harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathcal L(\b x,\frac{\partial}{\partial \b x})=\begin{pmatrix}L_{11}(\b x,\frac{\partial}{\partial\b x}) \ldots L_{1m}(\b x,\frac{\partial}{\partial\b x}) \vdots \vdots \vdots L_{M1}(\b x,\frac{\partial}{\partial\b x}) \ldots L_{Mm}(\b x,\frac{\partial}{\partial\b x})\end{pmatrix}.
P^{\lambda, -2\lambda-2a+1}_a(x-\sqrt{-1}y, x+\sqrt{-1}y) &=(\sqrt{-1})^{a-1}U_a(\lambda)\bigg(xC^{\lambda+\frac{1}{2}}_{a-1}(-x^2, y) +\sqrt{-1}\left( A_a(\lambda) y C^{\lambda+\frac{1}{2}}_{a-1}(-x^2, y)+ B_a(\lambda) (x^2 + y^2) C^{\lambda+\frac{3}{2}}_{a-2}(-x^2, y)\right) \bigg).
\beta_{e}^{( n )}\, = \, \int_{0}^{n T}\; \langle \nu_{e}( t ) | \; U^{- 1}( t ) \, i \partial_{t}\, \Big ( U ( t ) \; | \nu_{e}( t ) \rangle \Big ) \, d t \, = \, \int_{0}^{n T}\langle \tilde{\nu_{e}}( t ) | \; i \partial_{t}| \tilde{\nu_{e}}( t ) \rangle \, d t \, = \, 2 \pi \, n \, \sin^{2}\theta \, ,
P(V_n=0) \sum_{\substack{A\subset \{1,2,\cdots , n\}|A| = k-1}}\prod_{i\in A}\frac{b(i;n)}{1-b(i;n)}\frac{\beta_n -\sum_{i \in A}\frac{b(i;n)}{1-b(i;n)}}{k}& \leq P(V_n=0) \> \frac{\beta_n}{k}\sum_{\substack{A\subset \{1,2,\cdots , n\}|A| = k-1}}\prod_{i\in A}\frac{b(i;n)}{1-b(i;n)}.
{\phi(x)^{-1}+ \phi(y)^{-1}\over|x-y|}=\sum\limits_{k=1}^K \kappa_k\frac{|x-R_k|+|y-R_k|}{|x-y|}g_k(x)g_k(y)\ge\sum\limits_{k=1}^K \kappa_k g_k(x)g_k(y).
\left(\int_{B_{r_0/2}}g^{-\frac{1}{p-1}}\right)^{-(p-1)}=\left[\int_{B_{r_0/2}}\exp\left(\frac{(\alpha_+^\ast-2\varepsilon)(\alpha_+^\ast-\varepsilon)}{\varepsilon}|z_2|\right)\right]^{-(p-1)}\geq \left[\int_{B_{r_0/2}}\exp\left(\frac{2\pi}{\|s_-\|_{L^1(B_{r_0})}}|z_2|\right)\right]^{-(p-1)}\geq (2\pi r_0^2)^{-(p-1)}.
z^{-1}_{\beta}\partial^{(\alpha)}_{x}(z_\beta) =(x-\alpha)^{\beta+1}\binom{\alpha+\beta}{\alpha}(-1)^{\alpha}(x-\alpha)^{-\beta-1-\alpha}=(-1)^{\alpha}\binom{\alpha+\beta}{\alpha}(x-a)^{-\alpha}.
\tilde{A}(\varepsilon) \tilde{P}+ \tilde{P}\tilde{A}(\varepsilon)^\top + \tilde{B}\tilde{B}^\top + \tilde{H}(\tilde{P}\otimes \tilde{P}) \tilde{H}^\top + \displaystyle \sum_{j=1}^{n_{\rm in}}\tilde{N}_j \tilde{P}\tilde{N}_j^\top = 0 .
\mathbb{E}\left(e^{\gamma p\Gamma(\tau_{U,x_S})}\right)=\bigg(\frac{R(x_S,D)}{R(x_S,U)}\bigg)^{\frac{\gamma^2p^2}{2}}\le (2|D|)^{\frac{\gamma^2p^2}{2}}2^{l \frac{\gamma^2p^2}{2}},
\frac{1}{I_{q'^{2}}+R\left(W'\right)}\otimes W'-\left(\overline{\frac{1}{I_{q'^{2}}+R\left(W'\right)}\otimes W'}\right)^{t}=2\sqrt{-1}\left(V\otimes Z'\right)\overline{\left(V\otimes Z'\right)}^{t},
f_{1}= 1 +{\frac{r_{1}^{2}}{r^{2}}}, ~ f_{5}= 1 +{\frac{r_{5}^{2}}{r^{2}}}, ~ f_{K}= 1 +{\frac{r_{K}^{2}}{r^{2}}}, ~ d = 1 -{\frac{r_{0}^{2}}{r^{2}}},
Tr^2(\mathcal{M}\cdot{\mathcal{F}}_A)=\,4\,Tr(\mathcal{M}\,\cdot \mathcal{M}\cdot{\mathcal{F}}_A\cdot{\mathcal{F}}_A), & Tr^2(\mathcal{M}\cdot \tilde{\mathcal{F}}_A)=\,4\,Tr(\mathcal{M}\cdot \mathcal{M}\cdot\tilde{\mathcal{F}}_A\cdot \tilde{\mathcal{F}}_A).
T^{\alpha\beta\{ab\}}=\frac{1}{2}(-1)^{\varepsilon_\beta}C^{\alpha\{a}C^{\beta b\}},\;\;\;\;\hat{T}^{\alpha\beta}\varepsilon^{ab}=-\frac{1}{2}(-1)^{\varepsilon_\beta}\varepsilon_{cd}C^{\alpha c}C^{\beta d}\varepsilon^{ab}
\Gamma ={\frac{2 G_{F}^{2}\alpha E_{1}^{5}}{1 3 5 ( 2 \pi )^{4}}}\sin^{6}\theta \left[{\frac{1}{5 0}}({\frac{B}{B_{c}}})^{6}- \left({\frac{8}{1 0 5}}-{\frac{1}{4 9}}{\frac{E_{1}^{2}\sin^{2}\theta}{m^{2}}}\right) ({\frac{B}{B_{c}}})^{8}+ \cdots \right] .
p_{-3}(a)p_{-3}(1)-p_{-3}(a+1) =3p_{-3}(a)-p_{-3}(a+1) &=3\sum_{k=0}^a p_{-2}(k)p(a-k)-\sum_{k=0}^{a+1}p_{-2}(k)p(a+1-k) &=\left(\sum_{k=0}^a p_{-2}(k)(3p(a-k)-p(a+1-k))\right)-p_{-2}(a+1) &> 2p_{-2}(a)-p_{-2}(a+1) &=p_{-2}(1)p_{-2}(a)-p_{-2}(a+1)\ge 0.
{H}^{\prime}(t) \geq\frac{H^{2}(t)}{2B_{1}}-B_{2}& =\frac{1}{2B_{1}}\left( 1-\frac{2B_{1}B_{2}}{H^{2}(0)}\right)H^{2}(t)+B_{2}\left( \frac{H^{2}(t)}{H^{2}(0)}-1\right) & \geq\frac{1}{2B_{1}}\left( 1-\frac{2B_{1}B_{2}}{H^{2}(0)}\right)H^{2}(t) & =\frac{H^{2}(0)-2B_{1}B_{2}}{2B_{1}H^{2}(0)}H^{2}(t)
\rho_{\rm min}=\vert \beta \vert \biggl(\frac{\beta}{1+\beta}\biggr)^{1/4}\sqrt{\biggl(1+\frac{K_+}{4k_{\rm c}}\biggr)^2+\frac{1+\beta}{\beta}}\, , \quad \cos\Phi_{\rm min}\,=\,\frac{1+K_+/(4k_{\rm c})}{\sqrt{\left[1+K_+/(4k_{\rm c})\right]^2 +(1+\beta )/\beta}}.
\pi^{-s}\Gamma(s)\zeta(2s) E^P_{2s/r,\varphi}(g)=\pi^{-(\frac{r}{2}-s)}\Gamma\left(\frac{r}{2}-s\right)\zeta(r-2s){E^P_{1-2s/r,\varphi}\left((g^\top)^{-1}\right)}.
\Gamma(k) =\begin{bmatrix}\Gamma(k)_{1,1}\Gamma(k)_{2,1}\vdots \Gamma(k)_{n,1}\end{bmatrix}\otimes\begin{bmatrix}\Gamma(k)_{1,1}\Gamma(k)_{1,2}\ldots \Gamma(k)_{1,n}\end{bmatrix}.
\sum\limits_{i,j=1}^{n+1}[d_G(y,y_0)^{-1-2\alpha+\epsilon}Y_i Y_j \tilde{v}]_{C^{0,\epsilon}_\ast(\mathcal{N}_G(y_0))}&+ \sum\limits_{i,j=1}^{n+1}\|d_G(y,y_0)^{-1-2\alpha}Y_i Y_j \tilde{v}\|_{L^{\infty}(\mathcal{N}_G(y_0))}\leq C\|f\|_{Y_{\alpha,\epsilon}}.
\| \nu \varphi_{\varepsilon}(s) - \pi(\varphi_{\varepsilon}(s)) \|^2_V = \int_{\Omega}| \nu \varphi_{\varepsilon}(s) - \pi(\varphi_{\varepsilon}(s)) |^2 + \int_{\Omega}| \nu \nabla \varphi_{\varepsilon}(s) - \pi'(\varphi_{\varepsilon}(s)) \nabla \varphi_{\varepsilon}(s) |^2
\Gamma_{W Z}[ \bar{g}; \sigma ] = b \lim_{d \rightarrow 4}{\frac{\int \, d^{d}\, \, x \, \sqrt{- g}\, F_{d}- \int \, d^{d}\, \, x \, \sqrt{- \bar{g}}\, \overline{{F}}_{d}}{d - 4}}+ b^{\prime}\lim_{d \rightarrow 4}{\frac{\int \, d^{d}\, \, x \, \sqrt{- g}E_{d}- \int \, d^{d}\, \, x \, \sqrt{- \bar{g}}\, \overline{{E}}_{d}}{d - 4}}\, ,
\begin{array}{c}{\left( \Phi , \Psi ( n_{1}, n_{2}. . . . n_{N}) \right) = 0 , \quad}\\{\Psi ( n_{1}, n_{2}. . . . n_{N}) = a^{* n_{1}}a^{* n_{2}}. . . . a^{* n_{N}}\Omega , \quad n_{i}= 0 \o r \ 1}\\ \end{array}
X_m|_{C_k}=B|_{C_k}\lambda_k|_{C_k}\left(\Lambda+f_k^{m_k-1}|_{C_k}\right)\left(-\frac{\partial{f_k}}{\partial y}\Big|_{C_k}\partial_x+\frac{\partial f_k}{\partial x}\Big|_{C_k}\partial_y\right)\,.
\frac{\tilde{\cal H}_{k+1}^{(s,t)}(\kappa^{(t)})}{\tilde{\cal H}_{k-1}^{(s,t)}(\kappa^{(t)})}=\frac{\tilde{\cal H}_{k+1}^{(s,t)}(\kappa^{(t)})}{\tilde{\cal H}_{k}^{(s,t)}(\kappa^{(t)})}\frac{\tilde{\cal H}_{k}^{(s,t)}(\kappa^{(t)})}{\tilde{\cal H}_{k-1}^{(s,t)}(\kappa^{(t)})},
\begin{cases}x_i^{k+1}=\mathrm{prox}_{\frac{\alpha_i}{\beta}f_i}(x_i^k-\frac{\alpha_i}{\beta}A_i^\top y^k),~~i\in\mathbb{N}_s, y^{k+1}=y^k+\beta(\sum_{i=1}^s A_i(2x_i^{k+1}-x_i^k)-b), \end{cases}
V^2(r) \equiv{\Delta (r) \over R^2 (r)}\left\{\frac{\lambda}{R^2(r)}+ \frac{1}{R(r)}\frac{\mbox{d}}{\mbox{d}r}\left[ \frac{\Delta(r)}{R^2(r)}\frac{\mbox{d}R(r)}{\mbox{d}r}\right]- \frac{m^2 a^2}{R^6(r)}\left( r^2 + a^2 + 2 \mu r \cosh 2\beta \right)\right\}.
\begin{array}{c}C^{-1}\left( \begin{array}{c}\gamma_4 \gamma_5 \end{array}\right) C =R_C \left( \begin{array}{c}\gamma_4^T \gamma_5^T \end{array}\right), \Pi^{-1}\left( \begin{array}{c}\gamma_4 \gamma_5 \end{array}\right) \Pi= R_P \left( \begin{array}{c}\gamma_4 \gamma_5 \end{array}\right),\end{array}
N_3 =-\frac{1}{\sqrt{n}}\sum_{j=1}^m\sum_{k=1}^m\sum_{l=1}^{m}\frac{1}{\sqrt{p_jp_kp_l}}\mathbb{E}\tilde{I}_j(1)I_k(1)\mathbb{E}I_l(1)\mathbb{E}\frac{\partial^3g}{\partial s_j\partial s_k\partial s_l}(\mathbf{S}^{(1)})
H_k^{(s,t)}(z)=\left|\begin{array}{ccccc}f_s^{(t)}f_{s+1}^{(t)}\dots f_{s+k-1}^{(t)}1 f_{s+1}^{(t)}f_{s+2}^{(t)}\dots f_{s+k}^{(t)}z \vdots \vdots \ddots \vdots \vdots f_{s+k-1}^{(t)}f_{s+k}^{(t)}\dots f_{s+2k-2}^{(t)}z^{k-1}f_{s+k-1}^{(t+1)}f_{s+k}^{(t+1)}\dots f_{s+2k-2}^{(t+1)}(z-\mu^{(t)})z^{k-1}\end{array}\right|.
\int_M d \phi\wedge d^c_B \phi\wedge \wedge \omega_\phi^{n-1}\wedge \eta=\int_M \phi (-dd^c_B\phi)\wedge \omega_\phi^{n-1}\wedge \eta\leq \int_M \phi \omega^T\wedge \omega_\phi^{n-1}\wedge \eta\leq 1
{\frac{1}{a + b z}}\left( \begin{array}{l}{1}\\{z^{\prime}}\\{{z^{\prime}}^{2}}\\{\vdots}\\ \end{array}\right) ={\frac{1}{a + b z}}\left( \begin{array}{l}{1}\\{\left({\frac{c + d z}{a + b z}}\right)}\\{\left({\frac{c + d z}{a + b z}}\right)^{2}}\\{\vdots}\\ \end{array}\right) .
D_s X(t)(\omega) = \sigma(s,\omega, X(s)(\omega)) + \int_s^t U(s, r, \omega) dr + \int_s^t V(s, r, \omega) dW(r) &+\int_s^t \nabla_x b(r,\omega, X(r)(\omega)) D_s X(r)(\omega) dr + \int_s^t \nabla_x \sigma(r,\omega, X(r)(\omega)) D_s X(r)(\omega) dW(r),
{}^{M\!L\!}\textrm{F}_{p}(a,b;c;z) =\frac{(1-z)^{-a}}{B(b,c-b)}\int_{0}^{1}(1-t)^{b-1}t^{c-b-1}\left(1-\frac{z}{z-1}t\right)^{-a}E_{\alpha, \beta}^{\gamma}\left(-\frac{p}{t(1-t)}\right)dt.
\Phi_\epsilon(r) = \sum^2_{j=0}\alpha_j \left( \int_{\partial^*\!E_j\cap B_{r(1-\epsilon)}}\, d\mathcal{H}^{n-1}(x) + \int_{\partial^*\!E_j\cap(B_r \setminus B_{r(1-\epsilon)})}\left(\frac{1}{\epsilon}- \frac{|x|}{\epsilon r}\right)\, d\mathcal{H}^{n-1}(x) \right).
\rho_\lambda(\phi,\theta ,\psi)=\frac{2}{\pi[\mathbf K(\sqrt{\lambda})]^{2}}\frac{1}{1-\lambda\sin^2\phi\sin^2\theta-\lambda\cos^2\phi\sin^2\psi},\lambda\in(0,1);(\phi,\theta,\psi)\in[0,\pi/2]^3,
\frac{1}{\nu(\xi_{\Delta}\eta_{\Lambda_n \setminus \Delta})}\int \mathbf{1}_{\eta_{\Lambda_n}}(\omega) \frac{\gamma_{\Delta}(\xi_{\Delta}|\omega_{\Delta^c})}{\gamma_{\Delta}(\eta_{\Delta}|\omega_{\Delta^c})}\nu(d\omega) = 1,
\frac{\delta_0}{\nu^{2^j}}= q^{\log_{q}\delta_0 - \lceil \log_{q}\delta_{0}^{1/2^N}\rceil 2^{j}}\geq \delta_{0}^{1 - \frac{1}{2^{N - j}}}q^{-2^j}\geq \delta^{1 - \frac{1}{2^{N - j}}}q^{-2^j}
T_0 = 1; T_1(-;p) = p; T_2(a_2;p) = p^2+a_2; T_3(a_2, a_3;p) = p^3+p(a_2+a_3); T_4(a_2, a_3, a_4;p) = p^4+(a_2+a_3+a_4)p^2+a_2a_4; T_5(a_2, a_3, a_4, a_4;p) = p^5+(a_2+a_3+a_4+a_5)p^3+(a_2a_4+a_2a_3+a_3a_5)p.
(|D_v^2 u_{}|^2)_{Q_{2\nu r, (2\nu r) R}(z_0)}^{1/2}\le N \delta^{-\theta}\sum_{k=0}^\infty 2^{- k^2/8}F_k (R), (|(-\Delta_x)^{1/3}u_{}|^2)^{1/2}_{Q_{2\nu r, (2\nu r) R}(z_0)}\le N \delta^{-\theta}\sum_{k=0}^\infty 2^{-2k}F_k (R),
\tilde{f}_i^a(\pi_1 \otimes \pi_2) = \tilde{f}_i^M(\pi_1) \otimes \tilde{f}_i^{a-M}(\pi_2) \not\in \mathcal{B}_{s_i v}(\lambda) \otimes \mathcal{B}_w(\mu) \sqcup \{0 \}.
\begin{aligned}f_{m,p}(x) =xf_{m-1,p-1}(x)+x\sum_{i=1}^{m-p}f_{m,p+i-1}+\frac{x^{p+1}}{(1-x)^2}\sum_{i=1}^{m-p}\sum_{j=-1}^{i-2}x^{i-j}f_{m-i+j+1-p;j}(x) &+\frac{x^{3+p}}{(1-x)^3}\sum_{i=1}^{m-p}\sum_{k=0}^{i-1}\sum_{j=k}^{m-i-1+k-p}\frac{x^{i-k}}{(1-x)^{j-k}}f_{m-i+k-p,j}(x).\end{aligned}
K_{kl}(t,x)= \delta_{kl}\int\frac{d\omega d^2p}{(2 \pi)^3}\frac{e^{-i\omega t + i{\vec p}\cdot{\vec x}}}{(\frac{\hbar^2 p^2}{4m \rho_0}+ \lambda) p^2 - \frac{m}{\rho_0}\omega^2}
\mathbb{P}\left(d_\mathrm{N}>\psi(x)\right) =\mathbb{P}\left(\Phi_\mathrm{N}\cap\mathcal{B}(0,\psi_\mathrm{L}(x))=\emptyset\right) &=\mathrm{e}^{-2\pi\lambda\int_{0}^{\psi_\mathrm{L}(x)}(1-p(t))t\mathrm{d}t},
\Delta^a = \int d^2 \theta \frac{\partial^2 \delta_L}{\partial \theta^2 \delta \Phi^A(\theta)}\,\theta^a \frac{\delta}{\delta \bar{\Phi}_A(\theta)}= (-1)^{\epsilon_A}\frac{\delta_L}{\delta \phi^A}\, \frac{\delta}{\delta \phi_{A a}^*}.
P(l_{1},l_{2},l_{3}) & =\frac{(-1)^{(l_{1}+l_{2}-l_{3}+1)/2}\sqrt{2l_{3}+1}\Delta(l_{1},l_{2},l_{3})(L+1)((L-1)/2)!}{((-l_{1}+l_{2}+l_{3}-1)/2)!((l_{1}-l_{2}+l_{3}-1)/2)!((l_{1}+l_{2}-l_{3}-1)/2)!}\ ,
\beta_{2}^{\prime}(k)=(\begin{matrix}2k\\k\end{matrix})\sum_{j=0}^{k}(\begin{matrix}2j\\j\end{matrix})^{-1}(\begin{matrix}k\\j\end{matrix})(-1)^{k-j}\sum_{m=0}^{j}(\begin{matrix}j\\m\end{matrix})^{4}=1,0,12,24,564,2784,...
0= 2\langle \tau_{H}, \alpha_{2}\rangle \{(2m_{1}+m_{2}+2m_{3})(1-(\cot \langle 2\alpha_{2}, H\rangle )^{2}) & +n_{2}(1-(\tan \langle 2\alpha_{2}, H\rangle )^{2}) -4m_{3}\}(\alpha_{1}+ 2\alpha_{2}).
(-1)^{p(\eta,\ell_{m})-1}i_{\eta\cup \ell_{m}}= -(-1)^{p(\eta,\ell_{m})}(-1)^{|\pi(\eta\cup \ell_{m})|}{\prod_{r=1}^{d+1}U^{\eta\cup \ell_{m}}_{r}}&=-U{\prod_{r=m+1}^M U^{\eta\cup \ell_{m}}_{r}},
\Delta E_b=\alpha_*^2\left(O_0(\sqrt{M}\|p\|^{-3}+ \dfrac{\beta^{(4)}_b(\kappa,\alpha_*=0)}{\|p\|^{-4}}+O\left(M^{-1/2}\|p\|^{-5}\right)\right)+o(\alpha_*\sqrt{M}).
F_6(x,y) = 79202404017 x^5 - 159949407645 yx^4 + 109186695090 y^2x^3 - - 29296172862y^3x^2 + 2680421535 y^4x - 43464971 y^5 G_6(x,y) = 79202404017x^5 - 186350208984 yx^4 + 153702897192 y^2x^3 - 52808618772 y^3x^2 + 6928019514 y^4x - 229643804y^5
f_{22}^{>}(L,m,z, d)=-\frac{1}{4L^{d-2}}\sum_{k\geq \frac{d-3}{2}}^{\infty}C^{(1)}(d,k)(Lm)^{2k}\int_{Lm}^{\infty}du\, u^{d-3-2k}(\coth u -1)\cosh(\frac{2uz}{L}).
[\alpha_1] \mapsto [\alpha_1], [\alpha_2]\mapsto [\alpha_2], [\gamma_{-1}] \mapsto \; [\gamma_{-1}]-a_1[\alpha_1]-a_2[\alpha_2]-a_3[\alpha_3] =\; [\gamma_{-1}]+(a_3-a_1)[\alpha_1]+(a_3-a_2)[\alpha_2].
\frac{\partial}{\partial\overline{\tau}}\frac{v^2}{Q(\overline{\tau},1)}= \frac{iv^2Q_{\tau}}{Q(\overline{\tau},1)^2}, 2iv^2 \frac{\partial}{\partial\overline{\tau}}Q_{\tau}= Q(\tau,1), iv^2 \frac{\partial}{\partial\overline{\tau}}\frac{Q(\overline{\tau},1)}{v^2}= Q_{\tau}.
K_{\mu}^{i}= \frac 1 2 i \varepsilon^{i a b}\overline{{\psi}}^{a}\gamma_{\mu}\psi^{b}- i{\frac{\varphi^{a}\overline{{\psi}}^{a}\gamma_{\mu}\psi^{i}}{\sqrt{1 -{\varphi}^{2}}}}.
\hat{\mathbf{u}}_n=\underset{\mathbf{u}\in \mathrm{\Delta_M}}{\operatorname{argmin\,}}F_n(\mathbf{u}), F_n(\mathbf{u})=\sum\limits_{i=1}^M\sum\limits_{j=1}^M c_{ij}u_i u_j - 2 \sum\limits_{m=1}^M c_m u_m,
f^{\ast}(w^{-1}z) =f_{1}^{\ast}f_{2}^{\ast}(w_{1}^{-1}w_{2}^{-1}z_{2}z_{1}) &=f_{1}^{\ast}(w_{1}^{-1}z_{1}f_{2}^{\ast}(w_{2}^{-1}z_{2})) &=f_{1}^{\ast}(w_{1}^{-1}z_{1}w_{2}^{\ast}(z_{2}f_{2}^{-1})) &=w_{1}^{\ast}(z_{1}f_{1}^{-1}w_{2}^{\ast}(z_{2}f_{2}^{-1})) &=(w_{1}w_{2})^{\ast}(z_{1}z_{2}(f_{1}f_{2})^{-1}) &=w^{\ast}(zf^{-1})
\mathbb{P}(Y=y|X=x)=\frac{(\begin{matrix}3\\y\end{matrix})(\begin{matrix}7\\x-y\end{matrix})}{(\begin{matrix}10\\x\end{matrix})}=\frac{(\begin{matrix}x\\y\end{matrix})(\begin{matrix}10-x\\ 3-y\end{matrix})}{(\begin{matrix}10\\ 3\end{matrix})}
U^+(z) = U^+(y+x) \le U^-(y-x) &\le C(1+|y-x|^p) &\le C(1+2^{p-1}(|2y|^p+|y+x|^p)) & \le C(1+ 2^{p-1}|2y|^p+2^{p-1})(1+|y+x|^p) &= \tilde{C}(1+|z|^{p}),
g_\delta(b+c) vv^*g_\delta(b+c) =g_\delta(b)vv^*g_\delta(b)+g_\delta(b)vv^*g_\delta(c)+g_\delta(c)vv^*g_\delta(b)+g_\delta(c)vv^*g_\delta(c) &\leq 2(g_\delta(b)vv^*g_\delta(b)+g_\delta(c)vv^*g_\delta(c)).
\eta\left(\varphi_v\right)(\xi)=\frac{\left\langle \nu^H,\xi^H\right\rangle}{|\xi^H|^3}\left( -2k(\xi) |\xi^H|^2 -\left\langle \nu^H,\xi^H\right\rangle + 3k(\xi) |\xi^H|^2\right)=0.
\Omega_{2 g + k - l}={\frac{( - 1 )^{l + 1}}{2 g + 1 + k - l}}\sum_{j = 0}^{2 g + 1}( - 1 )^{j}( 2 g + 1 - j ) s_{j}{\frac{\partial}{\partial s_{l}}}\Omega_{2 g + k - j}.
(p,p') =(4,3)\ :G^{(2)}_{\mathrm{cl.}}(\zeta,Q)=\mathrm{const.}\times Q^2\left(T_4\left(\frac{Q}{\sqrt{2}}\right)-T_3(-\zeta)\right)\ , (p,p') =(5,2)\ :G^{(2)}_{\mathrm{cl.}}(\zeta,Q)=\mathrm{const.}\times \prod_{\pm}\left(T_5\left(\frac{2}{1\pm \sqrt{5}}Q\right)-T_2(\mp\zeta)\right)\ .
(E,|\cdot|^2) =-2\tilde{\mu}(Q_\mu,|\cdot|^2)+2\tilde{\mu}^2(\nabla\phi_{\partial_{\mu}Q_{\mu}}\cdot y,\partial_\mu Q_\mu) &=-2\mu(Q_\mu,|\cdot|^2)+2\tilde{\mu}^2(\nabla\phi_{\partial_{\mu}Q_{\mu}}\cdot y,\partial_\mu Q_\mu)+O(\mu) &=\frac{M\mu\log\mu}{\pi}+2\tilde{\mu}^2(\nabla\phi_{\partial_{\mu}Q_{\mu}}\cdot y,\partial_\mu Q_\mu)+O(\mu).
(H_1 \otimes F_1) \otimes w = ((1 \otimes F_1)(H_1 \otimes 1 + 1 \otimes \alpha(H_1))) \otimes w - (1 \otimes F_1\alpha(H_1)) \otimes w & \in (1 \otimes F_1) \otimes H_{1_{\Delta}}w - \frac{1}{2}(1 \otimes F_1) \otimes w + Z.
\kappa(\ell,h) \le 1+ \sum_{q=1}^{\ell-1}\frac{\ell}{\ell-q}\binom{\ell-1}{q}\frac{(e^{d+3}q^2 (h+1+q))^q}{d(1-\frac q\ell)}&\le \ell^2 \sum_{q=0}^{\ell-1}\binom{\ell-1}{q}(e^{d+3}(\ell-1)^2 (h+\ell))^q &\le \ell^2\big( e^{d+3}(\ell-1)^2(h+\ell)+1\big)^{\ell-1}
\|\nabla A(t)\|_2^2 = \langle - \Delta A(t), A(t) \rangle_{(H^1)',H^1}& \le \langle u(t) - \Delta A(t) , A(t) \rangle_{(H^1)',H^1}= \int_\Omega u^{in}A(t)\ \mathrm{d}x & \le \|u^{in}\|_\infty \|A(t)\|_1\,.
D_{a,b}= \tbinom p{a-b}\tbinom{p-(a-b)}{b}\tbinom{a-b}{2(a-b)-r-1}(-1)^{p - a}\sum_{z=0}^{a-b}(-1)^{z}\tbinom{a-b}{z}= \tbinom p{a-b}\tbinom{p-(a-b)}{b}\tbinom{a-b}{2(a-b)-r-1}(-1)^{p - a}(1-1)^{a-b}= 0.
C\Big[ \begin{array}{c}\alpha + \beta \cr \gamma \end{array}\Big] = (-1)^{\gamma(\psi^{\mu},\tilde{\psi}^{\mu})}\,\exp\Big\{{i\pi \over 2}\, n(\alpha \beta \gamma) \Big\}\, C\Big[ \begin{array}{c}\alpha \cr \gamma \end{array}\Big] C\Big[ \begin{array}{c}\beta \cr \gamma \end{array}\Big] .
h_\Pi(\theta(z),\theta(w)) =2\sinh^{-1}\frac{|\theta(z)-\theta(w)|}{2\sqrt{r_D(z)r_D(w)}}&\le2\sinh^{-1}\frac{c^2|z-w|}{2\sqrt{d_D(z)d_D(w)}}\le c^2s_D(z,w).
\left|B_{2R}\cap\left\{\left(w-2^{-k+1}t \right)_{-}\leq 2^{-k}t\right\}\right|=\left|B_{2R}\cap\left\{w\geq 2^{-k}t\right\}\right|\geq\left|B_{2R}\cap\{w\geq t\}\right| \geq \nu\left|B_{2R}\right|
\int\mathbf{1}_{\{k\geq0\}}e^{ikx}\partial_{x}\mathfrak{Q}(x,k)e^{ik^{2}t}h(k)\,dk = I_{1}(t,x)+I_{2}(t,x), & I_1(t,x) := \int\mathbf{1}_{\{0\leq k\leq \frac{1}{\sqrt{t}}\}}e^{ikx}\partial_{x}\mathfrak{Q}(x,k)e^{ik^{2}t}h(k)\,dk, & I_2(t,x) := \int\mathbf{1}_{\{k\geq \frac{1}{\sqrt{t}}\}}e^{ikx}\partial_{x}\mathfrak{Q}(x,k)e^{ik^{2}t}h(k)\,dk.
\begin{aligned}\theta_{00,00}^2(0,M_1\tau) = -\tau_{1,1}\theta_{01,00}^2(0,\tau), \theta_{00,00}^2(0,M_2\tau) = -\tau_{2,2}\theta_{10,00}^2(0,\tau), \quad \theta_{00,00}^2(0,N_{1,2}\tau) = (\tau_{1,2}^2-\tau_{1,1}\tau_{2,2}) \theta_{00,00}^2(0,\tau). \end{aligned}
I_{i n t}= \int d^{n}x \{\sum_{a}\big ({\frac{- 1}{2 ( p_{a}+ 1 ) !}}H_{\mu_{1}\ldots \mu_{p_{a}+ 1}}^{a}H^{a \mu_{1}\ldots \mu_{p_{a}+ 1}}+ g_{a}S^{a \mu_{1}\ldots \mu_{p_{a}}}B_{\mu_{1}\ldots \mu_{p_{a}}}^{a}\big ) \},
\|Tf_{r_n}\|_{\mathcal{Y}}=\|T(T_{r_n}f)\|_{\mathcal{Y}}\leq \|T\|_{\overline{\mathcal{P}}^{\mathcal{X}}\to \mathcal{Y}}\|T_{r_n}\|_{\mathcal{X}\to \mathcal{X}}\|f\|_{\mathcal{X}}&\leq \|T\|_{\overline{\mathcal{P}}^{\mathcal{X}}\to \mathcal{Y}}\sup_{0<r<1}\|T_r\|_{\mathcal{X}\to \mathcal{X}}\|f\|_{\mathcal{X}}.
f(x,t) = \frac{f_1+f_2}{2}+ \frac{\beta}{2}(f_1-f_2)^2 + \biggl(-\frac{\beta^3}{12}+c\biggr)(f_1-f_2)^4 & + C(x,t)(f_1-f_2)^6 + \frac{1}{N^{1/4}}y(x,t) - \frac{\log m(\beta N^{-1/4})}{\beta},
T_k(z)= :\Lambda'(q^{-n+k}s_1^{-1}z)\bar\Lambda'(q^{n-k}z) V_{\chi_{k,1}}(z):\,, &\bar{T}_k(z)= :\Lambda'(q^{n-k}z)\bar\Lambda'(q^{-n+k}s_1^{-1}z) V_{\chi_{\bar k, 1}}(z):\,,
\mathbf{S}^{k+1}:={\arg \min}_{\mathbf{S}}\left\{||\mathbf{S}||_1 \right. \left. +\frac{\lambda_1}{2}||\mathbf{A}(\mathbf{\rho}^{k+1}) +\mathbf{A}(\mathbf{S})- \mathbf{y}+ \mathbf{u}^k||_2^2 \right\}.
{\cal A}_{\Lambda}^a\left( \phi ,\phi^{*},\overline{\phi},\pi\right) =\left[ \overline{\Delta}^aW_\Lambda +\frac i{2\hbar}\left( W_\Lambda,W_\Lambda \right)^a\right]\left( \phi ,\phi^{*},\overline{\phi},\pi\right) =
\big\| \Delta^{\frac{1 + \sigma}{2}}\partial_l \Delta^{-1}\beta_j^{k,\lambda}\big\|_{L^p}\lesssim k^{-\frac{1}{2}}\lambda^{-1+ \frac{2}{r_j}}\sum_{m=1}^2 \left(\int_{\mathbb{R}^2}\lambda^{-2p'}|\xi|^{\sigma p'}\big| \hat{\rho}(\lambda^{-1}\xi_m^k) \big|^{p'}\, d\xi\right)^{1/p'}
A(x_0) = \ln |f^\prime(x_0)| + H\left( f(x_0) - [f(x_0)],\frac{f^{\prime \prime}(x_0)}{(f^\prime(x_0))^2}, \ldots,\frac{f^{(q)}(x_0)}{(f^\prime(x_0))^q}\right) + c_0 [f(x_0)]
\frac{\partial^{2}\ln I ( \phi , \chi )}{\partial \phi_{a}\partial \chi_{b}}= \sum_{n = 1}^{N}\left\langle \frac{P_{n}( x_{1}) \tilde{Q}_{n}( y_{1})}{( x_{1}- \phi_{a}) ( y_{1}- \chi_{b})}\right\rangle - \sum_{n , m = 1}^{N}\left\langle \frac{P_{n}( x_{1}) \tilde{Q}_{m}( y_{1})}{x_{1}- \phi_{a}}\right\rangle \left\langle \frac{P_{m}( x_{2}) \tilde{Q}_{n}( y_{2})}{y_{2}- \chi_{b}}\right\rangle
[e_3,e_1]+\lambda J[e_3,e_1] + J[e_3,Je_1]-\lambda [e_3,Je_1] = -e_1-\lambda Je_1 + J[e_3,Xe_1+Ye_2]-\lambda [e_3,Xe_1+Ye_2] = -e_1-\lambda Xe_1-\lambda Ye_2 - X(Xe_1+Ye_2)-Y\theta(\frac{-1-X^2}{Y}e_1-Xe_2)+\lambda Xe_1+\lambda\theta Ye_2
F\left({\cal A},G,p,t^{(A)}_{n,l}\right)= \sum_{n=-\infty}^{+\infty}F^{(\rm cov)}\left(G,p,r^{(A)}_{n,l}\right) \prod_{A=1}^p \prod_{l=1}^{\infty}\left[t_{n,l}^{(A)}\right]^{r^{(A)}_{n,l}}{\rm e}^{-|r|_n{\cal A}v(n)}~.
h(Y_1^n|W_1) =h(X_1^n+Z_1^n|W_1)\geq\frac{n}{2}\log\left[ 2^{\frac{2}{n}h(X_1^n+\tilde{Z}^n|W_1)}+ 2\pi e(1-\tfrac{1+bP_2}{a}) \right].
2(A_{3}C_{0}-C_{0}A_{3})+C_{0}e^{-xA_{3}}(-A_{3}A_{2}+A_{2}A_{3})e^{xA_{3}}-e^{-xA_{3}}(-A_{3}A_{2}+A_{2}A_{3})e^{xA_{3}}C_{0}=0.
\begin{aligned}\mu_0^{-(n-2s-1)}(t)\int_{\mathbb{R}^n}E_{0j}(y, t)Z_{n+1}(y)dy & \approx c_1\left[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)\right] &-\frac{2sc_1 A +c_2}{(n-4s)c_{n,s}^{n-4s}}b_j^{2s-1}\end{aligned}
\Sigma_4^{0}(r, s; \mu) \ll \frac{T^{3+\varepsilon}}{r_2s_0}\sum_{\substack{s_2 n_2 \leq r_1r_2s_0 T^{3+\varepsilon}M^{-2}n_2s_2 s_1 = m_2r_2 r_1}}T^{-\frac{3}{2}}\ll \frac{T^{\frac{9}{2}+\varepsilon}r_1}{M^2}\ll T^{\frac{3}{2}+ 3\lambda + 2\eta + \varepsilon}.
L_f^{(m)}(s)= \frac{1}{2\pi i}\left(\int_{(\frac{3}{2}-\sigma)}-\int_{(-\frac{1}{2}-\sigma)}\right)L_f^{(m)}(s+w)\frac{K_\varphi(w)}{w}\frac{\Gamma(s+w+\frac{k-1}{2})}{\Gamma(s+\frac{k-1}{2})}\times &\times\left(\frac{x}{2\pi}e^{-i\frac{\pi}{2}{\rm sgn\;}t}\right)^wdw.
L = \begin{pmatrix}1 -1 0 0 0 -1 2 -1 0 0 0 -1 2 -1 0 0 0 -1 2 -1 0 0 0 -1 1 \end{pmatrix}, P = \begin{pmatrix}1 0 1 0 1 0 0 1 0 1 \end{pmatrix}
(c_1, c_2, c_3,c_4) := \; 2^{k-5}(48,32,40,44) (c_1', c_2', c_3',c_4') := \; 2^{k-5}(48,40,44,48) (c_1'', c_2'', c_3'',c_4'') := \; 2^{k-5}(32,32,48,64) \mathcal Z_k:= \; \big(Z(s)\big)_{2^k\leq s< 2^{k+1}}.
(y^5 - 5y^4 + 14y^3 - 23y^2 + 23y - 11)(y^{n-5}- y^{n-6}+ y^{n-7}) &\qquad+ (y^4 - 4y^3 + 10y^2 - 13y + 8)y^{n-6}&= (y^7 - 6 y^6 + 21 y^5 - 46 y^4 + 70 y^3 - 70 y^2 + 42 y - 11)y^{n - 7}< (y+1)y^{n-3}(y-1)(y-2).
\sup_{t_1 < s < t}\int_{\Omega\,\cap\,B_{H_0}(x,R_2)}z(y,s)^2\zeta_x^2\,dy+\iint_{Q_2}H(\nabla (z\zeta_x))^2\,dy\,ds \qquad\le C[(R_2-R_1)^{-2}+(t_1-t_2)^{-1}]\iint_{Q_2}z(y,s)^2\,dy\,ds
\sum_{n=1}^{\infty}\frac{1}{F_{n}^{2}}=\frac{5}{24}[2\theta_{2}(\Phi^{-2})^{4}-\theta_{3}(\Phi^{-2})^{4}+1]=\frac{5}{24}[\theta_{3}(\Phi^{-2})^{4}-2\theta_{4}(\Phi^{-2})^{4}+1]
\sigma_\mu^{\alpha\beta'}= (\tau^a, i) \ , \qquad\bar\sigma_{\mu\, \alpha'\beta}= (\tau^a, -i) \ , \qquad\sigma_\mu^{\alpha\alpha'}= \bar\sigma_\mu^{\alpha'\alpha}= \epsilon^{\alpha'\beta'}\bar\sigma_{\mu\, \beta'\beta}\epsilon^{\alpha\beta}\ ,
\int_{m}^\infty dk\ (k^2-m^2)^{1-s}\partial_k\frac{t^j}{l^n}=-m^{2-2s}\frac{\Gamma(2-s)\Gamma\left(s+\frac j2-1\right)l^{j-n}}{\Gamma \left(\frac j2\right) (Rm)^{j}\left( 1+\left(\frac{l}{mR}\right)^2\right)^{s+\frac j2-1}}\ ,
d_n(b_1c_1a_1 \dots b_1c_1a_1b_1 \otimes 1) = b_1c_1a_1 \dots b_1c_1a_1 \otimes b_1 + b_1c_1a_1b_1 \dots c_1a_1b_1a_0 \otimes b_0c_0
xn \leq |X| = n - k!bdN \leq xn + k!bd, |X_1| = |V_1| - k!b\sigma dN \geq \phi n - (1-x)\sigma n = xn/k,\mbox{and}|X_j| = |V_j| - (k-2)!b(1-\sigma) kdN \geq \frac{(1-\phi)n}{k-1}- \beta n - \frac{(1-x)(1-\sigma)n}{k-1}= \frac{xn}{k}-\beta n,
\Phi_{\gamma,\delta}(\Phi_{\gamma,\delta}(d,e,s-r),e,t-s) = e^{\gamma (t-s)}\Phi_{\gamma,\delta}(d,e,s-r) + \varphi_{\gamma}(t-s)(e+\delta) = e^{\gamma (t-s)}\big( e^{\gamma (s-r)}d + \varphi_{\gamma}(s-r) (e+\delta) \big) + \varphi_{\gamma}(t-s)(e+\delta) = e^{\gamma (t-r)}d + \varphi_{\gamma}(t-r) (e+\delta) = \Phi_{\gamma,\delta}(d,e,t-r),
T = \frac{D-\rho^2 e R}{-\alpha S}-\frac{1}{\alpha}+\frac{D_1^3}{27\alpha^2 S^2}+\frac{D_2^3}{729 \alpha^4 S^2}+\frac{\rho^2 D_1D_2R}{9\alpha^2 S^2}.
\begin{array}{c l l c l l}{{{[ A_{m}^{i}, A_{n}^{j}]}}}&{{= \; m \delta^{i j}\delta_{m , - n}}}&{{,}}&{{[ C_{m}, C_{n}]}}&{{= \; m \delta_{m , - n}}}&{{,}}\\{{\{B_{r}^{i}, B_{s}^{j}\}}}&{{= \; \delta{i j}\delta_{r , - s}}}&{{,}}&{{\{D_{r}, D_{s}\}}}&{{= \; \delta_{r , - s}}}&{{,}}\\ \end{array}
e^{\sum V \left(x_i^{(a)}- y_j^{(b)}\right)-\frac{1}{2}\sum V \left(x_i^{(a)}- x_j^{(b)}\right)-\frac{1}{2}\sum V \left(y_i^{(a)}- y_j^{(b)}\right)}\left( \frac{1}{2\pi}\right)^{N(n+m)}\left[\frac{e^2 N}{4 \pi}e^{2\gamma}\right]^{\frac{N(n-m)^2}{2}}\; \;\; .
A =\nu\int (\partial_x^lu)^2\chi'(x+\nu t) \; dx+c(\epsilon;b)\int (\partial_x^lu)^2\chi'(x+\nu t;\epsilon/3,b+\epsilon) \; dx, B =2\int\partial_x^lu\partial_x^l(u\partial_x^3u)\chi(x+\nu t) \; dx.
\phi(\underline{x})=\frac{GM}{|\underline{x}|^{2}}\underline{x}-GM(\frac{1}{8}\frac{\underline{x}-\underline{b}}{|\underline{x}-\underline{b}|^{2}}+\frac{1}{64}\frac{\underline{x}-\underline{c}}{|\underline{x}-\underline{c}|^{2}})
\tilde{S}_{1}(\varphi)(x, T) + \tilde{S}_{2}(\mu_{1}, \mu_{2})(x, T) + \tilde{S}_{2}(U)(x, T) + \tfrac{\mu_{2}(T) - \mu_{1}(T)}{l}x = \Psi(x) 0 \leq x \leq l.
\partial_a \partial_\nu S_{a, [\nu]}^\times(x,y) = \partial_a \partial_\nu \big( \Theta(\xi^2-\nu) \:H_{a, [\nu]}(x,y) \big) &= \Theta(\xi^2-\nu) \:\partial_a \partial_\nu H_{a, [\nu]}(x,y) +\big( \partial_\nu \Theta(\xi^2-\nu) \big) \: \partial_a H_{a, [\nu]}(x,y) \:.