harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
J_{m, n, h}:= \begin{pmatrix}A_1 -A_2 0 0 \cdots 0 A_1 0 -A_3 0 \cdots 0 A_1 0 0 -A_4 \cdots 0 \vdots \vdots \vdots \vdots \ddots \vdots A_1 0 0 0 \cdots -A_m\end{pmatrix}.
w(x,y)=\frac{a^{2}}{2\pi^{2}D}\sum_{m=1}^{\infty}\frac{E_{m}}{m^{2}cosh\alpha_{m}}sin\frac{m\pi x}{a}(\alpha_{m}tanh\alpha_{m}cosh\frac{m\pi y}{a}-\frac{m\pi y}{a}sinh\frac{m\pi y}{a})
\alpha\theta\frac{\partial u^{\kappa}_{\alpha}}{\partial \theta}= \inf_{v}\left[ b(x,v)\cdot \nabla u^{\kappa}_{\alpha}+ \theta r(x,v)u^{\kappa}_{\alpha}\right] +\frac{1}{2}\mbox{trace}(a(x)\nabla^2 u^{\kappa}_{\alpha}) \mbox{in}(\kappa,1)\times D
\widetilde{U}(1_A^{(1)}\oplus 0) =\widetilde{W}(T \oplus 1_{X^\infty})(1_{H_A}\oplus \widetilde{W}^*)(1_A^{(1)}\oplus 0)=\widetilde{W}(T \oplus 1_{X^\infty})((1_A^{(1)}\oplus 0_{H_A}) \oplus 0_{X^\infty}) &=\widetilde{W}(1_A^{(1)}\oplus 0_{X^\infty}) =\widetilde{V}1_A^{(1)}=\xi_0^{(1)}.
\sum_{j=0}^{i}q^{jr}\chi^{(p,kp+1)}_{1,k(r+1)+1-i+2j}(q)=\displaystyle\sum_{n_1 \geq \cdots \geq n_{k}\geq 0}\frac{q^{n_1^2 +\cdots + n_{k-1}^2 + 2n_k^2 +r(n_1 + \cdots + n_{k-1}+2n_k ) -(n_1 + \cdots + n_i)}}{(q)_{n_1 - n_2}\cdots (q)_{n_{k-1}-n_{k}}(q)_{2n_{k}+r}}\overline{F}^{(2n_k +r)}_{p-1;r}(q).
(x_{d-i}u_i) \sim (x_{d-i}\tilde{u}_i) \exists (w_i)_i \in \hat{A}_{\sigma}^*, \; \frac{u_i}{\tilde{u}_i}= \frac{w_i^q}{w_{i+1}}& \exists w_1 \in \hat{A}_{\sigma}^*, \; w_1^NV((u_i)_i)= V((\tilde{u}_i)_i) \tilde{V}((u_i)_i) = \tilde{V}((\tilde{u}_i)_i)\pmod{({A}_{\sigma}^*)^N}
{}_{N}\! \langle \varrho_{n}^{\prime}, \bar{\varrho}_{n}^{\prime}| \varrho_{n}, \bar{\varrho}_{n}\rangle_{N}= \prod_{\mu = 0}^{p}\delta ( \mathrm{R e}\varrho_{n \mu}^{\prime}- \mathrm{R e}\varrho_{n \mu}) \delta ( \mathrm{I m}\varrho_{n \mu}^{\prime}- \mathrm{I m}\varrho_{n \mu}) .
\frac{\partial\varphi}{\partial x}=\frac{\partial}{\partial x}\int_{a,b}^{x,y}v\cdot dr=\frac{\partial}{\partial x}\int_{a,b}^{x_{1},y}v\cdot dr+\frac{\partial}{\partial x}\int_{x_{1},y}^{x,y}v\cdot dr=0+\frac{\partial}{\partial x}\int_{x_{1},y}^{x,y}v\cdot dr
N_{\sigma}(D) = \sum_{\Gamma\in S}\sigma_{\Gamma}(D) \cdot \Gamma = \sum_{\Gamma \in S}\lim_{\epsilon \to 0^+}\sigma_{\Gamma}(D+ \epsilon A) \cdot \Gamma = \lim_{\epsilon \to 0^+}( \sum_{\Gamma\in S}\sigma_{\Gamma}(D+ \epsilon A) \cdot \Gamma \ ) .
\begin{aligned}\|u\|_{H^2}+\|P\|_{H^1}\le C \|F\|_{L^2}(1+\|\nabla \rho\|_{L^q})^{\frac{q}{q-n}}& \le C \|-\rho u_t -\rho u \cdot \nabla u-{\rm div}(\nabla d \odot \nabla d)\|_{L^2}& \le C(\|\sqrt{\rho}\dot{u}\|_{L^2}+\left\| |\nabla d| |\nabla^2 d|\right\|_{L^2}).\end{aligned}
\bigl [ \cos \bigl ( \sum_{(i,j)\in C}\theta_{ij}\bigr ) - 1 \bigr] \prod_{(i,j)\in C}x_{ij}= 0 & c_{ij}= \sqrt{c_{ii}c_{jj}}\cos \theta_{ij}x_{ij}\qquad\qquad(i,j) \in C & s_{ij}= \sqrt{c_{ii}c_{jj}}\sin \theta_{ij}x_{ij}\qquad\qquad(i,j) \in C, & \eqref{dummy2}-\eqref{dummy5}.
d{\cal L}_{2}\vert_{\Psi_{-}^{I}= 0}= - 2 i E_{\dot{q}}^{-}\wedge E_{\dot{q}}^{-}\wedge E^{+ +}+{\frac{1}{2}}E^{i}\wedge ( E^{\mp \mp}\wedge \Omega^{\pm \pm i}- 4 i E_{q}^{+}\wedge E_{\dot{q}}^{-}\gamma_{q \dot{q}}^{i})
\left[ e^{i \gamma \varphi}\right]_{n}( \theta_{1 ,}\theta_{2 ,}\dots , \theta_{n}) = \left[ e^{i \gamma \varphi}\right]_{1}( \theta_{1}) \left[ e^{i \gamma \varphi}\right]_{n - 1}( \theta_{2 ,}\dots , \theta_{n}) + O ( e^{- \theta_{1}}) .
\bar{P}^{b}(x_{1}, x_{2}, x_{3}) = \left ( \begin{array}{ccc}0 \displaystyle\frac{b_{1}b_{2}}{b_{3}}x_{3}b_{1}x_{2}begin{align*}-0.2cm]-\displaystyle\frac{b_{1}b_{2}}{b_{3}}x_{3}0 0 begin{align*}-0.2cm]-b_{1}x_{2}0 0\end{array}\right ),
I_4 \lesssim \int_0^t e^{-c\lambda2^{j}\int_{0}^\tau \dot\theta(t'')d{t''}}2^j\dot\theta(\tau)d\tau\|{\Delta}_{k,j}e^{\alpha D_n}v_0^h\|_{L^p_h(L^2_v)}&\lesssim \frac{1}{\lambda}c_{k,j}2^{-k(\frac{n-1}{p}+1-s)}2^{-\frac{1}{2}j}\|e^{\alpha D_n}v_0^h\|_{\dot{B}_{p,1}^{\frac{n-1}{p}+1-s,\frac{1}{2}}}.
\epsilon^{+}( z ) = \frac{z}{2 \pi}+ \frac{i}{\pi}\sum_{n > 0}\frac{e^{- i n ( z - i \varepsilon )}}{n}, ~ \epsilon^{-}( z ) = \frac{z}{2 \pi}+ \frac{i}{\pi}\sum_{n < 0}\frac{e^{- i n ( z + i \varepsilon )}}{n}.
|\partial_{t}^{2}\left(\lambda(t) \langle \chi_{\geq 1}(\frac{2r}{\log^{N}(t)})F_{0,2}\vert_{r=R\lambda(t)},\phi_{0}\rangle\right)| \leq \frac{C}{t^{4}\log^{3b+1+2N-2b\alpha}(t)}+ \frac{C |e''''(t)|}{\log^{2b+2N-2b\alpha}(t)}
\leq \;\omega^{\sum_{a=1}^N 2 n_a}\; \left\langle \prod_{b=1}^N\left[\chi^{(b)}_+ [ Q^{M,S}] \right]^{q_b}\left[\chi^{(b)}_- [ Q^{M,S}] \right]^{2n_b-q_b}\right\rangle_{Q^{M,S}}\; .
\begin{aligned}|\tilde{\gamma}_{s}-\frac{1}{\|\Lambda Q\|_{2}^{2}}(\epsilon_{1},L_{+}\Lambda^{2}Q)|+|\frac{x_{s}}{\lambda}|\leq \delta(\alpha)(\int |\nabla \epsilon|^{2}e^{-2(1-\eta)\frac{\theta(b|y|)}{|b|}}+\int|\epsilon|^{2}e^{-|y|})^{\frac{1}{2}}&+C\int |\nabla\epsilon|^{2}+\Gamma_{b}^{1-C\eta}+C\lambda^{2}|E_{0}|.\end{aligned}
u_{0}=\begin{pmatrix}0 1 -{1}0\end{pmatrix},\,\,u_{1}=\begin{pmatrix}0{\bf i}{\bf i}0\end{pmatrix},\,\,u_{2}=\begin{pmatrix}0{\bf j}{\bf j}0\end{pmatrix},\,\,u_{3}=\begin{pmatrix}0{\bf k}{\bf k}0\end{pmatrix}.
(\varphi_L \circ \psi_{Col})([\pi_L]f) = \sum_{a \in LT_1}[\pi_L](a +_{LT}Z) f(a +_{LT}Z) = [\pi_L](Z) \sum_{a \in LT_1}f(a +_{LT}Z) = \varphi_L(Z) (\varphi_L \circ \psi_{Col})(f) = \varphi_L(Z \psi_{Col}(f)) \ .
\bigcup_{\lambda \in \Lambda^m}R_\lambda = \bigcup_{\lambda = \lambda_1 \lambda_2 \in \Lambda^m}\tau_{\lambda_1}(R_{\lambda_2}) = \bigcup_{d(\lambda_1) = \ell}\tau_{\lambda_1}\left( \cup_{\lambda_2 \in s(\lambda_1) \Lambda^{e_j}}R_{\lambda_2}\right)
f_Y(y)=\frac{\delta^{r\sigma^{-1}}}{\chi(\sigma^{-1})}\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\chi(\sigma^{-1}rs-\sigma^{-1}r+\sigma^{-1}) \left(y\delta^{r\sigma^{-1}}\right)^{-s}\,ds.
\sum_{j=1}^{n-1}\tfrac{(C\sqrt{M})^j}{\Gamma(\frac{j}{2})}\binom{n-1}{j}\leq (2C)^{n-1}\sqrt{M}\sum_{j=0}^{n-1}\tfrac{\sqrt{M}^j}{\Gamma(\frac{j+1}{2})}\leq 3(2C)^{n-1}\sqrt{M}^3e^M.
S_{10}=\left(E_{10},M, E_{10}\right), M =\begin{pmatrix}\lambda\pm\sqrt{3}0 0 0 \lambda 0 0 0 0 \end{pmatrix}S_{10}=\left(-E_{10},M, E_{10}\right), M =\begin{pmatrix}\lambda_1 \mp\frac{\sqrt 3}{2}0 \pm\frac{\sqrt 3}{2}\lambda_1 0 0 0 0 \end{pmatrix}
\| u - \Gamma_n u \| \leq \| u - \Phi_{r}\| + \| \Phi_{r}- \Gamma_n u \| &= \| u - \Phi_{r}\| + \| \Gamma_n \Phi_{r}- \Gamma_n u \| &\leq \| u - \Phi_{r}\| + \| \Phi_{r}- u \|.
a^{\dagger}( k , \rho ) \vert 1 \colon p_{i}\rangle = -{\frac{e}{( 2 \pi )^{3 / 2}}}{\frac{1}{\sqrt{2 k^{+}}}}f^{*}( k , \rho : p_{i}) \vert 1 \colon p_{i}\rangle + \vert 2 \colon p_{i}, k_{i}\rangle \; .
d v_{\underline{{\mu}}}^{\underline{{\alpha}}}= 1 / 4 v_{\underline{{\mu}}}^{\underline{{\beta}}}( \Gamma_{\underline{{a}}\underline{{b}}})_{\underline{{\beta}}}^{\underline{{\alpha}}}\Omega^{\underline{{a}}\underline{{b}}}( d ) \qquad
-\rho(s)^{-1}T_{s_1}-\rho(s)^{-1}\rho(s')^{-1}T_{s_2'}= -\rho(s)^{-1}\left[T_{b_{2}}-\rho\left(s\right)T_{b_{1}}\right]-\rho\left(s\right)^{-1}\rho\left(s'\right)^{-1}\left[T_{b_{3}}-\rho\left(s'\right)T_{b_{2}}\right] = T_{b_{1}}- \rho\left(s\right)^{-1}\rho\left(s'\right)^{-1}T_{b_{3}}
- \kappa_q \, e_{\delta, \gamma}(u) = \log(1 - \kappa_q \, e'_{\delta, \, \gamma}(u)), \kappa_q \, f_{\delta, \gamma}(u^{-1}) = \log(1 + \kappa_q \, f'_{\delta, \, \gamma}(u^{-1})),
\underset{p\rightarrow +\infty}{\lim}\sqrt{1-e^{-h_p(x)}}K(x,y) \sqrt{1-e^{-h_p(y)}}e_n(x)e_n(y) &=\sqrt{1-e^{-h(x)}}K(x,y) \sqrt{1-e^{-h(y)}}e_n(x)e_n(y)
\sum\limits_{j,l=1}^N |G_{j,l}|^{4}- \frac{K^2(K^2 + N -2K)}{N^2(N-1)}&=\tiny{\sum\limits_{\begin{array}{cc}j, l =1 j \neq l\end{array}}^N}\left(|G_{j,l}|^2 - C_{N,K}^2\right)^2+\sum\limits_{j=1}^N\left(G_{j,j}^2 - \frac{K^2}{N^2}\right)^2+\frac{2K(K-1)}{N(N-1)}\sum\limits_{j=1}^N\left(G_{j,j}- \frac{K}{N}\right)^2 \, .
\frac{d}{dt}\boldsymbol{P}_{\boldsymbol{\theta}}^t(\emptyset,\phi)=\frac{d}{dt}\:[\exp(t\boldsymbol{A}_{\boldsymbol{\theta}})\phi](\emptyset)=[\boldsymbol{A}_{\boldsymbol{\theta}}\exp(t\boldsymbol{A}_{\boldsymbol{\theta}})\phi](\emptyset).
{\cal R}_{0}^{*}=\left\{\mathbf{y}:~e^{n\alpha}\cdot\sum_{m=1}^{M}W(\mathbf{y}|\mathbf{x}_{m})+\max_{m}W(\mathbf{y}|\mathbf{x}_{m})\leq e^{n\beta}\cdot\sum_{m=1}^{M}V(\mathbf{y}|\mathbf{x}_{m})\right\}.
D \epsilon_{j}= \left( d + \frac{1}{2}[ e^{a}{\bf \Gamma}_{a}+ \frac{1}{2}\omega^{a b}{\bf \Gamma}_{a b}+ \frac{1}{r !}b^{[ r ]}{\bf \Gamma}_{[ r ]}] \right) \epsilon_{j}- a_{j}^{i}\epsilon_{i}.
\left[\begin{array}{ccc}0 r_2\bmod r_1 r_3 \bmod r_1 r_1 0 r_3 \bmod r_2 r_1 r_2 0\end{array}\right]\left[\begin{array}{c}x_1 x_2 x_3\end{array}\right]=\left[\begin{array}{c}r_1-1 r_2-1 r_3-1\end{array}\right] \, .
\frac{1}{\sqrt{n}}\sum\limits_{i=1}^{n}M_{i}{\mathop{\to}\limits^{{\rm{\mathcal D}}}}N\sim{\rm{\mathcal N}}\left(0,\left(I-\textbf{E}\left(\rho_{0}\right)\right)^{-1}\left(X_{1}-\textbf{E}\left(\rho_{0}\right)X_{0}\right)\right).
\displaystyle\frac{1}{k}({\boldsymbol \sigma}^l_h,\bar{\boldsymbol\sigma}_h)+ ( B_h{\boldsymbol \sigma}^l_h,\bar{\boldsymbol \sigma}_h) - 2(u^{l-1}_h\nabla u^l_h,\bar{\boldsymbol \sigma}_h)=0,\ \forall \bar{\boldsymbol\sigma}_h\in{\boldsymbol \Sigma}_h.
\mathcal{P}_s(\theta_s, \tau) = \sum_{k \in \mathcal{K}}\mathcal{A}_k \exp \bigg( -2 \pi \lambda_E \int_{0}^{\infty}\exp(-N_0 \gamma_{k}(r)) &\times \mathcal{L}_{I'_B}(\gamma_{k}(r)) \mathcal{L}_{I'_U}(\gamma_{k}(r)) \mathcal{L}_{I'_J}(\gamma_{k}(r)) r dr\bigg),
x = \left( \begin{array}{c c}{{u_{1}^{\ast}v_{1}+ v_{2}^{\ast}u_{2}}}&{{v_{1}u_{2}^{\ast}- v_{2}^{\ast}u_{1}}}\\{{v_{2}u_{1}^{\ast}- v_{1}^{\ast}u_{2}}}&{{v_{2}u_{2}^{\ast}+ v_{1}^{\ast}u_{1}}}\\ \end{array}\right) ( \mid u_{1}\mid^{2}+ \mid u_{2}\mid^{2})^{- 1}
n\gamma^{n-1}\cdot \lambda^{-[\frac{\log n\gamma^{n-k-1}}{\log \lambda}]}= \lambda^{\frac{\log n\gamma^{n-1}}{\log \lambda}-[\frac{\log n\gamma^{n-k-1}}{\log \lambda}]}= \lambda^{\frac{\log \gamma^{k}}{\log \lambda}+ \frac{\log n\gamma^{n-k-1}}{\log \lambda}-[\frac{\log n\gamma^{n-k-1}}{\log \lambda}]}
\begin{array}{cc}\varphi\left(\mathrm{Cap}_t(x_1,\dots,x_{t-1},x_t^g,y_1,\dots,y_{t+1})\right)= &\mathrm{Cap}_t(v_1,\dots,v_{t-1},v_t,w_1,\dots,w_{t+1})\otimes b=0. \end{array}
\begin{aligned}S_{1}(t)x =\int_{0}^{t}T(t-s,A)\tilde{B}T(s,A)x\,\textnormal{d}s S_{n}(t)x =\int_{0}^{t}T(t-s,A)\tilde{B}S_{n-1}(s)x\,\textnormal{d}s n> 1.\end{aligned}
{\rm Res}_{s=1}\left\{\frac{1}{2}\hat\tau(\pi(e^{-1})[D,\pi(e)](1+D^2)^{-s/2})\right\}=\lim_{s\to 1}\frac{1}{2}(s-1)\hat\tau(\pi(e^{-1})[D,\pi(e)](1+D^2)^{-s/2}) &=\frac{1}{2\pi i}\tau(e^{-1}\partial(e)).
M_{n,\mathbf p_k}= \frac1n \sum_{i\ne j}^n \prod_{\ell=1}^k I_{i,j}^{(p_\ell)}, N_{n,\mathbf p_k}= \frac1n \sum_{i=1}^n \prod_{\ell=1}^k I_{i,i}^{(p_\ell)}.
a = \frac{1}{2\sqrt{3}}\ , \delta= \frac{2\pi}{\sqrt{3}}\ , c_0= \frac{13}{4}\ , c_1= 4\ , c_2= 1\ , c_3 = \frac{1}{2}\ , c_4= \frac{1}{4}\ , c_5= \frac{1}{4}\ , c_6= 1
K_2 = K_1^{-1}\mathcal C_0^{-1}s^\beta 2^{1+\frac{1}{\mu_2}}c_2 c_3^{\frac{\mu_1+1}{\mu_2}}(\max_{i=1,2,3}\{\tilde d_i\})^{\frac{\mu_2+1}{\mu_2}}, \ K_3 = \left(1+\max\{1,\frac{\gamma\mu_2-\beta}{\alpha(\mu_2+1)}\}\right)\frac{\mu_2+1}{\mu_2}.
- \int_{0}^{t}\int_{\mathbb{T}}\rho_{n}W_{n}\underbrace{\left( \partial_{t}W_{n}+ u_{n}\partial_{x}W_{n}\right)}_{=0}~dxds + \int_{\mathbb{T}}\rho_{n}W_{n}^{2}(x,t)-\rho_{n}W_{n}^{2}(x,0)~dx = 0,
\chi(\epsilon + 4\pi i x)\frac{\Gamma(\frac{k}{2}- 2\pi i x)}{\Gamma(\frac{k}{2}+ 2\pi i x)}\ = \ \frac{\varphi(M)}{sM}\delta\left(\frac{m}{(m,M^\infty)},1\right)\left(\frac{k}{2}\right)^{-4\pi i x}+ O\left( \log (m \log M) \right).
0 \,\leq\, \frac{1}{k^2}|E(G)| - \frac{19}{12}(\gamma-1) \,\leq\, \frac{1}{6}\left(\gamma^2+4\gamma-2\right) - \frac{19}{12}(\gamma-1) \,=\, \frac{1}{12}(2\gamma^2 - 11\gamma + 15).
\frac{\partial^{2}\psi}{\partial x^{2}}+\frac{\partial^{2}\psi}{\partial y^{2}}=\frac{\partial^{2}\psi}{\partial r^{2}}+\frac{1}{r}\frac{\partial\psi}{\partial r}+\frac{1}{r^{2}}\frac{\partial^{2}\psi}{\partial\phi^{2}}
Q^k_{\ell}(z) = (-1)^k(1-z^2)^{k/2}\frac{d^k}{dz^k}Q_{\ell}(z) Q_{\ell}(z) = \frac{1}{2}P_{\ell}(z) \ln\left(\frac{1+z}{1-z}\right) - \sum_{j=1}^{\ell}\frac{P_{j-1}(z)P_{\ell-j}(z)}{j}P_{\ell}(z) = \frac{1}{2^{\ell}\ell !}\frac{d^\ell}{dz^\ell}\left[\left(z^2-1\right)^\ell\right],
\frac{1}{\lambda(t)}\|\ddot x(t)+\gamma(t)\dot x(t)\|^2 = \ \frac{1}{\lambda(t)}\|\ddot x(t)\|^2+\frac{\gamma^2(t)}{\lambda(t)}\|\dot x(t)\|^2+ &\frac{d}{dt}\left(\frac{\gamma(t)}{\lambda(t)}\|\dot x(t)\|^2\right)-\frac{\dot \gamma(t)\lambda(t)-\gamma(t)\dot\lambda(t)}{\lambda^2(t)}\|\dot x(t)\|^2.
E(\rho, \phi)= \int_{\Omega}\bigg( \sum_{i=1}^s \rho_i\log \rho_i+\frac{1}{2}\epsilon(x)|\nabla \phi|^2 \bigg )d x-\int_{\Gamma_D}\epsilon(x) \phi_D^b \partial_n\phi ds + \frac{1}{2\beta_R}\int_{\Gamma_R}|\phi|^2 ds.
\Theta_\nu(A,h_1,N;\tau) =\Theta_\nu(A,h_2,N;\tau), \Theta_\nu(A,-h,N;\tau)=(-1)^\nu\Theta_\nu(A,h,N;\tau), \Theta_\nu(A,N-h,2N;\tau) =(-1)^\nu \Theta_\nu(A,N+h,2N;\tau).
1 = \sum_{v_iv_j \in E}\alpha_{ij}(x_i^2x_j+x_ix_j^2+1) + \sum_{\substack{v_iv_j \in E v_k \notin \{v_i,v_j\}, \k \in V}}\beta_{ijk}(x_i^2x_k+x_ix_jx_k+x_j^2x_k).
\phi(x^{0}\,,\,x^{1}) = \frac{1}{2R}\sum_{n \geq 0}\left\{\left[ a_{n}(x^{0})+ib_{n}(x^{0}) \right]e^{i \omega_{n+\frac{1}{2}}x^{1}}+\left[ a_{n}(x^{0})-ib_{n}(x^{0})\right] e^{-i \omega_{n+\frac{1}{2}}x^{1}}\right\},
R_{d}=\mathrm{C}\left(\frac{p_{d}(1-2^{-C_{d}})\tilde{h}_{0}^{2}}{1+2p_{d}(1-2^{-C_{d}})\sum_{k>0}\tilde{h}_{k}^{2}+(2\beta_{ud}^{2}+\gamma_{ud}^{2})p_{u}+p_{d}2^{-C_{d}}(1+2\alpha^{2})}\right).
\frac{d \psi_{n}^{\lambda}(u)}{du}=-\frac{(n+\lambda)}{1-u^2}u\psi_{n}^{\lambda}(u) + \frac{n}{1-u^2}\frac{N_{n}^{\lambda}}{N_{n-1}^{\lambda}}\psi_{n-1}^{\lambda}(u)
h=\lambda^{1/2}|\omega|^{-1/2},b=-h \cdot \eta(\tilde{y}-\tilde{y}'), \alpha=\frac{(\xi_x-\xi'_x)\lambda^{-1}-\beta \lambda^{-1}\tilde{\lambda}^{-1}\eta (\tilde{y}-\tilde{y}')}{bh^{-1}}
I^qf(x, \xi) = \int_{0}^{l(\gamma_{x,\xi})}t^q\langle f(\gamma_{x,\xi}(t)), \dot{\gamma}_{x,\xi}^m(t) \rangle dt = \int_{0}^{l(\gamma_{x,\xi})}t^q f_{i_1\dots i_m}(\gamma_{x,\xi}(t))\dot{\gamma}_{x,\xi}^{i_1}(t)\cdots \dot{\gamma}_{x,\xi}^{i_m}(t) dt.
E_s = \ \frac1\pi \int_{-\infty}^\infty (1+\xi^2)^{s}(-\real \ln T(\xi/2)) d\xi = \ \frac{2\sin(\pi s)}\pi \int_1^\infty(\tau^2-1)^s \Big[ \real \ln T(i\tau/2) + \sum_{j=0}^N (-1)^j H_{2j}\tau^{-2j-1}\Big] d\tau + \sum_{j=0}^N \binom{s}{j}H_{2j}
P_{d}\left\{\mathbb{G}_{N}'f-\mathbb{G}_{N}'g>x\right\}=P_{d}\left\{e^{\lambda \sum_{i=1}^{N}Z_{i,N}(f,g)}>e^{\lambda x}\right\}&\leq e^{-\lambda x}E_{d}e^{\lambda \sum_{i=1}^{N}Z_{i,N}(f,g)}
\epsilon^i_1 = e^{{1\over 2}A}\tilde \eta^i_+\ , \qquad \epsilon_2^i = \left(e^{-{1\over 2}A}-2\kappa^{D-2}W(\phi_0)e^{{1\over 2}A}x^\mu\Gamma_{\underline\mu}\right)\tilde \eta^i_- \ .
\begin{aligned}g_3 g_2 \neq g_2 g_3 \mbox{and}g_3 g^{2}_1 \neq g^{2}_1 g_3 \,, \varphi (g_3 g_2) = \varphi (g_3) \varphi (g_2) \mbox{and}\varphi (g_3 g^{2}_1) = \varphi (g^{2}_1) \varphi (g_3) \,. \end{aligned}
\langle x^*,x(t) \rangle=\lim_{k\to \infty}\langle x^*,x_{n(k)}(t)\rangle =\lim_{k\to \infty}\int_0^t\langle x^*,x_{n(k)}'(s) \rangle ds+\langle x^*,\xi_0 \rangle & =\int_0^t\langle x^*,u(s) \rangle ds+\langle x^*,\xi_0 \rangle & =\left\langle x^*,\int_0^tu(s)ds+\xi_0 \right\rangle
h_k^{(H)}=\pi^{1/2}2^{-k}k!,h_k^{(L)}=\Gamma(k+1)\Gamma(a+k+1), &h_k^{(J)}=2^{a+b+1+2k}\frac{\Gamma(k+1)\Gamma(a+b+1+k)\Gamma(a+1+k)\Gamma(b+1+k)}{\Gamma(a+b+2k+1)\Gamma(a+b+2k+2)}, &h_k^{(C)}=\pi 2^{2k+2a+2}\frac{\Gamma(k+1)\Gamma(-2k-2a)\Gamma(-2a-2k-1)}{\Gamma(-2a-k)(\Gamma(-a-k))^2}.
h\alpha_n = \left( \begin{array}{cc}ap^n bp^{-n}cp^n dp^{-n}\end{array}\right) = \left( \begin{array}{cc}c^{-1}p^{-n}\ast 0 cp^{n}\end{array}\right) \left( \begin{array}{cc}0 -1 1 dc^{-1}p^{-2n}\end{array}\right) =: g_1g_2,
P_+=\int_{-L}^Ldx^-{\cal H}(x)=\sum_n p_{n+}\left(b_n^{a\dagger}b_n^a-d_n^a d_n^{a\dagger}\right)+ 2L \left(-{Nm_0 \over g_0}\sigma+{N \over 2g_0}\sigma^2\right).
\dfrac{p_2}{p_1}=\dfrac{(\gamma+1-2\tilde{b}\beta_i){\beta_r}-(\gamma-1)}{(\gamma+1)-{\beta_r}(\gamma-1+2\tilde{b}\beta_i)},~~~M^2_1=\dfrac{2{\beta_r}(1+\tan^2{\phi_r})(1-\tilde{b}\beta_i)}{(\gamma+1)-{\beta_r}(\gamma-1+2\tilde{b}\beta_i)},
\hat{a}^{\dagger}_a A^{ab}_{\;\;\;cb}\hat{a}^c + \hat{a}^{\dagger}_a A^{ab}_{\;\;\;bd}\hat{a}^d + \hat{a}^{\dagger}_b A^{ab}_{\;\;\;ca}\hat{a}^c + \hat{a}^{\dagger}_a A^{ab}_{\;\;\;ad}\hat{a}^d
\int_{\mathbb{G}}|\mathcal{R}_{1}^{\frac{a_{1}}{\nu_{1}}}\phi(x)|^{p}dx+\int_{\mathbb{G}}|\mathcal{R}_{2}^{\frac{a_{2}}{\nu_{2}}}\phi(x)|^{p}dx=\int_{\mathbb{G}}|\phi(x)|^{q}dx.
\beta_{-1}\cdot \alpha_0 = v_{(1,2)}\cdot (a_1 \cdot v_{(2,3)}) -\frac{3t}{2^3}a_1 + \frac{t}{2^2}(a_2 + a_{-2}) + \frac{3}{2^4}(a_1 + a_{-1}) \cdot v_{(2,3)}- \frac{1}{2^3}(a_2 + a_{-2}) \cdot v_{(1,3)}+ \frac{1}{2^2}(a_3 + a_{-3}) \cdot v_{(1,2)}.
2 g^{2}\Gamma ( a p ) [ \int_{- \pi}^{\pi}\frac{d^{4}k}{( 2 \pi )^{4}}\frac{1}{H^{2}( k + a p ) + ( a m \Gamma )^{2}( k + a p )}\frac{1}{H^{2}( k ) + ( a m \Gamma )^{2}( k )}] \Gamma ( a p ) .
\cos \gamma = \frac{|p_{0}-z|}{r}\le \frac{2r(1-d_{0}) +rd_{0}-d_{0}+3d_{0}^{2}/4}{2r(1-d_{0})}= 1 - \frac{d_{0}(1-r) - 3d_{0}^{2}/4}{2r(1-d_{0})}< 1 - d_{0}/4\,,
\begin{aligned}\mathcal{I}_{42}(t)+\mathcal{I}_{412}(t) = KR \int_{L^-_{\frac{\pi}{3}}(t)}\varrho(\theta, \omega, t)^2 \cos(\theta-\phi) ~ d\theta \leq-\frac{KR \sqrt{3}}{2}\Gamma^-_{\frac{\pi}{3},\omega}(t).\end{aligned}
T_{\pm}^* \kappa_0(\delta)(i\sigma)\kappa_0(\delta)T_{\pm}=e^{\pm\delta(t)e^{2i\alpha}}\Big[\pm \sin(\alpha)c(t)-s(t)\cos(\alpha)+i\Big(\cos(\alpha)c(t)\mp\sin(\alpha)s(t)\Big)\widetilde{I}\Big]~,
\inf_{\varepsilon>0}\lim_{\ell\to0}\frac{\int_{\Omega_\ell}(A\nabla \tilde v_\ell^\varepsilon). \nabla \tilde v_\ell^\varepsilon}{\int_{\Omega_\ell}|\tilde v_\ell^\varepsilon|^2}=\int_{\omega}(A_{22}\nabla W_1).\nabla W_1-\frac{|A_{12}\nabla W_1|^2}{a_{11}}\,.
\frac{\varphi(b)}{\varphi(b+1)}= p^{s[(n'+b+1)(b+1)-(n'+b)b]}\cdot\frac{[p,n'+b+1][p,b+1][p,m-b-1]}{[p,n'+b][p,b][p,m-b]}= \p^{s(n'+2b+1)}\cdot\frac{\left( 1-p^{-(n'+b+1)}\right)\left( 1-p^{-(b+1)}\right)}{1-p^{-(m-b)}}> p^{s(n'+2b+1)}\left( 1-p^{-1}\right)^2\,,
\phi^a(w)~\phi^b(z)={[I]\over(w-z)^{2\Delta_\phi}}~+~A\left\{{f^{abc}\over(w-z)^{\Delta_\phi}}\phi^c(z)~+~{1\over(w-z)^{\Delta_\phi-1}}J^a_{-1}\phi^b(z){}~+~...\right\}~+~...
d\delta\left(X^{(N)};t\right)=\frac{1}{N^2}\left[N^2dt+2\sqrt{N^2\delta\left(X^{(N)};t\right)}d\tilde{\beta}^N(t)\right]=dt+\frac{1}{N}2\sqrt{\delta\left(X^{(N)};t\right)}d\tilde{\beta}^N(t),
\frac{\partial}{\partial t_{2,1,b}}\Big(\frac{\partial H_k}{\partial t_{1,1,a}}(H_k)^{-1}\Big)+E_{a,a}H_{k+1}E_{b,b}(H_{k})^{-1}-H_kE_{b,b}(H_{k-1})^{-1}E_{a,a}=0.
\liminf_{k}{\Phi}_{\epsilon}^{k}(s)^{1/k}\geq\liminf_{k}k^{-1/k}{\Psi}_{\omega}^{d_{k}}(s,\epsilon)^{1/k}\geq\liminf_{k}k^{-1/k}{\Psi}_{\omega}^{d_{k}}(s,\epsilon)^{1/d_{k}}=1
S_3 = \sum_{d_1, \ldots, d_k, d_1', \ldots, d_k'}\frac{\lambda_{d_1, \ldots, d_k}\lambda_{d_1', \ldots, d_k'}}{\prod_{j = 1}^k |[d_j, d_j']|}S_4 = \sum_{d_2, \ldots, d_k, d_2', \ldots, d_k'}\frac{\lambda_{1, d_2, \ldots, d_k}\lambda_{1, d_2', \ldots, d_k'}}{\prod_{j = 2}^k \varphi([d_j, d_j'])}
\gamma(f;g_1,...,g_k)\triangleright n =(x_1\triangleleft g_1,...,x_k\triangleleft g_k)\triangleright n =((x_1\triangleleft g_1)\triangleright n,...,(x_k\triangleleft g_k)\triangleright n) =(x_1\triangleleft (g_1\triangleright n),...,x_k\triangleleft (g_k\triangleright n)) = \gamma(f;g_1\triangleright n,...,g_k\triangleright n).
\frac{(\Delta t)^{2}(s_{1}- s)^{2}}{2 \theta}\leq \frac{(t + \Delta t - s \Delta t)^{2}}{2 \theta}= U(k,s) - U^{\theta}(k,t + \Delta t) &\leq U(k,s) - U(k,s_{1}) + \frac{(\Delta t)^{2}}{2 \theta}&\leq L_{2}\tilde{L}_{c}\Delta t|s - s_{1}| + \frac{(\Delta t)^{2}}{2 \theta}.
\alpha = \sqrt{\frac{N_{k}}{\hbar \tau_{2}}}\left[ - i \tau \hat{A}_{1}+ i \hat{A}_{2}\right] \ \ , \ \ \alpha^{\dagger}= \sqrt{\frac{N_{k}}{\hbar \tau_{2}}}\left[ i \bar{\tau}\hat{A}_{1}- i \hat{A}_{2}\right] \ \ .
g(A_{ntX}X,Z) =g(\widetilde\nabla_{Z}X,\varphi tX-t^{2}X)=-g(\varphi\widetilde\nabla_{Z}X,tX)-g(\widetilde\nabla_{Z}X, t^{2}X) =-g(\widetilde\nabla_{Z}\varphi X-(\widetilde\nabla_{Z}\varphi)X, tX)-\lambda g(\widetilde\nabla_{Z}X, X).
\delta \eta_{\alpha_{2 k + 1}}^{*}= Z_{\alpha_{2 k + 1}}^{\alpha_{2 k}}\Phi_{\alpha_{2 k}}^{*}+ A_{\alpha_{2 k + 1}}^{\alpha_{2 k + 2}}\Phi_{\alpha_{2 k + 2}}^{*}, k = 0 , \cdots , b ,
d(s_0 a_{12}) = [a_{12},\a_{12},\s_0 a_{(12)},\s_0 a_1, \s_0 a_2,\a_{21}] d(s_1 a_{12}) = [s_0 a_2,\a_{12},\a_{12},\s_1 a_1,\a_{12},\s_1 a_{(12)}] d(s_2 a_{12}) = [s_1 a_{2},\s_1 a_{(12)},\a_{12}, \a_{12}, \s_1 a_1, a_{12}]
\begin{aligned}1\,030\,117 , 3\,259\,597 , 3\,928\,632 , 4\,593\,673 , 5\,327\,080, 5\,909\,813 , 7\,102\,277 , 7\,738\,629 , 7\,758\,589 , 9\,583\,736, \end{aligned}
\begin{aligned}\frac{d}{dt}\int |\nabla^{N+1}n|^2 dx+\!\int |\nabla^{N+2}n|^2 dx &\lesssim \sum_{k=1}^{N-1}\|\nabla^k u\|_{H^1}^2\|\nabla^{N+2-k}n\|_{L^2}^2 + \|\nabla n\|_{L^\infty}^2\|\nabla^N u\|_{L^2}^2.\end{aligned}
\|\boldsymbol{u}\|_{LPS}^2:=\nu\|\nabla\boldsymbol{u}\|_0^2+\gamma\|\nabla\cdot\boldsymbol{u}\|_0^2+\sum_{M\in\mathcal M_h}\tau_M \|\kappa((\boldsymbol{u}_M\cdot\nabla)\boldsymbol{u})\|_0^2
{Y}_{t n +i}= \begin{cases}\frac{f_i}{d_{\rm a,b}^{\alpha/2}}+ \frac{g_{t n + i}}{d_{\rm j,b}^{\alpha/2}}+{N_{t n + i}^{\rm (b)}}, \!\!\mbox{Alice transmits and}~t=0 \frac{g_{t n + i}}{d_{\rm j,b}^{\alpha/2}}+{N_{t n + i}^{\rm (b)}}, \!\!\mbox{else,}\end{cases}
Q'_1(0)=\sum\limits_{r=0}^{\min(n_1,\Delta n)}\sum\limits_{s=0}^{n_1}\sum\limits_{t=0}^{\Delta n-1}\sum\limits_{\substack{\mathbf{c}\in [1,n_1]_<^r \mathbf{c'}\in [1,n_1]_<^s}}\sum\limits_{\substack{\mathbf{d}\in [n_1+2,n_2]_<^r \mathbf{d'}\in [n_1+2,n_2]_<^t}}\det V(\mathbf{c},\mathbf{c'},\mathbf{d},\mathbf{d'})
e^{-itH_q}f(x) =e^{-itH_0}f(x) +(e^{-itH_0}(f\star\rho_q))(x)\cdot 1_{x\geq 0}(x) +(e^{-it H_0}(f\star\rho_q))(-x) \cdot 1_{x\leq 0}(x),
Y_{i_{1}j_{1}, \ldots , i_{p}j_{p}}^{k_{1}l_{1}, \ldots , k_{q}l_{q}}= \bar{a}_{i_{1}\ldots i_{p}}\bar{a}_{j_{1}\ldots j_{p}}b^{k_{1}\ldots k_{q}}b^{l_{1}\ldots l_{q}}\mathrm{~ - ~ t r a c e s ~}
g_2^{-1}g_1 \in_G(I_\eta) \iff I_\eta = g_2^{-1}g_1 I_\eta g_1^{-1}g_2 \iff g_2 I_\eta g_2^{-1}= g_1 I_\eta g_1^{-1}\iff I_{g_2}= I_{g_1}.
\int_{\mathbb{R}^d}\left(\sum_{j=1}^d \int_0^1 x_jU_j^k(\tau_{tx}\omega)dt \right)\partial_i \eta (x)dx = \int_0^1\sum_{j=1}^d \int_{\mathbb{R}^d}\frac{y_j}{t^{d+1}}U_j^k(\tau_y\omega)\partial_i \eta \left (\frac{y}{t}\right)dy dt
A ={\frac{\pi}{2}}\sum_{l = 1}^{h}\left\{a_{l}\left( m_{l}- \sum_{k = 1}^{h}\Omega_{l k}^{( 1 )}n_{k}\right) + \pi \sum_{k = 1}^{h}n_{l}\Omega_{l k}^{( 2 )}n_{k}\right\}.
g = c I + v^{n}t_{n}= \left( \begin{array}{c r}{c - v_{2}}&{- v_{1}- v_{0}}\\{- v_{1}+ v_{0}}&{c + v_{2}}\\ \end{array}\right) , \mathrm{w i t h}c^{2}+ v^{n}v_{n}= 1 ,
\mathcal{K}_{Y}(x, y)=\frac{c_{N, s}}{2}\left[\operatorname{div}Y(x)+\operatorname{div}Y(y)-(N+2 s) \frac{(Y(x)-Y(y)) \cdot(x-y)}{|x-y|^{2}}\right]|x-y|^{-N-2 s}.
S_n =\sum_{j=1}^A \sum_{m=0}^{n}p_{j,m}\sum_{k=1}^\infty \frac{d^2}{dk^2}\frac{1}{(k+m)^j}&=\sum_{j=1}^A \sum_{m=0}^{n}p_{j,m}\sum_{k=1}^\infty \frac{j(j+1)}{(k+m)^{j+2}}&=\sum_{j=1}^A \Big(\sum_{m=0}^{n}p_{j,m}\Big) j(j+1)\zeta(j+2)-\sum_{j=1}^A\sum_{m=0}^{n}\sum_{k=1}^{m}\frac{j(j+1)p_{j,m}}{k^{j+2}}.
\lim_{N \to \infty}\mathbb{P}\Big(2^{-\frac{4}{3}}N^{-\frac{1}{3}}\big(\lambda_{\mathrm{max}}(H^2)-4N\big) \leq x\Big)= F_{\mathrm{GUE}}(2^{-\frac{2}{3}}x)F_{\mathrm{GUE}}(2^{-\frac{2}{3}}x).
\prod_{j=1}^\infty \frac{1}{1-q^j}=\sum_{\substack{0\leqslant h<k (h,k)=1}}\sum_{\ell=1}^{\infty}\frac{C_{hk\ell}(\infty)}{(q-e^{2\pi ih/k})^\ell}\qquad(|q|<1)
C(\pi, s_i v) \subset C(\pi, v) \cup \left\{\tilde{f}_i^a (\pi_1 \otimes \pi_2 ) \left| \begin{array}{l}\pi_1 \otimes \pi_2 \in C(\pi, v), \tilde{e}_i(\pi_1 \otimes \pi_2) = 0, \tilde{f}_i^a(\pi_1 \otimes \pi_2) \not\in D(\pi, )\sqcup \{0 \}, a \geq 0. \end{array}\right. \right\}\setminus \{0 \}.
\beta^{p-1}\,\Big[N-\beta\,p+\beta-p+\gamma\Big] \int_{\mathbb{R}^N}v^p\,F_*(x)^{\gamma-p}\, dx &=\int_{\mathbb{R}^N}F_*(x)^\gamma\,F(\nabla u)^{p-1}\left\langle \nabla F(\nabla u),\nabla\left(\frac{v^p}{u^{p-1}}\right)\right\rangle\, dx.
\dot{r}^{2}\; - \; ( 1 - H^{2}r^{2})^{2}+ \frac{r^{2}}{E^{2}}\; ( 1 - H^{2}r^{2})^{3}( \mu + \frac{\Omega^{2}}{2 r^{2}})^{2}= 0 ,
x= \sum \limits_{j\in \mathbb{J}}\langle x, x_j\rangle \tau_j \iff \sum\limits_{j\in\mathbb{J}}(2e-c_j)\langle x, x_j\rangle\langle \tau_j, x\rangle = \langle x, x \rangle \iff \sum\limits_{j\in\mathbb{J}}c_j^2\langle x, x_j\rangle \langle x_j, x\rangle = \langle x, x \rangle.