harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{1}{U_{o}}=R_{fo}+R_{fi}\cdot\frac{D_{o}}{D_{i}}+\frac{D_{o}}{2k_{w}}\cdot ln\frac{D_{o}}{D_{i}}+\frac{1}{h_{o}}+\frac{1}{h_{i}}\cdot\frac{D_{o}}{D_{i}}
\lambda_{k,n}(\alpha_S) = \min_W \max_{w \in W}\left\{\frac{\int_0^{L_S}| w' |^2 (1+|F_S|^2)^{1/2}+ \frac{k^2 w^2}{(1+|F_S|^2)^{1/2}}\,dt}{\int_0^{L_S}w^2 (1+|F_S|^2)^{1/2}\,dt}\right\}
A_r(\theta) =m_r{\rm e}^{\theta}-\sum_{\ell =0,\bar{0}}\int_{-\infty}^{\infty}\frac{d\theta'}{2\pi}\Phi_{r,\ell}(\theta-\theta') \ln (i a_{\ell}+B_{\ell}(\theta') ) \quad(r=1,2), B_{\ell}(\theta) = -\sum_{r=1,2}\int_{-\infty}^{\infty}\frac{d\theta'}{2\pi}\Phi_{r,\ell}(\theta-\theta'){\rm e}^{-A_r(\theta')}\quad(\ell=0,\bar{0})
-\det(\eta_{ab}I + F_{ab}) = I + \frac{1}{2}F^2 + \frac{1}{3}F^3 -\frac{1}{4}\left[F^4 - \frac{1}{2}(F^2)^2\right] + \frac{1}{5}F^5 +\frac{1}{12}(F^2F^3 + F^3F^2),
C_{2k-1,k}= [(4k-2) a + 1, (4k-2) a + 3] \cup [(4k-2) a +5, (4k-1)a], D_{2k-1,k}= [(4k-1) a + 1, (4k-1) a + 3] \cup [(4k-1) a +5, (4k)a].
\Gamma = \frac{8}{m}\,\frac{k_{0}^{2}\,(V_{1}-k_{0}^{2})^{3/2}\,(V_{2}+ k_{0}^{2})^{1/2}}{V_{1}\,(V_{1}+V_{2})\,(1 + \sqrt{V_{1}- k_{0}^{2}}\,r_{1})}\, \exp\left\{-\,2\,\sqrt{V_{1}- k_{0}^{2}}\,\Delta r \right\}.
R_{\mu \nu \sigma}^{\lambda}= - \partial_{\sigma}\Gamma_{\mu \nu}^{\lambda}+ \partial_{\nu}\Gamma_{\mu \sigma}^{\lambda}+ \Gamma_{\rho \nu}^{\lambda}\Gamma_{\mu \sigma}^{\rho}- \Gamma_{\rho \sigma}^{\lambda}\Gamma_{\mu \nu}^{\rho}
\Im{\overline{v}\nabla v}- \Im{\overline{w}\nabla w}= \Im\left( \overline{w}|v|(w\nabla|v| + |v|\nabla w) - \overline{w}\nabla w \right) = \Im\left( |v|\nabla|v| + (|v|^2 - 1)\overline{w}\nabla w \right) = (|v|^2 - 1)\Im{\overline{w}\nabla w}
\overline\square_B\phi =\nabla_T^*\nabla_T\phi +\sum_a R^Q(V_a,\bar V_a)\phi +\operatorname{div}_\nabla (H^{0,1}) \phi = \bar\nabla_T^*\bar\nabla_T \phi -i \nabla_{J\kappa_B^\sharp}\phi +\operatorname{div}_\nabla (H^{0,1}) \phi.
\lambda^{(p+1)}(H)-\lambda^{(p+1)}(H') \geq r\sum_{e\in E(H)}\prod_{v\in e}y_v-r\sum_{e\in E(H')}\prod_{v\in e}x_v & =r(y_{v_0}-x_{u_0})\prod_{v\in e_0\backslash\{v_0\}}x_v+r(y_{v_0}-x_{v_0})\sum_{\substack{e\in E(H)\backslash e_0, v_0\in e}}\prod_{v\in e\backslash\{v_0\}}x_v & >0,
\tilde{\chi}( m , \bar{m}; \tau_{2}) = \int \int_{F}\frac{d^{2}m^{\prime}}{( m_{2}^{\prime})^{2}}\ll m , \bar{m}; \tau_{2}| m^{\prime}, \bar{m^{\prime}}; \tau_{1}\gg \tilde{\chi}( m^{\prime}, \bar{m^{\prime}}; \tau_{1}) .
c^nI_4^{(z)}(b) \leq\mathfrak{c}\cdot\left(\frac{1}{a+b}+\frac{1}{c+b}\right) &\cdot\int_{{\mathcal S^n}_z}\frac{(ab)^{n+\epsilon}}{\theta_x^{3n+2\epsilon}}\cdot\frac{(bc)^{n+\epsilon}}{\left(\angle(z,y)-R\right)^{3n+2\epsilon}}\cdot c^n\cdot \Xi b\,d\sigma(y)\cdot\nu(b),
\begin{cases}\partial_{t}V_{2}= D \partial_{xx}^{2}V_{2}+\beta_{1}(m^{\ast}- \delta_{0}) V_{2}+ (m^{\ast}- \delta_{0}) \frac{\beta_{2}}{(\delta_{0}+ K)}V_{4}- V_{2}(d+\gamma), \partial_{t}V_{4}= D_{4}\partial_{xx}^{2}V_{4}- U \partial_{x}V_{4}+ \xi V_{2}+ gV_{4}\left(1 - \frac{\delta_{0}}{K_{B}}\right) - \delta V_{4}.\end{cases}
\begin{array}{rcl}d{\tilde s}^2 = (H^2+\theta^2/(2\pi)^2)^{1/2}[H^{-1/2}ds_{1,7}^2-H^{1/2}d\omega d{\bar \omega}] \, , &{\tilde \lambda}= -2\pi i / \log{\omega}\, . \end{array}
L(f)(x,y) =[e_{xx},L(f)](x,y) &=\left([f,L(e_{xx})]+L([e_{xx},f])\right)(x,y) &=\left(fL(e_{xx})\right)(x,y)-\left(L(e_{xx})f\right)(x,y) &+L(e_{xx}f)(x,y)-L(fe_{xx})(x,y).
\tilde{\tau}_{t+1,i}^{2}\doteq\frac{1}{\sigma_{i}^{2}}\left(\overset{n}{\underset{j=1}{\sum}}\frac{\sigma_{j}^{2}}{m}\mathbb{E}\left\{\left|x_{j}-\eta_{t}\left(x_{j}+e_{j}^{t}\right)\right|_{2}^{2}\right\}+\sigma^{2}\right).
\begin{aligned}u_1(t)=\lambda_1 \int_{a}^{b}k_1(t,s)g_1(s)f_1(u_1(s),u_2(s))\,ds:=T_1(u_1,u_2)(t), u_2(t)= \lambda_2 \int_{a}^{b}k_2(t,s)g_2(s)f_2(u_1(s),u_2(s))\,ds:=T_2(u_1,u_2)(t),\end{aligned}
\varepsilon^-_u = \eta_{12}\gamma^{12}\varepsilon^-_l,\ \ \ \ varepsilon^-_u = -i \eta_{23}\gamma^{23}\varepsilon^-_l, \ \ \ \ varepsilon^-_u = i \eta_{1}\gamma^{1}\varepsilon^+_l, \ \ \ \ varepsilon^-_u = - \eta_{3}\gamma^{3}\varepsilon^+_l .
0\le\sqrt{t}\,\frac{w_n(t,\Lambda_t^n\pm y)}{y}\le\frac{\varepsilon}{2}\Big(1+\sup_{0<t\le\upsilon_0}\;|\sqrt{t}\,(\Lambda_t^n)'|^2\Big)\le\frac\varepsilon2(1+C_\varepsilon^2)\le C_\varepsilon,\; y\in(0,y_0],\; t\in(0,\upsilon\wedge\upsilon_0=\upsilon_0).
||\log \gamma_a||_{\rho-\delta/2}\leq C(d,\tau,\kappa)\delta^{-\tau}||\log a-\log\bar{a}||_{\rho}, ||\log \gamma_b||_{\rho-\delta/2}\leq C(d,\tau,\kappa)\delta^{-\tau}||\log b-\log\bar{b}||_{\rho}.
g(\nabla'_{Z_{i}}\overline{\nabla}_{\widehat{W}}\widehat{W},Z_{j}) =g(\mathcal{A}_{\widehat{W}}^{2}Z_{i},Z_{j})-\overline{g}(\overline{R}(Z_{i},\widehat{W})Z_{j}, \widehat{W}) -\overline{g}((\nabla'_{\widehat{W}}\mathcal{A}_{\widehat{W}})Z_{i},Z_{j})+g(\overline{\nabla}_{\widehat{W}}\widehat{W},Z_{i})g(Z_{j},\overline{\nabla}_{\widehat{W}}\widehat{W}).
(\mathcal{A}-\mathcal{B}\,{\mathcal{B}}^T \, X_{k+1}^{p})^T \, X\,+\,X \,(\mathcal{A}-\mathcal{B}\,{\mathcal{B}}^T \,X_{k+1}^{p})+X_{k+1}^{p}\, \mathcal{B}\mathcal{B}^T\,X_{k+1}^{p}+\mathcal{C}_{k+1}^T\mathcal{C}_{k+1}\,=0;
\Psi_{p_{m};{\bf k}^{1}, N_{1}; \dots ;{\bf k}^{l}, N_{l}}= e x p ( i p^{m}q_{m}\mathrm{d i m}( E ) ) \prod_{j = 1}^{l}( a^{\dagger}({\bf k}^{j}) )^{N_{j}}| 0 \rangle
F_{m,n}(t) = \frac{1}{\sqrt{n+1}}\frac{1}{\sqrt{m+1}}\left(1+i\log\frac{m+1}{n+1}\right)^{-t}G_{m,n}(t) = \frac{1}{\sqrt{m+1}}\int_n^{n+1}v^{-1/2}\left(1+i\log\frac{m+1}{v}\right)^{-t}dv.
D_s^{\ell}\alpha_s(\vec{a}) =(-i\partial_s)^{\ell}\sum_{j,m}e^{is\cdot m}a_{j,m}\prod_{g=1}^nU_g^{m_g}f_j &=\sum_{j,m}m^{\ell}e^{is\cdot m}a_{j,m}\prod_{g=1}^nU_g^{m_g}f_j &=\sum_{j,m}\delta^{\ell}e^{is\cdot m}a_{j,m}\prod_{g=1}^nU_g^{m_g}f_j &=\delta^{\ell}\alpha_s(\vec{a})
\bullet H_{\infty}(x_0,r):= \frac{1}{r^{N-1}}\int_{\partial B_r(x_0)}\sum_{i=1}^k v_{i}^2 \bullet E_{\infty}(x_0,r):= \frac{1}{r^{N-2}}\int_{B_r(x_0)}\sum_{i=1}^k |\nabla v_{i}|^2 \bullet N_{\infty}(x_0,r):= \frac{E_{\infty}(x_0,r)}{H_{\infty}(x_0,r)}().
\sum_{p_2 \in X^i_0}D_x^2 G^i(p_1,p_2) R(p_2,p_1) = D_x^2 G^i(p_1,p_1) R^i(p_1,p_1) + \sum_{0 < |p_2 - p_1| \leq r}D_x^2 G^i(p_2,p_1) \; R^i(p_1,p_2) +\sum_{|p_2 - p_1| > r}D_x^2 G(p_2,p_1) R^i(p_1,p_2)
\sum_{ij\in E^\prime,g_i>g_j}(g_i^p-g_j^p)=\sum_{ij\in E^\prime,g_i>g_j}p\int_{g_j}^{g_i}t^{p-1}dt=p\int_0^\infty t^{p-1}\sum_{ij\colon g_i>t\ge g_j}1dt
f_n^{(1)}(x) = \|(1-x)d_n\|; f_n^{(2)}(x) = \|x\|_2; f_n^{(3)}(x) = \| \alpha(x) -x\|; f_n^{(k+3)}(x) = \|c_n^{(k)}x - xc_n^{(k)}\|\;\;\;\;\;\;\;\; (k\geq 1).
\int_{\Sigma}\mu^{\rm eff}_{pqr}n_r \partial_{x_q}V_p \, ds = \int_{\Sigma}\mu^{\rm eff}_{pqr}n_q n_r n_l \partial_{x_l}V_p \, ds + \int_{\Sigma}\mu^{\rm eff}_{pqr}n_r \Pi_{lq}\partial_{x_l}V_p \, ds.
\Vert (f(n))_\beta \Vert = L(f(n))+1+R(f(n)) \leq L((1+\epsilon)a_g n^g) + 1 + R((1+\epsilon)a_g n^g) \leq \frac{g\log n + \log a_g}{\log \beta}+ O(1) + C' \log n \leq C \frac{\log n}{\log \beta},
S_{0}[B_{\mu\nu}(x),A_{\mu}(x)]=\int d^{4}x\; \Bigg\{\frac{1}{4}\epsilon^{\mu\nu\rho\sigma}B_{\mu\nu}(x)F_{\rho\sigma}(x) + \frac{1}{2}A_{\mu}(x)A^{\mu}(x) \Bigg\},
\partial_t{w}_{p\alpha +k-1}^{(i)}(x_i, t) + \Big( v^{(i)}_i(x_i) \, w^{(i)}_{p\alpha +k-1}(x_i, t) \Big)^\prime = \Big( w^{(i)}_{p\alpha +k-2}(x_i,t)\Big)^{\prime\prime}, (x_i, t) \in I_\varepsilon^{(i)}\times (0, T).
G^{\omega,\lambda}_{p,p,p}(z) =G^\omega_{p,p}(z)(I+\lambda C_pG^\omega_{p,p}(z))^{-1}, G^{\omega,\lambda}_{p,n,m}(z) =G^\omega_{n,m}(z)-\lambda G^\omega_{n,p}(z)(I+\lambda C_p G^\omega_{p,p}(z))^{-1}C_pG^\omega_{p,m}(z).
\sum_{m=1}^{d_0}e^{\varphi(m)}\le e^{\varphi(d_0)}\sum_{m=0}^{d_0}e^{-(d_0-m)\log3}\le \frac{3}{2}e^{\varphi(d_0)}\le \frac{3}{2}e^{\frac{3\log(2^{1/3}\cdot10\cdot e)d}{20}}\le \frac{3}{2}e^{\frac{3d}{5}},
\begin{pmatrix}\dot X_1 \dot X_2 \dot X_3 \end{pmatrix}=\left(\begin{array}{rrrr}0 2X_{3}2X_{1}-2X_{3}0 -2X_{2}-2X_{1}2X_{2}0 \end{array}\right)\begin{pmatrix}H_1 H_2 H_3\end{pmatrix}\,,
\frac{\partial^2 p}{\partial t^2}= -(c_1-c_2) \frac{\partial^2 p}{\partial x\partial t}+ c_1c_2\frac{\partial^2 p}{\partial x^2}-2\lambda(t)\Bigl[ \frac{\partial p}{\partial t}+\frac{(c_1-c_2)}{2}\frac{\partial p}{\partial x}\Bigr]
\{(v_i, w) \mid i=1,\ldots,m-1\}\cup\{(w, w_i) \mid i=1,\ldots,n-1 \}\cup \&\{(w_i, v) \mid i=1,\ldots,n-1 \}\cup\{(v, v_i) \mid i=1,\ldots,m-1 \}.
\lim_{\varepsilon\rightarrow 0}\mathbb{P}_{x}\Big(\sigma^\varepsilon_{\mathfrak{m}_{j}}\leq \sigma^\varepsilon_\mathcal{M}\Big)= 1 \quad\quad\lim_{\varepsilon\rightarrow 0}\mathbb{P}_{x}\Big(\sigma^\varepsilon_{\mathfrak{m}_{k}}\leq \sigma^\varepsilon_\mathcal{M}\Big)= 0, k\neq j.
((\phi\circ\psi)^{-1})^{*}A_{|P_{|U}}=A^{j}_{b}dx^{b}\otimes\tau_{j}, begin{align*}0.25cm] (\phi\circ\psi)_{*}E_{|P_{|U}}=E^{a}_{i}\frac{\partial}{\partial x^{a}}\otimes\tau^{*i}.
\textstyle \alpha \bigl( \{a_{j}, \dotsc, a_{j+1}-1 \}\bigr) = \bigl\{\sum_{j'=0}^{\beta(j)-1}a'_{\beta^{-1}(j')}, \dotsc, \sum_{j'=0}^{\beta(j)}a'_{\beta^{-1}(j')}- 1 \bigr\}
\sum_{n \geq 0}\frac{(x + \mathcal{U}+ \mathcal{B})^{n}}{n!}z^{n}= e^{zx + z \mathcal{U}+ z \mathcal{B}}= e^{zx}\frac{z}{e^{z}-1}\frac{e^{z}-1}{z}= e^{zx}
t_1 =2F+s-\frac{B_1+2B_2+B_3}{2}-\frac{3C_1+6C_2+9C_3+\sum_{j=1}^{12}jC_{16-j}}{4}t_2 =2F+s-\frac{\sum_{j=1}^7 j(C_j+C_{16-j})+8C_8}{2}t_3 =2F+s-\frac{B_1+2B_2+B_3}{2}-\frac{\sum_{j=1}^{12}jC_{j}+9C_{13}+6C_{14}+3C_{15}}{4}.
=I+[\begin{matrix}a&b\\c&d\\e&f\end{matrix}]\sum_{k=1}^{\infty}\frac{1}{k!}([\begin{matrix}0&1&0\\ 0&0&1\end{matrix}][\begin{matrix}a&b\\c&d\\e&f\end{matrix}])^{k-1}[\begin{matrix}0&1&0\\ 0&0&1\end{matrix}]
\nu_{\Phi^\alpha_{ij}}(z) \geqslant -\sum_{u=1}^3 \nu^{\infty}_{F_u^{ij}}(z)+\sum_{u=1}^3\nu^0_{F_u^{ij}}(z)+d_2+d_3-1+\chi_{\nu_j}&\geqslant -\sum_{u=1}^3 \nu^{\infty}_{F_u^{ij}}(z)+d_2+d_3-1+\chi_{\nu_j}\geqslant P(z).
\ell(x,w'\alpha) + \ell(x',\alpha) = \,\langle \mu,w'\alpha\rangle + \langle \mu',\alpha\rangle - \Phi^+(ww'\alpha) + \Phi^+(w'\alpha) - \Phi^+(w'\alpha) + \Phi^+(\alpha) = \,\langle (w')^{-1}\mu + \mu',\alpha\rangle -\Phi^+(ww'\alpha) + \Phi^+(\alpha)=\ell(xx',\alpha).
\langle \tilde{L}G_Df,\varphi\rangle =\langle G_Df,L\varphi\rangle=\lim_{n\to\infty}\langle G_{D_n}f,L\varphi\rangle &=\lim_{n\to\infty}- \langle f\mathbf{1}_{D_n},\varphi\rangle= -\langle f,\varphi\rangle.
\aligned (\hat d^2 \circ (\hat d^2\circ \widehat\psi - \widehat\psi\circ \hat d^1)+(\hat d^2\circ \widehat\psi - \widehat\psi\circ \hat d^1) \circ \hat d^1)_{\alpha_+,\alpha_-}&= d_0 \circ b(\alpha_+,\alpha_-) + b(\alpha_+,\alpha_-)\circ d_0.\endaligned
\overline{\tau}_{2n}:= (r_{2n}\cdots r_{n+1})\cdots (r_{2n}r_{2n-1}r_{2n-2})(r_{2n}r_{2n-1})r_{2n}(r_n\cdots r_{2n-2})\cdots (r_4r_5r_6)(r_3r_4)r_2
\left(\sum_{k\in\mathbb{Z}^d}\left|\langle f,\widetilde{\phi}_k\rangle\right|^\tau\right)^{\frac{1}{\tau}}+ \left( \sum_{i=1}^{2^{d}-1}\sum_{j\in\mathbb{N}_0}\sum_{k\in\mathbb{Z}^d}\left|\langle f,\widetilde{\psi}_{i,j,k}\rangle\right|^\tau\right)^{\frac{1}{\tau}}< \infty \, ,
{g^{\prime}}_{\mu \nu}( x , l ) = [ f ( l )^{\star}g_{\mu \nu}( l ) ] ( x ) = \frac{\partial{x^{\prime}}^{\alpha}( x , l )}{\partial x^{\mu}}g_{\alpha \beta}( x^{\prime}( x , l ) , l ) \frac{{x^{\prime}}^{\beta}( x , l )}{\partial x^{\nu}}
\Delta_k (x) = \exp \Biggl( \sum_{j \in \mathbb{Z}_+}a_j x^{- \frac{j}{k}}L(j) \Biggr) (k^\frac{1}{2})^{-2L(0)}\left(x^{\frac{1}{2k}\left( k-1\right)}\right)^{- 2L(0)}.
{\cal L}= \frac 1 4 F^{a \mu \nu}F_{\mu \nu}^{a}- \frac 1 2 F^{a \mu \nu}( \partial_{\mu}A_{\nu}^{a}- \partial_{\nu}A_{\mu}^{a}+ g f^{a b c}A_{\mu}^{b}A_{\nu}^{c}) + \frac 1 2
x\mapsto x y^3,y\mapsto y^4\Longrightarrow\mathcal{I}_{1,\beta_1,\beta_2}=4\int_{[0,1]^2}\tilde{f}_{t,s}(x,y)^s x^{\beta_1}y^{3\beta_1+4\beta_2+21s}\frac{dx}{x}\frac{dy}{y}
S_{n,k}(\beta,h) = \sum_{I \subset [1,n] \atop I=\{1 \le i_1,\ldots,i_k \le n \}}\prod_{a=1}^{k}\left(\beta^{n-k-i_{a}+a}+ h\frac{\beta^{n-k-i_{a}+a}-1}{\beta-1}\right).
LL(ZB) \le\sum_{i=0}^{\alpha}\sum_{j=0}^{e-1}{i(p^e-1)+p^j}=e(p^e-1)\Bigl(\sum_{i=0}^{\alpha}i\Bigr)+(\alpha+1)\frac{p^e-1}{p-1}&=e(p^e-1)\frac{\alpha(\alpha+1)}{2}+(\alpha+1)\frac{p^e-1}{p-1}&\le \Bigl(\frac{d}{e}+1\Bigr)\Bigl(\frac{d}{2}+\frac{1}{p-1}\Bigr)(p^e-1).
K ( z , \overline{{{z}}}, t , \overline{{{t}}}) = F ( z , \overline{{{z}}}, v , \overline{{{v}}}, w ) - t v - \overline{{{t}}}\overline{{{v}}}, \ \, \frac{\partial F}{\partial v}= t , \frac{\partial F}{\partial \overline{{{v}}}}= \overline{{{t}}}.
( \tilde{H}_r ,\langle \cdot, \cdot \rangle_{\tilde{H}_r},\left\| \cdot \right\|_{\tilde{H}_r})=( H_{\iota+r},\langle \cdot, \cdot \rangle_{H_{\iota+r}},\left\| \cdot \right\|_{H_{\iota+r}}).
J(u) = J(\bar u) + J'(\bar u)(u - \bar u) + \frac{1}{2}J''(u_\theta)(u - \bar u)^2 & \ge J(\bar u) + \frac{1}{2}[J'(\bar u)(u - \bar u) + J''(\bar u)(u - \bar u)^2] + \frac{1}{2}[J''(u_\theta) - J''(\bar u)](u - \bar u)^2 &\ge J(\bar u) + \frac{\gamma}{2}\|z_{\bar u,u - \bar u}\|^2_{L^2(\Omega)}- \frac{1}{2}|[J''(u_\theta) - J''(\bar u)](u - \bar u)^2|.
|\partial_t v|^2 = \varphi^{-1}|\partial_t u|^2 + \varphi^{-3}(M - t K)^2 |u|^2 + \varphi^{-2}(M- tK) \cdot 2\Re \langle\partial_t u,u\rangle &=\varphi^{-1}|\partial_t u|^2 + \varphi^{-3}(M - t K)^2 |u|^2 + \varphi^{-2}(M- tK) \partial_t \big(|u|^2\big),
v_{i}^{2}=v_{i},v_{i}v_{j}=v_{j}v_{i}u_{i}^{2}=u_{i}u_{i+1}u_{i}=u_{i+1}u_{i}u_{i+1}=0 v_{i}u_{i}=u_{i}v_{i+1}=u_{i},u_{i}v_{i}=v_{i+1}u_{i}=0 u_{i}u_{j}=u_{j}u_{i},v_{i}u_{j}=u_{j}v_{i}|i-j|>1\;
F_i(x_1,\dots,x_{n+1})=P_i( x_1,\dots,x_{n+1},j(iB(x_1)),\dots,j(iB(x_{n+1})), & ij'(iB(x_1)),\dots,ij'(iB(x_{n+1})),j''(iB(x_1)),\dots, j''(iB(x_{n+1})))
F_{d,\ell}\left(z; it\right)=\frac{1}{2}\left(2\ell^2 t\right)^{-\frac{1}{2}}\zeta^d-\zeta^d\sum_{a=0}^{N}\frac{B_{2a+1}\left(\frac{d}{\ell}\right)}{2a+1}\frac{\left(-2\pi\ell^2 t\right)^a}{a!}+ O\left(t^{N+1}\right).
\mathcal{F}_k := \{\mathcal{A}\subset \mathcal{S}:~ \mathcal{A}=h(\mathcal{S}^{k-1}), h: \mathcal{S}^{k-1}\to \mathcal{S}\,\}, \mathcal{S}:= \{u \in W_0^{1,p}(\Omega):~ \|u\|_{L^p(\Omega)}= 1 \}
P \equiv -\int_{-\infty}^{+\infty}\mathrm{d}x\, \frac{\partial \mathcal{L}}{\partial \varphi_t}\varphi_x \stackrel{\;\eqref{eq:lagrangian}}{=}- \frac{1}{2}\int_{-\infty}^{+\infty}\mathrm{d}x\, (\varphi_x)^2 = -\frac{1}{2}\int_{-\infty}^{+\infty}\mathrm{d}x\, u^2
\sum_{{V'}_{\mathbf{v}}}F(\mathbf{v'})\ \ge \ \sum_{U_0}P(\mathbf{u}) \sum_{{V'}_{\mathbf{v}}}F(\mathbf{v'|u}) &\overset{(c)}= \left( \sum_{U_0}P(\mathbf{u}) \right) \ \left( \sum_{{V'}_{\mathbf{v}}}F(\mathbf{v'|u}) \right)
\frac{\partial F}{\partial w_{n}^{j}}= - \frac{1}{2 \pi i}\oint_{c}\frac{\partial H^{\prime}}{\partial \alpha^{j}}\zeta^{n}\frac{d \zeta}{\zeta^{2}}- \frac{1}{2 \pi i}\oint_{0}\frac{\partial H}{\partial \alpha^{j}}\zeta^{n}\frac{d \zeta}{\zeta^{2}}
X_M := \left\{u = (u_0, u_1, \ldots, u_M)^T \in \mathbb{C}^{M+1}~|~u_0 = u_M\right\}, & Y_M := \left\{e^{i\mu_l (x-a)},\x\in\overline{\Omega},\l\in \mathcal{T}_M\right\}.
\Xi_0(p):= 1, &\Xi_1(p):= \theta_{a_1}(p)+\theta_{a_2}(p)+\theta_{a_3}(p), &\Xi_2(p):= \theta_{a_1}(p)\theta_{a_2}(p)+\theta_{a_2}(p)\theta_{a_3}(p)+\theta_{a_1}(p)\theta_{a_3}(p), &\Xi_3(p):=\theta_{a_1}(p)\theta_{a_2}(p)\theta_{a_3}(p).
\frac{d\hat{f}_{m+1}}{dv}(v) =\frac{1}{v^2}\left(\frac{df_{m+1}}{dv}(v)-\frac{2}{v}f_{m+1}(v)\right) &=\frac{\lambda}{18\kappa^2v^2}\left\{2v(y_m(v)+1)+\frac{1}{v^2}\int_0^vv'^3\frac{dy_m}{dv}(v')dv'-\frac{2}{v}(A_m+B_m)\right\}+\mathcal{O}_{\delta_2}(1),
\int_{t_i}^{t_{i+1}}{\mathcal{I}}_2''(k)\dt\leq\int_{t_i}^{t_{i+1}}dt\int_{A\cap D_{k}(t)}\ \frac{f(y,w,t)}{|X(t)-y|^2}\, dydw = &\int_{t_i}^{t_{i+1}}dt\int_{\widetilde{A\cap D_{k}(t)}}\ \frac{f(\bar y,\bar w,t_i)}{|X(t)-Y(t)|^2}\, \chi(|X(t)-Y(t)|>l_k) \, d\bar y d\bar w
A \nabla B \leqslant H_{\nu}(A, B) +(A-2A \sharp B + B ) -\sum_{k=0}^{\infty}r_{k}[ H_{\frac{m_k}{2^k}}(A,B) -2H_{\frac{2m_k+1}{2^{k+1}}}(A,B) + H_{{\frac{m_k+1}{2^k}}}(A,B)].
E_{k,n}=\frac{(-1)^{\frac{k(k-1)}{2}}}{(1-q^{-2})^{k-1}}\cdot \Upsilon\left([e_{n},[e_{n+2},\cdots,[e_{n+2(k-2)},e_{n+2(k-1)}]_{q^{-4}}\cdots]_{q^{-2(k-1)}}]_{q^{-2k}}\right) \,,
\int_{\Omega_{k}^{\prime}}d \mu_{b}^{\prime}( \eta_{k}^{\prime}) \widehat{\mu}_{a}^{\prime}( \eta_{k}^{\prime}) = \int_{\Omega^{k}}d \mu_{a}( \alpha_{k}) \widehat{\mu}_{b}^{\prime}( \alpha_{k}^{\prime}) .
iA=i-D_t\zeta \frac{\partial_{\alpha}\mathfrak{F}}{\zeta_{\alpha}}-\frac{i}{2\pi}\sum_{j=1}^2 \frac{\lambda_j(D_t\zeta(\alpha,t)-\dot{z}_j(t))}{(\zeta(\alpha,t)-z_j(t))^2}-(D_t^2\zeta+i)\Psi_{\zeta}\circ \zeta.
W_k = \gamma_1(W_k) = \langle x,y \rangle \; \gamma_2(W_k) W_k/\gamma_2(W_k) \cong C_{p^k}\times C_p, \gamma_i(W_k) = \langle [y,x,\overset{i-1}{\ldots},x] \rangle \; \gamma_{i+1}(W_k) \gamma_i(W_k)/\gamma_{i+1}(W_k) \cong C_p \text{for $2 \le i \le p^k$.}
\max_{1 \leq j \leq s}u_j = \frac{1}{\beta}\log \left[\exp \left(\beta \max_{1 \leq j \leq s}u_j\right) \right] & \leq \frac{1}{\beta}\log \left[\sum_{j=1}^s\exp\left(\beta u_j\right) \right] & \leq \frac{1}{\beta}\log \left[s \exp\left(\beta \max_{1 \leq j \leq s}u_j)\right)\right] & = \frac{1}{\beta}\log s + \max_{1 \leq j \leq s}u_j.
\begin{aligned}Lu = \sum_{i=1}^n\left(x_i\bar a_{ii}(z)u_{x_ix_i}+b_i(z) u_{x_i}\right) +\sum_{i,j=1}^n x_ix_ja_{ij}(z)u_{x_ix_j}+\sum_{l,k=1}^md_{lk}(z)u_{y_ly_k}&\quad+\sum_{i=1}^n\sum_{l=1}^mx_ic_{il}(z)u_{x_iy_l}+ \sum_{l=1}^me_l(z)u_{y_l}+c(z)u,\end{aligned}
F(\rho + \epsilon \nu) = \int_M \int_M W(x,y) d(\rho + \epsilon \nu)(x) d(\rho + \epsilon \nu)(y) &= F(\rho) + \epsilon^2 F(\nu) + \epsilon \left( \int_M \int_M W(x,y) d\rho(x) d\nu(y) + \int_M \int_M W(x,y) d\rho(y) d\nu(x) \right).
\Delta P^{\mathrm{\scriptsize{l e}}}( x , y ) \; = \; X \sum_{h = - 1}^{\infty}\: \sum_{\mathrm{\scriptsize{f i n i t e}}}\: \cdots \: ( T^{( h )}- T^{\mathrm{\scriptsize{r e g}}\: ( h )}) ( x , y ) .
h I_D \le \sum_{i=1}^a I_{A_i}-\sum_{j=1}^b I_{B_j}=\sum_{k_1=0}^{t_1-1}\sum_{k_2=0}^{t_2-1}d_{k_1,k_2}I_{R_{k_1,t_1}\times R_{k_2,t_2}}
W_1(u,z)=zI_\mu(uz)\left(\sum_{s=0}^{N-1}\frac{A_s(z)}{u^{2s}}+g_1(u,z)\right)+\frac{z}{u}I_{\mu+1}(uz)\left(\sum_{s=0}^{N-1}\frac{B_s(z)}{u^{2s}}+zh_1(u,z)\right),\!\!\!
V_F(t) = \int_{\Lambda^{*4}}\phi_t(p_1,p_2,q_2,q_1)a_{p_1}^* a_{p_2}^*a_{q_2}a_{q_1}\d p_1 \d p_2 \d q_1 \d q_2 & + \int_{\Lambda^{*2}}\bigg[\ \int_{\Lambda^{*2}}\phi_t(p_1,p_2,q_2,q_1) \delta(q_1 - p_2) \d p_2 \d q_1 \bigg]a_{p_1}^* a_{q_2}\d p_1 \d q_2 \ .
\theta(q,a)\theta(q,b)=(q^2;q^2){}_\infty^2(a,b;q){}_{\infty}{}_{2}\phi_1\bigg[\genfrac{}{}{0pt}{}{-a/q,-a}{a^2};q^2,q^2\bigg]{}_{2}\phi_1\bigg[\genfrac{}{}{0pt}{}{-b/q,-b}{b^2};q^2,q^2\bigg].
(g_1^{[n\lambda_1s_1]+[n\lambda_2s_1]}g_2^{[n\lambda_1s_2]+[n\lambda_2s_2]}\cdots g_k^{[n\lambda_1s_k]+[n\lambda_2s_k]})^{-1}(g_1^{[n\lambda_1s_1]}g_2^{[n\lambda_1s_2]}\cdots g_k^{[n\lambda_1s_k]})(g_1^{[n\lambda_2s_1]}g_2^{[n\lambda_2s_2]}\cdots g_k^{[n\lambda_2s_k]}) &=[f_{n,1},f_{n,1}^\prime]\cdots[f_{n,l(k)},f_{n,l(k)}^\prime].
\mathcal{S}_N\bigl(B^{(f)}_M(w)\,\log(w)\bigr)(q\,|\,b) =\log(q)\,\mathcal{S}_N\bigl(B^{(f)}_M(w)\bigr)(q\,|\,b) + O(q^{-1}),\;{\rm if}\; M<N, & = \log(q)\,\mathcal{S}_N\bigl(B^{(f)}_M(w)\bigr)(q\,|\,b) + O(1),\;{\rm if}\; M=N.
\prod_{v\in \mathcal{T}_i}\frac{d_{P(v)}}{d_v}=\frac{d_{R_i}d_{P(R_i)}}{4}2^{\#\{v\in \mathcal{T}_i~~\colon~~\textup{deg}_{\mathcal{T}_i}(v)=1,~d_{v}=1 \}}.
\mathcal{L}= -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}- A_{\mu}J^{\mu}+ i \overline{\psi}\partial_{0}\psi - \frac{1}{2m}\nabla \overline{\psi}\cdot \nabla \psi &= \frac{1}{2}(\partial_{0}A_{1}-\partial_{1}A_{0})^{2}- A_{0}J^{0}+ A_{1}J^{1}+ i \overline{\psi}\partial_{0}\psi - \frac{1}{2m}\partial_{1}\overline{\psi}\partial_{1}\psi,
\widetilde{c}_{j l}\equiv \cos \widetilde{\tau}_{j l}= \frac{( n_{j}\cdot n_{l})}{\sqrt{n_{j}^{2}n_{l}^{2}}}= \frac{( - 1 )^{j + l}D_{j l}^{( N - 1 )}}{\sqrt{D_{j j}^{( N - 1 )}D_{l l}^{( N - 1 )}}}.
\int_{0}^{t}(t-s)^{-\beta}\left(\frac{d}{ds}s^{k}\right)ds =t^{k + 1 - \beta}\int_{0}^{1}(1-z)^{-\beta}z^{k-1}dz =t^{k + 1 - \beta}B(1-\beta, k + 1),
\rho : = \frac{1}{2}\left( \rho_1 +\rho_2 + \frac{\kappa^2}{\rho_1}- \sqrt{\left( \rho_1 + \rho_2 + \frac{\kappa^2}{\rho_1}\right)^2 - 4 \rho_1 \rho_2}\right) >0.
\left( \begin{array}{c}{\chi^{( 1 )}}{\chi^{( 2 )}}\end{array}\right) = \frac{1}{\sqrt{2}}\left( \begin{array}{c c}{1}{1}{- 1}{1}\end{array}\right) \left( \begin{array}{c}{\chi^{L}}{\chi^{R}}\end{array}\right)
\alpha^1=\left(\begin{array}{ccc}1 0 0 0 0 1 0 1 0\end{array}\right),\ \ \alpha^2=\left(\begin{array}{ccc}0 1 0 1 0 0 0 0 1\end{array}\right),\ \ \alpha^3=\left(\begin{array}{ccc}0 0 1 0 1 0 1 0 0\end{array}\right),
\sum_{t\in \mathbb F_q}\nu^2(t) \leq q^{-1}|E|^2|F|^2 + q^{2d-1}|E| \sum_{x\in \mathbb F_q^d, s\in \mathbb F_q^*}E(x) |\widehat{F}(sx)|^2 &=q^{-1}|E|^2|F|^2 + q^{2d-1}|E| \sum_{x\in \mathbb F_q^d, s\in \mathbb F_q^*}E(sx) |\widehat{F}(x)|^2,
\phi_{m_{k}}^{2}( 0 ) = \frac{1}{{\cal A}_{n}R_{0}}\frac{\pi c}{8}\frac{( n + 1 )^{2}}{\Gamma^{2}[ ( n + 3 ) / 2 ]}\left( \frac{m_{k}}{c}\right)^{n}e^{- \frac{c}{2}\rho_{\mathrm{m a x}}},
d_H\bigl(R_{\tau}\bigl(X_0^{N}(\tau)\bigr),X\bigr)=d_H\bigl(X_0^{N}(\tau),R_{-\tau}(X)\bigr)=\min_{t\in\mathbb{R}}d_H\bigl(X_0^{N}(t),R_{-t}(X)\bigr).
0=c^{T}((\sum_{i=1}^{t}\lambda_{i}x_{i})-x^{*})=c^{T}(\sum_{i=1}^{t}\lambda_{i}(x_{i}-x^{*}))=\sum_{i=1}^{t}\lambda_{i}(c^{T}x_{i}-c^{T}x^{*}).
\mathbb{P}_{q,p}\{0<\left\vert \mathbf{C}_{\{0,x\}}\right\vert <\infty\}=\sum_{k\geq\varphi\left( x\right)}\mathbb{P}_{q,p}\left( \mathbf{A}_{k}\left( x\right) \right) \geq\mathbb{P}_{q,p}\left( \mathbf{A}\left(x\right) \right) \ ,
-K_S+\lambda C\sim_{\mathbb{Q}}\frac{1}{2}\big(C+\widetilde{C}\big)+\lambda C=\Big(\frac{1}{2}+\lambda\Big)C+\frac{1}{2}\widetilde{C}\sim_{\mathbb{Q}}\big(1+2\lambda\big)\Big(-K_S-\frac{\lambda}{1+2\lambda}\widetilde{C}\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) + 2 7 \right]}.
\begin{array}{ll}[\alpha ([x, z]), \alpha ([y, t])] = [[[x, y], \alpha (z)], \alpha^{2}(t)] + [[[y, z], \alpha (t)], \alpha^{2}(x)] & + [[[z, t], \alpha (x)], \alpha^{2}(y)] + [[[t, x], \alpha (y)], \alpha^{2}(z)].\end{array}
\pi(z,L,QNR)\geq \frac{z\left(\prod_{q|L}(\frac{q-1}{2})\right)}{2\phi(L)\log z}= \frac{z\left(\frac{\phi(L)}{2^{\omega(L)}}\right)}{2\phi(L)\log z}= \frac{z}{2^{\omega(L)+1}\log z}
|(K_1*\varphi_QT)(x)-\varphi_Q(x)(K_1*T)(x)| \leq|\langle T,\psi_Q(\varphi_Q-\varphi_Q(x))K_1^x\rangle| *[7pt] +\|\varphi_Q \|_{\infty}|\langle T,(1-\psi_Q)K_1^x\rangle|=A+B.
T_{\bar{u}}(\Psi_k^K u_k^K)-\Psi_k^K T_{\bar{u}}(u_k^K) =\Pi\left(\Psi_k^K (I-\Pi)(\bar{u}u_k^K)\right) &=\Pi\left(\Psi_k^K\bar{z}\overline{\Pi(\bar{z}u\overline{u_k^K})}\right) &=\Pi(\bar{z}\Psi_k^K\overline{K_u(u_k^K)}) &=\sigma_k\, \Pi(\bar{z}|\Psi_k^K|^2\overline{u_k^K}) &=\sigma_k\, \Pi(\bar{z}\overline{u_k^K}) &=0,
R(t) \leq\sin\frac{\pi}{3}(1-{\mathcal M}(L^+_{\frac{\pi}{3}}(t))-{\mathcal M}(L^-_{\frac{\pi}{3}}(t))+{\mathcal M}(L^+_{\frac{\pi}{3}}(t))-\sin\frac{\pi}{3}{\mathcal M}(L^-_{\frac{\pi}{3}}(t)) &= \frac{\sqrt{3}}{2}+{\mathcal M}(L^+_{\frac{\pi}{3}}(t)) \Big (1-\frac{\sqrt{3}}{2}\Big) - \sqrt{3}{\mathcal M}(L^-_{\frac{\pi}{3}}(t)).
r(E_i,E_j)-\rho^\nabla_M(E_i,E_j) =\sum_{l,k=1}^{6}g\left(JE_i,N(e_l,e_k) \right)g\left(E_j,N(e_l,e_k) \right) &+\sum_{l,k=1}^{6}g\left(N(E_i,e_k),e_l \right)g\left(N(E_j,e_k),Je_l \right), &=\sum_{l,k=1}^{6}g\left(N(E_i,e_k),e_l \right)g\left(N(E_j,e_k),Je_l \right).
[h_{\alpha}, h_{\alpha'}]=0,\left[h_{\alpha}, e^\pm_{\alpha'}\right]=\pm (\alpha|\alpha')e^\pm_{\alpha},\left[e^+_{\alpha},e^-_{\alpha'}\right]=\delta_{\alpha,\alpha'}h_{\alpha},(\, e^\pm_{\alpha})^{1-(\alpha|\alpha')}e^\pm_{\alpha'}= 0
1 \ge \mu (A_0 \oplus g_1A_1\oplus \ldots g_nA_n\oplus B_0 \oplus h_1B_1\oplus \ldots \oplus h_mB_m) & \ge \mu (A_0) + \sum_{k=1}^n \mu (g_kA_k) + \mu (B_0) + \sum_{l=1}^m \mu (h_lB_l) & \ge \sum_{k=0}^n \mu (A_k) + \sum_{l=0}^m \mu (B_l) = 2,
d_n^{\cal O}(R) =\frac{1}{n!}\frac{1}{(2 \pi)^n}\int d\theta_1\ldots d\theta_nF_{n,n}^{\cal O}(\theta_n,\ldots,\theta_1 |\theta_1,\ldots,\theta_n) e^{-E_n R}\,\,\,
\begin{aligned}\log L(1,{\rm sym}^m f) = \sum_{\substack{p\leq x p\nmid N}}\sum_{0\leq j\leq m}\log \left(1-\alpha_f(p)^{m-2j}p^{-1}\right)^{-1}& + \sum_{\substack{p\leq x p\mid N}}\log \left(1-\alpha_f(p)^mp^{-1}\right)^{-1}+ O\big(\log^{-1/2}(kN)\big),\end{aligned}
\lim_{R\to\infty}\inf_{v\in H_{i_0}(x_0)\setminus \{0\}}{1\over R}(D_{[R]}f_v)(0) = \chi_{i_0}= \lim_{R\to\infty}\sup_{v\in H_{i_0}(x_0)\setminus \{0\}}{1\over R}(D_{[R]}f_v)(0) .