harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Re(C_E) =\frac{z_1\bar{z}_2+\bar{z}_1z_2}{\sinh(2\beta)}\tanh(\beta-\sigma(\beta))\Re\circ\mathcal{H}([\theta\sigma(e_{\beta}),e_{\beta}]) &=\frac{z_1\bar{z}_2+\bar{z}_1z_2}{\cosh(2\beta)}\mathcal{H}\circ\Re([\theta\sigma(e_{\beta}),e_{\beta}]).
S_{\rm eff}^{d=2}|_{\mu=0}={4\over \left( \beta \sqrt{\pi}\right)^{3}}\sum_{k=1}^\infty{{\cal B}_{2k}(\beta^2 B / 2)^{2k}\over (2k)!}\sum_{j=0}^\infty{\left(-{\beta^2m^2\over 4}\right)^j \Gamma\left(\frac{3}{2}-j-2k\right)\eta(3-2j-4k)\over j!}
S_{i}=A_{i}^{\prime}(S_{i+1}-S_{i+1}B_{i}(B_{i}^{\prime}S_{i+1}B_{i}+R_{i})^{-1}B_{i}^{\prime}S_{i+1})A_{i}+Q_{i}+\tau_{\perp i}^{\prime}\Psi_{i+1}^{2}\tau_{\perp i},S_{N}=F
{\cal G}_{\rho 4}\left(\rho,z\right)={\displaystyle\frac{1}{4\pi A{\cal C}_{66}}\sum_{l=1}^{4}{\frac{\left(-z\right)\Gamma_{\rm{b4}}\left(a=-A_l\right)}{\rho\sqrt{A_l\rho^2+z^2}\prod_{j=1,(j\neq l)}^{4}{\left(A_j-A_l\right)}}}}
\begin{cases}dX(t) =\Pi(t)X(t)[b_0(t,Y)dt+\sigma_0(t,Y)dB(t)+\int_{\mathbb{R}}\gamma_0(t,Y,\zeta)\tilde{N}(dt,d\zeta)]; t\in[0, T] X(0) =x_0>0.\end{cases}
\rho(f)=Q_1\widehat{\rho}(E_1\widehat{f})Q_1\oplus Q_2\widehat{\rho}(E_2 \widehat{f})Q_2=\widehat{\rho}(E_1 \widehat{f})\oplus \widehat{\rho}(E_2 \widehat{f})
\psi=g\Big(\frac{x_1}{x_0},...,\frac{x_n}{x_0}\Big)~x_0^{-\frac{n+2}{2}}\delta\Big(\frac{x_{n+1}}{x_0}-f\Big(\frac{x_1}{x_0},...,\frac{x_n}{x_0}\Big)\Big)
\mathcal{N}_l(P):= \{y(\lambda)^T\in\mathbb{F}(\lambda)^{1\times m}\mbox{such that}y(\lambda)^TP(\lambda) = 0\}, \mathcal{N}_r(P):= \{x(\lambda)\in\mathbb{F}(\lambda)^{n \times 1}\mbox{such that}P(\lambda)x(\lambda) = 0\}.
H^{( 1 )}= \int d^{2}x \left[ m \Phi^{0}\partial_{i}B^{i}+ m \sqrt{m}\epsilon_{i j}\Phi^{i}B^{j}+ \sqrt{m}\Phi^{i}\partial_{i}B^{0}- \Phi^{3}( m B^{0}- \epsilon_{i j}\partial^{i}B^{j}) \right] .
P\left(\tilde{\omega}^{k}\right)=\left\Vert \mathbf{Q}\left(\tilde{\omega}^{k},b_{1}^{*}\left(\tilde{\omega}^{k}\right),b_{2}^{*}\left(\tilde{\omega}^{k}\right);\mathbf{t}\right)+\bar{p}^{*}\left(\tilde{\omega}^{k}\right)\mathbf{1}-\mathbf{f}\right\Vert_{2}^{2}
\begin{dcases}\rho_t+\rho_x +u_{x}=0 (0,T)\times (0,1), u_{t}-u_{xx}+u_x +\rho_x =0 (0,T)\times (0,1), \rho(t,0)=\rho(t,1)+ p(t) t\in (0,T) , u(t,0)=0, \u(t,1)=0 t\in (0,T) , \rho(0,x)=\rho_0(x),\u(0,x)=u_0(x) x\in(0,1).\end{dcases}
N_{\mathcal J}(u, v,w)= \langle N_{\mathcal J}(u, v), w\rangle =\eta ( \mathcal J u, v, w) -\eta ( \mathcal J v, u, w) +\eta (w,u, \mathcal J v) &- \eta (v, u, \mathcal J w) +\eta ( \mathcal J w, u, v) +\eta (u, v, \mathcal J w) .
(\mu+\Lambda_{C_\infty}(b))^{-1}|b_m|h = \Theta_p(\mu,b)|b_m|h & = (\mu - \Delta)^{-\frac{1}{2}-\frac{1}{q}}G_p(q,b_m)|b_m|^{\frac{2}{p}}h & - (\mu - \Delta)^{-\frac{1}{2}-\frac{1}{q}}Q_{p}(q,b) (1 + T_p(b))^{-1}T_p(b,b_m)|b_m|^{\frac{2}{p}}h.
\mathbf{d}_{\mathbf{D}}= (\boldsymbol{\delta}_{\mathbf{D}}-1)(k_{1}+1) \ \ , \ mathbf{d}_{l}> (\boldsymbol{\delta}_{l}-1)(k_{1}+1) \ \ , \ mathbf{\Delta}_{l}- \mathbf{d}_{l}+ \boldsymbol{\delta}_{l}- 1 \geq 0 \ \ , \ mathbf{\Delta}_{\mathbf{D}}= \mathbf{d}_{\mathbf{D}}- \boldsymbol{\delta}_{\mathbf{D}}+ 1
\begin{array}{c}{L = g_{\alpha \beta}( d X^{\alpha}\cdot d X^{\beta}) + 2 g_{\alpha a}( d X^{\alpha}\cdot d X^{a}) + g_{a b}( d \Phi^{a}\cdot d \Phi^{b}) +}\\{+ b_{\alpha \beta}d X^{\alpha}\wedge d X^{\beta}+ 2 b_{\alpha a}d X^{\alpha}\wedge d \Phi^{a}+ b_{a b}d \Phi^{a}\wedge d \Phi^{b}}\\ \end{array}
i_{R_{\theta'}}d\theta' = [i_{R_{\theta'}}d\rho]\theta -i_{R_{\theta'}}\theta\d\rho+ \rho i_{R_{\theta'}}d\theta \hphantom{i_{R_{\theta'}}d\theta'}{}=\left[\sum_i a_i(X_i\rho) + \sum_j b_j(P_j\rho) + c(R_\theta\rho)\right]\theta- c d\rho + \rho\left[\sum_i a_i\beta_i - \sum_j b_j\alpha_j\right],
V(f_1,f_2) = \left|\frac{\mu}{2}\right|^2 \frac{1}{f_1^i{\bar{f}}_{1i}+ f_2^i{\bar{f}}_{2i}}\left\{-|f_1^i{\bar{f}}_{2i}- f_2^i{\bar{f}}_{1i}|^2 +(f_1^i{\bar{f}}_{1i}+ f_2^i{\bar{f}}_{2i})^2 \right\}.
-\frac{1}{2}{\rm Tr}[\frac{M^2}{-\partial^2 +M^2{\tilde\omega}}\alpha \frac{M^2}{-\partial^2 +M^2{\tilde\omega}}\alpha ] = -{1\over 48\pi}\int d^2x{1\over\omega^2}\partial_{\mu}\omega\partial^{\mu}\omega \;.
\Delta^k_\tau(w(t)) =\tau(a)-\tau(y_0) -[\sum_{j=1}^{m_1}\tau(p_j)-\sum_{j=1}^{m_2}\tau(q_j)] &=\tau(a)-\tau(a_0)+\tau(b)-\tau(x_0)-\tau(b) &=\tau(a)-\tau(a_0)=0,\forall\,\tau\in T(A).
(2 \sin(\tfrac{\pi}{6}+ \tfrac{\pi(r-\frac{j}{6})}{12n+j}))^{-12n-j}\leq (2 \sin(\tfrac{\pi}{6}+\tfrac{\pi/3}{12n+j}))^{-12n-j}\leq (2 \sin(\tfrac{\pi}{6}+\tfrac{\pi/3}{12}))^{-12}\leq 0.192\dots,
(e^{-2\phi})_{\rm fix}={|q_0 q_1 + p^0 p^1| \over (q_1)^2 + (p^0)^2}\qquad(a)_{\rm fix}={p^1 q_1 - q_0 p^0 \over (q_1)^2 + (p^0)^2}\ .
S_{\Sigma}= \frac{1}{4 \pi \alpha^{\prime}}\int_{\Sigma}d^{2}\sigma \sqrt g [ g^{a b}G_{\mu \nu}( X ) \partial_{a}X^{\mu}\partial_{b}X^{\nu}+ i \epsilon^{a b}B_{\mu \nu}( X ) \partial_{a}X^{\mu}\partial_{b}X^{\nu}+ \alpha^{\prime}R \Phi ( X ) ]
\varphi^2(x) = \varphi\Big((y^{-1}-q^{-1}y)x^{-1}\Big) &= (-y+q^{-1}y^{-1})x(y^{-1}-q^{-1}y)^{-1}&= (q^{-1}y^{-1}-y)(q^{-1}y^{-1}-y)^{-1}x &= x
{\widehat T}={\widehat M}_1=\left( \begin{array}{cccccc}1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 -2 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 \end{array}\right)\; .
A(r)=\iint_{\Omega_{r}}|f^{\prime}(z)|^{2}dxdy=\frac{1}{2i}\int_{\partial\Omega_{r}}\overline{f(z)}f^{\prime}(z)dz=\frac{1}{2i}\int_{|w|=r}\overline{f(g(w)))}f^{\prime}(g(w))g^{\prime}(w)dw.
\dot{\bar{z}}= \bar{f}(\bar{z},e,v,w) \dot{\bar{\eta}}= M \bar{\eta}+M c_{1}e-\sum_{i=1}^{r}c_{i}\bar{g}_{i}(\bar{z},e,v,w) \dot{\bar{x}}= A_{d}\bar{x}+b(w)B\Psi\bar{\eta}+\bar{g}(\bar{z},e,v,w)+b(w)B\bar{u}+b(w)B\Psi\tilde{\eta}
k_{x}=k_{x}^{\prime}+ik_{x}^{\prime\prime}=[\frac{\omega}{c}(\frac{\epsilon_{1}^{\prime}\epsilon_{2}}{\epsilon_{1}^{\prime}+\epsilon_{2}})^{1/2}]+i[\frac{\omega}{c}(\frac{\epsilon_{1}^{\prime}\epsilon_{2}}{\epsilon_{1}^{\prime}+\epsilon_{2}})^{3/2}\frac{\epsilon_{1}^{\prime\prime}}{2(\epsilon_{1}^{\prime})^{2}}].
|q_k\nu-p_k-e|\leq\Delta(p_k,q_k)^{(p_k+q_k)L_{k}}, \Delta(p_k,q_k)=\min\Bigl\{\frac{1}{2}, |q\nu-p-re|\colon 0<|r|+|q|<3(q_k+1), (p,q,r)\neq 0, \pm(p_k,q_k,1), \pm2(p_k,q_k,1)\Bigr\}.
a_{11}=\frac{\sigma r_{1}^{2}+ \overline{\sigma}r_{2}^{2}}{r^{2}},a_{12}=a_{21}=\frac{r_{1}r_{2}(\sigma-\overline{\sigma})}{r^{2}}, a_{22}=\frac{\overline{\sigma}r_{1}^{2}+ \sigma r_{2}^{2}}{r^{2}}
{\cal F}= \sum_{n_{1}\geq n_{2}\geq \cdots \geq n_{N}\geq 0}^{\infty}\det\left[d_{n_{j}+N-j,i}\right]\det\left[t_{i}^{n_{j}+N-j+\frac{1}{2}}-t_{i}^{-(n_{j}+N-j+\frac{1}{2})}\right].
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] .
Z_V^{(2)}\left(v_{1},z_1; \ldots; v_{n},z_n\right) =\sum_{u\in V}Z_V^{(1)}\left(Y[v_{1},z_1]\ldots Y[v_{n},z_n]u;\tau_1\right) Z^{(1)}_V(\overline{u};\tau_2).
\frac{i}{\hbar}\Delta{\cal A}_{J}^{\prime}= - \Gamma_{\{\mu \nu \}}{}^{\mu}\Delta q^{\nu}+ \frac{1}{2}\left[ \partial_{\{\mu}\Gamma_{\lambda \nu \}}{}^{\mu}+ \frac{1}{3}\left( \Gamma_{\{\mu \kappa \}}{}^{\mu}\Gamma_{\{\lambda \nu \}}{}^{\kappa}- \Gamma_{\{\lambda \kappa \}}{}^{\mu}\Gamma_{\{\nu \mu \}}{}^{\kappa}\right) \right] \Delta q^{\lambda}\Delta q^{\nu}+{\cal O}( ( \Delta q )^{3}) .
=\sum_{\tau=p, N_1}\sum_{\mu, \nu, \kappa \in\{+,-\}}\mathfrak{Re}\big[\int \overline{\widehat{(SU)_{\mu}}}(\xi)\widehat{(SU)_{\nu}}(\xi-\eta) \widehat{U_{\kappa}}(\eta) p_{\mu, \nu}^{\kappa,\tau}(\xi-\eta, \eta) d \eta d \xi,
+\frac{x}{\ell_n^{p,q}(x)}\sum_{k=0}^{n-1}\left[\frac{px}{q};p^{n-k+1}\frac{[k]_{p,q}}{[n-k+1]_{p,q}q^{k}};f\right] \frac{1}{[n-k]_{p,q}}p^{\frac{(n-k)(n-k+1)}{2}+1}q^{\frac{k(k-3)}{2}-2}\left[\begin{array}{c}n k\end{array}\right]_{p,q}x^{k}.
(2k+3) \lambda_j - 2kq \sim_E\begin{cases}2\lambda_1 + \lambda_2 + 2q -q_1-q_2 j=1, \lambda_{j-1}-\lambda_j + \lambda_{j+1}+q + q'_3 j \leq n-2 , \lambda_{j-1}+\lambda_j + \lambda_{j+1}+q - q_3 j \geq 3 , \lambda_{n-1}- 2\lambda_n + q'_1+ q'_2 +2q j = n. \end{cases}
a_{n}\frac{d^{n}x(t)}{dt^{n}}+a_{n-1}\frac{d^{n-1}x(t)}{dt^{n-1}}+...+a_{1}\frac{dx(t)}{dt}+a_{0}x(t)=\sum_{k=0}^{n}a_{k}\frac{d^{k}x(t)}{dt^{k}}=Af(t).
\mathcal{{F}}(x) = \int_{\mathbb{T}}{\sum_{1 \leq l<k \leq N}\frac{m_k m_l}{2M}{[{|\sum_{l \leq j \leq k-1}\dot{x}_j|^2}+ \frac{2M}{|\sum_{l \leq j \leq k-1}{x}_j|}]}dt}
\widetilde{A}(\tilde{z}, p) = \left(\frac{1}{\ell}\left(k\tilde{z}+ k\left(p - \frac{1}{k}{}^t\!a\right)A^{-1}b + c\right), \, k\left(p - \frac{1}{k}{}^t\!a\right)A^{-1}\right).
\log{\cal M}= \log ( \eta^{- 2}( U ) \frac{1}{2}\eta^{- 2}( \frac{T}{2}) ) - B C \left( ( \partial_{U}\log \eta^{- 2}( U ) ) ( \partial_{T}\log \eta^{- 2}( \frac{T}{2}) ) \right) +{\cal O}( ( B C )^{2}) .
\delta_{\ell,n}= \int_{-1}^1 m_{n,2}(u) h_{\ell,2}(u) \,\d u = \sum_{k=0}^\infty a_k \frac2{\pi}\int_{-1}^1 \frac{U_{n+2k}(u)}{n+2k+1}h_{\ell,2}(u) \sqrt{1-u^2}\,\d u.
-te_{\lambda}^{-r}(t) =\sum_{k=1}^{\infty}H_{k,\lambda}^{(r)}(-1)^{k}k!\frac{1}{k!}\big(e_{\lambda}(t)-1\big)^{k}&=\sum_{k=1}^{\infty}H_{k,\lambda}^{(r)}(-1)^{k}k!\sum_{n=k}^{\infty}S_{2,\lambda}(n,k)\frac{t^{n}}{n!}&=\sum_{n=1}^{\infty}\bigg(\sum_{k=1}^{n}H_{k,\lambda}^{(r)}(-1)^{k}k!S_{2,\lambda}(n,k)\bigg)\frac{t^{n}}{n!}.
\int_{\vert x \vert\leq r}\vert x \vert^{\beta_k-2(N+2)}e^{\hat{u}_k}\leq C \left( \int_0^{r}t^{(\beta_k-2(N+2))q+1}\, dt \right)^{\frac1q}= C \left( \frac{r^{l_k}}{l_k}\right)^{\frac1q},
|\nabla_x p_D(s,x,y)| \le \frac{c_{12}}{r}p_D(s,x,y) \le \frac{c_{13}}{r}\left(1 \land{\frac{V(d_D(x))}{s^{1/2}}}\right) \left(1 \land{\frac{V(d_D(y))}{s^{1/2}}}\right) p(s, |x-y|/4) &\le \frac{c_{14}}{r}\frac{V(r)}{\sqrt s}p(s,|x-y|/4).
x^{n_1}+\tilde a_2x^{n_2}+\tilde a_3x^{n_3}=\tilde g(x) \circ \left ( x^l\left (x^{m_1}+ax^{m_2}+ \textnormal{lower degree terms}\right)\right) &=x^{n_1}+ta x^{n_1+m_2-m_1}+\textnormal{lower degree terms},
\times \prod_{b = 1}^{N}\prod_{j = 1}^{m_{b}}\exp \left( - 2 e U_{1 b}\Big ( \varepsilon_{\mu \nu}\partial_{\mu}A_{\nu}^{( 1 )}, \frac{\pi}{\pi + g N}K^{( 1 )}\delta ( y_{j}^{( b )}) \Big ) \right)
F:=\left\{f\in L^{1}(\Pi\times\Theta):~|x|^{\alpha}|f(x,\alpha)|\lesssim_{u}1~\mbox{uniformly on $\Pi_{0}\times\Theta$}\right\}, F^{(1)}:=\left\{f\in F\cap C^{1}_{0}(\Pi\times\Theta):|x|^{\alpha+1}\left|\frac{\mathrm{d}f}{\mathrm{d}x}(x,\alpha)\right|\lesssim_{u}1~\mbox{uniformly on $\Pi_{0}\times\Theta$}\right\},
\int\limits_{\Omega}L_cd\mu_{\infty}=\lim \limits_{n\to \infty}\frac{1}{t_n}\int\limits_{0}^{t_n}L_c(\Phi(s;M_0,N_0))ds &=\frac{U(M_0)-U(\Phi^1(t_n;M_0,N_0))-t_n\bar{H}(c)}{t_n}&=-\bar{H}(c).
\det(x^1, \dots, x^{i-1}, \tilde{y}, x^{i+1}, \dots, x^{n+1}) = \sum_{\sigma \in S_{n+1}}\textnormal{sign}(\sigma) \tilde{y}_{\sigma(i)}\prod_{\substack{j = 1 j \neq i}}^{n+1}x^j_{\sigma(j)}\, ,
Q_{ii}^{[2]}=\sum_{m_0+\cdots +m_{{D-1}}=m}{D_{m_0\cdots m_{{D-1}}}\over (m_i+1)(m_i+2)}x_0^{m_0}\cdots x_i^{m_i+2}\cdots x_{{D-1}}^{m_{{D-1}}},
\left (2p_i -q_i^2 (\xi_i + \epsilon) \right ) (\tau - \tau_0) \leq b_{i}(\tau) - b_{i}(\tau_0)\leq (2p_i -q_i^2 (\xi_i - \epsilon)) (\tau -\tau_0), & (E(\xi) - \epsilon) (\tau -\tau_0) \leq a(\tau) - a(\tau_0) \leq (E(\xi) +\epsilon) (\tau - \tau_0),
\int_{B^2_{\lambda_i}(x_i)}e_{\widetilde \epsilon_i}(\widetilde d_i)(x,0)\,dx=\frac{\delta_0^2}{C(3)}=\max_{z\in B_\frac12^2}\int_{B^2_{\lambda_i}(z)}e_{\widetilde \epsilon_i}(\widetilde d_i)(x,0)\,dx,
\mathcal{E}=\mathcal{A}\ast E_{r}f=\left\{\mathcal{F}:\mathcal{F}(z)= (f \ast E_{r}f)(z) = z +\sum_{n=2}^\infty\frac{(-1)^{n-1}a_n}{(2n-1)(n-1)!}z^{n}, f\in\mathcal{A}\right\},
|Z+\mathcal I_1|_k =|e_1z_1+e_2z_2+\mathcal I_1|_k &=|e_2z_2+\mathcal I_1|_k=\left\{|e_2z_2+e_1x_1|_k\;:\;x_1\in \mathcal I_1\right\}&=\left\{e_2|z_2| +e_1|x_1|\;:\;x_1\in \mathcal I_1\right\}&= e_2\;\;|z_2|+ e_1\;\left\{|x_1|\;:\;x_1\in \mathcal I_1\right\}= e_2|z_2|.
\Omega_s=\{\omega\in\Omega:\exists\:t_0\:\:\forall\:t\geq t_0,\:\:\forall\:x\in C(0,kt)\:\: \exists\:y\in B(x,\rho(t))\:\:\: B\left(y,R_{\rho(t)}+b\right)\subseteq K^c\},
D_{0}^{2}Q_{2}( T ) + \omega^{2}Q_{2}( T ) = - ( D_{1}^{2}+ 2 D_{0}D_{2}) Q_{0}( T ) - 2 D_{0}D_{1}Q_{1}( T ) - 1 2 Q_{0}^{2}( T ) Q_{1}( T ) ,
\frac{1}{Q(\pi(q))Q(\pi'(q))Q(\xi)}\sim_{E_q(\Pi) E_{q'}(\Pi')}\ \cfrac{P^{(q)}(\Pi,\imath_{v_0})P^{(n-q-2)}(\Pi',\imath_{v_0})}{P^{(q+1)}(\Pi,\imath_{v_0}) P^{(n-q-1)}(\Pi',\imath_{v_0})}
\frac{w(\Phi(\omega))}{w(\Phi(\eta))}= \frac{w(\Phi(\eta)) + \int_0^1 \frac{d}{dt}\big|_{t=s}\left[w(\Phi(\eta)+s\tau)\right]~ds}{w(\Phi(\eta))}\leq 1 +\frac{\sup_{\upsilon\in B_{2\delta}(\Phi(\eta))}\nabla_{\tau}w(\upsilon)}{w(\Phi(\eta))},
E(\prod_i N^{(\frac{1}{2})}_{\{x_i,y_i\}}\prod_l (N^{(\frac{1}{2})}_{z_l}+1))=E(\prod_i C_{(x_i,y_i)}{\varphi^{\mathbb{R}}_{x_i}\varphi^{\mathbb{R}}_{y_i}}\prod_l \frac{1}{2}\lambda_{z_l}({\varphi^{\mathbb{R}}_{z_l}})^2 ).
v_{i}\vert_{_{D S R 1}}= \frac{2 \lambda p_{i}}{1 - e^{- 2 \lambda p_{0}}+ \lambda^{2}p^{2}}, v_{i}\vert_{_{D S R 2}}= v_{i}\vert_{_{E x a m p l e}}= \frac{p_{i}}{p_{0}}.
f'_x (t, x) = - \left(1 - \frac{m}{2}\right) \frac{x^{- m / 2}}{(2t)^{(n- m )/2}}\psi \left(\frac{x}{2t}\right) - \frac{x^{1 - m / 2}}{(2t)^{(n- m )/2}}\psi' \left(\frac{x}{2t}\right) , t > 0, x > 0,
\sum^{2N-1}_{n=1}\frac{\cos^2\frac{n\pi}{2N}}{\sin^2\frac{n\pi}{2N}}=\sum^{N-1}_{n=1}\frac{\cos^2\frac{n\pi}{2N}}{\sin^2\frac{n\pi}{2N}}+\frac{\cos^2\frac{N\pi}{2N}}{\sin^2\frac{N\pi}{2N}}+\sum^{2N-1}_{n=N+1}\frac{\cos^2\frac{n\pi}{2N}}{\sin^2\frac{n\pi}{2N}}&=2\sum^{N-1}_{n=1}\frac{\cos^2\frac{n\pi}{2N}}{\sin^2\frac{n\pi}{2N}}\,.
\displaystyle{\mbox{$_4$F$_3$}(1,1,a+1,1-a;\,2,2,2;\,1)}=a\,{\mbox{$_5$F$_4$}(1,1,1,1+a,1-a;\,2,2,2,2;\,1)}-\left( a-1 \right)\,{\mbox{$_5$F$_4$}(1,1,1,1+a,2-a;\,2,2,2,2;\,1)}\,.
\begin{array}{lll}V_{n+1}(x) = [K+\min_{x\leq y \leq x+B}(vy + L_{n+1}(y) + \int_0^\infty C_{n}(y-\xi)f_{n+1}(\xi) \mathrm{d}\xi) & -(vx + L_{n+1}(x) + \int_0^\infty C_{n}(x-\xi)f_{n+1}(\xi) \mathrm{d}\xi)]^-.\end{array}
{\cal K}^{( 2 )}= 2 \partial \omega^{( 1 )}+{\frac{1}{2 \pi \alpha^{\prime}}}( i_{k}C^{( 3 )}) - 2{\cal K}^{( 1 )}\left( C^{( 1 )}+ ( i_{k}C^{( 1 )}) A^{( 1 )}\right) \, .
F_N(e^{itP}f)= a^{1/2}N^{-1/4}(\sum_{n=0}^{N/2-1}f(naN^{-1/2}+t\hbar)u(n)+ & \sum_{N/2}^{N-1}f((n-N)aN^{-1/2}+t\hbar)u(n))
\dim A\big(\mathcal{W}(p)^{A_m}\big)=\dim A_{-1}\big(\mathcal{W}(p)^{A_m}\big) + \dim A_0\big(\mathcal{W}(p)^{A_m}\big)+ \dim A_1\big(\mathcal{W}(p)^{A_m}\big) \phantom{\dim A\big(\mathcal{W}(p)^{A_m}\big)}{}\le \big(2 m^2 p + 2\big) p -1 + 2p=\big(2 m^2 + 4\big) p-1.
{\cal H}= - \frac{1}{4}\int_{- \infty}^{\infty}d \tau : [ P^{2}+{X^{\prime}}^{2}- \frac{\sqrt{\pi}}{\beta v^{2}}( P X^{\prime}P + \frac{1}{3}{X^{\prime}}^{3}) - \frac{1}{2 \beta \sqrt{\pi}}( \frac{v^{\prime \prime}}{3 v^{3}}- \frac{( v^{\prime})^{2}}{2 v^{4}}) X^{\prime}] : ,
[Tu,Tv,T_1(w)]+[Tu,T_1(v),Tw]+[T_1(u),Tv,Tw] = T\Big(D(Tu,T_1(v))w+D(T_1(u),Tv)w+\theta(Tv,T_1(w))u+\theta(T_1(v),Tw)u &-\theta(Tu,T_1(w))v-\theta(T_1(u),Tw)v\Big)+T_1\Big(D(Tu,Tv)w+\theta(Tv,Tw)u-\theta(Tu,Tw)v\Big),
A(1, \, (1,0,0)) =\frac{1}{5}\int_{0}^{1}(1-s)s^2ds\int_{0}^{1}sds = \frac{1}{5!}A(1, \, (0,1,0)) =\frac{1}{5}\int_{0}^{1}s^3ds\int_{0}^{1}(1-s)sds= \frac{1}{5!}A(1, \, (0,0,1)) =\frac{1}{5}\,\cdot\,\frac{1}{2}\,\int_{0}^{1}s^3ds\,\int_{0}^{1}s^2ds=\,\frac{1}{5!}.
\sum_{x=4}^{a}\sum_{y=a+1}^{b}u_{x,y,a,b}~=~ \sum_{x=4}^{a}\sum_{y=a+1}^{b}\Bigl( x \sqrt{(x \wedge a)(x \wedge b)}- y \sqrt{(y \wedge a)(y \wedge b)}\Bigr)^2 p(x) p(y) begin{align*}.5em] =~ \sum_{x=4}^{a}\sum_{y=a+1}^{b}\bigl( x^2 - y \sqrt{a y}\bigr)^2 p(x) p(y).
< \Psi_{1}| \Psi_{2}> = N \int_{p^{2}= - m^{2}}\frac{d{\bf p}}{p_{0}}\overline{{{\Theta}}}( p )_{1}^{A ( l_{1}) B ( l_{2}) . . . C ( l_{r - 1}) a ( l_{r})}\Theta ( p )_{2 A ( l_{1}) B ( l_{2}) . . . C ( l_{r - 1}) a ( l_{r})}
Q(T_1+T_2) =\{\,\{f,Q(h+k)\}:\, \{f,h\}\in T_1, \,\, \{f,k\}\in T_2\,\}=\{\,\{f,Qh+Qk)\}:\, \{f,h\}\in T_1, \,\, \{f,k\}\in T_2\,\}=QT_{1}+QT_{2}.
\sum_{\frac{ea^2}{L^2}< x_1 < \sqrt{a+dL}}2^{-\omega(x_1)}\frac{(\log x_1)^{-\frac12}}{x_1}\ll \sum_{k=\lfloor\log\frac{ea^2}{L^2}\rfloor}^{\lfloor\log\sqrt{a+dL}\rfloor}k^{-1}\leq 2 \log\log(a+dL) .
\mathbf{E}\left[\left\|\mathcal{X}^{t}-\mathcal{X}^{\star}\right\|_{F}^{2}|\mathcal{X}^{0}\right]\leq\left(1-\mathop{\min}_{k=1,2,\cdots,l}\frac{\lambda_{\min}\left(\widehat{\mathcal{A}}_{(k)}\widehat{\mathcal{A}}^{H}_{(k)}\right)}{\|\mathcal{A}\|_{F}^{2}}\right)^{t}\left\|\mathcal{X}^{0}-\mathcal{X}^{\star}\right\|_{F}^{2}
H = \frac{1}{2}p_{i}^{A}p_{i}^{A}+ \frac{1}{4}G_{i j}^{A}G^{i j A}- A^{0 A}\nabla^{i}p_{i}^{A}- \frac{m^{2}}{2}A^{0 A}A^{0 A}+ \frac{m^{2}}{2}A^{i A}A^{i A}.
\pi ( x ) ={\frac{\partial{\cal L}_{\sigma}}{\partial \partial_{+}\psi ( x )}}= i \bar{\psi}\gamma^{+}= \left( \begin{matrix}{\psi_{R}^{*}}&{\psi_{L}^{*}}\\ \end{matrix}\right) \left( \begin{matrix}{i \sqrt{1 + s}}&{0}\\{0}&{i \sqrt{1 - s}}\\ \end{matrix}\right) .
H^{d-2}(Y_1,\Omega^{d-2}_{Y_1}\otimes_{\mathcal{O}_{Y_1}}\mathcal{O}_{X_1}(Y_1)|_{\mathcal{O}_{Y_1}})\rightarrow H^{d-1}(Y_1,\mathcal{F}_i)\rightarrow H^{d-1}(Y_1,\mathcal{F}_{i+1}) \rightarrow H^{d-1}(Y_1,\Omega^{d-2}_{Y_1}\otimes_{\mathcal{O}_{Y_1}}\mathcal{O}_{X_1}(Y_1)|_{\mathcal{O}_{Y_1}})
I_t(\mu)=\int |\hat{\mu}(\omega)|^2|\omega|^{t-n}d\omega=\int_{|\omega|\leq 1}|\hat{\mu}(\omega)|^2|\omega|^{t-n}d\omega+\sum_{j\geq 0}\int_{2^j\leq |\omega|\leq 2^{j+1}}|\hat{\mu}(\omega)|^2|\omega|^{t-n}d\omega.
\sum_{x_1,x_2\in X}c(x_1,x_2) = \frac{1}{2}\left( \Delta\sum_{y\in Y}d_{X}(y) -\gamma\Delta^3 \right)-\frac{1}{2}m_{X,Y}= \frac{1}{2}\left( (\Delta-1) m_{X,Y}-\gamma\Delta^3 \right).
X_{1,V}=x^1\dfrac{\partial}{\partial y^1}+x^3\dfrac{\partial}{\partial y^3}, Y_{1,V}=-\dfrac{x^2}{x^1}\dfrac{\partial}{\partial y^3}, X_{2, V}=x^1\dfrac{\partial}{\partial y^2}, Y_{2, V}=\dfrac{\partial}{\partial y^3}.
G(t) = \prod_{j=1}^{\frac{n-1}{2}}(t - \lambda_{2j}) + \frac{4}{n + 3}\sum_{j=1}^{\frac{n-1}{2}}\sin^{2}\left(\frac{2j \pi}{n + 3}\right) (\lambda_{2j}- \lambda_{n+1}) \prod_{\substack{m=1 m \neq j}}^{\frac{n-1}{2}}(t - \lambda_{2m}).
G = (-1) Q^2 R^4+ (-1+i) Q^3 R^3+ Q^4 R^2+ Q^5 R+ (i) Q^6+ (-i) P Q R^4 + (-i) P Q^2 R^3 + (-1-i) P Q^4 R+ (-1-i) P Q^5+ P^2 R^4+ (-1) P^2 Q R^3 + (i) P^2 Q^3 R+ (-1) P^3 R^3 + (1+i) P^3 Q^2 R+ (-1+i) P^3 Q^3+ (-1) P^4 R^2 + P^5 R+ (1+i) P^5 Q+ P^6.
g_z(0) := 1-\frac{z}{2}, g_z(t) := 1-\frac{e^{\frac{z}{2}t}-e^{-\frac{z}{2}t}}{(e^{\frac{t}{2}}-e^{-\frac{t}{2}})(e^{\left(\frac{z-1}{2}\right)t}+e^{-\left(\frac{z-1}{2}\right)t})}.
-{\frac{\lambda^{2}}{3 2}}( F_{\mu \nu}^{2})^{2}+{\frac{\lambda^{2}}{8}}F^{\mu \nu}F_{\nu \alpha}F^{\alpha \beta}F_{\beta \mu}- i{\frac{\lambda}{2}}( \overline{{{\psi}}}_{+}\Gamma_{\nu}\partial_{\mu}\psi_{+}+ \overline{{{\psi}}}_{-}\Gamma_{\mu}\partial_{\nu}\psi_{-}) F^{\nu \mu}
\frac{\langle \langle \psi^{\prime}| e^{- \hat{H}\tau_{f}}\hat{O}_{1}( \tau_{1}) \hat{O}_{2}( \tau_{2}) \ldots \hat{O}_{n}( \tau_{n}) e^{\hat{H}\tau_{i}}| \psi \rangle \rangle}{\langle \langle \psi^{\prime}| e^{- \hat{H}( \tau_{f}- \tau_{i})}| \psi \rangle \rangle}.
L_{eff}= - R = \frac{1}{2}A^{\mu \nu}_{ij}\; \partial_{\mu}{\delta y}^i \, \partial_{\nu}{\delta y}^j + B^{\mu}_{ij}\;{\delta y}^i \, \partial_\mu \delta y^j + \frac{1}{2}C_{ij}\; \delta y^i \delta y^j \, ,
\langle \tilde{E}^{T}, e_{n}^{(2)}\rangle + \eta \, \langle \tilde{E}^{T}, \nabla M^{-ka}\left( e_{n}^{(2)}\right) \rangle - k^{2}\, \eta \, a^{2}\, \langle \tilde{E}^{T}, N^{-ka}\left( e_{n}^{(2)}\right) \rangle = \langle \tilde{E}^{Inc}, e_{n}^{(2)}\rangle.
+ ( U^{( 1 )}U^{( 2 )})^{- 1}d y_{3}^{2}+ ( H^{( 2 )}U^{( 2 )})^{- 1}d y_{4}^{2}+ ( H^{( 2 )}U^{( 1 )})^{- 1}( d y_{5}^{2}+ d y_{6}^{2}+ d y_{7}^{2}) + d x^{\alpha}d x^{\alpha}]
\sum_{i , j , k , l = 1}^{h}a_{i j}^{p}A_{i k}A_{j l}\overline{a}_{k l}^{q}= \sum_{i , j , k , l , m , n = 1}^{h}b_{i j m}^{p}A_{i k}A_{j l}A_{m n}\overline{b}_{k l n}^{q}.
I - \mathfrak{z}_1 T_{1,0}^* - \mathfrak{z}_2 T_{2,0}^* = \begin{pmatrix}1 \frac{1}{2}(\mathfrak{z}_1 + \mathfrak{z}_2) \frac{1}{2}(\mathfrak{z}_1 + \mathfrak{z}_2) 0 1 + \frac{1}{2}\mathfrak{z}_2 -\frac{1}{2}\mathfrak{z}_2 0 - \frac{1}{2}\mathfrak{z}_1 1 + \frac{1}{2}\mathfrak{z}_1 \end{pmatrix}.
U(t):=e^{\sum_{i=1}^N\left[\int_0^tB_i(s)d\beta_i+\beta_i\theta_i-\frac{1}{2}\int_0^tB_i^2(s)ds-\frac{1}{2}\theta_i^2t-\theta_i\int_0^tB_i(s)ds\right]}y(t).
\binom{n_{1}}{2}n_{k}a_{1}^{2}a_{k}+\binom{n_{k}}{2}n_{1}a_{k}^{2}a_{1}\leq n_{1}n_{k}\frac{l^{2}b^{2}}{n_{1}n_{k}}\left( lb-\frac{lb}{\sqrt{n_{1}n_{k}}}\right) & <l^{2}\left( l-1\right) b^{3},
E_1 := \overline{W_1}, E_2 := \overline{W_2}\setminus \left(\overline{W_1}\cup U_1\right), E_3 := \overline{W_3}\setminus \left(\overline{W_1}\cup U_1 \cup \overline{W_2}\cup U_2\right), \dots, E_m := \overline{W_m}\setminus \left(\overline{W_1}\cup U_1\cup \overline{W_2}\cup U_2\cup \dots \cup \overline{W_{m-1}}\cup U_{m-1}\right).
c^k_{\mathrm{s2b},2}\left(N_{\mathrm{sct}}\right) = \mathsf{E}\left\lbrace B \sum_{i=1}^{N_{\mathrm{sct}}}\log_2\left( \frac{P_{\mathrm{sc}}}{N_{\mathrm{sct}}}\left[B\sigma^2_{\mathrm{bs}}\|(\mathcal{H}^{\dagger}_{\mathrm{b2s}})^\mathrm{T}_j \|^{2}+ I_{\mathrm{u2b}}\left( i, k \right) \right]^{-1}+1 \right) \right\rbrace.
\begin{array}{l}\displaystyle{\vphantom{\Big(}\{w(f)_\lambda L_0\}_{z,\rho}=\sum_{j\in J^f_0}\{w(f)_\lambda w(q_j) \}_{z,\rho}w(q^j)}\displaystyle{\vphantom{\Big(}=\sum_{j\in J^f_0}\sum_{i\in J^f_{\leq-\frac12}}w(q_{i})w([q^{i},q_j]^\sharp)w(q^j)\,.}\end{array}
h(x, y)=J_p\big((m+\varepsilon)\,(\widetilde\Gamma(y)-\widetilde\Gamma(x))\big)-J_p\big((m+\varepsilon)\,(\widetilde\Gamma(y)-\widetilde\Gamma(x))-(\varepsilon+\varepsilon_0)\,\widetilde\Gamma(y)\big).
D(4i,\varepsilon) =\sigma_{\varepsilon}(4iI)=\sigma_{\varepsilon}([I\bullet iI,I]_{\ast})=\sigma_{\varepsilon}([\varphi(I)\bullet \varphi(iI),\varphi(I)]_{\ast}) &=\sigma_{\varepsilon}([\varphi(I)\varphi(iI)+\varphi(iI)\varphi(I)^{\ast},\varphi(I)]_{\ast})=\sigma_{\varepsilon}(4\varphi^{2}(I)\varphi(iI)).
U_{\theta}(r,r',r'') :=B^{\tilde{h}_{t_{\theta}}}(y_{\theta},a_{\theta}^{-\frac{2}{n-2}}e^{-f(t_{\theta})}r)\times [t_{\theta}-a_{\theta}^{-\frac{2}{n-2}}r'',t_{\theta}+a_{\theta}^{-\frac{2}{n-2}}r''] &\times B^{\sigma^{n-k-1}}(z_{\theta},a_{\theta}^{-\frac{2}{n-2}}r')
X_j \tilde{s}_{v_i}(g) = \frac{d}{dt}|_{t=0}\tilde{s}_{v_i}(g e^{tX_j}) &= (g, \frac{d}{dt}|_{t=0}g e^{tX_j}e_i) &= (g, g \rho(X_j) e_i).
{\cal L}_{\ast}= \omega \left( \frac 1{\alpha^{\prime}}V ( X ) + g^{i j}\partial_{i}X^{A}\partial_{j}X^{B}G_{A B}( X ) + \omega^{i j}\partial_{i}X^{A}\partial_{j}X^{B}H_{A B}( X ) + D ( X ) R + O ( \alpha^{\prime}) \right) .
\left[ d m\right]_{2}^{4}=\prod_{i=1}^{3}\frac{d z_i}{V_i^{'}(0)}\prod_{\mu=1}^{2}\frac{d k_{\mu}}{k_{\mu}^{2}}\frac{d \xi_1 d\eta_1}{(\xi_1 -\eta_1)^2}
\big|\mathbb{E}_{n,0}^Q(\varphi_n) - d(n)^{-1}\sum_{i = 1}^{d(n)}\mathbb{E}_{n,i}^Q(\varphi_n)\big|^2 = \left|\mathbb{E}^Q_{n,0}(\varphi_n(1 - L_n))\right|^2 \leq \mathbb{E}^Q_{n,0}\left((1 - L_n)^2\right) = \mathbb{E}^Q_{n,0}\left(L_n^2\right) -1,
\tilde M=\min\Big\{\frac{(\chi_2\lambda_2\mu_2-\chi_1\lambda_1\mu_1)_{-}+\chi_1\mu_1(\lambda_1-\lambda_2)_{-}}{\lambda_2}, \frac{\chi_2\mu_2(\lambda_1-\lambda_2)_{-}+ (\chi_2\mu_2\lambda_2-\chi_1\mu_1\lambda_1)_{-}}{\lambda_{1}}\Big\},
\sum_{\ell\le x}\frac{\phi(\ell)}{\ell}= \frac{6}{\pi^2}x + O\left( \log x \right) \sum_{\ell\le x}\phi(\ell) = \frac{3}{\pi^2}x^2 + O\left( x \log x\right) \sum_{\ell\le x}d(\ell) = O\left(x \log x\right)
\Psi(x_{1},x_{2})=\dfrac{1}{4a^{2}}\mathcal{E}(x_{1},x_{2},2)\doteq\frac{1}{2\pi}\log \left| \vartheta_{1}\left( \left. \frac{\pi}{2a}(x_{1}+ix_{2}) \right| i \right)\right| -\dfrac{x_{2}^{2}}{8a^{2}}
C\sum_{0<N\le A}e^{-\beta N + N^\kappa}\le CA \|e^{-\beta N + N^\kappa}\|_\infty \le \frac{C2^{1/(1-\kappa)}}{\beta^{1/(1-\kappa)}}\exp\Big(\Big((1-\kappa)\,\kappa^{-\frac{\kappa}{1-\kappa}}\Big) \beta^{-\frac{\kappa}{1-\kappa}}\Big) = \frac{a_0}{\beta^{c_0}}\exp\Big(b_1 \beta^{-c_1}\Big) \le a_2 \exp\Big(\beta^{-c_2}\Big) (\beta>0).
u_x = -\beta yz u_y = \beta xz u_z = \beta \left[ xy + \sum\limits_{n=1}^\infty \frac{32a^2(-1)^n}{\pi^3 (2n-1)^3}\sin \left( (2n-1) \frac{\pi x}{2a}\right) \frac{\sinh ((2n-1)\frac{\pi y}{2a})}{\cosh ((2n-1)\frac{\pi y}{2a})}\right]
\epsilon \alpha^{\tau}= -\zeta_{\ell}^{\tau^2}\epsilon^{1+\tau}- \zeta_{\ell}^{\tau^3}\epsilon^{1+\tau+\tau^2}- \cdots - \zeta_{\ell}^{\tau^{\ell}}\epsilon^{1+\tau + \cdots + \tau^{\ell-1}}& = - \zeta_{\ell}^{\tau^2}\epsilon^{1+\tau}- \zeta_{\ell}^{\tau^3}\epsilon^{1+\tau+\tau^2}- \cdots - \zeta_{\ell}^{\tau}\epsilon &= \alpha,