harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
H_{\beta \gamma}= \beta \delta_{\beta , \gamma}, \quad D_{\beta \gamma}= \delta_{\beta , \gamma}D_{\beta}, \quad D_{\beta}= - i g_{l}\left( z ( \beta \cdot q ) + \sum_{\kappa \in \Delta_{l}, \ \kappa \cdot \beta = 1}z ( \kappa \cdot q ) \right) ,
({\bf P1a}){\rm minimize}_{f\in L^2(0,T;{\mathbb R}^n)}\bar J_{i}'(u_i,f) =\{\mathbb E}\int_0^T \left\{-| x_i-(\Gamma m_i +\eta)|^2_Q +\frac{1}{\gamma}|f(t)|^2 \right\}dt -{\mathbb E}x_i^T(T)Hx_i(T).
A_{1}(\textbf{u}) = \left( \begin{array}{ccc}v_{1}\rho 0 f(\textbf{u}) v_{1}0 0 0 v_{1}\end{array}\right), \A_{2}(\textbf{u}) = \left( \begin{array}{ccc}v_{2}0 \rho 0 v_{2}0 f(\textbf{u}) 0 v_{2}\end{array}\right),
\begin{array}{l}\alpha(t)\left\|\nabla U_v(X(t))-\mathbf{1}\otimes\frac{1}{m}\sum_{i=1}^m \nabla u_i(X^{(i)}(t))\right\| \le \alpha(t)\left\|\nabla U_v(X(t))-\mathbf{1}\otimes\nabla U(\bar{X}(t))+\mathbf{1}\otimes\nabla U(\bar{X}(t))-\mathbf{1}\otimes\frac{1}{m}\sum_{i=1}^m \nabla u_i(X^{(i)}(t))\right\| \qquad\qquad\le 2\alpha(t) L \|\check{X}(t)\|\end{array}
P_{S,S_1|V,Y}(s,s_1|v,y)=P_{S|V,Y}(s|v,y)p_{S_1|S}(s_1|s),\quad\forall(v,y,s,s_1)\in\mathcal{V}\times\mathcal{Y}\times\mathcal{S}\times\mathcal{S}_1.
c^k_{\mathrm{b2s},3}\left( N_{\mathrm{scr}}\right) = \mathsf{E}\left\lbrace B N_{\mathrm{scr}}\log_2\left[ \frac{P_{\mathrm{bs}}}{BN_{\mathrm{bs}}\sigma^2_{\mathrm{sc}}}\left(\max \{\|\mathbf{g}_i \| \}_{i=1}^{N_{\mathrm{bs}}}\right)^{-2}+1 \right] \right\rbrace,
|I_3(x)|~\leq~\O(1),\big|I'_3(x)\big|~=~\left|\int_{3x/2}^{2}{g_b'(y)\over y-x}~dy+{3 \over 2}\cdot g'_b\left({3x\over 2}\right)\cdot\ln\left({x\over 2}\right)\right|~\leq~\O(1)\cdot\ln^2x,
4\int_{0}^{+\infty}F_U(x)\cosh\Big(\frac{x}{2}\Big)\d x = 4\int_{0}^{+\infty}F\Big(\frac{x}{U}\Big)\cosh\Big(\frac{x}{2}\Big)\d x &= 4U\int_{0}^1(1-5x^2+5x^3-x^5)\cosh\Big(\frac{Ux}{2}\Big)\d x &= 960\frac{\big((U-4)e^{\frac U4}+(U+4)e^{-\frac U4}\big)^2}{U^5}= S(U).
\begin{array}{rcl}D^{j}_{\mu,\nu}(\alpha,\beta,\gamma) = \displaystyle\sum_k\frac{(-1)^k\sqrt{(j+\mu)!(j-\mu)!(j+\nu)!(j-\nu)!}}{k!(j-\nu-k)!(j+\mu-k)!(k-\mu+\nu)!}& \displaystyle e^{-i\alpha\mu}(cos(\beta/2))^{2j-\nu+\mu-2k}(sin(\beta/2))^{2k-\mu+\nu}e^{-i\nu\gamma}\end{array}
\frac{R ( t )}{R_{0}}= \left\{\frac{\Omega_{M}}{1 - \Omega_{\Lambda}^{2}}\left[ \Omega_{\Lambda}+ \sin \left( \frac{3}{2}h_{0}^{2}\sqrt{1 - \Omega_{\Lambda}^{2}}t \right) \right] \right\}^{\frac{1}{3}}- \left( \frac{\Omega_{M}\Omega_{\Lambda}}{1 - \Omega_{\Lambda}^{2}}\right)^{\frac{1}{3}},
\langle \tilde{\Xi}_\pm\tilde{\Xi}_\pm D_a\varphi, \tilde{\Xi}_\pm\varphi\rangle =\pm \langle D_a\varphi, \tilde{\Xi}_\pm\varphi\rangle &=\pm \langle \varphi, D_a\tilde{\Xi}_\pm\varphi\rangle=\langle \varphi, \tilde{\Xi}_\pm D_a\varphi\rangle,
e^{-tD_{A_R}^2}- K_t^N = \frac{-i}{2\pi}\int_{C_t}e^{-tz}((D_{A_R}^2-z)^{-1}- Q^N_z)dz &= \frac{-i}{2\pi}\int_{C_t}e^{-tz}(D_{A_R}^2-z)^{-1}\epsilon_z^Ndz &= \frac{-i}{2\pi}\int_{C_t}e^{-tz}Q_z^N \epsilon_z^N dz+\frac{-i}{2\pi}\int_{C_t}e^{-tz}(D_{A_R}^2-z)^{-1}(\epsilon_z^N)^{2}dz.
d\alpha= (\xi\alpha)\eta-4c\eta^\perp\phi A\phi A- \frac\alpha 2(\phi A+ A\phi)-c(\phi+\phi^\perp) = c\sum^3_{a=1}\{\xi_a\otimes\eta_a\phi+\phi\xi_a\otimes\eta_a\}-2c(\xi\otimes\eta^\perp\phi+\phi\xi^\perp\otimes\eta).
(\phi^\circ_V)^* \mathrm{Res}_{U^\vee}\left( \frac{\omega_n}{F_1\dots F_k}\right) = (2\pi_i)^k \omega_{n-k}|_{U_V}(\phi^\circ_{V'})^*\mathrm{Res}_{U^\vee}\left(\frac{\omega_n}{F_1\dots F_k}\right) = (2\pi_i)^k \omega_{n-k}|_{U_{V'}}
\rho_{ik}=\frac{\text E (|\tilde h_{ik}| |\hat h_{ik}| ) -\text E(|\tilde h_{ik}|) \text E(|\hat h_{ik}|)}{\sqrt{\left(\text E( |\tilde h_{ik}|^2 ) - [\text E (|\tilde h_{ik}|)]^2 \right) \left( \text E( |\hat h_{ik}|^2 ) - [\text E (|\hat h_{ik}|)]^2 \right)}},
\sum_{l=1}^{h(m,n)}C_{m}^{l}(F^{0})(\lambda)(C_{m}^{l}(F^{0})(\lambda))^{*}=d_{m}^{0}(\lambda )^{\top}(\vec{\beta}\cdot \vec{\beta}^{*}+\Gamma_{m_{1}}(\lambda )+\Gamma_{m_{2}}(\lambda ))\overline{d_{m}^{0}(\lambda )}.
0 = 1 - g_{c}+ \frac{( D - 1 )}{2 N_{c}}\left[ 1 - \frac{1}{\tilde{g}_{s c}}\log ( 1 + \tilde{g}_{s c}) \right] - \frac{( D - 1 )^{2}( 4 - D )}{2 N_{c}( D - 2 )}\left[ 1 - \frac{1}{g_{V c}}\log ( 1 + g_{V c}) \right] \ ,
|M_k|_* -\epsilon< \left| S\cap \cup_{i=1}^N B_i \right|_*\leq \sum_{i=1}^N |S\cap B_i|_*\leq \left(1-\dfrac{1}{k}\right)\sum_{i=1}^N |B_i|<(1+\epsilon)\left(1-\dfrac{1}{k}\right)|M_k|_*,
(\nabla|u|^2u)\nabla u =\sum_{j,k,\ell=1}^n(\partial_ku^j)\partial_k(u^\ell)^2u^j &=\sum_{j,k,\ell=1}^n(u^\ell)^2(\partial_ku^j)^2+2\sum_{k=1}^n(u\cdot\partial_ku)^2.
lim_{x\rightarrow\infty}f(x)=lim_{x\rightarrow\infty}\frac{e^{x}\cdot f(x)}{e^{x}}=lim_{x\rightarrow\infty}\frac{e^{x}(f(x)+f^{\prime}(x))}{e^{x}}=lim_{x\rightarrow\infty}(f(x)+f^{\prime}(x))
J\left(\begin{array}{c}q^+z^a\bar{z}^b q^-z^b\bar{z}^a\end{array}\right)=\left(\begin{array}{c}-i(\eta_0)_tq^+z^b\bar{z}^a i (\bar{\eta}_0)_tq^-z^a\bar{z}^b\end{array}\right)+\frac{b}{a+1}\left(\begin{array}{c}-i(\bar{\eta}_0)_tq^+z^{b-1}\bar{z}^{a+1}i (\eta_0)_tq^-z^{a+1}\bar{z}^{b-1}\end{array}\right)
M_{2}(\mathcal{C}(A(n, p)))=(p^{n}+1) \dfrac{(p^{2n}-p^{n})(p^{2n}-p^{n}-1)^{3}}{2}=\dfrac{p^{n}(p^{2n}-1)(p^{2n}-p^{n}-1)^{3}}{2}.
C_L = \begin{pmatrix}0 0 \ldots 0 -\frac{\alpha_0}{\alpha_d}1 0 \ldots 0 -\frac{\alpha_1}{\alpha_d}\vdots \vdots \ddots \vdots \vdots 0 0 \ldots 1 -\frac{\alpha_{d-1}}{\alpha_d}\end{pmatrix}.
a_{nn}\!=\!1\!-\!t_{kn}^2(2)+t_{kn}^2(2)\alpha_{1k}(1)(1\!-\!c_{1n}^2)\!=\!1-t_{kn}^2(2)\frac{1+c_{1n}^2}{2}\!= \!1\!-\!\big((1\!+\!c_{2n})/2\big)^2\big(1\!+\!c_{1n}^2)/2.
\dim\mathbf{H}^1({\mathcal F}^{\bullet}_x) = \chi({\mathcal F}^1_x)-\chi({\mathcal F}^0_x)+2 =2r^2(g-1)+2+\sum_{i=1}^n\sum_{j'<j}2m_i r^{(i)}_{j'}r^{(i)}_j +\sum_{i=1}^n\sum_{j=0}^{s_i-1}m_i (r^{(i)}_j)^2 -\sum_{i=1}^n\sum_{j=0}^{s_i-1}m_ir^{(i)}_j =2r^2(g-1)+2+\sum_{i=1}^nm_ir(r-1)
\lim_{t\to t_{F_n}^+}\bigg(h_{\mu_t}(g)+t_{F_n}\int F_nd\mu_t\bigg) =\lim_{t\to t_{F_n}^+}\bigg(h_{\mu_t}(g)+t\int F_nd\mu_t\bigg) &=\lim_{t\to t_{F_n}^+}P(tF_n)=P(t_nF_n).
\left(G_0 (ST)^3 G^{-1}_0=S(ST)^3 S^{-1}\right)_\mathfrak{U}=\left(S^2 T\right)_\mathfrak{U}\left((ST)^3\right)_\mathfrak{U}\left(T^{-1}S^{-2}\right)_\mathfrak{U}
A=\int \frac{d\tau}{\tau^{2}}\sqrt{{1+\vec{\phi}_{\tau}^{\, 2}+\vec{\phi}_{\sigma}^{\, 2}+\vec{\phi}_{\tau}^{\, 2}\vec{\phi}_{\sigma}^{\, 2}-\left( \vec{\phi}_{\tau}\vec{\phi}_{\sigma}\right)^{2}}}.
\tilde{V}(x) := \frac{2\pi}{N(\pi + gN)}\;\left(\mbox{K}_0\left(\sqrt{\frac{e^2 N}{\pi\!+\!gN}}|x|\right)+ \ln\left(\frac{1}{2}\sqrt{\frac{e^2 N}{\pi\!+\!gN}}\right)+\gamma\right) \; .
-z(z^{-1})''= - \frac{9}{4}{\cal H}^2 + \frac{\varphi'''}{\varphi'}+ \frac{5}{2}{\cal H}' - 2 \biggl(\frac{\varphi''}{\varphi'}\biggr)^2 +\frac{\varphi''}{\varphi'}\biggl( 2 \frac{{\cal H}'}{\cal H}- 3{\cal H}+ \frac{{\varphi'}^2}{3{\cal H}}\biggr)
\left( \int_{\mathbb{S}^1_\rho}\left|\frac{\sin(\frac{\theta}{\rho})}{\cosh(\frac s\rho)-\cos(\frac{\theta}{\rho})}\right|^{q}\,d\theta \right)^{\frac 1q}\lesssim \begin{cases}s^{\frac 1q-1}s <1 e^{-\frac{s}{\rho}}s \geq 1\end{cases}
\begin{array}[c]{c}Ric_{\alpha\overline{\beta}}f^{\alpha}f^{\overline{\beta}}+\frac{n+1}{2}i(A_{\alpha\beta}f^{\alpha}f^{\beta}-A_{\overline{\alpha}\overline{\beta}}f^{\overline{\alpha}}f^{\overline{\beta}})+2Kf_{\alpha}f^{\alpha}=0,\end{array}
J_{o r b}= \sum \varepsilon_{i j}R_{p}^{i}\frac{\partial L_{e f f}^{( 1 )}}{\partial \dot{R}_{p}^{j}}= - 4 \pi \kappa \sum_{p}R_{p}^{i}\varepsilon_{i j}a_{j}( \vec{R}_{p}) \; \; .
\psi_{24}^b(m) = \sum_{p_1,p_2,p_3}\bigg\{\sum_{p_1p_2p_3^2 \le Y}+ \sum_{p_1p_2p_3^2 > Y}\bigg\}\psi(m/(p_1p_2p_3), p_3) = \psi_{25}(m) + \psi_{26}(m), .
i_{e}{}_{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}}^{N_{1}N_{2}N_{3}N_{4}}{}^{( a )}= \sum_{\Lambda}\M_{e}{}_{\Lambda}^{( a )}\{C}_{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}}^{N_{1}N_{2}N_{3}N_{4}\Lambda},
W^0_{e_i}(x)^2-(1-\alpha^2)xW^0_{e_i}(x)+\alpha^2 W_{e_i}^0(x)\sum_{j\neq i}\oint\frac{d\zeta_j}{2i\pi}\zeta_jW_{e_j}^0(\zeta_j)+(1-\alpha^2)=0.
\begin{array}{c c}{{A Q^{3}\leftrightarrow a q^{3}}}&{{Q^{6}\leftrightarrow q^{6}}}\\{{A \bar{Q}^{3}\leftrightarrow a \bar{q}^{3}}}&{{\bar{Q}^{6}\leftrightarrow \bar{q}^{6}}}\\{{A^{3}Q^{3}\leftrightarrow a^{3}q^{3}}}&{{Q \bar{Q}\leftrightarrow M_{0}}}\\{{A^{3}\bar{Q}^{3}\leftrightarrow a^{3}\bar{q}^{3}}}&{{Q A^{2}\bar{Q}\leftrightarrow M_{2}}}\\{{A^{4}\leftrightarrow a^{4}}}&{{}}\\ \end{array}
g_1(X) =\frac{f_1}{GCD(f_1,f_2)}=\prod\limits_{b\in D_1}(X+b), g_2(X) =\frac{f_2}{GCD(f_1,f_2)}=\prod\limits_{b\in D_2}(X+b), g_3(X) =GCD(f_1,f_2)=\prod\limits_{b\in D_3}(X+b).
r_{2}^{v a c , g l}( x - y ) = - 3 i g^{2}N ( N^{2}- 1 ) D_{0}^{-}( x - y ) \partial_{\nu}^{x}D_{0}^{-}( x - y ) \partial_{y}^{\nu}D_{0}^{-}( x - y ) \quad ,
h=\lambda e^2/2, begin{align*}0.2cm] 0<\beta<\min\left\{c,\ \lambda\right\}, begin{align*}0.2cm] \sigma>\max\left\{e^{2\beta/\lambda},\ \frac{f(1)hq(1)}{\left(c-\beta\right)\left(\lambda-\beta\right)\beta e}\right\}, begin{align*}0.2cm] r>\max\left\{h\sqrt{2/\lambda},\ \frac{4sh^2q(1)}{d q(0)}\left(\frac{7}{2e\lambda}\right)^{7/2}\right\},
L_{i j}= \delta_{i j}\xi_{i}+ ( 1 - \delta_{i j}) \lambda_{i \, x}\Phi ( \lambda_{i}- \lambda_{j}, z ) , \qquad \mathrm{w h e r e}\qquad \xi_{i}= \frac{\lambda_{i \, y}- \lambda_{i \, x x}- b_{i}}{2 \lambda_{i \, x}},
\Pr[K_{{\rm SVC},n,\ell}=k| B_{{\rm SVC},n,\ell,0}\ \ \mathbf x] = \sum_{\mathbf k\in\mathcal{SQ}_{\mathbf x,n,\ell}(k)}\prod_{m\in\mathcal N_{\mathbf x}}\Pr[v_m=k_m| B_{{\rm SVC},n,\ell,0}\ \ \mathbf x].
E_{tot}(x,t)=\sum_{n}\frac{E_{n}^{ret}(x,t)+E_{n}^{adv}(x,t)}{2}+\sum_{n}\frac{E_{n}^{ret}(x,t)-E_{n}^{adv}(x,t)}{2}=\sum_{n}E_{n}^{ret}(x,t).
B(\Phi(x),\Phi(y)) = B\left(x-\tau(x)-\frac{1}{2}B_{\mathfrak a}\big(\tau(x),\tau(\cdot)\big),y-\tau(y)-\frac{1}{2}B_{\mathfrak a}\big(\tau(y),\tau(\cdot)\big)\right) &=B_{\mathfrak a}\left(\tau(x),\tau(y)\right)-\frac{1}{2}B_{\mathfrak a}\big(\tau(y),\tau(x)\big)-\frac{1}{2}B_{\mathfrak a}\big(\tau(x),\tau(y)\big) &=0=B(x,y)
\widehat{S_0}(\textbf{m})=q^{-1}\delta_0(\textbf{m}) + q^{-d-1}\, G^d \sum_{\ell \in{\mathbb F}_q^*}\chi( \|\textbf{m}\| \ell)\mbox{for}~~\textbf{m}\in \mathbb F_q^d.
\gamma_{2n+1}^{\alpha,q}= (q+\mbox{$\frac 32$}-(n+q+\alpha+\mbox{$\frac 52$}))\frac{(n+q+\alpha+\frac 52-1)_{n}}{(2n+q+\alpha+\frac 52)(n+q+\alpha+\frac 52)_{n}}=-2\frac{(n+\alpha+1)(2n+2\alpha+2q+3)}{(4n+2\alpha+2q+3)(4n+2\alpha+2q+5)}.
H_NS_N(i,j) =\frac{1}{J_N}(2J_1(J_{N-2}+J_{N-1})+2J_0J_{N-2}-J_{N-1}) =\frac{1}{J_N}(2(J_{N-2}+J_{N-1})-J_{N-1}) =\frac{1}{J_N}(2J_{N-2}+J_{N-1}) =1.
q\cdot N_d =q^2+q\phi(b)+\frac{\phi(-b)}{q-1}\sum_{m=0}^{q-2}G_{-m+\frac{q-1}{2}}G_{md}G_{-m(d-1)}G_{\frac{q-1}{2}}T^m\left(\frac{b^{d-1}}{a^d}\right)\phi(b) &=q^2+q\phi(b)+\frac{\phi(-1)}{q-1}\sum_{m=0}^{q-2}G_{-m+\frac{q-1}{2}}G_{md}G_{-m(d-1)}G_{\frac{q-1}{2}}T^m\left(\frac{b^{d-1}}{a^d}\right).
\mathsf{C}_{\rho,1}:= p\mathsf{C}\left(h_{\rho}\mathsf{P}\right) + \bar{p}\mathsf{C}\left(h_1\mathsf{P}\right) + \frac{\bar{p}\log e}{2}\left(\frac{h_{\rho}}{1 + h_{\rho}\mathsf{P}}- \frac{h_1}{1+h_1 \mathsf{P}}\right)\mathsf{P}
R = \begin{pmatrix}1 -\tfrac12 0 0 \ldots -\tfrac12 1 -\tfrac12 0 \ldots 0 -\tfrac12 1 -\tfrac12 \ldots 0 0 -\tfrac12 1 \ldots \vdots \vdots \vdots \vdots \ddots\end{pmatrix}.
\quad\int_{|y-x|>s/4}\left(\left(2\frac{|4(y-x)|\wedge r_0}{s}\right)^{\eta}-1\right) J(x,y) \, dy & =\int_{|y-x|>r_0/4}( 2^{\eta}(r_0/s)^{\eta}- 1) J(x,y) \, dy + \int_{s/4<|y-x|\le r_0/4}\left(2^{\eta}\left(\frac{4|y-x|}{s}\right)^{\eta}-1\right) J(x,y) \, dy =: I_1 + I_2.
{}\dot{P}_{\mu}( x ) = \{P_{\mu}( x ) ,{\bf H}_{T}\}= \int_{M}d^{3}x^{\prime}\bar{N}^{\nu}( x^{\prime}) \{P_{\mu}( x ) , Q_{\nu}^{\sigma}( x^{\prime}) \}H_{\sigma}^{G}( x^{\prime}) ,
\begin{aligned}\Big( (U_2^pXU_1^{-p}-U_2^q XU_1^{-q})\,U_1^k e, U_2^l \bar e\Big)=\sum_{m=p}^{q-1}(U_2^{m}KU_1^{-m-1}\,U_1^k e, U_2^l \bar e) = \sum_{m=p}^{q-1}(KU_1^{k-m-1}e, U_2^{l-m}\bar e)=\sum_{m=p}^{q-1}\eta_m,\end{aligned}
(-1)^N 8\sqrt{\pi}N \sum_{\substack{n^2-m^2=N n,m\geq 1}}m \int_0^\infty \beta(m^2 y) e^{-4\pi N y}\, dy + \delta_\square(N) (-1)^N 4N \int_0^\infty y^{-\frac12}e^{-4\pi N y}\, dy.
\begin{aligned}\mathcal{H}/\mathcal{Z}\times \mathcal{H}/\mathcal{Z}\rightarrow \mathcal{Z}(h_1\mathcal{Z},h_2\mathcal{Z}) \mapsto \langle h_1\mathcal{Z},h_2\mathcal{Z}\rangle:=h_{1}h_{2}h_{1}^{-1}h_{2}^{-1}.\end{aligned}
W' \; : \; y'^2 = x'^3 + \frac{-2a}{\pi^{2\alpha}}x'^2 + \frac{a^2 - 4b}{\pi^{4\alpha}}x', \upsilon_F(\frac{-2a}{\pi^{2\alpha}})> 0, \upsilon_F(\frac{a^2 - 4b}{\pi^{4\alpha}}) = 0.
\langle\prod_{i=1}^{4}\phi(x_{i})\rangle=\frac{1}{|x_{12}|^{2\Delta}|x_{34}|^{2\Delta}}+\frac{1}{|x_{13}|^{2\Delta}|x_{24}|^{2\Delta}}+\frac{1}{|x_{14}|^{2\Delta}|x_{23}|^{2\Delta}}.
\frac{\partial}{\partial \bar{T}_{k}}d\Omega^{*}_{0}=\frac{\partial}{\partial X}d\Omega_{k}^{*},\;\;\;\frac{\partial}{\partial T_{k}}d\Omega_{0}^{*}=\frac{\partial}{\partial X}d\tilde{\Omega}_{k}^{*}.
(\partial_t - \partial_x^2)\Pi_{4 e_{(\xi,0)}+ e_{(0, 2 e_0 + 2e_{(0,1)})}}= \Pi_{e_{(\xi,0)}}^2 (\partial_x \Pi_{e_{(\xi,0)}})^2 & + c_{4 e_{(\xi,0)}+ e_{(0, 2e_0 + 2 e_{(0,1)})}}& + c_{2 e_{(\xi,0) + e_{(0, 2 e_{(0,1)})}}}\Pi_{e(\xi,0)}^2,
\widetilde{V}\big|_{\sigma_0}(z) = \theta^{2k+1}\Big(\frac{-1}{4z}\Big) \overline{\theta\Big(\frac{-1}{4z}\Big)}\Im^{\frac{k+1}{2}}\Big(\frac{-1}{4z}\Big) \frac{\lvert -2iz \rvert^k}{(-2iz)^k}
\lim_{t_1,t_2\rightarrow \infty}\left( \||u|^pu\|_{L_t^{l'}L_x^{m'}L^2_{y}([t_1,t_2]\times \mathbb{R}^d \times \mathbb{T})}+\||\nabla_x|^{\sigma}(|u|^pu)\|_{L_t^{l'}L_x^{m'}L^2_{y}([t_1,t_2]\times \mathbb{R}^d \times \mathbb{T})}\right. \left. +\||\partial_{y}|^{\sigma}(|u|^pu)\|_{L_t^{l'}L_x^{m'}L^2_{y}([t_1,t_2]\times \mathbb{R}^d \times \mathbb{T})}\right)=0.
\frac{\partial^{2}}{\partial \theta_{j}\partial \theta_{i}}f(\boldsymbol{\lambda},\boldsymbol{\theta}) = 4\log\left|2\cos\lambda_{i}-2u_{i}\right|\cdot \log\left|2\cos\lambda_{j}-2u_{j}\right| f(\boldsymbol{\lambda},\boldsymbol{\theta}).
A(\pi^i)=\begin{cases}\{1,2,\dots,i,n+1,n+2\}\textrm{if $1\le i \le k-2$,}\{1,2,\dots,k-2,n+1,n+2\}\textrm{if $i=n$,}\{1,2,\dots,k-2,n+2\}\textrm{if $i=n+1$.}\end{cases}
{\phi^{\mu_{1}}}_{\nu_{1}}{\phi^{\mu_{2}}}_{\nu_{2}}=\sum_{\mu,\nu,\gamma}(\begin{matrix}\mu_{1}&\mu_{2}&\gamma\\ \nu_{1}&\nu_{2}&\nu\end{matrix})\psi(\begin{matrix}\mu_{1}&\mu_{2}&\gamma\\ &&\nu\end{matrix}).
\begin{tabular}[c]{l}$\phi C_{n}\partial_{x}^{2}P_{n}-2x\left[ \left( \phi-1\right) C_{n}+\phi\right] \partial_{x}P_{n}+\left( n-2\gamma_{n}\right) \left[ \left(2x^{2}-1\right) C_{n}+2x^{2}\right] P_{n}$ $+\left[ 4\gamma_{n}\left( C_{n}+l_{n-1}\right) -\left( n-2\gamma_{n}\right)^{2}\right] C_{n}P_{n}=0.$\end{tabular}
\bar{G}(\theta ,\, p) = e^{\frac{\delta(1+\xi)}{8\pi\xi}\, (\theta+i\pi)^2}\, \exp\bigg\{\, \sum_{n=1}^{\infty}\, \frac{1}{n}\ \frac{\sin^2(\delta n (\theta+i\pi)/2)\, \sinh(\pi\delta(\xi+1) n/2)}{\sinh(\pi\delta n)\, \sinh(\pi\delta\xi n/2)\, \cosh(\pi \delta n/2)}\, \bigg\}\, ,
\begin{cases}\partial_t \bar{w}+ (u^s + \tilde{u}_1)\partial_x \bar{w}+ (u^s_{yy}+ \tilde{w}_{1, y})\bar{v}= \partial^2_y \bar{w}- \tilde{v}_2 \partial_y\bar{w}-(\partial_x\tilde{w}_2) \bar{u}, \partial_y\bar{w}|_{y=0}=0, \bar{w}|_{t=0}=\tilde{w}^1_0 - \tilde{w}^2_0 .\end{cases}
\det\left(\begin{array}{ccc}\scriptstyle{d_{i,j}}\scriptstyle{N-1}\scriptstyle{2N-d_{N,i}}& \scriptstyle{d_{N,j}d_{i,j}}\scriptstyle{2N}\sum_{l \not= i,N}\scriptstyle{d_{N,l}d_{i,l}}& \scriptstyle{d_{N,j}}\scriptstyle{N-1}\scriptstyle{2N-d_{N,i}}\end{array}\right)= 0,
n_k \geq w^k \left(n_0 + \frac{1}{1-w}\right) - \frac{1}{1-w}= w^k n_0 - \frac{1-w^k}{1-w}= w^k n_0 - \left(1+w+w^2+\dots+w^{k-1}\right) \geq w^k n_0 -k \geq w^k\left(w^{-k}(m+k)\right) -k \left( n_0 = \lfloor w^{-k}(k+m)+1\rfloor \right) =m.
\limsup_{r\to\infty}r^{p}\frac{\tilde q'(r)}{\tilde q(r)}= \limsup_{r\to\infty}\frac{c_1+c_2\int_1^r\rho^{n-3-\frac{2m \tilde\alpha}{\tilde\beta}}\tilde q^m(\rho)h_2(\rho)d\rho}{r^{n-p-1-\frac{2m\tilde\alpha}{\tilde\beta}}\tilde q^m(r)h_2(r)}\leq \limsup_{r\to\infty}\frac{c_2r^{n-3-\frac{2m \tilde\alpha}{\tilde\beta}}\tilde q^m(r)h_2(r)}{F(r)}
\frac{d}{d\beta}\P_{\boldsymbol{\beta}}(C^{}({\bf 0}) \cap S_n \neq \emptyset)=\sum_{i=1}^{k(\boldsymbol{x},\beta)}\frac{\partial}{\partial \beta_i}\P_{\boldsymbol{\beta}}(C^{}({\bf 0}) \cap S_n \neq \emptyset)
S_{\mathrm{p a r t}, 0}^{( \mathrm{N})}= \int d t \sum_{\alpha = 1}^{N}\left( \xi_{\alpha}^{\underline{{a}}}\left( E_{j , \alpha}^{\underline{{a}}}\dot{x}_{\alpha}^{j}+ E_{0 , \alpha}^{\underline{{a}}}\right) -{\frac{1}{2}}\xi_{\alpha}^{\underline{{a}}}\xi_{\alpha}^{\underline{{a}}}\right) .
{n_{\max ,i}}= \max \left\{{1,\sup \,\left|{y_{L,{N_y},i}^{(\alpha )}- \zeta_{L,{N_y},i}^{(\alpha ),y}}\right|}\right\}, D^{(\alpha)}= \frac{{\Gamma (1 + \alpha )}}{{2\,\Gamma (1 + 2\alpha )}},