harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{matrix}\geq 2 \geq 2 \geq 2 \geq 1 \geq 1 \geq 1 \geq 1 \geq 1 = 0\end{matrix}\begin{matrix}\geq 2 \geq 1 \geq 1 \geq 2 \geq 1 \geq 1 \geq 2 \geq 1 = 0\end{matrix}\begin{matrix}\geq 3 \geq 2 \geq 1 \geq 2 \geq 1 \geq 1 \geq 1 \geq 1 = 0\end{matrix}
\left| \{K_{(1-\sigma)\rho}: (n(\cdot, t)-\mu_{\pm}\pm \delta k)_{\pm}> 0 \}\right| &\leq \left[ \left(\frac{j}{j-1}\right)^2 (1-\nu) + \frac{C(c_0,\alpha) j}{\sigma^2 (j-1)^2}+ C(\delta) \right] |K_{\rho}|
{\cal M}^{(4+6+4|4)}= \{q^{{\cal M}}\}\equiv\{(Y^M; \l^A, \bar{\l}^{\dot{A}}) \}=\{(x^{A\dot{A}}, z^{AB},\bar{z}^{\dot{A}\dot{B}}; \l^A, \bar{\l}^{\dot{A}};\Theta^{A}, \bar{\Theta}^{\dot{A}})\},
R_{n,m}\left( q;z\right) =\frac{\left[ m-1\right]_{q}}{\left[n+m-1\right]_{q}\left[ n+1\right]_{q}}\left[ \left( 1+q^{m-1}\right)+\left( q{\displaystyle\sum\limits_{i=1}^{m-2}}\left[ i\right]_{q}+{\displaystyle\sum\limits_{i=1}^{m-2}}\left[ i\right]_{q^{-1}}\right) \left( z+1\right) \right]z^{m-2}\left( 1-z\right) .
E_n(x):=\det \begin{bmatrix}c_0 c_1 \cdots c_{n-2}1 1 c_1 c_2 \cdots c_{n-1}x_{k_0}x \vdots \vdots \vdots \vdots \vdots c_{n-1}c_n \cdots c_{2n-3}x_{k_0}^{n-1}x^{n-1}c_n c_{n+1}\cdots c_{2n-2}x_{k_0}^n x^n\end{bmatrix}
I_{\nu}(z) = -\frac{z^{\nu-3}}{\sqrt{\pi}\,2^{\nu-3}\Gamma(\nu-5/2)}* \times \int_{-1}^1 e^{-zs}\,\frac{s(s^2z^2+3sz+3)}{z^2}(1-s^2)^{\nu-7/2}\,ds,\nu>\frac52.
u(x,t)=A+O\left(t^{-\frac{1}{2}}e^{-8t\kappa_{\delta}(3\xi-\kappa_{\delta}^2)}\right), x>0, \t>0, \ \xi>\kappa^2. u(x,t)=O\left((-t)^{-\frac{1}{2}}e^{8t\kappa_{\delta}(3\xi-\kappa_{\delta}^2)}\right), x<0, \t<0, \ \xi>\kappa^2,
\overline{\overline{C}}_{X^{\ast}}(p_g,k_g) =\widetilde{C}_{X^{\ast}}(p_g,k_g)\widetilde{s}(p_g)\widetilde{s}(k_g)\widetilde{s}(g)^{-1}&=\widetilde{s}(p_g)\widetilde{s}(k_g)\widetilde{s}(g)^{-1}=1;
\ell_5 v_i =e_{\lambda(i)},\ \ell_5 v_{\overline{i}}=-e_{\overline{\lambda(i)}}\quad\mbox{for}i\in I_{(12)}\mbox{and that}\ell_5 v_i =e_{\eta_{13}(i)},\ \ell_5 v_{\overline{i}}=-e_{\overline{\eta_{13}(i)}}\quad\mbox{for}i\in I_{(13)}.
e_{(l+1)}= \Big(\frac{q-q^{-1}}{z-z^{-1}}\Big)^{l-1}\Big(\frac{q-q^{-1}}{z-z^{-1}}\Big)^{l-2}e_{(l-1)}H_{2l-2}H_{2l-1}H_{2l-3}^{-1}H_{2l-2}^{-1}e_{(l-1)}&\times H_{2l}H_{2l+1}H_{2l-1}^{-1}H_{2l}^{-1}e_{(l)}= \Big(\frac{q-q^{-1}}{z-z^{-1}}\Big)^{l-1}e_{(l)}H_{2l}H_{2l+1}H_{2l-1}^{-1}H_{2l}^{-1}e_{(l)}.
\begin{cases}|(u - w_n) (t)| \geq \theta_n g(b_n)\ \textrm{for}\t \in [x_n, x_n + b_n] b_n \geq a_n, |(u - w_n) (t)|\geq \theta_n g(a_n) \ \textrm{for}\t \in [x_n - a_n, x_n] a_n \geq b_n,\end{cases}
h_{N,s}(z_1,\dots,z_s) =\frac{\det\left[z_j^{k-1}(z_j-1)^{s-k}h_{N-k+1}(z_j)\right]_{j,k=1,\dots, s}}{\prod_{1\leq j<k \leq s}^{}(z_j-z_k)}.
\alpha M + R_C(M)= \alpha M + \frac{KN-KM}{KM+N}= \frac{\alpha N}{K}+ \frac{K(1-1/K)}{2}< \frac{\alpha(K^2+K)}{2\alpha K}+ \frac{K-1}{2}< \frac{K+1}{2}+ \frac{K-1}{2}< K.
{\frac{x}{r}}={\frac{2 \xi}{1 + \xi^{2}+ \eta^{2}}}\, , \,{\frac{y}{r}}={\frac{2 \eta}{1 + \xi^{2}+ \eta^{2}}}\, , \,{\frac{z}{r}}={\frac{\xi^{2}+ \eta^{2}- 1}{1 + \xi^{2}+ \eta^{2}}}\; .
{\mathcal L}_{eff}\;\sim\; h \rho_0 q \sum_{\alpha = 1}^M \epsilon_{kl}\, \dot{r}^{(\alpha)}_k R^{(\alpha)}_l \,-\, \frac{h^2\rho_0 q^2}{2 \pi m}\,\sum_{\alpha=1}^M \, \ln |\frac{r^{(\alpha)}}{\xi}|\;,
\int_0^T \sum_{i=1}^{q}\theta_{i, \lambda}(t) f_{i,\lambda}(\xi(t)) dt \geq \int_0^T \sum_{i=1}^{q}\theta_{i, \lambda}(t) f_{i,\lambda}(u_{\lambda}(t)) dt + \int_0^T \left\langle \sum_{i=1}^{q}\theta_{i, \lambda}(t) \nabla f_{i,\lambda}(u_{\lambda}(t)), \xi(t) - u_{\lambda}(t) \right\rangle dt.
d \widehat f!(h;\widehat{\frak S}^{\epsilon})=\widehat f!(dh;\widehat{\frak S}^{\epsilon})+\widehat f!(h\vert_{\partial_{\frak C^h}(X,\widehat{\mathcal U})};\widehat{\frak S}^{\epsilon}\vert_{\partial_{\frak C^h}(X,\widehat{\mathcal U})}).
\begin{array}{@{}l@{}}{\displaystyle \Psi_{0}(\mathbf{x}) = 2\sqrt{2}\sin(\pi x_1)\sin(\pi x_2)\sin(\pi x_3) , \mathbf{A}_{0}(\mathbf{x})=\mathbf{A}_{1}(\mathbf{x})=0,}begin{align*}2mm]{\displaystyle \mathbf{g}(\mathbf{x}, t) = \left(10\sin(1.5{\pi}^{2}t), 10\sin(1.5{\pi}^{2}t), 10\cos(1.5{\pi}^{2}t)\right).}\end{array}
Q(X\in A, \mathbf Y\in B) =\frac{1}{n}\sum_{i=1}^n Q_i(X\in A, \mathbf Y\in B) &=\frac{1}{n}\sum_{i=1}^n Q_i(X\in A)Q_i(\mathbf Y\in B) &=Q(X\in A)\frac{1}{n}\sum_{i=1}^n Q_i(\mathbf Y\in B)=Q(X\in A)Q(\mathbf Y\in B),
\prod_{j=1}^k(q;q)_{m+j-1}=\frac{\tilde H_q(k+m)}{\tilde H_q(m)}, \prod_{j=1}^k(q^{l+j};q)_{m}=\frac{\tilde H_q(l,k+l+m)}{\tilde H_q(k+l,l+m)}, \prod_{j=1}^k(q^{l+j};q)_{j-1}=\frac{\tilde H_{q^2}\left(\frac l2,\frac{l+1}2,\frac{l-1}{2}+k,\frac{l}{2}+k\right)}{\tilde H_q(l+k)}.
n_{1}^{2}+ n_{2}^{2}+ n_{3}^{2}={\frac{1}{3}}k_{1}^{2}+{\frac{2}{3}}( k_{2}^{2}+ k_{3}^{2}+ k_{2}k_{3}) , \qquad \quad n_{1}+ n_{2}+ n_{3}= k_{1}
\overline{v}_{i}\leftarrow\overline{v}_{i}-\gamma\{w_{internal}[\alpha\frac{\partial^{2}\overline{v}}{\partial s^{2}}(\overline{v}_{i})+\beta\frac{\partial^{4}\overline{v}}{\partial s^{4}}(\overline{v}_{i})]+\nabla E_{ext}(\overline{v}_{i})\}
i\frac{\partial}{\partial t}\bigl\{w_{k\tau}(t;\rho) - w_{\tau}(t;\rho)\bigr\}= H(t;\rho)\bigl\{w_{k\tau}(t;\rho) - w_{\tau}(t;\rho)\bigr\}+ \int_0^1 \frac{\partial H}{\partial \rho}(t;\rho+\theta\tau)d\theta\, \bigl\{u_k(t;\rho+\tau) - u (t;\rho+\tau)\bigr\}
\widetilde{Q^\omega}(t,D_{t'}) := \sum_{\alpha,\beta}h^{\alpha\beta}(t)D_{t_\alpha}D_{t_\beta}-i\sum_{\alpha,\beta}\left(\frac{1}{2}h^{\alpha\beta}(t)\partial_{t_\alpha}\log\delta(t)+\partial_{t_\alpha}h^{\alpha\beta}(t)-F^\alpha(t)h^\beta_\alpha(t)\right)D_{t_\beta}-\omega^2.
\Vert (v_{s},v_{u})\Vert^{2}=\Vert v_{s}\Vert^{2}+ \Vert v_{u}\Vert^{2}-2 \theta_{p}\Vert v_{s}\Vert\Vert v_{u}\Vert \geq \Vert v_{s}\Vert^{2}+ \Vert v_{u}\Vert^{2}+2(\mu_{i}-1)\Vert v_{s}\Vert\Vert v_{u}\Vert &\geq \mu_{i}(\Vert v_{s}\Vert^{2}+\Vert v_{u}\Vert^{2}).
\dim{\left(\overline{R}/q \right)_{m}}= \dim{\left(R/(q\cap R)\right)_{m\cap R}}+ \operatorname{trdeg}_{R/(q\cap R)}\overline{R}/q - \operatorname{trdeg}_{\kappa\left((m\cap R)/(q\cap R)\right)}\kappa(m/q) = \dim{R}+ 0 - 0 = \dim{\overline{R}},
j(\alpha)(X^1,\dots, X^q \vert \xi_1,\dots \xi_p) = \sum_{\sigma \in S_{p}}(-1)^{\sigma}\left.\frac{d}{ds_1}\right|_{_{s_1=0}}\cdots \left.\frac{d}{ds_p}\right|_{_{s_p=0}}\alpha(\Psi(\xi_1,\dots \xi_p))(X^1,\dots, X^q)
\begin{array}{l}\omega_{1\beta}=\displaystyle\frac 1{(\Lambda\Pi)^2}(\Lambda\Pi_\sigma)P^{+\sigma\gamma}\Lambda_n\Gamma^n\tilde\Gamma^m\Pi_{m\gamma}k_\beta, \omega_{2\beta}=\displaystyle\frac 1{(\Lambda\Pi)^2}(\Lambda\Pi_\sigma)P^{+\sigma\gamma}\Pi_{n\gamma}\Gamma^n\tilde\Gamma^m \Lambda_ms_\beta.\end{array}
\sum_{j=1}^{M_n}c_{j, 2n}\, u_0^{\beta_{0,1, j}/2}\, (1-u_0)^{\beta_{0,2, j}/2}\, \frac{u_1^{\beta_{1, j}}\, u_2^{\beta_{2,j}}\, \cdots \, u_{2n+2}^{\beta_{2n+2, j}}}{Q_{\alpha, 2n}^{\rho_{j,2n}}}\, \alpha^{k_{0,j}}\, \alpha_1^{k_{1,j}}\, \cdots \, \alpha_{2n}^{k_{2n,j}},
p_{|\dot{h}|,|h|}(\dot{x},x) =\sum_{i=1}^N p_{|\dot{h_i}|}(\dot{x}) begin{align*}-0.2em] \!\! \times \underbrace{\int_0^x\!\!\cdots\!\!\int_0^x}_{(N-1)-{\rm fold}}\!\!\!\! p_{|h_{1}|,\cdots,|h_{N}|}(x_1,\dots,x_i=x,\cdots,x_N) begin{align*}-0.2em] \times \underbrace{dx_1 \cdots dx_k\cdots dx_N}_{\substack{(N-1)-{\rm fold}k \neq i}},
I(\vec x) =\sum_{i<j}w_{ij}|x_i-x_j| &=\sum_{i< j}w_{ij}(|x_i|+|x_j| +||x_i|-|x_j||-|x_i+x_j|) &=\sum_{i=1}^n d_i|x_i| + \sum_{i< j}w_{ij}||x_i|-|x_j||- \sum_{i< j}w_{ij}|x_i+x_j| &= \|\vec x\|+\hat{I}(\vec x)-I^+(\vec x),
(Sf)(z)=(\frac{f^{\prime\prime}(z)}{f^{\prime}(z)})^{\prime}-\frac{1}{2}(\frac{f^{\prime\prime}(z)}{f^{\prime}(z)})^{2}=\frac{f^{\prime\prime\prime}(z)}{f^{\prime}(z)}-\frac{3}{2}(\frac{f^{\prime\prime}(z)}{f^{\prime}(z)})^{2}.
\bar{W}( \phi ) = \pm 4 \frac{1}{\sqrt 2}( \frac{1}{4}\mathrm{s i n h}2 \phi - \frac{3}{2}\phi ) , \bar{W}( \phi ) = \pm \frac{\phi^{2}}{2}( \frac{\phi^{2}}{2}- 1 ) .
\begin{array}{lcr}Region\,I\qquad \psi^{b}=s\left( \begin{array}{c}E+m -\kappa\end{array}\right) e^{\kappa x}\qquad x\leq -\xi begin{align*}0.5cm] Region\,III\qquad \psi^{b}=s^{\prime}\left( \begin{array}{c}E+m \kappa\end{array}\right) e^{-\kappa x}\qquad x\geq \xi\end{array}
m\left( r,\frac{f(z+c)}{f(z)}\right) =o\left(\frac{T(e^{\epsilon}r,f)}{r^{1-\varepsilon/2}}\right)=o\left(\frac{r^{\sigma+\varepsilon/2}}{r^{1-\varepsilon/2}}\right)=o(r^{\sigma-1+\varepsilon}),
A_{x1x2}(\tau,n)=\sqrt{\frac{1}{2\alpha_{u}}}exp[-\pi(\alpha_{u}\frac{(\tau-t_{d})^{2}}{2}+\frac{(n-f_{d})^{2}}{2\alpha_{u}})]exp[j2\pi(f_{u}\tau-t_{u}n+f_{d}t_{u})]
-\partial_{\nu^t_\theta,{\rm out}}V^\theta = \frac{1}{2}(N\cdot\nu^t_\theta) \frac{-\Delta h^0}{\partial_{ss}u^0}\,\frac{\lambda^t}{t}\circ \bar \pi_1 + \partial_{\nu^t_\theta,0}V^\theta\mbox{on}\Gamma^t_\theta,
\mathfrak{H}_N = \bigotimes_{j=1}^{N}V_j,V_j \simeq{\mathbb{C}}^2, \mathcal{H}_N = \frac{J}{4}\sum_{k=1}^{N}( \sigma_{k}^{x}\sigma_{k+1}^{x}+ \sigma_{k}^{y}\sigma_{k+1}^{y}+ \sigma_{k}^{z}\sigma_{k+1}^{z}-{\mathbb{I}}_N),\sigma_{N+1}^{a}=\sigma_{1}^{a}.
\langle\Delta V\rangle=\frac{4}{3}\frac{e^{2}}{4\pi\epsilon_{0}}\frac{e^{2}}{4\pi\epsilon_{0}\hbar c}(\frac{\hbar}{mc})^{2}\frac{1}{8\pi a_{0}^{3}}ln\frac{4\epsilon_{0}\hbar c}{e^{2}}=\alpha^{5}mc^{2}\frac{1}{6\pi}ln\frac{1}{\pi\alpha},
\big(\frac{v}{w-t\tau/2}\big)>0, \big(\frac{v}{w+t\tau/2}\big)>0, \big(\frac{v}{-t}\big)>0, \big(\frac{-t\tau/2}{w-t\tau/2}\big)>0 , \big(\frac{-t\tau/2}{w+t\tau/2}\big)>0, \big(\frac{-t\tau/2}{-t}\big)=(\tau/2)>0 .
f^{\kappa}_\infty(w)= ~\frac{C_{w_d,\varsigma,\kappa}}{(1-w^2)^2}\left(\frac{1+w}{1-w}\right)^{m/(2\varsigma)}\exp\left\{-\frac{1-m w}{\varsigma\left(1-w^2\right)}\left(1+\frac{2}{\kappa}\right)\right\}
g(A^*_V(X),A^*_V(X)) =g(A^*_U(X),A^*_U(X))-\delta g(A^*_U(X),N)^2-2\delta g(U,N)g(\nabla_XN,A^*_U(X)) &+g(U,N)^2g(\nabla_XN,\nabla_XN).
S_{\widehat{n}}^{e^{\prime}n^{\prime}/ I}= - 2 V \mathrm{s i n}( \pi e^{\prime}/ k ) \mathrm{e x p}( - 2 i \pi e^{\prime}n / k ) , \S_{\widehat{n}}^{e^{\prime}n^{\prime}/ I I}= 0
L = \mathrm{T r}\left\{- \frac{1}{4}G^{\mu \nu}G_{\mu \nu}+ \frac{1}{2}D_{\mu}S^{\dagger}D^{\mu}S \right\}+ \frac{1}{2}D_{\mu}\phi^{\dagger}D^{\mu}\phi - V ( S , \phi ) .
F^{}(x_{i+1/2}) = \underbrace{-\sum_{k=i+1}^{I}n_k(t) S(x_k)\Delta x_k \sum_{j=1}^i x_j b(x_j,x_k)\Delta x_j}_{=:J^{}_{i+1/2}(n)}+{\cal O}(\Delta x^2)
\overline U_{M_n,Q_n,n}(x,t) ={\bf\Phi}_{M_n,n}(\widetilde u_T )(x,t)-\widetilde \Phi_{M_n,n}( \widetilde g)(x,t)-\sum_{p=0}^{M_n}\Bigg[ \int_t^T e^{(s-t)p^{2\beta}}\overline F_{Q_n,p}(\overline U_{M_n,Q_n,n})(s)ds \Bigg]\phi_p(x).
R_{\mu\nu}=\frac{\partial^2\ln\sqrt{-g}}{\partial x^\mu\partial x^\nu}-\frac{\partial \Gamma_{\mu\nu}^\sigma}{\partial x^\sigma}+\Gamma_{\mu\sigma}^\lambda\Gamma_{\nu\lambda}^\sigma -\Gamma_{\mu\nu}^\sigma\frac{\partial\ln\sqrt{-g}}{\partial x^\sigma},
=(x_{N}+h_{N})^{\omega\gamma}\frac{|u(x^{\prime},x_{N}+h)-u(x^{\prime},x_{N})|}{h^{\gamma}}\leq C_{k}^{(1)}\left\langle \left\langle u\right\rangle \right\rangle_{\omega\gamma,x,\overline{H}}^{(\gamma)(k)}
\Gamma \lbrack A \rbrack = \! - 2 \! \int_{0}^{\infty}{\frac{d T}{T}}\mathrm{e}^{- m^{2}T}\int{\cal D}X \mathrm{e x p}\left\{- \int_{0}^{T}\! \! \! \! d \tau \int \! \! d \theta \, \left[ \! -{\frac{1}{4}}X \, D^{3}X \! - \! i e D X^{\mu}A_{\mu}( X ) \right] \right\}
u^\varepsilon (C_1\varepsilon |\log \varepsilon| ,x) \geq w_\varepsilon^- (C_1\varepsilon |\log \varepsilon| ,x) = Y \left( C_1 |\log \varepsilon| , u_0^{\varepsilon}(x) - h(x) ( \varepsilon^{1-C_1 \mu}-\varepsilon ) \right).
G_{R}=\sqrt{P_{R}/\mathsf{E}\left\{\left\Vert \mathbf{x}_{R}\right\Vert^{2}\right\}}=\sqrt{P_{R}/\left(P_{T}\mathrm{Tr}\left(\mathbf{\tilde{W}}\right)+\sigma_{R}^{2}\mathrm{Tr}\left(\mbox{\ensuremath{\mathbf{\tilde{W}}}}\mathbf{\tilde{W}}^{H}\right)\right)},
y=\langle X_1, \ell_1,\ell_2,\ell_3,\ell_4,X_2, \ell_5,\ell_6,\ell_7,\ell_8,X_3,\ell_9,\ell_{10},\ell_{11},\ell_{12},X_4,\dots\rangle= \langle a'_1,a'_2,a'_3,\dots\rangle.
\frac{x}{ct}=tanh\Psi=\frac{v}{c}\Rightarrow sinh\Psi=\frac{\frac{v}{c}}{\sqrt{1-\frac{v^{2}}{c^{2}}}},cosh\Psi=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}},
S_{\rm T}= \int d^4z\, d^4\bar{z}\,{\rm tr}\Bigr\{\hbox{$\frac{1}{12}$}\Bigr(\epsilon^{ab}F^{AB}_{\!~~~~ab}\epsilon^{cd}F_{AB cd}-\epsilon_{AB}F^{AB}_{\!~~~~ab}\epsilon_{CD}F^{CD ab}-F^{AB}_{\!~~~~ab}F_{AB}^{\!~~~~ba}\Bigr) \Bigr\},
\sup_{\gamma \in \mathcal{P}_{\mathrm{D}}}\sup_{\pi \in \Pi\left(\mu_1, \mu_2, \gamma\right)}\int_{\mathcal{V}\times \mathcal{V}}\varphi_\lambda \, d \pi = \sup_{\pi \in \mathcal{G}_{\mathrm{D}, \lambda}}\int_{\mathcal{V}\times \mathcal{V}}\varphi_\lambda \, d \pi.
\hat{r}( p )^{a n}= - 2 - \sqrt{1 - \frac{4 m^{2}}{p^{2}}}\log \frac{\sqrt{1 - \frac{4 m^{2}}{p^{2}}}- 1}{\sqrt{1 - \frac{4 m^{2}}{p^{2}}}+ 1}.
\frac{1}{2}n^{1 / 2}\Delta_n [V' (n^{- 1 / 2}v^n_t + \rho' (\lambda_0))] = \frac{1}{2}n^{1 / 2}\Delta_n \{V' (\rho' (\lambda_0)) + V'' (\rho' (\lambda_0)) (n^{- 1 / 2}v^n_t) + O (n^{- 1})\}= \frac{1}{2}V'' (\rho' (\lambda_0)) \Delta_n v^n_t + O (n^{- 1 / 2}),
\omega_1 + (\omega_3 - \omega_1) + (\omega_4 - \omega_3)+ (\omega_2 + \omega_5 - \omega_4) + (\omega_2 + \omega_6 - \omega_5) + (\omega_2 + \omega_7 - \omega_6) + (\omega_2 + \omega_8 - \omega_7)+ (\omega_2 - \omega_8) = 5\omega_2
\frac{\partial \tilde g_{\mathfrak c}}{\partial\rho}=\pm\frac{1}{g_{\mathfrak c}|\nabla V(\mathfrak c)|}\Big[-\frac{\partial g_{\mathfrak c}}{\partial x_1}\frac{\partial V}{\partial x_1}-\frac{\partial g_{\mathfrak c}}{\partial x_2}\frac{\partial V}{\partial x_2}\Big] \,.
\lambda_2 =-\frac{16}{11}a_j^{11/15}+\frac{10}{11}\,, \lambda_3 =-\frac{60}{11}a_j^{11/5}+\frac{75}{22}\,, \lambda_4 =\frac{60}{11}a_j^{11/5}-\frac{75}{22}\,,
\hat{K}_{p,q}(r) = \frac{{\pi_{p,q}}}{2}\sum\limits_{n = 0}^\infty{-\frac{1}{p}\choose n}\frac{1}{{(1 - \lambda )^{n + \frac{1}{p}}}}\sum\limits_{j = 0}^n (-1)^{j}{n \choose j}{- \frac{1}{q}\choose j}{\frac{1}{p}- 1 - \frac{1}{q}\choose j}^{-1}\lambda^{n - j}r^{qj}.
e_1(\boldsymbol{K}) = \boldsymbol{I}, e_2(\boldsymbol{K}) = - c_{1}\boldsymbol{I}+\boldsymbol{K}, \cdots \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots e_n(\boldsymbol{K}) = - c_{n-1}\boldsymbol{I}-c_{n-2}\boldsymbol{K}-\cdots-c_1\boldsymbol{K}^{n-2}+\boldsymbol{K}^{n-1},
\left( \begin{array}{c}{{\chi^{( 1 )}}}\\{{\chi^{( 2 )}}}\\ \end{array}\right) = \frac{1}{\sqrt{2}}\left( \begin{array}{c c}{{1}}&{{1}}\\{{- 1}}&{{1}}\\ \end{array}\right) \left( \begin{array}{c}{{\chi^{L}}}\\{{\chi^{R}}}\\ \end{array}\right)
\varepsilon_{\nu\tau}p^{\tau}(p_{\rho}p_{\sigma}-g_{\rho\sigma}p^{2}) =\varepsilon_{\rho\tau}p^{\tau}(p_{\nu}p_{\sigma}-g_{\nu\sigma}p^{2}) = \varepsilon_{\sigma\tau}p^{\tau}(p_{\nu}p_{\rho}-g_{\nu\rho}p^{2})
\frac{1}{2}\int_\Omega|\nabla u|^2\,dx+ \frac{1}{2}\int_\Omega V\,u^2\,dx-\langle f,u\rangle &\ge\frac{1}{2}\left(1-\frac{\|V_-\|_{L^{N/2}(\Omega)}}{T_N}-\delta\right)\int_\Omega|\nabla u|^2\,dx-\frac{1}{2\,\delta}\,\|f\|^2_{W^{-1,2}(\Omega)},
\lambda^+= sgn(\hat H^+_2) \frac{\hat a^+ [\hat u_2]}{\hat a^+ |\hat H^+_2| + \hat a^- |\hat H^-_2|}, \lambda^-=- sgn(\hat H^-_2) \frac{\hat a^- [\hat u_2]}{\hat a^+ |\hat H^+_2| + \hat a^- |\hat H^-_2|}.
\mathfrak{T}_n := |v_{13}|^\frac12 \tilde B_{12}^{-1}\bigl[ H_0 + k_n^2\bigr]^{\frac{-3+\delta}4}\mathfrak{D}_n:= |v_{12}|^\frac12 \tilde B_{13}^{-1}(k_n) \bigl[ H_0 + k_n^2\bigr]^{\frac{-1-\delta}4}.
\sum_{j=1}^M |\beta_j| \leq \frac{12}{\tilde{\varepsilon}\big|\bigcup_iB_{\ell(i)}\big|}\sum_{j\leq M}\Big(\int f_j - C\alpha_j\Big) \leq \frac{12}{\tilde{\varepsilon}\big|\bigcup_iB_{\ell(i)}\big|}( 3\varepsilon_1 M ),
\P \left[(X + \Delta X) \in dx \, | \, Y =i \right] = \sum_{j, i \in j}\P \left[X \in d(x+e_{j^c}), \Delta X= -e_{j^c}\, | \, Y=i \right] &=\sum_{j,i \in j}\frac{a(j)m(d(x+e_{j^c}))}{\hat a(i)}, & =\sum_{j,i \in j}\left(\frac{a(j)}{\hat a(i)}m\right)_{\sharp -e_{j^c}}(dx),
-m(-n\lambda^{-n}T_1+\lambda^{-n})T_{m,0}+m^2n\lambda^{-n}T_2\lambda^mM_0 -3m\frac{\partial T_{-n,0}}{\partial M_0}\lambda^mM_0 +n\lambda^mT_{-n,0}+nr_{m,1}\lambda^{-n}M_0 = -(m-n)T_{m-n,0}.
y_k^n e_k = \sum_{j=1}^{k-1}(-1)^{j+1}E_j(x_1, \ldots, x_{k-1}) . y_k^{k-j-1}(\psi_k \cdots \psi_n) (\psi_n \cdots \psi_k) e_k+ & \sum_{j=1}^{n-k+1}(-1)^{j+1}E_j(x_k, \ldots, x_n) .( y_k^{n-j}e_k).
[D_{\mathcal{K}_{\alpha}^{2}}(2\mathrm{i}z )]^{-1/2}=\exp\left(\frac{1}{2}\sum_{k=1}^{\infty}\frac{\mbox{Tr}\mathcal{K}_{\alpha}^{2k}}{k}(2\mathrm{i}z)^{k}\right)<\infty .
(\begin{matrix}F_{x}\\F_{y}\\F_{z}\\ \end{matrix})=q(\begin{matrix}E_{x}\\E_{y}\\E_{z}\\ \end{matrix})-q(\begin{matrix}0&-B_{z}&B_{y}\\B_{z}&0&-B_{x}\\ -B_{y}&B_{x}&0\\ \end{matrix})(\begin{matrix}v_{x}\\v_{y}\\v_{z}\\ \end{matrix})
p\left({{d_{p2}}\leq \delta}\right) = M\int_{\mathbf{\Omega}}{\prod\limits_{i = 1}^k{{{\left({{{\sin}^2}{\theta_i}}\right)}^{n - 2k}}}\prod\limits_{1 \leq i < j}^k{{{\left({{{\sin}^2}{\theta_i}-{{\sin}^2}{\theta_j}}\right)}^2}}d{{\sin}^2}{\theta_1}\cdots d{{\sin}^2}{\theta_k}}
f_1(x)-2f_2(x) =-x_1^2-2x_2^2+3x_1+4x_2=-x_1^2-2(x_2-2.5)^2+3x_1-6x_2+13.5 &\leq 3(x_1-2x_2)+13.5\leq -60+13.5\leq -20.
\begin{aligned}q_2:=b^{-1}(\widetilde{q}_1-\lambda (q,q_1)(dF)^{p-1}) \in w^p+b^{-1}v+b^{p-2}R[bx,y] &\subset x^p+R[bx,y]+R[x,y]\cap b^{-1}R[bx,y]. \end{aligned}
Tr\left({\tilde L}(\lambda|x_N,\partial_{x_N}) \ldots{\tilde L}(\lambda|x_1,\partial_{x_1})\right) =\prod_{i=1}^N -(\lambda +\beta_i)\frac{w_i(\lambda-1)}{w_i(\lambda )}+\prod_{i=1}^N\frac{w_i(\lambda+1)}{w_i(\lambda )}~.
\mathbb{S}^\rho(f_1,f_2,f_3) :=\sum_{Q\in\mathcal{D}}S_Q(f_1,f_2,f_3) :=\sum_{Q\in\mathcal{D}}\iiint_{Q\times Q\times Q}K_Q(x_1,x_2,x_3)\prod_{j=1}^3f_j(x_j)\,dx_1\, dx_2\, dx_3,
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{f o r I m}\ \nu < 0
D_{x_j}E_{ir}- D_{x_j}(Q_r)g_{i,z}= D_{x_j}\left(g_{i,r}- \frac{g_{i,z}g_{,r}}{g_z}\right) + D_{x_j}\left(\frac{g_{,r}}{g_z}\right)g_{i,z}= g_{ij,r}- g_{ij,z}\frac{g_{,r}}{g_z}.
I_{1,n}\leq \sum\limits_{k=1}^n{\left|\frac{\Psi_k \Gamma(\mu+\alpha+1)}{\Gamma(\mu+1)T^{\alpha+\mu}E_{\alpha,1+\frac{\beta}{\alpha},1+\frac{\beta+\mu}{\alpha}}(-\lambda_{k}T^{\alpha+\beta})}\right|^2}.
\psi_1(u,v,w)= \alpha(w/u)+u(1+\beta v), \psi_2(u,v,w)=\alpha (w/u)+v(1+\beta u), \psi_3(u,v,w)= \alpha (w/v)+v(1+\beta u)\psi_4(v,w)=\alpha (w/v)+u(1+\beta v).
\frac{u-(a_1-1)}{u-a_1}\frac{u-(a_2-1)}{u-a_2}\frac{u-(a_3-1)}{u-a_3}&\quad=1+3u^{-1}+(a_1+a_2+a_3+3)u^{-2}+(a_1^2+a_2^2+a_3^2+2a_1+2a_2+2a_3+1)u^{-3}&+(a_1^3+a_2^3+a_3^3+2a_1^2+2a_2^2+2a_3^2+a_1a_2+a_1a_3+a_2a_3+a_1+a_2+a_3)u^{-4}&\qquad+\ldots.
K_0+\dots+K_{2^J-1}=\sum_{j=0}^{J-1}\left |\{0\leq r < 2^J - 1 \;|\; K_r= 2^j K_0 \}\right | 2^jK_0+K_{2^J-1}&=\sum_{i=0}^{J-1}2^{J-1-j}2^j K_0+2^J K_0=(J+2)2^{J-1}K_0.
S [ \psi ] = V{\Biggr (}2 A \int_{0}^{1 / 2}W [ \psi ( x ) ] d x{\Biggl )}- 2 ( 1 - ( g + s + r / 2 ) ) \int_{0}^{1 / 2}d x \int_{0}^{1 / 2}d y \log | \psi^{2}( x ) - \psi^{2}( y ) | .
V [{\bf j}] = \sum_{n = - \infty}^{+ \infty}\int d^{3}x \left\{\Lambda |{\bf j}| \left[ \ln \left[ \frac{\Lambda}{2 \zeta}|{\bf j}| + \sqrt{1 + \left( \frac{\Lambda}{2 \zeta}|{\bf j}| \right)^{2}}\right] + 2 \pi i n \right] - 2 \zeta \sqrt{1 + \left( \frac{\Lambda}{2 \zeta}|{\bf j}| \right)^{2}}\right\}.
\begin{aligned}\dot q = -\frac{1}{4}(q+p)^3 \left(q+ \O((q+p)^3)\right), \dot p = \frac{1}{4}(q+p)^3 \left(p+ \O((q+p)^3)\right), \dot w = \frac{1}{4}(q+p)^2(p-q)\tilde w + \O((q+p)^5), \dot \phi = \omega^0 + \O((q+p)^6).\end{aligned}
\begin{aligned}\sum_l \left(S_{n - 1;l}(X) - \frac{c}{C^m_n}S_{m - 1;l}(X)\right) X^2_{l\bar l}&= S_n (X) \left(b_t f S_{n - 1}(X^{-1}) + (m + 1) \frac{c}{C^m_n}S_{n - m - 1}(X^{-1})\right) &\leq C(t) S_n (X) . \end{aligned}
\mathfrak{k}_{0}= \{X \in \mathfrak{k}_{1}\cap \mathfrak{k}_{2}\mid [\mathfrak{a}, X] =\{0\}\}, V(\mathfrak{k}_{1}\cap \mathfrak{m}_{2}) = \{X \in \mathfrak{k}_{1}\cap \mathfrak{m}_{2}\mid [\mathfrak{a}, X] =\{0\}\}, V(\mathfrak{m}_{1}\cap \mathfrak{k}_{2}) = \{X \in \mathfrak{m}_{1}\cap \mathfrak{k}_{2}\mid [\mathfrak{a}, X] =\{0\}\},
p_n^{\epsilon,\Delta t}=e^{-\frac{n\Delta t}{\epsilon^2}}p_0^{\epsilon}+\frac{\Delta t}{\epsilon}\sum_{k=0}^{n-1}e^{-\frac{(n-k)\Delta t}{\epsilon^2}}f(q_k^{\epsilon,\Delta t})+\frac{1}{\epsilon}\sum_{k=0}^{n-1}e^{-\frac{(n-k)\Delta t}{\epsilon^2}}\sigma(q_k^{\epsilon,\Delta t})\Delta\beta_k.
G_{\pm}( \tau , \theta ) = e^{\pm i \pi ( \omega + \nu ) / 2}( 2 / \omega )^{\nu - 1 / 2}\Gamma ( \nu ) \pi^{- 1 / 2}\exp \biggl \{- \frac{\omega}{2}[ \tau^{2}+ \theta^{2}] \biggr \}\ .
\left(C_1\cup C_2 \cup \bigcup_{i=1}^{g-1}R_i\, ;\, y_{g,1},y_{g,2},...,y_{11,1},y_{11,2}\right) \longrightarrow \left(C \cup \bigcup_{i=1}^{g-1}R'_i\, ; \, y_g,...,y_{11}\right),
\begin{aligned}D^{(1)}= D_1D_2^2D_3^3D_4^4, D^{(2)}= D_1^2D_2^4D_3^6D_4^8/D_3^5D_4^5=D_1^2D_2^4D_3D_4^3, D^{(3)}= D_1^3D_2^6D_3^9D_4^{12}/D_2^5D_3^5D_4^{10}=D_1^3D_2D_3^4D_4^2, D^{(4)}= D_1^4D_2^8D_3^{12}D_4^{16}/D_2^5D_3^{10}D_4^{15}=D_1^4D_2^3D_3^2D_4.\end{aligned}
\Sigma_{\mathrm{D}}(\delta) := \left\{\gamma \in \mathcal{P}(\mathcal{S}_1 \times \mathcal{S}_2): \boldsymbol{W}_{p_1}(\gamma_1, \mu_1) \leq \delta_1^{1/p_1}, \ \boldsymbol{W}_{p_2}(\gamma_2, \mu_2) \leq \delta_2^{1/p_2}\right\}.
\Omega(f \otimes \lambda^p \iota) = f(0) \Omega(1 \otimes \lambda^p\iota), \Omega^2(f \otimes \lambda^p \iota) = f(0) \otimes P^L_p(1, -1, \dots, (-1)^{p-1}; \lambda^1 \iota, \dots, \lambda^p \iota),
\nabla^{nc}_X (\alpha \cdot \psi) = (\nabla_X^{nc}\alpha ) \cdot \psi = \tau^{-1}\left({R}^{\nabla^{nc},\mathcal{T}(M)}(V_{\alpha},X) \right) \cdot \psi &= 2 \cdot{R}^{\nabla^{nc},\mathcal{S}}(V_{\alpha},X)\psi \stackrel{\psi \in \mathfrak{g}_1}{=}0,
\begin{array}{c c}{\displaystyle{\langle \psi^{i}( z_{1}) \psi^{j}( z_{2}) \rangle = C_{\psi}\frac{V^{i j}( z_{1 2})}{( \det{\cal X}_{1 2})^{\frac{1}{2}\eta}}}}&{V^{i j}( z_{1 2}) = V_{k}^{j}( z_{1 2}){\cal E}^{k i}}\\ \end{array}
\mathcal{CQ}f(x,u, b,\xi):=\inf\left\{\frac{1}{|D|}\int_{D}f(x,u, b+\eta,\xi+\nabla \varphi)\,dy:\eta\in L^{\infty}( D;\mathbb{R}^{m}), \int_{D}\eta dy=0,\varphi\in W_{0}^{1,\infty}(D;\mathbb{R}^d) \right\},
I = \int d x ( \Omega^{\prime}\rho^{\prime}+ \frac{3}{4}e^{2 \rho}\Omega^{- 1 / 3}- \frac{1}{4}\Omega f^{\prime 2}- \frac{1}{4}\Omega f_{1}^{\prime 2}- \frac{1}{4}\Omega f_{2}^{\prime 2}- \frac{1}{4}\Omega f_{1}^{\prime}f_{2}^{\prime}
|X^n_r-X^n_s|^{a}= |\psi_n(r, V^n_r)-\psi_n(s, V^n_s)|^{a}& \leq C ( |V_r^n-V_s^n| +|r-s|^\gamma )^{a}& \leq C (|V_r^n-V_s^n|^{a}+|r-s|^{a\gamma}),
\begin{aligned}\sum_{i=1}^NL_iv\frac{v_i}{u_{i,\infty}}= \sum_{r=1}^Rk_ru_{\infty}^{y_r}\sum_{i=1}^{N}\sum_{j=1}^{N}(y_{r,i}' - y_{r,i})y_{r,j}\frac{v_i}{u_{i,\infty}}\frac{v_j}{u_{j,\infty}}&= \sum_{r=1}^Rk_ru_{\infty}^{y_r}\sum_{i=1}^{N}\sum_{j=1}^{N}(y_{r,i}'y_{r,j}- y_{r,i}y_{r,j})\frac{v_i}{u_{i,\infty}}\frac{v_j}{u_{j,\infty}}.\end{aligned}
[\begin{matrix}1&0&-1\\ 2&0&-2\\ 1&0&-1\end{matrix}]=[\begin{matrix}1\\ 2\\ 1\end{matrix}][\begin{matrix}1&0&-1\end{matrix}]=[\begin{matrix}1\\ 1\end{matrix}]*[\begin{matrix}1\\ 1\end{matrix}][\begin{matrix}1&-1\end{matrix}]*[\begin{matrix}1&1\end{matrix}]
\lefteqn{(-1)^n\sum_{j=0}^k(\lambda-1)_{k-j}\binom{k}{j}\left(x^{j}f(x)\right)^{(j+n)}}&= (-1)^n\sum_{j=0}^k(\lambda-1)_{k-j}\binom{k}{j}\sum_{m=0}^{j}\binom{n+j}{j-m}\frac{j!}{m!}x^{m}f^{(n+m)}(x) &= (-1)^n\sum_{m=0}^k\left\{\sum_{j=m}^{k}(\lambda-1)_{k-j}\binom{k}{j}\binom{n+j}{j-m}\frac{j!}{m!}\right\}x^{m}f^{(n+m)}(x).
\begin{aligned}\Phi_k(z,s,a) =\sum_{m_1,\ldots,m_k=0}^\infty\frac{z^{m_1+\dots+m_k}}{(m_1+\dots+m_k+a)^{s}}&=\frac1{a^s}+\sum_{r=0}^{k-1}\sum_{\{m_{i_1},\ldots,m_{i_{k-r}}\}\subset S}\left[\sum_{m_{i_1},\ldots,m_{i_{k-r}}=1}^\infty\frac{z^{m_{i_1}+\cdots+m_{i_{k-r}}}}{(m_{i_1}+\cdots+m_{i_{k-r}}+a)^{s}}\right].\end{aligned}
\begin{array}{l l l l}{{T^{0}}}&{{=}}&{{J^{0}{\omega}^{0}\wedge{\omega}^{3}\nonumber}}\\{{T^{1}}}&{{=}}&{{J^{1}{\omega}^{3}\wedge{\omega}^{1}+ J^{2}{\omega}^{0}\wedge{\omega}^{1}\nonumber}}\\{{T^{2}}}&{{=}}&{{J^{1}{\omega}^{3}\wedge{\omega}^{2}+ J^{2}{\omega}^{0}\wedge{\omega}^{2}\nonumber}}\\{{T^{3}}}&{{=}}&{{J^{3}{\omega}^{0}\wedge{\omega}^{3}\nonumber}}\\ \end{array}
\sum_{\pi \in \mathcal{C}_n}f_{\vert \pi \vert}g_{\pi}= \sum_{p=1}^{n}f_p\sum_{\underset{k_{i}\ge1}{k_{1}+\dots+k_{p}=n}}g_{k_{1}}\dots g_{k_{p}}= \sum_{p=1}^{n}\left(\sum_{m=p}^{n}f_m \binom{m}{p}\right)\sum_{k_{1}+\dots+k_{p}=n}g_{k_{1}}\dots g_{k_{p}}.
H^1_{\mathcal{F}+\mathcal{G}}(K_v,E[p]) = H^1_\mathcal{F}(K_v,E[p]) + H^1_\mathcal{G}(K_v,E[p]) H^1_{\mathcal{F}\cap\mathcal{G}}(K_v,E[p]) = H^1_\mathcal{F}(K_v,E[p])\cap H^1_\mathcal{G}(K_v,E[p]).