harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
d\left(\underbrace{\left\lceil\frac{d_0}{d}\right\rceil + \cdots + \left\lceil\frac{d_0}{d}\right\rceil}_{\omega-1}+ \left\lceil\frac{(a-1)(b-1)d_0}{d}\right\rceil - \omega + 1 \right) + 2= \frac{p^L-1}{d'}d' + 2 > p^L,
\left( \begin{array}{c}{{{\mathbf{E}^{\prime}}}}\\{{\mathbf{B}^{\prime}}}\\ \end{array}\right) = \left( \begin{array}{c c}{{a}}&{{c}}\\{{b}}&{{d}}\\ \end{array}\right) \left( \begin{array}{c}{{\mathbf{E}}}\\{{\mathbf{B}}}\\ \end{array}\right)
\max_{P_i\ge 0, \forall i}~~ \mathbb{E}\left[\sum_{i=1}^K\ln\left(1+\frac{h_{i}P_i}{\sigma^2+\sum_{j=1,j\neq i}^K h_{j}P_j}\right)\right], \mbox{s.t.}~~ \eqref{Con-PIPC},~\eqref{Con-PTPC}.
\begin{array}{c}g(D)=\widetilde{B}^2+1+2d(g(C)-1)-(p_a(\widetilde{D})-g(D))\leq 2g(B)+dg(C)+d-1 begin{align*}2mm]\Leftrightarrow \widetilde{B}^2\leq 2g(B)+d(3-g(C))-2+(p_a(\widetilde{D})-g(D)).\end{array}
[\begin{matrix}1&F_{1}&P_{1}\\ 1&F_{2}&P_{2}\end{matrix}][\begin{matrix}y_{L}\\y_{F}\\y_{P}\end{matrix}]\ge[\begin{matrix}S_{1}\\S_{2}\end{matrix}],[\begin{matrix}y_{L}\\y_{F}\\y_{P}\end{matrix}]\ge0.
\alpha_j = \frac{e_j}{q_j}+ z_j (e_j, q_j) =1, 0 < q_j \leq \tau_j := Y^{j/2}&|z_j| \leq \frac{1}{q_j \tau_j}, z_j = \delta_j Y^{-j}(j=1,2, \dots, k) &\delta_0 = \max \{\delta_1, \dots , \delta_k \}\mathcal{Q}=[ q_1,\dots , q_k ]
\omega_{+}^{- Y^{\prime}}( x , \theta^{+}, u ) = \tau_{+}^{- Y^{\prime}}( x , u ) + \theta_{+}^{+ A^{\prime}}l_{A^{\prime}}^{-- Y^{\prime}}( x , u ) + i ( \theta_{+}^{+})^{2}\mu_{-}^{- 3 Y^{\prime}}( x , u ) .
r ={\sf c}_0(x\cdot{\sf c}_1y)\cdot{\sf c}_0(x\cdot -{\sf c}_1y), s ={\sf c}_0{\sf c}_1({\sf c}_1z\cdot{\sf s}^0_1{\sf c}_1z\cdot -{\sf d}_{01}) +{\sf c}_0(x\cdot -{\sf c}_1z), t ={\sf c}_0{\sf c}_1({\sf c}_1w\cdot{\sf s}^0_1{\sf c}_1w\cdot -{\sf d}_{01}) +{\sf c}_0(x\cdot -{\sf c}_1w),
E[(S_n)^2] = \dfrac{\Gamma(n+2\alpha)}{\Gamma(n)}\sum_{\ell=1}^n \dfrac{\Gamma(\ell)}{\Gamma(\ell+2\alpha)}=\begin{cases}\dfrac{n}{1-2\alpha}+ \dfrac{1}{2\alpha-1}\cdot \dfrac{\Gamma(n+2\alpha)}{\Gamma(n)\Gamma(2\alpha)}(\alpha \neq 1/2), \displaystyle n\sum_{\ell=1}^n \dfrac{1}{\ell}(\alpha=1/2).\end{cases}
\frac{a_{1}^{2}}{b_{1}}+\frac{a_{2}^{2}}{b_{2}}+\cdot\cdot\cdot+\frac{a_{n}^{2}}{b_{n}}\ge\frac{(a_{1}+a_{2}+\cdot\cdot\cdot+a_{n})^{2}}{b_{1}+b_{2}+\cdot\cdot\cdot+b_{n}}.
\displaystyle{\operatornamewithlimits{\mbox{minimize}}_{w \in \mathbb{R}^n}}\ \theta(w) \, \triangleq \, \underbrace{f(\Phi(w))}_{\mbox{denoted $\varphi(w)$}}+ \, \displaystyle{\sum_{i=1}^n}\, \alpha_i \, | \, w_i \, |
W( 01^l | 1^k0 ) = A(1^{k+1}01...) - A(10^\alpha1...) + \sum_{j=2}^l \left[ A(1^{k+j}0...) - A(1^j0^\alpha1...) \right] + &+ A(01^{k+l}0...) - A(01^l0^\alpha1...) &= c_{k+1}- d_\alpha + \sum_{j=2}^l(c_{k+j}- c_j) + (b_{k+l}- b_l).
| h , u > \equiv \left( \begin{array}{c}{{0}}\\{{x^{h +{\frac{n}{2}}}}}\\ \end{array}\right) \quad , \quad | \tilde{h}, d > = \left( \begin{array}{c}{{x^{\tilde{h}+{\frac{m}{2}}}}}\\{{0}}\\ \end{array}\right)
\begin{array}{l}d(x, e^{i \frac{(u - s)}{u}\theta}x)^2/4(u - s) +d(x,e^{- i \theta}e^{i \frac{(u - s)}{u}\theta}x)^2/ 4s - 4u^2/ 4s = d(x, e^{i \theta}x)^2/ 4u - 4u^2/4s =\dfrac{\theta^2}{u}- \dfrac{u^2}{s}\,\cdot \end{array}
\Delta_1 = n\left(\frac{m+1}{n}\wedge t - s \right) |X^n_{m+1}- X^n_m|, \Delta_2 = |X^n_k - X^n_{m+1}| \; \textnormal{if $k > m$}, \; \; 0 \textnormal{if $k= m$}\Delta_3 = n\left(t-\frac{k}{n}\vee s\right)|X^n_{k+1}- X^n_k|.
\int_{\mathbb{R}^{2}}((\partial_{11}^{2}v)^{2}+2(\partial_{12}^{2}v)^{2}+(\partial_{22}^{2}v)^{2})=\int_{\mathbb{R}^{2}}(\partial_{11}^{2}v+\partial_{22}^{2}v)^{2},
C_{m, n}= \frac{1}{{n\choose m}}\sum_{1\leq i_{1}<i_{2}<\ldots<i_{m}\leq n}\left(\frac{X_{i_{1}}+ \ldots + X_{i_{m}}}{m}- \theta (m)\right)\otimes\left(\frac{X_{i_{1}}+ \ldots + X_{i_{m}}}{m}- \theta (m)\right),
(\sigma_1-\sigma_2)^{2}+(\sigma_2-\sigma_3)^{2}+\cdots+(\sigma_n-\sigma_{n+1})^{2}=2\left(\mu_1\sigma_1+2\sum\limits_{i\in I\setminus\{1,n+1\}}\mu_i\sigma_i+\mu_{n+1}\sigma_{n+1}\right).
\frac{1}{s'}=\frac{a'}{p}+\frac{1-a'}{q}, \gamma'_1+\alpha'=a'(\gamma_2+\mu-1)+(1-a')(\gamma_3+\beta), \frac{1}{s''}=\frac{a''}{p}+\frac{1-a''}{q}, \gamma''_1+\alpha''=a''(\gamma_2+\mu-1)+(1-a'')(\gamma_3+\beta).
\frac{(n+\alpha+\beta+N+2)}{(2n+\alpha+\beta+2)}Q_n(x) - \frac{(N-n)}{(2n+\alpha+\beta+2)}Q_{n+1}(x)= \frac{(\alpha+x+1)}{(\alpha+1)}\hat Q_n(x), - \frac{(n+1)(n+\beta+1)}{(2n+\alpha+\beta+3)}\hat Q_n(x) + \frac{(n+\alpha+\beta+2)(n+\alpha+2)}{(2n+\alpha+\beta+3)}\hat Q_{n+1}(x) = (\alpha+1) Q_{n+1}(x).
A\cdot B=A^{\mu}\eta_{\mu\nu}B^{\nu}=A_{\nu}B^{\nu}=A^{\mu}B_{\mu}=\sum_{\mu=0}^{3}a^{\mu}b_{\mu}=a^{0}b^{0}-\sum_{i=1}^{3}a^{i}b^{i}=a^{0}b^{0}-\vec{a}\cdot\vec{b}
\hat{\mu}=\frac{1}{P_{}}\sum_{l=1}^{L}w_{l}\left(\hat{\mathbf{\Omega}}_{l}^{-1}-\left(\hat{\mathbf{\Omega}}_{l}+\mathbf{H}_{l,l}^{\dagger}\hat{\mathbf{\Sigma}}_{l}\mathbf{H}_{l,l}\right)^{-1}\right)
\int_{B^c_\varepsilon(x)}\frac{\, u(x) - u(y) \,}{|x - y|^{N + 2s}}\,dy = \int_{B_r \cap B^c_\varepsilon}\frac{\, u(x) - u(x+z) \,}{|z|^{N + 2s}}\,dz+\int_{B^c_r}\frac{\, u(x) - u(x+z) \,}{|z|^{N + 2s}}\,dz &= \int_{B_r \cap B^c_\varepsilon}\frac{\,u(x)-u(x-z) \,}{|z|^{N + 2s}}\,dz+\int_{B^c_r}\frac{\, u(x) - u(x-z) \,}{|z|^{N + 2s}}\,dz,
\begin{array}{c}\displaystyle g_3(\alpha) = \int_{0}^{1}p \phi^{(n)}_3 f^{(n)}_\alpha{\rm d}p = \frac{\Gamma(2n+2) \Gamma (x+y+2)}{\Gamma(2n+3) \Gamma(x+y+1)}= \displaystyle \frac{\alpha+\gamma}{2}+ \frac{1-\alpha-\gamma}{2n}+ O \left(\frac{1}{n^2}\right). \end{array}
\exp_{q}(x) =1+\sum_{k=1}^{\infty}\frac{x^{k}}{(k)_{q}!},\begin{pmatrix}n k\end{pmatrix}_{q}= \frac{(n)_{q}!}{(k)_{q}! (n-k)_{q}!}, (k)_{q}! =(1)_{q}(2)_{q}\cdots (k)_{q},(k)_{q}=\frac{1-q^{k}}{1-q}.
|G_{I(z_j')}(z_j', z_{j+1})|=\frac{|P_{|y_{j+1}-z_j'|}(\theta+(z_j'+1)\alpha)|}{|P_{|I(z_j')|}(\theta+z_{j+1}\alpha)|}= \frac{|\tilde{P}_{|y_{j+1}-z_j'|}(\theta+(z_j'+1)\alpha)|}{|\tilde{P}_{|I(z_j')|}(\theta+z_{j+1}\alpha)|}\prod_{\ell=z_{j+1}}^{z_j'}|\cos(\pi(\theta+\ell\alpha))|.
m_{\sigma}(B(0,r_{1}))\geqq m_{\sigma}(B(0,r_{2}))\geqq \cdot \cdot \cdot\geqq m_{\sigma}(\mathbb{R}^{N})=\underset{\left \Vert w\right \Vert_{L^{2}(\mathbb{R}^{N})}=\sigma}{\inf}J(w)>-\infty
S_1 =\sum_{i=1}^k (-1)^{k-i}\sum_{\substack{2\leq j\leq k j\equiv k\ ({\rm mod}\p)}}p(k,i,j) q^{i-1}, S_2 =\sum_{i=1}^k (-1)^{k-i}\sum_{\substack{1\leq j_1,j_2\leq k j_1+j_2\leq k j_1\equiv 1-a_1\ ({\rm mod}\p) j_2\equiv 1-a_2\ ({\rm mod}\p)}}p(k,i,j_1,j_2) q^{i-1}.
\dim_{\mathbb{C}}(\pi_{N,\psi_{A}}) = \frac{1}{q^{9}}\{S_{1}+ S_{2}+ S_{3}+S_{4]}\}&= \frac{1}{q^{9}}(q-1)^2(q^2-1)^2q^9 &= (q-1)^{2}(q^2-1)^2.
a_{m+1}\le a_{m}+ \Delta t f\left(\frac{K^{2}+1}{2}a_{m}\right) \le y(t_{m}) + \Delta t f\left(\frac{K^{2}+1}{2}y(t_{m})\right) &\le y(t_{m}) + \Delta t \frac{dy}{dt}(t_{m}) \le y(t_{m+1}),
0<x_0=\frac{(1-c_2)-\sqrt{(1-c_2)^2-4c_1a_0}}{2c_1}=\frac{1}{2c_1}\frac{4c_1 a_0}{(1-c_2)+\sqrt{(1-c_2)^2-4c_1 a_0}}<\frac{2}{1-c_2}a_0,
\tilde{\cal H}= \omega q \cdot \!{\frac{\partial}{\partial q}}-{\frac{1}{2}}\sum_{j = 1}^{r}{\frac{\partial^{2}}{\partial q_{j}^{2}}}- \sum_{\rho \in \Delta_{+}}{\frac{g_{| \rho |}}{\rho \cdot q}}\rho \cdot \!{\frac{\partial}{\partial q}}.
a(\vec{p})=\int{{d^{\nu}x}\over{(2 \pi)^{\nu\over 2}}}{e^{-i \vec{p}\cdot \vec{x}}\over{\sqrt{2 g(\vec{p})}}}\left(\int d^{\nu}y{1\over 2}G(x,y) \eta(y) +{\delta\over{\delta \eta(\vec{x})}}\right)
d_{\theta,{\rm MFTR}}^2={{\left({{V}_{1}}\cos{{\phi}_{1}}+{{V}_{2}}\cos{{\phi}_{2}}\right)}^{2}}+{{\left({{V}_{1}}\sin{{\phi}_{1}}+{{V}_{2}}\sin{{\phi}_{2}}\right)}^{2}}= V_1^2+V_2^2+2V_1V_2\cos{\left(\phi_2-\phi_1\right)}= V_1^2+V_2^2+2V_1V_2\cos{\theta}.
\left( \int_{\gamma_\rho^+}+\sum_{k}\int_{C_\rho (iy_k)}\right) \frac{H^{(1)}_{\mu}(\lambda z)}{H^{(1)}_{\nu}(z)}F(z)dz+\left( \int_{\gamma_\rho^-}+\sum_{k}\int_{C_\rho (-iy_k)}\right) \frac{H^{(2)}_{\mu}(\lambda z)}{H^{(2)}_{\nu}(z)}F(z)dz
\lim_{R \to \infty}\lim_{\delta \to 0}\lim_{i\to \infty}E_n(u_i; B_{R r_{2,i}}(x^{j}) \setminus B_{\delta r_{2,i}}(x^j)) =\lim_{R \to \infty}\lim_{\delta \to 0}\lim_{i\to \infty}E_n(\tilde u_{2,i}; B_{R}(0) \setminus B_{\delta}(0)).
\lim_{N \to \infty}\frac{1}{N}\log \left( \mathfrak{R}_{N}(B( \overline{\mathbf{P}}, \delta)) \right) = \lim_{N \to \infty}\frac{1}{N}\log \left( \mathfrak{Q}_{N}(B( \overline{\mathbf{P}}, \delta)) \right).
C_b\left(P_{\pi(i)},P_{-\pi(i)},H_{\pi(i)}\right)\triangleq \frac{1}{2}\log\left(1+\frac{H_{\pi(i)}P_{\pi(i)}(H_{\pi(i)})}{1+\sum_{j=i+1}^K H_{\pi(j)}P_{\pi(j)}(H_{\pi(j)})}\right).
\left. \begin{array}{c}\theta_{\beta \bar{\gamma}}=iu_{\beta \bar{\gamma}}^{\bar{\alpha}}\theta_{\alpha \delta}=iu_{\alpha \delta}^{\gamma}\end{array}\right\}\Leftrightarrow \left\{\begin{array}{c}\theta_{\alpha \beta}=iu_{\alpha \beta}^{\gamma}\theta_{\beta}=\theta_{\delta}\end{array}\right.
\left[ -\right] \circ \left( \left[ -\right] \otimes M\right) \circ \left[\mathrm{Id}_{\left( V\otimes V\right) \otimes V}+\left( M\otimes c\right)\left( c\otimes M\right) +\left( c\otimes M\right) \left( M\otimes c\right) \right] =0.
\|\lambda(\phi)\|_{K,m+n}=\sup_{\stackrel{|p|\leq m+n}{z \in K}}|D^p (\lambda(\phi))(z)|\,\,,\qquad D^p=\frac{\partial^{|q|+|r|}}{\partial x^{q_1}_1 \cdots \partial x^{q_m}_m\partial\theta^{r_1}_1 \cdots \partial\theta^{r_n}_n}\,\,.
+ \left.{\frac{1}{6 0}}F_{\mu \rho}~^{| | \rho}F^{\mu \lambda}~_{| | \lambda}-{\frac{1}{7 2}}( F^{\mu \nu}F_{\mu \nu}) \bar{R}-{\frac{1}{1 8 0}}\bar{R}^{\lambda \nu \mu \rho}F_{\lambda \nu}F_{\mu \rho}-{\frac{1}{9 0}}\bar{R}_{\mu \nu}F^{\rho \mu}F_{\rho}~^{\nu}\right) ~ ~ ,
PR^1(Rm)_{[0]}\otimes Q\big(R^2(Rm)_{[1]}\kappa \big)^1S^2(P) \otimes R \big(R^2(Rm)_{[1]}\kappa \big)^2 S^2(Q) \qquad{}=R^1 \big(QR^1m_{[0]}\big)_{[0]}\otimes R^2 \big(QR^1m_{[0]}\big)_{[1]}\kappa S^2(P) \otimes RR^2m_{[1]}\kappa,
c_{4}^{( 1 )}= - \frac{9}{2}\int_{0}^{1}\frac{d x}{x^{2}}\left[ \frac{1 - \frac{1 2 1}{7 2 9}x^{2}}{( 1 - \frac{2 5}{7 2 9}x^{2})^{1 3}}- 1 \right] \, .
\Delta(X) = \square(X) X=x_{i}^{\pm}, h_i, \Delta\big(x_{i,1}^+\big) = \square(x_{i,1}^+) - \hbar \big[1 \otimes x_i^+, \Omega_+\big], \Delta\big(x_{i,1}^-\big) = \square(x_{i,1}^-) + \hbar \big[x_i^- \otimes 1, \Omega_+\big], \Delta\big(\tilde{h}_{i,1}\big)= \square(\tilde{h}_{i,1}) + \hbar [h_i \otimes 1, \Omega_+].
T^{k_{n}(h_{n}+c_{n})}(I_{n,b}^{[i]}) = T^{k_{n}(h_{n}+c_{n})}(I_{n+1,b + i(h_{n}+c_{n})+\frac{1}{2}i(i-1)}) &= T^{k_{n}(h_{n}+c_{n})+ i(h_{n}+c_{n})+\frac{1}{2}i(i-1)}(I_{n+1,b})= T^{h_{n+1}}(I_{n+1,b+h_{n}+2c_{n}-\frac{1}{2}r_{n}(r_{n}-1)}).
\begin{aligned}y^{-1}h + q^{-1}g = (y+y^{-1})^{-1}\left(y^{-1}(x-\Lambda x^{-1}) + q^{-1}(xy + \Lambda x^{-1}y^{-1})\right) = (y+y^{-1})^{-1}\left(y^{-1}x - \Lambda y^{-1}x^{-1}+ yx + \Lambda y^{-1}x^{-1}\right) = (y+y^{-1})^{-1}(y^{-1}+ y)x &= x.\end{aligned}
\phi_k^n(x^n,z^n) \approx \sum_{m=1}^{N_{ep}}\hat{\phi}^n_{k,m}\psi_{m}(x^n,z^n)=\sum_{i=1}^{N_{ep}}\phi_k^n(x_i^n,z_i^n) h_{i}(x^n,z^n), (x^n,z^n) \in E^n,
\tilde{b}^{1}= - b^{1}= W - U \; , \qquad \tilde{b}^{2}= b^{2}- b^{1}= 2 W - U \; , \qquad \tilde{b}^{3}= b^{3}- b^{1}= T - \frac{5}{2}U + W \; .
\left|E\left[\int_s^t \partial_r (\beta V)(r,q_r^m)dr\right]-E\left[\int_s^t \partial_r (\beta V)(r,q_r)dr\right]\right| \leq T\sup_{r\in[0,T]}E\left[ |\partial_r (\beta V)(r,q_r^m)- \partial_r (\beta V)(r,q_r)|\right]dr \leq E[|\tilde C(1+\|q_t\|^{\tilde p}+\|q_t-q_t^m\|^{\tilde p})\|q_t-q_t^m\||] =O(m^{1/2})
\mathbb{P}(\zeta_{1}^{k}=0^{k-1}1|\zeta_{0}=1) =\prod_{i=1}^{k}P(\zeta_i=0|\zeta_{0}^{i-1}=10^{i-1})\times \mathbb{P}(\zeta_k=1|\zeta_{0}^{k-1}=10^{k-1}).
\mathrm{R}\left(f(x)-\frac{c}{\Phi_0(\nu;x)+\mathrm{MC}_k(x)}\right) = \nu-1+\mathrm{R}\left(\ln\frac{f(x)}{f(x+1)}+\ln\frac{\Phi_0(\nu;x)+\mathrm{MC}_k(x)}{\Phi_0(\nu;x+1)+\mathrm{MC}_k(x+1)}\right),k\ge 0.
E_{i}={\frac{1}{\kappa \left( 1 -{\frac{e^{2}}{\kappa^{2}}}| \Phi |^{2}\right)}}\left( -{\frac{i e}{2}}\right) \epsilon_{i j}\left[ \Phi^{\ast}\nabla_{j}\Phi - \Phi \left( \nabla_{j}\Phi \right)^{\ast}\right] .
S_{p}= - \frac{V_{p - 2}\tilde{V}_{2}}{( 2 \pi )^{p}\tilde{g}\tilde{b}}\int d \tau \left( \frac{u}{R}\right)^{7 - p}\left[ \sqrt{\tilde{f}}- 1 + \frac{u_{0}^{7 - p}}{2 u^{7 - p}}\right] .
( \bar{f}_{I}^{\Lambda}, f_{\bar{I}}^{\Lambda}) = ( \bar{f}_{i}^{\Lambda}P_{I}^{i}, f_{\bar{\imath}}^{\Lambda}\bar{P}_{\bar{I}}^{\bar{\imath}}) , \quad \quad P_{i}^{I}\bar{P}_{\bar{\jmath}}^{\bar{J}}\eta_{I \bar{J}}= g_{i \bar{\jmath}}
\psi_{v}( \vec{x^{\prime}}, t ) = \left( \frac{m \omega a^{2}}{\pi}\right)^{3 / 4}\exp \left[ - i \Phi_{v}( \vec{x^{\prime}}, t ) - \left( \frac{m}{2}\omega a^{2}\right) ( \vec{x^{\prime}}- \tau \vec{v})^{2}\right] ,
p_\beta (z_\Delta(\tau) \mid z(\tau)) =\prod^d_{i=1}\left\{\frac{\Gamma (z_i + (z_\Delta)_i + \beta_i)}{\Gamma (z_i + \beta_i) (z_\Delta)_i!}\frac{\{t_i(\tau)\}^{z_i + \beta_i}\{t_i(\tau+\Delta) - t_i(\tau)\}^{(z_\Delta)_i}}{\{t_i(\tau+\Delta)\}^{z_i + (z_\Delta)_i + \beta_i}}\right\},
f(x_1,x_2) = f(0,0) +2^{-1}[ f(x_1,0)+f(x_1+x_2,0)-f(0,0)-f(x_2,0) +f(0,x_2)+f(0,x_1+x_2)-f(0,0)-f(0,x_1) +f(x_1,x_1)+f(x_2,x_2)-f(0,0)-f(x_1+x_2,x_1+x_2) ].
G=\begin{pmatrix}1 0 \cdots 0 a_1 0 \cdots 0 1 0 1 \cdots 0 a_2 0 \cdots 1 0 \vdots \vdots \vdots \vdots \vdots \vdots \vdots 0 0 \cdots 1 a_k 1 \cdots 0 0 \end{pmatrix}_{k\times (2k+1)},
\langle\delta Y\rangle= \delta Y \bigg(\frac{\delta{\cal W}}{\delta{\cal J}_i}+\frac{\hbar}{i}\frac{\delta}{\delta{\cal J}_i},\,\psi^\alpha,\,\phi^*_{Aa},\, \bar{\phi}_A\,; (-1)^{\varepsilon_i}\frac{1}{i\hbar}{\cal J}_i,\,
[{L}_{u^{\alpha ( i , j )}}] : = \frac{\partial{{L}_{D}}^{( i , j )}}{\partial u^{\alpha ( i , j )}}- \Delta_{1}( \frac{\partial{{L}_{D}}^{( i - 1 , j )}}{\partial ( \Delta_{1}u^{\alpha ( i - 1 , j )})}) - \Delta_{2}( \frac{\partial{{L}_{D}}^{( i , j - 1 )}}{\partial ( \Delta_{2}u^{\alpha ( i , j - 1 )})}) ;
D_{p j}= \tilde{S}_{p j}+ s_{p j}^{2}( \tilde{S}_{p j}^{2}- \tilde{T}_{p j}) - \bar{S}_{p j}+ s_{p j}^{2}( \bar{S}_{p j}^{2}- 2 \tilde{S}_{p j}\bar{S}_{p j}+ \bar{T}_{p j}) \ .
\partial_t\bigg(\int_{[-t,0]}\varphi_i(x+t)d^-\eta(x)\bigg) \ = \ \partial_t\bigg(\eta(0)\varphi(t) - \int_{-t}^0\eta(x)\dot\varphi_i(x+t)dx\bigg) &= \ \eta(0)\dot\varphi(t) + \eta(-t)\dot\varphi_i(0) - \int_{-t}^0 \eta(x)\ddot\varphi_i(x+t) dx.
\mathbb{E}\left[ \mathbf{y'}^H\mathbf{y}'\right]=\sum_{l=1}^M\sum\limits_{\substack{j=1 j\neq l}}^M\mathbb{E}\left[y_l^* y_j\right]+2\sum_{i=1}^{M}\mathbb{E}\left[y_i^* y_i\right],
M_{0, \varphi}^{-}(x) =2 i \left(\begin{array}{cc}\frac{2 (e^{- i \varphi x}- e^{- \varphi})}{x + i}\frac{- 2 (e^{- i \varphi x}- e^{- \varphi})}{x + i}& \frac{2 (e^{- i \varphi x}- e^{- \varphi})}{x + i}\frac{- 2 (e^{- i \varphi x}- e^{- \varphi})}{x + i}\end{array}\right).
I_{Q,A'}\cap \sum_{Q_1\in \mathcal{P}}I_{Q_1}(e_{Q_1}(\sigma)) = I_{Q,A'}\cap I_{Q,A}\cap \sum_{Q_1\in \mathcal{P}}I_{Q_1}(e_{Q_1}(\sigma)) & =I_{Q,A'}\cap \sum_{Q_1\in \mathcal{P}}I_{Q_1,A}.
{\frac{u}{\cal R}}={\alpha^{\prime}}^{\frac{x + \tilde{d}- \Delta}{\Delta}}\, g_{s}^{\frac{4 ( k - 1 )}{\Delta}}\, \left( \frac{d_{p}}{c_{p}}g_{f}^{2}\right)^{\frac{- 2}{\Delta}}\, U^{\beta}\, .
Im(\Gamma^{\boxtimes m-k}\boxtimes (\Delta_2-\Gamma)^{\boxtimes k})^{S_{m-k}\times S_{k}}=\gamma^{\boxtimes m-k}\boxtimes Im((\Delta_2-\Gamma)^{\boxtimes k})^{S_{k}}Im(\Gamma^{\boxtimes \{1,\ldots,n\}\setminus B}\boxtimes (\Delta_2-\Gamma)^{\boxtimes B}) =\gamma^{\boxtimes \{1,\ldots,n\}\setminus B}\boxtimes Im((\Delta_2-\Gamma)^{\boxtimes B}).
(1+z)^{p+1}\Phi(p)= (1+z)^{a+p+1}{}_2F_1\bigg[\begin{matrix}-\frac{1}2(a+p) -\frac{1}2(a+p)+\frac12 &1\end{matrix}\bigg|\,\frac{4z}{(1+z)^2}\bigg] ={}_2F_1\bigg[\begin{matrix}-a-p -a-p &1\end{matrix}\bigg|\,z\bigg]+z^{a+p+1}{}_2F_1\bigg[\begin{matrix}-a-p -a-p &1\end{matrix}\bigg|\,\frac1z\bigg].
H(t) = \int d^3x \left\{-\frac{\hbar^2}{2 a^3(t)}\frac{\delta^2}{\delta\eta^2}+\frac{a(t)}{2}\left(\vec{\nabla}\eta\right)^2+a^3(t)\left(V(\phi)+V'(\phi)\eta+\frac{1}{2}V''(\phi)\eta^2+\cdots \right)\right\}
\varphi(x,T)-\varphi(y,\tau) =\int_0^\tau\int_{M}(W(T-s,x,Q)-W(\tau-s,y,Q))g(Q,s)\d\sigma \ds &\quad\quad+\int_\tau^T\int_{M}W(T-s,x,Q)g(Q,s)\d\sigma \ds
s^{2}=\frac{1}{n-1}\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}=\frac{\sum_{i=1}^{n}x_{i}^{2}}{n-1}-\frac{(\sum_{i=1}^{n}x_{i})^{2}}{(n-1)n}=(\frac{n}{n-1})s_{n}^{2}.
u(t) = U(t,s)P_0(s) u(s) + U(t,s)(I-P_0(s))u(s) + j^{-1}\int_s^t U^{\odot \star}(t,\tau) R_{\delta,s}(\tau,u(\tau)) d\tau = U(t,s)P_0(s) u(s) + (\mathcal{K}_s^\eta \tilde{R}_{\delta,s}(u))(t)
A_r(N,p)=\frac{(r-N+p)(p-r)(r+1)}{(2r-N)(N-2r-1)}, & B_r(N,p)=-\frac{r^2N-rN(N+1)+(2+N)(N-p)p}{(2r-N-2)(2r-N)}, & C_r(N,p)=-\frac{(r-N-1+p)(r-p-1)(N-r+2)}{(2r-N-2)(N-2r+3)}.
B_0=\frac{x\cdot y}{2}A_c+\frac{x^\perp\cdot y}{2}A_s =\frac{x\cdot y}{2}(x\otimes x)+\frac{x^\perp\cdot y}{2}(x\otimes x^\perp)+B_1=\frac{|x|^2(x\otimes y)}{2}+B_1
\bar{\nu}^{T}\bar{w}_{2^{k}}\left( t\right) =\left( -\tau^{2}\bar{\nu}^{T}\left( \alpha I_{2^{k}}+\beta \Lambda_{\bar{r}}\right) P^{2}+\tau \dot{x}_{0}e_{1}^{T}P+x_{0}e_{1}^{T}\right) \bar{w}_{2^{k}}\left( t\right)
H(\nu)=\ln\int_{R^d}e^{|\nu|(\dot{\nu},y)}a(y)dy=\ln\int_{-s^+}^{s^+}e^{|\nu|\tau}a_{\dot{\nu}}(\tau)d\tau,{\rm where}a_{\dot{\nu}}(\tau)=\int_{(\dot{\nu},y)=\tau}a(y)dy',
G_{\boldsymbol{Y}}(\boldsymbol{s}) = \operatorname{E}\left\lbrace (\boldsymbol{\pi}^t \boldsymbol{s})^{\vert \boldsymbol{Y}\vert}\right\rbrace = G_{\vert \boldsymbol{Y}\vert}\left( \boldsymbol{\pi}^t \boldsymbol{s}\right),
\tilde \psi \phi_1(v) = \phi_2 (v) \Longleftrightarrow \tilde \psi \left(\iota_1^{\ast}T_1 \iota_1\right) \iota_1^{\ast}ev^{\ast}(v) = \left( \iota_2^{\ast}T_2 \iota_2\right) \iota_2^{\ast}ev^{\ast}(v),
\mathbf u(t) = \mathbf S(t) \mathbf u_0 + i \int_0^t \mathbf S(t-s) \mathbf N\left( \Theta_R^{\mathbf u}(s)\mathbf u(s) \right) \, ds + i \int_0^t \mathbf S(t-s) \mathbf M\left( \mathbf u(s) \right) \, dW(s),
-\frac{1}{\omega}\int_{0}^{\frac{2}{\sqrt{\omega}\lambda(t)}}\widetilde{\phi_{0}}(r) \sqrt{r}F_{4}(t,r\lambda(t)) dr = \frac{-1}{2\omega}\int_{0}^{\frac{2}{\sqrt{\omega}\lambda(t)}}r \phi_{0}(r) F_{4}(t,r\lambda(t)) dr
E_\mathfrak{a}^k(\sigma_\mathfrak{b}z,w) = \delta_{[\mathfrak{a}= \mathfrak{b}]}y^w + \rho_{\mathfrak{a}, \mathfrak{b}}^k(0, w) y^{1-w}+ \sum_{h \neq 0}\rho_{\mathfrak{a}, \mathfrak{b}}^k(h, w) W_{\frac{|h|}{h}\frac{k}{2}, w - \frac{1}{2}}(4\pi \lvert h \rvert y) e^{2\pi i h x}
\lambda_k= \frac{2\cos \frac{2\pi}{n}}{\cos^2 \frac{\pi}{n}}-\frac{1}{\cos^2 \frac{\pi}{n}}(\omega_k+\omega_k^{n-1})=\frac{4\sin \frac{\pi(k-1)}{n}\sin \frac{\pi(k+1)}{n}}{\cos^2 \frac{\pi}{n}},
\Delta_1 =d_1-\nu_1+\frac12(d_1^2-2\nu_1 d_1-x)N_1^{-1/3}-\frac 12\nu_1d_1^2N_1^{-2/3}\Delta_2 =2(d_2+\nu_1)+(\eta_1-\nu_1^2-2\nu_1d_2)N_1^{-1/3}, \Delta_3 =d_3-\Delta\nu+\frac12(d_3^2-2\Delta\nu d_3+y)N_2^{-1/3}-\frac 12\Delta\nu d_3^2N_2^{-2/3}\Delta_4 =2(d_4+\Delta\nu)+(\Delta\eta-\Delta\nu^2-2\Delta\nu d_4)N_2^{-1/3}.
H=\int_{-\infty}^{\infty}dx\; \left[{1\over 2}\langle\vec{u}_x,\vec{u}_x\rangle -{1\over 2}\langle\vec{u}\,^\dag, \vec{u}\rangle^2 +V(x) \langle\vec{u}\,^\dag, \vec{u}\rangle \right].
b_{\scriptscriptstyle{L-1, L+1}}/b_{\scriptscriptstyle{L, L+1}}=\frac{(i ka -1)\,a\,\psi_{\scriptscriptstyle{L, L+1}}^>(k,a)}{(\gamma_{\scriptscriptstyle{L+1, L+1}}+ \frac{1}{2})\,\psi_{\scriptscriptstyle{L+1, L+1}}^>(k,a)}\,( 1+ \mbox{O}(ka)^2)\,,
\sum_{n \in A}\lambda_A A= \sum_i \sum_{(\epsilon_1,\ldots,\epsilon_{k+1})\in \{0,1\}^{k+1}}(-1)^{\sum \epsilon_i}B^i \bigsqcup \{c^i_{1,\epsilon_1},\ldots,c^i_{k+1,\epsilon_{k+1}}\}&+\sum_i \sum_{(\epsilon_1,\ldots,\epsilon_k)\in \{0,1\}^k}(-1)^{\sum \epsilon_i}\{r_i\}\cup B^i \bigsqcup \{c^i_{1,\epsilon_1},\ldots,c^i_{k,\epsilon_k}\},
P_n(f^{2^k}, \phi_{2^k}, \epsilon) = \sup_E \left\{\sum_{x \in E}\exp S_n[f^{2^k}] \phi_{2^k}(x) \right\}\leq\sup_E \left\{\sum_{x \in E}\exp \left( S_{n 2^k}[f]\left(\frac{1}{2^m}\phi_{2^m}\right)(x)+ 4nC_1 \right) \right\},
X(z)=c_{0}+c_{1}(1-z_{0}z^{-1})+c_{2}(1-z_{0}z^{-1})(1-z_{1}z^{-1})+\cdot\cdot\cdot+c_{N-1}\prod_{k=0}^{N-2}(1-z_{k}z^{-1}),
U =V_1^*V_1 U V_1^*V_1 \sim V_1^* V_0 \begin{pmatrix}U 0 0 1 \end{pmatrix}V_0^*V_1 &= V_1^*V_0 \begin{pmatrix}1 0 0 W \end{pmatrix}\begin{pmatrix}U 0 0 1_{\infty}\end{pmatrix}\begin{pmatrix}1 0 0 W^* \end{pmatrix}V_0^*V_1.
\begin{aligned}\lim_{N\to \infty}\frac{\theta_{N}}{N^{d}}\;=\;\frac{1}{2\, v_d}\, d\ge 3\;, \lim_{N\to \infty}\frac{\theta_{N}}{N^{2}\log N}\;=\; \frac 1{\pi}d=2\;.\end{aligned}
\psi(x) ={}\sum_{j=-\nu}^{\infty}b_j x^j = \frac{1}{(\mu-1)!}\sum_{j=-\nu}^{\infty}\frac{(j+\nu+\mu)!}{(j+\nu)!}x^j ={}\mu x^{-\nu}(1-x)^{-\mu-1}, 0 < |x| < 1.
-\frac{1}{2}\left(\sum_{i = 1}^{p - 1}b_i\right)^2 < -\frac{1}{8}(p^3 - pc - p^2 - 2)^2 < -\frac{1}{8}p^6 + \frac{1}{4}p^4c - \frac{1}{8}p^2c^2 + \frac{1}{4}p^5
u_{li}\nu^l = [c^{ij}+ \nu^i \nu^j ]\nu^l u_{lj}= c^{ij}[D_j(\nu^l u_{l}) -D_j \nu^l u_l ] + \nu^i \nu^j \nu^l u_{lj}= c^{ij}D_j\varphi - c^{ij}u_l D_j \nu^l + \nu^i \nu^j \nu^l u_{lj}.
{{\cal G}^{- 1}}_{\mu \nu}^{\alpha \beta}\eta_{\alpha \beta}^{\mathrm{T T}}= \left( k^{2}- \Lambda_{l}+ \frac{1}{2 r^{2}}N ( N - 1 ) - \frac{N + d}{N + d - 2}\Lambda \right) \eta_{\mu \nu}^{\mathrm{T T}}.
l(m_B \otimes 1_X) = l(1_B \otimes l), r(1_X \otimes m_A) = r(r \otimes 1_A), l(\iota_B \otimes 1_X) = 1_X = r(1_X \otimes \iota_A), r(l \otimes 1_A) = l(1_B \otimes r).
w_{n,l}\left( \frac{-\lambda}{\lambda-1}\right) =\frac{\left(\lambda-1\right)^{l}}{l!}\binom{n+l}{l}^{-1}\mathcal{B}_{n+l}^{\left(l\right)}\left( \lambda\right) ,\lambda\in\mathbb{C}\backslash\{1\}\alpha=l\in\mathbb{N}.
u_0^{ns}(x)=\begin{cases}\cos^2 4 \pi \big(x-\frac{1}{2}\big), x \in \big(\frac{3}{8},\frac{5}{8}\big), 0, x \in [0,1] \backslash \big(\frac{3}{8},\frac{5}{8}\big), \end{cases}
D_{n-1}(A^1,\ldots,A^{n-1}) = \sum_{i,j}A^1_{i,j}Q^{i,j}(A^2,\ldots,A^{n-1}) ~,~ Q^{i,j}(A^2,\ldots,A^{n-1}) := \frac{(-1)^{i+j}}{n-1}D_{n-2}(M^{i,j}(A^2),\dots,M^{i,j}(A^{n-1})) ,
V_{triv, even}= \langle v_0+v_2+v_4+v_6 \rangle V_{triv, odd}= \langle v_1+v_3+v_5+v_7 \rangle V_{ref, even}= \langle v_0-v_2+v_4-v_6 \rangle V_{ref, odd}= \langle v_1-v_3+v_5-v_7 \rangle V_{1, even}= \langle v_0-v_4, v_2-v_6 \rangle V_{1, odd}= \langle v_1-v_5, v_3-v_7 \rangle.
\ |\hat{v}_0|_{\mathbf{L}^1(\mathbb{R}^3 )}\le C\delta^{\frac{1}{2}}\ |\hat{v}_0|_{\mathbf{L}^2(\mathbb{R}^3 )}=C\delta^{\frac{1}{2}}\ |v_0|_{\mathbf{L}^2(\mathbb{R}^3 )}\le CM
\tilde{T}_{2}=\breve{T}_{2}+R_{3},\breve{T}_{2}=\int\alpha_{20}(x_{t},y_{2t})H_{p}(\gamma_{10}(x_{t+1}))[\hat{\gamma}_{1}(x_{t+1})-\gamma_{10}(x_{t+1})]F_{0}(dw), R_{3}=\int\alpha_{20}(x_{t},y_{2t})H_{pp}(\bar{\gamma}_{1}(x_{t+1}))[\hat{\gamma}_{1}(x_{t+1})-\gamma_{10}(x_{t+1})]^{2}F_{0}(dw),
F^2_i(x^0; u, v) :=\left(\langle \nabla f_i(x^0), u\rangle, \langle \nabla f_i(x^0), v\rangle+\langle\xi^{*i}, u\rangle\right),\ \i\in I, G^2_j(x^0; u, v) :=\left(\langle \nabla g_j(x^0), u\rangle, \langle \nabla g_j(x^0), v\rangle+\langle\zeta^{*j}, u\rangle\right),\ \j\in J.
i\left< a_\mu^i a_\nu^j A_\rho^a A_\sigma^b\right>_{\rm bare}=ig^2f^{aic}f^{cjb}\left[2g_{\mu\nu}g_{\rho\sigma}-\left(1-\frac1\alpha\right) (g_{\mu\rho}g_{\nu\sigma}+g_{\mu\sigma}g_{\nu\rho})\right] .
\tau (Q(X)S^{(i)}_jR(X)[(\partial_{X_i}P)(X)]\#S^{(i)}) &=\tau(R(X)P(X)Q(X) \xi^{(i)}_j)-\tau(S^{(i)}_j[(\partial_{X_i}R)PQ+RP(\partial_{X_i}Q)](X)\#S^{(i)})=0.
\left( \begin{matrix}{{\bf 1}}&{0}\\{{\tilde{\bf c}}}&{{\bf 1}}\\ \end{matrix}\right) \qquad \qquad{\tilde{\bf c}}= \sum_{i}N_{i}g_{i}^{t}{\tilde{\bf c}}^{v}g_{i}+ \sum_{j}M_{j}g_{j}^{t}{\tilde{\bf c}}^{h}g_{j}
r_\gamma(s) = r(s) := \sup\left\{r>0 \ : \begin{array}{c}\ |\gamma(s+h) -\gamma(s)|\ge \frac r2 \ \mbox{for all}|h|\ge r, \ \mbox{and}|\gamma'(s+h)-\gamma'(s) | \le \frac{|h|}r \ \mbox{for all}|h|\le r. \end{array}\right\}.
\int_{U_0}|\tilde{F}_j-(1-b_{1}(\Psi_1))f_jF^{2}|^2_he^{-\varphi_1+v_{1}(\Psi_1)-\Psi_1}c(-v_{1}(\Psi_1)) \le \left(c(T_1)e^{-T_1}+\int_{T_1}^{T_1+2}c(s)e^{-s}ds\right) \int_{U_0}\mathbb{I}_{\{-T_1-2<\Psi_1<-T_1-1\}}|f_j|^2_he^{-\Psi_1},
M^{2}=[\begin{matrix}a^{2}&b^{2}-a^{2}\\ 0&b^{2}\end{matrix}],M^{3}=[\begin{matrix}a^{3}&b^{3}-a^{3}\\ 0&b^{3}\end{matrix}],M^{4}=[\begin{matrix}a^{4}&b^{4}-a^{4}\\ 0&b^{4}\end{matrix}],...
\eta_{M, M-1}(q|a, b) = \exp\Bigl(\int\limits_0^\infty \frac{dt}{t}\Bigl[(e^{-tq}-1)e^{-b_0 t}\frac{\prod\limits_{j=1}^{M-1}(1-e^{-b_j t})}{\prod\limits_{i=1}^M (1-e^{-a_i t})}+ q e^{-t}\frac{\prod\limits_{j=1}^{M-1}b_j}{\prod\limits_{i=1}^M a_i}\Bigr]\Bigr).
\mathcal{R}\Phi(A\circ B) =\Phi(\mathcal{R}(A\circ B)) \geq\Phi(\mathcal{R}A\circ\mathcal{R}B) =\Phi(\mathcal{R}A)\circ\Phi(\mathcal{R}B) =\mathcal{R}(\Phi( A))\circ\mathcal{R}(\Phi( B)).