harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\cal G}_n(p,q,\ell ) = \frac{(2(n+1))^{1+\frac{\ell}{n+1}}}{4 \Gamma(2+\frac{\ell}{n+1})}\Gamma( 1+\frac{p+q+\ell}{2(n+1)})\Gamma( 1+\frac{-p+q+\ell}{2(n+1)}) &\times \Gamma( 1+\frac{p-q+\ell}{2(n+1)})\Gamma( 1+\frac{-p-q+\ell}{2(n+1)}).
\beta(t,q_t)H(t,x_t)-\beta(s,q_s)H(s,x_s) = \int_s^t \partial_r(\beta H)(r,x_r)dr+\int_s^t \nabla_q (\beta H)(r,x_r) \cdot dq_r+\int_s^t \nabla_p (\beta H)(r,x_r) \circ dp_r.
g_{k,\ell}(p^{\alpha}) := \sum_{j=0}^{\alpha}\Big( \frac{\sigma_{k}(p^j,1)\sigma_{\ell}(p^j,1)}{p^{j}}- \frac{\sigma_{k}(p^{j+1},1)\sigma_{\ell}(p^{j+1},1)}{p^{j+2}}\Big) \Big( \Big(1-\frac{1}{p}\Big)^{k}+ \Big(1-\frac{1}{p}\Big)^{\ell}- \Big( 1-\frac{1}{p}\Big )^{k+\ell}\Big)^{-1}.
\textsf{C}(I/(1+S)) = \min\{I(X_1;Y_1),\, I(X_1;Y_2)\}+ \min\{I(X;Y_1|X_1,W_1),\, I(X;Y_2|X_1)\}\le I(X_1;Y_2) + I(X;Y_2|X_1) =I(X;Y_2) \le \textsf{C}(I/(1+S)).
I_{\mu}^{\ \nu}\equiv \left[ \delta_{\mu}^{\nu}+ \phi^{- 1}\phi_{\mu \alpha}h^{\alpha \nu}\right] \; , \quad I^{- 1}\_{\mu}^{\ \alpha}I_{\alpha}^{\ \nu}= \delta_{\mu}^{\nu}\; , \quad I \equiv d e t I_{\mu}^{\ \nu}\; .
\int_{\mathbb{R}^{n}}e^{-\pi\epsilon^{2}|\xi|^{2}+2\pi ix\cdot\xi}(Ff)(\xi)d\xi=\int_{\mathbb{R}^{n}}\frac{1}{\epsilon^{n}}e^{-\frac{\pi}{\epsilon^{2}}|x-y|^{2}}f(y)dy=(\varphi_{\epsilon}*f)(x),
m\partial_{x_{j,m}}f =m\partial_{x_{j,m}}f'+m\partial_{x_{j,m}}g_{j,m}f_{j,m}-m\partial_{x_{j,m}}g_{j,m}(m\partial_{x_{j,m}}f') =m\partial_{x_{j,m}}f'+f_{j,m}-m\partial_{x_{j,m}}f' =f_{j,m},
L(X) \le S^T + \sum_{u= p_m^{T}}^{u = n-1}d(x_{u},x_{u+1}) = \sum_{1 \le a < b \le m}c_ad(x_{p^T_a}, x_{p^T_b}) + \sum_{u= p_m^{T}}^{u = n-1}d(x_{u},x_{u+1}).
\sum_{i=1}^n \alpha_n^i\partial_t^{i+n+1}u_{n}+\partial_t^{n+1}u_{n}-\sum_{i=1}^{2n}\beta_n^i\Delta \partial_t^i u_n &\quad-\omega_{n+1}\kappa_n \Delta u_{n}-(1-\omega_{n+1})\kappa_n \int_0^\infty g_{n+1}(s)\Delta u_{n}(t-s)ds=0.
nR \leq nC_1+nC_2 -I(X_1^n;X_2^n) &= nC_1+nC_2 -I(X_1^nX_2^n;U^n)+I(X_2^n;U^n|X_1^n)+I(X_1^n;U^n|X_2^n)-I(X_1^n;X_2^n|U^n).
\overline{G}=\overline{m(\frac{\mu^{n}}{\lambda^{n}}y)^{m-1}W}=\overline{m(\frac{\mu^{n}}{\lambda^{n}}y)^{m-1}}\,\overline{d\frac{\mu^{d-1}}{\lambda^{n}}s^{d-1}a(x)}.
\frac{h}{N}\Bigg(|F^{\delta,l}_0|+\sum_{n=2}^{\infty}|T_0^{\delta,l,(n)}|\Bigg)\le \alpha^{-4}, &c_0\alpha^2\Bigg(\|F_2^{\delta,l}\|_{0,r}+\sum_{n=2}^{\infty}\|T_2^{\delta,l,(n)}\|_{0,r}\Bigg)\le 1, &M^{-2l}\sum_{m=2}^Nc_0^{\frac{m}{2}}M^{\frac{l}{2}m}\alpha^m\Bigg(\|F_m^{\delta,l}\|_{0,r}+\sum_{n=2}^{\infty}\|T_m^{\delta,l,(n)}\|_{0,r}\Bigg)\le 1.
(-1)^F = (-1)^{1 + \sum_{j=2}^{n}\sum_{r>0}(b^{-j}_{-r}b_{r,j}+b^{j}_{-r}b_{r,-j}) + \sum_{j=n+1}^{5}\sum_{n>0}(b^{-j}_{-n}b_{n,j}+ b^{j}_{-n}b_{n,-j})}\cdot \!\!\! \prod_{k=n+1}^{5}(b_{0}^{k}b_{0}^{-k}- b_{0}^{-k}b_{0}^{k})
\mathcal{N}_{\rm cyl}(v_{\varepsilon}+t\psi)-\mathcal{N}_{\rm cyl}(v_{\varepsilon})=tP_{\rm cyl}\psi-tc_n\left(-P_{\rm cyl}v+\frac{n+6}{n-6}c_nv_{\varepsilon}^{\frac{12}{n-6}}\psi\right)+\mathcal{O}(t^2){\rm as}t\rightarrow 0
\widetilde V_1(\psi,z)=:\prod_{a=1}^M{i\over{(m_a-1)!}}{d^{m_a}\over{dz^{m_a}}}C_+^{i_a}(z)\prod_{b=1}^N{i\over{(n_b-1)!}}{d^{n_b}\over{dz^{n_b}}}C_-^{j_b}(z)e^{i{\bar\lambda\over\alpha}\cdot C_+(z)}e^{i{\lambda\over\alpha}\cdot C_-(z)}:\gamma_\lambda\,,
c^{(q_1,q_2)}= \sum_{k = 0}^{q_1}\sum_{l = 0}^{q_2}c^{(q_1,q_2)}b_{k,l}^{(q_1,q_2)}(x_1,x_2) & \leq \sum_{k = 0}^{q_1}\sum_{l = 0}^{q_2}c^{(q_1,q_2)}_{k, l}b_{k,l}^{(q_1,q_2)}(x_1,x_2) &= p(x_1, x_2),
\mathrm{d e t_{\it u}}[{\cal O}]^{{\frac{1}{2}}\sum_{i}c_{u_{i}}}\equiv \lim_{\{M_{u_{i}}^{2}\}\rightarrow \infty}\prod_{i = 1}^{n_{u}}\mathrm{d e t}\left[{\cal O}+ M_{u_{i}}^{2}1_{u}\right]^{{\frac{1}{2}}c_{u_{i}}},
\frac{\partial M_{pq}}{\partial x_{q}}=e_{prs}T_{sr}+e_{prs}x_{r}\frac{\partial T_{sq}}{\partial x_{q}}& =e_{pqr}x_{q}\mathcal{F}_{r}+\frac{\partial}{\partial t}\left(e_{pqr}x_{q}\mathcal{G}_{r}\right) ,
U ( \phi_{1}, \phi_{2}) = \frac{1}{2}\left( \frac{\partial W}{\partial \phi_{1}}\right)^{2}+ \frac{1}{2}\left( \frac{\partial W}{\partial \phi_{2}}\right)^{2}\qquad ; \qquad W ( \phi ) = 4 \sqrt{2}\left( \frac{1}{3}\phi_{1}^{3}- \frac{1}{4}\phi_{1}+ \frac{\sigma}{2}\phi_{1}\phi_{2}^{2}\right)
\mathbf{S}_h(\Omega) = \left\{\mathbf{v}\in [L^2(\Omega)]^2 \,:\, \mathbf{v}|_T\in\mathbf{S}_h(T); \right. \left. ~ \mathbf{v}|_{T_1}(N)=\mathbf{v}|_{T_2}(N)~ \forall N\in \mathcal{N}_h, ~ \forall T_1, T_2 \in \mathcal{T}_h ~\textrm{such~that}~ N\in T_1\cap T_2 \right\}.
\nabla_{A}W_{B C D}= \nabla_{\underline{{{A}}}}W_{\underline{{{B C D}}}}- \frac{1}{4}\varepsilon_{A B}\nabla^{E}W_{E C D}- \frac{1}{4}\varepsilon_{A C}\nabla^{E}W_{E B D}- \frac{1}{4}\varepsilon_{A D}\nabla^{E}W_{E B C}
(m_{1,i}-1)\alpha_r = (i-1)\left((r-1)-\frac{i}{2}\right) \frac{1}{r-2}\left(\frac{2}{r-1}+\frac{2}{r^2(r-1)}\right) = \frac{i-1}{r-2}\left( 2 - \frac{i}{r-1}+ \frac{2}{r^2}- \frac{i}{r^2(r-1)}\right).
=\int_{\theta_{0}}^{\frac{3\pi}{2}}|\sin\varphi|^{q-2}\Big(\int_{\frac{R}{\gamma}\sin\varphi+1}^{1}\frac{dt}{|t|^{\tau}|t-1|^{q-1}}\Big)d\varphi\quad\quad\quad\quad\quad\quad\quad\quad\quad\,
{\frac{\bar{A}_{1}^{T}( d )}{( 4 \pi )^{d / 2}}}={\frac{\Gamma \left( \frac d 2 - 1 \right)}{1 2 \gamma ( d - 3 ) !}}\left[ d ( d - 7 ) + ( d - 1 ) \left( \gamma^{2}- 1 \right) + 1 2 \left( 1 - \gamma \right) \right] + \delta_{d , 2},
\frac{v^{2}}{c^{2}}=\frac{c^{2}p^{2}}{E^{2}}\Rightarrow E^{2}=\frac{m_{0}^{2}c^{4}}{1-c^{2}p^{2}/E^{2}}\Rightarrow m_{0}^{2}c^{4}=E^{2}-c^{2}p^{2}.
\frac{f_{n_k}'(q)}{f_{n_k}'(p)}= \frac{f_{n_k}'(z)}{f_{n_k}'(x)}\cdot\frac{f_{n_k}'(x)}{f_{n_k}'(p)}\cdot\frac{f_{n_k}'(q)}{f_{n_k}'(z)}\le \frac{f_{n_k}'(z)}{f_{n_k}'(x)}e^{C(x-p)}e^{C(q-z)}\le \bar{H}e^{2C(x-z)}
W \equiv \left| \begin{matrix}{f_{1}}&{f_{2}}&{\cdots}&{f_{n}}\\{D f_{1}}&{D f_{2}}&{\cdots}&{D f_{n}}\\{\vdots}&{\vdots}&{}&{\vdots}\\{D^{n - 1}f_{1}}&{D^{n - 1}f_{2}}&{\cdots}&{D^{n - 1}f_{n}}\\ \end{matrix}\right| \quad .
\int_\Omega p_\varepsilon |\nabla u^\varepsilon|^2\,dx +\int_\Omega (\lambda+V_\varepsilon )u^\varepsilon(u^\varepsilon-u^0)\,dx-\int_\Omega (\lambda+V_0) u^0(Pu^\varepsilon-u^0)\,dx =\|u^\varepsilon-u^0\|_{X_\varepsilon^\frac{1}{2}}^2+\int_\Omega (V_\varepsilon-V_0)u^0(u^\varepsilon-u^0)\,dx.
S_{b}={\frac{- 1}{1 6 \pi G}}\int d^{4}x \sqrt{- \tilde{g}}\left\{( \varphi_{+}^{2}- \varphi_{-}^{2}) \ \tilde{\cal R}-{\frac{6 q}{q + 1 / 2}}\left[ ( \tilde{\partial}\varphi_{+})^{2}- ( \tilde{\partial}\varphi_{-})^{2}\right] \right\}.
\Gamma_{m,j}=e^{-\frac{2\pi i}{L}q\cdot c_{m}}[\frac{\partial C}{\partial f_{j}(r,k)}|_{ss}+\sum_{p=1}^{K}\frac{\partial C}{\partial f_{j}(r+c_{p},k)}|_{ss}e^{\frac{2\pi i}{L}q\cdot c_{p}}].
\langle i | P^{-}| j k \rangle \propto \int d p_{i}^{+}d p_{j}^{+}d p_{k}^{+}g ( p_{i}^{+}, p_{j}^{+}, p_{k}^{+}) \delta ( p_{i}^{+}+ p_{j}^{+}- p_{k}^{+})
\frac{d}{d \tau}\left( \frac{\dot{x}_{i}^{\mu}( \tau )}{\sqrt{\dot{x}_{i}^{2}( \tau )}}\right) = ( - 1 )^{i + 1}K_{i}( \tau ) \acute{x}_{i}^{\mu}( \tau ) \, \, \, \, \, ( i = 1 , 2 ) ,
R L_0\colon g\longmapsto R L_0 g = R \int_0^t e^{A(t-s)}B_0 g(s) ds - R A\int_0^t e^{A(t-s)}B_1 g(s) ds begin{align*}1mm] = R A^2 \int_0^t e^{A(t-s)}A^{-2}B_0 g(s) - R A^2 \int_0^t e^{A(t-s)}A^{-1}B_1 g(s)ds
A(\gg) \simeq \bigoplus_{j=1}^n J(\l_j,\sqrt{p})\bigoplus\Bigl(\bigoplus_{i=1}^{(p-1)n}J_0\Bigr), J(\l_j,\sqrt{p}) :=J(\{\l_j,0,0,\ldots\},\{\sqrt{p},1,1,\ldots\}).
\sum_{k=\lfloor \frac{N}{3}\rfloor+1}^\infty \left(5e^2 \|K\|_{L^\infty}\left(\sup_p \frac{M_p}{p}\right)\right)^{2k}\leq \sum_{k=1}^\infty \left(\frac{5}{8}\right)^{2k}\leq \frac{\left(\frac{5}{8}\right)^2}{1- \left(\frac{5}{8}\right)^2}< \infty .
\mathcal{Q}(u) - \mathcal{Q}(\hat w) = \hat{\rho}^\prime \left( U_2 + U_3 \frac{\hat \zeta}{|\zeta_0|}+ U_\beta \partial_{\hat \zeta}H - \lambda w \right) & \geq \hat{\rho}^\prime \left( U_2 + \max\{0, U_3\}+ |U_\beta| (1+\beta)(\zeta_0^2+(L/d)^2)^{\beta/2}+ \lambda \| w \|_{L^\infty(\mathcal{S})}\right).
\lambda^{^{S , A}}= \left( \begin{matrix}{{\zeta_{\lambda}\Theta_{[ j ]}\phi_{_L}^{\ast}( p^{\mu})}}\\{{\phi_{_L}( p^{\mu})}}\\ \end{matrix}\right) \, \, , \quad \rho^{^{S , A}}= \left( \begin{matrix}{{\phi_{_R}( p^{\mu})}}\\{{( \zeta_{\rho}\Theta_{[ j ]})^{\ast}\phi_{_R}^{\ast}( p^{\mu})}}\\ \end{matrix}\right) \, \, .
\int \left( u_j \Delta_h u_i - u_i\Delta_h u_j\right) \zeta \,dx =\frac1{2h}\int G_h(z) \int u_j [(u_i^z-u_i)-(u_i-u_i^{-z})] \zeta -u_i [(u_j^z-u_j)-(u_j-u_j^{-z})] \zeta \,dx\,dz,
\sum_{n=1}^{N}\left[\begin{matrix}N n\end{matrix}\right]\frac{(\frac{b}{a})_{n}(q)_{n}(a)_{N-n}a^{n}}{(1-cq^{n})(b)_n(a)_N}=\sum_{n=1}^{N}\left[\begin{matrix}N n\end{matrix}\right]\frac{(\frac{b}{c})_{n-1}(q)_n (cq)_{N-n}c^{n-1}}{(b)_{n-1}(cq)_N}\left(\frac{aq^{n-1}}{1-aq^{n-1}}-\frac{bq^{n-1}}{1-bq^{n-1}}\right).
\left( \begin{matrix}{{a}}\\{{b}}\\ \end{matrix}\right) = T \left( \begin{matrix}{{u^{+}}}\\{{u^{-}}}\\ \end{matrix}\right) = e^{i \chi}[ \tau^{4}\cos ( \kappa / 2 ) + i N^{k}\tau^{k}\sin ( \kappa / 2 ) ] \left( \begin{matrix}{{u^{+}}}\\{{u^{-}}}\\ \end{matrix}\right) = e^{i \chi}e^{i N^{k}\tau^{k}\kappa / 2}\left( \begin{matrix}{{u^{+}}}\\{{u^{-}}}\\ \end{matrix}\right) ,
M_6(p,H)={\pi^2\over 9}\left (1+\frac{c_a}{p}\right ), \hbox{where}c_a=\begin{cases}-2a-1 \hbox{if $a\equiv 0\pmod 6$}, -3 \hbox{if $a\equiv 2,3\pmod 6$}, 2a+1 \hbox{if $a\equiv 5\pmod 6$}.\end{cases}
e^{-\tau\phi_{-}}=\frac{\|x_{n+1}^{\frac{1-2s}{2}}\tilde{w}\|_{L^{2}(C_{s,1/2}^{+})}^{\frac{\phi_{-}}{\phi_{+}-\phi_{-}}(1-\alpha)}}{\lim_{x_{n+1}\rightarrow0}\|x_{n+1}^{1-2s}\partial_{n+1}\tilde{w}\|_{L^{2}(C_{s,1/2}')}^{\frac{\phi_{-}}{\phi_{+}-\phi_{-}}(1-\alpha)}}.
\int_{I_\alpha^{(k)}}\Big|\sum_{i=0}^{N-1}D\varphi(T^ix)\Big|\,dx =k\|Z(k+1)\|O(e^{(-\lambda_1(1-a)+\tau)k})(c_0+p_a)(D\varphi) &=O(e^{(-\lambda_1(1-a)+3\tau)k})(c_0+p_a)(D\varphi).
\biggl\{- 4z^4(1-z) \frac{d^2}{dz^2}+ 2[(4a+3)z^4 - (4a-4b_1-2l-d+3)z^3 - 4(b_1-2b_2)z^2 - 4(2b_2-3b_3)z -12b_3] \frac{d}{dz}+ [2a(2a+1) - A]z^3 + [- 4a^2 + 8ab_1 + 2a(2l+d-1) - 2b_1 - \epsilon]z^2 + [B_1 - 2b_1(4a-2b_1-2l-d-1) + 4b_2(4a-3)]z + B_2 - 4b_1^2 - 4b_2(4a-4b_1-2l-d-3) + 6b_3(4a-5)\biggr\}\phi(z) = 0.
e=\gcd(\frac{q^\ell-1}{q-1}, \frac{q^\ell-1}{\gcd(q^\ell-1,rn)})=\frac{(q^\ell-1) \gcd(q-1,rn)}{(q-1)\gcd(q^\ell-1,rn)}=\frac{q^\ell-1}{\gcd(q^\ell-1,(q-1)n)}.
(T_0 - Q_0)(T_0-Q_1) \dots (T_0-Q_{r-1})=0, (T_i -q )(T_i+q^{-1}) =0 ( 1 \leq i \leq n-1), & T_0 T_1 T_0 T_1=T_1 T_0 T_1 T_0, T_i T_{i+1}T_i = T_{i+1}T_i T_{i+1}( 1 \leq i \leq n-2), & T_i T_j = T_j T_i (|i-j| \geq 2).
c^{-2}v_{tt}-\Delta v- \beta \Delta v_t = 0,\quad(0,T)\times\Omega, \partial_\nu v+ \beta \partial_\nu v_t = g(\lambda),\quad(0,T)\times\partial\Omega, (v(0),v_t(0)) = (u_0,\lambda u_1) ,\quad\ \Omega,
\begin{aligned}\Delta r_i =\ln\Big(1+\frac{P_i^*+\Delta P_i}{\frac{N_0}{|h_i|^2}+\displaystyle{\sum_{j<i}}(P_j^*+\Delta P_j)}\Big)-r^* &=\ln\Bigg(1+\frac{\Delta P_i -(e^{r^*}-1)\displaystyle{\sum_{j<i}}\Delta P_j}{\frac{N_0}{|h_i|^2}+\displaystyle{\sum_{j\leq i}}P_j^* + e^{r^*}\displaystyle{\sum_{j<i}}\Delta P_j}\Bigg).\end{aligned}
\mathbf{C}^\prime_{(1,1,2,2)}=q^{-2/2}(\mathbf{T}_{(1,1,2,2)}+\mathbf{T}_{(+,-,1,1)}+\mathbf{T}_{(-,+,1,1)}+\mathbf{T}_{(1,1,+,-)}+\mathbf{T}_{(1,1,-,+)}&+\mathbf{T}_{(+,-,+,-)}+\mathbf{T}_{(-,+,-,+)}+\mathbf{T}_{(+,-,-,+)}+\mathbf{T}_{(-,+,+,-)}.
\frac{d}{dt_{2}}(\frac{\tilde{q}F}{y}) + q \dot{x}=0 & \dot{\tilde{q}}\frac{F}{y}= q\dot{x}q^{2}\dot{x}+ q \frac{d}{dt_{2}}(\frac{\tilde{q}F}{y}) - \dot{q}\frac{\tilde{q}F}{y}=0.
m_{p,q}(c)=\lim\limits_{n\rightarrow\infty}E_{p,q}(u_{n})=\lim\limits_{n\rightarrow\infty}E_{p,q}(u_{n}(\cdot+y_{n}))=\lim\limits_{n\rightarrow\infty}E_{p,q}(v_{n}^{1})+E_{p,q}(z^{1}).
a=n+\frac{3-j}{2},b= -n+\frac{i+j}{2},c=-\frac{j}{2},d=\frac{j-i+1}{2},e=1,f=\frac{1}{2}, g=\frac{3}{2}, h=\frac{\sqrt{2(n-i)+3}\sqrt{2(n-j)+3}(2n-i-j+1)}{\sqrt{2(n-i)+1}\sqrt{2(n-j)+1}(2n-j+2)}, &l=\frac{\sqrt{2(n-i)+3}\sqrt{2(n-j)+3}(2n-i-j-1)(2n-i-j+1)}{\sqrt{2(n-i)-1}\sqrt{2(n-j)-1}(2n-j)(2n-j+2)}.
(m-1) (2m - 1)(mn-n-1)^3 + m(2m - 1)(n - 1)^3 & > (m - 1)^2(mn-n-1)^3 + m^2(n-1)^2 + m(m-1)(mn-n-1)(n-1)(mn -2) & = ((m-1)(mn-n-1)+m(n-1))((m-1)(mn-n-1)^{2}+m(n-1)^{2}).
g_{20}(q_1, q_2) ={g \over 4}({q_1 q_2 \over q^2_1 + q^2_2 +q_1q_2}) [\hat{f}^2(q_1) C(g,q_1) + \hat{f}^2(q_2) C(g,q_2) + 2 \phi_0].
G_{L}\left( z \right) = \left( \begin{array}{c c}{{1}}&{{0}}\\{{\theta_{L}}}&{{1}}\\ \end{array}\right) \left( \begin{array}{c c}{{H_{L}}}&{{0}}\\{{0}}&{{g_{L}}}\\ \end{array}\right) \left( \begin{array}{c c}{{1}}&{{\tilde{\theta}_{L}}}\\{{0}}&{{1}}\\ \end{array}\right)
\begin{bmatrix}\pi(S_eS_e^*) 0 0 *\end{bmatrix}=\rho(S_eS_e^*)=\rho(S_e)\rho(S_e)^*=\begin{bmatrix}\pi(S_e)\pi(S_e)^*+X_eX_e^* * * *\end{bmatrix},
[ \bar{\Delta}_m^a, - i \hbar \frac{\delta}{\delta \eta_I}+ \phi^I ] = 0,\qquad[ \bar{\Delta}_A, - i \hbar \frac{\delta}{\delta \eta_I}+ \phi^I ] = 0,\qquad[ Y, - i \hbar \frac{\delta}{\delta \eta_I}+ \phi^I ] = 0,
\sum^{+\infty}_{k=1}\,\alpha_k \mathcal{G}_k =\, \alpha_{\mathcal{M}}\mathcal{M}+\,\tilde{\alpha}_{\tilde{\mathcal{M}}}\tilde{\mathcal{M}}+\,\alpha_A \mathcal{F}_A+\,\tilde{\alpha}_{\tilde{\mathcal{F}}}\tilde{\mathcal{F}}+ +\alpha_{\mathcal{M}\,\mathcal{F}}M\cdot \mathcal{F}_A\,+\alpha_{AB}\mathcal{F}_{AB}\,+...
v_p([z^n]F(z)) \geq\min_{n_0+n_1p+\dots=n}\sum_{i\geq0}\underbrace{v_p\left([z^{n_ip^i}]F_i(z)\right)}_{}&\geq \min_{n_0+n_1p+\dots=n}\sum_{b_i\leq 0}v_p\left([z^{n_ip^i}]F_i(z)\right) &=\min_{n_0+n_1p+\dots=n}\sum_{b_i\leq 0}\left(b_in_i-v_p\left(n_i!\right)\right)
\frac{d^{n}}{dz^{n}}\left\lbrace{}^{\Psi}\Phi_{p}^{(\alpha,\beta)}(b;c;z)\right\rbrace =\frac{(b)_{n}}{(c)_{n}}\left[{}^{\Psi}\Phi_{p}^{(\alpha,\beta)}(b+n;c+n;z)\right].
D_{m,n}(x,y) =\sigma_{m,2m;n,2n}(f;x,y)-f(x,y) & =\frac{4}{mn\pi^{2}}\int_{0}^{\pi}\int_{0}^{\pi}\phi_{x,y}(t_{1},t_{2})\frac{S(t_{1},t_{2})}{\left( 4\sin \frac{t_{1}}{2}\sin \frac{t_{2}}{2}\right)^{2}}dt_{1}dt_{2},
G_{\mathcal R,\zeta_o,1}(x,y):=\frac{i}{2\pi}\int_{\mathcal I_{\theta_c,\zeta_o}}\frac{2i\sin\zeta\mathcal S_{-}(\cos\zeta,n)}{n^2-1}{e^{ik_{+}\left(-\vert y \vert\cos(\zeta+\theta_{\hat y})+ \vert x \vert\cos(\zeta-\theta_{\hat x})\right)}}d\zeta.
\sum_{j=1}^{\infty}d_{i,j}=\mu(B(y,r_i))^{-1}\sum_{j\ge -\log_2 r_i}\mu(B(y,2^{-j+1})) &\le C_d^3\mu(B(y,r_i))^{-1}\sum_{j\ge -\log_2 r_i}\mu({B(y,2^{-j+1})\setminus B(y, 2^{-j})}) &\le C_d^3\mu(B(y,r_i))^{-1}\mu(B(y,2r_i)) &\le C_d^4,
\Delta_{\tilde{g}}\Phi = - \frac{n-4}{n-2}u^{-2\frac{n-3}{n-2}}\Delta_{\tilde{g}}u + 2\frac{(n-4)(n-3)}{(n-2)^2}u^{-\frac{3n-8}{n-2}}|\tilde{\nabla}u|^2_{\tilde{g}}.
V =u(V_1,s_1)\times\dots\times u(V_\ell,s_\ell) \quad\times (\nu^{\alpha_1}u(U_1,t_1)\times\nu^{-\alpha_1}u(U_1,t_1))\times\dots\times (\nu^{\alpha_m}u(U_m,t_m)\times\nu^{-\alpha_m}u(U_m,t_m))
\langle\Omega_{-\sqrt{(2p+1)/2}}^*(N),\prod_{i=1}^NV_{(0,0|1,1)}(z_i, \bar z_i)\Omega_0\rangle = \prod_{i<j}(z_i-z_j)^{2p+1}\exp\left(-\frac{1}{2}\sum_i|z_i|^2\right)\,,
\Gamma_m(z)=\frac i{2\pi}d\bar z_k\wedge\omega_{n-1}(z)= \frac 1{(1-|z|^2)^n}\bigl[(1-|z_m|^2) \gamma_m(z)-\sum_{j\neq m}\bar z_m z_j \gamma_j(z)\bigr]\ ,
r^{2\kappa_{x_0}+\epsilon+n+1}\sum_{j=0}^{\infty}2^{-j(2\kappa_{x_0}+\epsilon+n+1)}\leq \|u\|_{L^2(B_r^+(x_0))}^2 \left( =\sum_{j=0}^\infty \|u\|_{L^2(A_{2^{-(j+1)}r,2^{-j}r}^+)}^2\right) &\leq r^{2\kappa_{x_0}-\epsilon+n+1}\sum_{j=0}^{\infty}2^{-j(2\kappa_{x_0}-\epsilon+n+1)}.
\int (P_{t/2}\mathbf 1_E)^2 d\mu \le \left(\int_E\left(\int p_{t/2}(x,y)^2d\mu(y)\right)^{\frac{1}{2}}d\mu(x)\right)^2 & = \left(\int_E p_t(x,x)^{\frac{1}{2}}d\mu(x)\right)^2 \le\frac{A}{t^{Q/2}}\mu(E)^2.
{\cal{I}}(\rho) =\omega^\sigma\wedge\sum_{j=0}^{r=2}(-1)^jd_t^j\left(\frac{\partial}{\partial q^\sigma_j}\rfloor p_1\rho\right)=\big(A^0_\sigma-d_tA^1_\sigma+d_t^2A^2_\sigma\big)\omega^\sigma\wedge dt.
w_1(M) + (a-b)(s_1-t_{b-1}) = O(\geq 0) -\beta_{b-1}-\dots -\beta_{n-2}-\frac{1}{2}(\beta_{n-1}+\beta_n) +O(\geq 0) -\gamma_{a-1}-\dots-\gamma_{n-2}-\frac{1}{2}(\gamma_{n-1}+\gamma_n) .
\P\big(\sum_{j=1}^m \beta_j Z_j(p_j)>A \big)\geq \exp\big ( \omega A -\omega \sum_{j=1}^m \beta_j q_j - 2\omega^2 \sum_{j=1}^m \beta_j^2 \big) \P\big(\sum_{j=1}^m \beta_j Z_j(q_j)>A \big).
G^{\mathrm{g i}}( x_{1}, x_{2}; y_{1}, y_{2}) = - ( i \gamma_{1}^{\mu}\bar{\partial}_{1 \mu}+ m_{1}) ( i \gamma_{2}^{\nu}\bar{\partial}_{2 \nu}+ m_{2}) H^{\mathrm{g i}}( x_{1}, x_{2}; y_{1}, y_{2})
\Sigma_0 := S \cap \prod_{i\in \mathbf{I}}\Sigma_i= \Big\{(s_i)_{i\in \mathrm{\mathbf{I}}}\in\prod_{i\in \mathrm{\mathbf{I}}}\Sigma_i \colon \mathrm{F}(f)(s_i)=s_j f\colon i\to j \mathrm{\mathbf{I}}\Big\}.
\mu_\omega(B_\rho(\infty,R))=\mu(Z)\, \int_{H_R}^\infty y^{a-2\beta}\, dy =\frac{\mu(Z)}{2\beta-1-a}\, H_R^{1+a-2\beta}=\mu(Z)\, \frac{(\beta-1)^{\tfrac{2\beta-1-a}{\beta-1}}}{2\beta-1-a}\, R^{\tfrac{2\beta-1-a}{\beta-1}}.
\limsup_{\epsilon\to 0}\limsup_{L\to\infty}\frac 1{L^d}\sum_{i\in\mathbb T_L^d}\sum_{\eta\in\Omega_L}\mu^L_{[0,T]}(\eta)\;\!\Bigl|\hat \phi_i^{\lfloor \epsilon L\rfloor}(\delta_\eta) - \hat \phi_i(\nu_{\eta^{\lfloor \epsilon L\rfloor}(i)})\Bigr|= 0.
\frac{|S_n|}{|F_n|}\leq \frac{\binom{n^2+2n-1}{n^2-1}}{\binom{n^2+2n+1}{n^2}}=\frac{n^2(2n+1)}{(n^2+2n+1)(n^2+2n)}= O\bigg(\frac{1}{n}\bigg)\, .
\{x,y,z\}^{*}=\{z,y,x\}-\{z,x,y\}, [x,y,z]_C\;=\{z,y,x\}-\{z,x,y\}+\{x,y,z\}-\{y,x,z\}\quad\;\;\;\; =\{x,y,z\}^{*}+\{x,y,z\}-\{y,x,z\},
S = \frac{1}{1 6 \pi G_{N}}\int d^{4}x \sqrt{- g}e^{\psi}\left( - R + \frac{1}{2}\left( 3 \psi_{\phi}^{2}- 1 \right) ( \nabla \phi )^{2}\right) + S_{M}({\cal M}, g_{\mu \nu}) \; ,
Z ~ = ~ \int d x d \eta \ \exp \{- i T ( H + \frac{1}{2}\eta^{\mu}\omega_{\mu \nu}\eta^{\nu}) \}{\frac{1}{\sqrt{d e t | |{\delta^{\mu}}_{\nu}\partial_{t}- \frac{1}{2}({\Omega^{\mu}}_{\nu}+{R^{\mu}}_{\nu \rho \sigma}\eta^{\rho}\eta^{\sigma}) | |}}}
\begin{array}{l c r}{x_{0}= X_{0}[ R_{1}+ | Y_{0}|^{2}+ | Y_{1}|^{2}]^{-{\frac{1}{2}}}}{x_{1}= X_{1}[ R_{1}+ | Y_{0}|^{2}+ | Y_{1}|^{2}]^{-{\frac{1}{2}}},}\end{array}
\left( \sum\limits_{j_{1},...,j_{m}=1}^{n}\left\vert T\left(e_{j_{1}},...,e_{j_{m}}\right) \right\vert^{\rho}\right)^{\frac{1}{\rho}}\leq C_{\rho ,\mathbf{p}}^{\mathbb{K}}\Vert T\Vert
Y_{l_{1}}^{m_{1}}\star_{\vec{c}}^{n}Y_{l_{2}}^{m_{2}}=(-1)^{n+m}\sqrt{n+1}\sum_{l=0}^{n}\left[\begin{array}[c]{ccc}l_{1}l_{2}l m_{1}m_{2}-m\end{array}\right] \!\!{[j]}\ \frac{c_{l}^{n}}{c_{l_{1}}^{n}c_{l_{2}}^{n}}Y_{l}^{m}
\begin{aligned}\tilde{u}_n^{(1)}(x)-\tilde{u}_n^{(2)}(x)= -(\lambda_{n,j}^{(1)}-\lambda_{n,j}^{(2)}) +O\left( \sum_{i=1}^2e^{-\frac{\lambda_{n,j}^{(i)}}{2}}\right),\end{aligned}
2\mathrm{Re}(\bar\partial_{\Phi}^*\alpha,\alpha\llcorner(\bar\partial\eta)^{\sharp})_{D_j,\Phi}\geq -\int_{D_j}g^{-1}|\bar\partial_{\Phi}^*\alpha|^2e^{-\Phi}&+\sum_{j,k=1}^n\int_{D_j}-g(\partial_j\eta)(\bar\partial_k\eta)\alpha_{\bar j}\bar\alpha_{\bar k}e^{-\Phi},
\prod_{u = 1}^{N}d v_{u}^{l}d v_{u}^{l *}= \prod_{k = 1}^{l - 1}d \xi_{k}^{l}d \xi_{k}^{l *}\cdot 2^{N - l - 1}( \xi_{l}^{l})^{2 N - 2 l + 1}\, d \xi_{l}^{l}\, d^{2 N - 2 l + 1}\hat{\Omega}_{l}\ .
q_{1 , s 1}= \frac{q_{1 , 1}+ q_{1 , - 1}}{i \sqrt{2}}\hspace{2 e m}\mathrm{a n d}\hspace{2 e m}q_{1 , a 1}= \frac{q_{1 , 1}- q_{1 , - 1}}{\sqrt{2}}.
\vartheta^*(\psi) = \psi_1, \vartheta^*(\lambda) = \lambda=\frac{1}{10}\delta_0+\frac{1}{5}(\delta_{1,\{1\}}+\delta_{1,\{1,2\}}), \vartheta^*(\delta_0) = \delta_0, \vartheta^*(\delta_{1,1}) = \delta_{1,\{1\}}+ \delta_{0,\{1,2\}}, \vartheta^*(\delta_{2,1}) = -\psi_2 + \delta_{1,\{1,2\}}.
g(x,y+L_y) = \int_0^{y+L_y}\Big(f_{in}(x,z) - \bar{f_{in}}(x) \Big)dz = \int_{-L_y}^y f_{in}(x,z)dz - \bar{f_{in}}(x)(y+L_y) &= \int_0^y f_{in}(x,z)dz - \bar{f_{in}}(x)\, y + \int_{0}^{L_y}f_{in}(x,z)dz - \bar{f_{in}}(x)\, L_y = g(x,y)\,.
\deg\left(\frac{n^2\lambda x^{j+1}q'}{p^{m-1}}\right) = 0 &\deg\left((j+1)n\lambda x^j\frac{q}{p^{m-1}}\right) = 0\shortintertext{and}\deg\left(\frac{ntq'}{p^{m-1}}\right),\,\deg\left(t'\frac{q}{p^{m-1}}\right) < 0,
|\phi_{n}(\alpha_{a}(E_{\psi}))-\chi(E_{\psi})| &= | \phi_{n}(\alpha_{a}(E_{\psi}))-\chi(E_{f_{n}}E_{\psi})| & \leq \int_{b \in U_{n}}f_{n}(b)\Big(\int |\Delta(b^{-1}a)^{m}\psi(g_{1}b^{-1}a,\cdots,g_{m}b^{-1}a)-\psi(g_{1},g_{2},\cdots,g_{m})| dg_{1}\cdots dg_{m}\Big)db \leq \epsilon \int_{b \in U_{n}}f_{n}(b) & \leq \epsilon.
-\log \left[ (x)_{i\frac{\pi}{2}+\theta}(x)_{i\frac{\pi}{2}-\theta}\right] \, \sim \, -4\sin \pi x\, \mathrm{e}^{-\left| \theta \right|}+O\left( \mathrm{e}^{-2\left| \theta \right|}\right)\ .
\tilde H_\epsilon= \sum_i\alpha_i\int_0^{2\pi}\int_0^{2\pi}G_{\hat{g}}(z_1+\epsilon e^{i\theta_2},z_i+\epsilon e^{i\theta_2})\frac{_{d\theta_1}}{^{2\pi}}\frac{_{d\theta_2}}{^{2\pi}}.
t_1^{n_1}=1=t_2^{n_2}, t_1\theta_1=\theta_1 t_1, t_1\theta_2=\omega\, \theta_2t_1, t_1t_2=t_2t_1, & \theta_1^2=0=\theta_2^2, t_2\theta_2=\theta_2 t_2, t_2\theta_1=\omega^{-1}\theta_1 t_2, \theta_1\theta_2=-\omega\, \theta_2\theta_1.
J_{\varphi}(\mathfrak{m}_0, \rho, w):= \mathbb{E}_{\varphi}\left[ \sum_{t=0}^{T-1}f\left(t,X_{t},\mu_t,w\circ\varphi(t,X^{(t)},\mu^{(t)})\right) +F\left(X_T,\mu_T\right) \right],
\mathbb{B}'(\hat{{\mathbf U}}^{+},\hat{{\mathbf U}}^{-},\hat{\varphi})({\mathbf U}^{+},{\mathbf U}^{-},\varphi)=\left[\begin{array}{c}\partial_t\varphi+\hat{u}^+_2\partial_2\varphi-u^+_N \partial_t\varphi+\hat{u}^-_2\partial_2\varphi-u^-_N q^+-q^-\end{array}\right],
\varphi(4)=(16N)^{-\frac{1}{3}}, h(u)= e^{\frac{\eta_{-}}{\varphi(4)}(z_{0}+1)u}, T_{n}=e^{-2Nf(z_{0})}\prod_{k=1}^{n}\big(\frac{\sigma_k}{\tau}\big)^{2}.
\frac{2 C_G T}{L^d}\sum_{i\in\mathbb T_L^d}\sum_{k=1}^d \bigl(\cosh\bigl(\tfrac 12\nabla^{i,i+e_k}V(\cdot/L)\bigr)-1\bigr) \hat\chi_{i,i+e_k}^0\biggl(\frac 1T\int_0^T\mu^L_t\;\!\mathrm dt\biggr).
H_{0}^{c}= \frac{1}{2}p^{2}+ \frac{1}{2}\left( W^{\prime \prime}( x^{( i )}) \right)^{2}\left( x - x^{( i )}\right)^{2}- \frac{\hbar}{2}\left| W^{\prime \prime}( x^{( i )}) \right|
\lim_{N\to\infty}\big\|\sup_{0\leq t\leq T}|\hat\Psi_{k,N}^{(u)}(t)-\tilde\Psi_{k,N}^{(u)}(t)|\big\|_2=0 \lim_{N\to\infty}\big\|\sup_{0\leq t\leq T}|\hat\Psi_N^{(u)}(t)-\tilde\Psi_N^{(u)}(t)|\big\|_2=0
\begin{array}{lll}\lambda_3(\lambda_2-\lambda_3)b_2+2\lambda_3(\lambda_2-\lambda_3)a_2+\lambda_3b_2a_2-\lambda_3b_2a_3+\lambda_3a_2^2-\lambda_3a_2a_3\ & \ \ =(\lambda_2-\lambda_3)(1-\lambda_2).\end{array}
\begin{gathered}\overline{\partial_\mu l(\gamma)}=\frac 1\pi \int_{F_0}\frac{\phi(z)y^2}{(\bar z)^2}dxdy=\frac 1\pi \int_0^\pi \int_{0}^{l}\phi(e^t e^{i\theta}) \frac{(e^t\Im e^{i\theta})^2}{(e^t e^{-i\theta})^2}e^{2t}dt d\theta =-\frac 1\pi \frac 14\int_0^\pi \int_{0}^{l}\phi(e^t e^{i\theta}) (1-e^{2i\theta})^2 e^{2t}dt d\theta.\end{gathered}
(\Phi_1^{(1)}(\tilde{\gamma}_1;V_a+\widetilde{\mathcal{V}}(V_a,Z_{a}h)Z_{a}h),\Phi_1^{(2)}(\tilde{\gamma}_1;V_a+\widetilde{\mathcal{V}}(V_a,Z_{a}h)Z_{a}h)) \cdot \textbf{n}_{k}&=(\Phi_1^{(1)}(\gamma_1;V_a),\Phi_1^{(2)}(\gamma_1;V_a)) \cdot \textbf{n}_{k}.
\ell_{\rho}* \ell_{\omega^\beta}\ =\ \log(\ell_{\mu}* \ell_{\omega^\beta})\ =\ \log(\ell_{\mu}-1) =\ \log\big(\ell_{\mu}(1-\ell_{\mu}^{-1})\big)\ =\ \log(\ell_{\mu}) +\log(1-\ell_{\mu}^{-1}) &=\ \ell_{\rho}+ \epsilon_{\rho}, \epsilon_{\rho}\asymp \ell_{\mu}^{-1}\prec 1.
\mathcal{L}_{\gamma_{SCHCN}}(s) = \mathcal{L}_{\gamma_{coop}}(s)\cdot \mathcal{L}_{\gamma_{0}}(s) &=\frac{\frac{N!}{N_c!N_c^{N-N_c}}\lambda_{0}\lambda_{eq}^N}{(s+\lambda_{eq})^{N_c}(s+\lambda_{0}) \prod_{n=1}^{N-N_c}[s+(1+\frac{n}{N_c})\lambda_{eq}]}.
\partial_t V(t,z) = \partial_t \Phi_{\gamma}(t,x_t^*(z)) + \frac{1}{2}\xi''(t)\gamma(t) z^2 + \frac{1}{2}\nu(t), \partial^2_z V(t,z) = -\frac{1}{\partial_x^2\Phi_{\gamma}(t,x_t^*(z))}- \nu(t).
\int d^{4}x \sqrt{g_{{}_{E}\pm}}{\cal K}_{{}_{E}\pm}^{n}= \int d^{4}x \left({\frac{z_{\pm}}{z_{0}}}\right)^{( 4 - n ) \beta}{\frac{1}{z_{\pm}^{n}}}\propto \int d^{4}x \; r_{\pm}^{4 + ( 4 / 3 ) D_{2}- ( n / 3 ) D_{2}},
\lim_{x\to\infty}\dfrac{S(I_\epsilon,I_\epsilon)(x)}{\pi(x)}=\mu_{\mathrm{ST}}^{\otimes_2}\left(I_\epsilon\times I_\epsilon\right) \;\;\;\;\lim_{x\to\infty}\dfrac{S(I'_\epsilon, I'_\epsilon)(x)}{\pi(x)}=\mu_{\mathrm{ST}}^{\otimes_2}\left(I'_\epsilon\times I'_\epsilon\right).
K^x_{uv}\, k^u_\Lambda \, k^v_\Sigma \, - \,{\o{\lambda}{2}}\, \varepsilon^{xyz}\,{\cal P}_\Lambda^y \,{\cal P}_\Sigma^z\,= \,{\o{1}{2}}\,f^{\Delta}_{\phantom{\Delta}\Lambda\Sigma}\,{\cal P}_\Delta^{x}
K_{\theta}(x_1,x_2,x_3)=\frac{1}{\pi^2\theta^2}\,\exp\left[-\frac{2i}{\theta}\epsilon_{\alpha\beta}\Bigl(x_1^{\alpha}x_2^{\beta}+x_2^{\alpha}x_3^{\beta}+x_3^{\alpha}x_1^{\beta}\Bigr)\right]