harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\nabla_{\vartheta}\vartheta =\cos^2(t)\tan(t)\dot{\gamma}=\cos(t)\sin(t)\dot{\gamma}\,, \nabla_{\vartheta}\dot{\gamma}=-\tan(t)\vartheta \,, \nabla_{\dot{\gamma}}\vartheta =-\tan(t)\vartheta \,, \nabla_{\dot{\gamma}}\dot{\gamma}=0\,.
\mu(\{|f|\geq\beta\|f\|_\Psi\})=\mu\bigg(\bigg\{\Psi\bigg(\frac{|f|}{\|f\|_{L^\Psi(X,\mu)}}\bigg)\geq\Psi(\beta)\bigg\}\bigg)\leq\frac{1}{\Psi(\beta)}\int_X\Psi\bigg(\frac{|f|}{\|f\|_{L^\Psi(X,\mu)}}\bigg)d\mu.
D\big(p^{-1},{\rm sym}^m[g(\theta)]\big)^{-r}=\prod_{j=0}^{m}\left(1-\frac{2\cos(m-2j)\theta}{p}+\frac{1}{p^2}\right)^{r/2}& = \exp \left\{-\frac{r{\rm tr}({\rm sym}^m[g(\theta)])}{p}+O_m\bigg(\frac{r}{p^2}\bigg)\right\}& = \exp \left(-\frac{r{\rm tr}({\rm sym}^m[g(\theta)])}{p}\right)\left\{1+O_m\bigg(\frac{r}{p^2}\bigg)\right\}.
E\Big[\prod_{i=1}^{p-1}|S_{s_i}-y_i |_+^{-2}| S_{s_{p-1}}-y_p |_+^{-1}\Big] \le C^{p-1}\prod_{i=1}^{p-1}|s_i-s_{i-1}|_+^{-1/2}\prod_{i=1}^p |y_i-y_{i-1}|_+^{-1}\,.
W^{1,2}_{\langle 0\rangle}(\Omega) :=\Big\{g\in W^{1,2}(\Omega): \int_\Omega g(x)\, dx=0\Big\}, W^{2,2}_{\langle 0\rangle}(\Omega) :=\Big\{g\in W^{2,2}(\Omega): \int_\Omega g(x)\, dx=0 \mbox{and}\int_\Omega \nabla g(x)\, dx=0\Big\},
\begin{array}{l}\underset{i=1}{\overset{m+1}{\sum}}v_{i}=0,v_{m+1}\in N_{C}(x^{*})\mbox{and}v_{i}\in N_{C_{i}}(x^{*})\mbox{for all}i\in\{1,\dots,m\}\quad\mbox{implies}v_{i}=0\mbox{for all}i\in\{1,\dots,m+1\}.\end{array}
f'(x)\ =\ \frac{2\sqrt{b}\left(2x+1\right)\left(b+a\right)+2b\sqrt{4bx^{2}+4ax^{2}+4ax+4bx+b}}{a\sqrt{4bx^{2}+4ax^{2}+4ax+4bx+b}}+1 \ >\ 0
F^{(n)}_D (a;\alpha_1,\dots,\alpha_n;c;x_1,\dots,x_n) = \sum_{\beta_1,\dots,\beta_n\ge 0}\frac{(a)_{\beta_1+\cdots\beta_n}(\alpha_1)_{\beta_1}(\alpha_n)_{\beta_n}x_1^{\beta_1}\cdots x_n^{\beta_n}}{(c)_{\beta_1+\cdots\beta_n}\beta_1!\cdots \beta_n!},
\begin{aligned}\int_{B}(u_p-u_{p+N})(\Delta u_p)^k\wedge(\Delta u_{p+N})^{m-k}\wedge\beta_n^m \leq \sigma(v,p, N)+\int_B u_p(\Delta u_p)^m\wedge\beta_n^{n-m}-\int_B u_{p+N}(\Delta u_{p+N})^m\wedge\beta_n^{n-m}\end{aligned}
[\begin{matrix}\hat{S}_{1}\\ \hat{S}_{2}^{*}\end{matrix}]=[\begin{matrix}h_{1}&-h_{2}\\h_{2}^{*}&h_{1}^{*}\end{matrix}]^{-1}[\begin{matrix}x_{1}\\x_{2}^{*}\end{matrix}]=\frac{1}{h_{1}h_{1}^{*}+h_{2}h_{2}^{*}}[\begin{matrix}h_{1}^{*}&h_{2}\\ -h_{2}^{*}&h_{1}\end{matrix}][\begin{matrix}x_{1}\\x_{2}^{*}\end{matrix}]
\begin{aligned}D^{\beta_1}_ay_1(t) = y_2(t), D^{\beta_2}_ay_2(t) = y_3(t), \vdots D^{\beta_{n-1}}_ay_{n-1}= y_n(t), D^{\beta_{n}}_ay_{n}= f(t,y_1,y_2,\cdots,y_{n-2},y_{n-1}),\end{aligned}
\phi_n(w,u)= \displaystyle\sum_{i=1}^{\infty}\eta_i \left( \int\limits_T \ell( w(t)- x^n_i(t)) d \mu(t)\right) + \sum_{i=1}^{\infty}\eta_i \ell (u- y^n_i ) = \displaystyle\int\limits_T \left( \sum_{i=1}^{\infty}\eta_i \ell( w(t)- x^n_i(t)) \right) d \mu(t) + \sum_{i=1}^{\infty}\eta_i \ell (u- y^n_i ).
R_\lambda R_{\omega_r}=c_{\omega_r}(t)\sum_{\substack{J \subseteq \{1,\ldots ,n\},\, |J|=r \lambda+\bar{e}_J\in \Lambda^{(n)}}}R_{\lambda +\bar{e}_J}\prod_{\substack{1\leq j < k\leq n j\in J, \, k\not\in J \lambda_j=\lambda_k}}\frac{1-t^{k-j+1}}{1-t^{k-j}}.
x_i = x_{\epsilon_1 + \epsilon_i}+ \sum_{\substack{3 \leq r \leq i 2 \leq s < r}}c_{isr}x_{\epsilon_s + \epsilon_r}, y_j = x_{\epsilon_1 - \epsilon_j}+ \sum_{\substack{j \leq r \leq 4 2 \leq s < r}}d_{jsr}x_{\epsilon_s - \epsilon_r}+ \sum_{\substack{r = 3, 4 2 \leq s < j}}e_{jsr}x_{\epsilon_s + \epsilon_r}
k_{tr}(\rho , \theta , \lambda)=\frac{\dot \rho \ddot \theta - \dot \theta \ddot \rho}{(\dot \rho^2 + \dot \theta^2)^{\frac{3}{2}}}= \frac{1}{h}\frac{1-g \cos (t + \frac{\pi}{2})}{(1 + g^2 -2g \cos (t + \frac{\pi}{2}))^{\frac{3}{2}}}
\lim_{j\rightarrow\infty}\lim_{m\rightarrow\infty}\max\{\left|\frac{1}{3^l}\right|^k G\left(\frac{x}{3^{k+1}},\frac{x}{3^{k+1}}\right),j\leq k<m+j\}=\lim_{j\to\infty}\left|\frac{1}{3^l}\right|^j G\left(\frac{x}{3^{j+1}},\frac{x}{3^{j+1}}\right) & =0.
\left( \begin{array}{ccc}2 x_{1}^{2}-1 2 x_{1}x_{2}2 x_{1}x_{3}2 x_{1}x_{2}2 x_{2}^{2}-1 2 x_{2}x_{3}2 x_{1}x_{3}2 x_{2}x_{3}2 x_{3}^{2}-1 \end{array}\right)
\theta \varphi\theta^{-1}= (\tau,\boldsymbol \varepsilon ) (\sigma, R) (\tau, \boldsymbol \varepsilon )^{-1}&= (\tau \sigma \tau^{-1}, R^{\tau^{-1}}) &= (\tau \sigma \tau^{-1}, (\varphi_{\tau 1},\varphi_{\tau 2}, \varphi_{\tau 3})) &=(\tau \sigma \tau^{-1}, (\alpha_1,\alpha_2,\alpha_3)).
f_{+}^{\prime}(\overline{x},u) =\lim_{t\rightarrow0+}\frac{\varphi(t)-\varphi(0)}{t}=\lim_{t\rightarrow0+}\sum_{n\geq1}\frac{\varphi_{n}(t)-\varphi_{n}(0)}{t}=\sum_{n\geq1}\lim_{t\rightarrow0+}\frac{\varphi_{n}(t)-\varphi_{n}(0)}{t}& =\sum_{n\geq1}f_{n+}^{\prime}(\overline{x},u).
\lim_{n \to +\infty}\frac{a_n^\alpha}{n L^*(a_n)}\log \inf_{z \in [x a_n, y a_n]}\P_z\left( \frac{S_j}{a_n}\in \left[ f(j/n) ,g(j/n) \right], j \leq n, E_n\right) = -C_* \int_0^1 \frac{ds}{(g(s)-f(s))^\alpha}.
D(p_{Y,S|U}(\cdot,\cdot|u)\|p_{Y,S})=C,u\in\mathcal{U}\mbox{with}p_U(u)>0, &D(p_{Y,S|U}(\cdot,\cdot|u)\|p_{Y,S})\leq C,u\in\mathcal{U}\mbox{with}p_U(u)=0;
[s, x_1][ x_2, x_3, x_4] = [s, x_1][ x_2, x_3] x_4 - [s, x_1]x_4 [ x_2, x_3] \equiv \ [s, x_1][ x_2, x_3] x_4 - x_4 [s, x_1] [ x_2, x_3] \pmod{I'}\equiv - [s, x_2][ x_1, x_3] x_4 + x_4 [s, x_2] [ x_1, x_3] \pmod{I'}\equiv \ - [s, x_2][ x_1, x_3] x_4 + [s, x_2] x_4 [ x_1, x_3] \pmod{I'}= - [s, x_2][ x_1, x_3, x_4] ,
\int_{\Sigma}\frac{h(p)- h(x)}{|x-p|^{N}}dx=\sqrt{1+\alpha^2}A_{m-2}A_{n-2}\int_0^\infty (1-r^{-\beta})I(r) r^{N-2}dr &=\sqrt{1+\alpha^2}A_{m-2}A_{n-2}\int_0^1 (r^{N-2}-r^{N-2-\beta}+r^s- r^{\beta+s})I(r)d r
R_{n}(x)- R_{n+1}(x) = \frac{x(x+\gamma+\delta+1)}{N(\gamma+1)}\hat R_{n}(x), -(n+1)(N-n+\delta) \hat R_{n}(x)+(N-n-1)(n+\gamma+2) \hat R_{n+1}(x) = N(\gamma+1)R_{n+1}(x).
{\mathcal{F}_{B}^{0 0}}={\frac{1}{2}}\oint{d}\hat{s}( \xi ) d \hat{s}( \xi^{\prime}){\hat{H}_{D}^{0 0}}( \xi ){\hat{G}_{k}}( \xi , \xi^{\prime}){\hat{H}_{D}^{0 0}}( \xi^{\prime}) .
D_i(G_k) = \begin{cases}\langle [y,x,\overset{i-1}{\ldots},x] \rangle D_{i+1}(G_k) \langle [y,x,\overset{i-3}{\ldots},x,y], [y,x,\overset{i-1}\ldots,x] \rangle D_{i+1}(G_k) \end{cases}
-\left \Vert Dy_{j}\right \Vert^{2}=-y_{j}^{\mathrm{H}}D^{\mathrm{T}}Dy_{j}& =y_{j}^{\mathrm{H}}\left( A^{\mathrm{T}}P+PA\right) y_{j}& =2\operatorname{Re}\left( s_{j}\right) y_{j}^{\mathrm{H}}Py_{j}& =0,
\frac{\varphi_x(w_1)}{\phi(w_1)}\quad=2^{p/2}\frac{\Gamma(d/2)}{\Gamma((d-p)/2)}\left(||w_1||^2\right)^{-p/2}(1-\|x\|^2/||w_1||^2)^{\frac{d-p-2}{2}}e^{\frac{\|x\|^2}{2}}\quad=\frac{(d/2)^{-p/2}\Gamma(d/2)}{\Gamma((d-p)/2)}\left(\det S_1\right)^{-p/2}\left( 1-\frac{\|x\|^2}{d}\iota' S_1^{-1}\iota\right)^{\frac{d-p-2}{2}}e^{\frac{\|x\|^2}{2}},
(a_1, b_1).(b_2, a_2) = (a_3, b_3).(b_4, a_4) \,, (a_1, b_1).(a_2, -b_2) = (a_3, b_3).(a_4, -b_4), & \left|\left|(a_1, b_1)\right|\right|^2 - \left|\left|(a_3, b_3)\right|\right|^2 = 1\,, \left|\left|(a_4, b_4)\right|\right|^2 - \left|\left|(a_2, b_2)\right|\right|^2 = 1\,,
\frac{d^2}{dt^2}\lambda(ug+h,dK) = \int_M \frac{n-2}{2}u\triangle u-\frac{u^2}{2}|H|^2+u\langle N,H \rangle-\frac{1}{6}|N|^2-\frac{1}{2}|\nabla v|^2 dV_g &\kern2em+\int_M \langle \frac{1}{2}\triangle h,h \rangle- \langle\mathring{R}h,h \rangle-\mu |h|^2+\frac{3}{2}|\Lambda|^2 dV_g
AH_0 = c_H\, K_H \, \int{d^2 a_1 d^2 a_2 d^2 a_3 \,|a_{45}\,a_{46}\,a_{56}|^2 \over T^5 \prod_{i<j}^6 |a_{ij}|^2}\prod_{l=1}^4{d^2 z_l (r-z_l)\over y(z_l)}I(r) \bar{F}(\bar{a}, \bar{z}),
c_{1}t^{-\frac{n}{2}}\exp \bigg( -b_{1}\frac{|x-y|^{2}}{t}\bigg) \le k_{t}^{A}(x,y) = k_{t}^{A}(y,x) \le c_{2}t^{-\frac{n}{2}}\exp \bigg( -b_{2}\frac{|x-y|^{2}}{t}\bigg),
\left(\begin{array}{c}p'_{0}p'_{1}\end{array}\right)={1\over\sqrt{1-B_{01}^{2}}}\left(\begin{array}{cc}1 0 -B_{01}1-B_{01}^{2}\end{array}\right) \left(\begin{array}{c}p_{0}-B_{01}p_{1}p_{1}\end{array}\right) =\left(\begin{array}{c}\gamma p_{0}-\gamma vp_{1}-v\gamma p_{0}+\gamma p_{1}\end{array}\right)
\psi_{1 2}( p_{3 0}) = \,{\frac{\Lambda_{1 2}^{+ +}}{( P_{0}- S ) - ( p_{3 0}- h_{3}) + i \epsilon}}\, \left[ \, V_{1 2}\psi_{1 2}( p_{3 0}) + V_{2 3}\psi_{2 3}( h_{1}) + V_{3 1}\psi_{3 1}( h_{2}) \, \right]
\psi_{n,k}(z)=\begin{cases}z\int_{a_{k-1}}^{b_{k-1}}\frac{\psi_{n,k-1}(\tau)}{z-\tau}\,d\sigma_{n,k-1}(\tau), \ell<k, begin{align*}1em]\int_{a_{k-1}}^{b_{k-1}}\frac{\psi_{n,k-1}(\tau)}{z-\tau}\,d\sigma_{n,k-1}(\tau), k\leq \ell.\end{cases}
e^{\frac43 \phi}\simeq \frac{\left( 3\,r_3^2 +10\, r_3\, r_4 +3\,r_4^2 \right)^2\, \left( 5\,r_3^2 +6\, r_3\,r_4 +5\,r_4^2\right)}{128\,\left( r_3 +r_4 \right)}\, \, \sqrt{u_0 +2\,\kappa\,|u|^2}\;\kappa\, x + O(x^2)\,,
c^nI_2(b) \leq c^n\cdot\mathfrak{c}\cdot\left(\frac{1}{a+b}+\frac{1}{c+b}\right) &\cdot\frac{(ab)^{n+\epsilon}}{(a+b)^{3n+2\epsilon}}\cdot\frac{(bc)^{n+\epsilon}}{[\theta-\omega(a+4b/3)]^{3n+2\epsilon}}\cdot\Xi b\cdot(a+b)^n\cdot\nu(b).
\mathcal{K}_+(x,k) = \begin{cases}T(k) e^{ikx}k\geq0 e^{ikx}+ R_+(-k)e^{-ikx}k<0,\end{cases}\mathcal{K}_-(x,k) = \begin{cases}e^{ikx}+ R_-(k) e^{-ikx}k\geq0 T(-k) e^{ikx}k<0,\end{cases}
D_\Omega^*S(\bar x|\bar y)(\bar y^*):=w^*\mbox{-}\limsup\limits_{\stackrel{\stackrel{(x,y)\stackrel{\operatorname{gph}S|_\Omega}{\longrightarrow}(\bar x,\bar y)}{y^*\longrightarrow\bar y^*}}{\varepsilon\downarrow 0}}\widehat D_{\Omega,\varepsilon}^*S(x|y)(y^*).
( f \star g ) ( p^{( 3 )}) = \frac{1}{( 2 \pi )^{1 3}}\int d p^{( 1 )}\, d p^{( 2 )}\, \delta^{2 6}( p^{( 1 )}+ p^{( 2 )}- p^{( 3 )}) f ( p^{( 1 )}) g ( p^{( 2 )})
x_{1\,2}^-(1) =[\,x_{2}^-(0),\, x_{1}^-(1)\,]_{s^{3}}, x_{\theta}^-(1) =[\,x_1^-(0),\,x_{2}^-(0),\, x_{1}^-(1)\,]_{(s^{3},\,r^{-2}s^{-1})}.
L^{\pm}_{SCS}=STr\int_X d^3x\varepsilon^{ijk}\bigg({{\bf A}^{\pm}}_i\partial_j{{\bf A}^{\pm}}_k+\frac{2}{3}{{\bf A}^{\pm}}_i{{\bf A}^{\pm}}_j{{\bf A}^{\pm}}_k \bigg),
M_{U}^{\scriptstyle \mathrm{m i n}}= \frac{M_{p}\, g_{U}}{{2 \pi}\left| \eta \left( i \right) \right|^{4}}\times \frac{1}{\sqrt{1 + g_{U}^{2}\frac{Y ( 1 , 1 )}{1 6 \pi^{2}}}}\ .
var(\overline{y})=m_{x}^{2}var(r)=\frac{N-n}{N}\frac{\sum_{i=1}^{n}(y_{i}-rx_{i})^{2}}{n-1}=\frac{N-n}{N}\frac{(s_{y}^{2}+r^{2}s_{x}^{2}-2rs_{xy})}{n}
\left[ \begin{array}{c|c|c|c}\bar{b}_1 \bar{b}_2 \bar{b}_3 \bar{b}_4 \end{array}\right] \begin{bmatrix}c_i c_i^2 c_i^3 c_i^4 \end{bmatrix}= \begin{bmatrix}a_{i1}a_{i2}a_{i3}a_{i4}\end{bmatrix}
a_{1_\circ}\otimes (a_{2_\circ})_{1_\cdot}\otimes (a_{2_\circ})_{2_\cdot}=(a_{1\cdot})_{1_\circ}S(a_{2\cdot})(a_{3\cdot})_{1_\circ}\otimes (a_{1\cdot})_{2_{\circ}}\otimes(a_{3\cdot})_{2_\circ}
\sqrt{\frac{1}{2 \pi \hbar \epsilon ( 1 + f(q_j) \Delta q_j )}}\;\exp \left [ \frac{( \Delta q_j )^2}{2 \hbar \epsilon ( 1 + f(q_j)\Delta q_j)}\right ] \, \approx \, \sqrt{\frac{1}{2 \pi \hbar \epsilon}}\; \exp \left \{\frac{\epsilon}{\hbar}\left [\frac{1}{2}{\dot{q}}^2 + \frac{1}{2}\hbar f(q) \dot{q}\right ] \right \}\,,
\int_{\Sigma}hJf-fJh =-\int_{\partial \Sigma}\phi (fD_{\eta}h-hD_{\eta}f), S(u,v) = \int_\Sigma \phi \nabla f\nabla h-mfh-\alpha \int_{\partial \Sigma}\phi fh = -\int_\Sigma u Jv+\int_{\partial \Sigma}(D_\eta v-\alpha v) u\phi.
(H^\omega - z)^{-1}= (\tilde{H}^\omega-z)^{-1}+ (\tilde{H}^\omega-z)^{-1}K (\tilde{H}^\omega-z)^{-1}+ (H^\omega - z)^{-1}K (\tilde{H}^\omega-z)^{-1}K (\tilde{H}^\omega-z)^{-1}.
\Delta_n^t a^j :=\int_t^{t+\Delta_n}\int_t^u\frac d{du}\langle \alpha_s,\mathcal S(u-s)^*e_j\rangle dsdu \qquad\qquad+ \int_t^{t+\Delta_n}\int_t^u \langle \sigma_s, \mathcal A^*\mathcal S(u-s)^* e_j\rangle dW_sdu.
\mathbf \Phi(\mathbf V)(t) = \mathbf S(t-S) \mathbf U(S) + i \int_{S \wedge \tau_R}^{t \wedge \tau_R}\mathbf S(t-s) \mathbf N(\mathbf u(s)) \, ds + i \int_{S}^t \mathbf S(t-s) \mathbf M([\mathbf u,\mathbf V](s)) \, dW(s),
\kappa^2_{\mathsf{B}-\mathsf{F}+\mathsf{f}}-\kappa_{\mathsf{B}}^2=\sum\limits_{\mathsf{g}=1}^{\mathsf{F}-\mathsf{f}}\big[\kappa_{\mathsf{B}-\mathsf{g}}^2-\kappa^2_{\mathsf{B}-\mathsf{g}+1}\big]\leq \frac{\phi(\mathsf{F}-\mathsf{f})}{\mathsf{F}}\,\big[\kappa^{2}_{\mathsf{A}}-\kappa^{2}_{\mathsf{B}}\big]\,.
\sigma_{E}\longleftrightarrow ( - 1 )^{\sum J + j}\: \: \left\{\begin{array}{c c c}{J_{1}}&{j}&{J_{2}}\\{J_{3}}&{J_{4}}&{J_{5}}\\ \end{array}\right\}\doteq ( - 1 )^{\sum J + j}\: \{6 j \}( \sigma_{E}) ,
|\kappa_{\gamma_{j_l}}(\cdot,\tau)|^2\leq \frac{\bar c_0}{2 (-\tau)}\leq\begin{cases}\frac{\bar c_0}{2 (-\tau_0)}I_{j_l}\times [\tau_2,\tau_0] \tau_2<\tau_0, \frac{\bar c_0}{2 (-\tau_2)}I_{j_l}\times [\tau_0,\tau_2] \tau_0<\tau_2, \end{cases}
\begin{array}{c}p(x) + q(x) = \left(p(x) + q(x)\right)^{-}\cdot \left(p(x) + q(x)\right)^{+}; p(x) - q(x) = \left(p(x) - q(x)\right)^{-}\cdot \left(p(x) - q(x)\right)^{+}\end{array}
\begin{aligned}C_N^\pm \sim N^{N/2}\int_{\mathbb{R}^{2N}}\exp\Big( - \beta E_N^\pm - \frac{\beta}{2}\sum_{j=1}^N[ \frac{\tilde{p}_j^2}{N}+ \lambda_j^\pm(b_j^\pm)^2] \Big)d\tilde{p}_1\dots d\tilde{p}_N db_1^\pm\dots db_N^\pm \end{aligned}
\int_{-\infty}^{\infty}p_{_1}(y) dy = \frac{2}{\sqrt{2 \pi}\sigma_w}e^{\rho_0 +1}\int_{0}^{\infty}e^{-\left(\frac{1}{2 \sigma_w^2}+\rho_1\right) y^2}dy &=\frac{e^{-\rho_0 -1}}{\sqrt{1 + 2\rho_1 \sigma_w^2}},
\boldsymbol{\delta}_\emptyset^\star\mathbf{W}_{\mathbf{g}}^\star\mathbf{X}\mathbf{W}_{\mathbf{g}}\boldsymbol{\delta}_\emptyset= \boldsymbol{\delta}_\emptyset^\star\mathbf{W}_{\mathbf{g}^+}^\star\mathbf{X}\mathbf{W}_{\mathbf{g}^+}\boldsymbol{\delta}_\emptyset \equiv \boldsymbol{F}_\nu\mathbf{X}\boldsymbol{F}_\nu^\star,
\mu-{x}\mu\overline{x}+c_{1}{\bf i}+c_{2}{\bf j}+c_{3}{\bf k}+d_{2}{\bf j}+d_{3}{\bf k}=0, &-x\mu\overline{w}+c_{0}{\bf i}+c_{2}{\bf k}-c_{3}{\bf j}+d_{1}=0, &\mu -{w}\mu \overline{w}-c_{1}{\bf i}-c_{2}{\bf j}-c_{3}{\bf k}+d_{2}{\bf j}+d_{3}{\bf k}=0.
{}_4F_3\bigg[\begin{matrix}-n \alpha \beta \gamma &1 \delta \epsilon\end{matrix}\bigg|\,1\bigg]=\frac{(1-\alpha)_n(\epsilon-\alpha)_n}{n!\cdot(\epsilon)_n}\cdot{}_4F_3\bigg[\begin{matrix}-n \alpha \delta-\beta \delta-\gamma &\delta \alpha-n 1+\alpha-n-\epsilon\end{matrix}\bigg|\,1\bigg]
\frac{\kappa_{j}+ i k_{j}}{\kappa_{j}- i k_{j}}= - \frac{A_{a}( \bar{\sigma}_{N}\sigma^{\prime \prime}q_{j}^{\prime \prime}, i d )}{A_{a}( \sigma^{\prime \prime}q_{j}^{\prime \prime}, i d )}e^{- i k_{j}L}.
\widehat{N}_L(x) := \sum_{z \notin [x- 2 \ell_L, x+ 2\ell_L]}\; \sum_{i \le N(z,0)}\mathbf{1}_{\{\exists\; s\in [0,L^\alpha] \colon\, S^{z,i}_s \in [x-\ell_L, x+\ell_L]\}}
\Phi_h\bigg( s_h \bigg( \sum_{j=0}^{\rho(X)}x_j Q_j(y_1,\dots,y_{\rho(X)}) \bigg)^h \bigg) = s_h \Phi_h\bigg( \bigg( \sum_{j=0}^{\rho(X)}Q_j(y_1,\dots,y_{\rho(X)}) x_j \bigg)^h \bigg) &= s_h \bigg( \sum_{i=0}^{\rho(X)}\sum_{j=0}^{\rho(X)}Q_i(y_0,\dots,x_{\rho(X)}) Q_j(y_0,\dots,y_{\rho(X)}) z_{ij}\bigg)^{h/2}
\begin{cases}\nabla\lambda_{i}(u)\cdot r_{i}(u)<0 \text{if $j$ is even and$\omega_{i}^{j}[u]<0<\omega_{i}^{j+1}[u]$,}\nabla\lambda_{i}(u)\cdot r_{i}(u)>0 \text{if $j$ is odd and$\omega_{i}^{j}[u]<0<\omega_{i}^{j+1}[u]$}. \end{cases}
(1_1\otimes \vartheta)\circ(\epsilon\otimes 1_1)(v_b)= (1\otimes \vartheta)(\sum\limits_{a=-n}^nv_a\otimes v_{-a}\otimes v_b) = \sum\limits_{a=-n}^n(-1)^{[a]}v_a\otimes \epsilon(v_{-a}\otimes v_b) = \sum\limits_{a=-n}^n(-1)^{[a]}v_a\otimes \delta_{a,b}(-1)^{[-a]}= -v_b\otimes 1=-1_1(v_b).
\rho_i =\Big (1 -\sigma + \frac{\sigma}{2^{i}}\Big)R, \bar{\rho}_{i}= \frac{\rho_i + \rho_{i+1}}{2}, \xi_i (y)= \zeta\left(\frac{2^{i+1}}{\sigma R}(|y|-\rho_{i+1}) \right),
s = K - \sqrt{\frac{K}{\lambda/\mu}}+ \frac{2-K}{\lambda/\mu}+ O \left(\left(\frac{\lambda}{\mu}\right)^{-3/2}\right) P_b = 1 - \frac{K-1}{\lambda/\mu}+ \frac{2(K-1)}{\sqrt{K}}\left(\frac{\lambda}{\mu}\right)^{-3/2}+ O \left(\left(\frac{\lambda}{\mu}\right)^{-2}\right).
D^{2r+3}\Bigl(\frac{L_{r+1}(\lambda)}{H_\lambda}\Bigr) = | \lambda | D^{2r+2}\Bigl(\frac{L_{r}(\lambda)}{H_\lambda}\Bigr) +(2r+2)D^{2r+1}\Bigl(\frac{L_{r}(\lambda)}{H_\lambda}\Bigr) - D^-D^{2r+2}\Bigl(\frac{L_{r}(\lambda)}{H_\lambda}\Bigr) =0.
{\hat{\Pi}}_{2}( \vec{x}, t ) = 3 D - i ({\dot{\hat{\phi}}}_{2}( \vec{x}, t ) + \frac{1}{2 =}\hat{D_{{\bar{z}}_{1}}\phi_{1}}( \vec{x}, t ) + \hat{D_{3}\phi_{2}}( \vec{x}, t ) ) ,
x_A= (1,0)\wedge(1,1)\otimes(0,1)-(1,0)\wedge(0,2)\otimes(1,0)+(2,0)\wedge(0,2)\otimes(0,0) - (2,0)\wedge(0,1)\otimes(0,1)+(1,1)\wedge(0,1)\otimes(1,0)-(1,1)\wedge(1,1)\otimes(0,0).
\frac{1}{\left(\sqrt{2\pi}\right)^n}\int_{\mathbb{R}^n}e^{-\mathbf{x^TDx}-\mathbf{b^Tx}}\mathrm{d}\mathbf{x}=\frac{1}{\sqrt{2^n\det \mathbf{D}}}e^{\frac{\mathbf{b^TD^{-1}b}}{4}}.
\Sigma_{\mu \nu}\rightarrow \Sigma_{\mu \nu \rho}={\cal A}_{\mu \nu \rho}^{F}- 8{\cal A}_{\mu \nu \rho}^{V}, \qquad \Xi_{\mu \nu}\rightarrow \Xi_{\mu \nu \rho}={\cal A}_{\mu \nu \rho}^{F}+{\frac{1 6}{5}}{\cal A}_{\mu \nu \rho}^{V},
Q_j (x_3) = x_2^{2^{j-1}}Q_{j-1}x_3+x_3^{2^{j-1}}Q_{j-2}x_3, Q_j (x_4) = x_3 x_{4}^{2^{j-1}}+x_2^{2^{j-1}}Q_{j-1}x_4+x_3^{2^{j-1}}Q_{j-2}x_4.
\bar{a}_{d , 1}= r_{d}{\frac{\kappa^{2}}{2}}\left[ 1 + \rho^{2}\left( \frac 1 2 R_{i i}- \frac 1 2 R_{i j i j}- \frac 1 6 R - 2 m^{2}\right) + O ( \rho^{4}) \right] ,
\frac{1}{2}\Delta|\nabla^2 W|^{2}= |\nabla^{3}W|^{2}+ \langle \nabla^2 W, \nabla \Delta \nabla W \rangle +\frac{R}{4}|\nabla^2 W|^{2}+ 8 W_{ijkl,tr}W_{pjkl,ts}R_{pirs}+2W_{ijkl, tr}W_{ijkl, ps}R_{ptrs}\,,
X_{\langle Fz,\bar{z}\rangle}=(0,-(\sigma_{j_b}\partial_{x_b}\langle Fz,\bar{z}\rangle)_{1\leq b\leq n},-\mathbf{i}(\sigma_j\partial_{\bar{z}_j}\langle Fz,\bar{z}\rangle)_{j\in\mathbb{Z}_*},\mathbf{i}(\sigma_j\partial_{z_j}\langle Fz,\bar{z}\rangle)_{j\in\mathbb{Z}_*})^T
\sum_{s_i \in \left\{\frac{\dim V_i}{2}-1, \frac{\dim V_i}{2}-2,0\right\}}\mathrm{Res}_{s=s_i}\frac{e^{Ts}Z_{r_i}(\mathcal{F}_2(f),s+2-\tfrac{\dim V_i}{2})}{s}+o_f(1).
(T^1S^1f)(h)(\nu) =\sum_i\langle L^{**}(f'(g_i)), \nu_i\rangle =\sum_i\langle f'(g_i), \nu_iL \rangle &=\sum_i\sum_j \langle \pi^{**}(f(h_j)), (\nu_iL)_j \rangle &\quad\mbox{(Here, $(\nu_iL)_j:=(\nu_iL)|_{X_{g_i, h_j}}$, $X_{g_i, h_j}=\{x: c(g_i^{-1}, x)=h_j^{-1}\}.$)}&=\sum_i\sum_j \langle f(h_j), (\nu_iL)_j\pi\rangle.
G_1 (x,y,z;q):= \frac{j(yz;q^2)}{j(y;q)j(z;q)}\frac{J_1^4}{J_2^2}\sum_{k\in \mathbb{Z}}\frac{(-1)^kq^{k^2}(yz)^k}{1+q^{2k}x}&+ \frac{j(xy;q^2)}{j(x;q)j(y;q)}\frac{J_1^4}{J_2^2}\sum_{k\in \mathbb{Z}}\frac{(-1)^kq^{k^2}(xy)^k}{1+q^{2k}z}+ \frac{j(xz;q^2)}{j(x;q)j(z;q)}\frac{J_1^4}{J_2^2}\sum_{k\in \mathbb{Z}}\frac{(-1)^kq^{k^2}(xz)^k}{1+q^{2k}y},
\dfrac{w\!:\!1\!:\!\lozenge\phi\!=\!\frac{r}{m+1}}{w\mathsf{R}w'';w''\!:\!1\!:\!\phi\!=\!\frac{r}{m+1}}\dfrac{w\!:\!1\!:\!\lozenge\phi\!=\!\frac{r}{m+1};w\mathsf{R}w'}{w'\!:\!1\!:\!\phi\!=\!0\mid\ldots\mid w'\!:\!1\!:\!\phi\!=\!\frac{r-1}{m+1}}
\varTheta_p(\eta, 1,\chi) \ll X^{-1}(X^{1/2}\log X)^{(3+\eta)/2}\max_{t \le 2(X\log X)^{1/2}}\left|S_p(\chi;t)\right| + 1 & = X^{\eta/2}(\log X)^{(3+\eta)/2}\max_{t \le 2(X\log X)^{1/2}}\left|S_p(\chi;t)\right|
\begin{array}{c}{D^{B}H^{A}= 0 , \qquad \qquad D^{B}D^{A}\Phi_{A}= [ H^{A}, \Phi_{B}] ,}\\{D^{A}K^{B}= 0 , \qquad \qquad D^{A}D^{B}\Phi_{B}= [ K^{B}, \Phi_{A}] .}\\ \end{array}
\langle f, \vartheta_k P_{k,m}\rangle = \langle f, R_k P_{k,m}-\frac{k}{12}E_2^{*}P_{k, m}\rangle = \langle f, R_k P_{k,m}\rangle -\frac{k}{12}\langle f, E_2^{*}P_{k, m}\rangle &= -\frac{k}{12}\langle f, E_2^{*}P_{k, m}\rangle.
\lim_{x\uparrow 0}\frac{\partial}{\partial x}\Upsilon(s,x,u) =-\partial_x \Phi(s,0,u) \left(\frac{1}{2}\int_{s}^{u}\phi'(\omega)^2 d\omega\right) &=\frac{1}{\sqrt{2\pi(u-s)^3}}\mathbb{E}\left[ \left (-\int_{s}^{u}\phi''(z)\textbf{e}(z)dz \right)\right]
|b_{n,r,s}| \le C\sum_{j, k=1}^n(|j+r|+|k+s|)^{-\beta}L(|j+r|+|k+s|) &\le 4C\sum_{j, k=1}^{2n}(j+k)^{-\beta}L(j+k) &\ll \sum_{j=1}^{2n}j^{1-\beta}\max_{1\le k\le 2n}L(j+k) &\propto n^{2-\beta}\max_{1\le k\le 2n}L(2n+k)\propto n^{2-\beta}L(n).
\int_0^1 2\pi r\int_{\|\varphi\|=r}\begin{vmatrix}\varphi(z) \end{vmatrix}^{p-2}\mathrm{d}\omega_r\mathrm{d}r=\int_0^1 2\pi r\int_0^{2\pi}\begin{vmatrix}f'(re^{i\theta}) \end{vmatrix}^t\mathrm{d}\theta\mathrm{d}r
0=\nabla_0 W_{ijkl}=\kappa_i\Psi_{jkl}-\kappa_j\Psi_{ikl}+\kappa_k\Psi_{lij}-\kappa_l\Psi_{kij}\;; &0=\nabla_0 W_{01ij}=\kappa^m\Psi_{mij}-\kappa_i\Psi_j+\kappa_j\Psi_i\;; &0=\nabla_0 W_{0i1j}=-\kappa^m\Psi_{jmi}-\kappa_i\Psi_j\;.
I_2 ={2\pi^{1 \over 2}\over a\Gamma (s)}\sum_{n=1}^{\infty}n^2 \sum_{j=1}^{\infty}\cosh \left({2\pi jb \over a}\right) \int_0^{\infty}dt\t^{s -{3 \over 2}}\exp \Bigl[ -(c + \alpha n^2)\,t - \frac{j^2\pi^2}{a^2t}\Bigr]
{\cal L}=e\left({\frac{1}{2}}R+g^{2}V-{\frac{1}{4}}G_{IJ}F_{\mu \nu}{}^{I}F^{\mu \nu J}-{\frac{1}{2}}{\cal G}_{ij}\partial_{\mu}\phi^{i}\partial^{\mu}\phi^{j}+{\frac{e^{-1}}{48}}\epsilon^{\mu \nu \rho\sigma \lambda}C_{IJK}F_{\mu \nu}^{I}F_{\rho \sigma}^{J}A_{\lambda}^{K}\,\right) ,
\| F\|_{s}^2 = \frac{1}{8\pi^2}\iint_{[-\pi,\pi]^2}\left| \frac{F\big(s(e^{i\theta})\big) - F\big(s(e^{i\phi})\big)}{e^{i\theta}-e^{i\phi}}\right|^2 d\theta d\phi \ .
\sum_{k=0}^{m}\frac{(aq;q^2)_k (q/a;q^2)_k (q;q^2)_{2k}}{(q^2;q^2)_{2k}(aq^6;q^6)_k (q^6/a;q^6)_k}[8k+1]q^{2k^2}\equiv0\equiv q^{-(n-1)/2}[n]\biggl(\frac{-3}{n}\biggr) \pmod{[n]}
\begin{array}{l l l}{R}&{=}&{8 e^{- 2 \rho}\partial_{+}\partial_{-}\rho = 8 e^{- 2 \rho}{\left( 1 -{\frac{1}{2 4}}N e^{2 \Phi}\right)}^{- 1}\partial_{-}\Sigma}\\{\}&{=}&{4{\left( 1 -{\frac{1}{1 2}}N e^{2 \Phi}\right)}^{- 1}\left( 4 e^{- 2 \rho}\partial_{-}\Phi \Sigma + \lambda^{2}\right) ,}\\ \end{array}
\int_{U_+}|x-y|^{2-n}\partial_\nu \hat\zeta_\delta(y)\, dy\leq C\int_{U_+}|x-y|^{2-n}\frac{\delta^2}{(1+|y|)^{n-3}}\, dy= C\int_{U_+}|x-y|^{2-n}\frac{\delta^2}{(1+|y|)^{1+(n-4)}}\, dy.
{F}_w|_\mu=\begin{cases}s_i({F}_w|_{\nu})+\alpha_i\cdot s_i({F}_{s_iw}|_{\nu}) \mbox{if}\ \ell(s_iw)<\ell(w) , s_i({F}_w|_{\nu}) \mbox{if}\ \ell(s_iw)> \ell(w).\end{cases}
i \hbar{\frac{\partial \psi ( \vec{x}, t )}{\partial t}}= \hat{H}\psi ( \vec{x}, t ) = \left[ \hat{\vec{p}}\sp 2 -{\frac{l^{2}}{r^{2}}}+{\frac{\overline{{l}}\sp 2}{r^{2}}}\right] \Psi ( \vec{x}, t )
\limsup_{K_{0}\rightarrow +\infty}\left(\limsup_{j\rightarrow +\infty}J_{0, j}\right) =\limsup_{K_{0}\rightarrow +\infty}\left(\limsup_{j\rightarrow +\infty}\int_{A_{j,K_{0}}, \sigma_{0}}(v_j - K_{0})_{+}^{\gamma^*}dx \right)=0.
\frac{\partial \overline{{H}}_{0}}{\partial q_{n , \alpha_{n}}}\left( q_{n , \alpha_{n}}, p_{n , m_{n}- 1}\right) = - \frac{\partial L_{0}}{\partial q_{n , \alpha_{n}}}\left( q_{n , \alpha_{n}}, \dot{q}_{n , m_{n}- 1}( q_{n , \alpha_{n}}, p_{n , m_{n}- 1}) \right) \ ,
\big((\hat P^{[2]}(y))_{[n+1]}\big)^\top(\omega^{[2]})_{[n+1]}(H_{[n+1]})^{-1}(P^{[1]}(x))_{[n+1]}= \big((P^{[2]}(y))_{[n+1]}\big)^\top(H_{[n+1]})^{-1}(P^{[1]}(x))_{[n+1]}= K_{n}(x,y).
\min_{\mathfrak{E}}s_g= 8\pi\left(1+\sum a\right)\left(\sum a+\sum a^3+3\sum ab+6\sum a^2b+\sum a^3b\right)\Big/ begin{align*}0.1cm] \left(\sum a^2+\sum a^4+\sum ab+4\sum a^2b+4\sum a^3b+2\sum a^4b+3\sum a^2b^2+4\sum a^3b^2+\sum a^4b^2\right).
g_{1,[y,1]}(x) = \begin{cases}g_{1}(y)\cdot\frac{x}{y}, \forall x \in [0,y) g_{1}(x), \forall x \in [y,1],\end{cases}g_{2,[0,x]}(y) = \begin{cases}g_{2}(y), \forall y \in [0,x] g_{2}(x)\cdot\frac{1-y}{1-x}, \forall y \in (x,1],\end{cases}
\sup_{x\in\Omega}\int_{\Omega\,\cap\,B_{H_0}(x,1)}F(|z( y, t)|)\,dy\le M+CL^{1+\delta}\int_0^t s^{\sigma-1-\frac{\delta N}{2}}\,ds+CL\int_0^t s^{-\sigma}\,ds\le 2M
\{f_1, \dots, f_n\}=\det\left(\begin{array}{ccc}f_1 \dots f_n \frac{\partial f_1}{\partial x_1}\dots \frac{\partial f_n}{\partial x_1}\dots \dots \dots \frac{\partial f_1}{\partial x_{n-1}}\dots \frac{\partial f_n}{\partial x_{n-1}}\end{array}\right).
\prod_{i \notin I, j \in I \atop i < j}(-q_{ij}) \Phi_{I}=\Biggl\{\overrightarrow{\prod_{j \in I}}\Biggl\{\overrightarrow{\prod_{i \in I_{+}^{c}\atop i < j}}(\beta-u_{ij}) \Biggr\}\Biggr\}\Biggl\{\overrightarrow{\prod_{j \in I_{+}^{c}}}\Biggl\{\overrightarrow{\prod_{i \in I \atop i < j}}u_{ij}\Biggr\}\Biggr\}:= \widetilde{\Phi_I}\in 3HT_n.
\begin{aligned}\mathbb{E}^{u_1,u_2}[\Phi_1(x(T))\Gamma_1(T)]=\mathbb{E}^{u_1,u_2}\big[\int_0^T-\Gamma_1(t)l_1(t)dt]+\sum_{j=1}^2\mathbb{E}^{u_1,u_2}\big[\int_0^TQ_{j1}(t)\big(h_{jx}(t)x_1^1(t)+h_{jv_1}(t)v_1(t)\big)\big]dt.\end{aligned}
= \exp\left\{\left( \sum_{g=1}^{\infty}\frac{v^g}{g^q}\right) \left( \sum_{h=1}^{\infty}\frac{w^h}{h^r}\right) \left( \sum_{i=1}^{\infty}\frac{x^i}{i^s}\right) \left( \sum_{j=1}^{\infty}\frac{y^j}{j^t}\right) \left( \sum_{k=1}^{\infty}\frac{z^k}{k^u}\right)\right\}.
\mathcal{I}_{1}^{N_0}= \mathfrak{Re}\Big[ \int \frac{1}{2}\overline{\widehat{U^1}(\xi)}\tilde{q}_{N_0}^1 (\xi-\eta, \eta)\widehat{U^1}(\xi-\eta) \widehat{U^2}(\eta) + \overline{\widehat{U^2}(\xi)}\widetilde{q}_{N_0}^2 (\xi-\eta, \eta) \widehat{U^1}(\xi-\eta) \widehat{U^1}(\eta)
BE=E_{x}[\sum_{k=1}^{K}L(C_{k},\hat{C}_{B}(x))P(C_{k}|x)]=E_{x}[\sum_{k=1,C_{k}\ne\hat{C}_{B}(x)}^{K}P(C_{k}|x)]=E_{x}[1-P(\hat{C}_{B}(x)|x)]