harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sum_{\substack{0\leq i\leq k}}I_{\mathrm{dch}}^{\mathfrak{m}}(0';a_{1},a_{2},\dots,a_{i};1')I_{\mathrm{dch}^{-1}}^{\mathfrak{m}}(1';a_{i+1},\dots,a_{k};0')=I_{\mathrm{dch}\circ\mathrm{dch}^{-1}}^{\mathfrak{m}}(0';a_{1},\dots,a_{k};0')=\delta_{k,0},
M(\lambda) = \Sigma_{P}^{(\epsilon,\eta)}(\lambda)+ \left( \lambda \begin{bmatrix}0 D(\lambda) \end{bmatrix}+B \right)\left(L_\epsilon(\lambda^\ell)\otimes I_n\right) +\left(L_\eta(\lambda^\ell)^T \otimes I_m\right)\left( \lambda \begin{bmatrix}0 -D(\lambda) \end{bmatrix}+C\right),
\prod \limits_{i = 0}^{k-1}|c(\theta_i) \cdot p'(x_i)| = \prod \limits_{i = j}^{t-1}|c(\theta_i) \cdot p'(x_i)| \prod \limits_{i = t}^{k-1}|c(\theta_i) \cdot p'(x_i)| \leq 4^{3K_m}\cdot 4^{4K_m}\cdot (3/5)^{\left(1 - \frac1{M_0}\right)(k-t)/2},
2vg{h_{2}}^{2}=({h_{2}}^{2}-{h_{1}}^{2})ug+u\cdot0.5g(h_{2}+h_{1})\frac{(h_{2}-h_{1})^{2}}{h_{2}}+\frac{2U}{\rho\Delta t}+0.25h_{1}\frac{g^{2}({h_{2}}^{2}-{h_{1}}^{2})}{u}
- \frac{h^{2}}{2 \lambda}\int d t d^{2}x d^{2}x^{\prime}({\tilde{J}}_{k}- \frac{J_{k}^{0}}{\rho_{0}}{\tilde{J}}_{0}) ( t , x ) \Delta^{- 1}( x - x^{\prime}) ({\tilde{J}}_{k}- \frac{J_{k}^{0}}{\rho_{0}}{\tilde{J}}_{0}) ( t , x^{\prime}) .
\beta(dx,dy) = \delta_{(x_-,y_-)}+ \frac{z-y_-}{z-y_+}\delta_{(x_+,y_+)}+ \frac{y_--y_+}{z-y_+}\delta_{(x_+,z)}, \gamma(dx,dy) = \delta_{(x_+,y_-)}+ \frac{z-y_-}{z-y_+}\delta_{(x_-,y_+)}+ \frac{y_--y_+}{z-y_+}\delta_{(x_-,z)},
x_1'=\frac{x_1x_2x_3}{x_1x_3}=x_2; x_2'=\frac{x_1x_2x_3}{x_1x_2}=x_3; x_3'=\frac{x_1x_2x_3}{x_2x_3+x_3^2}=\frac{x_1x_2}{x_2+x_3}.
\begin{aligned}\widetilde B_h^n(\Theta, V_{\tau,h}) = Q_n (\langle \partial_t \theta^0-\theta^1,A_h w^0_{\tau,h}\rangle) + Q_n (\langle \partial_t \theta^1+ A_h \theta^0,w^1_{\tau,h}) =: T_1 + T_2\,.\end{aligned}
\displaystyle{\raisebox{0.9ex}{\scriptsize{$\diamond$}}}\mbox{\hspace{-0.33ex}}\rho_\tau (Q + P) \simeq \frac{1}{\pi}\, \frac{\gamma_q}{[q_0 - \tau q + p_0 - \tau{\bf p}\cdot \hat{{\bf q}}]^2 + \gamma_q^2}\, ,
{\cal L}_{B}= \frac{1}{2}B_{\mu}^{*}[ - i \epsilon_{\mu \nu \lambda}( \partial_{\nu}+ i a_{\nu}+ i C_{\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}\; ,
c_{Y_\ell}(y_\ell , y_\ell') = \min_{x_\ell, x_\ell' \in \mathcal{X}_\ell'}c_{\ell}(s_\ell, s_\ell') = (y_\ell - y_\ell')^{\top}V_{\ell, YY}^{-1}(y_\ell - y_\ell'),
e^{-e^{M-\sigma W(t)}}\sum_{n=0}^{\infty}\sum_{i=0}^{2n}\frac{(\frac{\sigma^{2}(T-t)}{2})^{n}(-1)^{i}}{n!}\begin{Bmatrix}2n i\end{Bmatrix}e^{i(M-\sigma W(t))}.
B(s) = \frac{6(n-1)^5(n^2 + 6n -12)(n-2)^2}{n^2}s^3 &+ \frac{2(n-1)(n-2)}{n^2}(9 n^7 -43 n^6 -24 n^5 + 502 n^4 -1242 n^3 +1424 n^2 - 816 n +192) s^2 &+ 2(n-2)^2(9 n^5 -59 n^4 + 131 n^3 -123 n^2 + 46n -8) s &+ 2 n^2(n-2)^3(n-4)(3n -4).
\begin{array}{rcl}\Psi_{M}\left(\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right],V\right) = \mathbb{E}^{\tilde{\mathbb{P}}}\left[P_{M}\left(\Pi,V\right)\right] = \mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]^{\top}\left(\sum_{c\in C}V_{c}+\sum_{p\in P}V_{p}\right).\end{array}
2 G^{i j i^{\prime}j^{\prime}}= ( h^{i i^{\prime}}h^{j j^{\prime}}+ h^{i j^{\prime}}h^{j i^{\prime}}- \frac{2}{D}h^{i j}h^{i^{\prime}j^{\prime}}) + C ~ h^{i j}h^{i^{\prime}j^{\prime}},
v^K=\left(\frac{\log{p_K}\cdot K\cdot v}{\log{p_K}\cdot K}\right)^K\geq\left(\frac{\log{m}}{\log{p_K}\cdot K}\right)^K=\frac{\log^K{m}}{(\log{p_K}\cdot K)^K}=c(K,p_K)\cdot\log^K{m},
\partial_{t}u = \mathrm{div}\, \big(\mathbf{g}(\nabla_{\sigma}u) \nabla u\big) (0, \infty) \times G, \mathbf{g}(\nabla_{\sigma}u) \cdot \mathbf{n}= 0 (0, \infty) \times \Gamma, u(0, \cdot) = \tilde{u}^{0}\Omega,
I(t)\le C_0\Vert{\mathbf F}\Vert_{L^2(\Omega_t)}^2+C_1\left\{\Vert\varphi(t)\Vert_{L^2(\mathbb R)}^2+\int_0^t (I+I_{1,n})(s)ds\right\}+C_2\int_0^t\Vert\varphi(s)\Vert^2_{L^2(\mathbb R)}ds
\binom{an}{bn}=\left(1 + o(1)\right) \frac{\sqrt{2 \pi an}e^{-an}(an)^{an}}{\sqrt{2 \pi bn}e^{-bn}(bn)^{bn}\sqrt{2 \pi (a-b)n}e^{-(a-b)n}((a-b)n)^{(a-b)n}}=\left(1 + o(1)\right) \sqrt{\frac{a}{2\pi b(a-b) n}}\left(\frac{a^a}{b^b (a-b)^{a-b}}\right)^n
W_{\frac{-2n_T-n+3}{2}, \frac{2n_T-n}{2}}\left(\frac{x}{\phi}\right) & = \exp\left(-\frac{x}{2\phi}\right)\left(\frac{x}{\phi}\right)^{\frac{-2n_T-n+3}{2}}\left( 1+ O \left(\frac{1}{x}\right) \right).
{\mathcal{A}}_{k}^{J}= \frac{2x_k\alpha(x_k)}{1+2x_k}\prod_{j\in J\setminus k}\mathbf{f}(x_k,x_{j})\,,\tilde{\mathcal{D}}_{k}^{J}(x) = \tilde{\delta}(x)\prod_{j\in J\setminus k}\mathbf{h}(x_k,x_{j})\,.
h_{\mathfrak{p}}(A(b,z))+\sum_{0\leq i,j<g}\left(\int^z_b \omega_i \right)\left(d_j +\sum_{0\leq k<g}a(Z)_{jk}\int^z_b \omega_k \right)B(\omega_i^* \otimes \omega_j^* )= \beta .
{\rm e}^{-A(\theta)}= -2\pi \frac{d}{dz}({\rm Ai}(z))^2, B(\theta) = 2\pi \frac{d}{dz}{\rm Ai}(z{\rm e}^{i\frac{\pi}{3}}){\rm Ai}(z{\rm e}^{-i\frac{\pi}{3}})
q(\rho_*,A^*_0,A_\epsilon):=\max\limits_{x\in \Omega}|{\mathbb B}^2(\rho_*,x,A^*_0,A_\epsilon)|\,=\min\limits_{\,A_0\in{\mathcal A}}\min\limits_{\rho\in{\mathbb R}_+}\, \max\limits_{x\in \Omega}|{\mathbb B}^2(\rho,x,A_0,A_\epsilon)|.
U S - S U = \left( \begin{array}{c c c c}{{0}}&{{0}}&{{0}}&{{0}}\\{{a - f}}&{{m}}&{{c - g}}&{{m}}\\{{0}}&{{0}}&{{0}}&{{0}}\\{{- a + f}}&{{- m}}&{{- c + g}}&{{- m}}\\ \end{array}\right) .
\begin{aligned}\langle \partial_t (\bar{J}_0 u_0 ) , \phi \rangle_{H^1(\Omega)',H^1(\Omega)}- \int_{\Omega}q u_0 \phi dx + \int_{\Omega}D_0^{\ast}\nabla u_0 \cdot \nabla \phi dx &= \int_{\Omega}\int_{Y^{\ast}}J_0 f_0 dy \phi dx - \int_{\Omega}\partial_t R_0 (u_0 - \rho) \phi \vert \Gamma(t,x)\vert dx .\end{aligned}
\hbox{$\frac{1}{2}$}\epsilon_{ab}\mathbf{s}_m^b \mathbf{s}_m^a A^i = R^i_{\alpha_0}B^{\alpha_0}+\hbox{$\frac{1}{2}$}\epsilon_{ab}R^i_{\alpha_0, j}R^j_{\beta_0}C^{\beta_0 b}C^{\alpha_0 a},
\vartheta_n= X_{0,0}, \vartheta_a=X_{a,b}, \vartheta_i=\sum_{j=1}^b X_{i,b}X_{i,b-1}\cdots X_{i,j}, \vartheta_i=\sum_{j=1}^a X_{a,n-i}X_{a-1,n-i}\cdots X_{j,n-i},
g_{1}\!-\!g_{20}\!-\!g_{24}\!-\!g_{32}\!-\!g_{29}\!+\!g_{9}\!+\!g_{4}\!+\!g_{13}\!=\!1, &g_{24}\!-\!g_{9}\!+\!g_{20}\!-\!g_{13}\!-\!g_{32}\!+\!g_{1}\!+\!g_{4}\!-\!g_{29}\!=\!1, &g_{1}\!-\!g_{13}\!-\!g_{32}\!-\!g_{4}\!+\!g_{29}\!+\!g_{9}\!-\!g_{24}\!+\!g_{20}\!=\!1, &g_{24}\!+\!g_{13}\!+\!g_{29}\!+\!g_{1}\!-\!g_{4}\!-\!g_{9}\!-\!g_{20}\!-\!g_{32}\!=\!1.
x = \frac{m \nu_{0}^{2}}{c \varepsilon^{2}H Z^{4}n}\int_{v}^{v_{0}}{\frac{v^{3}d v}{0 . 9 6 + \log \frac{h \nu_{0}v^{4}}{8 \pi c \varepsilon^{2}H Z^{4}}}}= 2 . 4 \frac{m \nu_{0}}{n h}\int_{\frac{0 . 1 0 4 h \nu_{0}v^{4}}{c \varepsilon^{2}H Z^{4}}}^{\frac{0 . 1 0 4 h \nu_{0}v_{0}^{4}}{c \varepsilon^{2}H Z^{4}}}{\frac{d u}{\log u}}~ ~ .
m \in \mathbb{N}: 1 \leq m-1 \leq \frac{1}{2a}\left( \sqrt{(1-a(N+1))^2 + \frac{Na}{1-a}}- (1 - a(N+1)) \right)= &\frac{N}{2(1-a)}\left( \frac{1}{1 - a(N+1) + \sqrt{(1-a(N+1))^2 + \frac{Na}{1-a}}}\right).
L_{Yuk}(\overline{\Psi^{\prime}}\Psi^{\prime}\varphi \,;\, \phi)=-\frac{h}{\sqrt{2}}\overline{\Psi}^{\prime}\Psi^{\prime}\varphi \frac{e^{\gamma\phi/M_{p}}}{\chi^{3/2}}= -\frac{h}{4}\overline{\Psi^{\prime}}\Psi^{\prime}\varphi\left[\frac{M^{4}+V_{1}(\phi)}{V_{2}(\phi)}\right]^{3/2}e^{\gamma\phi/M_{p}}.
\lvert u(x)\rvert =\lim_{i\to\infty}\lvert u(x)-u_{B_i}\rvert \le C(M,n,c_1,s,p) \limsup_{i\to\infty}\int_{D_i}\frac{g_i(y)}{\lvert x-y\rvert^{n-s}}\,dy &\le C(M,n,c_1,s,p) \int_D \frac{g(y)}{\lvert x-y\rvert^{n-s}}\,dy\,.
\alpha := f_{i,\ell_{1}}\oplus \ldots \oplus f_{\ell_{u-1},\ell_{u}}\,, \delta := f_{\ell_{u+1}}\oplus \ldots \oplus f_{\ell_{n-1},k}\, \beta := f_{\ell_{u},j}\, \gamma := f_{j,\ell_{u +1}}\,.
t_{\alpha}{}^{i}: = \stackrel{\mathrm{B}}{\Sigma}_{\alpha}{}^{\! i}( \stackrel{\mathrm{B}}{X}) \qquad \mathrm{a n d}\qquad \stackrel{\mathrm{B}}{\tau}_{\alpha \beta}{}^{i}= 0 ={\tau}_{\alpha \beta}{}^{\! i}- 2 \stackrel{\mathrm{B}}{X}_{[ \alpha \beta ]}{}^{\! i}
\epsilon(\iota_{E}(Q_{I}\eta_{I})) = \epsilon(Q_{I}( \iota_{E}(\eta_{I}))) = \sum_{\ell \notin I}(-1)^{\sigma_{I}(\ell)}Q_{I \cup \{\ell\}}\iota_{E}(\eta_{I}) = \iota_{E}(\sum_{\ell \notin I}(-1)^{\sigma_{I}(\ell)}Q_{I \cup \{\ell\}}\eta_{I}) = \iota_{E}(\epsilon (Q_{I}\eta_{I})).
(n_s^{-2}(\lambda_s)^{k}\cdot (n_s^{-2}c_{n_s}))\theta(n_s^{-2}(\lambda_s)^{k}n_s^{-1}(\lambda_s)^{-k}) &=(n_s^{-2}(\lambda_s)^{k}\cdot c_{n_s})\theta(n_s^{-2}(\lambda_s)^{k}n_s^{-2}n_s^{-1}(\lambda_s)^{-k}).
B(\varepsilon ;\vartheta ) =b(\mathbf{D})^*g(\mathbf{x})b(\mathbf{D})+\vartheta \varepsilon \sum \limits_{j=1}^d (a_j(\mathbf{x})D_j+D_j a_j(\mathbf{x})^*) &+\vartheta^2\varepsilon^2 Q(\mathbf{x})+\vartheta^2\varepsilon^2 c_5Q_0(\mathbf{x}).
\Delta \bar v^{i,(k+1)}= \Delta \bar v^{i,(k)}+ \left( \bar A^i\Delta w^{i,(k+1)}+ r_{}^{i,(l)}\right) , \Delta \bar v_c^{i,(k+1)}= \Delta \bar v_c^{i,(k)}+ \left( \Delta w^{i,(k+1)}- \Delta x_{J_i}^{(k+1)}+ r_c^{i,(l)}\right),
(w^r)\otimes L_0 \xrightarrow{\phi_r}L_{r-1}\xrightarrow[\sim]{\psi_{r-1}}(w)\otimes L_{r-2}\xrightarrow[\sim]{\mathrm{id}_{(w)}\otimes\psi_{r-2}}(w^2)\otimes L_{r-3}\xrightarrow[\sim]{\mathrm{id}_{(w^2)}\otimes\psi_{r-3}}\cdots \xrightarrow[\sim]{\mathrm{id}_{(w^{r-2})}\otimes\psi_1}(w^{r-1})\otimes L_0
\begin{array}{r l}{{{\cal H}= \frac{1}{2}\pi_{1}^{2}}}&{{+ \frac{\pi}{2 e^{2}}\left( \partial_{1}\pi_{1}\right)^{2}+ \frac{e^{2}}{2 \pi}\left( A^{1}- \frac{1}{2 e \lambda}\partial_{1}( \phi_{+}- \phi_{-}) \right)^{2}}}\\{{}}&{{+ \frac{\pi}{2 e^{2}}\phi^{2}- \frac{\pi}{e^{2}}\phi ( \partial_{1}\pi_{1}) \ .}}\\ \end{array}
\mathrm{tr}(T_{r}) =(-1)^{r}(n-r) S_{r}, \mathrm{tr}(\mathcal{A}_{\widehat{W}}\circ T_{r}) =(-1)^{r}(r+1)S_{r+1}, \mathrm{tr}(\mathcal{A}_{\widehat{W}}^{2}\circ T_{r}) =(-1)^{r+1}(-S_{1}S_{r+1}+ (r + 2) S_{r+2}), \mathrm{tr}(T_{r}\circ\nabla'_{X}\mathcal{A}_{\widehat{W}}) = (-1)^{r}X(S_{r+1})=(-1)^{r}\overline{g}(\nabla'S_{r+1},X),.
\Tilde{\Phi}_2(\beta_0,+\infty)=\tfrac{2}{L_{2m}^{2}}\bigg[\tilde{N_2}\tfrac{\beta_0^{3-2\mu-\nu}}{(3-2\mu-\nu)(1-\mu-\nu)}+ \tfrac{\tilde{L_2}N_{2M}}{L_{2m}}\tfrac{\beta_0^{\delta-3\mu+3-\nu}}{(\delta-3\mu+3-\nu)(1-\mu-\nu)}+\tfrac{\tilde{L_2}}{L_{2m}}\tfrac{\beta^{1-\mu-\nu}}{{\mu+\nu-1}}\bigg],
\int_{1/(N+1)}^\delta \frac{|Z_N(t)-\sigma B_N(t)|^p}{w(t)}dt &\leq \sup_{1/(N+1)<t<N/(N+1)}\left(\frac{|Z_N(t)-\sigma B_N(t)|}{t^\zeta}\right)^p\int_{1/(N+1)}^\delta \frac{t^{p\zeta}}{w(t)}dt &=O_P(N^{(-1/2+\zeta)p})(N+1)^{-p\zeta+p/2}\int_{0}^\delta \frac{t^{p/2}}{w(t)}dt.
h_B(\Gamma) = (u u')^4 +(f_0- 4) t t' (u u')^3+\beta t t' (u u')^3 +\gamma (t (t')^2 u^3 (u')^2 + t^2 t' u^2 (u')^3 - t^2 (t')^2 u^2 (u')^2),
v_{h_k}(\phi) \sim Ai\left( -h^{-\frac{2}{3}}\rho(\phi) \right)h^{-\frac{1}{6}}_k \sum_{n = 0}^\infty u_{0,n}(\phi)h^{n}+ Ai'\left(-h^{\frac{2}{3}}\rho(\phi)\right)h^{\frac{1}{6}}\sum_{n=0}^\infty u_{1,n}(\phi)h^n
H_{L C}^{( A d S )}= \frac{1}{2 p_{-}}\int_{0}^{l}d \sigma \left( \pi_{i}\pi_{i}+ \frac{1}{( 2 \pi \alpha^{\prime})^{2}}\frac{R^{4}}{Y^{4}}Z^{i \prime}Z^{i \prime}\right) \ .
2H=c^{2}=\frac{p_{t}^{2}}{c^{2}(1-\frac{r_{s}}{r})}-(1-\frac{r_{s}}{r})p_{r}^{2}-\frac{p_{\theta}^{2}}{r^{2}}-\frac{p_{\varphi}^{2}}{r^{2}sin^{2}\theta}
d_j+d_{j+1}\leq |N(x_j)\backslash\{x_1,\ldots,x_{j-1}\}|+| N(x_{j+1})\backslash\{x_1,\ldots,x_{j}\}| &=[((N(x_j)\backslash\{x_1,\ldots,x_{j-1}\})\cup( N(x_{j+1})\backslash\{x_1,\ldots,x_{j}\})| \leq n-j,
\L(\l)+\frac{16k(N-2+k)}{\left[(p-1)\left(2-N+\sqrt{(N-2)^2-4\l}\right)+4\right]^2}\le &\frac{1-p}{\left(\frac{M-2}2\right)^2}+o(1)=\frac{1-p}{\left(\frac{2}{p-1}\right)^2}+o(1)<0\quad\hbox{for $\l$ large enough.}
\big\vert \big\vert \big(\mu^{(n)}- \tilde{\mu}^{(n)}\big) - \big(\mu^{(n+1)}- \tilde{\mu}^{(n+1)}\big) \big\vert \big\vert_{L^2(\Sigma)}= O\left(\frac{1}{n^{3/2}\log^2 n}\right)
W^{(ab)}(r,r^{\prime})=\frac{\bar{I}_{\nu}^{(a)}(az)}{\bar{I}_{\nu}^{(b)}(bz)}\frac{\bar{K}_{\nu}^{(b)}(bz)}{\bar{K}_{\nu}^{(a)}(az)}\left[\frac{G_{\nu}^{(a)}(az,zr)G_{\nu}^{(b)}(bz,zr^{\prime})}{\bar{K}_{\nu}^{(a)}(az)\bar{I}_{\nu}^{(b)}(bz)-\bar{K}_{\nu}^{(b)}(bz)\bar{I}_{\nu}^{(a)}(az)}-I_{\nu}(zr^{\prime})K_{\nu}(zr)\right] .
\|\varphi\|_{L^2}= \|\hat{\varphi}\|_{\ell^2}\lesssim M^{1/2-1/p}\|\hat{\varphi}\|_{\ell^{p}}&\lesssim M^{1/2-1/p}\|\varphi\|_{L^{p'}}= M^{1/2-1/p}\,.
\sum_{j={\lfloor n/2 \rfloor +1}}^n \left ( E_{\mu}^Q h_{n-j}(Z_j) - E_{\pi^Q}^Q h_{n-j}(Z_j) \right)\ \le \c_1 \|\tilde g\|_\infty\|h\|_\infty \rho^{n/2}n/2 \to 0.
\left| A_k\right|\leq C\int\limits_{\beta_{kn}}\,\frac{\omega(d(M))}{d^{2}(M)}\cdot\frac{1}{\Lambda_n{|k-k_0|}^2}\, dm_3(M)\leq &\leq C\omega(2^{-n})\cdot 2^{-n}\cdot \frac{1}{\Lambda_n{|k-k_0|}^2}\leq C\frac{\omega (2^{-n})}{(k-k_0)^2},
= \int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}(\psi(x) - \psi(y))^{2}\left( \left( \frac{\tilde{u}(s,x)}{\psi(x)}\right)^{1-q}- \left( \frac{\tilde{u}(s,y)}{\psi(y)}\right)^{1-q}\right)k(x,y)dxdy.
\begin{array}{c}{{h = \left\{A^{-}, A^{+}\right\}_{q}= 2 \left\{\left( \frac{1}{\cos \frac{\omega x}{2}}\cdot \cosh \frac{i}{2}\frac{d}{d x}\right)^{2}- \cosh \frac{\omega}{4}\right\}=}}\\{{= \frac{\stackrel{\wedge}{P}}{2}+ V \left( x \right)}}\\ \end{array}
\mathbb{P}_\eta (\mathcal{G}^c) \leq e^{-c_3'}\mathbb{E}_\eta \left(\prod_{k=d\lfloor \alpha t \rfloor+1}^{\lfloor \beta t \rfloor}(\mathbb{1}_{\mathcal{U}_k}\mathbb{1}_{\mathcal{G}_{k}^c})\right),
f(x_{1},...,x_{n}|\mu,\sigma^{2})=\prod_{i=1}^{n}f(x_{i}|\mu,\sigma^{2})=(\frac{1}{2\pi\sigma^{2}})^{n/2}exp(-\frac{\sum_{i=1}^{n}(x_{i}-\mu)^{2}}{2\sigma^{2}}).
\begin{aligned}E^{*-}(f) =\int_{x_{j-1}}^{x_{j-1}+\alpha}(x-x_{j-1})(x-x_{j})(x-x_{j+1})f^-[x_{j-1}, x_{j}, x_{j+1},x]\dx &=f^-[x_{j-1}, x_{j}, x_{j+1},\xi_3]\int_{x_{j-1}}^{x_{j-1}+\alpha}(x-x_{j-1})(x-x_{j})(x-x_{j+1})\dx &=\frac{f^-_{xxx}(\eta_3)}{24}\left(\frac{\alpha^{4}}{4}-h{\alpha}^{3}+{h}^{2}{\alpha}^{2}\right),\end{aligned}
\frac{|\hat\eta_t(\hat{x}_t) - \mathbf{P}^{\hat{g}_t}_{\hat{x}'_t\hat{x}_t}(\hat\eta_t(\hat{x}'_t))|_{\hat{g}_t(\hat{x}_t)}}{d^{\hat{g}_t}(\hat{x}_t,\hat{x}'_t)^\alpha}=1.
\sigma \cdot T := (B^{\sigma(1)}, \cdots, B^{\sigma(n)}) S\cdot T := (B^1(- \delta_{1 \notin S}), \cdots, B^n(- \delta_{n \notin S}) ) S^{-1}\cdot T := (B^1(\delta_{1 \notin S}), \cdots, B^n(\delta_{n \notin S}) )
Z^{\cal N}\, = \, \left( \begin{array}{c l c l}{{Z_{1}\, \epsilon}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{Z_{2}\, \epsilon}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{Z_{3}\, \epsilon}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{Z_{4}\, \epsilon}}\\ \end{array}\right) \qquad , \qquad \epsilon \, = \, \left( \begin{array}{c l}{{0}}&{{1}}\\{{- 1}}&{{0}}\\ \end{array}\right)
\|\hat g_{n+1}- \hat g_n\|_S =\| F(\hat g_n) - F(\hat g_{n-1})\|_S < \frac{1}{2}\|\hat g_n - \hat g_{n-1}\|_S<\cdots <\left(\frac{1}{2}\right)^{n}\|\hat g_1 - \hat g_0\|_S.
0\le (-1)^k \frac{\partial^k}{\partial z^k_2}\Phi^1_p(x_1, x_2) = (-1)^k \frac{\partial^k}{\partial z^k_2}\left(\frac{\partial}{\partial z_1}\log p \right) (x_1, x_2) =\frac{\partial}{\partial z_1}\left((-1)^k \frac{\partial^k}{\partial z^k_2}\log p \right) (x_1, x_2).
\tilde A_0 := \begin{pmatrix}1 c^T c C \end{pmatrix}, \ \tilde A_1 := \begin{pmatrix}1 a_1^T a_1 A_m \end{pmatrix}, \ldots, \ \tilde A_m := \begin{pmatrix}1 a_m^T a_m A_m \end{pmatrix}.
P\left[ C_{H}^{\circ}\left(a,v,\varphi \right) \right] =\begin{cases}C_{V}^{\circ}\left(Pa,Pv,\varphi_1 \right), \varphi\leq\angle\left(v,V^{\bot}\right), V, \varphi > \angle\left(v,V^{\bot}\right)\mbox{.}\end{cases}
\|\hat{P}_-\psi_{\kappa\lambda}\|^2_{t\rightarrow\pm\infty}= \frac{1}{2}\mp \frac{1}{2}\int_0^\infty\frac{dp'}{p'}\int_0^\infty\frac{dp''}{p''}\psi_{\kappa\lambda}(p') \psi_{\kappa\lambda}(p'') \frac{p''-p'}{\log p'' - \log p'}\delta(p''-p').
2\Gamma(f,g)(y)=\sum\limits_{p}(\bar f(p)-\bar f(y))(g(p)-g(y))c_{py}\phantom{2\Gamma(f,g)(y)}=(T(\bar f g))(y)-\bar f(y)(Tg)(y)-(T\bar f)(y)g(y)+(C(\bar f g))(y) \phantom{2\Gamma(f,g)(y)}=-((C-T)(\bar f g))(y)+((C-T)\bar f)(y) g(y)+\bar f(y)((C-T)g)(y) \phantom{2\Gamma(f,g)(y)}=2\Gamma_{C-T}(f, g)(y).
m_{\mu}L_{N^\mu_{(i,k)}}^t [T; N_{(i,k)}^\mu, p]^- = q^{2 p -2}m_{\mu}\Phi^+_t (L_{N^\mu_{(i,k)}}, L_{N^\mu_{(i,k)}-1}, \dots, L_{N^\mu_{(i,k)}-p+1}).
X_{\textup{b}}:=\left\{w \in C^0_0(\overline{R}) \cap C^{9+\alpha}_{\textup{b}}(\overline{R^+})\cap C^{9+\alpha}_{\textup{b}}(\overline{R^-}): w|_{B}=0\right\}, Y_{\textup{b}}:= \left(C^{7+\alpha}_{\textup{b}}(\overline{R^+}) \cap C^{7+\alpha}_{\textup{b}}(\overline{R^-})\right) \times C^{8+\alpha}_{\textup{b}}(T)\times C^{8+\alpha}_{\textup{b}}(I).
\left( P_{b}^{i_{1}\ldots i_{p - 2 k - 1}}, b_{i_{1}\ldots i_{p - 2 k - 1}}\right) , \left( P_{b^{1}}^{i_{1}\ldots i_{p - 2 k - 1}}, b_{i_{1}\ldots i_{p - 2 k - 1}}^{1}\right) , k = 0 , \ldots , a ,
L_{h y b}= \tilde{L}\sp{C S}_{e x t}+ \frac{e \hat{B}\sp{e x t}}{2}( \epsilon_{i j}X_{i}\dot{X}_{j}- 2 e \theta \hat{A}_{0}\sp{C S}) + e \hat{E}_{i}\sp{e x t}\cdot \eta_{i}.
\chi_{m}^{\ell , s}= \chi_{m + 2 k + 4}^{\ell , s}= \chi_{m}^{\ell , s + 4}= \chi_{m + k + 2}^{k - \ell , s + 2}\; \; \mathrm{a n d}\; \; \chi_{m}^{\ell , s}= 0 \; \; \mathrm{u n l e s s}\; \; \ell + m + s \equiv 0 \; \; \mathrm{m o d}\; 2 ~ ,
\begin{array}{c}\displaystyle \int_{0}^{1}p \phi_1^{(n)}f^{(n)}_\alpha{\rm d}p = \frac{\Gamma(n+3)}{\Gamma(x+\gamma + 1) \Gamma(n-x-\gamma +2)}\int_0^1 p^{x+\gamma+1}(1-p)^{n-x+1-\gamma}{\rm d}p = \displaystyle \frac{\Gamma(n+3) \Gamma(x+\gamma + 2)}{\Gamma(n+4) \Gamma(x+\gamma + 1)}=g_1(\alpha). \end{array}
\int d^{1 0}x \, s q r t{- \hat{g}}\ \left[{\textstyle \frac{3}{4}}\hat{G}^{2}\right] = \int d^{9}x \sqrt{g}\ \left[{\textstyle \frac{3}{4}}k G^{2}-{\textstyle \frac{3}{4}}k^{- 1}\left( H^{( 2 )}- \ell H^{( 1 )}\right)^{2}\right] \, .
x =\frac{-1}{\sqrt{6}}(Y_{1,1}-Y_{1,-1}),y=\frac{i}{\sqrt{6}}(Y_{1,1}+Y_{1,-1}),z=\frac{1}{\sqrt{3}}Y_{1,0}(x\pm iy)^{2}=\sqrt{\frac{8}{15}}Y_{2,\pm2},xy=\frac{-i}{\sqrt{30}}(Y_{2,2}-Y_{2,-2}),x^{2}-y^{2}=\sqrt{\frac{2}{15}}(Y_{2,2}+Y_{2,-2})
\frac{\partial a_{5}}{\partial c}|_{c=0}=a_{3}\frac{(3+\alpha)_{2}(3+\beta)_{2}}{(1+3)_{2}\times1\times2}(\frac{1}{\alpha+3}+\frac{1}{\alpha+4}+\frac{1}{\beta+3}+\frac{1}{\beta+4}-\frac{1}{4}-\frac{1}{5}-\frac{1}{1}-\frac{1}{2}).
A_{\mu}^{( 3 )}= \mp{\frac{2 i}{g}}\, ( H_{2}\,{\frac{v_{\mu}^{1}}{\rho_{1}}}+ H_{3}\,{\frac{v_{\mu}^{2}}{\rho_{2}}}) + \kappa \, ( E_{2 1}^{\pm}\, R_{\mu}^{1}+ E_{3 1}^{\pm}\, R_{\mu}^{2}) \, \delta ( R^{1}\cdot R^{2}) .
{\cal I}_{2}= \left\{\begin{array}{c l}{2 | q_{1}|}&{\qquad 0 < | q_{1}| < | \tilde{a}_{1}| \quad \mathrm{a n d}\quad 0 < q \tilde{a}_{1}}\\{}&{}\\{0}&{\qquad \mathrm{o t h e r w i s e}}\\ \end{array}\right\}.
t_2 = \alpha_1 \frac{(\Upsilon_1^{-1}- \alpha_1) t_1^3 - \Upsilon_1^{-1}C_1 t_1 - \Upsilon_1^{-1}C_2}{t_1^2}+ O(T^{\varepsilon}B_3 E_1^{-1/2}).
\frac{P_*}{Q_*}={f(Q_*)\displaystyle{\int_0^m \exp\left\{-\int_0^s\displaystyle{\frac{\gamma_s(r,P_*,Q_*)+\mu(r,P_*,Q_*)}{\gamma(r,P_*,Q_*)}}\,dr\right\}\,ds}\over \displaystyle{\int_0^m F(s,P_*,Q_*) \exp\left\{-\int_0^s\displaystyle{\frac{\gamma_s(r,P_*,Q_*)+\mu(r,P_*,Q_*)}{\gamma(r,P_*,Q_*)}}\,dr\right\}\,ds}}, R(P_*,Q_*)=1,
{n-1\brack k}_{q^2}=\frac{(1-q^{2n-2})(1-q^{2n-4})\cdots(1-q^{2n-2k})}{(1-q^2)(1-q^4)\cdots(1-q^{2k})}begin{align*}7pt] \equiv \frac{(1-q^{-2})(1-q^{-4})\cdots(1-q^{-2k})}{(1-q^2)(1-q^4)\cdots(1-q^{2k})}\pmod{\Phi_n(q)}begin{align*}7pt] =(-1)^kq^{-k^2-k}.
\left|\frac{(d/dr)\Psi(re^{if(r)})}{\Psi(re^{if(r)})}\right| =\left|\frac{\Psi'(re^{if(r)})}{\Psi(re^{if(r)})}\right|\left|\frac{d(re^{if(r)})}{dr}\right| =\left|\frac{1}{re^{if(r)}}-2\beta\psi'_{\mu_{1}}(re^{if(r)})\right|\left|e^{if(r)}(1+irf'(r))\right| =\frac{1}{r}\left|1-2\beta re^{if(r)}\psi'_{\mu_{1}}(re^{if(r)})\right|\sqrt{1+r^{2}f'(r)^{2}}.
\Phi^{\Delta}=\left( A_{\alpha \beta}^{\;\;\;(\sigma )},\eta_{\alpha}^{\;(\sigma )},C^{(\sigma )}\right) ;\;\Phi_{\Delta}^{*}=\left(A_{\;\;\;\;(\sigma )}^{*\alpha \beta},\eta_{\;\;\;(\sigma )}^{*\alpha},C_{(\sigma )}^{*}\right) ,
A_{n}^{\lambda,\beta}(x)=\sum_{k=0}^{n}\binom{n}{k}\Bigg[\sum_{i=0}^{n-k}(-1)^{i}i!B_{n-k,i}((M_{\lambda,\beta}(j))_{j=1}^{n-k-i+1})\Bigg]x^{k}.
\begin{aligned}2\big<g ,\left(\nabla_{\xi}\psi(\xi), \sigma\nabla_{\xi}g\right)\big>_{\xi}= \int \sigma^{ij}\partial_{i}\psi \partial_{j}(g^2)d\xi &=-\int \nabla_{\xi}\cdot[\sigma\nabla_{\xi}\psi ]g^2 d\xi &=-\int \nabla_{\xi}\cdot[\lambda_1(\xi)\nabla_{\xi}\psi ]g^2 d\xi\,,\end{aligned}
\eqref{equ:Z3r1r202}= -\frac{(-1)^{r_2}}{r_2+1}\sum_{j=r_2+2}^{r_1+r_2+2}\binom{r_1}{j-r_2-2}\frac{B_j}{j}\zeta(s+j-r_1-r_2-2) &-\sum_{j=r_2+2}^{r_1+r_2+2}(-1)^{r_2}\sum_{k=1}^{r_2}\binom{r_2}{k-1}\binom{r_1}{j-r_2-2}\frac{B_kB_{j-k}}{k(j-k)}\zeta(s+j-r_1-r_2-2).
dL_{\gamma}(u)=\sum_{1\leq i<j\leq n}L_{\gamma}(\partial_{a_{i},a_{j}}u)\cdot d\log(a_{i}-a_{j})+\sum_{1\leq i\leq j\leq n}L_{\gamma}(\partial_{1-a_{i},a_{j}}u)\cdot d\log(1-a_{i}-a_{j})
\begin{cases}f(b_i)=0,\; f(u)>0 \mbox{for}u\in (b_i, q_{i-1}), \; f(u)\leq 0 \mbox{for}u\in (q_i, b_i), \int_{q_i}^u f(s)ds<0 \mbox{for}u\in (q_i, b_i].\end{cases}
\|a(x,\xi;h)u\|_{m_\mathcal{H}(x,\xi,\lambda;h)}\leqslant Cm(x,\xi,\lambda;h)\|u\|_{m_\mathcal{B}(x,\xi,\lambda;h)}, \lambda=h^{-\delta}d(\Sigma,\cdot), u\in\mathcal{B},
c^*_+-c^*_-= \inf\limits_{\mu>0}\Phi^+(\mu)-\inf\limits_{\mu>0}\Phi^-(\mu) \geq \inf\limits_{\mu>0}[\Phi^+(\mu)-\Phi^-(\mu)] = (\alpha-\beta)\inf\limits_{\mu>0}\frac{(e^{\mu}-e^{-\mu})}{\mu}>0.
Y_t^x=h(B_T+x)-\sum_{i=1}^m \int_t^T Z_s^{x,i}dB_s^i-\sum_{i=1}^m\frac{1}{2}\int_t^T \bar A(Y_s^x)(Z_s^{x,i},Z_s^{x,i})ds+\int_t^T \bar f (Y_s^x,Z_s^x)ds.
a^{\mathfrak{sp}, \theta_-^{\prime}}_{d^{\prime}}(v^2)= v^{\mathcal{E}(d^{\prime})}[m]_v \left[ \begin{matrix}n d_2 \end{matrix}\right]_v = v^{\mathcal{E}(d^{\prime})}P_{\mathfrak{M}_{d^{\prime}}^{\mathfrak{sp}, \theta^{\prime}_-}(Q^{\prime})}(v).
Q(P(i))-P(i) = |\{j\in J/\, P(j)>P(i), \psi(P(j),P(i))>0\}| &= |\{j\in J/\, P(j)>P(i), j<i\}| &= |\{j\in J/\, j<i,\phi(j,i)<0\}| &= i-P(i)\,\,
R_{N}(x)\propto\int_{x}^{\infty}t^{-2N}e^{-t^{2}}dt=e^{-x^{2}}\int_{0}^{\infty}(t+x)^{-2N}e^{-t^{2}-2tx}dt\le e^{-x^{2}}\int_{0}^{\infty}x^{-2N}e^{-2tx}dt\propto x^{-(1+2N)}e^{-x^{2}}.
\left\vert \tau \left( x,w;\alpha ,\beta \right) \right\vert =\left\vert\int\limits_{a}^{b}P(x,t)f^{\prime}(t)dt\right\vert \leq\int\limits_{a}^{b}\left\vert P(x,t)\right\vert \left\vert f^{\prime}(t)\right\vert dt,
z^2{\cal H}_{k-1}^{(s+1,t)}(z^2)={\cal H}_{k}^{(s,t)}(z^2)+q_{k}^{(s,t)}{\cal H}_{k-1}^{(s,t)}(z^2),k=1,2,\dots,m, &{\cal H}_k^{(s,t)}(z^2)={\cal H}_{k}^{(s+1,t)}+e_k^{(s,t)}{\cal H}_{k-1}^{(s+1,t)}(z^2),k=0,1,\dots,m.
{\cal L}_{b o s o n i c}= - \frac{1}{4 l^{2}}\partial_{\mu}l \partial^{\mu}l + \frac{1}{4 l^{2}}h^{\mu}h_{\mu}- G_{i{\bar{j}}}\partial_{\mu}z^{i}\partial^{\mu}{\bar{z}}^{\bar{j}}- \frac{i}{2}( G_{i j}\partial z^{j}- G_{i{\bar{j}}}\partial_{\mu}z^{j}- G_{i{\bar{j}}}\partial_{\mu}{\bar{z}}^{\bar{j}}) h^{\mu}.
\zeta(2)=\sum_{n=1}^{\infty}\frac{1}{n^{2}}=\frac{1}{1^{2}}+\frac{1}{2^{2}}+\frac{1}{3^{2}}+\frac{1}{4^{2}}+\cdot\cdot\cdot=\frac{\pi^{2}}{6}\approx1.644934.
\tilde{T}=\frac{S_{n}S_{n+3}}{S_{n+1}S_{n+2}}+\frac{S_{n+1}S_{n+4}}{S_{n+2}S_{n+3}}+\tilde\alpha(\frac{S_{n+1}S_{n+2}}{S_{n}S_{n+3}}+\frac{S_{n+2}S_{n+3}}{S_{n+1}S_{n+4}})+\tilde\beta\frac{S_{n+2}^2}{S_{n}S_{n+4}}.
\frac{1}{\rho^{3}}\frac{d}{d \rho}\left( \rho^{3}\frac{d f}{d \rho}\right) + \frac{3 a^{2}}{\rho^{2}}f - \frac{\overline{{a}}^{6}}{\rho^{6}f^{4}}f - f + f^{3}= 0
\left| D^n_s \left( \eta^{\rm slow}\circ \phi \right) \right|\lesssim_{n,k_{0}}\sum_{\left\{m_{\ell}\right\}}\left| (D^{m}_s \eta^{\rm slow}) (\phi) \right|\prod_{\ell=1}^{n}\left( \left( \log k \right)^{\ell}k^{\psi}\right)^{m_{\ell}}.
x\leftrightarrow n, \eta(x)\leftrightarrow\mathcal{E}(n), \eta(0)=0\leftrightarrow\mathcal{E}(0)=0, B(x)\leftrightarrow -A_n, D(x)\leftrightarrow -C_n,D(0)=0=C_0, \frac{\phi_0(x)}{\phi_0(0)}\leftrightarrow\frac{d_n}{d_0},
\int\psi^2\left(y(\partial_y\bar{w}_k)^2+(\partial_x\bar{w}_k)^2\right) &\qquad\leq C\int(\psi^2+y\psi_y^2+\psi_x^2+|\psi_y|)\bar{w}_k^2+C\theta_k^{-2}\int\psi^2|f_k|^2.
\sum_{\rho\in R_\epsilon}(\rho)= -\frac{\epsilon T}{2\pi}\log \pi + \frac{\epsilon T}{2\pi}\log \left(\frac{T}{2}\right) -\frac{\epsilon T}{2\pi}+\frac{1}{2\pi}\int_{\frac{1}{2}-\epsilon}^{\frac{1}{2}+\epsilon}\arg \zeta(\sigma+iT) \d\sigma+\left(\frac{\epsilon}{\pi}\right)\mathcal{O}(\log T)
\begin{array}[c]{clll}\mathbf{v}^{\nabla}:=\sum_{\ell=1}^{\infty}\sum_{m\in\iota_{\ell}}V_{\ell}^{m}\nabla_{\Gamma}Y_{\ell}^{m}\mathbf{v}^{\operatorname*{curl}}:=\sum_{\ell=1}^{\infty}\sum_{m\in\iota_{\ell}}v_{\ell}^{m}\overrightarrow{\operatorname*{curl}\nolimits_{\Gamma}}Y_{\ell}^{m}& \end{array}
\sum_{\mathbf{a}\in S_-}\mathbf{B}_2(\langle1/2-a_k+v_{k+g}\rangle)=\sum_{\mathbf{a}\in S_-}\mathbf{B}_2(\langle1/2-a_k+v_{k+g}'\rangle)\textrm{for each}~k=1,\ldots,g.