harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
F ( z ) \; = \; F ( 0 ) \, + \, \sum_{m = 1}^{\infty}\, \left\{\frac{b_{m}}{z - a_{m}}+ \frac{b_{m}}{a_{m}}\right\}\, + \, \sum_{m = 1}^{\infty}\, \left\{\frac{b_{- m}}{z - a_{- m}}+ \frac{b_{- m}}{a_{- m}}\right\}\, .
\exp\left[-\frac{c^2}{2\eta}t+o(t)\right]= P_0(A_t)\geq P_0(A_t\cap B_t)\geq P_0(A_t)-P_0(B_t^c) = \exp\left[-\frac{c^2}{2\eta}t+o(t)\right]-\exp\left[-\frac{k^2}{2\eta}t+o(t)\right] = \exp\left[-\frac{c^2}{2\eta}t+o(t)\right],
L =\frac{1}{2}(L+L^{\dag})=\frac{1}{2}\sum_{k=0}^{d}[a_{k}\partial^{k}+(-1)^{k}\partial^{k}M_{a_{k}}] & =\frac{1}{2}[a_{d}\partial^{d}+(-1)^{d}\partial^{d}M_{a_{d}}]+[\text{diff.operator of order}d-1].
f_{\mbox{\footnotesize 1-loop}}=- \frac{i}{2}\Bigg(\int\frac{d^4k}{(2\pi)^4}\frac{1}{k^2 (k+p)^2}- \sum\limits_i c_i \int\frac{d^4k}{(2\pi)^4}\frac{1}{(k^2-M_i^2)\Big((k+p)^2-M_i^2\Big)}\Bigg)
L_{0}:=\sup\limits_{l\in\{1,\dots,m\}}\|h_{l}^{(1)}\|_{H^{s}_{\overline{\Omega}}}\|h_{l}^{(2)}\|_{H^{s}_{\overline{\Omega}}}, \L_{1}:=\sup\limits_{l\in\{1,\dots,m\}}\|h_{l}^{(1)}\|_{L^2(\Omega)}\|h_{l}^{(2)}\|_{L^2(\Omega)},
\int_0^1 \left[K\left(\frac{x_1}{y}\right)-K\left(\frac{x_2}{y}\right)\right]\frac{\omega(y)}{y}~dy = \int_0^{2x_1}+\int_{2x_1}^{\frac{1}{2}x_2}+\int_{\frac{1}{2}x_2}^{2x_2}+\int_{2x_2}^1\left[K\left(\frac{x_1}{y}\right)-K\left(\frac{x_2}{y}\right)\right]\frac{\omega(y)}{y}~dy = I+II+III+IV.
{f_{Y\left| X \right.}}\left({y\left| x \right.}\right) = \frac{1}{{\sqrt{2\pi \left({1 + x{\varsigma^2}}\right){\sigma^2}}}}{e^{- \frac{{{{\left({y - x}\right)}^2}}}{{2\left({1 + x{\varsigma^2}}\right){\sigma^2}}}}},
\int_{\theta_{j}}\frac{\Gamma(\sum_{i=1}^{K}n_{j,(\cdot)}^{i}+\alpha_{i})}{\prod_{i=1}^{K}\Gamma(n_{j,(\cdot)}^{i}+\alpha_{i})}\prod_{i=1}^{K}\theta_{j,i}^{n_{j,(\cdot)}^{i}+\alpha_{i}-1}d\theta_{j}=1.
\hat{c}_{\mathit{4}y}(i_1,i_2,i_3) = \mathrm{E}\{y(i)y(i+i_1)y(i+i_2)y(i+i_3)\}- \mathrm{E}\{g(i)g(i+i_1)g(i+i_2)g(i+i_3)\},
X:=\big\{r'_1r'_2\in\sqrt{\mathbb{M}}{}^2:\ r'_1\cdot r'_2=r_1\cdot r_2,r'_1 r'_2\preccurlyeq r_1r_2,r'_1\perp r'_2, &D=\phi(\mathbb{M}^n;r'_1,r'_2,r_3,\dots,r_m)\big\}.
\sum_{n=0}^\infty p(25n+24)q^n \equiv \frac{(q^5)_\infty^6}{(q)_\infty^7}+q^2 \frac{(q^5)_\infty^{18}}{(q)_\infty^{19}}+q^4\frac{(q^5)_\infty^{30}}{(q)_\infty^{31}},
(D_t - Op_h^w(x\xi +p(\xi))v^{\Sigma}_{\Lambda}= h\, Op_h^w(\Gamma) Op_h^w(\Sigma) \Big[\sum_{|I|=3}Op_h(m_I)(v_I) + \sum_{|I|=3}Op_h(\widetilde{m}_I)(v_I)\Big] & + h Op_h^w\left(\Gamma_{-1}(\frac{x+p'(\xi)}{\sqrt{h}})(\frac{x+p'(\xi)}{\sqrt{h}})\right)v^{\Sigma}+ h^{\frac{3}{2}}Op_h^w(r)v^{\Sigma}\, ,
\|\zeta^1\|_{\widetilde{\mathcal{H}}}=\left\|\int_{0}^{T}S(T-\tau)(\xi\xi_x)(\tau)d\tau\right\|_{\widetilde{\mathcal{H}}}\leq C \sup_{t\in[0,T]}\left\|\int_{0}^{t}S(t-\tau)(\xi\xi_x)(\tau)d\tau\right\|_{\widetilde{\mathcal{H}}}\leq C\|\xi\|_{\mathbb{Z}_{\frac{1}{2},0}^T}^{2}.
\begin{bmatrix}1 0 0 0 0 b_{1,0}1 0 0 0 0 0 1 0 0 0 0 0 1 0 b_{4,0}b_{4,1}b_{4,2}b_{4,3}1 \end{bmatrix}\begin{bmatrix}x_0 x_1 x_2 x_3 x_4 \end{bmatrix}+\begin{bmatrix}0 \varepsilon_1 0 0 \varepsilon_4 \end{bmatrix}
p(x) = p_{10}(x) \cdot p_{20}(x) + \bigl(p_{\pi_11}(x) - p_{\pi_10}(x)\bigr) \cdot \prod_{i \neq \pi_1}p_{i0}(x) + \cdots + \bigl(p_{\pi_{2n}n}(x) - p_{\pi_{2n}(n-1)}(x)\bigr)\cdot \prod_{i \neq \pi_{dn}}p_{in}(x),
\lim_{T\rightarrow\infty}\left(\sum_{t=1}^{T}\frac{\gamma^{t}}{t}\right) \geq\int_{1}^{\infty}\frac{\gamma^{t}}{t}dt =\mathrm{Ei}\left(\infty\ln\gamma\right)-\mathrm{Ei}\left(\ln\gamma\right) = -\mathrm{Ei}\left(\ln\gamma\right) =\mathrm{E_{1}}\left(\ln\frac{1}{\gamma}\right)>0,
{\mathbf Y}_j(\theta,\ell) ={\mathbf T}_{j-1}(E,\ell){\mathbf T}_{j+1}(E,\ell) (1\le j\le k),{\mathbf Y}_t(\theta,\ell) ={\mathbf T}_{k}(E,\ell), &E=E_0^{(k)}\,{\rm e}^{\frac{2}{k+2}\theta}
p(x|\nu,\hat{\mu},\hat{\sigma})=\frac{\Gamma(\frac{\nu+1}{2})}{x\Gamma(\frac{\nu}{2})\sqrt{\pi\nu}\hat{\sigma}}(1+\frac{1}{\nu}(\frac{lnx-\hat{\mu}}{\hat{\sigma}})^{2})^{-\frac{\nu+1}{2}}
V(\vec{K}'_1) = \{x_1,x_2,y_1,y_2,y_3,y_4\}A(\vec{K}'_1) = \{x_1x_2, y_1y_2,y_2y_3,y_3y_4,y_4y_1, y_1x_1,y_2x_1,y_3x_1,y_4x_1, x_2y_1,x_2y_2,x_2y_3,x_2y_4\}
L^{0}\, = \, \eta_{i j}\Big ( \epsilon P_{0}^{i}\partial_{\tau}X_{0}^{j}, + \, P_{1}^{i}\partial_{\tau}X_{1}^{j}\, + . . . + P_{n}^{i}\partial_{\tau}X_{n}^{j}\, + \, . . . + P_{N - 1}^{i}\partial_{\tau}X_{N - 1}^{j}\, + \, \epsilon P_{N}^{i}\partial_{\tau}X_{N}^{j}\, \Big ) \, + V
E\left[( R_{[s-\delta_{k},s]}^{3})^{2}\right]\leq\sum_{i=0}^{3}E\left[ ( D_{s}^{6-i}F)^{2}\right]{\binom{3}{i}}^{4}\frac{i!}{\left( 3!\right)^{2}}\delta_{k}^{6-i}.
g^{(2)}=\frac{\langle\hat{E}^{-}(0)\hat{E}^{-}(\tau)\hat{E}^{+}(\tau)\hat{E}^{+}(0)\rangle}{\langle\hat{E}^{-}(0)\hat{E}^{+}(0)\rangle\langle\hat{E}^{-}(\tau)\hat{E}^{+}(\tau)\rangle}
t_{B}= \frac{1}{b^{2}}\left( \left[1 + \frac{b^{2}}{(a+b)^{2}}- \frac{(b-a)^{2}}{(a+b)^{2}}\right](a+b)^{2}- 4ab \right) &= \frac{1}{b^{2}}((a+b)^{2}+b^{2}- (b-a)^{2}- 4ab) = 1,
\frac{1}{2}ix\zeta^2-\frac{1}{2}ix\alpha^2\ln\zeta=\frac{i}{2}x\sqrt{\lambda}+\frac{i}{4}x-\frac{1}{4}\ln x-\frac{\pi i}{4}+\frac{1}{2}\ln(b{e}^{-2is})+o(1),
\psi\left(s_2^{m_1}s_1^{k_1}s_2^{m_2}\cdots s_1^{m_n}s_2^{k_n}s_1^{m_{n+1}}\right) = \psi\left(s_1^{k_1 + \cdots + k_n}\right) \prod^{n+1}_{j=1}\psi\left(s_2^{m_j}\right).
\begin{aligned}\|w\|_{C^{\frac{1+\beta_0}{2},1+\beta_0}(D)}\leq C_1(\|F_1\|_{L^p(D)}+ \|z\|_{L^p(D)}+ \|w_0\|_{W^{2,0}_{p}([0,1])}) & \leq C_1(\|F_1\|_{C^0(D)}+\|z\|_{C^0(D)}+ \|w_0\|_{C^{2}([0,1])}) \end{aligned}
(f_e\pi_{g^{-1}})(x) =f_e\left(\sum_{h\in G}f_h(x)\tau_{g^{-1}h}\right)=\sum_{h\in G}f_h(x)f_e(\tau_{g^{-1}h})=\sum_{h\in G}f_h(x)f_{g^{-1}g}(\tau_{g^{-1}h}) &=\sum_{h\in G}f_h(x)f_g(\tau_{h})=f_g\left(\sum_{h\in G}f_h(x)\tau_{h}\right)=f_g(x), \forall g \in G, \forall x \in \mathcal{X}.
\begin{array}{c c}{\mathrm{v}_{\alpha \beta}= \textstyle{\frac{1}{2}}\epsilon_{\alpha \beta \gamma \delta}\tilde{\mathrm{v}}^{\gamma \delta}}&{\tilde{\mathrm{v}}^{\alpha \beta}= \textstyle{\frac{1}{2}}\epsilon^{\alpha \beta \gamma \delta}\mathrm{v}_{\gamma \delta}}\\ \end{array}
\|B\|_{L^{\infty}(0,T; H^{\frac52-\alpha})}\geq \left(\sum_{q\geq0}\lambda_q^{2(\frac52-\alpha)}e^{-2\mu\lambda_q^\alpha t}\|B_q(0)\|^2_{L^2}\right)^{\frac12}-C \|B\|^2_{L^{2}(0,T; H^{\frac52-\frac{\alpha}{2}})}.
\rho\,u\,=\,\sqrt{\rho}\,u\,+\,\left(\sqrt{\rho}\,-\,1\right)\,\sqrt{\rho}\,u\quad\in\,L^\infty_T\bigl(L^2+L^{3/2}\bigr)\,\cap\,\Bigl(L^\infty_T\bigl(L^2\bigr)+L^2_T\bigl(L^2\bigr)\Bigr)\,.
\big|A_N\bigl(e^{i\lambda}\bigr)\bigl(1-e^{i\lambda\mu}\bigr)^{n}f^0(\lambda)-\lambda^{2n}C^{\mu,0}_{N}\bigl(e^{i\lambda}\bigr)\big|^2 &=\alpha_2\gamma(\lambda)\big|1-e^{i\lambda\mu}\big|^{2n}\bigl(f^0(\lambda)+\lambda^{2n}g^0(\lambda)\bigr)^2,
\frac{1}{2^{n - 1}}= \frac{b 1}{2 + 1}- \frac{b 2}{2^{2}+ 1}+ \frac{b 3}{2^{3}+ 1}- \cdots + ( - 1 )^{n}+ \frac{b^{n}}{2^{n - 1}+ 1}
\begin{array}{c}{\beta_{1}= \frac 2 N \delta_{a b}\delta_{c d}+ d_{a b e}d_{c d e}}\\{\beta_{2}= \frac 2 N \delta_{a d}\delta_{b c}+ d_{a d e}d_{b c e}}\\{\beta_{3}= \frac 2 N \delta_{a c}\delta_{b d}+ d_{a c e}d_{b d e}}\\ \end{array}
F_{PM}=\frac{P_{w}}{u}=\frac{C_{d}}{2C_{SO}}\rho u^{2}\pi R_{0}^{2}(\frac{P_{RMF}}{u^{2}R_{0}\rho\mu_{0}})^{1/f_{o}}(\frac{\mu_{0}^{2}}{4\pi\eta_{p}}\frac{2f_{o}-1}{f_{o}^{2}})^{1/f_{o}}
e^{-2\phi}\delta(H_3\wedge *H_3) =2 \delta dB_2\wedge e^{-2\phi}*H_3 &=2d\left(\delta B_2\wedge e^{-2\phi}*H_3\right)-2\delta B_2\wedge d\left(e^{-2\phi}*H_3\right).
\lim_{\stackrel{n\to\infty}{f(a_n)=1}}\frac{|H|}{n-1}\leq (1-\{\alpha\})\{\beta\}/((\beta-\alpha)\beta) &< (1-\{\alpha\})\{\beta\}=\lim_{\stackrel{n\to\infty}{f(a_n)=1}}\frac{|G|}{n-1}
\varphi_{C}= \frac{1}{g r}\left[{\cal{N}}{\sin}( \overline{{{\cal{N}}}}+{\cal{N}}) +{\cos}( \overline{{{\cal{N}}}}+{\cal{N}}) - 1 \right] +{\cal T}_{C}\left[{\cos}( \overline{{{\cal{N}}}}+{\cal{N}}) - 1 \right] +{\cal T}_{A}\,{\sin}( \overline{{{\cal{N}}}}+{\cal{N}}) \, .
E_\sigma = 2\left(k+ \sum_{i=1}^{k}\frac{k\!-\!i}{n\!+\!1}\cdot\frac{f^{{}^i\sigma}}{f^\sigma}- \sum_{i=1}^{k}\frac{k\!+\!1\!-\!i}{n\!+\!1}\cdot\frac{f^{\sigma^i}}{f^\sigma}\right).
N(X_{t}|\mu_{i,t},\sigma_{i,t})=\frac{1}{(2\pi)^{D/2}}\frac{1}{|\sigma_{i,t}|^{1/2}}exp(-\frac{1}{2}(X_{t}-\mu_{i,t})^{T}\sigma_{i,t}^{-1}(X_{t}-\mu_{i,t}))
t_1(m)^{\ell_1/2}t_2(m)^{\ell_2/2}\dotsm t_d(m)^{\ell_d/2}&= \left( \frac{t_1(m)}{t_2(m)}\right)^{\ell_1/2}\left( \frac{t_2(m)}{t_3(m)}\right)^{(\ell_1 + \ell_2)/2}\dotsm\left( \frac{t_d(m)}{t_{d+1}(m)}\right)^{(\ell_1 + \dotsb + \ell_d)/2}
B_{j,n}^{(\ell)}:=-\frac{\alpha^2 \sqrt{N_j}}{nF_j s_j}+ (-1)^\ell \frac{c_r\alpha^3 \sqrt{N_j}}{(nF_j)^{3/2}s_j}\mp \frac{\alpha\theta_j^0}{\sqrt{N_j}s_j}
e^{- \Phi}( - \det G_{\|})^{\frac{1}{2}}= g^{- 1}Z^{- 1}\Bigl ( 1 - | v |^{2}e^{\psi}Z \Bigr )^{\frac{1}{2}}\approx g^{- 1}Z^{- 1}- \frac{1}{2}g^{- 1}e^{\psi}| v |^{2}\ .
S_r =\frac{1}{r}\sum_{i,j,\alpha}(T_{r-1}^{\alpha})^i_j(A^{\alpha})^j_i \textrm{for $r$ even},{\bf S}_r =\frac{1}{r}\sum_{i,j,\alpha}(T_{r-1})^i_j(A^{\alpha})^j_ie_{\alpha}\textrm{for $r$ odd},
\sum_{\substack{d<X^{50/77-\epsilon}p|d\Rightarrow p>X^{\delta}}}\sum_{\substack{e<X^{\epsilon/2}(e,10)=1 p|e\Rightarrow p\le X^{\delta}}}R_d(e) \ll\sum_{\substack{q<X^{50/77-\epsilon/2}(q,10)=1}}\Bigl|\#\mathcal{A}'_{q}-\frac{\kappa\#\mathcal{A}{}}{q}\Bigr| &\ll_A \frac{\#\mathcal{A}}{(\log{X})^A}.
\begin{pmatrix}1-\varphi(x)\varphi(x)^* (1-\delta)(1-\varphi(x)\varphi(y)^*) (1-\delta)(1-\varphi(y)\varphi(x)^*) 1-\varphi(y)\varphi(y)^* \end{pmatrix}\geq 0.
d \hat{X}^{x, \kappa}_t = \big[ A \hat{X}^{x, \kappa}_t + F_{\kappa}( \hat{X}^{x, \kappa}_t ) \big] \, dt + B_{\kappa}( \hat{X}^{x, \kappa}_t ) \, dW_t , t \in [0,T] , \hat{X}_0^{x, \kappa}= x.
\eta_{a}(C(\mathsf{Q}))=\sum_{i,j,k}q^{(2\rho-2\alpha_{a},\lambda_{i})}(2c_{j}^{i}-\delta_{j}^{i})\varepsilon(F_{a}\triangleright\mathsf{P}_{k}^{j})\varepsilon(E_{a}\triangleright\mathsf{P}_{i}^{k}).
(u^2+8tu+12t^2+6u+30t+16)([4t+3u+3]c_3-[u+2t+5]b_3) & =(u^2+4tu+6t^2+10u+24t+24)([3u+2t+5]b_3-[u+4t+3]c_3) & +2(u^2+4tu+6u)c_3-2(u^2+2tu+4u)b_3
I_3 = \sum_{n=1}^\infty \left | \int^\infty_{n+1}f(\mu )\frac{1}{\mu -n}\big ( 1 -\psi (n^2 - \mu^2) \big) d\mu \right |^2 \leq \sum_{n=1}^\infty \int^\infty_{n+1}\frac{|f(\mu )|^2 \mu^{2\alpha}}{|\mu -n|^{2-2\beta}}d\mu \int^\infty_{n+1}\frac{1}{|\mu -n|^{2\beta}\mu^{2\alpha}}d\mu\, .
g_{{\bf{k}},{\bf{q}}}({\bf{x}}) = - e^{- i \mathrm{~}{\bf{q . x}}}( \frac{1}{2 \mathrm{~}N \mathrm{~}\epsilon_{{\bf{q}}}}) \Lambda_{{\bf{k}}}( -{\bf{q}}) \omega_{{\bf{k}}}({\bf{q}}) + i \mathrm{~}U_{{\bf{q}}}({\bf{x}}) \Lambda_{{\bf{k}}}( -{\bf{q}}) = - f_{{\bf{k}},{\bf{q}}}^{*}({\bf{x}})
aK_X-aR\sim af^*D+a\sum_{i=1}^rl_iF_i-a\sum_{i=1}^r(1-p^d)l_iF_i+a\sum_{i=1}^r \lfloor\frac{(1-p^d)l_i}{m_i}\rfloor m_iF_i = f^*(aD+\sum_{i=1}^r (\frac{al_ip^{d}}{m_i}+a\lfloor\frac{(1-p^d)l_i}{m_i}\rfloor)y_i).
{\tilde{\cal L}_{E}}= \frac{1}{4 \pi^{2}g}\bigg \{- \sqrt{R^{2}( 1 +{\dot{R}}^{2}+{R^{\prime}}^{2}) +{\tilde{B}}^{2}( 1 +{R^{\prime}}^{2})}+ \frac{h}{2}R^{2}\bigg \}.
\chi_{k}= A \frac{\left( 1 - i \cot \left( n_{1}\pi \right) \right)}{\Gamma \left( 1 + n_{1}\right)}\sqrt{\eta}\left( \frac{k \eta}2 \right)^{n_{1}}\left[ 1 - \frac{( k \eta )^{2}}{2 ( 2 + 2 n_{1})}\right] - i B \frac{\csc \left( n_{1}\pi \right)}{\Gamma \left( 1 - n_{1}\right)}\sqrt{\eta}\left( \frac{k \eta}2 \right)^{- n_{1}}\left[ 1 - \frac{( k \eta )^{2}}{2 ( 2 - 2 n_{1})}\right] .
\sum_{n=0}^{\infty}\Delta_{2}(5n+4)q^{n}=41\dfrac{E_{10}^{3}}{E_{1}E_{2}^{3}E_{5}}+860q\dfrac{E_{10}^{6}}{E_{1}^{4}E_{2}^{2}E_{5}^{2}}+6800q^{2}\dfrac{E_{10}^{9}}{E_{1}^{7}E_{2}E_{5}^{3}}\quad+24000q^{3}\dfrac{E_{10}^{12}}{E_{1}^{12}E_{5}^{4}}+32000q^{4}\dfrac{E_{2}E_{10}^{15}}{E_{1}^{15}E_{5}^{5}},
\begin{aligned}c_1 \left[t^{-d/2}\wedge \left(p_0(t, c_2x, c_2y)+\overline p_\beta(t, c_2x, c_2y)\right)\right] \leq q^b(t,x,y) \leq c_3 \left[t^{-d/2}\wedge \left(p_0(t, c_4x, c_4y)+\overline p_\beta(t, c_4x, c_4y)\right)\right] \end{aligned}
\begin{array}{c}S=\frac 1{16\pi G_{p+1}}\int d^{p+1}y\sqrt{-\tilde{g}}e^{-x^0}\{-\partial^Ax^0\partial_Ax^0+a^ia^j\partial^Aa_k\partial_Aa_l\eta_{ij}^{kl}-2\Lambda \}-\frac 1{16\pi G_{p+1}}\int d^{p+1}y\sqrt{-\tilde{g}}e^{-x^0}\tilde{R},\end{array}
\begin{aligned}|f(a,N,y)|=\frac1{|a+Nw_1+yw_2|^k}\rightarrow 0, &\left|\frac{\partial f}{\partial x}(N,y)\right|=k\frac{|w_1|}{|a+Nw_1+yw_2|^{k+1}}\rightarrow 0 \quad N\to \infty,\end{aligned}
\left|B_n P\left(V_n =k\right) - (\sqrt{2\pi})^{-1}\, e^{-\frac{(k-\lambda_n)^2}{2 B_n^2}}\right| < C\, \frac{\sum_{i=1}^n p_i q_i\left(p_i^2 + q_i^2\right)}{B_n^3}.
S_R(X; \pi, \hat{\phi},W) = \sum_{\substack{l>Z (l, 2)=1}}\mu(l)\sum_{(m,2)=1}W \left( \frac{l^2m}{X}\right) E(Y;\chi_{8l^2m}, \hat{\phi}) \ll \sum_{l>Z}\sum_{X/l^2 \leq m \leq 2X/l^2}\log^{3}(X) \ll \frac{X \log^{3}X}{Z}.
\frac{1}{2}\sum_{ij}\frac{\partial^{2}\epsilon_{n}}{\partial k_{i}\partial k_{j}}q_{i}q_{j}=\frac{\hbar^{2}}{2m}q^{2}+\sum_{n^{\prime}\ne n}\frac{|\int dru_{nk}^{*}\frac{\hbar^{2}}{m}q\cdot(-i\nabla+k)u_{n^{\prime}k}|^{2}}{\epsilon_{nk}-\epsilon_{n^{\prime}k}}
\tan(2\phi_{1}) =\frac{4\epsilon}{\epsilon^{1.5}}=\frac{4}{\sqrt{\epsilon}}, 0<\phi_{1}<\frac{\pi}{4}, \tan(2\psi_{1}) =-\frac{2\epsilon}{\epsilon^{1.5}}=-\frac{2}{\sqrt{\epsilon}}, -\frac{\pi}{4}<\psi_{1}<0.
x^2 = 1 - \frac{q\nu_y\nu_z - q^{-1}\nu_z\nu_y}{q-q^{-1}}, y^2 = 1 - \frac{q\nu_z\nu_x - q^{-1}\nu_x\nu_z}{q-q^{-1}}, z^2 = 1 - \frac{q\nu_x\nu_y - q^{-1}\nu_y\nu_x}{q-q^{-1}}.
\prod_{\substack{m,n \geq 1 m \leq n ; \, (m,n)=1}}\left( 1-z^n\right)^{\frac{m^2}{n^3}}= \sqrt{1-z}\; \exp\left\{\frac{-1}{6}Li_2(z) - \frac{1}{3}\frac{z}{1-z}\right\}.
\begin{aligned}\mathcal{I}(x)= \inf\left\{\int^{1}_{0}\Gamma(\phi(r))dr; ~\phi\in L^2([0,1]) \{\rm is \such \that \}\right. &\left.x(t)=x_{0}+\int^{t}_{0}b(x(s))ds+\int^{t}_{0}\sigma(x(s))\phi(s)ds \right\}.\end{aligned}
q_2 = q_2, q_1 = 2\alpha +\frac{u_t}{u}(1 + q_2),q_0 = 2\alpha\frac{u_t}{u}+ t + 2u^2, U(t) = 6\frac{d}{dt}\left(\log\frac{\kappa}{\sqrt{1-q^2_2}}\right) -\frac{t^2}{2}.
\partial_{\underline{{i}}}\epsilon_{I}-{\textstyle \frac{i}{4}}e^{2 \phi}\bigl ( \partial_{\underline{{i}}}a \bigr ) \epsilon_{I}+{\textstyle \frac{1}{2}}\sqrt 2 e^{- U - \phi}\biggl ( F_{0 i}^{+ ( 1 )}\alpha_{I J}+ i F_{0 i}^{+ ( 2 )}\beta_{I J}\biggr ) \gamma^{0}\epsilon^{J}= 0 \, .
\delta_{\kappa}A^{i j}= \frac 1 2 S t r \left( \left( \begin{array}{c c}{{0}}&{{\left( \not{X}_{k}\kappa^{( i}\right) \varepsilon^{j ) k}}}\\{{- \left( \tilde{\kappa}^{( i}\not{X}_{k}\varepsilon^{j ) k}\right)}}&{{0}}\\ \end{array}\right) \partial_{\tau}t \left( \Theta \right) t^{- 1}\left( \Theta \right) \right) .
\begin{array}{r c l}{{F^{( m )}( \zeta_{1}, \cdots , \zeta_{2 N})}}&{{= \displaystyle \sum_{\iota_{1}, \cdots , \iota_{2 N}}}}&{{v_{\iota_{1}}\otimes \cdots \otimes v_{\iota_{2 N}}F^{( m )}( \zeta_{1}, \cdots , \zeta_{2 N})_{\iota_{1}, \cdots , \iota_{2 N}}}}\\ \end{array}
\rho^{Z,f\oplus 0}_t(S_c S_d) =\rho^{A,f}_t(S_{a}), \rho^{Z,f\oplus 0}_t(S_d S_c) =\rho^{B,\phi(f)}_t(S_{b}), \rho^{Z,0\oplus g}_t(S_d S_c) =\rho^{B,g}_t(S_{b}), \rho^{Z,0\oplus g}_t(S_c S_d) =\rho^{A,\psi(g)}_t(S_{a})
I_0(t)\le \frac{C_3}{\varepsilon}\left\{\Vert{\mathbf F}\Vert^2_{L^2(\Omega_t)}+\Vert\partial_s{\mathbf F}\Vert_{L^2(\Omega_t)}^2+\int_0^t(I_{1,\ast}+I_{1,n})(s)ds+\int_0^t\Vert\varphi(s)\Vert_{L^2(\mathbb R)}^2 ds\right\}&+\varepsilon\left\{I(t)+I_{1,n}(t)\right\}+\frac{C_2}{\varepsilon}\Vert\varphi(t)\Vert_{L^2(\mathbb R)}^2\,.
H = \{v\in L^2(D)^n: \operatorname{div}v = 0 \;\;\mbox{in}\;\; L^2(D)\;, \gamma_0(v\cdot n) = 0 \;\;\mbox{in}\;\; H^{-\frac12}(\partial D) \}\;, V = \{v\in H^1_0(D)^n: \operatorname{div}v = 0 \;\;\mbox{in}\;\; L^2(D)\}\;,
\Phi(\zeta)=\frac{\langle A\zeta, \zeta\rangle}{\langle B\zeta, \zeta\rangle}=\frac{\langle BB^{-1}A\zeta, \zeta\rangle}{\langle B\zeta, \zeta\rangle}=\lambda\frac{\langle B\zeta, \zeta\rangle}{\langle B\zeta, \zeta\rangle}=\lambda.
R_\lambda(n) \le\sup_{\theta \in \left[ 0,1\right]}D_{1+\lambda}\left( P_{Y^n|V=\theta}\| Q_{Y^n}^{\epsilon}\right) &=\max \left\{\Xi_1(n,\lambda, \epsilon), \Xi_2(n,\lambda, \epsilon), \Xi_3(n,\lambda, \epsilon) \right\}
A_{\infty}= \frac{1}{( 1 + q ) Q}\left[ \sqrt{4 s^{2}- 2 ( 1 + q ) Q c}\frac{1 + \left( \frac{k - 1}{k + 1}\right)^{\sqrt{4 s^{2}- 2 ( 1 + q ) Q c}/ D}}{1 - \left( \frac{k - 1}{k + 1}\right)^{\sqrt{4 s^{2}- 2 ( 1 + q ) Q c}/ D}}- 2 s \right]
I = \int_{\Sigma}d^{3}\! x \sqrt{\gamma}\left( R ( \gamma ) -{\frac{1}{2}}\left( \left( D \tilde{U}\right)^{2}- e^{- 2 \tilde{U}}\left( D V \right)^{2}\right) -{\frac{1}{2}}\left( \left( D \tilde{\Phi}\right)^{2}- e^{- 2 \tilde{\Phi}}\left( D \Psi \right)^{2}\right) \right) \ ,
F_{\pm}^{\mu_{1}\ldots \mu_{d / 2}}= \pm \frac{1}{( d / 2 ) !}\frac{\epsilon^{\mu_{1}\ldots \mu_{d / 2}\nu_{1}\ldots \nu_{d / 2}}}{\sqrt{| g |}}F_{\pm \, \nu_{1}\ldots \nu_{d / 2}}
\lambda^{K-1}_K\sigma_K =\sum_{i=1}^m\lambda^{K-1}_i\sigma_i &=(Q^{K-1}-Q^K,Q^0\ominus Q^K) &=(Q^{K-1}-Q^K,Q^{K-1}\ominus Q^K)\ \{\rm (by\Lemma\ \ref{lem:27})}&>0\ \{\rm (by\Lemma\ \ref{lem:28})}.
\widetilde{\cal H}_{\scriptscriptstyle P}= \{Q , \widehat{X}_{( 1 )}\cdots \widehat{X}_{( p )}\}= \sum_{i = 1}^{p}( - 1 )^{i + 1}\widehat{X}_{( 1 )}\cdots{\widehat{\widehat X}}_{( i )}\cdots \widehat{X}_{( p )}\widetilde{\cal H}_{\scriptscriptstyle X_{( i )}}
D^{1}\varOmega_{2k-1}(u)={}\frac{2k-1}{2}\varOmega_{2k-1}(u)D^1\log\frac{1}{u^k \prod_{j=1}^k[(2j)^2-u]};\intertext{for $u\in(0,1) $, we have}D^{1}\omega_{2k}(u)={}k\omega_{2k}(u)D^1\log\frac{1}{u^k \prod_{j=1}^{k+1}[(2j-1)^2-u]}.
P_n^* = n \sum\limits_{r \geq 0}\sum\limits_{\sum k_i = n-1}{n-1 \choose k_0,k_1,\hdots,k_r}P_{k_{0}}^*\widetilde{P}_{k_{1}}\cdots \widetilde{P}_{k_{r}}k_1k_2\cdots k_r,
H ( t ) = \int d^{3}x \left\{- \frac{\hbar^{2}}{2 a^{3}( t )}\frac{\delta^{2}}{\delta \eta^{2}}+ \frac{a ( t )}{2}\left( \vec{\nabla}\eta \right)^{2}+ a^{3}( t ) \left( V ( \phi ) + V^{\prime}( \phi ) \eta + \frac{1}{2}V^{\prime \prime}( \phi ) \eta^{2}+ \cdots \right) \right\}
\widetilde{U}_{s,t}(\xi,\Delta)= \int_{\mathbb{R}}U_{s,t}(x, \Delta) e^{-ix \xi}\,dx, \widetilde{K}_{s,t}(\xi,\Delta)= \int_{\mathbb{R}}K_{s,t}(x, \Delta) e^{-ix \xi}\,dx
A_-(z) =\frac{e^{-\pi i/6}e^{\eta}}{2\sqrt{\pi}z^{1/4}}(1+\O{}(|z|^{-3/2})) A_-(-z) =\frac{e^{\pi i/12}e^{i\eta}}{2\sqrt{\pi}z^{1/4}}A_-'(z) =\frac{e^{-\pi i/6}z^{1/4}e^{\eta}}{2\sqrt{\pi}}(1+\O{}(|z|^{-3/2})) A_-'(-z) = \frac{e^{-5\pi i/12}z^{1/4}e^{i\eta}}{2\sqrt{\pi}}
g(\mathrm{T}^{\mathrm{w}}(J_{\varphi}^+X,Y),J_{\varphi}^+Z)-g(\mathrm{T}^{\mathrm{w}}(J_{\varphi}^+Z,Y),J_{\varphi}^+X)=0, g(\mathrm{T}^{\mathrm{w}}(J_{\varphi}^-X,Y),J_{\varphi}^-Z)-g(\mathrm{T}^{\mathrm{w}}(J_{\varphi}^-Z,Y),J_{\varphi}^-X)=0,
C [ \Delta , \omega ] ={\frac{1}{1 6 \pi^{2}}}\int_{S^{4}}\left\{\alpha_{0}\omega + \alpha_{1}\omega_{z z z z}+ \alpha_{2}\omega_{z z z}\omega_{z}+ \alpha_{3}\omega_{z z}^{2}+ \alpha_{4}\omega_{z}^{2}\omega_{z z}+ \alpha_{5}\omega_{z}^{4}+ \alpha_{6}\omega_{z}^{2}+ \alpha_{7}\omega_{z z}\right\}.
H_{\mu \nu \rho}= \epsilon_{\mu \nu \rho}\frac{r ~ r_{1}^{2}}{\left( r^{2}+ r_{1}^{2}\right)^{2}}\left( f_{1}f_{5}\right)^{1 / 4}; \; H_{a b c}= \epsilon_{a b c}\frac{r_{5}^{2}}{r^{3}}\left( f_{1}f_{5}\right)^{- 3 / 4}
z=x\left(C_1+C_2\int\frac{1}{x^2}e^{-\int \frac{\sigma'}{\sigma}{\rm d}r}{\rm d}r\right)=x\left(C_1+C_2\int(\rho''+\frac{\sigma'}{\sigma}\rho'-\frac{k^2\lambda\lambda'(\rho)}{\sigma^2})^{-2}\sigma^{-1}{\rm d}r\right),
\begin{aligned}\partial_t \rho_1=\nabla \cdot \left(\nabla \rho_1+\rho_1\nabla (U_1+\phi)\right), & \partial_t \rho_2=\nabla \cdot \left(\nabla \rho_2+\rho_2\nabla(U_2-\phi)\right), & -\Delta \phi=\rho_1-\rho_2,\end{aligned}
g(z)= g(z_*)+Dg(z_*)(z-z_*)+\frac{1}{2}\langle D^2g(z_*)(z-z_*) , (z-z_*) \rangle +\frac{1}{6}\sum \limits_{i,j,k=1}^{2n}D_{ijk}g((1-\tau_z) z_*+\tau_z z)(z-z_*)_i(z-z_*)_j(z-z_*)_k,
S_{\min}= \int_{t_{\mathrm{i}}}^{t_{\mathrm{f}}}d t \, \sum_{j = 1}^{3}r_{j}\left[ 1 - \frac{1}{6}( \dot{{\bf z}}_{j \mathrm{T}_{j}}^{2}+ \dot{{\bf z}}_{M \mathrm{T}_{j}}^{2}+ \dot{{\bf z}}_{j \mathrm{T}_{j}}\cdot \dot{{\bf z}}_{M \mathrm{T}_{j}}) + \ldots \right]
\begin{array}{cl}P_{m+1}^{\left( i\right)}=P_{m+1}^{\left( i+1\right)}+l_{(d+1)m+i+2}^{(i)}P_{m}^{\left( i+1\right)},\ \m=0,1,... P_{0}^{\left( i+1\right)}=1,\end{array}
Z ~=~ \int dx d\eta \ \exp\{-i T (H + \frac{1}{2}\eta^\mu\omega_{\mu\nu}\eta^\nu) \}{1 \over \sqrt{det ||{\delta^{\mu}}_{\nu}\partial_{t}- \frac{1}{2}({\Omega^{\mu}}_{\nu}+{R^{\mu}}_{\nu\rho\sigma}\eta^\rho\eta^\sigma) ||}}
\sum_{n \geq 0}\frac{a_{\alpha n+\beta}}{(\alpha n + \beta +1)^r}z^{\alpha n+\beta}= \frac{(-1)^{r-1}}{\alpha \cdot (r-1)!}\int_0^1 \log^{r-1}(t) \left( \sum_{0 \leq m < \alpha}\omega_\alpha^{-m\beta}A\left(\omega_{\alpha}^m tz\right) \right) dt
\mbox{div $\big(y^\perp\cdot S(v,q)\big)$}=\sum_{j,k}\partial_k\left[(y^\perp)_jS_{jk}(v,q)\right] &=y^\perp\cdot\mbox{div $S(v,q)$}-S_{12}(v,q)+S_{21}(v,q) &=y^\perp\cdot\mbox{div $S(v,q)$}-2a\, y\cdot v
U^{2a}_{2k,2a}(x;q) &=(xq)^{2a+1}(-xq^3;q^2)_\infty\sum_{h=1}^{k-a}[(xq^2)^{2h}\overline{Q}_{k,k-h}(x^2q^4;q)+(xq^2)^{2h-2}\overline{Q}_{k,k-h+1}(x^2q^4;q)] &+(xq)^{2a-1}(-xq^3;q^2)_\infty\sum_{h=1}^{k-a+1}[(xq^2)^{2h}\overline{Q}_{k,k-h}(x^2q^4;q)+(xq^2)^{2h-2}\overline{Q}_{k,k-h+1}(x^2q^4;q)].
\frac{\int_{}^{}\!\, d_i(\theta_i,\phi) N(\theta_i; m_i, \phi v_i) d \theta_i P(\delta_i=1)}{\int_{}^{}\!\, d_i(\theta_i,\phi) N(\theta_i; m_i, \phi v_i) d \theta_i P(\delta_i=1) + N(0; m_i, \phi v_i) P(\delta_i=0)}.
F^{\Psi}_{pqj}\bigl(e^*_{a_1}\otimes \ldots\otimes e^*_{a_p}\otimes\bigl( e_{c_1}\wedge\ldots \wedge e_{c_{j}}\bigr) \otimes e_{b_1}\otimes \ldots \otimes e_{b_{q-1}}\bigr) &=z_{a_1}\ldots z_{a_p}\Bar{z}_{b_1}\ldots\Bar{z}_{b_{q-1}}{\zeta}_{c_1}\wedge\ldots \wedge{\zeta}_{c_{j}}
\begin{aligned}I^2_{1,1}=\sum_{0<\tilde{\alpha}\leq \alpha}\begin{pmatrix}\alpha \tilde{\alpha}\end{pmatrix}\bigg\{(D^{\tilde{\alpha}}u\partial_x+D^{\tilde{\alpha}}v\partial_y)D^{\alpha-\tilde{\alpha}}u &-(D^{\tilde{\alpha}}h\partial_x+D^{\tilde{\alpha}}g\partial_y)D^{\alpha-\tilde{\alpha}}h\bigg\}.\end{aligned}
\;\sum_{x=1}^{t}\binom{n}{x}\left(\frac{exp}{2c}\right)^{cx}\le \sum_{x=1}^{t}\left(\frac{en}{x}\right)^x\left(\frac{exp}{2c}\right)^{cx}= \;\sum_{x=1}^{t}\left[\frac{en}{x}\cdot\left(\frac{exp}{2c}\right)^c\right]^{x}\le\sum_{x=1}^{t}\left(\frac{e^{c+1}np^ct^{c-1}}{(2c)^c}\right)^{x}=o(1)\,,
\psi_{\bf t}(s)=\sup_{{\bf b}\in \mathcal{D}(\psi_1^*)}\Big\{s\left\langle{\bf t},{\bf b}\right\rangle-\frac{1}{2}\sum_{i\in I}\Big[\big(1+b_i\big)\ln\big(1+b_i\big)+\big(1-b_i\big)\ln\big(1-b_i\big)\Big]\Big\}.
\int_I e^{\alpha |u|^{\frac{1}{1-s}}}\, dx=\sum_{k=0}^\infty\int_I \frac{\alpha^k}{k!}|u|^{\frac{k}{1-s}}\, dx\leq \sum_{k=0}^\infty\frac{1}{k!}\left(\frac{C}{1-s}\alpha k\right)^k,
B_{m+l}^{\left( m+l\right)}\left( m\right) =\left( -1\right)^{m+l}B_{m+l}^{\left( m+l\right)}\left( l\right) =\frac{\left(l+m\right)}{l}\binom{m+l}{l}\sum_{k=0}^{m}\left( -1\right)^{k}\genfrac{[}{]}{0pt}{}{m}{k}\binom{l+k}{l}^{-1}B_{l+k}^{\left( l\right)}.
\sum_{\substack{n=-\infty n\neq 0}}^\infty \frac{(3n-1)q^{3n^2}}{1-q^{2n}}=\sum_{n=1}^\infty (3n-1)q^{3n^2}+\sum_{n=1}^\infty\sum_{m=1}^\infty 6n q^{n(3n+2m)}.
A_{1,2}=\left(\begin{matrix}2 1\end{matrix}\right),A_{2,2}=\left(\begin{matrix}2 2 1 0 2 1 1 1\end{matrix}\right),A_{3,2}=\left(\begin{matrix}2 2 2 1 0 0 2 2 1 1 1 0 2 1 1 1 1 1 \end{matrix}\right).
{\cal C}_{a}( \nu_{I}) = \alpha^{\prime}\sum_{I , \overline{{{J}}}}2 \kappa_{a I}\overline{{{\kappa}}}_{a \overline{{{J}}}}G^{I \overline{{{J}}}}\left\{\gamma + \frac{1}{2}\Big ( \mathrm{\boldmath ~{\p s i}~}( \nu_{I}) + \mathrm{\boldmath ~{\p s i}~}( 1 - \nu_{I}) \Big ) \right\}~ ,
G^{ij}_\psi(\nu) \equiv (-1)^{i+j}\Sigma((-1)^i \bar R_i)\Sigma((-1)^j \bar R_j) \bar\gamma_i^{-2}\bar\gamma_j^{-2}\frac{\partial^2}{\partial r \partial r'}G(r,r',\nu)\Bigg|_{{r' = (-1)^i \bar R_i}\atop{r = (-1)^j \bar R_j}}\,.
u_k(t) - u_k(t') = (e^{ik^3t}- e^{ik^3t'})\big[u_k(0) + B_1(v,v)_k(0)\big] + B_1(v,v)_k(t') - B_1(v,v)_k(t) &+ \int_0^t e^{ik^3(t-r)}R_1(u,v,v)_k(r) \;\d r - \int_0^{t'}e^{ik^3(t'-r)}R_1(u,v,v)_k(r) \;\d r &+ \int_0^t e^{ik^3(t-r)}\rho_1(v,v)_k(r) \;\d r - \int_0^{t'}e^{ik^3(t'-r)}\rho_1(v,v)_k(r) \;\d r.
q_{3}\left( 4q_{1}q_{3}-q_{2}^{2}\right) \frac{B_{2}^{\prime}}{B_{2}}+q_{2}\left( 4q_{1}q_{3}-q_{2}^{2}\right) -q_{3}\left(4q_{1}q_{3}-q_{2}^{2}\right)^{\prime}+q_{3}^{\prime}\left(4q_{1}q_{3}-q_{2}^{2}\right) =0.
b_{\nu}(\zeta) = \left( q\frac{d}{dq}\right)^\nu F(q)\biggr\rvert_{q=\zeta}= \left( q\frac{d}{dq}\right)^\nu F(q, (\nu + 1)p-1) = \sum_{j=0}^\nu \sum_{i=0}^{p-1}C_{\nu,i,j}(p)\zeta^iA_p^{(j)}(p(\nu + 1)-1,i,1).