harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
f(t,x,v) \geq e^{-t/\kappa}f(0, x-vt, v) + \int_0^t e^{-(t-s)/\kappa}\int f(s, x-v(t-s), u) \mathrm{d}u\times \times \left( \alpha \mathcal{M}_{T(x-v(t-s))}(v) + (1-\alpha) \mathcal{M}_{\tau(x-v(t-s))}(v) \right)\mathrm{d}s.
f(A,B) +f(C,D) = p(X_I,X_O)+p(Y_I,Y_O) \ge\; p(X_I\cap Y_I,X_O\cap Y_O\setminus Z)+p((X_I\cup Y_I)\setminus Z,(X_O\cup Y_O)\setminus Z) =\; p(A\cap C, (A\cap C)\cup (B\cap D)) + p((A\cup C)\setminus (B\cup D),((A\cup C)\setminus (B\cup D))\cup((B\cup D)\setminus(A\cup C))) =\; f(A\cap C,B\cap D)+f((A\cup C)\setminus (B\cup D),(B\cup D)\setminus(A\cup C)),
\rho_i^* = \left[{\left({{X_i}{W_i}+{X_i}+{Y_i}{Z_i}+{Z_i}}\right) -}\right. &\left.{\sqrt{{{\left({{X_i}{W_i}+{X_i}-{Y_i}{Z_i}+{Z_i}}\right)}^2}+ 4{Y_i}Z_i^2}}\right]/\left({2{Y_i}{Z_i}}\right) &={{\mathcal B}_i^{DF}}\left(\pmb \rho \right),
\begin{aligned}3^{l_1+1}x^{r + 6}+ a(r)x^{15}+ a(r) x^{33}+3^{2l_1+1}x^{2r+3}+ 2 \cdot 3^{l_1+1}a(r) x^{18 + r}+ 2 \cdot 3^{l_1+1}b(r) x^{36 + r}\in \langle F \rangle . \end{aligned}
{f_Z}\left( z \right) = \frac{{{m_f}^{{m_f}}{{\left({{m_s}- 1}\right)}^{{m_s}}}{{\bar z}^{{m_s}}}{z^{{m_f}- 1}}}}{{B\left({{m_f},{m_s}}\right){{\left({{m_f}z + \left({{m_s}- 1}\right)\bar z}\right)}^{{m_f}+{m_s}}}}},
\tilde\Omega^{(j+1)}=\tilde\Omega^{(j)}- \frac{\omega_1\big(\tilde\Omega^{(j)}\big)-\omega_1^{*}}{\omega_1\big((1+\alpha)\tilde\Omega^{(j)}\big)- \omega_1\big(\tilde\Omega^{(j)}\big)}\, \alpha\tilde\Omega^{(j)}\ ,
\left\{j_{1},\cdots,j_{m_{1}}\right\}=\left\{\overline{n-j_{1}},\cdots,\overline{n-j_{m_{1}}}\right\},\left\{j_{1}^{\circ},\cdots,j_{m_{1}}^{\circ}\right\}=\left\{\overline{n-j_{1}^{\circ}},\cdots,\overline{n-j_{m_{1}}^{\circ}}\right\}.
Z^{(1)}= \binom{a_1-1,a_2-1,\ldots,a_{k-1}-1,a_{k+1},\ldots,a_{m+1}}{b_1-1,b_2-1,\ldots,b_{l-1}-1,b_{l+1},\ldots,b_m}, Z'^{(1)}=\binom{c_1-1,c_2-1,\ldots,c_{l'-1}-1,c_{l'+1},\ldots,c_{m'}}{d_1-1,d_2-1,\ldots,d_{k'-1}-1,d_{k'+1},\ldots,d_{m'}}.
b_{12}'= \frac{b_{12}b_{11}'-w_3 b_{11}b_{22}+c_{12}}{b_{11}}begin{align*}1mm] b_{13}'= \frac{b_{13}b_{11}'+w_2 b_{11}b_{33}-w_3 b_{11}b_{23}+c_{13}}{b_{11}}begin{align*}1mm] b_{23}'= \frac{b_{11}b_{23}b_{22}'-w_1 +b_{11}c_{23}-b_{12}c_{13}+b_{13}c_{12}}{b_{22}b_{11}}\ .
p_{1^{(n)}}=(x_{1}+x_{2}+\cdot\cdot\cdot+x_{k})^{n}=\sum\frac{n!}{d_{1}!d_{2}!\cdot\cdot\cdot d_{k}!}x_{1}^{d_{1}}x_{2}^{d_{2}}\cdot\cdot\cdot x_{k}^{d_{k}}
\int_0^T\langle\partial_t u_m,v\rangle\mathrm{dt}+\int_{Q_T}( \mathsf{A}\nabla u_m+T_m(u_m){\bf E}) \cdot \nabla v\mathrm{dx}\mathrm{dt}+ +\int_{\Sigma_T}b(u_m)u_mv\mathrm{ds}\mathrm{dt}=\int_{Q_T}{mf\over m+|f|}v\mathrm{dx}\mathrm{dt}+\int_{\Sigma_T}h v\mathrm{ds}\mathrm{dt},
C(q_{1}, \dots, q_{p}; M) = 0 \to M \to \iffalse \bigoplus_{1 \leq \ell \leq p}M[q_{\ell}^{-1}] \to \fi \cdots \to \bigoplus_{\substack{I \subseteq [p] \mid I \mid = t}}M[q_{I}^{-1}] \to \cdots \to M[q_{[p]}^{-1}] \to 0
G_n^{(N)}(L,r,l)=\widetilde{\sum}_{\tilde{m}\in{\bf Z}^{N+1}}q^{mC^{-1}m}(q)_{L-1}\prod_{i=0}^N \frac{1}{(q)_{m_i}}\left\{\sum_{k=n}^N q^{-e_{n-l}C^{-1}m}B^{(N)}(L,r,l|k)\right\}
2 h_{\cal O}\langle\!\langle{\cal O}|{\cal O}\rangle= 2 \langle\!\langle{\cal O}|L_0|{\cal O}\rangle= \langle\!\langle{\cal O}| [ L_1, L_{-1}]|{\cal O}\rangle= \| L_{-1}|{\cal O}\rangle \|^2 \geq 0,
ds_{str.}^{2}= -\left ( 1 - \frac{2\eta}{r}\right )^\frac{m+ \sigma}{\eta}dt^{2}+ \left ( 1 - \frac{2\eta}{r}\right )^{\frac{\sigma - m}{\eta}}dr^{2}+ \left ( 1 - \frac{2\eta}{r}\right )^{1+\frac{\sigma - m}{\eta}}r^{2}d\Omega^{2}
\int_\Omega\langle\nabla u_i,\nabla\varphi\rangle\,dx+\int_\Omega V_i\,u_i\,\varphi\,dx=\langle f,\varphi\rangle,\mbox{for every}\varphi\in W^{1,2}_0(\Omega)\cap L^2(\Omega;V_i),\i=1,2.
v^{\eta}(t,x;s)=v^{\eta}(s,x;s)e^{-\int_{s}^{t}(1-a^{\eta}(\tau,x;s))d\tau}+\int_{s}^{t}b^{\eta}(r,x;s)e^{-\int_{r}^{t}(1-a^{\eta}(\tau,x;s))d\tau}dr.
\langle x_j, x_k \rangle =\frac{\|x_j+x_k\|^2-\|x_j-x_k\|^2+i\|x_j+ix_k\|^2-i\|x_j-ix_k\|^2}{4}&=\frac{(|1|^2+|1|^2)-(|1|^2+|-1|^2)+i(|1|^2+|i|^2)-i(|1|^2+|-i|^2)}{4}=0.
S_{\rm{Sugra}}^{(D)}(\tilde{\hat{g}}, \tilde{\hat{B}}, \tilde{\hat{\phi}})=S_{\rm{Sugra}}^{(D)}(\hat{g}, \hat{B},\hat{\phi}) +\int d^{D}x\A(\hat{g},\hat{B},\hat{\phi}) \partial_{\underline{x}}B(\hat{g},\hat{B},\hat{\phi})\, ,
\big(\sum_{j=1}^d|x_j+y_j|^{2R_j}\big)^{1/(2p)}\le \big(\sum_{j=1}^d(|x_j|^{R_j/p}+|y_j|^{R_j/p})^{2p}\big)^{1/(2p)}&\le \big(\sum_{j=1}^d|x_j|^{2R_j}\big)^{1/(2p)}+\big(\sum_{j=1}^d|y_j|^{2R_j}\big)^{1/(2p)}.
E((z_1-b_1)\cdots(z_n-b_n)) = E( (z_1-b_1)\cdots (z_{j-1}-b_{j-1})z_j(z_{j+1}-b_{j+1})\cdots(z_n-b_n)) &- \hat b_jE( (z_1-b_1)\cdots(z_{j-1}-b_{j-1})(z_{j+1}-b_{j+1})\cdots(z_n-b_n)).
I \left( a \right) = \begin{cases}\infty a \in \left( -\infty,0 \right); \ell \left( a_{-}; a \right) a \in \left[ 0,a_{-}\right]; 0 a \in \left[ a_{-}, a_{+}\right]; \ell \left( a_{+}; a \right) a \in \left[ a_{+},\infty \right). \end{cases}
{A}^{[1]}_{1}= C_{1},\{A}^{[1]}_{2}= C_{2}+ t_{2}D - t_{2}^{2}E{B}^{[1],(0)}= ( - t_{1}+ c) C_{1}+(-\frac{t_{2}}{2}+ c_{0}) C_{2}+ t_{2}(-\frac{t_{2}}{2}+ c_{0}) D + t_{2}^{2}(\frac{t_{2}}{2}- c_{0})E &{B}^{[1], (1)}= \alpha C_{1}-\frac{1}{4}D + c_{0}E B^{[1],(k)}=0,\k\geq 2.
\Phi_{\left( k \right)}\left( \tau , \theta^{+}, \theta^{-}\right) = q_{0 \left( k \right)}\left( \tau \right) + \theta^{+}\lambda_{-}W^{- k}+ \theta^{-}\lambda_{+}W^{k}+ \theta^{+}\theta^{-}\left( - W + \nu \frac{W^{\prime}}{2 W}\right) .
dN_s =\varGamma_{t,s-}\int_U\bigl(V_s(e) + Y_{s-}\gamma_s(e)+V_s(e)\gamma_s(e)\bigr) \widehat{\pi}(de,ds) & + \varGamma_{t,s-}(Z_s+Y_s\beta_s) dW_s+\varGamma_{t,s-}dM_s.
x_{P_0}= \frac{\overrightarrow{S_2 P_0}(1)}{U_{1 1}}\Big[1+ \frac{U_{1 2}L_{2 1}}{U_{2 2}}\Big]-\frac{U_{1 2}}{U_{1 1}}\overrightarrow{S_2 P_0}(2),\;y_{P_0}= \frac{\overrightarrow{S_2 P_0}(2)- L_{2 1}\overrightarrow{S_2 P_0}(1)}{U_{22}},
\mu^{(1)}\; :=\; \langle u_0,(-\Delta+U_{\rm trap}^{(1)})u_0\rangle+c_1\langle u_0, V^{(1)}*|u_0|^2u_0\rangle+c_2 \langle u_0,V^{(12)}*|v_0|^2u_0\rangle, \mu^{(2)}\; :=\; \langle v_0,(-\Delta+U_{\rm trap}^{(2)})v_0\rangle+c_2 \langle v_0, V^{(2)}*|v_0|^2v_0\rangle+c_1 \langle u_0,V^{(12)}*|v_0|^2u_0\rangle\,.
\aligned T_{2n}:=\inf\{t>T_{2n-1},|X_t(x_0)-X_t(y_0)|\le|x_0-y_0|\}\wedge\zeta_{x_0}\wedge\zeta_{y_0}, & T_{2n+1}:=\inf\{t>T_{2n},|X_t(x_0)-X_t(y_0)|\ge 2|x_0-y_0|\}\wedge\zeta_{x_0}\wedge\zeta_{y_0}.\endaligned
|\check x^\perp| = |x^\perp - \check{\mathcal T}_{t,x_1,v_1}v^\perp| = \check{\mathcal T}_{t,x_1,v_1}|v_*^\perp(t,x,v_1) - v^\perp | \geq \frac 1 2 \check \tau_{t,x}|v_*^\perp(t,x,v_1) - v^\perp |.
\begin{aligned}(\theta)^{-n-4}\int_{B_{\theta}}(\tilde{f}_{>2})^2 \leq \theta^{2\alpha}C (2\theta)^{-n-4}\int_{B_{2\theta}}(\tilde{\beta}_{>2})^2 \leq (2\theta)^{2\alpha}\int_{B_{1}}\tilde{\beta}^2 (2\theta)^{-n-2}\int_{B_{2\theta}}(\tilde{y}_i)_{>1}^2 \leq (2\theta)^{2\alpha}\int_{B_1}\tilde{y}_i^2. \end{aligned}
| L , M , S \rangle = \sqrt{C}\sum_{l , m , s}{\frac{\sin ( L , l )_{k}}{~ ~ \left[ \sin ( l , 0 )_{k}\right]^{-{\frac{1}{2}}}}}\exp \left({\frac{i \pi M m}{k + 2}}\right) \exp \left( -{\frac{i \pi S s}{2}}\right) | l , m , s \rangle \rangle
\xi_k(t) \geq f(t)+\int_a^t k(t,s,v_k(s))\Delta s - f(t)-\int_a^t k(t,s,v_{k-1}(s))\Delta s &\geq \int_0^t k(t,s,v_{k-1}(s))\Delta s - \int_0^t k(t,s,v_{k-1}(s))\Delta s &=0,
F_1(q) = -q^{-1}+ q^{23}- 2q^{47}+ 3q^{71}- \cdots F_{25}(q) = -q^{-25}- 263 q^{23}+ 2781q^{47}- 17960q^{71}+ \cdots F_{49}(q) = -q^{-49}+ 3400q^{23}- 102060q^{47}+ \cdots F_{73}(q) = -q^{-73}-23374q^{23}+\cdots
M_{--}^{2}= g^{2}v^{2}\left( \begin{array}{c c c c c c c}{{2}}&{{- 1}}&{{0}}&{{\cdots}}&{{0}}&{{0}}&{{0}}\\{{- 1}}&{{2}}&{{- 1}}&{{\cdots}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{- 1}}&{{2}}&{{\cdots}}&{{0}}&{{0}}&{{0}}\\{{}}&{{}}&{{}}&{{\ddots}}&{{}}&{{}}&{{}}\\{{0}}&{{0}}&{{0}}&{{\cdots}}&{{- 1}}&{{2}}&{{- 1}}\\{{0}}&{{0}}&{{0}}&{{\cdots}}&{{0}}&{{- 1}}&{{2}}\\ \end{array}\right) .
L_{-1}^*\otimes X_{2, \, j+1, \, 2}= L_{-1}^*\otimes \tilde{X}_{2, \, j+1, \, 2}+ X_{2, \, j+2, \, 6}= X_{2, \, j+2, \, 4}+ X_{2, \, j+2, \, 5}+ X_{2, \, j+2, \, 6}
\begin{cases}- k_{\varepsilon}\Delta \tilde{\Psi}_{\varepsilon}+ |\nabla \tilde{\Psi}_{\varepsilon}|^2 - \tilde{\lambda}_0 \underline{m}= 0 \Omega \; , \tilde{\Psi}_{\varepsilon}= -k_{\varepsilon}\log(w)(\frac{\mathbf{x}-\mathbf{x}_{\varepsilon}}{k_{\varepsilon}}) \partial\Omega \; ,\end{cases}
\bigcup_{i=1}^{k+1}X_i =\biggl(\bigcup_{i=1}^{k}X_i\biggr) \bigcup X_{k+1}=\biggl(\bigsqcup_{i=1}^{k}X_i'\biggr) \bigcup X_{k+1}=\biggl(\bigsqcup_{i=1}^{k}X_i'\biggr) \bigsqcup \Biggl( X_{k+1}\backslash \biggl(\bigsqcup_{i=1}^{k}X_i'\biggr)\Biggr)
\alpha_t(\lambda) = \begin{pmatrix}1 0 0 1\end{pmatrix}, \beta_t(\lambda) = \begin{pmatrix}e^{-(2a+1)u_t+l\pi\sqrt{-1}}\frac{\sinh\bigl((2a+1)u_t\bigr)}{\sinh(u_t/2)}0 e^{(2a+1)u_t-l\pi\sqrt{-1}}\end{pmatrix}
|h(\theta)| \ge \left|\int_{\theta_0}^{\theta}h'(s)ds\right|-C\Delta=\int_{\theta_0}^{\theta}\left|h'(s)\right|ds-C\Delta =\int_{\theta_0}^\theta \int_{\theta_0}^s h''(u)duds-C\Delta\geq ct(\theta-\theta_0)^2-C\Delta.
u^s_{A,q}(x,y)=-\frac{k^2}{4\pi}\underset{\underset{(\ell_2,m_2)\in \Gamma}{(\ell_1,m_1)\in \Gamma}}{\sum}i^{\ell_1-\ell_2}\mu_{(\ell_1, m_1; \ell_2, m_2)}h^{(1)}_{\ell_1}(k\vert x\vert) h^{(1)}_{\ell_2}(k\vert y\vert) Y^{m_1}_{\ell_1}\left( \hat x\right)Y^{m_2}_{\ell_2}\left( \hat y\right),
P(X_n\notin A)^{1/r_n}\le\sum_{i=1}^\infty P(w_{i}(X_n,\delta_i)>\frac{1}{2^i})^{1/r_n}+\sum_{i=1}^\infty P(X_n(t_i)\notin K_i)^{1/r_n}<2\epsilon\,.
\log \Phi_{n+1}^* = \sum_{j=k}^n \log \left(1-\frac{\alpha_j z \Phi_j}{\Phi_j^*}\right) + \log \Phi_k^* = \sum_{j=k}^n \log\left(1-\alpha_je^{i\gamma_j(\theta)}\right) + \log \Phi_k^*.
\frac{1}{2}(1-\varepsilon_0)^{n-k}|K\cap F|=\frac{1}{2}e^{-\frac{n-k}{3}}|K\cap F|>e^{-\frac{n-k}{3}-1}|K\cap F|>e^{-(k+\sqrt{n})}|K\cap F|,
W^{1}= \phi_{1}\left( 1 - \frac{\phi_{1}^{4}}{5}+ 2 \phi_{1}^{2}\phi_{2}^{2}- \phi_{2}^{4}\right) \hspace{1 c m}W^{2}= \phi_{2}\left( 1 - \phi_{1}^{4}+ 2 \phi_{1}^{2}\phi_{2}^{2}- \frac{\phi_{2}^{4}}{5}\right)
\alpha_0=\Biggl( \prod_{\ell \in \mathcal{L}^+}\bigl( (1+\hat{H_{i+\ell}})(1+K_{i+\ell}) \bigr)^{\binom{k}{\ell}}- \prod_{\ell \in \mathcal{L}^-}\bigl( (1+\hat{H_{i+\ell}})(1+K_{i+\ell}) \bigr)^{\binom{k}{\ell}}\Biggr)
\begin{aligned}F^{kk}\nabla_{\bar 1}\nabla_1 W_{k\bar k}=\frac{1}{n-1}\sum_k \sum_{i\ne k}\widetilde{F}^{ii}\nabla_{\bar 1}\nabla_1 W_{k\bar k}&= \frac{1}{n-1}\sum_i \widetilde{F}^{ii}\sum_{k\ne i}\nabla_{\bar 1}\nabla_1 W_{k\bar k}&= \widetilde{F}^{ii}\nabla_{\bar 1}\nabla_1 Z_{i\bar i},\end{aligned}
\rho^{3/2}\,u\,=\,\sqrt{\rho}\,u\,+\,\left(\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)\,.
\nu^{}_{ab}\bigl( \tfrac{1}{\tau}\bigr) \, = \, \tfrac{1}{\tau}\bigl( \nu^{}_{aa}\bigl(- \tfrac{1}{\tau}\bigr) + \nu^{}_{ba}\bigl( -\tfrac{1}{\tau}\bigr) \bigr) \, = \, \tfrac{1}{\tau}\nu^{}_{ab}\bigl( \tfrac{1}{\tau}\bigr)
H = \sum_{\emptyset \ne I \subseteq [1,d]}\sum_{\substack{(j_i)_{i \in I}j_i \in [0,m]}}u_{I,(j_i)_{i \in I}}H_{I,(j_i)_{i \in I}}\big( (x_\mu)_{\mu \in [1,d]\setminus I}\big),
\!q(\!n_2 u\!-\!(\ell_\nu\!+\!\beta_3)\!)\!\equiv\!\!\!\!\sum_{w\in W_1}q(\ell_{\nu_{h_w}}+ h_w)+\sum_{w\in W_2\cup W_3}\nu_w+qi-q\ell_\nu\equiv \sum_{w\in W_1}(\nu_{h_w}+ qh_w)+\sum_{w\in W_2\cup W_3}\nu_w+qi-\nu\equiv\sum_{w=1}^{|V|}\nu_w+qi-\nu=\nu_i+q i-\nu=0;
\lim_{x \to \infty}\frac{1}{x}\#\bigg\{n \leq x \colon \frac{\log G(n) - A(\log\log n)^2}{\sqrt C(\log\log n)^{3/2}}< u \bigg\}= \frac{1}{\sqrt{2\pi}}\int_{-\infty}^u e^{-t^2/2}\, dt
\Xi=\frac{1}{2m}\left(\begin{array}{ccccccc}1+\epsilon 1-\epsilon 1 1 \cdots 1 1 1 1 1+\epsilon 1-\epsilon \cdots 1 1 \vdots \vdots \ddots \ddots \vdots \vdots 1 1 1 1 \cdots 1+\epsilon 1-\epsilon \end{array}\right),
|g(X_1,\dots, X_n)-g(X_1',\dots, X_n')|\leq &12 \frac{\|\Sigma\|^{1/2}+\sqrt{2\delta}}{\sqrt{n}}\sup_{\|u\|\leq 1}\biggl(\sum_{j=1}^n \langle X_j-X_j',u\rangle^2\biggr)^{1/2}\bigvee\biggl(\frac{12}{n}\sup_{\|u\|\leq 1}\sum_{j=1}^n \langle X_j-X_j',u\rangle^2\bigwedge 2\delta\biggr).
\sigma(x) = -\frac{1}{\pi i}\oint_{\Gamma_N}{\Bigg(\tilde \varphi(x, \mu)\varphi(x, \mu)-\frac{1}{2}\Bigg)\hat M(\mu)}d\mu - 2\sum_{k=N+1}^\infty \sum_{j = 0}^1 (-1)^j\alpha_{kj}\Bigg(\tilde \varphi_{k,j}(x)\varphi_{k,j}(x) - \frac{1}{2}\Bigg),
x = \frac{\left[D_{p,\lambda,t}\right]^{\frac{1}{\alpha}}b}{\left[C_{p,\lambda,t}\right]^{\frac{1}{\alpha}}a}, y= \frac{\left[D_{p,\lambda,t}\right]^{\frac{1}{\beta}}d}{\left[C_{p,\lambda,t}\right]^{\frac{1}{\beta}}c}.
( \exp [ \langle . . \rangle_{D_{2}}+ \langle . . \rangle_{A_{2}}] )_{\log \epsilon}\approx ( \log \epsilon ){\frac{1}{2 !}}( \phi \cdot \nabla_{Y})^{2}\exp [{\langle \cdots \rangle_{D_{2}}}+ \langle . . \rangle_{A_{2}}] .
\lambda V_n (z) - \frac{1}{2}\mathrm{Tr}\big(Q D^2 V_n (z) \big) - \langle Az, DV_n(z)\rangle - \langle B(z), DV_n(z)\rangle = B_n(z) - B(z) + \langle B_n (z) - B(z), D_vU_n (z)\rangle.
\tilde{h}=\sum_{i,j=1}^{\infty}a_{ij}\varphi_i v_j \cdot dr\otimes dr+\sum_{i,j=1}^{\infty}b_{ij}\varphi_i \cdot dr\odot f\omega_j+\sum_{i,j=1}^{\infty}c_{ij}\varphi_i\cdot f^2h_j.
e(\bold w,k) =e(\bold w,k+1)+\Big[e_{i_{k+1}},\big[[[[e_{i_1},e_{i_2}],e_{i_3}]\cdots,e_{i_k}],[e_{i_{k+2}},\dots[e_{i_{r-1}}, e_{i_r}]]\big]\Big] &=e(\bold w,k+1)+\big[e_{i_{k+1}},e(\widetilde{\bold w},k+1)\big]
C-\frac{\sqrt{ab}}{a}D=-\frac{\sqrt{ab}}{a}\frac{\alpha}{t^\alpha}\mathrm{e}^{-\frac{\alpha(x+y)}{t^\alpha}}\bigg(\frac yx\bigg)^{-\frac{1+\sqrt{ab}}{2}}I_{\sqrt{ab}+1}\bigg(\frac{2\alpha\sqrt{xy}}{t^\alpha}\bigg).
\sum_{k=0}^{\infty}\frac{\binom{2k}{k}^2}{32^k(1+k)}\bigg\{H_{2k}^{(2)}-\frac{1}{4}H_{k}^{(2)}\bigg\}=\Gamma\bigg(\frac{3}{4}\bigg)^2\frac{\pi^2+8G-16}{4\pi\sqrt{\pi}}.
\gamma_{n}\left( z\right) =\frac{n^{2}z^{2}}{4n^{2}-1}+\frac{4n^{3}z^{4}}{\left( 4n^{2}-1\right)^{2}\left( 4n^{2}-9\right)}+O\left(z^{6}\right) ,z\rightarrow0.
\begin{array}{rl}a(J,J,z)= q\cdot\frac{q^2-q+1}{(q-1)(q+1)}\left( \frac{\xi_3(J)\xi_3(J)\xi_3(z^{-1})}{q^3}+ \frac{\xi_4(J)\xi_4(J)\xi_4(z^{-1})}{q(q^2-q+1)}\right) begin{align*}10pt]= q\cdot\frac{q^2-q+1}{(q-1)(q+1)}\left(\frac{\xi_3(z^{-1})}{q}+q\cdot\frac{\xi_4(z^{-1})}{q^2-q+1}\right) begin{align*}10pt]\equiv -\xi_3(z^{-1}) begin{align*}10pt]\end{array}.
\Phi_{A,n}(t) = \lambda_n t, \Phi_{D_i,n}(t) = \frac{1}{n}\int_0^t\hat{Q}^n_i(s)ds-\frac{1}{n}\int_0^t\hat{Q}^n_{i+1}(s)ds, \Phi_{D_k,n}(t) = \frac{1}{n}\int_0^t\hat{Q}^n_{k}(s)ds,
L_{00}= L_{11}y^1y^1 + L_{22}y^2y^2 + 2L_{12}y^1y^2, = 2{w^1}_{\substack{x_1}}(y^1)^2 + 2({w^1}_{\substack{x_2}}+{w^2}_{\substack{x_1}}) y^1 y^2 + 2{w^2}_{\substack{x_2}}(y^2)^2.
P_{\mu\nu\rho\sigma}= \frac{k_\mu k_\nu k_\rho k_\sigma}{k^4}, \ \ \M_{\mu\nu\rho\sigma}= \frac{1}{2}\left(L_{\mu\rho}P_{\nu\sigma}+L_{\mu\sigma}P_{\nu\rho}+ L_{\nu\rho}P_{\mu\sigma}+L_{\nu\sigma}P_{\mu\rho}\right), \ \ \m^2 = \frac{\alpha\Lambda}{2}.
\begin{aligned}\frac{c^2(i-1)}{2(\sqrt{2}+\sqrt{k-1})^2}+ \frac{c^2}{4(\sqrt{2}+\sqrt{k-1})^2}< \frac{2i-1}{(\sqrt{2}+\sqrt{k-1})^2}&= \left(\frac{\sqrt{i-\frac{1}{2}}}{\sqrt{2}+\sqrt{k-1}}\right)^2 &< \frac{1}{2}\left( \sqrt{\frac{i - \frac{1}{2}}{k}}\right)^2 &\leq \frac{1}{2}\end{aligned}
[m ((0,1,0)\otimes(0,0,1)+(0,0,1)\otimes (0,1,0))] = [(0,N_b-y_b(1-b),0)\otimes(0,0,1)+(0,0,1)\otimes (0,N_b-y_b(1-b),0)] = [(N_ax_b,N_b,0)\otimes(0,0,1)+(0,0,1)\otimes (N_ax_b,N_b,0)],
\frac{(n...n-l+1)(n...n-(k-l)+1)}{n...n-k+1}=\frac{n!~n!}{n!}\cdot \frac{(n-k)!}{(n-l)! (n-(k-l))!}&=\frac{n!(n-k)!}{(n-l)!(n-(k-l))!}&=\frac{n!}{(n-l)!~l!}\cdot \frac{l!~(n-k)!}{n-(k-l)!}&=\frac{\binom{n}{l}}{\binom{n-k+l}{l}}
|\nabla^2\zeta_{j}(x)|\lesssim\begin{cases}\frac{j}{|x|}|\xi'(j|x|)| + j^2|\xi''(j|x|)| \ |x|\in [\frac{1}{2j},\frac{1}{j}], 0 \ |x|\in [0,\frac{1}{2j}]\cup [\frac{1}{j},j]\cup[2j,\infty], \frac{1}{j^2}\ |x|\in [j,2j].\end{cases}
\liminf_{\tau\to\infty}\mu_{1,\tau}\geq (2\pi)^n(\widehat{\mu}^{\infty}_1((2\pi)^n)^{\frac{2+4/\alpha-d}{d-4/\alpha}})=(2\pi)^n\widehat{\mu}^{\infty}_{(2\pi)^{-n}}.
\mathcal{F}_{b,\psi_t}= \bigoplus_{k=0}^\infty \mathcal F_{b,\psi_t}^{(k)}, \mathcal{F}_{b}^{(k)}= \left( L^2_{\perp \psi_t}(\mathbb{R}^3) \right)^{\otimes_s k}
\operatorname{Tr}[\pi_\lambda(x)\, \pi_\lambda(P) \,\widehat \varphi(\lambda)] = P_x \operatorname{Tr}[\pi_\lambda(x) \,\widehat \varphi(\lambda)], \operatorname{Tr}[\pi_\lambda \,\widehat \varphi(\lambda)] \in \mathcal O_\mathrm{M}(G).
\hat C_{y_k}= \sum_{m=1}^{K}({\bf h}^{H}_{kB}{\bf\hat{t}}_{B_m})^{2}S_{B_m}+\sum_{j=1}^{2}({\bf h}^{H}_{kj}{\bf\hat{t}}_j)^{2}S_j,\ \forall k\in\mathcal{C}, \hat C_{z_j}= ({\bf g}^{H}_{ji}{\bf\hat{t}}_i)^{2}S_i+ \sum_{m=1}^{K}({\bf g}^{H}_{jB}{\bf\hat{t}}_{B_m})^{2}S_{B_m}+\kappa({\bf g}^{H}_{jj}{\bf\hat{t}}_j)^{2}S_j, \ \forall j\in\mathcal{D},
q = \sum_{n=0}^\infty \sum_{y\in C_\delta}\P (S_\delta^{v^*}(T) \circ \theta_n \in J_\alpha^+, L =n, S_\delta^{v^*}(n)=y) &=\sum_{n=0}^\infty \sum_{y\in C_\delta}\P (S_\delta^{v^*}(T) \circ \theta_n \in J_\alpha^+, S_\delta^{v^*}(n)=y) &=\sum_{n=0}^\infty \sum_{y\in C_\delta}\P ( S_\delta^{v^*}(n)=y) \P ( S_\delta^y (T) \in J_\alpha^+ )
\left( \frac{\alpha}{\pi}\right)^{K^{2}}\, \int d \mu ( u ) \int d \mu ( \bar{v}) \exp \left( - \alpha{\mathrm{t r}\,}( u s \bar{v}- a ) ( \bar{v}^{\dagger}s u^{\dagger}- a ) \right) \ = \ \frac{1}{K !}\frac{\det \left[ \gamma ( s_{f}, a_{f^{\prime}}) \right]_{f , f^{\prime}= 1 , \ldots , K}}{\Delta_{K}( s^{2}) \Delta_{K}( a^{2})}\ .
\big(\ell_\gamma \circ (g\circ h)\big)'\ =\ \big(\ell_\gamma' \circ (g\circ h)\big)(g'\circ h)h'\ =\ \big((\ell_\gamma' \circ g)\circ h)\big)(g'\circ h)h' =\ \big(\big((\ell_\gamma' \circ g)g'\big)\circ h\big)h'\ =\ \big((\ell_\gamma \circ g)' \circ h\big)h'\ =\ \big((\ell_\gamma \circ g)\circ h\big)'
\left( \begin{array}{c c}{{\partial_{r}-{\frac{a + j - 1 / 2}{r}}}}&{{- \sqrt{2}e v f}}\\{{- \sqrt{2}e v f}}&{{\partial_{r}+{\frac{j + 1 / 2}{r}}}}\\ \end{array}\right) \left( \begin{array}{c}{{h_{2}( r )}}\\{{h_{3}( r )}}\\ \end{array}\right) = 0 ,
E(t)\geq \sum_{k=0}^s\int \frac{\inf_{\alpha\in \mathbb{R}}|z_{\alpha}|^{-2k+1}}{\sup_{\alpha\in \mathbb{R}}a|z_{\alpha}|}|\partial_{\alpha}^k u_t(\alpha,t)|^2 d\alpha \geq \sum_{k=0}^s \frac{(2C_2)^{-2k+1}}{2\|w_0\|_{H^s}+1}\int |\partial_{\alpha}^k u_t(\alpha,t)|^2 d\alpha \geq \frac{(2C_2)^{-2s+1}}{2\|w_0\|_{H^s}+1}\|z_{tt}(\cdot,t)\|_{H^s}^2.
\begin{cases}c_{p,q}^j=0, p\neq j q\neq j c_{p,q}^k=0, p\neq k q\neq k c_{p,j}^j=c_{p,k}^k, \forall p\in \{1,2,\cdots, j\}c_{j,r}^j=c_{r,k}^k, \forall r\in \{j+1,j+2,\cdots, k-1\}c_{j,q}^j=c_{k,q}^k, \forall q\in \{k,k+2,\cdots, n\}.\end{cases}
L_x^{}= \sum_{1\leq j\leq n}\Bigl( \frac{\partial^2}{\partial x_j^2}- g_S e^{-x_j}-a_n e^{-2x_j}\Bigr) - \sum_{1\leq j<k\leq n}\frac{\frac{1}{2}g_M(g_M-1)}{\sinh^{2}{\frac{1}{2}}(x_j-x_k)}.
\mathbb{E}[f(\zeta(t),y)] =\int\limits_{S_1}\mathbb{E}[f(u,\xi^{t,y}(t)) \exp \{\int\limits_0^t g(s,\xi^{t,y}(s))ds\}] m_0^1(du)=\mathbb{E}[f(\zeta'(t),y)]
M_{g}(S)\cong\{\begin{matrix}&stablecurves\pi:C\rightarrow S\\ &withaniso\\ &\mathbb{P}(\pi_{*}(\omega_{C/S}^{\otimes3}))\cong\mathbb{P}^{5g-6}\times S\end{matrix}\}/\sim\cong Hom(S,H_{g})
\begin{array}{rl}N even: \;\; \frac{1}{2}-\frac{1}{p_0}+\frac{1-v_s}{4}< \frac{s}{p_0}<\frac{1}{2}+\frac{1-v_s}{4}N odd: \;\;\;\; \frac{1}{2}-\frac{3}{4p_0}+\frac{1-v_s}{4}< \frac{s}{p_0}<\frac{1}{2}-\frac{1}{4p_0}+\frac{1-v_s}{4}\end{array}
\sum_{\substack{n\geq 0 k\geq 1}}\frac{u^{n+k}x^{k}(1-x)^{n}(1-u)^{2k}}{(1+u)^{n+2k}(1-ux)^{n+2k}}k^{\underline{d}}C_{k-1}\binom{n+2k-2}{n}2^{n}= (2d-2)^{\underline{d-1}}\frac{u^{d}x^{d}(1-u)}{(1-ux)^{2d}(1+u)}
\{e_1,\ldots,e_{\mu}\}=\{e_1,\ldots,e_{\mu'}\}\sqcup \{e_{\mu'+1},\ldots,e_\mu\}, \{v_1,\ldots,v_{m}\}=\{v_1,\ldots,v_{m'}\}\sqcup \{v_{m'+1},\ldots,v_m\},
c_1(\delta_k \otimes c_2 \otimes \delta_n \otimes b) = \delta_k \otimes \alpha_d^{-k}(c_1) c_2 \otimes \delta_n \otimes b \mapsto \alpha_d^k\!\left(\alpha_d^{-k}(c_1)c_2\right) \cdot \delta_{(n,k)}\otimes b = c_1 \left(\alpha_d^k(c_2)\cdot \delta_{(n,k)}\otimes b \right).
\nabla^{2}\equiv -{\frac{1}{\sqrt g}}\partial_{i}\sqrt g g^{i j}\partial_{j}={\frac{1}{\sqrt g}}\delta_{i j}\partial_{i}\partial_{j}= ({\frac{1 + \vec{x}^{2}}{2}})^{2}( \partial_{1}^{2}+ \partial_{2}^{2}) .
\limsup_{{\mathcal N}\ni n\to\infty}\left(M(P_n)\right)^{1/n}\le \limsup_{{\mathcal N}\ni n\to\infty}|a_n|^{1/n}\ \limsup_{{\mathcal N}\ni n\to\infty}\left(\prod_{k=1}^{K_n}|\alpha_{k,n}| \right)^{1/n}< \infty
\left(h_{n+1}(1), h_{n+1}(2)\right) =\left(h_{n}(1),h_{n}(2)\right)\left(\begin{array}{cc}b_n+4 b_n b_n+3 b_n+1\end{array}\right), \left(\begin{array}{c}\mu_{n}(1) \mu_{n}(2)\end{array}\right) =\left(\begin{array}{cc}b_n+4 b_n b_n + 3 b_n + 1\end{array}\right)\left(\begin{array}{c}\mu_{n+1}(1) \mu_{n+1}(2)\end{array}\right).
P^T L_3^\prime(\lambda) P = \left[\begin{array}{ccc|cc}\lambda P_5-P_4 \lambda P_4 0 -I_n 0 \lambda P_4 \lambda P_3+P_2 P_1 \lambda I_n -I_n 0 P_1 -\lambda P_1+P_0 0 \lambda I_n \hline-I_n \lambda I_n 0 0 0 0 -I_n \lambda I_n 0 0\end{array}\right]
\bar{R}(X,Y)Z =f_1\big\{g(Y,Z)X-g(X,Z)Y\big\}+f_2\big\{g(X,\phi Z)\phi Y & - g(Y,\phi Z)\phi X + 2g(X,\phi Y)\phi Z\big\}+f_3\big\{\eta(X)\eta(Z)Y & - \eta(Y)\eta(Z)X+g(X,Z)\eta(Y)\xi - g(Y,Z)\eta(X)\xi\big\}
0\le \int_{B_r}\big( |\nabla v_\rho|^2-|\nabla u_\rho|^2\big) \qquad\qquad+\frac{1}{\rho^n}\left[\Phi_0\left( \rho^n\lambda_2 \int_{B_r}Q(\rho x)\chi_{\{v_\rho>0\}}(x)\,dx +\Xi_\rho\right)-\Phi_0\left( \rho^n\lambda_2 \int_{B_r}Q(\rho x)\chi_{\{u_\rho>0\}}(x)\,dx +\Xi_\rho\right)\right].
\begin{aligned}\mathcal{C}[\mathcal{L}\chi](\zeta) =W^{(p-1)}[\mathcal{L}\chi](\zeta)=\psi(-\zeta)\ , \mathcal{C}[\mathcal{L}\psi](\eta) =W^{(p'-1)}[\mathcal{L}\psi](\eta)=\chi(-\eta)\ .\end{aligned}
\tau'_t(d) := \begin{cases}\frac{1}{d-2}(2(d-1)^{t/2+1}-d) 2\mid t \frac{1}{d-2}(d(d-1)^{(t+1)/2}-d) 2\nmid t\end{cases}\ \ \tau_t(d) :=\begin{cases}\frac{1}{d-2}(d(d-1)^{t/2}-2) 2\mid t \frac{1}{d-2}(2(d-1)^{(t+1)/2}-2) 2\nmid t\end{cases}.
V_C\left( x_1,\ldots,x_N;y_1,\ldots,y_K\right) = \sum_{1\leq i<j\leq N}v\left( x_i,y_j\right) - \sum_{i=1,\ldots,N \atop l=1,\ldots,K}Z_l v\left( x_i,v_l\right) + \sum_{1\leq l < k \leq K}Z_l Z_k v\left( y_l, y_k\right) \ .
\mathbb{S}_{f,g}= \frac{(-1)^{\sum_{i=0}^{n-1}(k_i(f)+k_i(g))}}{d^n}\prod_{i=0}^{n-1}\zeta^{k_i(f)k_i(\sigma\cdot g)}\sum_{\sigma\in\mathfrak{S}_{n+1}}\varepsilon(\sigma)\prod_{i=1}^{n+1}\zeta^{-f(i)(\sigma \cdot g)(i)},
\frac{1}{M_{p}^{2}p^{2}}\left( 1 + \langle T_{C F T}\left( p \right) T_{C F T}\left( - p \right) \rangle \frac{1}{M_{p}^{2}p^{2}}\right) + \frac{1}{M_{w}^{8}}\langle T_{C F T}\left( p \right) T_{C F T}\left( - p \right) \rangle ,
A_{+}= \frac{1}{2}\left( \begin{array}{c c}{{\sigma_{-}}}&{{0}}\\{{0}}&{{- \sigma_{+}}}\\ \end{array}\right) \; , \; \; A_{-}= \frac{1}{2}\left( \begin{array}{c c}{{\sigma_{+}}}&{{0}}\\{{0}}&{{- \sigma_{-}}}\\ \end{array}\right) .
(1) ~~\overline{C^*_{d}}=C^*_{d}, (2) ~~C^*_{d}=H_{d}+\sum\limits_{\substack{d'\in I_{k,n}\ell(d')<\ell(d)}}p^*_{d', d}H_{d'}, ~\mbox{where}p^*_{d', d}\in q\mathbb{Z}[q].
\sum_{\substack{b\leq a/\rho (a,b)=1 b\equiv\psi(a)\mod{n}}}\mu^2(b)=\frac{\phi(a)}{\rho an}\cdot\frac{1}{\zeta(2)}\cdot\prod_{p\mid an}\frac{p^2}{p^2-1}+O(2^{\omega(a)}\sqrt{a/\rho}).
x(t)\leq x(0)exp\Big\{2\epsilon\int_0^t (1+\frac{|\lambda|}{20\pi x(0)}\tau)^{-3/2}d\tau\Big\}\leq x(0)exp\Big\{4\epsilon \frac{20\pi x(0)}{|\lambda|}\Big\}\leq x(0)e^{2\epsilon \frac{40\pi}{200\pi\epsilon}}=e^{\frac{2}{5}}x(0)\leq \frac{3}{2}x(0).
\inf_{\varphi\in C^{\infty}_{cs}((0,\infty))}\frac{\int_{0}^{\infty}(\partial_r\varphi)^2\sinh^ndr}{\int_{0}^{\infty}\varphi^2\cosh^2\sinh^{n-2}dr}=\inf_{\varphi\in C^{\infty}_{cs}((0,\infty))}\frac{\int_{0}^{\infty}(\partial_s\varphi)^2(1+s^2)^{1/2}s^nds}{\int_{0}^{\infty}\varphi^2(1+s^2)^{1/2}s^{n-2}ds}.
0 = \langle A\varpi, \varpi \rangle=B [\varpi, \varpi]=\int_{\mathcal{Y}^d}e^{\frac{|\Lambda+\nabla_y\widetilde{w}|^2}{2}+\lambda V}\big( \big( (\Lambda+\nabla_y\widetilde{w})^T\nabla_y\varpi\big )^2+ |\nabla_y\varpi |^2 \big)dy.
\mu_{k}^i\left(B\right) = \frac{1}{Z_{k}^i}\int\limits_{\theta \in B}\prod\limits_{t=1}^{k}\prod\limits_{j=1}^{n}\ell^j(s_{t}^j|\theta)^{\left[A^{k-t}\right]_{ij}}d\mu_0^j\left(\theta\right)
\prod_{i=1}^n\sigma_{j-1}(\lambda|i)\geq C_{n,j}\sigma_{j}(\lambda)^{\frac{n(j-1)}{j}}\geq C_{n,j,k}\sigma_k(\lambda)^{\frac{n(j-1)}{k}}\geq C_{n,j,k}A^{-n(j-1)},
\widehat{L}_\beta (\lambda) = \frac{1}{\sin\gamma}\left( \begin{array}{cc}\sinh [\gamma(\lambda +{\rm i}S^3)]{\rm i}\, \sin\gamma \, e^{(\gamma-\beta) \lambda}\, S^-{\rm i}\, \sin\gamma \, e^{(\beta-\gamma) \lambda}\, S^+ \sinh [\gamma(\lambda -{\rm i}S^3)] \end{array}\right)