harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\cos\alpha(\cos^2\theta - \sin^2\theta) \frac{\cos^2\alpha \sin\theta}{\sin \alpha}- \sin\alpha \sin\theta \cos\alpha(\sin^2\theta - \cos^2\theta) = \cos\alpha \sin\theta (\cos^2\theta - \sin^2\theta)\left(\frac{\cos^2\alpha}{\sin\alpha}+ \frac{\sin^2 \alpha}{\sin\alpha}\right) = \cot\alpha \sin\theta(\cos^2\theta - \sin^2\theta).
g_2:=\frac{i}{\pi}\sum_{j=1}^N \lambda_j\Big(\frac{2z_{tt}+i-\partial_t^2 z_j}{(z(\alpha,t)-z_j(t))^2}-2\frac{(z_t-\dot{z}_j(t))^2}{(z(\alpha,t)-z_j(t))^3}\Big).
g_s=\xi_3\left(\tfrac{r_s}{s^{1/4}}\right)\left( \sum_{k=1}^{2n_0}f_k(\Phi_{s,k}^{-1})^*g_{0,k}(s) \right) + \left(1-\xi_3\left(\tfrac{r_s}{s^{1/4}}\right)\right)(\Psi_s^{-1})^*g_Z,
\!\!\!\!\!\! R_{,u}= S_{\mathrm{GA}}(R_{\mathrm{GA},u}^*)= \sum_{i=1}^{N_{max}}\frac{R_{\mathrm{GA},u}^*}{i\cdot\pi}\cdot \!\!\!\!\!\! \int_{0}^{\infty}\bigg[ \mathcal{I}m \Big\{\Phi_{\mathrm{GA}}(t;i, R_{\mathrm{GA},u}^*) - \Phi_{\mathrm{GA}}(t;i-1,R_{\mathrm{GA},u}^*) \Big\}\bigg] dt,
\alpha \cdot (x_{+}(h) - x_{+}(\tilde{h})) = \sqrt{1 + |m_{\alpha}|^{2}}\left[ g \left( \frac{h - h_{0}}{e_{2}\cdot \theta_{x_{0}}^{\perp}}\right) - g \left( \frac{\tilde{h}- h_{0}}{e_{2}\cdot \theta_{x_{0}}^{\perp}}\right) \right].
I ( n_{b}, n_{b}; a^{( b )}) = \int d \mu_{\tilde{Q}}[ \varphi ] d \mu_{\tilde{C}}[ \theta ] \; \exp \left( - 2 \sum_{b = 1}^{N}\sum_{j = 1}^{n_{b}}\Big ( \chi^{( b )}, \delta_{n}( x_{j}^{( b )}) \! - \! \delta_{n}( y_{j}^{( b )}) \Big ) \right) \times
\left[\frac{1}{2\hat{j}+ (2k + 1) \hat{I}}, \, \tau^\dagger_\theta\right] = \left(\frac{1}{2\hat{j}+ (2k + 1) \hat{I}}- \frac{1}{2\hat{j}+ (2(k-\theta)+1) \hat{I}}\right) \tau^\dagger_\theta.
v=\left(-2\frac{\theta^1_y}{\theta^1}x+\frac{\theta^0}{\theta^1}\right)_y, v=-2\frac{\theta^1_y}{\theta^1}\quad\mbox{with}\theta^i=\theta^i(t,y)\colon\ \theta^i_t=\theta^i_{yy},i=0,1.
A(x,v) =a_3(x)v^3+\frac{x}{1-v}(v^4A(x,1)-v^2A(x,v)), B(x,v) =b_3(x)v^3+xv(v^3A(x,v)+B(x,v)+C(x,v)-c_2(x)v^2), C(x,v) =c_2(x)v^2+\frac{x}{1-v}(v^3B(x,1)-vB(x,v))+\frac{x}{1-v}(v^3C(x,1)-vC(x,v)),
u_{t}^{i}- J_{2}^{i k}\delta_{u^{k}}H_{0}( u ) ={\cal O}_{j}^{i}( r^{l}) \Big [ r_{t}^{j}- J_{1}^{j k}\delta_{r^{k}}H_{0}\Big |_{u^{m}= M^{m}( r^{n})}\Big ]
\left[\overline{\Theta}_{\rm null}\right] \cdot \left[\overline{\mathcal{T}}\right] |_{V_2}= 16896, \left[ \overline{\mathcal{H}yp}_4\right] |_{V_2}= 30, \left[\overline{\mathcal{W}}_2 \right] |_{V_2}= 6, \gamma_1 |_{V_2}=0.
\mathrm{tr}(\log (E_q-\frac{1}{\lambda}A_q ))= \sum_{j=1}^\infty \frac{(-1)^{j-1}}{j}(-\frac{1}{\lambda})^j \mathrm{tr}(A_q^j) = \sum_{j=1}^\infty \frac{(-1)^{j-1}}{j}(-\frac{1}{\lambda})^j q^j \varphi(q) = \varphi(q) \log(1-\frac{q}{\lambda}).
\begin{aligned}k\sum_{l=1}^k\binom{c_nkn}l(k)_l\sum_{j=n-N+1}^{n-1}\frac{(k(j-1))_{c_nkn-l}}{(kn)_{c_nkn}}\frac 1{k(n-j+1)-l}= &k\sum_{l=1}^k\binom{c_nkn}l(k)_l\sum_{s=1}^{N-1}\frac{(k(n-s-1))_{c_nkn-l}}{(kn)_{c_nkn}}\frac 1{k(s+1)-l}.\end{aligned}
rE'(r) =(n-p+O(r))E(r)+(pr+O(r^2))A &\geq (n-p+O(r))E(r)+\frac{p+O(r)}{K}E(r) &=(n-p+\frac{p}{K}+O(r))E(r) &\geq (n-p+p\gamma)E(r)
\left[{\partial \over \partial t}+ \int dx \; \left( \Pi (x) \;{\delta \over \delta \Phi (x)}- \big( m^2 \Phi (x) - \nabla^2 \Phi (x) \big){\delta \over \delta \Pi (x)}+{\cal K}_I(x) \right) \right]W[\Phi,\Pi;t] = 0 \;,
T = \left( \begin{array}{c c c c}{0}&{- g_{z}}&{g_{y}}&{f_{x}}\\{g_{z}}&{0}&{- g_{x}}&{f_{y}}\\{- g_{y}}&{g_{x}}&{0}&{f_{z}}\\{- f_{x}}&{- f_{y}}&{- f_{z}}&{0}\\ \end{array}\right) .
(\widetilde{M}v)_i = (1-\widetilde{w}_i) v_i + \widetilde{w}_i v_{i+1}= \left(1-\frac{v_i}{v_{i+1}-v_i}\right) v_i + \frac{v_i}{v_{i+1}-v_i}v_{i+1}= v_i - \frac{v_i}{v_{i+1}-v_i}v_i + \frac{v_i}{v_{i+1}-v_i}v_{i+1}= 2v_i.
\sum_{\gamma\in\Gamma}\sum_{n=2}^{\lfloor T/|\gamma|\rfloor}\frac{n^{-1}}{1-E_{\gamma}^{-n}}+\sum_{\gamma\in\Gamma:|\gamma|\le T}\frac{E_{\gamma}^{-1}}{1-E_{\gamma}^{-1}}\le\sum_{n=2}^{\infty}\sum_{\gamma\in\Gamma:|\gamma|\le T/n}1+\sum_{\gamma\in\Gamma:|\gamma|\le T}2\cdot E_{\gamma}^{-1}.
{\rm Fil}^*{\rm THH}((A, M) ;{\Bbb Z}_p) := R\Gamma_{\rm lqsyn}((A, M), \tau_{\ge 2*}{\rm THH}((-, -) ;{\Bbb Z}_p)){\rm Fil}^*{\rm TC}^-((A, M) ;{\Bbb Z}_p) := R\Gamma_{\rm lqsyn}((A, M), \tau_{\ge 2*}{\rm TC}^-((-, -) ;{\Bbb Z}_p)){\rm Fil}^*{\rm TP}((A, M) ;{\Bbb Z}_p) := R\Gamma_{\rm lqsyn}((A, M), \tau_{\ge 2*}{\rm TP}((-, -) ;{\Bbb Z}_p))
z^2 H^{(1,2)}_{\nu}(z) = \pm{1\over \sin{\nu \pi}}\left[ -{2^{\nu}z^{2-\nu}\over \Gamma (1-\nu )}\left( 1 + O(z^2)\right) + e^{\mp\nu\pi i}{z^{\nu +2}\over 2^{\nu}\Gamma (\nu +1 )}\left( 1 + O(z^2)\right) \right]\,,
( V\tilde v_1(\eta^{-1}\lambda_iy,t), \psi_2 )_{L_y^2(B_{4{\sf R}_{\sf in}})}= \int_{|y|<4{\sf R}_{\sf in}}V(y) \tilde v_1(\eta^{-1}\lambda_iy,t) \psi_2(y) dy = \eta^n\lambda_i^{-n}\int_{|\xi|<4{\sf R}_{\sf in}\eta\lambda_i^{-1}}V(\eta\lambda_i^{-1}\xi) \tilde v_1(\xi,t) \psi_2(\eta\lambda_i^{-1}\xi) d\xi.
S=\sum_{n\in\mathcal{A}(x)}\Bigl(\sum_{i=1}^k\mathbf{1}_{\mathcal{P}}(L_i(n))-m-k\sum_{i=1}^k\sum_{\substack{p|L_i(n) p<x^\rho p\nmid B}}1\Bigr)w_n=S_1-S_2-S_3,
\partial_\mu D^\prime(p||p_{\mu,\sigma}) = \frac{1}{\mu_0^2}( \mu - \mathbb{E}_p[x]) + \frac{\mu}{\sigma_0^4}( \mu^2 + \sigma^2 - \mathbb{E}_p[x^2]), \partial_\sigma D^\prime(p||p_{\mu,\sigma}) = \frac{\sigma}{\sigma_0^4}(\mu^2 + \sigma^2 - \mathbb{E}_p[x^2]).
W_5(p,q,r; \psi, \phi, \eta) = \left( \begin{array}{ccccc}1+ \frac{p^2 + q^2 + r^2}{2}p q r -\frac{p^2 + q^2 + r^2}{2}p -p q R(\psi, \phi, \eta) -q r -r \frac{p^2 + q^2 + r^2}{2}p q r 1 -\frac{p^2 + q^2 + r^2}{2}\end{array}\right)
p(x) \geq a_{i_0}(x_{i_0}-x_{i_0}^0)^2+C &\geq a_{i_0}x_{i_0}^2-2a_{i_0}x_{i_0}x_{i_0}^0+\tilde{C}&\geq (a_{i_0}-\epsilon) x_{i_0}^2-C_\epsilon{x_{i_0}^0}^2+\tilde{C}&\geq C_0|x|^2-\tilde{C}_0,
I_{\mathcal{O}}(\alpha)=\left[ \mathcal{O}^{+}:\mathbb{Z}[\alpha]^{+}\right]=\left[ \mathcal{O}^{+}:\mathbb{Z}_{M}[\alpha]^{+}\right] \cdot\left[\mathbb{Z}_{M}[\alpha]^{+}:\mathbb{Z}[\alpha]^{+}\right] .
E=\left( \begin{array}{ccc}E_{0}E_{01}E_{01}E_{01}^T E_1 E_1 E_{01}^T E_1 E_1 \end{array}\right) \, \, \, \mbox{and}\, \, \, D=\left( \begin{array}{ccc}0 0 0 0 D_1 -D_1 0 -D_1 D_1 \end{array}\right).
\bar R_{Sk}=\lim_{N\to\infty}\frac{1}{N}\sum_{i=1}^N r_k^{\rm R}(i) R_{Sk}(i) &=\lim_{N\to\infty}\frac{1}{N}\sum_{i=1}^N r_k^{\rm R}(i) C_{Sk}(i), \bar R_{kD}=\lim_{N\to\infty}\frac{1}{N}\sum_{i=1}^Nr_k^{\rm T}(i) R_{kD}(i) &=\lim_{N\to\infty}\frac{1}{N}\sum_{i=1}^N r_k^{\rm T}(i) \min\{Q_k(i-1),C_{kD}(i)\}.
i^{3,3}(H^3(Y,V ;\mathbb{Q})) = 1, \, i^{2,2}(H^3(Y,V;\mathbb{Q})) = ph(Y,w) - 3, \, i^{1,1}(H^3(Y,V;\mathbb{Q})) = ph-2, i^{2,1}(H^3(Y,V;\mathbb{Q})) = \,i^{1,2}(H^3(Y,V;\mathbb{Q})) = h^{1,2}(Z), \, i^{2,0}(H^3(Y,V;\mathbb{Q})) = i^{0,2}(H^3(Y,V)) = 1.
\begin{aligned}w =ay_0y_1x_2x_3+by_0x_1y_2x_3+ax_0y_1y_2x_3+cx_0x_1x_2x_3 \quad+ax_0x_1y_2y_3+bx_0y_1x_2y_3+ay_0x_1x_2y_3+cy_0y_1y_2y_3. \end{aligned}
T r K_{B_{3}^{\alpha}}= \left({\frac{V_{B_{3}^{\alpha}}}{l^{3}}}+{\frac{A_{+}}{l}}( 2 \pi \alpha ) c_{2}( \alpha ) ~ \bar{s}~ \right){\frac{e^{- \mu \bar{s}}}{( 4 \pi \bar{s})^{3 / 2}}}~ + ~ E S ~ ~ ,
h_{\alpha_1, \alpha_2, \beta_1, \beta_2}(r_1,r_2,j_1,j_2) =\sum_{\gamma_1, \gamma_2}c_{\alpha_1, \alpha_2, \beta_1,\beta_2}^{\gamma_1, \gamma_2}\varphi_{\gamma_1, \gamma_2}(r_1,r_2,j_1,j_2).
\d(S_1,S_2)=\sup_{u\in\mathbb{S}^{n-1}}\max\left[\frac{r_{S_1}(u)}{r_{S_2}(u)},\frac{r_{S_2}(u)}{r_{S_1}(u)}\right]=\inf \left\{a\geq 1: \frac{1}{a}S_1\subseteq S_2 \subseteq a S_1\right\}.
C^{\mathrm{dl},k}_{2}\left( N_{\mathrm{sct}}, N_{\mathrm{scr}}\right) = \left(\frac{T_{\mathrm{dl}}}{T}\right) \min\left\lbrace c^k_{\mathrm{b2s},1}\left( N_{\mathrm{scr}}\right), c^k_{\mathrm{s2u},2}\left( N_{\mathrm{sct}}\right) \right\rbrace.
* j_{F1}= \frac{\tilde f 2 \kappa_{10}T_{F1}}{2}\delta^{(6)}(x) \wedge_{i=1}^6 dx_i \wedge\left( \delta( x_9 - \hat{x}_{9}(\rho)) \frac{d \psi}{2\pi}\wedge dx_9 + \delta(\rho - \hat{\rho}(x_9))d \rho \wedge \frac{d \psi}{2\pi}\right),
y^{(\ell)}(z) = \frac{(-1)^{\ell}\ell!}{z^{\ell+1}}+ \sum_{l=\lceil (\ell+1)/2\rceil}^{\infty}\frac{B_{2l}}{2l}\frac{z^{2l-\ell-1}}{(2l-\ell-1)!},0<|z|<2\pi.
\kappa(\mathbb{R}^n) = \sup\Bigg\{\|\nabla u\|_{L^2(\mathbb{R}^n)};~~ u\in W^{1,2}(\mathbb{R}^n,\mathbb{R}^n), ~\|D(u)\|_{L^2(\mathbb{R}^n)}=1 \Bigg\}= \sqrt{2}.
I_{k,j}'' \left( m_{1},\dots, m_{k}\mid n_{1}, \dots, n_{j}\right) = \overline{\left\{(X, \ell, p_{1}, \dots, p_{k}, \ell', q_{1}, \dots, q_{j}) \mid X\cap \ell = \sum_{i}m_{i}p_{i}, \, \, \, X \cap \ell' = \sum_{i}n_{i}q_{i}\right\}}
{C}_{X^n \rightarrow Y^n}^{FB,A.M}= \sup_{{\cal P}^{A.M}_{0,n}}\sum_{t=0}^n{\bf E}^{\pi}\left\{\log\Big(\frac{q_t(\cdot|Y_{t-M}^{t-1}, X_t)}{\nu_t^{\pi}(\cdot|Y_{t-M}^{t-1})}(Y_t)\Big)\right\}
\alpha^{\prime}m^{2}_{\rm vec}\equiv -\alpha^{\prime}\sum_{\sigma,\rho=0}^{p^{\prime}}G^{\sigma\rho}k_{\sigma}k_{\rho}=0~,\qquad \sum_{\sigma,\rho=0}^{p^{\prime}}G^{\sigma\rho}k_{\sigma}\zeta_{\rho}(k)=0~.
\mu_{\omega ,\mathbf{c}}:=\inf \{S_{\omega,\mathbf{c}}(\Psi)\ |\ \Psi \in \mathcal{H}^1\backslash \{(\mathbf{0},\mathbf{0},\mathbf{0})\},\K_{\omega ,\mathbf{c}}(\Psi )=0\}
\begin{cases}\widetilde g_\rho ''(t) + b(t) \widetilde g'_\rho (t)\leq \widetilde g_\rho (t)^p \ \mbox{for}t \in \lbrack 0,T_1), \widetilde g_\rho (0) = \delta_\rho, \widetilde g_\rho '(0) \leq A_0 \delta_\rho.\end{cases}
D_{P}=\beta \left( \left| k\right|^{2}-4p_{0}^{2}\right) \left( \left|k\right|^{2}-4\xi p_{0}^{2}\right) \quad ,\quad \widetilde{D}_{P}=2\beta\left( \left| k\right|^{2}-4\xi p_{0}^{2}\right)
\begin{aligned}2\langle L^+_sw, L^-_sw\rangle_{L^2_{\frac{1}{a}}(Q_T)}\le 2\langle L^+_sw, L^-_sw\rangle_{L^2_{\frac{1}{a}}(Q_T)}+\|L^+_sw \|_{L^2_{\frac{1}{a}}(Q_T)}^2 + \|L^-_sw\|_{L^2_{\frac{1}{a}}(Q_T)}^2 & =\| L_sw\|_{L^2_{\frac{1}{a}}(Q_T)}^2= \|he^{s\varphi}\|_{L^2_{\frac{1}{a}}(Q_T)}^2\end{aligned}
\|T(P(\omega))\|_X = \Big\| \int_{\mathbb{T}^m}{P(\omega r^{-1}) F_{m}(r \omega) \: d \omega}\Big\|_X &\leq \int_{\mathbb{T}^m}{\left| P(\omega r^{-1})\right| \| F_{m}(r \omega)\|_X \: d\omega}.
\frac{\partial^{2}\psi}{\partial r^{2}}-\frac{1}{r}\frac{\partial\psi}{\partial r}+\frac{\partial^{2}\psi}{\partial z^{2}}=-\mu_{0}r^{2}\frac{dp}{d\psi}-\frac{1}{2}\frac{dF^{2}}{d\psi},
\frac{D_\alpha(x)}{x}-\frac{D_\alpha(y)}{y}\asymp \sum_{\lambda=0}^\nu A_\lambda(\gamma(2^{\alpha}-1-\lambda)) j^{\gamma(2^\alpha-1-\lambda)-1}+ \mathcal{O}\left(j^{\gamma(2^\alpha-2-\nu)}\right), \int_y^x \frac{D_\alpha(z)}{z^2}\,dz \asymp \sum_{\lambda=0}^\nu A_\lambda j^{\gamma(2^\alpha-\lambda)-1}+ \mathcal{O}\left(j^{\gamma(2^\alpha-1-\nu)}\right).
\begin{gathered}\| u \|_{L^p(B_1)}^p + (1 - s) [u]_{W^{s, p}(B_1)}^p \le M, |B_1 \cap \{u \le 0 \}| \ge \gamma |B_1| \mbox{and}|B_1 \cap \{u \ge 1 \}| \ge \gamma |B_1|.\end{gathered}
\psi_9(m) = \sum_{p_1, p_2}\bigg\{\psi(m/(p_1p_2),z) - \sum_{z \le p_3 < p_2}\bigg( \sum_{p_1p_2p_3 \le Y}+ \sum_{p_1p_2p_3 > Y}\bigg) \psi(m/(p_1p_2p_3), p_3) \bigg\}= \psi_{14}(m) - \psi_{15}(m) - \psi_{16}(m), .
{\cal L}_{\mathrm{s p i n}}^{( 2 )}[ B ] ={( 4 \pi )}^{- D}e^{2}\int_{0}^{\infty}{\frac{d T}{T}}e^{- m^{2}T}T^{2 - D}\int_{0}^{\infty}d \hat{T}\int_{0}^{1}d u_{a}J ( z , u_{a}, \hat{T}, D ) \nonumber \,
U_{p,q}D_{1,p+q-1}((W_{1,p+q-1})\otimes(W_{1,p+q-1}')) =D_{p,q}(U_{p,q}\otimes U_{p,q})((W_{1,p+q-1})\otimes(W_{1,p+q-1}')) &=D_{p,q}((U_{p,q}W_{1,p+q-1})\otimes(U_{p,q}W_{1,p+q-1}'))
\|\Phi_{\#}^y\|^2 = O(\alpha^3\log(\alpha^{-1})), \|A^-_y\Phi_{\#}^y\|^2 = O(\alpha^3), \|P \Phi_{\#}^y\|^2 = O(\alpha^5 \log(\alpha^{-1})), \langle \alpha V_y\rangle_{\Phi_{\#}^y}= O(\alpha^5 \log(\alpha^{-1})) , \|V_y \Phi_{\#}^y\|^2 = O(\alpha^5 \log (\alpha^{-1})).
S= \int \frac{1}{16\pi G}R\sqrt{-g}d^4x - \int \left[ \frac{1}{4}F_{\mu\nu}^a F^{a\mu\nu}+ \frac{1}{2}D_\mu \phi^a D^\mu \phi^a + \frac{1}{4}\lambda (\phi^a \phi^a - v^2)^2 \right]\sqrt{-g}d^4x
M_h: =\max_l\left(\max_{\Omega_h}\left(\hat{u}_l(z)-\hat{u}_h^l(z)+V_h(z)\right)\right) &=\max_l\left(\max_{\{{u}_l(z)\leq u_h^l(z)\}}\left(\hat{u}_l(z)-\hat{u}_h^l(z)+V_h(z)\right)\right),
T^{\,n}\Bigl(\sum_{s=1}^{j_{m}}z_{s}\Bigr)-x_{l}=T^{\,n_{j_{m}}+k'\textrm{per}(x_{l})}\,\Bigl(\sum_{s=1}^{j_{m}}z_{s}\Bigr)-x_{l}\qquad\hbox{by (i)}&=T^{\,n_{j_{m}}+k'\textrm{per}(x_{l})}\,z_{j_{m}}-T^{\,k'\textrm{per}(x_{l})}\,\Bigl(x_{l}-\sum_{s=1}^{j_{m}-1}z_{s}\Bigr)\qquad\hbox{by (iv).}
\Delta_{F}^{\prime}( x^{2}) ={\frac{i \left( m^{2}\right)^{2}}{4 \pi^{2}}}\, \int_{0}^{\infty}\kappa \, d \kappa \, e^{- i \kappa ( m^{2}x^{2}- i \delta )}e^{- i ( 1 - i \epsilon ) / ( 4 \kappa )}.
P'Q =-\left(\sum_{i=1}^nP_i\right)\left(\sum_{j=1}^n-\lambda_j^2P_j\right) &=-\sum_{j=1}^n\lambda_j^2P_j^2-\sum_{\substack{1\leq i,j\leq n i\neq j}}\lambda_j^2P_iP_j
G_C=\begin{bmatrix}I_k\otimes M\tilde{D}_\alpha M^T+(A\widetilde{\otimes}M)(I_{N-k}\otimes\tilde{D}_\alpha)(A\widetilde{\otimes}M)^T (A\widetilde{\otimes}M)(I_{N-k}\otimes \tilde{D}_\alpha M_p^T) (I_{N-k}\otimes M_p\tilde{D}_\alpha)(A\widetilde{\otimes}M)^T I_{N-k}\otimes M_p\tilde{D}_\alpha M_p^T\end{bmatrix}.
\S_D^k[e^{i n \theta}]=\frac{(-1)i\pi R}{2}\times \begin{cases}J_n(k R) H_n^{(1)}(k r) e^{i n\theta}, r\geq R, H_n^{(1)}(k R) J_n(k r) e^{i n\theta}, 0\leq r < R. \end{cases}
[(2,0,0,0)\otimes(0,1,0,0)+(0,1,0,0)\otimes(2,0,0,0)] = [(1-b,0,0,0)\otimes(0,1,0,0)+(0,1,0,0)\otimes(1-b,0,0,0)] = [(1,0,0,0)\otimes(0,1-b,0,0)+(0,1-b,0,0)\otimes(1,0,0,0)].
R_{X Y Z}{}^{W}= f_{i A}^{W}f_{Z}^{i B}{\cal R}_{X Y B}{}^{A}\, , \qquad \delta_{j}^{i}\,{\cal R}_{X Y B}{}^{A}= f_{W}^{i A}f_{j B}^{Z}\, R_{X Y Z}{}^{W}\, ,
D_{\mu \nu \lambda \rho}^{\mathrm{g r a v}}= ( \eta_{\mu \lambda}\eta_{\nu \rho}+ \eta_{\mu \rho}\eta_{\nu \lambda}- \eta_{\mu \nu}\eta_{\lambda \rho}) \frac{- i}{( 2 \pi )^{4}}\int d^{4}k \frac{{\cal E}^{2}( k^{2})}{k^{2}- i \epsilon}\exp [ i k \cdot ( x - y ) ] ,
g_p(x)=\sum_{k=1}^\infty \left (\sum_{i=n_{k}}^{{n_{k+1}}}\frac{(2^i)^{\frac{1}{p-1}}\mu^{\frac{1}{1-p}}(A_{2^i})}{\sum_{i=n_k}^{n_{k+1}}(2^i)^{\frac{p}{p-1}}\mu^{\frac{1}{1-p}}(A_{2^i})}\chi_{A_{2^i}}(x)\right ) \big({\rm or\ \}g_1(x)=\sum_{k=1}^\infty 2^{-i_k}\chi_{A_{2^{i_k}}}(x)\big).
Q_{GL_{n-1}}^{GL_{n}}(a_1,\cdots,a_n;t_1,\cdots, t_{n-1}|\chi_n) =\chi_n\left((-1)^{n-1}\frac{a_1\cdots a_n}{t_1\cdots t_{n-1}}\right)\widetilde{Q}_{GL_{n-1}}^{GL_{n}}(a_1,\cdots,a_n;t_1,\cdots, t_{n-1}|\chi_n)
e_k(\lambda_k,\tilde{v},\tilde{w})=\frac{1}{m|g|\lambda_k}\sum_{i=0}^{m-1}[\tilde{v}_i\tilde{w}_i]_{k,k}\bigg(c_{i-1}+\sum_{r=0}^{i-1}[\tilde{v}_r\tilde{w}_r]_{k,k}-\sum_{s=0}^{m-1}\frac{m-s}{m}[\tilde{v}_s\tilde{w}_s]_{k,k}\bigg),
Z_\epsilon(t) = \left(\begin{array}{cc}\hat{Z}_\epsilon(t) A(t) 0 e^{-\epsilon t}I_{n-r}\end{array}\right),\mbox{and hence,}Z_\epsilon(T_{\epsilon}) = \left(\begin{array}{cc}\hat{Z}_\epsilon(T_\epsilon) A(T_\epsilon) 0 e^{-\epsilon T_\epsilon}I_{n-r}\end{array}\right).
\Delta t_0^{ij}~=~{\bf 1}\otimes t_0^{ij}~+~ t_0^{ij}\otimes{\bf 1}~,~~~\Delta t_n^{ij}~=~{\bf 1}\otimes t_n^{ij}~+~ t_n^{ij}\otimes{\bf 1}~+~ h\, \sum_{p+q = n-1}\, \sum_{k=1}^N ~ t_p^{ik}\otimes t_q^{kj}~,
|B_\delta(x) \cap \Sigma_N|\geq|I_0 \cap \Sigma_n| - \sum_{\ell = n}^{N-1}|I_0 \cap(\Sigma_\ell \setminus \Sigma_{\ell + 1})|>\frac{9 \delta}{10}- \frac{2 \delta}{5}=\frac{\delta}{2}.
K_2=1+T_{p-1}(u;x_0,\bar R_0)^{p-1}+T_{q-1}(u;x_0,\bar R_0)^{q-1}+\|u\|^{\frac{(\ell_1+j_\infty)(\ell_1-1)}{\ell_1-2}}_{L^\infty(B_{\bar R_0}(x_0))}+\| u \|^{q-1}_{L^\infty(B_{R_0})}+\|f\|_{L^\infty(B_{\bar R_0}(x_0))}
\tilde{T}(d\rightarrow\mu) :=\tilde{T}(\mu'\to\mu)\circ \tilde{T}(d\to \rho(G^{\mu'})), \tilde{T}(\mu\rightarrow d) :=\tilde{T}(\rho(G^{\mu'})\to d)\circ \tilde{T}(\mu\to\mu'),
ef(-es)^{k+1}+df(-es)^{k+2}=(ef(-es)^k+df(-es)^{k+1})(-es) &=(f(-es)^ke+f(-es)^{k+1}d)(-es). \intertext{Since $e^2=0$ and $d$ and $e$ anticommute, this is}=(-1)^{k+1}f(es)^{k+1}eds &=(-1)^{k+1}f(es)^{k+1}e(1-sd-\iota f).\intertext{Since $e$ and $\iota$ commute and $s\iota=0$, this is simply}=f(-es)^{k+1}e+f(-es)^{k+2}d
\frac1{(1+b^2)^{\frac12}}\Big( -b(1+b^2)\mathfrak{Re}\ddot{c}_0+(b^2+\frac12)\mathfrak{Re}\ddot{c}_1-\frac{\sqrt{2}}{4}b \mathfrak{Re}\ddot{c}_2 \Big) + \Sigma =0,
\frac{\partial}{\partial t}\!\left(\begin{matrix}v_1(t;z) v_2(t;z)\end{matrix}\right)\!=\!\left(\begin{matrix}0 \!\!\!\!\!\!\!\!q\left(t,z\right)e^{+2j\lambda t}-q^*\left(t,z\right)e^{-2j\lambda t}\!\!0\end{matrix}\right)\left(\begin{matrix}v_1(t;z) v_2(t;z)\end{matrix}\right),
\widetilde{\mathbf{y}}_{\scriptscriptstyle{A}}= \sqrt{\rho}\, \widetilde{\mathbf{H}}\, \mathbf{s}_{\scriptscriptstyle{A}}+ \begin{bmatrix}h^{\ast}_{\scriptscriptstyle{21}}n_{\scriptscriptstyle{A1}}- n_{\scriptscriptstyle{A2}}n_{\scriptscriptstyle{A3}}\end{bmatrix},
\sum_{i=0}^{n-1}(-1)^{(\deg \phi' -1) \epsilon_{i+1}}\phi([a_{1}| \dots | a_{i}| \phi'([a_{i+1}| \dots | a_{i+m}]) | a_{i+m+1}| \dots | a_{n+m-1}]) - \sum_{i=0}^{m-1}(-1)^{\epsilon_{i}'}\phi'([a_{1}| \dots | a_{i}| \phi([a_{i+1}| \dots | a_{i+n}]) | a_{i+n+1}| \dots | a_{n+m-1}]),
\chi^{(k)}_l(q) ={1 \over{\eta(\tau)^3}}\cdot 2(k+2) \sum_{m=\infty}^\infty \left( m +{{l+1}\over{2(k+2)}}\right) \exp \left( 2 \pi i (k+2) \tau \left( m +{{l+1}\over{2(k+2)}}\right)^2 \right),
l_1=\frac{1}{q_1}\sum_{j=3}^{n}l_j q_j \Big(\frac{\psi'(\beta_1 q_2)\psi'(\beta_2 q_j)+\psi'(\beta_2 q_2)\psi'(\beta_1 q_j)}{\psi'(\beta_1 q_1)\psi'(\beta_2 q_2)-\psi'(\beta_2 q_1)\psi'(\beta_1 q_2)}\Big),
\:\prod_{r=0}^{m-1}\:_{p}\Gamma_{k}(x+\frac{kr}{m})=\frac{p^{\frac{m-1}{2}}}{k^{m-1}}(2 \pi )^{\frac{(m-1)}{2}}m^{\frac{1}{2}-\frac{mx}{k}}\:_{p}\Gamma_{k}(mx); m=2,3,4,... \: .
\Big|f_\epsilon(V)-\sum_{i=1}^k(-1)^if_0(PU_{\mu_i,\xi_i}) \Big|_{\frac{2n}{n+2}}= \Big|f_\epsilon(V)-f_0(V)\Big|_{\frac{2n}{n+2}}+\Big|f_0(V)-\sum_{i=1}^k(-1)^if_0(PU_{\mu_i,\xi_i}) \Big|_{\frac{2n}{n+2}}.
\|Pe_\lambda\|_{L^1(\partial M)}\lesssim\|Pe_\lambda\|_{L^{1+\delta}(\partial M)}\lesssim\|e_\lambda\|_{L^{1+\delta}(\partial M)}\leq\|e_\lambda\|_{L^2(\partial M)}^\frac{2\delta}{1+\delta}\|e_\lambda\|_{L^1(\partial M)}^\frac{1-\delta}{1+\delta}&\lesssim\lambda^\frac{2(n-1)\delta}{4(1+\delta)}\|e_\lambda\|_{L^1(\partial M)}.
R_n(x, y)=\frac{\sigma^{n+1}}{\pi}\displaystyle e^{-\frac{\sigma^2}{2}Ay^2}\left(\sqrt{2\pi h}-\displaystyle \int_{-\infty}^{\infty}\left(\sum\limits_{m=2}^{\infty}\frac{t^m u^m}{m!}\right)\displaystyle \frac{e^{-\frac{1}{2}hu^2}}{u^2}\,du\right).
\cos(\theta) = \frac{\cos(d\sqrt{k}) - \cos(t\sqrt{k}) \cos(s\sqrt{k})}{\sin(t\sqrt{k})\sin(s\sqrt{k})}= \frac{\cos(d\sqrt{k}) - \cos(s\sqrt{k})}{\sin(t\sqrt{k})\sin(s\sqrt{k})}+ \frac{\cos(s\sqrt{k}) (1 - \cos(t\sqrt{k}))}{\sin(t\sqrt{k})\sin(s\sqrt{k})}.
\int_0^1 f(x)x^{n-1}\,dx > -M \int_0^a x^{n-1}\, dx+m\int_b^c x^{n-1}\, dx &=\frac{mc^n}{n}\left(1-\left(\frac{b}{c}\right)^n-\frac{M}{m}\left(\frac{a}{c}\right)^n\right).
Tr(T^{-1}, Sym^n(V)) = \sum_{a=0}^n \sum_{b=0}^a \sum_{c=0}^b \xi_3^{2n-a-b}= \xi_3^{2n}\sum_{a=0}^n \xi_3^{-a}\sum_{b=0}^a \xi_3^{-b}(b+1).
G_Y^0f(x) \leqslant2 \left(\int_{\mathbb{R}\setminus(-|x|,|x|)}H(x)f(y)\mu(d y)+\int_{-|x|}^{|x|}H(y)f(y)\mu(d y)\right) &=2\int_{0}^{|x|}H'(z)\left(\int_{\mathbb{R}\setminus(-z,z)}f(y)\mu(d y)\right)d z.
\delta^{k_{s}^{j}-1}(\varphi_{rest_{s-1}}^{j}) =\varphi_{rest_{s-1},[k_{s}^{j}-1]}^{j}(x,v_{1},v_{1,[1]},\ldots,v_{s-1},v_{s-1,[1]},\ldots) \delta^{k_{s}^{j}}(\varphi_{rest_{s-1}}^{j}) =\varphi_{rest_{s-1},[k_{s}^{j}]}^{j}(x,v_{1},v_{1,[1]},\ldots,v_{s-1},v_{s-1,[1]},\ldots,u_{rest_{s-1}})
H=\sum^{N-1}_{n=1}h_{n,n+1}+\frac{1}{2}(K^{-}_{1}(0)^{-1})\dot{K}^{-}_{1}(0)+\frac{tr_{0}[K^{+t}_{0}(-\eta)h_{N0}]}{tr[K^{+}(-\eta)]}\;\;\;,
s_k s_{k+1}\dots s_{n-1}s_n s_{n-1}\dots s_{k+1}s_k = s_{\alpha_k + \alpha_{k+1}+ \dots + \alpha_{n-1}+ \alpha_n}s_n s_{n-1}\dots s_{k+1}s_k s_{k+1}\dots s_{n-1}s_n = s_{\alpha_k + \alpha_{k+1}+ \dots + \alpha_{n-1}+ \alpha_n}
F_x =\Big((1-xK_u^2)^{-1}(u)\ |\u\Big)=\frac{1}{J(x)}\Big((1-xH_u^2)^{-1}(u)\ |\u\Big) &=\frac{1}{J(x)}\Big((1-xH_u^2)^{-1}H_u^2(1)\ |\ 1\Big)=\frac{J_x-1}{xJ_x}\ .
{\cal Z}_{2 N}= K ( 2 N ) ! \int_{\stackrel{\cdots}{R ( - \infty , x_{1}, \ldots , x_{2 N}, \infty )}}\int \prod_{i = 1}^{2 N}e^{- \frac{\beta}{2}\sum_{i = 1}^{2 N}V ( x_{i})}\times [ d e t ( x_{j}^{i - 1}) ]_{i , j = 1 , \ldots , 2 N}
+ i [ ({\frac{a}{2 \alpha^{2}}}-{\frac{1}{4 \alpha^{2}g \partial^{2}}}) g^{\mu \nu}\delta ( x - y ) - ({\frac{a}{2 \alpha^{2}}}+ g ){\frac{\partial^{\mu}\partial^{\nu}}{\partial^{2}}}\delta ( x - y ) +{\frac{1}{2 \alpha \partial^{2}}}( 1 +{\frac{a}{2 \alpha^{2}g}}) \epsilon^{\mu \nu \lambda}\partial_{\lambda}\delta ( x - y ) ] .
S=\int_{k}k^{2}|\phi(k)|^{2}+\frac{\lambda}{4!}\int_{k_{1}k_{2}k_{3}k_{4}}\phi(k_{1})\phi(k_{2})\phi(k_{3})\phi(k_{4})\delta(k_{1}+k_{2}+k_{3}+k_{4})=S_{F}+X.
\sup_{\psi\in\Psi}\mathcal{L}(z,\psi) \geq \mathcal{L}(z,\psi_\varepsilon) = f(z) + \psi_\varepsilon(Lz) - g^*_\Psi (\psi_\varepsilon) \geq f(x_\varepsilon) + g(Lx_\varepsilon) - 2\varepsilon.
\Psi_1\left(|\log\nu\left(\left\|F - G\right\|_{L_1}\right)|\right) = \mathcal{A}\left(\frac{p_1}{p_1 + 1}\, \left|\log \left\|F - G\right\|_{L_1}\right|\right)\ge \frac12\, \left\|F - G\right\|_{L_1}^{-\frac{p_1}{p_1 + 1}}.
f_k(z) = \sum_{i=0}^n 4^i c_i z^i; f_k^*(z) = \sum_{i=0}^{n-1}4^i z^i \int_{-1}^1 \bigg(\sum_{j\geq i+1}c_j (t^2-1)^{j-i-1}\bigg)\frac{t^2}{2}dt.
\frac{L_{\Gamma}'(s,\rho)}{L_{\Gamma}(s,\rho)}=\sum_{\overline{\gamma}\in \overline{\Gamma}_{p}}\sum_{m=1}^{\infty}\ell(\gamma) \chi_{\rho}(\gamma^{m}) \frac{e^{-sm\ell(\gamma)}}{1-e^{-m\ell(\gamma)}}.
I= \int_{\mathbb{Z}_{p}^{3}}e^{\left[w_{2}w_{3}x_{1}+w_{1}w_{3}x_{2}+w_{1}w_{2}x_{3}+w_{1}w_{2}w_{3}\left(y_{1}+y_{2}+y_{3}\right)\right]_{q}t}\times d\mu_{q^{w_{2}w_{3}}}\left(x_{1}\right)d\mu_{q^{w_{1}w_{3}}}\left(x_{2}\right)d\mu_{q^{w_{1}w_{2}}}\left(x_{3}\right).
N^{{\cal P}_0}~=~\sum_{m_1= -1/2}^{1/2}\cdots \sum_{m_{p_0}=-1/2}^{1/2}\left[ ~\delta_{(\sum_{n=1}^{p_0}m_n), 0}~-~\frac12~ \delta_{(\sum_{n=1}^{p_0}m_n), 1}~-~\frac12 ~\delta_{(\sum_{n=1}^{p_0}m_n), -1}~\right ]
c_{k,\delta}^\lambda(s) = (2\pi)^{n+1}\int_0^\infty f(x)g(x+|\lambda|) (x+|\lambda|)^{-n-k}x^k dx &= \frac{(2\pi)^{n+1}}{\Gamma\big(\frac12(n+1+s)\big)^2}\int_0^\infty e^{-\delta(2x+|\lambda|)}x^{\beta+k-1}(x+|\lambda|)^{\beta-k-n-1}dx.
\sum_{|\alpha|=m}|a_{\alpha}|^2 \leq \frac{L^L}{(L+m)^{L}}\frac{m^{2n}}{(1-r^2)^{L(1-2\delta)}}= \left[\frac{L m^{\frac{2n}L}}{(L+m) (1-r^2)^{1-2\delta}}\right]^L\ .
s=(\begin{matrix}0\\K\\ 0\end{matrix})and\hat{n}=\frac{(\begin{matrix}0\\K\\ 0\end{matrix})-(\begin{matrix}0\\ 0\\K\end{matrix})}{\|(\begin{matrix}0\\K\\ 0\end{matrix})-(\begin{matrix}0\\ 0\\K\end{matrix})\|}=\frac{(\begin{matrix}0\\K\\ -K\end{matrix})}{\sqrt{0^{2}+K^{2}+(-K)^{2}}}=(\begin{matrix}0\\ \frac{1}{\sqrt{2}}\\ -\frac{1}{\sqrt{2}}\end{matrix}).
v=c\sqrt{1-(\frac{m_{0}c^{2}}{K+m_{0}c^{2}})^{2}}=\frac{c\sqrt{K(K+2m_{0}c^{2})}}{K+m_{0}c^{2}}=\frac{c\sqrt{(E-m_{0}c^{2})(E+m_{0}c^{2})}}{E}=\frac{pc^{2}}{E}.
\begin{array}{l l}{\begin{array}{l l}{\frac{d}{d r}\sigma ( r ) = 0}{r = 0}{\sigma ( r ) = 0}{r \rightarrow \infty}\end{array}}{\begin{array}{l l}{A ( r ) \neq 0}{r \rightarrow \infty}{A ( r ) = 1}{r = 0 .}\end{array}}\end{array}
E_{P}\left[ \left. \frac{1}{\lambda_{\overline{\mathbf{a}}_{0}}\left( \overline{\mathbf{G}}_{0},\overline{\mathbf{B}}_{0}\mathbf{;}P\right)}\right\vert \overline{\mathbf{G}}_{0}\mathbf{,}\overline{\mathbf{A}}_{0}\mathbf{=}\overline{\mathbf{a}}_{0}\right] =\frac{1}{\pi_{a_{0}}\left( \overline{\mathbf{G}}_{0}\mathbf{;}P\right)}.
\mathbb{P}(A_{1})=\frac{4}{52},\mathbb{P}(A_{2}|A_{1})=\frac{3}{51},\mathbb{P}(A_{3}|A_{1}\cap A_{2})=\frac{2}{50},\mathbb{P}(A_{4}|A_{1}\cap A_{2}\cap A_{3})=\frac{1}{49}
\partial^{*h}:\mathcal{A}_{p,q}(H\mathcal{T}E)\rightarrow\mathcal{A}_{p,q-1}(H\mathcal{T}E),(\partial^h\Psi,\Phi) = (\Psi,\partial^{*h}\Psi), \bar{\partial}^{*h}:\mathcal{A}_{p,q}(H\mathcal{T}E)\rightarrow\mathcal{A}_{p-1,q}(H\mathcal{T}E),(\bar{\partial}^h\Psi,\Phi) = (\Psi,\bar{\partial}^{*h}\Psi),
\begin{cases}-\hat{v}_{\infty}\cdot L_{\Sigma}\hat{v}_\infty \leq 0\ \Sigma; -L_{\Sigma}\hat{v}_\infty = 0\ \big(\{\hat{u}_\infty^- < \hat{v}_\infty < \hat{u}_\infty^+\}\cap \Omega_0 \big) \cup \Sigma\setminus \Omega_0. \end{cases}
\max_{\substack{x,y \in X x\ne y}}\frac{\beta_{v_i}(x) - \beta_{v_i}(y)}{d(x,y)}= \max_{\substack{x',y' \in X x' \ne y'}}\frac{\beta_{v_i'}(x') - \beta_{v_i'}(y')}{d(x',y')}
d_T(t,t') \le \sum_{t_0=t,t_{1},\dots,t_{j-1}, t_j=t', (t_{i},t_{i+1}) \in F}d_T(t_i,t_{i+1}) = \sum_{t_0=t,t_{1},\dots,t_{j-1}, t_j=t', (t_{i},t_{i+1}) \in F}d(x_{t_i},x_{t_{i+1}}) \le |V(T)| d(x_t,x_{t'}),