harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\big\langle \xi_0|n_1^\perp, T(v-y)\big\rangle = \big| \xi_0|n_1^\perp \big|_2 \big| T(n_2|n_1^\perp) \big|_2^{-1}\big\langle T( n_2|n_1^\perp), T(v-y) \big\rangle &= \big| \xi_0|n_1^\perp \big|_2 \big| T(n_2|n_1^\perp) \big|_2^{-1}\big\langle n_2|n_1^\perp, v-y \big\rangle &= 0,
\left\vert \int(\psi\circ\mathsf{R}_{k})\,\frac{1}{M}{\textstyle\sum_{j=0}^{M-1}}\widehat{T}^{j}(u-u^{\ast})\,d\mu\right\vert \leq\sup\left\vert\psi\right\vert \,\left\Vert \frac{1}{M}{\textstyle\sum_{j=0}^{M-1}}\widehat{T}^{j}(u-u^{\ast})\right\Vert_{1}& <\frac{\varepsilon}{3}u,u^{\ast}\in\mathcal{U}
A_{3}= \frac{1}{\sqrt{2}}f_{L}^{\bar{A}\bar{B}\bar{C}}f_{R}^{A B C}\Phi^{\bar{A}A}\Phi^{\bar{B}B}\Phi^{\bar{C}C}= \frac{1}{\sqrt{2}}\mathrm{T r}\{[ \Phi , \Phi ] \Phi \},
\mathcal C : = \big\{c\in (0,+\infty)^{m^+}\times (-\infty,0]^{m-m^+};\; (H,B,c)\, \mathrm{satisfies}\, \mathrm{Condition\, (C)}\big\}= \big\{c\in D_1;\; (H,B,c)\, \mathrm{satisfies}\, \mathrm{Condition\, (C)}\big\},
x_3(z_1-z_2)+ x_1 x_3(y_1-y_2)+x_2 z_1 -x_2 z_3 &+x_1(z_3-z_2)+x_1 x_2(y_1 -y_2 +y_2 -y_3) &+(y_3 -y_2)x_1 x_2 +(y_3 -y_2)(x_2 x_3-x_1 x_3) &=x_3(z_1-z_2)+ x_1 x_3(y_1-y_2)+z_3(x_1 -x_2) &+x_2 z_1- x_1 z_2 +x_1 x_2(y_1 -y_2)+(y_3 -y_2)x_3 (x_2 -x_1).
S(\mathcal{A}_{m}) = S(I_q \otimes \mathcal{J}_{m-1}+ Q \otimes \mathcal{A}_{m-1}) = S( I_q \otimes (J_q \otimes \mathcal{A}_{m-2}) ) + S(Q)S(\mathcal{A}_{m-1}) = S(J_q)S(I_q)S(\mathcal{A}_{m-2}) = q^3 \, S(\mathcal{A}_{m-2}).
\underline{\underline{C}}=[\begin{matrix}C_{11}&C_{12}&C_{13}&0&0&0\\C_{12}&C_{22}&C_{23}&0&0&0\\C_{13}&C_{23}&C_{33}&0&0&0\\ 0&0&0&C_{44}&0&0\\ 0&0&0&0&C_{55}&0\\ 0&0&0&0&0&C_{66}\end{matrix}]
N_1(T,\chi) = \frac{T}{\pi}\log{\frac{qT}{2m\pi}}- \frac{T}{\pi}- A(t_q,\chi) - B(t_q,\chi) + A(T,\chi) + B(T,\chi) &\quad\quad+ A(-t_q,\chi) + B(-t_q,\chi) - A(-T,\chi) - B(-T,\chi) + O(m^{1/2}\log{q}),
H_{\mathrm{e f f}}= \sum_{p}\epsilon_{p}a_{p}^{\dagger}a_{p}+ \frac{3 \lambda}2 \frac 1 V \sum_{p_{1}+ p_{2}= p_{3}+ p_{4}}a_{p_{1}}^{\dagger}a_{p_{2}}^{\dagger}a_{p_{3}}a_{p_{4}}\; ,
\left[ \partial^{\tau}\partial_{\tau}+\kappa\left( u^{\tau}\partial_{\tau}\right)^{2}\right] \mathbf{Z}^{\left( e\right)}=4\pi\mu\mathbf{p},\qquad\left[ \partial^{\tau}\partial_{\tau}+\kappa\left( u^{\tau}\partial_{\tau}\right)^{2}\right] \mathbf{Z}^{\left( m\right)}=4\pi\mu\mathbf{m}
Z^{(0)}_1(\xi)=\begin{cases}\mathcal{O}(\exp(-2\pi \xi_2)), \xi_2 \to +\infty, begin{align*}3mm] \displaystyle{\Big(\!-\xi_1 + \frac12\Big)}+ \mathcal{O}(\exp(\pi h^{-1}_0 \xi_2)), \xi_2 \to -\infty,\end{cases}
\lambda ( p^{\mu}) \equiv \left( \begin{matrix}{\left( \zeta_{\lambda}\Theta_{[ j ]}\right) \phi_{L}^{*}( p^{\mu})}\\{\phi_{L}( p^{\mu})}\\ \end{matrix}\right) , \quad \rho ( p^{\mu}) \equiv \left( \begin{matrix}{\phi_{R}( p^{\mu})}\\{\left( \zeta_{\rho}\Theta_{[ j ]}\right)^{*}\phi_{R}^{*}( p^{\mu})}\\ \end{matrix}\right) \quad .
\pi_m^2[i,j] =\pi_m\left([i,j] - \sum_{k\not\in \{i,j\}}m_k \partial_2([i,j,k])\right) = \pi_m[i,j] - \sum_{k\not\in\{i,j\}}m_k \pi_m\partial_2[i,j,k] =\pi_m[i,j].
\int_{0}^{T}\int_{\mathbb{T}}[s|u_{xxxx}|^2+ s|u_{xxx}|^2+s^3|u_{xx}|^2+s^5|u_x|^2+s^7|u|^2]e^{2s\varphi}dxdt +s\int_{\mathbb{T}}[|u-b_1u_{xx}+bu_{xxxx}|^2e^{2s\varphi}]_{t=0}dx \leq C_2\int_{0}^{T}\int_{\omega}[s|u_{xxxx}|^2+s^3|u_{xx}|^2+s^7|u|^2]e^{2s\varphi}dxdt.
\lim_{\varepsilon\to 0}\Big{[}\inf_{\hat{\theta}}\sup_{\theta\in\mathcal{E}(\alpha_{0},B)}R(\theta,\hat{\theta})\Big{/}( \varepsilon / B )^{4\alpha_{0}/(2\alpha_{0}+1)}\Big{]}=c_{\mathrm{P}}(\alpha_{0}).
d_0(x){:=}g(x)-T(\omega,Bq^{-1})\nu_{\Omega_Q}(x) \forall x \in \partial \Omega_Q\, ,d_j(x) :=-b^\#_{j-l}(x)Bq^{-1}x \forall x \in \partial \Omega_Q \, , \forall j \in \mathbb{N}\setminus \{0\}\, ,
\sum_{j=0}^{\infty}\mu^{j/2}\exp\left( -\frac{2 \pi M \mu^j}{\tau_{0}^{\alpha}}\right) \geq \int_{0}^{\infty}\mu^{s/2}\exp\left( -\frac{2 \pi M \mu^s}{\tau_{0}^{\alpha}}\right) ds & =\int_{1}^{\infty}\exp\left( -\frac{2 \pi M}{\tau_{0}^{\alpha}}u^{2}\right) \frac{du}{\frac{1}{2}\ln\mu}.
\min(\mathcal{E}_1,\mathcal{E}_2)\leq \mathcal{E}_1^{12/17}\mathcal{E}_2^{5/17}= X^2(JK)^{-22/17}q^{-8/17}\ll X^2y^{-22/17}q^{-8/17}\ll X^{334/255}q^{-164/255}.
\Big\{\psi_1^{d_1}\cdots \psi_n^{d_n}\kappa_{e_1,\dots,e_{m}}\mid m\geq 0, d_i\geq 0, e_j\geq 1, \sum_{i=1}^n d_i+\sum_{j=1}^{m}e_j=g-1\Big\}
\tilde B_{Q'}(w,y) =2\tilde B_Q(v_1,v_1)\tilde B_Q(w,y)-2\tilde B_Q(v_1,w)\tilde B_Q(v_1,y)= &=2\tilde B_Q(\tilde B_Q(v_1,v_1)w-\tilde B_Q(v_1,w)v_1,y)=0
(\mathrm{E}^{-\mathrm{A}})^{-\alpha}_\lambda=\mathcal{M}_{\lambda}\left[\mathrm{E}^{-\mathrm{E}^{-\mathrm{A}}};\alpha\right]=e^{-\mathcal{M}_{\lambda}\left[\mathrm{E}^{-\mathrm{A}};\alpha\right]}=e^{-A_\lambda^{-\alpha}}\;,
<\int d^4z~\left[ \Theta (z)-\left( 1-{\frac{3N_c}{N_f}}-\gamma(g)\right) L_{matter}(z)\right] >=u_0\int \frac{d^4 k}{(2\pi)^4}\frac{1}{4}B(k)^{\mu\nu}{B}_{\mu\nu}(-k).
\widehat{\mathcal{A}}_\varepsilon := b(\mathbf{D})^* g^{\varepsilon}(\mathbf{x}) b(\mathbf{D}), \mathcal{A}_\varepsilon := (f^{\varepsilon}(\mathbf{x}))^* b(\mathbf{D})^* g^{\varepsilon}(\mathbf{x}) b(\mathbf{D}) f^{\varepsilon}(\mathbf{x}).
\left(z\alpha q^{1-n+k};q\right)_{\infty}=(-1)^{n}\alpha^{n}z^{n}q^{-\frac{1}{2}n(n-1)+kn}\left(q^{-k}z^{-1}\alpha^{-1};q\right)_{n}\left(q^{k+1}z\alpha;q\right)_{\!\infty}.
q_{\alpha}\left( \rho_{1}\right) :=\frac{\gamma_{1}^{\alpha-2}\Gamma\left(\alpha+1\right) \left( 1-\left( 1-\rho_{1}\right)^{\alpha}-\alpha\rho_{1}\left( 1-\rho_{1}\right)^{\alpha-1}\right)}{\rho_{1}^{2}\left(1-\rho_{1}\right)^{\alpha-2}},
| \psi_{+}({\bf a}) \rangle = \frac{1}{\sqrt{2}}\left( \begin{array}{c}{\sin \theta e^{- i \phi}}\\{- \cos \theta}\\{0}\\{- 1}\\ \end{array}\right) , | \psi_{+}({\bf a}) \rangle^{\prime}= \frac{1}{\sqrt{2}}\left( \begin{array}{c}{- \cos \theta}\\{- \sin \theta e^{i \phi}}\\{1}\\{0}\\ \end{array}\right) ,
I(t)\geq \frac{2(t+1)^2}{\gamma-1}|A_{0}|\int_{A(t)}A\rho^\gamma(x,t)\frac{x}{|A(t)|}\geq \frac{C(t+1)^2}{\gamma-1}|A_{0}|^{1-\gamma}m(0)^\gamma\geq C_0 (1+t)^2 ,
\int_{\mathbf{B}}\frac{(1-|y|^{2})^{n+\alpha-1}h_{j}^{p}(y)}{[x,y]^{n+|k|+\alpha-1}}d\tau(y) &\leq\int_{\mathbf{B}}\frac{(1-|y|^{2})^{pc+\alpha-1}}{[x,y]^{n+|k|+\alpha-1}}dv(y) &=I_{pc+\alpha,-pc+|k|}(x) &\leq C(pc+\alpha,-pc+|k|) &\leq \left(\frac{j}{j-1}\right)^{pc}C(pc+\alpha,-pc+|k|)h_{j}^{p}(x),
A_{2-n}A_{4-n}A_{6-n}\psi_{n-5}^{(0)}= A_{2-n}A_{4-n}A_{6-n}\alpha_{n-5}^{(0)}+(B_{2-n}A_{4-n}A_{6-n}+A_{2-n}B_{4-n}A_{6-n}+A_{2-n}A_{4-n}B_{6-n})\beta_{n-5}^{(0)}.
\dfrac{d\psi}{dt}+\left\Vert a\right\Vert_{L^{\infty}}\psi \left(h^{-1}\right)^{\prime}\left( \psi \right) =0,~0<\psi \left( 0\right) \leq 4\sqrt{E_{\partial_{t}u,\partial_{t}v}\left( 0\right) E_{u,v}\left(T_{0}\right)}.
|A_{\ell+1}|=|X_1| \geq \frac 12 |X| \geq \frac \gamma 2 |A_{\ell}| \geq \frac \gamma 2 \left( \frac \gamma 2 \right)^\ell |A|=\left( \frac \gamma 2 \right)^{\ell+1}|A|,
\Phi(t,\eta)= \left\{{2(m-1)\over(m-2)(m-4)}\left[ h(\xi_{0}) -\frac{m-2}{4(m-1)}S_{g}(\xi_{0}) +{3 (m-2)\over 2(m-1)}\frac{\Delta_g a(\xi_0)}{a(\xi_{0})}\right] +{1\over2}\frac{D^2_g a(\xi_0)[\eta,\eta]}{a(\xi_{0})}\right\}t -{\varepsilon\over|\varepsilon|}{(m-2)^2\over8}\log t
b(x,\alpha) =\exp{\Bigg(2\pi i (x,0)(A^{-1}(\tau_0'+\Im(\tau')t)\delta^{-1}-1/2S\mathfrak{d}^{-1})\begin{pmatrix}\alpha 0\end{pmatrix}\Bigg)}&=b_0b'(x,\alpha)b_t(x,\alpha),
\langle \Upsilon \rangle_{\Sigma}\ \longrightarrow l a n g l e \Upsilon \rangle_{\Sigma}+ \delta \! \langle \Upsilon \rangle_{\Sigma}, \delta \! \langle \Upsilon ( z_{0}) \rangle_{\Sigma}= \frac{1}{\pi}\int_{\Sigma}\langle \Psi ( z ) \Upsilon ( z_{0}) \rangle_{\Sigma}d^{2}\! z .
\begin{gathered}\widetilde q(x_1,\ldots,x_{i-1},x_{i+1},\ldots,x_k,y) = q(x_1,\ldots,x_{i-1},\phi(y),x_{i+1},\ldots,x_k), x = (x_1,\ldots,x_k) \in \mathbb R^k_+, y \in \mathbb R^m_+, \end{gathered}
P(X_{n+1}= \pm 1 \mid \mathcal{F}_n) = \dfrac{\#\{i=1,\ldots,n : X_i=\pm 1\}}{n}\cdot p+ \dfrac{\#\{i=1,\ldots,n : X_i= \mp 1\}}{n}\cdot (1-p) & = \dfrac{n \pm S_n}{2n}\cdot p+ \dfrac{n \mp S_n}{2n}\cdot (1-p) = \dfrac{1}{2}\left( 1 \pm \alpha \cdot \dfrac{S_n}{n}\right).
\Gamma^X_{\gamma_n}\Gamma^X_{\widetilde \gamma_n}\rho^X_\gamma(A)\Gamma^X_{\gamma_n}\Gamma^X_{\widetilde \gamma_n}=\Gamma^X_{\gamma_n}\Gamma^X_{\widetilde \gamma_n}\Gamma^X_{\gamma_n}A\Gamma^X_{\gamma_n}\Gamma^X_{\gamma_n}\Gamma^X_{\widetilde \gamma_n}=\Gamma^X_{\widetilde \gamma_n}A \Gamma^X_{\widetilde \gamma_n}=A.
\rho-w(\rho)+w(\gamma) =\rho-s_{\alpha}u(\rho)+s_{\alpha}u(\gamma) &=\rho-u(\rho)+u(\gamma)+2\frac{(\rho,u^{-1}\alpha)}{(\alpha,\alpha)}\alpha-2\frac{(\gamma,u^{-1}\alpha)}{(\alpha,\alpha)}\alpha.
\mathcal L_{K,F}^{\beta}\Phi(m) = \sum_{1\leq i < j < \infty}K(m_i,m_j) \left[ \Phi\left(c_{ij}(m)\right) - \Phi(m) \right] + \sum_{i\geq 1}F(m_i) \int_{\Theta}\left[ \Phi\left(f_{i\theta}(m)\right) - \Phi(m) \right] \beta(d\theta).
h(Z,\lambda)=\int_{G^{\mathbb{N}}}\int_{X}\log\frac{d\omega_{1}.\eta}{d\eta}\left(x\right)d\eta_{\omega}(x)d\mathbb{P}_{\mu}(\omega)+ \int_{G^{\mathbb{N}}}\int_{X}\log\frac{d\left(\omega_{1}\lambda\right)_{x}}{d\lambda_{x}}\left(\psi(x,\omega)\right)d\eta_{\omega}(x)d\mathbb{P}_{\mu}(\omega)={\rm I}+{\rm II}.
e^{-\mu}(\partial_{1}+iA_{1}+i\partial_{2}-A_{2}+{1 \over 2}\partial_{1}\mu +{i \over 2}\partial_{2}\mu)\psi_{1}+e^{-\nu}(-\partial_{3}-iA_{3}+i\partial_{4}-A_{4}-{1 \over 2}\partial_{3}\nu +{i \over 2}\partial_{4}\nu)\psi_{2}=0.
\textrm{Pr}\!\! \left\{\left(d^{\max}_i\right)^2\delta_1\cdot\mathsf{SNR}\le \epsilon'\right\}\!\! = \!\! \prod_{k=1}^{S}\!\!\textrm{Pr}\! \left\{\left(d^{[k,i]}\right)^2 \!\! \delta_1 \!\cdot \! \mathsf{SNR}\le \epsilon'\right\},
\alpha := nK + n|\log p| + n|\log (1-p)|\, , \beta_i := \frac{2K b_i}{\delta}+ |\log p| + |\log(1-p)|\, , \gamma_{ij}:= \frac{2K c_{ij}}{\delta}+\frac{6K b_i b_j}{n\delta^2}\, .
\gamma_Q(1)= 1 \gamma_Q (X_{i_1}\cdots X_{i_s})= Q_{i_1,\cdots ,i_s}\gamma_Q (X_{i_1}\cdots X_{i_k}) = 0 \gamma_Q (X_{A(1)}\cdots X_{A(ps)})= \sum_{\{S_1,\cdots ,S_p\}\in \mathfrak{P}([1,ps]),\#S_i=s}Q_{A|S_1}\cdots Q_{A|S_p}& A:[1,ps]\to[1,n].
(S)\int_{t_{0}}^{t}g(t^{\prime})dB(t^{\prime})=(I)\int_{t_{0}}^{t}g(t^{\prime})dB(t^{\prime})+\frac{1}{2}\sqrt{\gamma}N\int_{t_{0}}^{t}dt^{\prime}[g(t^{\prime}),c(t^{\prime})].
H_{\mathrm{M}}= \int_{- \infty}^{\infty}\! d k \sum_{\{M , N \}}\sum_{A , B}\omega_{k}\cdot a_{k \{M , N \}}^{A \dagger}a_{k \{M , N \}}^{B}\cdot ( - )^{\sum_{n}M_{n}}\eta_{A B}\ ,
\sum_{D \in \mathcal{H}_{2g+1}}L ( \tfrac{1}{2},\chi_D)^4 = q^{2g+1}(a_{10}g^{10}+a_9 g^9+a_8 g^8) + O(q^{2g+1}g^{7+\frac{1}{2}+\epsilon}),
\phi_1 := \phi_{2B}(\tau, z) = 2 \zeta^{\pm 1}- 4+ (-4 \zeta^{\pm 2}+ 8) q + (2 \zeta^{\pm 3}+ 8 \zeta^{\pm 2}- 2 \zeta^{\pm 1}- 16) q^2 + O(q^3)
K_{a,\pm}(\lambda )=K_{a}(\lambda q^{(1\pm 1)/2};\zeta_{\pm},\kappa_{\pm},\tau_{\pm})=\left( \begin{array}{cc}a_{\pm}\left( \lambda \right) b_{\pm}\left( \lambda \right) c_{\pm}\left( \lambda \right) d_{\pm}\left( \lambda \right)\end{array}\right)_{a},
q_{\sigma'}(f(x_{i_1}\cdots x_{i_s})+x'_{i_1}\cdots x'_{i_s}) =q_\sigma(x_{i_1}\cdots x_{i_s})+q_{\sigma'}(x'_{i_1}\cdots x'_{i_s}) &=(x_{i_1}\cdots x_{i_s})^2+(x'_{i_1}\cdots x'_{i_s})^2.
b_i^{S \cup \{j\}}= b_i^S + w^S_j \, x_{ij}^{1, S \cup \{j\}},j \in N^{\{0, 1\}}\setminus S b_i^S = b_i^{S \setminus \{j\}}+ w^S_j \, x_{ij}^{0, S \setminus \{j\}},j \in S.
\omega_{2}( t , r , \theta ) = - \frac{F ( r , \theta )}{\rho_{0}C^{5 / 3}}\frac{\left( \frac{\rho}{\rho_{0}}\right)^{2 / 3}\ln \frac{\rho}{\rho_{0}}}{\left( \ln \frac{\rho}{\rho_{0}}+ 1 \right)^{8 / 3}}.
\langle a_{i}^{\epsilon}| G^{(n)}_{0}|a_{j}^{\epsilon}\rangle =G^{(n)}_{0}(a_{i}^{\epsilon},a_{j}^{\epsilon}) = \int_{M \times M}d_g \mu(x) \; d_g \mu(y) \; K_\epsilon(x,a_i) G^{(n)}_{0}(x,y) K_\epsilon(a_j,y) \;.
S_{\mathrm{b o u n d}}={\frac{3 M_{\mathrm{P}}^{2}}{8 \pi}}\int d \Omega \int_{0}^{\infty}d r{\frac{\partial}{\partial r}}\Bigl [ R^{2}( r ) ( 1 - R^{\prime}( r ) ) \Bigr ] \ .
N_0(\omega,x,\omega',x') =\sum_{n\in N}f_n(\omega)\;\!v(\omega,x)\;\!v(\omega',x')\;\!\overline{f_n(\omega')},\quad(\omega,x),(\omega',x')\in\Omega, N_1(0)(\omega,x,\omega',x') =-\frac1{2}\sum_{n\in N}f_n(\omega)\;\!v(\omega,x)\;\!|x-x'|\;\!v(\omega',x')\;\!\overline{f_n(\omega')},\quad(\omega,x),(\omega',x')\in\Omega.
[L:K]=\frac{[H^1_{et}(U,j_{*}\hat{T}')][H^0_{et}(U,\mathcal{R})]}{[H^1_{et}(U,\mathcal{Q})]}=\frac{[H^1_{et}(U,j_{*}\hat{T}')][Ext^2_{U}(\mathcal{R},\mathbb{G}_m)]}{[H^1_{et}(U,\mathcal{Q})]}
q_{k,\ell}(t,h) = \sum_{\substack{i_1,i_2,i_3 \ge 0 i_1+i_2+i_3 = k-1}}\frac{\alpha_{i_1,k}(\log t)^{i_3}}{i_2! i_3!}\sum_{\substack{j_1,j_2,j_3 \ge 0 j_1+j_2 +j_3 = \ell-1}}\frac{\alpha_{j_1,\ell}\mathcal{D}_{k,\ell}^{(i_2,j_2)}(0,0) (\log (t+h))^{j_3}}{j_{2}! j_3!}.
\lim_{t \to \infty}\mathcal{M}\bigg(\bigg\{\theta\in\mathbb{T}~ : ~ \cos(\theta-\phi(t))\leq\sqrt{1-\bigg(\frac{2/3+1}{(2/3)^{2}}\frac{M}{K}+\frac{1-2/3}{2/3}\bigg)^{2}}-\eta \bigg\}\bigg) = 0,
\frac{2}{\pi}\int_{-(\log{t})^{2}}^{(\log{t})^{2}}\log\zeta(\sigma+i(t+u))\left( \frac{\sin{(u/2)}}{u}\right)^2 (1+\cos(\theta+u\log{x}))du &= \frac{1}{2}e^{i\theta}\sum_{e^{-1/2}x\le n \le e^{1/2}x}\frac{\Lambda(n)}{n^{\sigma+it}\log{n}}\left( \frac{1}{2}- \left| \log\frac{n}{x}\right| \right) + O\left( \frac{x}{(\log{t})^2}\right).
T^{H,H_t,H}_{f^{[2]}}(V_k-W,W+t(V_k-W)) &=f(H)(V_k-W)(H_t-iI)^{-1}(W+t(V_k-W))(H-iI)^{-1}&\quad-T_{(fu)^{[1]}}^{H,H_t}\big((V_k-W)(H_t-iI)^{-1}\big)(W+t(V_k-W))(H-iI)^{-1}&\quad-T_{(fu)^{[1]}}^{H,H}\big((V_k-W)(H_t-iI)^{-1}(W+t(V_k-W))(H-iI)^{-1}\big) &\quad+T_{(fu^2)^{[2]}}^{H,H_t,H}\big((V_k-W)(H_t-iI)^{-1},(W+t(V_k-W))(H-iI)^{-1}\big).
V (D)=\left( \begin{array}{cc}e^{-D^- D^+}e^{-\frac{1}{2}D^- D^+}\left( \frac{I- e^{-D^- D^+}}{D^- D^+}\right) D^- e^{-\frac{1}{2}D^+ D^-}D^+ I- e^{-D^+ D^-}\end{array}\right)
C = \frac{1}{k-1}(C_{\lambda_2 , \ldots , \lambda_k}+ \ldots + C_{\lambda_1 + \ldots + \lambda_{k-1}}+ 1)|\lambda_1 \cdots \lambda_k|^{(|\lambda_1 \cdots \lambda_k|)^2 + 1}&\leq \frac{1}{3}{k+1 \choose 2}|\lambda_1 \cdots \lambda_k|^{(k-1)(|\lambda_1| + \ldots + |\lambda_k|)^2 + k-1}.
(\gamma_{5})^{T}=\alpha(\begin{matrix}0.8673\\ 0.1327\end{matrix})\circ(\begin{matrix}1.0000\\ 1.0000\end{matrix})=\alpha(\begin{matrix}0.8673\\ 0.1327\end{matrix})=(\begin{matrix}0.8673\\ 0.1327\end{matrix})
S_{worm}= - 6 \pi^5 \int d\rho \big[{R^7 \over N}(\partial_\rho R)^2 + N R^7-{1 \over 144}{R^9 \over N}(\partial_\rho \phi)^2 -{Q^2 \over \pi^{10}}{N \over R^9}e^{-2 \phi}\big].
B = \left( \begin{array}{c c c c c c c c r}{{0}}&{{x}}&{{z}}&{{y}}&{{0}}&{{- y}}\\{{- x}}&{{0}}&{{x}}&{{z}}&{{y}}&{{0}}\\{{- z}}&{{- x}}&{{0}}&{{x}}&{{z}}&{{y}}\\{{- y}}&{{- z}}&{{- x}}&{{0}}&{{x}}&{{z}}\\{{0}}&{{- y}}&{{- z}}&{{- x}}&{{0}}&{{x}}\\{{y}}&{{0}}&{{- y}}&{{- z}}&{{- x}}&{{0}}\\ \end{array}\right) \; ,
T_{n}(x) =\frac{\alpha^{n+1}(x)}{(\alpha(x)-\omega_{1}(x))(\alpha(x)-\omega_{2}(x))}-\frac{\omega^{n+1}(x)}{(\alpha(x)-\omega_{1}(x))(\omega_{1}(x)-\omega_{2}(x))}&\ \ +\frac{\omega_{2}^{n+1}(x)}{(\alpha(x)-\omega_{2}(x))(\omega_{1}(x)-\omega_{2}(x))},\n\geq0
\frac{d}{dt}(|\rho|_{D^{2,q}}+|H|_{D^{2,q}}+|P|_{D^{2,q}}) \leq C(1+|\nabla u|_\infty+F)(1+|\rho|_{D^{2,q}}+|P|_{D^{2,q}}+|H|_{D^{2,q}})+C(1+F).
\mathcal{I}^{G, \mu}_{M_{S_2}, b_{S_2}}=\{(M_{S_2}, \mu_{S_2}) \in \mathcal{C}_{M_{S_2}}: (M_{S_2}, \mu_{S_2}) \in \mathcal{I}^{M_{S_1}, \mu_{S_1}}_{M_{S_2}, b_{S_2}}\,\ \,\ (M_{S_1}, \mu_{S_1}) \in \mathcal{I}^{G, \mu}_{M_{S_1}, b_{S_1}}\}.
\lim_{n\to\infty}\left\{\log[\frac{\lambda_n^j}{\lambda_n^k}]+\frac{|x_n^j-x_n^k|^2}{\lambda_n^j\lambda_n^k}+\frac{|t_n^j(\lambda_n^j)^2-t_n^k(\lambda_n^k)^2|}{\lambda_n^j\lambda_n^k}\right\}=\infty.
{\bf w}_t\;\!+\,\mbox{\boldmath $u$}\cdot \nabla \mbox{\bf w}\;=\;\gamma \;\! \, \Delta \mbox{\bf w}\,+\, \nabla \:\!(\:\!\nabla \cdot\;{\bf w}\:\!)\,+\,\chi \, \nabla \times\, \mbox{\boldmath $u$}\,-\, 2 \, \chi \,{\bf w},
[..H_sH_t]_n=\begin{bmatrix}[..G_sG_t]_n * 0 0 q^n 0 0 * [..F_sF_t]_n\end{bmatrix}=\begin{bmatrix}[..H_s''H_t'']_n 0 * [..F_sF_t]_n\end{bmatrix}=\begin{bmatrix}[..G_sG_t]_n * 0 [..H_s'H_t']_n\end{bmatrix}
f\left(R_{n}\right)=2\pi p_{0}d_{k}\log\frac{1}{\varepsilon_{n}}+2\pi p_{0}\left(d_{k}^{2}-d_{k}\right)\log\frac{1}{R_{n}}+\frac{\pi}{2}\alpha_{k}\left(1-\alpha\right)R_{n}^{s_{k}}\log\frac{1}{\varepsilon_{n}}
< 0 |{\cal T}\left[ \varphi ( y , t ) \varphi ( y_{0}, t_{0}) \right] | 0 >_{E}= \int_{0}^{\infty}\, \frac{\mathrm{d}\theta}{2 \pi}\, e^{- m | x - x_{0}| \cosh \theta + i m ( y - y_{0}) \sinh \theta}\, ,
<{\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}
\hat{\mathbf{V}}^{\mathrm{h}}_{k_l}= \frac{1}{\left\|\breve{\mathbf{D}}_{k_l}^\mathrm{h}\right\|_\mathrm{F}}\breve{\mathbf{S}}^{\mathrm{h}}_{k_l}\breve{\mathbf{D}}_{k_l}^\mathrm{h},
W_{n,m}:= \sum_{j=1}^n \log R^2_{0,j},E_{n,m}:=\sum_{j=1}^n \log R^2_{m,j}, N_{n,m}:=\sum_{i=1}^m \log R^1_{i,n}, S_{n,m}:=\sum_{i=1}^m \log R^1_{i,0}.
= \frac{\hbar}{8\pi^{2}}\alpha^{4}\phi_{c}^{4}\sum_{N=1}^{m}\frac{K_{2}\left( 2N\alpha a\phi_{c}\right)}{\left( N\alpha a\phi_{c}\right)^{2}}+ \frac{\hbar}{8\pi^{2}}\alpha^{4}\phi_{c}^{4}\sum_{N=m+1}^{\infty}\frac{K_{2}\left( 2N\alpha a\phi_{c}\right)}{\left( N\alpha a\phi_{c}\right)^{2}}.
Z_{\scriptscriptstyle S D}= \frac{1}{N_{\scriptscriptstyle S D}}e^{- \frac{i \pi}{4}\eta \bigl ( 0 , ( \mathrm{1 \m k e r n - 5 m u I \!}+ \ast d_{\scriptscriptstyle 1}) \bigr )}v o l \Bigl ( k e r ( \mathrm{1 \m k e r n - 5 m u I \!}+ \ast d_{\scriptscriptstyle 1}) \Bigr ) d e t^{\prime}( \mathrm{1 \m k e r n - 5 m u I \!}+ \ast d_{\scriptscriptstyle 1})^{\scriptscriptstyle - 1 / 2}
\frac{\partial^{2}p}{\partial z\partial\tau}=\frac{c_{0}}{2}\nabla_{\perp}^{2}p+\frac{\delta}{2c_{0}^{3}}\frac{\partial^{3}p}{\partial\tau^{3}}+\frac{\beta}{2\rho_{0}c_{0}^{3}}\frac{\partial^{2}p^{2}}{\partial\tau^{2}}
i \Pi^{\underline{a}\underline{b}}_{S \mu \nu}(q) = Ng^2\delta_{\underline{a}\underline{b}}\mu^{2\epsilon}\int \frac{d^n \ell}{(2\pi)^n}\left\{\frac{\ell_{\mu}\ell_{\nu}- m^2_G g_{\mu \nu}}{(\ell + q)^2 (\ell^2 -m^2_G)}- \frac{1}{8}\frac{(2 \ell + q)_{\mu}(2 \ell + q)_{\nu}}{\ell^2 (\ell + q)^2}\right\}
Q^\eta(y) =i^s(2\pi)^{s/2}\int_{\mu^{-1}(0)}q^*(\alpha_0 \wedge \eta_0)\wedge e^{iq^*d\alpha_0 - iy(d\theta)}\Omega = i^s(2\pi)^{s/2}\int_{\mu^{-1}(0)}q^*(\alpha_0 \wedge \eta_0) \wedge e^{iq^*d\alpha_0 - iy(F^\theta)}\Omega,
\sum\limits_{k=0}^{n}a_{k}x_{k}=\sum\limits_{k=0}^{n}a_{k}\left( \sum\limits_{j=0}^{k}\frac{1}{r}\left( -\frac{s}{r}\right)^{k-j}\frac{f_{k+1}^{2}}{f_{j}f_{j+1}}y_{j}\right) =\sum\limits_{k=0}^{n}\left( \sum\limits_{j=k}^{n}\frac{1}{r}\left( -\frac{s}{r}\right)^{j-k}\frac{f_{j+1}^{2}}{f_{k}f_{k+1}}a_{j}\right)y_{k}=D_{n}(y)
t=\tau_0<\ldots <\tau_N=T,\ \ \ Delta_N=\hbox{\vtop{\offinterlineskip\halign{\hfil#\hfil\cr{\rm max}\cr$\stackrel{}{{}_{0\le j\le N-1}}$\cr}}}\Delta\tau_j\to 0\ \ \hbox{if}\ \N\to \infty,\ \ \ Delta\tau_j=\tau_{j+1}-\tau_j.
\det ( \mathcal{G}_{\mu \nu}+ ( 2 \pi \alpha^{\prime}) \mathcal{F}_{\mu \nu}) = \det \left( \begin{array}{c c}{{\mathcal{G}_{\alpha \beta}+ ( 2 \pi \alpha^{\prime}) \mathcal{F}_{\alpha \beta}}}&{{0}}\\{{0}}&{{1}}\\ \end{array}\right) = \det ( \mathcal{G}_{\alpha \beta}+ ( 2 \pi \alpha^{\prime}) \mathcal{F}_{\alpha \beta}) .
c^k_{\mathrm{s2b},3}\left(N_{\mathrm{sct}}\right) = \mathsf{E}\left\lbrace B \sum_{i=1}^{N_{\mathrm{sct}}}\log_2\left[ \frac{P_{\mathrm{sc}}}{N_{\mathrm{sct}}}\right. \right. \left. \left. \,\,\, \times \left(B\sigma^2_{\mathrm{bs}}\|\mathbf{g}^\mathrm{T}_j \|^{2}\right)^{-1}+1 \right] \right\rbrace,
J_{\lambda/\nu}(1;\beta)=\prod_{s\in C_{\lambda/\nu}}\left( \frac{\beta l(s)+a(s)+1}{\beta (l(s)+1)+a(s)}\right)_\lambda\left( \frac{\beta (l(s)+1)+a(s)}{\beta l(s)+a(s)+1}\right)_\nu .
i \left( k - 2 \right) : \partial_{z}g g^{- 1}: \, = \left( \begin{array}{c c}{{- J^{2}( z )}}&{{J^{0}( z ) + J^{1}( z )}}\\{{- J^{0}( z ) + J^{1}( z )}}&{{J^{2}( z )}}\\ \end{array}\right)
x\mu \overline{x}=-{\bf i}w\mu \overline{w}{\bf i}=-{\bf i}\left(\frac{\mu}{4}-\frac{{\bf i}\mu{\bf i}}{4}-{\frac{d_{1}}{2}}{\bf i}\right){\bf i}=\frac{\mu}{4}-\frac{{\bf i}\mu{\bf i}}{4}-{{red}\frac{d_{1}}{2}}{\bf i}=w\mu\overline{w}.
T = - \frac{1}{2}( \partial X )^{2}- \frac{1}{2}( \partial \phi )^{2}- \frac{1}{\sqrt{2 k}}\partial^{2}\phi - \frac{1}{2}( \partial u )^{2}- \frac{1}{2}( \partial v )^{2}- \frac{1}{2}( \partial H )^{2}- \eta \partial \xi ,
\dfrac{\partial}{\partial v}\Big(\dfrac{1}{\hat{H}}\big(\lambda \nu_u+F\nu\times\nu_u-E\nu\times\nu_v\big)\Big)=\dfrac{\partial}{\partial u}\Big(\dfrac{1}{\hat{H}}\big(\lambda \nu_v+G\nu\times\nu_u-F\nu\times\nu_v\big)\Big),
( i \hat{\Pi}_{1 , \pm}^{a}( n ) + \frac{e_{\pm}\mathrm{L}}{\sqrt{2 \pi}}A_{1 , \pm}^{a}( n ) ) | \mathrm{v a c}; A \rangle = 0 \hspace{5 m m}\mathrm{f o r}\hspace{5 m m}n \in \cal Z ,
X^{\mu}(\sigma,\tau)=x_{0}^{\mu}+2\alpha^{\prime}p_{0}^{\mu}\tau-i\sqrt{2\alpha^{\prime}}\sum_{n=1}(a_{n}^{\mu*}e^{in\tau}-a_{n}^{\mu}e^{-in\tau})\frac{cosn\sigma}{\sqrt{n}},
\psi_{+}= \left[ \begin{array}{l}{\xi}\\{0}\\ \end{array}\right] , \psi_{-}= \left[ \begin{array}{c}{0}\\{\left( \frac{1}{i \partial^{+}}\right) [ \sigma^{i}( i \partial^{i}+ g A^{i}) + i m_{F}] \xi}\\ \end{array}\right] .
q^{\frac{1}{2}(\ell(x)-\ell(y)+\ell(yx))}T_y T_{x\i}\i= q^{\frac{1}{2}(\ell(sx)-\ell(ys)+\ell(yx))}T_{ys}T_{(sx)\i}\i +q^{\frac{1}{2}(\ell(sx)-\ell(y)+\ell(yx)-1)}(1-q) T_y T_{(sx)\i}\i .
\frac{dt_{e}}{dt_{a}}=\frac{|C|^{2}-C\cdot v}{|C|^{2}-C\cdot u}=\frac{C\cdot(C-v)}{C\cdot(C-u)}=\frac{\frac{1-C\cdot v}{|C|^{2}}}{\frac{1-C\cdot u}{|C|^{2}}}
B_{0}(M_{*}^{2})= \frac{\Gamma (\frac{d}{2})\,\Gamma (d-2+2k)\,\Gamma(k+\frac{1}{2})}{\sqrt{\pi}\,\Gamma (d-2)\,\Gamma (2k+1)\,\Gamma(k+\frac{d}{2})}M_{\ast}^{2k}\ \cdot
\lefteqn{\| (f - Mn d^{2\theta-1})^+\|_{L^n(\Omega)}}\le{}\left( \int_{\{f \ge Mnd^{2\theta-1}\}}(f^+)^n \right)^{1/n}\le{}\left( \int_{\{f \ge Mnd^{2\theta-1}\}}(f^+)^n \frac{(f^+)^{q-n}}{(Mnd^{2\theta -1})^{q-n}}\right)^{1/n}.
Z_{\mu \nu}\equiv - ( k^{2}+ i \epsilon ) \biggl [ g_{\mu \rho}+ \biggl ({\frac{1}{\lambda}}( 1 - \kappa )^{2}- 1 \biggr ){\frac{k_{\mu}k_{\rho}}{k^{2}+ i \epsilon}}- i{\frac{\kappa ( 1 - \kappa )}{\lambda}}{\frac{k_{[ \mu}\eta_{\rho ]}}{k^{2}+ i \epsilon}}+{\frac{\kappa^{2}}{\lambda}}{\frac{\eta_{\mu}\eta_{\rho}}{k^{2}+ i \epsilon}}\biggr ] ,
\tilde{c}_i(t)= ~ (\cosh t)^{-2-i+s}F(1+i, 1-s, r+2, \tanh^2 t) = ~(\cosh t)^{-2r-2+i-s}F(1+r-i, 1+r+s, r+2, \tanh^2 t),i=0,\ldots, r, \psi_\lambda(t)= ~\sum^r_{i=0}\tilde{c}_i(t).
\begin{aligned}\int_0^T \int_{B_{2R}^d}\int_{\mathbb{S}^{d-1}}\mathbf{1}_{\left(\tau,v_{s+k+1},\omega_{k+1}\right) \in \mathcal{B}^-_{III}}d\omega_{k+1}dv_{s+k+1}d\tau \leq & \leq C_d \left( s+k-1 \right) T R^d \theta^{d-1}\end{aligned}
\det\begin{pmatrix}t_1^{k_1}t_2^{k_1}\dots t_n^{k_1}F(t_1)t_1^{k_2}F(t_2)t_2^{k_2}\dots F(t_n)t_n^{k_2}t_1^{k_3}t_2^{k_3}\dots t_n^{k_3}\vdots \vdots \ddots \vdots t_1^{k_n}t_2^{k_n}\dots t_n^{k_n}\end{pmatrix}
\Sigma_{BAE}=\left\{\{\lambda_{1},...,\lambda_{\mathsf{N}}\}\in \mathbb{C}^{\mathsf{N}}:\mathbf{A}(\lambda_{a})Q_{\mathbf{\lambda}}(\lambda_{a}-\eta )+\mathbf{A}(-\lambda_{a})Q_{\mathbf{\lambda}}(\lambda_{a}+\eta)=0,\forall a\in \{1,...,\mathsf{N}\}\right\}.
\int_{\mathbb{R}^{N}}\nabla u\cdot \nabla h\,dx+\int_{\mathbb{R}^{N}}V\left(\left| x\right| \right) uh\,dx=\int_{\mathbb{R}^{N}}g\left( \left| x\right|,u\right) h\,dx,\forall h\in H_{V}^{1}
h_1^2+\frac{\partial_x\xi(a_N',b_N)}{\gamma_N}=\frac{a_N'}{(1-a_N')^2}, h_2^2+\frac{\partial_y\xi(a_N,b_N')}{1-\gamma_N}=\frac{b_N'}{(1-b_N')^2}
t r \: e^{- \beta \triangle_{0}}= F + ( 4 \pi \beta )^{- \frac{1}{2}}e^{- \beta / 4}\sum_{n = 1}^{\infty}\sum_{p}\frac{l ( p ) / 2}{\sin l ( p^{n}) / 2}e^{- l^{2}( p^{n}) / 4 \beta}
\gamma_i^{AF}= \frac{{{\rho_i}\left({1 -{\rho_i}}\right){X_i}{Z_i}}}{{{\rho_i}\left({1 -{\rho_i}}\right){Y_i}{Z_i}+ \left({1 -{\rho_i}}\right)\left({{X_i}+{Y_i}}\right)\left({{W_i}+ 1}\right) +{\rho_i}{Z_i}+{W_i}+ 1}}
\mathbf{W}(\mathbf{z},\phi) =\mathbf{W}_0(\mathbf{z})+\mu^{2M+2}\mathbf{w}^{\mathbf{0}}_1(\phi)+\mathcal{O}(\mu^{2M+3}), \mathbf{R}(\mathbf{z},\Omega t) =\mathbf{R}_0(\mathbf{z})+\mu^{2M+2}\mathbf{r}^{\mathbf{0}}_1(\Omega t)+\mathcal{O}(\mu^{2M+3}),
\mathcal{Y}^{t,p}_s=\left\{\psi\in\mathcal{X}^p_s\, :\, \exists h_0>0 \mbox{such that}\sup_{0<|h|<h_0}\int_{\mathbb{R}^N}\left|\frac{\psi(x+h)-\psi(x)}{|h|^t}\right|^p\, \frac{dx}{(1+|x|)^{N+s\,p}}<+\infty\right\}.