harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
-h(A_{2}\varphi_{n+1}, B_{1}v_{n+1})_{H}= -(A_{2}\varphi_{n+1}, B_{1}\varphi_{n+1}-B_{1}\varphi_{n})_{H}= - \frac{1}{2}(A_{2}\varphi_{n+1}, B_{1}\varphi_{n+1})_{H}+ \frac{1}{2}(A_{2}\varphi_{n}, B_{1}\varphi_{n})_{H}\,\quad- \frac{1}{2}(A_{2}(\varphi_{n+1}-\varphi_{n}), B_{1}(\varphi_{n+1}-\varphi_{n}))_{H}.
f(2n,r)= -f(2n-2,r)-f(2n-2,r-1)+2\sum_{k=0}^{2n}\sum_{j=1}^{k-r}(-1)^k \binom{k-j-1}{r-1}\binom{2n-k-1}{r-1}= (-1)^r\binom{n-1}{r-1}-(-1)^r\binom{n-1}{r}+2\sum_{j=1}^{2n}\sum_{k=j+r}^{2n}(-1)^k \binom{k-j-1}{r-1}\binom{2n-k-1}{r-1}= (-1)^r\binom{n-1}{r-1}-(-1)^r\binom{n-1}{r}-2\sum_{j=1}^{2n}(-1)^j f(2n-2-j,r-1).
\hat{O}W_n = \frac{n+1}{(n+1)! 0!}\{\hat{O}^{0}x, \hat{O}^{n+2}x \}+ \sum^{\frac{n-1}{2}}_{j=1}\frac{(n+1)(n - 2j + 2)}{(n+2-j)! j!}\{\hat{O}^j x, \hat{O}^{n+2-j}x \}+
\int_{- t}^{t}\frac{d x}{( x^{2}+ \lambda^{2})^{\kappa + 1}}= \frac{1}{\kappa !}\left( - \frac{1}{2 \lambda}\frac{d}{d \lambda}\right)^{\kappa}\int_{- t}^{t}\frac{d x}{x^{2}+ \lambda^{2}}
\mathbb{P}_{\widetilde{g}^{(n)}}^{(s,n)}: = \frac{\Psi_{\widetilde{g}^{(n)}}\mathbb{B}^{(s,n )}}{\int_{\big((0, + \infty)\big)}\Psi_{\widetilde{g}^{(n)}}d \mathbb{B}^{(s,n )}},
\int_a^{b}(A + x)^{\xi}x^{\lambda}\, dx \simeq\begin{cases}(b-a) b^{\lambda}(b+ A)^{\xi}\xi + \lambda > -1, b^{\lambda}(b+ A)^{-\lambda}\log \Big( 1 + \frac{b-a}{a+A}\Big) \xi + \lambda = -1, (b-a) b^{\lambda}(a + A)^{\xi + \lambda + 1}(b + A)^{- \lambda - 1}\xi + \lambda < -1,\end{cases}
E\{\bar W(\psi)\}=E\{W(\psi)\}-2\lambda^{ar}E(b_a l_r)+\lambda^{ar}\lambda^{st}\beta_a(2\lambda_{rs,t}+\lambda_{rst})-\lambda^{ab}(\beta_{ab}+\beta_a\beta_b)+O(n^{-3/2}) &=E\{W(\psi)\}+\lambda^{ar}\lambda^{st}\beta_a(2\lambda_{rs,t}+\lambda_{rst})-2\lambda^{ar}\beta_{a/r}+\lambda^{ab}(\beta_{ab}-\beta_a\beta_b)+O(n^{-3/2}),
{\cal M}_{1 - l o o p}( \theta ) = \frac{i g^{4}}{8 \pi m \Theta^{2}}[ \varepsilon^{c a b}T_{c}\otimes T_{b}T_{a}] = - \frac{i g^{4}}{8 \pi m \Theta^{2}}[ T^{a}\otimes T_{a}] .
F^{kl}W_{ij;kl}- \dot{W}_{ij}= -F^{kl}h_{rk}h^r_l W_{ij}- F^{kl}g_{kl}W_{ij}+ 2F h_i^k W_{kj}&\, - F^{kl,rs}W_{kl;i}W_{rs;j}+2F \epsilon g_{ij}&\,- (1 - \epsilon) \{F^{kl}h_{rk}h^r_l - 2F + F^{kl}g_{kl}\}g_{ij}.
\begin{cases}\partial_z^2\phi_2 = -\dfrac{\zeta}{h_b}\big( 1 + \dfrac{h}{h_b}\big) \Delta_X \psi \ \ \mathrm{in}\ \ \mathcal{S}_b, \phi_2|_{z=0}= 0, \ \ \partial_z \phi_2|_{z=-1 + \beta b}= 0. \end{cases}
I_4\le \sum^{\infty}_{m=0}\dfrac{1}{\rho_{m}}\left( \mu \left( B\left( 0,\rho_{m}\right) \right) \right)^{\frac2{n-2}}\leq C\int^{1}_{0}\dfrac{\left( \mu \left( B\left( 0,t\right) \right) \right)^{\frac2{n-2}}}{t^{2}}dt.
\mathcal Q(\mathbf h_1, \mathbf h_2,\mathbf g_1, \mathbf g_2,\pmb{\phi}_1, \pmb{\phi}_2) = \frac{1}{4}\frac{L(1/2, \pi \times \mathrm{ad}\tau)}{L(1,\pi,\mathrm{ad})L(1,\tau,\mathrm{ad})}\prod_v \mathcal I_v(\mathbf h_{1,v},\mathbf h_{2,v}, \mathbf g_{1,v}, \mathbf g_{2,v},\pmb{\phi}_{1,v}, \pmb{\phi}_{2,v}),
\int_{- \frac{d}{2}}^{\frac{d}{2}}2 \pi s \sqrt{1 + \left( \frac{\mathrm{d}s}{\mathrm{d}t}\right)^{2}}\mathrm{d}t = \frac{\pi}{\alpha}\left( d + \frac{\operatorname{s i n h}\alpha d}{\alpha}\right)
\alpha_{0}=\frac{r_{0}^{T}r_{0}}{p_{0}^{T}Ap_{0}}=\frac{[\begin{matrix}-8&-3\end{matrix}][\begin{matrix}-8\\ -3\end{matrix}]}{[\begin{matrix}-8&-3\end{matrix}][\begin{matrix}4&1\\ 1&3\end{matrix}][\begin{matrix}-8\\ -3\end{matrix}]}=\frac{73}{331}\approx0.2205
\mathcal{I}_{2,2,n}= \frac{\pi_1^n}{\mu}\underbrace{\int_{0}^{\theta_{\mathrm{S}}}\frac{\exp({-Sz})}{\left( z + \pi_1 \right)^{n+1}(z + \tau)}}_{\mathcal{J}_{2,2,n}}\mathrm{d}z,
\lim_{\lambda\to0^+}\lambda\left\|\left[\int_{\mathbb{R}^n}\mathbf{1}_{E_{\lambda,\frac{\gamma}{q}}[f]}(\cdot,y)\left|\cdot-y\right|^{\gamma-n}\,dy\right]^\frac{1}{q}\right\|_{L^p(\mathbb{R}^n)}=\left[-\frac{\kappa(q,n)}{\gamma}\right]^\frac{1}{q}\left\|\,\left|\nabla f\right|\,\right\|_{L^p(\mathbb{R}^n)}.
S = - \frac{1}{2 \kappa_{5}^{2}}\int_{{\cal M}_{5}}d^{5}x \sqrt{- g}\left[ R + \frac{1}{2 V^{2}}\partial_{\alpha}V \, \partial^{\alpha}V + \frac{1}{3 V^{2}}\alpha^{2}\right] - \sum_{i = 1}^{2}\frac{\sqrt{2}}{\kappa_{5}^{2}}\alpha_{i}\int_{{\cal M}_{4}^{( i )}}d^{4}\xi_{( i )}\sqrt{- g}\, V^{- 1}\, ,
\sum_{k = 0}^{a-1}\bigl(A_kf_1^{(2a-1-k)}f_{12}^{(k+1)}f_2^{(2a-1-k)}+ B_kf_1^{(2a-2-k)}f_{12}^{(k+2)}f_2^{(2a-2 -k)}+ C_kf_1^{(2a-k)}f_{12}^{(k)}f_2^{(2a - k)}\bigr),
_d:=H^\cdot(\mathcal{B}PS_d)\hookrightarrow H^\cdot \left(^d(\mathbb{C}^2),{}^p\tau^{\leq 0}R\pi_*\omega_{\mathcal{C}(d)}\right)\hookrightarrow H^{\mathrm{BM}}_\cdot(\mathcal{C}(d)).
\begin{cases}x_j^{k+1}= \arg\min\{f_j(x_j)+\langle J_j(x_1^{k+1}, \dots, x_{j-1}^{k+1}, x_j^k, \dots, x_s^k, y^k), x_j-x_j^k\rangle +\frac{\beta}{2\alpha_j}\|x_j-x_j^k\|_{2}^2: x_j\in\mathbb{R}^{n_j}\},~~j\in\mathbb{N}_s, y^{k+1}= y^k+\beta(\sum_{i=1}^sA_ix_i^{k+1}-b). \end{cases}
H_{i_{1}i_{2}\cdots i_{m-1}}H_{i_{2}i_{3}\cdots i_{m}}\cdots H_{i_{m}i_{1}\cdots i_{m-2}}=-1\qquad\forall\left(i_{1},\cdots,i_{m}\right)\notin\left\{\left(0,0,\cdots,0\right),\,\left(1,1,\cdots,1\right)\right\}
\begin{array}{c}u''(t) + g_1(t)f_1(t, u(t),v(t)) = 0,\t \in (0,1),\t\neq \tau_1, v''(t) + g_2(t)f_2(t, u(t),v(t)) = 0,\t \in (0,1),\t\neq \tau_2,\end{array}
\begin{aligned}\|(z-T)^{-1}x_r\|_p =\Big(\sum\limits_{n=0}^{\infty}\big|f_n\big((z-T)^{-1}x_r\big)\big|^p\Big)^\frac{1}{p}&\leq \big|f_0\big((|z|-T)^{-1}x_r\big)\big|\bigg(1+\sum\limits_{n=1}^{\infty}\Big(\frac{r^n}{(n+m)!\prod\limits_{j=1}^n w_j}\Big)^p\bigg)^\frac{1}{p}&=\big|f_0\big((|z|-T)^{-1}x_r\big)\big|\|x_r\|_p.\end{aligned}
g ( x_{1}, x_{2}) = - \frac{1}{8 \pi}\frac{1}{\beta}\sum_{p_{0}}\frac{1}{( x_{1}+ p_{0})^{2}- ( x_{2}+ p_{0})^{2}}\log \left( \frac{m^{2}- ( x_{1}+ p_{0})^{2}}{m^{2}- ( x_{2}+ p_{0})^{2}}\right) .
e\triangleq e_1+ie_2,~~~n\triangleq e_3+ie_4,~~~\o\triangleq \o_1+i\o_2,~~~\O\triangleq \o_3+i\o_4,~~~~~~ \mathrm{II}_1\triangleq(\o_{13}+i\o_{23})+i(\o_{14}+i\o_{24}),~~~\mathrm{II}_2\triangleq(\o_{13}-i\o_{23})+i(\o_{14}-i\o_{24}).
\lim_{l\to\infty}\Phi_{p_0,q_0,v_0,w_0}(2^lQ)=\infty,(Q\in\mathcal{Q}),\Phi_{p_0,q_0,v_0,w_0}(Q):=v_0(Q)^{\frac{1}{p_0}-\frac{1}{q_0}}w_0(Q)^\frac{1}{q_0}.
\chi(k_0,\xi) =\lim\limits_{k\to k_0}(k-k_0)^{\i\nu}\cdot \exp\left[\frac{1}{2\pi\i}\int\limits_{-\infty}^{k_0}\frac{\ln(1+|r(s)|^2)\ \d s}{s-k}\right] &= (k+N)^{\i\nu}\cdot\exp\left[\frac{1}{2\pi\i}\int\limits_{-N}^{k_0}\frac{\ln\frac{1+|r(s)|^2}{1+|r(k_0)|^2}\ \d s}{s-k_0}+\frac{1}{2\pi\i}\int\limits_{-\infty}^{-N}\frac{\ln(1+|r(s)|^2)\ \d s}{s-k}\right]
\lceil\frac{s}{3}\rceil+(\lceil\frac{s}{5}\rceil-\lfloor\frac{\lceil\frac{s}{5}\rceil}{3}\rfloor)+(\lceil\frac{s}{7}\rceil-\lfloor\frac{\lceil\frac{s}{7}\rceil}{3}\rfloor-\lfloor\frac{\lceil\frac{s}{7}\rceil}{5}\rfloor+\lceil\frac{\lceil\frac{s}{7}\rceil}{15}\rceil)\ge S
B-\hat \nu_m(s) = \begin{cases}A_1\xi'(s) +A_2[\xi'(q)-\xi'(b)]m + A_2[\xi'(b)-\xi'(q)] +\frac1{|\nu|}, 0\le s \le q, A_1\xi'(q)+A_2[\xi'(b)-\xi'(q)]+A_2[\xi'(s)-\xi'(b)]m +\frac1{|\nu|}, q\le s\le b, A_1\xi'(q) + A_2[\xi'(s)-\xi'(q)]+\frac1{|\nu|}, b\le s\le 1.\end{cases}
\begin{aligned}q_\alpha(x) = C ( a_1 x_1^\alpha+\cdots+ a_n x_n^\alpha)^{\frac 1 \alpha}, \alpha \in (-\infty,1] \setminus 0, q_{-\infty}(x) = C \min(a_1 x_1,\ldots, a_n x_n), q_0(x) = C x_1^{a_1}\cdots x_n^{a_n}, \end{aligned}
T_{3 , 7}T_{4 , 1 1}T_{7 , 1 0}T_{1 1 , 1 2}T_{1 , 3}T_{4 , 4}( 7 , 5 , 4 , 3 , 0 ; 9 , 6 , 4 , 4 , 2 , 2 , 1 , 1 , 1 ) = ( 5 , 4 , 2 , 1 , 0 ; 9 , 7 , 6 , 4 , 4 , 3 , 2 , 1 , 1 ) = \Omega
\limsup_{{\mathcal N}\ni n\to\infty}\left(\prod_{k=1}^{K_n}|\alpha_{k,n}| \right)^{1/n}\le \limsup_{{\mathcal N}\ni n\to\infty}|a_n|^{1/n}\ \limsup_{{\mathcal N}\ni n\to\infty}\ \left(S_m(|\alpha_{1,n}|,\ldots,|\alpha_{n,n}|)\right)^{2/n}< \infty,
0\geq \nabla_{ii}E =-w^{11}\nabla_{ii}w_{11}+2w^{11}\sum w^{kk}(\nabla_{1}w_{ik})^{2}-(w^{11})^{2}(\nabla_{i}w_{11})^{2}-d\left(\frac{h_{ii}}{h}-\frac{h^{2}_{i}}{h^{2}}\right)+l\rho^{2}_{i}+l\rho \rho_{ii}.
\lambda\varphi_1(t)+(1-\lambda)\varphi_2(t)= C(t,0)x_0+S(t,0)y_0+\int_0^t S(t,s)[\lambda f_1(s) +(1-\lambda)f_2(s)]ds &+\int_0^t S(t,s)BB^{*}S^{*}(b,t) R(a,\Upsilon_0^b)\Bigg[x_b-C(b,0)x_0-S(b,0)y_0 &-\int_0^b S(b,s)[\lambda f_1(s) +(1-\lambda)f_2(s)]ds\Bigg](s)ds.
\delta \Xi_{sut}- \delta \tilde{\Xi}_{sut}= -\sum_{n=1}^p Y_{su}^{(n) \sharp}X_{ut}^{(n)}+ \sum_{n=1}^p \tilde{Y}_{su}^{(n) \sharp}\tilde{X}_{ut}^{(n)}& = -\sum_{n=1}^p Y_{su}^{(n) \sharp}(X_{ut}^{(n)}- \tilde{X}_{ut}^{(n)}) + \sum_{n=1}^p (Y_{su}^{(n) \sharp}- \tilde{Y}_{su}^{(n)\sharp})\tilde{X}_{ut}^{(n)}
Q_1 =\mathrm{P}\left( z_t>\frac{\epsilon_1}{\rho}, z_m<\max\left\{\frac{\epsilon_2}{\rho\xi_2},\cdots, \frac{\epsilon_i}{\rho\xi_{i}}\right\}\right) &=\mathrm{P}\left( z_t>\frac{\epsilon_1}{\rho}, z_m<\frac{(1+\epsilon_1)}{\phi_i \left( \rho -\frac{\epsilon_1}{z_t}\right)}\right).
t^2\left\|u_{tt}(t)\right\| + t^3\left\|u_{ttt}(t)\right\| + t\left\|\mathbf{B}_qu_t(t)\right\| + t^2 \left\|\mathbf{B}_qu_{tt}(t)\right\| \leq \mathcal{C}t^{1+ \alpha_{*}}, t\rightarrow 0^+,
a\bigwedge\Bigl\{\frac{\sin\theta-\theta\cos\theta}{2\theta^2}e_1+\Bigl[ \frac{\theta-\sin\theta\cos\theta}{4\theta^2}-\frac{\sin\theta}{\theta}\cdot \frac{\sin\theta-\theta\cos\theta}{2\theta^2}\Bigr]b \Bigr\}=\beta e_{23}.
q\cdot N_{d}^{\prime}=q^2+q\phi(b)+\frac{\phi(-1)}{q-1}\sum_{n=0}^{q-2}G_{-(d-1)n}G_{nd}G_{-n+\frac{q-1}{2}}G_{\frac{q-1}{2}}T^n\left(\frac{b}{a^d}\right)\phi(b) &=q^2+q\phi(b)+\frac{\phi(-b)}{q-1}\sum_{n=0}^{q-2}G_{-(d-1)n}G_{nd}G_{-n+\frac{q-1}{2}}G_{\frac{q-1}{2}}T^n\left(\frac{b}{a^d}\right).
x_1 =\cos\theta_1, x_2 =\sin\theta_1\cos\theta_2, x_3 =\sin\theta_1\sin\theta_2\cos\theta_3, &\dots x_{n-1}=\sin\theta_1\sin\theta_2\dots\sin\theta_{n-2}\cos\theta_{n-1}, x_n =\sin\theta_1\sin\theta_2\dots\sin\theta_{n-2}\sin\theta_{n-1}\cos\varphi, x_{n+1}=\sin\theta_1\sin\theta_2\dots\sin\theta_{n-2}\sin\theta_{n-1}\sin\varphi.
V_{\mathrm{e f f}}= ( | \sigma + m |^{2}- D ) \, | \phi |^{2}+{\frac{D}{2 \pi}}\ln{\frac{\mu}{| \Lambda |}}+{\frac{| \sigma |^{2}}{4 \pi}}\ln{\frac{| \sigma |^{2}}{e \mu^{2}}}-{\frac{| \sigma |^{2}- D}{4 \pi}}\ln{\frac{| \sigma |^{2}- D}{e \mu^{2}}}\, \cdotp
c_{jk}+d_{jk}e^{i b_k \l}+ \sum_{l=1}^n \bigl(c_{jl}\cdot o(1) + d_{jl}\cdot o(1) \cdot e^{i b_k \l}\bigr) = (d_{jk}+ o(1)) e^{i b_k \l}, j \in \{1,\ldots,n\},
\left( \begin{array}{l l l l l}{{1}}&{{x_{1 1}}}&{{\cdots}}&{{\cdots}}&{{x_{1 n}}}\\{{}}&{{1}}&{{\ddots}}&{{}}&{{\vdots}}\\{{}}&{{}}&{{\ddots}}&{{\ddots}}&{{\vdots}}\\{{}}&{{}}&{{}}&{{\ddots}}&{{x_{n n}}}\\{{0}}&{{}}&{{}}&{{}}&{{1}}\\ \end{array}\right) .
\begin{array}{c c}{\theta^{i}= \left( \begin{array}{c}{\vartheta_{-}^{a}}\\{( \bar{\vartheta}_{a}^{-})^{t}}\\ \end{array}\right)}&{\tilde{\theta}^{i}= \left( \begin{array}{c}{\vartheta_{+}^{a}}\\{( \bar{\vartheta}_{a}^{+})^{t}}\\ \end{array}\right)}\\ \end{array}
P_\ell = 2 P_{\ell-1}+ 2 (C_0 n_0 + 6(\ell-1)) n_{\ell-1}+ 4 n_{\ell-1}-1 < 2 P_{\ell-1}+ (C_0 n_0 + 6 \ell - 6) n_\ell + 2 n_\ell = 2 P_{\ell-1}+ 3 (2 \ell - 1) n_\ell + (C_0 n_0 - 1) n_\ell.
E_{0,0}=[x^-_{n-1,0},\ldots,[x^-_{2,0},x^-_{1,-1}]_q\ldots]_qK_1\cdots K_{n-1}\,, F_{0,0}=(K_1\cdots K_{n-1})^{-1}[\ldots [x^+_{1,1},x^+_{2,0}]_{q^{-1}},\ldots,x^+_{n-1,0}]_{q^{-1}}\,.
f_\epsilon(x_1, \dots, x_k) - f_\epsilon(x'_1, \dots, x'_k) = \sum_{i=1}^k \frac{e^{-\frac{x'_i}{\epsilon}}- e^{-\frac{x_i}{\epsilon}}}{\left ( 1 + \sum_{r =1}^k e^{-\frac{x_r}{\epsilon}}\right )\left ( 1 + \sum_{s =1}^k e^{-\frac{x'_s}{\epsilon}}\right )}.
C_{0}\simeq \bigoplus_{\alpha\in Q_{1},\,t(\alpha)=j}\mathbb{S}_{2}^{\deg(\alpha)}(M^{s(\alpha)}), C_{1}\simeq \bigoplus_{\alpha\in Q_{1},\,s(\alpha)=j}\mathbb{S}_{2}^{1-\deg(\alpha)}(M^{t(\alpha)}).
e^{-\mu - 1}= \frac{\mathrm{Tr}\left[e^{-\frac{1}{k_BT}H_S(z_S)}\right]}{\mathrm{Tr}\left[e^{-\sum_{i=1}^k \lambda_i F_i(z_S) -\frac{1}{k_BT}H_S(z_S)}\right]}.
\delta w_\delta(y) + \inf_{\beta \in \mathbb{R}^r}\left\{\left(2\tau(y) \beta -2\tau(y)\sigma(\bar{x},y)^T \bar{p}\right)\cdot D_y w_\delta+|\beta|^2\right\}- |\sigma(\bar{x},y)^T \bar{p}|^2=0.
h=\frac{1}{Q^3}\left(\begin{array}{c}\begin{array}{cc}-1-{y^{\prime}}^2 y^{\prime}{\phi}^{\prime}y^{\prime}{\phi}^{\prime}1-{{\phi}^{\prime}}^2 \end{array}\end{array}\right),\;\;{\hat H}^{\dagger}={\hat H},
u(t,x) = \Re\left[\frac{\varepsilon}{\sqrt{t}}a_{\varepsilon}\left(\frac{x}{t}\right)\exp\left[it \varphi\left(\frac{x}{t}\right)+i\varepsilon^2\left|a_{\varepsilon}\left(\frac{x}{t}\right)\right|^2\Phi_1\left(\frac{x}{t}\right)\log t\right] \right] + \frac{\varepsilon}{t^{\frac{1}{2}+\sigma}}r(t,x) \,,
\begin{aligned}\mathbb{E}^{\boldsymbol{\xi}}[|r^{N,i}(t,\boldsymbol{x}^*_t)|^2]\leq \frac{C}{N^2}\Big(1+\frac{1}{N}\underset{j=1}{\overset{N}\sum}\mathbb{E}^{\boldsymbol{\xi}}[|x^{*,i}_t-x^{*,j}_t|^2]\Big) \leq \frac{C}{N^2}\Big(1+\frac{1}{N}\underset{j=1}{\overset{N}\sum}|\xi^{i}-\xi^{j}|^2]\Big).\end{aligned}
h''(\theta) = \big[D_{ij}u(x_{\theta})-g_{ij}(x_{\theta},y_0,z_0) \quad-D_{p_k}g_{ij}(x_{\theta},y_0,z_0)(D_ku(x_\theta)-D_kg(x_{\theta},y_0,z_0))\big](\dot{x_\theta})_i(\dot{x_\theta})_j \quad\quad+g_{j,z}(E^{m,j}q_mh'+E^{n,j}q_nh').
\mathrm{Gap}(x, \bar \zeta, \bar \omega) = \frac 12 x^\top Q x + f(A^f x) + G(x) + G^*(-(A^f)^\top \bar \zeta - \bar \omega) + \frac 12 \bar \omega^\top Q^{\dagger}\bar \omega + f^*(\bar \zeta) \;.
\mathbb{A}^{-1}_L=\Delta^{-1}\left(\begin{matrix}-\Delta(bd^{-1}c-a)^{-1}\Delta(c-db^{-1}a)^{-1}-c-[a,c](a-bd^{-1}c)^{-1}a-[c,a](db^{-1}a-c)^{-1}\end{matrix}\right).
{\mathcal L}_{eff}\;=\; \frac{h \rho_0 q}{2}\epsilon_{kl}[\dot{x}^{(1)}_k x^{(1)}_l - \dot{x}^{(2)}_k x^{(2)}_l]\,-\,\frac{h^2\rho_0 q^2}{2\pi m}\, \ln | \frac{x^{(1)}- x^{(2)}}{\xi}| \;,
\Phi_{\ell}^{(m)}(r, \varphi) = \mathrm{e}^{\mathrm i m \varphi}(-1)^m J_{|m|}( r \varkappa \ell) = (-1)^m \frac{\ell^{|m|}\varkappa^{|m|}}{2^{|m|}}\cdot \Psi_m (r, \varphi) + \mathcal O(\ell^{|m|+1}) ,
(1-zqb)p_n(z;a,\,bq|q)={bq^{n+1}(aq^{n+1}-1)\over abq^{2n+2}-1}p_{n+1}(z;a,b|q)+{bq^{n+1}-1\over abq^{2n+2}-1}p_{n}(z;a,b|q)
\Gamma_{i n v}= \int \left\{- \frac{1}{4}F_{\mu \nu}F^{\mu \nu}+ \frac{1}{2}( D_{\mu}\underline{{\varphi}}) ( D^{\mu}\underline{{\varphi}}) - \frac{1}{8}\frac{m_{H}^{2}}{m^{2}}e^{2}( \varphi_{1}^{2}+ 2 \frac m e \varphi_{1}+ \varphi_{2}^{2})^{2}\right\}
2 \sum_{k = 1}^{n_{c}- \tilde{n}_{c}}\frac{\delta_{\ell}- \gamma_{k}}{1 - \omega^{2 k - 2 \ell + 1}}= ( n_{c}- \tilde{n}_{c}) \delta_{\ell}- 2 \sum_{k = 1}^{n_{c}- \tilde{n}_{c}}\frac{\gamma_{k}}{1 - \omega^{2 k - 2 \ell + 1}}.
\|a\|^2 =\sum_{i=1}^n\left(\sum_{k=1}^m\bar w_k c_{k,i}\right)^2 &=\sum_{k_1=1}^m\sum_{k_2=1}^m\bar w_{k_1}\bar w_{k_2}\sum_{i=1}^nc_{k_1,i}c_{k_2,i}&=\sum_{k_1=1}^m\sum_{k_2=1}^m\bar w_{k_1}\bar w_{k_2}\bar c_{k_1,k_2}\,,
(e_i^{\pm})^3 e_j^{\pm}- [3]_q (e_i^{\pm})^2 e_j^{\pm}e_i^{\pm}+ [3]_q e_i^{\pm}e_j^{\pm}(e_i^{\pm})^2 - e_j^{\pm}(e_i^{\pm})^3 = 0, i \neq j.
C_{N}( x_{1}, x_{2}) = \pi N \Xi \frac{x_{1}J_{1}( 2 N \Xi x_{1}) J_{0}( 2 N \Xi x_{2}) - x_{2}J_{0}( 2 N \Xi x_{1}) J_{1}( 2 N \Xi x_{2})}{x_{1}^{2}- x_{2}^{2}}\; ,
\begin{bmatrix}p +1 0 a_{21}a_{22}+p\end{bmatrix}*\begin{bmatrix}p +a_{21}1 0 a_{22}+p\end{bmatrix}&=\begin{bmatrix}p 1 a_{21}a_{22}+p\end{bmatrix}+ v^{a_{22}-1}\overline{[a_{22}+p+1]}\begin{bmatrix}p +1 0 a_{21}-1 a_{22}+p+1\end{bmatrix}.
\begin{cases}-(K_H(x)\nabla v)=\frac{1}{\varepsilon^2}\left (v-q_{\hat{x}}\ln\frac{1}{\varepsilon}\right )^{p}_+,\ \ \ \ \ \Omega_{\hat{x}}. v=0,\ \ \ \ \ \partial \Omega_{\hat{x}}.\end{cases}
{\cal S}\; i \left( \begin{array}{c c}{{0}}&{{1}}\\{{1}}&{{0}}\\ \end{array}\right) \;{\cal S}^{- 1}= ( 1 \; \vert \;{\cal J}) \; \left( \begin{array}{c c}{{1}}&{{0}}\\{{0}}&{{1}}\\ \end{array}\right) \; \; ,
F(x) =\operatorname{I}_{\exp\left(-\frac{\theta}{x^2}\right)}(a,b)=\frac{1}{\operatorname{B}(a,b)}\int_0^{\exp\left(-\frac{\theta}{x^2}\right)}\omega^{a-1}(1-\omega)^{b-1}\mathrm{d}\omega,
\|h_n(\cdot, t_2)-h_n(\cdot, t_1)\|_{L^2(Q)}=\|\Delta w\|_{L^2(Q)}\leq C\|D^3 w\|^{\frac12}_{L^2(Q)}\|Dw\|^{\frac12}_{L^2(Q)}&\leq C\|Dh(\cdot, t_2)-Dh(\cdot, t_1)\|^{\frac12}_{L^2(Q)}\|h(\cdot, t_2)-h(\cdot, t_1)\|^{\frac12}_{H^{-1}(Q)}&\leq C(\Lambda_0)(t_2-t_1)^{\frac14}\,,
\langle \bar{\psi}\psi \rangle = \mathrm{t r}S ( x = 0 ) = \frac{2}{\pi^{2}}\int_{0}^{\infty}d p \frac{p^{2}B ( p^{2})}{p^{2}A^{2}( p^{2}) + B^{2}( p^{2})},
\nabla_{\mu}\nabla^{\mu}T + \nabla_{\mu}T \nabla^{\mu}\Phi - \frac{e^{- 2 \Phi}\left( \nabla_{\mu}\nabla_{\nu}T - \nabla_{\mu}\Phi \nabla_{\nu}T \right) \nabla^{\mu}T \nabla^{\nu}T}{1 + e^{- 2 \Phi}\nabla_{\alpha}T \nabla^{\alpha}T}- \frac{e^{2 \Phi}}{V}\frac{d V}{d T}= 0 ,
x_iL^{\alpha_i}_{k_i}(x_i)=-(k_i+1)L^{\alpha_i}_{k_i+1}(x_i)+(2k_i+\alpha_i+1)L^{\alpha_i}_{k_i}(x_i)-(k_i+\alpha_i)L^{\alpha_i}_{k_i-1}(x_i)
\sum_{i=1}^d \| |D_{q_i}| e^{-t(K_V+\sqrt{A})}\|_{{\cal L}(L^{2}(\mathbb{R}^{2d}))}+ \||\partial_{q_i}V(q_i)|e^{-t(K_V+\sqrt{A})}\|_{\mathcal{L}(L^2(\mathbb{R}^{2d}))}\le\frac{c}{t^{\frac{3}{2}}}
\frac{\delta S}{\delta \phi^{A}}\frac{\delta S}{\delta \phi_{A a}^{\ast}}+ \varepsilon^{a b}\phi_{A b}^{\ast}\frac{\delta S}{\delta \overline{{\phi}}_{A}}= i \hbar \frac{\delta^{2}S}{\delta \phi^{A}\delta \phi_{A a}^{\ast}}
{\rm i}\,{\cal P}_\Lambda \, = \,{\o{1}{2}}\left ( k^{i}_\Lambda \, \partial_i{\cal K}-k^{i^\star}_\Lambda \, \partial_{i^\star}{\cal K}\right )\, = \, k^{i}_\Lambda \, \partial_i{\cal K}\, = \, - k^{i^\star}_\Lambda \, \partial_{i^\star}{\cal K}
\lim_{r \rightarrow 0}G_{i}( r ) \sim \frac{q_{i}}{\pi r}\frac{1}{1 + \exp \varepsilon_{i}( 0 , r , 0 )}\left. \frac{d \varepsilon_{i}( 0 , r , \mu_{i}/ 2 )}{d \mu_{i}}\right\vert_{\mu_{i}= 0}.
{\frac{d{e}_{1}( s )}{d s}}= -{e}_{2}^{*}( s ) e_{3}( s ) , \quad{\frac{d{e}_{2}( s )}{d s}}= -{e}_{1}^{*}( s ){e}_{3}( s ) , \quad{\frac{d{e}_{3}( s )}{d s}}= -{e}_{1}( s ) e_{2}( s ) .
B_{ij}\;=\; B_{ij}^{(k)}+ \frac{B_{ik}B_{kj}}{B_{kk}}\, , \frac1{B_{ii}}\;=\; \frac1{B_{ii}^{(k)}}- \frac{B_{ik}B_{ki}}{B_{ii}^{(k)}B_{ii}B_{kk}}\, ,
\theta^{*}(1^{(a)}t^{(2)}i_1...i_{k-2}\otimes j_1...j_{b-k}) = 1^{(a-1)}t^{(2)}i_{1}...i_{k-2}\otimes1j_1...j_{b-k}+1^{(a)}ti_1...i_{k-2}\otimes tj_1...j_{b-k}+\sum_{u=1}^{k-2}1^{(a)}t^{(2)}i_1...\widehat{i_u}...i_{k-2}\otimes i_uj_1...j_{b-k}),
\begin{aligned}d_{0,v}^*f_+ -d_{1,v}^*f_+ = d_{0,h}^*\delta -d_{1,h}^*\delta +d_{2,h}^*\delta -d_{3,h}^*\delta d_{0,v}^*g_+ -d_{1,v}^*g_+ = -\delta + \tau^*\delta. \end{aligned}
\chi = 2 \lim_{n \to \infty}\mathbb{E}[\mathcal{D}^{ij}(0) \mathcal{D}^{lm}(-n,-1)] = \begin{cases}- \frac p{k-1}i = l, j = m, - \frac p{2(k-1)}i = l, j \neq m, 0 i, j, l, m .\end{cases}
|g_{\lambda_i}(z)^{-\frac{1}{2}}-\lambda_i^{-\frac{1}{2}}| =\frac{|\lambda_iz^2-2z|}{|\lambda_ig_{\lambda_i}(z)|^{\frac{1}{2}}|g_{\lambda_i}(z)^{\frac{1}{2}}+\lambda_i^{\frac{1}{2}}|}&\leq \frac{|\lambda_iz^2-2z|}{|\lambda_i\lambda_i/2|^{\frac{1}{2}}\lambda_i^{\frac{1}{2}}}=\sqrt{2}\lambda_i^{-\frac{3}{2}}|\lambda_iz^2-2z|,
h^{d-1}(X_{-i}):=\begin{pmatrix}h^{d-1}\big(x^{(1)}_1,\dots,x^{(i-1)}_1,x^{(i+1)}_1,\dots,x^{(d)}_1\big) h^{d-1}\big(x^{(1)}_2,\dots,x^{(i-1)}_2,x^{(i+1)}_2,\dots,x^{(d)}_2\big) \vdots h^{d-1}\big(x^{(1)}_n,\dots,x^{(i-1)}_n,x^{(i+1)}_n,\dots,x^{(d)}_n\big) \end{pmatrix}
\sum_{q\leq x^{\kappa/2}}\max_{(a,q)=1}\left\lvert \#\mathcal{P}_{L_i,\mathcal{A};q,a}(x)-\frac{\#\mathcal{P}_{L_i,\mathcal{A}}(x)}{\phi(q)}\right\rvert \ll_A\frac{N}{(\log N)^A}\,.
D :=\{(x,t)\::\:t<0,\: x_N <\frac{\delta_1}{2},\: |x-\delta_1 e_N|^2 + t^2 < \delta_1^2 \}; \Gamma_1 := \{(x,t)\::\: t \le 0,\: x_N <\frac{\delta_1}{2},\: |x-\delta_1 e_N|^2 + t^2 = \delta_1^2 \}; \Gamma_2 := \{(x,t)\::\:t \le 0,\: |x-\delta_1 e_N|^2 + t^2 \le \delta_1^2, \: x_N =\frac{\delta_1}{2}\}.
\left( - i \varepsilon \delta \theta \int d^{4}z \{\partial^{0}A_{0}+ \frac{1}{\theta^{2}}\partial_{i}A_{i}\}( z ) \int d^{4}x \{( 1 - \theta ) \partial^{0}A_{0}+ ( \frac{1}{\theta}- 1 ) \partial_{i}A_{i}\}^{\alpha}( x ) M_{0}^{' - 1 \alpha \beta}( x , z ; \alpha_{0}) \right)
S_{c l}= - \frac{T}{\pi \alpha^{\prime}}{\sqrt{1 - \frac{4 \pi^{2}}{T^{2}}}}; \qquad W ( m ) = \frac{4 \pi}{T \alpha^{\prime}}\frac{1}{\sqrt{1 - \frac{4 \pi^{2}}{T^{2}}}}
U'_f(\alpha) := \frac{\partial}{\partial \alpha}U_f(\alpha) = \int_{\mathbb{D}}|f(z)|^{2\alpha}(1-|z|^2)^{\alpha}d\mu(z) & - (\alpha-1) \int_{\mathbb{D}}|f(z)|^{2\alpha}(1-|z|^2)^{\alpha}\log\frac{1}{|f(z)|^{2}(1-|z|^2)}d\mu(z).
\lim_{n\rightarrow \infty}\nu_{x_n-x, z}(t) = 1\Rightarrow \displaystyle\lim_{n\rightarrow \infty}\frac{t}{t+\|x_n-x,z\|}= 1 & \Rightarrow \displaystyle\lim_{n\rightarrow \infty}\|x_n-x, z\|=0\Rightarrow x_n\rightarrow x\textrm \quad{~~ for~~ the ~~ 2-norm ~~}.
\mathcal{F}_{i+\frac{1}{2}}^{\tau}=\frac{1}{2}(-u_{i}^{n}-u_{i+1}^{n})-\frac{\lambda}{2}(\tau_{i+1}^{n}-\tau_{i}^{n}), &\mathcal{F}_{i+\frac{1}{2}}^{u}=\frac{1}{2}(p(\tau_{i}^{n})+p(\tau_{i+1}^{n}))-\frac{\lambda}{2}(u_{i+1}^{n}-u_{i}^{n}).
\partial_Y\left(\Phi(\mathcal{P}(X))\right)=\Phi(\nabla_Y\mathcal{P}(X))-\sum_{i=1,2}\sqrt{c_i}\ \langle Y_i,\mathcal{P}(X)\rangle N_i +\Phi(B(Y,\mathcal{P}(X)_T)-B^*(Y,\mathcal{P}(X)_N))
2\;\nabla\cdot(\mu\;\varepsilon(\delta\b u)_i)-\rho(\delta u_i)_{tt}=\nabla\cdot(\mu\begin{pmatrix}\partial_1 \delta u_i \partial_2 \delta u_i \partial_3 \delta u_i \end{pmatrix}) + \nabla\cdot(\mu \begin{pmatrix}\partial_i \delta u_1 \partial_i \delta u_2 \partial_i \delta u_3 \end{pmatrix})-\rho(\delta u_i)_{tt}.
{\cal Q}_{3 A B}= Q_{3 A ; B}+ Q_{B A ; 3}- \frac 1 3 \eta_{A B}Q_{3}{^\nu}{_{; \nu}}={Q}_{3 A | | B}- \frac{1}{3}\eta_{A B}\left( Q_{3}{^C}{_{| | C}}+ v^{- 1}\dot{Q}_{0 3}\right) +
\frac{\partial\Pi_{1}}{\partial q_{1}}=\frac{\partial P(q_{1}+q_{2})}{\partial q_{2}}\cdot\frac{\partial q_{2}(q_{1})}{\partial q_{1}}\cdot q_{1}+\frac{\partial P(q_{1}+q_{2})}{\partial q_{1}}\cdot q_{1}+P(q_{1}+q_{2}(q_{1}))-\frac{\partial C_{1}(q_{1})}{\partial q_{1}}.
\psi^{( + )}\stackrel{r \sim 0}{\rightarrow}\left[ c_{1}r^{\left({\frac{7}{4}}\right)}\left( 1 +{\cal O}( r ) \right) + c_{2}r^{\left( -{\frac{3}{4}}\right)}\left( 1 +{\cal O}( r ) \right) \right] .
\sum_{\ell=1}^{k}(-1)^{\ell}(n+\ell)!L(k,\ell)=\sum_{\ell=1}^{k}(-1)^{\ell}(n+\ell)!\binom{k-1}{\ell-1}\frac{k!}{\ell!}=\sum_{\ell=1}^{k}(-1)^{\ell}(n+\ell)!\frac{(k-1)!}{(\ell-1)!(k-\ell)!}\frac{k!}{\ell!}=\sum_{\ell=1}^{k}\binom{k}{\ell}(-1)^{\ell}\frac{(n+\ell)!(k-1)!}{(\ell-1)!},
(DF)_1 =\frac{\partial h}{\partial t}-Jk+(a_{01}x(1-x)D+a_{00})h, (DF)_2 =\frac{\partial k}{\partial t}+H_1\mathcal{L}h+(a_{11}x(1-x)D+a_{10})h+ &+(a_{21}x(1-x)D+a_{20})k.
\left(\hat{f}^{*}\hat{\bf \omega}^{n}\right)\left.\left(\hat{\bf A}_{1},\ldots,\hat{\bf A}_{n}\right)\right|_{\Phi_{p}}=\hat{f}_{*}\hat{\bf \omega}^{n}\left.\left(\hat{f}_{*}\hat{\bf A}_{1},\ldots,\hat{f}_{*}\hat{\bf A}_{n}\right)\right|_{f\left(\Phi_{p}\right)}.
Z^{W}= \int{\cal D}\gamma_{+ +}{\cal D}c{\cal D}b^{--}\mathrm{e x p}\left[ i \Gamma^{W}( \gamma ) + i \int \! d^{2}\xi b^{--}\hat{\nabla}_{-}( g_{- \alpha}\epsilon^{\alpha \beta}\partial_{\beta}c ) \right] .
f(\eta) = C_Q \bigg(\frac{1+\cos\alpha}{1+\cos\eta}\bigg)^{\frac{d-1}{2}}\bigg(\frac{1+\cos\alpha}{\cos\eta-\cos\alpha}\bigg)^{\frac{1}{2}}\,{}_2 F_1\bigg (1, \frac{d-1}{2};\frac{1}{2}; \frac{\cos\eta-\cos\alpha}{1+\cos\eta}\bigg) + F(\eta), 0 \leq \eta \leq \alpha.
\varphi - \tilde\varphi = f_1, &\rho_1\tilde\varphi + a A^\frac12(A^\frac12\varphi + \psi)= \rho_1 f_2, &\psi - \tilde\psi = f_3, &\rho_2\tilde\psi + A (b\psi - \delta A^{\gamma-1}\theta) + a (A^\frac12\varphi + \psi) = \rho_2 f_4, &\rho_3\theta + A(c\theta + \delta A^{\gamma-1}\tilde\psi )= \rho_3 f_5.
da = \sum_{x \in X^{\underline{n+1}}}a_x \sum_{i=1}^{n-1}(-1)^i x_0 \cdots \hat{\bar{x}}_i \cdots x_n &= \sum_{y \in X^{\underline{n}}}\sum_{i=1}^{n-1}(-1)^i \sum_{z \in X}a(y_0 \cdots y_{i-1}\bar{z}y_i \cdots y_{n-1}) \: y.
F_{\kappa}(q) := \kappa K \Big(\sqrt{1-q^{2}}-\varepsilon_{\kappa}\Big)\sqrt{1-q^{2}}, q \in [-1, 1], D(\varepsilon,\kappa) :=\frac{2\big(\sqrt{1-\varepsilon_{\kappa}^{2}}-\varepsilon\big)}{F_{\kappa}\bigg(\sqrt{1-\varepsilon_{\kappa}^{2}}-\varepsilon\bigg)}.
\varphi(u(t)) = \varphi(u_0) + \int_0^t \varphi'(u(s))b(s)\,ds + \int_0^t \varphi'(u(s))G(s)\,dW(s) + \frac12 \int_0^t \operatorname{Tr}_{G(s)}\varphi''(u(s))\,ds.
\left( \sum_{j_{1}=1}^{\infty}\left( \ldots\left( \sum_{j_{m}=1}^{\infty}\left\Vert T\left( e_{j_{1}},\dots,e_{j_{m}}\right) \right\Vert^{s_{m}}\right)^{\frac{s_{m-1}}{s_{m}}}\dots\right)^{\frac{s_{1}}{s_{2}}}\right)^{\frac{1}{s_{1}}}\leq\left( \sqrt{2}\right)^{m-1}\Vert T\Vert
\int_{S^2}(h_0^{(3)}- h^{(3)}) dS^2 = \frac{1}{2}\int_{S^2}|\AA^{'(3)}|^2_{\tilde{\sigma}}dS^2-\frac{1}{4}\int_{S^2}(\tilde{\Delta}X_0^{(3)})^2 dS^2-\int_{S^2}\tilde X^i \tilde{\Delta}(X_i^{(5)}-X_i^{'(5)}) dS^2+\int_{S^2}(k^{(3)}- h^{(3)}) dS^2.