harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{lll}{\rm (a)~~}\{f,g\}_\kappa = - \{f,g\}_\kappa\,,{\rm antisymmetry\,,}{\rm (b)~~}\{af+bg,h\}_\kappa = a\{f,h\}_\kappa + b\{g,h\}_\kappa\,, ~~~{\rm linearity\,,}{\rm (c)~~}\{f,\{g,h\}_\kappa\}_\kappa + \{g,\{h,f\}_\kappa\}_\kappa +\{h,\{f,g\}_\kappa\}_\kappa = 0\,,{\rm Jacobi~identity}\end{array}
\frac{1}{|n|}\langle\gamma'',\gamma'\times n\rangle = \langle\gamma'', J^{90}_\gamma(\gamma') \rangle = \frac{1}{\kappa}\langle\gamma'', \nabla_{\gamma'}\gamma' \rangle = \frac{1}{\kappa}|\nabla_{\gamma'}\gamma'|^2 \; .
a_{0}+ \sum_{n=1}^{\infty}a_{n}\sum_{k=1}^{\infty}d_{n,k}\varepsilon^{k}= a_{0}+\sum_{k=1}^{\infty}\sum_{n=1}^{\infty}a_{n}d_{n,k}\varepsilon^{k}= (P\circ Q)(\varepsilon).
\frac{1}{a_{0}}+\frac{x}{a_{0}a_{1}}+\frac{x^{2}}{a_{0}a_{1}a_{2}}+\cdot\cdot\cdot+\frac{x^{n}}{a_{0}a_{1}a_{2}...a_{n}}=\frac{1}{a_{0}-}\frac{a_{0}x}{a_{1}+x-}\frac{a_{1}x}{a_{2}+x-}\cdot\cdot\cdot\frac{a_{n-1}x}{a_{n}+x}.
|a(n)| \leq \sqrt{\log k}\left(11 \sqrt{\sum_{m=1}^\ell \frac{|a(m)|^2}{m^{k-1}}}+ \frac{e^{18.72}(41.41)^{k/2}}{k^{\frac{k-1}{2}}}\cdot \left|\sum_{m=1}^\ell a(m) e^{-7.288m}\right| \right) \cdot d(n) n^{\frac{k-1}{2}},
\int\limits_{\mathbb{R}^{N}}\left\vert V_{0}\right\vert^{r}\frac{dx}{\left\vert x\right\vert^{s}}=d^{1+\frac{p-1}{p}r}\int\limits_{\mathbb{R}^{N}}\frac{\left\vert U_{0}\right\vert^{r}}{\left\vert x\right\vert^{N+sd-Nd}}dx
\frac{z\mathtt{g}_{a, \nu}'(z)}{\mathtt{g}_{a, \nu}(z)}=a (1-\nu)+ a^{a/2}\frac{z\;{}_{a}\mathtt{B}'_{2a-1, a \nu-a+1,1}(a^{a/2}z)}{{}_{a}\mathtt{B}_{2a-1, a \nu-a+1,1}(a^{a/2}z)}.
g_\infty = - \: 16 du^2 + u^2 d\theta^2 + \left[ 4^{1+\nu}c_1 c_\lambda^{-1-\nu}d\widehat{\sigma}^2 + 4^{1- \nu}c_1^{-1}c_\lambda^{-1+\nu}d\widehat{\delta}^2 \right].
\left( A^{\prime}e^{n A}\right)^{\prime}= e^{n A}\left( \frac{1}{6}T_{0}^{0}+ \left( \frac{n}{1 2}- \frac{1}{6}\right) T_{5}^{5}+ \left( n - 1 \right) \bar{\Lambda}e^{- 2 A}\right) .
a_k\leq Cb^{-\frac{s}{Q}}\bigg(\int_{2B_0}\Big(\sum_{j}g_j^q\Big)^{\frac{p}{q}}\,d\mu\bigg)^{\frac{s}{Q}}\sum_{n=k_0}^{k-1}2^{n(1-\frac{sp}{Q})}+C'r_0^{s}2^{k_0}.
-g^{-1}(\d g) g^{-1}\mathcal{L}_{Jv}g+g^{-1}(\mathcal{L}_{Jv}g)g^{-1}\d g = \d(g^{-1})\mathcal{L}_{Jv}g-\mathcal{L}_{Jv}(g^{-1})\d g = 0.
\begin{array}{rcl}\mathcal L_\partial(\theta_*(s \otimes p)) = \mathcal L_\partial(se_S\otimes p) = \partial(s)e_S\otimes p + s \mathcal L_\partial(e_S) \otimes p + s e_S \otimes \partial^\bullet(p) = \theta_*(\mathcal L_\partial(s \otimes p)) +\lambda\theta_*(s \otimes p)\,. \end{array}
dX^{(n)}_i(t)=\sqrt{2}|X_i^{(n)}(t)|d\beta^{(n)}_i(t)+\left[\left(2-2n-2\Re(s)\right)X_i^{(n)}(t)\right]dt+\frac{1}{2}dK_i^{(n),-}(t)-\frac{1}{2}dK_i^{(n),+}(t).
u(x,t)=\frac{4 a_2 xf\Big((a_2x+a_1)e^{a_2t}\Big)+g\Big((a_2x+a_1)e^{a_2t}\Big)}{(a_2x+a_1)^2}\exp\left(\frac{(a_2x+a_1)^2}{a_2}\right),
nR_1- n \epsilon_n & = H(W_1) - n \epsilon_n &= H(W_1|\Omega^{n}) - n \epsilon_n = I(W_1;y^{n}|\Omega^{n})+\underbrace{H(W_1|y^{n},\Omega^{n})}_{\leq n \epsilon_n}- n \epsilon_n &\leq I(W_1; y^{n}|\Omega^{n}) &= h(y^{n}|\Omega^{n}) - h(y^{n}|W_1, \Omega^{n})
( \kappa_{\tau}\frac{\partial}{\partial \kappa_{\tau}}+ \gamma_{\phi^{2}( \tau )}^{*}( 2 + t \frac{\partial}{\partial t}) ) \Gamma_{R ( \tau )}^{( 0 , 2 )}= ( \kappa_{\tau}^{- 2 \tau})^{\frac{\epsilon_{L}}{2}}B_{\tau}( u_{\tau}^{*}) ,
\tau_\pm^m \sim 2 \pi \frac{\prod_{j=2}^N \sqrt{\lambda_j^s}}{\prod_{j=1}^N \sqrt{\lambda_j^\pm}}\frac{(1-\frac{\beta E_N^\pm}{N})^{N-1}}{(1-\frac{\beta E_N^s}{N})^{N-1}}.
| g . s . > = | 0 , 0 , - 3 J , 0 > = \frac{1}{\sqrt{1 2}}( 2 | \uparrow \downarrow \uparrow \downarrow > + 2 | \downarrow \uparrow \downarrow \uparrow > - | \uparrow \uparrow \downarrow \downarrow > - | \uparrow \downarrow \downarrow \uparrow > - | \downarrow \downarrow \uparrow \uparrow > - | \downarrow \uparrow \uparrow \downarrow > ) \quad .
\sum_{2r_0+r_1=2d_1-d_2}{d_1+1 \choose r_0}{d_1+1-r_0 \choose r_1}\frac{(d_2+2)!}{2^{r_0+d_2-d_1+1}}{n_1 \choose d_1+1}{n_2 \choose d_2+2}.
B_s^{(\bf p)}\,{\bf x}=\left(\prod_{k=2}^{s-1}\left( A - \gamma_1 \lambda_k I\right)\right)\,\left( A - \gamma_1 \lambda_1 I\right){\bf x}=\prod_{k=2}^{s-1}\left( A - \gamma_1 \lambda_k I\right){\bf 0}={\bf 0}.
v^{\lambda}_{3,1}(t,r)= \frac{-1}{r}\int_{t}^{\infty}ds \int_{0}^{s-t}\frac{\rho d\rho}{(s-t)}\lambda''(s) \left(\frac{-1-\rho^{2}+r^{2}}{\sqrt{(1+\rho^{2}-r^{2})^{2}+4r^{2}}}+ F_{3}(r,\rho,\lambda(s))\right)
P_{2k}(m_r+1,n_1,m_1,\dots,m_{k-1},n_k) =n_kP_{2k-1}(m_r+1,n_1,m_1,\dots,n_{k-1},m_{k-1}) &\quad+P_{2k-2}(m_r+1,n_1,m_1,\dots,m_{k-2},n_{k-1}) &=n_k\left(1+m_r+\sum\limits_{i=1}^{k-1}m_im_{1,N_i}\right)+m_{1,N_{k-1}}&\stackrel{(\ref{relfan1})}{=}m_{1,N_k}.
H_{x}=L\frac{\partial T}{\partial y}^{TM}+\frac{1}{j\omega\mu}\frac{dL}{dz}\frac{\partial T}{\partial x}^{TE}=L\frac{\partial T}{\partial y}^{TM}-\frac{k_{z}}{\omega\mu}L\frac{\partial T}{\partial x}^{TE}(30)
\begin{aligned}-\iint \epsilon v_{sy}|g^\epsilon|^2 \leq \epsilon \Vert v_{sy}\Vert_{L^\infty}\Vert g^\epsilon\Vert_{L^2}^2\leq \epsilon L^2\Vert v_{sy}\Vert_{L^\infty}\Vert g^\epsilon_x\Vert_{L^2}^2.\end{aligned}
V^{p}=\bigoplus_{j\in p{\Bbb Z}_{p^n}}{\rm E}(\lambda^j)=\bigoplus_{e\in{\cal E}^p}{\Bbb C}e\quad\mbox{and}V^{*}=\bigoplus_{j\in{\Bbb Z}_{p^n}^*}{\rm E}(\lambda^j)=\bigoplus_{e\in{\cal E}^*}{\Bbb C}e
\frac{d\|\boldsymbol{w}(t)\|_p^p}{dt}=\left(\frac{\partial \|\boldsymbol{w}(t)\|_p^p}{\partial \boldsymbol{w}(t)^\ast}\right)^{\!H}\,\frac{d\boldsymbol{w}(t)}{dt}=:\boldsymbol{b}(t)^H\frac{d\boldsymbol{w}(t)}{dt}=0
{1 \over 2}(h_1-h_2)' =\left [{d' \over d}+{{12 /r^2 - (2{f'}_1^2 /f_1^2 +{f'}_K^2 / f_K^2 )}\over{f'/f + 6/r}}-{4 \over r^6 d}{{Q_K^2/f_K^2 - Q_5^2 / f_1^2}\over{f'_1 / f_1 - f'_K /f_K}}\right ] h_2.
V_{b}= \frac{e^{2}}{2 \pi}\frac{1}{{q^{+}}^{4}}\int_{0}^{q^{+}}\frac{d k^{+}}{q^{-}- \frac{m^{2}}{2 k^{+}}- \frac{m^{2}}{2 ( q^{+}- k^{+})}}\stackrel{q^{+}\rightarrow 0}{\longrightarrow}- \frac{e^{2}}{\pi}\frac{1}{6 m^{2}}\frac{1}{{q^{+}}^{2}}
\frac{dc}{dt}=J_{\rm flux}-J_{\rm pump}+J_{\rm leak}\tau_h\frac{dh}{dt}=\frac{k^2_2}{k^2_2+c^2}-h, \textrm{where}J_{\rm flux}=k_f\mu(p)h\frac{bk_1+c}{k_1+c},\,\,\,J_{\rm pump}=\frac{\gamma c}{k_{\gamma}+c},\,\,\,J_{\rm leak}=\beta,
{\Phi_5}({\rho}) =4\pi^2\,g_s\left({\frac{m^I d_I}{|\Gamma|}}+{g_s\over 2\pi}{\widehat C_{IJ}m^I m^J}\ln{\frac{\rho}{\rho_0}}\right)= 4\pi^2\,g_s\,{Q}({\rho})~,
\langle A^{-1}_{\varepsilon, \delta}\overline{\sigma_{j}}, \overline{\sigma_{s}}\rangle_{\omega_{\varepsilon}}= \langle \biggl( \sum_{\substack{1 \leq m \leq k m \neq j}}\frac{1 + \delta \lambda_{m}}{\varepsilon + \lambda_{m}}+ \frac{n-k}{\varepsilon}\biggl)^{-1}\overline{\sigma_{j}}, \overline{\sigma_{s}}\rangle_{\omega_{\varepsilon}}\leq \frac{\varepsilon}{n-k}\delta_{j s}
\lim_{t \to 0}\frac{\mathcal F(\rho_t^\xi)-\mathcal F(\rho)}t = \int_\Lambda \nabla [ L_f(\rho) ]\cdot \nabla\xi\;\!\mathrm du\quad\textrm{and}\lim_{t \to 0}\frac{\mathcal V(\rho_t^\xi)-\mathcal V(\rho)}t = \int_\Lambda \rho\nabla V\cdot \nabla\xi \;\!\mathrm du.
{}(S)= a'(0)^3[b'(0)+4c'(0)]+a'(0)^2[-2b'(0)^2+b'(0)c'(0)-8c'(0)^2] &+a'(0)[5b'(0)^2c'(0)+4c'(0)^3]-b'(0)^2c'(0)^2-b'(0)^4.
h_{L_x,v_1}(P_m) =O(1),h_{L_x,v_2}(P_m)=m+O(1), h_{L_x,v_i}(P_m)=\frac{m}{n-2}+O(1), i=3,\ldots, n, h_{L_y,v_1}(P_m) =m+O(1),h_{L_y,v_2}(P_m)=O(1), h_{L_y,v_i}(P_m)=\frac{m}{n-2}+O(1), i=3,\ldots, n, h_{L_z,v_1}(P_m) =m+O(1),h_{L_z,v_2}(P_m)=m+O(1), h_{L_z,v_i}(P_m)=O(1), i=3,\ldots, n.
\mathbb{P}_{M}^{N\,F}(x_{1},x_{2},\ldots,x_{M})=Z^{-1}\,\int_{-1}^{+1}\,P_{t}^{M}(x_{1},x_{2},\ldots ,x_{M})\,\frac{e^{-N\,F(t)\,/2}}{1-t^{2}}\,dt~
\frac{(d_{1}\cdot d_{2})^{2}}{c^{4}}+\frac{2(d_{1}\cdot d_{2})(d_{1}\cdot w)(d_{2}\cdot w)}{c^{6}}\le\frac{d_{1}^{2}\cdot d_{2}^{2}}{c^{4}}+\frac{d_{1}^{2}(d_{2}\cdot w)^{2}+d_{2}^{2}(d_{1}\cdot w)^{2}}{c^{6}}
f_Z-\partial_{\dot \vec v}L\cdot{\vec \xi}_Z={}\frac{\mu}{\lVert \vec r\rVert}\vec r\cdot\vec u -\vec v\cdot (\vec r\times \vec v)\times\vec u= ={}-\Bigl(\vec v\times (\vec r\times \vec v) -\frac{\mu}{\lVert \vec r\rVert}\vec r\Bigr)\cdot\vec u= N.
1= \psi \left( c^2 [a_1,b_1] \right)=x \sigma_1 x \sigma_1 y g z y^{-1}g^{-1}z^{-1}=x \ldotp \sigma_1 x \sigma_1^{-1}\ldotp A_{1,2}\ldotp y \ldotp g z g^{-1}\ldotp g y^{-1}g^{-1}\ldotp z^{-1}.
d_{TV}(X,\pi)\leq\sup_{h\in\mathcal{H}}\sup_{l\in\mathbb{Z}^+}|m_l(h)|\sum_{i=0}^\infty\mathbb{P}(X=i)\sum_{j=0}^\infty\left|\sum_{k=j+1}^\infty\left(Q_{i,k}-P_{i,k}\right)\right|.
A_n(t) := 2^{(6s+12)n}\cdot (4t+2n)^{\delta}\cdot \frac{\left(t+\frac{1}{4}\right)_{n}^{s+2}\left(t+\frac{3}{4}\right)_{n}^{s+2}}{(t)_{n+1}^{2s+4}}, B_n(t) := 2^{(3s+6)n}\cdot\frac{\left(t+\frac{3}{4}\right)_{n}^{s+2}}{(t)_{n+1}^{s+2}}.
< \! \mu(z,\bar z) \mu(0) \! >^2 =< \! \cos({\Phi(z,\bar z) \over 2}) \cos({\Phi(0) \over 2}) \! > ={1 \over 2}\left( G ({1 \over 2},-{1 \over 2}) +G ({1 \over 2},{1 \over 2}) \right) ,
\frac{C_{k-2-l,n}}{l!}= - 2 \sum_{m=2}^{k-1-l}a_{l+m+1}\biggl[(n-1) m_{(m-1,\dot{0}}+ \sum_{p=1}^{[(m-1)/2]}m_{(m-1-p,p,\dot{0})}\biggr] - \sum_{m=1}^{k-2-l}b_{l+m+1}m_{(m,\dot{0})}- \frac{n(n-1)}{k(k-1)}(k-l-2)(k+l+1) a_{l+2}- \frac{n}{k-1}(k-l-2) b_{l+1}, l=0, 1, \ldots, k-3.
C_{\sigma}= C_{\varphi}^{*}M_{1/G\circ\varphi}^{*}= \Big(M_{1/G\circ\varphi}C_{\varphi}\Big)^{*}= \Big(C_{\varphi}M_{1/G}\Big)^{*}= M^{*}_{1/G}C_{\varphi}^{*}.
\delta (2H) = \delta(g^{ij}h_{ij}) = (\delta g^{ij})h_{ij}+ g^{ij}(\delta h_{ij}) &= 2uh^{ij}h_{ij}+ g^{ij}(u_{ij}- uh_{il}h_j^l + g_{ij}k_0u) &= \Delta{u}+u(4H^2-2K+4k_0).
\left[ \phi ( t , \, z ) , \dot{\phi}( t , \, z^{\prime}) \right] = i \delta ( z - z^{\prime}) \, , \quad \left\{\psi_{\alpha}( t , \, z ) , \bar{\psi}_{\beta}( t , \, z^{\prime}) \right\}= \left( \gamma^{0}\right)_{\alpha \beta}\delta ( z - z^{\prime})
G^{1,0}_{0,1}\left(\begin{array}{c}\_\beta \end{array}\Bigg|\, z \right)= z^{\beta}e^{-z},~~~~~~G^{1,2}_{2,2}\left(\begin{array}{c}1,\, 1 1,\, 0 \end{array}\Bigg|\, z \right)= \ln(1+z),
\begin{array}{rcl}\overline{G}^{(m)}_i (x|z) = (-\tau)^{i-1}\displaystyle\sum_{1\leq \mu_1<\cdots <\mu_{n-1}\leq M}(-1)^{\sum_{i=1}^{n}mn-m-n+i +\mu_i}& \times \det\left( h_{\alpha}^{(m-1)}(x_{\mu_i}|z')\right)_{1\leq i, \alpha \leq n-1}\overline{G}^{(m-1)}_i (\stackrel{\mu}{\hat{x}}| z).\end{array}
M(u(t)) = \int |u(t,x)|^2 dx = M(u_0), E(u(t)) = \frac{1}{2}\int |\nabla u(t,x)|^2 dx +\frac{c}{2}\int |x|^{-\sigma}|u(t,x)|^2 dx \pm \frac{1}{\alpha+2}\int |u(t,x)|^{\alpha+2}dx = E(u_0).
\frac{f_{\lambda m}^{( q )}( z , z x )}{\bar{J}_{m}^{2}+ \bar{Y}_{m}^{2}}- f_{m}^{( q )}( z x ) = - \frac{1}{2}\sum_{n = 1 , 2}\Omega_{\lambda m}^{( n )}( z ) f_{m}^{( n q )}( z x ) ,
\int d^{4}x(-\frac{1}{2}\partial_{\mu}{\hat{A}}_{\nu}^{a}\, \partial^{\mu}{\hat{A}}^{a\nu}-\partial_{\mu}{\hat{\overline{\eta}}}^{a}\, \partial^{\mu}{\hat{\eta}}^{a}-\frac{1}{2}B_{\mu}^{a}{\mathcal{O}}_{A}^{-1ab\mu \nu}B_{\nu}^{b}+{\overline{\psi}}^{a}{\mathcal{O}}_{\eta}^{-1ab}\psi^{b}
|u(x)-u(y)|^{q-2}\, (u(x)-u(y)) \left[\frac{v(x)^q}{u(x)^{q-1}}-\frac{v(y)^q}{u(y)^{q-1}}\right] &=u(x)^q\,\left(\frac{v(y)}{u(y)}\right)^q\left[ \left(1-\frac{u(y)}{u(x)}\right)^{q-1}\, \left(\left(\frac{v(x)\,u(y)}{v(y)\,u(x)}\right)^q-\frac{u(y)}{u(x)}\right)\right]
S_c(t{\setlength\arraycolsep{0.3pt}+}1){\setlength\arraycolsep{0.3pt}=}\begin{cases}\max (S_c(t){\setlength\arraycolsep{0.3pt}-}1, 0) u_c^1(t){\setlength\arraycolsep{0.3pt}=}0 ~ ~u_c^2(t){\setlength\arraycolsep{0.3pt}=}0, N_c^1(t)-1 u_c^1(t)=1, N_c^2(t)-1 u_c^2(t)=1. \end{cases}
K_{1}=1-\cos^{2}\varphi,K_{2}=-2\cos\varphi(1-\cos\theta),K_{3}=\frac{1}{2}(1-\cos^{2}\theta), & K_{4}=2(-\cos\theta+\cos^{2}\varphi), K_{5}=\cos^{2}\theta-\cos^{2}\varphi.
=4\pi\Biggl[\left(\frac1c\left(-{\bf j}_m\times{\bf E}\right)+\rho_m{\bf B}\right)\otimes\varepsilon^1\vee\varepsilon^1+\left(\frac1c\left({\bf j}_e\times{\bf B}\right)+\rho_e{\bf E}\right)\otimes\varepsilon^2\vee\varepsilon^2+
\mathbb{E}\left[\mathbf{y'}^H\mathbf{s}^{\left(\right)}\right]=a_c\sum_{j=1}^{M}\left(\mathbf{h}_{l,*}\mathbf{p}_c\right)^{*}+\sum_{l=1}^M a_l \left(\mathbf{h}_{l,*}\mathbf{p}_l\right)^{*}.
\widetilde{O}_{p}\left[\left(\frac{n}{b}+\sqrt{\kappa}+\frac{1}{b}\sqrt{n\kappa d}\right) \left(Z\frac{b}{n}+d^{2}\right)+d^{\omega}\right] & = \widetilde{O}_{p}\left[Z\left((1+ \sqrt{\frac{\kappa d}{n}}\right) + d^{\omega}+ d^2 \sqrt{\kappa}+ \frac{d^2 \sqrt{n}}{b}\sqrt{\kappa d + n}+ Z \sqrt{\kappa}\frac{b}{n}\right] ~.
\sum_{\mu + \beta + \delta = \alpha}\int_0^s \int_{\Sigma_s}(\partial^\mu f) m(d \partial^\beta \phi,d \partial^\gamma \phi) r L(r \partial^\alpha \phi) d x d t = \int_0^s \int_0^\infty \int_{S^2}\partial^\alpha (f m(d \phi,d \phi)) r^3 L(r \partial^\alpha \phi) d \omega d r d t.
\tilde{f}(\tau_1, \xi_1 , \eta_1) = f (N_1^3 \tau_1 , N_1 \xi_1, N_1 \eta_1), \tilde{g}(\tau_2, \xi_2, \eta_2) = g (N_1^3 \tau_2, N_1 \xi_2, N_1 \eta_2), \tilde{h}(\tau, \xi, \eta) = h (N_1^3 \tau, N_1 \xi, N_1 \eta).
I_{1}= \int_{\partial \Omega}T_{jk}n_{k}d\sigma =\int_{\partial \Omega}\left( \partial_{x_{j}}u\ \partial_{x_{k}}u-\frac{1}{4}\delta_{jk}\Delta\left( u^{2}\right) \right) n_{k}d\sigma = \int_{\partial \Omega}\left( \partial_{x_{j}}u\ \partial_{x_{k}}u-\frac{1}{2}\delta_{jk}|\nabla u|^{2}\right) n_{k}d\sigma
\mathbb{E}\left[ \sup\limits_{t\in\lbrack0,T]}\left( |X_{1}(t)|^{\beta}+|Y_{1}(t)|^{\beta}\right) \right] +\mathbb{E}\left[ \left( \int_{0}^{T}|Z_{1}(t)|^{2}dt\right)^{\beta/2}\right] =O(\epsilon^{\beta/2}),
\left(\mathbf{L}_{\mu}f\right)(g)= \left.\frac{\partial}{\partial q^{\mu}}f\left(k(q)g\right)\right|_{q=0},\left(\mathbf{R}_{\mu}f\right)(g)= \left.\frac{\partial}{\partial q^{\mu}}f\left(gk(q)\right)\right|_{q=0},
\rho(\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z})=\rho g-\frac{\partial P}{\partial x}+\mu(\frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}v}{\partial y^{2}}+\frac{\partial^{2}w}{\partial z^{2}}).
\mathrm{det}\left( \begin{array}{c}\rho_{W}(W, \overline{W}) \Lambda_{1}\rho_{W}(W, \overline{W}) \cdots \Lambda_{n}\rho_{W}(W, \overline{W}) \end{array}\right)=\pm\frac{i}{2}\phi_{n\overline{n}}+O(2).
d\big(\Psi_m(x),\Psi_k(x)\big) \le ~ d\big(\Psi_m(x),\Psi_{m-1}(x)\big) + \dots + d\big(\Psi_{k+2}(x),\Psi_{k+1}(x) \big)+d\big(\Psi_{k+1}(x),\Psi_k(x) \big) \le ~ \big(\prod_{i=1}^{m-1}s_i\big) d\big(T_m(x),x\big)+\dots + \big(\prod_{i=1}^{k+1}s_i\big) d\big(T_{k+2}(x),x \big)+ \big(\prod_{i=1}^{k}s_i\big) d\big(T_{k+1}(x), x \big).
1=\sum_{j=1}^n \frac{a_j}{d}= \sum_{m=1}^k \sum_{j\in I_m}\frac{a_j}{d}= \sum_{m=1}^k \sum_{j\in I_m}\frac{a_j}{u_m(d\wedge a_j)}= \sum_{m=1}^k S_{{\bf A_m},d}(u_m).
(-1)^{|a||c|}\tilde{\pi}_1(a\bar{b}c+\bar{a}b\bar{c})+(-1)^{|b||a|}\tilde{\pi}_1(b\bar{c}a+\bar{b}c\bar{a})+(-1)^{|c||b|}\tilde{\pi}_1(c\bar{a}b+\bar{c}a\bar{b})=0,
\begin{array}{r c l}{i T^{\mathrm{1 P I}}}&{=}&{\displaystyle i \lambda^{2}v \int \frac{d^{4}k_{1}}{( 2 \pi )^{4}}\int \frac{d^{4}k_{2}}{( 2 \pi )^{4}}T ( k_{i}) ,}\\{T ( k_{i})}&{=}&{\displaystyle T_{a}+ T_{b}+ T_{c},}\\ \end{array}
R + \frac{\gamma}{4}( \frac{( D \Omega )^{2}}{\Omega^{2}}- 2 \frac{D \cdot D \Omega}{\Omega}) + ( 1 - \lambda ) \mu \Omega^{- \lambda}- \frac{1}{4}( 1 - \frac{\epsilon}{2}) \Omega^{- \epsilon / 2}F^{2}= T_{\Omega}^{X},
\big( \, \mathfrak{J}^L \, U^{L}_{g_0}f , g \,\big) = (\mathfrak{J}^L \, f, k), \,\,\, \textrm{i. \, e.}\int \, \Big( \mathfrak{J}_L \, \big( \, U^{L}_{g_0}\, f \big)_x , g_x \, \Big) \, d\mu_{\mathfrak{G}/H}(x)= \int \, (\mathfrak{J}_L \, f_x , k_x ) \, d\mu_{\mathfrak{G}/H}(x);
\int_{\mathbb{R}}|\partial_z \hat{G}(u,z,t,y)|dy \le \frac{2}{\sqrt{2\pi(t-u)^3}}\int_{0}^{\infty}\omega e^{-\frac{\omega^2}{2(t-u)}}d\omega=\frac{2}{\sqrt{2\pi(t-u)}}
[L_{1},L_{2},[{\textstyle \frac{k+1}{2}},+]]\oplus[L_{1},L_{2},[{\textstyle \frac{k+1}{2}},-]] longrightarrow \ \bigoplus_{J}\N^{(k)}_{L_{1}\frac{k+1}{2}}{}^{J}\ [J,L_{2},0]\ \ .
X :M_{k}\rightarrow{M_{k}}Xe_{n}= \delta_{n}e_{n}\ ~ \delta_{0}=1, ~\delta_{n}=w_{0}w_{1}\cdots w_{n-1}~ ~1\leq n\leq k; T_{1}^{*}:M_{k}\rightarrow{M_{k}}T_{1}^{*}e_{0}=0, \T_{1}^{*}e_{n}=e_{n-1}~ ~ 1\leq n\leq k.
\begin{array}{l}(m,\tilde{m},n=0,p)=(m+\frac{k}{(k,2k')},\tilde{m}-\frac{k}{(k,2k')},n=0,p-\frac{2k'}{(k,2k')}), (m,\tilde{m},n=1,p)=(m+\frac{k}{(k,2k')},\tilde{m}-\frac{k}{(k,2k')},n=1,p+\frac{2k'}{(k,2k')}) \end{array}
\begin{aligned}\int_\Omega B_j v_j\cdot v_k = \int_\Omega \Re\nabla u_j \cdot \Re\gamma_k(\nabla u_k) + \Im\gamma_j(\nabla u_j) \cdot \Im \nabla u_k &= \Re\int_\Omega \div(\gamma_k(\nabla u_k) \Re u_j) + \Im\int_\Omega \div(\gamma_j(\nabla u_j) \Im u_k) &=\int_{\partial\Omega}( \Re u_j \Re h + \Im u_k \Im h).\end{aligned}
\sum_{a=1}^x c_q(m-a) c_r(a-n) =\begin{cases}x c_q(m-n), ; 0, , \end{cases}& \sum_{a=1}^x S(m,a;q) S(a,n;r) =\begin{cases}x c_q(m-n), ; 0, . \end{cases}
\ln W \equiv \left(-\int_x \frac{1}{\tilde W^2}\frac{\delta \tilde W}{\delta f(x)}\frac1i \frac{\delta \tilde W}{\delta j_3(x)}+\int_x\frac{1}{\tilde W}\frac{\delta^2 \tilde W}{\delta j_3(x)\delta f(x)}\right)_{f=f^{(0)}}
\begin{gathered}\frac{\sigma}{2}C_\omega\bigl(\tau^{(k_n)}-\tau^{(k_n)}_\varepsilon \bigr)^{H-\delta}>\frac{k_n}{4\varepsilon}\bigl(\tau^{(k_n)}-\tau^{(k_n)}_\varepsilon\bigr). \end{gathered}
\log \rho_k = \log \rho_{\hat{k}}+ \sum_{i=\hat{k}}^k \log(1 + L_i^{-1/16}) \leq \log \rho_{\hat{k}}+ \sum_{i=\hat{k}}^\infty L_i^{-1/16}< \log (\iota_{\hat{k}}\rho_{\hat{k}}) < \infty
(a_{1},...,a_{n})\mapsto[\begin{matrix}1&a_{1}&a_{2}&\cdot\cdot\cdot&a_{n-1}&a_{n}\\ 0&1&0&\cdot\cdot\cdot&0&0\\ \vdots&\vdots&\vdots&&\vdots&\vdots\\ 0&0&0&\cdot\cdot\cdot&1&0\\ 0&0&0&\cdot\cdot\cdot&0&1\end{matrix}]
G \cap H = ((D \setminus \{\bar{b}_n\}) \cap (E \setminus \{\bar{b}_n\}))\cup &\qquad\qquad\cup ((D \setminus \{\bar{b}_n\}) \cap [\bar{a}_n \mapsto \bar{b}_n](K)) &\qquad\qquad\cup ((E \setminus \{\bar{b}_n\}) \cap [\bar{a}_n \mapsto \bar{b}_n](J)) &\qquad\qquad\cup ([\bar{a}_n \mapsto \bar{b}_n](J) \cap [\bar{a}_n \mapsto \bar{b}_n](K))
\begin{aligned}\sum_{r=1}^{k-1}\sum_{s=1}^{k-1}b^{(k)}_{ir}a^{(m)}_{rs}w^{(m)}_{sj}=\sum_{r=1}^{k-1}\sum_{s=1}^{k-1}(-1)^{r-i+j-s}\binom{k-i}{r-i}\binom{m-r}{s-r}\binom{m-k+j-s-1}{j-s}&=\sum_{s=1}^{k-1}(-1)^{j-s}\binom{m-k+j-s-1}{j-s}\sum_{r=1}^{k-1}(-1)^{r-i}\binom{k-i}{r-i}\binom{m-r}{s-r}.\end{aligned}
< 0 \mid m^{2}\alpha^{\prime}\mid 0 > = ( D - 1 ) [ - \frac{1}{6}+ \frac{( D - 1 )^{2}}{1 4 4}\zeta ( 3 ) \lambda^{2}+ \frac{( D - 1 )^{3}}{8 6 4}\zeta ( 5 ) \lambda^{3}+{\cal O}( \lambda^{4}) ] .
\lambda_{1}\geqslant \frac{\pi^{3}}{d^{2}\int_{- \pi / 2}^{\pi / 2}z(t) dt}= \frac{\pi^{2}}{d^{2}(1 - \delta)}{\rm and}\lambda_{1}\geqslant \frac{\pi^{2}}{d^{2}}+ \frac{1}{2}K.
\sum_{m=1}^{\infty}I_{m,p}^{\star}(s,x)=\sqrt{\pi}2^{\frac{1}{2}-s}x^{s+\frac{1}{2}}\intop_{0}^{\infty}\,\frac{y^{s-\frac{1}{2}}J_{s-\frac{1}{2}}(xy)}{\sigma\left(\frac{y}{2\pi}\right)e^{y}-1}\,dy.
\begin{array}{c}{{{\displaystyle < \prod_{< i j > \in \omega}U_{i j}> = \hspace{6 c m}}}}\\{{{\displaystyle \frac{1}{Z_{D}}\sum_{k = 0}^{\alpha - 1}\sum_{\{\tau_{m}, \tau_{m}^{'}\}}\exp \sum_{< m n > \in L_{D}^{1}}\left( J_{D}{\tilde{V}}_{m n}( \gamma_{k}) \tau_{m}\tau_{n}+ J_{D}^{'}{\tilde{V}}_{m n}( \gamma_{k}+ \omega ) \tau_{m}^{'}\tau_{n}^{'}\right)}}}\\ \end{array}
K_{p}=\frac{{p_{S}}^{\sigma}{p_{T}}^{\tau}}{{p_{A}}^{\alpha}{p_{B}}^{\beta}}=\frac{{X_{S}}^{\sigma}{X_{T}}^{\tau}}{{X_{A}}^{\alpha}{X_{B}}^{\beta}}P^{\sigma+\tau-\alpha-\beta}=K_{X}P^{\sigma+\tau-\alpha-\beta}
\tau = \begin{cases}|B| - |A| + |O_A| - |O_B'| - |D'\cap X| - |E_D'| |D'\setminus X| = 0 |B| - |A| + |O_A| - |O_B'| - |D'\cap X| - |E_D'| - (|P_D'\cap B| - |P_D'\cap A|) |D'\setminus X| \geq 1. \end{cases}
m_{3 / 2}= \left( \frac{5 \pi^{\frac{1}{2}}\lambda}{2^{\frac{3}{2}}}\right)^{\sqrt 3}\left( \alpha_{G U T}\right)^{\frac{\sqrt 3}{2}}\left( \frac{M_{G U T}}{M_{p l}}\right)^{3 \sqrt 3}M_{p l},
e^{\tau\Phi}D_j^{\alpha_j}e^{-\tau\Phi}= (e^{\tau\Phi}D_je^{-\tau\Phi})(e^{\tau\Phi}D_je^{-\tau\Phi})\cdots (e^{\tau\Phi}D_j e^{-\tau\Phi}) (\alpha_j).
\int\limits_{\psi_0}^{\pi-\psi_0}Cf(0, e_n, \psi)g(\psi_0,\psi)d\psi &=\frac{(\cos\psi_0)^{n-3}}{|S^{n-3}|}\int\limits_{S^{n-2}}\int\limits_{\mathbb{R}^n}f(x) \delta(x\cdot ((\cos\psi_0)\omega,\sin\psi_0))dx d\omega.
\lim_{t\to t_+-0}\arg \left ( \frac{t-t_+}{\alpha(t)-t_+}\right)=\lim_{\theta\to0}\arg(i(\overline{\beta}t-1) ) =\lim_{\theta\to 0}\arg( \sqrt{|\beta|^2-1}\,(\sin\theta- \cos\theta) + i(\cos \theta+\sin \theta - 1))=\pi ,
\frac{\langle n\rangle}{V}=\frac{1}{V}\lambda \frac{\partial}{\partial \lambda}\ln Z\left[\lambda,T\right]=\lambda \langle{\rm Tr}g^{\dagger}{\rm Tr}g-1\rangle\rightarrow \lambda\left|\left<{\rm Tr}g\right>\right|^2 +O(N^0) ~~({\rm as}~N\rightarrow \infty)
| S^{i}\rangle = ( \alpha_{- 1}^{p}\alpha_{- 1}^{p}- \alpha_{- 1}^{i}\alpha_{- 1}^{i}- \alpha_{0}^{p}\alpha_{- 2}^{p}+ \alpha_{0}^{i}\alpha_{- 2}^{i}) | 0 \rangle , \quad p - d + 1 \le i \le p - 1
c_*(A)=\gamma_A(X)\leqslant\sup_{\nu\in\widehat{\mathcal E}'_A}\,\nu(X)\leqslant\sup_{\nu\in\widehat{\mathcal E}'_A}\,G(\nu)\leqslant\max_{\nu\in\mathcal E'_A}\,G(\nu)=G(\gamma_A)=c_*(A),
P_\lambda(\lambda)=-\frac{1}{m\,\Delta_p(\{\hat{v}^{-1}\})}\, \sum_{\mu=1}^m (-1)^{\mu}\frac{\lambda^{m-\mu}}{\Gamma(m-\mu+1)}\det\begin{bmatrix}\left[\frac{\exp(-\lambda/\hat{v}_j)}{\hat{v}_j^m}\right]_{j=1,...,p}\left[\hat{v}_j^{-i+1}\right]_{\substack{i,j=1,...,p (i\neq \mu)}}\end{bmatrix}.
\int_{R^{8 + 1}}{\hat{C}}_{{\hat{\mu}}_{1}{\hat{\mu}}_{2}{\hat{\mu}}_{3}}D{\hat{X}}^{{\hat{\mu}}_{1}}D{\hat{X}}^{{\hat{\mu}}_{2}}D{\hat{X}}^{{\hat{\mu}}_{3}}\wedge \mathrm{T r}({\hat{F}}\wedge{\hat{F}}\wedge{\hat{F}}) .
u_{2k}^{(s,t)}=e_k^{(s,t)}\frac{\kappa^{(t)}{\cal H}_{k-1}^{(s+1,t)}(\mu^{(t)})}{{\cal H}_{k}^{(s,t)}(\mu^{(t)})}&=-e_k^{(s,t)}\kappa^{(t)}\frac{H_{k-1}^{(s+1,t+1)}H_k^{(s,t)}}{H_{k-1}^{(s+1,t)}H_k^{(s,t+1)}}.
D_{a}X_{\mu}( \delta_{\mu \nu}I - i [ X_{\mu}, X_{\nu}] )^{- 1}D_{b}X_{\nu}= ( I - [ X , Y ]^{2})^{- 1}( D_{a}X_{\mu}D_{b}X_{\mu}+ i D_{a}X_{\mu}[ X_{\mu}, X_{\nu}] D_{b}X_{\nu}) .
\begin{aligned}\|x(2 \bar{t}/\gamma)\|_{\mathcal{A}}\le \beta(\|x(\bar{t}/\gamma)\|_{\mathcal{A}},\bar{t}) + \tilde{d}&\le \beta(\underline{\rho},\bar{t}) + \tilde{d}&\le \beta(\rho,\bar{t}) + \tilde{d}& \le \underline{\rho}\end{aligned}
F(H,W)=\exp\left(W^*H\left(I_N+H\right)^{-1}W\right)-\det\left(I_N+H\right) \times\sum_{K}\prod_{1\le j,\ell\le N}\frac{\left(h_{j,\ell}\right)^{k_{j,\ell}}}{k_{j,\ell}!}H_{r_1,c_1}\left(\overline{w_{1}},w_{1}\right)\dotsm H_{r_N,c_N}\left(\overline{w_N},w_N\right).
\begin{aligned}\eta(w,z) =\frac{z+\lambda}{(w+\lambda)(z+\rho)}+\frac{z+\lambda}{(w+\lambda)w}\frac{1}{1-z/w}&=\frac{1}{w-z}+\frac{\lambda-\rho}{(w+\lambda)(z+\rho)}.\end{aligned}
\|f_1f_2\|_{L^2}= \big\|\sum\limits_{m\in\mathbb{Z}}e^{2\pi imz}f_{1,m}(y)f_2\big\|_{L^2}\leq \sum\limits_{m\in\mathbb{Z}}\|e^{2\pi imz}f_{1,m}(y)f_2\|_{L^2}\leq C\sum\limits_{m\in\mathbb{Z}}\|f_{1,m}\|_{L^2}\| f_2\|_{H^1}\leq C\big(\|f_1\|_{L^{2}}+\|\partial_zf_1\|_{L^{2}}\big)\|f_2\|_{H^{1}},
(X_0 + X_0) \cap(X_1+X_1^c) \cap (X_2 + X_2^c) = \varnothing & (X_0 + X_0^c) \cap (X_1 + X_1) \cap (X_2 + X_2^c) = \varnothing &(X_0 + X_0^c) \cap (X_1 + X_1^c) \cap (X_2 + X_2) = \varnothing
p_s(x^N|\theta) =c(\emptyset)p(x_1|\theta)c(x_1)p(x_2|x^1,\theta)\ldots c(x^{N-1})p(x_N|x^{N-1},\theta)(1-c(x^N)) &\propto \Pi_{i=1}^N p(x_i|x^{i-1},\theta) &=p(x^N|\theta)
\sum_{\mu\in\Lambda^{(m,n)}_{\texttt{a}}}P_{\texttt{a};\mu}\bigl( \boldsymbol{\xi}^{(m,n)}_{\texttt{a};\lambda};q \bigr) \overline{P_{\texttt{a};\mu}\bigl( \boldsymbol{\xi}^{(m,n)}_{\texttt{a};\kappa};q \bigr)}\delta^{(m,n)}_{\texttt{a};\mu}(q) = 0 \quad\ \lambda\neq\kappa
= m(B_{n-t,x,y,c}^{(0)}-v,k)+\sum\limits_{r\in N_{B_{3}}(v)}m(B_{n-t,x,y,c}^{(0)}-v-r,k-1)+tm(T,k-1) = m(T,k)+\sum\limits_{r\in N_{B_{3}}(v)}m(T-r,k-1)+tm(T,k-1).
\phi(\tilde X_1,\tilde Y-(\tilde t-t)X_1,\tilde t)+\phi(X_1,\tilde Y-(\tilde t-t) X_1,\tilde t)-2\phi(X,Y,t) &= \langle \nabla_X^2\phi(X,Y,t)(X_1-X), (X_1-X)\rangle-2(X\cdot\nabla_Y-\partial_t)\phi(X,Y,t)(\tilde t-t)+{o}(\epsilon^2), \mbox{as}\epsilon\to 0.
\Theta(\varphi):=(\underbrace{\Xi^{\omega}(\varphi)}_{=:\Theta_1(\varphi)},\underbrace{(\varphi_i^{-1}-(\Phi_i\circ\Phi_j^{-1})\circ\varphi_j^{-1})_{i<j}}_{=:\Theta_2(\varphi)}, \underbrace{(\lambda^2(\varphi_i)-1)_i)}_{=:\Theta_3(\varphi)})=0\in Y_{\infty}.
J^{-{\frac{1}{n}}}\equiv f_{n}= \left| \begin{matrix}{{s_{1}^{'}}}&{{\cdots}}&{{s_{1}^{( n - 1 )}}}\\{{\vdots}}&{{\ddots}}&{{\vdots}}\\{{s_{n - 1}^{'}}}&{{\cdots}}&{{s_{n - 1}^{( n - 1 )}}}\\ \end{matrix}\right|^{-{\frac{1}{n}}}