harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\lefteqn{\log L({\rm Sym}_f^{\mu},\sigma)-\log L({\rm Sym}_f^{\nu},\sigma)}&=-\sum_{p\neq q}\bigl(\log(1-\alpha_f^{\mu}(p)p^{-\sigma})+\log(1-\beta_f^{\mu}(p)p^{-\sigma})\bigr).
N (|A|^2) = 6h^S_{22}H h_{11}+2 (h^S_{\nu\nu}-2 h^S_{22}) |A|^2 -4 h^S_{22}h^2_{11}& +2\nabla^S_\nu h^S_{22}(h_{11}-h_{22}) +4 h_{12}\nabla_1 h_{12}- 4 (h^S_{\nu\nu}-2 h^S_{22}) h^2_{12}
P^+A^k+SP^+B^k-S(P^+B^k-\bar{S}{P^+A^k}) = (1+|S|^2)P^+A^k=O_{L^2}(\varepsilon)+P^+\Big(\frac{\zeta_{1,\varepsilon}f^k}{a^*}\Big)+\bar{S}P^-\Big(\frac{\zeta_{1,\varepsilon}f^k}{a^*}\Big).
IV=\left\{-\frac{1}{4}\overline{Ric}_{yy,kk}\left((n+2)I_3+I_5\right)-\frac{1}{24}\left(\overline{Ric}_{yy,yy}+2(\bar{R}_{yiyj})^2\right)(I_4+I_6)\right\}\mu^4+\mu^4o(1),
0< \Delta_2:=\sum_{i=\ell+2}^{\infty}x^{i^2-(\ell+1)^2}\prod_{j=i+1}^{\infty}\left(1-x^j\right)^2<\sum_{i=\ell+2}^{\infty}x^{i^2-(\ell+1)^2}<\sum_{i=2\ell+3}^{\infty}x^{i}=\frac{x^{2\ell+3}}{1-x}<x^{2\ell}
< \alpha_{p}( t ) > ~ ={}~ ( - i )^{p /{\bar{n}}}{\frac{\lambda ( p )}{\lambda ( p ) - 1}}({\frac{C_{\bar{n}}^{\frac{1}{2}}}{{C_{\bar{n}}^{\frac{1}{2}}}^{*}}})^{{\frac{p}{\bar{n}}}}~ J_{p /{\bar{n}}}( \Phi ({\bar{n}}, p ) )
{\mathbb P}=\left\{\begin{bmatrix}q_{11}q_{13}q_{21}q_{23}\end{bmatrix},\begin{bmatrix}q_{11}q_{14}q_{21}q_{24}\end{bmatrix},\begin{bmatrix}q_{12}q_{13}q_{22}q_{23}\end{bmatrix},\begin{bmatrix}q_{12}q_{14}q_{22}q_{24}\end{bmatrix}\right\}.
\|F\|^2_{E^2}= \sum_n \bigg\{\frac{|F(s_n)|^2}{K_2(s_n,s_n)}+ \frac{|F(t_n)|^2}{K_2(t_n,t_n)}\bigg\}= \frac{\pi}{2}\sum_n \bigg\{\frac{|F(s_n)|^2}{|A'(s_n)B(s_n)^3|}+ \frac{|F(t_n)|^2}{B'(t_n)A(t_n)^3}\bigg\}.
X ( p_{0}) = \frac{i \Gamma}{2 \pi}\left[ \left( p_{0}^{2}- \frac{3 m^{2}}{2}+ \frac{m^{4}}{2 p_{0}^{2}}\right) \log \left| \frac{p_{0}^{2}- m^{2}}{m^{2}}\right| + \frac{m^{2}}{2}- \frac{5 p_{0}^{2}}{4}\right] .
\frac{\binom{n+k_1-1}{k_1}}{\binom{n+2k_1-1}{2k_1}-1}<\frac{(2k_1)!}{k_1!}n^{-k_1}\frac{\binom{m+k_2-1}{k_2}}{\binom{m+2k_2-1}{2k_2}-1}<\frac{(2k_2)!}{k_2!}m^{-k_2}),
uf_3(x)=u+uh_1^{\prime}x+\cdots+uh_{k-1}^{\prime}x^{k-1}+ux^k, &2uf_4(x)=2u+2uh_1^{\prime\prime}x+\cdots+2uh_{l-1}^{\prime\prime}x^{l-1}+2ux^l,
\hat{\cal A}_{2}= \theta^{\alpha \beta}\partial_{\beta}[ < \partial_{\alpha}\hat{\bar{\psi}}( x_{+}) \gamma_{( 2 n + 1 )}\gamma_{\mu}\hat{\psi}( x_{-}) > \{g ( \hat{A}^{\mu}( x_{+}) - \hat{A}^{\mu}( x_{-}) ) - i \partial^{\mu}\hat{U}\}] .
\mathbb{E}_{\underline{Z}}\exp\left\{\frac{1}{\gamma}\sum_{i=1}^nZ_i(g^2(X_i)-g^2(X^{\prime}_i))\right\}= \prod_{i=1}^n\mathbb{E}_{Z_i}\exp\left\{\frac{1}{\gamma}Z_i(g^2(X_i)-g^2(X^{\prime}_i))\right\}.
\int_{0}^{\infty}\frac{t^{-1-\alpha/2}}{\omega(B^g(2^k\sqrt{t}))}\sum_{j\geq 0}\| g_k^{j,t}\|^2_{L^2 (C_0^{j,t}, \mu_2)}dt \leq C 2^{k(2\kappa+\alpha)}\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\frac{|f(x)-f(y) |^2}{\omega(B^g_{xy})\cdot d_g( x,y)^{\alpha}}d\mu_2(x)d\mu_2(y).
\int_X \int_X u(x) v(x-y) w(y) \, dx \, dy = \int_G \left( \int_X \int_X u(g.x) v(g.(x-y)) w(g.y) \, dx \, dy \right) dg = \int_X \int_X \left( \int_G u(g.x) v(g.(x-y)) w(g.y) \, dg \right) dx \, dy \leq \int_X \int_X M_p(u)(x) M_q(v)(x-y) M_r(w)(y) \, dx \, dy.
\zeta_{F}( 2 | x_{1}, x_{2}) = \sum_{( n_{1}, n_{2}) \neq ( 0 , 0 ) ( 0 , - 1 ) ( - 1 , 0 ) ( - 1 , - 1 )}\left[ ( n_{1}a + x_{1})^{2}+ ( n_{2}b + x_{2})^{2}\right]^{- 2}
\Sigma_{A}( \bar{p}_{\parallel}) ~ \simeq ~ e^{2}( 4 | e H | ) \int \frac{d^{4}\hat{q}}{( 2 \pi )^{4}}\frac{\mathrm{e}^{- \hat{q}_{\perp}^{2}}}{\hat{q}^{2}}\frac{\Sigma_{A}( \bar{p}_{\parallel}- q_{\parallel})}{( \bar{p}_{\parallel}- q_{\parallel})^{2}+ \Sigma_{A}^{2}( \bar{p}_{\parallel}- q_{\parallel})},
\left\{\begin{array}{c}{{\left( \begin{array}{c l c}{{\frac{\delta - \sqrt{\delta^{2}+ \omega^{\ast}\omega}}{\omega}}}&{{,}}&{{1}}\\ \end{array}\right)}}\\{{\left( \begin{array}{c l c}{{\frac{\delta + \sqrt{\delta^{2}+ \omega^{\ast}\omega}}{\omega}}}&{{,}}&{{1}}\\ \end{array}\right)}}\\ \end{array}\right\}.
k_0(\tilde f) :=\tilde f_0^{(n_{\tilde\varepsilon})}(1), k_{i+1}(f) :=\tilde f_i^{(n_{\tilde\varepsilon})}(k_i(f)) \tilde f(n) :=\max\{f^M(16dn^2),16dn^2\}, n_{\tilde\varepsilon}:=\lceil 8d^4/\tilde\varepsilon^4\rceil, i_0 :=\lceil\log_{\tau}(\tilde\varepsilon/6d^2)-1\rceil, \tilde\varepsilon :=\frac{(1-\tau)^2}{6+8d}\varepsilon.
-(n-\delta-N)R_{n}(x)+(n-N) R_{n+1}(x) =\frac{(x+\gamma+1)(x+\delta)}{(\gamma+1)}\hat R_n(x), -(n+1) \hat R_n(x)+ (n+\gamma+2) \hat R_{n+1}(x) = (\gamma+1) R_{n+1}(x).
\frac{\partial}{\partial t_{0}}R_{n + 1}[ u ] = \frac{n}{2 n + 1}\left( \frac{1}{2}\lambda^{2}\frac{\partial^{3}}{\partial t_{0}^{3}}+ 2 u ( t_{0}) \frac{\partial}{\partial t_{0}}+ \frac{\partial}{\partial t_{0}}u ( t_{0}) \right) R_{n}[ u ] .
I (f) := c_{\alpha}\; \int_{\mathbb{T}}\frac{(f(x)-f(y))^2}{|x-y|^{1+\alpha}}dy+c_{\alpha}\sum_{k\in\mathbb{Z}\setminus\{0\}}\int_{\mathbb{T}}\frac{(f(x)-f(y))^2}{|x-y+2k\pi|^{1+\alpha}}dy.
L_I(s) =\exp \left(-\rho_I \int\limits_{\delta_0}^{\infty}(1-\frac{1}{1+s \gamma_0 (r^2+L^2)^{-1}}) d r \right) = \exp \left(-\rho_I \frac{s \gamma_0 ArcCot(\frac{\delta_0}{\sqrt{L^2+s \gamma_0}})}{\sqrt{L^2+s \gamma_0}}\right)
\Psi_{(\Sigma, U)}(f) = \Psi_{(\Sigma, U)}\left( \mathfrak{D}^{(\mu)}(i, i_U) (f|_{\Sigma}) \right) = \mathfrak{P}_{\ell}(i, i_U) \left( \Psi_{(\Sigma_f, U_f)}(f|_{\Sigma}) \right).
\Big\{a+\frac{1}{\lambda_2}\left[(\chi_2\mu_2\lambda_2-\chi_1\mu_1\lambda_1)_{+}+\chi_{1}\mu_{1}(\lambda_1-\lambda_2)_{+}\right]\overline{u}&- \frac{1}{\lambda_2}\left[(\chi_2\mu_2\lambda_2-\chi_1\mu_1\lambda_1)_{-}+\chi_1\mu_1(\lambda_1-\lambda_2)_{-}\right]\underline{u}\Big\}_+=0,
S_n(\kappa,t) = \frac{\kappa}{i\pi}\int_{\gamma_{\kappa}}\sum_{j=1}^n[(1-w^2)e^{-t}]^j \sum_{m= 0}^{j-1}L_{j-m-1}^{(m+1)}(2jt) \left\{P_{n-j}^{0,2j}(1-2w^2) - P_{n-1-j}^{0,2j}(1-2w^2)\right\}\frac{(-2)^mw^{m-1}}{(w-\kappa)^{m+1}}dw.
[\frac{\partial f}{\partial(\nabla^{(i)}\rho)}]_{\alpha_{1}\alpha_{2}\cdot\cdot\cdot\alpha_{i}}=\frac{\partial f}{\partial\rho_{\alpha_{1}\alpha_{2}\cdot\cdot\cdot\alpha_{i}}}where\rho_{\alpha_{1}\alpha_{2}\cdot\cdot\cdot\alpha_{i}}\equiv\frac{\partial^{i}\rho}{\partial r_{\alpha_{1}}\partial r_{\alpha_{2}}\cdot\cdot\cdot\partial r_{\alpha_{i}}},
\mathbb{P}\bigg(\mathbb{P}_{Q^{(\mathsf{B}_n)}}\Big(\mathbf{U}(I,\mathsf{B}_n)\notin\mathcal{T}_\epsilon^n(Q_U)\Big)>2e^{-n\eta(\epsilon)}\bigg)\leq e^{\frac{2^{nR_1}e^{-n\eta(\epsilon)}}{3}}=e^{-\frac{1}{3}e^{n\big(R_1\ln2-\eta(\epsilon)\big)}},
K(z,z'')= \Big(\int_{r'<\frac{r}{2}}+\int_{\frac{r}{2}\leq r'\leq2r}+\int_{r'>2r}\Big)G(z,z')Q(z',z'')\;d\mu(z') = K_{3,1}(z,z'')+K_{3,2}(z,z'')+K_{3,3}(z,z'').
\sum_{j}\;{\frac{t_{j}^{k}\, d t_{j}}{s_{j}}}= 4 \, d \tau \, u ( \lambda ) \, \sum_{j}\; ( - 1 )^{N - 1 - j}{\frac{t_{j}^{k}}{t_{j}- \lambda}}{\frac{D_{j}}{D}}= - 4 \,{\frac{\lambda^{k}}{\prod_{l}\, ( \lambda - a_{l})}}\, d \tau
\ll \nabla_T^*\nabla_T\phi,\psi\gg =\int_M \langle\nabla_{\rm T}\phi,\nabla_{\rm T}\psi\rangle &=\int_M\langle\phi,-\sum_a \nabla_{V_a}\nabla_{\bar V_a}\psi\rangle +\int_M\langle\phi,\nabla_{H^{0,1}}\psi\rangle =\ll\phi,\nabla_T^*\nabla_T\psi\gg,
\sum_{p=0}^{r-3}(-1)^p \left[a_{k-r+2+p}(n-p)(n-p-1) + b_{k-r+1+p}(n-p) + c_{k-r+p}\right] e_p + (-1)^{r-2}[a_k (n-r+2)(n-r+1) + b_{k-1}(n-r+2) + c_{k-2}] e_{r-2}= 0, r=3, 4, \ldots, k.
\int_0^\infty e^{2\pi i nt}e^{-2^k\pi(m+1)t}dt =\int_0^\infty e^{2\left(\pi i n-2^{k-1}\pi(m+1)\right)t}dt=-\dfrac{1}{2\pi(i n-2^{k-1}(m+1))}\int_0^\infty e^{-2\pi i nt}e^{-2^k\pi(m+1)t}dt =\int_0^\infty e^{-2\left(\pi i n+2^{k-1}\pi(m+1)\right)t}dt=\dfrac{1}{2\pi(i n+2^{k-1}(m+1))}
\begin{aligned}|\langle\boldsymbol{\nu}, G(\lambda|\bar U,x,\tau)\rangle| \le c_3 \langle \boldsymbol{\nu}, \eta (\lambda | \bar U;x,t ) \rangle = c_3 \mathcal{H}( \boldsymbol{\nu},U,\bar{U};x,t)\;,\end{aligned}
(\begin{matrix}f_{1}(x,y,y^{\prime},y^{\prime\prime},...,y^{(n)})\\f_{2}(x,y,y^{\prime},y^{\prime\prime},...,y^{(n)})\\ \vdots\\f_{m}(x,y,y^{\prime},y^{\prime\prime},...,y^{(n)})\end{matrix})=(\begin{matrix}0\\ 0\\ \vdots\\ 0\end{matrix})
g(z)=g[A_{w}^{-1}]_{2m+2}(\zeta)j(A_{w},z)^{-2m-2}=\frac{(2it)^{m+1}}{(z-\overline{w})^{2m+2}}\sum_{n\gg-\infty}a_{n}(w)A_{w}(z)^{n}.
\left| \frac{1}{H(H+1)^2}\sum_{h, k = 0}^{H-1}(H + 1 - k) a_h b_k c_{h+k}\right|^2 &\leq \| a \|_\infty^2 \left( \frac{1}{H}\sum_{h=0}^{H-1}\left| \frac{1}{(H+1)^2}\sum_{k=0}^{H-1}(H + 1 - k) b_k c_{h + k}\right|^2 \right).
S_{1 ( 2 )}= \frac 1 2 \int_{0}^{T}d \tau \left( e_{1}\dot{x}_{1}^{2}- \frac{m_{1}^{2}}{e_{1}}\right) + \lambda_{2 \mu}\int_{0}^{T}d \tau \dot{x}_{1}^{\mu}A_{1}
\delta(t_{k+1}) =\delta(t_{k})+\int^{t_{k+1}}_{t_{k}}(\lambda-u(t))dt &=\delta(t_{k})+\lambda\mathsf{T}_{k}-0.5(\beta^{\top}u(t_{k})+\mathbf{1}^{\top}u(t_{k+1}))\mathsf{T}_{k}.
\begin{array}{l}\displaystyle{d u_s= -H(t_0)\left(u_s^2-1 \right)ds + \frac{3 \sigma^2}{2}u_s ds+ \sigma \sqrt{u_s^2-1}dB_s}, \displaystyle{d v_s= -H_{\infty}\left(v_s^2-1 \right)ds + \frac{3 \sigma^2}{2}v_s ds+ \sigma \sqrt{v_s^2-1}dB_s}. \end{array}
[\begin{matrix}y_{1,t}\\y_{2,t}\end{matrix}]=[\begin{matrix}c_{1}\\c_{2}\end{matrix}]+[\begin{matrix}a_{1,1}&a_{1,2}\\a_{2,1}&a_{2,2}\end{matrix}][\begin{matrix}y_{1,t-1}\\y_{2,t-1}\end{matrix}]+[\begin{matrix}e_{1,t}\\e_{2,t}\end{matrix}],
\int_{A_0}(u^{\prime}(t))^{2}dt=\int_{\sigma}^{\tau}(u^{\prime}(t))^{2}dt-\int_{(\sigma,\tau)\setminus A_0}\!\!\!\!\!\!\!\!\!\!\!\!(u^{\prime}(t))^{2}dt\geq\frac{1}{\tau-\sigma}-\frac{1}{4}\frac{1}{\tau-\sigma}\geq\frac{3\cdot2^{4}}{\varepsilon\delta_{\!J}}~.
\begin{aligned}A_2^{(N)}(0,x;1+N-n)= N-n, &A_2^{(N)}(1,x;1+N-n)= -(x-1), &A_3^{(N)}(0,x;1+2N-n)= \frac12(2N-n)(N-n), &A_3^{(N)}(1,x;1+2N-n)= -\frac12(2N-n)(x-1)-\frac12(x-2)(N-n+1), &A_3^{(N)}(2,x;1+2N-n)= \frac12(x-2)(x-1),\end{aligned}
h_i f_i = ~ \int_{I_i}f(x) ~\mathrm{d}x = ~ \int_{I_i}\Big[ \alpha_i f \big(L_i^{-1}(x)\big)+ q_i \big(L_i^{-1}(x)\big) \Big]~\mathrm{d}x = ~ \alpha_i a_i \int_{I}f(x)~ \mathrm{d}x + a_i \int_{I}q_i(x)~ \mathrm{d}x = ~ \alpha_i a_i \sum_{i=1}^N h_i f_i + a_i \int_{I}q_i(x)~ \mathrm{d}x, ~i=1,2,\dots, N.
\sum_{\mu\in\Lambda}\gamma_\mu\langle\mu|:e^{B_1(z)}:e^{A_1(z)}|\chi\rangle V_0(\psi,z)=V_1(\psi,w-z)\sum_{\mu\in\Lambda}\gamma_\mu\langle\mu|:e^{B_1(z)}:e^{A_1(z)}|\chi\rangle\,,
F_{n}^{\mathcal{O}|\mu_{1}\ldots \mu_{n}}(\theta_{1},\theta_{2}\ldots,\theta_{n}):=\langle 0|\mathcal{O}(0)|Z_{\mu_{1}}^{\dagger}(\theta_{1})Z_{\mu_{2}}^{\dagger}(\theta_{2})\ldots Z_{\mu_{n}}^{\dagger}(\theta_{n})\rangle .
\frac{\partial}{\partial z}= \frac{1}{2}\left( \frac{\partial}{\partial x}+ \frac{1}{\i}\frac{\partial}{\partial y}\right), \frac{\partial}{\partial z^\rho}= \frac{1}{2}\left( \frac{\partial}{\partial x}- \frac{1}{\i}\frac{\partial}{\partial y}\right)
\begin{aligned}\mathcal{Q}_1 =2\sum\limits_{i\in I}\sigma_i^2-2\sum\limits_{i=1}^{n-1}\sigma_{i}\sigma_{i+1}-2\sigma_{n}\sigma_{n+1}-2\sigma_{n+1}\sigma_{1}&=(\sigma_1-\sigma_2)^{2}+(\sigma_2-\sigma_3)^{2}+\cdots+(\sigma_n-\sigma_{n+1})^{2}+(\sigma_{n+1}-\sigma_1)^{2}.\end{aligned}
\Gamma:=(c\,\Im m_{\mu_A\boxplus\mu_B}+\hat{\Lambda})^{-2}\big(|\Lambda_A|^2+|\Lambda_B|^2\big)+\Big(\frac{N^{5\varepsilon}}{(N\eta)^{\frac13}}\Big)^{-2}|\Upsilon|^2+\Big(\frac{N^{5\varepsilon}}{\sqrt{N\eta}}\Big)^{-1}\frac{1}{N}\sum_{i=1}^N (| T_{i}|^2+N^{-1})^{\frac12},
\frac{1}{s}=\sum_{k=2}^{\infty}{e^{sk}[\frac{(k-1)e^{-s}+s(k-1)^2e^{-s}-\frac{s(k-1)^2e^{s(k-2)}}{1+e^{s(k-1)}}}{(\alpha+s(k-1))e^{s(k-1)}+\alpha}}
S^{( 2 )}( p_{2}, p_{3}) S^{( 2 )}( p_{3}, p_{1}) S^{( 2 )}( p_{1}, p_{2}) = S^{( 2 )}( p_{1}, p_{2}) S^{( 2 )}( p_{1}, p_{3}) S^{( 2 )}( p_{2}, p_{3}) \; ,
\left( \sum_{j_{1},\ldots,j_{m}=1}^{n}\left\vert T(e_{j_{1}},\ldots,e_{j_{m}})\right\vert^{\rho}\right)^{\frac{1}{\rho}}\leq C_{m}\sup_{\left\Vert x_{1}\right\Vert ,\dots,\left\Vert x_{m}\right\Vert \leq 1}\left\vert T(x_{1},\dots,x_{m})\right\vert,
(\frac{\partial\xi}{\partial\zeta})^{-1}=\frac{4\pi}{2n-1}\mbox{diag}(j_b^{-1}:1\leq b\leq n) \begin{pmatrix}\frac32-n 1 \cdots 1 1 \frac32-n \cdots 1 \cdots \cdots \cdots \cdots 1 1 \cdots \frac32-n\end{pmatrix}.
\frac{\prod_{i=1}^{n}p_i}{\prod_{i=2}^{n}(p_i-2)}\cdot\prod_{i=2}^{m}(p_i-2)\cdot(1-\frac{1}{p_{m+1}})(1-\frac{1}{p_{m+2}})\cdots(1-\frac{1}{p_{m+k}})\leqslant\pi(X)-m.
\Phi_{p,q}(\lambda)={\lambda}^{5}- \left( 2\,q-10+2\,p \right){\lambda}^{4}&- \left( -3\,pq+16\,p+16\,q-40 \right){\lambda}^{3}&- \left( -18\,pq+44\,p+50\,q-74\right){\lambda}^{2}&- \left( -30\,pq+45\,p+63\,q-53 \right) \lambda+12\,pq-10\,p-22\,q+2.
|Q_{\Sigma}(\sum_i a_i \phi_i, \sum_i a_i\phi_i) -Q_{\Sigma}(\sum_i a_i \psi_i, \sum_i a_i\psi_i)| =\ |\sum_{1\leq i,j\leq I}a_ia_j(Q_{\Sigma}(\phi_i, \phi_j) - Q_{\Sigma}(\psi_i, \psi_j))|\leq C(\Sigma, I)\varepsilon\cdot \sum_i a_i^2
k_1\mathbf{q}\bigl(\tfrac{\partial_1a_\gamma}{a_\gamma},\dots,\tfrac{\partial_ma_\gamma}{a_\gamma}\bigr)=k_2\mathbf{q}\bigl(\tfrac{\partial_1e_\gamma}{e_\gamma},\dots,\tfrac{\partial_me_\gamma}{e_\gamma}\bigr)
1 = \int D S_{\mu \nu}\delta \left( S_{\mu \nu}- \Sigma_{\mu \nu}^{\mathrm{g a s}}\right) = \int D S_{\mu \nu}D l_{\mu \nu}\exp \left[ - i \pi \int d^{4}x l_{\mu \nu}\left( S_{\mu \nu}- \Sigma_{\mu \nu}^{\mathrm{g a s}}\right) \right]
\widehat r = \bigl\{\, \bigl(\vec a,f_1(\vec a),\dotsc,f_m(\vec a)\bigr) \bigm| \vec a \in r \,\bigr\}\quad \widehat s = \bigl\{\, \bigl(\vec a,h(\vec a),f_{\theta(1)}(\vec a),\dotsc,f_{\theta(m)}(\vec a)\bigr) \bigm| \vec a \in r \,\bigr\}.
\sum_{\substack{k \geq 1 P_1 \cup \cdots \cup P_k = \{1,\ldots, n\}P_i}}b''_{|P_1|}\cdots b''_{|P_k|}\frac{(k-1)!}{2}\sum_{j=0}^{k-1}a_ja_{k-1-j}\left(\sum_{i=1}^{k}\binom{|P_i|}{2}+ \binom{n}{2}\right)\frac{(j+1)(k-j)}{\binom{n}{2}}= 0
[\begin{matrix}1&2&1\\ 0&0&0\\ -1&-2&-1\end{matrix}]=[\begin{matrix}1\\ 0\\ -1\end{matrix}][\begin{matrix}1&2&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}]
\widehat{x}\left( z\right) =\frac{\widehat{\alpha}_{0}\left( z\right) \widehat{\beta}\left( z\right)}{1-\widehat{\alpha}\left( z\right)}=\frac{\widehat{\alpha}_{0}\left( z\right)}{\widehat{\alpha}\left( z\right)}\widehat{x_{b}^{.}}\left( z\right) .
\prod_{A=1}^{N^2-1}\frac{d X^A_D}{\sqrt{2 \pi}}=\frac{2^{\frac{N(N-1)}{2}}}{\prod_{i=1}^{N}i!}\sqrt{N}\prod_{k=1}^{N}\frac{d \lambda_k}{\sqrt{\pi}}\Delta^2(\lambda_i)\sqrt{\pi}\delta(\lambda_1+ \ldots + \lambda_N)
\Phi^{\prime}\ = \ \Phi ( \chi^{\prime}, \chi^{*}) \ = \, s u m a_{m_{1}m_{2}n_{1}n_{2}}{\chi^{\prime}}_{1}^{* m_{1}}{\chi^{\prime}}_{2}^{* m_{2}}{\chi^{\prime}}_{1}^{n_{1}}{\chi^{\prime}}_{2}^{n_{2}}\ .
\left\lbrace\begin{aligned}X_{n+1}^{\epsilon,\Delta t}=\Phi(\frac{\Delta tm_{n+1}^{\epsilon,\Delta t}}{\epsilon},X_n^{\epsilon,\Delta t}) m_{n+1}^{\epsilon,\Delta t}=m_n^{\epsilon,\Delta t}-\frac{\Delta t}{\epsilon^2}m_{n+1}^{\epsilon,\Delta t}+\frac{\Delta\beta_n}{\epsilon}\end{aligned}\right.
T \| S \Longleftrightarrow \pi(T) \| \pi(S) &\Longleftrightarrow (\pi(T)^* \pi(T) \| \pi(T)^*\pi(S) ~~~ ~~~ \| \pi(T)^* \pi(S)\| = \| \pi(T)\|\| \pi(S)\|) &\Longleftrightarrow (T^*T \| T^*S \| T^*S\| = \| T\|\| S\|).
T^{\psi_2,\psi_1}_{\psi_3,\psi_4}(t+\tfrac{i}{2})=\overline{(T^*)^{J_H \psi_2,\psi_3}_{\psi_1,J_H\psi_4}(t)}=T^{\psi_1,J_H\psi_4}_{J_H\psi_2,\psi_3}(-t).
\langle p|j_{\mu}|p+k \rangle A^{\mu}(k)=\bar{\Psi}_{in}(p)\Biggl( \gamma_{\mu}F_{1}(k^{2})+\frac{i}{2}[\gamma_{\mu},\gamma_{\nu}]k^{\nu}F_{2}(k^{2}) \Biggr)\Psi_{in}(p+k)A^{\mu}(k)
F_N(a,b;t) =\frac{(1-b)(1-tq^{N+1})}{(1-t)(1-bq^{N+1})}+\frac{(1-q^{N+1})(b-atq)(1-bq^{N})}{(1-bq^{N+1})(1-bq^{N})(1-t)}\sum_{n=0}^{N}\frac{(q)_N(aq)_n(q)_n(tq)_{N-n}(tq)^n}{(q)_{N-n}(bq)_n(q)_n(tq)_N}&=\frac{(1-b)(1-tq^{N+1})}{(1-t)(1-bq^{N+1})}+\frac{(1-q^{N+1})(b-atq)}{(1-bq^{N+1})(1-t)}F_N(a,b;tq).
cov\{\hat{m}_{2y}(\lambda)\hat{m}_{2y}(\lambda),\hat{m}_{2y}(\beta)\hat{m}_{2y}(\beta)\}&= cov\{\hat{m}_{2y}(\lambda),\hat{m}_{2y}(\beta)\}cov\{\hat{m}_{2y}(\lambda),\hat{m}_{2y}(\beta)\}\ + cov\{\hat{m}_{2y}(\lambda),\hat{m}_{2y}(\beta)\}cov\{\hat{m}_{2y}(\lambda),\hat{m}_{2y}(\beta)\}.
\begin{cases}\left(i\partial_t + i\xi\cdot \nabla_x -i\nabla_x V(t,x) \cdot\nabla_{\xi}-\frac{1}{2}|\xi |^2-\widetilde{V}(t,x)\right) \times &\phantom{xxxxxxxxxxxxxxxxxxxxxxxxxxx}W_{\varphi^{(t)}}u(t,x,\xi ) =Ru(t,x,\xi ), &W_{\varphi^{(0)}}u(0,x,\xi ) = W_{\varphi_0}u_0(x,\xi ),\end{cases}
dist_{g_{X}}(X,Y) =(1+O^{(4)}(|x-y|^2))dist_{\tilde g}(\exp_{\bar X}x,\exp_{\bar X}y) &=(1+O^{(4)}(|x-y|^2))(1+O^{(4)}(|x|^2)+O^{(4)}(|y|^2))|x-y|,
\sum_{i=1}^{u}(-1)^{i-1}a_{[\ell_1\ell_2\dots \ell_{k-u+1}s_i]}^{s_1\dots s_{u}\setminus s_i}=\sum_{i=1}^{u}(-1)^{i-1}a_{[\ell_2\dots \ell_{k-u+1}s_i]}^{\ell_1s_1\dots s_{u}\setminus s_i}=a^{s_1\dots s_u}_{\ell_1\dots\ell_{k-u+1}}.
Z_{DD}={1\over N}\sum_{p=0}^{N-1}\chi^f_p(q){1\over \eta(q)}\int_{-\infty}^\infty dx q^{x^2/2N}\sum_{m=-\infty}^\infty [|a_m|^2e^{2\pi imx}+(-1)^p|b_m|^2e^{2\pi i(m+1/2)x}]
F^C_{\tiny\mbox{div}}[g,\beta,D]=-\eta\,{\Gamma\left(1-\frac D2\right) \over (4\pi)^{D/2}}\,\,{\pi^2 m^{D-4}\over 3\kappa \beta^2}\,\,\int_{\Sigma}\left[f_1 m^2- \left(p_1{4\pi^2 \over \kappa^2\beta^2}{\cal P}+p_2 R+p_3{\cal Q})\right)\right]~~~.
\tilde{\nabla}\tilde{X}^i \cdot \tilde{\nabla}W_0 = 2 \bar W_{0m0n}\tilde{X}^m (\delta^{in}- \tilde{X}^n \tilde{X}^i) = -2 W_0 \tilde{X}^i +2\bar W_{0m0i}\tilde{X}^m,
\Big\langle \frac{1}{4}\int d^4x\,d^4\theta\,\Big(\phi^* e^{2V}\phi+ \tilde\phi^* e^{-2V}\tilde\phi\Big)\Big\rangle= - \frac{1}{16\pi^2}\mbox{Re}\int d^4x\,d^2\theta\,W_a C^{ab}W_b.
T(22,30) = \left[ \begin{array}{cccc}1 0 0 0 0 1 0 0 0 1 0 1 0 0 1 0\end{array}\right] T(30,22) = \left[ \begin{array}{cccc}0 1 0 0 1 0 1 0 0 0 1 0 0 0 0 1\end{array}\right]
\mathcal{T}:= (\mathbb{R}\times \mathcal{T}_0) \, \bigcup \, \bigl((0,\infty) \times \{-\infty\}\times \mathbb{R}\bigr) \, \bigcup \, \bigl((-\infty,0) \times \mathbb{R}\times \{\infty\}\bigr).
\frac{\partial U_{r}(X_{r})}{\partial X_{r}}F_{r}(X_{r},\tilde{\vartheta}_{r},\tilde{\chi},\mu) \leq-2\|X_{r}\|^{2}+\delta^{2}\leq-\|X_{r}\|^{2},~\forall \|X_{r}\|\geq\delta.
\begin{aligned}T_4(x,y):= \frac{\partial P_4}{\partial x}+ \frac{\partial Q_4}{\partial y}, D_4(x,y):= \frac{\partial P_4}{\partial x}\frac{\partial Q_4}{\partial y}-\frac{\partial P_4}{\partial y}\frac{\partial Q_4}{\partial x}\end{aligned}
A=\left( \begin{array}{cccc}\frac{1}{4}1 0 0 \frac{1}{27}\frac{1}{4}1 0 \frac{1}{256}\frac{1}{27}\frac{1}{4}1 \frac{1}{3125}\frac{1}{256}\frac{1}{27}\frac{1}{4}\end{array}\right)
g(x) = x^{\lambda -k_2}\frac{\big( q^{\lambda -h_1 +1/2}x /t_1 ; q\big)_{\infty}}{\big( q^{-h_1 +1/2}x /t_1 ; q\big)_{\infty}}{}_3\phi_2 \biggl(\!\! \begin{array}{c}q^{-h_1 +1/2}x /t_1, q^{\lambda -h_1 + l_1},q^{\lambda -h_1 + l_2}t_2 /t_1 q^{\lambda -h_1 +1/2}x /t_1, q^{1 -h_1 + h_2}t_2 / t_1 \end{array}\! ;q,q^{\lambda +k_1 -k_2}\biggr)
K_{\mu}(s):=e^{-A_n^2(t-s)}(-A_n)^{1+\mu}\left\{F_n(\mathbb U(s))-F_n(U(s))\right\}, L_{\mu}(s):=\sqrt{n}(-A_n)^{\mu}e^{-A_n^2(t-s)}\left\{\Sigma_n(\mathbb U(s))-\Sigma_n(U(s))\right\}.
R_{abc}^{i_{1}i_{2}i_{3}}R_{i_{1}i_{2}d}^{i_{4}i_{5}i_{6}}R_{i_{4}i_{3}i_{6}}^{ei_{7}i_{8}}R_{i_{5}i_{7}i_{8}}^{fgh}= R_{bcd}^{i_{1}i_{2}i_{3}}R_{ai_{2}i_{3}}^{i_{4}i_{5}i_{6}}R_{i_{4}i_{1}i_{6}}^{i_{7}i_{8}h}R_{i_{7}i_{8}i_{5}}^{efg}
\overline{S_2}(p, N) = N \sum\limits_{i = (1-\varepsilon)\mu}^{(1+\varepsilon)\mu}{\left[ \binom{N}{i}{{p}^i}{{\left({1 - p}\right)}^{N - i}}\overline{L}\left( \frac{i + 1 - \mu}{\mu}\right) \right]},
0 = 1 + \sum_{i=0}^{k-1}(e(a_i) + w_i)^2\quad\mbox{and}0 = \sum_{i=0}^{j-1}(e(a_i) + w_i)(\alpha e(a_{k-j+i}) + w_{k-j+i}) + \sum_{i=j}^{k-1}(e(a_i) + w_i)(e(a_{i-j})+ w_{i-j})
\begin{pmatrix}I \ast \ast \dots \ast \ast 0 M_1 \ast \dots \ast \ast 0 0 M_2 \ddots \vdots \vdots 0 0 0 \ddots \ast \ast \vdots \vdots \vdots \ddots M_{d-1}\ast 0 0 0 \dots 0 M_d\end{pmatrix},
\bar{v}_t^\epsilon (\bar{x},\bar{t})+\int \left(1-\frac{\exp(\bar{v}^\epsilon(\eta^\epsilon (\bar{x},y),\bar{t})/\epsilon)}{\exp(\bar{v}^\epsilon(\bar{x},\bar{t})/\epsilon)}\right) K dy +\mathcal{J}_\epsilon = \mu(\hat{\bar{x}}|\bar{x}|^{1/\epsilon}) - E e^{-\bar{v}^\epsilon(\bar{x},\bar{t})/\epsilon},
x_1x_{i_2-t}\cdots x_{i_d-t}\ge x_1x_{n-(d-1)t-1}\cdots x_{n-\ell t-1}\cdot x_{n-(\ell-1)t}\cdots x_{n-t}&=x_1(v_{\ell}/x_{\max(v_\ell)})\ge x_1(v_{\ell+1}/x_{\max(v_{\ell+1})}).
P_{n}^{\left( r+1\right)}\left( x\right) =\left( u_{0}^{\left( r\right)}\theta_{0}P_{n+1}^{\left( r\right)}\right) \left( x\right) ;u_{n}^{\left( r+1\right)}=\left( xu_{n+1}^{\left( r\right)}\right) \left(u_{0}^{\left( r\right)}\right)^{-1},\n,r\geq 0.
\mathbb{P}_{n_{l_q}}\left( \left| \log(\frac{d\tilde{\mathbb{Q}}_{n_{l_{q}}}}{d\mathbb{P}_{n_{l_{q}}}}) - \sum_{k=2}^{K}\frac{2\mu_{k}(C_{n_{l_q},k}-(n-1)\mathbb{I}_{k=2})-\mu_{k}^{2}}{4k}\right|\ge \frac{\epsilon}{2}\right) \le \frac{\delta}{2}.
\left\Vert \mathbf{F}\right\Vert_{1,\mathfrak{L}}:=\underset{y\in \mathfrak{L}}{\sup}\sum_{x\in \mathfrak{L}}\mathbf{F}\left( \left\vert x-y\right\vert\right) =\sum_{x\in \mathfrak{L}}\mathbf{F}\left( \left\vert x\right\vert\right) <\infty \ .
S_{0}^{( l )}[ Q ] ={\frac{l}{2 g^{2}\beta}}\sum_{{\vec{n}}}\sum_{{\hat{e}}}\varphi_{{\vec{n}}}[ \varphi_{{\vec{n}}+ 2{\hat{e}}}- 2 \varphi_{{\vec{n}}+{\hat{e}}}+ \varphi_{\vec{n}}] .
|\varphi(t)-\varphi(s)| =| \, \|x(t+t_0)-y(t+t_0)\|-\|x(s+t_0)-y(s+t_0)\| \, | & \le \| x(t+t_0)-x(s+t_0)+y(s+t_0)-y(t+t_0)\| &= \left\| \int_{s+t_0}^{t+t_0}[x'(r)-y'(r)] \, dr \right\| =\left\| \int_{s+t_0}^{t+t_0}[f(r,x(r))-f(r,y(r))] \, dr \right\| & \le (t-s) \, \max_{s+t_0 \le r \le t+t_0}\|f(r,x(r))-f(r,y(r))\|,
( \delta_{1}X_{2})_{\mathrm{e x t r a}}={\frac{\tau_{1}\tau_{2}}{1 - \tau_{1}\tau_{2}}}F_{1}M^{- 1}\eta_{1}^{\dagger}M X_{2}+{\frac{t_{1}t_{2}}{t_{1}t_{2}- 1}}X_{2}M_{0}^{- 1}\epsilon_{1}^{\dagger}M_{0}.
\varphi_{-1}(x) \ = \ \frac{1}{T}, \varphi_i(x) \ = \ \tilde e_i(0)-\tilde e_i(x-T)- \frac{1}{T}\int_{-T}^0 xe_i(x)dx, x\in[0,T], a_{-1}\ = \ -\frac{1}{T}, \qquad\;\; a_i \ = \ \frac{1}{T}\int_{-T}^0 xe_i(x)dx.
\begin{array}{c c}{{\displaystyle{\{D_{i \alpha}, D_{j \beta}\}= - 2 i \bar{\cal E}_{i j}\gamma_{\alpha \beta}^{A}\partial_{A}}}}&{{\displaystyle{\{\tilde{D}_{\tilde{\imath}}^{\alpha}, \tilde{D}_{\tilde{\jmath}}^{\beta}\}= - 2 i \bar{\tilde{\cal E}}_{\tilde{\imath}\tilde{\jmath}}\tilde{\gamma}^{A \alpha \beta}\partial_{A}}}}\\ \end{array}
\psi(\begin{matrix}\mu_{1}&\mu_{2}&\gamma\\ &&\nu\end{matrix})=\sum_{\nu_{1},\nu_{2}}(\begin{matrix}\mu_{1}&\mu_{2}&\gamma\\ \nu_{1}&\nu_{2}&\nu\end{matrix}){\phi^{\mu_{1}}}_{\nu_{1}}{\phi^{\mu_{2}}}_{\nu_{2}}
H = t_r \sum_{i=0}^{d-r}\frac{a^i q^{i(d-2r)}(A-\theta_r I )( A-\theta_{r+1}I ) \cdots (A-\theta_{r+i-1}I)}{(q^2;q^2)_i}, H^{-1}= t^{-1}_r \sum_{i=0}^{d-r}\frac{a^{-i}q^{i(2r-d)}(A-\theta_r I )( A-\theta_{r+1}I ) \cdots (A-\theta_{r+i-1}I)}{(q^{-2};q^{-2})_i}.
{\cal L}~=~ -{\scriptstyle \frac{1}{2}}\, \partial_{\mu}\, \varphi\,\partial^{\mu}\, \varphi -{\scriptstyle \frac{1}{2}}m^2\varphi^2 -{\scriptstyle \frac{1}{4}}\, \lambda\, \left( \partial_{\mu}\, \varphi\, \partial^{\mu}\, \varphi\right)^2
F^{(q)}_{\delta,k}u(z,t):=\frac{1}{2\pi}\int e^{i<t-\theta,\eta>}u(z,\theta)\tau_\delta(\frac{\eta}{k})d\eta d\theta\in\Omega^{0,q}(X,L^k),\ \u(z,t)\in\Omega^{0,q}_0(X,L^k).
\langle H_y\rangle_{R^{\#}_y}= \|R^{\#}_y\|^2_{\#}+ (e_{\alpha}+\alpha \|\lambda_y\|^2 )\|R_y^{\#}\|^2 + \alpha \|A^-_y R^{\#}_y\|^2 + \alpha \langle V_y\rangle_{R^{\#}_y}.
\sigma_{j}^{a}= \mathbf{1}^{\otimes (j-1)}_{V}\otimes \sigma_{}^{a}\otimes\mathbf{1}^{\otimes (N-j)}_{V}, \qquad{\cal V}= V^{\otimes N}=\underbrace{V\otimes V\otimes \ldots \otimes V}_{\mbox{$N$ factors}},
J(0)\int_{-X}^{X}h(\frac{q}{Q},t)dt = J(0)\left(1+O\left(X\left(\frac{q}{Q}\right)^{N-1}+\left(\frac{q}{Q}X^{-1}\right)^{N}\right)\right) =J(0)+O\left(\left(\frac{q}{Q}\frac{x_{i_{0}}}{|x_{j_{0}}|}\right)^{\frac{N}{2}}\right).
d(f(x),f(a))\le d(f(x),f(x_{T_0}^a))+d(f(x_{T_0}^a),f(a))< <d(f(x),f(x_{\{t_1\}}^a))+d(f(x_{\{t_1\}}^a),f(x_{\{t_1,t_2\}}^a))+\dots+ +d(f(x_{\{t_1,\dots,t_{n-1}\}}^a),f(x_{\{t_1,\dots,t_n\}}^a))+\frac{\varepsilon}{2}<\frac{\varepsilon}{2n}\cdot n+\frac{\varepsilon}{2}=\varepsilon.
\frac{1}{h_n(v)+j}\!\!\!\!\sum_{k=0}^{h_n(v)+j-1}\!\!\!\!\!{\bf 1}_{B_m(u)}(T^{k}x^{(n)}) = \frac{1}{h_n(v)+j}\sum_{k=0}^{j-1}{\bf 1}_{B_m(u)}(T^{k}x^{(n)}) + \frac{1}{h_n(v)}\!\!\!\!\sum_{k=j}^{h_n(v)+j-1}\!\!\!\!\!{\bf 1}_{B_m(u)}(T^{k}x^{(n)}) - \frac{j}{h_n(v)(h_n(v)+j)}\!\!\!\!\sum_{k=j}^{h_n(v)+j-1}\!\!\!\!\!{\bf 1}_{B_m(u)}(T^{k}x^{(n)}).
{\cal U}_{k}( x ) ={\cal P}\left\{U_{k}( x ) U_{k}( x + \hat{k}) + \alpha_{f}\sum_{\pm ( j \neq k )}U_{j}( x ) U_{k}( x + \hat{j}) U_{k}( x + \hat{j}+ \hat{k}) U_{j}^{\dagger}( x + 2 \hat{k}) \right\}.