harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
-\varphi_2\nu\big(1\otimes(r_1-r_2)\big) =\varphi_2\big(\alpha_B\pi^\flat(r_2)\otimes\frac{r_1}{r_2}\big)=\alpha_B\pi^\flat(r_2)\cdot(u-1)=\alpha_C\varphi^\flat(r_2)\cdot(u-1) &=u\alpha_C\varphi^\flat(r_2)-\alpha_C\varphi^\flat(r_2)=\alpha_C\varphi^\flat(r_1)-\alpha_C\varphi^\flat(r_2)
\begin{aligned}(u,v).f(A,C,S) =f(u^{-1}Av, u^{-1}Cv, u^{-1}Sv) = (a_{11}c_{22}- a_{22}c_{11})(s_{12}- s_{22}u_{12}+ s_{11}v_{12}) + (s_{11}a_{22}- s_{22}a_{11})(c_{12}- c_{22}u_{12}+ c_{11}v_{12}) + (c_{11}s_{22}- c_{22}s_{11})(a_{12}- a_{22}u_{12}+ a_{11}v_{12}) =f(A,C,S). \end{aligned}
\mathrm{E}\bigg[\Big\|\int_t^{t+h}\Psi \Big( \int_t^s \Phi \, \mathrm{d}W_r^K\Big) \,\mathrm{d}W_s^K - \sum_{i,j \in \mathcal{J}_K}\hat{I}_{(i,j)}^{Q,(D)}(h) \; \Psi \big(\Phi \tilde{e}_i, \tilde{e}_j\big)\Big\|_H^2\bigg] \leq 2 C \sum_{i=1}^L\mathrm{E}\Big[\big(\tilde{A}_{(i)}^Q(h) -\hat{A}_{(i)}^{Q,(D)}(h)\big)^2\Big]
(a \prec b) \prec c =~ a \prec (b * c), (a \succ b ) \prec c =~ a \succ (b \prec c), (a * b ) \succ c =~ a \succ ( b \succ c), (a \curlyvee b ) \prec c + (a * b ) \curlyvee c =~ a \succ (b \curlyvee c) + a \curlyvee (b * c),
\int d^2w \frac{1}{[z_1-w]^{1-\beta_1}[z_2-w]^{1-\beta_2}[z_3-w]^{1-\beta_3}}=\frac{\pi a(1-\beta_1\,,1-\beta_2\,,1-\beta_3)}{[z_2-z_1]^{\beta_3}[z_1-z_3]^{\beta_2}[z_3-z_2]^{\beta_1}}\,
\sum_{i = k}^{n^{2/3}}\ell_{\check \mu - i}\le \ell_{\check \mu}\sum_{i = k}^{n^{2/3}}2^{-i^2 + 2i}= 2\ell_{\check \mu}\sum_{i = k}^{n^{2/3}}2^{-(i - 1)^2}\le 2\ell_{\check \mu}\left (2^{-(k - 1)^2}+ \sum_{i = k}^{\infty}2^{-i^2}\right) =2\ell_{\check \mu}\left (2^{-(k - 1)^2}+ \frac{2^{-(k-1)^2}}{2(k-1) \ln 2}\right).
\frac{1}{\pi}=16i\sum_{k=0}^{\infty}s_{4C}(k)\frac{1-2\sqrt{2}(k+\frac{1}{2})}{(-16U_{2}^{4})^{k+\frac{1}{2}}},j_{4C}(\frac{1+\sqrt{-4}}{2})=-16(1+\sqrt{2})^{4}=-16U_{2}^{4}
\begin{cases}(v,u,\mathbf{w},\mathbf{h},\theta,\psi)|_{t=0}=(v_0,u_0,\mathbf{w}_0,\mathbf{h}_0,\theta_0,\psi_0)(y), y\in\Omega, (u,\mathbf{w},\mathbf{h},\theta_y,\psi)|_{\partial\Omega}=0.\end{cases}
\mathcal{R}= \int_1^{\xi}\left(\int_{1}^{\hat{X}}\frac{(X-\lambda/3 -\sqrt{X(X-\lambda)})dX}{2\sqrt{X(X-1)(X-\lambda)}}\right)\frac{d\hat{X}}{2\sqrt{\hat{X}(\hat{X}-1)(\hat{X}-\lambda)}}\mathcal{R}_\varphi = \lambda\int_1^{\xi}\frac{R_{\varphi}d\hat{X}}{2\sqrt{\hat{X}(\hat{X}-1)(\hat{X}-\lambda)}}.
a_j = \frac{1}{j!}\left.\frac{d^j}{dz^j}\right|_{z=q}\left(\frac{f(z)}{z^m}\right) = \sum_{k=0}^j\left(\frac{(-1)^k}{q^{m+k}}\binom{k+m-1}{k}\frac{f^{(j-k)}(q)}{(j-k)!}\right)
\sum_{m=-\infty}^{\infty}\sum_{n=0}^{\infty}\overline{N}_{t-1}(m,n)z^{m}q^{n}=\dfrac{(-zq,-z^{-1}q,\cdots,-z^{s}q,-z^{-s}q;q)_{\infty}}{(zq,z^{-1}q,\cdots,z^{s}q,z^{-s}q;q)_{\infty}},
F(g \kappa_h) = \prod_{\sigma\in \Sigma_\infty}z_\sigma^r \cdot \prod_{\sigma \in \Sigma_\infty \smallsetminus \Sigma_B}e^{i k_\sigma \theta_\sigma}\cdot \bigotimes_{\sigma \in \Sigma_{B,\infty}}\rho_{\sigma,k_\sigma,r}^\vee (e^{-i\theta_\sigma}) F(g).
\tilde{\Phi}= ( - 1 )^{j_{1}^{2}- j_{1}^{3}+ j_{2}^{2}- j_{2}^{3}+ r_{2}^{2}- r_{2}^{3}}\sqrt{\frac{( 2 r_{2}^{1}+ 1 ) ( 2 j_{2}^{1}+ 1 )}{( 8 \pi^{2})^{2}( 2 j_{1}^{1}+ 1 )}}\prod_{i = 1}^{3}( - 1 )^{j_{1}^{i}- r_{2}^{i}}\sqrt{( 2 r_{2}^{i}+ 1 ) ( 2 j_{1}^{i}+ 1 ) ( 2 j_{2}^{i}+ 1 )}.
S_{\Sigma}=\frac{1}{4\pi\alpha^\prime}\int_\Sigma d^2\sigma\sqrt g[g^{ab}G_{\mu\nu}(X)\partial_a X^\mu\partial_b X^\nu+i\epsilon^{ab}B_{\mu\nu}(X)\partial_a X^\mu\partial_bX^\nu+\alpha^\prime R\Phi(X)]
L\varphi_{\lambda}+ hB_{1}\varphi_{\lambda}+ h^2A_{2}\varphi_{\lambda}+ h^2\Phi_{\lambda}\varphi_{\lambda}+ h^2{\cal L}\varphi_{\lambda}+ \eta h^2 B_2 (I+hA_1)^{-1}\varphi_{\lambda}= g \mbox{in}\H.
dX_t=-\frac{\sigma\left(e^{X_t}\right)}{2}dt +dW_t+ \frac{\sigma_2-\sigma_1}{\sigma_1+\sigma_2}dL_{t}^{\left(0\right)}(X)=\mu(X_t) dt +dW_t+(p-q)dL_{t}^{\left(0\right)}(X)
\sum_{j=1}^n \alpha_j \gamma_j(\alpha) =\sum_{j=1}^{n-1}w_j\bar w_j \frac{d\psi}{2\pi}\wedge d\beta_{n-1}(w) +\sqrt{1-|w|^2}e^{i\psi}\sqrt{1-|w|^2}e^{-i\psi}\, \frac{d\psi}{2\pi}\wedge d\beta_{n-1}(w) =(|w|^2+1-|w|^2) \, \frac{d\psi}{2\pi}\wedge d\beta_{n-1}(w)= \frac{d\psi}{2\pi}\wedge d\beta_{n-1}(w),
\real\frac{z \mathtt{f}'_{a, \nu}(z)}{\mathtt{f}_{a, \nu}(z)}\geq a^{a/2}+\frac{a^{a/2}}{a\nu-a+1}\sum_{n=1}^\infty \frac{2|z|^2}{\mathtt{j}^2_{\nu, n}+|z|^2}=\frac{i|z| \mathtt{f}'_{a, \nu}(i|z|)}{\mathtt{f}_{a, \nu}(i|z|)}.
\mathbb{E}\left[e^{\alpha M_{s,t}^{(n)}}\right]=\mathbb{E}\left[e^{\alpha\max_{1\leq i\leq n}Z_{i}}\right]\leq\mathbb{E}\left[e^{\alpha\max_{1\leq i\leq n}Y_{i}}\right]=\mathbb{E}\left[e^{\alpha N_{s,t}^{(n)}}\right],
\begin{aligned}P = m_1 n_1 m_2 n_2 n_3^2 + m_2 n_2 m_3 n_3 n_1^2 + m_3 n_3 m_1 n_1 n_2^2 Q = m_1 n_2 n_3 + m_2 n_3 n_1 + m_3 n_1 n_2 \end{aligned}
p_1+p_2+\dots+p_l + r = p; q_1+q_2+\dots+q_l + s = q; p_i=0\ \Rightarrow\q_i=1;q_j=0\ \Rightarrow\p_j=1; r=p \Rightarrow\s=q-1; s=q \Rightarrow\r=p-1; r = 0 \Rightarrow\s \leq 1; s= 0 \Rightarrow\r \leq 1; r > 0 \Rightarrow\n_p = 0; s > 0 \Rightarrow\n_q = 0; n_p = 1 \Rightarrow\r = 0, p_1 \geq 1; n_q = 1 \Rightarrow\s = 0, q_1 \geq 1.
\mathcal{L}^{\langle 2\rangle}\cong\left(\bigotimes_{j=1}^g pr_j^*T\otimes \bigotimes_{j=1}^g pr_{j,g+1}^*\O(\Delta)^{\vee}\otimes\bigotimes_{j<k}^{g}pr_{j,k}^*\O(\Delta)\right)^{\otimes 8}\otimes \left(pr_{g+2}^* T\right)^{\langle 2\rangle},
\sum_{k=0}^{p-1}\frac{(\frac{d+r}{d})_k^{d-r-1}(\frac{r}{d})_k^{r+1}}{k!^d}\equiv -\big(\tfrac{r}{d}\big)^{r+1}\Gamma_p \big(-\tfrac{r}{d}\big)^d\pmod{p^2}.
\mathcal{W}(L\downarrow\mathfrak{g})=\mathcal{W}(L\downarrow\mathfrak{k})\downarrow\mathfrak{g}\subseteq BC_{r}\cup\{0\}\downarrow\mathfrak{g}=\Gamma_{\mathfrak{k}}\downarrow\mathfrak{g}=\Gamma_{\mathfrak{g}}=\Theta_{r}.
\int_0^r \rho^{n-1}\,E_-(\rho)\,|B[v](\rho)|\,d\rho=\sum_{j=0}^\infty \int_{r/2^{j+1}}^{r/2^j}\rho^{n-1}\,E_-(\rho)\,|B[v](\rho)|\,d\rho,
\frac{d}{dt}\int (\partial_x^lu)^2\chi(x+\nu t) \; dx+ \int (\partial_x^{l+2}u)^2 \chi'(x+\nu t) \; dx & \qquad\qquad\lesssim \int (\partial_x^lu)^2\chi'(x+\nu t;\epsilon/3,b+\epsilon) \; dx+ \int\partial_x^lu\partial_x^lQ(u)\chi(x+\nu t) \; dx & \qquad\qquad=: A+B
\frac{8\pi^{2}}{g^{2}(\mu)}+ C_{A}\log g^{2}(\mu) = \frac{8\pi^{2}}{g^{2}(M)}+ C_{A}\log g^{2}(M) + b_{0}\log \frac{\mu}{M}- \sum_{f}T_{f}\log Z_{f}(\mu,M) ,
J_{0}= \left( \begin{array}{c c}{\beta_{-}^{\star}}&{0}\\{0}&{\beta_{+}^{\star}}\\ \end{array}\right) , J_{+}= \left( \begin{array}{c c}{0}&{0}\\{N_{0}}&{0}\\ \end{array}\right) , J_{-}= J_{+}^{\dag},
\delta_{0,P}^{+},\delta_{0,P}^{-}=m-1,0 &\delta_{1,P}^{+},\delta_{1,P}^{-}=m,-1 &\delta_{2,P}^{+},\delta_{2,P}^{-}=\frac{m-1\pm \sqrt{(m-1)^{2}+8(m-1)}}{2},
\mu'(A_{i'j'}=1|A_{ij}=a_{ij},(i,j)\in E)=\frac{|\Sigma'(a':a_{i'j'}'=1)|^{1/2}}{|\Sigma'(a':a_{i'j'}'=0)|^{1/2}+|\Sigma'(a':a_{i'j'}'=1)|^{1/2}}
\begin{array}{ccl}Aw_{\tilde{l}_{m-1}+s,\tilde{l}_{n-1}+t}= \mu_1^{(mn)}w_{\tilde{l}_{m-1}+s,\tilde{l}_{n-1}+t}+ \displaystyle \sum\limits_{e=1}^{l_m}\sum\limits_{f=1}^{l_n}b^{st}_{ef}\u_{\tilde{l}_{m-1}+e,\tilde{l}_{n-1}+f}.\end{array}
\left( f_{\omega}^{R *}\, \tilde{\beta}_{\omega^{\prime \prime}\omega}^{R *}+ f_{\omega}^{R}\, \tilde{\alpha}_{\omega^{\prime \prime}\omega}^{R}\right) \left( \tilde{\beta}_{\omega^{\prime \prime}\omega^{\prime}}^{R}\, f_{\omega^{\prime}}^{R}+ \tilde{\alpha}_{\omega^{\prime \prime}\omega^{\prime}}^{R *}\, f_{\omega^{\prime}}^{R *}\right) \, ,
\lim_{\beta\rightarrow 2\pi}\mbox{Tr}_if_\beta^{\pm}(x,x)=\lim_{\beta\rightarrow 2\pi}\mbox{Tr}_i\bar{f}_\beta^{\pm}(x,x)=\frac 12\left(\delta(x,x_1)+\delta(x,x_2)\right)~~~,
R^{D}( \lambda ) = R^{C}( \lambda ) \ ; \L_{1 \ell}^{D}( \lambda ) = R_{1 \ell}^{D}( \lambda - i / 2 ) = F L_{1 \ell}^{A}( \lambda ) F \ ; \, t i l d e{R}^{D}( \lambda ) = \tilde{R}^{B}( \lambda ) \ ;
(M_0f,\bar{f}) =(T_0(U_\theta(f)),U_\theta(\bar{\tilde{f}})) =\tau_\theta(T_0(U_\theta(f))U_\theta(\bar{\tilde{f}})) \leq \|T_0(U_\theta(f))\|_{L_{p^\prime}(\mathcal{R}_\theta^d)}\|U_\theta(\bar{\tilde{f}})\|_{L_{p}(\mathcal{R}_\theta^d)}\lesssim \|U_\theta(f)\|_{L_{p}(\mathcal{R}_\theta^d)}^2.
\left( \partial_{\mu}\phi \right) \left( \partial_{\nu}\phi \right) = e^{2 \phi}\left( \partial_{\mu}a \right) \left( \partial_{\nu}a \right) \ , \qquad \partial_{\mu}\left( e^{2 \phi}\partial_{\mu}a \right) = 0 \ , \qquad \partial^{2}\phi = - e^{2 \phi}\left( \partial_{\mu}a \right) \left( \partial_{\mu}a \right) \ .
\int_{\mathcal{V}\times \mathcal{V}}\varphi_\lambda d \widetilde{\pi}\geq \int_{\mathcal{V}}g \, d \gamma - \lambda_1 \boldsymbol{K}_{1}(\mu_1, \gamma_1) - \lambda_2 \boldsymbol{K}_{2}(\mu_2, \gamma_2) - \epsilon = I_{\mathrm{D}, \lambda}[\gamma] - \epsilon.
\alpha_\rho(q_1 ,q_2) = \rho(\chi(q_1,q_2))M_{q_2}^{-1}M_{q_1}^{-1}M_{q_1q_2}&= \widetilde{\rho}\left(\sigma(q_{1}q_{2})^{-1}\sigma(q_{1})\sigma(q_{2})\right) \widetilde{\rho}(\sigma(q_2))^{-1}\widetilde{\rho}(\sigma(q_1))^{-1}\widetilde{\rho}(\sigma(q_1q_2))=1
\left(\hat{\Gamma}_0 \hat{\Gamma}_i \hat{\Gamma}_j \right)^{\dagger}= \hat{\Gamma}_j^{\dagger}\hat{\Gamma}_i^{\dagger}\hat{\Gamma}_0^{\dagger}= \hat{\Gamma}_j \hat{\Gamma}_i (-\hat{\Gamma}_0) = \hat{\Gamma}_0 \hat{\Gamma}_i \hat{\Gamma}_j
\textstyle \mu_{_{\rm SM}}= \frac{\mathrm{tr}\left(\mathbf{H}\mathbf{S_{_\mathrm{SM}}}\mathbf{H}^{\rm H}\right)}{\mathrm{tr}\left(\frac{\mathbf{H}\mathbf{S_{_\mathrm{SM}}}\mathbf{H}^{\rm H}}{\sigma^2\,\ln2}\left(\mathbf{I}_{N_R}+\left(1-\rho_{_{\rm SM}}\right)\sigma^{-2}\mathbf{H}\mathbf{S_{_\mathrm{SM}}}\mathbf{H}^{\rm H}\right)^{-1}\right)}.
\begin{array}{c}U_0(z)=\frac 12(u_0+u_{-1})+\frac 1{\sqrt{2}}(u_0-u_{-1})\,z_{per}V_0(z)=\frac 12(v_0+v_{-1})+\frac 1{\sqrt{2}}(v_0-v_{-1})\,z_{per},\end{array}
S_{F , n}= \int d^{3}x \left[{\bar{\chi}}_{n}^{( 1 )}( \not \! d + \omega_{n}+ e \varphi ) \chi_{n}^{( 1 )}+{\bar{\chi}}_{n}^{( 2 )}( x ) ( \not \! d - \omega_{n}- e \varphi ) \chi_{n}^{( 2 )}( x ) \right] .
< \psi_{f}| e^{- i / \hbar ( t_{2}- t_{1}) \hat{H}_{\mathrm{e f f}}}| \psi_{i}> = i \delta_{n , m}( t_{2}- t_{1}) \int_{- \infty}^{\infty}d \lambda \, \tilde{\psi}_{f}^{*}( \lambda ) \, e^{- i / \hbar \lambda ( t_{2}- t_{1}) ( \hbar \omega ( n + 1 / 2 ) - \Lambda )}\, \tilde{\psi}_{i}( \lambda ) \ ,
j_0:=\max\Bigg\{\frac{2}{sq}\log_4\left(\frac{32^{q+1}N(1+\left|B_1\right|)}{s \delta^{q-1}}\right), \frac{2\gamma(p-1)}{p\gamma - N}\log_4\left(\frac{4}{\delta|B_1|^{\frac{1}{\gamma(p-1)}}}\right) \Bigg\}.
|F(z,\sigma + it)| \ll_{\sigma,\lambda}\frac{y^{1-\sigma-\lambda}t^{\lambda}}{|\zeta(2\sigma + 2it)|}\sum_{n=1}^{\infty}n^{\sigma-\frac12}\tau_0(n) n^{\frac12-\sigma-\lambda}\ll_{\sigma,\lambda}\frac{y^{1-\sigma}(t/y)^{\lambda}}{|\zeta(2\sigma + 2it)|},
X_i(A_j f) = 0 (Y_i + 2\delta_{ij}) (A_j f) = 0 Z (A_j f) = 0\intertext{If $A_j f = 0$ for all $j$ then}X_i(B_jf) = 0 (Y_i + \delta_{ij}) (B_j f) = 0 (Z+1) (B_j f) = 0\intertext{If $B_j f = 0$ for all $j$ then}X_i (C_j f) = 0 Y_i(C_j f) = 0 (Z+2) (C_j f) = 0
\langle Re\,T_{\widetilde{\varphi},\pm}u_1, w_1 \rangle_{L^2}+ \langle Im\,T_{\widetilde{\varphi},\pm}u_2, w_2 \rangle_{L^2}= \langle Re\, v, w_1\rangle_{L^2}+\langle Im\,v, w_2\rangle_{L^2}
\langle y^*-\tilde{y}^k,\tilde{\lambda}^k-\beta(\tilde{y}^k-y^k)-\beta(y^k-\tilde{y}^k)\rangle_{\mathcal{L}^2}=\langle y^*-\tilde{y}^k,\tilde{\lambda}^k\rangle_{\mathcal{L}^2}\geq0.
\sum_{k=0}^\infty\frac{(a;q)_k (1-aq^{3k})(d;q)_k(q/d;q)_k(b;q^2)_k}{(q^2;q^2)_k (1-a)(aq^2/d;q^2)_k (adq;q^2)_k (aq/b;q)_k}\frac{a^k q^{k+1\choose 2}}{b^k}=\frac{(aq;q^2)_\infty (aq^2;q^2)_\infty (adq/b;q^2)_\infty (aq^2/bd;q^2)_\infty}{(aq/b;q^2)_\infty (aq^2/b;q^2)_\infty (aq^2/d;q^2)_\infty (adq;q^2)_\infty}.
\delta^N(x,v,t) = & \, \left| \int_0^t dt_1 \int_0^{t_1}dt_2 \, \Big[E^N\left(X^N(t_2), t_2\right) - E^{N+1}\left(X^{N+1}(t_2), t_2\right)\Big] \right|\leq &\int_0^t dt_1 \int_0^{t_1}dt_2 \, \left[ \mathcal{F}_1(X^N(t_2), X^{N+1}(t_2),t_2) + \mathcal{F}_2(X^N(t_2), X^{N+1}(t_2),t_2) \right],
\begin{array}{l c l}{{{\widehat \Sigma}_{m}^{( r , s )}}}&{{=}}&{{\oplus_{k = r}^{s}{\widehat \Sigma}_{m}^{( k , k )}( \theta )}}\\{{}}&{{=}}&{{\oplus_{k = r}^{s}\oplus_{l = 0}^{k}{\theta}^{l}{\Sigma}_{m}^{( k - l , k - l )}}}\\ \end{array}
\int_{0}^{1}x^{-\alpha}(1-x)^{\alpha-\beta-1}G_{p,q}^{m,n}(\begin{matrix}a_{p}\\b_{q}\end{matrix}|zx)dx=\Gamma(\alpha-\beta)G_{p+1,q+1}^{m,n+1}(\begin{matrix}\alpha,a_{p}\\b_{q},\beta\end{matrix}|z),
\overline{\gamma}_{( X,\alpha_{X},\beta_{X}) ,( Y,\alpha_{Y},\beta_{Y})}^{{c,d,\nu}}=\tau \big[ \big( \alpha_{X}\beta_{X}^{-1}\big) \otimes \big( \alpha_{Y}\beta_{Y}^{-1}\big)^{-1}\big] =\tau \big[ \big( \alpha_{X}\beta_{X}^{-1}\big) \otimes \big(\alpha_{Y}^{-1}\beta_{Y}\big) \big] ,
\frac{{\partial^2_{y}{\partial^n_x u}}}{u^s_y +{u}_y}= \partial_y g_n + 2(g_n \eta_2 - 2 \partial^{-1}_y g_n \eta_2^2) + \partial^{-1}_y g_n \left(\frac{u_{yyy}^s +{u}_{yyy}}{(u^s_y + \tilde{u}_y)}\right),
\overline{p}=D_{1}^{-1}pD: N_{out}\rightarrow \operatorname{ker}(\hat{\gamma}) \overline{\sigma}_{2}=C_{2}\sigma_{2}\underline{C}^{-1}: \frac{\operatorname{ker}(\hat{\alpha})}{\operatorname{im}(\hat{\gamma})}\rightarrow \operatorname{ker}(\hat{\alpha})
\int_{Q}f( v_h + \theta h \partial_{t}v_h) \, dx dt = \int_{Q}(\partial_{t}u_h v_h + \theta h \partial_{t}u_h \partial_{t}v_h + \nabla_x u_h \cdot \nabla_x v_h + \theta h \nabla_x u_h \cdot \nabla_x \partial_{t}v_h)\,dxdt - \int_{\partial Q}n_x \cdot \nabla_x u \,(v_h + \theta h \partial_{t}v_h)\, ds.
\mathcal{A}(\mu) := \sum_{l=1}^{Q}\frac{\alpha_{l}}{|\Omega_{1D}|}A(\mathcal{P}(\mu) ;\mu_{l}), \mathcal{A}^{h}(\mu) := \sum_{l=1}^{Q}\frac{\alpha_{l}}{|\Omega_{1D}|}A(\mathcal{P}^{h}(\mu) ;\mu_{l}).
V_{2}( t ) = \exp i{\frac{e}{\hbar c}}\int \! \int d \vec{x}d \vec{y}J_{0}^{( 2 )}( \vec{x}, t ) \left[ g_{2}( \vec{x}, \vec{y}){\theta_{2}^{( 2 )}( \vec{y}, t )}+ G_{p}( \vec{x}, \vec{y}) \partial_{k}a_{k}^{( 2 )}( \vec{y}, t ) \right] \, .
|q_{t,s}(x,y)-q_{t,s}(x',y)| \leq c_2\left( |x-x'|^{\beta-\gamma}\wedge 1\right) \times\Big( (\varrho^0_\gamma+\varrho^\beta_{\gamma-\beta})(s-t,x-y)+(\varrho^0_\gamma+\varrho^\beta_{\gamma-\beta})(s-t,x'-y)\Big) .
{\rm tr}\left[ Q\,\Gamma^{,\rho}\Gamma^{,\sigma}\right] =-4k^{\rho}k^{\sigma}t^{2}\xi^{-2}-\frac{2}{k^{2}}P^{\rho \sigma}\left[ \cosh\left( k^{2}(-1+\xi^{-1})t\right) -1\right]
\sum_{k=1}^{K}\Vert \widehat{\beta}_{-k}(\widehat{\lambda})-\beta \Vert_{2,n,k}^{2}\lesssim \sum_{k=1}^{K}\Vert \widehat{\beta}_{-k}(\bar{\lambda}_{0})-\beta \Vert_{2,n,k}^{2}& +\sqrt{\frac{\log^{r+1}n}{n}}\sum_{k=1}^{K}\Vert \widehat{\beta}_{-k}(\widehat{\lambda})-\widehat{\beta}_{-k}(\bar{\lambda}_{0})\Vert_{2,n,k},
\begin{array}{l}s_0(z_1,\dots,z_n,q/2) = (q-z_1,z_2,\dots,z_n,q/2), begin{align*}.5ex] s_n(z_1,\dots,z_n,q/2) = (z_1,\dots,z_{n-1},-z_n,q/2),\end{array}
\frac{\int_0^{L_{m-1}}\frac{|v'|^2 V_{\beta_m}}{|\beta_m'|}+ \frac{k^2 |v|^2 | \beta_m'|}{V_{\beta_m}}\,dt}{\int_0^{L_{m-1}}|v|^2 V_{\beta_m}|\beta_m'| \,dt}= \frac{\int_0^{L_m^*}\frac{|w'|^2 V_{\gamma_m}}{|\gamma_m'|}+ \frac{k^2 |w|^2 | \gamma_m'|}{V_{\gamma_m}}\,dt}{\int_0^{L_m^*}|w|^2 V_{\gamma_m}|\gamma_m'| \,dt}
\tilde p_0 = \tilde p, \tilde p_i = \frac{d\tilde p_{i-1}}{2d - \tilde p_{i-1}}, \tilde p_i > \tilde p_{i-1}\{\rm for}\i =1,2,3,..,N, 2d \geq \tilde p_{N}> \frac{4d}{3}\geq \tilde p_{N-1}.
M_p^2 = M_5^3 \int_{-\infty}^{\infty}e^{2A_0(y)}\, d y =\frac{4 M_5^3 \sqrt{\pi}}{a(k^2-1)}\frac{\Gamma\left((k^2+3)/4)\right)}{\Gamma\left((k^2+1)/4\right)}\;,
Q =\frac{1}{12}\left(\frac{\sigma^2}{4}+ 6\sigma H^2 +36H^4 +\vert \mathring{A}\vert^4 - 3\sum_i\mu_i^4 +6\sum_i \mu_i^2\left(a_{ii}-\frac{\sigma}{3}\right)\right. \left.\ \ -12H\sum_i \mu_i a_{ii}-18H^2\vert \mathring{A}\vert^2 +12H\sum_i \mu_i^3 -3 \vert\mathring{a}\vert^2\right).
\left(\begin{matrix}x=0 x=\infty\,\left( \frac12 \right) \begin{matrix}0 \theta^0\end{matrix}\overbrace{\begin{matrix}1 0 -\frac{3t_2}{2}0 \frac{t_1}{2}+\frac{3{t_2}^2}{8}\theta^\infty_1/2 -1 0 \frac{3t_2}{2}0 -\frac{t_1}{2}-\frac{3{t_2}^2}{8}\theta^\infty_1/2\end{matrix}}\end{matrix}\right) ,
\sum_{j=0}^{n-1}\phi(F_j(x)) = \sum_{j=0}^{n-1}\phi \circ h_j \circ f^j (h_0^{-1}(x)) & = \sum_{i=0}^{N-1}\sum_{\ell=0}^{\big[\frac{n}{N}\big]}\sum_{0\le \ell N+i \le n-1}(\phi \circ h_i) ( f^{\ell N+i}(y)) & = \sum_{i=0}^{N-1}\sum_{\ell=0}^{\big[\frac{n}{N}\big]}\sum_{0\le \ell N+i \le n-1}\psi_i( f^{\ell N}(y))
\mathbf{V}_{i'}^{*}=\left\{\mathbf{v}_{i',1}^{*},...,\mathbf{v}_{i',K}^{*}\right\}=\arg\min_{\mathbf{V}_{i'}\in\mathcal{C}_{i'}}\rho_{K}\left(\mathbf{A}_{i'}\left(\mathbf{V}_{i'}\right)\right).
\frac{1}{\psi_1}= \frac{1}{\sigma_h^2+\sigma_\xi^2}+ \frac{\sigma_v^2}{\sigma_n^2}\frac{1}{\psi_{\ell+1}}= \frac{1}{\alpha^2\psi_\ell+\sigma_\xi^2}+ \frac{\sigma_v^2}{\sigma_n^2},
\mathbb{Q}_{x\to 0}^{\delta,T}*\mathbb{Q}_{x'\to 0}^{\delta',T}=\mathbb{Q}_{x+x'\to 0}^{\delta+\delta',T}\mathbb{Q}_{0\to x}^{\delta,T}*\mathbb{Q}_{0\to x'}^{\delta',T}=\mathbb{Q}_{0\to x+x'}^{\delta+\delta',T}.
S = - \frac{1}{2}\int \! d \sigma d \tau [ \partial_{\mu}\chi_{1}\partial^{\mu}\chi_{1}+ 2 \chi_{2}\partial_{\mu}\chi_{1}\partial^{\mu}\chi_{3}+ \frac{1}{1 +{\chi_{1}}^{2}}\partial_{\mu}\chi_{2}\partial^{\mu}\chi_{2}+ ({\chi_{2}}^{2}+ \frac{1}{1 +{\chi_{1}}^{2}}) \partial_{\mu}\chi_{3}\partial^{\mu}\chi_{3}+ \partial_{\mu}t \partial^{\mu}t ] .
\lambda^{k-1}_k\sigma_k+\sum_{i=k+1}^K\lambda^{k-1}_i\sigma_i =\sum_{i=1}^m\lambda^{k-1}_i\sigma_i &=(Q^{k-1}-Q^K,Q^0\ominus Q^K) &=(Q^{k-1}-Q^K,Q^{k-1}\ominus Q^K) &>0.
\int_{0}^{t}\langle J\circ \phi(s,\tilde{X})\circ J^{-1}(\cdot), a \rangle_{\bar{U}}\,\bar{\tilde{W}}_s = \sum_{k=1}^{\infty}\int_{0}^{t}\langle J \phi(s,\tilde{X}) J^{-1}(Je_k),a\rangle_{\bar{U}}\,\tilde{\beta}_k(s),
\begin{aligned}\frac{\partial^2 (P_k, P_{RF})}{\partial (P_k)^2}= \frac{\partial^2 \frac{R_{ave}}{P_k + P_{RF}}}{\partial (P_k)^2}&= \frac{\frac{-\ln2}{(P_k\ln2)^2}-\frac{1}{P_k\log_2\log_2K\ln2}}{(P_k + P_{RF})^2}&- \frac{2[\frac{1}{P_k\ln2}-\log_2(\frac{P_k}{N_0}\log_2\log_2K)]}{(P_k + P_{RF})^3}.\end{aligned}
\frac{\sin^{2}A + \sin^{2}B}{\sin^{2}C}= \frac{\frac{a^{2}}{4 R^{2}}+ \frac{b^{2}}{4 R^{2}}}{\frac{c^{2}}{4 R^{2}}}= \frac{a^{2}+ b^{2}}{c^{2}}= 2
dC_4-C_2\wedge db_2=\frac{i}{6}\bar{L}\tau_3\tau_1 \hat{L}^3 L +\frac{1}{30}(\varepsilon^{a_1...a_5}L^{a_1}\wedge \cdots \wedge L^{a_5}+\varepsilon^{a'_1 ...a'_5}L^{a'_1}\wedge \cdots \wedge L^{a'_5}).
\frac{a_{1}\vee a_{2}\vee\cdot\cdot\cdot\vee c,b_{1}\vee b_{2}\vee\cdot\cdot\cdot\vee\neg c}{a_{1}\vee a_{2}\vee\cdot\cdot\cdot\vee b_{1}\vee b_{2}\vee\cdot\cdot\cdot}
\alpha_0^{*}=\alpha_0^{*}(\beta_0)=\bigg(\frac{\theta_b-\theta_m}{2l_b\gamma_b}\bigg[\dfrac{2l_m\gamma_m(\beta_0)^{\nu+1}}{\theta_b-\theta_m}-\dfrac{u_c}{\Phi_2[\beta_0,+\infty,u_2]}\bigg]\bigg)^{\frac{1}{{\nu+1}}}
\operatorname{sq}^t(S^\delta(V))\cong \bigoplus_{\begin{subarray}{c}\boldsymbol{b}=(b_1,\ldots,b_r)\in\mathbb N^r |\boldsymbol{b}|=b_1+\cdots+b_r=\delta b_1\mu_1+\cdots+b_r\mu_r=t\end{subarray}}S^{b_1}(V_1/V_0)\otimes\cdots\otimes S^{b_r}(V_r/V_{r-1})
\dot{\Lambda}^{\frac{d}{p}-1}e^{(t - \tau)\Delta}\mathbb{P}\nabla .\big(u(\tau) \otimes v(\tau)\big) =\frac{1}{(t - \tau)^{\frac{d}{2}(\frac{1}{p}+1)}}K\Big(\frac{.}{\sqrt{t - \tau}}\Big)*\big(u(\tau) \otimes v(\tau)\big),
R_{12}(\frac{z_\pm}{w_\pm}) L_1^{\pm}(z) L_2^{\pm}(w) = L_2^{\pm}(w) L_1^{\pm}(z) R_{12}(\frac{z_\pm}{w_\pm}) \,, R_{12}(\frac{z_\pm}{w_\mp}) L_1^{+}(z) L_2^{-}(w) = L_2^{-}(w) L_1^{+}(z) R_{12}(\frac{z_\mp}{w_\pm}) \,,
P_{\min}^{\rm{S-N}}\left({\overline S,\overline \varepsilon ,\overline R}\right) =\mathop{\arg \min}\limits_{{W_m}\in \left[{0,W}\right]}\;\left({{p_K}\left({\bar S,\bar \varepsilon ,{W_m}}\right) + p_{b,m}^* + p_{b,o}^*}\right).
\mbox{heterogamous if$\pi_{11}+ \pi_{22}< \pi_{12}+ \pi_{21}$,}&\mbox{panmictic if\qquad\ \ $\pi_{11}+ \pi_{22}= \pi_{12}+ \pi_{21}$,and}&\mbox{homogamous if\quad\ $\pi_{11}+ \pi_{22}> \pi_{12}+ \pi_{21}$.}
\lim_{n\to\infty}\frac{(\mathcal{X}^n,\mathcal{Y}^n) - (\mathcal{X},\mathcal{Y})}{t_n}=\big(\Pi'_{K}(\mathcal{X}+\mathcal{Y};\mathcal{Z}_1+\mathcal{Z}_2),\Pi'_{K^{\circ}}(\mathcal{X}+\mathcal{Y};\mathcal{Z}_1+\mathcal{Z}_2)\big) = \big(\mathcal{Z}_1,\mathcal{Z}_2\big).
z = \frac{3 \left( 1 + \frac{2{\bf Q}^{2}}{a^{2}}\right)^{2}}{\left[\rho^{2}\left( 1 + \frac{2{\bf Q}^{2}}{a^{2}}\right)^{2}+ 6 \rho \left( 1 + \frac{2{\bf Q}^{2}}{a^{2}}\right) + 27 \right]}.
\begin{aligned}\langle\mathbf{u}_h^{n}\cdot\mathbf{n}\hat{c}_h^{n},\zeta^{n}_c\rangle_{\partial\mathcal{T}_h}=-\langle|\mathbf{u}_h^{n}\cdot\mathbf{n}|\lambda_h^{c,{n}},\zeta^{n}_c\rangle_{\partial\mathcal{T}_h^{in}}+\langle|\mathbf{u}_h^{n}\cdot\mathbf{n}|c_h^{n},\zeta^{n}_c\rangle_{\partial\mathcal{T}_h^{out}}=0.\end{aligned}
\Delta U(M_s,M^d_s)-\langle \partial_x U(M_{s-},M^d_{s-}),\Delta M_s\rangle - \langle \partial_y U(M_{s-},M^d_{s-}),\Delta M^d_s\rangle = V(M_{s-}+M^d_{s-}+2\Delta M_s,M^d_{s-}-M_{s-})- V(M_{s-}+M^d_{s-},M^d_{s-}-M_{s-}) \quad-\langle \partial_x V(M_{s-}+M^d_{s-},M^d_{s-}-M_{s-}),2\Delta M_s\rangle \leq 0,
F_{\ast}\left( j_{F}^{p}(x) - j_{F}^{p}(y) \right) = \left(\sum_{i}\rho_{\ast}\left( j_{\rho}^{p}(x_{i}) - j_{\rho}^{p}(y_{i}) \right)^{p^{\prime}}\right)^{\frac{1}{p^{\prime}}}.
\lim_{c_1,c_2\to\infty}F_{\lambda,k,r}(c_1,c_2;b_1,\dots,b_\lambda;a;q)=\lim_{c_1,c_2\to\infty}G_{\lambda,k,r}(c_1,c_2;b_1,\dots,b_\lambda;a;q).
\hat{S}= \int d^{1 0}\hat{x}\sqrt{| \hat{g}|}\left\{e^{- 2 \hat{\phi}}\left[ \hat{R}- 4 ( \partial \hat{\phi})^{2}\right] +{\textstyle \frac{1}{2 \cdot 7 !}}e^{2 \hat{\phi}}\hat{\tilde{H}}{}^{2}\right\},
\begin{aligned}\underset{t\in\left[ t_k, t_{k+k_T}\right]}{\sup}\!\left\| \bar{\boldsymbol{\psi}}(t) \!-\! \tilde{\boldsymbol{\psi}}^k(t)\!\right\|_2 \!\le\! \frac{C_{\mathbf{f}}b_{S,k+1}}{2}\!+\! C e^{L (T+b_{S,1})}\!,\!\end{aligned}
\zeta= \left(\dfrac{1}{1-\omega}y_3, \dfrac{1}{1-\omega^2}y_4\right)\tau_B^{-1}= \frac{1}{2}\left( \frac{y_3}{(1-\omega)y_1}+ \frac{y_4}{(1-\omega^2)y_1}, \frac{y_3}{(1-\omega)y_2}- \frac{y_4}{(1-\omega^2)y_2}\right).
|e_t^n|^2 = |e_{t_0}^n|^2+2 \int_{t_0}^{t}e^n_s \bar b_s^n ds + 2\int_{t_0}^{t}e^n_s \bar \sigma^{n}_s dw_s +\int_{t_0}^{t}|\bar{\sigma}^n_s|^2ds &+ 2\int_{t_0}^{t}\int_{Z}e^n_{s}\bar\gamma_s^n(z) \tilde N(ds,dz) +\int_{t_0}^{t}\int_{Z}|\bar \gamma_s^n(z)|^2 N(ds,dz)
\left\| \nabla u - (\nabla u)_{B_{R/2}}\right\|_{\underline{L}^2(B_{R/2})}= \inf_{\ell \in \mathcal{P}_1}\left\| \nabla u - \nabla \ell \right\|_{\underline{L}^2(B_{R/2})}\leq \frac{C}{R}\inf_{\ell \in \mathcal{P}_1}\left\| u - \ell \right\|_{\underline{L}^2(B_{R})}.
t_{\mu\nu}(x;\gamma) ={2\over3}\left( t^1_\mu t^2_\nu + t^2_\mu t^1_\nu -{1\over2}g_{\mu\nu}(t^1\cdot t^2) \right)-{1\over3}\left(t^1_\mu t^1_\nu + t^2_\mu t^2_\nu -{1\over2}g_{\mu\nu}\right)_.
\big\langle g(t -1,x) \, g(s-1,y) \big\rangle = \frac12 \int_{|t-s|}^{t+s}\frac{1}{\sqrt{4 \pi a_0 \tau}}\exp \Big(- \frac{(x-y)^2}{4 a_0 \tau}\Big) \, d \tau \,
\begin{aligned}D\lambda_1c_1+D\lambda_1c_2x^2+D\lambda_1|x|^{m}\ge Dc_2\gamma_1+Dc_2x^2+D(\lambda_1-\delta_1)\gamma_1|x|^{m-2}+D(\lambda_1-\delta_1)|x|^{m}.\end{aligned}
\liminf_{t\to\infty}\frac{\varphi(t)}{g(G^{-1}(t))}= b\liminf_{t\to\infty}\frac{g(x(t-\tau(t)))}{g(G^{-1}(t))}&\geq b\liminf_{t\to\infty}\frac{g(\lambda(1-\epsilon)^2\{(1+\epsilon)(1-q)\}^{-1/(\beta-1)}G^{-1}(t))}{g(G^{-1}(t))}&=b\lambda^\beta(1-\epsilon)^{2\beta}\{(1+\epsilon)(1-q)\}^{-\beta/(\beta-1)}.
{\bar{\Delta}}_{{i}_{\alpha}{{j}_{\beta}}}=\overbracket[0.5pt]{{\varphi}^{\left(+ \right)}_{{i}_{\alpha}}{{\varphi}}^{\left( - \right)}_{{j}_{\beta}}}={\left\langle T\left({{\varphi}_{{{i}_{\alpha}}}^{\left( +\right)}{\varphi}_{{{j}_{\beta}}}^{\left( -\right)}}\right) \right\rangle}_{0}\left( \alpha ,\beta =1,2,\ldots ,n\right) \ \ .
\Gamma(x, y)=P(x^2+y^2)^3+Q(x^3-3xy^2)^2-\frac{9}{4L}(x^2+y^2)(x^3-3xy^2) &+(\frac{27}{16L^2}-L)(x^2+y^2)^2+(2-\frac{27}{64L^3})(x^3-3xy^2)-\frac{9}{8L}(x^2+y^2)+\frac{27}{256L^3},
s''(t+k\bar{\tau}) = \bigoplus_{i = t + k\bar{\tau}}^{t + (k+1)\bar{\tau}-1}\bar{s}(i) = \bar{\nu}^{k}\left(\bigoplus_{i = t}^{t+\bar{\tau}-1}\bar{s}(i)\right) = \bar{\nu}^{k}s''(t),
R_{\;B}^A\;=\; \left( \begin{array}{cc}\gamma^{\mu\nu}\,R_{\mu\nu}^{(1)}\;_{\;B}^A k\,\gamma^\mu\,\gamma^5\,e^{-\sigma}\,P_\mu\;_{\;B}^A k^*\,\gamma^\mu\,\gamma^5\,e^{-\sigma}\,Q_\mu\;_{\;B}^A\quad \gamma^{\mu\nu}\,R_{\mu\nu\;B}^{(2)\,A}\,+\,(k^*k-1)\,L_{\;B}^A\end{array}\right) \;.
\|f\circ \phi\|^2_{g_\delta}= \int |f\circ \phi(x)|^2 |g_\delta(x)|^2 dx \leq |\tfrac{1}{\phi'}|_{C^0}|\tfrac{g_\delta}{g_\delta\circ \phi}|^2_{C^0}\int |f\circ \phi(x)|^2 |g_\delta\circ \phi(x)|^2 |\phi'(x)| dx\leq C \|f\|^2_{g_\delta},
\mathbb{P}_x(\tau^{(-a,a)^{{\rm c}}}<\tau^{\{0\}}) = \mathbf{P}_{\log|x|,{\rm sign}(x)}(T^+_{\log a}<\infty) = \mathbf{P}_{0,{\rm sign}(x)}(T^+_{\log (a/|x|)}<\infty)