harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\max_{j=1,\dots,M}\ell_j\le N+1, \max_{s\in\{m_1,\dots,m_{\ell_j}\},\,j=1\dots,M}y^{s}_j\le L,\quad \sum_{j=1}^N\sum_{s\in\{m_1,\dots,m_{\ell_j}\}}n^{s}_j=N.
\theta(n_s(\lambda))T_{n_s}- T_{n_s}\theta(\lambda) = -\sum_{k = 0}^{-\langle\nu(\lambda),\alpha\rangle-1}c_{n_s,-k}\theta(\mu_{n_s}(k)\lambda) & = -\sum_{k = 1}^{-\langle\nu(\lambda),\alpha\rangle}\theta(n_s(\lambda)\mu_{n_s}(-k))c_{n_s,-k}.
\int_{b_{\ell+1}}^1 E_1(x) \, dx = \int_{T^{-m}[b_{\ell+1}, 1]}\frac{|u'(x)|}{(T^m)'x}\, dx = \sum_{j=0}^\infty \int_{T^{-m}[b_{\ell+1}, 1] \cap [b_{j+1}, b_j]}\frac{|u'(x)|}{(T^m)'x}\, dx + \int_{T^{-m}[b_{\ell+1}, 1] \cap [\frac12, 1]}\frac{|u'(x)|}{(T^m)'x}\, dx.
Y_\nu (z) = \frac{2 \, (\frac z2)^\nu}{\sqrt{\pi}\ \Gamma(\nu+\frac 12)}\, \Big[\int_0^1 (1-t^2)^{\nu-\frac 12}\, \sin(zt)\, dt ) -\int_0^\infty e^{-zt}\, (1+t^2)^{\nu-\frac 12}\, dt\Big].
\left( \begin{array}{c}{< W ( L_{\mp}) >}\\{< W ( L_{W \mp}) >}\\ \end{array}\right) = \left( \begin{array}{c c}{q^{\pm{\frac{1}{4}}}}&{( q^{\mp{\frac{1}{4}}}- q^{\pm{\frac{1}{4}}})}\\{0}&{q^{\mp{\frac{1}{4}}}}\\ \end{array}\right) \left( \begin{array}{c}{< W ( L_{I}) >}\\{< W ( L_{W}) >}\\ \end{array}\right) .
W_{94,1}=1+2\alpha x^{16}+(134044-2\alpha+128\beta)x^{18}&\quad+(2010660-30\alpha-896\beta)x^{20}+\cdots, W_{94,2}=1+2\alpha x^{16}+(134044-2\alpha+128\beta)x^{18}&\quad+(2018852-30\alpha-896\beta)x^{20}+\cdots, W_{94,3}=1+2\alpha x^{16}+(134044-2\alpha+128\beta)x^{18}&\quad+(2190884-30\alpha-896\beta)x^{20}+\cdots,
B = \biggl(N_D - N_G - \sum_{\begin{subarray}{l}\widetilde{\Sigma}< D \vert\widetilde{\Sigma}\rvert=2\end{subarray}}N_{\widetilde{\Sigma}}\biggr)B \subseteq B^G+\sum_{\begin{subarray}{l}\widetilde{\Sigma}< D \vert\widetilde{\Sigma}\rvert=2\end{subarray}}B^{\widetilde{\Sigma}}\subseteq B,
\left| \sum_{\alpha \beta}\langle \Phi_{\alpha}\Phi_{\beta}\rangle_{c}\left. \right/ \sum_{\alpha \beta}\langle \Phi_{\alpha}\Phi_{\beta}\rangle \right| = \left| \langle \Phi^{2}( x ) \rangle_{c}\left. \right/ \langle \Phi^{2}( x ) \rangle \right| \; ( n = 2 ) \; ,
A=\begin{pmatrix}c_1 -1 begin{align*}4pt]1 \phantom{-}0\end{pmatrix}\begin{pmatrix}c_2 -1 begin{align*}4pt]1 \phantom{-}0\end{pmatrix}\cdots\begin{pmatrix}c_N -1 begin{align*}4pt]1 \phantom{-}0\end{pmatrix}.
g_{jk}= \begin{cases}\int_{\Omega}\phi_{j}(x)\epsilon\phi_{k}(x) d\nu(x), 1 \leq j \leq N, 1 \leq k \leq M, -\int_{\Omega}\epsilon\phi_{j}(x)\phi_{k}(x) d\nu(x), N < j \leq M, 1 \leq k \leq N, \alpha_{jk}, N < j, k \leq M. \end{cases}
\varphi_1(s/N) = \pi_1\frac{B_1(s/N)}{-(B_2(s/N) + B_3(s/N))}= \pi_1 \frac{c_\infty s/N^2 + o(N^{-2})}{(s/N)^2 \sum_{d=1}^D \pi_d[\lambda_d h_{d2}/2 + a_d(\lambda_dh_{d1}- c_d)] + c_\infty s/N^2 + o(N^{-2})}= \pi_1 \frac{1 + o(1)}{1 + c_\infty^{-1}\sum_d \pi_d \left[\lambda_d h_{d2}/2 + a_d (\lambda_d h_{d1}- c_d)\right]s + o(1)}.
\sum_{i = 0}^{s}\frac{(q^{-s})_{i}\, q^{i}}{(q)_{i}(c q^{i})_{1 - s}}\phi_{D}\biggl(\genfrac..{0pt}{}{q^{1 - s};\, a q^{-i}, b_{1}q^{i}, \dotsc, b_{r}q^{i}}{c q^{i + 1 - s}}; x q^{i}, x_{1}, \dotsc, x_{r}\biggr) = 0.
F'(\tilde{x}) =\frac{k}{4\varepsilon}-\frac{\sigma(H-\delta)}{2}C\tilde{x}^{H-\delta-1}=0 \implies\tilde{x}= \biggl(\frac{k}{2\sigma\varepsilon C(H-\delta)}\biggr)^{1/(H-\delta-1)}&= \biggl(\frac{2\sigma C (H-\delta)}{k}\biggr)^{1/(1+\delta-H)}\varepsilon^{1/ (1+\delta-H )}.
x_r = \A \phi_r = y y_r = \A \phi_{rr}= A \left( A \phi_r \phi - \frac{\gg^2}{\phi^2}\right) = xy - z^2 z_r = \A^{\tfrac{1}{2}}\psi_r = - AA^{\tfrac{1}{2}}\left( \frac{\gg \phi_r - A \gamma \phi^2}{\phi^2}- \frac{\gg \phi_r}{\phi^2}\right) = xz.
z \partial_{i}\tilde{T}+ \sum_{j=1}^{m}(\partial_{i}{h^{j}})( A_{j}\circ h ) \tilde{T}- \tilde{T}\tilde{A}_{i}=0,\ \forall i & z^{2}\partial_{z}\tilde{T}+ (B\circ h) \tilde{T}- \tilde{T}\tilde{B}=0,
k_1+ k_2 + \cdots + k_j + \cdots + k_{\ell}= \ell(\lambda)=\lambda \;, \noalign{and}k_1 + 2 k_2 + \cdots + j k_j + \cdots + \ell k_{\ell}= |\lambda| = \sum_{i=1}^{\ell(\lambda)}\lambda_i = \lambda \;.
m^{\varphi^{*}}\in \arg\max\limits_{m}\Biggl\{(\lambda_{\ell}\tilde{r}_{i}- \nu_{\ell})\frac{\sum\limits_{n=0}^{m-1}\frac{(\lambda_{\ell})^n}{n! (\mu_{i})^{n}}}{\sum\limits_{n=0}^{m}\frac{(\lambda_{\ell})^n}{n! (\mu_{i})^n}}\Biggr\}
H^K_{\theta}\Theta^K_{\mu}(\theta) = E^K_{\mu}\Theta^K_{\mu}(\theta), H^K_{\theta}= - \partial_{\theta}^2 - \cot\theta \partial_{\theta}+ \frac{K^2}{\sin^2\theta}, K = k \epsilon_{\nu}, \mu=0, 1, 2, \ldots.
\liminf_{R\to \infty}\limsup_{k\to \infty}\bigg\|\Phi_{R}u_{k}^{\frac{\alpha(\cdot)}{q(\cdot)}}v_{k}^{\frac{\beta(\cdot)}{q(\cdot)}}\bigg\|_{L^{q(\cdot)}(\mathbb{R}^{N})}\geqslant \min \big\{\nu_{\infty}^{\frac{1}{q_{\infty}+\epsilon}}, \nu_{\infty}^{\frac{1}{q_{\infty}-\epsilon}}\big\}.
g_t(\hat{x}_1+z,\hat{x}_1+z+t^{-1/d}(\hat{x}_2-\hat{x}_1+y_2),\hdots,\hat{x}_1+z+t^{-1/d}(\hat{x}_k-\hat{x}_1+y_k)) & =g_1(\hat{x}_1,\hat{x}_2+y_2,\hdots,\hat{x}_k+y_k)
\begin{aligned}x_0^2 + x_1^2 + x_2^2 + x_3^2 = 0 & x_0^2 + b^4 x_1^2 + b^2 x_2^2 + b^2 x_3^2 = 0 x_0^2 - b^2 x_1^2 = 0 x_0^2 - b^2 x_1^2 = 0.\end{aligned}
D_{\vec{z}}\vec{F}_0 \vec{g}\left({\vec{F}_0,\vec{z},0}\right) = D_{\vec{x}}\vec{f}\left({\vec{F}_0,\vec{z},0}\right)\vec{F}_1 + \frac{\partial \vec{f}}{\partial \varepsilon}\left({\vec{F}_0,\vec{z},0}\right),
\varphi_1(0,\rho) = (\tfrac{T}{T_0})^{\frac{2}{p-1}}c_p + T^{\frac{2}{p-1}}f(T\rho ) - c_p \varphi_2(0,\rho) = (\tfrac{T}{T_0})^{\frac{p+1}{p-1}}\tfrac{2}{p-1}c_p + T^{\frac{p+1}{p-1}}g(T \rho)- \tfrac{2}{p-1}c_p.
\lim_{N\rightarrow\infty}{\frac{P_{\lambda(N)}\big(R^{\pm 1}, t^{-1}, \dots, t^{1-N}\big)}{P_{\lambda(N)}\big(1, t^{-1}, \dots, t^{1-N}\big)}}= \Phi^{\nu}(R^{\pm 1}; q, t),
\sum_{1 \le m \le \frac{1}{T_2 - T_1}}\frac{\log^{d-1}(m+1)}{m \prod_{k \neq j}\left\| m \frac{a_j}{a_k}\right\|}= O \left( (T_2 - T_1)^{1 - \gamma_{d-1}- \varepsilon}\right) .
\frac{{\rm d}^2}{{\rm d}x^2}H(x) =\frac{{\rm d}^2}{{\rm d}x^2}\left\{- (2x)^{\log_3(\sqrt{2}+1)}\right\}&=-2^{\log_3(\sqrt{2}+1)}\log_3(\sqrt{2}+1)(\log_3(\sqrt{2}+1)-1)x^{\log_3(\sqrt{2}+1)-2},
F^e_*\widehat{R}\otimes_{\widehat{R}}T = F^e_*\widehat{R}\otimes_{\widehat{R}}\varinjlim_{\lambda \in \Lambda}T_{\lambda}\cong \varinjlim_{\lambda \in \Lambda}(F^e_*\widehat{R}\otimes_{\widehat{R}}T_{\lambda}) \cong \varinjlim_{\lambda \in \Lambda}F^e_*T_{\lambda}\cong F^e_* T.
\frac{d}{dt}\sum_{ij}g_{ij}(t) V^i(t) W^j(t) = - 2 \sum_{ij}R_{ij}V^i W^j +\sum_{ij}g_{ij}(\partial_t V^i) W^j + \sum_{ij}g_{ij}V^i (\partial_t W^j).
\|\mathcal{S}_{0,2}(U,U)\|_{C^{0,\alpha}}\geq \delta, \quad\pi_{\mathcal{K}}(\mathcal{S}_{0,2}(U,U))=0,\|\pi_{\mathcal{K}}(\mathcal{S}_{0,3}(U,U,U))\|_{C^{0,\alpha}}\geq\delta,
\beta_\delta(\lambda,n)-w^{-1}(\beta_\delta(\mu,n)) =(\delta-|\lambda|-\lambda_j+j+rp)(\varepsilon_0-\varepsilon_j) &=\langle\hat\lambda+\rho_n(\delta+rp),\varepsilon_0-\varepsilon_j\rangle(\varepsilon_0-\varepsilon_j).
-\log\big(y_{p}\big(e^{4\pi\slash y_{p}}-1\big)\big)=-\log\bigg(\sum_{n=1}^{\infty}\frac{(4\pi)^{n}}{n!\,y_{p}^{n-1}}\bigg)= &-\log\bigg(4\pi+\sum_{n=1}^{\infty}\frac{(4\pi)^{n+1}}{(n+1)!\,y_{p}^{n}}\bigg)=-\log( 4\pi) + O\bigg(\frac{1}{y_{p}}\bigg).
x\left(\begin{array}{c}Q_0(x) Q_1(x) Q_2(x) \vdots\end{array}\right)=\left(\begin{array}{ccccc}0 1 \zeta_1 0 1 &\xi_1 0 1 & \ddots \ddots \ddots\end{array}\right)\left(\begin{array}{c}Q_0(x) Q_1(x) Q_2(x) \vdots\end{array}\right)
\Big(\sum_{d=1}^\infty d^3N_d e^{dy_1}\frac{y_2^{3d-1}}{(3d-1)!}q^d\Big)\Big(\sum_{d=1}^\infty dN_d e^{dy_1}\frac{y_2^{3d-3}}{(3d-3)!}q^d\Big)
\phi(x) =a_0x^n+a_1x^{n-1}+\cdots+a_{n-1}x+a_n, u(x) =u_0x^{n-2}+u_1x^{n-3}+\cdots+u_{n-3}x+u_{n-2}, v(x) =v_0x^{n-1}+v_1x^{n-2}+\cdots+v_{n-2}x+v_{n-1},
\mathfrak{g}_{0, \epsilon}=( \mathfrak{g}(0, \epsilon) + \mathfrak{g}(0, \epsilon^{-1}) \cap \mathfrak{g}, &\mathfrak{g}_{\alpha, \epsilon}=(\mathfrak{g}(\alpha, \epsilon) + \mathfrak{g}(- \alpha, \epsilon^{-1})) \cap \mathfrak{g}.
\left\{\prod_{j=1}^k\left[h^{(j)}(x)\right]^{m_j}\right\}'=\left[\sum_{i=1}^km_i\frac{h^{(i+1)}(x)}{h^{(i)}(x)}\right]\prod_{j=1}^k\left[h^{(j)}(x)\right]^{m_j},
\frac{d w}{d r}= \frac{4 R}{3 \sqrt{M}}\left[ \frac{1}{1 6 R^{4}}( w^{3}- 3 w + 4 \kappa ) ( 1 - w^{2}) + \frac{1}{2 R^{2}}( 2 \kappa - w^{3}) - w \right] \, ,
0=(\Delta_{E}h,h)_{L^2}=\left\|D_1 h\right\|^2_{L^2}+2\mu \left\| h\right\|_{L^2}^2-4(h,\mathring{R}h)_{L^2}\geq0+2\mu \left\| h\right\|_{L^2}^2-2\mu \left\| h\right\|_{L^2}^2=0.
S^{\left( \Sigma_{1 + 1}\right)}= - 2 \pi i \theta f \left( \overline{{{t_{*}}}}\right) g \left( t_{*}\right) \int_{\Sigma_{1 + 1}}\left[ i \overline{{{\psi}}}^{L}\sigma^{a}D_{a}\psi^{L}- \overline{{{\psi}}}^{L}m \psi^{L}\right] ,
\mathbb{E}\left(\prod_{j=1}^{k}|p_{N}(x_{j})|^{2\alpha_{j}}\right) = \prod_{j=1}^{k}C(\alpha_{j})(1-x_{j}^{2})^{\alpha_{j}^{2}/2}(N/2)^{\alpha_{j}^{2}}e^{(2x_{j}^{2}-1-2\log(2))\alpha_{j}N}&\times \prod_{1 \leq i < j \leq k}(2|x_{i}-x_{j}|)^{-2\alpha_{i}\alpha_{j}}\left[1+O\left(\frac{\log N}{N}\right)\right]
h(k, \ell) = \begin{cases}f^\ast(k) \mbox{if}k > 1 \ell = 1 2f^\ast(1) - 1 \mbox{if}k = 1 \ell = 1 f^\ast(\ell) \mbox{if}k = 1 \ell > 1 0 \mbox{else.}\end{cases}
S^{\delta_n^k+v-\alpha_{k+1-v}}\left (\alpha \right )= \alpha \iff \begin{array}{c}\{\alpha_{j+(k+1-v)}= \alpha_j+\alpha_{k+1-v}\}_{j=1}^{v-1}\mbox{and}\ \ \delta_{n}^k=\alpha_{v}+\alpha_{k+1-v}\mbox{and}\ \ \{\alpha_{j+v}= \alpha_{j}+ \alpha_{v}\}_{j=1}^{k-v}. \end{array}
U_{d e f g}^{a b c}= - \frac{1}{3 !}\eta_{d i j k}P_{e}^{( 2 ) i}P_{f}^{( 2 ) j}P_{g}^{( 2 ) k}\eta^{( 2 ) a}\eta^{( 2 ) b}\eta^{( 2 ) c}.
D_{+ +}^{( M L ) c d}( x ) = D_{+ +}^{M L}( x ) \delta^{c d}={\frac{i \delta^{c d}}{\pi^{2}}}\int d^{2}k \, e^{i k x}{\frac{k_{+}^{2}}{( k^{2}+ i \epsilon )^{2}}}={\frac{\delta^{c d}}{\pi}}{\frac{( x^{-})^{2}}{( - x^{2}+ i \epsilon )}}\
\begin{aligned}err_1 = \int \left( a_t+ba_{\alpha}\right) |w|^2 + M (1+a)\vert w\vert^2 \, d\alpha- 2 \Re \int a \bar w \, (\left[\bar Y,P \right] (r_\alpha+R_\alpha w) + [P,b] w_\alpha) \, d\alpha.\end{aligned}
\frac{\det\left(\frac 1{w_k-z_j}\right)}{1-\frac{z_1\cdots z_n}{w_1\cdots w_n}}\prod\limits_{j=1}^{n}\frac{w_j^n(1-z_j)^{n-1}}{z_j^{n-1}(1-w_j)^{n-1}}\sum\limits_{k,\ell=1}^n\frac{(-1)^{n-1}(1-z_k)\prod\limits_{j=1}^n(w_\ell-z_j)(w_j-z_k)}{z_k(1-w_\ell)(w_\ell-z_k)\prod\limits_{j\neq \ell}(w_\ell-w_j)\prod\limits_{j\neq k}(z_k-z_j)}.
\hat{\mathbf{h}}_{eff}^{(g)}\approx\sum_{l=0}^{L_g-1}\left( \underbrace{\operatorname{pinv}\left\{\mathbf{X}^{(g)}\left[\mathbf{I}_{K_g}\otimes \mathbf{E}_{L_g,l}\right]\right\}}_{\textrm{(temporal) correlator}}\otimes \underbrace{\left[\mathbf{S}_D^{(g)}\mathbf{E}_{D,l}\right]^H}_{\textrm{pre-beamformer}}\right)\mathbf{y}
\{J , H_{c}\}= \frac{1}{k}\int d^{2}\vec{x}d^{2}\vec{y}d^{2}\vec{z}\{G ( \vec{x}- \vec{y}) G ( \vec{y}- \vec{z}) - G ( \vec{x}- \vec{z}) G ( \vec{y}- \vec{z}) \}f_{a b c}A_{2}^{a}( \vec{x}) J_{2}^{b}( \vec{y}) J_{\circ}^{c}( \vec{z})
\prod_{i<j}(-1)^{\nu_i-\nu_j}\prod_{i\neq j}\frac{\Gamma(1-r_i+r_j+\nu_i-\nu_j)}{\Gamma(1+\nu_j+r_i-r_j)}\frac{\Gamma(r_{i}-r_j-\nu_{i}+\nu_j)}{\Gamma(r_{j}-r_{i}-\nu_j)}=1.
g(x,y)=\gamma_{i+1}(x,y)y^{e_{i+1,\theta}}+\pi^{m(D_{i+1}-D_i)}(), g_r(x,y)=\gamma_{r+1}(x,y)y^{e_{r+1,\theta}}+\pi^{m(D_{r+1}-D_i)}(), \intertext{and}G(x,y)=\sum_{\widetilde{w}_{i,\theta}(\mathcal{V}_i)=0}\gamma_i(x,y)y^{e_{i,\theta}},
\lVert AB-A_0B_0 \rVert^2 = \sum_{j=1}^N \left( \sum_{i=1}^{M-1}\{(a_{i1}-a_{i2})b_j -(a^0_{i1}-a^0_{i2})b^0_j+a_{i2}-a^0_i\}^2 \right. &\left. + \left[ \sum_{i=1}^{M-1}\{(a_{i1}-a_{i2})b_j-(a^0_{i1}-a^0_{i2})b^0_j +a_{i2}-a^0_i\}\right]^2 \right).
R ( a_{j}, a_{j}) = \sum_{k \neq j}( - 1 )^{k}{\frac{( q_{j}p_{k}- p_{j}q_{k})^{2}}{a_{j}- a_{k}}}+ p_{j}^{2}- a_{j}q_{j}^{2}- 2 p_{j}q_{j}u + 2 q_{j}^{2}v .
I = \frac{M}{2}\int_{0}^{s}d \tau \dot{x}_{i}^{2}+ \frac{M}{2}\int_{0}^{s^{\prime}}d \tau^{\prime}\dot{y}_{j}^{2}+ \frac{M}{2}( s + s^{\prime}) + \frac{K_{R}}{2}\int_{0}^{s}d \tau \epsilon_{i j}x_{i}\dot{x}_{j}+ \frac{K_{R}}{2}\int_{0}^{s^{\prime}}d \tau^{\prime}\epsilon_{i j}y_{i}\dot{y}_{j}.
\lambda_l (t, \boldsymbol{\theta}) = \gamma_l (\boldsymbol{\theta}) t^2 + \mu_l (\boldsymbol{\theta}) t^3 + \ldots, l = 1, \ldots, n, \varphi_l (t, \boldsymbol{\theta}) = \omega_l (\boldsymbol{\theta}) + t \psi^{(1)}_l (\boldsymbol{\theta}) + \ldots, l = 1, \ldots, n.
R_{12}(z)=\rho_N(z)\left[\sum_i e_{i,i}\otimes e_{i,i}+ \frac{q(1-z)}{1-q^2z}\sum_{i \ne j}e_{i,i}\otimes e_{j,j}+ \frac{(1-q^2)}{1-q^2z}\left(\sum_{i<j}+z\sum_{i>j}\right)e_{i,j}\otimes e_{j,i}\right] \,.
\left|\nabla\left(e^{-\frac{1}{2}\int_0^tB_i^2(s)ds}f\right)\right|_q\leq \mathcal{B}|\nabla f|_q,\ \forall f\in e^{\int_0^tB_i^2(s)ds}[W^{1,q}(\mathbb{R}^d)],\t\geq0,\i=1,2,...,N.
Z_0=\frac 1{(2\pi )^N}\int_M{\rm e}^{H(x)}\sqrt{det\omega_{ij}}d^{2N}x=\frac 1{(\pi )^N}\int_{{\cal M}}\exp ({H(x)+\frac 12\omega_{ij}\theta^i\theta^j)}d^{2N}xd^{2N}\theta ,
\frac{\Gamma(k-1/2)}{\Gamma\left(\tfrac{k}{2}+\tfrac{-z-1}{4}\right)\Gamma\left(\tfrac{k}{2}+\tfrac{z-1}{4}\right)}=2^{k-3/2}\sqrt{\pi}^{-1}\times \frac{\Gamma(k/2-1/4)}{\Gamma\left(\tfrac{k}{2}+\tfrac{-z-1}{4}\right)}\times \frac{\Gamma(k/2+1/4)}{\Gamma\left(\tfrac{k}{2}+\tfrac{z-1}{4}\right)}
J_\mu(U_*,a) \leq \max\{f(1),f(T)\}= \max\bigg\{J_\mu(U_1,a)+\sigma|U_1|^{-\frac{2}{d}},T^{-2}\lambda_1(U_1,a) + \mu T^{d}|U_1|+\sigma|U_2|^{-\frac{2}{d}}\bigg\}\leq\max\bigg\{J_\mu(U_1,a),T^{-2}\lambda_1(U_1,a)+ \mu T^{d}|U_1|\bigg\}+ \sigma |U_1|^{-\frac{2}{d}}\leq\max\bigg\{J_\mu(U_1,a),J_\mu(U_2,a)\bigg\}+2 \sigma |U_1|^{-\frac{2}{d}}.
K_{\lambda \nu}^{\hspace{.3 true cm}\alpha}=-\frac{1}{2}(S_{\lambda \hspace{.2 true cm}\nu}^{\hspace{.1 truecm}\alpha}+ S_{\nu \hspace{.2 true cm}\lambda}^{\hspace{.1 truecm}\alpha}- S_{\lambda \nu}^{\hspace{.3 true cm}\alpha})
{\displaystyle \Psi(\mathbf{x},0) = \Psi_0(\mathbf{x}),\quad\mathbf{A}(\mathbf{x},0)=\mathbf{A}_{0}(\mathbf{x}),\mathbf{A}_{t}(\mathbf{x},0)=\mathbf{A}_{1}(\mathbf{x}),}
(i\partial_t +\partial_x^2 +i\gamma)T[W_t^\gamma u_0, v, \cdots, v] = -\widehat{\mathcal{HL}_B}[W_t^\gamma u_0, v \cdots, v] &\qquad- \frac{p-1}{2}\widehat{T}[W_t^\gamma u_0, (i\partial_t+\bigtriangleup)v,v, \cdots, v] &\qquad+\frac{p-1}{2}\widehat{T}[W_t^\gamma u_0, v, (i\partial_t+\bigtriangleup)v,v, \cdots, v],
f =KS_{\xi\chi}f &=K\int_{\Theta}v^{2}(w)s^{2}(w)\pi_{G(w)}\xi_{w}^{\ast}\chi_{w}\pi_{F(w)}fd\mu(w) &=\int_{\Theta}v^{2}(w)s^{2}(w)K\pi_{G(w)}\xi_{w}^{\ast}\chi_{w}\pi_{F(w)}fd\mu(w) &=\int_{\Theta}T_{w}fd\mu(w).
\begin{cases}u_{t}(x,t)+v_{x}(x,t)+v_{xxx}(x,t)+\lambda u(x,t) =\sum_{i=1}^3\Psi_i(x,t), , v_{t}(x,t)+u_{x}(x,t)+u_{xxx}(x,t)+\lambda v(x,t) =\sum_{i=1}^3\Phi_i(x,t), , u(0,t)=u(L,t)=u_{x}(0,t)=0, , v(0,t)=v(L,t)=v_{x}(L,t)=0, ,\end{cases}
\int_{-\infty}^{\infty}q_t(x,y)f(y)dy =\int_{-\infty}^{x}dz\int_{-\infty}^{\infty}-\partial_yp_t(y,z)f(y)dy=\int_{-\infty}^{x}dz\int_{-\infty}^{\infty}p_t(y,z)f'(y)dy &=\int_{\infty}^{x}dz\int_{-\infty}^{\infty}\frac{m(z)}{m(y)}\frac{m(y)}{m(z)}p_t(y,z)f'(y)dy &=\int_{\infty}^{x}m(z)dz\int_{-\infty}^{\infty}p_t(z,y)\frac{f'(y)}{m(y)}dy.
\begin{array}{l}R_{12}e_{1}e_{2}= c e_{2}e_{1}\; , \;\;R_{12}A_{1}R_{21}A_{2}+ A_{2}R_{12}A_{1}R^{-1}_{12}=0 \; , e_{1}A_{2}= (\pm) R_{21}A_{2}R_{12}e_{1}\; , \;\;R_{12}F_{1}R_{21}F_{2}= F_{2}R_{12}F_{1}R_{21}\; , e_{1}F_{2}= R_{21}F_{2}R_{12}e_{1}\; , \;\;R_{21}A_{2}R_{12}F_{1}= F_{1}R_{21}A_{2}R_{12}\; .\end{array}
\begin{array}{c}lim \varepsilon \rightarrow 0\end{array}\frac{\partial}{\partial \theta}I(p,m,\theta ,\varepsilon )-\frac{\partial}{\partial \theta}\begin{array}{c}lim \varepsilon \rightarrow 0\end{array}I(p,m,\theta ,\varepsilon )\mid_{_{\theta^{\pm}_{0}}}\neq 0
\mathfrak{B}_1\times \mathfrak{B}_{2,\emptyset}=[123,\bar1\,32], \mathfrak{B}_1\times \mathfrak{B}_{2,\{1\}}=[1\bar{2}3,\bar1\,\bar32], \mathfrak{B}_1\times \mathfrak{B}_{2,\{2\}}=[13\bar{2},\bar12\bar{3}]
\left\Vert\Delta_jf\right\Vert_{L^{\infty}}\lesssim 2^{\frac{jd}p}\left\Vert \Delta_jf\right\Vert_{L^p}=2^{\alpha j}\left\Vert \Delta_jf\right\Vert_{L^p}\leq \left\Vert f\right\Vert_{\dot{F}^{\alpha,p}_q},
\mathcal{F}_2(\alpha^{(2)})=\begin{pmatrix}1 0 0 \cdots 0 0 1 0 \cdots 0 0 \alpha_{3}1 \cdots 0 \vdots \vdots \vdots \ddots \vdots 0 \alpha_{r}0 \cdots 1\end{pmatrix}.
p_1 = (0:0:1), p_2 = \left(\tfrac{64}{3}:\tfrac{256}{3}:1\right), p_3 = \left(\tfrac{49}{24}+i\tfrac{77\sqrt{7}}{24}:\tfrac{-637}{48}+i\tfrac{343\sqrt{7}}{48}:1\right), p_4 = \left(\tfrac{49}{24}-i\tfrac{77\sqrt{7}}{24}:\tfrac{-637}{48}-i\tfrac{343\sqrt{7}}{48}:1\right).
0\longrightarrow\mathbf{R}\longrightarrow A\cdot N\longrightarrow\mathbf{R}\longrightarrow0 & b\mapsto\begin{pmatrix}1 b & 1\end{pmatrix}\qquad\begin{pmatrix}e^{r}b & e^{-r}\end{pmatrix}\mapsto r
\psi_{i+1}^*(f) \le\max\left\{f^M\left(16d\cdot(\tilde f^{(i)}(1))^2\right),16d\cdot\left(\tilde f^{(i)}(1)\right)^2\right\}&=\tilde f\left(\tilde f^{(i)}(1)\right) &=\tilde f^{(i+1)}(1).
2 i H_{A}^{\ \C}\sigma_{C \dot{B}}^{\ \ \m}+ 2 i H_{\dot{B}}^{\ \ \dot{C}}\sigma_{A \dot{C}}^{\ \ \m}- 2 i H_{A \dot{B}}^{\ \ \m}- \nabla_{A}H_{\dot{B}}^{\m}- \nabla_{\dot{B}}H_{A}^{\m}= 0
F_{\alpha,\beta}(x)=\frac{1}{|x|_*^{\alpha+\beta-d}}1_{\alpha\vee \beta<d}+\frac{1}{|x|_*^{\alpha\wedge\beta}}1_{\alpha\vee\beta>d}+\frac{\log|x|_*}{|x|_*^{\alpha\wedge\beta}}1_{\alpha\vee\beta=d}.
P_{2n}^{\alpha,q}(x)=\frac{(-1)^{n}(q+\frac{3}{2})_{n}}{(n+q+\alpha+\frac{3}{2})_{n}}\_{2}F_{1}\Big(-n,n+q+\alpha+\frac{3}{2};q+\frac{3}{2};x^{2}\Big).
X_\tau^{1}- X_\tau^{2}= e^{\tau A}(x_1 - x_2) + e^{\tau A}[v^{(1)}(0,x_1) - v^{(1)}(0, x_2)] - [v^{(1)}(\tau,X_\tau^{1}) - v^{(1)}(\tau,X_\tau^{2})]+\int_0^\tau e^{(\tau-s)A}[ \nabla^G v^{(1)}(s,X_s^1) - \nabla^G v^{(1)}(s,X_s^2) ]\;dW_s, \;\; \tau \in [T_0,(2T_0) \wedge T].
\mathbb{E}\left[ \left\| \Gamma_{m}\left( \overline{Z}_{n}- m \right) \right\|^{2}\right] = \frac{1}{n^{2}}\mathbb{E}\left[ \left\| \sum_{k=1}^{n}\xi_{k+1}\right\|^{2}\right] + \frac{1}{n^{2}}\mathbb{E}\left[ \left\| R_{n}\right\|^{2}\right] + \frac{2}{n^{2}}\mathbb{E}\left[ \left\langle \sum_{k=1}^{n}\xi_{k+1}, R_{n}\right\rangle \right]
\frac{1}{M^l}\int_{Q_R}f^l \, d\mu \leq (c_0 \lambda_0 )^l \mu (Q_R)+ \Bigg[ (c_0 \lambda_0)^l \mu(Q_{2 R}) + \int_{Q_{2R}}g^l d\nu +\frac{M^l-1}{(M^{\frac{l}{\hat p}}-1)^{\hat p}}\Big(\int_{Q_{2R}}\hat g^{\frac{l}{\hat p}}d\hat \nu \Big)^{\hat p}\Bigg] \sum_{j=1}^\infty (\alpha M^l )^j,
R^i(\alpha) =R^i(z^{i})+\alpha (R^{i})'(z^{i})\Delta z^{i}+\alpha\left(\int_{0}^1 \left( (R^{i})'(z^{i}+t\alpha\Delta z^{i})-(R^{i})'(z^{i}) \right)\Delta z^{i}\right) &=R^i(z^{i})(1-\alpha)+\alpha\hat{r}^i_1+\alpha\left(\int_{0}^1 \left( (R^{i})'(z^{i}+t\alpha\Delta z^{i})-(R^{i})'(z^{i}) \right)\Delta z^{i}\right),
\lim_{\lambda \downarrow 0}\frac{J(\theta^{\lambda})-J(\theta)}{\lambda}= E_x\left[\int_{0}^{\infty}p_{s}^{\theta}e^{-rs}\,d(\eta_s-\theta_s)\right]+\lim_{\lambda \downarrow 0}\frac{1}{\lambda}E_x\left[\int_0^{\infty}k_{s}e^{-rs}\,d(\theta^{\lambda}_s -\theta_s) \right] = E_x\left[\int_0^{\infty}(p_{s}^{\theta}+k_{s})e^{-rs}\,d(\eta_s-\theta_s)\right].
\phi_1 := \phi_{6B}(\tau, z) = 2 \zeta^{\pm 1}- 4 + (-4 \zeta^{\pm 2}+ 12 \zeta^{\pm 1}- 16)q + (2 \zeta^{\pm 3}- 16 \zeta^{\pm 2}+ 40 \zeta^{\pm 1}- 52) q^2 + O(q^3)
Y_{t}Y_{t+10}= Y_{t+3}Y_{t+7}Z_{t}Z_{t+1}Z_{t+2}+ Y_{t+2}^2Y_{t+5}^2Y_{t+8}^2, Z_{t}Z_{t+1}Z_{t+2}Z_{t+3}= Y_{t+2}Y_{t+3}Y_{t+5}Y_{t+6}Y_{t+8}Y_{t+9}+ Y_{t+1}Y_{t+4}Y_{t+7}Y_{t+10}.
c_{1}:=\frac{5}{4}\left(\frac{\alpha^{4}\beta}{3}\right)^{1/5},c_{2}:=20\left(\frac{2\alpha^{6}}{3\beta}\right)^{1/5},c_{3}:=\frac{5}{4}\left(\frac{3\alpha^{6}}{\beta}\right)^{1/5}.
\begin{aligned}I_{hn}=\frac{25h\sqrt{5\pi}\exp\bigl(\frac{5h^2}{4n}\bigr)}{4n^2}\sum_{m\ge 1}\Bigl(\frac{5 h^{2}}{2n}m^2-3\Bigr)m^2 \exp\Bigl(-\frac{5h^2}{4n}m^2\Bigr) =\frac{25h\sqrt{5\pi}\exp(\alpha^2)}{4n^2}G(\alpha). \end{aligned}
(x+\beta)\frac{\Gamma(x+\alpha)}{\Gamma(x+\beta+1)}=\frac{(x+\beta)}{\Gamma(\beta+1-\alpha)}\int_0^{\infty}e^{-(x+\alpha)v}(1-e^{-v})^{\beta-\alpha}\,dv &\le (x+\beta) (x+\alpha)^{-(\beta-\alpha)-1}.
h = \frac{1}{Q^{3}}\left( \begin{array}{c}{{\begin{array}{c c}{{- 1 -{y^{\prime}}^{2}}}&{{y^{\prime}{\phi}^{\prime}}}\\{{y^{\prime}{\phi}^{\prime}}}&{{1 -{{\phi}^{\prime}}^{2}}}\\ \end{array}}}\\ \end{array}\right) , \; \;{\hat{H}}^{\dagger}={\hat{H}},
( *) = t^{k-6n+1}B_{2^{k-8n+2}-u_{n-1}}( t^{-1}) + ( t^{4n-1}+t^{4n-2}+\ldots+t+1) t^{k-6n}B_{2^{k-4n}-v_{n-1}}( t^{-1}) &= t^{k-2n-1}\left( t^{2-4n}B_{2^{k-8n+2}-u_{n-1}}( t^{-1}) +( t^{4n-1}+t^{4n-2}+\ldots+t+1) t^{1-4n}B_{2^{k-4n}-v_{n-1}}( t^{-1}) \right) &= t^{k-2n-1}B_{2^k-u_n}( t^{-1}) .
\int_\Omega |\nabla \overline{w}|^2 = \int_\Omega r\cdot\nabla \overline{w}=\int_{B_{\frac{s_0\rho_0}{2}}(x_0)}r\cdot\nabla \overline{w}+\int_{\Omega\setminus B_{\frac{s_0\rho_0}{2}}(x_0)}r\cdot\nabla \overline{w}=\int_{\Omega\setminus B_{\frac{s_0\rho_0}{2}}(x_0)}r\cdot\nabla \overline{w},
\int_0^t \int_\mathbb{R}w_{p\tau\tau}w_p g^2 dxd\tau \leqslant O(1) \bigg(\|w_p(t)\|^2 + \|w_{pt}(t)\|^2+ \|w_p(0)\|^2 + \|w_{pt}(0)\|^2+ \int_0^t \|w_{p\tau}\|^2 d\tau+ \varepsilon_0 \delta^{1/2}\bigg).
\dfrac{\partial \vec{h}}{\partial \vec{x}}(\vec{x}_0) :=\left[\begin{array}{ccc}\dfrac{\partial{h}_1}{\partial x_1}(\vec{x}_0) \cdots \dfrac{\partial{h}_1}{\partial x_n}(\vec{x}_0) \vdots \ddots \vdots \dfrac{\partial{h}_m}{\partial x_1}(\vec{x}_0) \cdots \dfrac{\partial{h}_m}{\partial x_n}(\vec{x}_0)\end{array}\right],
T_{\mu \nu}^{e f f}= \phi_{, \mu}\phi_{, \nu}- K \tilde{g}_{\mu \nu}+ \frac{b_{g}M^{4}}{( \zeta + b )^{2}}e^{- 2 \alpha \phi / M_{p}}\tilde{g}_{\mu \nu}+ T_{\mu \nu}^{( f , c a n o n i c a l )}- m F ( \zeta ) \overline{{{\Psi}}}^{\prime}\Psi^{\prime}\tilde{g}_{\mu \nu},
R_\ell\!\geq I(X;A,V,Y) \!-\!I(X;Y|A)\!+\!I(X;Z|A) &\;\qquad+ I(U;Y|A)-I(U;Z|A) &\;\overset{(a)}{=}I(X;A,V,Y) - I(X;Y|A,U) + I(X;Z|A,U)
A_n :=\frac{3 n^2-6 n-32}{4 n(n-1)}B_n :=\frac{3 n^4-12 n^3-52 n^2+128 n+192}{16 n^2(n-1)^2}C_n :=\frac{(n^4-20 n^2+64)(n-6)}{64 n^2(n-1)^3}
\Phi_1(t)=\left\lbrace\begin{aligned}\Phi(t) t>t_0 & \Phi(t_0) t\leq t_0\end{aligned}\right.\qquad\hbox{and}\Phi_2(t)=\left\lbrace\begin{aligned}0 t>t_0 & \Phi(t_0)-\Phi(t) t\leq t_0\end{aligned}\right. .
(\gamma^{\sharp}\otimes_{\mathbb{Z}}\mathbb{Q})\left(\pi_A^{-1}\sum_{i=0}^{j-1}\ell_i^{{A}}(v_{j-i}^{{A}})^{q^i}\right) = \pi_B^{-1}\sum_{i=0}^{j-1}\ell_i^{{B}}(v_{j-i}^{{B}})^{q^i}
\frac{1}{N^{\mathrm{in}}}\sum_{h=0}^{\infty}h \ \# \{\mbox{inner nodes in $\mathcal{T}$ of depth}h \}\geq \frac{1}{N^{\mathrm{in}}}\sum_{h=0}^{\lfloor log_2 N^{\mathrm{in}}\rfloor}h 2^h
\bigl(y_{u_1, \dots, u_k}- y_{u_1 p, u_2, \dots, u_k}\bigr)^2\! =\! F\! \left(\!\frac{\log u_1}{\log R}, ..., \frac{\log u_k}{\log R}\!\right)^{\!2}\!-\! F\!\left(\! \frac{\log u_1 \! +\! \log p}{\log R},..., \frac{\log u_k}{\log R}\! \right)^{\! 2}&\ll_F \frac{\log p}{\log R}\ll \frac{\log p}{\log R},
\int\limits_0^\infty\frac{d\tau}{\tau^2}\,\mathrm{exp}\left(-\frac{p}{2\tau}\right)\Gamma\left(q,\frac{c\tau}2\right)= \left(\frac p4\right)^{\frac{q-2}2}c^{\frac q2}K_q(\sqrt{pc})\,.
{}_2F_1\left(1,\tfrac{1}{2}+\nu;\tfrac{1}{2};-x^2\right)= (1+x^2)^{-\nu-\tfrac{1}{2}}{}_2F_1(\tfrac{1}{2}+\nu,-\tfrac{1}{2};\tfrac{1}{2}; \tfrac{x^2}{1+x^2}).
h\big([m_a^R(\sigma_{-\mathrm i /p}(y^*))]^*x\big) = \hat h \Big( \big([m_a^R(\sigma_{-\mathrm i /p}(y^*))]\,\hat{}\,\big)^*\hat x\Big)=\hat h \Big( \big([a(\sigma_{-\mathrm i /p}(y^*))\,\hat{}\,]\big)^*\hat x\Big) &=\hat h \Big( [(\sigma_{-\mathrm i /p}(y^*))\,\hat{}\,]^*a^*\hat x\Big) = h\big( (\sigma_{-\mathrm i /p}(y^*))^* m_{a^*}^Rx \big).
0= \int d^5 x \left( ie \frac{\delta\Gamma}{\delta \Phi^*_c}\Phi^*_cw_c - ie \frac{\delta\Gamma}{\delta \Phi_c}w_c \Phi_c + D_\alpha\frac{\delta\Gamma}{\delta\Gamma_\alpha^c}w_c + \frac{1}{\xi}\frac{\delta\Gamma}{\delta u_c}D^\alpha \Gamma_\alpha^c \right).
\delta_L\Theta^1(x,y) =[x,\Theta^1(y)]-[y,\Theta^1(x)]-\Theta^1[x,y] &=\sum_{i+j=N+1,~i,j>0}[f_0^x,[f_i^y,\alpha_j]]-[f_0^y,[f_i^x,\alpha_j]]-[f_i^{[x,y]},\alpha_j] ~~\mbox{(using equation \eqref{c3})}.
\begin{aligned}f_2(x_2) -f_2(\breve{x}_2^k)+(x_2-\breve{x}_2^k)^T( -A_2^T\widetilde{\lambda}^k &+\beta A_2^T A_2[\widetilde{x}_2^k-x_2^k]-rA_2^T[\widetilde{\lambda}^k-\lambda^k] )\geq 0,~\forall x_2\in\mathcal{X}_2. \end{aligned}
IV_{A}=-\varepsilon\int_{0}^{t}\int_{\mathbb{T}^{d}}\left(\partial^{\alpha}\mu^{n+1}\right)(\mu^{n}+\bar{m})\sum_{i=1}^{d}\sum_{j=1}^{d}\left[\left(\Theta_{p_{i}p_{j}}(\tau,x,\mu^{n},Dw^{n})\right)\left(\partial^{\alpha}\partial_{x_{i}x_{j}}^{2}w^{n}\right)\right]\dx d\tau,
\Theta^{i j}= \frac{2 \pi \alpha^{\prime}e}{g_{1}( 1 - e^{2})}( \delta_{0}^{i}\delta_{1}^{j}- \delta_{1}^{i}\delta_{0}^{j}) + \frac{2 \pi \alpha^{\prime}b}{g_{2}( 1 + b^{2})}( - \delta_{2}^{i}\delta_{3}^{j}+ \delta_{3}^{i}\delta_{2}^{j}) ,