right).
\widetilde{\pi}_{i,j}(\lambda) \, = \, \frac{f(\lambda, i)}{\lambda}\frac{1}{\theta_i+f(\lambda, i)}\delta_{ij}+ \sum_{k \in \mathcal{S}}\frac{\theta_i}{\theta_i+f(\lambda, i)}h_{ik}\widetilde{\pi}_{k,j}(\lambda).
\left\|\mathfrak{X}_{[0,r_\varepsilon]}\mathbb{P}(z) \mathcal{A}(z) \mathfrak{X}_{(R, \infty)}\right\|_{HS}^2 = \left\|\mathbb{P}(z) G(z) M' (z) \mathfrak{X}_{(R, \infty)}\bigl(1- M_d (z)\bigr)^{-1/2}\right\|_{HS}^2 \leq c' \left\|G(z) M' (z) \mathfrak{X}_{(R, \infty)}\right\|_{HS}^2 ,
\|u\|_{1,\infty,Q_T}\leq \|u_0\|_{1,\Omega}+\|f\|_{1,Q_T}+\|h \|_{1,\Sigma_T}:= \mathcal{Z}; \|\nabla u\|_{p,Q_T}^p \leq \alpha_\ell +\beta_\ell\left(\mathcal{B}+2^{\ell-2}\beta\left((S_{1}^{n(p-1)\over n-p(n-1)}|\Omega|^{n(p-1)^2\over (n-p(n-1))p}+S_{1}^{n(p-1)\over p})T\mathcal{Z}\right)^{\ell-1}\right),
\widetilde{\nabla}_{X}Y =\nabla_{X}Y+\sigma(X,Y), \widetilde{\nabla}_{X}V =-A_{V}X+D_{X}V, \widetilde{\nabla}_{X}^{\ast}Y =\nabla_{X}^{\ast}Y+\sigma^{\ast}(X,Y), \widetilde{\nabla}_{X}^{\ast}V =-A_{V}^{\ast}X+D_{X}^{\ast}V
\omega^{2}_{\sigma}(q) = q^{2}\frac{1 + \frac{2}{\pi}V_{(0)}^{11}+ \frac{1}{2 \pi^{2}}\{(U_{(0)}^{11})^{2}- 2 U_{(0)}^{11}U_{(1)}^{11}\}}{1 + \frac{2}{\pi}V_{(1)}^{11}- \frac{1}{2 \pi^{2}}(U_{(1)}^{11})^{2}},
-\Delta_m f(z) = \left(\frac{4}{pq}\right)^m (2f(z) - f(y_1) - f(y_2)) i([z, y_2]) = i([y_1, z]), -\Delta_m f(z) = \left(\frac{4}{pq}\right)^m (2f(z) - 2qf(y_1) - 2pf(y_2)) i([z, y_2]) = i([y_1, z])+1, -\Delta_m f(z) = \left(\frac{4}{pq}\right)^m (2f(z) - 2pf(y_1) - 2qf(y_2)) i([z, y_2]) = i([y_1, z])-1.
{1\over{\Gamma(s)}}\int_0^{\infty}t^{s-1}{{e^{-{a\over t}}}\over t}e^{-\mu t}dt={1\over{(s-1)}}\mu^{1-s}+{{\Gamma(1-s)}\over{\Gamma(s)}}({a})^{s-1}+{\cal O}(\mu^{2-s},\mu)
\rho :=\left[\Gamma^2 -\frac{1}{\kappa^2}\Big(\big(\sum_{i\in I}\frac{\lambda_i}{2-\lambda_i}\big) -\min_{i\in I}\frac{\lambda_i}{2-\lambda_i}\Big)^{-1}\right]_+^\frac{1}{2(m-1)}<1.
\frac{1}{z-H_g}= R_g(z) = \left( \begin{array}{cc}0 0 0 R_{\Omega}(z)\end{array}\right) +\left( \begin{array}{c}1 R_{\Omega}(z)|g \rangle\end{array}\right)\frac{1}{C_g(z)}\left( 1, \langle g | R_{\Omega}(z)\right )
(-\Delta)^s R_{2n+1}^\lambda(x)= (2s+1)\,b_n^\lambda\, x \times \sum_{k=0}^{n}\frac{(-n)_{k}(n+\lambda+1)_{k}}{(\lambda+\frac{1}{2})_{k}k!}\, A_s^{k+\frac{\lambda}{2}+1}{}_{2}F_{1}\Big(s+k+\frac{\lambda}{2}+1,s+\frac 3 2;\frac{3}{2};-x^2\Big),
V(\mathbf{j}') =3j_{n-2}'-s(j_{n-2}')+2j_{n-1}'-s(j_{n-1}')-c(j_n,-\alpha_{\mathbf{j}'}(n)-1) & = 3(1)-s(1)+2(16q+5)-s(16q+5)-c(1,8q+2) & = 32q+12-s(q)-s(5) & = 32q+10-s(q).
\mu_{j}\cdot \bar{q}={\frac{\pi}{2}}-{\frac{\pi ( 2 j - 1 )}{2 ( r + 1 )}}= q_{j}, \quad -{\frac{\pi}{2}}< \mu_{j}\cdot \bar{q}<{\frac{\pi}{2}}\quad ( j = 1 , \ldots , r + 1 ) .
N(\mathrm{x}^{E^{(1)}_i}=u_{1i}, 1\leq i\leq r_1\{\rm and}\x_{n_1+1}...x_{n_3}=0) &=\#\{(x_1,...,x_{\max\{n_2,n_4\}})\in (\mathbb{F}_q)^{\max\{n_2,n_4\}}: \mathrm{x}^{E^{(1)}_i}=u_{1i}, 1\leq i\leq r_1\{\rm and}\x_{n_1+1}...x_{n_3}=0\}.
Z^{*}W_{\pi^{0}}( K ) Z = e^{i 9 0 m}b_{1}z_{1}^{*}z_{1}+ e^{i 9 0 m}b_{2}z_{2}^{*}z_{2}+ \frac 1 2 e^{i 9 0 m}b_{3}[ z_{1}^{*}( z_{1}+ z_{2}) + z_{2}^{*}( z_{1}+ z_{2}) ]
\left( -\frac{d^{2}}{d\alpha^{2}}+\frac{D_{j,m,n_1,n_2}-\frac{1}{4}}{\sin^{2}\alpha}\right) G_{\ell,j,m,n_1,n_2}(\alpha )=A_{\ell,j,m,n_1,n_2}G_{\ell,j,m,n_1,n_2}(\alpha ),
\langle J^{\mu A B}\rangle_{b a r e}= \lim_{x \rightarrow y}T r \left\{\gamma^{\mu}( x ){\frac{i}{2}\sigma^{A B}}P_{L}\left[ \frac{1}{i{{\cal D}\kern - 0 . 1 5 e m \raise 0 . 1 7 e x \llap{/}\kern 0 . 1 5 e m \relax}}\frac{1}{2}\left( 1 - \sigma^{3}\right) \right] \delta ( x - y ) \right\}
0 = s_0^{i+1,j}d_{X,0}^{ij}+d_{Y,0}^{i-1,j-1}s_0^{ij}f_{n}^{i j}=\sum_{p+q=n+1}(s_p^{i+1,j+q}d_{X,q}^{ij}+d_{Y,p}^{i-1,j+q-1}s_q^{ij})
\left[ \begin{array}{cccc}1 1 \dots 1 \lambda_{1,a_1}\lambda_{2,a_2}\dots \lambda_{n,a_n}\vdots \vdots \vdots \vdots \lambda_{1,a_1}^{r-1}\lambda_{2,a_2}^{r-1}\dots \lambda_{n,a_n}^{r-1}\end{array}\right]\left[ \begin{array}{c}c_{1,a}c_{2,a}\vdots c_{n,a}\end{array}\right]=0
C_{2}^{a b}( n_{1}" n_{2}" n " k " n_{1}^{\prime}n_{2}^{\prime}n^{\prime}k^{\prime}n_{1}n_{2}n k ) = C_{2}^{a b}( n_{1}^{\prime}n_{2}^{\prime}n^{\prime}k^{\prime}n_{1}" n_{2}" n " k " n_{1}n_{2}n k )
\begin{cases}-\Delta u + \nabla p = - \partial_{t}u+ G^{1}\Omega \{u}=G^{2}\Omega (pI - \mathbb{D}u ) e_{3}= (\eta I + \sigma_{0}\Delta_{*}\eta) e_{3}- \sigma_{0}^{\prime}\nabla_{*}c + G^{3}\Sigma u = 0 \Sigma_{b},\end{cases}
\lim_{|x|\rightarrow \infty}|x|^{-\alpha}|\phi_0(x)|=\lim_{|x|\rightarrow \infty}|x|^{1-\alpha}|\nabla \phi_0(x)| =\lim_{|x|\rightarrow \infty}|x|^{2-\alpha}| \nabla^2 \phi_0(x)|=0.
l_3(\xi,\psi,\phi)(X_1,\dots, X_{k+l+m-1})= (-1)^{m+k(l+m)}\cdot \sum_{\tau\in S_{l,m,k-1}}(-1)^{\tau}\xi\Big(\mathrm{pr}_K \Big[ \psi(X_{\tau(1)},\dots,X_{\tau(l)})\,,\,\phi(X_{\tau(l+1)},\dots,X_{\tau(l+m)})\Big],,\dots,X_{\tau(l+m+k-1)}\Big) \pm\circlearrowleft
\big\| (A_{I_r,[r+2k_0]})^\top \,v\big\|_2 = \Bigg\| \Bigg( \begin{matrix}(A_{I_r,[r]})^\top v (A_{I_r,[r+1,r+2k_0]})^\top v \end{matrix}\Bigg) \Bigg\|_2 &\ge \max \Big\{n^{-\beta/(50p)}\,\big\|P_{F_r^\perp}\,v\big\|_2 ,\, n^{-\beta/(40p)}\,\big\|P_{F_r}\,v\big\|_2 - 3\sqrt{\beta n}\,\big\|P_{F_r^\perp}\,v\big\|_2\Big\},
U_{9}= \left( \begin{matrix}{0}&{1}&{0}&{\cdots}&{0}\\{0}&{0}&{1}&{}&{0}\\{\vdots}&{}&{\ddots}&{\ddots}&{\vdots}\\{0}&{0}&{}&{0}&{1}\\{1}&{0}&{\cdots}&{0}&{0}\\ \end{matrix}\right) .
p \equiv \frac{2 B_{3}( B_{1}- \alpha B_{2})}{( \alpha - \beta ) ( 1 + B_{1}^{2}- B_{2}^{2})}, \hspace{0 . 3 c m}q \equiv \frac{2 B_{3}( B_{1}- \beta B_{2})}{( \alpha - \beta ) ( 1 + B_{1}^{2}- B_{2}^{2})}, \hspace{0 . 3 c m}a \equiv \frac{( 1 + B_{1}^{2}- B_{3}^{2})}{( 1 + B_{1}^{2}- B_{2}^{2})}.
E_{p}^{s}=\{v\in T_{p}M_{i}: ( \Vert D (\textbf{\textit{f}}^{n}_{i})_{p}(v)\Vert)_{n\geq1}\}\quad\quad E_{p}^{u}=\{v\in T_{p}M_{i}: (\Vert D (\textbf{\textit{f}}^{-n}_{i})_{p}(v)\Vert)_{n\geq1}\}.
\frac{1}{2}\|w(t)\|_2^2+\int_{\bar t}^t\|\nabla w\|_2^2 d\tau &=\frac{1}{2}\|w(\bar t)\|_2^2+\int_{\bar t}^t \Big[\langle (u_s+\widetilde U)\otimes w, \nabla w\rangle+\langle f,w\rangle\Big] d\tau
\renewcommand{\arraystretch}{1}\sum_{\begin{array}{c}{{\scriptstyle \sigma_{1}, \ldots , \sigma_{L - 1}= 0 , 1}}\\{{\scriptstyle \sigma_{j}\sigma_{j + 1}= 0}}\\ \end{array}}q^{\; \; \displaystyle \sum_{k = 1}^{L - 1}k \sigma_{k}}\qquad \qquad \sigma_{0}= a , \; \sigma_{L}= 0 ,
\left| \left. q \left( T \right) \right\rangle \right\rangle = \prod_{k = - \infty}^{\infty}e^{i G_{k}^{1}\left( T \right)}\prod_{s = - \infty}^{\infty}e^{i G_{s}^{2}\left( T \right)}\left| \left. q \right\rangle \right\rangle ,
\begin{bmatrix}\dot{x_1}\dot{v_1}\dot{x_2}\dot{v_2}\end{bmatrix}= \begin{bmatrix}v_1 -\frac{k_1}{m_1}x_1 - \frac{k_2}{m_1}(x_1-x_2) v_2 -\frac{k_2}{m_2}(x_2-x_1)\end{bmatrix}
F ( \sigma ) \ast G ( \sigma ) = \int{\frac{d^{2}\xi}{( 2 \pi )}}\exp \left( i{\frac{\hbar}{2}}\omega_{m n}\sigma^{m}\xi^{n}\right) F \left({\frac{\sigma}{2}}+ \xi \right) G \left({\frac{\sigma}{2}}- \xi \right) \ ,
\frac{1}{n}\max_{x_2^n}C(\mathcal G_{x_2^n}) = \frac{1}{n}\max_{x_2^n}\log \mathsf{IP}(\mathcal G_{x_2^n}) \leq \frac{1}{n}\max_{x_2^n}\log \mathsf{LP}(\mathcal G_{x_2^n})+\log (1+\epsilon) &= \max_{x_2}\bar{C}(\mathcal G_{x_{2}})+\log (1+\epsilon).
J_2^{k}= \big( |\Psi_{h}^{k}|^{2}- |\mathcal{I}_{h}\Psi^{k}|^{2},\; \overline{\theta}_{\phi}^{k}\big) - \big( |\Psi_{h}^{k-1}|^{2}- |\mathcal{I}_{h}\Psi^{k-1}|^{2},\; \overline{\theta}_{\phi}^{k}\big),
\sum_{k=0}^{M}[4k-1]\frac{(q^{-1};q^2)_k^4(cq^{-1},dq^{-1};q^2)_k}{(q^2;q^2)_k^4(q^2/c,q^2/d;q^2)_k}\bigg(\frac{q^7}{cd}\bigg)^k begin{align*}5pt]\: \:\equiv\frac{\Omega_q(n)(1-q)^2(q^3/cd;q^2)_2}{(1+q)^2(q^2/c,q^2/d;q^2)_2}\sum_{k=0}^{(n-3)/2}\frac{(q^3;q^2)_k^3(q^7/cd;q^2)_k}{(q^2,q^6,q^6/c,q^6/d;q^2)_k}q^{2k},
e_{\hat{0}}^{\;\;\mu}e_{\alpha}^{\;\; \nu}\left. \frac{\partial \Gamma_{R}}{\partial b^{\mu \nu}}\right|_{g_{\mu\nu}}={\bar{e}}_{\alpha}^{\;\; i}{e}_{\hat{0}}^{\;\;0}a \frac{\partial \Gamma_{R}}{\partial w^{i}},
\sum_{j=0}^{n}\left( -1\right)^{j}Y_{j}\left( \lambda \right) Y_{n-j}\left(\lambda \right) =\frac{4\lambda^{2n}\left( 1+\left( -1\right)^{n}\right)\left( n+1\right) !}{\left( n+2\right) \left( \lambda -1\right)^{n+2}}.
\frac{-1}{2\pi i}\int_{(5)}L^s\tilde{w}(s)\frac{L'(s-ir_1-ir_2,\xi\chi_1\chi_2^{-1})}{L(s-ir_1-ir_2,\xi\chi_1\chi_2^{-1})}ds = \frac{-1}{2\pi i}\int_{\mathcal{C}_2}L^s\tilde{w}(s)\frac{L'(s-ir_1-ir_2,\xi\chi_1\chi_2^{-1})}{L(s-ir_1-ir_2,\xi\chi_1\chi_2^{-1})}ds.
x_\alpha(r) x_\alpha(s) = x_\alpha(r+s), [x_\alpha(r), x_\beta(s)] = x_{\alpha+\beta}(N_{\alpha,\beta}\cdot rs), \ \alpha,\beta\in \Phi,\ \alpha+\beta\in\Phi, [x_\alpha(r), x_\beta(s)] = 1, \ \alpha,\beta\in \Phi,\ \alpha+\beta \not \in \Phi \cup \{0\}.
-k\int_{B_R}\left[1-c_1^{-2}(y)\right]\cdot \hat{u}_1(y, k)\cdot\Phi(x-y)\dy & -\frac{i}{2\pi}\int_{B_R}c_1^{-2}(y)\cdot f_1(y)\cdot \Phi(x-y)\dy = -k\int_{B_R}\left[1-c_2^{-2}(y)\right]\cdot \hat{u}_2(y, k)\cdot\Phi(x-y)\dy & -\frac{i}{2\pi}\int_{B_R}c_2^{-2}(y)\cdot f_2(y)\cdot \Phi(x-y)\dy.