harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
A_{k(2j-1)}^* = (A_{k(2j-1)}^2 + A_{k(2j)}^2)A_{k(2j-1)}+ A_{k(2j)}[\sum_{t=1}^{j-1}(A_{k(2t-1)}A_{k(2t)}+ A_{k(2t)}A_{k(2t-1)}) \ \ \ + \sum_{t=j+1}^n(A_{k(2t-1)}A_{k(2t)}+ A_{k(2t)}A_{k(2t-1)})] \ \ \forall \k,j
\begin{aligned}T(q,z) =\sum_{l=-\infty}^{\infty}q^{p\frac{l(l+1)}{2}+l+1}z^{lp+1}=(qz)\sum_{l=-\infty}^{\infty}(q^p)^{\binom{l+1}{2}}(qz^p)^l & =(qz)\prod_{n=1}^{\infty}(1+z^pq^{np+1})(1+z^{-p}q^{(n-1)p-1})(1-q^{np}),\end{aligned}
\ell(\nu) :=\frac{1}{2}\log{(2\pi e)^2N^2(X,Y)}- \nu R_p -\nu \log{(2 \pi e \Delta)}& + \frac{\nu}{2}\log{\frac{\nu^2}{2\nu-1}}-\frac{1-\nu}{2}\log{(2\pi e)^2\frac{(1-\rho)^2}{2\nu-1}},
\textbf{c}_{q,p}^{(n)}=\frac{\binom{n}{q-1}}{(n!)^2}\sum_{i=p+2}^{2n}(-1)^iQ_i(q)(n+1-q)^{i-2-p}\widetilde{\textbf{c}}_{q,p}^{(n)}=\frac{\binom{n}{q-1}}{(n!)^2}\sum_{i=p+2}^{2n}P_i(q)(n+1-q)^{i-2-p},
e^2_{k-1}(w)+D_{k-1}\cdot e_{k-1}(w) = \left(\sum_{i=0}^{k-2}B_{k-1,i}w^{2^i}\right)^2+D_{k-1}\left(\sum_{i=0}^{k-2}B_{k-1,i}w^{2^i}\right)\cr =\sum_{i=1}^{k-1}B^2_{k-1,i-1}w^{2^i}+\sum_{i=0}^{k-2}D_{k-1}B_{k-1,i}w^{2^i}.
\sup_{0 \leq j \leq k-1}\, \sup_{\theta \in [0, 2 \pi)}|\psi_{j}(\theta) - \psi_{j}( 2 \pi k^{-D}\lfloor k^D \theta/ 2 \pi \rfloor ) | = \sup_{0 \leq j \leq k-1}\, \sup_{0 \leq s \leq k^D - 1}\psi_j (2 (s+1)\pi k^{-D}) - \psi_j (2 s \pi k^{-D}).
D=E[(x-Q(x))^{2}]=\int_{-\infty}^{\infty}(x-Q(x))^{2}f(x)dx=\sum_{k=1}^{M}\int_{b_{k-1}}^{b_{k}}(x-y_{k})^{2}f(x)dx=\sum_{k=1}^{M}d_{k}
\det \left( ({\mathbf{I}}-({\mathbf{T}}_{a-1})^{2}\,{\mathbf{T}}_{b,a})|{\mathcal{B}}_{Q}^{[\underline{j}]_a^b}\right) = 1- \prod_{0\le i\le b-a}d_{i}= 1- \prod_{i\neq j\in T}q_{ij}= 1- \prod_{\{i,j\}\subset T}\sigma_{ij}
\left( \sum_{i_{1},\ldots ,i_{m}=1}^{N}\left\vert T(e_{i_{1}},\ldots,e_{i_{m}})\right\vert^{\frac{2mp}{mp+p-2m}}\right)^{\frac{mp+p-2m}{2mp}}\leq C_{m,p}^{\mathbb{K}}\left\Vert T\right\Vert .
U_{( 0 , r - 1 )}^{2}+ \Lambda_{1}^{4}\Omega_{( 2 , r - 1 )}+ \Lambda_{r}^{4}\Omega_{( 1 , r - 2 )}- M_{0}\Omega_{( 1 , r - 1 )}- \Lambda_{( 1 , r )}^{2}= 0 .
D_{\mu \nu}^{( 0 )}= \frac{1}{p^{2}+ \kappa^{2}}\left[ ( \delta_{\mu \nu}- \frac{p_{\mu}\, p_{\nu}}{p^{2}}) - \kappa \epsilon_{\mu \nu \lambda}\,{\frac{p_{\lambda}}{p^{2}}}\right] + \xi \, \frac{p_{\mu}p_{\nu}}{( p^{2})^{2}},
\mathbb{E}\left[ GF(B_{t})-GF(0)\right] =\int_{\mathbb{R}^{N(n+1)}}F(x)g(y)p_{t,\tau}(x,y)dxdy-\mathbb{E}(G)F(0) & =\int_{0}^{t}\int_{\mathbb{R}^{N(n+1)}}F(x)g(y)\frac{\partial p_{s,\tau}}{\partial s}(x,y)dxdyds.
e_{ij}=(\begin{matrix}e_{11}\\e_{22}\\e_{33}\\ 2e_{12}\\ 2e_{13}\\ 2e_{23}\\ \end{matrix}),(\begin{matrix}k_{1}\\k_{2}\\k_{3}\\ \end{matrix})=(\begin{matrix}e_{11}-e_{33}\\e_{22}-e_{33}\\e_{12}C_{3312}+e_{13}C_{3313}+e_{23}C_{3323}\\ \end{matrix})
V^{(1)}_2(r) = \lambda A - \frac{\lambda A}{1+\lambda r^2}+ \lambda B_1 (1+\lambda r^2) + \lambda B_2 (1+\lambda r^2)^2 + \lambda B_3 (1+\lambda r^2)^3 + \lambda B_4 (1+\lambda r^2)^4
S[q] = \lim_{T \rightarrow \infty}\left\{\int_{-T}^T dt \; \frac12 m \dot{q}^2 - \frac14 m \omega_0^2 \ell_0^2 T \left({1 \over T}\int_{-T}^T dt\;{q^2 \over \ell_0^2}\right)^2 \right\}\; ,
\Psi_p(x\otimes g) \Psi_q(y\otimes h) = \psi_p(x) U_g \psi_q(y) U_h = \psi_p(x) \psi_q(\beta_g(y)) U_{g|_q}U_h = \psi_{pq}(x\beta_g(y)) U_{g|_q h}= \Psi_{pq}(x\beta_g(y)\otimes g|_q h) = \Psi_{pq}((x\otimes g)(y\otimes h)).
\lim_{s \rightarrow 0}\mathcal{M}_1\left(s-r+1\right) &=L\sum_{j=1}^{r}\sum_{i=1}^{m}\mathcal{D}\left(i,j\right)\frac{\left(-1\right)^{r-j}}{\left(r-j\right)!}\mathbf{b}_i^t\mathbf{\Psi}^{-1}\mathbf{D}_i\mathbf{a}_j .
n(r;\sigma+it) \leq \sum_{i=1}^{1+[r]}n_L(t_i) \leq \alpha_8 \sum_{i=1}^{1+[r]}\{\log d_L +n_L \log(|t_i|+2)\}\leq \alpha_8 (1+r) \{\log d_L +n_L \log(|t|+r+2)\}.
\nabla \hat{J}(u_{0}) = \mathbb{E}_{v_0 \in \mathbb{B}^d}\left[ \nabla J(u_{0}+ \epsilon v_0) \right] = \frac{d}{\epsilon}\cdot \mathbb{E}_{v_0 \in \mathbb{S}^{d-1}}\left[ J(u_{0}+ \epsilon v_0)v_0 \right],
H^{(0)}(q,{\bf n})-m{\quad}\rightarrow{\quad}-\frac{1}{{2m}{q}^2}\frac{\partial}{{\partial}{q}}{q}^{2}\frac{\partial}{{\partial}{q}}+\frac{\bf L^{2}}{2mq^2},{\quad}{\bf P}^{(0)}(q,{\bf n})\rightarrow{\quad}-i\nabla_{\bf q}.
S_1 \leq \sum_{i= 1}^I \sum_{k=1}^{i-1}\frac{x_k}{x_i}\Delta x_k \sum_{j=\alpha_{i-1,k}}^{\alpha_{i,k}-1}\beta_{j,k}\Delta x_j |n_j n_k-\hat{n}_j \hat{n}_k|.
\eta^{-1}\left\{\P[1, \ldots, 1+\eta \delta,\ldots, 1] - \P[1, \ldots, 1,\ldots, 1]\right\}= \eta^{-1}\left\{\sum_{\Phi_k}\cdots \sum_{\Phi_i}\eta \delta(f_C) \sum_{\Phi_{i-1}}\cdots \sum_{\Phi_{1}}1 \right\}= \lambda.
\left(U^T\left(\frac{3n+2}{6}- \int_0^\infty Tr[\gamma e^{-2y \gamma}] e^{-y \gamma}dy\right)U\right)^{ij}=\left(\frac{3n+2}{6}-\sum_l \frac{\lambda_l}{2\lambda_l+\lambda_i}\right) \delta^{ij}.
\Phi(E_{13}+E_{31})= \begin{bmatrix}-18 -22 -100 -22 52 161 -100 161 -26 \end{bmatrix}, \Phi(E_{23}+E_{32})=\begin{bmatrix}4 -30 -165 -30 20 -50 -165 -50 -20 \end{bmatrix}.
D_{\varphi(t)}[\ddot{\varphi}(t) - \langle \partial \dot{\varphi}(t), \bar{\partial}\dot{\varphi}(t) \rangle_{\varphi(t)}] = 0, \int [\ddot{\varphi}(t) - \langle \partial \dot{\varphi}(t), \bar{\partial}\dot{\varphi}(t) \rangle_{\varphi(t)}] \omega_{\varphi(t)}^n = 0.
W^{\beta}_{k=0}= \mbox{}-\frac12 \int_0^{\infty}\! \frac{{\mathrm{d}}s}{s}\,{\mathrm e}^{-s m^2}\Big({\mathrm Tr}{K}^{\beta}(s)|_{k=0}-{\mathrm Tr}K (s)|_{\phi=0}\Big) \Big|_{m^2=0}.
\frac{J_\infty^{\mathsf{MPC}}- J_\infty^\star}{J_\infty^\star}J_\infty^{\mathsf{MPC}}= \sum_{k=0}^\infty \ell \left( x_k^\mathsf{MPC},\tilde \mu \left(x_k^\mathsf{MPC}\right) \right),
I_{1,2}:= \int_0^t\! \frac{\mathrm{d}s}{\sqrt{t-s}}\int_{\mathbb{T}\times \mathbb{R}}\! \left\vert{\boldsymbol{\tilde{\varphi}}}^{\nu_s}(x) \right\vert \nu_s(\mathrm{d}x, \mathrm{d}v) \left\vert \int_{\mathbb{T}\times \mathbb{R}}\! \varphi(x-y) (\nu_t -\mu_t)(\mathrm{d}y,\mathrm{d}w) \right\vert\,.
\hat{R}d K_{1}^{\dagger}\hat{R}K_{1}= K_{1}\hat{R}d K_{1}^{\dagger}\hat{R}^{- 1}\; , \quad \hat{R}d K_{1}^{\dagger}\hat{R}d K_{1}^{\dagger}= - d K_{1}^{\dagger}\hat{R}d K_{1}^{\dagger}\hat{R}^{- 1}\; .
\left[{\partial^2\over \partial u^2}+{u^2+3\Lambda^4\over 4(u^2-\Lambda^4)^2}\right]\sqrt{\Lambda^4-u^2}{\partial a_D\over \partial u}=0=\left[{\partial^2\over \partial u^2}+{u^2+3\Lambda^4\over 4(u^2-\Lambda^4)^2}\right]\sqrt{\Lambda^4-u^2}{\partial a\over \partial u}.
\sigma\in \widetilde{L}^{\infty}(\mathbb{R}^{+};\widetilde{B}^{\frac{N}{2}+1-\alpha,\frac{N}{2}}), u\in \widetilde{L}^{\infty}(\mathbb{R}^{+};\dot{B}^{\frac{N}{2}+1-\alpha}_{2,1})\cap L^{1}(\mathbb{R}^{+};{\dot{B}^{\frac{N}{2}+1}_{2,1}}).
\sigma_{y}^{2}(\tau)=\langle\sigma_{y}^{2}(2,\tau,\tau)\rangle=\frac{1}{2}\langle(\overline{y}_{n+1}-\overline{y}_{n})^{2}\rangle=\frac{1}{2\tau^{2}}\langle(x_{n+2}-2x_{n+1}+x_{n})^{2}\rangle
\sqrt{det[\begin{matrix}x_{1}\cdot x_{1}&x_{1}\cdot x_{2}&\cdot\cdot\cdot&x_{1}\cdot x_{k}\\x_{2}\cdot x_{1}&x_{2}\cdot x_{2}&\cdot\cdot\cdot&x_{2}\cdot x_{k}\\ \cdot\cdot\cdot&\cdot\cdot\cdot&\cdot\cdot\cdot&\cdot\cdot\cdot\\x_{k}\cdot x_{1}&x_{k}\cdot x_{2}&\cdot\cdot\cdot&x_{k}\cdot x_{k}\end{matrix}]}
x = \frac{m\nu_0^2}{c\varepsilon^2HZ^4n}\int\limits_v^{v_0}{\frac{v^3dv}{0.96 + \log \frac{h\nu_0 v^4}{8\pi c\varepsilon^2HZ^4}}}= 2.4\frac{m\nu_0}{nh}\int\limits_{\frac{0.104h\nu_0 v^4}{c\varepsilon^2HZ^4}}^{\frac{0.104h\nu_0 v_0^4}{c\varepsilon^2HZ^4}}{\frac{du}{\log u}}~~.
\|e_F^{\mathcal{N}}\|\leq \|e_F^{\check{s}}\| \leq \tau_F\sum^{\mathcal{N}}_{n=1}\|g(u^n)-g(U_C^n)-g'(U_C^n) \left(U_F^n-U_C^n\right)\| +\bar{C}\tau_F\sum^{\mathcal{N}}_{n=2}\|e_F^{n-1}\|+ 2\sum^{\mathcal{N}}_{n=1}\tau_F\|(R)^{n}\|.
I \leq 2 \sum_{m=1}^K\, ||m^q h_m||_{H_{\Lambda}^s}^2 \cdot \sum_{n=1}^K\, ||n^q h_n||_{H_{x,v}^s}^2\,, II \leq c\, \sum_{k=1}^K\, ||k^q f_k||_{\Lambda}^2\,,
2\sum_{i=0}^{k-1}(k-1-i)b^i \le (k-1)k+\frac{2(k-2)}3\Biggl\{\sum_{i=1}^{k-1}b^i-(k-1)\Biggr\}&=(k-1)k+\frac{2(k-2)}3\Biggl\{\sum_{i=0}^{k-1}a^i-ka^{k-1}\Biggr\}&<(k-1)k+\sum_{i=0}^{k-2}(k-1-i)a^{i}\,.
\left( \frac{\mathsf{c}_{0}\alpha_{n}}{q^{1/2}a_{n}}\right)^{p}+\left( \frac{q^{1/2}\mathsf{c}_{0}\alpha_{n}}{c_{n}}\right)^{p}=k\left( 1+\left( \frac{\mathsf{c}_{0}^{2}\alpha_{n}^{2}}{c_{n}a_{n}}\right)^{p}\right) .
e^{-\frac{5}{3}}\sum_{x=0}^{4}\frac{1}{x!}(\frac{5}{3})^{x}=e^{-\frac{5}{3}}(1+\frac{5}{3}+\frac{25}{18}+\frac{125}{162}+\frac{625}{1944})=\frac{10009}{1944*e^{\frac{5}{3}}}
T = \frac{d - 2 - 2 \kappa}{4 \pi r_{H}}\frac{r_{0}^{\frac{d - 2}{2}}}{h^{\frac{d - 2}{2}}}\sp \Omega_{i}= \frac{l_{i}}{( l_{i}^{2}+ r_{H}^{2})}\frac{r_{0}^{\frac{d - 2}{2}}}{h^{\frac{d - 2}{2}}}
1 = \langle x , s_j \rangle = \langle x_s, s_j \rangle = 2b_{j,j}+\sum_{\substack{1 \leq i \leq 2s i\neq j}}b_{i,j}+ \sum_{\substack{1 \leq i \leq 2s i\neq j}}b_{j,i}= 2 \sum_{1 \leq i \leq 2s}b_{i,j}.
G ( z , z^{\prime}, t ) = \alpha \sum_{n}\frac{e^{i n ( \phi - \phi^{\prime})}}{2 \pi}\int_{o}^{\infty}\frac{r^{( \alpha - 1 )}}{\alpha}J_{\frac{| n |}{\alpha}}( \frac{k}{\alpha}r^{\alpha}) ( k^{2})^{- i c t / l_{P}}k d k \frac{{r^{\prime}}^{( \alpha - 1 )}}{\alpha}J_{\frac{| n |}{\alpha}}( \frac{k}{\alpha}{r^{\prime}}^{\alpha}) .
\max_{1\leq j,k \leq n, j\neq k}|f_j(\tau_k)f_k(\tau_j)|^m \geq \frac{n-{d+m-1 \choose m}}{{d+m-1 \choose m}(n-1)}=\frac{n-(^m(\mathcal{X}))}{(^m(\mathcal{X}))(n-1)}=\frac{1}{n-1}\left[\frac{n}{{d+m-1\choose m}}-1\right]
P(r,\vartheta,\varphi)=\frac{\lambda(1-r^2)}{\pi}\int_0^\pi\frac{\sin^{2\lambda-1}\xi}{\left(1-2r(\cos\vartheta\cos\varphi+\sin\vartheta\sin\varphi\cos \xi)+r^2\right)^{\lambda+1}}d\xi.
M := \displaystyle\max_{i=0,1}(M_{u_i}+ [w_i]_{2\alpha}+ \|u_i\|_{\alpha}+ \| w_i \|_{\alpha}) + N_0 + N_0^{int}\delta M := M_{u_1 - u_0}+ [w_1 - w_0]_{2\alpha}+ \| u_1 - u_0\|_{\alpha}+ \| w_1 - w_0\|_{\alpha}& + (N_0 + N_0^{int}) \|a_1 - a_0\|_{\alpha}+ \delta N_0 + \delta N_0^{int}.
\| E_{1,cone}\|_{N^* \to N}\lesssim \sum_{k < 0}\sum_{j < 0}\sum_{\theta < 2^j}2^{-(k+2j)/2}\| Q_{k+2j}G_{out,k}^{(\theta)}\|_{DL^2L^\infty}\lesssim \sum_{k < 0}\sum_{j < 0}\sum_{\theta < 2^j}2^{-\frac{(k+2j)}{2}}\theta \theta^{\frac32}2^{2k}\big\|P_kQ_{k+2j}A_x\big\|_{L^2L^2}+ 2^{-(k+2j)/2}\theta^{\frac32}2^{2k}\big\|P_kA_0\big\|_{L^2L^2}
\mathbf{W}_{\mathrm{u},i}= \frac{1}{\sqrt{\zeta_{\mathrm{u},i}}}\cdot \hat{\mathbf{H}}_i^{\mathrm{H}}\left( \hat{\mathbf{H}}_i \hat{\mathbf{H}}_i^{\mathrm{H}}\right)^{-1},
A^{-1}(-\omega_1) = (-4/3,-1,-5/3,-2,-4/3,-2/3) A^{-1}(-\omega_1) + \alpha_1 = (-1/3,-1,-5/3,-2,-4/3,-2/3) \vdots A^{-1}(-\omega_1) + \alpha_1 + \alpha_3 + \alpha_4 + \alpha_5 + \alpha_6 = (-1/3,-1,-2/3,-1,-1/3,1/3).
\Psi_*(X\star_2 f)(x,g) = (X\star_2 f)(xg^{-1},g) = \frac{d}{dt}|_{t=0}f(xg^{-1}, g \exp(tX)) = \frac{d}{dt}|_{t=0}f (x \exp(tX) (g \exp(tX))^{-1}, g\exp(tX) ) = \frac{d}{dt}|_{t=0}\Psi_*(f)(x \exp(tX), g \exp(tX) ) = X \star_{2,3}\Psi_*(f)(x,g)
H_{m, n}^+ (x) = 2 i \int_{-\infty}^\infty J_{2 i y + 1}(x) \frac{k\Bigl(- \dfrac{1}{2}i + y\Bigr) V\Bigl(m^2 n, - \dfrac{1}{2}i + y\Bigr) \Bigl(- \dfrac{1}{2}i + y\Bigr)}{\cosh\Bigl(\pi \Bigl(- \dfrac{1}{2}i + y\Bigr)\Bigr)}\, d y
b_2 e_1 - \frac{b_1 - \frac{p-1}{2}b_2}{X}e_2 = b_2 (\frac{p-1}{2}, 1) + \frac{b_1 - \frac{p-1}{2}b_2}{X}(X,0) & = (b_1, b_2) & = (0,0) \in \mathcal{Y}'_n
F_{r, m,n,s}(x; q, c) = \frac{(-\ln q)^{m+n}}{c^2}\sum^{\infty}_{k=2}q^{kx}\sum^{k-1}_{j=1}\frac{(1-q^{jc})(1-q^{(k-j)c})}{(1-q^j)(1-q^{k-j})}(j^{m-1}(k-j)^{n-1}-\alpha_{r,m,n,s}j^{r-1}(k-j)^{s-1}),
\varepsilon_{\mu\nu\lambda\rho}\partial_\lambda\partial_\rho\theta_a^{\rm sing.}(x)=2\pi\Sigma_{\mu\nu}^a(x)\equiv2\pi\int\limits_{\Sigma_a}^{}d\sigma_{\mu\nu}\left(x^{(a)}(\xi)\right)\delta\left(x-x^{(a)}(\xi)\right).
\beta_{n}^{R^{[1]}}= q+\alpha_{1}+\alpha_{2}+ \left(p+\beta \right)\beta \; , n \geq 0 \alpha_{n}^{R^{[1]}}= \frac{n(n+3)}{(n+1)(n+2)}\left( \alpha_{1}\alpha_{2}+ \gamma(p+2\beta)\right) \; , \quad \gamma_{n}^{R^{[1]}}= \frac{n(n+5)}{(n+2)(n+3)}\gamma^{2}\; , \; n \geq 1.
e^{\beta^n}(1-\varepsilon_n)^a < S_n \varphi(x) < e^{\beta^n}(1+\varepsilon_n)^a + \sum_{i=1}^{n-1}(1+\varepsilon_i)^a \cdot e^{\beta^i}< e^{\beta^n}(1+a\varepsilon_n+o(1)).
v_T= \begin{pmatrix}e^{-iN \xi}1 0 e^{iN\xi}\end{pmatrix}=\begin{pmatrix}1 0 e^{iN \xi}1 \end{pmatrix}\begin{pmatrix}0 1 -1 0 \end{pmatrix}\begin{pmatrix}1 0 e^{-iN \xi}1 \end{pmatrix}=: v_l v_0 v_u.
E \left[\left(\mathbf{M}\right)\right] = \left[\left( E \left( \Gamma_{11}\right) \mathbf{W}^H \mathbf{R}_{\scriptscriptstyle{A}}\mathbf{W}+ E \left( \Gamma_{22}\right) \mathbf{W}^H \mathbf{R}_{\scriptscriptstyle{B}}\mathbf{W}\right) \mathbf{P}^2_{\scriptscriptstyle{A}}\right]
=\frac{F_{1}^{2}}{F_{1}F_{2}(F_{1}-F_{2})^{2}F_{3}^{2}}\left((F_{3}-F_{2})F_{1}-(F_{1}-F_{2})F_{3}\right)\left((F_{3}-F_{2})F_{1}+(F_{1}-F_{2})F_{3}\right)=
s^{(-)}_m(y)=\big[y+2m(\lambda+m) +\tfrac{\beta}{2}(\lambda-\tfrac{\beta}{2}-1) \big]s^{(-)}_{m-1}(y) -(m-1)(\lambda+m)(\lambda-\tfrac{\beta}{2}+m-1)(\tfrac{\beta}{2}+m)s^{(-)}_{m-2}(y),
e^{2k}= e^{({{\alpha^2+1}\over{2 \alpha^2}})2k_g}e^{-{(\gamma q_1 q_2 -{\tau_o^2 \over{2 \alpha^2}})}2k_e}e^{-({\tau_o \over{2 \alpha^2}})2k_s}\equiv e^{2K_2 k_g}e^{2K_1 k_e}e^{2K_3 k_s},
T f(D) h=T \bigl(\underset{n\rightarrow\infty}{\mathrm{lim}}{f_{n}(D) h}\bigr)=\underset{n\rightarrow\infty}{\mathrm{lim}}{T f_{n}(D) h}=\underset{n\rightarrow\infty}{\mathrm{lim}}{\tilde{f_{n}}(D) T h}={\tilde{f}(D) T h},
T_{1 2}\, ( \, p_{1 2 0}^{\prime}, \, p_{1 2 0}, \, P_{0}- p_{3 0}) \, \to \, \, T_{1 2}\, ( \, p_{1 2 0}^{\prime}, \, s_{1 2}\, \mp \,{\frac{P_{0}- S}{2}}\, \pm \,{\frac{p_{3 0}- h_{3}}{2}}, \, P_{0}- p_{3 0}) \,
\sum_{m = 0}^{+ \infty}\sum_{\ell = - \infty}^{+ \infty}\psi_{m , \ell , p}( r_{f}, \theta_{f}, z_{f}) e^{- \frac{i}{\hbar}\Delta t \left[ \frac{1}{2}\hbar^{2}g^{2}\ell^{2}+ \hbar \omega ( 2 m + | \ell | + 1 ) \right]}\psi_{m , \ell , p}^{*}( r_{i}, \theta_{i}, z_{i}) \ ,
P = P_{U n t w i s t e d}+ P_{T w i s t e d}, \; P_{U}= \sum_{n = 0}^{n - 1}I ( 1 , \theta^{n}) , \; P_{T}= \sum_{n = 0}^{n - 1}\sum_{[ h , g ] = 0 , m \neq 1}I ( \theta^{m}, \theta^{n}) .
\bar{\mathbf{a}}+\bar{\mathbf{b}}=(\bar{\alpha}_1+\bar{\beta}_1,a_1+\bar{\beta}_2,a_2+b_1,a_3+b_2,\cdots,a_{2k-2}+b_{2k-3},a_{2k-1}+b_{2k-2},\bar{\alpha}_2+b_{2k-1},\bar{\alpha}_3+\bar{\beta}_3,\bar{\alpha}_4+\bar{\beta}_4).
\hat{\hat{S}}= \int d^{11}x sqrt{|\hat{\hat{g}}|}\ \left[\hat{\hat{R}}-{\textstyle\frac{1}{2\cdot 4!}}\hat{\hat{G}}{}^{2}-{\textstyle\frac{1}{6^{4}}}\frac{1}{\sqrt{|\hat{\hat{g}}}|}\hat{\hat{\epsilon}}\partial\hat{\hat{C}}\partial \hat{\hat{C}}\hat{\hat{C}}\right]\, .
\frac{\partial \tilde{N}_{\pm}( x ; \pm{\bf p})}{\partial x_{0}}\pm \frac{{\bf p}}{\omega_{\pm}}\cdot \frac{\partial \tilde{N}_{\pm}}{\partial{\bf x}}\simeq \frac{- i \Sigma_{o f f}^{( b a r e )}( x ; \omega_{\pm},{\bf p})}{2 \omega_{\pm}}.
\begin{array}{l}{{F_{1 2}( x_{1}, x_{2}) = 3 D - | \phi_{1}( x_{1}, x_{2}) |^{2}}}\\{{= 7 F ( D_{1}+ i D_{2}) \phi_{1}( x_{1}, x_{2}) = 3 D 0}}\\ \end{array}\hspace{5 . 9 c m}( 7 )
\mathbb{E}[\ln(1+\mathit{SINR})] = \ \mathbb{E}_N{\left[ \sum_{k=1}^{n}\binom{n}{k}(-1)^{k+1}\int_0^\infty \frac{\mathrm{e}^{- (1-f_{N_i}(0)) \sqrt{kx}\arctan(\sqrt{kx})}\,\mathrm{d}x}{x+1}\right]}.
S=\displaystyle\int d^{8}z E^{-1}\left[\bar{Q}Q+\bar{Q}^{c}Q^{c}+\lambda_{0}(\bar{Q}^{c}Q)(\bar{Q}Q^{c}) +\bar{\xi}_{1}(\bar{Q}Q) +\bar{\xi}_{2}(\bar{Q}^{c}Q^{c})\right],
\alpha_A(l_\Theta(e_i)) =l_\Theta(Ae_i) =l_\Theta(a_{1i}e_1+ a_{2i}e_2) =\exp(\pi i \theta a_{1i}a_{2i}) l_\Theta(e_1)^{a_{1i}}l_\Theta(e_2)^{a_{2i}}\, ((\ref{expansion})).
S^{( N )}\left[{\bf r}_{1}, \dots{\bf r}_{N - 1}\right] ({\bf r}^{\prime \prime},{\bf r}^{\prime}; t^{\prime \prime}, t^{\prime}) = \sum_{j = 0}^{N - 1}\left[ \frac{M \left({\bf r}_{j + 1}-{\bf r}_{j}\right)^{2}}{2 \epsilon}- \epsilon V ({\bf r}_{j}, t_{j}) \right] \; ,
A_1=-\frac{i\sqrt{2}G\hbar D_1\sin(\frac{1}{\hbar}p_3^{(1)}L)}{(E+p_3^{(1)})\left[ e^{-\frac{i}{\hbar}p_3^{(1)}L}-\frac{(G\hbar D_1)^2}{(E+p_3^{(1)})^2}e^{\frac{i}{\hbar}p_3^{(1)}L}\right]}
\Gamma_0(n) =\left\{\left(\begin{matrix}a b c d\end{matrix}\right)\in \mathrm{SL}(2,\mathbb Z);\, c\equiv 0\ \operatorname{mod}\n \right\}, \Gamma_0(m,n) =\left\{\left(\begin{matrix}a b c d\end{matrix}\right)\in \mathrm{SL}(2,\mathbb Z);\, c\equiv 0\ \operatorname{mod}\m,\b\equiv 0\ \operatorname{mod}\n\right\}.
\frac{\xi_{i_{l_{k-1}+1}}}{\xi_{i_{l_{k-1}+2}}}=\frac{\xi_{i_{l_{k-1}+2}}}{\xi_{i_{l_{k-1}+3}}}= \cdots =\frac{\xi_{i_{l_k-1}}}{\xi_{i_{l_k}}}= \alpha_k
\Delta-\Delta' =a'E\otimes\frac{1}{aa'}-ap^{e-c}E\otimes\frac{1}{aa'}&=(m(p^{e}+1)-mp^cp^{e-c})E\otimes\frac{1}{aa'}=mE\otimes\frac{1}{aa'}=\frac{m}{a'}\Delta=\frac{1}{p^e+1}\Delta \ge0,
f(t) = \left( \left( 2 + \frac{4s}{N}\right) \log \left( 1+|t|^{\alpha_{N,s}}\right) + \frac{\alpha_{N,s}|t|^{\alpha_{N,s}}}{1+|t|^{\alpha_{N,s}}}\right) |t|^{\frac{4s}{N}}t
\frac{\partial}{\partial \eta^{\prime \prime}{}^{\mu}}\frac{\partial}{\partial \overline{{\psi}}^{\prime \prime}}\Omega ( \zeta^{\prime \prime}) = \frac{\partial}{\partial \overline{{\psi}}^{\prime \prime}}\frac{\partial}{\partial \eta^{\prime \prime}{}^{\mu}}F ( \zeta^{\prime \prime}) \, ,
\begin{aligned}x_5 =-x_4,\; x_6 =-x_3,\;\; x_7 =-x_2,\;\; x_8 =-x_1, y_5 =-y_4,\; y_6 =-y_3,\;\; y_7 =-y_2,\;\; y_8 =-y_1. \end{aligned}
a_{1, 0, 0, 2}=a_{0, 1, 0, 2}= a_{0, 0, 1, 2}= a_{0, 0, 0, 3}= 0, a_{2,0,0,1}a_{0,1,1,1}^2-a_{1,0,1,1}a_{1,1,0,1}a_{0,1,1,1}+a_{0,2,0,1}a_{1,0,1,1}^2+a_{0,0,2,1}a_{1,1,0,1}^2-4 a_{0,0,2,1}a_{0,2,0,1}a_{2,0,0,1}=0,
G_{\gamma}^{t}\left( \begin{array}{c}{0}\\{1}\\ \end{array}\begin{array}{c c}{1}\\{0}\\ \end{array}\right) G_{\gamma}- \left( \begin{array}{c}{0}\\{1}\\ \end{array}\begin{array}{c c}{1}\\{0}\\ \end{array}\right) = 2 \left( \begin{array}{c}{a_{1 1}a_{2 1}}\\{a_{1 2}a_{2 1}}\\ \end{array}\begin{array}{c c}{a_{1 2}a_{2 1}}\\{a_{1 2}a_{2 2}}\\ \end{array}\right) .
F_{a b}= \int_{a}^{b}d x e^{i \beta f ( x )}= \frac{1}{i \beta}\left[ \frac{1}{f{'}}e^{i \beta z}|_{f ( a )}^{f ( b )}- \int_{f ( a )}^{f ( b )}d z e^{i \beta z}\frac{d}{d z}( f{'})^{- 1}\right] .
f_+^\delta(u) \geq \sup_{|v|\leq \delta}f_+^\delta(u+v),\f_+^\delta(\pm 1+\delta) =0,\f_+^\delta(-\delta)=0,\ \frac{d}{du}f_+^\delta (-\delta)=\mu , f_-^\delta(u) \leq \inf_{|v|\leq \delta}f_-^\delta(u+v),\f_-^\delta(\pm 1-\delta) =0,\f_-^\delta(\delta)=0,\ \frac{d}{du}f_-^\delta (\delta)=\mu ,
e^{C_2s}D^2 = \exp\left( C_2\left( \log \frac{1}{D}\right)^{\alpha}\right)D^2 = \exp\left( C_2\left( \log \frac{1}{D}\right)^{\alpha}\right)e^{2\log D}= \exp\left( -2\left( \log \frac{1}{D}\right)+ C_2\left( \log \frac{1}{D}\right)^{\alpha}\right).
\Phi_{k-1}(\xi_{i_1},\ldots,\xi_{i_k})=\emph{pr}\circ[\cdots[[Q',\iota_{\xi_{i_1}}],\iota_{\xi_{i_2}}],\ldots,\iota_{\xi_{i_k}}]\subset\mathfrak{X}(E)[1],\;k\in\mathbb{N},
(c\tau+d)^{-k}F(\gamma\tau) = (c\tau + d)^{-k}\Big(f(\gamma\tau) - f_r(\gamma\tau)\Big(\frac{E_2(\gamma\tau)}{12}\Big)^r\Big) &= (c\tau + d)^{-k}f(\gamma\tau) - \frac{1}{12^r}(c \tau + d)^{-k+2r}f_r(\gamma\tau)\big((c\tau + d)E_2(\gamma\tau)\big)^r
\left( \begin{array}{cc}\partial_1 \partial_2 \partial_4 \partial_3 \end{array}\right) \mapsto\left( \begin{array}{cc}\partial_1 q^{\alpha L_6}\partial_2 \partial_4 \partial_3 q^{- \alpha L_6}\end{array}\right) \equiv \tilde{\partial}\quad ,
C^{(-)}_r(m-l)=\frac{{m-l\choose r}(\tfrac{\beta}{2}-\lambda-m+l)_{m-l-r}(-\tfrac{\beta}{2}-m+l-1)_{m-l-r}}{{m\choose k}(\tfrac{\beta}{2}-\lambda-m)_{m-k}(-\tfrac{\beta}{2}-m-1)_{m-k}}C^{(-)}_k(m)
0 = t(t^2+1)(t^4-2\beta t^2+1)y'' - ((1-2 n) t^6+(2 n+3) t^4+t^2 (-4 \beta +(4 \beta -2) n-1)-2 n+1)y' -(\beta +1) n t \left(n t^2+n+t^2-1\right)y.
G_{\hat{\mu}\hat{\nu}}^{1 1}= \left( \begin{array}{c c}{e^{- \frac{2}{3}\phi}G_{\mu \nu}+ e^{- \frac{2}{3}\phi}C_{\mu}C_{\nu}}&{e^{\frac{1}{3}\phi}C_{\mu}}\\{}&{}\\{e^{\frac{1}{3}\phi}C_{\nu}}&{e^{\frac{4}{3}\phi}}\\ \end{array}\right) .
\frac{u_0^{\sf T}v_{-d}}{a}+\frac{u_{\ell d}^{\sf T}v_{(\ell-1)d}}{b}= \frac{1}{c}\;, \frac{u_{(\ell-1)d}^{\sf T}v_{\ell d}}{a}+\frac{u_{-d}^{\sf T}v_0}{b}= \frac{1}{c}\;.
\|W_mx_iW_m^* - W_nx_iW_n^*\|_2 = \|w_{n+1}\cdots w_{m}x_i w_m^* \cdots w_{n+1}^* - x_i\|_2\le \sum_{j = n+1}^m \frac{1}{2^j}< \frac{1}{2^n}
u(t,x) = \int_F p^b_t(x,y) u_0(y) \mu(dy) + \int_0^t \int_F p^b_{t-s}(x,y) f(s,u(s,y)) \mu(dy)ds &+ \int_0^t \int_F p^b_{t-s}(x,y) g(s,u(s,y)) \xi(s,y) \mu(dy)ds
\begin{aligned}\sum_{\begin{subarray}{c}0{\le}m_1<_{c_1}\cdots<_{c_{p-1}}m_p<\infty\end{subarray}}z^{m_p}\frac{{m_p}!}{(\alpha)_{m_p}}\left\{\prod_{i=1}^{p}\frac{1}{(m_i+\alpha)^{a_i}(m_i+1)^{b_i}}\right\},\end{aligned}
\mathcal B:T\mathcal S\times C^{m,\alpha}_{\delta}(M) \rightarrow \mathbb B D\mathcal B(h^{(4)},G)= (~ h_{\partial M}&(H_{\partial M})'_{h^{(4)}}+O_0(G) &tr_{\partial M}K'_{h^{(4)}}+O_0(G) &(\omega_{\mathbf n})'_{h^{(4)}}+Nd_{\partial M}G+O_0(G) &\beta_{g^{(4)}}h^{(4)}~).
\sum_{Q}\frac{1}{|Q|^{\alpha}}\left[ \left(\int_{Q}f \right)^2 - \frac{\Lambda}{a}\int_{Q}f \right] \ge 0, a:= \frac{2^d}{2}\left(1 + \sqrt{1 + \frac{1-2^{-d}}{2^{2s}-1}}\right).
R_n(s,P_n,F_s)=- \sum_{x}P_n({x})\left[ \mu_{S_{x},F_s}(s)+(1-s) \mu_{S_{x},F_s}'(s)\right]+\frac{1}{\sqrt{n}}(1-s)\sqrt{2\sum_{x}P_n({x}) \mu_{S_{x},F_s}''(s)}+ \frac{1}{n}\log 8.
S ={\frac{1}{\sqrt{M^{2}}}}\varepsilon_{i j}L^{i}\tilde{D}L^{j}={\frac{L^{*}{}^{2}}{\sqrt{M^{2}}}}\varepsilon_{i j}\hat{k}^{i}\tilde{D}\hat{k}^{j}={\frac{L^{*}{}^{2}}{\sqrt{M^{2}}}}\overline{\kappa}_{2}.
\displaystyle \displaystyle \int_{\Omega}f(y) dy =\int_{Z_G(x)(F)\backslash G(F)}\int_{\Omega_x}f(g^{-1}yg)\eta^G_x(y)dydg =[Z_G(x)(F):G_x(F)]^{-1}\int_{G_x(F)\backslash G(F)}\int_{\Omega_x}f(g^{-1}yg)\eta^G_x(y)dydg
\frac{d^{n-1-l}A}{dv^{n-1-l}}=P_{l-1}+\frac{v^{l-1}}{2}\left\{(-1)^l\frac{(n-l-1)!}{(n-1)!}\hat{I}_{n,n-1}-\sum_{m=1}^l\frac{(-1)^m}{(m-1)!(l-m)!(n-m)}\hat{I}_{n,m-1}\right\}.
i{\tilde{A}}^{+}= \left[ \begin{array}{l l}{{< \psi_{3}^{( 0 )}| d \psi_{3}^{( 0 )}>}}&{{< \psi_{4}^{( 0 )}| d \psi_{3}^{( 0 )}>}}\\{{< \psi_{3}^{( 0 )}| d \psi_{4}^{( 0 )}>}}&{{< \psi_{4}^{( 0 )}| d \psi_{4}^{( 0 )}>}}\\ \end{array}\right] = i{\tilde{A}}_{a}d x^{a}.
\langle \Phi_0(x)^{(0)}(\alpha),e\rangle =\Phi_0(x)^{(0)}[\langle \alpha,e\rangle]-[\Phi_0(x)^{(0)}, \iota_e]^{(-1)}(\alpha) &=\varrho(x)[\langle\alpha,e\rangle]-\langle\alpha,\nabla_x(e)\rangle.
\int_{0}^{t}J \phi(s,\tilde{X})\tilde{W}_s= \int_{0}^{t}J \chi(s,\tilde{X}) \sigma(s,\tilde{X})\tilde{W}_s = \int_{0}^{t}J \chi(s,\tilde{X})N_s,\,\,\,t \geq 0
\sqrt[4]{1+8x}\cdot{}_2F_1\left(\frac{1}{3},\frac{2}{3};1,x\right)={}_2F_1\left(\frac{1}{6},\frac{5}{6};1;\frac{1}{2}-\frac{1-20x-8x^2}{2(1+8x)^{3/2}}\right).
E = \int d^{2}x \left({\frac{\kappa^{2}- e^{2}| \phi |^{2}}{2 e^{2}| \phi |^{2}}}B^{2}+{\frac{1}{2}}\left[ | \partial_{i}\phi |^{2}+{\frac{\kappa^{2}e^{2}| \phi |^{2}}{\kappa^{2}- e^{2}| \phi |^{2}}}\tilde{A}_{i}^{2}\right] + V ( \phi ) \right) .
\Lambda^2_7 V_7^\ast = \{\alpha^2 \in \Lambda^2 V_7^\ast \mid \ast(\alpha^2 \wedge \varphi) = 2 \alpha^2\}, \mbox{and}\Lambda^2_{14}V_7^\ast = \{\alpha^2 \in \Lambda^2 V_7^\ast \mid \ast(\alpha^2 \wedge \varphi) = - \alpha^2\}.
f_{n+1}(C_{f_{n+1},\delta}\cap I_L) \subset f_{n+1}(C_{f_{n+1},L^{-1}}\cap I_L) = f_{n}(C_{f_{n},L^{-1}}\cap I_L) + c_{I}\subset f_{n}(C_{f_{n},L^{-1}}) + c_{I}
\left. \frac{d^{2}}{dt^{2}}{\rm Vol}[\iota_{t}] \right|_{t=0}= \int_{L}\left(\frac{1}{4}(\Delta f)^{2}- 2 g(Y, Y) -{\rm Ric}(\phi Y, \phi Y) \right. &\left. -2 g(\nabla_{Y}Y, H)+ g(Y, H)^{2}\right){\rm vol}_{\iota^{*}g},
\rho^{0}= \left( \begin{array}{c c}{{0}}&{{- i}}\\{{i}}&{{0}}\\ \end{array}\right) \, \, \, \mathrm{a n d}\, \, \, \rho^{1}= \left( \begin{array}{c c}{{0}}&{{i}}\\{{i}}&{{0}}\\ \end{array}\right) .
\|t\|_p \le \left( \sum_k |t_k|^2 \|t\|_{\infty}^{p-2}\right)^{1/p}\le \left( p^{-1 - (p-2)/r}\right)^{1/p}= \left( p^{-p/r}p^{(2-r)/r}\right)^{1/p}& \le \frac{e^{(2-r)/r}}{p^{1/r}}= \frac{e^{(2-r)/r}}{p^{1/r}}\left\{\sqrt{p}\|t\|_2 + p^{1/r}\|t\|_\infty \right\},
f(\mathfrak{M}_1) \subset (\mathfrak{M}_2\otimes_{\mathfrak{S}}\mathcal{O}_{\mathcal{E}}) \cap (\mathfrak{M}_2\otimes_{\mathfrak{S}}\mathfrak{S}_g) = \mathfrak{M}_2\otimes_{\mathfrak{S}}(\mathcal{O}_{\mathcal{E}}\cap \mathfrak{S}_g) = \mathfrak{M}_2.
[\begin{matrix}ct^{\prime}\\x^{\prime}\\y^{\prime}\\z^{\prime}\end{matrix}]=[\begin{matrix}cosh\phi&0&-sinh\phi&0\\ 0&1&0&0\\ -sinh\phi&0&cosh\phi&0\\ 0&0&0&1\\ \end{matrix}][\begin{matrix}ct\\x\\y\\z\end{matrix}],
|f_{\gamma, I}(\mathbf{t})[\alpha] - f_{\gamma, I}^{[N]}(\mathbf{t})[\alpha]| \preceq \sum_p\|\alpha\|^{|T(\gamma)|}_{O(\gamma)}\sum_{i = 1}^{|E(\gamma)|}t_i^{N - n/2 - p_i}\prod_{j \neq i}t_j^{- n/2 - p_j}