harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{d}{d\Lambda}S_{int\Lambda}=\frac{1}{2}\frac{\delta S_{int\Lambda}}{\delta\phi}\cdot(\frac{d}{d\Lambda}R_{\Lambda}^{-1})\cdot\frac{\delta S_{int\Lambda}}{\delta\phi}-\frac{1}{2}Tr[\frac{\delta^{2}S_{int\Lambda}}{\delta\phi\delta\phi}\cdot R_{\Lambda}^{-1}].
A_\mathrm{L}=B_\mathrm{L}\mathbb{P}\left(C_\mathrm{L}d_\mathrm{L}^{-\alpha_\mathrm{L}}>C_\mathrm{N}d_\mathrm{N}^{-\alpha_\mathrm{N}}\right) &\stackrel{(a)}=B_\mathrm{L}\int_{0}^{\infty}\mathbb{P}\left(d_\mathrm{N}>\psi_\mathrm{L}(x)\right)f_\mathrm{L}(x)\mathrm{d}x,
\phi_{3,i}= -\frac{n-2}{2}u_{a_{i}}\frac{\lambda_{i}x}{1+\lambda_{i}^{2}r^{2}}(\frac{\lambda_{i}}{1+\lambda_{i}^{2}r^{2}})^{\frac{n-2}{2}}& +O(r^{n-2}(\frac{\lambda_{i}}{1-\lambda_{i}^{2}r^{2}})^{\frac{n-2}{2}})+O(\frac{r}{\lambda_{i}}(\frac{\lambda_{i}}{1+\lambda_{i}^{2}r^{2}})^{\frac{n-2}{2}})
\frac{k}{2 \pi}\partial_{+}g g^{- 1}\equiv a^{I}G^{I}\equiv \left( J^{3}\frac{\sigma^{3}}{2}+ J^{+}\sigma^{+}+ J^{-}\sigma^{-}\right) + \left( \frac{d - 1}{2 C_{\rho}}B^{a}T^{a}\right) + \left( F_{+ \alpha}R^{+ \alpha}+ F_{- \alpha}R^{- \alpha}\right)
c\Vert u \Vert_{L^{\frac{2n}{n-2}}}^{\frac{2n}{n-2}}\leq k_{u}=\int Ku^{\frac{2n}{n-2}}d\mu_{g_{0}}\leq C\Vert u \Vert_{L^{\frac{2n}{n-2}}}^{\frac{2n}{n-2}}.
\partial_1 f = \nabla_1 \tilde{h}_{22}= h^S_{22}H + (h^S_{\nu \nu}-3 h^S_{22}) h_{22}- \nabla_\nu^S h^S_{22}\geq \frac{11}{6}h^S_{22}\tilde{D}+ \frac{1}{2}(h^S_{\nu \nu}-3 h^S_{22}) \tilde{D}-\nabla_\nu^S h^S_{22}\geq \frac{1}{3}H^S \tilde{D}-\nabla_\nu^S h^S_{22}\geq 0
D_l = \mathfrak{H}R(z)=\bigg\{\xi\in \mathbb{C}\oplus \mathcal{L}:\xi = c(1,(E|R(z))+(0,(f|)\bigg\}D_r = R(z)\mathfrak{H}=\bigg\{\xi\in \mathbb{C}\oplus \mathcal{L}:\xi = c\binom{1}{-R(z)|E)}+\binom{0}{R(z)f}\bigg\}
{X_j (-\Delta_{\mathbb H^n})^{-\frac{1}{2}}(g)}=Cd_K^{-Q-1}(g)\bigg[x_j\int_{\mathbb R}\Big({\lambda\over\sinh\lambda}\Big)^{-{1\over 2}}\cosh\lambda \big(\cosh(\lambda-i\phi)\big)^{-n-{3\over 2}}d\lambda &\hskip4cm -iy_j\int_{\mathbb R}\Big({\lambda\over\sinh\lambda}\Big)^{-{3\over 2}}\lambda \big(\cosh(\lambda-i\phi)\big)^{-n-{3\over 2}}d\lambda\bigg].
\int_{0}^{\infty}\frac{1}{T}e^{-\frac{\epsilon}{T}}\cdot e^{-\lambda\epsilon}d\epsilon=\frac{1}{T}\int_{0}^{\infty}e^{-(\lambda+\frac{1}{T})\epsilon}d\epsilon=-\frac{1}{T(\lambda+\frac{1}{T})}e^{-(\lambda+\frac{1}{T})\epsilon}|_{0}^{\infty}=\frac{1}{\lambda T+1}=g(\lambda)
\int_{- \infty}^{\infty}d \tau{\cal L}= \frac{1 6 C_{4}}{1 5}\frac{v^{3}}{b^{6}}+ \frac{2 5 6 C_{6}}{3 1 5}\frac{v^{5}}{b^{1 0}}+ \frac{2 0 4 8 C_{8}}{3 0 0 3}\frac{v^{7}}{b^{1 4}}+ O ( v^{9}) .
\int_{T-\tau}^{T}\mod{\widetilde{U}^p(t)}^2dt=\int_{0}^{\tau}\mod{\widetilde{U}^p(T-t)}^2dt \geq C\sum_{k\geq N}\mod{a_k}^2 |1-e^{\tilde{\lambda}_k}|^2 e^{2\Re(\tilde{\lambda}_k)\tau}&\geq C\sum_{k\geq N}\mod{a_k}^2e^{2\Re(\tilde{\lambda}_k)\tau},
x^{\frac{\alpha}{\alpha+1}}+y^{\frac{\alpha}{\alpha+1}}\le(\frac{x}{s}+\frac{y}{t})^{\frac{\alpha}{\alpha+1}}(s^{\alpha}+t^{\alpha})^{\frac{1}{\alpha+1}}=(\frac{x}{s}+\frac{y}{t})^{\frac{\alpha}{\alpha+1}}
\int_{-r}^r |\eta_2(x_1,t)|\, dx_1 = \int_{-r}^r \eta_2(x_1,t)\, dx_1 = \int_0^t \eta_1(-r, x_2) - \eta_1(r, x_2)\, dx_2\, \le 2t\|\eta\|_\infty\,.
\int_0^x e^{-\beta_0/y}dy=x-\beta_0 \log (x)+c_0-\frac12\beta_0^2x^{-1}+\frac{\beta_0^3}{3!2}x^{-2}-\cdots \hphantom{\int_0^x e^{-\beta_0/y}dy=}{}-(-1)^n\frac1{n!(n-1)}\beta_0^{n}\frac1{x^{n-1}}-\cdots,
\left\lbrace \begin{array}{lllll}\alpha_i = \frac{\zeta^{\gamma_i-\gamma_1}}{\sum_{i=1}^r \zeta^{\gamma_i-\gamma_1}}1\leqslant i \leqslant r, \sum_{i=1}^r \gamma_i \zeta^{\gamma_i-\gamma_1}= 0. \end{array}\right.
C \ge\max_{p(x_1,x_2,u_2)}\min \bigl\{I(X_1, X_2; Y_3), begin{align*}-1em] I(X_1; U_2, Y_3 | X_2) - I(U_2; X_1 | X_2, Y_2) \bigr\}= \max_{p(x_1,x_2,u_2)}\min \bigl\{I(X_1, X_2; Y_3), begin{align*}-1em] I(X_1; Y_3 | X_2, U_2) + I(U_2; Y_2 | X_2) \bigr\}.
E\big[B^H(s) (B^H(t) - B^H(s))\big] = E\big[B^H(s)B^H(t)\big] - E\big[B^H(s)B^H(s)\big] = \frac{1}{2}\big( t^{2H}+ s^{2H}- (t-s)^{2H}- 2s^{2H}\big) = \frac{1}{2}\big( t^{2H}- s^{2H}- (t-s)^{2H}\big).
a ( \vec{p}) = \int{\frac{d^{\nu}x}{( 2 \pi )^{\frac{\nu}{2}}}}{\frac{e^{- i \vec{p}\cdot \vec{x}}}{\sqrt{2 g ( \vec{p})}}}\left( \int d^{\nu}y{\frac{1}{2}}G ( x , y ) \eta ( y ) +{\frac{\delta}{\delta \eta ( \vec{x})}}\right)
\Psi (BL) =\ 4\cdot \Psi(H_{n-2}\setminus \{a,f\}) + 3\cdot \Psi(H_{n-2}\setminus \{c,f\})+14\cdot \Psi(H_{n-2}\setminus c) &+4\cdot \Psi(H_{n-2}\setminus \{e,h\})+3\cdot \Psi(H_{n-2}\setminus \{c,h\})+9\cdot \Psi(H_{n-2}\setminus \{a,e\}) &+7\cdot \Psi(H_{n-2}\setminus \{a,c\})+7\cdot \Psi(H_{n-2}\setminus\{c,e\}).
1 = \left(\frac{-2^{3r+1}P_1^2P_2xz_1^3}{2^{r+1}P_1P_2P_3xz_1-y^2}\right) = \left(\frac{-2^{3r+1}P_2xz_1}{2^{r+1}P_1P_2P_3xz_1-y^2}\right).
T_k\bigg(\frac1{T_k(1)}\bigg)(x) =\int_{\Bbb R^N}\theta(r^k \|x-y\|)\frac1{T_k(1)(y)}d\omega(y) &\sim\int_{\Bbb R^N}\theta (r^k \|x-y\|) \frac 1{V_k(y)}d\omega(y) &\sim \frac 1{V_k(x)}\int_{\Bbb R^N}\theta(r^k \|x-y\|)d\omega(y) &= \frac 1{V_k(x)}T_k(1)(x)\sim 1.
\tilde{g}^{i\overline{i}}V_{1}V_{1}(\tilde{g}_{i\overline{i}})=\tilde{g}^{p\overline{p}}\tilde{g}^{q\overline{q}}|V_{1}(\tilde{g}_{p\overline{q}})|^{2}+V_{1}V_{1}(F)+V_{1}V_{1}\left(\frac{\partial\varphi}{\partial t}\right).
H(\boldsymbol{\lambda}_{1},\boldsymbol{\lambda}_{2},\boldsymbol{\lambda}_{3}, \boldsymbol{\lambda}_{4})=\sum_{n=1}^{\infty}\mu_{n}(\mathcal{H})\varphi_{n}(\boldsymbol{\lambda}_{1},\boldsymbol{\lambda}_{2})\varphi_{n}(\boldsymbol{\lambda}_{3},\boldsymbol{\lambda}_{4}).
\begin{array}{ccll}\dot{x}_1 = x_1^2 +7x_2^2 - 4x_1x_2 + 5x_1 \dot{x}_2 = -2x_2^2+x_1x_2 + 2x_1 x_3 - 7 x_2x_3 + 2x_2 \dot{x}_3 = -x_2^2-7x_3^2 - 4x_2x_3 - x_3\end{array}
T r D_{\mu \nu}( x - x^{\prime}) ={\int}d^{4}x \left[ 3{\int}\frac{d^{4}p}{( 2 \pi )^{4}}\frac{1}{p^{2}+ e^{2}< \rho >^{2}}+ \frac{\delta^{4}( 0 )}{e^{2}< \rho >^{2}}\right] .
B_0(\hat{{\mathbf U}})\partial_t\dot{{\mathbf U}}+\tilde{B}_1(\hat{{\mathbf U}},\hat{\Psi})\partial_1\dot{{\mathbf U}}+B_2(\hat{{\mathbf U}})\partial_2\dot{{\mathbf U}}+\tilde{C}(\hat{{\mathbf U}},\hat{\Psi})\dot{{\mathbf U}}=\tilde{\mathbf F}(\hat{{\mathbf U}}),
b_N = f(a_N) + b_{N-1}+a_N-a_{N-1}&= ([N\beta]-[N\alpha]) -([(N-1)\beta]-[(N-1)\alpha])+ b_{N-1}+ a_N-a_{N-1}&= ([N\beta]-[N\alpha]) -([(N-1)\beta]-[(N-1)\alpha])+ [(N-1)\beta] + a_N-[(N-1)\alpha] &= [N\beta] + a_N-[N\alpha]
{\frac{\Theta_{i}}{\Theta_{i_{0}}}}={\frac{\det_{k \neq i_{0}, i \to i_{0}; j = 1 , \dots , N - 1}\theta_{\frac{k}{N}}( N q_{j}| N \tau )}{\det_{k \neq i_{0}; j = 1 , \dots , N - 1}\theta_{\frac{k}{N}}( N q_{j}| N \tau )}}.
\mathcal{A}_{}\left(P_{1}\right)= \left\{A^{}_{}+a(r)\left(\theta^{2}e_{3}-\theta^{3}e_{2}+\theta^{4}e_{1}\right)+b(r)\left(\theta^{5}e_{3}-\theta^{6}e_{2}+\theta^{7}e_{1}\right)|a,b\in C^{\infty}\left(0,\infty\right)\right\}.
( 0 ~ 0 ~ 4 ) \longrightarrow ( 0 ~ 4 ) \oplus ( 0 ~ 3 ) , \, ( 0 ~ 0 ~ 2 ) \longrightarrow ( 0 ~ 2 ) \oplus ( 0 ~ 1 ) , \, ( 0 ~ 0 ~ 1 ) \longrightarrow ( 0 ~ 1 ) \oplus ( 0 ~ 0 ) .
\sum_{r \in \mathbb{Z}}c^F_{-r}\big(r_l(\pi_F)^{(l)}(w_n)W,\sigma_F^{(l)}(w_{n-1})W'\big)q_F^{-\frac{r}{2}}X^{-r}= \gamma\big(X, r_l(\pi_F)^{(l)}, \sigma_F^{(l)},\overline{\psi}_F^l\big)\sum_{r \in \mathbb{Z}}c^F_r(W,W')q_F^{\frac{r}{2}}X^{r}.
\mathcal{I}\mathfrak{w}\mathcal{I}=(B_n\cap K)\mathfrak{w}N_{\alpha_k}(\mathcal{O})\prod_{\beta>0:\beta\ne\alpha_k}N_{\beta}^-(\varpi\mathcal{O})\subset (B_n\cap K)\mathfrak{w}N_{\alpha_k}(\mathcal{O})(H\cap K).
\eta(s)=\left (n_1\,s\,\left [\sigma^{2}\,\log\left (1+\frac{\sigma^{q}\,n_2}{s}\right )\right ]^{1-q/2}\right )\vee\left ( n_1\,s^{2/q}\right )\vee \left (n_1\,n_2\,\sigma^{2}\right )
{\frac{1}{2 i}}\, ( T - T^{*}) ={\frac{1}{1 6 \sqrt{s}}}\, \int_{0}^{2 \pi}\,{\frac{d \theta}{2 \pi}}\, | T ( s , \theta ) |^{2}, \quad 4 \mu^{2}\le s < 1 6 \mu^{2}.
n(2\ln n)^{\frac{1}{2}}\int_ID_n(\sqrt{4-y^2}/S(I) \cdot G_n(x)/2)dy = \sum_{j=1}^2n(2\ln n)^{\frac{1}{2}}\int_{I_j}D_n(\sqrt{4-y^2}/S(I_j) \cdot G_n(x)/2)dy \to M(I_1)e^{c_0-x}+M(I_2)e^{c_0-x}=M(I)e^{c_0-x},\,\,\,n\to+\infty,
\frac{d\xi^{n}_m(t)}{dt}\leq e^{-t}\left[\sum_{i=1}^{m-1}\sum_{j=i}^{n-1}j(i+j)\psi^{n}_i \psi^{n}_j+(m-1) \psi^{n}_{m-1}\sum_{j=1}^{m-1}j(m-1+j) \psi^{n}_j-2\mu^{n}_1(0)^2-2m\mu^{n}_1(0)^2\right].
G_{T}(y) =\frac{Q_{T,0}(x)+(Q_{T,1}(x)-x^{2}Q_{T,0}(x))y+(Q_{T,2}(x)-x^{2}Q_{T,1}(x)-xQ_{T,0}(x))y^{2}}{1-x^{2}y-xy^{2}-y^{3}}&=\frac{y+\textbf{i}+(x^{2}+xy+y^{2})\textbf{j}+(x^{4}+x+x^{3}y+y+x^{2}y^{2})\textbf{k}}{1-x^{2}y-xy^{2}-y^{3}}.
\mathrm{T r}\Big ({\frac{1}{2 ! 2^{2}}}\left( - i [ \phi^{i_{1}}, \phi^{i_{2}}] \right) \left( - i [ \phi^{i_{3}}, \phi^{i_{4}}] \right) C_{i_{1}i_{2}i_{3}i_{4}}^{( 4 )}\Big )
\tilde{R}^{+}_{\xi}=\left[\frac{(L-2)(L+1)(L+4)(L+2)}{27 \sqrt{3}L^4\sin\left(\frac{\pi (2L+1)}{3(L+2)}\right)\sin\left(\frac{\pi(L-2)}{3L}\right)}\right]^{\frac{1}{2}}_{L=1}=\left[\frac{40}{27\sqrt{3}}\right]^{\frac{1}{2}}.
q(\alpha,\beta;t_1,y_1;t_2,y_2)=q(1-\beta,1-\alpha;t_2,y_2;t_1,y_1)\cdot \left(\frac{\sin(\pi\beta)}{\sin(\pi\alpha)}\right)^{\epsilon}\cdot\frac{|t_2 y_2|}{|t_1 y_1|},
\bigl( \mathcal{T}_{k_1}^{A,d}\times \mathcal{T}_{k}^{A,d}\bigr)\cap ( \mathfrak{D}_{j_1}^A \times \mathfrak{D}_{j}^A )\not= \emptyset, | F (\xi_1, \eta_1, \xi ,\eta)| \geq A^{-1}d^{-1}N_1^2 \ \ \textnormal{for any}\ (\xi_1, \eta_1)\times (\xi,\eta) \in\mathcal{T}_{k_1}^{A,d}\times \mathcal{T}_{k}^{A,d}.
|\big((\phi \otimes f_{n})* (\psi \otimes f)\big)(x,g)-(\phi R_{a}\psi \otimes L_{a}(f))(x,g)| = \big|\int_{u \in U_{n}}\phi(x)\psi(xu)f_{n}(u)f(u^{-1}g)du - \int_{u \in U_{n}}\phi(x)\psi(xa)f_{n}(u)f(a^{-1}g)du \big | &\leq |\phi(x)|\int_{u \in U_{n}}|\psi(xu)f(u^{-1}g)-\psi(xa)f(a^{-1}g)|f_{n}(u)du \leq \epsilon ||\phi||_{\infty}.
\sum_{\substack{C \in \mathcal{M}_i C | l^{\infty}}}\frac{1}{|C|^2}= \frac{1}{2 \pi i}\oint_{|u|=r_1}\frac{1}{q^{2i}u^{i+1}\prod_{P|l}(1-u^{d(P)})}\, du,
\Psi(0)={}_2F_1\bigg[\begin{matrix}-a \beta &\gamma\end{matrix}\bigg|\,z\bigg]=\frac{\Gamma_p(\gamma)\Gamma_p(\gamma+a-\beta)}{\Gamma_p(\gamma+a)\Gamma_p(\gamma-\beta)}\cdot{}_2F_1\bigg[\begin{matrix}-a \beta &1-a+\beta-\gamma\end{matrix}\bigg|\,1-z\bigg]=\Omega(0)\Phi(0).
\left\langle R_1,\,S_1,\,P\ \Big\vert\ \begin{array}{c}id=R_1^p=P^7=(P^{-1}R_1)^3=P^2S_1^{-1}P^2R_1P^{-2}R_1 id =S_1^{-2}R_1=(P^2S_1)^p =(P^{-3}R_1)^{4p/(p-4)}=(P^{-2}S_1)^{6p/(p-6)}\end{array}\right\rangle
\int f(x) (\alpha \cdot x) d\mu(x) - |\alpha |^2 \sigma^2 = \int (f(x)-\alpha \cdot x)(\alpha \cdot x)d\mu(x) &\leq \left( \int |f(x)-\alpha \cdot x|^2 d\mu(x) \right)^{1/2}\left( \int |\alpha \cdot x|^2 d\mu(x) \right)^{1/2}&\leq |\alpha| \sigma \left( C_p \int |\nabla f -\alpha |^2 d\mu \right)^{1/2}.
\sum_{M}\frac{|W_{0}^{M}|}{|W_{0}^{G}|}\sum_{\gamma\in(M(Q))}a^{M}(\gamma)I_{M}(\gamma,f)=\sum_{M}\frac{|W_{0}^{M}|}{|W_{0}^{G}|}\int_{\Pi(M)}a^{M}(\pi)I_{M}(\pi,f)d\pi
\int_{\mathbb{R}^N}(I_\mu*|t_\nu\star{u_0}|^{p})|t_\nu\star{v_0}|^{q}\leq\mathbb{A},\ \ \mathbb{A}=\Bigl(\frac{22^*_\mu-\gamma_p-\gamma_q}{22^*_\mu-2}\nu(\gamma_p+\gamma_q)D^{\frac{2}{\gamma_p+\gamma_q}}_0\Bigr)^{\frac{\gamma_p+\gamma_q}{2-\gamma_p-\gamma_q}}.
\frac{\hat{\lambda}_j^{(n)}-\lambda_k^{(n)}}{\sqrt{\lambda_j^{(n)}\lambda_k^{(n)}}}\langle \hat{u}_j^{(n)},u_k^{(n)}\rangle =\bar{\eta}_{kj}^{(n)}\langle\hat{u}_j^{(n)},u_j^{(n)}\rangle+\sum_{l\neq j}\bar{\eta}_{kl}^{(n)}\sqrt{\frac{\lambda_l^{(n)}}{\lambda_j^{(n)}}}\langle\hat{u}_j^{(n)},u_l^{(n)}\rangle.
\left(\sum_{j=1}^n z_{j,n}\right)^m = \sum_{k_1+k_2+\ldots+k_n=m}\binom{m}{k_1,k_2,\ldots,k_n}z_{1,n}^{k_1}z_{2,n}^{k_2}\ldots z_{n,n}^{k_n}= m!\,\sigma_m(z_{1,n}, z_{2,n}, \ldots, z_{n,n}) + \sum_{k_1+k_2+\ldots+k_n=m \atop{\exists\k_j \ge 2}}\binom{m}{k_1,k_2,\ldots,k_n}z_{1,n}^{k_1}z_{2,n}^{k_2}\ldots z_{n,n}^{k_n}.
f(x^0)=\frac{1}{1+ (\lambda_1+\lambda_2) \frac{\sin (\pi \sqrt{\lambda_1^2-\lambda_2^2})}{\sqrt{\lambda_1^2-\lambda_2^2}}e^{x^0}}+\frac{1}{1+ (\lambda_1-\lambda_2) \frac{\sin (\pi \sqrt{\lambda_1^2-\lambda_2^2})}{\sqrt{\lambda_1^2-\lambda_2^2}}e^{-x^0}}-1.
\lim_{\mu \to 0}\lambda_{min}(\delta I + (\Theta^{-1}+\rho I )^{-1})^{-1}= \frac{\rho}{\delta \rho +1}& \lim_{\mu \to 0}\lambda_{max}(\delta I + (\Theta^{-1}+\rho I )^{-1})^{-1}= \frac{1}{\delta}.
D( f_{\circ}g_{\circ})= D((\varphi\circ f)(\varphi\circ g)) = D(\varphi\circ fg) = D_{\varphi}(fg) = \varphi\circ f(x)D_{\varphi}g+ \varphi\circ g(x)D_{\varphi}f = \varphi\circ f(x)D(\varphi\circ g)+ \varphi\circ g(x) D(\varphi\circ f) = f_{\circ}(x)Dg_{\circ}+g_{\circ}(x)Df_{\circ}
X(bc, a_{k+1}, \dots , a_{l}) = (-1)^{p(c)(\bar{p}(a_{k+1}) + \dots + \bar{p}(a_{l}))}X(b, a_{k+1}, \dots , a_{l})c &+ (-1)^{p(b)\bar{p}(X)}b X(c, a_{k+1}, \dots , a_{l}) &-(-1)^{p(bc)(\bar{p}(a_{k+1}) + \dots + \bar{p}(a_{l}))}X(1, a_{k+1}, \dots , a_{l})bc.
(P^L) \sup_{f\in L^p(\Phi)}\int_{\Phi}c(x)f(x)dx \int_{\Phi}A(y, x)f(x)dx \leq a(y), y\in \Gamma \int_{\Phi}B(z, x)f(x)dx = b(z), z\in \Sigma f \geq 0.
L(u,v) = \left(u-\frac 1 6 (u+i v)^{3}\right)w_1 + \left(v-\frac i 6 (u+i v)^{3}\right)w_2 + \frac 1 2 (u+i v)^{2}w_3, e_3(u,v) = -(u+i v) w_1 - (i u-v) w_2 + w_3,
\left( \begin{array}{c c}{{\{\Phi_{0}, \Phi_{0}\}}}&{{\{\Phi_{0}, \Phi_{1}\}}}\\{{\{\Phi_{1}, \Phi_{0}\}}}&{{\{\Phi_{1}, \Phi_{1}\}}}\\ \end{array}\right) ={\cal D}\delta ( z - z^{\prime}) ={\cal D}\delta ( x - x^{\prime}) \delta ( \theta - \theta^{\prime})
\liminf_{n \to \infty}\mathbb{P}_n \left(\beta_{\hat{\mathbb{M}}_n, n}^{*, (j)}\in \mathrm{CI}_{1-\alpha, \hat{\mathbb{M}}_n}^{(j), \mathrm{lm}}j = 1, \hdots, m(\hat{\mathbb{M}}_n) \right) \geq 1-\alpha.
w(C(X),\tau_{\mathcal{B}}^{sw})=w_X(\mathcal{B})\cdot ib(C(X),\tau_{\mathcal{B}}^{sw})=w_X(\mathcal{B})\cdot c(C(X),\tau_{\mathcal{B}}^{sw}) =w_X(\mathcal{B})\cdot L(C(X),\tau_{\mathcal{B}}^{sw})=w_X(\mathcal{B})\cdot d(C(X),\tau_{\mathcal{B}}^{sw})=w_X(\mathcal{B})\cdot nw(C(X),\tau_{\mathcal{B}}^{sw}).
\sum_{k_{1}+\dots+k_{p}=n}\frac{\left(-1\right)^{p}}{\left(k_{1}+1\right)!\dots\left(k_{p}+1\right)!}=\left(-1\right)^{p}\frac{p!}{\left(n+p\right)!}\left\{\begin{array}{c}n+p p\end{array}\right\}
Z_{O ( 2 ) N L S}= \sum_{n = 0}^{\infty}{\frac{\zeta^{n}}{( n ! )^{2}}}\int \prod_{j = 1}^{n}d^{2}z_{j}\exp \left[ ( 2 \pi )^{2}\beta \sum_{i , j}Q_{i}Q_{j}{\frac{1}{2 \pi}}\ln{\frac{R_{0}}{| z_{i}- z_{j}|}}\right] ,
\begin{array}{l}{t \operatorname{l n}t > ( t - 1 ) \operatorname{l n}( t + 1 ) \qquad \mathrm{f o r}\quad t > 1 \, ,}{t \operatorname{l n}t < ( t - 1 ) \operatorname{l n}( t + 1 ) \qquad \mathrm{f o r}\quad 0 < t < 1 \, ,}\end{array}
g( x_2, x_3, x_4, x_5) + g( x_2, x_3, x_5, x_4) + g(x_2, x_4, x_5, x_3) = \bigl( [x_2, x_3] [x_4, x_5] + [x_2, x_4] [x_3, x_5] \bigr) & + \ \bigl( [x_2, x_3] [x_5, x_4] + [x_2, x_5] [x_3, x_4] \bigr) + \bigl( [x_2, x_4] [x_5, x_3] + [x_2, x_5] [x_4, x_3] \bigr) = 0
\sum_{i = 1}^{n}q_{a}^{i}[ \varphi_{i}^{\alpha}\overline{{{\varphi}}}_{i \beta}+ \varphi_{i \beta}\overline{{{\varphi}}}_{i}^{\alpha}] = \vec{\xi_{a}}\vec{\sigma}_{\beta}^{\alpha}\qquad ; a = 1 , . . . , r .
\varphi_{s^{\prime},i,j}\left(\omega_{i},\omega\right)=f_{s^{\prime},i,j}\left(\omega_{i},X_{s_{i}^{\prime}}\left(\omega\right)\right)=f_{s^{\prime},i,j}\left(\omega_{i},X_{s_{i}^{\prime}}\left(\omega^{\prime}\right)\right)=\varphi_{s^{\prime},i,j}\left(\omega_{i},\omega^{\prime}\right).
\langle 2 n , 0 | B | 2 n , 0 \rangle ={\frac{1}{c_{n}}}\langle 0 | A^{n}B ( A^{\dagger})^{n}| 0 \rangle ={\frac{1}{c_{n}}}\langle 0 | ( 2 n + B ) A^{n}( A^{\dagger})^{n}| 0 \rangle = 2 n + 3
\max_{1\leq j,k \leq n, j\neq k}|f_j(\tau_k)f_k(\tau_j)|\geq \frac{\frac{1}{(G_{f,\tau})}\left(\sum_{j=1}^nf_j(\tau_j)\right)^2-\sum_{j=1}^n|f_j(\tau_j)|^2}{n^2-n},
\sum_{l=1}^{h(m,n)}C_{m}^{l}(F^{0})(\lambda)(C_{m}^{l}(F^{0})(\lambda))^{*}=(\beta_{m}^{2}+\gamma_{m_{1}}(\lambda )+\gamma_{m_{2}}(\lambda ))d_{m}^{0}(\lambda )^{\top}B_{2}\overline{d_{m}^{0}(\lambda )};
A_{10}E_{ij}= E_{ij}A_{10}= \theta_{ij}E_{ij}, A_{01}E_{ij}= E_{ij}A_{01}= \mu_{ij}E_{ij}, A_{10}^\star E_{ij}^\star = E_{ij}^\star A_{10}^\star = \theta_{ij}^\star E_{ij}^\star, A_{01}^\star E_{ij}^\star = E_{ij}^\star A_{01}^\star = \mu_{ij}^\star E_{ij}^\star.
\Psi(x_0, x_1, x_2, x_3) =\frac{(2x_0, 2x_1, 2x_2, x_0^2 + x_1^2 + x_2^2 + x_3^2- 1)}{x_0^2 + x_1^2 + x_2^2 + x_3^2 - 2x_3 + 1}.
\displaystyle{\mbox{$_4$F$_3$}(a,b,c,-n;\,d,e,f;\,1)}={\frac{\Gamma \left( n+e-a \right) \Gamma \left( n+f-a \right) \Gamma \left( e \right) \Gamma \left( f \right){\mbox{$_4$F$_3$}(a,-n,d-b,d-c;\,d,a+1-n-f,a+1-n-e;\,1)}\mbox{}}{\Gamma \left( e-a \right) \Gamma \left( f-a \right) \Gamma \left( n+e \right) \Gamma \left( n+f \right)}}
d s^{2}= 4 \frac{f^{\prime}( u ) g^{\prime}( v )}{( f ( u ) - g ( v ) )^{2}}( d y^{2}- d t^{2}) + \frac{1}{( f ( u ) - g ( v ) )^{2}}\delta_{i j}d x^{i}d x^{j}, \,
{\cal L}_B =\frac{1}{2}B^*_{\mu}[-i\epsilon_{\mu\nu\lambda}(\partial_\nu + ia_\nu+iC_\nu)+M\delta_{\mu\lambda}]B_\lambda- \frac{i}{8\pi(\alpha-\frac{1}{2})}\epsilon_{\mu\nu\lambda}a_\mu\partial_\nu a_\lambda\;,
\hat{A}\Psi_{0}^{( 3 )}= \Biggl [ \frac{\partial}{\partial a^{*}}+ \sum_{k = 0}^{3}b_{k}^{( 3 )}a^{* ( 3 - k )}\Bigl ( \frac{\partial}{\partial a^{*}}\Bigr )^{k}\Biggr ] \Psi_{0}^{( 3 )}= 0 .
p^{r_0}(t;x,y) = \sum_{n=1}^\infty \left[ \left(\frac{2 \sin\left(\frac{n \pi x}{r_0}\right) \sin\left(\frac{n \pi y}{r_0}\right)}{r_0 \sinh (\mu x) \sinh (\mu y)}\right) \exp\left(-\frac{(n^2 \pi^2 / r_0^2 +\mu^2)t}{2}\right) \right].
d\theta_{p}^0 = 0, d\theta_p^i = 0 d\theta_{x}= 0, d\theta_y^0 = \theta^{x}\wedge \theta_{p}^0, d\theta_y^i = \theta_{x}\wedge \theta_{p}^i, d\theta_z^i = \theta_y^i \wedge \theta_{p}^0 + \theta_y^0\wedge \theta_{p}^i.
I_4(x,t) =-\frac{4\lambda (\lambda +1)}{\pi}t\int_0^\infty \frac{u^{2\lambda -1}}{(1+u^2)^{\lambda +2}}du\int_{x/2}^{3x/2}\frac{(xy)^\lambda (x-y)}{(xy)^\lambda ((x-y)^2+t^2)^2}dy &=-\frac{2t}{\pi}\int_{-x/2}^{x/2}\frac{z}{(z^2+t^2)^2}dz=0,x,t\in (0,\infty ).
G_{\, +}^{\ell k}( q^{0}) = i \int_{- \infty}^{\infty}d x^{0}e^{i q^{0}x^{0}}\int d \vec{x}\left( \Phi_{p}, \left( j^{\ell}( \vec{x}, x^{0}) j^{k}( \vec{y}, 0 ) \right)_{+}\Phi_{p}\right)
{\cal Y}^{\overline{\alpha V}}_{\overline{\beta V}}= \xi^{\overline{\alpha V}}_{\overline{\beta V}}Y^{\overline{\alpha V}}_{\overline{\beta V}}\prod_{\ell=1}^3{\overline X}^{{\overline{\alpha T}}^{\,\ell}}_{{\overline{\beta T}}^{\, \ell}}\prod_{\ell=4}^{14}X^{{\overline{\alpha T}}^{{\,\ell}}}_{{\overline{\beta T}}^{{\, \ell}}}
u_{t_{2}}=\frac{1}{4}u_{xxx}+\frac{3}{8}(uu_{x}v_{x}+uw_{x}^{2})_{x}, v_{t_{2}}=\frac{1}{4}v_{xxx}+\frac{3}{8}(vu_{x}v_{x}+vw_{x}^{2})_{x}.
w ( i \rightarrow f ) \simeq 2 \exp \left\{-{\frac{2 \pi}{\hbar v g^{2}[ V_{1}^{\prime}( 0 ) - V_{2}^{\prime}( 0 ) ]}}\right\}\left( 1 - \exp \left\{-{\frac{2 \pi}{\hbar v g^{2}[ V_{1}^{\prime}( 0 ) - V_{2}^{\prime}( 0 ) ]}}\right\}\right)
E_{c}={\frac{Q^{2}}{2 \pi \epsilon_{0}}}({\frac{1}{2}}+ l n [ \sqrt{{\frac{\epsilon_{0}}{2 \pi m}}}{\frac{J_{\theta}}{Q r_{0}}}] ) ={\frac{Q^{2}}{2 \pi \epsilon_{0}}}({\frac{1}{2}}+ l n{\frac{r}{r_{0}}}) ,
g_{a b}= \left( \begin{array}{c c c c c}{{0}}&{{- \omega_{1}}}&{{}}&{{}}&{{}}\\{{\omega_{1}}}&{{0}}&{{}}&{{}}&{{}}\\{{}}&{{}}&{{\ddots}}&{{}}&{{}}\\{{}}&{{}}&{{}}&{{0}}&{{- \omega_{\frac{p + 1}{2}}}}\\{{}}&{{}}&{{}}&{{\omega_{\frac{p + 1}{2}}}}&{{0}}\\ \end{array}\right) .
\frac{1}{N}\sum_{i,j}q^{-ij}\sum_{a,b}q^{ia+jb}\delta_a\otimes \delta_b =\frac{1}{N}\sum_{i,j,a,b}q^{ab}q^{-(a-i)(b-j)}\delta_a\otimes \delta_b &=\frac{1}{N}\sum_{a,b}q^{ab}\delta_a\otimes \delta_b\sum_{i,j}q^{-(a-i)(b-j)}&=\sum_{a,b}q^{ab}\delta_a\otimes \delta_b,
\frac{p[(N-\sigma)r-(N-s)q]}{(\mu+p-\sigma)r+(p-q)(N-s)}+\frac{q[(N-s)p-(N-\mu-p)r]}{(\mu+p-\sigma)r+(p-q)(N-s)}& = \frac{r[(N-\sigma)p-(N-\mu-p)q]}{(\mu+p-\sigma)r+(p-q)(N-s)},
(-1)^{d_1 d_2}q^{-d_1 d_2}\prod_{\begin{subarray}{c}1\leq i\leq d_1 d_1<j\leq d\end{subarray}}z_i^{-1}z_j \cdot \prod_{\begin{subarray}{c}1\leq i\leq d_1 d_1<j\leq d\end{subarray}}\xi'(z_i z_j^{-1})=\prod_{\begin{subarray}{c}1\leq i\leq d_1 d_1<j\leq d\end{subarray}}\xi(z_i z_j^{-1}).
\sigma_{t,R}=\delta_{(-t,\dots,-t)}\ast \frac{1}{a_1\,\dots\,a_s\,R}\,\sum_{0\le k_1\le a_1 R-1}\,\dots\,\sum_{0\le k_s\le a_s R-1}\,\delta_{(-k_1/a_1,\dots,-k_1/a_s)}\ast \nu_\mu,
{R}_{1 2}( u - v ) \eta_{1 2}L_{1 j}( u ) \eta_{1 2}L_{2 j}( v ) = \eta_{1 2}L_{2 j}( v ) \eta_{1 2}L_{1 j}( u ){R}_{1 2}( u - v ) \ll{b q y b e l s}
\Lambda_{\nu}^{\mu}( x , y ) = \delta_{\nu}^{\mu}\, \delta^{3}( x - y ) - J^{\mu \alpha}\int d^{\, 3}\! \rho \; d^{\, 3}\! \sigma \, \delta_{\alpha ( x )}\varphi^{i}( \rho ) \, J_{i j}( \rho , \sigma ) \, \delta_{\nu ( y )}\varphi^{j}( \sigma )
{P}_{2n}(q) = \frac{(-1)^n}{\sqrt{2}}{Q}_n\big(q^2;\alpha,\beta,N\big), n=0,1,\ldots,N, P_{2n+1}(q) = - \frac{(-1)^n}{\sqrt{2}}\frac{1}{(\alpha+1)N}\sqrt{\frac{(N-n)(n+\alpha+1)(n+\alpha+\beta+1)(2n+\alpha+\beta+2)}{(2n+\alpha+\beta+1)}}\hphantom{P_{2n+1}(q) =}{}\times q{Q}_n\big(q^2-1;\alpha+1,\beta,N-1\big), n=0,1,\ldots,N-1.
\prod_{i=-2}^2b_{j_\sigma-i}\times P_2\Big(I_5,(1/b_{j_\sigma-2},1/b_{j_\sigma-1},1/a_m,1/b_{j_\sigma+1},1/b_{j_\sigma+2})\Big) &=\prod_{i=0}^4 a_{m-i}\times P_2\Big(I_5,(1/b_{j_\sigma-2},1/b_{j_\sigma-1},1/a_m,1/b_{j_\sigma+1},1/b_{j_\sigma+2})\Big).
D_y\tilde{H}(x,y)=\frac{2a_0}{a_0+1}\sum_{k=1}^\infty(-\alpha^{2k-1}D_y\log|\Phi(\Phi(x)-{\bf 2k+1})-\overline{y}| +\alpha^{2k}\log|\Phi(\Phi(x)+{\bf 2k})-y|)-CD_y\log|y|+\frac{a_0-1}{a_0+1}D_y\log|x-y|,
\overline{\Theta}(u,\mathcal{O})(x)\,:=\,\inf_{M\in\mathbb{R}}\left\lbrace\exists \,P_M\,|\,P_M(x)=u(x)\;\;\mbox{and}\;\;P_M(y)\geq u(y),\;\;\forall y\in \mathcal{O}\right\rbrace
\begin{gathered}\left (x^{k-1}\cdot \frac{y^k-1}{y-1}-y^k\cdot\frac{x^{k-1}-1}{x-1}\right )\cdot B_3(k) =\left [x^{k-1}(\underline{{\bf y^{k-1}}}+y^{k-2}+y^{k-3}+\cdots)-y^k(\underline{{\bf x^{k-2}}}+x^{k-3}+\cdots) \right ](xy-1),\end{gathered}
[ S_{a}] = e^{\mathrm{\it ~ i{\m a t h v e r s i o n{b o l d}~ A \cdot \s i g m a ~}}}, \hspace{3 m m}[ S_{b}] = e^{\mathrm{\it ~ i{\m a t h v e r s i o n{b o l d}~ B \cdot \s i g m a ~}}}
A_{i_1 \ldots i_p}^{(g)}(x_0,{\bf x}) = x_0^{d\over 2}\int{d^dk\over (2\pi)^d}e^{-i{\bf k}\cdot{\bf x}}a_{i_1 \ldots i_p}({\bf k}) K_\nu(|{\bf k}| x_0).
Y_u(s,\tau) =Z_u(s,\tau)-r_u(s,\tau,s_l,\tau_n)\frac{\sigma_u(\tau)}{\sigma_u(\tau_n)}Z_u(s_l,\tau_n), h(u,y) =r_u(s,\tau,s_l,\tau_n)\frac{\sigma_u(\tau)}{\sigma_u(\tau_n)}\left(m(u)-\frac{y}{m(u)}\right)-m(u).
[f,g]_\Omega= \ \Big(0,\ldots,0,{[f,g]_{\Omega}}^A_{\{1,\ldots,n+m-1\}},{[f,g]_{\Omega}}^V_{\emptyset},{[f,g]_{\Omega}}^V_{\{1\}},\ldots,{[f,g]_{\Omega}}^V_{\{2,\ldots,n+m-1\}},0\Big) \in\frak{L}'(n+m-1) ,
J^{w_1}(X_{i:n}) =-\frac{n c_{i,n}}{2}\int_{0}^{1}\Lambda_{X}^{w_1}(u)\phi_{2i-1:2n-1}(u)du &=-\frac{c_{i,n}(2i-1)}{4}\int_{0}^{1}w_1(F^{-1}(u))r\left(F^{-1}(u)\right)\phi_{2i:2n}(u)du &=-\frac{c_{i,n}(2i-1)}{4}E\left(M(B_{2i:2n})\right),
c^{\rm (in)}_{n,d}= c^{\rm (in)}_{n-2,d+2}+ c^{\rm (out)}_{n-1,d-1}+ 2 \cdot c^{\rm (out)}_{n-2,d}+ \sum\limits_{i=0}^{n-2}\sum\limits_{j=0}^{d-2}\bigl (2 \cdot \tilde{s}_{i,j}\cdot c^{\rm (in)}_{n-2-i,d-2-j}+ d_{i,j}\cdot d_{n-2-i,d-2-j}\bigr ).
\left( P^{( 2 )}+ P^{( 1 )}+ P^{( 0 - s )}+ P^{( 0 - w )}\right)_{\mu \nu , \kappa \lambda}= \frac 1 2 \left( \eta_{\mu \kappa}\eta_{\nu \lambda}+ \eta_{\mu \lambda}\eta_{\nu \kappa}\right) .
\psi_\rho'(\alpha_\rho) = 1+(2-\rho)\frac{\rho-1-\alpha_\rho}{\alpha_\rho}- (\rho-1)\frac{\rho-1-\alpha_\rho}{1-\alpha_\rho}= \frac{(\rho-1)(2-\rho)}{\alpha_\rho(1-\alpha_\rho)},
\Delta \left(X^-_\pm\right)\left(V_\alpha(z_1)V_\beta(z_2)\right) =X^-_\pm\left(V_\alpha(z_1)\right)V_\beta(z_2) +k^{-1}_\pm\left(V_\alpha(z_1)\right)X^-_\pm\left(V_\beta(z_2)\right){}~.
(\begin{matrix}z\\k\end{matrix})\approx\frac{(-1)^{k}}{\Gamma(-z)k^{z+1}}and(\begin{matrix}z+k\\k\end{matrix})=\frac{k^{z}}{\Gamma(z+1)}(1+\frac{z(z+1)}{2k}+O(k^{-2}))
f(\delta_{M}>\delta_{c})=\int_{\delta_{c}}^{\infty}d\delta_{M}\frac{1}{\sqrt{2\pi}\sigma(R)}exp(-\frac{1}{2}\frac{\delta_{M}^{2}}{\sigma^{2}(M)})=\frac{1}{2}erfc(\frac{1}{\sqrt{2}}\frac{\delta_{M}}{\sigma(M)}).
R_{\Lambda^{(n)}}^{(k)}\big(M(\Lambda^{(n)};f_1),\dots,M(\Lambda^{(n)};f_k)\big)=\frac1{V^{(n)}}\int_{\Lambda^{(n)}}f_1(x)\dotsm f_k(x)\,d\sigma(x)+\sum_{j=2}^k\frac{c^{(n)}_j}{(V^{(n)})^j}.
\sum_{n=1}^{N}\left[\begin{matrix}N n\end{matrix}\right]\frac{(-1)^{n-1}z^nq^{\frac{n(n+1)}{2}}(q)_{n}}{(1-q^{n})(zq)_n}= \sum_{n=1}^N \frac{z q^n}{1- z q^n}.
\sum_{j=1}^{n}\varepsilon_j^2 = \frac{1}{n}\sum_{k=1}^{n-1}\left| \sum_{j=1}^{n}\varepsilon_j e^{-2 \pi i k \frac{j}{n}}\right|^2 \leq \max_{1 \leq k \leq n-1}\left| \sum_{j=1}^{n}\varepsilon_j e^{-2 \pi i k \frac{j}{n}}\right|^2.