harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
G(\kappa,\lambda;x):=\mathrm{MC}_{k-1}(x)\begin{array}{cc}\kappa \cline{2-2}+ x+\lambda\end{array}=\begin{array}{ccccc}\kappa_1 \kappa_{k-1}\kappa \cline{1-1}\cline{3-3}\cline{5-5}x+\lambda_1 +\cdots+ x+\lambda_{k-1}+ x+\lambda\end{array}.
m_1 =\max\{i_1:\lambda_{i_1,\dotsc,i_s}\ne 0i_2,\dotsc,i_s\}, m_2 =\max\{i_2:\lambda_{m_1,i_2,\dotsc,i_s}\ne 0i_3,\dotsc,i_s\}, m_3 =\max\{i_3:\lambda_{m_1,m_2,i_3,\dotsc,i_s}\ne 0i_4,\dotsc,i_s\}, & \vdotswithin{=}m_s =\max\{i_s:\lambda_{m_1,\dotsc,m_{s-1},i_s}\ne 0\}.
v^{(s)}(\vec{p})=\frac{-p/+m}{\sqrt{2m(E+m)}}v^{(s)}(\vec{0})=\sqrt{\frac{E+m}{2m}}[\begin{matrix}\frac{\vec{\sigma}\cdot\vec{p}}{E+m}\chi^{(s)}\\ \chi^{(s)}\end{matrix}]
\partial_{t}\int_{\Omega}-\tfrac{\gamma \tau_{0}}{\kappa}v \cdot q \mathrm{d}x= -\tfrac{\gamma \tau_{0}}{\kappa}\int_{\Omega}v_{t}\cdot q \mathrm{d}x+ \tfrac{\gamma \delta}{\kappa}\int_{\Omega}v \cdot q \mathrm{d}x + \gamma \int_{\Omega}v \cdot \nabla \theta \mathrm{d}x.
W_{0}(z) =\frac{1}{z-a_{0}^{(0)}-\sum_{j=1}^{p}\sum_{k=0}^{j}a_{-k}^{(j)}\,A_{j-k-1}^{(1)}(z)\,B_{k-1}^{(1)}(z)}W_{j}(z) =W_{0}(z)\,A_{j-1}^{(1)}(z)1\leq j\leq p.
d(\gamma_{\tau}\zeta_{\tau(1)},\ldots,\zeta_{\tau(n)})= d(\gamma_{\tau})\zeta_{\tau(1)},\ldots,\zeta_{\tau(n)}+\gamma_{\tau}\sum\limits_{i=0}^{n-1}\zeta_{\tau(1)}\ldots ,\ldots,d(\zeta_{\tau(j)}),\zeta_{\tau(j+1)}\ldots,\zeta_{\tau(n)}
\lim_{m_{1}\rightarrow 0}\Bigl | \langle ( \bar{\psi}_{1}\psi_{1}\rangle_{A}|_{h = 0}\Bigr | = \lim_{m_{2}\rightarrow 0}\Bigl | \langle ( \bar{\psi}_{2}\psi_{2}\rangle_{A}|_{h = 0}\Bigr |
\check{Q}_{i b}( \tau ) \check{\vec{A}}_{a \perp}( \tau ,{\vec{\eta}}_{i}( \tau ) ) = Q_{i a}( \tau ) [ P \, e^{\Omega_{s a}^{( \hat{\gamma})}( \eta^{( A )}( \tau , -{\vec{\eta}}_{i}( \tau ) ) ){\hat{T}}^{a}}]_{a b}\check{\vec{A}}_{b \perp}( \tau ,{\vec{\eta}}_{i}( \tau ) ) ,
\r^{k+\alpha}\bigg[r^{- (|q| + 2\kappa + 1)}(q_n +1) (q_n +2) \tilde b_{q+ \overline{n}, \kappa}+ r^{- (|q| + 2\kappa + 1)}\sum_{i \neq n}(q_i + 1) (q_i + 2) \tilde b_{q+ 2 \overline{i}- n,\kappa}- \r^{- (|q| + 2\kappa + 1)}(\kappa+1) \tilde b_{q- \overline{n}, \kappa+1}+ r^{-(|m|+ 2\ell)}c^{m,\ell}_{q,\kappa}\tilde b_{m,\ell}\bigg] = 0.
\begin{aligned}\frac{1}{\tau^k}[f(x)-f(\breve{x}^{k})]-\frac{1}{\tau^{k-1}}[f(x)-f(\breve{x}^{k-1})]+(u-\widetilde{u}^k)^TF(\widetilde{u}^k) + (2-\gamma)(1-\tau^k)\beta^k(A(x-\widetilde{x}^k))^T(A\breve{x}^{k-1}-b)\geq(v-\widetilde{v}^k)Q^k(v^k-\widetilde{v}^k),~\forall u,\end{aligned}
a(Q)=\sum_{j=0}^m\:\sum_{\alpha,\beta,\gamma}a_{j,\alpha,\beta,\gamma}(Q) \Big(\frac{y}{yy'-v^2}\Big)^\alpha\Big(\frac{v}{yy'-v^2}\Big)^\beta\Big(\frac{y'}{yy'-v^2}\Big)^\gamma S^{m-j}T^j.
\left|\int_0^t \langle (hu_s+\widetilde U)\otimes w, \nabla w\rangle d\tau\right| &\leq \frac{1}{4}\int_{T_1}^t\|\nabla w\|_2^2 d\tau+C\left(|h|_\infty\|u_s\|_3+\|v_0\|_{3,\infty}+M_3\right)\int_0^{T_1}\|\nabla w\|_2^2 d\tau
\big({}^{lr}R(u)\big)^{-1}=\big({}^{rl}R(u)\big)^{-1}=\big(1-hNu^{-1}\big)^{-1}\big(R(-u)-hNu^{-1}\big), begin{align*}0.4em] \big({}^{ll}R(u)\big)^{-1}=\big({}^{rr}R(u)\big)^{-1}=R(u)^{-1}=\big(1-h^2 u^{-2}\big)^{-1}R(-u),
\nabla_{(a_1, \cdots, a_{r})}fs = \rho(a_1 , \cdots, a_{r})(f)~ s + (-1)^{\widetilde{f}(\widetilde{a_{1}}+ \cdots \widetilde{a_{r}}+r)}f~\nabla_{(a_1, \cdots, a_{r})}s, \nabla_{(a_1, \cdots, f a_{r})}s = (-1)^{\widetilde{f}(\widetilde{a_{1}}+ \cdots \widetilde{a_{r}}+r +1)}f~ \nabla_{(a_1 , \cdots, a_{r})}s,
f(k;r,p)=\frac{\Gamma(k+r)}{k!\cdot\Gamma(r)}(1-p)^{k}p^{r}=\frac{\lambda^{k}}{k!}\cdot\frac{\Gamma(r+k)}{\Gamma(r)(r+\lambda)^{k}}\cdot\frac{1}{(1+\frac{\lambda}{r})^{r}}
2 \partial_{\sigma}\left( N g_{s}e^{\Phi_{D}}\rho^{\prime}\right) - \frac{\partial e^{\Phi_{D}}}{\partial \rho}\left( e^{2}- r^{2}e^{2}\omega^{2}+ ( r^{\prime})^{2}+ N g_{s}( \rho^{\prime})^{2}\right) = 0 \, ,
\displaystyle\sum\limits_{m_1,m_2}\langle m_1,l_1,m_2,l_2 | m_{12}, l_{12}\rangle P_{m_1}^{l_1}(x) P_{m_2}^{l_2}(y) = x^{-m_{12}-k}\displaystyle\sum\limits_{m=0}^{m_{12}+k}\frac{\langle m,l_1,m_{12}+k-m,l_2 | m_{12}, l_{12}\rangle}{\sqrt{m! (2 l_1 )_m}\sqrt{(m_{12}+k-m)! (2 l_2 )_{m_{12}+k-m}}}(x^2)^m.
S_n' \Big( \sigma'_1 \circ \sigma'_k(y') - \sigma_{k-1}'(y') \Big) = S_n' \Big( \sigma'_1 \circ \sigma'_k(y') - \sigma'_1(y') + \sigma'_1(y') - y' + y' - \sigma_{k-1}'(y') \Big) &= \sigma'_1 (S_n' (\sigma'_k(y') - y')) + S_n' (\sigma'_1(y') - y') + S_n' (y' - \sigma_{k-1}'(y')).
f(x)=\frac{\prod_{k=1}^{m^+}(x-\eta_k)^{m_k}\prod_{k}^{n^-}(x+\vartheta_k)^{n_k}(l_0+l_1x+\cdots+l_{M+1}x^{M+1})}{\prod_{i=1}^{M}(x-\beta_{i,\xi})\prod_{i=1}^{N}(x+\gamma_{i,q})\prod_{i=1}^{M}(x-\beta_{i,q})},
\Biggl [ \cdots \biggl [ \Bigl [ [ j_{+}{}^{n}, ( j_{+}{}^{m})^{\dagger}] , ( j_{+}{}^{i_{1}})^{\dagger}\Bigr ] , ( j_{+}{}^{i_{2}})^{\dagger}\biggr ] , \cdots ( j_{+}{}^{i_{k + 1}})^{\dagger}\Biggr ] \left|{\cal E}_{N}, 2 N \right\rangle = 0 ,
\varphi\left( \frac{\rho_{K_1}(u)}{\rho_{\widetilde{+}_{\varphi}(K_1,\dots,K_m)}(u)},\dots, \frac{\rho_{K_m}(u)}{\rho_{\widetilde{+}_{\varphi}(K_1,\dots,K_m)}(u)}\right) = 1.
\begin{array}{c}h^{k-1}(\tilde{y}^{(k-1)})\geq h^{k}(\tilde{y}^{(k)})+\frac{1}{2}\underset{i=1}{\overset{m+1}{\sum}}\|y_{i}^{(k)}-y_{i}^{(k-1)}\|^{2},\end{array}
I = \int_{\cal{M}}d^{d}\! x \sqrt{\tilde{g}}e^{\Phi}\left( \tilde{R}( \tilde{g}) + \left( \nabla \Phi \right)^{2}-{\frac{1}{1 2}}H_{\mu \nu \rho}H^{\mu \nu \rho}+ \Lambda \right) \ ,
n_{\gamma}=\frac{1}{\pi^{2}}(\frac{k_{B}T}{\hbar c})^{3}\times\int_{0}^{\infty}\frac{x^{2}}{e^{x}-1}dx=\frac{1}{\pi^{2}}(\frac{k_{B}T}{\hbar c})^{3}\times2\zeta(3)\approx20.3(\frac{T}{1K})^{3}cm^{-3},
\int_{\mathbb{R}^n}|V(x,0)|^p dx =\frac{1}{c}\int_{\mathbb{R}^{n+1}_{+}}c^2 |\nabla V(x,y)|^2 + m^2 c^4 V(x,y)^2 dx dy + (-mc^2 +\mu) \int_{\mathbb{R}^n}V(x,0)^2 dx.
= \;\frac{1}{(2\pi)^l}\left( \frac{-1}{2L}\right)^l\exp \left( \frac{1}{2}l^2 e^2 \frac{1}{\pi^2}\int d^2 p \;\frac{1}{p^2 + m_d^2}\; \frac{1}{p^2}\; \sin^2(p_1 L) \right)
E_M(yg^{-1}) =w^*\lim_\mu E_M(\phi(x_\mu)g^{-1})=w^*\lim_\mu \phi(x_\mu)_g &=w^*\lim_\mu \theta(E_M(x_\mu \sigma(z_{g^{-1}}g^{-1})))=0,
\left(-zq^{1/2},-zq;q\right)_{\infty}=\left(-zq^{1/2};q^{1/2}\right)_{\infty}=\sum_{n=0}^{\infty}\frac{q^{n(n+1)/4}z^{n}}{\left(q^{1/2};q^{1/2}\right)_{n}}=\sum_{k=0}^{\infty}z^{k}q^{k^{2}/2}S_{k}\left(-q^{-k+1/2};q\right)
y^{\alpha}\left(b^{-1}y^{\alpha}\cdot x\right) =x\left(y^{\alpha}\right)^{2}\cdot b^{-1}&=\left(y^{\alpha}\right)^{2}x\cdot b^{-1}=\left(y^{\alpha}\right)^{2}\cdot xb^{-1}=y^{\alpha}\left(y^{\alpha}\cdot xb^{-1}\right) &b^{-1}y^{\alpha}\cdot x=y^{\alpha}\cdot xb^{-1}&y^{\alpha}b^{-1}\cdot x=y^{\alpha}\cdot b^{-1}x
P^\varepsilon_\mu \:P^\varepsilon_{\mu'}\;=\; \frac{1}{4}\left( X\: t^\varepsilon_\mu\: t^\varepsilon_{\mu'}\:X^* + X\: t^\varepsilon_\mu\: X\: t^\varepsilon_{\mu'}+t^\varepsilon_\mu\: X^*\: t^\varepsilon_{\mu'}\:X^* \right).
H=-\frac{\hbar^{2}}{2M_{tot}}\nabla_{X}^{2}+H^{\prime}withH^{\prime}=-\frac{\hbar^{2}}{2}\sum_{i=1}^{N_{tot}-1}\frac{1}{m_{i}}\nabla_{i}^{2}+\frac{\hbar^{2}}{2M_{tot}}\sum_{i,j=1}^{N_{tot}-1}\nabla_{i}\cdot\nabla_{j}+V(t).
\begin{cases}\pi_i^2 = \pi_i 1 \leq i \leq n-1, \pi_i \pi_j = \pi_j \pi_i |i - j| > 1, \pi_i \pi_{i+1}\pi_i = \pi_{i+1}\pi_i \pi_{i+1}1 \leq i \leq n-2.\end{cases}
\rho_{j_{k},j_{l}}^{[n]}=\frac{1}{2N}\cdot\frac{\rho_{j_{k},j_{l},2}^{[n]}\rho_{j_{k},j_{l},1}^{[n]}}{\rho_{j_{k},j_{l},2}^{[n]}+3\left(\mathrm{SNR}_{T}+N\right)},
\textrm{P}\frac{1}{z^{2}}= \frac{1}{2}\left[ \frac{1}{( z + i \epsilon )^{2}}+ \frac{1}{( z - i \epsilon )^{2}}\right] = -{\frac{\partial}{\partial z}}\left[ \textrm{P}\left({\frac{1}{z}}\right) \right] .
\frac{1}{N}\sum_{n=0}^{N-1}\prod_{k=1}^{j-1}f_k(T_{1}^{p_{1,k}(n)}\cdots T_{d}^{p_{d,k}(n)}x)(f_j-\mathbb{E}(f_j|\mathcal{A}_{g_{j_0}}))(T_{1}^{p_{1,j}(n)}\cdots T_{d}^{p_{d,j}(n)}x)\cdot &\ \ \ \ \ \ \ \ \prod_{l=j+1}^{m}\mathbb{E}(f_{l}|\mathcal{A})(T_{1}^{p_{1,l}(n)}\cdots T_{d}^{p_{d,l}(n)}x)\rightarrow 0.
\mathbf{H}(q,k)=\sum_{p=1}^{P}\sum_{r=1}^{R_p}\beta_{r,p}\mathbf{a}_{\mathrm{r}}(\theta_{r,p})\mathbf{a}^T_{\mathrm{t}}(\phi_{r,p})\exp(jq\gamma_{r,p}-jk\eta_p)
\Psi(W_{v_m},\phi^m,\tilde f_s^i)-\Psi(W_{v_m'}, \phi^m, \tilde f_s^i) = q_F^{-3i+m}\int_{T}( W_{v_m}(t(a)w)-W_{v_m'}(t(a)w))(\omega_{\psi^{-1}}(w)\phi^m)(a)\chi(a)\eta^{-1}(a)|a|^{-s}da.
\left.\begin{array}{ll}\displaystyle\int_{\mathbb{R}^N}\left(a_2|\phi(t,x)|^2+a_1|\psi(t,x)|^2\right)dx=\int_{\mathbb{R}^N}\left(a_2|\phi_0(x)|^2+a_1|\psi_0(x)|^2\right)dx;\end{array}\right.
\mathcal{D}(A)=\{U\in \mathcal{H}_{\boldsymbol\Gamma}^m\,:\, \mathcal{A}^0(\widehat U)U=\mathcal{E}_1^0(\widehat U)U_x + \mathcal{E}_2^0(\widehat U) U_y\in\mathcal{H}_{\boldsymbol\Gamma}^m\}.
y^{2}= \left( \sum_{i = 0}^{N}s_{i}x^{N - i}+ ( - 1 )^{N}\left( \Lambda_{1}^{2 N}+ \Lambda_{2}^{2 N}\right) \right)^{2}- 4 \Lambda_{1}^{2 N}\Lambda_{2}^{2 N},
n(R_2-\epsilon_n) \leq I(X_1^n;Y_2^n) = h(\sqrt{a}X_1^n+\sqrt{b}X_2^n+Z_2^n) - h(\sqrt{b}X_2^n+Z_2^n) \leq \frac{n}{2}\log(1+aP_1+bP_2) - n\xi(b),
R \sqrt{\lambda}\left( \ln{\lambda}+ \sigma \right)^{\prime}\varepsilon_{u l}^{\pm}= i{\bf P}_{i}^{N K}\gamma^{i}\varepsilon_{l u}^{\mp}\pm{\frac{1}{2}}{\bf P}_{i j}^{R F}\gamma^{i j}\varepsilon_{l u}^{\pm},
R_i(\l)=R_i(\l_*)\frac{\l-\l^{(i)}_0}{\l_*-\l^{(i)}_0}\sqrt{\frac{\l_*}{\l}}\prod_{j\ge 1}\frac{(\l-\l^{(i)}_j)\sqrt{(\l_*-a_j)(\l_*-b_j)}}{(\l_*-\l^{(i)}_j)\sqrt{(\l-a_j)(\l-b_j)}},
(p-1)\eta^2 u_k^{p-2}(a\nabla u_k,\nabla u_k)+2u_k^{p-1}\eta (a\nabla u_k,\nabla \eta) = \frac{4(p-1)}{p^2}(a\nabla (\eta u_k^{p/2}),\nabla (\eta u_k^{p/2})) \;\;\;\;-\frac{4(p-2)}{p^2}u_k^{p/2}(a\nabla (\eta u_k^{p/2}),\nabla \eta)-\frac{4}{p^2}u_k^p(a\nabla \eta,\nabla \eta).
\phi_{k_i}^{\alpha_i,\beta_i}(x_i)=2^{(\alpha_i+\beta_i+1)/2}\tilde{P}_{k_i}^{\alpha_i,\beta_i}(\cos x_i)\,\bigg(\sin \frac{x_i}{2}\bigg)^{\alpha_i+1/2}\,\bigg(\cos \frac{x_i}{2}\bigg)^{\beta_i+1/2},
\mathcal{M}(\tilde u_2,a_2,b_2)u-\mathcal{M}(\tilde u_1,a_1,b_1)u= [\mathcal{M}(\tilde u_2,a_2,b_2)u-\mathcal{M}(\tilde u_1,a_2,b_2)u ] &+[\mathcal{M}(\tilde u_1,a_2,b_2)u-\mathcal{M}(\tilde u_1,a_1,b_2)u] &+[\mathcal{M}(\tilde u_1,a_1,b_2)u-\mathcal{M}(\tilde u_1,a_1,b_1)u].
\tilde{\Phi}= (-1)^{j_1^2 -j_1^3 +j_2^2 -j_2^3 +r_2^2 -r_2^3}\sqrt{\frac{(2r_2^1 + 1)(2 j_2^1 + 1)}{(8 \pi^2 )^2 (2j_1^1 +1)}}\prod_{i = 1}^{3}(-1)^{j_1^i -r_2^i}\sqrt{(2r_2^i +1)(2j_1^i +1)(2j_2^i +1)}.
\gamma_{j,i}^{(k+1)}= \gamma_{j,i}^{(k)}- v_{j,i}^{(k)}= \cdots = \gamma_{j,i}^{(1)}- \sum_{\ell=1}^{k}v_{j,i}^{(\ell)}= 1 - \epsilon_{j,i}^{(k)}, j = 2, 3, \ldots, d.
G_\ell(n) =\frac{1}{6}\cdot\begin{cases}\zeta_2(n)+3\zeta_1(n)-12\zeta_0(n) \zeta_2(n)+\zeta_1(n)-2\zeta_0(n) \zeta_2(n)-\zeta_1(n)+4\zeta_0(n) \text{if $n\equiv 2\mod 3$}\rlap{\,.}\end{cases}
\sigma_{\ell - \ell_P}(E_o^P(w)) = \sigma_{\ell - \ell_P}(E^P_o(s))\sigma_{\ell - \ell_P}(E^P_{o\cdot s}(s^{-1}w) & = (-1)^{\ell(s) - \ell_P(s)}\sigma(E^P_o(s))(-1)^{\ell(s^{-1}w) - \ell_P(s^{-1}w)}\sigma(E^P_{o\cdot s}(s^{-1}w)).
i A^{+}= \left[ \begin{array}{l l}{{< \psi_{1}^{( 0 )}| d \psi_{1}^{( 0 )}>}}&{{< \psi_{2}^{( 0 )}| d \psi_{1}^{( 0 )}>}}\\{{< \psi_{1}^{( 0 )}| d \psi_{2}^{( 0 )}>}}&{{< \psi_{2}^{( 0 )}| d \psi_{2}^{( 0 )}>}}\\ \end{array}\right] = i A_{a}d x^{a},
E_{C}^{\mathrm{f i x e d}}( R \to \infty , T ) = \frac{R T}{2}\sum_{n = - \infty}^{+ \infty}\Omega_{n}= 2 \pi R T^{2}\sum_{n = 1}^{\infty}n = 2 \pi R T^{2}\zeta ( - 1 ) = - \frac{\pi R T^{2}}{6}\,{,}
\gamma_F(\frac{1}{2}\psi \circ q_{10}) = \gamma_F(\frac{1}{2}\psi)^2 \cdot \gamma_F(-J_1, \frac{1}{2}\psi) \cdot (J_2, \frac{2 (a-ct)}{b-dt})_F = \gamma_F(J_1, \frac{1}{2}\psi)^{-1}\cdot (J_1, \frac{2 (a-ct)}{b-dt})_F.
\big( 1-z^{2}\big) \frac{d^{2}}{dz^{2}}P_{n}^{(N,M)}( z ) + [ (M-N)-z ( N+M+2 ) ] \frac{d}{dz}P_{n}^{(N,M)}(z) \qquad{}= -n ( N+M+n+1 ) P_{n}^{(N,M)}(z) ,
\lim_{L\to\infty}\frac 1{L^d}\int_0^T\Psi^\star_L\bigl(\mu^L_t,F^V(\mu^L_t)\bigr)\;\!\mathrm dt = \frac 12\int_0^T \|\Delta\phi(\rho_t)+\nabla\cdot(\chi(\rho_t)\nabla V)\|_{-1,\chi(\rho_t)}^2\;\!\mathrm dt.
| \hat{g}(j) | \leq h ! g \left| \frac{\sin ( \frac{\pi}{hN}( \frac{h! hg N - |A|^h}{h! g}+ 1 ) )}{\sin ( \frac{\pi}{hN})}\right|= h! g \left| \frac{\sin ( \pi Q_h - \frac{\pi}{hN})}{\sin ( \frac{\pi}{hN})}\right|.
J_{42}= -\frac{4}{mn\pi^{2}}\int_{0}^{\pi -h_{1}}\int_{h_{2}}^{\pi}H_{x,z_{1},y,z_{2}}(t_{1}+h_{1},t_{2})S(t_1,t_2) & \times \left( \frac{1}{\left( 2\sin \frac{t_{2}}{2}\right)^{2}}-\frac{1}{t_{2}{}^{2}}\right) \frac{1}{\left( t_{1}+h_{1}\right){}^{2}}dt_{1}dt_{2}.
\frac{f(x+y)}{f(x)}= \frac{\sup_{t\ge x+y}f (t)}{\inf_{s\in [x_0,x]}f(s)}\frac{f(x+y)}{\sup_{t\ge x+y}f(t)}\frac{\inf_{s\in [x_0,x]}f(s)}{f(x)}& \ge \frac{F(x+y+\Delta)}{F(x-c+\Delta)}\frac{f(x+y)}{\sup_{t\ge x+y}f(t)}\frac{\inf_{s\in [x_0,x]}f(s)}{f(x)}
e^{ih}=e^{i(h_{1,n}+h_{2,n}+h_{3,n}+h_{4,n})}e^{ic_n}&=e^{i(h_{1,n}+h_{2,n})}e^{i(h_{3,n}+h_{4,n})}e^{ic_n'}&= e^{ih_{1,n}}e^{i h_{2,n}}e^{ih_{3,n}}e^{i h_{4,n}}e^{ic_n''},
\int_{\{-t_3\le \Psi<-t_4\}}|F_{t_0,\tilde{c}}|^2_hc(-\Psi)= \int_{\{-t_3\le \Psi<-t_4\}}|\tilde{F}|^2_hc(-\Psi) + \int_{\{-t_3\le \Psi<-t_4\}}|F_{t_0,\tilde{c}}-\tilde{F}|^2_hc(-\Psi)
\hat{P}= - \frac{2 \lambda^{2}}{4 !}\left( \begin{array}{c c}{3{\varphi_{1}}^{2}+{\varphi_{2}}^{2}}{2{\varphi_{1}}{\varphi_{2}}}{2{\varphi_{1}}{\varphi_{2}}}{3{\varphi_{2}}^{2}+{\varphi_{1}}^{2}}\end{array}\right) ,
\| \partial_{\Lambda}\partial^{p}C_{2 m}^{( p , q )}\| \leq{\frac{1}{\Lambda}}P \left( l n{\frac{\Lambda}{\Lambda_{R}}}\right){\frac{1}{\Lambda^{2 m - 6}}}\left({\frac{\Lambda}{M}}\right)^{2 - \epsilon}\left({\frac{M}{\Lambda_{R}}}\right)^{\epsilon^{\prime}}, \qquad q > 1 , M > \Lambda_{R}
|\frac{-1-\rho^{2}+r^{2}}{\sqrt{(1+\rho^{2}-r^{2})^{2}+4r^{2}}}+F_{3}(r,\rho,\lambda(s))| \leq \begin{cases}2, \rho \leq 4r |\frac{-1-\rho^{2}+r^{2}}{\sqrt{(1+\rho^{2}-r^{2})^{2}+4r^{2}}}+1|+|-1+F_{3}(r,\rho,\lambda(s))|, \rho > 4r\end{cases}&\leq \begin{cases}2, \rho\leq 4r \frac{r^{2}}{\rho^{2}}, \rho > 4r\end{cases}
d_n(c_1a_1b_1 \dots c_1a_1b_1 \otimes 1) = c_1a_1b_1 \dots c_1a_1b_1c_1 \otimes a_1b_1 + c_1a_1b_1 \dots c_1a_1b_1a_0 \otimes b_0c_0
\displaystyle MAE=\displaystyle \frac{1}{N}\sum_{i=0}^{N}|[\sum_{\substack{j\in J\setminus J_{K(n)}}}\frac{B_{t_{j+1}}+ B_{t_{j}}}{2}(B_{t_{j+1}}-B_{t_{j}})]-\frac{1}{2}B_{T}^2 |.
E_{A_{j}}(au_{k}au_{k}(1-a^{2})u_{k}^*au_{k}^*a) =aE_{A_{j}}(u_{k}au_{k}(1-a^{2})u_{k}^*au_{k}^*)a &=a E_{A}(u_{k}au_{k}(1-a^{2})u_{k}^*au_{k}^*)a=0
=(-1)^{n-1}n^{n-2}(n+1)^{n-1}\times\left|\begin{array}{cccccc}n(n+1)/2 0 0 \cdots 0 0 0 1 0 \cdots 0 0 0 0 1 \cdots 0 0 0 0 0 \cdots 0 0 \cdots \cdots \cdots \cdots \cdots \cdots 0 0 0 \cdots 0 1 \end{array}\right|
L^{\rm resc,h}_n(u):=\frac{L_{(0,0)\to(\gamma^2 n+ \beta_1 u n^{2/3},n)}- \left((1+\gamma)^2 n + 2 u (1+\gamma)^{5/3}\gamma^{1/3}n^{2/3}-\beta_2 u^2 n^{1/3}\right)}{\beta_2 n^{1/3}},
\frac{d}{dx}\bigg(\frac{\Gamma(a+b-c-x)}{\Gamma(a-c-x)}\bigg)\bigg|_{x=0}= \frac{d\big((a-c-x)_b\big)}{dx}\bigg|_{x=0}= -\prod_{\substack{0\leq j<b j\neq c-a}}(a-c+j)=(-1)^{b-1}\cdot\frac{\Gamma_p(a+b-c)}{\Gamma_p(a-c)}.
\beta = \frac{1 6 \sigma^{2}\eta^{2}}{\alpha}\int_{0}^{\infty}\mathrm{d}\xi \; \xi^{2}\phi_{\sigma}^{2}\left( \xi \right) \, , \quad \phi_{\sigma}\left( \xi \right) = \mathrm{e}^{- 2 \sigma \xi}\int_{0}^{\infty}\mathrm{d}s \; \mathrm{e}^{- 2 \xi s}\,{\cal V}_{\sigma}\left( \eta , \zeta , s \right)
E_i(h_{12}^3) =0, i=1,2,3, h_{12}^3\left(2(\omega_{13}^2+\omega_{21}^3)+\frac{1}{\sqrt{3}}\right) =\frac{1}{3}\sin(2(\theta_1-\theta_2)), h_{12}^3\left(2(\omega_{12}^3+\omega_{31}^2)-\frac{1}{\sqrt{3}}\right) =\frac{1}{3}\sin(2(\theta_1-\theta_3)), h_{12}^3\left(2(\omega_{21}^3+\omega_{32}^1)+\frac{1}{\sqrt{3}}\right) =\frac{1}{3}\sin(2(\theta_2-\theta_3)).
\gamma_{s}= \frac{\left( - 1 \right)^{s}}2 \int_{\Sigma_{s}}\int_{\partial \Sigma_{s}={\bf c}}\frac{\left( \vec{R}- i \frac 1 2 \vec{\Gamma}\right) \cdot d \vec{S}}{\left[ \left( \vec{R}- i \frac 1 2 \vec{\Gamma}\right)^{2}\right]^{\frac 3 2}},
\frac{dP^{\mu}|_{\mathcal{G}_{1}}|\mathcal{G}_{2}}{dP^{\nu}|_{\mathcal{G}_{1}}|\mathcal{G}_{2}}=\frac{E^{\nu}[\frac{d\mu}{d\nu}(X_{0})|\mathcal{G}_{1}\vee \mathcal{G}_{2}]}{E^{\nu}[\frac{d\mu}{d\nu}(X_{0})|\mathcal{G}_{2}]}~~~P^{\mu}~a.s.
\alpha_n^{-1}B_\lambda(s) =i^{nk}\,2^{-n(2n+1)(s+\frac12)+n}\times \int\limits_{\mathfrak{a}^+}\prod_{1\leq i<j\leq n}(\cosh(a_i)^2-\cosh(a_j)^2)\prod_{j=1}^n \cosh(a_j)^{-(2n+1)s+\frac12-n-k}\prod_{j=1}^n\sinh(a_j)\,dH.
(u_{p_{i_1},j_1}\cdot u_{p_{i_2},j_2})(a) = u_{p_{i_1},j_1}(a)\cdot u_{p_{i_2},j_2}(a) = \begin{cases}\frac{\displaystyle n}{\displaystyle{p_{i_1}^{e_{i_1}}}}a^{j_1}\cdot \frac{\displaystyle n}{\displaystyle{p_{i_2}^{e_{i_2}}}}a^{j_2}p_{i_1}p_{i_2}\:|\:a 0 \mbox{otherwise.}\end{cases}
\|f\|_{1+k_{\gamma,}}^2 =\left| \int^1_0 f(y) \,{\rm d}y \right|^2+ \frac{1}{\gamma}\left(\sum_{\nu=1}^{r-1}\left|\int_0^1 f^{(\nu)}(y) \,{\rm d}y \right|^2 + \int^1_0 |f^{(r)}(y)|^2 \,{\rm d}y\right),
\big[S_j,\big(I_k(\kappa)+S_k\big)^{-1}\big] =\big(I_k(\kappa)+S_k\big)^{-1}\big[I_k(\kappa)+S_k,S_j\big]\big(I_k(\kappa)+S_k\big)^{-1}&=\big(I_k(\kappa)+S_k\big)^{-1}\big[I_k(0)+\O(\kappa)+S_k,S_j\big]\big(I_k(\kappa)+S_k\big)^{-1}&=\big(I_k(\kappa)+S_k\big)^{-1}\big[\O(\kappa),S_j\big]\big(I_k(\kappa)+S_k\big)^{-1},
C(s,N,\mu,r)=C(N,\mu)=\pi^{\frac{\mu}{2}}\frac{\Gamma(\frac{N-\mu}{2})}{\Gamma(N-\frac{\mu}{2})}\Bigl(\frac{\Gamma(\frac N2)}{\Gamma(N)}\Bigr)^{-1+\frac{\mu}{N}}.
\frac{1}{8 \pi}\int_{\partial{\cal M}}T r{\cal K}= \frac{1}{8 \pi}\frac{\partial}{\partial{\bf n}}\, ( \mathrm{V o l}\, \partial{\cal M}) = \frac{1}{8 \pi}\frac{\partial}{\partial \, t}\, \left[ \left( \frac{3}{2}\, t \right)^{1 / 3}\,{\cal V}\right] \ .
eV^{\rm QED_{3}}(\vec r)=\frac{e^2}{2\pi}\left(1+\frac{e^2}{8\pi m}\right)K_{0}\left(\frac{e^2}{8\pi}r\right)+\frac{8}{mr^{2}}\left[1-\frac{e^2}{8\pi}r K_{1}\left(\frac{e^2}{8\pi}r\right)\right]L.
\begin{aligned}\|\sqrt{\rho_{n}}\partial_{x}p_{n}(\rho_{n})\|_{L^{\infty}_{t}L^{2}_{x}}= \| \gamma_{n}\rho_{n}^{\gamma_{n}-\frac{1}{2}}\partial_{x}\rho_{n}\|_{L^{\infty}_{t}L^{2}_{x}}\ge \gamma_{n}\underline{\rho_{n}}^{\gamma_{n}-\frac{1}{2}}\|\partial_{x}\rho_{n}\|_{L^{\infty}_{t}L^{2}_{x}}. \end{aligned}
\begin{cases}-\Delta \eta_t-k^2\eta_t=0\,\,\,\,\Omega\backslash K_t begin{align*}0.3cm] \frac{\partial \eta_t}{\partial n}=0\,\,\,\,\partial K_t begin{align*}0.3cm] \eta_t=0\,\,\,\,\partial\Omega.\end{cases}
L_{p_{\epsilon(1)}\left(T_1E(S_1)\right)}= L_{p_{\epsilon(1)}\left(T_1R_{E(S_1)}1_B\right)}= L_{p_{\epsilon(1)}\left(T_11_B\right) E(S_1)}= L_{p_{\epsilon(1)}\left(T_11_B\right)}L_{E(S_1)},
S^{(s)}_n =-i \lambda_s \sum_{k=s}^n k! [\rho^k] \phi^s(\rho) \cdot B_{n,k}\left( b_1, b_2, \ldots \right) = -i\lambda_s[\phi^n] \left( \big(\phi (\rho(\phi)) \big)^s \right) =-i\lambda_s [\phi^n]\phi^s = -i\lambda_s \delta_{ns}.
|I_2| \leq\int_{\frac{1}{\beta}\leq|x|\leq\frac{2}{\beta}}\frac{1}{|x|^{n-\beta}}\,dx+\int_{\frac{1}{2\beta}\leq|x-z|\leq\frac{1}{\beta}}\frac{1}{|x-z|^{n-\beta}}\,dx &\leq C\frac{2^\beta-1}{\beta}\beta^{-\beta}+C\frac{1-2^{-\beta}}{\beta}\beta^{-\beta}.
\begin{array}{l}\mu^{(0)}_{n'}=\frac{l_{n'}}{l_{m'}}\mu_{m'}^{(0)}+\left(1-\frac{l_{n'}}{l_{m'}}\right)\mu a_{m'n'}=\sqrt{\frac{l_{m'}}{l_{n'}}}(\mu_{n'}^{(0)}-\mu)=\sqrt{\frac{l_{n'}}{l_{m'}}}(\mu^{(0)}_{m'}-\mu),\end{array}
[u]_{\frac{1+s}{2}}^2-\l_1^s\,\|u\|_{L^2}^2\ge\frac{1}{4}\,[u]_{\frac{1+s}{2}}^2+\l_1^s\bigg(\frac12\,\sum_{k=2}^{\infty}\sum_{i=1}^{d(k)}|a^i_k|^2-\sum_{i=1}^n|a^i_1|^2-|a_0|^2\bigg)\,.
\mathbf{j}_{\operatorname*{m}}=\mathbb{A}_{j}^{\operatorname*{ext}}\nabla_{\operatorname*{pw};j}u_{\operatorname*{m}}=\mathbb{A}_{j}^{\operatorname*{ext}}\nabla_{\operatorname*{pw};j}u_{\operatorname*{uw}}.
\frac{S_{L_{1}l}^{( k - 2 )}S_{L_{2}l}^{( k - 2 )}}{S_{0 l}^{( k - 2 )}}= \sum_{L = 0}^{k - 2}N_{L_{1}, L_{2}}^{L}S_{L l}^{( k - 2 )}~ ,
r := \begin{cases}0,{\mathcal E}_1^2(1+b_1^2) \leq \frac{1}{10}\sum_{j > 1}{\mathcal E}_j^2(1+b_j^2) 1,{\mathcal E}_1^2(1+b_1^2) > \frac{1}{10}\sum_{j > 1}{\mathcal E}_j^2(1+b_j^2),\;{\mathcal E}_2^2(1+b_2^2) \leq \frac{1}{10}\sum_{j > 2}{\mathcal E}_j^2(1+b_j^2) 2, \end{cases}
\sup_{t\in[0,T]}\left|\mathbf{A}_n(t)\right|\leq\max_{j=0,1,\dots,\lfloor T/h\rfloor}\left(\left|\mathbf{A}_n(u_j)\right|+\sup_{t\in[u_j,u_{j+1}]}\left|\mathbf{A}_n(t)-\mathbf{A}_n(u_j)\right|\right).
S_\omega := \{(k_1,s_1) \dots (k_n,s_n) \in (\omega \times S)^* \ | \s_i{R}s_{i+1}~(0 \leq i < n)\}, & R_\omega[(k_1,s_1) \cdots (k_n,s_n)] := \{(k_1,s_1)\cdots(k_n,s_n)(k_{n+1},s_{n+1}) \ : ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ : \k_{n+1}\in \omega, s_n{R}s_{n+1}\}.
{\frac{1}{1 6 \pi G_{r e n}}}={\frac{1}{1 6 \pi G_{B}}}+{\frac{1}{1 2}}{\frac{1}{( 4 \pi )^{3 / 2}}}\int_{\epsilon^{2}}^{\infty}{\frac{d s}{s^{3 / 2}}}e^{- \mu s / l^{2}}
\Upsilon_i(\epsilon_i,\nu_i) = \frac{(1-(-1)^k)n}{2k}(\delta_{i,3}+\delta_{i,4})+(-1)^i\frac{1-(-1)^n}{4}+\frac{1}{2}(-1)^i \displaystyle{\sum_{i=n+3}^{2n}}c^{n+3-2i}P_{i-n-3,-i}.
|Df^{m_q}(x)(v)| =\prod_{i=0}^{q-1}|Df^{m_{i+1}-m_i}(f^{m_i}(x))(v)| &\geq \prod_{i=0}^{q-1}\lambda_0^{2m_{i+1}-2m_i}\frac{|\xi_{m_{i+1}}(x)-c(\lambda^{m_{i+1}-1}z_0)|}{|\xi_{m_{i}}(x)-c(\lambda^{m_i-1}z_0)|}&=\lambda_0^{2m_q}\frac{|\xi_{m_q}(x)-c(\lambda^{m_q-1}z_0)|}{|w_0-c(z_0/\lambda)|},
\frac{-1}{2\omega}\int_{0}^{\frac{2}{\sqrt{\omega}\lambda(t)}}r \phi_{0}(r) F_{4}(t,r\lambda(t)) dr = \frac{1}{2\omega}\int_{\frac{2}{\sqrt{\omega}\lambda(t)}}^{\infty}\phi_{0}(r) F_{4}(t,r\lambda(t)) r dr
i_{n_k, n_{k+1}}(\Theta_{k,\Lambda}) = i_{n_k, n_{k+1}}\Big(\sum_{\eta \in E^{<n_k},s(\eta) \in \Lambda}\theta_{n_k,\eta}\Big) = \sum_{\zeta \in E^{<n_{k+1}}, s([\zeta]_{n_k}) \in \Lambda}\theta_{n_{k+1},\zeta}= \sum_{\zeta \in E^{<n_{k+1}}, s(\zeta) \in \Lambda}\theta_{n_{k+1},\zeta}= \Theta_{k+1,\Lambda}.
\frac{1-w}{u}=N(x,t_{m})\mathrm{exp}(w_{0}^{\infty-})\frac{\theta(\underline{z}(P_{\infty-},D_{\underline{\hat{\nu}}(x,t_{m})}))}{\theta(\underline{z}(P_{\infty-},D_{\underline{\hat{\mu}}(x,t_{m})}))},
p x E\big[XY^{p-\gamma}]E\big[Y^{p}\big]^{\frac{\beta}{p}}\leq p x E\big[X\big]_{\frac{p}{\gamma}}E\big[Y^{p}\big]^{1 - \frac{\gamma - \beta}{p}}&\leq \big[X\big]_{\frac{p}{\gamma}}\big((\gamma - \beta)x^{\frac{p}{\gamma-\beta}}+ (p + \beta - \gamma)E\big[Y^{p}\big]\big)
[t^k]\circ \mathcal{T}_{\bold{c}}(t^m) =\dfrac{t^{k+m}}{c_m}=\dfrac{1}{c_{m+k}}\dfrac{A(m+k-1)\cdots A(m)}{B(m+k)\cdots B(m+1)}t^{k+m}&=\mathcal{T}_{\bold{c}}\circ A(\theta_t-1)\circ \cdots \circ A(\theta_t-k)\circ B(\theta_t)^{-1}\circ \cdots \circ B(\theta_t-k+1)^{-1}\circ [t^k](t^m) \enspace.
Q_j(\alpha_p,\beta_p,\gamma_p,\delta_p)=\lambda_{(\pi_1\boxtimes Sym^2\pi_2) \times(\widetilde\pi_1\boxtimes Sym^2\widetilde\pi_2)}(p^j)-|\lambda_{\pi_1}( p^j)|^2 |\lambda_{\pi_2}( p^{2j})|^2
a^{-1}K^{-2}- \frac{a^{-1}q - a q^{-1}}{q-q^{-1}}K^{-1}B^{-1}- \frac{a q - a^{-1}q^{-1}}{q-q^{-1}}B^{-1}K^{-1}+ a B^{-2}= 0.
\left(\frac{d}{dt}-\Delta_{\omega_{\varphi_{\epsilon}}}\right)\varphi_{\epsilon}=\dot{\varphi}_{\epsilon}-{\rm tr}_{\omega_{\varphi_{\epsilon}}}(\omega_{\varphi_{\epsilon}}-\omega_{\epsilon})=\dot{\varphi}-n+{\rm tr}_{\omega_{\varphi_{\epsilon}}\omega_{\epsilon}}\geq{\rm tr}_{\omega_{\varphi_{\epsilon}}}\omega_{\epsilon}-(C+n).
P_p(x_0)\ge \left(\frac{\varepsilon}{2l}\right)^{K_1}\prod_{i=1}^{K_1}\left(1+\kappa (i-1)\right), P_e(x_0)\ge \left(\frac{\delta}{2l}\right)^{K_2}\prod_{i=1}^{K_2}\left(1+\kappa (i-1)\right).
\varphi(r) = \frac 1{\mathsf{d}(r)}\varphi(\mathsf{n}(r)) = \frac 1{\mathsf{d}(r)}\sum_{i=1}^k c_i \varphi(n_i) = \frac 1{\mathsf{d}(r)}\sum_{i=1}^k c_i n_i \frac{\varphi (n_j)}{n_j}= rq.
f_k(q) = (q;q)_\infty (q;q^2)_\infty \sum_{n=0}^\infty a_k(n)q^n \qquad\ =(q;q^2)_\infty (q^{k};q^{k})_\infty\sum_{n=1}^{\infty}\frac{q^{kn}+ 2q^{2kn}+\dots+ (k-1)q^{(k-1)kn}}{1 + q^{kn}+ q^{2kn}+\dots+ q^{(k-1)kn}}.
\partial^{\mu}{\cal{G}}_{\mu \mu_{2 n}. . . \mu_{s}}^{\{n - 1 \}}\; \; = \; \;{\frac{( s + 1 - 2 n )}{( s + 2 - 2 n )}}\, \partial_{( \mu_{2 n}}\,{\cal{G}}_{\mu_{2 n + 1}. . . \mu_{s})}^{\{n \}}
\dot{x}= \lambda-dx+ rx \left( 1- \frac{x}{K}\right)- \frac{\beta x v}{\alpha y+ \gamma x}, \dot{y}= \frac{\beta x(t- \tau_1) v(t- \tau_1)}{\alpha y(t- \tau_1)+ \gamma x(t- \tau_1)}e^{- \alpha_1 \tau_1}- a y - pyz, \dot{v}= k e^{- \alpha_2 \tau_2}y(t- \tau_2)-uv, \dot{z}= cyz-bz,
\Pr\{X(\tau +t)-X(\tau)=n \}= \begin{cases}\sum_{r=1}^{\infty}\frac{(-1)^{n+r}}{r!}\int_{[\tau,\tau +t]^r}\lambda^{\beta_r(s)}\binom{\beta_r(s)}{n}ds_1...ds_r, n \geq 1, e^{-\int_{\tau}^{\tau +t}\lambda^{\alpha (s)}ds}, n=0,\end{cases}.
\hat{H}(k_x, k_y, k_z) =\left(\begin{array}{cc}\sin k_y \cos k_x + \cos k_y + e^{ik_z}\cos k_x + \cos k_y + e^{-ik_z}- \sin k_y\end{array}\right) &=\hat{H}_{\mathrm{loc}}(a(k_x, k_y), b(k_x, k_y), k_z),