harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
1 = \frac{e^{2}}{4 \pi N}\int \frac{k d k}{[ k^{2}+ \Pi_{0 0}( k , \beta , \tilde{A}_{0}) ] \sqrt{k^{2}+ \Sigma^{2}}}\left[ \frac{\sinh \beta \sqrt{k^{2}+ \Sigma^{2}}}{\cosh \beta \sqrt{k^{2}+ \Sigma^{2}}+ \cos ( e \beta \tilde{A}_{0}/ \sqrt{N})}\right]
C=AB=(\sum_{k=1}^{p}A_{ik}B_{kj})_{\begin{matrix}1\le i\le m\\ 1\le j\le n\end{matrix}}=[\begin{matrix}&&\\a_{1}^{col}&\cdot\cdot\cdot&a_{p}^{col}\\ &&\end{matrix}][\begin{matrix}&b_{1}^{row}&\\ &\vdots&\\ &b_{p}^{row}&\end{matrix}]=\sum_{k=1}^{p}a_{k}^{col}\otimes b_{k}^{row}
\underbrace{s\left([x,y] \right)+s'\left( \theta(x,y)\right)}_{\in J}+\underbrace{i\left( [x,y]\right) +i'\left( \theta(x,y)\right)}_{\in V}=\underbrace{\left[s(x),s(y) \right]}_{\in J}+ \underbrace{\rho\left(s(x) \right)i(y)+\rho\left(s(y) \right)i(x) + \theta'\left(s(x),s(y) \right)}_{\in V}.
B_{1}= \left( \begin{array}{c c}{{0}}&{{0}}\\{{0}}&{{v t I}}\\ \end{array}\right) , B_{5}= \left( \begin{array}{c c}{{0}}&{{0}}\\{{0}}&{{P_{2}}}\\ \end{array}\right) , B_{9}= \left( \begin{array}{c c}{{0}}&{{0}}\\{{0}}&{{Q_{2}}}\\ \end{array}\right) .
-g_{2}((\nabla \pi_{\ast})(U,\varphi V),\pi_{\ast}\mathcal{C}X) =g_{1}(V,\hat{\nabla}_{U}\phi\mathcal{B}X+\mathcal{T}_{U}\omega\mathcal{B}X)+g_{1}(V,\varphi U)\eta(X),
\mathrm{` ` a "}= \sum ( - 1 )^{r + z + w + p - m}\left( \begin{array}{c c c}{{j_{1}}}&{{j_{4}}}&{{j_{1 2}}}\\{{o}}&{{r}}&{{z}}\\ \end{array}\right) \left( \begin{array}{c c c}{{j_{1}^{\prime}}}&{{j_{1 3}}}&{{j_{1 4}}}\\{{- u}}&{{- w}}&{{- p}}\\ \end{array}\right)
n_{k}= \frac{1}{2}\, \langle \,\frac{p_{k}^{2}}{\omega_{k}}+\omega_{k}q_{k}^{2}\, \rangle \approx \frac{{\cal B}_{k}}{\omega_{k}}+ \frac{\omega_{k}}{16{\cal A}_{k}}- \frac{1}{2}\,,
\int_{\tilde{H}_{e_1}}...\int_{\tilde{H}_{e_{d-k}}}1\wedge \left(\frac{\rho}{|\sum_{i=1}^{d-k}(x_i-y_i)|}\right)^{d-k+\frac{1}{4}}d\omega(y_{d-k},...,y_1)\lesssim\rho^{d-k}\mathbf{M}(\omega,\bigotimes_{i=1}^{d-k}\tilde{H}_{e_i})(\sum_{i=1}^{d-k}x_i),
I_{i,n}:=] R_n - e^{- \lambda \delta n}i \ell, R_n- e^{- \lambda \delta n}(i-1)\ell] ,I'_{i,n}:=] R'_n - e^{- \lambda \delta n}i \ell, R'_n- e^{- \lambda \delta n}(i-1)\ell].
\begin{multlined}\varphi (\xi) =\dfrac{(n-1)}{(n-6)(n^2-4)}(Tr_g (A_g- A_{paneitz}))(\xi) \ \ \ \ \ \ \ \ \ \ \ \ \ \ + \dfrac{2^{n}u_0(\xi)\omega_{n-1}}{(n+2) (n (n-4)(n^2-4))^{\frac{n-4}{8}}\omega_n}1_{n=8},\ \xi\in M,\end{multlined}
c_{1}\cdots c_{n-k}\left(\sum_{\substack{\left|\delta\right|=n \delta_{1}=\dots=\delta_{n-k}=1}}\alpha_{\delta}\right)+\dots+c_{k+n+1}\cdots c_{2n}\left(\sum_{\substack{\left|\delta\right|=n \delta_{k+n+1}=\dots=\delta_{2n}=1}}\alpha_{\delta}\right)=0.
P^{\mu_1\nu_1\,\mu_2\nu_2}=c_A{\cal T}_A^{\mu_1\nu_1\,\mu_2\nu_2}+c_B{\cal T}_A^{\mu_1\nu_1\,\mu_2\nu_2}+c_C{\cal T}_A^{\mu_1\nu_1\,\mu_2\nu_2}+\sum_{i=4}^{14}c_i T_i^{\mu_1\nu_1\,\mu_2\nu_2}.
\mathrm{Mant}_{G,b, \mu}(\rho)= \sum\limits_{i, j \geq 0}(-1)^{i+j}\varinjlim\limits_{U \subset G(\mathbb{Q}_p)}\mathrm{Ext}^i_{J_b(\mathbb{Q_p})}(H^j_c(\mathbb{M}^{rig}_{b, \mu, U}\times \overline{\hat{\mathbb{Q}^{ur}_p}}, \overline{\mathbb{Q}_l}), \rho)(- \mathrm{dim}\mathbb{M}^{rig}_{b, \mu, U}).
(\cosh\tfrac{\ell_\nu}{2}+\cosh\ell_\mu)(\cosh\tfrac{\ell_\nu}{2}+\cosh\ell_|{\mu'}) &=(1+\cosh\tfrac{\ell_\nu}{2})(2\cosh\tfrac{\ell_\nu}{2}+\cosh\ell_\mu+\cosh\ell_{\mu'}).
f_3^{p^2-1}= \sum_{i=0}^{\frac{p^2-1}{3}}\mu_i x^{3(p^2-1-i)}y^{3i}+ \sum_{i=\frac{p^2-1}{3}+1}^{p^2-1}\mu_i x^{3(p^2-1-i)}y^{3i}=: T_1 + T_2.
c_{0}=\frac{H}{P}\sum_{j=1}^{3}\frac{n_{j}}{p_{j}}+l,\;P=p_{1}p_{2}p_{3},\;H=P\sum_{j=1}^{3}\frac{q_{i}}{p_{i}}=p_{1}p_{2}q_{3}+p_{1}q_{2}p_{3}+q_{1}p_{2}p_{3}.
Var(\hat{\pi}_{h,n}(x)) \le \frac{c}{T_n}\big[ \frac{1}{\prod_{l =1}^d h_l}(\delta_1 + \Delta_n) + \frac{\delta_2^{\frac{1}{2}}}{\prod_{l \ge 2}h_l}+ \frac{1}{\prod_{l \ge 4}h_l \, \delta_2^{\frac{1}{2}}}+ D + \frac{1}{(\prod_{l = 1}^d h_l)^2}e^{- \rho D}\big].
F_{\Gamma}^{p}(z) - F_{\Gamma'}^{p}(z) = \frac{1}{(1-z^2)^{n-1}}\sum_{\ell=0}^{n}\left(\vartheta_{\mathcal L}^{(\ell)}(z) - \vartheta_{\mathcal L'}^{(\ell)}(z)\right)A_{p+1,\ell}(z)
\Psi^{m}_{j-\tfrac{1}{2}}=\frac{1}{\sqrt{2j}}\begin{pmatrix}\sqrt{j+m}Y^{m-\tfrac{1}{2}}_{j-\tfrac{1}{2}}\sqrt{j-m}Y^{m+\tfrac{1}{2}}_{j-\tfrac{1}{2}}\end{pmatrix}\Psi^{m}_{j+\tfrac{1}{2}}=\frac{1}{\sqrt{2j+2}}\begin{pmatrix}\sqrt{j+1-m}Y^{m-\tfrac{1}{2}}_{j+\tfrac{1}{2}}-\sqrt{j+1+m}Y^{m+\tfrac{1}{2}}_{j+\tfrac{1}{2}}\end{pmatrix}.
\left[ A^{i}( x ) , \partial_{0}A^{* j}( y ) \right]_{x^{0}= y^{0}}= i \delta^{3}( x - y ) \delta^{i j}, \qquad \left\{\psi_{\alpha}^{i}( x ) , \bar{\psi}_{\dot{\beta}}^{j}( y ) \right\}_{x^{0}= y^{0}}= - \delta^{3}( x - y ) \delta^{i j}\sigma_{\alpha \dot{\beta}}^{0}.
\binom{n}{k}^2(1-p)^{k^2}\leq \left(\frac{en}{k}\right)^{2k}e^{-pk^2}=\left(\frac{e^2n^2e^{-pk}}{k^2}\right)^k\leq \left(\frac{e^2k}{n}\right)^k=o(1).
\# \left\{x,y,z \in \mathbb{Z}: \frac{N}{2}=4x^{2}+y^{2}+32z^2 \right\}\ne\frac{1}{2}\# \left\{x,y,z \in \mathbb{Z}: \frac{N}{2}=4x^{2}+y^{2}+8z^2 \right\},
\frac{3}{2}n \lambda_{n}\widehat{Q}_{\beta}^{1}\psi_{0}= - P_{0}\bigl ( Q_{\beta}^{1}\overline{{P}}_{0}\psi_{n}+ Q_{\beta}^{2}\psi_{n - 1}+ \ldots + Q_{\beta}^{n + 1}\psi_{0}\bigr ) , \nonumber
z.x =\big([z_{p^k-1},\ldots, z_0]\cdot({A^{-1}})^{\textrm{tr}}\big)\cdot\big(A^{\textrm{tr}}\cdot [x_{p^k-1},\ldots,x_0]^{\textrm{tr}}\big) =[z_{p^k-1},\ldots, z_0]\cdot[x_{p^k-1},\ldots,x_0]^{\textrm{tr}}=0.
E_{K}=\sum_{i=1}^{N}\frac{1}{2}m_{i}v_{i}\cdot v_{i}=\sum_{i=1}^{N}\frac{1}{2}m_{i}(\omega r_{i})^{2}=\frac{1}{2}\omega^{2}\sum_{i=1}^{N}m_{i}r_{i}^{2}.
{\tilde z}= \frac{\big( \cos \frac{\theta}{2}+ i \omega_3 \sin\frac{\theta}{2}\big) z + \big(\omega_2 - i \omega_1 \big) \sin\frac{\theta}{2}}{\big(-\omega_2 - i \omega_1 \big) \sin\frac{\theta}{2}z +\big( \cos \frac{\theta}{2}- i \omega_3 \sin\frac{\theta}{2}\big)}.
\;e^{-ith^{-1}(-ihX-i(q_1+W))}- e^{-ith^{-1}(-ihX-iW)}= \;-h^{-1}\int_0^t e^{-i(t-s)h^{-1}(-ihX-iW)}q_1 e^{-ish^{-1}(-ihX-i(q_1+W))}ds.
S(\omega_{{\sf f_1}\cdots{\sf f}_n}\|\omega) = S(\tilde{\omega}\|\omega) = i\left.\frac{d}{dt}\right|_{t = 0}(\Omega_\omega,F_\omega \Delta^{it}F_\omega^*\Omega_\omega) = i\left.\frac{d}{dt}\right|_{t = 0}\omega(B{\tt T}_t(B^*)) = i\left.\frac{d}{dt}\right|_{t = 0}{\rm Tr}(\varrho F U_tF^*U_t^*) ={\rm Tr}(\varrho F[\boldsymbol{H},F^*])
d_{G_{2}}(u_{1}) - d_{G_{1}}(u_{1}) = \big( d_{G}(u_{1}) - d_{G_{1}}(u_{1})\big) - d_{G_{1}}(u_{1}) &\ge \big( s_{1}+ s_{2}+ 2l - (s_{1}+ l)\big) - (s_{1}+ l) = s_{2}- s_{1}.
\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)
\textnormal{Area}\left(\textnormal{Sect}^{+}_{\alpha,\epsilon}(R')\right) = \theta \cdot (R')^{2}= \theta \cdot \left(R + \frac{\sqrt{2}}{2\theta}+ O(1)\right)^{2}&= \textnormal{Area}(\textnormal{Sect}_{\alpha, \epsilon}(R))+O\left(R\right),
\mathbb{E}[\theta(\int_I x^G q(y^G)dE)-\theta(\int_I x^V q(y^V)dE)]=\sum_{\gamma=1}^{\gamma_{\max}}\mathbb{E}[\theta(\int_I x^{S}q(y^{S})dE)-\theta(\int_I x^{T}q(y^{T})dE)].
I_{1, \alpha}(0, I_{\beta, \gamma}(0, 1)) = I_{1, \alpha}(0, 1 - I_{\gamma, \beta}(0, 1)) = I_{1, \alpha}(0, 1) - I_{1, \alpha}(0, I_{\gamma, \beta}(0, 1)) = 1 - I_{1, \alpha}(0, I_{\gamma, \beta}(0, 1))
\frac{\partial \overline{MG}}{\partial q_{j}}=\frac{1-\alpha}{\sum_{j}q_{j}}\left[\overline{p}^{\alpha}_{j}\overline{q}^{-\alpha}_{j}-\sum_{i}\overline{p}^{\alpha}_{i}\overline{q}^{1-\alpha}_{i}\right] =\frac{1-\alpha}{\sum_{j}q_{j}}\left[ \frac{\overline{MG}_{j}}{\overline{q}_{j}}-\sum_{i}\overline{MG}_{i}\right]
([f,g])(x)=\left(\begin{matrix}-\frac{\partial a}{\partial x_3}(x_1,x_2,x_3)x_3^L-La(x_1,x_2,x_3)x_3^{L-1}-\frac{\partial b}{\partial x_3}(x_1,x_2,x_3)x_3-b(x_1,x_2,x_3) 0\end{matrix}\right)
\dot{\iota}_t(x)=X_{E_t}(\iota_t(x)) &\dot{\phi}_t=\left(\vartheta(X_{E_t})-E_t\right)(\iota_t(p)) &\frac{\rm{d}}{\rm{d}t}\left(\iota_t^*\left(\rho_t\right)\mu_t\right)=0,
\int_{0}^{x}f(t) h(t/q) \left( -\frac{q(1+q)+t}{q^2 (1-q)t}u(t/q) +\frac{1}{q(1-q)^2 t}\right)y(t)\, d_qt &= \frac{f(x)}{cq (1-q)}\sum_{n=0}^{\infty}q^{\frac{n(n-3)}{2}}x^{n-1}\left(1-q+qc+q^{n+2}x\right) y(q^n x)= G(x).
K_{\mu}^{i}= \frac 1 2 ~ i ~ \varepsilon^{i a b}\overline{{{\psi}}}^{a}\gamma_{\mu}\psi^{b}- i ~{\frac{\varphi^{a}\overline{{{\psi}}}^{a}\gamma_{\mu}\psi^{i}}{\sqrt{1 -{\varphi}^{2}}}}~ .
-\left(\alpha\varphi_k + d \psi_k\right)'' = \mu_1 \left(1 - 2 u_{1,k}\right) \alpha\varphi_k + \mu_2\left(1 - 2 u_{2,k}\right) \psi_k + \lambda_{1,k}\left(\alpha\varphi_k + \psi_k\right).
A :=i(0,\beta(X))\cap \cap_{n\in L^0(\mathbb{N})}(([i(x,X)]^{\beta[\frac{n}{x}]})^c \cup [i(x+1,X)]^{\beta[\frac{n}{x}]}) B :=\cap_{n\in L^0(\mathbb{N})}[i(x,X)]^{\beta[\frac{n}{x}]}
\mathit{P}_{n}^{( \alpha ,\beta )}(z)=\frac{( -1)^{n}\Gamma ( n+\beta +1)}{n!\Gamma ( n+\alpha +\beta +1)}\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}\frac{\Gamma ( n+\alpha +\beta +1+k)}{2^{k}\Gamma ( \beta +1+k)}( 1+z)^{k},
\bigsqcup_{k=1}^{K}\bigsqcup_{\ell=1}^{m_{k}}\Gamma z_{(k,\ell)}\Gamma=\bigsqcup_{k=1}^{K}\bigsqcup_{\ell=1}^{m_{k}}\bigsqcup_{n=1}^{d_{k}}\gamma^{k}_{n}g_{i(k,\ell)}h_{j(k,\ell)}\Gamma.
\sum_{j=0}^m\binom mj(1-q)^{m-j}q^j(1-|H_1|^{-1})^{j}+|H_1|^m\exp(-r'e(H_1)^{-2}) &=(1-q|H_1|)^m+|H_1|^m\exp(-r'e(H_1)^{-2}) &\leq\exp(-mq|H_1|)+|H_1|^m\exp(-r'e(H_1)^{-2}),
\tau + F(\mathcal{H}^p(Tfv,Tv)) \leq F \Big(\varphi\Big(\max\Big\{d^p(fv,Tfv),0,0,\frac{1}{2}d^p(fv,Tfv),0,0,0\Big\}\Big) \Big) = F \Big(\varphi\big(d^p(Tfv,fv)\big)\Big).
pdf_{MJ}(\beta;\theta)d\beta_{1}d\beta_{2}d\beta_{3}=\frac{4}{\pi}\frac{1}{\theta}K_{2}(\frac{1}{\theta})^{-1}e^{-\frac{\gamma(\beta)}{\theta}}\gamma(\beta)^{5}d\beta_{1}d\beta_{2}d\beta_{3}
\left(\begin{matrix}x=0 x=t_1 x=\infty \, \left( \frac12 \right) \overbrace{\begin{matrix}0 0 -t_2 \theta^0\end{matrix}}\begin{matrix}0 \theta^{t_1}\end{matrix}\overbrace{\begin{matrix}1 \theta^\infty_1/2 -1 \theta^\infty_1/2\end{matrix}}\end{matrix}\right),
{n\choose j-1}=\sum_{i=1}^{d}{c-1\choose i-1}{n-c+1\choose j-i}+\sum_{i=d+1}^{c}{c-1\choose i-1}{n-c+1\choose j-i}&=\sum_{i=1}^{d}{c-1\choose i-1}{n-c+1\choose j-i}+\sum_{\ell=d}^{c-1}{\ell-1\choose d-1}{n-\ell\choose j-d-1}.
\begin{cases}\|\dot{U}^{a,\pm}\|_{s_1+2, X}+\|\dot{\Psi}^{a,\pm}\|_{s_1+3, X}+\|f_a^\pm\|_{s_1+1, X}\le \delta, \|f_a^\pm\|_{\alpha+1, X}/\delta{\rm is~small}, \|f_a^\pm\|_{\alpha+2, X}/\delta{\rm is~bounded}\end{cases}
\begin{gathered}A^{(j)}_{x_{i}}\left(x_{i_1},u_j,(u_j)_{x_{i_1}},(u_j)_{x_{i_1}^{j_1},x_{i_2}^{j_2}},\dots\right)=A^{(j')}_{x_{i'}}\left(x_i,u_j,(u_j)_{x_{i_1}^{j_1}},(u_j)_{x_{i_1}^{j_1},x_{i_2}^{j_2}},\dots\right), x_{i}\neq x_{i'},A^{(j)}\neq A^{(j')}, 1\leq i\leq n,j_1+\dots+j_n\leq p, 1\leq j,j' \leq 2n.\end{gathered}
D_-^{\alpha}\equiv \partial_z+\frac{i}{2}(A_z^1+\frac{1}{\sqrt{3}}A_z^2+\cdots +\frac{1}{\sqrt{(\alpha-1)(\alpha-2)/2}}A^{\alpha-2}_z+\frac{\alpha}{\sqrt{\alpha(\alpha-1)/2}}A^{\alpha-1}_z)
\mathcal W_\infty^{p-2-\alpha}\,\lim_{r\to \infty}\frac{u''(r)}{u_0''(r)}=\lim_{r\to \infty}\frac{g_1(v(r))}{h_1(v_0(r))}=\lim_{r\to \infty}Q_1(\mathcal V(r), v_0(r))
{n \choose k}p^{k\choose 2}={n \choose k}\left(1-n^{-3/4}\right)^{k\choose 2}& \le n^k \left(1-n^{-3/4}\right)^{k^2/4}& \le 2^{k\lg n}\left(e^{-n^{-3/4}}\right)^{k^2/4}& \le e^{k\lg n-k^2 n^{-3/4}/4}& \le e^{-k \lg n}.
\int_{C_W}e^{-\d a^*M_xa}(\det M_x)^{q-1}dx=2^{q-1}\Gamma(q)e^{\d a^*Wa}\prod_{i=1}^na_i^{q(s(i)-2)}\prod_{i<j}|w_{ij}|^qK_q(a_ia_j|w_{ij}|)
y[n]=\frac{b_{0}K+b_{1}}{a_{0}K+a_{1}}\cdot x[n]+\frac{-b_{0}K+b_{1}}{a_{0}K+a_{1}}\cdot x[n-1]-\frac{-a_{0}K+a_{1}}{a_{0}K+a_{1}}\cdot y[n-1].
\mathbb{P}(h^{U}(x)\geq a) \leq\mathbb{P}(h^{U}(x)\geq a,\varphi^{V,U}(x)\geq0)+\mathbb{P}(h^{U}(x)\geq a,\varphi^{V,U}(x)\leq0) &=2\mathbb{P}(h^{U}(x)\geq a,\varphi^{V,U}(x)\geq0)\leq2\mathbb{P}(h^{V}(x)\geq a).
Q \ = \ \{(x,y,z) : (x,z), (z,y) \in R \}\ \cup \{(x,y,z) : (y,z),(z,x) \in R \}, Q^{\circ}\ = \ \{(x,y,z) : (x,z), (z,y) \in R^{\circ}\}\ \cup \{(x,y,z) : (y,z),(z,x) \in R^{\circ}\}
S'_1=S^{\pm 1}_1,S'_{\alpha_i}=S_{\alpha_i},S'_{\beta_i}=S_{\beta_i}S^{-1}_1, S'_{\gamma_i}=S_1 S_{\gamma_i}, S'_{\delta_i}=S_1 S_{\delta_i}S^{-1}_1.
4D\delta(\zeta/\chi)A_{p}(\chi)v - \delta(\zeta/\chi)B_{p}(\chi)v &=(\chi/\zeta)(1+\zeta/\chi)^2\left(2(p+1)D\delta(\zeta/\chi)R_{p+1}(\chi)v-2\delta(\zeta/\chi)D_{\chi}R_{p+1}(\chi)v\right)
\bigsqcup_{t\in H_P}\sigma_t(U_P) = \bigcup_{t\in \mathcal{O}_S}\sigma_t(U_P) \subseteq \bigcup_{\widetilde{t}\in \mathcal{O}_P}\sigma_{\widetilde{t}}(U_P) = \bigsqcup_{\widetilde{t}\in G_P}\sigma_{\widetilde{t}}(U_P).
I_{1}= \int \frac{d^{4}k}{( 2 \pi )^{4}}\, \frac{d^{4}l}{( 2 \pi )^{4}}\, \frac{d^{4}q}{( 2 \pi )^{4}}\, \frac{2 ( q^{2}+ q_{\mu}k_{\mu})}{k^{2}\Big ( 1 + k^{2 n}/ \Lambda^{2 n}\Big ) \, l^{2}\Big ( 1 + l^{2 n}/ \Lambda^{2 n}\Big ) \, q^{4}( q + k )^{4}( q + l )^{2}}
= \int_{Q}(\partial_{t}u_h v_h + \theta h \partial_{t}u_h \partial_{t}v_h + \nabla_x u_h \cdot \nabla_x v_h - \theta h \partial_{t}\nabla_x u_h \cdot \nabla_x v_h)\,dxdt + \theta h\int_{\Sigma \cup \Sigma_T}n_t \nabla_x u_h \cdot \nabla_x v_h\, ds - \theta h\int_{\Sigma}n_x \cdot \nabla_x u_h \partial_{t}v_h\, ds.
\operatorname{Hyp}\big(S^1,G\big)=\bigsqcup_{\lambda\in \operatorname{Hom}(S^1,T)}\Sigma^{\rm hyp}_{\lambda},\qquad\Sigma_{\lambda}^{\rm hyp}=H^0 (\Delta^{*},G )\cdot\lambda\cdot H^0(\Delta ,G).
-\frac{1}{6}T_{7,r}=\left(\frac{\eta(q^4)^5\eta(q^{28})^5}{\eta(q^2)^2\eta(q^8)^2\eta(q^{14})^2\eta(q^{56})^2}+4\frac{\eta(q^8)^2\eta(q^{56})^2}{\eta(q^4)\eta(q^{28})}\right)^2.
I_2(\varepsilon) = \langle\log |J(\lambda,z)|\widetilde{W}_{\mathcal{M}}, dd^c \phi \rangle = \int_{\mathrm{Supp}\mathcal{M}\setminus S_\varepsilon}d\mathcal{M}(\gamma) \left(\int_B\log |J(\lambda,\gamma(\lambda))| dd^c(\phi\circ \gamma) \right) = 0,
S=\frac{1}{8\pi}\int_{\Sigma}d^2x\left\{\partial\varphi\bar{\partial}\varphi +e^{-\sqrt{\frac{2}{k}}\varphi}\left((\partial_t \tilde{\gamma}^1)^2+ (\partial_x \tilde{\gamma}^1)^2 \right) + 2i \partial_t \tilde{\gamma}^1 \right\},
ds^2 ={U^2\over \sqrt{4\pi gN}}\left[-\left(1-{U_0^4\over U^4}\right) dt^2 + dy_idy^i\right]+{\sqrt{4\pi gN}\over U^2}\left(1-{U_0^4\over U^4}\right)^{-1}dU^2 + \sqrt{4\pi gN}d\Omega_5.
E_{j}\triangleright \tilde{C}= \sum_{i = 1}^{N}q^{-3i}E_{j}\triangleright (E_{\xi_{i}}^{*}E_{\xi_{i}})= q^{-3j}q^{-2}E_{\xi_{j + 1}}^{*}E_{\xi_{j}}- q^{-3(j + 1)}q E_{\xi_{j + 1}}^{*}E_{\xi_{j}}= 0.
{\bf \Psi}( x ) = \left( \begin{array}{c}{\varphi ( x )}\\{\Psi ( x )}\\ \end{array}\right) , \hat{x}^{\mu}= x^{\mu}, \hat{\pi}_{\mu}= - i \partial_{\mu}, \hat{\psi}^{n}= \frac{i}{2}\Gamma^{n},
\frac{\sigma_{P_n,j}(\theta_n)}{\hat\sigma^M_{n,j}(\theta_n)}-1=\frac{\sigma_{P_n,j}(\theta_n)}{\hat\sigma_{n,j}(\theta_n)}\Big(\frac{\hat\sigma_{n,j}(\theta_n)}{\hat\sigma^M_{n,j}(\theta_n)}-1\Big)+\Big(\frac{\sigma_{P_n,j}(\theta_n)}{\hat\sigma_{n,j}(\theta_n)}-1\Big)=O_{P_n}(1)\Big(\frac{\hat\sigma_{n,j}(\theta_n)}{\hat\sigma^M_{n,j}(\theta_n)}-1\Big)+o_{P_n}(1),
\int_B |\varphi| \,dx \le |B|^{\frac{p-1}{p}}\|d_\Omega\|_{L^{\infty}(B)}^{s}\, \left(\int_\Omega \frac{|\varphi|^p}{d_\Omega^{s\,p}}\,dx\right)^{\frac{1}{p}}\le |B|^{\frac{p-1}{p}}\,\|d_\Omega\|_{L^{\infty}(B)}^{s}\, \left(\frac{1}{\mathfrak{h}_{s,p}(\Omega)}\right)^{\frac{1}{p}}\,[\varphi]_{W^{s,p}(\mathbb{R}^N)},
\begin{array}{lcl}v = G_{\varepsilon,R,a,r}\left(L_{\varepsilon,R,a}(v)- L_{g}^{u_{\varepsilon,R,a}}(v)- Q_{\varepsilon,R,a}(r^{-\overline\gamma}|x|^{\overline\gamma}v_\phi+h+v) \right. & + \left. H_\delta(u_{\varepsilon,R,a})-H_g(u_{\varepsilon,R,a})- L_g^{u_{\varepsilon,R,a}}( r^{-\overline\gamma}|x|^{\overline\gamma}v_\phi+h)\right).\end{array}
{\cal P}_{R}= ( q^{I}, n^{i}, m_{j}) \left( \begin{array}{c}{l_{I}\cdot{\bf e}_{\mu}^{( R )}}\\{{\bar{k}}_{i}\cdot{\bf e}_{\mu}^{( R )}}\\{k^{j}\cdot{\bf e}_{\mu}^{( R )}}\\ \end{array}\right){\bf e}_{\mu}^{R}.
\sum_{i=1}^N\sum_{j=1}^N\widehat\Gamma_i\widehat\Sigma_{ii}^{-1}\bigl(\widehat BM_{z z}^{i j}\widehat B{}'-\widehat\Sigma_{zz}^{i j}\bigr)\widehat\Sigma_{j j}^{-1}\widehat\Gamma_j'=0.
V ( e^{t}\sigma ; \lambda , e^{t}\Lambda ; e^{t}M ) = V ( e^{t}\sigma Z_{S}^{1 / 2}( t ) ; \lambda Z_{Q}^{- 1}( t ) Z_{S}^{- 1 / 2}( t ) , e^{t}\Lambda Z_{Q}( t )^{1 / 2}; M ) ,
\left\langle d , x^{\rho_1 + \cdots + \rho_i}\right\rangle = \left\langle d \cap x^{\rho_1}, x^{\rho_2 + \cdots + \rho_i}\right\rangle = \left\langle 0 , x^{\rho_2 + \cdots + \rho_i}\right\rangle = 0,
\dot{\omega}_{I}= \sigma_{I}\frac{d}{dt}\sum_{p_{I}}Arg(\vec{x}-\vec{R}_{p_{I}})= \sigma_{I}\sum_{p_{I}}\varepsilon_{ij}\dot{R}^{i}_{p_{I}}\partial_{j}\ln\mid\vec{x}-\vec{R}_{p_{I}}\mid \;\;
\Big|u\Big|_{\mathrm{L}^2\Big(\mathbb{R};\mathrm{H}^{\mathrm{r}}(\partial\Omega)\Big)}^2 := \int_{\partial\Omega}\int_{\partial\Omega}\dfrac{\Vert u(\mathrm{x},\cdot)-u(\mathrm{y},\cdot)\Vert_{\mathrm{L}^2(\mathbb{R})}^2}{|\mathrm{x}-\mathrm{y}|^{n-1+2\mathrm{r}}}d\sigma_\mathrm{x}d\sigma_\mathrm{y},
\sup_{t\in [0,T]}t^{-\frac r2}\|(\mathcal S(t)-I)Q^{\frac 12}\|_{L_{}(U,H)}= \sup_{t\in [0,T]}t^{-\frac r2}\|A^{-\frac r2}(\mathcal S(t)-I)A^{\frac r2}Q^{\frac 12}\|_{L_{}(U,H)}\leq C \|A^{\frac r2}Q^{\frac 12}\|_{L_{}(U,H)}= C \|Q^{\frac r2}\|_{L_{}(U,\dot H^r)}<\infty.
\mathcal{D}^{\star}(fg) = - fg \mathcal{D}1 + fg \mathcal{D}1 + g \mathcal{D}^{\star}f+ f( \cdot + l) \mathcal{D}^{\star}g & = g \mathcal{D}^{\star}f+ f( \cdot + l) \mathcal{D}^{\star}g.
\beta_g(\chi_\mu \chi_\nu) =\delta_{s(\mu),r(\nu)}\, \chi_{g\cdot(\mu\nu)}=\delta_{s(\mu),r(\nu)}\chi_{g\cdot\mu (g|_\mu\cdot \nu)}=\delta_{s(g\cdot\mu),r(g_\mu\cdot \nu)}\, \chi_{g\cdot\mu}\chi_{(g|_\mu\cdot \nu)}&=\beta_g(\chi_\mu)\beta_{g|_\mu}(\nu)=\beta_g(\chi_\mu)\beta_{g|_p}(\nu).
{1\over 2\sqrt{2}}\int_{{\cal C}^3}d^3\xi \sqrt{\det h_{ij}}=\int_{{\cal C}^3}d^3\xi\frac{|\phi|}{\|\Omega\|}\,\geq\,\frac{1}{\|\Omega\|}\left|\int_{{\cal C}^3}d^3\xi \phi\right|=\frac{1}{6\|\Omega\|}\left|\int_{{\cal C}^3}f^*\Omega\right|=V_{\rm CY}^{1/2}\left|\int_{{\cal C}^3}f^*\widehat{\Omega}\right|\ .
x^{4}=2\sum_{l=1}^{n}\Bigl(\sum_{j=1}^{n}\Theta_{j}\gamma_{jpl}+\sum_{u=1}^{\nu}\widetilde{\Theta}_{u}\gamma_{iul}\Bigr)e_{l}+2\sum_{s=1}^{\nu}\Bigl(\sum_{j=1}^{n}\Theta_{j}\widetilde{\gamma}_{jps}+\sum_{u=1}^{\nu}\widetilde{\Theta}_{u}\widetilde{\gamma}_{ius}\Bigr)\widetilde{e}_{s}.
\left\lbrace \begin{aligned}xy \equiv -(n'+4k-4)t +Ct^4 \mod t^5 x^2-4txy \equiv \left(8k^2-5k+4n'k+\binom{n'}{2}-3 \right)t^2 \mod t^5 y^2 \equiv 1+C't^3+C''t^4 \mod t^5 \end{aligned}\right.
P_{A,l}(z)= \sum_{\alpha \in \Lambda_{A,l}}x_\alpha z^\alpha=\sum_{\gamma \in \Gamma_{A,l}}\sum_{\beta \in B_{A,l}}x_{2\beta+2\gamma+1_A}z^{2\beta+2\gamma+1_A}=\sum_{\gamma \in \Gamma_{A,l}}\left(\sum_{\beta \in B_{A,l}}x_{2\beta+2\gamma+1_A}z^{2\beta}\right)z^{2\gamma+1_A}.
{\cal{L}}_{A'}=-\frac{1}{2g}\left\{A_\mu^{'a}\left[ -\left(D^2 \right)^{ac}g^{\mu \nu}+\left(F_{\rho \sigma}^b{\cal{J}}^{\rho \sigma}\right)^{\mu \nu}\left(t^b\right)^{ac}\right] A_\nu^{'c}\right\}
\det \left( \det X \begin{pmatrix}( i_\alpha ) \sqcup K ( i'_\beta ) \sqcup K' \end{pmatrix}\right)_{1 \le \alpha, \, \beta \le p}=\det X \begin{pmatrix}I \sqcup K I' \sqcup K' \end{pmatrix}\cdot\left( \det X \begin{pmatrix}K K' \end{pmatrix}\right)^{p-1}.
\tilde{\mathcal{V}}= \{(\varphi, \psi)' \in \mathcal{V}\,|\, \forall (\tilde{\varphi}, \tilde{\psi})' \in \mathcal{V}:\int_{\Omega}\nabla \varphi \cdot \mathrm{rot}' \tilde{\psi}\mathrm{d}x =\int_{\Omega}\nabla \tilde{\varphi}\cdot \mathrm{rot}' \psi \mathrm{d}x = 0 \}
\int_{-\infty}^{\frac{-\log (2r\sqrt{k})}{k-1}}\int_{x_{k-1}}^{\frac{-\log (2r\sqrt{k})}{k-1}}\cdots \int_{x_{2}}^{\frac{-\log (2r\sqrt{k})}{k-1}}\left(e^{\sum_{i=1}^{k-1}2ix_i}-2re^{\sum_{i=1}^{k-1}(2i+1)x_i}\right) \,dx_1\,dx_2\cdots dx_{k-1}.
T ( 0 ) = \prod_{j = l + 1}^{P}\; \prod_{i = l}^{j - 1}R_{i j}^{\iota^{P - j + i - l + 1}}( ( \epsilon^{- j}- \epsilon^{- i}) \theta^{\prime}) \; \; \prod_{j = l}^{P - 1}\; \prod_{i = j + 1}^{P}R_{i j}^{\iota^{P - j + i}}( ( \epsilon^{- j}- \epsilon^{- i}) \theta^{\prime})
(\nu'_1\nu_1\diamond\dots\diamond\nu'_n\nu_n)= (\nu'_1\diamond\dots\diamond\nu'_n)\circ(\nu_1\diamond\dots\diamond\nu_n)\ , \rho^{(\vec l)}_i\circ(\nu_1\diamond\dots\diamond\nu_n)= \nu_i\circ\rho^{(\vec k)}_i\ ,
(D+1) + r(D-d+1) = (D+1)(r+1) - d\cdot r &= (s-r+1)(1-\alpha)q (r+1) - d r &> (s-r+1)\left(\frac{\ell}{r+1}+ \frac{r}{r+1}\frac{d}{(s-r+1)q}\right)q (r+1) - d r &= (s-r+1) \ell q + dr - dr &= (s-r+1) \ell q.
\sum_{i'=0}^{\ell'-1}y_{i'}'\cdot \frac{q^{i'}}{q^{\ell'}-1}= y = \sum_{i=0}^{\ell-1}y_i \cdot \frac{q^i}{q^\ell-1}= \sum_{i=0}^{\ell-1}y_i \cdot \sum_{j=0}^{\ell'-1}\frac{q^{i+\ell j}}{q^{\ell\ell'}-1}= \sum_{i''=0}^{\ell\ell'-1}y_{i''}'' \cdot \frac{q^{i''}}{q^{\ell\ell'}-1},
\Vert \Psi\Vert_V^2 = \Vert \Psi\Vert_H^2 + \Vert \Psi_x \Vert_H^2\leq \left(1 + 2 \frac{c^2}{\nu^2}\right)\int e^{-2\frac c\nu x}\, \hat{v}_x^2 \, dx + 2 \int e^{-2\frac c\nu x}\, \hat{v}_{xx}^2 \, dx
g(\mathrm{T}^{0}(X,Y),Z)-g(\mathrm{T}^{0}(Z,Y),X)+g(\mathrm{T}^{0}(J_{\varphi}X,Y),J_{\varphi}Z)-g(\mathrm{T}^{0}(J_{\varphi}Z,Y),J_{\varphi}X)=\frac{1}{2}g( N_{J_{\varphi}}(X,Z),Y),
T_{+ p \pm 2}^{+ q}= \frac 1 2 \gamma_{q \dot{q}}^{i}\Omega_{+ p}^{+ 2 i}\psi_{\pm 2 \dot{q}}^{1 -}, \quad T_{- p \pm 2}^{+ q}= \frac 1 2 \gamma_{q \dot{q}}^{i}\Omega_{- p}^{+ 2 i}\psi_{\pm 2 \dot{q}}^{1 -};
A_{1}= \left({\frac{\omega - m \Omega_{H}}{4 \pi T_{H}}}\right)^{2}, B_{1}= -{\frac{\rho_{-}^{2}}{\rho_{+}^{2}}}\left({\frac{\omega - m \Omega_{H}\rho_{+}^{2}/ \rho_{-}^{2}}{4 \pi T_{H}^{\mathrm{B T Z}}}}\right)^{2}.
\sum_{k=0}^{a_z+1}\alpha_ks_{\mathcal{A}}(r+k,m)=c_1\sum_{k=0}^{a_z+1}\alpha_k\left|\mathcal{A}\right|^{r+k}+0=c_1\left|\mathcal{A}\right|^rg\left(\left|\mathcal{A}\right|\right).
P = P_{0}( l ) \left( \frac{2}{\lambda}\right)^{2 l + 1}\frac{\Gamma^{4}( 1 + l / 2 )}{\pi}= P_{0}( l ) \left( \frac{4 \pi T_{H}}{\omega}\right)^{2 l + 1}\frac{\Gamma^{4}( 1 + l / 2 )}{\pi}\, .
\Phi_1(x) = \frac{1}{8}\langle \omega_0(x)\rangle^{-4}\left[3P_0(1, \omega_0(x)\omega_1(x),\omega_1(x)^2;\,\omega_0(x),\omega_1(x)) \right. &\left. + P_2(1, \omega_0(x)\omega_1(x),\omega_1(x)^2;\,\omega_0(x),\omega_1(x))\right]\, ,
2\left(1-{1\over12}N\,e^{2\Phi}\right)\partial_-\Sigma-4\left(1-{1\over24}N\,e^{2\Phi}\right)\partial_-\Phi\,\Sigma=\left(1-{1\over24}N\,e^{2\Phi}\right)\lambda^2e^{2\rho}\;,
\begin{aligned}\mathrm{lo}(\pi{G})=\mathrm{lo}(G/\gamma_c{G})=\mathrm{lo}(G)-\mathrm{lo}(\gamma_c{G})=\mathrm{lo}(G)-1, \mathrm{cl}(\pi{G})=\mathrm{cl}(G/\gamma_c{G})=\mathrm{cl}(G)-1, \mathrm{cc}(\pi{G})=\mathrm{lo}(\pi{G})-\mathrm{cl}(\pi{G})=\mathrm{lo}(G)-1-(\mathrm{cl}(G)-1)=\mathrm{lo}(G)-\mathrm{cl}(G)=\mathrm{cc}(G).\end{aligned}
H ={\bf p}_{1}\cdot \dot{\bf q}_{1}+{\bf p}_{2}\cdot \dot{\bf q}_{2}- L ={\bf p}_{1}\cdot{\bf q}_{2}-{\frac{m}{2}}{\bf q}_{2}^{2}-{\frac{\omega^{2}}{2 m}}{\bf p}_{2}^{2}.
\L_3(18,2,1) = \{13_g+4_r+1_r,~12_g+5_r+1_r,~12_r+5_g+1_r,~11_g+6_r+1_r,~11_r+6_g+1_r, &10_r+7_g+1_r,~11_g+5_r+2_r,~10_g+6_r+2_r,~10_r+6_g+2_r,~9_r+6_r+3_g\},
\langle \widetilde{S}_{ij}^{*}(\mathbf{x},t) \rangle = 0, &\langle \widetilde{S}_{ik}^{*}(\mathbf{x},t) \widetilde{S}_{lm}^{*}(\mathbf{x^{'}},t^{'})\rangle = \dfrac{2k_{B}T\rho_{f}^{2}\widetilde{\mathit{x}}^{4}}{\mu^{3}}(\delta_{il}\delta_{km}+ \delta_{im}\delta_{kl})\delta(\mathbf{x}- \mathbf{x^{'}})\delta(t - t^{'}),
Au^n=\left[\begin{array}{c}\langle A_1^n,u^n \rangle \langle A_2^n,u^n \rangle \vdots \langle A_m^n,u^n \rangle \end{array}\right], \;\;\;\;Av^n=\left[\begin{array}{c}\langle A_1^n,v^n \rangle \langle A_2^n,v^n \rangle \vdots \langle A_m^n,v^n \rangle \end{array}\right],
\psi(E_{ij})=\psi(E_{ji}^{*})=\psi(E_{ji})^{*}=\overline{\psi(E_{ji})^{t}}=\overline{\psi(E_{ji})}\quad; &\psi(E_{ii})=\psi(E_{ii})^{*}=\overline{\psi(E_{ii})^{t}}\quad.
\sigma^0=\left( \begin{array}{rr}1 0 0 1\end{array}\right),\quad\sigma^1=\left( \begin{array}{rr}0 1 1 0\end{array}\right),\quad\sigma^2=\left( \begin{array}{rr}0 -i i 0\end{array}\right),\quad\sigma^3=\left( \begin{array}{rr}1 0 0 -1\end{array}\right)
\{p\in Spl(f) \mid g_i( r_{\mu(1)}) \equiv r_{\mu(i)}\bmod p\,\,(1\le{}^\forall i\le n)\}= \left\{p\in Spl(f) \left| \{g_{\mu^{-1}(1)}(r_{\mu(1)})/p\}\le\dots\le\{g_{\mu^{-1}(n)}(r_{\mu(1)})/p\}\right.\right\}
a_{\lambda p(p+1)+\rho}=\frac{P_{\lambda p(p+1)+\rho,0}(z)}{P_{\lambda p(p+1)+\rho-p,0}(z)}-\frac{P_{\lambda p(p+1)+\rho+1,0}(z)}{P_{\lambda p(p+1)+\rho-p,0}(z)},z\in\mathbb{C}\setminus [a_{0},b_{0}].