harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
F(U_0) := \int_{U_0}^{U_+}\frac{(1 - \sqrt{3}u^2)}{(1 + \sqrt{3}u^2)^2}\frac{du}{\sqrt{E + A(u)}}=\frac{3 \sqrt{3}U_0^2 \sqrt{1 - U_0^4}}{4 (1 - 3 U_0^4)}=: G(U_0).
\int_{0}^{\infty}\mathrm{d}r \; \left[ u_{k 0}^{2}\left( r \right) - v_{k 0}^{2}\left( r \right) \right] = 2 \eta_{0}^{\prime}\left( k \right) + 2 \pi \delta \left( k \right) \sin^{2}\eta_{0}\left( k \right)
I m \int_{0}^{\infty}d \nu \frac{e^{- 2 \pi \nu + i \pi \alpha + 2 i \nu t}}{1 + e^{- 2 \pi \nu + i \pi \alpha}}= \frac{1}{4 t}- \frac{e^{- \alpha t}}{4 \operatorname{s i n h}t}\ .
f_{e^{\prime}}(x) = f_{e}(x),\f_{\bar{e}^{\prime}}(x)=f_{\bar{e}}(x)\x \in G_{e^{\prime}}=G_e \, , & f_{e^{\prime \prime}}=id_{G_{e''}}, \f_{\bar{e}^{\prime \prime}}=id_{G_{e''}}.
M^{\prime}_{R,p,q}({\mathbf{v}}) := \frac{1}{[p+q]!}\sup_{R/2 \leq r < R}(R - r)^{p+q+2}\|\partial_{y}^{q}\nabla_{x}^{p}{\mathbf{v}}\|_{L^{2}(B_{r}^{+})}.
\dot{P}_{\alpha}(\theta) =\sum_{l=0}^{N}\left(\dot{p}_{\alpha}(l+1)-\dot{p}_{\alpha}(l)\right) \Theta \left(\theta-\theta_l \right) -\sum_{l=0}^{N}\left(p_{\alpha}(l+1)-p_{\alpha}(l) \right) \delta(\theta-\theta_l)\dot{\theta_l}
h(\sigma,\epsilon) = \max_{\mu_1 \in \bar{M}_{\epsilon}}\biggl[ H_2(\sigma) -\gamma\frac{\delta-1}{\delta}H_2 (\mu_1) \quad\qquad+\inf_{u>0}\{\sigma \log p(u) + (1-\sigma) \log q(u) - \mu_1 \gamma \log u \}\biggr].
\nabla_{j}= \left( \begin{array}{c c}{i \theta_{j k}^{- 1}\hat{x}^{k}}{0}{0}{\partial_{j}}\end{array}\right) + i \left( \begin{array}{c c}{( D_{j})_{i}^{l}}{0}{0}{( \hat{D}_{j})_{\beta}^{\alpha}}\end{array}\right)
\mathbb{I}^m_n=\left\{(X_1,\ldots,X_m)\in M_n^m\: \left|\:\: \begin{array}{l}X_j X_k-X_k X_j=\mathbf{0}_n, X_j-X_j^\ast=\mathbf{0}_n, \|X_j\|\leq 1\end{array}1\leq j,k\leq m\right.\right\}
\begin{array}{l}x^ix^j=qx^jx^i~,~~~~~{\rm for}~i<j~ ~{\rm and}~~i\not=-j~, qx^{+2}x^{-2}-q^{-1}x^{-2}x^{+2}=\frac{q^{1/2}-q^{-1/2}}{q-1+q^{-1}}\frac{1}{a^2}~, qx^{+1}x^{-1}-q^{-1}x^{-1}x^{+1}=(1-q^2)x^{+2}x^{-2}+q\frac{q^{1/2}-q^{-1/2}}{q-1+q^{-1}}\frac{1}{a^2}~.\end{array}
\phi(\boldsymbol{\lambda}(a,b)) =\phi(\mathbf{h}_{11}(a,b)-(-1)^{|a||b|}\mathbf{h}_{11}(1,ba)) &=e_{11}([a,b]-\overline{[a,b]})\oplus(2\langle a,b\rangle_{\mathsf{d}}^--2\langle1,ba\rangle_{\mathsf{d}}^-).
e^{i{\bf k}\cdot ({\bf x}-{\bf x_{0}})}= \sum_{m , n = - \infty}^{\infty}J_{m}( k \rho ) J_{n}( k \rho_{0}) ( - 1 )^{n}e^{i m \varphi}e^{i n \varphi_{0}}e^{i ( m + n ) \theta},
\begin{aligned}A_1^{(N)}(0,x;s)=1 &A_r^{(N)}(i,x;s)=\frac{s-1}{r-1}A_{r-1}^{(N)}(i,x;s-N)-\frac{x-(r-1)}{r-1}A_{r-1}^{(N)}(i-1,x;s-N+1).\end{aligned}
I_{1\epsilon}(t,x,\xi) = \epsilon\chi\,'(\epsilon(\mu +h))\sum_{|\alpha| = 1}\Bigl\{h^{(\alpha)}h_{s(\alpha)}- h^{(\alpha)}_sh_{(\alpha)}\Bigr\}= \epsilon\chi\,'\bigl(\epsilon(\mu +h(t,x,\xi))\bigr)\sum_{|\alpha| = 1}\frac{i}{2m^2}(\xi_{\alpha}- A_{\alpha}(t,x))(-i\partial_x)^{\alpha}\nabla\cdot A(t,x).
\vec{B}=\frac{\Phi_0}{2\pi\lambda^2}\,\vec{b}\ \ ,\ \ \frac{\vec{E}}{c}=\frac{\Phi_0}{2\pi\lambda^2}\,\vec{e}\ \ ,\ \F_{\mu\nu}=\frac{\Phi_0}{2\pi\lambda^2}\,f_{\mu\nu}\ \ ,\ \A^0=\frac{\Phi_0}{2\pi\lambda}\,\varphi\ \ ,\ \ \vec{A}=\frac{\Phi_0}{2\pi\lambda}\,\vec{a},
\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}
\frac{\delta \Sigma}{\delta c}+ \varepsilon^{a b}\overline{{{c}}}^{a}\frac{\delta \Sigma}{\delta b^{b}}= - \partial^{2}\, \overline{{{c}}}- \partial^{\mu}A_{\mu}^{*}+ \varepsilon^{a b}A_{\mu}^{a *}A^{b \mu}- \varepsilon^{a b}c^{a *}c^{b}+ ( \xi - \alpha ) \varepsilon^{a b}b^{a}\overline{{{c}}}^{b}\; .
n_{d}+N_{d}\cdot(n_{d-1}+N_{d-1}\cdot(n_{d-2}+N_{d-2}\cdot(\cdot\cdot\cdot+N_{2}n_{1})\cdot\cdot\cdot)))=\sum_{k=1}^{d}(\prod_{l=k+1}^{d}N_{l})n_{k}
\left. -\frac{8}{9}{\rm Tr}(T^{a}_{f}T^{b}_{f}T^{c}_{f}T^{d}_{f}) \right]F^{a}_{\mu\nu}F^{b}_{\nu\mu}F^{c}_{\rho\sigma}F^{d}_{\sigma\rho}+ \left[ \frac{41}{60}f^{efa}f^{fgb}f^{ghc}f^{hed}+ \right.
\sum_{b = 1}^{N}m^{( b )}c^{( b )}: \cos \left( 2 \sqrt{\pi}\sqrt{\frac{\pi}{\pi \! + \! g N}}U_{1 b}\Phi^{( 1 )}( x ) + 2 \sqrt{\pi}\sum_{I = 2}^{N}U_{I b}\Phi^{( I )}( x ) - \frac{\theta}{N}\right) :
\begin{array}{c}\mu(x, y)=0.24, \mathbf{b}(x, y)=[20, 2 \sin(6x)]^T, \sigma(x, y) = 0.1, begin{align*}2mm]f(x,y)=1000\chi_{R_1}(x, y) + 1000\chi_{R_2}(x, y), \end{array}
\kappa(\Omega) = \sup\Bigg\{\min_{A\in L_\Omega}\|\nabla u - A\|_{L^2(\Omega)};~~ u\in W^{1,2}(\Omega,\mathbb{R}^n), ~ u\cdot\vec n = 0 \mbox{on}\partial\Omega \mbox{and}\|D(u)\|_{L^2(\Omega)}=1 \Bigg\},
A e ( n_{1}n_{2}n k ; i ) = \sum_{n_{1}^{\prime}n_{2}^{\prime}n^{\prime}k^{\prime}i^{\prime}}A ( n_{1}^{\prime}n_{2}^{\prime}n^{\prime}k^{\prime}; i^{\prime}| n_{1}n_{2}n k ; i ) e ( n_{1}^{\prime}n_{2}^{\prime}n^{\prime}k^{\prime}; i^{\prime})
[\Psi_G]_q=q^{N_G-1}-[\Psi^1]_q+ q^2 [\Psi^{12},\Psi^1_2,\Psi^2_1,\Psi_{12}]_q+q [\Psi^{1,2}]_q- q[\Psi^{12},\Psi^2_1]_q.
q_2(D)u = p(D)^{-1}[((1-\chi) p)(D),x^\beta]\partial^\alpha u &=-\sum_{0\not=\gamma\leq\beta}i^{|\gamma|}\binom{\beta}{\gamma}p(D)^{-1}\big(\partial^\gamma_\xi ((1-\chi)p)\big)(D)(x^{\beta-\gamma}\partial^\alpha u).
\Pr\big(\Delta(C_M)>\delta\big) =\Pr\big(\Delta(C_M)>\delta\mid{\rm rank}(M)=k\big) \Pr\big({\rm rank}(M)=k\big) ~~ +\Pr\big(\Delta(C_M)>\delta\mid{\rm rank}(M)<k\big) \Pr\big({\rm rank}(M)<k\big).
Z \left({\cal A}, G , p , t_{n , l}^{( A )}\right) = \sum_{\{r_{n , l}^{( A )}\}}Z \left({\cal A}, G , p , r_{n , l}^{( A )}\right) \prod_{A = 1}^{p}\prod_{l = 1}^{\infty}\prod_{n = - \infty}^{+ \infty}\left[ t_{n , l}^{( A )}\right]^{r_{n , l}^{( A )}}.
\lambda_4 = \gamma_4 - \tfrac{3}{5}A^2, &\lambda_6 = \gamma_6 - \tfrac{4}{5}A \gamma_4 - \tfrac{2}{25}A^3, &\lambda_8 = -\tfrac{6}{5}A \gamma_6 - \tfrac{3}{25}A^2 \gamma_4 + \tfrac{12}{125}A^4, &\lambda_{10}= \tfrac{9}{25}A^2 \gamma_6 + \tfrac{18}{125}A^3 \gamma_4 + \tfrac{72}{3125}A^5.
\aligned \partial \mathcal N(\alpha_-,\alpha_+) &\supseteq (\mathcal M(\alpha_-,\alpha_+) \times_{R_{\alpha_+}}\mathcal N(\alpha_+,\alpha_+)) \cup (\mathcal N(\alpha_-,\alpha_-) \times_{R_{\alpha_-}}\mathcal M(\alpha_-,\alpha_+)) & = \mathcal M(\alpha_-,\alpha_+) \sqcup \mathcal M(\alpha_-,\alpha_+)\endaligned
w^\beta_+ w^\gamma_- = ( \theta^\beta_+ \theta^\gamma_- b_{\beta,\gamma}+ b_{\alpha+\beta, \alpha+\gamma})e^{\beta+\gamma}+ (\theta^\beta_+ b_{\beta, \alpha + \gamma}+ \theta^\gamma_- b_{\alpha+\beta, \gamma})e^{\alpha+\beta+\gamma}
\hat{\gamma}_{sa,br}(n) =\displaystyle \sum_{w \in L(k)}s^{\ast}(r^{-1}w) \displaystyle \sum_{b_{1}ha_{1},r',r''}w^{\ast}(r'r'')C_{r''a,ha_{1}}Y_{[b_{1}ha_{1}]}(P)D_{b_{1},br'}n &=\displaystyle \sum_{b_{1}ha_{1},r',r''}s^{\ast}(r^{-1}r'r'')C_{r''a,ha_{1}}Y_{[b_{1}ha_{1}]}(P)D_{b_{1},br'}n
\begin{array}{r l}{{N}}&{{e v e n : \; \; \frac{1}{2}- \frac{1}{p_{0}}+ \frac{1 - v_{s}}{4}< \frac{s}{p_{0}}< \frac{1}{2}+ \frac{1 - v_{s}}{4}}}\\{{N}}&{{o d d : \; \frac{1}{2}- \frac{3}{4 p_{0}}+ \frac{1 - v_{s}}{4}< \frac{s}{p_{0}}< \frac{1}{2}- \frac{1}{4 p_{0}}+ \frac{1 - v_{s}}{4}}}\\ \end{array}
W = a_1 \frac{H^2}{H^k}+ 2! a_2\, \frac{H^3}{H^k}+ \dots + (k-1)!\, a_{k-1}\,\frac{H^k}{H^k}& = a_1\, \delta^{(k-2)}+ 2!\, a_2\, \delta^{(k-1)}+ \dots + (k-1)!\, a_{k-1}\, \delta.
{\widetilde{\bf s}}_{\textrm{ind},m}[f_k]= (\widetilde{\alpha}_{m,2}{\bf E}_{m,k}{\bf\Phi}_m{\bf G}_{m,k}+\widetilde{\alpha}_{m,3}{\bf D}_{m,k}{\bf\Phi}_m{\bf B}_{m,k}+\widetilde{\alpha}_{m,4}{\bf D}_{m,k}{\bf\Phi}_m{\bf W}_{m,k}{\bf\Phi}_m{\bf G}_{m,k}){\bf F}_k{\bf s}_k,
\mathfrak{p}_{N}^{R}(\mathbf{s}_{N};\mathbf{p}_{N}):=\begin{cases}\frac{\prod_{i=1}^{N}p_{i,N}^{s_{i}}(1-p_{i,N})^{1-s_{i}}}{\sum_{\mathbf{s}_{N}'\in\Omega_{N,n}}\prod_{i=1}^{N}p_{i,N}^{s_{i}'}(1-p_{i,N})^{1-s_{i}'}}\mathbf{s}_{N}\in\Omega_{N,n}, 0 ,\end{cases}
\int_0^T\int_\Omega\Big(\lambda_1(f_1^*-r_1^*)f^\epsilon_{1,1}+\lambda_2(f_2^*-r_2^*)f^\epsilon_{2,1}-\beta_1^0\dfrac{\partial z_{g_1}}{\partial t}-\beta_2^0\dfrac{\partial z_{g_2}}{\partial t}-s_1^0\dfrac{\partial z_{g_3}}{\partial t}-s_2^0\dfrac{\partial z_{g_4}}{\partial t}\Big)dxdy d t=-\varpi(\epsilon, T).
\frac{d^{2}x^{\lambda}}{dt^{2}}(\frac{dt}{dT})^{2}+\frac{dx^{\lambda}}{dt}\frac{d^{2}t}{dT^{2}}=-\Gamma_{\nu\alpha}^{\lambda}\frac{dx^{\nu}}{dt}\frac{dx^{\alpha}}{dt}(\frac{dt}{dT})^{2}.
\theta_1({\bf v};z):=\sum_{p\geq0}\theta_{1,p}({\bf v}) z^p = -2e^{zv}\sum_{m\geq0}\Bigl(\gamma-\frac12 u+\psi(m+1)\Bigr)e^{m u}\frac{z^{2m}}{m!^2}, &\theta_2({\bf v};z):=\sum_{p\geq0}\theta_{2,p}({\bf v}) z^p = z^{-1}\biggl(\sum_{m\geq0}e^{m u+z v}\frac{z^{2m}}{m!^2}-1\biggr),
2M \sum_{n=1}^{\infty}q_n \sum_{k=1}^n p_{x,k-1}\leq 2M \cdot \left[ \sum_{n=1}^{n_0}q_n \sum_{k=1}^{n_0}p_{x,k-1}~+~ \sum_{n > n_0}q_n \sum_{k=1}^n p_{x,k-1}\right] & \leq 2M \cdot \left[ \sum_{k=1}^{n_0}p_{x,k-1}~+~ \sum_{n > n_0}q_n \cdot n\right].
\beta(v_{n+1},v_{i})+\beta(v_{n+1},v_{i+2})+\beta(v_{i},v_{i+2})+\beta(v_{i+2},v_{i}) =1 \beta(v_{n+1},v_{i})+\beta(v_{n+1},v_{i+3})+\beta(v_{i},v_{i+3})+\beta(v_{i+3},v_{i}) =1
\hat{\epsilon}_+(w)=-4p_0I(\hat{\rho}_+^{(0)}(w)+\hat{\rho}_+^{(1)}(w)+\dots)+\frac{\tilde{z}h}{i(w-i\epsilon)}\frac{\hat{J}_-(0)}{\hat{J}_+(w)}+\sum_j\frac{\tilde{b}_j\tilde{z}H\hat{J}_-(0)\hat{J}_-(iw_j)}{iw_j\hat{J}_+(w)i(w-iw_j)}e^{-2Bw_j}+\dots
tg'(t)= \big[\big(2^*(s_2)-2\big)t^{2^*(s_2)-2^*(s_1)}-\big(2^*(s_1)-2\big)\big]\lambda_1 b t^{2^*(s_1)-2}&+\big[\big(2^*(s_2)-2\big)t^{2^*(s_2)-p-1}-(p-1)\big]\lambda_3dt^{p-1}&-a\big(2^*(s_2)-2\big)t^{2^*(s_2)-2}=: I+II+III.
\tilde{\phi}_n^k (t,z) - \tilde{\phi}_n^k (t,y) = 5\theta_n |g_n(t)|\psi_n^k(t) - \psi_n^k(t)|y| \geq 0; \shortintertext{and so}|\tilde{\phi}_n^k (t, z) - \tilde{\phi}_n^k (t,y)| = \psi_n^k (t)(5\theta_n |g_n(t)| - |y|) \leq \psi_n^k(t)(|z| - |y|) \leq \psi_n^k(t)| z-y|.
\frac{d}{dt}\begin{bmatrix}u \phi_0 \vdots\end{bmatrix}=\begin{bmatrix}A W_\infty 0 H_\infty\end{bmatrix}\begin{bmatrix}u \phi_0 \vdots\end{bmatrix}, \begin{bmatrix}u(0) \phi_0(0) \vdots\end{bmatrix}=\begin{bmatrix}u_0 e_1 \end{bmatrix}.
P^*(t) = \begin{cases}0 \textrm{for}\E_b(t)< E_{b,}(t). \frac{V}{\Delta t(V\zeta_{\max}+\Delta tP_{\max}+E_{\min}-E_b(t))}-\frac{1}{\gamma(t)}\textrm{for}\E_{b,}(t) \le E_b(t) \le E_{b,}(t) P_{\max}\textrm{for}\E_b(t)>E_{b,}(t)\end{cases}
\rho_{\mu}( x ) = \rho_{\nu}( x ) \frac{q^{C ( \mu ) / 2}+ \epsilon ( \mu ) \epsilon ( \nu ) x q^{C ( \nu ) / 2}}{x q^{C ( \mu ) / 2}+ \epsilon ( \mu ) \epsilon ( \nu ) q^{C ( \nu ) / 2}}, ~ \forall \mu \neq \nu .
u_n \Phi(x) u_n^{-1}- \Phi (x) = \Phi\left(v_n x v_n^{-1}-x\right)=-2\Phi(y_n) + \sum\limits_{k=1}^{n-1}\left(u_n \Phi(y_k) u_n^{-1}- \Phi(y_k)\right).
\begin{array}{l}\Phi_{1}\Phi_{2}=p^{-1}\Phi_{2}\Phi_{1}, \Phi_{1}\bar{\Phi}_{1}=1+pq\bar{\Phi}_{1}\Phi_{1}+(pq-1)\bar{\Phi}_{2}\Phi_{2}, \Phi_{2}\bar{\Phi}_{2}=1+pq\bar{\Phi}_{2}\Phi_{2}, \Phi_{1}\bar{\Phi}_{2}=p\bar{\Phi}_{2}\Phi_{1}, \Phi_{2}\bar{\Phi}_{1}=q\bar{\Phi}_{1}\Phi_{2}, \bar{\Phi}_{1}\bar{\Phi}_{2}=q\bar{\Phi}_{2}\bar{\Phi}_{1}.\end{array}
\frac{1}{2}\frac{1}{\Gamma(\frac{1}{\alpha}+1)}e^{-|z|^{\alpha}}=\frac{1}{2}\frac{1}{\Gamma(\frac{1}{\alpha}+1)}e^{-(z^{2})^{\alpha/2}}=\int_{0}^{\infty}\frac{1}{s}(\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}(z/s)^{2}})V_{\alpha}(s)ds,
I_{1}\ll\frac{1}{n^{5/2}}\sum_{v=2}^{a+1}\sum_{r\in I_{v}^{\prime}}\Vert\big ({\widetilde{X}}_{ij}{\widetilde{X}}_{i\ell}-{\mathbb{E}}({X}_{ij}{X}_{i\ell})\big )\big({\widetilde{X}}_{ir}-{\mathbb{E}}({\widetilde{X}}_{ir}|{\mathcal{F}}_{u-v}^{X}\big )\Vert_{1}\ll\frac{1}{n^{5/2}}(Aap)\sigma^{2}\,.
\frac{d \hat{N}(t)}{dt}\leq \sum_{i=1}^I \sum_{k= i+1}^I \hat{N}_k(t) S(x_k) b(x_i,x_k) \Delta x_i - \sum_{i=1}^I \hat{N}_i(t) S(x_i) \sum_{j= 1}^{i-1}\frac{x_j}{x_i}b(x_j,x_i) \Delta x_j.
\rho_{j+1,t}\circ \psi_j(s_{\theta_j}^m i_{\theta_j}(f)s_{\theta_j}^{*n}) = e^{it(Nm-Nn)/N^{j+1}}s_{\theta_{j+1}}^{Nm}i_{\theta_{j+1}}(f)s_{\theta_{j+1}}^{*Nn}= e^{it(m-n)/N^j}s_{\theta_{j+1}}^{Nm}i_{\theta_{j+1}}(f)s_{\theta_{j+1}}^{*Nn}= \psi_j\circ\rho_{j,t}(s_{\theta_j}^m i_{\theta_j}(f)s_{\theta_j}^{*n}).
A=circshift(A,[\lfloor \frac{y_1}{4}\rfloor~~-\lfloor\frac{y_2}{4}\rfloor~~\lfloor\frac{y_3}{4}\rfloor]), &A=circshift(A,[\lfloor \frac{y_1+y_2}{10}\rfloor~~\lfloor\frac{y_2+y_3}{10}\rfloor~~-\lfloor \frac{y_3+y_1}{10}\rfloor]),
g^{2}(0)=1+\frac{Var(\hat{a}^{\dagger}\hat{a})-\langle\hat{a}^{\dagger}\hat{a}\rangle}{(\langle\hat{a}^{\dagger}\hat{a}\rangle)^{2}}=1+\frac{Var(n)-\overline{n}}{\overline{n}^{2}}
\prod_{\substack{1 \leq k \leq r, 1 \leq l \leq N_k, i_{k,l}= s}}e^{\mathbf{a}_{k}(l) \varpi_s}(b_{k,l})= \prod_{j=1}^r e^{d_{j,s}\varpi_s}(b_{k(j,s), m(j,s)})
\sum_{p \leq x}\sum_{1 \leq i \leq j \leq M}\alpha_{\pi}(p, i)\alpha_{\pi}(p, j)\log p= \frac{1-\delta(\pi)}{2}x+O(x^{1/2}\log^{2}x), \mbox{and}\sum_{p \leq x}\sum_{1 \leq i < j \leq M}\alpha_{\pi}(p, i)\alpha_{\pi}(p, j)\log p= \frac{\delta(\pi) + 1}{2}x+O(x^{1/2}\log^{2}x),
\check{\tilde{R}}_{12}(u,v)\check{R}_{23}^{\iota_1\iota_2}(u)\check{\bar{R}}_{12}^{\iota_1}(v)=\check{\bar{R}}_{23}^{\iota_1\iota_2}(v)\check{R}_{12}^{\iota_1}(u)\check{R}_{23}(u,v),
I_3 \leq |\partial_{\phi}\mathcal{F}_h(u^*)|\cdot |\tilde \phi| & \leq \frac{1}{h}(|\epsilon|+\beta_a^{-1}+\beta_b^{-1})C_\phi^* \cdot \gamma |z|(|c|+k|d|)h^2 =C_0C^*_\phi h.
\Gamma_1^0=\Gamma_1(\mathcal{N})=\left\{\gamma=\left[\begin{array}{ccc}a b c g e f h l d\end{array}\right]\in G(\mathcal{O}):\gamma\equiv\left[\begin{array}{ccc}1_m \ast \ast 0 1_r \ast 0 0 1_m\end{array}\right] \mathcal{N}\right\}.
\vartheta_{1}=\int_{-\infty}^{+\infty}\cdot\cdot\cdot\int_{-\infty}^{+\infty}|\int_{0}^{1}e^{2\pi i(\alpha_{n}x^{n}+\alpha_{m}x^{m}+\cdot\cdot\cdot+\alpha_{r}x^{r})}dx|^{2k}d\alpha_{n}d\alpha_{m}...d\alpha_{r},
H(\theta_{ij})=-J\sum_{<i,j>}[S_{i}^{z}S_{j}^{z}+\frac{1}{2}(S_{i}^{+}e^{i\theta_{i}}S_{j}^{-}e^{-i\theta_{j}}+S_{i}^{-}e^{-i\theta_{i}}S_{j}^{+}e^{i\theta_{j}})]
\psi_{b}( w , z ) = \psi \left( \frac{w}{z}\right) \psi \left( w z \right) = \frac{1}{( x w / z ; x^{4})_{\infty}( x z / w ; x^{4})_{\infty}( x w z ; x^{4})_{\infty}( x / z w ; x^{4})_{\infty}}.
\gamma(x)e=T_{\tau(x)^{-1}}e=\lambda(\tau(x))e, \gamma(x)f=T_{\tau(x)^{-1}}T_{\dot{u}}e=T_{\dot{u}\tau(x)^{-1}}e=T_{\dot{u}}T_{\dot{u}\tau(x)^{-1}\dot{u}^{-1}}e=T_{\dot{u}}\lambda(\dot{u}\tau(x)\dot{u}^{-1})e=\lambda(\dot{u}\tau(x)\dot{u}^{-1})f
\mathcal{L}(x_i;\lambda,\mu_i)= \sum_{i=1}^N\frac{-Mx_i}{\sigma_n^2d_i^{\alpha}+\sigma_{v,i}^2M x_i}+\lambda\left(\sum_{i=1}^{N}x_i-P\right)-\sum_{i=1}^N\mu_ix_i\;,
\begin{array}{c c c c}{{\Gamma_{1 , \epsilon}}}&{{\Gamma_{2 , \epsilon}}}&{{\epsilon = 1 , \; 2 \nonumber}}\\{{\Gamma_{1 , \epsilon}}}&{{n_{1}\in Z \ + \; \frac{1}{2}}}&{{n_{2}\; o d d \nonumber}}\\{{\Gamma_{2 , \epsilon}}}&{{n_{1}\in Z}}&{{n_{2}\; e v e n \nonumber}}\\{{\Gamma_{3 , \epsilon}}}&{{n_{1}\in Z + \frac{n_{2}+ 1}{2}}}&{{n_{2}\; \in \; Z ,}}\\ \end{array}
g_6(x)=6x+\frac{15 x^2}{2^{19/20}}+\frac{10 x^3}{3^{19/20}}+\frac{5x^4}{40 \cdot 2^{9/10}}+\frac{x^5}{4 5^{19/20}}+\frac{x^6}{120 \cdot 6^{19/20}}
\tilde{\Omega}= \big\{\omega \in \Omega \colon \big( \, \forall \, q_1, q_2 \in \{6,7,8, \ldots \}, t \in [0,T] \colon \tilde{X}_{q_1,t}(\omega) = \tilde{X}_{q_2,t}(\omega) \big) \big\}
\prod_{j=1}^k e^{ih_j}= \prod_{j=1}^k e^{ih_{j,0}}\prod_{j=0}^{k-1}e^{ih_{k-j,1}}\cdots\prod_{i=1}^{\lfloor \frac{m+1}2\rfloor(k-1)}(u_i,v_i)
\; \; + 2 K_{a}^{b}K_{b}^{a}\: s \partial_{s}\phi^{( 2 )}+ 4 ( K_{a}^{b}K_{b}^{a})^{2}\: s^{3}\partial_{s}\phi_{+}^{( 0 )}+ 4 K_{c}^{a}K_{a}^{b}K_{b}^{c}\: s^{2}\partial_{s}\phi^{( 1 )}.
\mathcal{L}\left(z,\psi_\varepsilon\right)=f\left(z\right)+\psi_\varepsilon\left(Lz\right)-g^{*}_\Psi\left(\psi_\varepsilon\right)\geq f\left(x_\varepsilon\right)+g\left(Lx_\varepsilon\right)-2\varepsilon.
E\left[\int_s^t(\nabla_q\chi)(r,q_r^m,z_r^m)\cdot z_r^m dr\right] = E\left[\int_s^t \int \left((\nabla_q\chi)(r,q_r,z)\cdot z\right) h(r,q_r,z)dzdr\right]+O(m^{1/2}).
\Gamma_{P,j}:= \Gamma_{0, k_1(j), k_2(j), 1, \{(a_i, b_i)\}_{i \in I_j}}; O_j(P):= \left\langle\prod_{i\in{I_j}}\tau^{(a_i,b_i)}_{d_i}\sigma_1^{k_1(j)}\sigma_2^{k_2(j)}\sigma_{12}\right\rangle^{\mathbf{s}^{\Gamma_{P,j}},o}
\begin{array}{r c l}{\vec{\alpha}_{1}}&{=}&{( a_{1 1}, a_{1 2}, \ldots , a_{1 d})}\\{\vec{\alpha}_{2}}&{=}&{( a_{2 1}, a_{2 2}, \ldots , a_{2 d})}\\{\multicolumn{3}{c}{\vdots}}\\{\vec{\alpha}_{d}}&{=}&{( a_{d 1}, a_{d 2}, \ldots , a_{d d}) .}\\ \end{array}
\|\hat{p}^{(k)}(\cdot|Z^n)-\mu_k^{(b)}\|_1 \leq \sum_{a^k}\Big|{t\over n-k}\sum_{i=1}^{k+g}\hat{p}^{(1)}(a^k|S^{(i),t})-\mu_k^{(b)}(a^k)\Big|+{k+g\over n-k}.
\pi_{k}(x,y_{0:k},A)\coloneqq\frac{\mathbf{E}_{x}\left[\mathbf{1}_{A}(X_{k\Delta})\prod_{j=0}^{k}g_{j}(X_{j\Delta},y_{j})\right]}{\mathbf{E}_{x}\left[\prod_{j=0}^{k}g_{j}(X_{j\Delta},y_{j})\right]},A\in\mathcal{B}(\mathbb{R}^{p}),
\begin{array}{l l}{e^{2 \varphi}}&{= 1 - \frac{\mu^{2}}{r^{2}}\cosh^{2}\! \alpha}\\{e^{- 2 \lambda}}&{= 1 - \frac{\mu^{2}}{r^{2}}\cosh^{2}\! \beta ,}\\ \end{array}
u_1(t,x,z) = - \left( G_1 \partial_x f(t,x) + G_2\mu \partial_x g(t,x) -\sigma_2^c\mu\partial_x^3(f+g)(t,x)-\sigma_1^c\partial_x^3f(t,x)\right) begin{align*}5pt] \quad\times \left(f(t,x)z-\frac{1}{2}z^2\right)+\displaystyle{\mu}\partial_z u_2(t,x,f)z.
\hat{A}\geq \frac{1-(2\theta-1)\vert 1-\alpha \vert}{\alpha}D_{A\Omega_1}-\frac{2\theta\alpha-2(\theta-1)\beta}{\alpha}\vert L_{A\Omega_1}\vert -2\theta\vert U_{A\Omega_1}\vert \geq \frac{1-(2\theta-1)\vert 1-\alpha \vert}{\alpha}D_{A\Omega_1}-2\theta\vert B_{A\Omega_1}\vert\:= T.
\Psi(u,q,v)=\sum\limits^{\infty}_{s,j\in \mathbb{N}/2\cup\{0\}}\sum\limits^{s}_{m,n=-s}\sum\limits^{j}_{k,l=-j}D^{s}_{mn}(u)f^{sj}_{nk|ml}(q)D^{j}_{kl}\left(v^{-1}\right),
{\rm and}r_{\alpha,\lambda}(x) = - \frac{1}{\pi}\sum_{k=1}^\infty \frac{(-1)^k}{k}\sin(\pi k \alpha)\frac{\Gamma(k \alpha+1)}{\lambda^k x^{k \alpha}}= -\alpha \sum_{k=1}^\infty \frac{(-\lambda x^\alpha)^{-k}}{\Gamma(1-k \alpha)}
{}a_1^p + (\frac{1}{2}\displaystyle\sum_{k=1}^{2}a_k )^p+\cdots+(\frac{1}{N}\displaystyle\sum_{k=1}^{N}a_k )^p +(\displaystyle \sum_{k=1}^{N}a_k )^p\{\displaystyle\sum_{n={N+1}}^{\infty}\frac{1}{n^p}\}< (\frac{p}{p-1})^{p}(\sum_{k=1}^{N}a_{k}^p ).
I_1= \bigg\| 1_{\{|T_{n}-1|\le 1\}}\sup_{t\in[0,1]}|X_{n}(t)-B(T_{k})|\bigg\|_{L^{p}}\le \bigg\| 1_{\{|T_{n}-1|\le 1\}}\max_{0\le k\le n-1}|\zeta_{k+1}|\bigg\|_{L^{p}}\le \bigg\| \max_{0\le k\le n-1}|\zeta_{k+1}|\bigg\|_{L^{p}}\le Cn^{-\left(\frac{1}{2}-\frac{1}{p}\right)}.
\left( q_{\mathrm{r e d}}, q_{\mathrm{g r n}}, q_{\mathrm{b l u}}\right) \rightarrow \left( q_{\mathrm{r e d}}, q_{\mathrm{g r n}}, q_{\mathrm{b l u}}\right) \exp ( - \mathrm{i}\theta_{a}\lambda_{a}^{\ast}/ 2 )
m_{G}^{2}\; = \; - 2 e^{2}\int_{{\mathcal R}}\frac{d^{2}p}{( 2 \pi )^{2}}\, \frac{\partial}{\partial p_{\mu}}\left[ \frac{\Lambda^{2}}{p^{2}( p^{2}+ \Lambda^{2})}p_{\mu}\right] \; ,
\Psi(\theta):= \mathrm{i}\theta\gamma- \frac{1}2\sigma^2 \theta^2 +\int_{\mathbb{R}\setminus\{0\}}\bigl(e^{\mathrm{i}\theta x}-1-\mathrm{i}\theta x{\mathbf{1}}_{\{|x|\le1\}}\bigr) \Pi(\mathrm{d}x).
\Phi ( r ) = m ( r ) \left( \begin{array}{c c c}{1}&{0}&{0}\\{0}&{- \frac{1}{2}}&{0}\\{0}&{0}&{- \frac{1}{2}}\\ \end{array}\right) + q ( r ) \left( \begin{array}{c c c}{0}&{0}&{0}\\{0}&{\frac{3}{2}}&{0}\\{0}&{0}&{- \frac{3}{2}}\\ \end{array}\right)
-\sum_{r=0}^p(-1)^{p+r}\sigma(i_0, \dots, \widehat{i_r}, \dots, i_p, j_0, \dots, j_q) - \sum_{r=0}^q (-1)^{r+1}\sigma(i_0, \dots, i_p, j_0, \dots, \widehat{j_r}, \dots, j_q)
u_\lambda \cdot x_{\alpha_4}= x_{\alpha_4}- \lambda z_2 + \lambda^2 z_3, u_\lambda \cdot x_{\alpha_j+\alpha_4}= x_{\alpha_j+\alpha_4}- \lambda \sum_{i \in \{1,2,3\}\setminus \{j\}}x_{\alpha_i+\alpha_j+\alpha_4}(j=1,2,3), v_\lambda \cdot x_{\alpha_j}= x_{\alpha_j}+ \lambda x_{\alpha_j+\alpha_4}(j=1,2,3),
\frac{1}{n}\sum\limits_{k=0}^{m^n_0-1}R^{n,k}= \frac{1}{n}\sum\limits_{k=0}^{N_1-1}R^{n,k}+ \frac{1}{n}\sum\limits_{k=N_1}^{k_2-1}R^{n,k}+ \frac{1}{n}\sum\limits_{k=k_2}^{m^n_0-1}R^{n,k}&\le \left|\log\left(a_1^*\delta_0^A\right)\right|\frac{k_1}{n}+ |\log\delta_1|\frac{m^n_0}{n}.
J_{n}( k_{1}, \cdots , k_{n}) = \int_{0}^{1}d \tau_{1}\cdots \int_{0}^{1}d \tau_{n}\; \exp \left[ - \frac{i}{2}\sum_{i < j}^{n}( k_{i}\times k_{j}) ( 2 \tau_{i j}- \epsilon ( \tau_{i j}) ) \right]
<{\cal J}_{\mu_{1}\cdots \mu_{s}}^{( s )}( x ){\cal J}_{\nu_{1}\cdots \nu_{s}}^{( s )}( 0 ) > ={\frac{c_{s}( t )}{| x |^{4 + 2 s}}}{\cal I}_{\mu_{1}\cdots \mu_{s}, \nu_{1}\cdots \nu_{s}}^{( s )}( x ) + \sum_{s^{\prime}= 0}^{s - 1}{\prod}
S_{N,M,1}:=\sum_{k=1}^{M}\left(\frac{2^{n_{k}}}{N}\right)^{1+s}\lambda_{N,k},S_{N,M,2}:=\sum_{k=M+1}^{\tau(N)}\left(\frac{2^{n_{k}}}{N}\right)^{1+s}\lambda_{N,k},
{\partial{\cal O}[\tilde\phi(\kappa),\kappa]\over{\partial\kappa}}+\int \tilde\delta_{i\BRS}[\tilde\phi(\kappa),\kappa]\Theta^\prime[\tilde\phi(\kappa)]{\delta{\cal O}[\tilde\phi(\kappa),\kappa]\over{\delta\tilde\phi_i(\kappa)}}=0,
\begin{array}{c}\mathcal{E}(x)\rightarrow \mathcal{E}^V(x)=\Lambda \mathcal{E}(\Lambda^{-1}(x-a)), \mathcal{E}(x)\rightarrow \mathcal{E}^{TS}(x)=F(\Lambda )\mathcal{E}(\Lambda^{-1}(x-a)), \mathcal{E}(x)\rightarrow \mathcal{E}^S(x)=S(\Lambda )\mathcal{E}(\Lambda^{-1}(x-a)),\end{array}
{\frac{d}{d t}}\left( \begin{matrix}{{p}}\\{{q}}\\ \end{matrix}\right) ={\frac{1}{\hbar}}\left( \begin{matrix}{{0}}&{{-{\cal H}}}\\{{{\cal H}}}&{{0}}\\ \end{matrix}\right) \left( \begin{matrix}{{{\frac{\delta H_{0}}{\delta p}}}}\\{{{\frac{\delta H_{0}}{\delta q}}}}\\ \end{matrix}\right)
|S_{mnk}|\leq (C_b + \xi C_z)\, ||\psi_n||_{L^{\infty}}\, \langle |\psi_m|, \, |\psi_k|\rangle_{L^2_z}\leq (C_b + \xi C_z)\, ||\psi_n||_{L^{\infty}}\, ||\psi_m||_{L^2_z}\, ||\psi_k||_{L^2_z}=\widetilde C\, n^p\,,
t^N\langle 1 \cdots M|\mathcal{B}^\prime(z_1) \cdots\mathcal{B}^\prime(z_N)| \overline{x_1}\cdots \overline{x_N}\rangle = \prod_{j=1}^N z_j^{j-1-N}(1+t^{-1}z_j^2)\prod_{1 \le j<k \le N}(1+t^{-1}z_j z_k)(t^{-1}+z_j z_k^{-1})sp_{\overline{\lambda}}( \{z \}_N ).
\langle h_2 | \phi^{(h)}(z) | h_1 \rangle= \lim_{\rule{0mm}{1.7mm}u \rightarrow \infty \atop w \rightarrow 0}u^{2 h_2}\langle 0| \phi^{(h_2)}(u) \phi^{(h)}(z) \phi^{(h_1)}(w) |0\rangle= \frac{\sqrt{C_{h_2,h,h_1}}}{z^{h+h_1-h_2}}
[\begin{matrix}Y_{r}\\U_{r}\\V_{r}\end{matrix}]=[\begin{matrix}\frac{1}{4}&\frac{1}{2}&\frac{1}{4}\\ 1&-1&0\\ 0&-1&1\end{matrix}][\begin{matrix}R\\G\\B\end{matrix}];[\begin{matrix}R\\G\\B\end{matrix}]=[\begin{matrix}1&\frac{3}{4}&-\frac{1}{4}\\ 1&-\frac{1}{4}&-\frac{1}{4}\\ 1&-\frac{1}{4}&\frac{3}{4}\end{matrix}][\begin{matrix}Y_{r}\\U_{r}\\V_{r}\end{matrix}]
(T+S)^{n+m-1}=\sum_{i=0}^{n+m-1}C_{n+m-1}^{i}T^{n+m-1-i}S^{i}&=\sum_{i=0}^{m-1}C_{n+m-1}^{i}T^{n+m-1-i}S^{i}+\sum_{i=m}^{n+m-1}C_{n+m-1}^{i}T^{n+m-1-i}S^{i}&=T^{n}A+S^{m}B,
d_\mathcal{D}C_k(u_1,\cdots,u_k) =(-\frac{1}{2})k\sum_{j=1}^k(-1)^{j-1}d_\mathcal{D}(u_j)\bullet C_{k-1}(u_1,\cdots,\widehat{u_j},\cdots,u_k) &=(-\frac{1}{2})k\sum_{j=1}^k(-1)^{j-1}d_\mathcal{D}(u_j)\wedge C_{k-1}(u_1,\cdots,\widehat{u_j},\cdots,u_k).
\mathbf{S}(\mathcal{M}_{2}(\ast D))_{x_0}\simeq\mathbf{S}(p_{+}p^{\dag}(\mathcal{M}_2(\ast D)))_{x_0}\simeq (p_{\ast}\mathbf{S}(p^{\dag}(\mathcal{M}_{2}(\ast D))))_{x_0}[1-d] \indent \simeq\displaystyle{\bigoplus_{j=1}^{n}}\mathbf{S}(\mathcal{M}_{2,j}(\ast x_j))_{x_j}[1-d]
|\{J_k\}\cup N \cup (B|_{J_1 \cap J_2})_{\rm max}\cup N' \cup(B|_{(J_1 \triangle J_2) \setminus \{j_1, j_2\}})_{\rm max}| &=1+|J_1 \cap J_2|+|(J_1 \triangle J_2) \setminus \{j_1, j_2\}| &=|J_1 \cup J_2|-1
\mathbf{x}_{j\Omega}=\begin{cases}\sum_{m=0}^{M}\rho^{2m+j}e^{ji\psi(\rho)}\mathbf{w}_0^{(m+j,m)}+\delta_{1j}\varepsilon\mathbf{W}^+, 0 \leq j \leq M, begin{align*}0.2mm]\sum_{m=0}^{M}\rho^{2m+j}e^{-ji\psi(\rho)}\mathbf{w}_0^{(m,m+j)}+\delta_{-1j}\varepsilon \mathbf{W}^-, -M \leq j<0,\end{cases}
\left\| R_{\alpha}^{i}( A ) \right\|_{A = A_{0}}= \left( \begin{array}{c c c}{0}&{\bar{\partial}- h \partial + h^{\prime}}&{B^{\prime}- 2 B \partial}\\{0}&{0}&{\bar{\partial}- h \partial + 2 h^{\prime}}\\ \end{array}\right) ,
C(z)-D(z)=\mathrm{i}g\sum_{\substack{j,k=1 (j\neq k)}}^n\frac{-1}{\lambda_j-\lambda_k}\prod_{\substack{\ell=1 (\ell\neq k)}}^n(z-\lambda_\ell)=\mathrm{i}g\sum_{\substack{j,k=1 (j<k)}}^n\prod_{\substack{\ell=1 (\ell\neq j,k)}}^n(z-\lambda_\ell).
\phi( \exp_U (t X_0)) \, \phi( \exp_U (s Y)) \, \phi( \exp_U (t X_0))^{-1}=\phi ({\rm Ad}(\exp_{U}(tX_0)) sY) = \phi(\exp_U (sY + ts [X_0,Y] + \sum_{n \geq 1}( t^n s / n!){\rm ad}(X_0)^n Y)).
\P\Biggl[ 1+\sum_{i=1}^k\tau_i > Ck \Biggr] \leq \exp\biggl[ -k\biggl(\frac{C(1-e^{-\beta})}{2}-1\biggr)\biggr] \leq \exp\biggl( -\frac{C(1-e^{-\beta})}{2}+1\biggr).