harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\boldsymbol\beta= \frac{\boldsymbol\sigma^\bot\cdot\boldsymbol\beta}\delta \boldsymbol\gamma -\frac{\boldsymbol\gamma^\bot\cdot\boldsymbol\beta}\delta \boldsymbol\sigma=-\frac{\boldsymbol\sigma \cdot\boldsymbol\beta}\delta \boldsymbol\gamma^\bot+\frac{\boldsymbol\gamma \cdot\boldsymbol\beta}\delta \boldsymbol\sigma^\bot.
- i ( 2 \pi )^{- 4}{\frac{\delta_{\mu \nu}+ \mu^{- 2}k_{\mu}k_{\nu}}{k^{2}+ \mu^{2}- i \epsilon}}, \qquad - i ( 2 \pi )^{- 4}{\frac{- i \gamma^{\mu}k_{\mu}+ m}{k^{2}+ m^{2}- i \epsilon}}.
\left( \begin{array}{l}{\psi_{R}}\\{\psi_{G}}\\{\psi_{B}}\\ \end{array}\right) \rightarrow \exp ( - i \theta_{a}\lambda_{a}/ 2 ) \left( \begin{array}{l}{\psi_{R}}\\{\psi_{G}}\\{\psi_{B}}\\ \end{array}\right) .
\label[Ieq]{eq:mckeanvlasov-pde}\partial_t f_t( x ) = -\nabla_x\cdot\{b( x ,f_t)f_t\}+\frac{1}{2}\sum_{i,j=1}^d \partial_{x_i}\partial_{x_j}\{a_{ij}(x,f_t)f_t \}
V\left( z\right) =-\left( \alpha \left( 0\right) -\beta \left( 0\right)\right) \frac{\pi^{3}}{a^{4}}F\left( \frac{\pi z}{a}\right) +\left( \alpha\left( 0\right) +\beta \left( 0\right) \right) \frac{\pi^{3}}{360a^{4}}.
\psi\delta_0 |_{S_2}= -2, \psi\delta_{1,1}|_{S_2}= -1, \psi\delta_{2,1}|_{S_2}= 1, \lambda \delta_0 |_{S_2}= -2, \lambda\delta_{2,1}|_{S_2}= 1, \delta_0^2 |_{S_2}= -48, \delta_0\delta_{1,1}|_{S_2}= -10, \delta_0\delta_{2,1}|_{S_2}= 12, \delta_{1,1}^2 |_{S_2}= 2, \delta_{1,1}\delta_{2,1}|_{S_2}= -1.
M_{1}= -{\frac{1}{4 \pi}}\int d^{2}x \left\lbrace{\frac{( a - 1 )}{2}}\partial_{\mu}\theta \partial^{\mu}\theta + \theta \left\lbrack ( a - 1 ) \partial_{\mu}A^{\mu}+ \epsilon^{\mu \nu}\partial_{\mu}A_{\nu}\right\rbrack \right\rbrace
\check{I}_{j}(e^{x-v_{j+1}})-e^{-v_{j}}\check{I}_{j}(e^{x})=((1-q)e^{-v_{j+1}}+\alpha)e^{x}=\check{I}_{j}(e^{x-v_{j}})-e^{-v_{j+1}}\check{I}_{j}(e^{x})
\| I_{h,T}\mathbf{u}- \mathbf{u}\|_{0,T_{\ast}^s}+ h| I_{h,T}\mathbf{u}- \mathbf{u}|_{1,T_{\ast}^s}+ h^2| I_{h,T}\mathbf{u}- \mathbf{u}|_{2,T_{\ast}^s}\leqslant C h^2 (|\mathbf{u}|_{1,T}+|\mathbf{u}|_{2,T}), ~s=\pm, ~\forall T \in \mathcal{T}_h^i.
i \partial_{\tau}\psi = \mathrm{K}\psi = \left[ \frac{1}{2 m}( \mathrm{p}- e \mathrm{A}^{( n )})^{2}+ \frac{1}{2 \lambda}( \pi^{\mu}+ y^{T}S^{\mu}\mathrm{p}- e \chi^{\mu}) ( \pi_{\mu}+ y^{T}S_{\mu}\mathrm{p}- e \chi_{\mu}) + V \right] \psi
\sum_{i=1}^{m}\frac{1}{a_i}\leq \sum_{i=1}^{m}\frac{1}{x_i}- \left( \sum_{i=m+1}^{n}\frac{1}{a_i}- \sum_{i=m+1}^{n}\frac{1}{x_i}\right) & < \sum_{i=1}^{m}\frac{1}{x_i}\leq \sum_{i=1}^{n}\frac{1}{x_i}< \frac{p}{q}.
\hat{\alpha}^* - \hat{\alpha}= \frac{1}{2x_m^T x_m}x_m^T \log \hat{f}_p^m - \frac{1}{p x_m^T x_m}x_m^T \log \hat{\gamma}_p^m = \frac{1}{2x_m^T x_m}x_m^T \log \left( \frac{\hat{f}_p^m}{(\hat{\gamma}_p)^{2/p}}\right),
\begin{aligned}\tilde{w}_{2s}=\frac{\kappa_N (-j)_s(2a)_s(a+1)_s(a-c-j)_s(a+c)_s}{(a+c)_{j+1}(c-a)_{j+1}(2a+1)_j s!(a)_s(2a+1+j)_s(a-c+1)_s(a+c+j+1)_s}, \tilde{w}_{2s+1}=\frac{-\kappa_N (-j)_s(2c)_s(c+1)_s(c-a-j)_s(a+c)_s}{(a+c)_{j+1}(a-c)_{j+1}(2c+1)_j s!(c)_s(2c+1+j)_s(c-a+1)_s(a+c+j+1)_s}\end{aligned}
\lim_{\delta \to 1}\sup_{\tilde{\sigma}^1,\tilde{\sigma}^2}\bigg| \mathbb{E}_{\tilde{\sigma}^1,\tilde{\sigma}^2}(1 - \delta) \bigg[\sum_{t=0}^{\infty}\delta^t u^1(a^1_t,a^2_t)\bigg] & - \limsup_{N \to \infty}{1 \over N}\mathbb{E}_{\tilde{\sigma}^1,\tilde{\sigma}^2}\bigg[ \sum_{t=0}^{N-1}u^1(a^1_t,a^2_t)\bigg] \bigg| = 0
\frac{\alpha u}{\mathbb{Y}_{\nu,\alpha}(u)}\left[-\left( 1-\frac{V}{u}\right)\frac{\nu+1}{\mathbb{Y}_{\nu,\alpha}(V)}+\frac{\partial}{\partial V}\frac{\mathbb{Y}_{-\nu-1,\alpha}(V)}{1-\alpha uV}\right]=\frac{\partial}{\partial u}\left[ \frac{\alpha V}{1-\alpha uV}\frac{\mathbb{Y}_{- \nu-1,\alpha}(u)}{\mathbb{Y}_{\nu,\alpha}(V)}\right]
\mbox{Im}\int_{r_o}^{r_{f}}P_{-}dr = -\pi\int^{-\omega}_{0}{\frac{d{\cal H}- \Phi dQ}{\kappa(M+{\cal H},Q-e)}}~=~-\frac{1}{2}\int_{M,Q}^{M-\omega, Q-e}dS_{BH},
\nabla_{Z}\nabla_{Z}(\phi e_{i})= Z(g(Z, e_{i})) \xi - g(Z, e_{i}) \phi (Z)+\{g(\phi(Z), e_{i}) - \eta (\nabla_{Z}e_{i}) \}Z &- \eta (e_{i}) \nabla_{Z}Z +g(Z, \nabla_{Z}e_{i}) \xi - \eta (\nabla_{Z}e_{i}) Z +\phi (\nabla_{Z}\nabla_{Z}e_{i}), \nabla_{Z}\nabla_{Z}\xi= - g(Z, Z) \xi + \eta (Z) Z - \phi (\nabla_{Z}Z),
\int_{0^-}^{\infty}e^{-s x}dG_3(x)=\frac{\mathbb E\left[e^{s \underline{X}_{e(q)}}\right]}{\mathbb E\left[e^{s \underline{X}_{e(p+q)}}\right]}=e^{\int_{0}^{\infty}(e^{-s x}-1) \Pi_3(dx)}, \ \s\geq 0,
T_{F,G,N}^L(f_1,\dots,f_d) = \sum\limits_{\widetilde{\mathbf{r}}\in \widetilde{R}}\frac{1}{N^{d-m}}\sum\limits_{\mathbf{n}\in \mathbb{Z}^{d-u}}\Big(\prod\limits_{j=1}^d f_j(\xi_j(\mathbf{n}) + \widetilde{\mathbf{r}}_j) \Big)F(\Xi(\mathbf{n}) + \widetilde{\mathbf{r}}) G(L\Xi(\mathbf{n}) + L\widetilde{\mathbf{r}}),
(\sigma^{ab}n_{ab})^{(1)}= \rho (\sigma^{ab}n_{ab})^{(2)}= \frac{2}{3}D\rho (\sigma^{ab}n_{ab})^{(3)}= \frac{3}{8}D^2\rho+\frac{1}{30}|\alpha|^2- \frac{11}{45}|\beta|^2 .
\frac{\partial v}{\partial t}(P)=(k_g-p_1)|\nabla \phi|^2+p_1\left(\frac{\partial \phi}{\partial x}\right)^2+\frac{\partial\phi}{\partial x}\frac{\partial^2\phi}{\partial t\partial x}+\frac{\partial f}{\partial t}\left(\frac{\partial \phi}{\partial t}\right)^2+\frac{\partial f}{\partial x}\frac{\partial \phi}{\partial x}\frac{\partial \phi}{\partial t}.
\mathrm{WCE}(n, \mathbb{M}^2_{\mathrm{e}^2}) := \sup_{\substack{f\in \mathbb{M}^s_{\mathrm{e}^2}, \|f\|_{\mathbb{M}^s_{\mathrm{e}^2}}\leq 1}}\left| \int\limits_{\,-\infty}^\infty f(x) W(x) \mathrm{d}x- \sum_{x\in X_n}\omega(x)f(x)\right|^2 = \sum_{x,y\in X_n}\omega(x)\omega(y)K_{t}(x,y)-\frac{1}{\sqrt{2}\, t}.
\left ({V_2 \over V_1}\right )^2 ={n_1 \bar n_1 \over n_2 \bar n_2}\ ,\qquad\left ({V_3 \over V_2}\right )^2 ={n_2 \bar n_2 \over n_3 \bar n_3}\ ,\qquad\left ({V_4 \over V_3}\right )^2 ={n_3 \bar n_3 \over n_4 \bar n_4}\ ,
\overline{{\omega}}_{r}( \varepsilon^{s}) ={\frac{\partial{\cal M}( \tau )}{\partial \overline{{\cal S}}^{r}}}= F_{r s}( \overline{{\cal S}}^{u}, q^{\mu}, p_{\mu}) \overline{{\cal S}}^{s}+ G_{r}( \overline{{\cal S}}^{u}, q^{\mu}, p_{\mu}) ,
dg =dg_{K}dk &= \det((e_{ij}))\;d\phi_1\ldots d\phi_{N-1}d\psi_1\ldots d\psi_{N-1}dg_{SU(N-1)}d\omega_N &=2\cos(\psi_{N-1})\sin^{2(N-1)-1}(\psi_{N-1})\left[\prod_{j=1}^{N-2}\cos^{2j-1}(\psi_{j})\sin(\psi_j)\right]d\phi_1\ldots d\phi_{N-1}d\psi_1\ldots d\psi_{N-1}dg_{SU(N-1)}d\omega_{N-1}.
I_{2}=(-1)\int\, \delta^{3}\!\left({\vec{r}_{1}+ \vec{r}_{2}-\vec r}\right)\,\delta(r_{1}+r_{2}-t)\,\frac{d^{3}r_{1}}{4\pi\,r_{1}}\,\frac{d^{3}r_{2}}{4\pi\,r_{2}\,{\ell_{0}}^{2}}
X = \frac{\partial}{\partial y^1_{k+1}}+ \sum_j \bigg(g^k_j + \frac{\partial a_j^k}{\partial y^1_k}\bigg) \frac{\partial}{\partial y^j_{k+1}}+ \sum_\rho \bigg(\gamma^k_\rho + \frac{\partial \beta^k_\rho}{\partial y^1_k}\bigg) \frac{\partial}{\partial \eta^\rho_{k+1}}+ Z,
L=-\frac{1}{4}Z_{3}F_{\mu\nu,r}F_{r}^{\mu\nu}+Z_{2}\overline{\psi}_{r}(i\partial/-m_{r})\psi_{r}-\overline{\psi}_{r}\delta m\psi_{r}+Z_{1}e_{r}\overline{\psi}_{r}\gamma^{\mu}\psi_{r}A_{\mu,r}
\mathrm{tr}\,\left(\hat{Q}(A_{x_n}\cdots A_{x_1})^TQA_{x_n}\cdots A_{x_1}\right) = \mathrm{tr}\,\left(\hat{U}^T\hat{U}(A_{x_n}\cdots A_{x_1})^TU^TUA_{x_n}\cdots A_{x_1}\right) &=\left\|UA_{x_n}\cdots A_{x_1}\hat{U}^T\right\|_F^2
\lambda\int_{\Omega}\vert y_1\vert^{\kappa_1 + 1}\vert y_2\vert^{\kappa_2 + 1}dz\leq \frac{p - 1 + \nu}{(\kappa_1 + \kappa_2 + 2)(\kappa_1 + \kappa_2 + 1 + \nu)}\| (y_1, y_2)\|_{1,p}+ \frac{q - 1 + \nu}{(\kappa_1 + \kappa_2 + 2)(\kappa_1 + \kappa_2 + 1 + \nu)}\| (\nabla y_1, \nabla y_2)\|_{q,\eta}.
\mathbb{E}\left[ \sup\limits_{t\in\lbrack0,T]}\left( |X^{\epsilon}(t)-\bar{X}(t)|^{\beta}+|Y^{\epsilon}(t)-\bar{Y}(t)|^{\beta}\right) \right]+\mathbb{E}\left[ \left( \int_{0}^{T}|Z^{\epsilon}(t)-\bar{Z}(t)|^{2}dt\right)^{\frac{\beta}{2}}\right] =O\left( \epsilon^{\frac{\beta}{2}}\right) .
\sum_{j}\left( \frac{\partial X^{\mu}}{\partial x_{i}}\frac{\partial X^{\mu}}{\partial x_{j}}+ F_{i j}\right) \hat{L}_{j k}\, + \, \left( \frac{\partial X^{\mu}}{\partial x_{i}}\frac{\partial X^{\mu}}{\partial x_{j}}- F_{i j}\right) \hat{L}_{k j}\, = \, 2 L \delta_{i k}
\frac{1}{2}\int_{\mathbb{R}^{n+1}_+}y^{1-2\gamma}|\nabla u|^2\,dxdy\leq \int_{\mathbb{R}^{n+1}_+}\tilde{\rho}^{1-2\gamma}_{x_0,\mu}|\nabla u|^2_{\tilde{g}_{x_0,\mu}}\,dv_{\tilde{g}_{x_0,\mu}}\leq 2\int_{\mathbb{R}^{n+1}_+}y^{1-2\gamma}|\nabla u|^2\,dxdy,
x_1^3 y_1^3 + x_1 y_1 x_2 y_2 (ax_1 y_1 + b x_2 y_2) + c x_2^3 y_2^3 +x_1^3 y_1 y_2^2 + x_1 x_2^2 y_1^3 + + d x_2^3 y_1^2 y_2 + e x_1^2 x_2 y_2^3 = 0 \, .
F_{4,14}(\tau)=\left(\begin{smallmatrix}0 0 0 2(u-1/u) (u+1/u)\end{smallmatrix}\right)XY^3-\left(\begin{smallmatrix}(u+1/u) 2(u-1/u) 0 0 0\end{smallmatrix}\right)X^3Y+\ldots
F_\Gamma^{p-1}(z) = \sum_{\ell=0}^n\sum_{j=1}^{p}(-1)^{j-1}\sum_{t=0}^{\lfloor\frac{p-j}{2}\rfloor}\binom{n-p+j+2t}{t}&\sum_{\beta=0}^{p-j-2t}2^{p-j-2t-\beta}\binom{n-\ell}{\beta}\binom{\ell}{p-j-2t-\beta}\sum_{\alpha=0}^\beta \binom{\beta}{\alpha}\sum_{i=0}^{j-1}G(z),
- 2 \sqrt{h}D_{j}\Bigl ( \frac{1}{\sqrt{h}}\frac{\delta S}{\delta h_{\hspace{0 . 5 e m}j}^{i}}\Bigr ) + \partial_{i}\phi^{I}\frac{\delta S}{\delta \phi^{I}}+ D_{i}\tilde{A}_{j}^{a}\frac{\delta S}{\delta \tilde{A}^{a j}}- \sqrt{h}D_{j}\Bigl ( \frac{1}{\sqrt{h}}\frac{\delta S}{\delta \tilde{A}_{j}^{a}}\tilde{A}_{i}^{a}\Bigr ) = 0 .
_cD_{0t}^{\alpha}\left(\int_0^x (x-t)^{\alpha-1}E_{\alpha, \alpha}[\lambda(x-t)^{\alpha}]f(t)dt\right) = f(x)-\int_0^xf(z)\frac{d}{dz}[E_{\alpha, 1}(\lambda (x-z)^{\alpha})]dz.
\alpha_j^2 = r(x,\xi'+i\tau d_{x'}\varphi)+(-1)^j\sigma^2 = r(x,\xi') +\int_{0}^1 d_{\xi'}r(x,\xi'+i \tau s\, d_{x'}\varphi) (i \tau d_{x'}\varphi)ds +(-1)^j\sigma^2.
{\cal L}_{v e c}^{\prime}\, = \, \mathrm{i}\, \Bigl [{\cal F}^{- T}\, \bigl ( A + B{\bar{\cal N}}\bigr )^{T}{\bar{\cal N}}^{\prime}( A + B{\bar{\cal N}}\bigr ){\cal F}^{-}\, - \,{\cal F}^{+ T}\, \bigl ( A + B{\cal N}\bigr )^{T}{\cal N}^{\prime}( A + B{\cal N}\bigr ){\cal F}^{+}\Bigr ]
\Vert \nabla\varphi \Vert_{\mathbb{L}^2(\Omega)}= \Vert \nabla \hat{\varphi}\Vert_{\mathbb{L}^2(\mathrm{B})}\Vert \nabla\varphi \Vert_{\mathbb{L}^4(\Omega)}= \delta^{-\frac{1}{2}}\Vert \nabla \hat{\varphi}\Vert_{\mathbb{L}^4(\mathrm{B})}.
(\prescript{RL}{x_L}{\mathcal{D}}_{x}^{\mu}f)(x)=\frac{d}{dx}\left[(\prescript{}{x_L}{\mathcal{I}}_{x}^{1-\mu}f)(x)\right]=\frac1{\Gamma(1-\mu)}\frac{d}{dx}\int_{x_L}^x\frac{f(s)}{(x-s)^{\mu}}ds,x>x_L,
as_1(F_1, F_2, \psi) = \int_{X_\psi}e^{-\psi^*(\nabla \psi(x)}\det \, \nabla^2 \psi(x) dx = \int_{\Omega_\psi}e^{-\psi^*(\nabla \psi(x)}\det \, \nabla^2 \psi(x) dx
x_{\mathbf{M}}^n+a_{n-1}x_{\mathbf{M}}^{n-1}+\dots +a_{k+1}x_{\mathbf{M}}^{k+1}=a_k x_{\mathbf{M}}^k+a_{k-1}x_{\mathbf{M}}^{k-1}+\dots+ a_l x_{\mathbf{M}}^l.
\frac{A^{2 n + 2}}{V^{2 n + 1}}= \frac{2 \pi}{( n + 1 )}{\mathrm{V o l{( M )}}}2^{2 n + 1}( n + 1 )^{2 n + 1}\frac{1}{\left( 1 - \frac{a^{2 n + 2}}{r^{2 n + 2}}\right)^{n}}
\frac{1}{\left( 1+x\right)^{s+1}}w_{n}^{\left( s+1\right)}\left(\frac{-x}{1+x};\alpha,\beta,r\right) =\frac{1}{2\pi i\Gamma\left(s+1\right)}{\displaystyle\int\limits_{a-i\infty}^{a+i\infty}}\left( r-\beta t\mid\alpha\right)_{n}x^{-t}\Gamma\left( t\right)\Gamma\left( s+1-t\right) dt,
I=m[\begin{matrix}n_{1}&n_{2}&n_{3}\end{matrix}][\begin{matrix}y^{2}+z^{2}&-xy&-xz\\ -yx&x^{2}+z^{2}&-yz\\ -zx&-zy&x^{2}+y^{2}\end{matrix}][\begin{matrix}n_{1}\\n_{2}\\n_{3}\end{matrix}].
\frac{\int_0^{L_m^*}\frac{|v'|^2 V_{\eta}}{| \eta'|}+\frac{k^2|v|^2 | \eta'|}{V_{\eta}}\,dt}{\int_0^{L_m^*}|v|^2 V_{\eta}| \eta'| \,dt}=\frac{\int_0^{L_m}\frac{|w'|^2 V_{\chi_m}}{| \chi_m'|}+\frac{k^2|w|^2 | \chi_m'|}{V_{\chi_m}}\,dt}{\int_0^{L_m}|w|^2 V_{\chi_m}| \chi_m'| \,dt}
(g\circ f)^{*}H ={\pi_{X}}_{*}\widetilde{f}^{*}\widetilde{g}^{*}H f^{*}(g^{*}H) ={\pi_{X}}_{*}\widetilde{f}^{*}\pi_{Y}^{*}{\pi_{Y}}_{*}\widetilde{g}^{*}H.
\left(\frac{e^{\theta_n}-e^{\theta_n u}}{e^{\theta_n}-1}\right)^{\alpha-\beta-1}=(1-u)^{\alpha-\beta-1}e^{u\theta_n(\alpha-\beta-1)}\left( \frac{e^{(1-u)\theta_n}-1}{(1-u)\theta_n}\right)^{\alpha-\beta-1}\left( \frac{\theta_n}{e^{\theta_n}-1}\right)^{\alpha-\beta-1}.
a_i=\frac{i}{2}\Psi^\dagger\partial^{\hspace{-2.9mm}\raisebox{1.2mm}{$\leftrightarrow$}}_i\Psi - \Psi^\dagger\frac{\sigma^3}{2}\Psi A_i = \frac{i}{2}\Psi^\dagger\nabla^{\hspace{-3.8mm}\raisebox{1.2mm}{$\leftrightarrow$}}_i\Psi.
\partial_j \widetilde T_n(x) := \frac{1}{2\pi}\sqrt{\frac{2}{n}}\frac{\partial}{\partial x_j}T_n(x) = \sqrt{\frac{2}{n}}\frac{i}{\sqrt{\mathcal N_n}}\sum_{\lambda\in \Lambda_n}\lambda_j\,a_{\lambda}e_\lambda(x).
\varepsilon_{\psi}\sum_{l = h}^{\infty}( w - z )^{l - h}\langle \varphi , V ( \psi , \zeta ) \; V ( S_{- l}\chi , z ) \; \Omega \rangle - \sum_{m = - \infty}^{h - 1}( w - \zeta )^{m - h}\langle \varphi , V ( S_{- m}\psi , \zeta ) V ( \chi , z ) \; \Omega \rangle ;
\begin{array}{rcl}\frac{\partial^3}{\partial \rho^3}H(t,\rho)|_{\rho=0}= 3!\int\limits_{-1}^tf(s)H^2(s,0)ds + 3!\int\limits_{-1}^tf(s)\frac{\partial}{\partial \rho}H(s,0)ds &= 3!\int\limits_{-1}^tf(s)G^2(s)ds + 3!\int\limits_{-1}^tf(s)F(s)ds.\end{array}
\frac{\Gamma_k(\rho)}{\Gamma_k\left(\frac{n+1+\lambda}{2}\right)}=\frac{\Gamma\left(\frac{n-k+1}{2}\right)}{\Gamma\left(\frac{n+1+a}{2}\right)}\prod_{j=2}^k \frac{\Gamma\left(\frac{n-j+2}{2}\right)}{\Gamma\left(\frac{n-j+2+\lambda_j}{2}\right)}.
\begin{aligned}l^{(1)}=\frac{1}{2}(c_2+u/a,-c_1u-1/a,c_1), &l^{(2)}=(1-c_2,c_1u,-c_1), &l^{(3)}=\frac{1}{2}(c_2-u/a,-c_1u+1/a,c_1), \end{aligned}
d^2 \big( \Psi(y_1,s),\Psi(y_2,t) \big) \ge \max\big\{d_Y^2 \big( \pi \circ \Psi(y_1,s),\pi \circ \Psi(y_2,t) \big), |u \circ \Psi(y_1,s) -u \circ \Psi(y_2,t)|^2 \big\}= \max\big\{d_Y^2(y_1,y_2),|s-t|^2 \big\}\ge \frac{1}{2}\Big( d_Y^2(y_1,y_2) +|s-t|^2 \Big).
((1-t)\otimes_{p,s}f\oplus_{p,s}t\otimes_{p,s}g)(z)= \sup_{0<\lambda< 1}\left( \sup_{z=C_{p,\lambda,t}x + D_{p,\lambda,t}y}\left[C_{p,\lambda,t}f(x)^{\frac{1}{s}}+D_{p,\lambda,t}g(y)^{\frac{1}{s}}\right]^s\right)
\sum_{p=1}^{q/2 - 1}\frac{(q!)^4}{(p!)^2}\left(\frac{2}{(q-p)!^2}- \frac{1}{2\big((q/2)!(q/2 - p)!\big)^2}\right) \|f_n\star_p^p f_n \|^2 + \sum_{p=1, p\neq q}^{2q-1}p! \|G_p^q\,f_n \|^2 \ge \sum_{p=1}^{q/2 - 1}\frac{(q!)^4}{(p!)^2}\left(\frac{6}{(q-p)!^2}- \frac{1}{2\big((q/2)!(q/2 - p)!\big)^2}\right) \|f_n\star_p^p f_n \|^2.
\int \frac{{\rm d}^3 Q}{|{\bf Q}|}N(|{\bf Q}|)\left\{\frac{{\rm d}}{{\rm d}Q_0}\left(\frac{1}{Q_0}{\rm ln}(k\cdot Q)\right)\right\}_{Q_0=|{\bf Q}|}=\pi{\rm ln}(T){\rm ln}(k^2)+\cdots ,
\alpha=0 \beta = -640,-650,-656,-660,-662,-664,-668,-672,-676, &-678,-680,-684,-686,-688,-690,-692,-694,-696, &-698, \alpha=1 \beta= -650,-668,-680,-682,-686,-688,-692,-694,-696, & -698,-700,-702,-704,-712,-722,-738,-748, \alpha=2 \beta = -672,-720,-732,-734
\mathring H_{min}\cong\bigoplus_{j=\frac{1}{2},\frac{3}{2},\dots}^\infty\,\bigoplus_{m_j=-j}^j\,\bigoplus_{k_j=\pm(j+1/2)}\,h_{m_j,k_j}, H_{max}\cong\bigoplus_{j=\frac{1}{2},\frac{3}{2},\dots}^\infty\,\bigoplus_{m_j=-j}^j\,\bigoplus_{k_j=\pm(j+1/2)}\,{h}^*_{m_j,k_j},
(g_{n}+g_{n+1})^{\ast}(u^{\prime},v^{\prime}) =g_{n}^{\ast}(n\gamma_{n}^{\prime},(-1)^{n}\varsigma_{n}\gamma_{n}^{\prime})+g_{n+1}^{\ast}((n+1)\gamma_{n+1}^{\prime},(-1)^{n+1}\varsigma_{n+1}\gamma_{n+1}^{\prime}) & =\gamma_{n}^{\prime}(\ln\gamma_{n}^{\prime}-1)+\gamma_{n+1}^{\prime}(\ln\gamma_{n+1}^{\prime}-1).
a_1 (2 - b) = a_0 (1 - b + a), &\cdots a_{n-1}(n - b)(n-1) = a_{n-2}(1-b+a+n-2), a_n (n + 1 - b)n = a_{n-1}(1 - b + a + n -1), a_{n+1}(n+2-b)(n + 1) = a_n(1-b+a+n), &\cdots
\sqrt{2}=\prod_{k=0}^{\infty}\frac{(4k+2)^{2}}{(4k+1)(4k+3)}=(\frac{2\cdot2}{1\cdot3})(\frac{6\cdot6}{5\cdot7})(\frac{10\cdot10}{9\cdot11})(\frac{14\cdot14}{13\cdot15})\cdot\cdot\cdot
\begin{array}{l}\lambda=a+\alpha b-\beta^2+\beta+(2n-1)\alpha^2 \mu=b+\beta b+(\alpha-\alpha\beta)(2n-2) \nu=\alpha b-(2n+1)\beta+\beta^2+(1-2n)\alpha^2+c-\beta b+(\alpha\beta-\alpha)(2n-2) \end{array}
S^{2}= \Delta \left( Y^{\prime}- \frac{2 C_{2}+ \frac{1}{\Delta}( a C_{3}+ \frac{\tilde{d}\, C_{4}}{D - 2}) r^{\tilde{d}}}{r ( C_{2}+ C_{1}r^{2 \tilde{d}})}\right)^{2}+ \frac{K}{( C_{2}+ C_{1}r^{2 \tilde{d}})^{2}}r^{2 \tilde{d}- 2},
I_4\leq C\prod\limits_{k=1}^2\omega_k(B)^{\lambda_k}\prod\limits_{k=1}^2\|f_k\|_{\dot{B}^{p_k,\lambda_k}}\prod_{k=1}^2\|b_k\|_{C\dot{M}O^{q_k}_{\omega_k}}&\times\int_{[0,1]^n}\prod_{k=1}^2|s_k(t)|^{(d+\alpha_k)\lambda_k}\left(1+\sum\limits_{k=1}^2\log\frac2{|s_k(t)|}\right)\psi(t)dt.
E\mathcal{P}_{n}^{\left( +\right)}\left( E\right) =b_{n-1}^{\left( +\right)}\mathcal{P}_{n-1}^{\left( +\right)}\left( E\right) +a_{n}^{\left( +\right)}\mathcal{P}_{n}^{\left( +\right)}\left( E\right) +b_{n}^{\left( +\right)}\mathcal{P}_{n+1}^{\left( +\right)}\left( E\right)
( \partial_{-})^{- 1}( x^{-}- y^{-}) = i \int_{- \infty}^{\infty}d k \exp{- i k_{-}( x^{-}- y^{-})}\ \mathrm{\large C P V}\frac{1}{k_{-}}= \frac 1 2 \mathrm{s g n}( x^{-}- y^{-}) .
\int_{\Gamma}\frac{f(\zeta)}{\zeta-T}d\zeta+\int_{\Omega^{\prime}}\frac{f(\zeta)}{\zeta-T}d\zeta=\int_{\Gamma}\frac{f(\zeta)}{\zeta-T}d\zeta-\int_{\Omega}\frac{f(\zeta)}{\zeta-T}d\zeta=0.
\xi_2(T) =\lim_{n\to +\infty}\frac{1}{n}\log c_2(T^n) \ge -\lim_{n\to +\infty}\frac{1}{n}\log \lVert ((T_{\mid \mathcal{N}_{\lambda_1}})^{-1})^n\rVert =\log |\lambda_1|,
v_{f}^{i}( \sigma , t ) = \sum_{n = 1}^{\infty}W ( n + \frac{\epsilon}{\tau}) \Biggl [ \frac{c_{n}^{i}}{\sqrt{n}}\dot{x}_{n}^{+}+ \frac{\tilde{c}_{n}^{i}}{\sqrt{n}}\dot{x}_{n}^{-}+ h . c . \Biggr ] .
\sum_{i=1}^{n}{^\beta}\gamma_{in}\frac{\partial F^{n}_{\beta\alpha}}{\partial z_{\beta}^{i}}\biggr|_{z_\beta(\gamma p)}={^\alpha}\gamma_{nn}\frac{\partial F^{n}_{\beta\alpha}}{\partial z_{\beta}^{n}}\biggr|_{z_\beta(p)}.
\left|\mu_A(E_n)-\frac{\mu(A')}{\mu(A)}\mu(E_n)\right| =\frac1{\mu(A)}\left|\mu(A\cap E_n)-\mu(A')\mu(E_n)\right| &=\frac1{\mu(A)}\left|\mu(A'\cap E_n)-\mu(A')\mu(E_n)\right|
0\ge H_y ={}u_{xxy}+ \tau^2 u_{yyy}+ 2\tau u_{xyy}={}(\tau^2-1) u_{yyy}- 2\tau u_{xxx}- f'(u) u_y ={}- \frac{(1-\tau^2)^2 + 4\tau^2}{2\tau}u_{xxx}- (\tau^2-1) f'(u) u_y - f'(u)u_y ={}- \frac{(1+\tau^2)^2}{2\tau}u_{xxx}- \tau^2 f'(u) u_y.
{\cal A}_{L}^{( 1 ) \mathrm{l e a d}}= \frac m{8 \pi}\left( \frac{\lambda^{2}}{1 6 m^{4}}- \frac{4 e^{4}}{m^{2}\Theta^{2}}\right) \left[ \ln \left( \frac{\Lambda^{2}}{{\bf p}^{2}}\right) + i \pi \right] .
H_{c}^{r}\, =\,{1\over 2}\int \left[ \left( \pi_1 \phi_2 - \pi_2 \phi_1 \right)^2\, -\,{1\over \phi_{1}^{2}+ \phi_{2}^{2}}\left(\phi_1 \partial_i\partial_i \phi_1 + \phi_2 \partial_i \partial_i \phi_2\right)\right] dx\, .
\Gamma^{\mu_{1}. . . \mu_{n}}\Gamma_{1 1}= ( - )^{n}\Gamma_{1 1}\Gamma^{\mu_{1}. . . \mu_{n}}={\frac{\epsilon^{\mu_{1}. . . \mu_{1 0}}}{( 1 0 - n ) !}}\; \Gamma_{\mu_{1 0}. . . \mu_{n + 1}}
g = \frac{4 \pi^{2}l}{T^{2}}= \frac{4 \times 3 . 1 4 \times 3 . 1 4 \times 1 0 7 . 4 3}{0 . 0 0 3 2 \times 0 . 0 3 2}= 9 . 7 7 m / s
T_{\widetilde{m}^{\varepsilon}}^{(2)}(f_{1},f_{2})(x):=\int_{\mathbb{R}^{4}}\widetilde{m}^{\varepsilon}(\xi)\hat{f_{1}}(\xi_{1})\hat{f_{2}}(\xi_{2}) e^{2\pi ix\cdot(\xi_{1}+\xi_{2})}d\xi.
\forall f\in C(G,C(G)):\ \ \I(f)(h,g) := f(\alpha_{g}(h^{-1}),g)=f(gh^{-1}g^{-1},g), I^{-1}(f)(h,g) \ =f(\alpha_{g^{-1}}(h^{-1}),g),\ \ \h,g\in G.
Z_{\alpha,\beta}(\tau)=\frac{1}{2\pi}\int_{\mathbb{R}^2}'' |x_1x_2|^{-\alpha/2}\int_{\mathbb{R}}\frac{e^{i\tau(x_1+u)}-1}{i(x_1+u)}\frac{e^{i\tau(x_2-u)}-1}{i(x_2-u)}|u|^{-\beta}du~ W(dx_1)W(dx_2),
\frac{\sum_{k=0}^n V_k - \frac{\mu_Q}{\mu_{\sigma}}n}{n^{1/2}}\leq \frac{\sum_{i = 1}^{i_n+1}Z_i}{n^{1/2}}+ \frac{\frac{\mu_Q}{\mu_{\sigma}}(\sigma_{i_n+1}- \sigma_0 - n) + Q_0}{n^{1/2}}\equiv (I') + (II').
R^L \asymp n^{-1}\int_{\prod_{k=1}^rx_{p+k}\leq (nb)^{1/m},x_{p+k}\geq 1}\left(1-\frac{1}{nb}\prod_{k=1}^rx_{p+k}^m\right) & \asymp [\log(nb)]^{r-1}n^{-1+1/m}b^{1/m}\asymp [n(\log n)^{1-r}]^{-2m/(2m+1)},
[y_1, y_1]=y_4+\theta_1y_5, [y_1, y_2]=\frac{\alpha_3}{\alpha_1}y_4+\theta_2y_5, [y_2, y_1]=\frac{\alpha_6\gamma_1}{\alpha_1(\gamma_2+\gamma_4)}y_5, [y_1, y_3]=\frac{\beta_4}{\gamma_2+\gamma_4}y_5, [y_2, y_3]=y_4+y_5, [y_3, y_2]=-y_4.
E |\mu^N(t)- m(t)| + \frac CN \sum_{i=1}^N E|X^N_i(t) - Y^N_i(t)| &\leq E |\overline{\mu}^N(t)- m(t)| + \frac{2C}{N}\sum_{i=1}^N E|X^N_i(t) - Y^N_i(t)| &\leq \frac{C}{\sqrt{N}}+ 2K_1 \frac1N \sum_{i=1}^N \int_0^t \left[ E|X^N_i(s) - Y^N_i(s)| + E |\mu^N(s)- m(s)|\right] ds .
\begin{aligned}A_n(y) =\sqrt{\frac{n}{\pi}}\int_\mathbb{R}{e^{-n(t-y)^2}\alpha_t(A) dt}&=\sqrt{\frac{n}{\pi}}\int_\mathbb{R}{e^{-n{t^\prime}^2}\alpha_{t^\prime+y}(A) dt}&=\alpha_y\left(A_n\right). \end{aligned}
M_{s}=\frac{\left(M_{s}^{n_0-1}\backslash \phi_{n_0}( (0,s^{1/4}]\times X\times [-\eta/2,\eta/2])\right) \bigsqcup\left( \{\mathbf{r}_s\leq 1\}\times [L/2-\eta,L/2]\right)}{\thicksim_{n_0}},
\|v_j\|_{\ell_2}^2 = \sum\limits_{n=1}^{\infty}\left|\Delta^{k-j}c_{n-j}\Delta^j f(n) \right|^2=\sum\limits_{n=1}^{\infty}\frac{\left| n^{j}\Delta^j f(n) \right|^2 \left|\Delta^{k-j}c_{n-j}\right|^2}{n^{2j}}&\leq C(j)^2\sum\limits_{n=1}^{\infty}\frac{\left|\Delta^{k-j}c_{n-j}\right|^2}{n^{2j}}= C(j)^2 \Omega_j\leq 4^j M C(j)^2 \|x\|_k^{2}.
d \hat{s}( \xi ) \oint{d}\hat{s}( \xi^{\prime \prime}){\hat{K}_{D}}( \xi , \xi^{\prime \prime}){\delta_{\rho}}{\hat{G}_{K}}( \xi^{\prime \prime}, \xi^{\prime}) = -{\frac{\rho ( \xi ) d \hat{s}( \xi )}{2 \oint{d}\hat{s}( \eta )}}+{\frac{d \hat{s}( \xi ) \oint{d}\hat{s}( \zeta ) \rho ( \zeta )}{2{{\left[ \oint{d}\hat{s}( \eta ) \right]}^{2}}}}.
C(m)=Am=[\begin{matrix}1&a_{1}&a_{1}^{2}&...&a_{1}^{k-1}\\ 1&a_{2}&a_{2}^{2}&...&a_{2}^{k-1}\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ 1&a_{n-1}&a_{n-1}^{2}&...&a_{n-1}^{k-1}\end{matrix}][\begin{matrix}m_{0}\\m_{1}\\ ...\\m_{k-1}\end{matrix}]
c_{\boldsymbol{r}}= \displaystyle\int_{-s/N^{1/d}}^{s/N^{1/d}}\cdots\int_{-s/N^{1/d}}^{s/N^{1/d}}e{\Big(-\sum_{i\leq d}r_i\alpha_i\Big)}d\alpha_1\cdots d\alpha_d = c_{r_1}\cdots c_{r_d},
\sum_{j=1}^{2}(-1)^{j-1}S_{j}\le...\le\sum_{j=1}^{n-3}(-1)^{j-1}S_{j}\le\sum_{j=1}^{n-1}(-1)^{j-1}S_{j}\le\sum_{j=1}^{n}(-1)^{j-1}S_{j}=\mathbb{P}(\bigcup_{i=1}^{n}A_{i})
D^+_iw_h(x) \leq \sup_{x \in U_i}\frac{1}{n}f(x)^{\frac{1}{n}-1}(D^+_if(x))_+ &\leq \sup_{x \in U_i}\left( \frac{1}{n}f(x)^{\frac{1}{n}-1}f_{x_i}(x) + Ch\right)_+ &=\sup_{U_i}\left( (f^\frac{1}{n})_{x_i}+ Ch\right)_+ &\leq [f^\frac{1}{n}]_{1;[0,1]^n}+ Ch.
\left\|[H_j^0,P_N^0]\right\|_p,~\left\|P_N^0H_j^0P_N^0 - H_j^{0(N)}P_N^0\right\|_p \longrightarrow 0 ~~N\longrightarrow \infty ~~ p\geq 2,~ j = 1,2~~
n^{\alpha \beta}= \frac{\partial W}{\partial \alpha_{\alpha \beta}}= h \, K^{\alpha \beta}\, H^{\alpha \beta \gamma \delta}\, \alpha_{\gamma \delta}\, , m^{\alpha \beta}= \frac{\partial W}{\partial \beta_{\alpha \beta}}= \frac{h^3}{12}\, K^{\alpha \beta}\, H^{\alpha \beta \gamma \delta}\, \beta_{\gamma \delta}\, .
A_{n}= \left[ \begin{array}{c c c c c}{u_{0}}&{q_{1}}&{0}&{. . .}&{0}\\{- 1}&{u_{1}}&{q_{2}}&{. . .}&{0}\\{0}&{- 1}&{u_{3}}&{. . .}&{0}\\{}&{.}&{.}&{.}&{}\\{0}&{. . .}&{0}&{- 1}&{u_{n}}\\ \end{array}\right]
\underset{[0,T]}{}(\rho_2^2 \|y(t)\|^2) + \iint_Q \rho_2^2 |y_x|^2 dxdt \leq C \left(\|y_0\|^2 + \iint_Q \rho^2 |G|^2 dxdt + \iint_{\mathcal{O}\times (0,T)}\rho_1^2 |f|^2 dxdt \right. &+ \left. \iint_Q \rho_0^2 |y|^2 dxdt + \sum_{i=1}^2 \iint_Q \rho_0^2 |p^i|^2 dxdt \right).
+\frac{1}{2e^{2}\mid\varphi\mid^{2}}\{(\partial_{\nu}\Lambda^{\nu\mu})+\frac{e}{2}(\varphi^{\dagger}\partial_{\mu}\varphi-\partial_{\mu}\varphi^{\dagger}\varphi )\}^{2}+\frac{1}{2}\mid\partial_{\mu}\varphi\mid^{2}.
R_g(N_{W/X})\cdot i_*x= R_g(N_{W/X}){\rm ch}_g(i_*x)=R_g(N_{W/X})i_*\big({\rm Td}_g^{-1}(N_{X/Y}){\rm ch}_g(x)\big) = i_*\big(i^*R_g(N_{W/X}){\rm Td}_g^{-1}(N_{X/Y}){\rm ch}_g(x)\big)=0.
A_{n,s}= \sum_{k=\frac{n}{2}}^{\infty}\left( \frac{1}{(2k)^s}- \frac{1}{(2k+1)^s}\right) \ \ \frac{1}{2(n-\frac{1}{4})^s}= \sum_{k=\frac{n}{2}}^{\infty}\left( \frac{1}{2(2k-\frac{1}{4})^s}- \frac{1}{2(2k+\frac{7}{4})^s}\right),
\overrightarrow{G}\otimes \overrightarrow{H}=\left(\oplus_i \overrightarrow{C}(i)\right)\otimes\left(\oplus_j \overrightarrow{C}'(j)\right) &=\oplus_i\oplus_j\left(\overrightarrow{C}(i)\otimes \overrightarrow{C}'(j)\right)
u = \Phi ( \chi_{x}, \chi_{x x}, \chi_{x x x}) \equiv \frac{\chi_{x x x}}{\chi_{x}}- \frac{\chi_{x x}^{2}}{\chi_{x}^{2}}+ 2 \alpha \chi_{x x}+ \alpha^{2}\chi_{x}^{2}+ 2 \alpha \varepsilon + \frac{\varepsilon^{2}}{\chi_{x}^{2}}
V_p=-(1+\frac{2\mu}{z^{7-p}})^{\alpha+\beta_+}(1+\frac{\mu}{z^{7-p}})^{2(p-8)/(7-p)}+\frac{(8-p)(6-p)}{4z^2}- \mu^2\frac{(7-p)^2}{z^2(z^{7-p}+2\mu)^2}