harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
b_{jN}= M\int_0^M\phi_{N+j}(\eta)\,d\eta\int_0^M\phi_N(\eta')\,d\eta'- \int_0^M \eta\phi_{N+j}(\eta)\,d\eta\int_0^{\omega_1}\phi_N(\eta')\,d\eta'- \beta_{1N}- \beta_{NN},
f^{(p)}(x) = \sum_{k=0}^p \binom{p}{k}(p)_k (x - \lambda)^{p-k}g^{(p-k)}(x) = (x - \lambda) \sum_{k=0}^{p-1}\binom{p}{k}(p)_k (x - \lambda)^{p-1-k}g(x) + p! g(x)
\exp \left( - \frac{1}{2}\int_{\Lambda}d^{4}x \, B_{\Vert}^{2}\right) = \int{\cal D}\nu \, \exp \left( \int_{\Lambda}d^{4}x \, \left( - \frac{1}{2}\nu^{2}+ i \, \nu \, B_{\Vert}\right) \right) ,
\frac{d}{d\tau}\tilde{F}^\varepsilon (\mathbf{x},\tau)= \frac{1}{2}\frac{d}{d\tau}\left\langle \tilde{\boldsymbol{\nu}}^\varepsilon , \tilde{X}^\varepsilon\right\rangle (\mathbf{x},\tau)= \frac{-1}{2\sqrt{\varepsilon^2+|D{v^\varepsilon}(\mathbf{x})|^2}}\,.
(\tan \vartheta)^2 =\frac{m_{1}+n_{1}+6m_{2}\pm \sqrt{(m_{1}+n_{1}+6m_{2})^2- 4(n_{1}+m_{2})(m_{1}+m_{2})}}{2(n_{1}+ m_{2})}&=\frac{(c-1)+b +3 \pm \sqrt{((c-1)+b +3)^2- (2b +1)(2(c-1) +1)}}{2b + 1}.
\widehat{\lambda}_{n}(\lambda) = \lambda^{n/(n+1)}\widehat{\lambda}_{n,n+1}(\lambda)^{1/(n+1)}, \widehat{\lambda}_{n+1}(\lambda) = \lambda^{(n+1)/(n+2)}\widehat{\lambda}_{n+1,n+2}(\lambda)^{1/(n+2)}.
\bigg| \frac{\pi_a}{\pi_1}\bigg|^2 = 1 + \frac{(M^2/\Lambda^2)(\big| \pi_1\big|^2-\big| \pi_a\big|^2)}{G_SW_{{\rm eff};\lambda}^\ast \left( 1-\alpha \right)+\beta X_B +(M^2/\Lambda^2)\big| \pi_a\big|^2}.
J^{*}[\psi^{(4)}]_{T,t}={\int\limits_t^{*}}^T\psi_4(t_4){\int\limits_t^{*}}^{t_4}\psi_3(t_3){\int\limits_t^{*}}^{t_3}\psi_2(t_2){\int\limits_t^{*}}^{t_2}\psi_1(t_1)d{\bf w}_{t_1}^{(i_1)}d{\bf w}_{t_2}^{(i_2)}d{\bf w}_{t_3}^{(i_3)}d{\bf w}_{t_4}^{(i_4)}
\Sigma_{1,M,B}= \{M + \frac{B-R}{2}\le t \le M, \, |x| + t = 2 M + B\}, \Sigma_{2,M,B}= \{t=M, \, M + B <|x| < R + M\}, \Sigma_{3,M,B}= \{M + \frac{B-R}{2}\le t \le M, \, |x| - t = R\}.
u(x,t)=-2\kappa P_{12}(x,t)+2i\lim_{k\rightarrow\infty}k\breve{M}^{r}_{12}(x,t,k), x>0, \t<0, u(x,t)=-2\kappa P_{21}(-x,-t)+2i\lim_{k\rightarrow\infty}k\breve{M}^{r}_{21}(-x,-t,k), x<0, \t>0,
L\big( xww_{1}^{-1}\big) \tau(w_{1}) =L\big(xw_{x}^{-1}w_{0}^{m}\big) \tau\big( w_{0}^{-m}w_{x}w\big)=\upsilon^{-m}L\big( xw_{x}^{-1}\big) \upsilon^{m}\tau\big( w_{0}^{-m}w_{x}w\big) \hphantom{L\big( xww_{1}^{-1}\big) \tau(w_{1})}{}=\upsilon^{-m}L\big( xw_{x}^{-1}\big) \tau(w_{x}w) =\upsilon^{-m}L(x) \tau(w)
\partial_{\vec{z}}^{2}H_{F}+ H_{3}\partial_{\vec{x}}^{2}H_{F}= 0 , \ \partial_{\vec{z}}^{2}H_{1}+ H_{3}\partial_{\vec{x}}^{2}H_{1}= 0 , \ \partial_{\vec{z}}^{2}H_{3}= 0 .
{\frac{\partial S_{0}}{\partial \Omega_{p + 1}}}= -{\frac{B}{2 \Omega_{p + 1}( p + 1 ) !}}\left( \sigma_{\mu_{1}\dots \mu_{p + 1}}- \sigma_{0 ) \mu_{1}\dots \mu_{p + 1}}\right)^{2}- \frac{m_{p + 1}}{2}p \Omega_{p + 1}\,
\zeta_{Q}\left(s\right):=\zeta_{2}(s,c)=\frac{\pi}{\sqrt{c}}\,\frac{1}{s-1}+\frac{\pi}{\sqrt{c}}\,\left(2\gamma-\log\left(4c\right)-4\log\left(\left|\eta\left(i\sqrt{c}\right)\right|\right)\right)+O(s-1).
I_2= f v^3-(1+c^{11}\alpha_0^2) u_1^2\log|Dw|^2+\big[-1-c^{11}\alpha_0^2+\alpha_0\sum_{2\leq i\leq n}(\gamma^i)_i\big] u_1^2 &-2\alpha_0\gamma^1 u_1\log|Dw|^2-2\alpha_0\gamma^1u_1-\alpha_0^2+\alpha_0\sum_{1\leq i\leq n}(\gamma^i)_i \geq f v^3-(1+c^{11}\alpha_0^2) u_1^2\log|Dw|^2-C_1u_1^2.
\theta^0=-\varepsilon^{-1},\theta^t=\tilde{\theta}^0+\varepsilon^{-1},t_2=-\varepsilon\tilde{t}_2, q_2=\varepsilon\tilde{t}_2\tilde{p}_2,p_2=-\frac{\tilde{q}_2}{\varepsilon\tilde{t}_2},H_{t_2}=-\varepsilon^{-1}\left( \tilde{H}_{2}-\frac{\tilde{p}_2\tilde{q}_2}{\tilde{t}_2}\right).
\begin{array}{lccl}(0~0~0~0~0~0~0~-1) \mid \mbox{gnd}> (0~0~0~0~0~0~0~-1) = ({\bf 1},~{\bf 1}) \Downarrow \beta_8^- \mid \mbox{1st}> (0~0~0~0~0~0~1~-1) = ({\bf \overline 8},~{\bf 1}). \end{array}
(\widetilde{\phi_{n}}*\widetilde{\psi_{n}}) (X,s) = \int \widetilde{\phi_{n}}(X,t)\alpha_{(X,t)}\big(\widetilde{\psi_{n}}(X.t,t^{-1}s\big)1_{X}(t^{-1})dt &= \int_{t \in Int(P)}(y^{*}+I_{X})(\alpha_{t}(z_{n})+I_{X})f_{n}(t)g(t^{-1}s) \big[ \textrm{For $supp(f_n) \subset Int(P) \subset X^{-1}$}\big] & = \int (y^{*}+I_{X})\widetilde{F_{z_n,f_n}}(X,t)g(t^{-1}s)dt
d\rho_t = \underbrace{\left[-i\left[H_t,\rho_t\right] + \mathbf{L}^{\top}\rho_t\mathbf{L}^{\ast}- \frac{1}{2}\mathbf{L}^{\dagger}\mathbf{L}\rho_t - \frac{1}{2}\rho_t\mathbf{L}^{\dagger}\mathbf{L}\right]dt}_{}+ \underbrace{\zeta_{\rho}^{\top}\Sigma^{-\top}d\mathbf{W}}_{},
J_{N}(x,y)=p^{N}{\textstyle\sum\limits_{J\in G_{N}^{0}}}{\textstyle\sum\limits_{K\in G_{N}^{0}}}A_{JK}\Omega\left( p^{N}\left\vert x-J\right\vert_{p}\right) \Omega\left(p^{N}\left\vert y-K\right\vert_{p}\right)
J_1=g(\epsilon)^{n-1}\epsilon^{-1/2}\int_{\mathbb{R}^n}{\frac{u(\bar{y},y_n)-u(\bar{y},0)}{(g(\epsilon)^2|\bar{y}|^2+\frac{1}{\epsilon}y_n^2)^{\frac{n+2s}{2}}}dy}\leq \epsilon^s C_1C_2,
\widehat L(z) =\begin{vmatrix}1+(a_q-1)\, r_{q,q+1}(z) (a_q-1)\, r_{qq}(z) (a_q-1)\, r_{q+1,q+1}(z) 1+(a_q-1)\, r_{q+1,q}(z)\end{vmatrix}&=1+(1-a_q^2) z^2\,\frac{z^{2q}-1}{z^2-1}\,.
w_{=n}(\zeta)= w_{n}(\zeta), \widehat{w}_{=n}(\zeta)= \widehat{w}_{n}(\zeta), w_{=n}^{\ast}(\zeta)=w_{n}^{\ast}(\zeta),\widehat{w}_{=n}^{\ast}(\zeta)= \widehat{w}_{n}^{\ast}(\zeta)
S(0,y,t)= S(1,y,t)= S(x,0,t)= S(x,1,t)= 0 I(0,y,t)= I(1,y,t) = I(x,0,t)= I(x,1,t)= 0 V(0,y,t)= V(1,y,t)= V(x,0,t)= V(x,1,t)= 0 .
3\sum_{\substack{\substack{(a,b,c)\equiv(1,1,0)\bmod{2}a+b+c=w+2,\,b>1}}}(p_{a,1,b}+\delta_{a,1}\delta_{b,3})\cdot I_{{\rm bl}}^{\mathfrak{m}}(a,b,c)-\sum_{\substack{\substack{(a,b,c)\equiv(0,0,0)\bmod{2}a+b+c=w+2}}}p_{a,b,c}\cdot I_{{\rm bl}}^{\mathfrak{m}}(a,b,c) \overset{?}{=}\frac{w^{2}(w-2)(5w-11)}{24}I_{{\rm bl}}^{\mathfrak{m}}(w+2).
\|\mathcal{P}(\mathbf{x}_k) -\mathbf{x}_k \| \leq \|\mathcal{P}(\mathbf{x}_k) -\mathcal{P}(\mathbf{x}_{k-1}) \|+ \|\mathcal{P}(\mathbf{x}_{k-1}) -\mathbf{x}_{k}\| & \leq \eta \|\mathbf{x}_k -\mathbf{x}_{k-1}\|+ \|\mathcal{P}(\mathbf{x}_{k-1}) -\mathbf{x}_{k}\|,
\begin{aligned}u(t) = F(t;q)v + A(q) \int_0^t E(s;q)D(q)b \d s + \int_0^t E(s;q)f \d s &= F(t;q)v + (I-F(t;q))D(q)b +(I-F(t;q)) A(q)^{-1}f,\end{aligned}
\begin{array}{r c l c r c l c r c l c r c l}{f_{1}}&{=}&{- 6 i}&{;}&{f_{3}}&{=}&{- 6 i}&{;}&{f_{2}}&{=}&{- \frac{4}{3}a_{5}}&{;}&{f_{4}}&{=}&{- 8 i a_{5}}\\ \end{array}
\gamma^2_{{d_1^h}^*E}\circ ({d^v_2}^*\gamma^1{d^v_0}^*\gamma^1) = (1\times c \times 1) \circ \gamma^1_{{d_1^v}^*E}\circ ({d^h_2}^*\gamma^2{d^h_0}^*\gamma^2)
\sigma_z^2= 2|d_{\mathrm{min}}|^2\left( \alpha_0 b_0^{\mathrm{opt}}|y_0[k-1]|^2\right.\left.+ \sum \limits_{i=1}^{R}\alpha_i b_i^{\mathrm{opt}}|y_i[k-1]|^2 \right).
A_{\mu \nu \eta \sigma \rho}^{( 2 )}( x ) = \frac{e^{2}}{2}( A_{\mu}A_{\eta}\partial_{\sigma}F_{\nu \rho}^{0}- \partial_{\nu}A_{\rho}\partial_{\eta}A_{\mu}A_{\sigma}+ A_{\mu}F_{\nu \eta}^{0}F_{\sigma \rho}^{0}) ,
S_{{\mathrm{\scriptsize{g a u g e d}}}}= \frac 1 2 \int_{M}d^{2}x \left( G_{\mu \nu}D^{a}X^{\mu}D_{a}X^{\nu}+ i Y_{\alpha}\epsilon^{a b}( \partial_{a}V_{b}^{\alpha}- \partial_{b}V_{a}^{\alpha}) \right) ,
\frac{d}{dt}T_{j,t}h(x)=\frac{d}{dt}[\nu_{j,t}\ast h(x)] =\frac{d}{dt}\bigg[\int_{2^jt<|y|\le 2^{j+1}}\frac{\Omega(y')}{|y|^n}h(x-y)dy\bigg] &=\frac{d}{dt}\bigg[\int_{\mathbf S^{n-1}}\Omega(y')\int_{2^jt}^{2^{j+1}}\frac{1}{r}h(x-ry')drd\sigma(y')\bigg] &=-\frac{1}{t}\int_{\mathbf S^{n-1}}\Omega(y')h(x-2^jty')d\sigma(y').
{\frac{2 p^{-}}{\mu}}= \sum_{n}\left[ e^{U_{0}}{\frac{s}{J_{V}}}N_{n}^{r , w}| n | + \sqrt{1 + e^{2 U_{0}}{\frac{s^{2}}{J_{V}^{2}}}n^{2}}N_{n}^{y}\right] \ .
E_{M}^{( 1 1 ) \, A}= \left( \begin{array}{l l}{e^{-{\frac{\Phi}{3}}}e_{\breve{\mu}}^{( 1 0 ) \, \breve{\alpha}}}{e^{{\frac{2}{3}}\Phi}B_{\breve{\mu}}}{0}{e^{{\frac{2}{3}}\Phi}}\end{array}\right) ,
P_{\alpha}= \frac{( \lfloor n^{4d}\rfloor )(\lfloor n^{4d}\rfloor - 1) \dots (\lfloor n^{4d}\rfloor - k + 1)}{n^4 (n^4 - 1) \dots (n^4 - k +1)}< \frac{n^{k 4 d}}{n^{4k}}
B =\{x\in X\colon \|x\|_X + \|Rx\|_Y\leq 1\}, A =\{x^{**}\in X^{**}\colon \|x^{**}\|_{X^{**}}+ \|R^{**}x^{**}\|_{Y^{**}}\leq 1\}, A_0 =\{x^{**}\in X^{**}\colon \|x^{**}\|_{X^{**}}+ \|R^{**}x^{**}\|_{Y^{**}}< 1\}.
\sum_{k: M_k\in B_{j+1}^{j+1}}\lambda_{k}|A-M_{k}|\lesssim \sum_{k: M_k\in B_{j+1}^{j+1}}\lambda_{k}j=\nu(B_{j+1}^{j+1})j\lesssim j^{2m_1'+3m_2'-3n'+1}i^{m_1'}\leq j^{-2}.
\lim_{N \to \infty}\big(2^{\frac{4}{3}}N^{\frac{1}{3}}\big)^k R_{N}^{(k)}\Big(4N+2^{\frac{4}{3}}N^{\frac{1}{3}}u_1, \ldots, 4N+2^{\frac{4}{3}}N^{\frac{1}{3}}u_k\Big)= \mathrm{Pf}\big[{K^{(\mathrm{soft})}(\kappa, \pi; u_i, u_j)}\big]_{i,j = 1}^{k},
S_{\Lambda}^k:= \{i \in \mathbf{m}| i \neq k_1 + \ldots + k_j \mbox{for all}j \}, \, \, \, \, \, \,S_{\Lambda}^m:= \{i \in \mathbf{m}| i \neq m_1 + \ldots + m_j \mbox{for all}j \}.
\Pr_{c\sim \mathcal{D}(C)}\Big[|c| = \alpha N\Big] =q^{-\eta N}\cdot \binom{\eta N}{(1-\alpha) \eta N}(q-1)^{\alpha\eta N}&\geq q^{-\eta N}\cdot \sqrt{\frac{8\pi}{e^4\eta N}}\cdot 2^{h(\alpha)\eta N}\cdot q^{\alpha \eta N \log_q (q-1)}&= \sqrt{\frac{8\pi}{e^4\eta N}}\cdot q^{-(1-h_q(\alpha))\eta N}
u=(u^{(1)}, \ldots, u^{(i)}, \ldots, u^{(\mu_1)},\ldots, u^{(\mu_{p''})}, \ldots, u^{(\nu_1)},\ldots, u^{(\nu_{q''})}, \ldots, u^{(Q_0)})\in \mathbb{U}\cong U^{\oplus Q_0},
\begin{aligned}\phi_p(x, -y) = \dfrac{x^p + y^p}{x+y}= \sum_{i=0}^{p-1}x^{p-i-1}(-y)^i = A_p(x, -y) + (-1)^k(xy)^k = D_p(x, -y) + (xy)^k, \end{aligned}
+\sin\psi\cot\eta\cdot\left(\frac{F[p,q](\frac{i}{\mu_{0}})w_{1}[p,q](\frac{i}{\mu_{0}})w_{3}[p,q](\frac{i}{\mu_{0}})}{w_{2}[p,q](\frac{i}{\mu_{0}})}\right)^{-\frac{1}{2}}\gamma^{0}\gamma^{2}\gamma^{0}\partial_{\psi}\left(-\gamma^{0}u_{n}(\frac{1}{\mu})\right)\mid_{\mu=\mu_{0}}
\lim_{n \to \infty}- \frac{s_{j,n}}{\mu_{j,n}}= + \infty, \lim_{n \to +\infty}\frac{s_n/2 - s_{j,n}}{\mu_{j,n}}= +\infty; & \lim_{n \to \infty}- \frac{s_{j,n}}{\mu_{j,n}}= - \infty, \lim_{n \to +\infty}\frac{s_n/2 - s_{j,n}}{\mu_{j,n}}= -\infty.
g_{\mu \nu}\, d x^{\mu}\, d x^{\nu}\simeq - d t^{2}+ \sum_{i = 1}^{d}\, t^{2 p_{i}( x )}\, ( \omega^{i})^{2}\, , \; \; \varphi \simeq p_{\varphi}\, ( x ) \, \ln \, t \, + \psi \, ( x ) \, ,
\mathbb{E}[(\mathbf{W}^T_1(\Gamma),\mathbf{W}^T_2(\Gamma))^T(\mathbf{W}^T_1(\Gamma),\mathbf{W}^T_2(\Gamma))] &=\frac{1}{n}\mathbb{E}[(\mathbf{W}^T_1,\mathbf{W}^T_2)^T(\mathbf{W}^T_1,\mathbf{W}^T_2)].
\Psi_{n_{12}}^{\mu_{12},\epsilon_{12}}(x,y) = \langle x,y|n_{12},\mu_{12},\epsilon_{12}\rangle, H_{xy}\Psi_{n_{12}}^{\mu_{12},\epsilon_{12}}(x,y) = (n_{12}+ \mu_{12}+ 1/2)\Psi_{n_{12}}^{\mu_{12},\epsilon_{12}}(x,y).
\Delta^{\xi,\delta}:=\frac14\,\left(B_{1,1}^{\xi,\delta}-B_{2,2}^{\xi,\delta}+\delta^2\,\left(\alpha_1^{\xi,\delta}B_{1,3}^{\xi,\delta}-\alpha_2^{\xi,\delta}B_{2,3}^{\xi,\delta}\right)\right)^2+\left(B_{1,2}^{\xi,\delta}+\delta^2\,\alpha_2^{\xi,\delta}B_{1,3}^{\xi,\delta}\right)\left(B_{2,1}^{\xi,\delta}+\delta^2\,\alpha_1^{\xi,\delta}B_{2,3}^{\xi,\delta}\right)\,.
A_0 \to A_1, A_2, A_4; A_1 \to A_2, A_3, A_5; A_2 \to A_3, A_4, A_6; A_3 \to A_0, A_4, A_5; A_4 \to A_1, A_5, A_6; A_5 \to A_0, A_2, A_6; A_6 \to A_0, A_1, A_3.
J_h(w_h) = J_h(u_h) + \frac{d h^{d-2}\epsilon^2}{8}- \frac{h^d \epsilon}{2}\Delta_h u_h(x_0) = J_h(u_h) + \frac{h^{d-2}\epsilon}{2}( \frac{d \epsilon}{4}- h^2 \Delta_h u_h(x_0)) & < J_h(u_h),
X_{z_{\pm}}= - i Q_{\pm}^{- 1}\displaystyle \displaystyle \frac{\partial}{\partial \overline{{{z}}}_{\pm}}~ , X_{p_{\pm}}= i \displaystyle \displaystyle \frac{\partial}{\partial z_{\pm}}- i Q_{\pm}^{- 1}\displaystyle \frac{\partial p_{\pm}}{\partial z_{\pm}}\displaystyle \displaystyle \frac{\partial}{\partial \overline{{{z}}}_{\pm}}~ ,
F=F_{1}+\frac{F_{2}-1}{G_{1}}+\frac{F_{3}-1}{G_{1}G_{2}}+\frac{F_{4}-1}{G_{1}G_{2}G_{3}}+\cdot\cdot\cdot+\frac{F_{n}-1}{G_{1}G_{2}G_{3}\cdot\cdot\cdot G_{n-1}},
Z = \int{\cal D}A_{-}{\cal D}\pi^{-}{\cal D}\varphi{\cal D}\pi_{\varphi}{\cal D}\theta^{1}{\cal D}\pi^{1}{\cal D}\theta^{2}{\cal D}\pi^{2}\prod_{i , j = 1}^{2}\delta ( \widetilde{\top}_{i}) \delta ( \Gamma_{j}) \mathrm{d e t}\mid \{\widetilde{\top}_{i}, \Gamma_{j}\}\mid e^{i S},
\prod_{\substack{p\leq z, p\equiv 7, 13\pmod{24}}}\left(1-\frac{1}{p}\right)^{-1}= \exp\sum_{\substack{p\leq z, p\equiv 7, 13\pmod{24}}}\left(\frac{1}{p}+\frac{1}{2p^2}+\cdots \right) > \exp 0.015815+\sum_{\substack{p\leq z, p\equiv 7, 13\pmod{24}}}\frac{1}{p}> 0.95442\log^\frac{1}{4}z,
\psi_{i,0}e_{j,l}=q^{a_{i,j}}e_{j,l}\psi_{i,0},\ [h_{i, k}, e_{j,l}]=d^{-km_{i,j}}\gamma^{-|k|/2}\frac{[ka_{i,j}]_q}{k}e_{j,l+k}\ (k\ne 0),
(C\Gamma^{\mu_6})_{\alpha{'}\beta{'}}(C\Gamma_{\mu_1\dots\mu_6})_{\delta{'}\epsilon{'}}=(C\Gamma^{\mu_6})_{\alpha{'}\beta{'}}(C\Gamma_{\mu_1\dots\mu_5}\Gamma_{11})_{\delta{'}\lambda}{(\Gamma_{11}\Gamma_{\mu_6})^\lambda}_{\epsilon{'}}
\frac{\partial^{2}z}{\partial t^{2}}=\frac{\partial}{\partial t}\frac{\partial z}{\partial t}=\frac{\partial}{\partial t}[(\frac{\partial}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial}{\partial y}\frac{\partial y}{\partial t})z]=(\frac{\partial}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial}{\partial y}\frac{\partial y}{\partial t})\frac{\partial z}{\partial t}
a_{n}( f , D ) \simeq ( 4 \pi )^{- ( m - 1 ) / 2}\int_{\partial{\cal M}}d z \, \left( c_{0}S^{n - 1}+ c_{1}f_{; m}S^{n - 2}+ c_{2}L_{a a}S^{n - 2}+ \dots \right) \, ,
\begin{array}{cc}\Lambda(g)^{-1}\tilde{f}^{|u\rangle}(\Lambda(g)p) =\tilde{f}^{|u'\rangle}(p) = \frac{u'}{u' \cdot p}, \,\,\,u' = \Lambda(g)^{-1}u, g \in SL(2, \mathbb{C}),\end{array}
b =k_1-(k_2-\ldots (k_{2m-1}-k_{2m}).\,\!.\,\!.)=\bigvee_{l=1}^m (k_{2l-1}-k_{2l}) =c_{1,j}-(c_{2,j}-\ldots (c_{2m-1,j}-c_{2m}).\,\!.\,\!.)=\bigvee_{l=1}^m (c_{2l-1,j}-c_{2l,j}).
\frac{1}{8 \pi}\int^{r}d^{3}x ( K - K_{0}) = - \frac{\pi}{\kappa}\Biggl [ r - M + \frac{r ( r - r_{+}) ( r - r_{-})}{R^{2}}- 2 [ ( r - r_{+}) ( r - r_{-}) ]^{\frac{1}{2}}\Biggr ] \ .
v^{(k)}:= \mathcal{Q}_- \omega^{(k)}\mathrm{and}\tilde u^{(k)}:= (\Phi_-)^{\lambda_0}\sum_{\ell =1}^k \frac{(\log \Phi_-)^{k-\ell}}{(k-\ell)!}\tilde v^{(\ell)}.
\mathcal{E}(g) =-\lim_{r\rightarrow\infty}\int_{S^{n-1}_1}g(\nabla R_g,\nu)r^{n-1}d\omega_{n-1}, &=-\lim_{r\rightarrow\infty}\int_{S^{n-1}_1}\nabla R_g\cdot \nu^e r^{n-1}d\omega_{n-1}, &=-\lim_{r\rightarrow\infty}\int_{S^{n-1}_1}\partial_r R_g r^{n-1}d\omega_{n-1}.
\tilde{S}_{r s}= \oint \frac{d \xi}{2 \pi \imath}\frac{d \xi^{\prime}}{2 \pi \imath}\xi^{- r - \frac{1}{2}}\xi^{\prime}{}^{- s - \frac{1}{2}}\frac{2 \imath}{\sqrt{1 + \xi^{2}}\, \sqrt{1 + \xi^{\prime}{}^{2}}}\ln \left( \frac{( 1 + i \xi ) ( 1 - i \xi^{\prime})}{( 1 - i \xi ) ( 1 + i \xi^{\prime})}\right) .
{\hat{\Pi}}= D{\hat{X}}^{\hat{\mu}}D{\hat{X}}^{\hat{\nu}}{\hat{g}}_{{\hat{\mu}}{\hat{\nu}}}= \partial{\hat{X}}^{\hat{\mu}}\partial{\hat{X}}^{\hat{\nu}}\left({\hat{g}}_{{\hat{\mu}}{\hat{\nu}}}+ |{\hat{k}}|^{- 2}{\hat{k}}_{\hat{\mu}}{\hat{k}}_{\hat{\nu}}\right) ,
\begin{aligned}R(M_s,M_p) = \frac{M_s}{M}\frac{\displaystyle \sum_{n=1}^{\Lambda-t_s}\mathcal{L}_n \binom{\Lambda-n}{t_s}}{\binom{\Lambda}{t_s}}+ \frac{M_p}{M}\frac{K-t_p}{t_p+1}.\end{aligned}
\| \lambda_{k+1}\|_2^2 \le \| \lambda_1 \|_2^2 + \alpha^2 \sigma_g^2 k + \alpha^2 \sum_{i=1}^k \| \delta_i - \delta \|_2^2 & + 2 \alpha^2 \sum_{i=1}^k (\delta_i - \delta)^T (g(x_i) + \delta) & + 2\alpha \sum_{i=1}^k \lambda_i^T (g(x_i) + \delta_i)
J=\left(\begin{matrix}0 -b_1 \cdots 0 0 0 b_1 0 \cdots 0 0 0 \vdots \vdots \ddots \vdots \vdots \vdots 0 0 \cdots 0 -b_s 0 0 0 \cdots b_s 0 0 0 0 \cdots 0 0 \tilde J\end{matrix}\right)
{\ddot{a}\over a}=-{1\over L^2}\left[p(\bar{E}^2-\bar{P}^2_{\phi})\left({L\over a}\right)^{2(p+1)}-(p-1)\bar{E}^2\left({L\over a}\right)^{2p}+(p-1)\bar{P}_{\phi}\left({L\over a}\right)^{p+1}\right],
X=X\cap X=(L_{1}\cup L_{2})\cap(R_{2}\cup R_{2})=(\bigcup_{i\in I}L_{i})\cap(\bigcup_{i\in I}R_{i})\ne\bigcup_{i\in I}(L_{i}\cap R_{i})=(L_{1}\cap R_{1})\cup(L_{2}\cap R_{2})=\emptyset\cup\emptyset=\emptyset.
\begin{aligned}{\bf{a}}_{{\rm{R}},n,{l_n}}^H{{\bf{a}}_{{\rm{R}},m,{l_m}}}= \left|{{\bf{a}}_{{\rm{R}},n,{l_n}}^H{{\bf{a}}_{{\rm{R}},m,{l_m}}}}\right| \times{e^{- j\frac{{{N_{\rm{R}}}- 1}}{2}\Phi_{n,l_n}^{\rm{A}}}}{e^{j\frac{{{N_{\rm{R}}}- 1}}{2}\Phi_{m,l_m}^{\rm{A}}}}, \end{aligned}
\alpha_1 = \min_{\prec_\chi}(W_0), \alpha_2 = \max_{\prec_\chi}(\{k \in W_0 \, \mid \, k \preceq_{\chi}\theta \}), \beta_1 = \max_{\prec_\chi}(W_0), \mbox{and}\beta_2 = \min_{\prec_\chi}(\{k \in W_0 \, \mid \, \gamma \preceq_\chi k \}).
\bar{R}_{ij}(q)\int_{\mathbb{R}^{n-1}}U(0,z)^{\frac{2(n-1)}{n-2}}z_{i}z_{j}dz =\sum_{i=1}^{n-1}\bar{R}_{ii}(q)\int_{\mathbb{R}^{n-1}}U(0,z)^{\frac{2(n-1)}{n-2}}z_{i}^{2}dz = \frac{\bar{R}_{ii}(q)}{n-1}\int_{\mathbb{R}^{n-1}}\frac{|z|^{2}dz}{(1+|z|^{2})^{n-1}}=\frac{\bar{R}_{ii}(q)}{n-1}\omega_{n-1}I_{n-1}^{n}.
\begin{aligned}\mathbf{P}_{x_0}(T_{x_n}>T_x-\varepsilon_0/2, T_x<\infty) =\mathbf{P}_{x_0}(T_{x_n}>T_x-\varepsilon_0/2, T_{x_n}<\infty) =\mathbf{P}_{x_0}(T_x\circ \theta_{T_{x_n}}<\varepsilon_0/2, T_{x_n}<\infty) =\mathbf{P}_{x_0}(T_{x_n}<\infty) \mathbf{P}_{x_n}(T_x<\varepsilon_0/2)<\frac{2c}{3},\end{aligned}
( \, \Lambda \, \vert \Lambda \, ) = 1 \, , \, \, ( \, \Phi \vert X \Psi \, ) = ( \, X^{*}\Phi \vert \Psi \, ) \, \, \mathrm{f o r}\, \, \, \Phi \, , \Psi \, \in{\cal V}_{\Lambda}\, , \, \, X \in U_{q}^{( n )}\, .
+ \left( \frac{1}{2}\bar{C}_{\gamma}^{\dagger}U_{\alpha \beta}^{\dagger \gamma}C^{\alpha}C^{\beta}+ \frac{1}{2}C^{\dagger \beta}C^{\dagger \alpha}U_{\alpha \beta}^{\gamma}\bar{C}_{\gamma}+ C^{\dagger \alpha}\bar{U}_{\alpha \beta}^{\gamma}\bar{C}_{\gamma}C^{\beta}+ C^{\dagger \beta}\bar{C}_{\gamma}^{\dagger}\bar{U}_{\alpha \beta}^{\dagger \gamma}C^{\alpha}\right) ( - 1 )^{\varepsilon_{\beta}}.
+ \sum_{n}\pi_{n , 1}\frac{\partial \overline{{H}}_{0}}{\partial q_{n , 0}}+ \sum_{n}\sum_{j_{n}= 2}^{m_{n}- 1}\pi_{n , j_{n}}\left[ \frac{\partial \overline{{H}}_{0}}{\partial q_{n , j_{n}- 1}}- \mu_{n , j_{n}- 1}\right] \ .
\Pi_{0 0}^{\Sigma}={\frac{4 \alpha}{\beta}}\int_{0}^{1}d x \int{\frac{d |{\bf l}|}{( 2 \pi )}}\left[ |{\bf l}| S_{1}^{\Sigma}+ ( |{\bf l}|^{2}+ \Sigma^{2}){\frac{\partial S_{1}^{\Sigma}}{\partial |{\bf l}|}}+{\frac{p_{0}^{2}}{p}}{\frac{\partial S_{1}^{\Sigma}}{\partial p}}|{\bf l}| - ( 1 - 2 x ){\frac{\beta p_{0 b}}{2}}{\frac{\partial S_{1}}{\partial \omega}}\right]
C^{\alpha \beta}( x , y ) = \left( \begin{array}{c c}{\{\chi^{i j}( x ) , \chi^{m n}( y ) \}}&{\{\chi^{i j}( x ) , \phi^{m n}( y ) \}}\\{\{\phi^{i j}( x ) , \chi^{m n}( y ) \}}&{\{\phi^{i j}( x ) , \phi^{m n}( y ) \}}\\ \end{array}\right) ,
\{\pi_{\mu}( \textbf{x}, t ) , \pi_{\nu}( \textbf{y}, t ) \}^{*}= \{A_{\mu}( \textbf{x}, t ) , A_{\nu}( \textbf{y}, t ) \}^{*}= 0 ( \mu , \nu = 1 , 2 , 3 , 4 ) .
\begin{array}{r c l c r c l}{{H_{1 7}( r )}}&{{=}}&{{1 + \frac{\vert g^{1 7}\vert}{r}}}&{{;}}&{{H_{2 4}( r )}}&{{=}}&{{1 + \frac{\vert g^{2 4}\vert}{r}}}\\{{H_{1 8}( r )}}&{{=}}&{{1 + \frac{\vert e_{1 8}\vert}{r}}}&{{;}}&{{H_{2 3}( r )}}&{{=}}&{{1 + \frac{\vert e_{2 3}\vert}{r}}}\\ \end{array}
\begin{aligned}|p\pm v|^2 - |\tilde{p}\pm v|^2 = |p|^2-|\tilde{p}|^2 \pm 2(\overline{p-\tilde{p}}v) &=(1-|z|^2)|E_{\pm}|^2 + (1-|w|^2)|F_{\pm}|^2\end{aligned}
\lambda_{\phi}(0)=\int_{\mathbb{R}^d}\left(\frac12|\nabla\phi_g|^2+V|\phi_g|^2+\beta|\phi_g|^4-\Omega\,\overline{\phi}_g L_z\phi_g\right)\mathrm{d}\mathbf{x}=\mu(\phi_g)=:\mu_g,
\zeta^m(s_{m,0}) = \zeta^{m-1}(s_{m-1,-1})\zeta^{m-2}(s_{m-2,-1})...\zeta^2(s_{2,-1})\zeta(s_{1,-1})\zeta(s_1,0) & = \zeta^{m-1}(s_{m-1,-1})\zeta^{m-2}(s_{m-2,-1})...\zeta^2(s_{2,-1})\zeta(s_{1,-1})\overline{a_{-n+1}}a_{-n+1}.
0=\int_\Omega \varphi(x)\left[\int_\Omega G^1_{|\Omega}(x,\xi)\left[h(\xi)-\int_{\partial\Omega}P^s_{|\Omega}(\xi,y)d\zeta(y)\right]\:d\xi\right]dx=\int_\Omega \psi(\xi)\left[h(\xi)-\int_{\partial\Omega}P^s_{|\Omega}(\xi,y)\;d\zeta(y)\right]d\xi
E\Big[\Big(\int_0^t \int_{\mathbb R_0}Y(s)\lambda_{\varepsilon}^{i,j}(z) \tilde N^j(ds,dz)\Big)^2 \Big]= E\Big[\int_0^t Y(s)^2ds \Big]\| \lambda^{i,j}_{\varepsilon}\|^2_{L^2(\nu_j)}=E\Big[\Big(\int_0^t Y(s)\Lambda^{i,j}(\epsilon) dB^j(s)\Big)^2 \Big],
\| b(t,h) - b(t,g) \| \leq \| a(h) - a(g) \| + \sum_{j=1}^r \| \sigma^j(h) - \sigma^j(g) \| \, |\dot{B}_m^j(t)| \leq L \| h-g \| + L \| h-g \| \sum_{j=1}^r Y_m^j = (L + Z_{r,m}) \| h-g \|.
\bigg(\frac{1}{n(n-1)}\sum_{i_1\neq i_2}X_{i_1}^\top X_{i_2}\bigg)=\frac{2}{n(n-1)}(\Sigma_{1}^{2})+\frac{4}{n}\mu_{1}^{\top}\Sigma_{1}\mu_{1}=O(p N^{-1}).
0\, \longrightarrow\, (E_H)\, \stackrel{\iota'}{\longrightarrow}\,(E_H)/\theta'({\mathcal F})\,\stackrel{\widehat{\mathrm{d}}p}{\longrightarrow}\, TX/{\mathcal F}\,=\,{\mathcal N}_{\mathcal F}\, \longrightarrow\, 0\, ,
I=\int_{\Sigma}d^3 \! x \sqrt{\gamma}\left(R(\gamma) -{1 \over 2}\left( \left(D\tilde{U}\right)^2 - e^{-2\tilde{U}}\left(DV \right)^2 \right)-{1 \over 2}\left( \left(D \tilde{\Phi}\right)^2 - e^{- 2\tilde{\Phi}}\left(D \Psi \right)^2 \right) \right) \ ,
\int_{\mathbb{R}^n}|\nabla \tilde u| = \int_{B}|\nabla \tilde u| + \int_{\mathbb{R}^n \setminus \bar B}|\nabla \tilde u| \leq \liminf_{\varepsilon \to 0}\int_B |\nabla u_\varepsilon| + \liminf_{\varepsilon \to 0}\int_B |\nabla u_\varepsilon| \leq \int_{\mathbb{R}^n}|\nabla \tilde u|,
\sigma(t,\omega)=\mathbf{s}(\omega(t))=(\mathbf{s}_{ij}(\omega(t)))_{1\le i,j\le n}\mathrm{and}\vartheta(t,\omega)=\boldsymbol{\theta}(\omega(t))=(\boldsymbol{\theta}_1(\omega(t)),\dots,\boldsymbol{\theta}_n(\omega(t)))'
{\overline{\Gamma}^{i}}_{kl}=\frac{\partial\overline{x}^{i}}{\partial x^{m}}\frac{\partial x^{n}}{\partial\overline{x}^{k}}\frac{\partial x^{p}}{\partial\overline{x}^{l}}{\Gamma^{m}}_{np}+\frac{\partial^{2}x^{m}}{\partial\overline{x}^{k}\partial\overline{x}^{l}}\frac{\partial\overline{x}^{i}}{\partial x^{m}}
Lf =\lim_{t\to 0}\frac 1 t(Ue^{-tL_1}U^{-1}f-f) &=U\lim_{t\to 0}\frac 1 t(e^{-tL_1}U^{-1}f-U^{-1}f) &=UL_1U^{-1}f &=L_2f.
f(T)h=\overset{\infty}{\underset{j=-\infty}{\sum}}f_jT^jh=\overset{\infty}{\underset{j=-\infty}{\sum}}f_j(P_\mathcal{H}N^jh)=P_\mathcal{H}\overset{\infty}{\underset{j=-\infty}{\sum}}f_jN^jh=P_\mathcal{H}f(N)h.
\sum_{k=0}^{n-1}[3k+1]\frac{(q;q^2)_k^3 q^{-{k+1\choose 2}}}{(q;q)_k^2 (q^2;q^2)_k}\equiv q^{(1-n)/2}[n]+\frac{(n^2-1)(1-q)^2}{24}\,q^{(1-n)/2}[n]^3 \pmod{[n]\Phi_n(q)^3}.
(b_{\ell+1,i}^x - a_{\ell+1,i}^x) \leq \overline q (b_{\ell,i}^x - a_{\ell,i}^x), i=1,\ldots,d, \ell=-L,\ldots,L-1, (b_{\ell+1,i}^y - a_{\ell+1,i}^y) \leq \overline q (b_{\ell,i}^y - a_{\ell,i}^y), i=1,\ldots,d, \ell=-L,\ldots,L-1.
\left||\eta|^2 -(\tau-i\gamma)^2\right|^2 = (\tau^2-\gamma^2-|\eta|^2)^2 + 4\gamma^2\tau^2 = (\tau^2 - |\eta|^2)^2 +\gamma^4 + 2\gamma^2|\eta|^2 + 2\tau^2 \gamma^2 \ge \gamma^2(\gamma^2 + 2\tau^2 + 2|\eta|^2) \ge \gamma^2
\phi^{\prime}( x ) \; \mathrm{e x p}\left[ -{\frac{i}{\hbar}}\int d x \; \Pi ( x ) \phi^{\prime}( x ) \right] = i \hbar{\frac{\delta}{\delta \Pi ( x )}}\; \mathrm{e x p}\left[ -{\frac{i}{\hbar}}\int d x \; \Pi ( x ) \phi^{\prime}( x ) \right] \; ,
\lim\limits_{x\to \pm\infty}\frac{1}{|x|^{\frac{1}{\beta}}\ell^{\frac{1}{\beta}}(|x|^{\frac{1}{\beta}})}\eta_K(x)=\int^{\infty}_0 \big(K_{\infty}(\pm t^{-\beta})-K_{\infty}(0)\big)dt:=C^{\pm}_K.
\vert\widetilde{T}_{F,G,N}^{L,\Xi,\widetilde{\mathbf{r}}}(g_1,\dots,g_d)\vert \ll \Big\vert \frac{1}{N^{h-m}}\int\limits_{\mathbf{x}\in \mathbb{R}^{h-m}}F_1(\mathbf{x}+ \sum\limits_{k=1}^{s+1}w_k \mathbf{f_k})\prod\limits_{j=1}^d g_j(\psi_j^\prime(\mathbf{x},\mathbf{w}) + a_j) \, d\mathbf{x}\Big\vert,
u_1:f_1\in \mathfrak{A}_1 \mapsto u_1(f_1)=f_1\otimes 1\otimes \cdots \otimes 1 \in \textstyle\bigotimes \mathfrak{A}_i. \vdots u_n:f_n\in \mathfrak{A}_n \mapsto u_n(f_n)=1\otimes \cdots \otimes 1\otimes f_n \in \textstyle\bigotimes \mathfrak{A}_i.
\frac{\partial^{3}\varphi_{1}}{\partial x_{1}^{3}}+\frac{\partial^{3}\varphi_{1}}{\partial x_{1}\partial x_{2}^{2}}+\frac{\partial^{3}\varphi_{2}}{\partial x_{1}^{2}\partial x_{2}}+\frac{\partial^{3}\varphi_{2}}{\partial x_{2}^{3}}=\frac{q}{D}.
\begin{array}{l}||\tilde{f}^{j}-f(t_{j})||_{\mathbb{L}^{1}}\leq (1+\tilde{K}_{1}\Delta t) ||\tilde{f}^{j-1}-f(t_{j-1})||_{\mathbb{L}^{1}}+\tilde{K}_{2}\Delta t (\Delta t+\Delta \mathbf{x}), j=1,2,\ldots,[[T/\Delta t]].\end{array}
\Pi^{1}= R \Psi^{1}/ \Psi^{0}, \Pi^{2}= R \Psi^{2}/ \Psi^{0}, . . . , \Pi^{i}= R \Psi^{i}/ \Psi^{0}, . . . , \Pi^{N}= R \Psi^{N}/ \Psi^{0}
\sum_{i=1}^{p}J_{[X,S_i]}S_i = -16\sum_{i=1}^{p}\Im(XS_i^*)S_i = -16\sum_{i=1}^{p}(XS_i^*)S_i+16\sum_{i=1}^{p}\langle X,S_i\rangle S_i = -16X\sum_{i=1}^{p}S_i^*S_i + 16X = -16(q+1)X + 16X = -16qX,
\phi \sim \sqrt{2 p \pi}\ \phi_{0}\C^{\frac{3 p + 1}{2}}\ \exp \left( - \,{\frac{C \pi p}{2}}\right) \ \left({\frac{p}{t m_{\phi}}}\right)^{\frac{3 p}{2}}\sin \left( m_{\phi}t +{\frac{( 2 - 3 p ) \pi}{4}}\right) \ .