harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Gamma_{2}= \frac{1}{C_{2}}= \frac{3}{a^{2}( 4 - a^{2})}, \ \ \ \Gamma_{4}= - \frac{C_{4}}{( C_{2})^{4}}= \frac{9 ( 1 - a^{2}) ( 5 - 2 a^{2})}{a^{4}( 4 - a^{2})^{4}}
\mathcal{L}_{\epsilon}^{\mathcal{K}}(\cdot)(x) = \Bigl \langle \bigtriangledown(\cdot), \Bigl(A x + \bigl(B, \mathcal{K}\bigr)x \Bigr) \Bigr\rangle + \frac{\epsilon}{2}\operatorname{tr}\Bigl \{\sigma(x)\sigma^T(x)\bigtriangledown^2(\cdot) \Bigr\},
\tilde{I}'(g_t)|_{t=0}=\frac{2-n}{n}V(t)'|_{t=0}(V(0))^{\frac{2-n}{n}-1}I(g_0)+v(0)^{\frac{2-n}{n}}I'(g_t)|_{t=0}, &=V(0)^{\frac{2-n}{n}}\{\frac{2-n}{2n}\frac{I(g_0)}{V(0)}\int_{SM}tr(h)\eta+\int_{SM}A_{ij}h^{ij}\eta\}|_{t=0}.
E_{2}(f):=-\frac{1}{\pi i}\int_{0}^{2\pi}\frac{f(\beta)\left(\frac{\epsilon(\beta)}{z(\beta)}-\frac{\epsilon(\alpha)}{z(\alpha)}\right)^{2}(z(\alpha)z(\beta))^{2}}{(z(\beta)-z(\alpha))|z(\beta)-z(\alpha)|^{2}}\partial_{\beta}\left(\frac{\epsilon(\beta)}{z(\beta)}\right)d\beta.
M_{n}^{\left( \alpha\right)}\left( \upsilon\right) =\gamma_{1}^{\alpha}\mu_{\alpha}^{\left( 1\right)}+\int_{1}^{\infty}\mathbf{L}_{n}\left(x;\upsilon\right) d\log^{\alpha}x+\left( 1+o_{\mathbf{P}}\left( 1\right)\right) \alpha\gamma_{1}^{\alpha-1}\mu_{\alpha}^{\left( 2\right)}\left(\rho_{1}\right) \mathbf{A}_{0}\left( n/\upsilon\right) ,
\overline{\nabla}{\phi}=\frac{2g'(\overline{\nabla}{f},\overline{\nabla}\,\overline{\nabla}{f})}{(1-f)^{2}}+\frac{2\left\|\overline{\nabla}{f}\right\|^{2}\overline{\nabla}{f}}{(1-f)^{3}}=\frac{2f_{i}f_{ij}}{(1-f)^{2}}+\frac{2f_{i}^{2}f_{j}}{(1-f)^{3}}.
\Psi'_{\underline{o}}(\vec{\sigma}_{2o})\sum_{\underline{n}}\,c_{\underline{n}}\Psi'_{\underline{n}}(\vec{\sigma}_{1o})=-\Psi'_{\underline{o}}(\vec{\sigma}_{2o})\frac{R}{\cal N}=-\frac{n_o(\vec{\sigma}_{2o})}{\cal N}.
\frac{W (a, b, c\,; x)}{W (a + k, b + l, c + m\,; x)}= \frac{(-1)^{k + l - m}(c - a)_{m - k}(c - b)_{m - l}}{(a)_{k}(b)_{l}}x^{m}(1 - x)^{k + l - m}.
K_{\theta_1,\theta_2}\bigl((y,\omega),(z,\eta)\bigr) = K_{\theta_1,\theta_2}\bigl((-z,\omega),(-y,\eta)\bigr) = \overline{K_{\theta_2,\theta_1}\bigl((-y,\eta),(-z,\omega)\bigr)}= \overline{K_{\theta_2,\theta_1}\bigl((z,\eta),(y,\omega)\bigr)}.
c~_2F_1(a,b;c;x) = \sum_{m=0}^\infty \frac{c (a)_m (b)_m}{(c)_m}\frac{x^m}{m!}&= \sum_{m=0}^\infty \frac{(c+m) (a)_m (b)_m}{(c+1)_m}\frac{x^m}{m!}&= (\theta+ c)\bullet~_2F_1(a,b;c+1;x)
R_0=\|\varrho_0\|_{B^1_{\infty,1}},U_0=\|u_0\|_{B^1_{\infty,1}}, &R(t)=\|\varrho\|_{L^\infty_t(B^1_{\infty,1})},S(t)=\|\varrho\|_{L^1_t(B^3_{\infty,1})},U(t)=\|u\|_{L^\infty_t(B^1_{\infty,1})}.
\mathrm{\Lambda_{\m u}( p^{\prime}, p ) = \frac{- \a l p h a_{s}}{3 \p i}\g a m m a_{\m u}\Gamma ( 2 - \o m e g a - \s i g m a ) \left|^{\a l p h a \to ( 1 / 2 )^{+}}_{\o m e g a \to ( 3 / 2 )^{+}}\right. .}
a^{k}b^{l}[f] = \Delta_{11}(0) \, \int_{C}d^{2}x\, F^{(k,l)}[f](x) +\int_{C}d^{2}x d^{2}y \, \Delta_{C}(x;y)\, B^{(k,l)}[f](x,y) =0\, .
\theta_{s,a^1}^1=r^1(s,a^1,a_s^2)+\beta\sum_{s'\in S}p(s'|s,a^1,a_s^2)\frac{r^1(s',a_{s'}^1,a_{s'}^2)}{1-\beta}-\frac{r^1(s,a_{s}^1,a_{s}^2)}{1-\beta}.
\mathfrak M^0_{i,i}(x,y) = 1-\frac{(x_i-y_i)^2}{|x-y|^2}, 1\le i \le d, \mathfrak M_{1,j}^0 (x,y) = -\frac{x_j-y_j}{2|x-y|^2}\big(2x_1-(1+\cos S_c)y_1\big), j\ge 2, begin{align*}2pt] \mathfrak M_{i,1}^0(x,y) = \mathfrak M_{1,i}^0(y,x), i\ge 2, begin{align*}2pt] \mathfrak M_{i,j}^0(x,y) = -\frac{(x_i-y_i)(x_j-y_j)}{|x-y|^2}, i,j\ge 2,\,\, i\neq j .
\begin{array}{r c l}{{\bf{Q}}_{-{\frac{1}{k}}}^{-}}&{=}&{[{\bf{Q}}_{-{\frac{1}{k}}}^{+}]^{+}}\\{{\bf{Q}}_{1 -{\frac{1}{k}}}^{-}}&{=}&{[{\bf{Q}}_{1 -{\frac{1}{k}}}^{+}]^{+}}\\ \end{array}
\int_{0}^{\frac{w}{2\lambda(t)}}\frac{R dR}{(1+R^{2})^{3}}\frac{4 R^{2}\lambda(t)^{2}}{2w\left(\sqrt{(1+w^{2}-R^{2}\lambda(t)^{2})^{2}+4R^{2}\lambda(t)^{2}}+1+w^{2}-R^{2}\lambda(t)^{2}\right)}&\leq C \int_{0}^{\infty}dR \frac{R}{(1+R^{2})^{3}}\frac{R^{2}\lambda(t)^{2}}{w(1+w^{2})}\leq \frac{C \lambda(t)^{2}}{w (1+w^{2})}, w \geq 1
\sum_{j=1}^k e^{t_j}= \sum_{t=1}^{M-2T}b_te^{T+t}\leq \sum_{t=1}^{M-2T}T(M-2T-t+1)e^{T+t}= \sum_{s=1}^{M-2T}Tse^{M-T+1-s}< Te^{M-T+1}
E\max_{2\leq k\leq n}\left\vert \sum\nolimits_{i=1}^{k-1}P^{i}(S_{i})\right\vert^{p}\ll E\left\vert \sum\nolimits_{i=1}^{n-1}P^{i}(S_{i})\right\vert^{p}& +E\max_{1\leq k\leq n-1}\left\vert \sum\nolimits_{i=k}^{n-1}P^{i}(S_{i})\right\vert^{p}.
v_l=\sum_{k\in S}\frac{\omega(l,k)}{\lambda_0-\omega(l,l)}v_k+\sum_{l_1\in\overline S,l_1\neq l}\frac{\omega(l,l_1)}{\lambda_0-\omega(l,l)}v_{l_1}-\frac{u_l}{\lambda_0-\omega(l,l)}.
\left( \begin{matrix}{{\Gamma_{S}^{1 1}( p )}}&{{\Gamma_{S}^{1 2}( p )}}\\{{\Gamma_{S}^{2 1}( p )}}&{{\Gamma_{S}^{2 2}( p )}}\\ \end{matrix}\right) ={\bf \sf M}^{- 1}\left( \begin{matrix}{{\Gamma_{R F}^{\phi_{S}}( p )}}&{{0}}\\{{0}}&{{{\Gamma_{R F}^{\phi_{S}}}^{*}( p )}}\\ \end{matrix}\right){\bf \sf M}^{- 1},
\nabla_{X_i}X_j = \nabla^B_{X_i}X_j, & \nabla_{X_i}Y_p = \nabla_{Y_p}X_i = f^{-1}f_iY_p, & \nabla_{Y_p}Y_q = \nabla^F_{Y_p}Y_q -f\,h_{pq}\,{\rm grad}\: f,
\Bigl ([X \xleftarrow{f_1}M_1 \xrightarrow{g_1}Y] + [X \xleftarrow{f_2}M_2 \xrightarrow{g_2}Y] \Bigr) \circ [Y \xleftarrow h N \xrightarrow k Z] & = [X \xleftarrow{f_1}M_1 \xrightarrow{g_1}Y] \circ [Y \xleftarrow h N \xrightarrow k Z] + [X \xleftarrow{f_2}M_2 \xrightarrow{g_2}Y] \circ [Y \xleftarrow h N \xrightarrow k Z],
\boldsymbol{b}^{(4)}=u\boldsymbol{a}^{(4)}+r(a_2, -a_1, a_4, -a_3)+s(a_3, -a_4, -a_1, a_2)+t(a_4, a_3, -a_2, -a_1) &=(ua_1+ra_2+sa_3+ta_4, ua_2-ra_1-sa_4+ta_3, ua_3+ra_4-sa_1-ta_2, ua_4-ra_3+sa_2-ta_1).
s_1(U) = \sum_{j,k,l,m=1}^nR(e_j, \bar e_k, e_l, \bar e_l) R(e_k, \bar e_j, e_m, \bar e_m)= \sum_{j,k=1}^n|r(e_j, \bar e_k)|^2= |r|^2 s_2(U) = \sum_{j,k,l,m=1}^nR(e_j, \bar e_k, e_l, \bar e_m) R(e_k, \bar e_j, e_m, \bar e_l)= |R|^2.
[\begin{matrix}y_{1}-x_{0}\\y_{2}-x_{0}\\y_{4}-x_{0}\\y_{3}-x_{0}\end{matrix}]=[\begin{matrix}w&w^{3}&w^{4}&w^{2}\\w^{2}&w&w^{3}&w^{4}\\w^{4}&w^{2}&w&w^{3}\\w^{3}&w^{4}&w^{2}&w\end{matrix}][\begin{matrix}x_{1}\\x_{3}\\x_{4}\\x_{2}\end{matrix}]
\left(\lambda_{G/H_1}(k)\oplus\lambda_{G/H_2}(k)\right)\left(\varphi(\delta_{g(H_1\cap H_2)})\right) =\left(\lambda_{G/H_1}(k)\oplus\lambda_{G/H_2}(k)\right)\left(\delta_{(gH_1,gH_2)}\right) &=\delta_{kgH_1,kgH_2}&= \varphi(\delta_{kg(H_1\cap H_2)}) &=\varphi\left( \lambda_{G/(H_1\cap H_2)}(k)(\delta_{g(H_1\cap H_2)})\right).
\delta\tbinom{m_2,m_3,\ldots,m_{i},m_{i+2},\ldots,m_r}{n_1,n_{\sigma(2)},\ldots,n_{\tau(i-1)},n_{\tau(i+2)},\ldots,n_{\tau(r)}}b_{n_{\tau(i)},n_{\tau(i+1)}}^{m_1}&+ \delta\tbinom{m_2,m_3,\ldots,m_{i},m_{i+2},\ldots,m_r}{n_1,n_{\tau'(2)},\ldots,n_{\tau'(i-1)},n_{\tau'(i+2)},\ldots,n_{\tau'(r)}}b_{n_{\tau'(i)},n_{\tau'(i+1)}}^{m_1}=0.
K_{23}(t,x,y)|_{t=\log\left(\frac{1+s}{1-s}\right)}\lesssim(1-s)^{n+1}\frac{|x+y|+|x-y|}{{s}^{(n-1)/2}}& \times\exp\Big( -\frac{1}{8}(s|x+y|^2+\frac{1}{s}|x-y|^2)\Big)\,e^{-(|y|^2-|x|^2)/4}\, e^{|y|^2-|x|^2}& \lesssim\frac{(1-s)^{n}}{{s}^{n/2}}\exp\Big(-\frac{|(1+s)y-(1-s)x|^2}{16s}\Big) \,e^{|y|^2-|x|^2}, \quad(x,y)\in N^c.
F(a+1, b, c, z)= ~\frac{\Gamma(c)}{\Gamma(a+1)\Gamma(b)}\sum^\infty_{n=0}\frac{\Gamma(a+n+1)\Gamma(b+n)}{n!\Gamma(c+n)}z^n = ~\frac{\Gamma(c)}{\Gamma(a)\Gamma(b)}\sum^\infty_{n=0}\bigg(1+\frac{n}{a}\bigg)\frac{\Gamma(a+n)\Gamma(b+n)}{n!\Gamma(c+n)}z^n = ~F(a,b,c,z)+\frac{z}{a}\frac{d}{dz}F(a,b,c,z).
\begin{aligned}g_1(a+b,c;d) + g_1(a,b;d)^{x_{cd}}= g_1(a,b+c;d) + g_1(b,c;d) +f_+(a,b,c)\,d, g_2(a;b+c,d) + g_2(a;b,c)^{x_{ad}}= g_2(a;b,c+d) + g_2(a;c,d) +a\,f_+(b,c,d), \end{aligned}
\omega(\hat{x}) - f(\hat{x}, \hat{y})= \Big( \underbrace{\omega(\hat{x}) - f ( \hat{x}, q^{(k)}(\hat{x}, \hat{y}) )}_{S_1}\Big) \,+ \, \Big(\underbrace{f( \hat{x}, q^{(k)}(\hat{x}, \hat{y}) ) - f(\hat{x}, \hat{y})}_{S_2}\Big).
{\sigma}^{(s,0)}_{\varepsilon}\Big ( \big ( \theta^{(s,u)}\big )^{-1}(A) \cap \big ( \theta^{(u,0)}\big )^{-1}(B) \Big ) ={\sigma}_\varepsilon^{(s, u)}\Big ( \big ( \theta^{(s,u)}\big )^{-1}(A) \Big ){\sigma}_\varepsilon^{(u, 0)}\Big ( \big( \theta^{(u,0)}\big )^{-1}(B) \Big ).
\lim_{k\to\infty}\Vert\prod_{n=3}^k\frac{I+T_n}{2}{\bf 1}\Vert^2=\Big( \prod_{n=3}^\infty\frac{\sqrt{\alpha_{1n}(0)}+\sqrt{\alpha_{1n}(1)}}{\sqrt{2}}\Big)^2=H^2(\mu_{\alpha},\mu_{inv}).
v_{1}^{1}=\varphi_{1,[2]}^{1}=q^{2}+2T\omega^{2}+T^{2}\left(a_{1}\sin(q^{2})+a_{2}\cos(q^{2})+b_{2}\cos(q^{3})u^{1}\right) v_{1,[1]}^{1}=\varphi_{1,[3]}^{1}(q^{2},q^{3},\omega^{2},\omega^{3},u^{1},u_{[1]}^{1})\,, \:\:\vdots
L^1(\sqrt{1+v^2}\,\mathrm{d}x\, \mathrm{d}v) := \left\{f\colon{\mathbb{T}}_x \times \mathbb{R}_v \to \mathbb{R}\:\colon \int_{\mathbb{T}}\!\mathrm{d}x \int_{\mathbb{R}}\! \mathrm{d}v\, \left\vert f \right\vert \sqrt{1+v^{2}}< \infty \right\}\,,
F_{l}(\Gamma_{t},x)\triangleq\sum_{k=1}^{l}\sum_{i_{1},\ldots,i_{k}=1}^{d}V_{(i_{1},\ldots,i_{k})}(x)\int_{0<t_1<\cdots<t_k<t}dB_{t_1}^{i_{1}}\cdots dB_{t_k}^{i_{k}},
u^{\beta ( \alpha )}( \vec{p}) = ( - 1 )^{\alpha}\gamma^{2}\gamma^{5}u^{\alpha *}( \vec{p}) , \qquad \mathrm{a n d}\qquad e_{\mu}^{- \ell}( \vec{q}) = e_{\mu}^{\ell *}( \vec{q}) ,
\frac{\partial a}{\partial k_z}(k^0_x, k^0_z) = 0, \frac{\partial a}{\partial k_z}(k^0_x, k^0_z) = 1, \frac{\partial b}{\partial k_z}(k^0_x, k^0_z) = 1, \frac{\partial b}{\partial k_z}(k^0_x, k^0_z) = 0.
\begin{cases}u_t(t,x)=\frac{1}{2}\;\mbox{\rm Tr}\;[Q(x)D^2u(t,x)]+\langle Ax+ b(x),Du(t,x) \rangle, (t,x) \in [0,+\infty)\times H, u(0,x)=\phi(x), \ \x\in H.\end{cases}
e^{-i\phi}\partial\partial\psi= -\partial\bar{\partial}|\sigma|-i|\sigma|\partial\bar{\partial}\phi+ie^{-i\phi}\partial\phi\partial\psi -ie^{-i\phi}\partial|\sigma|\bar{\partial}\phi \qquad+\frac{2\xi\partial|\sigma|}{1+\xi\bar{\xi}}+\frac{2|\sigma|}{(1+\xi\bar{\xi})^2},
\frac{\sqrt{n}}{\sin^{2}\left( n ( y_{0}- | y | ) \right)}{}_{2}F_{1}\left( 5 / 4 \mp 3 \delta / 4 , 5 / 4 \pm 3 \delta / 4 ; 3 ; \sin^{- 2}\left( n ( | y | - y_{0}) \right) \right)
\mathrm{pr}_{\dot{v}}\left( \iota_k(S', N)(\Lambda) \circ\mathrm{loc}_{k,v}(S', N) \right) = \iota_{k,v}(N)(l_{k,v}(\Lambda)) \times\mathrm{d}\left( \mathrm{pr}_{\dot{v}}(\sqrt[N]{\beta_k})\underset{E_k/F}{\sqcup}N \Lambda \right).
(\nabla k^j) \cdot (\mathbf y^j T(\mathbf y^{-1}))(\nabla k) = k^{j-1}([1,\mathbf y] z^j) (\nabla k) \cdot (\mathbf y^j T(\mathbf y^{-1}))(\nabla k) = k^{j-1}([1,\mathbf y_1] z^j) (\mathbf y_2^j T(\mathbf y_2^{-1})) (\nabla k \cdot \nabla k)
|\Gamma_a^{113}(x,y)|\leq C|y|^2|x|^2\int_0^\infty \int_t^\infty\left(|x|^2s^{-5}+s^{-4}\right) e^{-|x|^2/16s}\,ds\, dt= \frac{C|y|^2}{|x|^2}
U_1(q) = 2\sum_{k,\ell, \, \mathrm{ord}(\omega) \mid k}(q{n \brack k}'_q)^2 q^{f(n,k,\ell)}{n \brack \ell}_q{k \brack \ell}_q{k+\ell \brack n}_q, U_2(q) = 2\sum_{k,\ell, \, \mathrm{ord}(\omega) \nmid k}(q{n \brack k}'_q)^2 q^{f(n,k,\ell)}{n \brack \ell}_q{k \brack \ell}_q{k+\ell \brack n}_q.
|M_m(x)| =\prod_{n=m^3}^{\infty}\left|\frac{\sin(\frac{x}{n^2})}{\frac{x}{n^2}}\right|\leq \prod_{n=m^3}^{[|x|^{\frac{1}{2}}]}\frac{n^2}{|x|}=\exp\left(\sum_{n=m^3}^{[|x|^{\frac{1}{2}}]}\ln\frac{n^2}{|x|}\right)\leq\exp\left(\sum_{n=m^6}^{[|x|]}\ln\frac{n^2}{|x|}\right) =\exp\left(-\int_{m^6}^{|x|}\frac{B(u)-m^6+1}{u}du\right).
\max_{\pm}\| \sum_{n \in B}\pm e_n \| \le \sum_{j=1}^N \max_{\pm}\| \sum_{n \in B_j}\pm e_n \| &\le N\overline{D}\min_{\pm}\| \sum_{n \in A}\pm e_n \| &\le \frac{2\overline{D}}{\alpha}\min_{\pm}\| \sum_{n \in A}\pm e_n \|
\left\| \nabla^{k}_{x}\left( \partial_{t}K_{1}(t)g+\nabla_{x}\mathcal{F}^{-1}\left[e^{-\frac{\nu t |\xi|^{2 \sigma}}{2}}\sin (t |\xi|)\right] \ast g\right)\right\|_{2}& \le C(1+t)^{-\frac{n}{4 \sigma}-1-\frac{k}{2 \sigma}}\|g \|_{1}+ C e^{-ct}\| \nabla^{(k+2(1-\sigma))}_{x}g\|_{2},
z^{(i+1)}=z^{(i)}-\frac{\mathcal{W}_\sigma^\prime(z^{(i)})\pm \left[\left(\mathcal{W}_\sigma^\prime(z^{(i)})\right)^2-2\mathcal{W}_\sigma(z^{(i)})\,\mathcal{W}_\sigma^{\prime\prime}(z^{(i)})\right]^{1/2}}{\mathcal{W}_\sigma^{\prime\prime}(z^{(i)})}\,,
^0\langle \, m\neq p \mid p \, \rangle = \{\sum_n \!^\prime \, [1 + Q(H_0-E_p)^{-1}QVQ]^{-1}_{mn}\frac{V_{np}}{(E_p-E^0_n)}\}^0 \langle \, p \mid p \, \rangle
\frac{d}{dt}\sum_{q\geq -1}\lambda_q^{2s}\left(\|u_q\|_2^2+\|b_q\|_2^2\right)\lesssim -\sum_{q\geq -1}\lambda_q^{2s}\left(\nu\|\nabla u_q\|_2^2+\mu\|\nabla b_q\|_2^2\right) &+f(t)\sum_{q\geq -1}\lambda_q^{2s}\|u_q\|_2^2+Qf(t)\sum_{q\geq -1}\lambda_q^{2s}\|b_q\|_2^2.
K_{T^{N+1}}\left( \mathbb{P}^N \right) = \mathbb{Z}[\Lambda_0^{\pm 1},\dots,\Lambda_N^{\pm 1}][P^{\pm 1}] \left/\left( \left(1-\Lambda_0 P^{-1}\right) \cdots \left(1-\Lambda_N P^{-1}\right) \right) \right.
\Psi(1,1) =(\alpha, \alpha), \Psi(2,1) =(\beta, \beta, \alpha), \Psi(3,1) =(\beta, \alpha, \alpha), \Psi(1,2) =(\alpha, \beta), \Psi(2,3) =(\beta, \beta, \beta), \Psi(3,3) =(\beta, \alpha, \beta)
{\cal E}_{\nu}(\eta ,\eta') = (r r')^{-\frac{d-1}{2}- i \nu}{W}_{\nu}^{\infty}({\omega},{\omega}') =\frac{\Gamma\left(\frac{d-1}{2}+i\nu\right)\Gamma(-i\nu)}{4\pi^{\frac{d+1}{2}}}\ \ \left[-(\eta - \eta')^2\right]^{-i\nu-\frac{d-1}{2}}
S_{6}^{(1)}({{p}^{r}}) \equiv -\frac{20}{3}p^{r-1}B_{p-3}^2 \pmod{p^r}, S_{6}^{(2)}({{p}^{r}}) \equiv \frac{80}{3}p^{r-1}B_{p-3}^2 \pmod{p^r}, S_{6}^{(3)}({{p}^{r}}) \equiv -40p^{r-1}B_{p-3}^2 \pmod{p^r}.
E^{0}(S_{n_{i}}-S_{n_{i-1}})^{2}=E^{0}(E((S_{n_{i}}-S_{n_{i-1}})^{2}|\mathcal{K}_{n_{i-1}}))=K^{n_{i-1}}(E^{0}(S_{n_{i}-n_{i-1}}^{2})) & \leq K^{n_{i-1}}\left( E^{0}(S_{2^{i}}^{2})\right) .
\theta_{n+k}(\pi) =\pi(\lambda_{d},\Psi,\mu,\nu,(\varepsilon\mid((-1)^{\frac{p-q}{2}},(-1)^{\frac{p-q}{2}},\dots,(-1)^{\frac{p-q}{2}})), \qquad(\kappa\mid(1+n-\frac{p+q}{2},2+n-\frac{p+q}{2},\dots,k+n-\frac{p+q}{2})))
2 P^{+}P^{-}| \varphi \rangle = \sqrt{2}P^{+}( Q^{-})^{2}| \varphi \rangle = \sqrt{2}P^{+}( g Q_{\mathrm{S Y M}}^{-}+ Q_{\perp}^{-}+ i \kappa Q_{\mathrm{C S}}^{-})^{2}| \varphi \rangle = M_{n}^{2}| \varphi \rangle
G_n(z, \tau) = J_{4,n}(2z, 2 \tau) & = -B_n \Big( 1 - \frac{1}{2^{n - 1}}\Big) - n \sum_{k , r \geq 1}(r - 1/2)^{n - 1}(y^{2k}+ (-1)^n y^{-2k}) q^{k (2 r - 1)}.
\phi \left( \left[ \delta_{a,c}\right] +\alpha^{\prime}\left[ \delta_{b,d}\right] +\left[ \delta_{a,d}\right] \right) =\lambda \left( \alpha \left[ \delta_{a,c}\right] +\beta \left[ \delta_{b,d}\right] +\left[\delta_{b,c}\right] +\left[ \delta_{a,d}\right] \right) .
\frac{\phi^{n+1}-\phi^{n}}{\tau}=\left(\frac12\Delta-\vartheta^n\right)\phi^{n+1}+\bigg(\vartheta^n-V-\beta|\phi^n|^2+\Omega L_z+\chi^n-\sigma R^\prime(\|\phi^n\|^2-1)\bigg)\phi^n, \frac{\chi^{n+1}-\chi^{n}}{\xi^n\tau}= 1-\|\phi^{n+1}\|^2,n=0,1,\ldots
\partial(\overline{\varphi}\frown\mu)=(\overline{\varphi}\circ\partial)\frown\mu=[(\overline{\varphi}\circ\partial)\frown\mu]=\sum_ka_k\overline{\varphi}(\partial[v_0^k,\ldots,v_{i+1}^k])[v_{i+1}^k,\ldots,v_d^k].
A=\begin{bmatrix}X_0 X_1 \cdots X_{n-2}X_{n-1}X_{-1}X_0 \cdots X_{n-3}X_{n-2}\vdots \vdots \ddots \vdots \vdots X_{2-n}X_{3-n}\cdots X_0 X_1 X_{1-n}X_{2-n}\cdots X_{-1}X_0\end{bmatrix}\in \mathbb{C}^{nk \times nk},
\|\pi_{n-}^{\mu}-\pi_{n-}^{\gamma}\|_{TV}= &\frac{E^{\gamma}\left[\left.|E^{\gamma}[\frac{d\mu}{d\gamma}(X_{0})|Y_{[0,n-1]},X_{n}]-E^{\gamma}[\frac{d\mu}{d\gamma}(X_{0})|Y_{[0,n-1]}]|\right|Y_{[0,n-1]}\right]}{E^{\gamma}\left[\left.\frac{d\mu}{d\gamma}(X_{0})\right|Y_{[0,n-1]}\right]}
F_{3n}^{\left(3\right)}=F_{n}^{\left(3\right)},\thinspace\thinspace F_{3n+1}^{\left(3\right)}=F_{n}^{\left(3\right)},\thinspace\thinspace F_{3n+2}^{\left(3\right)}=2F_{n}^{\left(3\right)}
\begin{array}{rl}\phi= g_1(z)+x^0g_2(z)+sg_3(z)+zg_4(x^0)+sg_5(r) &+\displaystyle\int^s_0\left(\int^\eta_0g_6(r^2-\xi^2)d\xi\right)d\eta+\int^r_0\xi g_6(\xi^2)d\xi,\end{array}
P(\vec{\rho})(L,L',R,R';k)\ :=\ \lim_{n \to \infty}P_n(\vec{\rho})(L,L',R,R';k) &\ =\ \mathbb{P}_{2k}[(L, L', R, R';k)] - \sum_{j > k}^{\infty}\mathbb{P}_{j+k}[(L, L', R, R';k), [k+1, j-1] \in A+B; \, j \not \in A+B].
r\dot{r}\left[ \frac{d}{df}\left( \frac{f^{3}}{p\dot{p}}\right) \right] -2g\left[ \frac{d}{df}\left( \frac{fp}{\dot{p}}\right) \right] +\left[ \frac{d}{df}\left( \frac{p^{3}}{f\dot{p}}\right) \right] \left[ \left( \frac{g^{2}}{r}\right) \frac{d}{dg}\left( \frac{g^{2}}{r}\right) \right] =0.
F ( x , y ) \; = \; \frac{H^{2}}{2 \pi \mu}\ln \frac{{\cal T}_{B G V}}{{\cal T}}\; = \; 2 ( - 1 + \sqrt{1 - x^{2}}) \; + \; y \ln \frac{1 + \sqrt{1 - x^{2}}}{x},
\left\langle u\right\rangle_{\omega\gamma,x,\overline{Q}}^{(\gamma)}\equiv\sup_{x,\overline{x}\in\overline{Q}}\left( x_{N}^{\ast}\right)^{\omega\gamma}\frac{|u(x,t)-u(\overline{x},t)|}{|x-\overline{x}|^{\gamma}},\,x_{N}^{\ast}=\max\{x_{N},\overline{x}_{N}\},
\eta_{3}\leq \eta - \eta_{2}\leq 0 \begin{cases}\eta \geq \eta_{0}0 < s \leq -1+\sqrt{\cosh 1}, 0 \leq \eta -\eta_{0}\leq \eta_{1}-1+\sqrt{\cosh 1}< s \leq 1/\sqrt{2}, \end{cases}
D_{c,h}\left(D_s^k c \right) = D_s^k h - \sum_{j=0}^{k-1}\sum_{i=0}^j \binom{j}{i}D_s^i \langle D_s h, v\rangle D_s^{k-j+j-i}c &= D_s^k h - \sum_{i=0}^{k-1}\sum_{j=i}^{k-1}\binom{j}{i}D_s^i \langle D_s h, v\rangle D_s^{k-i}c &= D_s^k h - \sum_{i=0}^{k-1}\binom{k}{i+1}D_s^i \langle D_s h, v\rangle D_s^{k-i}c\,.
S^{2N_0n}\subset (H\overline{P}(L))^{2N_0n}= H\overline{P}(L)^{2N_0n}\subset H\overline{P}(2N_0nL)\qquad\qquad\qquad\qquad\qquad\qquad\qquad \subset\overline{P}(nL)X_n\subset H(\overline{P}(L))^{Mn}X_n\subset S^{M^2n}X_n=S^{N_0n}X_n,
\begin{aligned}W^u(D_3)=\{(a,r,W, \xi,b): w_1=w_{L1}, \w_2=w_{L2}-\lim_{\varepsilon\rightarrow 0}\varepsilon^5 \int_{-\infty}^{\zeta}\frac{y_2(t)^2}{(y_1(t)^{16/15}+\varepsilon^{10})^{3/2}}\dt, \ \ \ \ \ \xi=s_{}, \b=b(\zeta), \ \zeta\in\mathbb{R}\}.\end{aligned}
\begin{aligned}z-y = r^p, \phi_p(z, y) = r_1^p, x = r r_1, z-x = s^p, \phi_p(z, x) = s_1^p, y = s s_1, x+y = t^p, \phi_p (x, -y) = t_1^p, z = t t_1.\end{aligned}
\| u(\mu,\cdot) - \mathcal{I}_{L}[P_{k}[u]](\mu,\cdot) \|_{L^{2}({\omega})}^{2}= \sum_{I \in \mathfrak{I}}\| u(\mu,\cdot) - \mathcal{I}_{L}[P_{k}[u]](\mu,\cdot) \|_{L^{2}(I)}^{2}= \sum_{I \in \mathfrak{I}}\| u(\mu,\cdot) - P_{k}[u](\mu,\cdot) \|_{L^{2}(I)}^{2}.
p_{Y|U,V}(y|u,v)=\mspace{-8mu}\sum_{(s,x,z)\in\mathcal{S}\times\mathcal{X}\times\mathcal{Z}}\mspace{-8mu}p_{S|U,V}(s|u,v)p_{X|S,U,V}(x|s,u,v)p_{Y,Z|X,S}(y,z|x,s),\quad\forall(u,v,y)\in\mathcal{U}\times\mathcal{V}\times\mathcal{Y}.
f(a_{\max}) = a_{\max}\left( 1 + \frac{2}{Q\sqrt \kappa}- \frac{a_{\max}}{Q^2}\right) = \frac{Q^2}{4}\left( 1 + \frac{2}{Q\sqrt \kappa}\right)^2 = \frac{4}{\kappa}+ \frac{\kappa}{16}+ 1 = q
Q_R = \frac{1}{\sqrt{2 T_2 U_2}}(n_2{\bar U}\tau+ n_1 \tau -{\bar U}l_1 + l_2),\;\;\;{\bar Q}_R = \frac{1}{\sqrt{2 T_2 U_2}}(n_2 U \tau + n_1 \tau - U l_1 +l_2).
\xi = \sum_{\star =1}^\ell \,\, \sum_{1 \leq i_1 \leq \ldots \leq i_\star \leq n}\,\,\sum_{n+1 \leq i_{\star+1}\leq \ldots \leq i_\ell \leq n+k}\xi_{i_1 \ldots i_\star \ldots i_\ell}dx^{i_1}\wedge \ldots \wedge dx^{i_\star}\wedge \ldots \wedge dx^{i_\ell}.
\delta_{B}^{A}{\overline{{V}}}\, = \, - 1 2 ( S^{M A})^{\star}\, S_{M B}\, + \, g_{i j^{\star}}W^{i M A}\, W_{M B}^{j^{\star}}\, + \, 2 N_{\alpha}^{A}\, N_{B}^{\alpha}
\mathbf{\hat{Q}}\mathbf{\hat{R}}= \mathbf{Q}\mathbf{R}+\mathbf{\Delta H}\Leftrightarrow\mathbf{\hat{Q}}^{\mathcal{H}}=(\mathbf{Q}\mathbf{R}\mathbf{\hat{R}}^{-1})^{\mathcal{H}}+(\mathbf{\hat{R}}^{-1})^{\mathcal{H}}\mathbf{\Delta H}^{\mathcal{H}}.
{\cal A}^{*}({\cal E}) = \sum_{i}\sum_{j}\int_{- \infty}^{\infty}d \tau \, e^{- i{\cal E}\tau}\, u_{i}[{\tilde{x}}( \tau ) ] \, v_{j}^{*}[{\tilde{x}}( \tau ) ] \; \langle 0 \vert{\hat{a}}_{i}{\hat{b}}_{j}\vert \Psi \rangle .
\delta(u(t)) = \Big| \| \nabla u(t) \|_2^2 - \| \nabla Q \|_2^2 \Big| = \Big| 2 \alpha \| \nabla Q \|_2^2 +2 \int Q h_1 dx \Big| + O(\widetilde{\delta}^2) =2 \left( \| \nabla Q \|_2^2 + \| Q\|_2^2 \right) |\alpha(t)| + O(\widetilde{\delta}^2).
i_{abc}(t)=\frac{3}{2}[\begin{matrix}\frac{2}{3}&0\\ -\frac{1}{3}&\frac{\sqrt{3}}{3}\\ -\frac{1}{3}&-\frac{\sqrt{3}}{3}\end{matrix}][\begin{matrix}i_{\alpha}(t)\\i_{\beta}(t)\end{matrix}].
\theta^{\ast}:\Omega\rightarrow C_{2}\ltimes(C_{n}\ltimes C_{p}) \theta^{\ast}(x_{1}^{\prime}) =sy^{-1}\theta^{\ast}(c_{0}^{\prime})=s\theta^{\ast}(c_{1}^{\prime})=sx,\theta^{\ast}(c_{2}^{\prime})=sy^{-2},\theta^{\ast}(e^{\prime})=sy^{-1}
\mathcal{Z}^n(A,\rho):= (d_{A,\rho}\colon C^n(A,\rho)\longrightarrow C^{n+1}(A,\rho)) \mathcal{B}^n(A,\rho):= (d_{A,\rho}\colon C^{n-1}(A,\rho)\longrightarrow C^n(A,\rho))
\begin{cases}(\partial_t+\mathbf{u}\cdot\nabla)p+\rho c^2\mathrm{div}\mathbf{u}=0, \rho(\partial_t+\mathbf{u}\cdot\nabla)\mathbf{u}-(\mathbf{H}\cdot\nabla)\mathbf{H}+\nabla q=0, (\partial_t+\mathbf{u}\cdot \nabla) \mathbf{H}-(\mathbf{H}\cdot\nabla)\mathbf{u}+\mathbf{H}\mathrm{div}\mathbf{u}=0, (\partial_t+\mathbf{u}\cdot\nabla)S=0, \end{cases}
G(s, \mu) := \frac{1}{\Gamma(s_1 + s_2)}\prod_{j=1}^3 \Gamma(s_1 - \mu_j) \Gamma(s_2 + \mu_j), \tilde{G}^{\pm}(s, \mu) := \frac{\pi^{-3s}}{12288 \pi^{7/2}}\Biggl(\prod_{j=1}^3\frac{\Gamma(\frac{1}{2}(s-\mu_j))}{\Gamma(\frac{1}{2}(1-s+\mu_j))}\pm i \prod_{j=1}^3\frac{\Gamma(\frac{1}{2}(1+s-\mu_j))}{\Gamma(\frac{1}{2}(2-s+\mu_j))}\Biggr),
\frac{\varphi(VP_f)}{\varphi(V)(VP_f)}=\frac{1}{V}\cdot\frac{V}{\varphi(V)}\cdot\frac{\varphi(VP_f)}{(VP_f)}&=\frac{1}{V}\prod_{p\mid V}\frac{p}{p-1}\prod_{p\mid VP_f}\frac{p-1}{p}&=\frac{1}{V}\prod_{\substack{p\mid P_f p\nmid V}}\frac{p-1}{p}.
r({\mathcal{K}}) = \limsup_{T \rightarrow \infty}\,\inf_{\substack{\varphi(t) \in C_{0T}([0,T], \mathbb{R}^d) \varphi(0)=x_0}}\frac{1}{T}\biggl \{S_{0T}^{\mathcal{K}}(\varphi(t)) \, \Bigl \vert \,\varphi(t) \in D \cup \partial D,\,\, \forall t \in [0, T] \biggr\}.
k (Z^0,z^a) =|Z^0|^2\left(4h(t)-\frac{\chi \zeta(3)}{(2\pi)^3}+\frac{2}{(2\pi)^3}\sum_{q_a\gamma^a \in \Lambda^+}n_{\gamma}(\mathrm{Li}_3(e^{2\pi iq_az^a}))+\frac{2}{(2\pi )^2}\sum_{q_a\gamma^a \in \Lambda^+}n_{\gamma}(\mathrm{Li}_2(e^{2\pi iq_az^a}))q_at^a\right)\,,
W_j - \eta_j = \frac{g^{\prime \prime}(0)}{2 g^\prime (0)}\frac{W_j \eta_j}{\sqrt{n}}+ \bigg( \frac{g^{(3)}(0 )}{6 g^\prime (0)}- \frac{g^{\prime \prime}(0 )^2}{4 g^\prime (0)^2}\bigg) \frac{W_j \eta_j^2}{n}+ O (n^{- 3/2}) .
\left\|d_G(\cdot,y_0)^{-2(1-\frac{n+1}{p})}\hat E^{y_0,ij}\right\|_{L^\infty(\mathcal{B}_{1/2}^+(y_0))}+\left[d_G(\cdot,y_0)^{-2(1-\frac{n+1}{p}-\epsilon)}\hat E^{y_0,ij}\right]_{\dot{C}^{0,\epsilon}_\ast(\mathcal{B}_{1/2}^+(y_0))}\leq C.
\nabla_{\phi}\mathbb{E}_{z\sim q_{\phi}(\cdot|x)}[ln\frac{p_{\theta}(x,z)}{q_{\phi}(z|x)}]=\mathbb{E}_{\epsilon}[\nabla_{\phi}ln\frac{p_{\theta}(x,\mu_{\phi}(x)+L_{\phi}(x)\epsilon)}{q_{\phi}(\mu_{\phi}(x)+L_{\phi}(x)\epsilon|x)}]
\omega_0^{+}+\omega_0^{-}\;, \;\;\Omega_0^{+}+\Omega_0^{-}\;,\;\;\omega_n^{+}|_{+}\;, \;\; \omega_n^{-}|_{-}\; , \;\;\Omega_n^{+}|_{+}\;, \;\; \Omega_n^{-}|_{-}\quad \mbox{for all}n\geq 1 \quad ,
a_l^0(\Upsilon_{\!\rho}) =\left<Y_l^0,\Upsilon_{\!\rho}\right> =\left<Y_l^0,\int_{SO(n)}\sum_{l=0}^\infty\sum_{k\in\mathcal M_{n-1}(l)}a_l^k(\Psi_\rho^L)Y_l^k(\tilde g^{-1}\circ)\,d\nu(\tilde g)\right>.
\delta_{\bf a}{\cal L}_{0}= \mu \dot{\bf x}\cdot \delta_{\bf a}\dot{\bf x}+ \frac{\mu}{R^{2}}\,{\bf x}\cdot \delta_{\bf a}{\bf x}= \frac{\mu}{R}\,{\bf a}\cdot \frac{d}{d t}\left[{\bf x}\sinh \left( \frac{t}{R}\right) \right] .
Z ( s ) = \int{\cal D}\psi{\cal D}\bar{\psi}{\cal D}b_{\mu}~ F ( \partial b ) \delta ( \varepsilon^{\mu \nu \rho}f_{\nu \rho}( b ) ) ~ e^{- \int d^{3}x ~ \bar{\psi}( \partial \! \! \! / + i ( s \! \! \! / + b \! \! \! / ) + m ) \psi}\; .
\|h\|_{C^{0+\alpha}(\mathbb R^d)}:= \| h \|_{L^\infty(\mathbb R^d)}+ \sup_{x\neq y \in \mathbb R^d}\frac{|h(x)-h(y)|}{|x-y|^\alpha}& \|h\|_{C^{1+\alpha}(\mathbb R^d)}:= \| h \|_{L^\infty(\mathbb R^d)}+ \| \nabla h \|_{L^\infty(\mathbb R^d)}+ \sup_{x\neq y \in \mathbb R^d}\frac{|\nabla h(x)-\nabla h(y)|}{|x-y|^\alpha},
\zeta^i_{t|s}= \sum_{j=0}^nF^\infty_{ij}(t-s)\zeta^j_s & = \sum_{j=0}^nf^\infty_{ij}(t-s)\nu^j_s\pi & = \sum_{j=0}^i{n-j \choose n-i}\exp(-(n-i)\lambda_\infty\,(t-s))(1-\exp(-\lambda_\infty\,(t-s)))^{i-j}\nu^j_s\pi & ={n-N_s \choose n-i}\exp(-(n-i)\lambda_\infty\,(t-s))(1-\exp(-\lambda_\infty\,(t-s)))^{i-N_s}\pi,
P_{0,x}e^{-\left\langle f,X_{Q}\right\rangle}=e^{-w\left( 0,x\right)}=\Pi_{0,x}\left[ e^{-k\tau}e^{-f\left( \tau ,\xi_{\tau}\right)}+\int_{0}^{\tau}dske^{-ks}\varphi \left( s,\xi_{s};e^{-w\left( \tau-s,\xi_{s}\right)}\right) \right]
\begin{aligned}-\iint g_s \partial_y h^\epsilon u^\epsilon \leq \Vert g_s \Vert_{L^\infty}\Vert u^\epsilon\Vert_{L^2}\Vert \partial_y h^\epsilon\Vert_{L^2}&\leq L\Vert g_s \Vert_{L^\infty}\Vert v^\epsilon_y\Vert_{L^2}\Vert h^\epsilon_y\Vert_{L^2},\end{aligned}
[{\bf n}^{\Sigma}({\bf u}_0)]^{-1}\cdot du^i({\bf u}_0) =\nabla_{{\bf n}^{\Sigma}}u^i({\bf u}_0)=[x_0,[x_0,\nabla u^i(x_0)]] &={\bf v}_{u^i}({\bf u}_0,{\bf 0}),~~~\mbox{all}~~~i=\overline{1,\dim \Sigma}.