harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Psi [ A_{i}] = \int D s_{i}D \phi \exp \left\{i \kappa [ s_{i}\epsilon_{i j}A_{j}- \partial_{i}\phi \epsilon_{i j}A_{j}+ \epsilon_{i j}\partial_{i}\phi s_{j}] - \frac{1}{2}A_{i}^{\phi , s}G^{- 1}A_{i}^{\phi , s}\right\}
\sum_{\eta \in s(\lambda) \Lambda^{(1,1)}}\mu_2(Z(\lambda \eta)) = \mu_2(Z(\lambda)) \cdot \begin{cases}(1/2 - \delta_{2N+1}) + (1/2 + \delta_{2N+1}), \;\; s(\lambda) = v (1/2 - \delta_{2N+2}) + (1/2 + \delta_{2N +2}), \;\; s(\lambda) \in \{u, w\}.\end{cases}&= \mu_2(Z(\lambda)).
m_A =\min\{|A_i| : 1\leq i\leq r\}, \Gamma_A = \{i : |A_i|= m_A, 1\leq i\leq r\}, \lambda_A = j\quad\Longleftrightarrow\min\left(\bigcup_{i\in\Gamma_A}A_i\right)\in A_j.
~ - \partial_{y}( K ( \chi ) \partial_{y}\chi ) - \frac{K^{\prime}( \chi )}{2}( \partial_{y}\chi )^{2}+ V^{\prime}( \chi ) + v_{i}^{\prime}( \chi ) \delta ( y - y_{i}) = \sum_{n > 0}\frac{1}{M^{n}}\frac{\delta S_{h . d .}^{( n )}}{\delta \chi}.
\Phi_{A_3}= -\frac{1}{2}t_0^2t_2 - \frac{1}{2}t_0t_1^2 - \frac{1}{4}t_1^2t_2^2 - \frac{1}{60}t_2^5 \Phi_{D_3}= -\frac{1}{2}t_0^2 t_2 + \frac{1}{2}t_0 t_\ast^2 - \frac{1}{4}t_2^2 t_\ast^2 -\frac{1}{60}t_2^5
W_{\nu}(\zeta) \sim\frac{2^{i\nu}\Gamma\left(\frac{d-1}{2}-i\nu\right)\Gamma(i\nu)}{2(2\pi)^{\frac{d+1}{2}}}\ \ \zeta^{-\frac{d-1}{2}+i\nu}\ \ \ \ \ \ \ \ \makebox{forIm}\ \nu < 0
\Lambda^i_{1, \, R}(f_0^i) = 0, \Lambda^i_{2, \,R}(f_0^i, \, f_1^i) = - f_1^i \circ R \, D^2R, \Lambda^i_{L, \,R}(f_0^i, \, \dots, \, f_{L-1}^i) = D [\Lambda^i_{L-1, \, R}(f_0^i, \, \dots, \, f_{L-2}^i )] & - (L-1) \, f_{L-1}^i \circ R \, (DR)^{L-2}\, D^2R, L \in \{3, \, \dots, \, r \},
\varepsilon^{-\tilde{s}}\geq \varepsilon^{-q s - s\log_2(C)}= \varepsilon^{-q s}\cdot \left(\frac{1}{\varepsilon}\right)^{s\log_2(C)}\geq \varepsilon^{-q s}\cdot 2^{s\log_2(C)}= \varepsilon^{-q s}\cdot C^s = \left(\frac{\varepsilon^{q}}{C}\right)^{-s},
{\textrm{Optimal User}}= \arg \mathop{\max}\limits_{i \in{\cal{S}}}{C_{\textrm{secrecy}}}(i) &= \arg \mathop{\max}\limits_{i \in{\cal{S}}}\dfrac{{1 + \frac{{|{h_{is}}{|^2}P_i}}{{{N_0}}}}}{{1 + \frac{{|{h_{ie}}{|^2}P_i}}{{{N_0}}}}},
\begin{array}{c}{( a_{v}+ \{\epsilon_{v}, . \}) R_{v}( \epsilon , a ) \equiv ( a_{v}+ \{\epsilon_{v}, . \}) ( a H_{v}+ \{\epsilon , H_{v}\})}\\{= R_{v}( \{\epsilon_{v}, \epsilon \}, \{\epsilon_{v}, a \}- \{\epsilon , a_{v}\}) ,}\\ \end{array}
q_{\beta}(n,t) =\sum_{k=1}^{n}\mathrm{Pr}\{X_{1}+X_{2}+\dots+X_{k}=n\}\mathrm{Pr}\{N_{\beta}(t)=k\}&=\sum_{k=1}^{n}\sum_{\Theta_{n}^{k}}k!\prod_{j=1}^{n}\frac{(\theta^{j}/j!)^{x_{j}}}{x_{j}!}\left(\alpha t^{\beta}\right)^{k}E_{\beta,k\beta+1}^{k+1}\left(-\alpha(e^{\theta}-1)t^{\beta}\right),
F_{1}(b_{1},b_{2},b_{3};c_{1};x,y)=\frac{\Gamma(c_{1})}{\Gamma(b_{1})\Gamma(c_{1}-b_{1})}\int_{0}^{1}t^{b_{1}-1}(1-t)^{c_{1}-b_{1}-1}(1-xt)^{-b_{2}}(1-yt)^{-b_{3}}dt,
\frac{d}{dt}P_{\lambda,d}(\beta_t^{\delta_1}(O)=1) =-P_{\lambda,d}(\beta_t^{\delta_1}(O)=1) &+\lambda\sum_{y:y\sim O}P_{\lambda,d}(\beta_t^{\delta_1}(O)=0,\beta_t^{\delta_1}(y)=1) &\leq -P_{\lambda,d}(\beta_t^{\delta_1}(O)=1)+\lambda\sum_{y:y\sim O}P_{\lambda,d}(\beta_t^{\delta_1}(y)=1).
\sum_{j=1}^{p-1}\sum_{t=1}^{p-j}x^{-2j+2t}=\frac{x^{-2p+4}(1-x^{2p-2})(1-x^{2p})}{(1-x^2)(1-x^4)}&=\begin{cases}\displaystyle\sum_{j=1}^{p/2}\sum_{t=1}^{p-1}x^{2-4j+2t}(p: ), \displaystyle\sum_{j=1}^{p}\sum_{t=1}^{(p-1)/2}x^{-2j+4t}(p: ).\end{cases}
W_0(\psi \tilde h) =\int_{B_1}|\nabla (\psi \tilde h)|^2-\int_{\partial B_1}|c|^2 &\le \left(1+\frac{C_0\rho^{2(\alpha_k-1)+d}}{\alpha_k}\right)\int_{B_1}|\nabla \tilde h|^2-\int_{\partial B_1}|c|^2 &=\left(1+\frac{C_0\rho^{2(\alpha_k-1)+d}}{\alpha_k}\right)W_0(\tilde h)+\frac{C_0\rho^{2(\alpha_k-1)+d}}{\alpha_k}\int_{\partial B_1}|c|^2.
p_\mu(I,X)= \frac{1}{\omega_{2n-3}}\bigg[ (n-1)((2n-3)c_1-2(n-2) c_0) X^{\overline 1}_{\overline 1}+ (2(n-2) c_0-(n-3)c_1) \sum_{i=2}^n\left(X_{i}^{i}+ X_{\overline i}^{\overline i}\right)\bigg]=0
\begin{aligned}\ge \sum_{m=1}^{\infty}\frac{1}{A_{m+1}}\Bigl( \sum_{j=0}^{m-1}\int_{A_j s}^{A_{j+1}s}P(v) dv\Bigr)\Bigl(F_1(A_{m+1}-F_1(A_m))\Bigr) \ge \ge \sum_{m=1}^{\infty}\frac{sm}{2\pi (m+1)}2\pi a_0 \Bigl(F_1(A_{m+1}-F_1(A_m))\Bigr) \ge \ge \frac{s a_0}{2}\Bigl(1-F_1(A_1)\Bigr)= \frac{s a_0}{2}\Bigl(1-F(2 \pi /s)+F(-2 \pi /s)\Bigr).\end{aligned}
H=H_{elastic}+H_{enthalpic}+H_{external}=\frac{1}{2}k_{B}T\int_{0}^{L_{0}}P\cdot(\frac{\partial^{2}\vec{r}(s)}{\partial s^{2}})^{2}ds+\frac{1}{2}\frac{K_{0}}{L_{0}}x^{2}-xF
D_{\tau}^{\alpha}\omega^n:= \frac{\tau^{-\alpha}}{\Gamma(2-\alpha)}\sum_{j=1}^na_{n-j}{\delta_t\omega^j}= \frac{\tau^{-\alpha}}{\Gamma(2-\alpha)}\sum_{j=0}^n b_{n-j}\omega^j, n=1,\cdots,N,
\nu(J)\int_{G_{\theta(J)}'}\sum_{z\in Z_\delta}{\bf 1}_{\Gamma^{\delta}_z}(\theta(J),y) dy = \int_J \int_{G_{\theta(J)}'}\sum_{z\in Z_\delta}{\bf 1}_{\Gamma^{\delta}_z}(\theta(J),y) dy d\nu(\theta) &\geq \frac{1}{2}\int_J \int_{G_{\theta}'}\sum_{z\in Z_\delta}{\bf 1}_{\Gamma^{\delta}_z}(\theta,y) dy d\nu(\theta).
\int_{B_{r}(x)}\sum_{i,j=1}^{m}\left( |D u|^{2}\delta_{ij}- 2 \sum_{\ell=1}^{Q}\langle D_{i}u_{\ell}, D_{j}u_{\ell}\rangle \right) D_{i}X^{j}\, \mathrm{d}y = 0.
A_{\delta}:= \mathop{\int_X\int_X}_{\{|u(x)-u(y)|>\delta\}}\frac{\delta^p}{\rho(x,y)d(x,y)^{p}}\, d \mu(x) \, d \mu(y), B_{\delta, r}:= \mathop{\int_X\int_X}_{\{|u(x)-u(y)|>\delta,\, d(x,y) \leq r \}}\frac{\delta^p}{\rho(x,y)d(x,y)^{p}}\, d \mu(x) \, d \mu(y).
\lim_{q\to 0}[\theta_{k_1,k_2}(q,u,q^{-2m-1}v)]_{(k_1,k_2)\in H}=[\underset{k_2=0}{\underbrace{T^*_{m\tau_{12}}\vartheta_{k_1}}},\underset{k_2=1}{\underbrace{(T^*_{(m+1)\tau_{12}}\vartheta_{k_1})v}}]
\int_{J_{3}}\frac{|I_{l}|dv}{((1-v)|I_{l}|+|I_{l+1}|+\cdots+|I_{k}|)^{H+\frac{1}{2}}}= \frac{1}{\left(H-\frac{1}{2}\right)}\left(\frac{1}{(|I_{l+1}|+\cdots+|I_{k}|)^{H-\frac{1}{2}}}-\frac{1}{\left(\frac{1}{3}|I_{l}|+|I_{l+1}|+\cdots+|I_{k}|\right)^{H-\frac{1}{2}}}\right)\leq \frac{C_{H}}{|I_{k}|^{H-\frac{1}{2}}},
\nabla^{2}\nabla^{2}w=w_{,1111}+2w_{,1212}+w_{,2222}=[\frac{\partial^{4}W}{\partial x_{1}^{4}}+2\frac{\partial^{4}W}{\partial x_{1}^{2}\partial x_{2}^{2}}+\frac{\partial^{4}W}{\partial x_{2}^{4}}]F(t)
\int_{[0,t]^n\times E_{n,j}}\left (A^2_{t,x}(u) +A^2_{t,x}(v)\right)\prod_{j=1}^n|u_j-v_j|^{-\beta_0}dvdu \le \frac{2^{\beta_0}}{1-\beta_0}(t-s)^{1-\beta_0}D_t^{n-1}\int_{[0,t]^n}A^2_{t,x}(u)du+D_t^n \int_{E_{n,j}}A^2_{t,x}(v)dv.
\frac{\partial X}{\partial t}=-XZ+L_{1}(1-Z-\mu(1-Y))+\delta\Delta X \frac{\partial Y}{\partial t}=-L_{3}YZ+\frac{L_{4}}{L_{2}}(1-Y)+\delta\Delta Y \frac{\partial Z}{\partial t}=\frac{1}{L_{2}}((-XZ+1-Z-\mu(1-Y))+\mu)-L_{3}YZ+\frac{L_{4}}{L_{2}}(1-Y)))+\delta\Delta Z
\begin{array}{l}Q_{p}(J,u) := \| u \|_{L_{t}^{\infty}\tilde{H}^{p}(J)}+ \| D u \|_{L_{t}^{\frac{2(n+2)}{n}}L_{x}^{\frac{2(n+2)}{n}}(J)}+ \| D^{p}u\|_{L_{t}^{\frac{2(n+2)}{n}}L_{x}^{\frac{2(n+2)}{n}}(J)}\cdot\end{array}
{\cal M}_{4 , 2}= \frac{S O ( 4 , 2 )}{S O ( 4 ) \otimes S O ( 2 )}\mathrm{~ a n d ~}K = - \log \left( ( T + \overline{{{T}}}) ( U + \overline{{{U}}}) - \frac{1}{2}( B + \overline{{{C}}}) ( C + \overline{{{B}}}) \right)
d^2(x_{i,j}) = d \left( \sum_{i<r<j}x_{i,r}x_{r,j}\right) = \sum_{i<r<j}d(x_{i,r})x_{r,j}- \sum_{i<r<j}x_{i,r}d(x_{r,j}) = \sum_{i<r<j}\sum_{i<t<r}x_{i,t}x_{t,r}x{r,j}- \sum_{i<r<j}\sum_{r<t<j}x_{i,r}x_{r,t}x_{t,j}= 0;
v^{E,-}_i=\lim_{\nu_m\downarrow 0,\epsilon_m\downarrow 0}h_i\ast_{sp}G_{\nu_m}+\lim_{\nu_m\downarrow 0,\epsilon_m\downarrow 0}\delta v^{\nu_m,\epsilon_m,-,(k+1)},~ 1\leq i\leq D,
\Big(t_1 \cos(\frac{2s\pi}{N})+ t_2 \sin(\frac{2s\pi}{N})\Big)^2 = \Big(t_1 \cos(\frac{2s(N-1)\pi}{N})+ t_2 \sin(\frac{2s(N-1)\pi}{N})\Big)^2 &= \Big(t_1 \cos(\frac{2s\pi}{N})- t_2 \sin(\frac{2s\pi}{N})\Big)^2.
- \frac{1}{2}\sum_{j = 1}^{N}D_{j}^{2}= - \frac{1}{2}\sum_{j = 1}^{N}\partial_{x_{j}}^{2}+ \sum_{1 \leq j < k \leq N}\frac{\beta ( \beta - K_{j k})}{( x_{j}- x_{k})^{2}}\, .
\frac{\theta}{4}\sqrt{\frac{i\pi}{2}}\,e^{- i\frac{\Phi}{2}}\,\, k^{3/2}\,\,\left( \pi\, \delta [\varphi] -{\cal P}\left[ \frac{1}{1-e^{-i \varphi}}\right] - \frac{\Phi}{2 \pi}\log [1 - e^{-i(-i \epsilon + \varphi)}] +\dots\right).
S_{\sigma}= \left\{x \in \mathbb{R}^d \,\, \middle| \,\, \sigma_1 x_1 \ge 0 , \dots , \sigma_d x_d \ge 0 , \,\, \frac{\sigma_1 x_1}{a_1}+ \cdots + \frac{\sigma_d x_d}{a_d}\le 1 \right\}.
\begin{matrix}a_1^2+b_1c_1=\theta_1^2,\ \ \a_2^2+b_2c_2=\theta_2^2\ \ \ 2a_1a_2+2\theta_4(a_1+a_2)+b_1c_2+b_2c_1+\theta_1^2+\theta_2^2-\theta_3^2+\theta4^2=0.\end{matrix}
\begin{array}{l}\mbox{equation~}\eqref{09_07_2020_eq_12}=g_0(x)(x-1)^{n-k_1}\hat f_{00}(x), \mbox{equation~}\eqref{09_07_2020_eq_13}=g_0(x)(x-1)^{n-r}(u\tilde f_{01}(x)), \mbox{equation~}\eqref{09_06_2020_eq_2}=g_1(x)(x-1)^{n-r_1}\tilde f_{10}(x), \mbox{equation~}\eqref{09_06_2020_eq_1}=g_0(x)(u^2f_{02}(x)), \mbox{equation~}\eqref{09_06_2020_eq_3}=g_1(x)(u f_{11}(x)),\end{array}
\tilde{H}= \frac{(u')^2}{2}+\frac{\left(h'u-\left(h+\frac{\omega}{\Lambda-1}\right)u'\right)^2}{2u^4}&-\frac{\left(1-u^2-\frac{1}{u^2}\left(h+\frac{\omega}{\Lambda-1}\right)^2\right)^2}{4}-\frac{\Lambda-1}{2}h^2
\rightarrow \bigoplus_{\substack{\sigma \in W^{aff}, l(\sigma)=r}}P_{\sigma(\rho)-\rho}\rightarrow \bigoplus_{\substack{\sigma \in W^{aff}, l(\sigma)=r-1}}P_{\sigma(\rho)-\rho}\rightarrow \dots\bigoplus_{\substack{\sigma \in W^{aff}, l(\sigma)=1}}P_{\sigma(\rho)-\rho}\rightarrow P_0 \rightarrow V_0.
\mathbf{\Sigma}^1(x_1)=\frac{(-x_1)^{-3/2}}{2\sqrt{2\pi}}\begin{pmatrix}i\beta(H_{24}^2-H_{22}H_{44})[(-x_1)^{i\epsilon}- (-x_1)^{-i\epsilon}] (H_{44}H_{12}-H_{14}H_{24})[(-x_1)^{i\epsilon}+ (-x_1)^{-i\epsilon}] (H_{14}H_{22}-H_{24}H_{12})[(-x_1)^{i\epsilon}+ (-x_1)^{-i\epsilon}]\end{pmatrix},
g(x)= (x-1)^{(\tau-1)\rho+i'}f(x) = (x^{\rho}-1)^{\tau-1}\left(\sum_{\theta=0}^{i'}\binom{i'}{\theta}(-1)^{i'-\theta}x^{\theta}\right) \left(\sum_{j=0}^{\rho-1}\bar{f_j}(x^{\rho})x^j\right)
|\mathcal{F}(g_a)( b^{-1}\xi )|^2 ={}b^{d+1}|\xi|^{-d-1}R^{d-1}{\tfrac{1}{\pi^2}}\Big(f(\beta)\sin(2\pi b^{-1}(R+a\varphi(\beta))|\xi| + \nu_d) &- f(\omega)\sin(2\pi b^{-1}(R+a\varphi(\omega))|\xi| + \nu_d) + O(a+a^{-1}b|\xi|^{-1})\Big)^2.
[y_1, y_1]=y_4, [y_1, y_2]=\frac{\alpha_3}{\alpha_5}y_4+\theta_1y_5, [y_2, y_1]=y_4+\theta_2y_5, [y_2, y_2]=\frac{\beta_2\gamma_6}{\gamma^2_2}y_5, [y_2, y_3]=y_5, [y_3, y_3]=y_5.
\begin{multlined}\alpha\sigma\alpha\sigma\alpha\sigma(A)((i,(j,k))) =(-1)^{|i|(|j|+|k|)}(-1)^{|j|(|k|+|i|)}(-1)^{|k|(|i|+|j|)}A((i,(j,k))) =A((i,(j,k)))\;.\end{multlined}
D_{ad}^{2}(b)=2\cot \left( \frac{\pi x}{2h}\right) \tan \left( \frac{\pi\left( 1-x\right)}{2h}\right) \tan \left( \frac{\pi}{2h}\right) \mathcal{T}_{0}\mathcal{T}_{q-1}\mathcal{T}_{h/2-q}
H:=\int_\Omega\int_\Omega\phi\left(\frac{|u (z)-u (w)|}\lambda\right)\frac{dzdw}{|z-w|^{n+\beta}}\le \int_W\int_{W}\phi\left(\frac{| z-w|}{\lambda/ L}\right)\frac{dzdw}{|z-w|^{n+\beta}}+ 2\int_{\Omega\setminus W}\int_{V}\phi\left(\frac{L}{\lambda}\right)\frac{dzdw}{|z-w|^{n+\beta}}.
\begin{aligned}\int_{Q_T}\frac{1}{a}L^+_s w L^-_s w dxdt \ge Cs\int_{Q_T}\Theta(w_x)^2 dxdt &+ Cs^3 \int_{Q_T}\Theta^3 \left(\frac{x-x_0}{a}\right)^2w^2 dxdt &-sd_1\int_0^T\Theta(t)\Big[(x-x_0)e^{R(x-x_0)^2}(w_x)^2\Big]_{x=0}^{x=1}dt.\end{aligned}
f ( y ) = D_{1}\, y^{- 2 \mu}\,{_2 F_{1}}( - \mu - \nu - \lambda , - \mu + \nu - \lambda , 1 - 2 \mu ; y ) + D_{2}\,{_2 F_{1}}( \mu - \nu - \lambda , \mu + \nu - \lambda , 1 + 2 \mu ; y )
\begin{pmatrix}{\dot{f}}_1^{\alpha+1}\vdots{\dot{f}}_N^{\alpha+1}\end{pmatrix}=\begin{pmatrix}\lambda_{11}^{\alpha+1}\cdots \lambda_{1N}^{\alpha+1}\vdots \ddots \vdots \lambda_{N1}^{\alpha+1}\cdots \lambda_{NN}^{\alpha+1}\end{pmatrix}\begin{pmatrix}f_1^{\alpha+1}\vdots f_N^{\alpha+1}\end{pmatrix},
\sum_{|I|=j, I \subseteq \{1,\ldots k\}}e^{t \sum_{i\in I}\lambda_i^{(k)}(A)}= \sum_{\substack{|I|=|J|=j I,J\subseteq \{n-k+1,\ldots,n\}}}P_{I,J}\left(\Delta_i^{(k)}(b_t), \overline{\Delta}_i^{(k)}(b_t)\right).
\Lambda^rV\otimes \Lambda^s V^* = V_{\underbrace{{\scriptstyle{1,\dots,1}}}_{\scriptstyle{r}}}^{\overbrace{\scriptstyle1,\dotsc, 1}^{\scriptstyle{s}}}+ V_{\underbrace{\scriptstyle1,\dots,1}_{\scriptstyle{r-1}}}^{\overbrace{\scriptstyle1,\dotsc, 1}^{\scriptstyle{s-1}}}+\dots + V^{\overbrace{\scriptstyle1,\dotsc, 1}^{\scriptstyle{s-r}}};
\phi^{\mu_{0}\mu_{1}\ldots\mu_{p}}(X)\rightarrow\phi^{\mu_{0}\mu_{1}\ldots\mu_{p}}(X)+ \sum_{i=0}^{p}(-1)^{p\cdot i}\frac{\partial}{\partial X_{\mu_{i}}}\Lambda^{\mu_{i+1}\mu_{i+2}\ldots\mu_{p}\mu_{0}\mu_{1}\ldots\mu_{i-1}},
\Gamma_{t}[ \phi ] = \int \! \! d^{D}x \, \left\{V ( \phi^{2}, t ) + \frac{1}{2}K ( \phi^{2}, t ) \left( \partial_{\mu}\phi^{a}\right)^{2}+ \frac{1}{2}Z ( \phi^{2}, t ) \left( \phi^{a}\partial_{\mu}\phi^{a}\right)^{2}+ \cdots \right\}
\ell(x,\alpha) + \ell(x,-\alpha) = \langle \mu,\alpha\rangle +\Phi^+(\alpha) - \Phi^+(w\alpha) + \langle \mu,-\alpha\rangle + \Phi^+(-\alpha) - \Phi^+(-w\alpha) = \Phi^+(\alpha) + \Phi^+(-\alpha) - (\Phi^+(w\alpha) + \Phi^+(-w\alpha)) = 1-1=0.
\left|\bigcup_{g\in G(q)}X(q_0)^g \right| \le |X(q_0)|\,|G(q):G(q_0)| \le c_{10}q_0^{s/2}q^{d/2}q_0^{-d/2}= c_{10}q^{\frac{d}{2}+\frac{s-d}{2k}}\le c_{10}q^{\frac{s}{2}-\frac{1}{2}+\frac{1}{2k}}.
H^{(6)}_{2B,r}(\tau) :=[-\tfrac{1}{24}]\frac12\left(H^{(2)}_{6A,r}(\tau/3)+H^{(3)}_{4C,r}(\tau/2)\right) H^{(6)}_{4A,r}(\tau) :=[-\tfrac{1}{24}]\frac12\left(H^{(2)}_{6A,r}(\tau/3)+H^{(3)}_{4B,r}(\tau/2)\right) H^{(6)}_{8AB,r}(\tau) :=[-\tfrac{1}{24}]\frac12\left(H^{(2)}_{12A,r}(\tau/3)+H^{(3)}_{8CD,r}(\tau/2)\right)
V = \frac{1}{2{b}_{1 1}}\left( \tilde{m}^{\dagger}\tilde{m}- m^{\dagger}m \right)^{2}+ \frac{2}{{b}_{1 1}}\left| a_{D}\right|^{2}\left( m m^{\dagger}+ \tilde{m}\tilde{m}^{\dagger}\right) + \frac{{b}_{0 1}}{{b}_{1 1}}D_{0}\left( m m^{\dagger}- \tilde{m}^{\dagger}\tilde{m}\right)
[e_i,e_j][u,w_2]+[e_j,w_2][e_i,u]=[e_i,e_j][u,e_k v]+[e_j,e_k v][e_i,u]= [e_i,e_j][u,e_k]v+[e_i,e_j]e_k[u,v]+e_k[e_j,v][e_i,u]+[e_j,e_k]v[e_i,u]= e_k([e_i,e_j][u,v]+[e_j,v][e_i,u])+v([e_i,e_j][u,e_k]+[e_j,e_k][e_i,u])
\dot{\theta}= \left(\frac{x}{2}- \frac{m-1}{x}\right)\sin \theta + \left(\frac{n-1}{y}- \frac{y}{2}\right)\cos \theta + \lambda &\leq \frac{x}{2}- (m-1)\left( \frac{\dot{x}}{x}\tan \theta \right) &\leq \frac{1}{2}+ (m-1)\left( \delta \frac{\dot{x}}{x}\right),
\delta G^{(2,2,2,2)}(1\vert 2\vert 3\vert 4) = -\lambda\cdot\partial G^{(2,2,2,2)}(1\vert 2\vert 3\vert 4) + 2(\Lambda(1) + \Lambda(2) + \Lambda(3)+ \Lambda(4)) G^{(2,2,2,2)}(1\vert 2\vert 3\vert 4) \;.
E^a_{Q,+,h}(\sigma_0) =\mathbb{P}^a_{Q,\bar{w},h}(0\longleftrightarrow g)\geq\mathbb{P}^a_{Q,\bar{w},h}(0\overset{a\mathbb{Z}^2}\longleftrightarrow \partial_{in}Q^a) &\geq \mathbb{P}^a_{Q,w,h}(0\overset{a\mathbb{Z}^2}\longleftrightarrow \partial_{in}Q^a)=\tilde{\mathbb{P}}^a_{Q,w,h}(0\longleftrightarrow\partial_{in}Q^a).
\int_{\mathbb{Z}^N}(I_\alpha\ast u)(x)v(x)\,d\mu:=\underset{x,y\in\mathbb{Z}^N, x\neq y}{\sum}\frac{u(y)v(x)}{|x-y|^{N-\alpha}}\leq C_{r,s,\alpha,N}\|u\|_r\|v\|_s,
\mathbf{h}=\left[\mathbf{h}^T(1,1),\cdots,\mathbf{h}^T(Q,1),\mathbf{h}^T(1,2),\cdots,\right. \qquad\left.\mathbf{h}^T(Q,2),\cdots,\mathbf{h}^T(1,3),\cdots,\mathbf{h}^T(Q,K)\right]
\|(\mathcal{P}_{\lambda_n})^{1/2}v_n\|_{L^2}^2-\||\nabla|^\sigma v_n\|_{L^2}^2 &=\int_{|\xi|\leq C\lambda_n}+ \int_{|\xi|\geq C\lambda_n}\big(|\xi+\lambda_n|^{2\sigma}-\lambda_n^{2\sigma}-2\sigma\lambda_n^{2\sigma-1}\xi-|\xi|^{2\sigma}\big)|\hat{v}_n(\xi)|^2d\xi &=I_n+II_n.
u^{\prime\prime}=D(x,u)(u^\prime-w_1(x,u))(u^\prime-w_2(x,u))(u^\prime-w_3(x,u)) &=:A(x,u)+B(x,u)u^\prime+C(x,u)u^{\prime2}+D(x,u)u^{\prime3},\;A, B, C, D\in\mathcal Q,\;w_1,w_2,w_3\in\mathcal{O},
\begin{aligned}\partial_{t_e}\vec q_h \cdot (\partial_{t_e}\lambda_{v_e}\vec q_h (v) + \partial_{t_e}\lambda_v \vec q_h (v_e)) =\ \partial_{t_e}\vec q_h \cdot(\partial_{t_e}(\lambda_v + \lambda_{v_e})\vec q_h((v + v_e)/2) + \frac{h_e}{2}\partial_{t_e}(\lambda_v - \lambda_{v_e}) \partial_{t_e}\vec q_h). \end{aligned}
T_{32}=\tfrac{2}{\bar{\Delta}_M(\theta)}\sum_{I\in \bar{\mathcal{S}}^c_M}r^2(I,\theta) \mathrm{E}_{\theta}\hat{\pi}(I|X) &\le \tfrac{2}{\bar{\Delta}_M(\theta)}\sum_{I\in \bar{\mathcal{S}}^c_M}r^2(I,\theta) \big(\tfrac{ne}{|I|}\big)^{-c_1|I|}\exp\big\{-2C_1\sigma^{-2}r^2(I,\theta)-2C_2M\big\}\le C_5 \exp\{-2C_2 M\},
\left[ - \frac{d^{2}}{d w^{2}}- \frac{2 4}{1 + e^{\sqrt{8}w}}+ \frac{3 0}{\left( 1 + e^{\sqrt{8}w}\right)^{2}}\right] \chi_{n}( w ) = \nu_{n}^{2}\chi_{n}( w ) \; ,
\displaystyle \varphi^{-1}(C_{r''a,ha_{1}}) =\displaystyle \sum_{u \in L(\sigma(a))}(r'')^{\ast}(hu)\varphi^{-1}(C_{ua,a_{1}}) \varphi^{-1}(D_{b_{1},br'}) =\displaystyle \sum_{v \in L(\tau(b_{1}))}(r')^{\ast}(v)\varphi^{-1}(D_{b_{1},bv})
\sum_{c = 1 , 2}\Gamma_{\phi^{-}}^{b c}( p ) [ \delta^{c a}- L^{c a}( p ) G{( - 1 )}^{a + 1}] = i \frac{G}{2}\delta^{b a}{( - 1 )}^{( a + 1 )}, b , a = 1 , 2 ,
Aw_{q}=Q\left[ a(x)\left\{(t\phi_{1}+w)^{q}\left( \log(t\phi_{1}+w)+\frac{qw_{q}}{t\phi_{1}+w}\right) -w_{q}\right\}\right], & Aw_{t}=Q\left[ a(x)\left\{q(t\phi_{1}+w)^{q-1}(\phi_{1}+w_{t})-(\phi_{1}+w_{t})\right\}\right] .
\begin{array}{rcl}\Leftrightarrow x^{(1)}{}_{(2)}\otimes x^{(2)'}\widehat{x^{(1)}{}_{(1)}}(a) = x_{(2)}{}^{(1)}\otimes x_{(1)}(a) x_{(2)}{}^{(2)'}\Leftrightarrow x^{(1)}{}_{(2)}\otimes x^{(2)'}<x^{(1)}{}_{(1)},a_{(1)}>a_{(2)}= x_{(2)}{}^{(1)}\otimes x_{(1)}(a) x_{(2)}{}^{(2)'}\end{array}
d\omega_{\pm}=\omega_{\pm}^{\prime}(I_{\pm})dI_{\pm}\backsim dI_{\pm}, \frac{d}{d\omega_{\pm}}=\frac{dI_{\pm}}{d\omega_{\pm}}\frac{d}{dI_{\pm}}=\frac{1}{\omega_{\pm}^{\prime}(I_{\pm})}\frac{d}{dI_{\pm}}\backsim\frac{d}{dI_{\pm}},
\left\langle \xi ( z ) \frac{1}{2}c \partial c \partial^{2}c ( w ) e^{- 2 \phi ( y )}\right\rangle = \langle 0 | \xi_{0}c_{1}c_{0}c_{- 1}e^{- 2 \phi ( 0 )}| 0 \rangle = 1 .
\left\lVert F \right\rVert_{X,s}^2 := \left\lVert F\right\rVert^2_{X,0}+ \left\lVert\left(D^{\mathrm{s}}\right)^s F\right\rVert^2_{X,0}\left\lVert G \right\rVert_{X,s}^2 := \left\lVert G\right\rVert^2_{X,0}+ \left\lVert\left(D^{\mathrm{c}}\right)^s G\right\rVert^2_{X,0}
\int_{\mathbb{R}^n}\prod_{j_1=1}^\infty(1+4\lambda_{j_1}^2)^{-\frac14}dp \leq c\,\int_{\mathbb{R}^n}e^{-c_{H_1}\max_{1\leq j\leq n}|s_j-s_{j-1}||\xi_j|^{\frac1{H_1}}}d\xi &\leq c\,\int_{\mathbb{R}^n}\left(e^{-c_{H_1}\sum_{j=1}^n|s_j-s_{j-1}||\xi_j|^{\frac1{H_1}}}\right)^{\frac1n}d\xi &\leq c_{H_1,n}\prod_{j=1}^n|s_j-s_{j-1}|^{-H_1}.
s_{m,\mu}(x) =(-i)^{m-1}x^{(m-1)/2}h_{m-1,\mu+1}((2\sqrt{x}i)^{-1}) &=(-1)^{m-1}(\mu+1)_{m-1}{\;}_2 F_3 \left({-(m-1)/2,-(m-2)/2 \atop \mu+1,-m+1,1-m-\mu}\Big{|}4x \right) &=(-1)^{m+1}\sum_{j=0}^{\lfloor (m-1)/2 \rfloor}\binom{m-j-1}{j}(\mu+j+1)_{m-2j-1}x^{j}.
H(t,\xi)=E\bigg(\varepsilon_0+\frac{\varepsilon_0\eta_*}{t^2}\bigg)+C_0^{1/3}\sum_{|k|> \eta_*/(2T_0)}H(t,k,\xi),H(t,k,\xi)=\sum_{q\neq 0}e^{-2|k-q|}\frac{\varepsilon_0\eta_*}{(\xi-tq)^2+q^2}.
b_{j}= \left( \sinh(2\alpha x_j)-2\alpha h_j - 2\tanh(\alpha x_{j-1})(\cosh^2(\alpha x_j)-\alpha^2 h_{j}^2) \right) / F_{j-1}&-\left( \sinh(2\alpha x_j)+2\alpha h_{j+1}- 2\tanh(\alpha x_{j+1})(\cosh^2(\alpha x_j)-\alpha^2 h_{j+1}^2) \right) / F_{j}= \dfrac{h_{j}+h_{j+1}}{3}+O\left((\alpha \overline{h})^2\right); \alpha\to 0.
\,\cos^3\alpha\,\bigg(\!(1+\Delta/R)^3 \bigg(1 + \frac{3}{2}\tan^2\alpha\,(\Delta/R)\big(2 + O(\Delta/R)\big)+ \tan^4\alpha\,O(\Delta^2/R^2)\!\bigg) - 1\bigg) & = \cos^3\alpha\,\Big(\!(1+\Delta/R)^3 -1 + (1+\Delta/R)^3(\Delta/R)\tan^2\alpha\,\big(3 + O(\Delta/R) + \tan^2\alpha\,O(\Delta/R)\big)\!\Big).
n^{\epsilon_1}< \sum_{j=1}^{N_n^+}\Delta_{\tau,j}< n^{\epsilon_1}+ 2n^{\frac{1}{2}(\epsilon_1 + \epsilon_2)}~~~\mbox{and}~~~n^{\epsilon_1}- 2n^{\frac{1}{2}(\epsilon_1 + \epsilon_2)}< \sum_{j=1}^{N_n^-}\Delta_{\tau,j}< n^{\epsilon_1}.
\begin{aligned}\lambda_{ni}^{(1)}= \langle \psi^{(0)}_{n}, \Lambda_{n}^{(1)}\psi^{(0)}_{n}\rangle ~~~~~~~~\psi^{(0)}_{n}\in P_{nij}^{(2)}\mathcal{H}\subseteq P_{ni}^{(1)}\mathcal{H}\lambda_{nij}^{(2)}= \langle \psi^{(0)}_{n}, \Lambda_{ni}^{(2)}\psi^{(0)}_{n}\rangle ~~~~~~~~\psi^{(0)}_{n}\in P_{nij}^{(2)}\mathcal{H}\end{aligned}
\int_{-1}^{1}C_{n}^{\lambda}(t)C_{m}^{\lambda}(t)\omega_\lambda(t)\, dt=\gamma_{n}^{\lambda}\delta_{nm}, \gamma_{n}^{\lambda}=\frac{\pi 2^{1-2\lambda}\Gamma(n+2\lambda)}{n!\left(n+\lambda\right)\Gamma^2(\lambda)}.
\big|A_N\bigl(e^{i\lambda}\bigr)\bigl(1-e^{i\lambda\mu}\bigr)^{n}\beta^2f^0(\lambda)-\lambda^{2n}C_{\mu,N}^{\beta,0}\bigl(e^{i\lambda}\bigr)\big|^2 &=\alpha_2\lambda^{2n}\gamma(\lambda)\big|1-e^{i\lambda\mu}\big|^{2n}\bigl(p^0(\lambda)\bigr)^2,
\frac{d}{dx}\left(x^2 \frac{\theta_2'(x)}{\theta_2(x)}\right) = -\frac{1}{2}- \, \underbrace{\frac{d}{dx}\left(\frac{\theta_4'(\tfrac{1}{x})}{\theta_4(\tfrac{1}{x})}\right)}_{> 0}< 0.
\begin{array}{l c r}{{M^{+{\frac{n ( n + 1 )}{2}}}= ( \eta^{+})^{{\frac{n ( n + 1 )}{2}}}= N^{+{\frac{n ( n + 1 )}{2}}}}}\\{{S^{+ ( n + 1 )}= ( \eta^{+})^{n + 1}.}}\\ \end{array}
W_{\mathcal{C}}^r = \frac{1}{q^{n-k}}\sum_{j=0}^n W_{\mathcal{C}^{\perp}}^j\sum_{i=0}^r(-1)^i\binom{n-j}{r-i}\binom{j}{i}(q-1)^{r-i}, r = 0,1,\cdots,n
\Sigma_{\pm}(q_3,\left|{\bf q}\right|) = - \frac{1}{4\pi}\ln \left[ \frac{1}{2m}\left( -\frac{1}{2}\delta \pm iq_3 + \frac{\left|{\bf q}\right|^2}{2m}\right) \right] \, + \, O\left(\frac{1}{m^2}\right).
\mathbb E \bigl[(\xi_i-\eta_i)^3 \bigr] &\quad{}=6\Biggl(\sum_{j=1}^{k_1}p_{1,j}\,\mu_{1,j}^3 -\mu_2\sum_{j=1}^{k_1}p_{1,j}\,\mu_{1,j}^2+\mu_1\sum_{j=1}^{k_2}p_{2,j}\,\mu_{2,j}^2 -\sum_{j=1}^{k_2}p_{2,j}\,\mu_{2,j}^3 \Biggr).
Q_g =-\frac{1}{6}\Delta_gR_g - \frac{1}{2}|\mathrm{Ric}_g|^2_g + \frac{1}{6}R_g^2, P_gu =\Delta^2_gu + \mathrm{div}_g\left(\left( 2\mathrm{Ric}_g - \frac{2}{3}R_g\:g \right)(\nabla u,\cdot)\right),
\mathrm{E}_{\theta,\varepsilon^2}\|\theta-\hat{\theta}_{\mathrm{MA},\beta}\|^2= \mathrm{E}_{\theta,\varepsilon^{2}}\sum_{i=1}^{D}(\theta_{i}- \hat{\theta}_{\mathrm{MA},\beta, i})^{2}+ \mathrm{E}_{\theta,\varepsilon^{2}}\sum_{i=D+1}^{\infty}(\theta_{i}- \hat{\theta}_{\mathrm{MA},\beta,i})^{2}.
S(x) :={\sum_n}^* \bigl|e(\lambda_1 n)\Delta_1(\log(n+\alpha))+\cdots+e(\lambda_m n)\Delta_m(\log(n+\alpha))\bigr|^2 &=S_1(x) + S_2 (x) \gg \frac{e^{x(1-\sigma_2)}}{x^{2m}}\sum_{j=1}^m \bigl|\Delta_j(x^{j}_0)\bigr| + O\left(e^{-2\sigma_1 x}\right)
\alpha_2 = (y_1+y_2)^{n_2}y_2 &= (y_1+y_2)^{n_2-1}(y_1+y_2)y_2 &= (y_1^{n_2-1}+y_2^{n_2-1})(y_1+y_2)y_2 ~(\mbox{as}~ \binom{n_2-1}{i}~\mbox{is even for}~ 0<i<n_2-1) &= (y_1+y_2)y_2^{n_2}~( y_1^{n_2-1}=0, ~ \mbox{since}~ n_2-1 \geq n_1+1 ~\mbox{and}~ y_1^{n_1+1}= 0) &= y_1y_2^{n_2}+ y_2^{n_2+1}.
\begin{cases}dQ^{i,N}_t=P^{i,N}_t\,dt, dP^{i,N}_t=- \beta\, Q^{i,N}_t\,dt-A(Q^{i,N}_t)\,dt-\frac{1}{N}\sum_{j=1}^N B(Q^{i,N}_t-Q^{j,N}_t)\,dt+\lambda\,Z^{i,N}_t\,dt, dZ^{i,N}_t=-\lambda\,P^{i,N}_t\,dt-\alpha\, Z^{i,N}_t\,dt+\sqrt{2}\,dW^{i}_t, (Q^{i,N}_0,P^{i,N}_0,Z^{i,N}_0)=(Q^{i}_0,P^{i}_0,Z^{i}_0).\end{cases}
\Lambda\equiv\int\limits_{-\infty}^{\infty}e^{a_0 x_{+}^{5/4}}|u_1(x,0)-u_2(x,0)|^2\,dx= \int\limits_{-\infty}^{\infty}e^{a_0 x_{+}^{5/4}}|u_{01}(x)-u_{02}(x)|^2\,dx <\infty,
( \varphi_{X,Y}^{\sharp}F )_\beta= \sum_{\lambda \in \Lambda_{n,m}(\beta)}\frac{\beta !}{\lambda !}F_{\sum_{\alpha}\lambda_\alpha}\circ \varphi \prod_{\alpha}\frac{\left(\partial^\alpha \varphi\right)^{\lambda_\alpha}}{(\alpha!)^{\sum_i \lambda_{i,\alpha}}},
\displaystyle{\frac{m_{\pi}^2f_{\pi}j_{\pi}(q^2)}{m_{\pi}^2-q^2}= \partial^{\mu}J_{\mu}(q) =\frac{q^2f_{\pi}j_{\pi}(q^2)}{m_{\pi}^2-q^2}+\partial^{\mu}\widehat J_{\mu}(q).}
P_{\beta,h}^{+,\pm}(\pm \mid \omega^{-1}_{-\infty}) := \lim_{n \rightarrow \infty}\lim_{l\rightarrow \infty}\nu_{\beta,h}^+ \left( \eta_0=\pm \mid \eta^{-1}_{-(n+l)}= \pm^{-(n+1)}_{-(n+l)}\omega^{-1}_{-n}\right)
a = \frac{l-n^2/l}{2n}= \frac{l^2-n^2}{2ln}, a' = \frac{l+n^2/l}{2n}= \frac{l^2+n^2}{2ln}, b = \frac{m-n^2/m}{2n}= \frac{m^2-n^2}{2mn}, b' = \frac{m+n^2/m}{2n}= \frac{m^2+n^2}{2mn},
1-\pi_{11}> (1 - \bar\alpha_1)(1 - \bar\beta_1),1 - \pi_{12}= (1 - \bar\alpha_1)(1 - \bar\beta_2), 1-\pi_{21}= (1 - \bar\alpha_2)(1 - \bar\beta_1)\quad\mbox{and}1 - \pi_{22}> (1 - \bar\alpha_2)(1 - \bar\beta_2).
N_{f}=\frac{1}{C(\Delta\sigma)^{m}(\sqrt{\pi})^{m}}\int_{a_{0}}^{a_{c}}\frac{da}{a^{\frac{m}{2}}[0.265[1-\frac{a}{W}]^{4}+\frac{0.857+0.265\frac{a}{W}}{[1-\frac{a}{W}]^{\frac{3}{2}}}]^{m}}
\int_{\Omega_T}v\cdot\left([u_n\cdot\nabla]u_n-[u\cdot\nabla]u\right)\,dx=\int_{\Omega_T}v\cdot([(u_n-u)\cdot\nabla]u_n)\,dx+\int_{\Omega_T}v\cdot([u\cdot\nabla](u_n-u))\,dx, &\int_{\Omega_T}v\cdot(\phi_n\nabla\mu_n-\phi\nabla\mu)\,dx=\int_{\Omega_T}v\cdot\nabla\mu_n(\phi_n-\phi)\,dx+\int_{\Omega_T}v\phi\cdot(\nabla\mu_n-\nabla\mu)\,dx