harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\gamma_1 =\{(E^u_0(\lambda\cdot t_0)),E^s_0(-\lambda\cdot t_0)\}_{\lambda\in I},\gamma_2=\{(E^u_\lambda(t_0),E^s_\lambda(-t_0))\}_{\lambda\in I}, \gamma_3 =\{(E^u_1((1-\lambda)t_0), E^s_1(-(1-\lambda)t_0))\}_{\lambda\in I}.
(I-\mathcal{H})A= 1+i[D_t\zeta,\mathcal{H}]\frac{\partial_{\alpha}\mathfrak{F}}{\zeta_{\alpha}}+i[D_t^2\zeta,\mathcal{H}]\frac{\bar{\zeta}_{\alpha}-1}{\zeta_{\alpha}}-(I-\mathcal{H})\frac{1}{2\pi}\sum_{j=1}^2 \frac{\lambda_j(D_t\zeta(\alpha,t)-\dot{z}_j(t))}{(\zeta(\alpha,t)-z_j(t))^2}.
\frac{B^{'^{\prime}}}{A B}- \frac{B^{'}}{2 A B}\left( \frac{A^{'}}{A}+ \frac{B^{'}}{B}\right) + \frac{2}{r A}\left( \frac{B^{'}}{B}- \frac{A^{'}}{A}\right) + \frac{2}{A r^{2}}- \frac{2}{r^{2}}= 0 .
\displaystyle v_{t}+\frac{1-{p}^{\prime}/c^{2}}{1-{p}^{\prime}v^{2}/c^{4}}vv_{r}+\frac{(1-v^{2}/c^{2})^{2}{p}^{\prime}}{q(1-{p}^{\prime}v^{2}/c^{4})}\rho_{r}=\frac{4\pi n(\sqrt{1-v^{2}/c^{2}})^{3}}{q(1-{p}^{\prime}v^{2}/c^{4})}\phi_{r}+\frac{2(1-v^{2}/c^{2})v^{2}{p}^{\prime}/c^{2}}{r(1-{p}^{\prime}v^{2}/c^{4})}
\langle P,P\rangle=P^{\alpha}\eta_{\alpha\beta}P^{\beta}=(\begin{matrix}\frac{E}{c}&p_{x}&p_{y}&p_{z}\end{matrix})(\begin{matrix}-1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\\ \end{matrix})(\begin{matrix}\frac{E}{c}\\p_{x}\\p_{y}\\p_{z}\end{matrix})=-(\frac{E}{c})^{2}+p^{2},
\log P(n-a_n-S_n(n-h(n))\geq\lceil\varepsilon f_4(n)\rceil) =\log P(\mathrm{Bin}(n-a_n,1-\pi_n(n-h(n)))\geq\lceil\varepsilon f_4(n)\rceil) &\sim_e\varepsilon f_4(n)\log(b_c^{(n)'}/f_4(n)).
\Re \left\{\frac{-\nu^{-s}}{\pi}\, e^{\displaystyle{- i \frac{\pi}{2}(s+1)}}\int_{- i z}^{\infty- i z}t^{-s}\, \displaystyle{\frac{d\left(\ln{I_\nu\left(\nu\, t\right)}\right)}{d\, t}}\, dt \right\}.
[y_1, y_1]=y_4, [y_1, y_2]=y_4+\theta_1y_5, [y_2, y_1]=\frac{\alpha_5}{\alpha_3}y_4+\theta_2y_5, [y_1, y_3]=\frac{\alpha_3\beta_4}{\alpha_1\gamma_4}y_5, [y_2, y_3]=\frac{\gamma_2}{\gamma_4}y_5, [y_3, y_2]=y_5.
g_{x}\left( s\right) ={\textstyle\sum\limits_{n=1}^{\infty}}TP_{x=ne}s^{n}e={\textstyle\sum\limits_{n=1}^{\infty}}TP_{x=ne}s^{x}& =T\left[ \left({\textstyle\sum\limits_{n=1}^{\infty}}P_{x=ne}\right) s^{x}\right] =Ts^{x}.
L^{\vee}(x_i|t_j)=\left( \begin{array}{cc}\sigma^{+}\sigma^{-}+x_{i}\ominus t_{j}~\sigma^{-}\sigma^{+}\sigma^{+}(1+\beta x_{i}\ominus t_{j})\sigma^{-}\sigma^{-}\sigma^{+}\end{array}\right)
\sum_{i=1}^q m_{L_i,S}(P)=\sum_{v\in S}\sum_{i=1}^q h_{L_i,v}(P)\leq \sum_{v\in S}\sum_{i=1}^2 h_{L_{i,v},v}(P)+\sum_{v\in S}h_{L,v}(P)+O(1)
\Sigma(\kappa;\eta,P) =-i e^{\pi i \kappa}\sum_{T<\gamma \le 2T}\zeta'(1-\rho) \, \zeta\Big(\rho+\frac{2\pi i \kappa}{\log T}\Big)\, M\Big(1-\rho-\frac{2\pi i\eta}{\log T},P\Big) \, M\Big(\rho+\frac{2\pi i\eta}{\log T},P\Big) &+O\big(T\log T\big) &= -i e^{\pi i \kappa}\, (\log T) \, I(2\pi i\kappa,0,g_1,g_2,1,-x)+O\big(T\log T)
\| F \|_{\infty,\epsilon}\leq\sum_{|\alpha |\leq \sigma}\frac{1}{\alpha !}A_{\epsilon,|\alpha |}|F_{\alpha}(0)|+\sum_{|\alpha |=\sigma+1}\frac{1}{\alpha !}B_{\epsilon, \sigma+1}\| F_\alpha \|_{\infty,\epsilon},
\int_0^1 e^{b(t^{m-1}- 1)}(1 - \sqrt{t})dt \leq \int_0^1 e^{\l b(t - 1)}(1 - t)dt & = \frac{1}{\l^2b^2}- \left(\frac{1}{\l b}+\frac{1}{\l^2b^2}\right) e^{-\l b}= O\left(\frac{1}{b^2}\right) \quad~b\rightarrow\infty.
f(\zeta)=\zeta+2+3\pi\mathrm{i}+\boldsymbol L_1^{-1}+\zeta^{-1}\boldsymbol L_2^{-1}\boldsymbol L_3+ 4\zeta^{-1}\boldsymbol L_1^{-1}\boldsymbol L_2^{-1}\boldsymbol L_3+ &+\zeta^{-2}+\mathrm{e}^{-\zeta}\boldsymbol L_1^2+\mathrm{e}^{-2\zeta},
\begin{cases}^{C}_{0}D_{t}^{q}\varphi(x,t)= A \varphi(x,t) \hbox{in}Q_{T}\varphi(\xi,t) = 0 \hbox{on}\Sigma_{T}\varphi(x,0) = \varphi_{_{0}}(x) \hbox{in}\Omega.\end{cases}
\begin{aligned}Y(t,\zeta) := \Xi(t,\zeta) \Psi(t) + t (Z \Xi(t,\zeta) + \Xi(t,\zeta) Z^*)(t^3 K + \Psi(t)) & - t^3 \Xi(t,\zeta) N \Xi(t,\zeta) (t^3 K + \Psi(t)) + \mathcal{J}(t,\zeta) \left( t^2 SP + t^3 K + \Psi (t) \right).\end{aligned}
l^{j}- \theta^{j i}\hat{F}_{i i^{\prime}}l^{i^{\prime}}= i \theta^{j j^{\prime}}( i k_{j^{\prime}}+ i \partial_{j^{\prime}}\hat{A}_{i^{\prime}}l^{i^{\prime}}) + \theta^{j j^{\prime}}l^{i^{\prime}}D_{i^{\prime}}\hat{A}_{j^{\prime}},
Q(z; \tau, \sigma) = \frac{z^3}{3\tau \sigma}- \frac{\tau + \sigma - 1}{2 \tau \sigma}z^2 + \frac{\tau^2 + \sigma^2 + 3 \tau \sigma - 3 \tau - 3 \sigma + 1}{6 \tau \sigma}z + \frac{1}{12}(\tau + \sigma - 1)(\tau^{-1}+ \sigma^{-1}- 1).
\sum_{i\in I_1}\left(|\tilde{\xi}^{(k,\ell)}_{i,1}(t)|^2+|\tilde{\xi}^{(k,\ell)}_{i,2}(t)|^2 \right)+ \sum_{i\neq 0}|\tilde{\xi}^{(k,\ell)}_i(t)|^2=\|\tilde{q}^{(k,\ell)}\|^2_G(t) \leq C \|\tilde{u}^\perp\|^2_{C^{s+1}}(t).
\frac{\Gamma(s+w+\frac{k-1}{2})}{\Gamma(s+\frac{k-1}{2})}\left(\frac{x}{2\pi}e^{-i\frac{\pi}{2}{\rm sgn\;}t}\right)^w &\ll\begin{cases}|t|^{\frac{1}{2}-\sigma-\frac{k-1}{2}}(1+|t+v|)^{\sigma+u-\frac{1}{2}+\frac{k-1}{2}}(x/2\pi)^u, w\in L_{\pm 1, \pm 2}, |t|^u(x/2\pi)^u, w\in\mathcal{F}\end{cases}
\mathbf r_m = \sum_{n\in\mathcal N}\alpha_n g_n^{\frac{1}{2}}\mathbf H_{n,m}\mathbf s_n + \mathbf n_m = \sum_{n\in\mathcal N}\alpha_n g_n^{\frac{1}{2}}\mathbf H_{n,m}\mathbf{F}^{H}\tilde{\mathbf s}_n + \mathbf n_m, m\in\mathcal{M},
P(\lambda^\prime)K^\pm_{\lambda,\nu}(x^\prime,x_n) =(\lambda+\nu-n)(-2\lambda^\prime+\lambda+\nu+1)K^\mp_{\lambda-1,\nu}(x^\prime,x_n) -2\nu(2\lambda+2\lambda^\prime-2n)K^\mp_{\lambda,\nu+1}(x^\prime,x_n).
d s^{2}= R^{2}\left[ ( 1 + b^{2}-{\frac{2 M G_{5}}{R^{2}b^{2}}}) d t^{2}- ( 1 + b^{2}-{\frac{2 M G_{5}}{R^{2}b^{2}}})^{- 1}d b^{2}- b^{2}d \Omega^{2}\right]
\chi = \frac{C_{F}g^{2}}{8 \pi}\Biggl ( 1 - \frac{g^{2}}{8 \pi^{2}}\biggl [ \frac{1 1 C_{A}- 2 n_{f}}{3}\Biggl ( \frac{1}{n - 4}+ l n \; \frac{m^{2}}{4 \pi \nu^{2}}\Biggr ) + \Phi + O ( g_{0}^{2}; n - 4 ) \biggr ] \Biggr ) ,
\langle \phi \mid \eta , \psi\rangle = \int_{F}\phi(\tau,lz) \overline{\psi(\tau,z)}e^{- 4 \pi m l^2 y^2/v}v^k dV &= l^{-2}\int_{F'}\phi(\tau,z) \overline{\psi(\tau,z/l)}e^{- 4 \pi m y^2/v}v^k dV &=l^{-2}\int_{F'}\phi(\tau,z) \overline{\psi(\tau,z) \mid \eta^{-1}}e^{- 4 \pi m y^2/v}v^k dV,
\ln G_\beta(x) = \frac{x-Q/2}{2}\ln (2 \pi) + \int_0^\infty \frac{dt}{t}\left( \frac{e^{- \frac{Q}{2}t}- e^{- x t}}{(1-e^{-\beta t})(1-e^{-t/\beta})}+\frac{e^{-t}}{2}(Q/2-x)^2 + \frac{Q/2-x}{t}\right)
\int_{|y| > t^{\beta/\alpha}}\left|\nabla I_{A}\right|dy \le C \int_{|y| > t^{\beta/\alpha}}t^{1 + q}|y|^{-d + d - 1 - 1 - q\alpha/\beta - \alpha/\beta}dy = C t^{1 + q - \beta/\alpha - q - 1}= C t^{-\beta/\alpha}.
\displaystyle\lim_{\varepsilon\to 0}\int_{0}^{T}\,\int_{\Omega}\,f_{j}(u^{\varepsilon})\,\frac{\partial \phi}{\partial x_{j}}\,dx\,dt = \int_{0}^{T}\,\int_{\Omega}\, f_{j}(u)\,\frac{\partial\phi}{\partial x_{j}}\,dx\,dt
q_i = \left(\sum \limits_{j=i}^{n}(-1)^{j-i}\prod\limits_{k=j+1}^{n}a_k\prod\limits_{k=1}^{i-1}a_k + \sum\limits_{j=1}^{i-1}(-1)^{n+j-i}\prod\limits_{k=j+1}^{i-1}a_k\right)\bigg/\left(\prod\limits_{k=1}^n a_k+(-1)^{n+1}\right).
(a(1),a(2),\ldots) = (b_1+2,b_1+2\cdot 2,\ldots,b_1+2 \cdot d_1,b_2+2^2,b_2+2^2\cdot 2,\ldots,b_2+2^2 \cdot d_2,b_3+2^3, b_3 + 2^3 \cdot 2,\ldots)
d\hat{X}= \pi_t \left[ -i \left[X_t,H_t\right] + \mathcal{L}_L(X_t)\right] dt+ \beta_t^{\top}\left[d\mathbf{Y}_t - \pi_t \left(d\mathbf{Y}_t\right)\right], = \pi_t \left[ -i \left[X_t,H_t\right] + \mathcal{L}_L(X_t)\right] dt+ \beta_t^{\top}d\mathbf{W}_t.
\begin{array}{l}\mathcal V=\Bigl\{v(\cdot,\cdot)\in C([0,t_f]\times \overline{D})\Bigl|v(t,x)>S\ast (\frac{\kappa^2}{4}|v|^r-\frac{c_1^2+c_2^2}{2}v)(t,x)+I(t,x)\Bigr \},\end{array}
f_{T T T}= -{\frac{2 i}{\pi}}\frac{j_{T}( T )}{j ( T ) - j ( U )}\left\{\frac{j ( U )}{j ( T )}\right\}\left\{\frac{j_{T}( T )}{j_{U}( U )}\right\}\left\{\frac{j ( U ) - j ( i )}{j ( T ) - j ( i )}\right\}.
\left( \begin{array}{c c}{{\hat{h}+ \Sigma}}&{{\beta \Delta^{\dagger}\beta C}}\\{{- C \, \Delta}}&{{\hat{h}+ \Sigma}}\\ \end{array}\right) \left( \begin{array}{c}{{u}}\\{{v}}\\ \end{array}\right) = E \left( \begin{array}{c}{{u}}\\{{v}}\\ \end{array}\right)
\exp \left\{- \Gamma_{\mathrm{W}}[ \tilde{A}] \right\}\propto \prod_{k}\frac{\lambda_{k}}{\lambda_{k}+ i \Lambda_{0}}\left( \prod_{l = 1}^{\infty}\frac{\lambda_{k}^{2}+ M_{l}^{2}}{\lambda_{k}^{2}+ ( M_{l}+ \Lambda )^{2}}\right) ,
{\cal L}=\frac{1}{2}\rho x_t^2-\varepsilon(\rho,S),{\cal L}_0{\bf (x,m,t)}=x_m{\cal L}=\frac{1}{2}x_t^2-F(x_m,m),F(x_m,m)=\frac{\varepsilon(\rho,S)}{\rho}.
N^{2k+1}T_V^{(k)}(e_{q/N,1})\sim \frac{N^{2k+1}}{(e_{q/N,1})^{k+\frac{1}{4}}}\bigg(\begin{smallmatrix}\!-\frac{1}{2}\!~~k\end{smallmatrix}\bigg)\int_0^{+\infty}\frac{dx}{(1+x^4)^{k+1/2}}~~{q}/{N}=T_{V}(e_{q/N,1})\sim\frac{1}{e_{q/N,1}^{{1}/{4}}}\int_0^{+\infty}\frac{dx}{(1+x^4)^{1/2}}.
(1 + z^{-1}\sum\limits_{n,l=0}^{\infty}z^{-n}q\bar{x}^{n}x^{l}p)^{-1}=(1 - z^{-1}\sum\limits_{n,l=0}^{\infty}z^{-n}qx^{l}\bar{x}^{n}p)^{-1}.
\lambda_1 : w_1\mapsto -w_1,\w_2\mapsto -w_2,\w_3\mapsto \frac{1}{w_3}, -I_3 : w_1\mapsto w_1,\w_2\mapsto w_2,\w_3\mapsto \frac{1}{w_3}, \rho_2=\rho_3 : w_1\mapsto -w_1,\w_2\mapsto w_2,\w_3\mapsto -w_3.
\Pi_{i}^{a}\left({\bf x}\right) \left| \Omega \right\rangle = \overline{\psi}\left({\bf y}\right) U ({\bf y},{\bf 0}) \psi \left({\bf 0}\right) \Pi_{i}^{a}\left({\bf x}\right) \left| 0 \right\rangle + g T^{a}\int_{\bf 0}^{\bf y}{d z_{i}}\delta \left({{\bf x}-{\bf z}}\right) \left| \Omega \right\rangle .
a(n) \hat{b}(n-1) + \hat{a}(n) b(n) +{\hat{A}}(n) +{\hat{C}}(n) = d(x){\hat{d}}(x)- \hat{\Lambda}( x), a(n) \hat{b}(n-1) + \hat{a}(n-1) b(n-1) + A(n) + C(n) ={\hat{d}}(x) d(x)- \Lambda ( x).
\begin{aligned}00000 \longleftrightarrow 000000011 &00001 \longleftrightarrow 000000101 &00011 \longleftrightarrow 000001001 &\vdots \quad\quad\quad\ \vdots \quad\quad\quad\vdots &11111 \longleftrightarrow 100000001.\end{aligned}
-(p-1) \int_M |\nabla u|^{p-1}g^{ij}\nabla_i u \nabla_j u_t d\mu = (p-1) \int_M g^{ij}\nabla_i( |\nabla u|^{p-1}\nabla_i u ) u_t \displaystyle = (p-1) \int_M \Delta_p u u_t d\mu
\mu ={\frac{2 \pi \epsilon \, ( 4 - 5 \epsilon + 2 \epsilon^{2})}{( 1 - \epsilon ) \, ( 2 + \epsilon - 2 \epsilon^{2})}}\, , \hspace{0 . 5 c m}\alpha ={\frac{\epsilon \, ( 1 - \epsilon )}{( 1 - \epsilon + \epsilon^{2})^{3 / 2}}}\, .
0= (m_{1}+m_{2})(\cot \vartheta)^{4}-\left( \frac{3}{2}(m_{1}+n)+8m_{2}\right)(\cot \vartheta)^{2}+(m_{2}+n) = ((m_{2}+m_{2})(\cot \vartheta)^{2}-(m_{2}+n))((\cot \vartheta)^{2}-1)-\left( \frac{m_{1}+n}{2}+6m_{2}\right)(\cot \vartheta)^{2}.
\begin{array}{l l}{{g_{\mu \nu}= \eta_{\mu \nu}+ h_{\mu \nu},}}\\{{\Lambda ( \phi ) = \Lambda ( \phi_{0}) + \Lambda^{\prime}( \phi_{0}) \phi_{( 1 )},}}\\{{T_{\mu \nu}= T_{( 0 ) \mu \nu}+ T_{( 1 ) \mu \nu},}}\\{{\phi = \phi_{0}+ \phi_{( 1 )},}}\\ \end{array}
F_{t,\varsigma}\big(\sqrt{t}D^F_{Z_{j,R}}\big)(x,y), G_{t,\varsigma}\big(D^F_{Z_{j,R}}\big)(x,y) & \in\left(\Lambda^\bullet\big(T^*Z_{j,R}\big) \otimes F\right)_x \otimes\left(\Lambda^\bullet\big(T^* Z_{j,R}\big) \otimes F\right)^*_y, x,y\in Z_R,
- \left. \frac{i}{4}\mathrm{T r}\log \widehat{\Omega}^{2}\right|_{d i v}= \left. \frac{i}{4}\mathrm{T r}\log \left( - \Delta +{\frac{R}{4}}\hat{1}\right) \right|_{d i v}= -{\frac{1}{2 \epsilon}}\int d^{2}x \sqrt{g}{\frac{n}{1 2}}R \ .
W_{\mathrm{t r e e}}={\frac{1}{3}}h_{\alpha \beta \gamma}( M^{i}) Q^{\alpha}Q^{\beta}Q^{\gamma}+{\frac{1}{3}}h_{i \alpha \beta}( M^{i}) Q_{\mathrm{h i d}}^{\alpha}Q_{\mathrm{h i d}}^{\beta}.
B\, =\,\frac{1}{2}dz^M dz^N B_{NM}=\frac{1}{2}E^A{E_A}^M\,E^B\,{E_B}^N\,B_{NM}\,=\, (-)^{b(m+a)}\frac{1}{2}E^AE^B \,{E_A}^M{E_B}^N B_{NM},
\Phi_{1, t, \alpha}(f)(x_1, x_2) = \lim_{\epsilon \to 0}\frac{1}{d_{\alpha}}\int_{|y|\geq \epsilon}\frac{e^{it(\varphi(x_1 + y,x_2)- \varphi(x_1,x_2))}- 1}{|y|^{1 + \alpha}}\, f(x_1 + y, x_2)\, dy
\Big \| M_{\chi_{B(0,R)}}\Big ( C_{z_\gamma}(A P_\alpha + Q_\alpha) C_{-z_\gamma}- (A_{x}P_\alpha + Q_\alpha) \Big ) \Big \| &= \Big \| M_{\chi_{B(0,R)}}P_\alpha ( C_{z_\gamma}A C_{-z_\gamma}- A_{x}) P_\alpha \Big \| & \to 0
v_{i}\circ h=(\overline{h_{i}})^{-1}\cdot v_{i}, \forall 1\leq i\leq r, v_{i}^{'}\circ h=h_{i}\cdot v_{i}^{'}, \forall 1\leq i\leq r, u\circ h=u, \forall u\in V^{R-r},
Y_{\kappa,\varepsilon}(t) :=\bigg\{\theta\in\mathbb{T}~ : ~ \cos(\theta-\phi(t))>\sqrt{1-\varepsilon{}_{\kappa}^{2}}-\varepsilon \bigg\}, \cr\tilde{Y}_{\delta}(t) :=\bigg\{\theta\in\mathbb{T}~ : ~ \cos(\theta-\phi(t))\leq-\delta \bigg\},
\| B \|_2 = \sqrt{\sum_{\stackrel{j :}{x_j < x_{1}}}\sum_{\stackrel{k :}{y_k > x_{1}+ \delta/2}}| \langle \delta_{x_j}, \rho(s_j,t_k) \delta_{y_k}\rangle |^2}\leq \sum_{\stackrel{j :}{x_j < x_{1}}}\sum_{\stackrel{k :}{y_k > x_{1}+ \delta/2}}\hat \rho( x_j,y_k) \, ,
H^{1}_{0, \mathrm{rot}}(\Omega) = \big\{u \in (H^{1}_{0}(\Omega))^{2}\,|\, \nabla u = (\nabla u)'\big\}=\big\{u \in (H^{1}_{0}(\Omega))^{2}\,|\, \mathrm{rot}u = 0 \big\}
\frac{d}{dr}W_{k,l}^-+\frac{(k+\frac{1}{2})}{r}W_{k,l}^-+ 2lW^+_{k,l}+P^+_{k,l}(f)=0; \frac{d}{dr}W_{k,l}^+-\frac{(k+\frac{1}{2})}{r}W_{k,l}^++2lW^-_{k,l}+P^-_{k,l}(f)=0
\vartheta^{\Gamma}\cdot (\pi, \delta) = (\pi\circ \vartheta, (\vartheta^{\Gamma})^{-1}(\delta)) \leftrightarrow ({^\vee}\vartheta^{\Gamma}\cdot(y,\Lambda), \tau \circ{^\vee}\vartheta^{-1}) ={^\vee}\vartheta^{\Gamma}\cdot ((y,\Lambda), \tau).
\sum_{n=1}^{N}\left[\begin{matrix}N n\end{matrix}\right]\frac{q^{\frac{n(n+1)}{2}}(q)_{n}}{(1-cq^{n})(-q)_n}=\frac{1}{c}\sum_{n=1}^{N}\left[\begin{matrix}N n\end{matrix}\right]\frac{(-q/c)_{n-1}(q)_{n}(cq)_{N-n}(cq)^{n}}{(-q)_{n}(cq)_{N}}.
f_{\mathbf{M^{\ast}}_{-\xi_{\boldsymbol{s}k}}}(\boldsymbol{w})=c_{n-1,\xi_{\boldsymbol{s}k}}^{\ast}\left(1+\sqrt{\boldsymbol{w}^{T}\boldsymbol{w}+\xi_{\boldsymbol{s},k}^{2}}\right)\exp\left\{-\sqrt{\boldsymbol{w}^{T}\boldsymbol{w}+\xi_{\boldsymbol{s},k}^{2}}\right\},~\boldsymbol{w}\in \mathbb{R}^{n-1},
\frac{1}{24}\left(\frac{\pi}{\sqrt{r}}+12\frac{1}{K}\frac{dK}{dk_r}\frac{dk_r}{dr}+\frac{4}{k_r}\frac{dk_r}{dr}+\frac{4}{k'_r}\frac{dk'_r}{dr}\right)=2q^{-1}P^{*}(q^2)\frac{q\pi}{2\sqrt{r}}
\gamma'(t) G(t,\sigma(t)) + \gamma(t) \partial_t G(t,\sigma(t)) + \gamma(t) \sigma'(t) \partial_s G(t,\sigma(t)) = G(t,\sigma(t))\left(\gamma'(t) + \varphi(t) \gamma(t)\right) + \gamma(t)\partial_sG(t,\sigma(t)) \left(\sigma'(t) + q + \varphi(t)\sigma(t) \right) = 0.
{\cal V}= ( U^{\Lambda \Sigma}- 3 \bar{L}^{\Lambda}L^{\Sigma}) P_{\Lambda}^{x}P_{\Sigma}^{x}+ \frac 1 3 \bar{L}^{\Lambda}L^{\Sigma}h^{u v}\nabla_{u}P_{\Lambda}^{x}\nabla_{v}P_{\Sigma}^{x}\, ,
I_{31}= \int_{0}^{2\pi}\log\left(\sin^{2}\left(\frac{x-y}{2}\right)\right)h(x)(-r\cos(y)\sin(x)+\cos(x)\sin(y))dy I_{32}= \int_{0}^{2\pi}\log\left((1+r^2)+(r^2-1)\cos(x+y))\right)h(x)(-r\cos(y)\sin(x)+\cos(x)\sin(y))dy
u(z) =1-\alpha e^{\lambda_1z}-\beta e^{\lambda_2z}-\gamma e^{\lambda_3z}+h.o.t., begin{align*}0.2cm] w(z) =0-\alpha \lambda_1 e^{\lambda_1z}-\beta \lambda_2 e^{\lambda_2z}-\gamma \lambda_3 e^{\lambda_3z}+h.o.t., begin{align*}0.2cm] v(z) =0-\alpha \psi\left(\lambda_{1}\right) e^{\lambda_1z}-\beta\psi\left(\lambda_{2}\right) e^{\lambda_2z}+h.o.t..
\omega^{1 2}= - i \{\frac{1}{2}d \ln ( \frac{d \overline{{g}}_{1}}{d \overline{{z}}_{1}}\frac{d z_{1}}{d g_{1}}) + \frac{( g_{1}d \overline{{g}}_{1}- \overline{{g}}_{1}d g_{1})}{( 1 - g_{1}\overline{{g}}_{1})}\},
\begin{aligned}-h^2\Delta_x\Phi-{\partial^2\Phi\over\partial z^2}=0, \ -D(x)<z<0\cr h^2{\partial^2\Phi\over\partial t^2}+g{\partial\Phi\over\partial z}=f^+(x,t) \ @ \z=0\cr{\partial \Phi\over\partial z}+h^2\langle\nabla_x D,\nabla_x\Phi\rangle=f^-(x,t) \ @ \z=-D(x)\cr\end{aligned}
= ( 2\nu^2 + 4 C^2_{\pi}) \int_{\Omega}| \varphi_{\varepsilon}(s)|^2 + (2 \nu^2 + 2 C_{\pi}^2) \int_{\Omega}|\nabla \varphi_{\varepsilon}(s) |^2+ 4 |\pi(0)|^2 |\Omega|\leq c ( \| \varphi_{\varepsilon}(s) \|^2_V + 1).
\prod_{(r,c) \in \mu}\prod_{j=1}^{u_\mu(r,c)}\frac{1- q^{\mu_r-c+1}t^{v_\mu(r)-j+1}}{1-q^{\mu_r-c+1}t^{v_\mu(r)-j}}=\prod_{(r,c) \in \mu}\frac{1- q^{\mu_r-c+1}t^{v_\mu(r)}}{1-q^{\mu_r-c+1}t^{v_\mu(r)-u_\mu(r,c)}}= \prod_{(r,c)\in \mu}\frac{1-q^ct^{v_\mu(r)}}{1- q^{\mu_r-c+1}t^{v_\mu(r)-u_\mu(r,c)}}.
a_h(\tilde u - u_h, v_h) = (-\Delta\tilde u + \tilde u, v_h)_{\Omega_h}+ (\partial_{n_h}\tilde u, v_h)_{\Gamma_h}- (\tilde f, v_h)_{\Omega_h}- (\tilde\tau, v_h)_{\Gamma_h}= \operatorname{Res}_u(v_h).
\begin{array}{l l l}{{d \left( \lambda x^{-}\right)}}&{{=}}&{{e^{- \lambda \sigma^{-}}\, d \left( \lambda \sigma^{-}\right) = - \lambda \left( x^{-}+ \Delta_{q}\right) \, d \left( \lambda \sigma^{-}\right)}}\\{{}}&{{=}}&{{{\frac{N}{4 8 \lambda x^{+}}}\, d \left( \lambda \sigma^{-}\right) \; .}}\\ \end{array}
(K + \mu) (D_1 \Lambda_{11}+ 1) = \underline{(K + \mu)}, \mu \left( \frac{1}{2}D_1 \Lambda_{22}+ 1 \right) = \underline{\mu}, (K + \mu) D_1 \Lambda_{13}+ K - \mu = \underline{(K + \mu)}\overline{\left(\frac{K - \mu}{K + \mu}\right)}.
\begin{array}{cc}\mathring{K}_5[H_0]=0 \textup{in}\ [2M,R_1]\cup [2R_1,\infty), \mathring{K}_5[H_0]\geq -\frac{3C_0}{(\log 2) r}|\nabla_L H_0|^2 \textup{in}\ [R_1,2R_1].\end{array}
\frac{1}{N}\widehat G_{2}\sum_{i=1}^N\sum_{j=1}^N\widehat\Gamma_i\widehat\Sigma_{ii}^{-1}\bigl(\widehat BM_{zz}^{ij}\widehat B{}'-\widehat\Sigma_{z z}^{i j}\bigr)\widehat\Sigma_{j j}^{-1}I_{K+1}^{1}\hat\psi_j'=0.
\delta_{t}(U_F)^{1}_{ij}-\mu\Delta_{h}(U_F)^{1}_{ij}- \mathfrak{w}_{1,1}\Delta_{h}(U_F)^{1}_{ij}=b^{1}_{ij}+\frac{1}{2}g((U_F)^{0}_{ij})+ \frac{1}{2}\Big[g((U_C)^1_{ij})+g'((U_C)^1_{ij})\left((U_F)_{ij}^1-(U_C)_{ij}^1\right)\Big], (x_i,y_j)\in\Omega_{h},
g(A)=|\bigcap_{i\in\underline{m}\backslash A}A_{i}|,g(\underline{m})=|\bigcup_{i\in\underline{m}}A_{i}|andg(A)=\mathbb{P}(\bigcap_{i\in\underline{m}\backslash A}A_{i}),g(\underline{m})=\mathbb{P}(\bigcup_{i\in\underline{m}}A_{i})
A\bigl((T_{a_{0}}\circ f_{0}) \times \cdots \times (T_{a_{r}}\circ f_{r})\bigr) =A(T_{a_{0}}\circ f_{0})\cup \cdots \cup A(T_{a_{r}}\circ f_{r}) &=A(f_{0})\cup \cdots \cup A(f_{r})=A(f).
\delta{\tilde{e}}_{\mu}^{\ \underline{\nu}}= - \bar{\tilde{\epsilon}}_{+ i}\tilde{\Gamma}^{\underline{\nu}}\tilde{\psi}_{\mu +}^{i}- 2 \kappa^{D - 2}W \bar{\tilde{\epsilon}}_{+ i}\tilde{\Gamma}^{\underline{{\nu \lambda}}}\tilde{\eta}_{-}^{i}\tilde{e}_{\mu \underline{\lambda}}+ c . c . \ .
C^{\mathrm{ul},k}_{2}\left( N_{\mathrm{sct}}, N_{\mathrm{scr}}\right) = \left(\frac{T_{\mathrm{ul}}}{T}\right) \min\left\lbrace c^k_{\mathrm{u2s},1}\left( N_{\mathrm{scr}}\right), \right. \,\,\,\,\,\, \left. c^k_{\mathrm{s2b},2}\left(N_{\mathrm{sct}}\right) \right\rbrace.
\frac{\partial{u_3}}{\partial{t}}=\frac{\partial}{\partial{t}}\Gamma_{k}^{(6)}[\varphi=0]=\frac{1}{2}{T}{r}\frac{\delta^6}{\delta\varphi_1(q=0)\cdots\delta\varphi_6(q=0)}\left\{(\Gamma_{k}^{(2)}[\varphi]+{R}_{k})^{-1}\frac{\partial}{\partial{t}}{R}_{k}\right\}\Bigg\vert_{\varphi=0}
G^{+}( x , x^{\prime}) = \frac{1}{\pi^{2}}\int \frac{d^{D - 1}{\mathbf k}}{( 2 \pi )^{D - 1}}\, e^{i{\mathbf k}({\mathbf x}-{\mathbf x^{\prime}})}\int_{0}^{\infty}d \omega \sinh ( \pi \omega ) e^{- i \omega ( \tau - \tau^{\prime})}Z_{i \omega}( \lambda \xi , \lambda a ) Z_{i \omega}^{*}( \lambda \xi^{\prime}, \lambda a ) .
\sum_{y\in\mathbb{F}_q}\chi_4((y-1)(y-a))=\sum_{y'\in\mathbb{F}_q}\chi_4(y'(y'+1-a)) &=\sum_{y''\in\mathbb{F}_q}\chi_4((1-a)y'')\chi_4((1-a)(y''+1))=\varphi(1-a)\sum_{y''\in\mathbb{F}_q}\chi_4(y''(y''+1)) &=\varphi(1-a)\sum_{y''\in\mathbb{F}_q}\chi_4(-y''(-y''+1)) &=\varphi(1-a)J(\chi_4,\chi_4),
\|\Psi^{n}(dd^{c}\varphi)\|_U =\left(\frac{d^{(k-1)m}}{N}\right)^n\int_Udd^{c}\varphi\wedge (\omega+d^{-nm}dd^{c}u_{nm})^{k-1}& \leq \left(\frac{d^{(k-1)m}}{N}\right)^n C'\|dd^c\varphi\|_W.
\|1_{\varepsilon A}\|^p \ =\ \left\|\frac{1}{2}(1_{\varepsilon A}+1_{B\backslash A})+\frac{1}{2}(1_{\varepsilon A}-1_{B\backslash A})\right\|^p &\ \le\ \frac{1}{2^p}\|1_{\varepsilon A}+1_{B\backslash A}\|^p + \frac{1}{2^p}\|(1_{\varepsilon A}-1_{B\backslash A})\|^p\ \le\ 2^{1-p}(\mathbf h_r(N))^p.
\mathrm{Hom}(V,W) = \bigoplus_{\alpha \in K(\mathcal{I})}\mathrm{Hom}(V,W)_{\alpha}, \mathrm{Hom}(V,W)_{\alpha}=\{f \in \mathrm{Hom}(V,W)|f(V_{i}) \subset W_{j}, [(j,i)]=\alpha\}.
S=\int L=\lambda^2_u(u)\lambda^2_v(u)+\lambda^2_u(u)\lambda^2_v(v)+\lambda^2_u(v)\lambda^2_v(w)+\lambda^2_u(w)\lambda^2_v(u)+\lambda^2_u(w)\lambda^2_v(w)+\lambda^2_u(v)\lambda^2_v(v)\qquad
\begin{aligned}-\int \nabla^2{\rm curl}\left[g(\varrho)\left(B\cdot \nabla B -\nabla(\frac{1}{2}|B|^2)\right)\right]\nabla^2 B~dx &\lesssim \|\nabla^2 \varrho\|_{L^2}^2\|\nabla B\|_{H^1}^2+\delta\|\nabla^3 B\|_{L^2}^2. \end{aligned}
0 = \int_{\partial X}d^2 B^{f(\xi)}_{F_p(x)}( DF_p(v), w) > \tau(\xi) d\mu(\xi) + p \int_{\partial X}< - \overrightarrow{F_p(x)f(\xi)}, w) > ( < \overrightarrow{x\xi}, v > - < \overrightarrow{F_p(x)f(\xi)}, DF_p(v) > ) \tau(\xi) d\mu(\xi)
m^{\delta}_{i,j}(K^{i,j}) :=\max\left\{0,-\gamma^{-1}|I|K^{i,j}\right\}, M^{\delta}_{i,j}(a) :=\gamma^{-1}|I|\cdot\left(\min\left\{\bar k^{i,j-1}_{+}(a),\bar k^{i,j-1}_{-}(a)-K^{i,j}\right\}-\delta\right),
c_{li+1}\dots c_{l(i+1)}=\, c_1\dots c_{l(i+1)}-c_1\dots c_{li}\times10^l =\, \lfloor10^{l(i+1)}a/b\rfloor -\lfloor10^{li}a/b\rfloor\times10^l =\, 10^{l(i+1)}a/b -s_{i+1}/b - (10^{li}a/b -s_{i}/b)10^l =\, (10^ls_i-s_{i+1})/b,
e(\bar{S})+e(\bar{S},S) =\sum_{x\in \bar{S}}d_{S}(x) +\frac{1}{2}\sum_{x\in \bar{S}}d_{\bar{S}}(x) &=\frac{1}{2}\sum_{x\in \bar{S}}d_{S}(x)+\frac{1}{2}\sum_{x\in \bar{S}}\left(d_S(x)+d_{\bar{S}}(x)\right) &=\frac{1}{2}\sum_{x\in \bar{S}}\left(d_S(x)+d_{G'}(x)\right).
\nabla \bigl ( \, U ( t ) \, \tilde{\alpha}\, U ( t )^{*}\bigr ) \; = \; \bigl ( U ( t ) \, \otimes \, U ( t ) \bigr ) \; \nabla \tilde{\alpha}\; \bigl ( U ( t )^{*}\, \otimes \, U ( t )^{*}\bigr ) ,
b_{\frac{n+1}{2}}= x_{\frac{n+1}{2}}\left(x_{\frac{n-1}{2}}x_{\frac{n+3}{2}}\right) = \left(x_{\frac{n+1}{2}}x_{\frac{n-1}{2}}\right)x_{\frac{n+3}{2}}= (-1)^{\frac{n+3}{2}}zx_{\frac{n+3}{2}}= (-1)^{\frac{n+3}{2}}c_{\frac{n+3}{2}}= 0
\frac{c}{b}=\frac{1}{2}(\frac{m}{n}+\frac{n}{m})=\frac{m^{2}+n^{2}}{2mn},\frac{a}{b}=\frac{1}{2}(\frac{m}{n}-\frac{n}{m})=\frac{m^{2}-n^{2}}{2mn}.
E\left[\exp\left\{\frac{c_q}{2}\left(\frac{|\Delta_{i,j}|}{\sigma_{i,j}}\right)^{2/q}\right\}-1\right] =\int_0^\infty P\left(\frac{|\Delta_{i,j}|}{\sigma_{i,j}}>\{\log((1+u)^{2/c_q})\}^{q/2}\right)du &\leq\int_0^\infty(1+u)^{-2}du=1.
\pi_{TM,X_{1,C},Y_{1,V},X_{2,C},Y_{2,V},c}({\bf x},{\bf y})=\left( \begin{array}{ccc|ccc}0 0 0 0 0 0 0 0 0 0 x^3 x^1 0 0 0 0 0 0 \hline 0 0 0 0 0 0 0 -x^3 0 0 0 y^1 0 -x^1 0 0 -y^1 0 \end{array}\right).
\left\|Y_1\left(\begin{array}{cc}z_1 0 0 z_2\end{array}\right)Y_2\right\|^2 &=\frac{1}{2}\left( (|c_{11}d_{11}|^2 +|c_{11}d_{12}|^2)| z_1|^2+|c_{22}d_{22}|^2|z_2|^2\right) &+\frac{1}{2}\sqrt{( (|c_{11}d_{11}|^2+|c_{11}d_{12}|^2)| z_1|^2+|c_{22}d_{22}|^2|z_2|^2)^2-4|c_{11}d_{11}|^2 |z_1|^2 |c_{22}d_{22}|^2|z_2|^2}.
\mu_F(E)= \mu_F(F'\cap E)=\nu_F^+(F'\cap E)+\nu_F^-(F'\cap E) = \big|x^*_F\big(s_h(F'\cap E\cap P)\big)\big|+\big|x^*_F\big(s_h(F'\cap E\setminus P)\big)\big| \leq \big\|s_h(F'\cap E\cap P)\big\|+\big\|s_h(F'\cap E\setminus P)\big\| \leq \widetilde{\varphi}(E)+\widetilde{\varphi}(E),
P_1 =\frac1{(1-t_{11})}P_2 =\frac1{(1-t_{11})(1-t_{22})(1-t_{12}t_{21})}P_3 =\frac{1-t_{12}t_{13}t_{23}t_{32}t_{31}t_{21}}{\prod_i(1-t_{ii})(1-t_{12}t_{21})(1-t_{13}t_{31})(1-t_{23}t_{32})(1-t_{12}t_{23}t_{31})(1-t_{13}t_{32}t_{23})}
\int_{-M}^0\int_{-\infty}^0|x|^\sigma J(x-y)\psi(y)dydx = \int_{-\infty}^0\int_{-y}^{M-y}(x+y)^{\sigma}J(y)\psi(-x)dxdy+\int_0^M\int_{0}^{M-y}(x+y)^{\sigma}J(y)\psi(-x)dxdy.
\left(\begin{array}{c}\hat{a}_i(t) \hat{a}_i^\dag(t)\end{array}\right)=\sum_{j=1}^2\left( \begin{array}{cc}\alpha_{ij}(t) \beta_{ij}(t) \beta_{ij}^\ast(t) \alpha_{ij}^\ast(t)\end{array}\right)\left(\begin{array}{c}\hat{a}_j \hat{a}_j^\dag\end{array}\right)\;,
\beta_\nu^{(1)}=\varepsilon^{\mu ,[\mu \nu ]}+\varepsilon^{[\mu \nu ],\mu},\hspace{0.3in}\beta_\nu^{(\widetilde{1})}=\frac 12e_{\mu \nu \rho \omega}\left(\varepsilon^{\widetilde{\mu},[\rho \omega ]}+\varepsilon^{[\rho\omega ],\widetilde{\mu}}\right) ,
\left({\cal H}^{a, b, c}\,\oplus\,{\cal H}^{a^{\prime}, b^{\prime},c^{\prime}}\right) \,\textcircled{\scshape A}\, \left({\cal H}^{a, b, c}\,\oplus\,{\cal H}^{a^{\prime},b^{\prime}, c^{\prime}}\right)
\{Q_{\alpha}, Q_{\beta}\}= ( C \Gamma^{\mu})_{\alpha \beta}P_{\mu}+ ( C \Gamma_{\mu}\Gamma_{1 1})_{\alpha \beta}{\hat{Z}}^{\mu}+ ( C \Gamma_{\mu \nu})_{\alpha \beta}{\hat{Z}}^{\mu \nu}+ ( C \Gamma_{1 1})_{\alpha \beta}{\hat{Z}}\quad .
\phi_{x_1x_1x_1}-\phi_{x_3}+3H_{x_1}\phi_{x_1}-\frac{k-5}{2}H_{x_1x_1}\phi=0, &\phi_{x_1x_2}+H_{x_2}\phi+\frac{k-5}{6}\frac{H_{x_1x_2}}{H_{x_2}}\phi_{x_2}=0.
\xi = \big|\{j\,:\, w_j \neq x_j^\star = 0\}\big| - \big|\{j\,:\, w_j = x_j^\star \neq 0\}\big| \overline{h}= \big|\{j\,:\, x_j^\star > 0, \,\,\epsilon_j > 0\}\cup \{j\,:\, x_j^\star < 0, \epsilon_j < 0\}\big|\,,