harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
v_{b^*}'(b^*) = - \theta C^1_{b^{*}}W^{(\theta)}(b^*) + \beta Z^{(\theta)}(b^*) &\quad+ \lambda \left[\left( \theta \dfrac{{C^{(\theta,r)}(b^*;w)}+ \rho_{b^{*}}^{(\theta)}(b^*;w)}{Z^{(\theta)}(b^*)}-w(0)\right)W^{(\theta)}(b^*) - \rho^{(\theta)}_{b^*}(b^*;w'_{+}) \right]=1 .
D_{n,k}\Big(1,\frac{1}{4}\Big)\,=\,\frac{k(p^{l_1}+p^{l_2}+\cdots +p^{l_i}-1)+2}{2^{p^{l_1}+p^{l_2}+\cdots +p^{l_i}}}=\frac{2-k}{2^i}.
{\bf L}_{n}=\int{dz\over{2i\pi}}\lbrack a_{0}{{\gamma}^{n}}J_{sl(2)}^{0}-{a_{-}\over 2}{{\gamma}^{n+1}}J_{sl(2)}^{-}+{a_{+}\over 2}{{\gamma}^{n-1}}J_{sl(2)}^{+}\rbrack,
\left(1-pqx+\frac{qx}{1-u}-\frac{pxyu}{1-yu}+\frac{xyu}{(1-u)(1-yu)}\right)G^t(u) &=\frac{xy}{1-yu}+\left(\frac{xy^2u^2}{(1-yu)^2}+\frac{qxyu}{1-yu}\right)G^{bt}(u)+\frac{xyu+qx(1-yu)}{(1-u)(1-yu)}G^t(1),
\begin{array}{ccl}\mu:R[x]/\langle x^n-1\rangle \longrightarrow R/\langle x^n-\lambda \rangle c(x) \mapsto \mu(c(x))=(\mu_1(c(x),\ldots,\mu_k(c(x)))=(c(\delta_1^{-1}x),\ldots,c(\delta^{-1}_k x)).\end{array}
G_{z}^{-}=-P^{-}_{z}\Lambda\,, M_{z}^{-}=(D^{-}_{0}-D_{z}^{-})\Lambda\,, G_{z}^{+}=-P^{+}_{z}(\Lambda\oplus\Lambda_{\partial})\,, M_{z}^{+}=(D^{+}_{0}-D_{z}^{+})(\Lambda\oplus\Lambda_{\partial}).
\leq (-1-d)|I| - 1 - k + (-1-d)\sum_{i=1}^{2^n-1}|I_i| - (2^n - 1) + d\left(|I| + \sum_{i=1}^{2^n - 1}|I_i| - |I|\right) + n &\leq (-1-d)|I| - 1 -k - \sum_{i=1}^{2^n-1}|I_i| - 2^n +1 + n &< (-1-d)|I| - 1 - k.
I=\frac{\alpha}{6\pi}\int_0^{+\infty}\mathrm{d}\rho \;\rho^2 \, \chi_{\Lambda}^2(\rho)\, \left\| \, \frac{(h_{1}-e_{1})^{1/2}}{(\alpha^2(h_{1}-e_{1})+\rho)^{1/2}}\, x u_{1}\right\|^2 \left\{\left(\frac{i}{\rho y}- \frac{1}{\rho^2 y^2}- \frac{i}{2 \rho^3 y^3}\right) e^{2 i \rho y}+ \right\}.
{\frac{\Theta_{i}}{\Theta_{i_{0}}}}={\frac{\operatorname*{d e t}_{k \neq i_{0}, i \to i_{0}; j = 1 , \dots , N - 1}\theta_{\frac{k}{N}}( N q_{j}| N \tau )}{\operatorname*{d e t}_{k \neq i_{0}; j = 1 , \dots , N - 1}\theta_{\frac{k}{N}}( N q_{j}| N \tau )}}.
\mathcal{L}_m(g)=\frac{(-1)^m}{2^{m}}\delta_{\mu_1\nu_1\ldots\mu_m\nu_m}^{\rho_1\sigma_1\ldots\rho_m\sigma_m}{R^{\mu_1\nu_1}}_{\rho_1\sigma_1}\cdots{R^{\mu_m\nu_m}}_{\rho_m\sigma_m}\;,
U_{i+1,j+3}=\left\{g+3(i+1)+5,\frac{1}{2}(2g-3(j+3)+6(i_1+1)+10),\right. &\phantom{\;\;\;\;\;}\left.\frac{1}{2}(2g+3(j+3)-12(i_1+1)-20)\right\}_{g}&= \left\{g+3i+8,\frac{1}{2}(2g-3j+6i+7),\frac{1}{2}(2g+3j-12i-23)\right\}_{g}.
\varphi = \pi_1(f_0) \varphi + \pi_1(f_1) P(X^M_1) \varphi = \pi_1(f_0) \pi_2(f_0) \varphi + \pi_1(f_0) \pi_2(f_1) P(X^M_2) \varphi &+ \pi_1(f_1) \pi_2(f_0) P(X_M^1) \varphi + \pi_1(f_1)\pi_2(f_1)P(X^M_2)P(X^M_1) \varphi \vdots = \sum_{i_1, \ldots,i_k \in \{0,1\}}\pi_1(f_{i_1}) \cdots \pi_k(f_{i_k}) P(X^M_k)^{i_k}\cdots P(X^M_1)^{i_1}\varphi
\nabla.\mathbf{\sigma}=0\Rightarrow \nabla.(\underbrace{\xi_1 \mathbf{e}_t+\xi_2 \theta_t \mathbf{I}}_+\underbrace{E'(\mathbf{e}+\nu' \theta \mathbf{I})}_-\underbrace{\tau(c)\mathbf{I})}_=0,
|\Lambda_{P_1,P_2}(f_0,f_1,f_2)|\leq q^{\delta_1}\min_{i=1,2}\|f_i\|_{U^1}^{1/2}+O_{P_1,P_2}(q^{\delta_1-1/4}+q^{-\delta_2}+q^{3\delta_3/4-\delta_4/4}+q^{\delta_3-1/4})
\begin{array}{lcl}\tau_3(t) = \frac{\sqrt{6}}{7\,y(t)^5}(-f^{135}+f^{146}+f^{236}+f^{245})+\frac{4\sqrt{6}}{21y(t)^5}(f^{127}+f^{347}+ f^{567}), \end{array}
\mathcal{F}_g(T^0,\dots,T^{N},\mathsf{q})=\sum_{k\geq 0}\sum_{\beta}\frac{1}{k!}\big\langle \sum_{i=0}^N \gamma_i T^i,\dots,\sum_{i=0}^N \gamma_i T^i\big\rangle_{g,k,\beta}\mathsf{q}^{\beta},
\begin{array}{lll}vu = f_{vu}:= q^{-1}uv, v'u' = f_{v'u'}:= q^{-1}u'v' uu' = f_{uu'}:= qu'u, vu' = f_{vu'}:= q^2 u'v + (q-q^3) v'u, uv' = f_{uv'}:= q^2 v'u, vv' = f_{vv'}:= qv'v.\end{array}
\frac{\binom{K-2}{s-1}}{\binom{K-2}{s-1}+\sum_{i=0}^{K-1-s}\binom{K-1}{s+i}(N-1)^iN}\geq \frac{\binom{K-2}{s-1}+\binom{K-2}{s-2}}{\binom{K-2}{s-1}+\binom{K-2}{s-2}+\sum_{i=0}^{K-s}\left[ \binom{K-1}{s+i}+\binom{K-1}{s+i-1}\right](N-1)^iN},
\log L(s, \pi_p \times \bar \pi_p) = \sum_{1 \le i, j \le 2}\log \frac 1{1-\alpha_{i,p}\alpha_{j,p}p^{s}}= \sum_{1 \le i, j \le 2}\sum_{n \ge 1}\frac{(\alpha_{i,p}\alpha_{j,p})^n}{np^{ns}}&= \frac{c_p}{p^{s}}+ O(p^{-2s}),
a^{10}_{00}=a^{11}_{00},a^{10}_{01}=a^{11}_{01},a^{10}_{10}=a^{11}_{10},a^{10}_{11}=a^{11}_{11}, a^{00}_{10}=a^{00}_{11},a^{01}_{10}=a^{01}_{11},a^{10}_{10}=a^{10}_{11},a^{11}_{10}=a^{11}_{11}.
\sum_{i=0}^{n-1}(\phi_1+\phi_2)(x_i,i+\ell) =\sum_{i=0}^{n-1}\phi_1(x_i,i+\ell) +\sum_{i=0}^{n-1}\phi_2(x_i,i+\ell) &\leq \sup_{k \in \mathbb{Z}}\sum_{i=0}^{n-1}\phi_1(x_i,i+k)+ \sup_{k \in \mathbb{Z}}\sum_{i=0}^{n-1}\phi_2(x_i,i+k),
L_{M}= - \frac{1}{4}e^{2 \kappa \Psi}F_{\mu \nu}^{a}F^{\mu \nu , a}- \frac{1}{2}\partial_{\mu}\Psi \partial^{\mu}\Psi - \frac{1}{2}e^{- 4 \kappa \Psi}D_{\mu}\Phi^{a}D^{\mu}\Phi^{a}- e^{- 2 \kappa \Psi}V ( \Phi^{a}) \ ,
e_i = b_{i-1}^+ b_{i}^- ,{\hskip 0.5cm}f_i = b_{i-1}^- b_{i}^+ ,{\hskip 0.5cm}h_i = b_{i-1}^+ b_{i-1}^- - b_{i}^+ b_{i}^-,i = 1, 2,\dots, r.
D^{(\alpha )}(S_{1}||S_{2}^{-1}) =\frac{1}{2\pi \alpha^{2}}\int_{-\pi}^{\pi}\big\{(S_{1}S_{2})^{-\alpha}-1+\alpha \log{(S_{1}S_{2})}\big\}dw &=D^{(-\alpha )}(S_{0}||S_{1}S_{2})=D^{(\alpha )}(S_{1}S_{2}||S_{0})
{\cal W}\equiv{\frac{1}{4 \rho^{2}}}\Big [ \cosh ( 2 \varphi_{2}) ( \rho^{6}- 2 ) - ( 3 \rho^{6}+ 2 ) \Big ] , \qquad \rho \equiv e^{{\frac{1}{\sqrt{6}}}\varphi_{1}}.
\frac{\eta_n}{n}=\frac{\eta_{3N}}{3N}=\frac{1}{3}\Biggl(\frac{1}{N}\sum_{i=1}^N(Z_{3i-2}-1) +\frac{1}{N}\sum_{i=1}^N(Z_{3i-1}-1)+\frac{1}{N}\sum_{i=1}^N(Z_{3i}-1)\Biggr).
\int_{t+2(r+1)}^{\infty}ds \frac{\lambda''(s)}{1+s-t}= \int_{t}^{\infty}ds \frac{\lambda''(s)}{1+s-t}- \int_{t}^{t+2(r+1)}ds \frac{\lambda''(s)}{1+s-t}&=\int_{t}^{\infty}ds \frac{\lambda''(s)}{1+s-t}+E_{3,\partial_{r}v_{1}}(t,r)
u_{n+1}= 2\sum_{k=0}^{n+1}\cal R^k \tilde{\cal A}_{-1}{\rm RHS}= \cal R \Big(2 \sum_{k=0}^{n}\cal R^k\tilde{\cal A}_{-1}{\rm RHS}\Big) + 2\tilde{\cal A}_{-1}{\rm RHS}&= \cal R u_{n}+u_{0}, u_{0}=2\tilde{\cal A}_{-1}{\rm RHS}\, .
\widehat{\xi}_{[(\sigma,k,l)]}(s) =\frac{(s-l_\alpha)^{\frac{k-(m_\sigma-2)}{m_\sigma}}}{m_\sigma^2 k!}k\neq m_\sigma-2\ \operatorname{mod}m_\sigma \widehat{\xi}_{[(\sigma,k,l)]}(s) =-\frac{(s-l_\alpha)^{\frac{k-(m_\sigma-2)}{m_\sigma}}\log(s-l_\alpha)}{m_\sigma^2 k!}k= m_\sigma-2\ \operatorname{mod}m_\sigma;
L ={1\over 2}\int d^D{\bf x}\, \dot \phi^2 -{1\over 2}\int d^D{\bf x}d^D{\bf y}\, \phi({\bf x}) F({\bf x}-{\bf y}) \phi({\bf y}) = \int{d^D{\bf k}\over (2\pi)^D}{1\over2}\left[ |\dot Q_{\bf k}|^2 - \omega_{\bf k}^2 |Q_{\bf k}|^2\right]
\int_0^T \left[DE(\eta(t)) + D_2R(\eta(t), \partial_t \eta(t))\right] \langle \varphi(t)\rangle\,dt = \int_0^T \langle \sigma(t) ,\varphi(t)\rangle\,dt + \int_0^T \langle f(t), \varphi(t) \rangle_{L^2}\,dt
\frac{dx}{dt}=-0.5x^3y+0.5x^3z^2-x^3+y^4z+y^4-yz^3+3yz^2+z^3-3z^2 \frac{dy}{dt}=y^4z-y^4-2y^3-z^3+3z^2 \frac{dz}{dt}=z^2-3z
\begin{aligned}\mathbb{E}\left[ \rho_2 (Y_2, y_2) \right] - 1 \leq \mathbb{E}\left[ \boldsymbol{d}_{\mathcal{S}_2}(S_2, s_2)^{p_2}\right] = \boldsymbol{W}_{p_2}\left(\gamma_{2,3}^{\eta_1, \delta}, \delta_{s_2}\right)^{p_2}\leq \left[ \delta^{1/p_2}+ \boldsymbol{W}_{p_2}\left(\mu_2 , \delta_{s_2}\right) \right]^{p_2}, \end{aligned}
N^{(d)}_{m}(n) =m^d\#\mathrm{RC}^{(d)}_m(n) &=m^d\sum_{p\in\mathbb{Z}^d}\#\overline{\mathrm{RP}}^{(d)}_{m}\bigl(n;p\bigr) &=m^d\sum_{0\le h\le \frac{n}{m}\atop mh\equiv n \!\!\!\!\! \pmod{2}}\sum_{\mu\vdash h \atop l(\mu)\le d}m(\mu)N^{(d)}_{m,h}(n;\mu),
||P_{\lambda^t}(\vec{a})||_{s-t}^2 = \sum_{j,m}(1+|m_1|^2+\cdots+|m_n|^2)^{s-t}\lambda^{2t}(m)|a_{j,m}|^2 &= \sum_{j,m}(1+|m_1|^2+\cdots+|m_n|^2)^{s-t}(1+|m_1|^2+\cdots+|m_n|^2)^t|a_{j,m}|^2 &= \sum_{j,m}(1+|m_1|^2+\cdots+|m_n|^2)^s|a_{j,m}|^2 &= ||\vec{a}||_s
\delta_\phi\left(P_\phi^m\right)^k = \left(P_\phi^m\right)^{k-1}\left(\delta_\phi P_\phi^m\right) + \sum_{j=1}^{k-1}\left(P_\phi^m\right)^{k-1-j}\left((\nabla P_\phi^m)\cdot\left(P_\phi^m\right)^j\right) .
S_n = \sum_{p=1}^{n-1}A_p+b_p-\epsilon_p &= \int_1^n \log x \, dx +\frac{1}{2}\log n -\sum_{p=1}^{n-1}\epsilon_p &= \big(n+\frac{1}{2}\big) \log n -n +1 -\sum_{p=1}^{n-1}\epsilon_p.
{-1\over \alpha_{6,4}}\Gamma_1\Gamma_2\Gamma_3\Gamma_4 C^{-1}(Q^+)^* = Q^-\, ,\hskip 2truecm{-1\over \alpha_{6,4}}\Gamma_1\Gamma_2\Gamma_3\Gamma_4C^{-1}(Q^-)^* = \,- Q^+\,,
\left(\displaystyle\sum_{j=1}^3 |x_{j1}|^2a_j-b_1\right)i,\left(\displaystyle\sum_{j=1}^3 |x_{j2}|^2a_j-b_2\right)i,\left(\displaystyle\sum_{j=1}^3 |x_{j3}|^2a_j-b_3\right)i
\lim_{\varepsilon\to +0}\langle \varphi(\varepsilon), \mathrm i\sigma_2\psi(\varepsilon)\rangle_{\mathbb C^2}=0 \quad\textrm{for all}\varphi\in \mathfrak D(D^\nu_{\varkappa,\,\min}),\ \psi\in \mathfrak D(D^\nu_{\varkappa,\,\max}).
[\mathbb{E}^V-\mathbb{E}^G]\theta(\frac{N^{2/3}}{\varpi}(\lambda_{\alpha^{\prime}}-a_{2k-1}))=-[\mathbb{E}^V-\mathbb{E}^G] \int_I \frac{N^{2/3}}{\varpi}\theta^{\prime}(\frac{N^{2/3}}{\varpi}(E-a_{2k-1})) f_{l}(\mathcal{N}(E, E_U))dE+O(N^{-D_1}).
||[D^{\alpha}_{\ast},\mathcal{A}_j]\partial_j{\mathbf V}||^2_{L^2(\Omega_t)}\leq \sum_{0<\beta\leq\alpha}||D^{\beta}_{\ast}\mathcal{A}_jD^{\alpha-\beta}_{\ast}\partial_j{\mathbf V}||^2_{L^2(\Omega_t)}\,.
\langle \mathcal{E}_k^{\alpha}\varphi, \psi \rangle_{L^2}= \int_{\mathcal{D}\times \mathcal{D}}k(x,y)\partial^{\alpha}\varphi(x)\partial^{\alpha}\psi(y)dydx = \langle \varphi, \mathcal{E}_k^{\alpha}\psi \rangle_{L^2}
\lim\limits_{n \to \infty}\frac{1}{n}\sum\limits_{k=0}^{n-1}\ln{w_i(\theta^{k}\omega)}= \int \ln{w_i(\cdot)}\, d\mathbf{P}(\cdot) \ ({}> -\infty) \lim\limits_{n \to \infty}\frac{1}{n}\sum\limits_{k=0}^{n-1}\ln{d_i(\theta^{k}\omega)}= \int \ln{d_i(\cdot)}\, d\mathbf{P}(\cdot) \ ({}= \ln{\bar{d}_i})
x_ix_j =(-1)^{|x_i||x_j|}x_jx_i, y_ix_j =(-1)^{|x_i||x_j|}x_jy_i+\{x_i, x_j\}, y_iy_j =(-1)^{|x_i||x_j|}y_jy_i+\psi(\{x_i, x_j\}),
1+R=\frac{M_{1}-C_{1}}{M_{0}}\times\frac{M_{2}-C_{2}}{M_{1}}\times\frac{M_{3}-C_{3}}{M_{2}}\times\cdot\cdot\cdot\times\frac{M_{n-1}-C_{n-1}}{M_{n-2}}\times\frac{M_{n}-C_{n}}{M_{n-1}}
K = \left[{\frac{d T}{d z}}\right]_{\bar{z}^{+}}- \left[{\frac{d T}{d z}}\right]_{\bar{z}^{-}}= \left[{\frac{d T}{d s}}\right]_{s ( \bar{z})}\, \left[ \left({\frac{d s_{2}}{d z}}\right)_{\bar{z}^{+}}- \left({\frac{d s_{1}}{d z}}\right)_{\bar{z}^{-}}\right] .
t^{\lambda,n}\,g_1= \mathop{\underbrace{\mathop{\left( \sum_{u=0}^{n-1}t_1^u \cdot t^{\lambda,n}\cdot t_{p+1}^{-u}\right)}}}_{f_1}(1-t_1)\, g_2 + t_1^{n}\cdot t^{\lambda,n}\cdot t_{p+1}^{-n}\cdot g_1.
\,K_{il}^{X_1,0}\leq \,V_{X_2}(-(i+1)A-(l+1)B)+\,V_{X_3}(-(d-l+1)B) &=\,V_{X_2}(-(i+1)A-lB)-1+\,V_{X_3}(-(d-l+1)B) &=\,K_{il}^{X_1,0}-1,
\langle \nabla f_i(x^0), v\rangle+\langle \xi^{*i}, u\rangle \epsilon \leq 0,\ \ \ \i=1, 2, ..., l, \langle \nabla f_i(x^0), v\rangle+\langle \xi^{*i}, u\rangle\epsilon < 0 \ \ \ \ \ \i, \langle \nabla g_j(x^0), v\rangle+ \langle \zeta^{*j}, u\rangle\epsilon \leq 0,\ \ \ \j\in J(x^0; u), \epsilon >0.
\bar{g}_1(x) = (q\!-\!1)\alpha_1^{-\mu}\frac{\big(q^{\lambda+1}/(\alpha_1 x), q, \alpha_2/\alpha_1;q\big)_{\infty}}{\big(q/(\alpha_1 x), \beta_1/\alpha_1, \beta_2/\alpha_1 ;q\big)_{\infty}}\,{}_3\phi_2 \left(\!\! \begin{array}{c}q/(\alpha_1 x), \beta_1/\alpha_1, \beta_2/\alpha_1 q^{\lambda+1}/(\alpha_1 x), \alpha_2/\alpha_1 \end{array}\! ;q,q^{\mu}\right)\! .
c^{1, v_i}= (4, -4, 0,0,0,0,0,0) c^{2, v_i}= (0, 0, 4, -4, 0, 0, 0, 0) c^{3, v_i}= (0, 0, 0, 0, 4, -4, 0, 0) c^{4, v_i}= (0,0,0,0,0,0,4, -4) c^{5, v_i}= \sqrt{2}(2,2, -2, -2, 0,0,0,0) c^{6, v_i}= \sqrt{2}(0,0,0,0,2,2,-2,-2) c^{7, v_i}= (2,2,2,2,-2,-2,-2,-2).
\ln\Delta_l\tilde\Delta_l=\ln\left[(\tilde s_l(x')\tilde e_l'(x)-\tilde s_l'(x')\tilde e_l(x))^2-\xi^2(\tilde s_l(x')\tilde e_l'(x)+\tilde s_l'(x')\tilde e_l(x))^2\right]+\mbox{constant}.
\prod_{a = 1}^{\tilde{n}_{c}}( \Lambda \omega^{2 \ell - 1}+ \delta_{\ell}- \phi_{a})^{2}= ( \Lambda \omega^{2 \ell - 1})^{\tilde{n}_{c}}\left[ 1 + 2 \sum_{a = 1}^{\tilde{n}_{c}}\frac{\delta_{\ell}- \phi_{a}}{\Lambda \omega^{2 \ell - 1}}\right] .
K_2:= \max\left( L_{m,2}|\S^{d-2}|^{\frac{m+2}{2m+2}}\, \left( 2^{\frac{d-2}{2}}\Lambda_0^{2-d}B^2 \right)^{\frac{m}{2m+2}}, A_m |\S^{d-2}| e^{1+\Lambda_0^2}\right)
f(a)=\lim_{k\rightarrow\infty}f_k(a)=\lim_{k\rightarrow\infty}\frac{1}{|\Omega|}\int_\Omega{f_k}=\frac{1}{|\Omega|}\int_\Omega{f}=\frac{1}{|\Omega_1|}\int_{\Omega_1}f,
M_{2N}(\mathbb{R})\to M_{2N}(\mathbb{C}),g=\begin{bmatrix}A B C D\end{bmatrix}\mapsto g^c=\begin{bmatrix}\frac{A+D}{2}+\sqrt{-1}\frac{C-B}{2}\frac{A-D}{2}+\sqrt{-1}\frac{C+B}{2}\frac{A-D}{2}-\sqrt{-1}\frac{C+B}{2}\frac{A+D}{2}-\sqrt{-1}\frac{C-B}{2}\end{bmatrix}.
d\psi_s = \rho \left( \frac{\lambda}{\lambda+\rho}+ \theta_s \right) \psi_s ds + \frac{\rho \lambda}{\lambda + \rho}\sqrt{\gamma_0}\zeta_s (1-K_s) ds + \sum_{j=1}^m \phi_s^j dW_s^j, s\in [0,T], \psi_T = -\frac12\sqrt{\gamma_0}\xi.
E\left[\int_0^T h(t,X(t)) \int_A \varphi(t,X(t),a) [\widehat{\gamma}(t,X(t))](da) dt \right] &= T \int_{[0,T]\times \Sigma}h(t,x) \int_A \varphi(t,x,a) [\widehat{\gamma}(t,x)](da) \Theta_1(dt,dx) &=T \int_{[0,T]\times \Sigma\times A}h(t,x) \varphi(t,x,a) \Theta(dt,dx,da) &= E\left[\int_0^T h(t,X(t)) \int_A \varphi(t,X(t),a) \rho_t(da) dt\right],
\theta_n = \sum_{i=1}^r b_{ni}h_i = \sum_{i=1}^r b_{ni}\left( \nu_i + \sum_{j=1}^s \alpha_{ij}e_j \right) & = \sum_{i=1}^r b_{ni}\nu_i + \sum_{j=1}^s \left(\sum_{i=1}^r b_{ni}\alpha_{ij}\right)e_j = \sum_{i=1}^r b_{ni}\nu_i.
\Big|\int_{[- \pi, \pi] \times \mathbb{T}^d}e^{it\Delta}\phi_1 \overline{e^{it\Delta}\phi_2}e^{\pm i t |\nabla|}\phi_3\, dtdx\Big|\lesssim N^\alpha \prod_{j=1}^3 \| \phi_j \|_{L^2(\mathbb{T}^d)},
\bigg ( \langle \langle F_{\mu \nu}( z_{1}) F_{\rho \sigma}( z_{1}^{\prime}) \rangle \rangle - \langle \langle F_{\mu \nu}( z_{1}) \rangle \rangle \langle \langle F_{\rho \sigma}( z_{1}^{\prime}) \rangle \rangle \bigg ) = 0
T^{n}[\begin{matrix}x_{1}^{n-1}\\x_{2}^{n-1}\\ \vdots\\x_{n-1}^{n-1}\\ 0\end{matrix}]=[\begin{matrix}y_{1}\\y_{2}\\ \vdots\\y_{n-1}\\ \epsilon_{x}^{n-1}\end{matrix}].
V_{2}( s ) = \frac{1}{8}[ \zeta ( 1 + s ) + 1 ] \Gamma \left( \displaystyle{\frac{3 - s}{2}}\right) \Gamma \left( \displaystyle{\frac{1 + s}{2}}\right) \left[ 3 - 5 ( 1 + s ) + \frac{7}{8}( 1 + s ) ( 3 + s ) \right] \, .
\widetilde{\mathcal{J}}(\delta):=\left\{(\mathcal{X},\lambda,\mathcal{Y})\in \mathbb{X}\times\mathbb{R}^{p}\times\mathbb{X}\ |\ \widetilde{\Psi}\big(\delta,\mathcal{X},\lambda,\mathcal{Y}\big)=0\right\}.
p_r(h)=\inf\Bigl\{\sum_{i=1}^n\rho_{g_i}(x_i,y_i):h=g_1x_1^{\varepsilon_1}y_1^{-\varepsilon_1}g_1^{-1}\cdots g_nx_n^{\varepsilon_n}y_n^{-\varepsilon_n}g_n^{-1}, n\in{\mathbb N}\Bigr\}
\alpha_1 =\gamma_i + \beta_i=\epsilon_{ij}\alpha_j +\delta_{ij}\beta_{\sigma_j(i)}+ \beta_i \alpha_1 =\gamma_{\sigma_j(i)}+ \beta_{\sigma_j(i)}=\epsilon_{\sigma_j(i)j}\alpha_j +\delta_{\sigma_j(i)j}\beta_{i}+ \beta_{\sigma_j(i)}.
F_{\mathrm{M 2}}= - \frac{2^{7 / 2}\pi^{2}}{3^{4}}N^{3 / 2}V_{2}T^{3}\left[ 1 + \frac{9}{8 \pi^{2}}\sum_{i = 1}^{4}\omega_{i}^{2}- \frac{2 7}{1 2 8 \pi^{4}}\left( \sum_{i = 1}^{4}\omega_{i}^{2}\right)^{2}+ \frac{8 1}{6 4 \pi^{4}}\sum_{i = 1}^{4}\omega_{i}^{4}+ \ldots \right]
G^{[B;m]}(\tau,u,v,t) := e^{2\pi i \left(m+\frac{1}{2}\right)t}\sum\limits^{2m}_{j=0}{a_j(\tau,v) \left(\Theta^{-}_{j+\frac{1}{2},m+\frac{1}{2}}+ \Theta^{-}_{-\left(j+\frac{1}{2}\right),m+\frac{1}{2}}\right) (\tau,2u)}
\frac{1}{T_{\mathrm{D R}}^{\mathrm{o n}}( E )}= \frac{1}{C_{\mathrm{D R}}}+ \frac{2 \mu}{( 4 \pi )^{w}}( - 2 \mu E )^{w - 1}\Gamma ( 1 - w ) + i \frac{2 \pi \mu}{\Gamma ( w ) ( 4 \pi )^{w}}( 2 \mu E )^{w - 1}.
\left[ \begin{smallmatrix}T; N, \mu d \end{smallmatrix}\right]^+ = \left[ \begin{smallmatrix}T; N, \mu -1 d \end{smallmatrix}\right]^+ + (T; N,d)^+ (q^{\mu-d}T_{N+d}T_{N+d+1}\dots T_{N+\mu-1}) \left[ \begin{smallmatrix}T; N, \mu -1 d -1\end{smallmatrix}\right]^+
\textnormal{(I)}:= \frac{4(p-2)}{p}\int u_k^{p/2}(a\nabla (\eta u_k^{p/2}),\nabla \eta)\;dx+\frac{4}{p}\int u_k^{p}(a\nabla \eta,\nabla \eta)\;dx, \textnormal{(II)}:= \int u_k^p (\nabla a,\nabla (\eta^2))\;dx+(p-1)\int u\eta^2 u_k^p\;dx+pk\int u\eta^2 u_k^{p-1}\;dx.
b(z_1\otimes\dots\otimes z_n) = z_1\otimes\dots\otimes z_n &+\sum_{1}^{n-1}(-1)^i z_1\otimes\dots \otimes z_iz_{i+1}\otimes \dots\ \otimes z_n &+ (-1)^nz_n\otimes z_1\otimes\dots\otimes z_{n-1}
\begin{array}{ll}\nabla_{e_1}e_1 = \nabla_{X}X = e_1, \nabla_{e_2}e_2 = \nabla_{U}U = 0, \nabla_{e_1}e_2 = \nabla_{X}U = 0, \nabla_{e_2}e_1 = \nabla_{U}X = 0.\end{array}
\psi_r\, \mapsto \begin{cases}B_r^{(0)}r\in \mathbb Z B_{r-\frac12}^{(1)}r\in \mathbb Z+\tfrac12 \end{cases}\psi_s^*\, \mapsto \begin{cases}K_s^{\perp (0)}s\in \mathbb Z K_{s+\frac12}^{\perp (1)}s\in \mathbb Z+\tfrac12 \end{cases}
\begin{cases}\frac{\partial g_{i\bar{j}}}{\partial t}=-R_{i\bar{j}}+\beta g_{i\bar{j}}+\theta_{i\bar{j}}\frac{\partial f}{\partial t}\ =\beta n\tau^{-1}-R+tr_{g(t)}\theta-\triangle f+|\nabla f|^{2}\frac{\partial\tau}{\partial t}\ =\beta(\tau-1) \end{cases},
\min\{p^{d-1},4p^{d-2}\}\le LL(ZB)=p^{a_1}+\ldots+p^{a_s}-s+1 &\le p^{a_1}+p^{a_2}+p^{a_3+\ldots+a_s}-2\le p^{d-2}+2(p-1).
V = -{\frac{1}{2 \bar{\bf Z}}}\left[ \left( \begin{matrix}{{{\bf 0}}}&{{{\bf I}}}\\{{-{\bf I}}}&{{{\bf 0}}}\\ \end{matrix}\right) \cdot{\bf M}({\cal F}) + i \left( \begin{matrix}{{{\bf I}}}&{{{\bf 0}}}\\{{{\bf 0}}}&{{{\bf I}}}\\ \end{matrix}\right) \right] \cdot{\cal Q},
\begin{array}{r c l}{\nu}&{=}&{\nu_{o}+ \frac{\displaystyle \left( E_{q - \mathrm{r o t}}( J + 2 ) - E_{q - \mathrm{r o t}}( J ) \right)}{\displaystyle h c},}\\{}&{=}&{\nu_{0}+ B \left( [ J + 3 ]_{q}[ J + 2 ]_{q}- [ J + 1 ]_{q}[ J ]_{q}\right) ,}\\ \end{array}
\frac{p}{2}- \frac{1}{s'}+\frac{Np}{2}- N = \frac{1}{s}+ \left( \frac{p}{2}-1 \right) \left( N+1 \right) \in \Bigg[ \left( \frac{p}{2}-1 \right) \left( N+1 \right), \frac{N-\gamma}{N}+ \left( \frac{p}{2}-1 \right) \left( N+1 \right) \Bigg),
\int\limits_{B_{2^{-n+3}\cdot |\Lambda |}(M_0)\setminus\widetilde{\Omega}_{n-2}}\, \frac{|\Phi_n(M)|}{\rho^2_{M_0}(M)}\, dm_3(M)\leq &\leq C\cdot 2^{2n}\int\limits_{B_{2^{-n+3}\cdot |\Lambda |}(M_0)}\, |\Phi_n(M)|\, dm_3(M)\leq &\leq C\cdot 2^{n}\omega (2^{-n}).
(\sigma \circ\pi )(\mu(a)) =\sigma(\phi(a))=\sigma(\oplus_{n=1}^\infty \mu_n(E_na E_n)) &=\oplus_{n=1}^\infty P_n \sigma_n ( \mu_n(E_na E_n))P_n=\oplus_{n=1}^\infty P_n \psi_n(E_na E_n)P_n &=\oplus_{n=1}^\infty P_n \mu(E_na E_n)P_n=\oplus_{n=1}^\infty \mu(E_na E_n) &=\lim_{N\to\infty}\mu\left( \oplus_{n=1}^N E_n a E_n\right) &=\mu(a)
P(x)-S(x)^2=\sum_{k=1}^{n-1}P(t_k)S_k(x)^2+\sum_{k=1}^{n-1}\left(P'(t_k)-P(t_k)\frac{S''(t_k)}{S'(t_k)}\right)(x-t_k)\,S_k(x)^2,
D_{q_kq_l}X^\alpha = \frac{g_{j,z}}{g_z}D_{q_k}X^\alpha D_{q_l}X^j + \frac{g_{i,z}}{g_z}D_{q_k}X^iD_{q_l}X^\alpha-D_{p_\alpha}g_{ij}D_{q_k}X^iD_{q_l}X^j.
\begin{aligned}z(t_i)-z(t_j) =z'(t_i-t_j)+\frac12z''(t_i^2-t_j^2)+\frac16z'''(t_i^3-t_j^3)+... =(t_i-t_j)(z'+\frac12z''(t_i+t_j)+\frac16z'''(t_i^2+t_it_j+t_j^2))+...\end{aligned}
\mathcal{BL}_{p,q}^{\mathcal{T}}(F,f)=\sup\{\left\Vert M_{\mathcal{T}}\phi\right\Vert_{L^{p,q}(X,\mu)}^{q}:\phi\geq0\text{is measurablewith}\left\Vert \phi\right\Vert_{L^{1}(X,\mu)}=f,\left\Vert \phi\right\Vert_{L^{p,q}(X\mu)}^{q}=F\}.
\left\|V(\gamma_1,\sigma_1,t) - V(\gamma_2,\sigma_2,t)\right\|_{L^2_N}\leq \|\psi(t)\|_{W^{1,\infty}}\left(\|\gamma_1 - \gamma_2\|_{L^2_N}+ \frac{1}{\sqrt{N}}|\sigma_1 - \sigma_2|\right)
\int_{|t-t_0|=|t|=|w'/\pi|}\eta_1\bigl((t-t_0)t\bigr)\,dt = |w'/\pi|^{-2}\int_{|t-t_0|=|t|=|w'/\pi|}\eta(1-t_0/t)\,dt = -q^{-1}|w'/\pi|^{-1}.
\sum_{\lambda = j+n}^{m+n+p-2}= \sum_{\lambda = j+n}^{m+n+p-2}(-1)^\lambda~ h ( a^{m+n-1}_{1, i-1}, ~ f^n, ~ a^{m+n-1}_{i+m, j-1}, ~ g^m,~ a^{m+n-1}_{j+n, \lambda -1},~ \alpha^{m+n-2}(a_\lambda \cdot a_{\lambda +1}),~ a^{m+n-1}_{\lambda +2, m+n+p-1}).
\big(M_{+}^\nu(s)\big)^*M_{+}^\nu(s)= \begin{pmatrix}1 + \nu^2s^{-2}-\dfrac{\nu(1 -2\mathrm is)}{s^2 +\mathrm is}P(s) -\dfrac{\nu(1 +2\mathrm is)}{s^2 -\mathrm is}\overline P(s) 1+\nu^2(1+ s^2)^{-1}\end{pmatrix}
\xi = - \sqrt{\alpha^{\prime}}\frac{1}{\partial_{\tau}}( u_{2}\psi^{2}+ i u_{4}\psi^{4}) \ , \overline{{{\xi}}}= - \sqrt{\alpha^{\prime}}\frac{1}{\partial_{\tau}}( u_{2}\psi^{2}- i u_{4}\psi^{4}) \ ,
\sup_{s \in{\mathbb{R}^{\ge 0}}, \Delta s \in [0, \eta L ]}\left| \sqrt{s + \Delta s}- \sqrt{s}\right| \cdot \left\| b (t) \right\|_{{C}_{0}(Y)}= \ \sqrt{\eta L}\cdot \left\| b (t) \right\|_{{C}_{0}(Y)}\; .
\begin{cases}\frac{\partial x}{\partial s_2}=\Im(f_1(0,0,s_3)+f_2(0,0,s_3)) \frac{\partial y}{\partial s_2}=\Re(f_2(0,0,s_3)-f_1(0,0,s_3)) \frac{\partial t}{\partial s_2}=\Im g(0,0,s_3)\end{cases}
E ={\rm Tr}\; \sqrt{\det (\delta_{\mu \nu}+ 2 \pi \alpha' F_{\mu\nu})}= 2 \sqrt{(1 + 4 \pi^2 \alpha'^2 F_{12}^2)(1 + 4 \pi^2 \alpha'^2 F_{34}^2)}= 2(1 + 16 \pi^4\alpha'^2)
\begin{aligned}W^{(M)}_\varnothing[\psi](t;\zeta) =a_\varnothing(\partial)\left.\prod_{k=1}^M\psi^{(j_k)}(t;\zeta_k)\right|_{\zeta_1=\zeta_2=\dots\zeta_M=\zeta}\ , W^{(M)}_\varnothing[\chi](t;\eta) = a_\varnothing(\partial)\left.\prod_{k=1}^M\chi^{(j_k)}(t;\eta_k)\right|_{\eta_1=\eta_2=\dots\eta_M=\eta}\ .\end{aligned}
( \delta g^{( 1 )}, \delta g^{( 2 )}) = \int \! \sqrt{g}\; \delta g_{\mu \nu}^{( 1 )}\left( g^{\mu \alpha}g^{\nu \beta}+ g^{\mu \beta}g^{\nu \alpha}+ C g^{\mu \nu}g^{\alpha \beta}\right) \delta g_{\alpha \beta}^{( 2 )},
V_{\alpha}= D ( \alpha ) V_{Q - \alpha}, D ( \alpha ) = - [ \mu \pi \gamma ( b^{2}) ]^{( Q - 2 \alpha ) / b}\frac{\Gamma ( ( 2 \alpha - Q ) b ) \Gamma ( ( 2 \alpha - Q ) / b )}{\Gamma ( - ( 2 \alpha - Q ) b ) \Gamma ( - ( 2 \alpha - Q ) / b )}.
T_{k}= \left( \begin{array}{c c c c}{{0}}&{{\ldots}}&{{}}&{{1}}\\{{1}}&{{\ldots}}&{{}}&{{0}}\\{{0}}&{{1}}&{{\ldots}}&{{0}}\\{{\vdots}}&{{}}&{{}}&{{}}\\{{0}}&{{\ldots}}&{{1}}&{{0}}\\ \end{array}\right)
C(l)_{n,n'}^{m,m'}= \left (\begin{array}{cc}n l \end{array}\right ) \left [\begin{array}{c}m' + n' \end{array}\right ]_{l}- \left (\begin{array}{cc}n' l \end{array}\right ) \left [\begin{array}{c}m + n \end{array}\right ]_{l}
\left(\begin{array}{@{\,}c@{\,}}Y_{k+1}\vdots Y_{m}Z \varepsilon \end{array}\right) = \left(\begin{array}{@{\,}c@{\,}}u_{k+1}w^{\lambda_{k+1}}\vdots u_{m}w^{\lambda_{m}}\zeta w^{r}w^{s}\end{array}\right).
\limsup_{z\to 0^+}\left| z^{\mu-1}W_2(u,z)-\Gamma(\mu)2^{\mu-1}u^{-\mu}\left(1-2\mu\sum_{s=0}^{N-1}\frac{B_s'(0)}{u^{2s+2}}\right)\right|=O\left(\frac{u^{-\mu}}{u^{2N+2}}\right)
0=2\left\langle \sum_{x \in H^G}b_n(x)\theta^{(k)}(x),v\right\rangle=\sum_{x \in H^G}\left\langle b_n(x),v-\pi_n(x)v\right\rangle \theta^{(k)}(x) = \sum_{x \in H^G}|| b_n(x)||^2_{V_n}\theta^{(k)}(x)>0,
A_{\mu}^{(N)}dx^\mu = \frac{1}{2e}\left( \begin{array}{ccccc}(N-1)\cos\theta d\phi \omega_1 \Theta 0 \ldots 0 \omega_1 \bar \Theta (N-3)\cos\theta d\phi \omega_2 \Theta \ldots 0 \vdots \ddots \vdots 0 \ldots 0 \omega_{N-1}\bar \Theta (1-N)\cos\theta d\phi\end{array}\right)
\det\left( M_{f}\right) =\sum_{s\in T\left( n,m\right)}\left( -1\right)^{\left\vert s\right\vert +\left\vert t\right\vert}\det\left( M_{f}\left(s,t\right) \right) \det\left( \left( M_{f}\left( s,t\right) \right)^{c}\right)
I_{51}= \int_{0}^{2\pi}\log\left(\sin^{2}\left(\frac{x-y}{2}\right))\right)(h'(x)-h'(y))(r\cos(x)\cos(y)+\sin(x)\sin(y))dy I_{52}= \int_{0}^{2\pi}\log\left(((1+r^2)+(r^2-1)\cos(x+y))\right)(h'(x)-h'(y))(r\cos(x)\cos(y)+\sin(x)\sin(y))dy
e_{1,[2]}^{1}+a_{1}^{1,1}e_{1,[1]}^{1}+a_{1}^{1,0}e_{1}^{1}=0 e_{2,[2]}^{1}+a_{2}^{1,1}e_{2,[1]}^{1}+a_{2}^{1,0}e_{2}^{1}=0
\left\{L_{\alpha}, L_{\beta}\right\}= 2 i \left[ ( \hat{p}^{\mu}+ \Pi_{1}^{\mu}){( C \Gamma^{\mu}S^{+})}_{\alpha \beta}- ( \hat{p}^{\mu}- \Pi_{1}^{\mu}){( C \Gamma^{\mu}S^{-})}_{\alpha \beta}\right] ,
\left(\sum_{q\in A_2}\left(\sum_{j\in E_q}|\lambda_j|^{p_{1}}\right)^{p_\zeta/p_{1}}\right)^{1/p_\zeta}< 2^{1/p_{1}}\left(\sum_{j=1}^m|\lambda_j|^{p_{1}}\right)^{1/p_{1}}.
\mathcal{K}_s(1,w)= \frac{\Gamma(n+s+1)}{4^n\pi^{n+1}}\int_0^{\infty}\frac{\cos \Big((n+s+1)\arctan\Big(\frac{4w}{\coth t}\Big)\Big)}{\Big(w^2+\frac{1}{16}\coth^2t\Big)^{\frac{n+s+1}{2}}}(\sinh t)^{-(n+s+1)}\,dt.
\widetilde{\rho}_j(\xi,w)=\frac{1}{T_j(\xi)^{\delta_j}}\rho\left(\xi,(\xi-1)^{\delta_1}(\xi+1)^{\delta_{-1}}w+ w_1 \frac{1+\psi(\xi)}{2}+ w_{-1}\frac{1-\psi(\xi)}{2}\right)
0 = \left(-n^3 - 20n^2 - 133n - 294\right) c_{n+6}+ \left(4n^2+ 52n + 168\right) c_{n+5}+ \left(52n^3 + 816n^2 + 4304n +7620\right) c_{n+4}&+ \left(96n + 384\right) c_{n+3}+ \left(-624n^3 - 5952n^2 - 19008n - 20304\right) c_{n+2}+ \left(-576n^2 - 2880n - 3456\right) c_{n+1}&+ \left(1728n^3 + 6912n^2+ 8640n + 3456\right)c_n.
{\cal L}_{\mathrm{g a u g e}}\sim{\frac{( r M_{h})^{6}}{g_{h}^{2}}}\mathrm{t r}F^{2}\mathrm{a n d}\{\cal L}_{\mathrm{g r a v}}\sim{\frac{r^{6}M_{h}^{8}}{g_{h}^{2}}}{\cal R}\ ,