harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathcal{L}=\sum_{i=1}^{k}\sum_{j=1}^{N_{i}}V(\lambda_{i},N_{i};\lambda,N;\varepsilon )\bar{\psi}_{(i)}^{j}\left( \partial \!\!\!\slash+\lambda_{i}U(\lambda_{i},N_{i};\lambda ,N;\varepsilon )\sigma \right) \psi_{(i)}^{j},
\Delta{(\Psi_{\{[\{1,3\},3],[\{2\},4],[\{4\},1]\}})}= 1 \otimes \Psi_{\{[\{1,3\},3],[\{2\},4],[\{4\},1]\}}+ \Psi_{\{[\{1,3\},3],[\{2\},4]\}}\otimes \Psi_{\{[\{1\},1]\}}& + \Psi_{\{[\{1,3\},3],[\{2\},4],[\{4\},1]\}}\otimes 1 .
C_{+}(\mathbb{R}^n) = \Bigl\{\mu=(\mu_1, \mu_2, \ldots, \mu_n) \,\bigl \vert \,\mu \in C(\mathbb{R}^n), \mu_k(0)=0, \,\, \,\, 1 \le k \le n, \mu_k(t) \,\, \,\, \sum\nolimits_{k=1}^n \mu_k(t) = t, \,\, t \in [0, T] \Bigr\}.
U_{n}(x):=\frac{(n+1)(2n)!!}{(2n+1)!!}\sum_{k=0}^{n}\binom{n+\frac{1}{2}}{n-k}\binom{n+\frac{1}{2}}{k}\left(\frac{x+1}{2}\right)^{n-k}\left(\frac{x-1}{2}\right)^{k},
j = \left( \begin{array}{c c c c c c}{\theta_{1}}&{}&{}&{}&{}&{}\\{}&{\ddots}&{}&{}&{}&{}\\{}&{}&{\theta_{q + 1}}&{}&{}&{b}\\{}&{}&{}&{\ddots}&{}&{}\\{}&{}&{}&{}&{\theta_{p}}&{}\\{}&{}&{c^{t}}&{}&{}&{d}\\ \end{array}\right) .
\sigma^2(f)= \frac{1}{(2\pi)^{n}}\*\sum_{\pi}^* \prod_{B\in \pi}\*\int_{L_{\pi}}\f(t^{(1)})\*f(t^{(2)})\*\prod_{B\in\pi}c_{|B|}(t_i: i\in B) \ \* d\lambda,
s_{\mu,\upsilon}(z)=\frac{z^{\mu+1}}{(\mu-\upsilon+1)(\mu+\upsilon+1)}{}_{1}F_{2}\left(\ \begin{array}{lll}~~~~~~~~~~~~~~~~~~~~~~1;~ \frac{\mu-\upsilon+3}{2},\frac{\mu+\upsilon+3}{2};~\end{array}\frac{-z^{2}}{4}\right),
P_{j}(y,D_y)U_j-qU_j=F_{j}(y), \y\in K_j^\varepsilon; U_j(y) -\sum\limits_{k=1}^{N_\nu}\sum\limits_{s=1}^{S_{j\sigma k}}B_{j\sigma ks}(y)U_k({\mathcal G}_{j\sigma ks}y) =0, \y\in\gamma_{j\sigma}^\varepsilon.
\begin{array}{rcl}\langle y-\tilde{x}_{i},d-\tilde{x}_{i}\rangle = \langle y-\tilde{x}_{i},e_{i-m}+e_{i-m+1}+\cdots+e_{i}\rangle = \underset{k=i-m}{\overset{i}{\sum}}\langle y-\tilde{x}_{i},e_{k}\rangle = \underset{k=i-m}{\overset{i}{\sum}}\langle y-\tilde{x}_{k},e_{k}\rangle+\underset{k=i-m}{\overset{i}{\sum}}\langle\tilde{x}_{k}-\tilde{x}_{i},e_{k}\rangle.\end{array}
(LG)_{ij}=\dfrac{[c+d+(j-i-1)\delta]_N[-(i+j)\delta]_N[2u+\delta]_N[d-c]_N}{[u+d+j\delta]_N[u-c+(1-j)\delta]_N[u+c-i\delta]_N[-u+d-(1+i)\delta]_N}\dfrac{[c\pm u]_j}{[d\pm u]_j}
G_\varphi'(s)= \int_X AT_s \varphi |DT_{t-s}f|^2\,d\mu -2\int_X T_s \varphi DT_{t-s}f\cdot DAT_{t-s}f \,d\mu \geq 2K \int_X T_s \varphi |DT_{t-s}f|^2\,d\mu = 2K G_\varphi(s).
c_{2t}=c_t=1; &a_k=a_{[g/2]+1-k}^{-1}\ \ \mathrm{and}\ \b_k=a_{[g/2]+1-k}b_{[g/2]+1-k}^{-1}a_{[g/2]+1-k}^{-1}, \ \ 1\leq k\leq t, a_{[g/2]+1-k}= a_k^{-1}\ \ \mathrm{and}\ \b_{[g/2]+1-k}=a_{[g/2]+1-k}^{-1}b_k^{-1}a_{[g/2]+1-k}, \ \ 1\leq k\leq t,
y_{(2r'_0 -1)m+m^{+}_0,j}= MN - x_{m^{+}_0,j}+(r'_0 -\tfrac{1}{2})(mn) +\tfrac{3}{2}\mbox{\if}m^{+}_0+j \mbox{is odd and}y_{(2r'_0 -1)m+m^{+}_0,j}= MN - x_{m^{+}_0,j}- (r'_0 -\tfrac{1}{2})(mn) +\tfrac{3}{2}\mbox{\if}m^{+}_0+j \mbox{is even.}
\Gamma^{(\mathcal{C})}(P(A),\tau_A(\tilde{\mathcal{L}}))=\sum_{\eta \in \mathcal{P}}[ \sum_{\gamma \in Merge^-{(\{\tilde{\mathcal{L}},\eta}\})}\tau_A(\gamma)-\sum_{\gamma \in Merge^+(\{\tilde{\mathcal{L}},\eta\})}\tau_A(\gamma)].
\widehat{R}_{j,j+1}(\zeta)\widehat{R}_{j+1,j+2}(\xi\zeta) \widehat{R}_{j,j+1}(\xi) = \widehat{R}_{j+1,j+2}(\xi)\widehat{R}_{j,j+1}(\xi\zeta) \widehat{R}_{j+1,j+2}(\zeta).
\mathbb{Q}_{M}(M=1)=\mathbb{P}_{k^{*}}=\begin{cases}\frac{1}{2}+\frac{2}{n-1}-\frac{2}{(n-1)^2}\,\, n \,; \frac{1}{2}+\frac{2}{n-1}-\frac{5}{2(n-1)^2}.\end{cases}
{\mathrm{}^{*}\mathcal{L}}_{e f f}= - \frac{1}{4}F_{\mu \nu}F^{\mu \nu}+ \beta^{2}\sqrt{1 + \frac{1}{6 m^{2}\beta^{2}}H_{\mu \nu \rho}H^{\mu \nu \rho}}- \frac{\chi}{2 \theta}\varepsilon^{\mu \nu \rho \lambda}B_{\nu \rho}\partial_{\mu}A_{\lambda}\; .
_d \texttt{M}_{_p A}' =\begin{cases}{}_d \texttt{M}_{_pA}- \sum_{B\in \mathcal J_d}c_{B, p,n_p}(v^{n_p}+v^{-n_p}) \{_p B\}_d, n_p>0,{}_d \texttt{M}_{_pA}- \sum_{B\in \mathcal J_d}c_{B, p,n_p}\{_p B\}_d, n_p =0. \end{cases}
\left( \begin{array}{c}{m^{\prime}}\\{N^{\prime}}\\ \end{array}\right) = \left( \begin{array}{c c}{a}&{b}\\{c}&{d}\\ \end{array}\right) \left( \begin{array}{c}{m}\\{N}\\ \end{array}\right) , \qquad \Theta^{\prime}= \frac{c + d \Theta}{a + b \Theta},
1 5 \mathrm{C l}_{2}( \theta_{1 1}) - 1 0 \mathrm{C l}_{2}( 2 \theta_{1 1}) + \mathrm{C l}_{2}( 5 \theta_{1 1}) = Z_{1 1}= 1 1 \sum_{k = 1}^{5}\left( \frac{k}{1 1}\right) \mathrm{C l}_{2}\left( \frac{2 \pi k}{1 1}\right)
\| A^{r-\frac{1}{2}}e^{\tau A}\partial_t \overline{v}\| \leq C_r \Big( \|e^{\tau A}\overline{v}\|_{H^{r}}\|e^{\tau A}\overline{v}\|_{H^{r+\frac{1}{2}}}+ \|e^{\tau A}\widetilde{v}\|_{H^{r}}\|e^{\tau A}\widetilde{v}\|_{H^{r+\frac{1}{2}}}\Big),
V_1=\langle u_{\chi_1}(d_1^{2^{n-2}})\rangle \times\prod_{2l+1\in A_0\setminus\{1\}}\langle u_{\chi_1}(d_{2l+1}^{-1})u_{\chi_1}(d_{2^{n-1}-(2l+1)})\rangle\times &\times\prod_{k=1}^{n-3}\prod_{2l+1\in A_k}\langle u_{\chi_1}(d_{2l+1}^{-2^k})u_{\chi_1}(d_{2^{n-1}-(2l+1)}^{2^k}).
\Pr\big(\big||S_{n}|^{2}-a_{n}\big|\geq t\big)\geq \frac{1}{6}\sum_{k=1}^{n}\Pr(\tau=k)=\frac{1}{6}\Pr\Big(\max_{1\leq k\leq n}\big||S_{k}|^{2}-a_{k}\big|\geq 3t\Big)
\sum\limits_{s \in^{2}\Gamma^{j}_{\alpha'}}\widehat{\gamma}^{s}_{\beta_{s}(\alpha')}\le q \sum\limits_{s=0}^{i}\sum\limits_{\substack{m' \in \mathbb{Z}^{n}Q_{l_{i},m}\subset Q_{l_{s},m'}}}\widehat{\gamma}_{l_{s},m'}=:C t_{l_{i},m}.
\partial_r \Bigl( \Delta(F_0(rx_1, \frac 1 r x_2) \Bigr) \Bigr|_{r=1}= \; 48 (x_1^5 - 10 x_1^3 x_2^2 + 5 x_1 x_2^4)/(x_1^2 + x_2^2)^5 =:H_0(x).
B_{ij}':=B_{ij}-\Big(\frac{(\lambda_{n+3}-\lambda_i)x_i}{(\lambda_{n+3}-\lambda_1)x_1}\Big)B_{1j}=(\lambda_{n+3}s-t)\frac{(\lambda_1-\lambda_i)(\lambda_{n+2}-\lambda_j)x_ix_j}{(\lambda_{n+3}-\lambda_{n+2})(\lambda_{n+3}-\lambda_1)x_{n+3}^2}
div(M^{\Lambda^1(t_1)}\nabla(d(t_1)-d(t_2)))= div((M^{\Lambda^1(t_2)}-M^{\Lambda^1(t_1)})\nabla d(t_2)) +div( (M^{\Lambda^2(t_2)}-M^{\Lambda^1(t_2)})(\nabla v_2(t_1)-\nabla v_2(t_2))) +div ([(M^{\Lambda^2(t_1)}-M^{\Lambda^1(t_1)})-(M^{\Lambda^2(t_2)}-M^{\Lambda^1(t_2)})]\nabla v_2(t_2)).
\begin{cases}\gamma_1=K_{11}\alpha_5+\alpha_1+\beta_1+O(1)\Delta''(\boldsymbol{\alpha}^{*},\beta_{1}), &\sigma'_{i}=K_{1i}\alpha_5+\alpha_i+\sigma_i+O(1)\Delta''(\boldsymbol{\alpha}^{*},\beta_{1}),i=2,3, &\gamma_4=\alpha_4+\beta_4, &\gamma_5=K_{15}\alpha_5+\beta_5+O(1)\Delta''(\boldsymbol{\alpha}^{*},\beta_{1}),\end{cases}
\tilde \phi_{}(\mathcal Z, n) := \sup_{k\geq 0, \atop f \in BV_1}\big\| \mathbb{E}\big( f(Z_k) \big| A_{k+n}^{\infty}\big) - \mathbb{E}f(Z_k) \big\|_{\infty}&= \sup \big \{\mathrm{cor}(h \circ Z_{k}, \varphi) : k \geq 0, \varphi \in B_{L_1(\mathcal A_{k+n}^\infty, \mu)}, h \in B_{BV(Z)}\bigr\}\, ,
\| \Theta - f_N \|_2^2 = \| \widehat{\Theta}- \widehat{f}_N \|_2^2\ge \| \widehat{\Theta}\|_{L^2(\mathcal{W}^c_N)}^2\ge \sum_{J\in \mathbb{J}^c_N}\| \widehat{\Theta}W^{-}_J \|^2_2\gtrsim \sum_{J\in \mathbb{J}^c_N}2^{-js(2-\alpha)}.
P_i^\ell (x_{n+\ell},x_{n-\ell})\triangleq\Pr(X_n=i|X_{n-\ell}=x_{n-\ell},X_{n+\ell}=x_{n+\ell}) &=\frac{P(X_{n+\ell}=x_{n+\ell},X_n=i,X_{n-\ell}=x_{n-\ell})}{P(X_{n-\ell}=x_{n-\ell},X_{n+\ell}=x_{n+\ell})}&=\frac{P(X_{n+\ell}=x_{n+\ell}|X_n=i)P(X_n=i|X_{n-\ell}=x_{n-\ell})}{P(X_{n+\ell}=x_{n+\ell}|X_{n-\ell}=x_{n-\ell})},
\alpha_1 = 1 + \frac{1+i}{4}(q-1) + o(q-1) = 1 \left(1 + \frac{1+i}{4}(q-1) + o(q-1) \right) \alpha_2 = -i + \frac{i}{2}(q-1) + o(q-1) = -i \left(1 - \frac{1}{2}(q-1) + o(q-1) \right) \alpha_3 = -1 - \frac{1-i}{4}(q-1) + o(q-1) =-1\left(1 + \frac{1-i}{4}(q-1) + o(q-1) \right)
\frac{L_{i}}{n_{i}}\sum_{l=1}^{n_{i}}n_{x_{l}}DB_{x_{l}}\hat{u}=-\frac{L_{t}}{n_{t}}\sum_{j=1}^{n_{t}}\overline{t}_{x_{j}}-\frac{S}{n_{\Omega}}\sum_{k=1}^{n_{\Omega}}b_{x_{k}}
Z_N(\boldsymbol{\lambda};\boldsymbol{\nu})=\frac{\prod_{j,k=1}^{N}a(\lambda_j,\nu_k)b(\lambda_j,\nu_k)}{\prod_{1\leq j<k \leq N}d(\lambda_k,\lambda_j)d(\nu_j,\nu_k)}\det[\varphi(\lambda_j,\nu_k)]_{j,k=1,\dots,N},
\begin{bmatrix}x^{(1)}y^{(1)}x^{(2)}y^{(2)}x^{(3)}y^{(3)}x^{(4)}y^{(4)}x^{(5)}y^{(5)}\end{bmatrix}=\left[\begin{array}{rrr|rrr}1 1 -1 1 1 -1 1 -1 1 1 -1 1 -1 1 1 -1 1 1 1 1 1 1 1 1 2 -3 3 -2 0 2 \end{array}\right],
\delta_{\kappa}S = \delta_{\kappa}^{1}S + \delta_{\kappa}^{2}S = \int d^{p + 1}\sigma \; \bar{\kappa}( 1 + \Gamma ){\cal P}\Delta + \int d^{p + 1}\sigma \; \bar{\kappa}( 1 + \Gamma ) ( 1 -{\cal P}) \Delta \, .
|\Theta_1|_r= |D^\alpha(BH)-BD^\alpha H|_r\leq C(|\nabla^2 v|_r |\nabla H|_{\infty}+|\nabla^3 v|_r| H|_\infty) \leq C\|\nabla^2 v\|_{H^1\cap W^{1,q}}\|H\|_{H^2 \cap W^{2,q}}.
H_{n-2}\left( A_{i,k},\Gamma_{i};\mathbb{Z}G_{i}\right) =\ker\partial\cong\left(\mathbb{Z}G_{i}\right)^{3}H_{n-3}\left( A_{i,k},\Gamma_{i};\mathbb{Z}G_{i}\right) =\operatorname*{coker}\partial\cong\left(\mathbb{Z}G_{i}\right)^{3}
\mu_3^2\geq \nu_2\mu_4^2 \nu_2 :=\left(\sqrt{1 + \frac{c_{1,\infty}}{2c_{3,\infty}}}- 1\right)^2\left(-1 + \sqrt{1 + \frac{c_{3,\infty}+ c_{2,\infty}}{c_{4,\infty}}}\right)^{-2}.
T^{(2)}(M_{KC}, \phi \circ \psi, \gamma \circ \psi)(t) \ \dot{=}\ \max(1,t)^{(e-1) | \phi(\psi(a_{e+1}))|}& = \max(1,t)^{(e-1) |n_{e+1}+ k n_1 + \ldots + k n_e|}.
\mu_{k}^{\mathrm{( i n )}}( \eta ) = C^{( \mathrm{i n )}}\left( 1 + \mathrm{e}^{- x}\right)^{d +{\frac{1}{2}}}{}_{2}F_{1}\left( b + d +{\frac{1}{2}}, - b + d +{\frac{1}{2}}; 2 d + 1 ; 1 + \mathrm{e}^{- x}\right)
\mathfrak{d}_2 = [\mathfrak{d}_1,\mathfrak{n}] +J[\mathfrak{d}_1,\mathfrak{n}] = \mathrm{span}\left\{[T,X] + J[T',X'] : \forall T,T' \in \mathfrak{d}_1, \forall X,X' \in \mathfrak{n}\right\}.
W^{1}( x_{A}, \theta_{2}, \theta_{3}, \theta_{4}, \bar{\theta}^{1}, u ) \quad \mathrm{o r}\quad W^{1 2 3}( x_{A}, \theta_{4}, \bar{\theta}^{1}, \bar{\theta}^{2}, \bar{\theta}^{3}, u ) \equiv \bar{W}_{4}
< \overline{{{\psi}}}_{L}^{({\cal N})}\ldots \overline{{{\psi}}}_{L}^{( 1 )}\psi_{R}^{( 1 )}\ldots \psi_{R}^{({\cal N})}> = ( \frac{e^{\gamma}}{4 \pi})^{\cal N}( \sqrt{\frac{\cal N}{\pi}}e_{c})^{\cal N}\quad .
\sum_{i_0=0}^\infty \sum_{i_1\geq i_0}^\infty \cdots \sum_{i_{d}\geq i_{d-1}}^\infty 2^{-(i_0+i_1+\cdots +i_{d})}~U(s, i_0, \ldots, i_d, \ell_0, \ell_1, \ldots, \ell_s)~q^{-s(d-k)}\lesssim 1.
{\Gamma^{1}}_{a}= \left\{\left( \begin{matrix}{{0}}&{{\sigma_{1}}}\\{{- \sigma_{1}}}&{{0}}\\ \end{matrix}\right) , \quad \left( \begin{matrix}{{0}}&{{- I}}\\{{- I}}&{{0}}\\ \end{matrix}\right) , \quad \left( \begin{matrix}{{I}}&{{0}}\\{{0}}&{{- I}}\\ \end{matrix}\right) \right\},
e^{-\frac{x^{2}}{2}}\cdot H_{n}(x)\sim(\frac{2n}{e})^{\frac{n}{2}}\sqrt{2}cos(x\sqrt{2n+1-\frac{x^{2}}{3}}-\frac{n\pi}{2})(1-\frac{x^{2}}{2n+1})^{-\frac{1}{4}}.
\begin{array}{rl}\psi_j(y_i,z_i,\theta,\hat\eta^j) = z_{ij}(y_i-z_{ij}\theta-z_{i}^T\hat \beta_{-j})+\Gamma_{jj}\theta & -(\hat \mu^j)^T\{z_{i,-j}(y_i-z_{ij}\theta-z_{i,-j}^T\hat \beta_{-j})+\Gamma_{-j,-j}\hat \beta_{-j}\}& = (e^j-\hat\mu^j)^T\{z_{i}(y_i-z_{ij}\theta-z_{i,-j}^T\hat \beta_{-j})+\Gamma(\theta e^j+\hat\beta_{-j})\}\end{array}
L_{D}^{\ast}= \int_{\cal M}d^{3}x \varepsilon^{i j k}\bigg \{-{\frac{3}{4 a}}( a \partial_{i}B_{j A B}+ b G_{i j A B}) [{\bf R}^{- 1}]_{k n}^{A B C D}\varepsilon^{l m n}( a \partial_{l}B_{m C D}+ b G_{l m C D}) + c \alpha_{i}^{A B}G_{j k A B}\bigg \},
\tau^{j-1}\cdot n^{\l_j-2}\geq\tau^{\binom{\l_j}{2}-1}\cdot n^{\l_j-2}=C_0^{\binom{\l_j}{2}-1}n^{-\frac{(\l_j-2)(\l_j+1)}{k+1}+\l_j-2}=C_0^{\binom{\l_j}{2}-1}n^\frac{(\l_j-2)(k-\l_j)}{k+1}\,.
|V(x)u(x)| \leq C_{k}V(x) \int_{B_{0}}{1\over{\big( 1+{{|x-y|}\over{\rho(x)}}\big)^{k}}}{1\over{|x-y|^{n-2}}}\Big( |Lu(y)| + + \sum_{i,j=1}^{n}|a_{ij}(y)-a_{ij}(x)| |u_{x_{i}x_{j}}(y)| \Big) dy.
\sum_{n=i}^{\infty}\frac{1}{n(n+1/2)}\le\sum_{n=i}^{\infty}\frac{1}{(n-1/2)(n+1/2)}=\sum_{n=i}^{\infty}(\frac{1}{n-1/2}-\frac{1}{n+1/2})=\frac{1}{i-1/2}
\left( 1 - \frac{\mu+\tau}{\mu_1(\lambda) + \tau}\right) \sum_{k=1}^\infty \tilde{c}_k^2 \; \leq \; \sum_{k=1}^\infty \left( 1 - \frac{\mu+\tau}{\mu_k(\lambda)+ \tau}\right) \tilde{c}_k^2 \; =\; \ell(\hat{u})
\varphi_{\alpha_{1}, \alpha_{2}, \alpha_{3}}( z_{1}, z_{2}, z_{3}) \equiv z_{1}^{\alpha_{1}- h}z_{2}^{\alpha_{2}- h}z_{3}^{\alpha_{3}- h}z_{1 2}^{\alpha_{3}}z_{2 3}^{\alpha_{1}}z_{3 1}^{\alpha_{2}},
\alpha = \sum_{i\ne j\ne k}\lambda_{ij}\hat{F}_{ijk}F_k'dF_i\wedge dF_j + \sum_{i\ne j}2\lambda_{ij}\hat{F}_{ij}dF_i'\wedge dF_j + \beta &= d\rho(\lambda,\underline{F})(0,(F_i')_{i=1}^m) + \beta .
\lambda(z)= \bigg(|h(z)|\Pi_{l=1,2}\dfrac{\Pi_{j=1}^{q_l}|G^l_j(z)|^{(1-\delta)p_l}}{e^{p_l\sum_{j=1}^{q_l}u^l_j(z)}|W_z(g^l_0,g^l_1)(z)|^{p_l}}\bigg)^{\frac{1}{1-p_1 -p_2}}& \times \bigg(|h(1/z)|\Pi_{l=1,2}\dfrac{\Pi_{j=1}^{q_l}|G^l_j(1/z)|^{(1-\delta)p_l}}{e^{p_l\sum_{j=1}^{q_l}u^l_j(1/z)}|W_z(g^l_0,g^l_1)(1/z)|^{p_l}}\bigg)^{\frac{1}{1-p_1 -p_2}}.
{\cal E}_{\rm log}=- V(t)^2\int\!\frac{{\rm d^3}k}{(2\pi)^3}\frac{1}{16 (\omega_k^0)^3}=-\frac{\lambda^2\mu^\epsilon}{256\pi^2}\left( \phi(t)^2 - \phi(0)^2\right)^2\left\{\frac{2}{\epsilon}+\ln\frac{4\pi\mu^2}{m_0^2}-\gamma \right\}\;.
B(b) = \frac{\pi}{2}\left| \sqrt{b}\, \vartheta_3(0, i b) \vartheta_3(0, i b) \right|^2 = \frac{\pi}{2}\left| \vartheta_3(0, i b^{-1}) \vartheta_3(0, i b) \right|^2 = \frac{\pi}{2}\, \vartheta_3(0,ib^{-1})^2 \vartheta_3(0,i b)^2.
\rho \ddot{u}_i = \left( A_{iJKj}u_{j,K}- \beta_{Ji}\dot{\tau}-(C_{iJKLIj}u_{j,IL}+ M_{iJKL}\tau_{,L})_{,K}\right)_{,J}-E(|\dot{u}|) \dot{u}_i, a \ddot{\tau}=-\beta_{Ki}\dot{u}_{i,K}+ m_{IJ}q_{I,J}+ M_{jLKI}u_{j,LKI}+K_{IJ}\tau_{,IJ}, \kappa \dot{q}_i = \dot{\tau}_{,i}- q_{i},
\beta_{1}\beta_{2}K^{T}( s_{1}, s_{1}) \, \equiv \! \int d p_{0}^{\prime}d p_{0}\delta ( p_{0}^{\prime}\! - \! s_{1}) \beta_{1}\beta_{2}K^{T}( p_{0}^{\prime}, p_{0}) \delta ( p_{0}\! - \! s_{1}) .
V_{p}=\int_{a}^{b}|\frac{d^{m}P_{N}(x)}{dx^{m}}-\frac{1}{b-a}\int_{a}^{b}\frac{d^{m}P_{N}(z)}{dz^{m}}dz|^{p}dx-\sum_{i=1}^{n}\lambda_{i}(P_{N}(x_{i})-f(x_{i})),
\langle z, g \rangle_{L^2,L^2}= \langle f, \varphi \rangle_{(X_T)',X_T}+ \langle z^0, \varphi(0, \cdot) \rangle_{H^{-1}(0,1),H^1_0(0,1)}+ \nu \langle v_0, \varphi_x(\cdot, 0) \rangle_{H^{-1/4}(0,T),H^{1/4}(0,T)}- \nu \langle v_1, \varphi_x(\cdot, 1) \rangle_{H^{-1/4}(0,T),H^{1/4}(0,T)},
\begin{array}{r c l}{v_{\mu}A_{\nu}^{a}= i \mathrm{s g n}( k ){\epsilon}_{\nu \mu \rho}{\partial}^{\rho}c^{a}}&{,}&{v_{\mu}c^{a}= 0 ,}\\{v_{\mu}{\bar{c}^{a}}= A_{\mu}^{a}}&{,}&{v_{\mu}B^{a}= - D_{\mu}c^{a},}\\ \end{array}
x_{1,1}= \mathbf{e}_1(a+\frac{h^{(1)}}{2}) + \mathbf{e}_2(a+\frac{h^{(2)}}{2}), x_{i,j}= x_{1,1}+ \mathbf{e}_1(i-1)h^{(1)}+ \mathbf{e}_2(j-1)h^{(2)}, i=1,\dots,L,~j=1,\dots,M,
u_{\beta}^{(1)}\left( 0,x\right) +\Pi_{0,x}\int_{0}^{\tau}ds\psi_{\beta}^{(1)}\left( s,\xi_{s};u_{\beta}^{(1)}\right) =\Pi_{0,x}f_{\beta}^{(1)}\left( \tau ,\xi_{\tau}\right)
S= \sum_{\ell=-\lfloor 1/2\delta_K\rfloor}^{1/\delta_K-1-\lfloor 1/2\delta_K\rfloor}h_K G\big(h_K\ell \big)e^{h_K\ell\partial_x\varphi(t_K,x_K)+C\frac{(h_K\ell)^2}{\log K}}.
< T_{--}^{\mathrm{f}}( \sigma^{+}, \sigma^{-}) > |_{\sigma^{+}\rightarrow \infty}= \frac{\lambda^{2}}{4 8}\left[ 1 - \frac{1}{( 1 + \frac{m}{\lambda}e^{\lambda ( \sigma^{-}- \sigma_{0}^{+})})^{2}}\right]
\mathcal{N}\big( \mathcal{C}_{\alpha}(r,1) \big) = n \big( [r,1] \big) \big( 1 + \mathcal{O}\big( \delta \vert \ln \delta \vert^{2}\big) \big) & + \mathcal{O}\big( \vert \ln \delta \vert^{2}\big) n \big( [r(1-\delta),r(1+\delta)] \big) + \mathcal{O}_{\delta}(1),
E(\phi) = \int_{\mathbb{R}^d}\left( \frac12|\nabla\phi(\mathbf{x})|^2+V(\mathbf{x})|\phi(\mathbf{x})|^2+\frac{\beta}{2}|\phi(\mathbf{x})|^4 -\Omega\overline{\phi}(\mathbf{x})L_z\phi(\mathbf{x})\right) \mathrm{d}\mathbf{x}
\delta = \epsilon^{0}\underline{\epsilon}_{0}+ \epsilon^{+}\underline{\epsilon}_{+}+ \epsilon^{-}\underline{\epsilon}_{-}+ \alpha^{I}\underline{\alpha}_{I}+ \beta_{I}\underline{\beta}^{I}+ \hat{\alpha}{}^{I}\underline{{\hat{\alpha}}}_{I}+ \hat{\beta}_{I}\underline{{\hat{\beta}}}{}^{I}+ \omega^{i}\underline{\omega}_{i}\ .
\sum_{j=k}^{k+1}(-1)^{j-k}(j)_{k}(n-j)_{n-k-1}f_j(x) y^{k-j+1}= \k!(n-k)! f_k(x)y - (k+1)! (n-k-1)! f_{k+1}(x) = \ (k+1)!(n-k-1)! \left( \frac{n-k}{k+1}f_k(x)y - f_{k+1}(x) \right).
||w^{-\frac{1}{p}}\mathcal{A}(x,\nabla u_{n_k})||^\frac{p}{p-1}_{L^\frac{p}{p-1}(\Omega)}=\int_{\Omega}w^{-\frac{1}{p-1}}(x)\big|\mathcal{A}\big(x,\nabla u_{n_k}(x)\big)\big|^\frac{p}{p-1}\,dx &\leq\;\int_{\Omega}w^{-\frac{1}{p-1}}(x)w^\frac{p}{p-1}(x)|\nabla u_{n_k}(x)|^p\,dx &\leq\;||u_{n_k}||_{X}^p &\leq\; c^p
f_2 = -\frac{3 e_{-1}}{2},f_{-1}= -3 g_{-2}^2, f_{-4}=-\frac 34 (2 g_{-2}g_{-5}+ e_{-7}),f_{-7}= -\frac 37 (g_{-5}^2 + 2 e_{-10}), &b_1 = -6 g_{-2},b_{-2}= 3 g_{-5}.
g^{\mu\alpha}[\frac{\partial g_{\gamma\mu}}{\partial x^{\beta}}]=g^{x\alpha}\frac{\partial g_{\gamma x}}{\partial x^{\beta}}+g^{y\alpha}\frac{\partial g_{\gamma y}}{\partial x^{\beta}}+g^{z\alpha}\frac{\partial g_{\gamma z}}{\partial x^{\beta}}+g^{t\alpha}\frac{\partial g_{\gamma t}}{\partial x^{\beta}}
B^{(n)}(t) =B_0^{(n)}+\sum_{k=1}^{\lfloor t/\Delta t^{(n)}\rfloor}\delta B_k^{(n)}&=B_0^{(n)}+\sum_{k=1}^{\lfloor t/\Delta t^{(n)}\rfloor}\left[p^{(n)}\left(S^{(n)}_{k-1}\right)\Delta t^{(n)}+r^{(n)}\left(S^{(n)}_{k-1}\right)\delta Z^{(n)}_k\right]
s_{\alpha}\left( \rho_{1}\right) :=\frac{\rho_{1}^{2}\left( 1-\left(1-\rho_{1}\right)^{2\alpha}-2\alpha\rho_{1}\left( 1-\rho_{1}\right)^{2\alpha-1}\right)}{\left( 1-\left( 1-\rho_{1}\right)^{\alpha+1}-\left( \alpha+1\right) \rho_{1}\left( 1-\rho_{1}\right)^{\alpha}\right)^{2}}.
\left|B_{0,N}A_N-B_{0,n}A_n\right| \le C_{|A|}\sum_{j=n+1}^{N}\frac{c^{A}_J}{\lambda}\hat{b}_{j-1,j}\exp \left(\frac{c^{A}_J}{\lambda}\sum_{k=1}^N\hat{b}_{k-1,k}\right) \quad.
D_{\alpha,+}(z)D_{\alpha,-}(z) = 1, \mbox{for}z \in (-\infty,-x_{1}), & D_{\alpha,+}(z) = D_{\alpha,-}(z)e^{-\pi i \alpha}, \mbox{for}z \in (-x_{1},0), & D_{\vec{s},+}(z)D_{\vec{s},-}(z) = s_{j}, \mbox{for}z \in (-x_{j},-x_{j-1}), \, j = 2,...,m+1.
\langle L ( u , p_{3}, p_{4}) \rangle = \left[ \begin{array}{l l}{c_{3}c_{4}- b_{3}b_{4}u}&{- u ( b_{3}d_{4}- c_{3}a_{4})}\\{\vspace{- 2 m m}}\\{d_{3}b_{4}- a_{3}c_{4}}&{d_{3}d_{4}- a_{3}a_{4}u}\\ \end{array}\right] .
I_3= -4(n-6)(n-4)\Big[(n-6)\sigma_1(A)_{,ij}(p)x^ix^j r^{2-n}+4(n-2)r^{-n}(A_{ij,kl}(p)x^ix^jx^kx^l)\Big] &-4(n-6)(n-4)(n-10)r^{2-n}\sigma_1(A)_{,i}x^i+O(r^{5-n}) = -8(n-8)(n-6)(n-4)\sigma_1(A)_{,ij}(p)x^ix^j r^{2-n}-16(n-6)(n-4)(n-2)r^{-n}(A_{ij,kl}(p)x^ix^jx^kx^l)+O(r^{5-n}).
V / N_{f}= ( Q_{0}^{2}+ \bar{Q}_{0}^{2}) \bigl [ Q_{0}\bar{Q}_{0}\bigr ]^{- 2 \gamma}+ \frac{g^{2}}{4 \gamma}( Q_{0}^{2}- \bar{Q}_{0}^{2})^{2}- m_{Q}^{2}Q_{0}^{2}+ m_{\bar{Q}}^{2}\bar{Q}_{0}^{2}
p^{\gamma,\nu}_{k_1,\ldots,k_n}= \sum_{g\geq 0, b\geq 1}\eta^{g,b}_{k_1,\ldots, k_n}\gamma^g\nu^b, p^\nu_{k_1,\ldots,k_n}= \sum_{b\geq 1}\eta^{*,b}_{k_1,\ldots, k_n}\nu^b.
\phi_i(x) =(x_1,\dots,x_{i-1},1,x_{i+1},\dots,x_n), \phi_{i,j}(x,A) =\begin{cases}\phi_i(x) \mbox{if}x_i=j\mbox{and}\phi_i(x)\notin A x \mbox{otherwise,}\end{cases}\phi_{i,j}(A) =\{\phi_{i,j}(x,A)\; :\; x\in A\}.
m > 0 \psi ( x ) = \sum_{n = 0}^{\infty}\int_{- \infty}^{+ \infty}d p a_{n p}\psi_{n p}^{( + )}( x ) + \sum_{n = 1}^{\infty}\int_{- \infty}^{+ \infty}d p b_{n p}^{\dag}\psi_{n p}^{( - )}.
H = \int \! d^{2}\!{\bf r}\, \left\{\sum_{p = 1}^{M}\frac{\hbar^{2}}{2 m_{p}}\mathrm{\boldmath ~ \Pi ~}_{p}^{\dagger}({\bf r}, t ) \cdot \mathrm{\boldmath ~ \Pi ~}_{p}({\bf r}, t ) + U ( \Psi^{\dagger}, \Psi ) \right\},
\mathbf{x}^{k+1}:= (\mathbf{W}\otimes \mathbf{I}_p)\mathbf{x}^{k}- \xi \nabla f(\mathbf{x}^{k+1}) + \sum_{t=0}^{k}(\mathbf{W}- \tilde{\mathbf{W}})\otimes \mathbf{I}_p \mathbf{x}^t,
\partial_{-}h_{m+}= - \partial_{i}h_{im}-e^A \left(\frac{D_1-2}{2}A' h_{rm}+ h_{m\underline l ;}^{\quad\underline l}-\kappa^2\left(\phi^{\dagger}D_m\Phi + (D_m \Phi)^{\dagger}\phi + V_{\underline l}F_n^{\quad\underline l}\right)\right)~.
I(w_1,w_2)- \dfrac{1}{2}\sum\limits_{i=1}^{2}{||\nabla \tilde{w}_i||^2_2}=\dfrac{4k_{11}}{|K|}\eta_1\bigg(1-\ln(\eta_1)+\ln\bigg(\int_{\Omega}{e^{u_0^{'}+\sqrt{|K|}\tilde{w}_1}}dx\bigg)\bigg) &+\dfrac{4k_{11}}{|K|}\eta_2\bigg(1-\ln(\eta_2)+\ln\bigg(\int_{\Omega}{e^{u_0^{''}+(|K|\tilde{w}_2+k_{21}\sqrt{|K|}\tilde{w}_1)/k_{11}}}dx\bigg)\bigg).
c^k \cdot \|A\| = \sum_{i=0}^k c^{i-1}(c \cdot \|L_\eta(A,i)\| - \|L_\eta(A,i-1)\|) \leq \sum_{i=0}^k (c \cdot \|L_\eta(A,i)\| - \|L_\eta(A,i-1)\|).
{\mathbb E}{\rm e}^{-\alpha \bar S_{N_{\beta,\omega}}- \gamma G_{N_{\beta,\omega}}}= \frac{\beta}{\psi(\beta)}\frac{\psi(\beta+\gamma)-\alpha}{\beta+\gamma-\varphi(\alpha)}\frac{\psi(\beta+\omega)}{\beta+\omega}\frac{\beta+\omega+\gamma-\varphi(\alpha)}{\psi(\beta+\omega+\gamma)-\alpha}.
\prod_{\alpha = 1}^{3}\left[{\frac{P^{\alpha}! Q^{\alpha}! R^{\alpha}! S^{\alpha}! U^{\alpha}! V^{\alpha}! ( U^{\alpha}! + 2 I^{\alpha}! 1 ) ( 2 I^{\alpha}! + 1 )}{( V^{\alpha}! + 2 I^{\alpha}! + 1 )}}\right]^{1 / 2}.
\hat \delta^{f}_r(c_i,c_j)= \mathbb F^f_r((c_{i-1}, c_i]\times [c_j, c_{j+1}))= \mathbb F^f_r (c_{i}, c_{j})/ \mathbb F^f_r(c_{i-1}, c_{j}) + \mathbb F^f_r (c_{i}, c_{j+1})
\sup_{||v||\leq 1}\left|\Big(I_\infty'(u_n)-{I}'({u}_n)\Big)v\right| = \sup_{||v||\leq 1}\left|\int_{\mathbb{R}^N}\Big(V_{0,\infty}- V_0(x)\Big){u}_n(x)v(x)\;dx\right| \to 0.
\psi_{r,k_{r}}^{\epsilon}=\begin{cases}s_{k_{r}-1}s_{k_{r}-2}...s_{r}\mbox{if}\epsilon=1,\cr s_{k_{r}}s_{k_{r}+1}\cdots...s_{n-1}s_{n}s_{n-1}\cdots s_{k_{r}}s_{k_{r}-1}\cdots s_{r}\mbox{if}\epsilon=2,\cr \mbox{empty string}\mbox{if}\epsilon=0,\cr \end{cases}
B_0(N) =-\frac{\alpha_{2}}{4}\left((N+u)^{2}-\frac{1}{4}\right)+\left(\frac{-\beta \alpha_{1}+ \alpha_{2}\delta}{2\beta^{2}}\right) + \epsilon \quad{}-\left(\frac{-2\beta \alpha_{1}\delta +\alpha_{2}\delta^{2}}{4\beta^{4}}\right) \frac{1}{\left((N+u)^{2}-\frac{1}{4}\right)},
A_{\mu}^{( 2 )}= \mp{\frac{2 i}{g}}( H_{3}{\frac{v_{\mu}^{1}}{\rho_{1}}}+ H_{1}{\frac{v_{\mu}^{2}}{\rho_{2}}}) + \kappa ( E_{3 2}^{\pm}R_{\mu}^{1}+ E_{1 2}^{\pm}R_{\mu}^{2}) \delta ( R^{1}\cdot R^{2}) .
\frac{1}{(x+u)\left((x+v)^2+w^2\right)}= \frac{1}{(x+u)(x+v-w~i)(x+v+w~i)}= \frac{1}{2w~i}\left(\frac{1}{(x+u)(x+v-w~i)}-\frac{1}{(x+u)(x+v+w~i)}\right).
(I+\mathcal{H})\partial_{\alpha}^k\tilde{\theta}= (I+\mathcal{H})\partial_{\alpha}^k(I-\mathcal{H})(\zeta-\bar{\zeta})=-[\partial_{\alpha}^k,\mathcal{H}]\tilde{\theta}= -\sum_{m=0}^{k-1}\partial_{\alpha}^m [\zeta_{\alpha}-1,\mathcal{H}]\frac{\partial_{\alpha}\partial_{\alpha}^{k-m-1}\tilde{\theta}}{\zeta_{\alpha}}.
E ( L ) = E_{\mathrm{b u l k}}L + E_{\mathrm{b o u n d a r y}}- \frac{\pi c ( m L )}{2 4 L}\; , \quad c ( l ) = \frac{6 l}{\pi^{2}}\int_{- \infty}^{\infty}d \theta \: L ( \theta ) \cosh \theta \ ,
\mathrm{U}:=\left\{\left( x_{1},\thinspace \mathellipsis ,x_{d}\right)\in R_{+}^{d}:\thinspace \thinspace h_{i}\le x_{i}\le \frac{b_{i}}{c_{ii}},\thinspace \forall i=1,\thinspace 2,\thinspace \mathellipsis ,\thinspace d \right\}.
[x,x] = \frac{\left[ (A - \gamma)^{2m}\,x - \ell\,,x \right]}{(\lambda - \gamma)^{2m}}= \frac{\left[(A - \gamma)^{2m}\,x\,,x \right]}{(\lambda - \gamma)^{2m}}= \frac{[y,y]}{(\lambda - \gamma)^{2m}}> 0,
\det\nabla (\phi_m\circ\cdots\circ \phi_i)= \det \nabla \phi_m (\phi_{m-1}\circ\ldots \phi_i) \cdot\ldots \cdot \det \nabla \phi_{i+1}(\phi_i) \cdot \det \nabla \phi_i \geq \prod_{k=i}^m (1-C\tau_0|M_k|^{1/2})\geq 1-C\tau_0\sum_{k=i}^{m}|M_k|^{1/2}\geq 1-C\tau_0\frac{\|2f\|_{L^p}^{p/2}}{1-(1+\tau_0)^{-(p-1)/2}}.
P(A_{0}) = 1-\lambda(t)\Delta t + o(\Delta t) = e^{-(m(t+\Delta t) - m(t))}P(A_{1}) = \lambda(t)\Delta t + o(\Delta t) = (m(t+\Delta t) - m(t))e^{-(m(t+\Delta t) - m(t))}\vdots P(A_{k}) = \frac{(m(t+\Delta t) - m(t))^{k}}{k!}e^{-(m(t+\Delta t) - m(t))}
F(s) = \frac{1}{48}\int_0^\infty\! \! \! d\mu \, \tilde{c}(\mu) \mu \sqrt{s}\left( ({\mu}^{3}{s}^{3/2}\!+\!24\mu \sqrt{s}) K_0(\mu \sqrt{s})\!+\!(8 \mu^2 s\!+\!48) K_1(\mu \sqrt{s})\right )
A^{1 1}= \frac{b^{\prime}r - b}{2 r^{3}}+ \frac{1}{r \left( e^{2 \psi}\alpha^{2}- \beta^{2}r^{2}\right)}\left[ e^{2 \psi}\psi^{\prime}\left( 1 - \frac{b}{r}\right) \alpha^{2}- \beta^{2}b \right]
\sigma_i(u(x))f(\xi,\xi'_1,\xi'_2) =\int_F \left(\int_{F}f(\xi-Jx\xi_1',\xi_1',v)\bar{\psi}((x^t\xi-Q_i(x)\xi'_1+u)v)dv\right) \psi(\xi_2'u)du &=\int_F \left(\int_{F}f(\xi-Jx\xi_1',\xi_1',v)\bar{\psi}(uv)dv\right) \psi(\xi_2'(-x^t\xi+Q_i(x)\xi'_1+u))du &=f(\xi-Jx\xi_1',\xi_1',\xi'_2) \psi(\xi_2'(-x^t\xi+Q_i(x)\xi'_1)).
\phi(r)=\frac{1}{4\pi\epsilon_{0}}\int\frac{\rho(r_{0})}{|r-r_{0}|}d^{3}r_{0}+\frac{1}{4\pi\epsilon_{0}}\int\frac{p(r_{0})\cdot(r-r_{0})}{|r-r_{0}|^{3}}d^{3}r_{0},
\sigma(\lambda \circ b^{*}) - \sum_{s=0}^{N-1}\sum_{(h+1)|\alpha| + i + \ell = s}\frac{1}{\alpha!}\partial_{\xi'}^{\alpha}\lambda_{-\ell/(h+1)}(x', \xi') D_{x'}^{\alpha}\overline{b_{-i/(h+1)}}(x',\xi', x_1) \in H_{h}^{m' + m'' -\frac{N}{h+1}}.