harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{\lambda_2^{K_{\ell}}}{\lambda_2^L}=\Bigl\{2^{(\dim M_2^{K_{\ell}}-m)/2}\cdot\frac{\displaystyle{\prod_{i=1}^{m}}(2^{2i}-1)}{|M_2^{K_{\ell}}|}\Bigr\}\Bigl\{2^{(\dim M_2^L-m)/2}\cdot\frac{\displaystyle{\prod_{i=1}^{m}}(2^{2i}-1)}{|M_2^L|}\Bigr\}^{-1}\leq 2^{1/2}(2^{n_{2,\nu_2}}-1).
\sum_{r=0}^{\infty}\frac{a^r q^{r^2}}{(q)_r}=\frac{1}{(aq)_{\infty}}\sum_{r=0}^{\infty}\frac{1-a q^{2r}}{1-a}\, \cdot\frac{(a)_r}{(q)_r}\,\cdot (-1)^r a^{2r}q^{5\binom{r}{2}+2r}.
G_{\beta}(\vec{x}, \tau) ={1\over V}\sum_{\vec{p}}{1\over{2 E_p}}\left[{{e^{i \vec{p}\cdot \vec{x}-E_p \tau}}\over{1- e^{-\beta E_p}}}+{{e^{-i \vec{p}\cdot \vec{x}-E_p \tau}}\over{e^{\beta E_p}-1}}\right] \; .
\mathbb{P}\left\{\left\vert \sum_{i=1}^na_iX_i -\mathbb{E}\sum_{i=1}^na_iX_i\right\vert >C_{q}\left(t\left\vert a \right\vert_2 +e^{t^2/(2q)}\left\vert a \right\vert_q\right)\right\}\leq Ce^{-t^2/2}
[\begin{matrix}N_{11}\\N_{22}\\N_{12}\end{matrix}]=\frac{2hE}{(1-\nu^{2})}[\begin{matrix}1&\nu&0\\ \nu&1&0\\ 0&0&1-\nu\end{matrix}][\begin{matrix}u_{1,1}^{0}\\u_{2,2}^{0}\\ \frac{1}{2}(u_{1,2}^{0}+u_{2,1}^{0})\end{matrix}]
B_{\mathbf{d}, \mathcal{A}}(ab) = \sum\limits_{c \in \mathcal{A}(a) \setminus \mathcal{A}(b)}P_{\mathbf{d}, \mathcal{A}}(ac) + \sum\limits_{c \in \mathcal{A}(a) \cap \mathcal{A}(b)}Y_{\mathbf{d}, \mathcal{A}}(acb) .
\lbrack -{\vcenter{\mathrm{\hrule ~ h e i g h t . 4 p t \mathrm{\vrule ~ w i d t h . 4 p t h e i g h t 8 p t \kern 8 p t \vrule ~ w i d t h . 4 p t}\hrule ~ h e i g h t . 4 p t}}}+ m^{2}+ \overline{A}( x ) - \overline{{\partial^{\mu}\gamma_{\mu}}}- \overline{\gamma}^{\mu}\partial_{\mu}\rbrack = 0 .
(\partial^{2(p+1)}_y \tilde{u})(t, x, 0)=\sum^p_{q=2}\sum_{(\alpha, \beta)\in \Lambda_q}C_{\alpha,\beta}\prod\limits_{j=1}^q \partial_x^{\alpha_j}\partial_y^{\beta_j +1}\Big( u^s(t, 0) + \tilde{u}(t, x, 0) \Big),
\begin{gathered}\int_0^\infty u^{2\alpha}\left[C_1(u)+C_2(u)\right] J_\nu (u\rho; q^2) \,d_q u - \int_0^\infty \dfrac{w(u)}{1+w(u)}u^{2\alpha}\left[C_1(u)+C_2(u)\right] J_\nu (u\rho; q^2) \,d_q u = f_3 (\rho), \rho\in B_{q,b}. \end{gathered}
A ( e_{1}, p_{1}; e_{2}, p_{2}; e_{3}, p_{3}) = \frac{1}{\alpha^{'}G_{0}^{2}}\langle : c V_{- 1}( p_{1}, x_{1}) : : c V_{- 1}( p_{2}, x_{2}) : : c V_{0}( p_{3}, x_{3}) : \rangle + 1 \leftrightarrow 2
L:=2 \cdot 4^{\frac{s j_0}{2}}\|u\|_{L^{\infty}(B_{4 R(x_0)})}+\operatorname{Tail}(u ; x_0, 4 R) +\|f\|^{\frac{1}{p-1}}_{L^\gamma(B_{4 R(x_0)})}.
\frac{1}{r^{d - 1}}\, \left( r^{d - 1}\, B^{p + 1}\, F^{d - 2}\, \eta_{a}^{\prime}\right)^{\prime}= \frac{1}{2}\, B^{p + 1}\, F^{d}\, T ( x ) \, \delta^{d}( \vec{x}) ~ ~ ,
\psi\left(-k'_{1},s'_{1}\right)\left(\mathcal{T}^{*}\right)^{-1}\overline{f}\left(\mathbf{k}',\mathbf{s}'\right)=\left(\mathcal{T}^{*}\right)^{-1}\overline{h}\left(\mathbf{k}',\mathbf{s}'\right),
R^{(\rho,\ell)}= R' \cdot \left( \frac{\prod_{i=1}^{k-1}(1-R_{i,m})(1-R_{i,m+1})}{\prod_{i=1}^{\rho_m}(1+R_{i,m})(1+R_{i,m+1})}\right) \times \left( \prod_{i=k}^{m-1}\prod_{j=m+2}^\ell (1-R_{i,m})(1-R_{i,m+1})(1-R_{m,j})(1-R_{m+1,j}) \right) (1-R_{m,m+1}),
Z = \sum_{J_{+}, J_{-}}\frac{1}{J_{+}! J_{-}!}( \frac{V}{V_{c}})^{J}\exp - [ J R e \; I_{c}] e^{i k \pi ( J_{+}- J_{-})}= \exp \{e^{i \pi k}2 ( \frac{V}{V_{c}}) \exp - [ R e \; I_{c}] \}; \; \; ( J = J_{+}+ J_{-}) .
\Gamma_E (t) = \frac{\ 1 \}{t \wedge r}\, \delta \left( \frac{U(\cdot) \, \Lambda (t)}{A_1 \big( X (\cdot -r) \big) \, X( \cdot )}\, \mathbb{I}_{[0 \vee (t-r),t]}( \cdot ) \right),
\mathcal{E}^{\epsilon,\kappa}(\alpha,\beta):=\widetilde{\Lambda}^{\epsilon,\kappa}(\alpha,\beta)\Lambda^\epsilon(\alpha,\beta)\, \mathfrak{h}^\epsilon(\alpha-\beta)=\Lambda^{{\epsilon,\kappa}}(\alpha,\beta)\, \mathfrak{h}^\epsilon(\alpha-\beta)
(\delta_{c_1}\circ \delta_{c_3}) \circ \delta_{c_4}|_{c_0}= \delta_{c_1}\circ (\delta_{c_3}\circ \delta_{c_4})|_{c_0},~~~(\delta_{c_1}\circ \delta_{c_4}) \circ \delta_{c_3}|_{c_0}=\delta_{c_1}\circ (\delta_{c_4}\circ \delta_{c_3})|_{c_0},
\Pr\left(\max_{m\in S}(X_m)<a\right) \le 2\, \Pr\left(\max_{m \in S}(Y_m) < a\right) = 2 \Pr\left(Y_1 < a\right)^{|S|}\le 2\left(1-e^{-\theta}\frac{\theta^a}{a!}\right)^{|S|}&\le 2\left(1- \frac{\theta^a}{ea!}\right)^{|S|}\le 2\exp\left(-\frac{\theta^a|S|}{ea!}\right)\ .
\sum_{k\geq1}\mathbb{E}\sup_{n}|\sum\limits_{i=1}^{n\wedge N_{k}}e_{-N_{k}+i}|\mathbf{1}_{A_{k}}=\sum_{k\geq1}\mathbb{E}\max_{1\leq j\leq N_{k}}|\sum\limits_{i=1}^{j}e_{-N_{k}+i}|\mu(A_{k}) & \leq\sum_{k\geq1}\mathbb{E}\max_{1\leq j\leq N_{k}}|\sum\limits_{i=1}^{j}e_{-N_{k}+i}|\rho_{k}\leq\sum_{k\geq1}\sqrt{N_{k}}\rho_{k}.
R=R^x+\langle R^y,y\rangle+\langle R^z,z\rangle+\langle R^{\bar{z}},\bar{z}\rangle+\langle R^{zz}z,z\rangle+\langle R^{z\bar{z}}z,\bar{z}\rangle+\langle R^{\bar{z}\bar{z}}\bar{z},\bar{z}\rangle,
P_1 =R_0\cup S_0\cup T_0 \cup U_0\cdots R_{\frac{d-1}{2}-1}\cup S_{\frac{d-1}{2}-1}\cup T_{\frac{d-1}{2}-1}\cup U_{\frac{d-1}{2}-1}\cup R_{\frac{d-1}{2}}&= \cup_{i=0}^{\frac{d-1}{2}}R_i\bigcup \cup_{i=0}^{\frac{d-1}{2}-1}S_i \bigcup \cup_{i=0}^{\frac{d-1}{2}-1}T_i\bigcup \cup_{i=0}^{\frac{d-1}{2}-1}U_i,
\mathcal{O}^- = \frac{1}{2}\sum_{m \geq 1}\sum_{n \geq 1}\frac{A(n, m)}{(m^2 n)^{\frac{1}{2}}}\sum_{c > 0}\frac{S(-n, p; c)}{c}H_{m, n}^- \Bigl ( \frac{4 \pi \sqrt{n p}}{c}\Bigr ).
\|g^{-1}f^*h(x)\|^k =\|D_Hf(x)\|^{2k}\leq K^{2n/(n+1)}J(x,f)^{2n/(n+1)}&=K^{2n/(n+1)}HJ(x,f)^2=K^{2n/(n+1)}\det(g^{-1}f^*h(x)).
\tau_{m_1h_1+\cdots+m_sh_s}(f) = \sum_{k_s=0}^{n_s-1}\sum_{k_{s-1}=0}^{n_{s-1}-1}\cdots \sum_{k_{1}=0}^{n_{1}-1}a_{s,m_s,k_s}a_{s-1,m_{s-1},k_{s-1}}\cdots a_{1,m_{1},k_{1}}(\tau_{h_s})^{k_s}\cdots (\tau_{h_1})^{k_1}(f).
e^{\tau \tilde{\phi}(\ln(r_2))}r_2^{-1}\left\| w \right\|_{L^2(A^+_{r_2,2r_2}(x_0))}& \leq C(n,\mu, \epsilon)\left( e^{\tau \tilde{\phi}(\ln(r_1))}r_1^{-1}\left\| w \right\|_{L^2(A^+_{r_1,2r_1}(x_0))}+ e^{\tau \tilde{\phi}(\ln(r_3))}r_3^{-1}\left\| w \right\|_{L^2(A^+_{r_3,2 r_{3}}(x_0))}\right).
\Big| F'_{k,j}(x,y) \Big|  \leq C_1 j^{\alpha -1}\frac{k^{j-1}}{(j-1)!}\frac{2^{2k+3j}\pi^{2k+2j}\Bigl(x^2+y^2\Bigr)^{k+2j}}{\Gamma(2k_j+5)\prod\limits_{\ell = 1}^{j-1}\Gamma(2 (k_{\ell}-k_{\ell+1})+5)}\leq C_1 j^{\alpha -1}\frac{k^{j-1}}{(j-1)!}\frac{2^{2k+3j}\pi^{2k+2j}\Bigl(x^2+y^2\Bigr)^{k+2j}}{\Bigl( \Gamma \left( 2 k/j +5\right) \Bigr)^j},
D^{(\beta)}_N(k,\alpha)=\begin{cases}\displaystyle\frac{\partial^{k-1}M_N^{(\beta)}(s)}{\partial s^{k-1}}\biggl|_{s\to 0^{-}}\ , \displaystyle\frac{(-1)^{|k|}}{|k|!}\int_{-\infty}^{0}M_N^{(\beta)}(s) s^{|k|}\,\mathrm{d}s \ . \end{cases}
f\left(\boldsymbol{P}\right)=\ln\left(1+\frac{h_{n}P_n}{\sigma^2+\sum_{j=1,j\neq n}^Kh_{j}P_j}\right)+\sum_{i=1,i\neq n}^K\ln\left(1+\frac{h_{i}P_i}{\sigma^2+\sum_{j=1,j\neq i}^K h_{j}P_j}\right).
\Phi (u,v) = \Phi_0 (u,v) + \Phi_r(u,v), \Phi_0 (u,v) = \int_M \textbf{A}(\nabla u, \nabla v) d\mu, \,\,\,\, \Phi_r(u,v) = \int_\Omega \textbf{b}(\nabla u) v d\mu + \int_\Omega c u v d\mu.
\mathfrak{G}_{1 \times w_{0}^{(7)}}^{(\beta)}(x_1=q, \,x_i=1,\,\forall\, i \ge 2) = (0,1,15,50,50,15,1)_{\beta+1}t^6 +(0,2,24,60,40,6)_{\beta+1}t^5 \qquad{}+ (0,3,27,45,15)_{\beta+1}t^4+ (0,4,24,20)_{\beta+1}t^3+(0,5,15)_{\beta+1}t^2+ 6(\beta+1) t+1.
X_H=\sum_{i=1}^n\left(p_i\frac{\partial H}{\partial p_i}-\sum_{i=1}^nH\right)\frac{\partial}{\partial t}-\sum_{i=1}^n\left(p_i\frac{\partial H}{\partial t}+\frac{\partial H}{\partial q^i}\right)\frac{\partial}{\partial p_i}+\sum_{i=1}^n\frac{\partial H}{\partial p_i}\frac{\partial}{\partial q^i}
\tilde{\mathbf{h}}_k\left(n\right) = \sqrt{\frac{M}{N_{\mathrm{cl},k}N_{\mathrm{ray},k}}}\sum_{i=1}^{N_{\mathrm{cl},k}}\sum_{j=1}^{N_{\mathrm{ray},k}}\tilde{\alpha}_{ijk}\Delta\left(\phi_{ijk},\theta_{ijk}\right)\mathbf{a}\left(\phi_{ijk},\theta_{ijk}\right)\delta\left(n-\tau_{ijk}\right),
J(d_1 ,\ldots ,d_n ;e_1 ,\ldots ,e_n ) \leq \sum_{0\leq j\leq n}\frac{\beta^{n+1}-\beta^j}{\beta -1}\sum_{0\leq k\leq j}r^k \sum_{(m_\ell )\in S^{(2)}_{j-k}}\prod_{\ell=2}^n d_\ell^{m_\ell}& \leq \sum_{0\leq j\leq n}\frac{\beta^{n+1}-\beta^j}{\beta -1}\sum_{0\leq k\leq j}r^k \cdot \alpha^{j-k}\cdot \# S^{(2)}_{j-k}.
{\frac{\rho_{+}}{\rho ( \rho^{*})}}={\frac{Y}{1 - \sigma^{2}}}\sum_{n = 0}^{\infty}a_{n}( \sigma ) Y^{2 n}, ~ ~ Y = \tanh \lambda \rho^{*}, ~ ~ \sigma ={\frac{\rho_{-}}{\rho_{+}}}, ~ ~ \lambda ={\frac{\rho_{+}}{R^{2}}}( 1 - \sigma^{2}) , ~ ~ a_{0}= 1 .
S_{1 1}^{A_{2}^{( 1 )}}\left( \theta \right) = S_{2 2}^{A_{2}^{( 1 )}}\left( \theta \right) = \left\{1 \right\}_{\theta}\qquad S_{1 2}^{A_{2}^{( 1 )}}\left( \theta \right) = \left\{2 \right\}_{\theta}\qquad S^{A_{2}^{( 2 )}}\left( \theta \right) = \left\{1 \right\}_{\theta}\left\{2 \right\}_{\theta}\, .
\bar{V}_{i}= ( \bar{a}_{4}^{i}- \bar{a}_{5}^{i}) \, \bar{Q}^{1 / 2}\Big [ \begin{array}{c c}{0}{0}{1}{0}\end{array}\Big ] \, \bar{Q}^{1 / 2}\Big [ \begin{array}{c c}{0}{1}{1}{0}\end{array}\Big ] ,
\partial_t\vert\vec u\vert^2 = \Delta(\vert \vec u\vert^2)-2\vert\vec\nabla\otimes\vec u\vert^2 &- 2\lim_{\epsilon\rightarrow 0}\left(\vec u_\epsilon\cdot\varphi_\epsilon*(\vec u\cdot\vec\nabla\vec u)+\lim_{\eta\rightarrow 0}( (p*\xi_{\eta,\epsilon})\vec u_{\epsilon,\eta})\right)+2\vec u\cdot\vec f.
{}^{M\!L\!}\Gamma_{p}(x){}^{M\!L\!}\Gamma_{p}(y) =4\int_{0}^{\infty}\int_{0}^{\infty}\eta^{2x-1}\xi^{2y-1}E_{\alpha, \beta}^{\gamma}\left(-\eta^2-\frac{p}{\eta^2}\right)E_{\alpha, \beta}^{\gamma}\left(-\xi^2-\frac{p}{\xi^2}\right) d\eta d\xi.
A_{N}= 2 \pi \, \sqrt{l}\, \left( \frac{1}{\pi}\right)^{{\textstyle \frac{k + 1 - D}{2}}}\Gamma \left( \frac{D - 3 - k}{2}\right) \left( \frac{1}{Y^{2}}\right)^{{\textstyle \frac{D - 3 - k}{2}}}
p_r(\cos\vartheta)=\frac{1}{\Sigma_n}\cdot \frac{1-r^2}{(1-2r\cos\vartheta+r^2)^{(n+1)/2}}=\frac{1}{\Sigma_n}\cdot\sum_{l=0}^\infty r^l\cdot\frac{\lambda+l}{\lambda}\,\mathcal C_l^\lambda(\cos\vartheta).
A_{n_{k-1}}(p^*_{n_k, n_{k-1}}(m))(\{\mu\}) = p^*_{n_k, n_{k-1}}(m)(\{\mu_2\dots\mu_{|\mu|}\}) = \sum_{\tau \in E^{<n_k}, [\tau]_{n_{k-1}}= \mu_2\dots\mu_{|\mu|}}m(\{\tau\}) = \sum_{\eta \in E^{<n_k}, [\eta]_{n_{k-1}}= \mu}A_{n_k}(m)(\{\eta\}) = p^*_{n_k, n_{k-1}}(A_{n_k}(m))(\{\mu\}).
\delta^{( 4 )}( x - x^{\prime}) = \frac{1}{\rho}\delta \left( \overline{{t}}- \overline{{t^{\prime}}}- S \frac{( \phi - \phi^{\prime})}{\alpha}\right) \delta ( \rho - \rho^{\prime}) \delta ( \phi - \phi^{\prime}) \delta ( z - z^{\prime}) .
\Phi_{g,n,\beta}((\iota_*\alpha)\boxtimes\gamma)=\sum_{\beta_1+\beta_2=\beta}\left(\Phi_{g_1,n_1+1,\beta_1}\otimes \Phi_{g_2,n_2+1,\beta_2}\right)(\alpha\boxtimes (\gamma\boxtimes\eta)),
\begin{cases}D^{\alpha_1}_0y(t) = w(t), ~y(0) = y_0,\,y(b) = y_b, D^{1-\alpha_1}_0w(t) = z(t), ~w(0) = 0, D^{\alpha_2-1}_0z(t) = f(t,y(t),w(t)), ~z(0) = y^{(1)}(0).\end{cases}
\omega = - \displaystyle \frac{1}{C_{K}}\left( R_{1}\, d R_{2}\wedge d L + R_{2}\, d L \wedge d R_{1}+ \displaystyle \frac{( \lambda - 8 E ) L + 4 \lambda L^{3}}{( \lambda - 8 E ) + 8 \lambda L^{2}}d R_{1}\wedge d R_{2}\right) + \sum_{i = 1}^{3}d \bar{C}_{i}\wedge d C^{i}~ ,
\big\lvert\Psi\big(u+\mathrm{i}a(u)+\mathrm{i}\tau\big)- \alpha\big\rvert = \lvert h_j(u)- z_{0j}\rvert \geqslant \frac12\big(\lvert\mathrm{Re}\, h_j(u)-x_{0j}\rvert +\varepsilon\big),
\mathop{\operatorname{res}}\limits_{s = z_1 + z_2 + 1/\beta}f(z_1,z_2,s;\beta )\frac{{{x^s}}}{s}= \frac{{\zeta (z_1 + z_2 + \tfrac{1}{\beta})\zeta (\beta z_2 + 1)\zeta (\beta z_1 + 1)}}{{(\beta z_1 + \beta z_2 + 1)\zeta (2 + \beta z_1 + \beta z_2)}}{x^{z_1 + z_2 + \tfrac{1}{\beta}}}.
E\left[S^{part,m}_{s,t}\right]=E\left[S^{part,0}_{s,t}\right] &- E\left[\int \ln(p^0(t,q_t,z))p^0(t,q_t,z)dz\right]+ E\left[\int \ln(p^0(s,q_s,z))p^0(s,q_s,z)dz\right]
\begin{aligned}\dot{x}= -y - \frac{4}{3}x^2 - 2 k_1 x y - \frac{x}{3}\left(4 k_1 x^2 + 3 k_1^2 x y \right), \dot{y}= x + k_1 x^2 - \dfrac{16}{3}x y - k_1 y^2 - \frac{y}{3}\left(4 k_1 x^2 + 3 k_1^2 x y \right),\end{aligned}
F_{k}( x^{\omega_{1}}\Phi^{1}, x^{\omega_{2}}\Phi^{2}, . . . , x^{\omega_{N}}\Phi^{N}) = x^{1 - \rho_{k}}F_{k}( \Phi^{1}, \Phi^{2}, . . . , \Phi^{N}) , \quad 1 \le k \le M \, ,
M_{\varepsilon}: = \int\limits_X\widetilde{\alpha}_{m,\,\varepsilon}^{n-1}\wedge\beta_{\varepsilon}= \int\limits_X(\alpha + \varepsilon\omega)^{n-1}\wedge(\beta + \varepsilon\omega)\downarrow\int\limits_X\alpha^{n-1}\wedge\beta:=M_0\geq 0,
\left\langle Z_{*},i_{1},i_{1}+1,\dots,\widehat{i_{j}+\epsilon},\dots,i_{\frac{m}{2}},i_{\frac{m}{2}}+1\right\rangle =\det\left(\begin{matrix}C V_{*}I_{i_{1},i_{1}+1,\ldots,\widehat{i_{j}+\epsilon},\ldots,i_{\frac{m}{2}},i_{\frac{m}{2}}+1}\end{matrix}\right)\det(\mathcal{Z}),
C_{p,q}^s(\cup_{j\in I_i}B(x_j,r_j)) \le \|u_i\|_{N^s_{p,q}(X)}^p &\le C \, (1+ ||u_i||_{\infty})^p (1+2^{is})^p \mu(\cup_{j\in I_i}B(x_j,2^{-i+2})) &\le C \, 2^{isp}\mu(\cup_{j\in I_i}B(x_j,2^{-i+2})) &\le C\sum_{j\in I_i}\frac{\mu(B(x_j,r_j))}{r_j^{sp}}.
m = m_{0}\left[ 1 + \frac{\alpha}{4 \pi}\log^{2}\frac{| e B |}{m_{0}^{2}}+ \frac{5}{2 4}\left( \frac{\alpha}{2 \pi}\log^{2}\frac{| e B |}{m_{0}^{2}}\right)^{2}+ \frac{6 1}{7 2 0}\left( \frac{\alpha}{2 \pi}\log^{2}\frac{| e B |}{m_{0}^{2}}\right)^{3}+ \cdots . \right]
b_j^{n-2s-1}H(q_j, q_j) - \sum_{i\neq j}b_j^{\frac{n-2s}{2}-1}b_i^{\frac{n-2s}{2}}G(q_j, q_i) = \frac{2sb_j^{2s-1}}{n-2s}j = 1,\cdots, k.
\left(L(f|_x^y)g\right)(x,y) =\sum_{x\le z\le y}L(f|_x^y)(x,z)g(z,y) &=\sum_{x\le z\le y}L\left((f|_x^y)|_{x}^{z}\right)(x,z)g(z,y) &=\sum_{x\le z\le y}L(f|_{x}^{z})(x,z)g(z,y) &=\sum_{x\le z\le y}\hat{L}(f)(x,z)g(z,y)=(\hat{L}(f)g)(x,y).
\begin{array}{rcl}\omega^{x}= 2 \, f_{tt}\, d f_{tx}\wedge d f_{tt}+ \frac{3}{2}\, d f_{x}\wedge d f_{tx}, begin{align*}2mm] \omega^{t}= 2 \, f_{tx}\, d f_{tt}\wedge d f_{tx}- \frac{1}{2}\, d f_{x}\wedge d f_{xx}\end{array}
D\left(\Xi_{\mathbf{x},\mathbf{t}}\right)=\left\{f\in\mathcal{A}_{n}:\xi\left(-\sum_{i=1}^{n}k_{i},\sum_{i=1}^{n}s_{i}\right)\overline{f}\left(\mathbf{k},\mathbf{s}\right)=\overline{h}\left(\mathbf{k},\mathbf{s}\right),\exists h\in\mathcal{A}_{n}\right\}.
n^{\gamma-2}\sum_{|x| \leq 2 b_G n}\sum_{|y| \leq 2 b_G n}\sup_{s \in [0,T]}C_{G,\delta}|x-y|^{\delta}n^{-\delta}p(y-x) \lesssim n^{\gamma-2-\delta}\sum_{|x| \leq 2 b_G n}\sum_{z \neq 0}|z|^{\delta-\gamma-1}= 2 (2b_G +1) n^{\gamma-1-\delta}\sum_{z = 1}^{\infty}z^{\delta-\gamma-1}.
\tilde{\Pi}_{R{( 2 )}}^{( \beta )}( \lambda , p ) = - \frac{8 e^{4}}{p^{2}}m^{2}I ( T / \lambda ) \int d^{2}q \, n_{F}( | q^{0}| ) \left( q^{2}+ m^{2}\right) \delta^{\prime \prime}( q^{2}- m^{2}) .
\begin{cases}\Vert \Phi_1\Vert_{L^{\infty}(S')}\leq C \lambda^{1-\frac{m-1}{p}}\Vert \Phi\Vert_{W^{1,p}(S')}\Vert \Phi_2\Vert_{L^{\infty}(S')}\leq C \lambda^{-\frac{m-1}{p}}\Vert \Phi\Vert_{L^p(S')}\Vert \nabla^{'}\Phi_2\Vert_{L^{\infty}(S')}\leq C \lambda^{-\frac{m-1}{p}}\Vert \Phi\Vert_{W^{1,p}(S')}.\end{cases}
l_{1}= k_1+k_0t^3/k_4 l_{2}= k_4/t^3 b_{15}= k_6t^2 b_{25}= k_5t^2 b_{12}= k_3/t b_{13}= -k_0k_6t^2+k_7/t b_{23}= -k_0k_5t^2+k_8/t
\left(\sum\limits_{i=1}^{l}m_{i}B_{\sigma ir}Q_{ir}-B_{2\sigma r}\right)\odot Q_{1r}=\frac{1}{2}\sum\limits_{i=1}^{l}\sum\limits_{j=1,\textrm{}j\neq i}^{l}\frac{m_{i}m_{j}(Q_{ir}+Q_{jr})\odot Q_{1r}}{(\sigma-\sigma(Q_{ir}\odot Q_{jr})^{2})^{\frac{1}{2}}\sigma(1+Q_{ir}\odot Q_{jr})}.
\sum_{M\geq0}g_1(M)\,q^M =\sum_{M\geq1}\frac{z^{2M}}{(1-z)^{M+1}}+\sum_{M\geq0}\frac{z^{2M+2}M}{(1-z)^{M+2}}&=\frac{z^2}{(1-z)(1-z-z^2)}+\frac{z^4}{(1-z)^3\left(1-\frac{z^2}{1-z}\right)^2}=\frac{z^2}{(1-z-z^2)^2}.
\Psi(0)=\frac{(1+\alpha)_b(1+\alpha-\gamma-\delta)_b}{(1+\alpha-\gamma)_b(1+\alpha-\delta)_b}\cdot{}_4F_3\bigg[\begin{matrix}1-\epsilon -b \gamma \delta 1 \alpha-\epsilon+1 \gamma+\delta-b-\alpha\end{matrix}\bigg|\,1\bigg].
\left[ q_{1}^{\prime}+\frac{q_{2}R}{2B_{2}q_{3}}-\frac{B_{2}^{\prime}}{B_{2}}q_{1}\right] f+\left[ 2q_{1}+q_{2}^{\prime}-\frac{1}{2}\frac{B_{2}^{\prime}}{B_{2}}q_{2}-\frac{1}{2}\left( q_{2}+q_{3}^{\prime}\right) \frac{q_{2}}{q_{3}}+\frac{R}{B_{2}}\right] f^{\prime}=0.
\frac{d}{dt}(\iint_{\Sigma(t)}F(r,t)\cdot dA)=\iint_{\Sigma(t)}(F_{t}(r,t)+(F\cdot\nabla)v+(\nabla\cdot F)v-(\nabla\cdot v)F)\cdot dA-\oint_{\partial\Sigma(t)}(v\times F)\cdot ds.
\varphi\circ T_a(g)(b)=1 \iff{(T_a(g))}_\ell+b\not\equiv 0\bmod \mathfrak{b}_\ell \ell\geq 1 &\iff g_{\ell}+a+b\not\equiv 0\bmod \mathfrak{b}_\ell \ell\geq 1 &\iff \varphi(g)(b+a)=1 &\iff S_a\circ \varphi(g)(b)=1.
\int_{SO(2N+1)}\left(\frac{P'}{P}\Big(1-\frac{a}{N}\Big)\right)^{K}= (-1)^K \left[ \left(\frac{N}{a}\right)^K - \frac{N^K}{a^{K-1}}K \right] + O\left(\frac{N^{K-1}}{a^{K-1}}+ \frac{N^{K}}{a^{K-2}}\right)
f(y,x) =\bigg\langle \dfrac{y}{1+\|y\|^2},x-y\bigg\rangle &\leq\dfrac{1}{1+\|y\|^2}(\langle y, x-y\rangle-\langle x, x-y\rangle) &\leq-\dfrac{1}{2}\|x-y\|^2 &=-\gamma\|x-y\|^2,
A^K_i(t)=\widetilde A^K_i(t)+2C(\bar b+\bar d) t + 2 \bar p t\leq \max_{0\leq j\leq 1/\delta_K}\widetilde A^K_j(0)+2 C(\bar b+\bar d) t + 2 \bar p t = \max_{0\leq j\leq 1/\delta_K}\beta^K_j(0)+2C(\bar b+\bar d) t + 2 \bar p t.
\beta_i|_{\frak{t}^\sigma}=\alpha_{i+1}|_{\frak{t}^\sigma}\ (1\leq i\leq n-1),\beta_n|_{\frak{t}^\sigma}=\beta_{n+1}|_{\frak{t}^\sigma}=\alpha_{n+1}|_{\frak{t}^\sigma}
g(x) \coloneqq \begin{cases}f(x) x < x_{i-1}, y_{i-1}+ a_i (x-x_{i-1}) x_{i-1}\leq x < \hat{x}_{i+2}, f(\frac{a_i}{a_{i+2}}(x-\hat{x}_{i+2})+x_{i+2}) \hat{x}_{i+2}\leq x \end{cases}
\hat{H}\Psi=[\hat{T}+\hat{V}+\hat{U}]\Psi=[\sum_{i=1}^{N}(-\frac{\hbar^{2}}{2m_{i}}\nabla_{i}^{2})+\sum_{i=1}^{N}V(r_{i})+\sum_{i<j}^{N}U(r_{i},r_{j})]\Psi=E\Psi,
a-b = \int_0^1 \int_0^1 (1-py)^{-2}(1-px)^{-1}y^jx^{j-1}(y-x)dydx = \int_0^1 \int_0^x (1-py)^{-2}(1-px)^{-1}y^jx^{j-1}(y-x)dydx \quad+ \int_0^1 \int_x^1 (1-py)^{-2}(1-px)^{-1}y^jx^{j-1}(y-x)dydx.
x_1x_{n+1}= z, ~~~ x_2x_{n+2}=z, ~~~ \cdots ~~~ x_nx_{2n}= z, x_{n+1}x_1 = \lambda z, ~~~ x_{n+2}x_2=\lambda z, ~~~ \cdots ~~~ x_{2n}x_n = \lambda z, x_{n+1}x_2 = z, ~~~ x_{n+2}x_3 = z, ~~~ \cdots ~~~ x_{2n-1}x_n = z,
M_{i,r}^{n^{-1}}=\frac{8}{\pi(1+\lambda_{i})^{2}}\left(\frac{2^{i}(i+1)!}{(2i+1)!!}\right)^{2}\left(\Lambda_{r,n}^{i}+\sum_{d=0}^{i}\lambda_{d}\Lambda_{r,n}^{d}\right).
H = \frac{1}{( X^{2}+ Y^{2}+ Z^{2})^{3 / 2}}\cdot \left( X^{\prime \prime}( Y^{2}+ Z^{2}) + Y^{\prime \prime}( X^{2}+ Z^{2}) + Z^{\prime \prime}( X^{2}+ Y^{2}) \right)
\zeta ( \nu | \beta ) ={\frac{\varrho^{- 2 \nu}}{2 \pi i}}\int_{\mu}^{\infty}d \omega \int_{C}d z \sum_{\sigma_{l}}{\frac{\breve{\Phi}( \omega ; z )}{z - \sigma_{l}}}( z^{2}+ \omega^{2})^{- \nu},
\varphi ( \theta ) = - \frac{2 \cosh \theta \sin \frac{\pi}{\lambda}}{\sinh^{2}\theta + \sin^{2}\frac{\pi}{\lambda}}\sim - 4 \sin \frac{\pi}{\lambda}\mathrm{e}^{- \left| \theta \right|}+ O \left( \mathrm{e}^{- 2 \left| \theta \right|}\right) ,
\bigl\langle T(x\otimes w_1), y\otimes w_2\bigr\rangle =\int_{\Omega}\bigl\langle [\Delta(t)](w_1),w_2\bigr\rangle\, x(t)y(t)\, \mu(t) & = \bigl\langle M_\Delta (x\otimes w_1), y\otimes w_2\bigr\rangle
\int_0^t \mu x_{\ell - 1}(u)e^{s \ell (t - u)}\: du = \frac{N \mu^{\ell}e^{s \ell t}}{s^{\ell - 1}(\ell - 1)!}\int_0^t (e^{su}- 1)^{\ell - 1}e^{- s \ell u}\: du = x_{\ell}(t).
\lim_{r\to 0^+}\frac{A(r)^2}{A'(r)}\cdot \left[ \mathcal W(r)^{p-1-\alpha}\right]' = \lim_{r\to 0^+}r^\delta u_0'(r)^{p-1-\alpha}\cdot \frac{g_1(v(r))}{h_1(v_0(r))}= 0
\tilde{Y}^b_t=0-\int_t^Tg^b(s,\tilde Y^b_s + \xi_s^a, \tilde Z^b_s+\gamma_s)-D_s+\lambda_s \tilde f(\tilde{Y}^b_s) ds -\int_t^T \tilde{Z}^b_s \cdot dW_s,
0 = \tau_0(z_0) \{\alpha^3 \tau_1(z_0) \tau_2(z_0) L(\tau(z_0)^3) - (1+a) \tau_1(z_0) L(\tau(z_0)^2) + (a+b) \alpha L(\tau(z_0)) & + (a+b) \alpha L(\tau(z_0))\}- b L(z_0) & = -b,
\xi=\left(\begin{array}{cc}0 1 0 0 \end{array}\right)=\frac{1}{2}(\sigma_1+i\sigma_2) \ , \overline{\xi}=\left(\begin{array}{cc}0 0 1 0 \end{array}\right)=\frac{1}{2}(\sigma_1-i\sigma_2) \ .
Q = x^1 P_1 + x^2 P_2 + x^3 P_3 + \theta^1 J_1 + \theta^2 J_2 + \theta^3 J_3= \begin{pmatrix}0 0 0 0 x^1 0 -\theta^3 \theta^2 x^2 \theta^3 0 -\theta^1 x^3 -\theta^2 \theta^1 0\end{pmatrix}.
\lim_{n\to\infty}\mathbf E[H_q^\circ(n)] = \lim_{t\to\infty}\mathbf E[H_q^\circ(t)] = \lim_{t\to\infty}\mathbf E[\widetilde H^\circ_{x}(t)] = \lim_{t\to\infty}\mathbf E[\widetilde H^\circ_{X_t}(0)|X_0=x] = \mathbf E[\widetilde H_0^\circ(0)] = \mathbf E[H_1^\circ(0)] = 1.
\begin{array}{l}\displaystyle a(r)=2\pi^2\, r\frac{d}{dr}G_{\rho m_{_W}}(r)+\frac{1}{\nu}f^{a}(r), \displaystyle h(r)=-2\pi^2\,G_{\rho m_{_H}}(r)+\frac{1}{\nu}f^{h}(r), \displaystyle f^{a}(r)=-\pi^2\,r\frac{d}{dr}G_{\rho \mu_{_W}}(r), \displaystyle f^{h}(r)=\pi^2\,G_{\rho \mu_{_H}}(r), \end{array}
\Bigl|\exp\Bigl(\sum\alpha_k\Bigr)-\prod_{k=1}^n\exp(\alpha_k)\Bigr| =\Bigl|\exp\Bigl(\sum_{k=1}^n\alpha_k+\sum_{k=n+1}^\infty\alpha_k\Bigr)-\exp\Bigl(\sum_{k=1}^n\alpha_k\Bigr)\Bigr| &=\Bigl|\exp\Bigl(\sum_{k=1}^n\alpha_k\Bigr)\Bigr|\Bigl|\exp\Bigl(\sum_{k=n+1}^\infty\alpha_k\Bigr)-1\Bigr|\xrightarrow{n\to\infty}0.
d s_{B T Z}^{2}= - \frac{\Delta^{2}}{l^{2}\rho^{2}}d t^{2}+ \frac{l^{2}\rho^{2}}{\Delta^{2}}d \rho^{2}+ \rho^{2}\left( d \phi - \frac{\rho_{+}\rho_{-}}{l \rho^{2}}d t \right)^{2}
LHS =\sum_{k=1}^n\sum_{j=1}^k(-1)^{k+j}[a_k(x)P_s^{(k)}(x)]^{(k-j)}f^{(j-1)}(x), RHS =\sum_{k=1}^n\sum_{j=1}^k(-1)^{k+j+1}[a_k(x)f^{(k)}(x)]^{(k-j)}P_s^{(j-1)}(x).
\sum_{p,q=0,0}^{m,n}a_{p,q}z_{1}^{-p}z_{2}^{-q}Y(z_{1},z_{2})=\sum_{p,q=0,0}^{m,n}b_{p,q}z_{1}^{-p}z_{2}^{-q}X(z_{1},z_{2})
(\begin{matrix}ct^{\prime}\\x^{\prime}\\y^{\prime}\\z^{\prime}\end{matrix})=(\begin{matrix}\gamma&-\beta\gamma&0&0\\ -\beta\gamma&\gamma&0&0\\ 0&0&1&0\\ 0&0&0&1\end{matrix})(\begin{matrix}ct\\x\\y\\z\end{matrix})=(\begin{matrix}\gamma ct-\gamma\beta x\\ \gamma x-\beta\gamma ct\\y\\z\end{matrix}).
S_{[ 1 ]}^{c}= e^{i \vartheta_{[ 1 ]}^{c}}\left( \begin{array}{c c}{0}&{\Theta_{[ 1 ]}}\\{- \Theta_{[ 1 ]}}&{0}\\ \end{array}\right){\cal K}\equiv{\cal C}_{[ 1 ]}{\cal K}\quad .
\|s^{\Re(z)-1}\Omega(c)(1+D^2)^{1-\frac{p}{2}}e^{-s^2(1+D^2)}\|_1 = s^{\Re(z)-2}\|\Omega(c)h_1(s(1+D^2)^{1/2})\|_1 = O(s^{\Re(z)-3}),s\downarrow 0.
\left[ N\frac{\partial}{\partial N}+ 2 \right] F(N,0,1,g,0,0,\ldots) = \sum_{(n_1,\ldots,n_k)}\beta_{n_1\cdots n_k}(g) \frac{\partial F}{\partial g_{n_1\cdots n_k}}(N,0,1,g,0,0,\ldots).
\|i\omega(i\omega+\varepsilon-A)^{-1}x\|={}\|(i\omega-(A_{-1}-\varepsilon))^{-1}(A_{-1}-\varepsilon)x+x\| \leq{}\|(i\omega-(A_{-1}-\varepsilon))^{-1}A_{-1}x\| + \varepsilon\|(i\omega-(A-\varepsilon))^{-1}x\|+\|x\| \leq{}(C+M+1)\|x\|,
X = vv^T \geq 0 F_0 = \left(\frac{1}{2^2},\cdots,\frac{\mathbf 1_{n-1}\mathbf 1^T_{n-1}}{(n)^2},\frac{\mathbf 1_{n}\mathbf 1^T_{n}}{(n+1)^2}\right) &F_i = \left( e_{n,i},e_{n-1,i},\cdots, e_{n-i+1,i}, 0_{n-i}, \cdots , 0_{1}\right)
\int\limits_{\mathbb{R}^d}a_{\rm sym}(\xi-q) \, \mu(\xi,q) \, \big( \varkappa_{\rm sym}(q) - \varkappa_{\rm sym}(\xi) \big) \, dq \ = \ \int\limits_{\mathbb{R}^d}a_{\rm sym}(\xi-q) \, (\xi-q) \, \mu(\xi,q) \, dq,
{\rm Var}\left( \mathcal{N}^{c}(f_{\ell})\right) =\sum_{\xi_{i,\varepsilon}, \xi_{j,\varepsilon}\in \Xi_\varepsilon}{\rm Cov}\left( \mathcal{N}^{c}(f_{\ell};{\cal V}(\xi_{i,\varepsilon}) ),\mathcal{N}^{c}(f_{\ell};{\cal V}(\xi_{j,\varepsilon}) )\right),
\sum_{n = 1}^{\infty}e^{- \beta k E_{n}}= \frac{1}{\sqrt{\pi}}\int_{0}^{\infty}e^{- \frac{x^{2}}{4}- \frac{1}{x^{2}}{\beta}^{2}k^{2}m^{2}}\frac{1}{e^{\frac{2{\beta}^{2}e B k^{2}}{x^{2}}}- 1}\; .
\Big(\langle[\Phi^{\sqrt{\Delta_{\Phi}^{\Psi}}}(x)],[\Phi^{\sqrt{\Delta_{\Phi}^{\Psi}}}(y)]\rangle\Big)_{ij}=\sum_{r=1}^{n}(\Phi^{\sqrt{\Delta_{\Phi}^{\Psi}}}_{ri}(x))^*\Phi_{rj}^{\sqrt{\Delta_{\Phi}^{\Psi}}}(y) &=(S_{i}^{\Phi})^*\Delta_{1}\pi^{\varphi}(\langle x,y\rangle)S_{j}^{\Phi}&=\varphi_{ij\Delta_{1}}(\langle x,y\rangle).
\partial_{t}A_{0}(\textbf{w}+\bar{\textbf{u}}) = A'_{0}(\textbf{w}+\bar{\textbf{u}})(- A (\textbf{w}+\bar{\textbf{u}}) \partial_{x}{\textbf{w}}+ D(\textbf{u}, \bar{\textbf{u}}, \textbf{w}){\textbf{w}}).
\begin{array}{c}{A^{\left( * \right)}( p ) = \sum_{m}v ( p , m ) a^{*}( p , m )}\\{B^{\left( * \right)}( p , ) = \sum_{m}v ( p , m ) b^{*}( p , m )}\\{A ( p ) = \sum_{m}u ( p , m ) a ( p , m )}\\{B ( p ) = \sum_{m}u ( p , m ) b ( p , m )}\\ \end{array}
b_{j,m}= \Big[ \prod\nolimits_{i=0}^{m-2}y_{j+i}^{\, (-1)^{m+i}\binom{m-2}{i}}, y \Big] = \prod\nolimits_{i=0}^{m-2}e_{j+i}^{\, (-1)^{m+i+1}\binom{m-2}{i}}.
\int_0^Tg_s\,\mathrm{d}^-R_s^H = \int_0^Tg_s\delta R_s^H + 2c_H^{B,R}\int_0^T (\nabla^\frac{H}{2}g_s)(s)\delta{B}_s^{\frac{H}{2}+\frac{1}{2}}+ c_H^R\int_0^T(\nabla^{\frac{H}{2},\frac{H}{2}}g_s)(s,s)\,\mathrm{d}{s}.
\frac{\rho(s+1)}{\rho(s)}=\frac{\sigma(s)+\tau(s)\Delta x(s-\frac 12)}{\sigma(s+1)}=\frac{\phi(s)}{\sigma(s+1)}, \frac{\rho_{\nu}(s+1)}{\rho_{\nu}(s)}=\frac{\sigma(s)+\tau_{\nu}(s)\Delta x_{\nu}(s-\frac 12)}{\sigma(s+1)}=\frac{\phi_{\nu}(s)}{\sigma(s+1)},