harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\log(e^{Q_1}e^{Q_2})=\left(\begin{matrix}-0.0571752 0.0718248 0.0498348 0.0498348 0.0291051 -0.0998949 0.0200951 0.0200951 0.0109967 0.0109967 -0.0947047 0.0158953 0.0170734 0.0170734 0.0247748 -0.0858252\end{matrix}\right).
\partial(a \cdot b) = \cdot \sigma \Delta(a \cdot b) \overset{\eqref{eq:asiifs}}{=}\cdot \sigma \bigg(\sum_i a_i^1 \cdot b \otimes a_i^2 + \sum_i b_i^1 \otimes a \cdot b_i^2\bigg) = \sum_i a_i^2 \cdot a_i^1 \cdot b + \sum_i a \cdot b_i^2 \cdot b_i^1 = \partial(a) \cdot b + a \cdot \partial(b),
M_2 = T^2 - 2D \le \left(\alpha_0 + \alpha_1 t + \alpha_2 t^2\right)^2 - 2\left(\beta_0 + \beta_1 t + \beta_2 t^2+ \beta_3 t^3 + \beta_4 t^4 \right) = \gamma_0 + \gamma_1 t + \gamma_2 t^2 +\gamma_3 t^3 +\gamma_4 t^4
P_{n}(a,q;0)=(-1)^{n}q^{n/2}a^{-n/2}\,\frac{1-a^{n+1}}{1-a},\Q_{n}(a,q;0)=(-1)^{n+1}q^{n/2}a^{-n/2}\,\frac{1-a^{n}}{1-a}.
{\mathcal U}_n(t,s) = \exp \left(t A_n^{1/n}\right)= \exp \left( t ~ \left\{\Pi_{k=1}^n \left[ ( I - \kappa e^{-a_k(t,s)})^{-1}\partial_t a_k(t,s) \right] \right\}^{1/n}\right),
\frac{\partial U_{L}}{\partial P_{BS}}= \frac{\mu (1 - \theta) \frac{\xi \theta \|\mathbf{h}\|^{2}|h_{s}|^{2}}{(1-\theta)\sigma_{s}^{2}}}{1 + \frac{\xi \theta \|\mathbf{h}\|^{2}|h_{s}|^{2}}{(1-\theta)\sigma_{s}^{2}}P_{BS}}- \lambda \theta \|\mathbf{h}\|^{2}= 0.
\mathcal{C}(p,0,s_1,s_2) = 1- \frac{k\ell}{p^{s_1+s_2+2}}+ O_{k,\ell}( p^{-2.7}) \sigma_1, \sigma_2 \ge -10^{-1}, \mathcal{C}(p,j,s_1,s_2) = \frac{\tau_{k}(p^j) \tau_{\ell}(p^j)}{p^{j(s_1+s_2+1)}}\Big(1+O_{k,\ell}(p^{-0.9}) \Big) \sigma_1, \sigma_2 \ge -10^{-1}.
\frac{(1-\epsilon')(n - n_0)}{2 (1+ \delta_n) k_n}\log \left( 1 + (1+ \delta_n) k_n P \right) - (1-\epsilon) B_1(n) \geq_n \left[ (1-\epsilon')^2 (1 - n_0 /n) - (1-\epsilon) \right] B_1(n) = \left[ (1-\epsilon')^3 - (1-\epsilon) \right] B(n) ,
\begin{cases}\partial_t u_t(x)= m u_t(x)- (m^{\frac{2}{\alpha}}-\Delta)^{\frac{\alpha}{2}}u_t(x)+\xi\sigma(u_t(x)) \dot F(t,x), \ \ \x\in B_R(0), \ \t>0 u_t(x)= 0, \ \ \x\in{B_R(0)^c}. \end{cases}
\frac{d}{dt}(\lambda-\mu) =\,2(\lambda-\mu)\big((1-2\rho)(\lambda+\mu)-(1+2\rho)\nu\big) \frac{d}{dt}(\mu-\nu) =\,2(\mu-\nu)\big((1-2\rho)(\mu+\nu)-(1+2\rho)\lambda\big)\,.
F(x)=(\begin{matrix}&&\cdot\cdot\cdot\\ &&\alpha_{22}\cdot\cdot\cdot\alpha_{2k}\cdot\cdot\cdot\\ &\alpha_{11}&\alpha_{12}\cdot\cdot\cdot\alpha_{1k}\cdot\cdot\cdot\\ \alpha_{00}&\alpha_{01}&\alpha_{02}\cdot\cdot\cdot\alpha_{0k}\cdot\cdot\cdot\end{matrix})
\lambda = \frac{4(1 - c^2)+c^2(2c^2 - 1)^2 +3 |\xi|^2}{2(3-2 c^2)}+ \frac{3-2 c^2}{2}\pm \frac{1}{2}\Big( \Big( \frac{4(1 - c^2)+c^2(2c^2 - 1)^2+ 3|\xi|^2}{3-2 c^2}- (3-2 c^2)\Big)^2 + 4 c^2(2c^2 - 1)^2 \Big)^\frac{1}{2}.
\mathcal{K}^{-1}= \begin{bmatrix}S_M^{-1}-W^\top W S_N^{-1}\end{bmatrix}=\begin{bmatrix}S_M^{-1/2}0 0 S_N^{-1/2}\end{bmatrix}\begin{bmatrix}I -V^\top V I \end{bmatrix}\begin{bmatrix}S_M^{-1/2}0 0 S_N^{-1/2}\end{bmatrix},
g(\lambda) =B^\frac{-1}{2}(B^\frac{1}{2}A^\lambda B^\frac{1}{2})^\frac{\delta+m}{pk\lambda+m}B^\frac{-1}{2}&= |A^{*m}|^{-1}(|A^{*m}||A^p|^{2k\lambda}|A^{*m}|)^\frac{\delta+m}{pk\lambda+m}|A^{*m}|^{-1}
\begin{array}[c]{ll}\operatorname*{curl}\operatorname{curl}\mathbf{u}^{+}-k^{2}\mathbf{u}^{+}=0 \Omega^{+}, \left[ \left( \mathbf{u},\mathbf{u}^{+}\right) \right]_{0,\Gamma}=0 \Gamma\end{array}
\begin{array}{l}\displaystyle \# \{\sum_{\substack{j\in J\setminus J_{k}}}\Phi(B_{t_{j}},t_{j})(B_{t_{j+1}}-B_{t_{j}})|\forall J_{k}=\{i_1,\cdots,i_k\}\subset J, 1\leq k\leq K(n)=n^r, \ \ \ \ \ \ \forall r\in(0,1), \displaystyle \lim_{\substack{n\rightarrow +\infty}}\displaystyle \frac{K(n)}{n}=0 \}\end{array}
f(\mathbf{x}) = \sum_{i=1}^{m-1}a_{i}x_{\pi(i)}x_{\pi(i+1)}+\sum_{k=1}^{m}d_{k}x_{k}^{2}&= 2x_{1}x_{2}+x_{2}x_{4}+x_{4}x_{3}+2x_{1}^{2}+x_{3}^{2}+x_{4}^{2}
\Big \langle \prod_{a = 1}^{N}\prod_{i = 1}^{n_{a}}\overline{{\psi}}^{( a )}( x_{i}^{( a )}) P_{+}\psi^{( a )}( x_{i}^{( a )}) \prod_{i = 1}^{m_{a}}\overline{{\psi}}^{( a )}( y_{i}^{( a )}) P_{-}\psi^{( a )}( y_{i}^{( a )}) \Big \rangle
\bf{A}+\bf{B}+\bf{C}+\bf{A}\times\bf{B}+\bf{A}\times\bf{C}+\bf{B}\times\bf{C}-(\bf{A}\cdot\bf{B})\,\bf{C}+(\bf{A}\cdot\bf{C})\,\bf{B}-(\bf{B}\cdot\bf{C})\,\bf{A}= 0.
\langle V_{N}^{( 0 )}| ={}_{1 \dots N}\langle 0 | \exp \left\{\frac{1}{2}\, \sum_{I , J = 1}^{N}\int_{- \infty}^{\infty}d \kappa \, a^{- ( I )}( \kappa ) \, \mu_{N}^{I J}( \kappa ) \, C a^{- ( J )}( \kappa ) \right\}.
e^{\delta (t-y_0)}= \frac{e^{\delta v}}{1-e^{\delta v}}= \frac{1}{e^{-\delta v}-1}, e^{\delta y_0}= e^{\delta (t-v)}(1-e^{\delta v}) = e^{\delta t}(e^{-\delta v}- 1), \intertext{and to}e^{\delta v}= \frac{1}{e^{-\delta (t-y_0)}+ 1}
\varphi\bigl(s\sigma^{\mu}_{-i}(x)\bigr) =(\mu\otimes\varphi)\bigl((\Delta s)(\sigma^{\mu}_{-i}(x)\otimes1))\bigr)=(\mu\otimes\varphi)\bigl((x\otimes1)(\Delta s)\bigr) &=(\mu\otimes\varphi)\bigl(\Delta(xs)\bigr)=\varphi(xs),
(\nabla v)(x) =(\partial_rv^\sharp)(H_0(x))\nabla H_0(x), H((\nabla v)(x))\nabla_\xi H((\nabla v)(x)) =(\partial_r v^\sharp)(H_0(x))\nabla_\xi H(\nabla H_0(x))=\frac{(\partial_rv^\sharp)(H_0(x))x}{H_0(x)},
V(x(t_1, j_1)) \leq e^{-d (j_1 - j_0) - c (t_1 - t_0)}V(x(t_0, j_0)) \leq e^{-\eta (j_1 - j_0) - \lambda (t_1 - t_0) + \mu}V(x(t_0, j_0)),
0= \langle\nabla\nabla S\vert k \vert m_{3}km_{2}m_2\rangle_{3}=-\nabla_{3}\nabla_{3}S_{22}+\nabla_{1}\nabla_{0}S_{22}=-\nabla_{1}S_{22}\nabla_{3}k_3, 0= \langle\nabla\nabla S\vert k \vert m_{2}km_{2}m_{2}\rangle_{3}=-\nabla_{2}\nabla_{3}S_{22}=-\nabla_{1}S_{22}\nabla_{2}k_{3}, 0= \nabla_{0}\nabla_{3}S_{22}=\nabla_{1}S_{22}\nabla_{0}k_{3}.
\begin{array}{c}H^{\ast}\left( x\right) =x\left( \left( h^{-1}\right)^{\prime}\right)^{-1}\left( x\right) -h^{-1}\left( \left( \left( h^{-1}\right)^{\prime}\right)^{-1}\left( x\right) \right) ,~\left[ 0,\left(h^{-1}\right)^{\prime}\left( r_{0}\right) \right] .\end{array}
S(t)v(x)=\int_{\mathbb{R}^n}e^{-\frac{|\xi|^2 t}{2(1+|\xi|^2)}}\tfrac{2(1+|\xi|^2)}{|\xi|^2\sqrt{3+4|\xi|^2}}\sin\left(\tfrac{|\xi|^2 \sqrt{3+4|\xi|^2}t}{2(1+|\xi|^2)}\right)\widehat{v}(\xi)e^{ix\cdot\xi}d\xi,
dY_t = -h\nabla U(\bar{X}(T(t)))dt-h\left(\frac{1}{m}\sum_{i=1}^m \nabla u_i(X^{(i)}(T(t)))-\nabla U(\bar{X}(T(t)))\right)dt+\frac{\sqrt{2\sigma\alpha(T(t))h}}{\alpha(T(t))}d\bar{B}(T(t))
\begin{array}{rcl}\alpha \cdot ((v \otimes n) \otimes u) = \alpha \cdot (v \otimes n) \otimes u + (v \otimes n) \otimes \alpha u = -(v\alpha \otimes n )\otimes u + (v \otimes \alpha \cdot n) \otimes u + (v\otimes n) \otimes \alpha u\,. \end{array}
\begin{array}{rrcll}\min~ \multicolumn{4}{l}{c^\top x+\sqrt{x^\top Qx}}begin{align*}1.5ex] \textnormal{s.t.}\sum_{e\in \delta(i)}x_e = 2 \forall i\in V begin{align*}1ex] \sum_{e\in \delta(X)}x_e \ge 2 \forall \emptyset\neq X\subsetneq V begin{align*}1ex] x \in \{0,1\}^{E}\end{array}
(\exists x < |\sigma|)\Big( (\forall s)\big( \{e\}^{\sigma}_s(x) \uparrow \big) \vee (\exists s,y) \big( \{e\}^{\sigma}_s(x) \downarrow = y &\wedge (y = \{x\}_s(x) \downarrow \vee y \geq n + 1) \big) \Big).
\begin{array}{c c c c c c c}{{}}&{{\circ_{0}}}&{{\left( \frac{1}{g}\right)}}&{{}}&{{}}&{{}}&{{}}\\{{}}&{{\mid}}&{{}}&{{}}&{{}}&{{}}&{{}}\\{{\left( \frac{R_{1}}{g}\right) \; \; \circ_{1}-}}&{{\circ_{2}}}&{{-}}&{{\circ_{3}}}&{{- \dots -}}&{{\circ_{d - 1}\; \; \left( \frac{1}{R_{d}}\right)}}\\ \end{array}
\alpha^k_l+\beta^k_l=1+1-\frac{G_l(x_k)+H_l(x_k)}{\sqrt{G_l^2(x_k)+H_l^2(x_k)+2t_k}}=1-\frac{\varphi^t_\textup{FB}(G_l(x_k),H_l(x_k))}{\sqrt{G_l^2(x_k)+H_l^2(x_k)+2t_k}}\geq1
0 = \begin{bmatrix}m_1 m_1^2 \cdots m_1^{k-1}m_2 m_2^2 \cdots m_2^{k-1}\vdots \vdots \ddots \vdots m_{k-1}m_{k-1}^2 \cdots m_{k-1}^{k-1}\end{bmatrix}\begin{bmatrix}pd_1 p^2d_2 \vdots p^{k-1}d_{k-1}. \end{bmatrix}
\begin{aligned}(M-L_m)D^s_{L_m,M}(x_{n,i}) \:\:\: \leq \:\:\: L_mD^s_{L_m}(x_{n,i})+MD^s_M(x_{n,i}) \leq \:\:\: \sqrt{s}\sqrt{L_m}+\sqrt{s}\sqrt{M}\leq 2\sqrt{s}\sqrt{M}. \end{aligned}
\int d q_{j}\, \exp \left[ - \epsilon \frac{1}{2}{q_{j}}^{2}+ q_{j}( \Delta K_{j}+ \Delta p_{j}) \right] = \sqrt{\frac{2 \pi}{\epsilon}}\exp \left[ - \frac{( \Delta p_{j}+ \Delta K_{j})^{2}}{\epsilon}\right] \, .
\frac{1}{n^2}\sum\limits_{i=1}^{\lfloor n\tau\rfloor}\sum\limits_{j=\lfloor n\tau\rfloor +1}^{\lfloor n\lambda \rfloor}1_{\left\{G(\xi_i)\leq G(\xi_j)+\Delta\right\}}\overset{P}{\longrightarrow}\tau(\lambda-\tau)\int_{\mathbb{R}}F(x+\Delta)dF(x)
Q_{\eta,d,t}(G) = \{R(G,\mathcal{P},d) : \mathcal{P}\textnormal{is an}\eta\textnormal{-regular equipartition for}G\textnormal{and}\mathcal{P}\textnormal{has order}t\},\textnormal{and}Q_{\eta, d}(G) = \bigcup_{t=\frac{1}{\eta}}^{CM(\frac{1}{\eta}, \eta)}Q_{\eta,d,t}(G).
\kappa_3 = \frac{1}{2}\min(\epsilon_0, \kappa',\kappa''', \frac{\frac{1}{100}\epsilon_2}{k ( \log(\Vert f\Vert_{C^{0,1}}+1)+1 )}\kappa''), N_5 = 2N' + 2N'' + 100 \frac{k}{\epsilon_2}(\log(\Vert f\Vert_{C^{0,1}}+1) + 1)N_1(H,\epsilon_0) + 100 \frac{k}{\epsilon_2}.
T^{R_1}_{J_1}\star T^{R_2}_{J_1}= \sum_{R, J}\frac{1}{N}\left ( \begin{array}{ccc}R R_1 R_2 J J_1 J_2\end{array}\right ) \left \{\begin{array}{ccc}R R_1 R_2 \frac{p}{2}\frac{p}{2}\frac{p}{2}\end{array}\right \}T^R_J,
\lim_{\epsilon\to +0}\left[K_{\mu\nu}\right]_{x^\perp =-\epsilon}^{x^\perp =\epsilon}=\left.\frac{1}{m_d^{d-1}}\left(T_{\mu\nu}-\frac{1}{d-1}g_{\mu\nu}g^{\alpha\beta}T_{\alpha\beta}\right)\right|_{x^\perp =0}
\lim_{\beta\to 0}\sum_{n=0}\frac{e^{-\beta(n+1/2)}}{n!}H_{n}(\xi_{1}x_{1})H_{n}(\xi_{2}x_{2}) \exp\Bigl[-\frac{\xi_{1}^{2}x_{1}^{2}+\xi_{2}^{2}x_{2}^{2}}{4}\Bigr]=\sqrt{\frac{\pi}{2}}\delta\Bigl(\frac{\xi_{1}}{2}x_{1}-\frac{\xi_{2}}{2}x_{2}\Bigr).
H = \int{d^3}x\left({- \frac{1}{2}\Pi_i^a \Pi_a^i +\frac{1}{4}F_{ij}^a F^{aij}+ c^a \left( x \right)\left({\partial^i \Pi_i^a + gf_{abc}A_b^i \Pi_i^c-J^0}\right)}\right) ,
\chi({\bf x},t)=\int \frac{d^3{\bf k}}{[(2 \pi)^3 2 \omega_{\bf k}]^{1/2}}[c({\bf k}) e^{i({\bf k}_n \cdot x -\omega_{\bf k}t)}+c^{\dagger}({\bf k}) e^{-i({\bf k}_n \cdot x -\omega_{\bf k}) t)}]
\begin{cases}v_{\mathfrak{P}}(\mathfrak{D}_{E/F})=e-1 v_{\mathfrak{P}}(e)=0 , e\le v_{\mathfrak{P}}(\mathfrak{D}_{E/F})\le e-1+v_{\mathfrak{P}}(e) v_{\mathfrak{P}}(e)\ge 1 .\end{cases}
I = \frac{m_{P}^{2}}{1 6 \pi}\int d^{4}x \sqrt{- g}\Bigl [ R - 2 \Lambda \Bigr ] + \frac{m_{P}^{2}}{8 \pi}\int d^{3}x \sqrt{h}K + \int d^{4}x \sqrt{- g}\Bigl [ \frac{1}{2}g^{\mu \nu}\partial_{\mu}\phi \partial_{\nu}\phi - V ( \phi ) \Bigr ] ,
\varphi^a=\Bigl(\delta^a_b-\frac{1}{2}\xi^{ae}\S_{ec}\delta^c_jG^{jk}\delta^d_k\xi_{db}\Bigr)\delta^b_i\,\phi^i-\Bigl(\xi^{ac}\S_{cb}\delta^b_jG^{ji}\Bigr)\,{\bar\phi}_i\;\; ,
=\underset{t^{q_{i}-i}}{coef}\left[exp\left(\frac{-pt^{p}\theta}{e^{px}-t^{p}}\right)\cdot\frac{(1-t^{p}e^{-px})^{k+g_{Y}-1}}{(1-t^{p}e^{-2px})^{2g_{Y}-2}}\cdot \prod_{j=0}^{p-1}\left\{\frac{(1-\nu^{j}te^{-2x})^{m_{j,2}+m'_{p-j,-2}}}{(1-\nu^{j}te^{-x})^{m_{j,1}}}\right\}\right].
\int_0^T \int_{\Omega}|\partial_t w^i| \, dxdt \leq \sigma^{-1}\int_0^T \int_{\Omega}\Big( \frac{\varepsilon_i |\partial_t \varphi^i |^2}{2}+\frac{W (\varphi^i)}{\varepsilon_i}\Big) \, dxdt \leq D_1(1+T).
\mathbf{d}_{p+1}=(-a_{1}^{(p-1)},\ldots,-a_{p-1}^{(1)},z-a_{p}^{(0)}), \mathbf{v}_{p+1}=(-a_{1}^{(p)}A_{0}^{(p+1)},-\sum_{j=p-1}^{p}a_{2}^{(j)}A_{j-p+1}^{(p+1)},\ldots,-\sum_{j=1}^{p}a_{p}^{(j)}A_{j-1}^{(p+1)}),
\begin{aligned}\mathbb{E}^{u_1,u_2}\bigg[\Phi_{ix}(x(T))x_i^1(T)+\gamma_{iy}(y(0))y_i^1(0)+\Phi_i(x(T))\Gamma_i(T)+\int_0^T\Gamma_i(t)l_i(t)dt + \int_0^T[l_{ix}(t)x_i^1(t)+l_{iy}(t)y_i^1(t)+l_{iz}(t)z_i^1(t)+\sum_{j=1}^{2}l_{iz_j}(t)z_{ji}^1(t)]dt+\int_0^Tl_{iv_i}(t)v_i(t)dt\bigg]\geq 0.\end{aligned}
\frac{\mathrm{d}}{\mathrm{d}z}\, \phi_{\, 1}( z ) = - F^{\prime}\, \phi_{\, 1}( z ) -{\frac{1}{2}}F^{'^{\prime}}\, \langle \, \chi^{2}( z , \, t ) \, \rangle_{\mathrm{s o l}}\; .
\begin{aligned}\rho(v_n)-\rho(v_n-v_\mu) =\rho(v_\mu)+o_n(1) \int_{\Omega}b(x)\left(|v_{n}|^{\alpha(x)}-|v_{n}-v_\mu|^{\alpha(x)}\right)dx =\int_{\Omega}b(x)|v_\mu|^{\alpha(x)}dx+o_n(1), \end{aligned}
-\frac{\partial L_{d}D_{\alpha}(p\|q)}{\partial q_{j}}=\frac{1}{\alpha}\left[\frac{a-1}{a-b}A^{a-2}-\frac{b-1}{a-b}A^{b-2}\right]\left(p^{\alpha}_{j}q^{-\alpha}_{j}-1\right)
\tilde\tau_D(x,y)=\frac{1}{k}\sum_{i=1}^{k}\tilde\tau_{p_i}(x,y)=\frac{1}{k}\log\Bigg(\prod_{i=1}^{k}\frac{\mu_{p_i}(x,y)}{\sqrt{d(x,p_i)d(y,p_i)}}\Bigg).
\begin{array}{l}\mathbf{E}(\mathbf{x},t) =\displaystyle\frac{1}{4\pi}\int\rho_e(\mathbf{x}',t-r)\frac{\mathbf{r}}{r^3}\,d^3x' ~ +\displaystyle\frac{1}{4\pi}\int\mathbf{J}_m(\mathbf{x}',t-r)\,{\bf\times}\,\frac{\mathbf{r}}{r^3}\,d^3x'-\displaystyle\frac{1}{4\pi}\int\frac{1}{r}\frac{\partial\mathbf{J}_m(\mathbf{x}',t-r)}{\partial t}\,d^3x' ,\end{array}
\sum_{n = - \infty}^{+ \infty}z^{n}q^{n^{2}}= \prod_{n = 0}^{\infty}\left( 1 - q^{2 n + 2}\right) ~ \left( 1 + z q^{2 n + 1}\right) ~ \left( 1 + z^{- 1}q^{2 n + 1}\right) ,
I_1^\delta(a) =\int_\delta^\infty x e^{-ax^2}dx = -\frac{1}{2a}\int_\delta^\infty \frac{d}{dx}\left[ e^{-ax^2}\right] dx = \frac{1}{2a}e^{-a\delta^2}\leq \frac{1}{2a}e^{-\frac{a\delta^2}{2}}.
m_{p}= \sqrt{\alpha_{G}}\sim 2 . 7 2 \times 1 0^{- 2 0}\approx 5 . 9 2 \times 1 0^{- 2 8}\, \mathrm{k g}\approx 0 . 3 5 4 \, m_{p \ \mathrm{e x p}}\, .
z \approx{\frac{1}{2}}({\frac{h^{2}}{2 m \pi k T}})^{\frac{3}{2}}{\frac{\langle N \rangle}{V}}- \delta ( p , q ) ({\frac{h^{2}}{2 m \pi k T}})^{3}({\frac{\langle N \rangle}{V}})^{2}.
(t^{k_{0}}-1)A^{*}=[t^{k_{0}}A_{0}(\frac{h}{t})-A_{0}(h)]+(t^{k_{0}}a_{1}(\frac{h}{t})^{k_{1}}-a_{1}h^{k_{1}})+(t^{k_{0}}a_{2}(\frac{h}{t})^{k_{2}}-a_{2}h^{k_{2}})+O(h^{k_{3}}).
ds^{2}=g(x,dx)=\frac{\|dx\|^{2}}{1-\|x\|^{2}}+\frac{(x\cdot dx)^{2}}{(1-\|x\|^{2})^{2}}=\frac{(1-\|x\|^{2})\|dx\|^{2}+(x\cdot dx)^{2}}{(1-\|x\|^{2})^{2}}
-{\frac{1}{4 \mu_{0}l_{\gamma}}}\int d s \sqrt{\overline{y}^{\prime 2}( s )}{\frac{\delta^{2}\Psi [ \sigma^{\mu \nu}\ ; A ]}{\delta \sigma^{\mu \nu}( s ) \sigma_{\mu \nu}( s )}}= i{\frac{\partial}{\partial A}}\Psi [ \sigma^{\mu \nu}\ ; A ]
M_{nm}=b_{nm}\times\underset{\mathcal{R}(\mathbf{F},\mathbf{M})\,\cap\,\mathcal{X}^0_{nm}}{\max}(\theta_n-\theta_m) & M_{mn}=b_{nm}\times\underset{\mathcal{R}(\mathbf{F},\mathbf{M})\,\cap\,\mathcal{X}^0_{nm}}{\max}(\theta_m-\theta_n)
{\bf E}\Bigl[\Bigl(M_\mu^{(c=1/2)}\bigl(0,\frac{1}{2}\bigr)\Bigr)^{n}\Bigl( M_\mu^{(c=1/2)}\bigl(\frac{1}{2}, 1\bigr)\Bigr)^{m}\Bigr] = n!m! \int\limits_{\substack{0<t_1<\cdots<t_{n}<1/2 1/2<t_{n+1}<\cdots<t_{n+m}<1}}\prod\limits_{k<p}^{n+m}|t_p-t_k|^{-\mu\,2^{k-p-1}}\,dt.
m_{x_{k,j}}^{post}= \frac{m_{x_{k,j}}^{prio}/ v_{x_{k,j}}^{prio}+ \mathbf{h}_{k,j}^H \mathbf{V}_{\xi_{k,j}}^{-1}\left( \mathbf{y}- \mathbf{H}\mathbf{m}_\mathbf{x}^{prio}+ \mathbf{h}_{k,j}m_{x_{k,j}}^{prio}\right)}{1/v_{x_{k,j}}^{prio}+ \mathbf{h}_{k,j}^H \mathbf{V}_{\xi_{k,j}}^{-1}\mathbf{h}_{k,j}}.
<\chi\mid\chi>=\S_{\begin{array}{l}{\scriptstyle n_{1},...,n_{N}=0}{\scriptstyle m_{1},...,m_{M}=0}{\scriptstyle q_{1},...,q_{Q}=0}{\scriptstyle t_{1},...,t_{T}=0}\end{array}}^{1}{\left| c'({\{n_{r}\}}^{N}_{1};{\{m_{r}\}}^{M}_{1};{\{q_{r}\}}^{Q}_{1};{\{t_{r}\}}^{T}_{1}) \right|}^{2}.
\nonumber d s^{2}= - V [ ( 1 - \Omega w ) d t - w d \varphi ]^{2}+{\frac{1}{V}}\left[ \rho^{2}( d \varphi + \Omega d t )^{2}+ e^{2 \gamma}( d \rho^{2}+ d z^{2}) \right]
\sum_{\pi(1,2,...n)}\mbox{Tr}\Big[H^{(1)}H^{(l_{\pi(1)})}.....H^{(l_{\pi(n)})}\Big]\frac{1}{\tilde{x}_0 - \tilde{x}_{\pi(1)}}\frac{1}{\tilde{x}_{\pi(1)}- \tilde{x}_{\pi(2)}}.....\frac{1}{\tilde{x}_{\pi(n)}- \tilde{x}_0}\; .
\beta^\frac{1}{q}\,[u]_{W^{\beta,q}(\mathbb{R}^N)}\le C\,\left(\frac{\alpha}{\alpha-\beta}\right)^\frac{1}{q}\,\|u\|_{L^p(\mathbb{R}^N)}^{(1-\theta)\,\left(1-\frac{\beta}{\alpha}\right)}\,\|u\|^\theta_{L^r(\mathbb{R}^N)}&\times \left((1-t)^\frac{1}{p}\,[u]_{W^{t,p}(\mathbb{R}^N)}\right)^{\frac{\beta}{\alpha}\,(1-\theta)}.
\begin{array}{c c}{{{\cal Q}^{a}= \left( \begin{array}{c}{{{\cal Q}_{+}^{a}}}\\{{{\cal Q}_{-}^{a}}}\\ \end{array}\right) ~}}&{{~ \bar{{\cal Q}}_{a}={{\cal Q}^{a}}^{\dagger}\Gamma^{0}= ( \bar{{\cal Q}}_{a}^{-}, \bar{{\cal Q}}_{a}^{+})}}\\ \end{array}
\Vert \gamma\Vert^2 = a^2 + b^2 + c^2 + d^2 &= (a+d)^2 + (b-c)^2 - 2 &\geqslant (a+d)^2 - 2 &\geqslant q^{4}/4 - 2 &\geqslant q^4/8.
\left. (-1)^{i}\,\,{^{\mathbf{x}_i}\Delta_{k_1}^{i}}\alpha(n;k_1,2d,3d,\ldots,n d) \right|_{k_1=l}= \left.{^{\mathbf{y}_i}\delta_{k_n}^{i}}\alpha(n;d,2d,\ldots,(n-1) d,k_n) \right|_{k_n=r}.
\tilde{\theta}_{T}=\frac{\int_{0}^{T}f(X_{t})(\mathrm{d}X_{t}-\mathrm{d}L_{t}+\mathrm{d}R_{t})}{\int_{0}^{T}f^{2}(X_{t})\mathrm{d}t}=\theta_{0}+\frac{\sigma\int_{0}^{T}f(X_{t})\mathrm{d}W_{t}}{\int_{0}^{T}f^{2}(X_{t})\mathrm{d}t}.
\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) .
\ddot{\varphi}_{c}+ 3 \frac{\dot{R}}{R}\dot{\varphi}_{c}+ ( m^{2}+ 3 \lambda \langle \hat{\phi}_{f}^{2}\rangle ) \varphi_{c}+ \lambda ( \varphi_{c}^{*}\varphi_{c}) \varphi_{c}= 0 .
\lim\limits_{t\rightarrow\infty}\left\{\sup\limits_{|x|>(2\sqrt{ds}+\epsilon)t}v(x,t)\right\}=0, \epsilon>0, begin{align*}0.2cm] \liminf\limits_{t\rightarrow\infty}\left\{\inf\limits_{|x|<(2\sqrt{ds}-\epsilon)t}v(x,t)\right\}>\mu, \epsilon\in(0,2\sqrt{ds}),
\gamma_k(s) := \log \sum_A \exp\left(s \phi_k(a)\right) \tilde{P}_k(a) \gamma(s) := \sum_{k=1}^K \frac{n_k}{n}\gamma_k(s) Q_k(a) := \exp\left(s \phi_k(a) - \gamma_k(s)\right) \tilde{P}_k(a)
\dot x = n(x)-f(x(t),y(t),v(t))v(t), \dot y = \int^{\infty}_{0}f_{1}(\tau)f(x(t-\tau),y(t-\tau),v(t-\tau))v(t-\tau)e^{-\alpha_1 \tau}d\tau -a\varphi_{1}(y(t))-pw(y(t),z(t)), \dot v = k\int^{\infty}_{0}f_{2}(\tau)e^{-\alpha_{2}\tau}\varphi_{1}(y(t-\tau))d \tau-uv(t), \dot z = c\int^{\infty}_{0}f_{3}(\tau)w(y(t-\tau),z(t-\tau))d \tau -b\varphi_{2}(z(t)).
\Delta f=\frac{1}{1-|\bar{\nabla}f|^2}(g^{kl}\bar{\nabla}_k\bar{\nabla}_l f)=\frac{1}{1-|\bar{\nabla}f|^2}\left(\bar{\Delta}f+\frac{1}{{1-|\bar{\nabla}f|^2}}(\bar{g}^{kp}\bar{g}^{lq}f_p f_q)(\bar{\nabla}_k\bar{\nabla}_l f)\right),
( \psi^{1}, \psi^{2}, \psi^{3}, \psi^{4}) \mapsto ( \psi^{3}, \psi^{1}, \psi^{2}, \psi^{4}) , \quad ( j^{1}, j^{2}, j^{3}, j^{4}) \mapsto ( j^{3}, j^{1}, j^{2}, j^{4}) .
\frac{1}{4!}(\int_{D})^{4}\langle F_{c}^{2}F_{t}^{2}\rangle =\frac{1}{4!}r^{4}(-e^{4})\langle n^{2}\rangle (\int_{D})^{4}\;e^{2}[\delta^{2}(x-y)+5\;{\rm permut.}]=-\frac{e^{6}}{4}r^{4}\langle n^{2}\rangle\Delta^{3}
g^{\prime \prime}g-\frac{\left( f^{\prime}\right)^{2}}{ff^{\prime \prime}}\left( g^{\prime}\right)^{2}+m_{0}\frac{g}{f}+m_{0}\frac{g^{\prime \prime}}{f^{\prime \prime}}=\frac{n_{0}}{ff^{\prime \prime}}.
\sum\limits_{\substack{r \bmod{q}(r,q)=1}}\left|\sum\limits_{\substack{n\in \mathbb{Z}[i] \mathcal{N}(n)\le N}}a_n \cdot e\left(\mbox{\rm Tr}\left(\frac{nr}{2q}\right)\right)\right|^2 \ll (\mathcal{N}(q)+N)Z,
\mathcal{A}(t)(v_{n,\varepsilon}^p) = pv_{n,\varepsilon}^{p-1}\mathcal{A}(t)v_{n,\varepsilon}+p(p-1)v_{n,\varepsilon}^{p-2}\langle Q(t)\nabla_x v_{n,\varepsilon}, \nabla_x v_{n,\varepsilon}\rangle ,
\|\varrho\|_{L^\infty(0,T; L^\gamma(\Omega))}+\|\varrho|\mathbf{u}|^2\|_{L^\infty(0,T;L^1(\Omega))}\leq c(E, \varepsilon), \|\mathbf{u}\|_{L^2(0,T; W^{1,2}_0(\Omega))}\leq c(E, \varepsilon).
K_{+}( \omega ) = \sqrt{- i \frac{\Delta ( 1 - 2 \Delta ) \omega}{2 \pi}}e^{- i \omega \Delta \ln ( - i \omega ) + i \mu \omega}\frac{\Gamma ( - i \omega \Delta ) \Gamma ( - i \omega ( 1 - 2 \Delta ) )}{\Gamma ( - i \omega )}
{\bf E}\bigl[\beta_{2,2}(a, b)^q\bigr] = \frac{\Gamma_2(q+b_0\,|\,a)}{\Gamma_2(b_0\,|\,a)}\frac{\Gamma_2(b_0+b_1\,|\,a)}{\Gamma_2(q+b_0+b_1\,|\,a)}\frac{\Gamma_2(b_0+b_2\,|\,a)}{\Gamma_2(q+b_0+b_2\,|\,a)}\times \times \frac{\Gamma_2(q+b_0+b_1+b_2\,|\,a)}{\Gamma_2(b_0+b_1+b_2\,|\,a)}.
\varphi \big( g_2 g^{j}_1 \big) = \varphi (g_2) \varphi \big( g^{j}_1 \big) \mbox{and}\varphi \big( g_3 g^{j}_1 \big) = \varphi \big( g^{j}_1 \big) \varphi (g_3) \,.
E=-a_{1}a_{2}\int\frac{d^{3}k}{(2\pi)^{3}}\frac{\vec{v}_{1}\cdot[1-\hat{k}\hat{k}]\cdot\vec{v}_{2}}{\vec{k}^{2}+m^{2}}exp(i\vec{k}\cdot(x_{1}-x_{2}))
\begin{array}{c}{{\{p_{i}( x ) , p_{j}( y ) \}= - \delta_{i j}\delta^{\prime}( x - y ) \ 1 \leq i , j \leq l}}\\{{\{p_{i}( x ) , p_{j}( y ) \}= \delta_{i j}\delta^{\prime}( x - y ) \l + 1 \leq i , j \leq k + 1 + 2 l}}\\{{\{p_{i}( x ) , p_{j}( y ) \}= 0 \ 1 \leq i \leq l , \l + 1 \leq j \leq k + 1 + 2 l}}\\ \end{array}
\begin{cases}-d_1\Delta w=wq(\rho w)\left[f(\rho w)-v\right], x\in \Omega, -d_2\Delta v=v\left[h(v) +q(\rho w)w\right], x\in\Omega, \partial_\nu u=\partial_\nu v=0, x\in \partial \Omega. \end{cases}
\chi_{1}^{m}\to g^{m \bar{n}}\frac{\partial}{\partial ( d \bar{z}^{\bar{n}})}, \quad \chi_{2}^{m}\to d z^{m}, \quad \bar{\chi}^{\bar{m}, 1}\to d \bar{z}^{\bar{n}}, \quad \bar{\chi}^{\bar{m}, 2}\to g^{n \bar{m}}\frac{\partial}{\partial ( d z^{n})}
\mathbf{1}_N=\begin{pmatrix}[F_{N-1}(\phi_N,\ldots,\omega_{N-2})]^{-1}0 0 1\end{pmatrix}\begin{pmatrix}e^{-i\omega_{N-1}}&\ddots & e^{-i\omega_{N-1}}& e^{i(N-1)\omega_{N-1}}\end{pmatrix}\begin{pmatrix}X &u^{(N-1)}_{NN}\end{pmatrix}.
\pi_{i}^{( 3 ) a}= - \frac{1}{3 m^{2}}\eta^{1 b}G_{1 i}^{( 2 ) b a}+ \frac{g}{6 m^{2}}\bar{\eta}^{b c}( D_{j}\pi^{j})^{b}G_{1 i}^{( 2 ) c a}- \frac{1}{3}\eta^{2 b}G_{2 i}^{( 2 ) b a},
\begin{array}{cc}\hat{M}_{0}(z)Z_{0}=Z_{0}H(z)~~~~ ~~~~H(z)=\left(\begin{array}{cc}\tilde{\hat{\omega}}(z)-\frac{1}{2}\hat{\lambda}(z) 2\hat{\rho}^{j}(z) 0 \hat{T}_{i}^{~j}(z)\end{array}\right)\end{array}
\begin{array}{l l l}{{}}&{{\}}&{{\left( \sigma_{i}, \sum_{j = 1}^{\infty}\sigma_{j}\, S a_{i n}\left( \overline{{{\sigma}}}_{j}\right) S^{- 1}+ \overline{{{\sigma}}}_{j}\, S a_{i n}^{\dagger}\left( \sigma_{j}\right) S^{- 1}\right)_{K . G .}}}\\{{}}&{{=}}&{{S a_{i n}\left( \overline{{{\sigma}}}_{i}\right) S^{- 1}\; .}}\\ \end{array}
G_{1 2 2}^{\mu}( p , q , - p - q ) = ( - i g ) ( i g )^{2}\int \frac{d^{4}s}{( 2 \pi )^{4}}\, [ i D_{1 2}( s ) ] \gamma_{\alpha}[ i S_{2 2}( s + q ) ] \gamma^{\mu}[ i S_{2 1}( s - p ) ] \gamma^{\alpha}\, .
K(w,\bar w) = \sum_{\ell=1}^{n}d_\ell K_\ell(w,\bar w) = \sum_{\ell=1}^{n}d_\ell \ln \Delta_\ell(w,\bar w) = \ln \left[\prod_{\ell=1}^{n}(\Delta_\ell(w,\bar w))^{d_\ell}\right] .
d\sqrt{n}n^{1/4}2^{\sum_{l=1}^dJ_{n,l}\left[\frac{1}{2}-\frac{\alpha^{*}}{2d}\right]}\epsilon_n^{1/2}\lesssim\underbrace{(n/\log{n})^{-\frac{(2\alpha^{*}-3d)(\alpha^{*}+d)}{8d(2\alpha^{*}+d)}}(\log{n})^{-1/2}}_{\tau_n}\sqrt{n\log{n}}.
\limsup_{k \to \infty}\int_{\mathbb{R}^{N}}\bigg[|u_k|^{\overline{p}}+|v_k|^{\overline{p}}+ \int_{\mathbb{R}^{N}}\frac{|u_k(x)-u_k(y)|^{p(x,y)}}{|x-y|^{N+sp(x,y)}}\,dy +\int_{\mathbb{R}^{N}}\frac{|v_k(x)-v_k(y)|^{p(x,y)}}{|x-y|^{N+sp(x,y)}}\,dy \bigg]\,dx= \mu (\mathbb{R}^{N}) + \mu_{\infty}.
\Gamma [ \bar{\Phi}, \Phi ] = \int d^{8}z{\cal L}_{e f f}( \Phi , D_{A}\Phi , D_{A}D_{B}\Phi ; \bar{\Phi}, D_{A}\bar{\Phi}, D_{A}D_{B}\bar{\Phi}) + ( \int d^{6}z{\cal L}_{e f f}^{( c )}( \Phi ) + h . c . ) + \ldots
n\sum_{j=1}^{\left\lfloor\frac{n}{k_n}\right\rfloor}\mathbb{P}((x,\omega_1,\dots,\omega_j): x\in U_n, f_{\underline{\omega}}^j(x)\in U_n)\leq n\left\lfloor\frac{n}{k_n}\right\rfloor\mu_\epsilon(U_n)^2\leq \frac{(n\mu_\epsilon(U_n))^2}{k_n}.
\frac{\|dx_n\|}{\|dx\|}=\frac{\|dx_{n}\|_p}{\|dx\|_p}\cdot \frac{\cos\varphi}{\cos\varphi_{n}}\cdot\sqrt{\frac{1+(d\varphi_{n}/dr_{n})^2}{1+(d\varphi/dr)^2}}&\leq C\cos\varphi_1\cdot n^{-\frac{\beta}{2\beta-1}}N^{\frac{\beta(\beta-2)}{(2\beta-1)(\beta-1)}}\cdot \frac{\cos\varphi}{\cos\varphi_{n}}\cdot\sqrt{\frac{1+(d\varphi_{n}/dr_{n})^2}{1+(d\varphi/dr)^2}},
x_0^T Q x_0 \int_0^t \left\|u(s)\right\|_2^2 \exp\left(\int_s^t \left\|u(w)\right\|_2^2dw\right) ds= x_0^T Q x_0 \left[-\exp\left(\int_s^t \left\|u(w)\right\|_2^2dw\right)\right]_{s=0}^t &=x_0^T Q x_0\left(\exp\left(\int_0^t \left\|u(s)\right\|_2^2ds\right)-1\right).
{F^{ij}}{v_{ij}}- v_t ={F^{ij}}{u_{ij\xi \xi}}-{u_{t\xi \xi}}- \left({F^{ij}}v{'_{ij}}-{v{'_t}}\right) +2{u_k}\left({F^{ij}}{u_{ijk}}-{{u_{tk}}}\right) + 2{F^{ij}}{u_{ik}}{u_{jk}}+{K_1}\sum\limits_{i = 1}^n{{F^{ii}}},
\mathfrak{D}=\Bigg\lfloor m\,\frac{p^{\gamma}\nu_1-(p+1)\nu_1-1}{p^{\gamma}\nu_1}\Bigg\rfloor=\Bigg\lfloor m\Bigg(1-\frac{1}{p^{\gamma-1}}-\frac{1}{p^{\gamma}}-\frac{1}{p^{\gamma}\nu_1}\Bigg)\Bigg\rfloor
\begin{pmatrix}\overline c_{1,1}.. .. \overline c_{1,T}.. .. .. .. \cdot \cdot \cdot \cdot \overline c_{T-1,1}.. \overline c_{T-1,T-1}\end{pmatrix}\begin{pmatrix}a_{1,T}a_{2,T}\cdot a_{T-1,T}\end{pmatrix}=-a_{T,T}\begin{pmatrix}a_{1,T}a_{2,T}\cdot a_{T-1,T}\end{pmatrix}
a ( r , r^{\prime}) = - \left( \frac{z + m}{2 i k}\right) \left[ \Theta ( r - r^{\prime}) e^{i k ( r - r^{\prime})}+ \Theta ( r^{\prime}- r ) e^{- i k ( r - r^{\prime})}+ \alpha \, e^{i k ( r + r^{\prime})}\right]