harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\vartheta_{2,4}= (G, G)_4, \vartheta_{3,2}= (G, \vartheta_{2,4})_4, \vartheta_{8,2}= (\vartheta_{2,4},\vartheta_{3,2}^2)_3, \vartheta_{15,0}= ((G, \vartheta_{2,4})_1,\vartheta_{3,2}^4)_8.
m' = \psi(m) = \psi\Big( \sum_{i,j}\theta(u_{i,j}) \cdot (a_{i,j}\cdot g_i) \Big) = \sum_{i,j}\psi(\theta(u_{i,j})) \cdot \psi(a_{i,j}\cdot g_i) = \sum_{i,j}\theta'(\phi(u_{i,j})) \cdot \psi(a_{i,j}\cdot g_i).
\nabla^* \Psi^{(i)}= \beta\nabla V\cdot \Psi^{(i)}- \mbox{div}\,\Psi^{(i)}= \beta\nabla V\cdot (a\Pi \nabla u_i) - \mbox{div}(a\Pi \nabla u_i) = -\beta \mathcal{L}_0 u_i \,.
\frac{\alpha \gamma_j \Delta_j^\beta}{f_0(x_{j-1})}\Big((K\circ v_j) f_0^{(r)}+ \sum_{q=1}^r \sum \binom{r}{q}c_{m_1, \ldots, m_q}\frac{K^{(M_q)}\circ v_j}{F_j^{M_q}}f_0^{(r-q)}\prod_{s=1}^q \big(f_0^{(s-1)}\big)^{m_s}\Big),
V = \left( \begin{array}{c | c}{{\begin{array}{c c}{{0}}&{{B}}\\{{- B}}&{{0}}\\ \end{array}}}&{{M}}\\ \hline{{- M^{T}}}&{{\begin{array}{c c}{{0}}&{{\tilde{B}}}\\{{- \tilde{B}}}&{{0}}\\ \end{array}}}\\ \end{array}\right) \; .
e^{(\delta_n-L+50\varepsilon)tq_n}r_{\ell_t}\leq e^{(\delta_n-L+50\varepsilon)t_0 q_n}\max_{1\leq j\leq y_n-1}r_j\leq C_0 y_n q_n e^{(\delta_n-L+50\varepsilon)t_0 q_n}\leq e^{\varepsilon y_nq_n}e^{(\delta_n-L+50\varepsilon)\frac{y_n}{2}q_n}= e^{(\delta_n-L+52\varepsilon)\frac{y_n}{2}q_n}\leq e^{(\delta_n-L+52\varepsilon)\ell_0 q_n}.
\int_{-\alpha_{N}}^{\alpha_{N}}\nu_{N}(z) \, \mathrm{d}z = 2(N+1) \iff \int_{-\infty}^{\infty}\nu_{N}(z) \, \mathrm{d}z = 2(N+1) & \iff \lim_{\omega \to 0}\mathcal{F}[\nu_{N}](\omega)= 2(N+1).
\mathbf u(t) = \mathbf S(t-S) \mathbf u(S) + i \int_S^t \mathbf S(t-s) \mathbf N\left( \Theta_R^{\mathbf u}(s)\mathbf u(s) \right) \, ds + i \int_S^t \mathbf S(t-s) \mathbf M\left( \mathbf u(s) \right) \, dW(s)
v_k^{(s,t+1)}+V_{k+2}^{(s,t)}=v_{k+2}^{(s,t)}+V_{k+1}^{(s,t)},k=0,1,\dots,2m-2, &v_{k}^{(s,t+1)}V_{k}^{(s,t)}=v_{k}^{(s,t)}V_{k+1}^{(s,t)},k=1,2,\dots,2m-1.
\begin{array}{l}\displaystyle{\vphantom{\Big(}(u'+2u\partial+c\partial^3)f_n+v\partial g_n=(\alpha_1\partial+c_1\partial^3)f_{n-1}+(\beta_1\partial+\gamma_1\partial^2)g_{n-1}\,,}\displaystyle{\vphantom{\Big(}\partial(vf_n)=(\beta_1\partial-\gamma_1\partial^2)f_{n-1}+\epsilon_1\partial g_{n-1}\,.}\end{array}
f^{o,\kappa,h}_C(\rho,j) \coloneqq w_{j,i_{\kappa}}\sum\limits_{n=0}^{n_{\kappa}+1}\biggl(\rho\pi^{o,\kappa,h,+}_{i_{\kappa},k}(n) + \frac{1}{h}\pi^{o,\kappa,h}_{\rho}(n)\biggr)n
\beta_i=\begin{cases}0 i\in A \frac{\alpha_i}{2}i\in A^c\end{cases}, \gamma_i=\begin{cases}\frac{\alpha_i-1}{2}i\in A 0 i\in A^c\end{cases}1_{A,i}=\begin{cases}1 i\in A 0 i\in A^c\end{cases},
\phi ( x ) = - \int_{\sigma}{d \Sigma_{\nu}( x^{\prime \prime}) \left[{d ( \partial ) \Delta ( x - x^{\prime \prime}) \cdot \Gamma^{\nu}( \partial^{\prime \prime}, \mathord{\frac{\buildrel{\lower 3 p t \mathrm{~ \scriptscriptstyle \leftarrow ~}}}{- \partial^{\prime \prime}}}) \phi ( x^{\prime \prime})}\right]}.
\begin{aligned}x_i =F_{x_i}(\varphi,\dot{\varphi},\ldots,\varphi^{(r-1)})\,, i =1,\ldots,n u_j =F_{u_j}(\varphi,\dot{\varphi},\ldots,\varphi^{(r)})\,, j =1,2\,.\end{aligned}
\mathcal{L}\left(\sum_{\beta=1}^{\kappa}N_{\beta,\rm{s}}(t), \omega \right) = \sum_{\beta=1}^{\kappa}N_{\beta,\rm{s}}(t) - \omega \left( \sum_{\beta=1}^{\kappa}\Delta \Phi_{\rm{st},\beta}(t) - \Delta \Phi_{\rm{st},\max}(t) \right).
\mathbf{\Phi}_{\mathrm{t}, g}^\star = [\mathbf{\Phi}_{g}^\star]_{1:\bar{M},:}, ~~~ \mathbf{\Phi}_{\mathrm{r}, g}^\star = [\mathbf{\Phi}_{g}^\star]_{\bar{M}+ 1 : 2\bar{M},:}, \forall g \in \mathcal{G}.
\int_{0}^{l}\exp(i\xi^2)\widehat h(-k+\xi/\sqrt t)d\xi=\int_{0}^{l}\exp(i\xi^2)\widehat h(-k+\xi/\sqrt t)\mu_0d\xi+\int_{0}^{l}\exp(i\xi^2)\widehat h(-k+\xi/\sqrt t)(1-\mu_0)d\xi\,.
|\beta\wedge d\beta| >|F^2\widehat \beta \wedge d\widehat \beta + (1-F)^2\alpha\wedge d\alpha + 2F(1-F)\widehat \beta \wedge d\widehat\beta|-C\varepsilon_0 &=|F(2-F)\widehat \beta \wedge d\widehat \beta+(1-F)^2\alpha\wedge d\alpha|-C\varepsilon_0>c_0>0\,,
(w_{\Delta_2}\tilde w_{\Delta_2})(t) = \prod_{k=1}^{g+1}\frac{(1-x_0^2)^2}{(1-a_kx_0)(1-b_kx_0)}\frac{(w_{\Delta_1}\tilde w_{\Delta_1})(x)}{(1-xx_0)^{2g+2}}.
Z_f(s, \chi, A_4)= \sum_{a=0}^{\omega-1}q^{-a-2-aps}\int_{B_4^a}\chi(ac(x^p+\pi^{k+r+l-ap}y^rz^l\mathbb{H}_r^k(y, tz-y)))|dxdydz| := F_4(q^{-s}),
\left( \mathcal{X}^{K\textnormal{th,eq}}\left(q,\varphi_{q,z}^{-1}(Q)\right) P_{q,z}\right)_{0i}= \Lambda_i^{-\ell_q(Q)}\sum_{d \geq 0}\frac{1}{z^{d(N+1)}}\frac{(1-q)^{d(N+1)}Q^d}{\left(q\Lambda_0 \Lambda_i^{-1}, \dots q, \dots ,q\Lambda_N \Lambda_i^{-1};q \right)_d}
\psi_{1,n}(i_{\theta_1}(1) - s_{\theta_1}s^*_{\theta_1}) = i_{\theta_n}(1 \circ \iota_{N^n}) - s_{\theta_n}^{nN}s_{\theta_n}^{*nN}= i_{\theta_n}(1) - s_{\theta_n}(s_{\theta_n}^{nN-1}s_{\theta_n}^{*(nN-1)}) s_{\theta_n}^* \ge i_{\theta_n}(1) - s_{\theta_n}s_{\theta_n}^*,
\mathrm{LF}_{\tau}:=\{k\in\mathbb Z,\; k^{2}\leq \frac{\tau^{2}}{2}\inf |\partial_{x_2}\varphi |^{2}\},\mathrm{HF}_{\tau}:=\{k\in\mathbb Z,\; k^{2}> \frac{\tau^{2}}{2}\inf |\partial_{x_2}\varphi |^{2}\}.
\begin{cases}\underbar r_1 -\epsilon \le u^*(x,t;t_0,u^*(\cdot,t_0),v^*(\cdot,t_0)) \le \bar r_1+\epsilon\cr\underbar r_2 -\epsilon \le v^*(x,t;t_0,u^*(\cdot,t_0),v^*(\cdot,t_0)) \le \bar r_2+\epsilon\end{cases}
\exp(P_X(u))= \prod_{[P]}\exp(u^{\ell(P)})= \prod_{[P]}\prod_{n=1}^{\infty}\left( 1-u^{n \ell(P)}\right)^{-{\mu(n)}/{n}}= \prod_{n=1}^{\infty}Z_{X}(u^{n})^{{\mu(n)}/{n}},
Q ( z ) = \sum_{n}\left[ \frac{1 - \mu_{n}}{4 ( z - z_{n})^{2}}+ \frac{\beta_{n}}{2 ( z - z_{n})}\right] + \sum_{B}\left[ - \frac{3}{4 ( z - z_{B})^{2}}+ \frac{\beta_{B}}{2 ( z - z_{B})}\right] ,
\varepsilon(z)\Lambda(z^{-1})=-\Lambda(-z^{-1}) , \dot{s}\Lambda(z)=-\Lambda(-z^{-1}), \quad \dot{s}\varepsilon(x)\Lambda(y)=\varepsilon(-x^{-1})\Lambda(x(xy-1))+\Lambda(x)-\Lambda(y^{-1}(xy-1)),
\varphi(x) = c_k^{-\frac{1}{2}}\left| \frac{uJ}{4}\right|^{\frac{1}{2}}\int_{-\infty}^\infty (x_2' - v \sqrt{-1}x_1')^k e^{- \frac{\pi}{2}(v x_1'^2 + \frac{1}{v}x_2'^2)}e^{2 \pi \sqrt{-1}x_2'y_2'}\, dx_2' \int_{-\infty}^\infty e^{- \frac{\pi}{2}(v x_3'^2 + \frac{1}{v}x_4'^2)}e^{2 \pi \sqrt{-1}x_4'y_4'}\, dx_4'.
\begin{array}{rllr}p_{\delta}^{(0)}= \displaystyle{\sum_{I \le \delta}b_{I,\delta}z_{I,\delta}}& z_{I,\delta}\ge 0 I \le \delta, & \displaystyle{\sum_{I \le \delta}z_{I,\delta}=1}& z_{I,\delta}\in \mathbb{R},\; I\le \delta. \end{array}
\beta =\xi_1(p-t+1)+\sum_{r=2}^{\lceil s\rceil-1}(t-1)\xi_r\binom{p-t+1}{(r-1)t+1}+(t-1)\xi_{\lceil s\rceil}\gamma =(p-t+1)\left(\sum_{r=1}^{\lceil s\rceil -2}\xi_{r+1}\binom{p-t}{rt}+\xi_{\lceil s\rceil}\right).
\frac{\partial}{\partial t}\begin{bmatrix}x_1(\xi,t) x_2(\xi,t)\end{bmatrix}=\begin{bmatrix}0 1 1 0\end{bmatrix}\frac{\partial}{\partial\xi}\Bigl(\begin{bmatrix}\rho(\xi)^{-1}0 0 T(\xi)\end{bmatrix}\begin{bmatrix}x_1(\xi,t) x_2(\xi,t)\end{bmatrix}\Bigr)
F_{T,R}(y) = \mathcal{\tilde{M}}+ \sum_{m'=0}^k \sum_{l'=1}^3 \mathcal{M}_{m',l'}+ \sum_{m, m'=0}^{k, k'}\sum_{l=1}^3 \sum_{l' \neq l}\mathcal{R}_{m,l,m',l'}
r_{2^{n-2}+j}=s_{2^{n-2}+j}+s_{2^{n-1}-(2^{n-2}+j)}=-s_{-2^{n-1}+(2^{n-2}+j)}+s_{-j}= &=s_j-s_{-2^{n-2}+j}=s_j-s_{2^{n-2}-j}=r_j-2s_{2^{n-2}-j}.
S = \frac{1}{1 6 \pi}\int_{B}d^{D}x \, \sqrt{| g |}\left( R + \alpha^{2}\left( d ( d - 1 ) \right) \right) - \frac{1}{8 \pi}\int_{\partial{B}}d^{d}x \sqrt{| \gamma |}K + \frac{1}{8 \pi}S_{( c t )}( \gamma ) .
{\cal S}= \frac{1}{4 \pi \alpha^{\prime}}\int d \tau d \sigma \left\{\Phi_{( 1 )}^{1}\left( - \partial_{\tau}^{2}+ \partial_{\sigma}^{2}+ \frac{2}{\cos^{2}\tau}\right) \Phi_{( 1 )}^{1}+ \Phi_{( 1 )}^{2}\left( - \partial_{\tau}^{2}+ \partial_{\sigma}^{2}\right) \Phi_{( 1 )}^{2}\right\}
\begin{array}{ll}\left\lbrace \begin{array}{l}\frac{d^2 x_c}{dt^2}(t)+ \nabla V[x_c(t)]=E(t)\,, x_c(0)=x_0\,, \frac{dx_c}{dt}(0)=\dot{x}_0\,,\end{array}\right. \left\lbrace \begin{array}{l}\frac{dQ}{dt}(t) + Q(t)^2 + V''[x_c(t)]=0\,, Q(0)=ib I_N\,,\end{array}\right.\end{array}
H_{n c}= \left( \nu_{n c}\: M + \overline{{\nu_{n c}}}\: \overline{{M}}+ L + \overline{{L}}\right) \: T_{[ 0 ]}^{( - 1 )}\: T_{[ 0 ]}^{( 0 )}\: \overline{{T_{[ 0 ]}^{( - 1 )}}},
\tilde{\gamma}(t,s)=\begin{dcases}a^++E(s)\, \frac{\gamma(t,s)-a^+}{|\gamma(t,s)-a^+|}s>s^+|\gamma(t,s)-a^+|>E(s), a^-+E(s)\, \frac{\gamma(t,s)-a^-}{|\gamma(t,s)-a^-|}s<s^-|\gamma(t,s)-a^-|>E(s), \gamma(t,s) \end{dcases}
\max_{1\leq j,k \leq n, j\neq k}|f_j(\tau_k)f_k(\tau_j)|\geq \frac{\frac{1}{d}\left(\sum_{j=1}^n f_j(\tau_j)\right)^2-\sum_{j=1}^n |f_j(\tau_j)|^2}{n^2-n}.
\left[\left(2\lambda K+\frac{4(\lambda-1)}{n}(2\left\vert b \right\vert -b)\right)\omega^{2}+(2(2-\lambda)\left\vert b\right\vert-\lambda b)p\omega -2\lambda \left\vert b \right\vert p(p-1)\right]a\omega^{p-2}\geq 0.
f(x,n) \ =\C(0,h_0(x,n)) begin{align*}12pt] =\ \bigg[\frac{0}{h_0(x,n) + 1}+ \frac{1}{2}\bigg] begin{align*}12pt] =\ \bigg[\frac{1}{2}\bigg] begin{align*}12pt] =\ 0.
\textstyle \epsilon\int_{\Omega}|v|^{2}= \Im \int_{\Omega}f \overline{v}, \int_{\Omega}a(\nabla^{b}v,\nabla^{b}v)= \lambda\int_{\Omega}|v|^{2}-\Re\int_{\Omega}f \overline{v}- \int_{\Omega}c|v|^{2}.
\sqrt{1+x}=\sum_{n=0}^{\infty}\frac{(-1)^{n}(2n)!}{(1-2n)(n!)^{2}(4^{n})}x^{n}=1+\frac{1}{2}x-\frac{1}{8}x^{2}+\frac{1}{16}x^{3}-\frac{5}{128}x^{4}+\cdot\cdot\cdot,
\frac{3}{2}( \partial^{\mu}\partial^{\nu}h_{\mu \nu}- \partial_{\mu}\partial^{\mu}\tilde{h}) - \frac{1 5}{2}\partial_{4}\sigma \partial_{4}\tilde{h}- \frac{3}{2}\frac{\partial^{2}\tilde{h}}{\partial{x^{4}}^{2}}- \frac{3}{2}\partial_{\mu}\partial^{\mu}\phi + 3 0 k^{2}\phi - 1 2 k \phi \tilde{\delta}= 0 .
\frac{1}{2\pi}\int_{\mathbb{R}}\varphi_k(\mu)e^{iy\mu}\frac{e^{i\mu\rho}-1}{\rho}\,d\mu=\frac{2^k\big[\chi_0(2^k(y+\rho))-\chi_0(2^ky)\big]}{\rho}=:K_k(y,\rho),
G [ x ( s ) , x_{0}( s ) ; m^{2}] ={\frac{i}{\hbar}}\int_{0}^{\infty}d A \langle x ( s ) | e^{i H A / \hbar}| x_{0}( s ) \rangle = \langle x ( s ) |{\frac{1}{H}}| x_{0}( s ) \rangle \ ,
\eta ( \tau ) = q^{1 / 2 4}\prod_{n = 1}^{\infty}( 1 - q^{n}) = \sum_{- \infty}^{\infty}( - 1 )^{n}q^{\frac{3}{2}( n - 1 / 6 )^{2}}= q^{1 / 2 4}( 1 - q - q^{2}+ \cdots ) ,
\mathfrak{G}_{k}:=\left\{\mathcal{G}\subset\mathfrak{G}:\mathcal{G}=G\left[\left( \partial G^{\prime}\right)^{\ast}\right] \ ,\G^{\prime}\in\mathbb{G}_{c}^{d}\ ;\ \left\vert V\left( \mathcal{G}\right) \right\vert=k\right\}\ .
\mathcal{B}_{(p,\infty),(q,r)}^{\mathcal{T}}(F,f,L)=\sup\{\left\Vert\max(M_{\mathcal{T}}\phi,L)\right\Vert_{L^{q,r}(X,\mu)}^{r}:\phi\geq0\text{is measurable with}\left\Vert \phi\right\Vert_{L^{1}(X,\mu)}=f,\left\Vert \phi\right\Vert_{L^{p,\infty}(X\mu)}^{p}=F\}
\mathsf{IP}(\mathcal G_{x_2^n}) \leq \prod_{i=1}^r \mathsf{IP}(\mathcal G_{i}^{\otimes n_i}), \mathsf{LP}(\mathcal G_{x_2^n}) =\prod_{i=1}^r \mathsf{LP}(\mathcal G_{i}^{\otimes n_i}).
(\Psi\otimes\cdots\otimes\Psi)\rho(u) =\varepsilon(u)(\Psi\otimes\cdots\otimes\Psi), \rho(u)(\Psi^*\otimes\cdots\otimes\Psi^*) =\varepsilon(u)(\Psi^*\otimes\cdots \otimes\Psi^*)
|\phi_{}(1/h)|^2 = \Big(\ln(1/h) \int_0^{1/h}\limits \frac{1}{q_{}(x)}\d x \Big)^k |\phi_{}(1/h)|^2 = \Big( (1/h)^{2\alpha - 1}\int_0^{1/h}\limits \frac{1}{q_{}(x)}\d x \Big)^k.
d\Theta_{t}= (de^{{\bf e}_{t}})e^{{\bf h}_{t}}e^{{\bf f}_{t}}+e^{{\bf e}_{t}}(de^{{\bf h}_{t}})e^{{\bf f}_{t}}+e^{{\bf e}_{t}}e^{{\bf h}_{t}}(de^{{\bf f}_{t}}) &+(de^{{\bf e}_{t}})(de^{{\bf h}_{t}})e^{{\bf f}_{t}}+(d e^{{\bf e}_{t}})e^{{\bf h}_{t}}(de^{{\bf f}_{t}})+e^{{\bf e}_{t}}(de^{{\bf h}_{t}})(de^{{\bf f}_{t}}).
\sum_{i \in \mathbb{Z}}M_i \left[\psi(\nu,\cdot)\right](\vec{x}) =\sum_{k\in \ell(\vec{x})}t \Big( \psi(\nu, \vec{x}_k^{-}) - \psi(\nu,\vec{x}) \Big)+\sum_{k\in r(\vec{x})}\Big( \psi(\nu, \vec{x}_k^{+}) - \psi(\nu,\vec{x}) \Big),
F_{\beta}(t) \equiv E_{\beta}^{{\rm soft}}\bigl(0; (t, \infty)\bigr) = \lim_{N\rightarrow \infty}E_{\beta N}^{{\rm soft}}\left(0;\left(\sqrt{2N}+\frac{t}{\sqrt{2}N^{1/6}}, \infty\right)\right),
\mathbf{G}^{(t)}=\frac{t-1}{t}\mathbf{G}^{(t-1)}+\frac{1}{t}\sum_{n=1}^{N}\mathbf{g}_{n}^{(t)}(\mathbf{g}_{n}^{(t)})^{T}, \mathbf{b}^{(t)}=\frac{t-1}{t}\mathbf{b}^{(t-1)}+\frac{1}{t}\sum_{n=1}^{N}y_{n}^{(t)}\mathbf{g}_{n}^{(t)}.
b_{0}= 0 , \b_{j}= - \left[ \epsilon_{j k}\frac{( x - x^{( l )})_{k}}{| x - x^{( l )}|^{2}}\, + \, \epsilon_{j k}\frac{( x - x^{( r )})_{k}}{| x - x^{( r )}|^{2}}\right]
\begin{matrix}I_{1}\dot{\omega}_{1}+(I_{3}-I_{2})\omega_{2}\omega_{3}&=&0\\I_{2}\dot{\omega}_{2}+(I_{1}-I_{3})\omega_{3}\omega_{1}&=&0\\I_{3}\dot{\omega}_{3}+(I_{2}-I_{1})\omega_{1}\omega_{2}&=&0\end{matrix}
\beta_1=\frac{1}{2}\left\langle\sum_{X_i\in ({\it G}_1)_{X'_{i'}}}\sum_{\stackrel{i_{\alpha_{1}}< j_{\alpha_{1}}<i_{\alpha_{2}}< j_{\alpha_{2}}}{\left\{i_{\alpha_{1}},j_{\alpha_{1}},i_{\alpha_{2}},j_{\alpha_{2}}\right\}\in J_{X_{i}}}}t_{i_{\alpha_{1}}}t_{j_{\alpha_{1}}}{\bf 1}_{(w(t_{i_{\alpha_{1}}})=w(t_{j_{\alpha_{1}}}))}\right. \times
\lim_{u\to\infty}\limsup_{N\to\infty}\sup_{0\leq t\leq T}|\frac1NE[\big(\sum_{n=1}^{[Nt]}Z^{(u)}_{N,i,n}\big)(\sum_{n=1}^{[Nt]}Z^{(u)}_{N,i',n}\big)]-\frac t{c^{(k)}}D_{i,i'}|=0.
c_1 c_5 -c_2 c_4 + c_4 c_9 -c_6 c_7 =0\;, c_1 c_8 -c_2 c_7 + c_5 c_9 -c_6 c_8 =0\;, c_1 c_9 -c_2 c_6 + c_3 c_5 -c_3 c_7 =0\;.
F(x) = \begin{pmatrix}f_{11}(x) \cdots f_{1r_2}(x) \vdots \ddots \vdots f_{r_11}(x) \cdots f_{r_1r_2}(x)\end{pmatrix}G(x) = \begin{pmatrix}g_{11}(x) \cdots g_{1q_2}(x) \vdots \ddots \vdots g_{q_11}(x) \cdots g_{q_1q_2}(x)\end{pmatrix},
i\xi_j \widehat{p}(\xi,y_d) = \frac{\xi_j \xi}{|\xi|}e^{-|\xi|y_d}\cdot \partial_{y_d}\widehat{u'}(\xi,0), \partial_{y_d}\widehat{p}(\xi,y_d) = i\xi e^{-|\xi| y_d}\cdot \partial_{y_d}\widehat{u'}(\xi,0),
\times\,\sum_{\ell=-\infty}^{+\infty}\,e^{-i\frac{\pi}{2}|\ell|}\,e^{i\ell(\varphi_f-\varphi_i)}\,e^{-\frac{i}{2}\hbar\Delta t g^2\ell^2}\,J_{|\ell|}\left(\frac{\omega r_f\,r_i}{\hbar\sin\omega\Delta t}\right)\ \ \ ,
RHS> \frac{\eta R^2}{12}(3\eta^2 + 1) - \frac 18 (1-\eta)^3 R^2 = \frac{R^2}{24}\Big[ 2\eta\cdot ( 3\eta^2 + 1 ) - 3 (1-\eta)^3 \Big] = \frac{R^2}{24}( 9\eta^3 - 9 \eta^2 + 11\eta -3 ) = \frac{R^2}{24}( \eta - \frac 13) \Big [( 3\eta - 1 )^2 + 8 \Big ] = C_3(\eta_0, c_0)R.
\nabla_k \nabla_{}^G R_t[ \Phi](x) \in L_2(K,U) \;\; \| \nabla_k \nabla_{}^G R_t[ \Phi](x) \|_{L_2(K,U)}= \Big (\sum_{m \ge 1}\sup_{|a|_U =1}| \nabla_k \nabla_{a}^G P_t[\langle \Phi , f_m \rangle_K ](x)|^2 \Big)^{1/2}\le \frac{C}{t^{\frac{4 - 3\alpha}{2}}}|k|_K \, \| \Phi\|_{C^{\alpha}_b(H, K),}\;\; t \in (0,T];
Y^p \frac{\partial\tilde{\Gamma}_{\,\,ij}^k}{\partial u^p}=Y^p \frac{\partial\Gamma_{\,\,ij}^k}{\partial x^p}(V)+ (\nabla^V_YV)^l\frac{\partial\Gamma_{\,\,ij}^k}{\partial y^l}(V) -Y^mN^l_{\,\,m}(V)\frac{\partial\Gamma_{\,\,ij}^k}{\partial y^l}(V).
A =\ \sum_{i=3}^{n}\sum_{j=3}^{m}(A_{i,j}+A_{i-1,j-1}-A_{i-1,j}-A_{i,j-1})L^{(i,j)}+ &\sum_{j=3}^{m}(A_{2,j}-A_{2,j-1})L^{(2,j)}+\sum_{i=3}^{n}(A_{i,2}-A_{i-1,2})L^{(i,2)}+A_{2,2}^{(2,2)}L^{(2,2)}
\Lambda(A_t,\Theta_t)= \int_{\Theta_1\times A_2}\left(g_2 \log |t|+O(1)\right)\tfrac{g_1}{g \cdot g_1! g_2!}\nu_1^{g_1-1}\nu_2^{g_2}&+\int_{A_1\times \Theta_2}\left(g_1\log |t|+O(1)\right)\tfrac{g_2}{g \cdot g_1! g_2!}\nu_1^{g_1}\nu_2^{g_2-1}= \tfrac{2g_1 g_2}{g}\log |t|+O(1).
\begin{aligned}(A+BK^{(i)})^{\top}P^{(i+1)}+P^{(i+1)}(A+BK^{(i)})+(C+DK^{(i)})^{\top}P^{(i+1)}(C+DK^{(i)}) &~~~\qquad\qquad\qquad+K^{(i)\top}RK^{(i)}+S^{\top}K^{(i)}+K^{(i)\top}S+Q=0,\end{aligned}
\nabla_{t}(R^P(\psi,\psi)(Ns)^{2-n})\partial_t\cdot\psi= (\nabla_{d\phi(\partial_t)}R^P)(\psi,\psi)(Ns)^{2-n}\partial_t\cdot\psi +2R^P(\nabla_t\psi,\psi)(Ns)^{2-n}\partial_t\cdot\psi &+(2-n)R^P(\psi,\psi)(Ns)^{2-n}\partial_t\log N\partial_t\cdot\psi &+(2-n)\dot ss^{1-n}N^{2-n}R^P(\psi,\psi)\partial_t\cdot\psi.
\Bigg\vert\Bigg\vert \prod_{\ell=0}^j \big(e^{i\eta n_{k_\ell}}-1\big)e_\eta-\prod_{\ell=0}^j\big(e^{i\xi n_{k_\ell}}-1 \big)e_\xi \Bigg\vert\Bigg\vert \le 2^{j+1}\vert\vert e_\eta-e_\xi\vert\vert &+\sqrt{\frac{\pi}{2}}\Bigg\vert \prod_{\ell=0}^j \big(e^{i\eta n_{k_\ell}}-1\big)-\prod_{\ell=0}^j\big(e^{i\xi n_{k_\ell}}-1\big) \Bigg\vert.
\prod_{j=1}^{m}\sum_{b=1}^{\alpha_j+1}\frac{(-i)^{b-1}L_{\alpha_{j+1},b}}{b!}\, x^{b - 1}\frac{\sin bx}{\sin^{b}x}\ =\ \prod_{j=1}^{m}\Bigg( \frac{(-1)^{\alpha_j}}{(2\alpha_j +1)!!}\, x^{2\alpha_j}+ O(x^{2\alpha_j+1}) \Bigg).
\Phi_{00}:=\frac{1}{2}R_{ab}l^{a}l^{b},\Phi_{11}:=\frac{1}{4}R_{ab}(l^{a}n^{b}+m^{a}\overline{m}^{b}),\Phi_{22}:=\frac{1}{2}R_{ab}n^{a}n^{b},\Lambda:=\frac{R}{24};
\lim_{n\rightarrow \infty}\sqrt{n}\mathbf{E}_{\mathbf{e_{i}},a}\left[Y_{n};\tau >n\right] =\frac{2}{\sigma \sqrt{2\pi}}\mathbf{\hat{E}}_{\mathbf{e}_{i},a}\left[ Y_{\infty}\right] h(\mathbf{e}_{i},a)>0.
F_0(\bar{u}) \le \liminf_{\varepsilon \to 0}F_0(u_\varepsilon) & \le \liminf_{\varepsilon \to 0}F_\varepsilon(u_\varepsilon) & \le \liminf_{\varepsilon \to 0}F_\varepsilon(u_0) & = F_0(u_0).
\sum_{|\alpha|=m}\left(\sum_{i=1}^n\frac{\partial c_\alpha(z)}{\partial z_i}d^1z_i(d^{1}z)^{\alpha_1}\cdots (d^{k}z)^{\alpha_k}+\sum_{j=1}^k\sum_{i=1}^nc_\alpha(z) \alpha_j^id^{j+1}z_i(d^{1}z)^{\alpha_1}\cdots (d^jz)^{\alpha'_{j,i}}\cdots (d^{k}z)^{\alpha_k}\right),
- \frac{1}{4}( k^{4}- 6 k^{2}\tilde{k}^{2}+ \tilde{k}^{4}) \sinh 2 \theta \sin 2 \tilde{\theta}+ k \tilde{k}( k^{2}- \tilde{k}^{2}) [ 1 + \cosh 2 \theta \cos 2 \tilde{\theta}] .
(x,y)\Psi_{[[X_1, X_2],X_3]}^{(f_1, f_2,f_3)}(t_1,t_2,t_3) = (x, y) &+\Big(t_1^3t_2^3\left(e^{-t_3}- 1\right)x + t_1t_2^2\left(e^{t_3}- 1\right)\left(t_1t_2 - 1\right)y, - t_1^2t_2e^{-t_3}\left(e^{t_3}- 1\right)\left(t_1^2t_2^2 + t_1t_2 + 1\right)x+t_1^3t_2^3\left(e^{t_3}- 1\right)y\Big)
L_{c}^{- 1}[ \psi_{1}\otimes \cdots \otimes \psi_{m}]_{x_{1}^{0}, . . . , x_{l}^{0}}( \underline{{{x}}}; \underline{{{n}}}, \underline{{{T}}}) ) = \bigl ( \prod_{i = 1}^{l}L_{c}^{- 1}[ \psi_{i}]_{x_{i}^{0}}( x_{i}; n_{i}, T_{i}) \bigr ) \bigl ( \prod_{i = l + 1}^{m}\psi_{i}( x_{i}) \bigr )
\big\|\sum_kg_k\big\|_{L^2}^2 =\sum_{\nu, \nu'}c_\nu c_{\nu'}\sum_{k,l}\big\langle b_k,e^{-(\nu r_k^2+\nu' r_l^2)\mathcal L_a}b_l\big\rangle &\lesssim\sum_{r_k\geq r_l}r_k^{-d}\int_{Q_l}\int_{Q_k}\big(\tfrac{r_k}{|x|}\vee1\big)^\sigma |b_k(x)|e^{-\frac{|x-y|^2}{cr_k^2}}\big(\tfrac{r_k}{|y|}\vee1\big)^\sigma |b_l(y)|\,dx\,dy.
\sigma_{1}= \left( \begin{array}{c c}{{0}}&{{1}}\\{{1}}&{{0}}\\ \end{array}\right) \, , \sigma_{2}= \left( \begin{array}{c c}{{0}}&{{- i}}\\{{i}}&{{0}}\\ \end{array}\right) \, , \sigma_{3}= \left( \begin{array}{c c}{{1}}&{{0}}\\{{0}}&{{- 1}}\\ \end{array}\right) \, ,
L(fe_{xx})(x,y) =\left(L(fe_{xx})e_{yy}\right)(x,y) &=-L\left([e_{yy},fe_{xx}]\right)(x,y)+\left(L(e_{yy})fe_{xx}\right)(x,y) &-\left(fe_{xx}L(e_{yy})\right)(x,y)+\left(e_{yy}L(fe_{xx})\right)(x,y) &=-L(e_{yy}fe_{xx})(x,y)-f(x,x)L(e_{yy})(x,y) &=-f(y,x)L(e_{yx})(x,y)-f(x,x)L(e_{yy})(x,y).
G = i g_{\mu \bar{\nu}}\, d z^{\mu}\wedge d \bar{z}^{\bar{\nu}}={\frac{i}{\sqrt{F_{2}F_{6}}}}h_{\mu \bar{\nu}}\, d z^{\mu}\wedge d \bar{z}^{\bar{\nu}}\equiv{\frac{K}{\sqrt{F_{2}F_{6}}}}
S(z,w) =\sum_{n=0}^{\infty}e_n(z)\overline{e_n(w)}=g_{}(z)\left(\sum_{n=0}^{\infty}\frac{e_n(z)}{g_{}(z)}\overline{\frac{e_n(w)}{g_{}(w)}}\right)\overline{g_{}(w)}&=g_{}(z)S_{\mu}(z,w)\overline{g_{}(w)}.
T_0^{1}\;=\; \alpha^{-\frac{L+1}{2}}\;\sum_{j=1}^{L}\alpha^{j}\,\tau_{j}^{1}\,,\hspace{1.5cm}T_0^{2}\;=\; \beta^{-\frac{L+1}{2}}\;\sum_{j=1}^{L}\beta^{j}\,\tau_{j}^{2}\,,\hspace{1.5cm}E\;=\;L\,,
\Bigl(\exp\bigl(-\rho\rho_0 \cos(\varphi - \varphi_0)\bigr) \cdot w_{(p_1, p_2)}(\rho, \varphi)\Bigr)^{(2l-1)}_\alpha = 0\; \mbox{\rm for all}\; \alpha \in \Pi\; \mbox{\rm and}\;1 \le l \le \mu_\alpha,
\frac{d\!\Pr\!\left[ Y_{j}=I\right]}{dt}=\beta\sum_{k=1}^{N}a_{kj}\!\Pr\!\left[ Y_{k}=I\right] -\delta\Pr\!\left[ Y_{j}=I\right] -\beta\sum_{k=1}^{N}a_{kj}\!\left\{\Pr\!\left[ Y_{j}=I,\!Y_{k}=I\right]+\Pr\!\left[ Y_{j}=R,\!Y_{k}=I\right] \right\}
\partial_{x_1}^{e_s}\big\{\psi(x_1,x_1+x_2,\ldots,x_1+x_M,x_{M+1},\ldots,x_N)\big\}&=\sum_{j=1}^{M}\big(\partial_{x_{j,s}}\psi\big)(x_1,x_1+x_2,\ldots,x_1+x_M,x_{M+1},\ldots,x_N) &=\sqrt{M}\big(\partial_{x_Q}^{e_s}\psi\big)(x_1,x_1+x_2,\ldots,x_1+x_M,x_{M+1},\ldots,x_N)\,,
= - ( 2 \pi \sqrt{\alpha^{\prime}})^{2}\, \left( \frac{1}{2}+ \frac{1}{( 2 \pi \sqrt{\alpha^{\prime}})^{2}}\tilde{b}\right) + ( 2 \pi \sqrt{\alpha^{\prime}})^{2}= ( 2 \pi \sqrt{\alpha^{\prime}})^{2}\, \left( \frac{1}{2}- \frac{1}{( 2 \pi \sqrt{\alpha^{\prime}})^{2}}\tilde{b}\right)
\sum_{\gamma \in \Gamma}\frac{| \gamma^{\prime}( f ({\bf y}) ) |^{\Delta}}{| f ({\bf x}) - \gamma f ({\bf y}) |^{2 \Delta}}\; = \; | f^{\prime}({\bf x}) |^{- \Delta}| f^{\prime}({\bf y}) |^{- \Delta}\sum_{\gamma \in f^{- 1}\Gamma f}\frac{| \gamma^{\prime}({\bf y}) |^{\Delta}}{|{\bf x}- \gamma{\bf y}|^{2 \Delta}}.
h(Y)= \frac{1}{2}\pi\cdot\sum_{j=1}^r (\|\exp(-Y)( E_j-F_j i)\|_{z_0}^2+\|\exp(-Y) (E_j+F_j i)\|_{z_0}^2) = \pi\cdot \sum_{j=1}^r (\|\exp(-Y)v_j\|_{z_0}^2+\|\exp(-Y) v_{n+j}\|_{z_0}^2).
\sum_{j=a}^{a+b}\binom b{j-a}\sum_{\ell=0}^{k-2i+j}\binom{{n-1}+k-2i-\ell}{{n-1}-j-\ell,k-2i+j-\ell,\ell}&=\sum_{\ell=0}^{k-2i+a+b}\binom{{n-1}+k-2i-\ell}{\ell}\sum_{j=a}^{a+b}\binom b{a+b-j}\binom{{n-1}+k-2i-2\ell}{k-2i+j-\ell}&=\sum_{\ell=0}^{k-2i+a+b}\binom{{n-1}+k-2i-\ell}{\ell}\binom{{n-1}+k+b-2i-2\ell}{k+a+b-2i-\ell}.
{\mathbf b}= \left( \, \begin{array}{c c c}{{0}}&{{b_{1 2}}}&{{0}}\\{{0}}&{{0}}&{{b_{2 3}}}\\{{b_{3 1}}}&{{0}}&{{0}}\\ \end{array}\, \right) \ , \qquad{\mathbf d}= \left( \, \begin{array}{c c c}{{0}}&{{d_{1 2}}}&{{0}}\\{{0}}&{{0}}&{{d_{2 3}}}\\{{d_{3 1}}}&{{0}}&{{0}}\\ \end{array}\, \right)
(D_{\mu,u_{x}})_{x}\psi_{x}=\lim_{\Delta u_{x}^{\mu}\rightarrow 0}\frac{(U_{u_{x},u_{x}+\Delta u_{x}^{\mu}})_{x}\psi_{x}(u_{x}+\Delta u_{x}^{\mu})-\psi_{x}(u_{x})}{\Delta u^{\mu}_{x}}.
\left|\underbrace{\frac{1}{\sqrt{M}}\sum_{i=1}^{M}\left(\sqrt{\frac{K \overline{\gamma}_{sp}}{K+1}}e^{j (\theta_{n^{*},i}(k)+\phi_{i,n^{*}})}\right)}_{\mbox{Artificial Fading}}+ \sqrt{\frac{1}{K+1}}c_{n^{*}}(k)\right|^{2},
S_{W}( \vec{y}_{j}, g_{j}) = N \frac{m_{W}}T - \frac{g^{2}}{2 \pi T}\sum_{i j}g_{i}g_{j}\ln ( T | \vec{y}_{i}- \vec{y}_{j}| ) \, , \qquad g_{i}= \pm 1 \, , \qquad \sum_{i}g_{i}= 0 \, .
X: \sum_{i_0 + i_1 + i_2 + i_3 = 4}a_{i_0 i_1 i_2 i_3}x_{0}^{i_0}x_{1}^{i_1}x_{2}^{i_2}x_{3}^{i_3}= 0, a_{i_0 i_1 0 0}= 0 ,
\int_{0}^{2\pi}e^{iky}\log\left(\left|\sin\left(\frac{y}{2}\right)\right|^{2}\right)dy = -\frac{1}{i k}\int_{0}^{2\pi}\frac{e^{iky}- 1}{1-\cos(y)}\sin(y) dy= -\frac{1}{k}\int_{0}^{2\pi}\frac{e^{iky}- 1}{e^{iy}- 1}(e^{iy}+ 1) dy
\left[\begin{matrix}\mathbf{a}_1 \cdots \mathbf{a}_{2g}\end{matrix}\right]=\left[\begin{matrix}\mathbf{b}_1 \cdots \mathbf{b}_{2g}\end{matrix}\right]\alpha\quad\textrm{for some}~\alpha\in M_{2g}(\mathbb{Z})\cap\mathrm{GL}_{2g}(\mathbb{Q}),
\lim_{z\rightarrow \infty}z^{-2}\log \P\left\{\log u_t(x) \ge z\right\}=\lim_{z\rightarrow \infty}z^{-2}\log \P \left\{\log\left[\sup_{y\in Q(x,1)}u_t(y) \right] \ge z \right\}&=-(2f(0)t)^{-1}.
X_{\left( \eta ,\upsilon ,\zeta \right)}^{\mathfrak{z}_{d}}\left(Ad_{g^{-1}}^{\ast}\lambda ,\mu ,\xi \right) =\left( ad_{\eta}^{\ast}\circ Ad_{g^{-1}}^{\ast}\lambda ,\upsilon +ad_{\eta}^{\ast}\mu ,\zeta +\left[ \xi ,\eta \right] \right)
R(e_1,d,p_d) =\frac{p_d}{2d}+P\big(2\leq\tau(d,p_d)<+\infty\big|S_0(d,p_d)=e_1\big) &\leq\frac{p_d}{2d}+P\big(2\leq\tau(d,1)<+\infty\big|S_0(d,1)=e_1\big).
B_{hk}= \frac{1}{p^2}\left[ q - \sum_{j=1}^{p-1}0\cdot e(-jh) - \sum_{\ell=1}^{p-1}0 \cdot e(-\ell k) + \sum_{j=1}^{p-1}\sum_{\ell=1}^{p-1}e(-jh -\ell k) \{1+K(\chi_p, j\ell u)\}\right] ~~.
\begin{aligned}|P_n(\cos t)|^2 = |P_{n}^{*}(e^{it})|^2 + O(q^n), t \in [\pi/2-\delta,\pi/2 + \delta], |P_n(\cos t)|^2 = |P_{n}^{*}(e^{-it})|^2 + O(q^n), t \in [-\pi/2 - \delta, -\pi/2 + \delta], |P_n(\cos t)|^2 = O(q^n) \textnormal{otherwise,}\end{aligned}
\hat W_{n,m_1,\ldots,m_n}^T(k_1,\ldots,k_n)=\left[\sum_{j=1}^n \prod_{l=1}^{j-1}\delta^+(k_l^2-m_l^2)\frac{1}{k_j^2-m_j^2}\prod_{l=1}^n \delta^-(k_l^2-m_l^2)\right]\delta\left(\sum_{l=1}^nk_l\right).
\mathbb{P}\left({\gamma}_{\mathrm{u},ik}\geq \theta \right) &\approx\mathbb{E}\left[ \mathbb{P}\left( \left. R_k \leq \frac{\left( 1 - \tau_{{iik}}^2\right)(1-\beta) N t^\alpha}{\left(\tau_{iik}^2 + \frac{2 \pi \lambda t^{2}}{\alpha-2}\right) \theta}- t^\alpha \right\vert r_{iik}= t \right) \right].
\left(\sum_{i=0}^{p-1}\frac{p(-1)^{i}}{2i-3}\right)^2\equiv 1+\frac{16p^2}{9}+\frac{10p(-1)^{\frac{p-1}{2}}}{3}+\frac{2p(-1)^{\frac{p-1}{2}}}{2p-3}-p^2\sum_{k=1}^{\frac{p-3}{2}}\frac{(-1)^k}{k^2}\pmod{p^3}.
Q_{20}= \frac{1/20}{r^{19}}*(-16796 r^9+2645370 r^8 -68643960 r^7 +686439600 r^6 - 3442004280 r^5 + 9704539845 r^4 - 16087739850 r^3 + 15557374800 r^2 - 8119857900 r + 1767263190)