harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\arraycolsep=2pt\left[\begin{matrix}\,1 begin{align*}6pt] L\ \end{matrix}\right]\left[\begin{matrix}\,f(x) 0 begin{align*}6pt]\widetilde{F}G \ \end{matrix}\right]=\left[\begin{matrix}\,f(x) 0 begin{align*}6pt]\Phi LG\ \end{matrix}\right]=\left[\begin{matrix}\,f(x) \ 0\ begin{align*}2pt]\Phi' 0 \Phi'' M\end{matrix}\right].
(x^1,\ldots,x^{2n},x^{2n+1})(y^1,\ldots,y^{2n},y^{2n+1})= \Big( x^1+y^1,\ldots,x^{2n}+y^{2n},x^{2n+1}+y^{2n+1}&+\frac{1}{2}\sum_{i=1}^n(x^iy^{i+n}-y^ix^{i+n})\Big)
\mathcal{A}\left(\sum_{k=1}^{n+1}P_{k}^{(l)}\right)=\sum_{k=1}^{n+1}\mathcal{A}(P_{k}^{(l)})=\sum_{k=1}^{n+1}\int_{P_{0}}^{P_{k}^{(l)}}\underline{\omega}=\underline{\rho}^{(l)},
P=\frac{1}{4}[\begin{matrix}1+\sigma^{3}&1+\sigma^{3}\\ 1+\sigma^{3}&1+\sigma^{3}\end{matrix}]=\frac{1}{2}[\begin{matrix}1&0&1&0\\ 0&0&0&0\\ 1&0&1&0\\ 0&0&0&0\end{matrix}]
q^{2}\widetilde \Sigma ( q^{2}) ={\frac{a}{\pi^{2}}}\int{\frac{d^{4}l}{l^{2}- l^{2}\widetilde \Sigma ( l^{2})}}{\frac{q \cdot l}{( q + l )^{2}}}- \mathrm{s u b t r a c t i o n s}
\boldsymbol{\rho}^{\star}=P_{\operatorname{Box}[\boldsymbol{0}, \boldsymbol{1}]}\left(\boldsymbol{\rho}\left(i+\frac{1}{2}\right)-\begin{bmatrix}\boldsymbol{\lambda}^{\star}\boldsymbol{\lambda}^{\star}\end{bmatrix}\right),
\tilde{m}_2 ={\frac{2\int d\tilde{p}(\Sigma(\tilde{p}))^2 f(\tilde{p})-{\frac{\beta^2}{16\pi}}(\tilde{\mu}^2(\varphi_0)-\tilde{m}_0^2)(\int d\tilde{p}{\frac{\Sigma(\tilde{p})}{f(\tilde{p})}})^2}{\int d\tilde{p}(\Sigma(\tilde{p}))^2}}\;,
\sum_{n = 1}^{\infty}e^{- \beta k E_{n}}= \frac{1}{\sqrt{\pi}}\int_{0}^{\infty}e^{- \frac{x^{2}}{4}- \frac{1}{x^{2}}{\beta}^{2}k^{2}m^{2}}\frac{1}{e^{\frac{2{\beta}^{2}e B k^{2}}{x^{2}}}- 1}.
\Gamma^{\alpha}{}_{\beta}= \frac{(-1)^{d(d-3)/4}}{d!{\sqrt-\gamma}}\epsilon^{i_{1}..i_{d}}E_{i_{1}}{}^{a_{1}}E_{i_{2}}{}^{a_{2}}\ldots E_{i_{d}}{}^{a_{d}}\Gamma_{a_{1..a{_d}}}~~.
\sum_{\substack{z_1<q\le p\le z_2 q\le (X/p)^{1/2}}}S_{p q}(q) =\sum_{\substack{z_1<q\le p\le z_2 q\le (X/p)^{1/2}z_6< p q}}S_{p q}(q)+\sum_{\substack{z_1<q\le p\le z_2 q\le (X/p)^{1/2}z_3\le p q<z_5}}S_{p q}(q) &\qquad+\sum_{\substack{z_1<q\le p\le z_2 z_1\le p q<z_2}}S_{p q}(q)+o\Bigl(\frac{\#\mathcal{A}{}}{\log{X}}\Bigr).
a_{j,k}=-\lfloor \frac{j-k+2}{4}\rfloor ,\b_{j,k}=\lfloor \frac{j+k+2}{4}\rfloor, \c_{j,k}=\lfloor\frac{j+k}{4}\rfloor,\d_{j,k}=-\lfloor \frac{j-k}{4}\rfloor
\mathcal{M}[(\hat{x}_t,\hat{x}_{t+1});(\theta_t,\theta_{t+1})]:=\beta_t\,\frac{\partial{U(\hat{x}_t,\theta_t)}}{\partial{\hat{x}}}\left[\beta_{t+1}\frac{\partial{U(\hat{x}_{t+1},\theta_{t+1})}}{\partial{\hat{x}}}\right]^{-1}.
\mathsf H_W = \nu \sum_{f}\left( 1 - \mathsf T_f^2 \right), \mathsf H_V = \mu \sum_e \left( 2 - \mathsf T_e^2 \prod_{f:\,e \in \partial f}\left( \mathsf T_f + \mathsf T_f^3 \right) \right)
M_{R}\equiv \left( \begin{array}{r l}{{\cos \theta}}&{{\sin \theta}}\\{{- \sin \theta}}&{{\cos \theta}}\\ \end{array}\right) , \hspace{0 . 5 c m}M_{H}\equiv \left( \begin{array}{c c}{{1}}&{{\beta}}\\{{0}}&{{1}}\\ \end{array}\right) , \hspace{0 . 5 c m}M_{S}\equiv \left( \begin{array}{c c}{{e^{\alpha}}}&{{0}}\\{{0}}&{{e^{- \alpha}}}\\ \end{array}\right) .
t_\mathrm{w,2}(n) =g_\mathrm{UL}^{-1}N l_\mathrm{n}+ g_\mathrm{DL}^{-1}N l_r + g_\mathrm{UL}^{-1}l_\mathrm{n}+ g_\mathrm{DL}^{-1}n l_\mathrm{h}.
\psi_{klm}(\vec{r})=\int\psi_{\vec{k}}(\vec{r})Y_{l}^{m}(\hat{k})d\hat{k}=R_{kl}(r)Y_{l}^{m}(\hat{r}),R_{kl}(r)=4\pi i^{l}w_{l}(\eta,\rho)/r.
2^{n}\widehat{A}_{n+1}(x)=(1-x)^nP_n'\left(\frac{1+x}{1-x}\right)=\sum_{k=0}^{\lfloor(n+1)/2\rfloor}(n-2k+1)p(n,n-2k+1)(1-x)^{2k}(1+x)^{n-2k}.
b_{l}^{n}=\sqrt{\frac{n(n-1)...(n-l+1)}{(n+2)(n+3)...(n+l+1)}}=\frac{\sqrt{\binom{n}{l}}}{\sqrt{\binom{n+l+1}{l}}}= \frac{n!\sqrt{n+1}}{\sqrt{(n+l+1)!(n-l)!}}
E(f)(t) =\sum_{|\alpha|+|\beta|\le N}\| \partial^\alpha_\beta f\|^{2}_{L^2_{x,{p,\mathcal{I}}}},\cr\mathcal{D}(f)(t) =\|\{I-P\}f\|^2_{L^2_{x,p,\mathcal{I}}}+\sum_{1\le |\alpha|+|\beta|\le N}\|\partial^\alpha_\beta f\|^2_{L^2_{x,p,\mathcal{I}}},
\alpha_{a}^{- 1}( \mu ) = \alpha_{a}^{- 1}( \Lambda ) +{\frac{{\widetilde b}_{a}}{4 \pi}}\int_{( \xi \Lambda )^{- 2}}^{( \xi \mu )^{- 2}}{\frac{d t}{t}}\sum_{{\bf m}}\exp ( - \pi t M_{\bf m}^{2}) .
\mathbb{E}_0\left[ \left(U_h^{(13)}\right)^2 \left(U_h^{(23)}\right)^2 \left(U_h^{(14)}\right)^2 \left(U_h^{(24)}\right)^2 \right] ={k \choose d}^8 \left(\frac{d!\zeta_{d}}{n^{d}}\right)^4 + O\left(n^{-4d -1}\right).
\mathrm{P}^o_n = \mathrm{P}\left( \log\left( \frac{x_n\rho+1}{x_n \alpha_{1,n}^2\rho + 1}\right) <R_{1,k}\right) + \underset{Q_3}{\underbrace{\mathrm{P}\left( \log\left( \frac{x_n\rho+1}{x_n \alpha_{1,n}^2\rho + 1}\right) >R_{1,k},\log\left(1+\rho x_n\alpha_{1,n}^2<R_{1,n}\right) \right)}}.
\nabla^a\left(r^{p-2}Q_{ab}T^b\right)= -6Mr^{p-5}D^\dagger H_2 \nabla_T H_1 +\left(\frac{2-p}{4}\right)r^{p-3}\left(2|\nabla_{\underline{L}}H_1|^2-2|\nabla_L H_1|^2+|\nabla_{\underline{L}}H_2|^2-|\nabla_L H_2|^2\right).
C_2 (z) = \chi_{(r, \infty)}(|p|) C_2 (z) \chi_{(r, \infty)}(|p|)+ \bigl\{\chi_{[0, r]}(|p|) C_2 (z) \chi_{(r, \infty)}(|p|) + \chi_{(r, \infty)}(|p|)C_2 (z) \chi_{[0, r]}(|p|) + \chi_{[0, r]}(|p|) C_2 (z) \chi_{[0, r]}(|p|)\bigr\}.
T_{t}^{t}= T_{z}^{z}= \varphi^{2}e^{- 2 g M}+ \frac{A^{2}\varphi^{2}}{r^{2}}e^{- 2 g M}+{\cal P}M^{2}+ \left( \frac{A^{\prime}}{r h}\right)^{2}+{\cal R}\left( \frac{G}{g r}\right)^{2}+ U .
\left[ \begin{array}{c}p_{n+1}(\tau) q_{n+1}(\tau)\end{array}\right] ={\cal D}(\tau)\left[\begin{array}{c}p_{n}(\tau) q_{n}(\tau)\end{array}\right], \left[\begin{array}{c}p_{1}(\tau) q_{1}(\tau)\end{array}\right]= \left[\begin{array}{c}0 1\end{array}\right],
e^{i S_{\Lambda - \delta \Lambda}( \phi_{+}, \phi_{-})}\equiv \int \prod_{\Lambda_{0}> |{\vec{q}}| > \Lambda - \delta \Lambda}{\cal D}[ \phi_{+}({\vec{q}}, t ) ]{\cal D}[ \phi_{-}({\vec{q}}, t ) ] \; e^{i S_{c l}[ \phi_{+}, \phi_{-}]}
osc_{\mathcal{U},\Gamma}(x,y) := \sup_{z\in Q_y}|\langle A^{-1}_\sigma\pi(\sigma(x))\psi,\pi(\sigma(y))\psi-\Gamma(y,z)\pi(\sigma(z))\psi\rangle| & = \sup_{z\in Q_y}|\mathcal{R}(x,y)-\Gamma(y,z)\mathcal{R}(x,z)|
\operatorname{det}\left(\mathbf{R_e}^{mmse}\right)=\frac{\operatorname{det}\left(\mathbf{R}_{full}^{(g)}\right)}{\operatorname{det}\left(\mathbf{I}_{TD}+\sum_{l=0}^{L_g-1}\mathbf{R}^{(g)}_{code}(l) \otimes \mathbf{SNR}^{(g)}_{mimo}(l)\right)}
P_{k}^{S}\approx \left. \frac{1 6}{9}\frac{G^{2}V^{2}}{c_{s}\dot{\phi}^{2}F_{, X}}\right|_{c_{s}k = a H}= \left. \frac{1 6}{9}\left( \frac{3}{8 \pi}\right)^{2}\frac{H^{4}}{( c_{s}F_{, X}) \dot{\phi}^{2}}\right|_{c_{s}k = a H}~ ,
\mathrm{det}_{q}K_{+}(\lambda )\mathrm{det}_{q}\,\mathcal{U}_{-}(\lambda)=4(\lambda^{2}-\eta^{2})(\lambda^{2}-p^{2})((1+\xi^{2})\lambda^{2}-q^{2})\prod_{b=1}^{\mathsf{N}}(\lambda^{2}-(\xi_{n}+\eta)^{2})(\lambda^{2}-(\xi_{n}-\eta )^{2}).
\tau_x=\tau_u=\xi_u=\eta_{uu}=0, GA(2\xi_x-\tau_t)- G_xA\xi- GA_u\eta=0, \eta_t- W_t\tau-GA\eta_{xx}-G_xA\eta_x+W(\eta_u-\tau_t)=0, GA_u(2\xi_x-\eta_u-\tau_t)-GA\eta_{uu}- GA_{uu}\eta- G_xA_u\xi=0, GA(\xi_{xx}+\xi_x-2\eta_{xu})-G_xA\tau_t-\xi_t-G_xA_u\eta -G_{xx}A\xi -2 GA_u\eta_x=0.
b_{0}( K , t ) = b ( K , t ) + \int \int \int \overline{{{\mu}}}_{K , P}^{( - )}( K^{\prime}, P^{\prime}) b^{\ast}( P , t ) b ( P^{\prime}, t ) b ( K^{\prime}, t ) d K^{\prime}d P^{\prime}d P + O_{2 , 3}
{\bf b}= Q \circ \pi_{\mathbb{R}^{k'_1}\times \mathbb{W}_1 \times \mathbb{R}^{k'_2}\times \mathbb{W}_2}\circ (\Psi_1 \times \Psi_2) &\,\,\,\,\,\,\,\,\circ ((h_1 \times h_2) - (\mathfrak{s}_1 \times \mathfrak{s}_2)) \circ (\psi_1 \times \psi_2)^{-1}\circ q^{-1}&= Q \circ ({\bf b}_1 \times{\bf b}_2) \circ q^{-1}.
\eta = h'(v) \mathrm{d}v, \T_0= T_1 =0, \T_2=\frac{h'(v)}{\sqrt{c^2 u^2 a(h(v))^2+h'(v)^2}}, \T_3=\frac{c u a(h(v))}{\sqrt{c^2 u^2 a(h(v))^2+h'(v)^2}}.
R_{K}=\log_{2}\bigg(1+\frac{P_{K}{\lvert \mathbf{d}_{K}^{H}\mathbf{h}_{K}\rvert}^{2}}{\sum_{\pi_{k',i'}>\pi_{K}}P_{k',i'}{\lvert \mathbf{d}_{k',i'}^{H}\mathbf{h}_{k'}\rvert}^{2}+\sigma_{\textrm{ul}}^{2}}\bigg).
\frac{L[C]}{\epsilon}+\mathcal{A}[C]\mathop{=}_{\epsilon ,\epsilon '\to 0}\frac{L[f(C)]}{\epsilon '}+\mathcal{A}[f(C)]+\oint_{C}\left| \frac{df(x(s))}{ds}\right| ds\int_{\epsilon /x(s)^{2}}^{\epsilon '}\frac{dy}{y^{2}}\; .
\left( \begin{array}{c c}{\frac{\tilde{B}_{\lambda}}{( k^{2})^{3 / 2 - \beta_{\lambda}}}}&{i \vartheta \frac{\tilde{B}_{\lambda \sigma}}{( k^{2})^{\beta_{\lambda \sigma}}}}\\{i \vartheta \frac{\tilde{B}_{\lambda \sigma}}{( k^{2})^{\beta_{\lambda \sigma}}}}&{\frac{\tilde{B}_{\sigma}}{( k^{2})^{3 / 2 - \beta_{\sigma}}}}\\ \end{array}\right)
W^{\mu \nu}= \frac{\partial^{2}L}{\partial \dot{x}_{\mu}\partial \dot{x}_{\nu}}= - \frac{m}{\sqrt{\dot{x}^{2}}}\left( g^{\mu \nu}- \frac{\dot{x}^{\mu}\dot{x}^{\nu}}{\dot{x}^{2}}\right) = - m \left( g^{\mu \nu}- \dot{x}^{\mu}\dot{x}^{\nu}\right) ,
( M_{\mathrm{P V}})^{3 T_{G}- \sum T ( R_{i})}\left( \frac{1}{g^{2}}\right)^{T_{G}}\! \exp \left( - \frac{8 \pi^{2}}{g^{2}}\right) \, \prod_{i}\left( Z_{i}\right)^{- T ( R_{i})}= \mathrm{c o n s t}\, .
\prod_{1\le i_1<i_2\le k}(y_{i_2}-y_{i_1}) =\biggl(\prod_{\substack{1\le i_1<i_2\le k i_1\ne j, i_2\ne j}}(y_{i_2}-y_{i_1})\biggr)\biggl(\prod_{i=j+1}^k (y_i-y_j)\biggr)\biggl(\prod_{i=1}^{j-1}(y_j-y_i)\biggr) &=(-1)^{k-j}\varphi'(y_j)\prod_{\substack{1\le i_1<i_2\le k i_1\ne j, i_2\ne j}}(y_{i_2}-y_{i_1}).
e_0^{+}\mapsto q^{(1-N)/2}\sum_{m=1}^N \alpha_mR_m, e_1^{+}\mapsto q^{(N-1)/2}\sum_{m=1}^N (L_mR_m)^{-1}L_m, &e_0^{-}\mapsto \sum_{m=1}^N \alpha_m^{-1}L_m, e_1^{-}\mapsto \sum_{m=1}^N (R_mL_m)^{-1}R_m, &k_0 \mapsto \prod_{m=1}^N K_m^{-1}, k_1 \mapsto \prod_{m=1}^N K_m.
\mathcal{M}_{}:=\{[A,B,C]\in\mathcal{M}_{VW}:\}, \mathcal{M}_{RVW}:=\{[A,B,C]\in\mathcal{M}_{VW}:\}, \widetilde{\mathcal{M}}_{VW}:=\{[A,B,C]\in\mathcal{M}_{VW}:\}.
- \frac{5 \cdot 2^{3}\sqrt{2}}{3^{5}}\mathrm{T r}\; \left( B_{\mu}^{\; \mu}A_{\nu}A^{\nu}\right) + \frac{2^{7}\sqrt{2}}{3^{5}}\mathrm{T r}\; \left( B^{\mu \nu}A_{\nu}A_{\mu}+ B^{\mu \nu}A_{\mu}A_{\nu}\right) - \frac{1 1 \cdot 2^{4}}{3^{5}}\mathrm{T r}\; \left( \beta A_{\mu}A^{\mu}\right)
\nabla(\nabla\cdot\Pi_{e})=\frac{(-Il/\omega)cos[\omega t]_{t^{\prime}}}{4\pi\epsilon}\sum_{l=0}^{\infty}\frac{r^{\prime l}(l+1)P_{l}(cos\gamma)}{r^{l+3}}[(l+2)cos(\theta)\hat{r}+sin(\theta)\hat{\theta}]
A_k=(a^k_{ij})_{i,j}\in C^1\left(\overline{\Omega}; \mathbb{R}^{N\times N}\right), a^k_{ij}=a^k_{ji}, 1\leq i,j\leq N, D_k=(d^k_{ij})_{i,j}\in C^1\left(\Gamma; \mathbb{R}^{N\times N}\right), d^k_{ij}=d^k_{ji}, 1\leq i,j\leq N,
I_1+I_*=\bigcap\bigcap_{j_i\not=1}\left[(X_1^{\alpha_1},X_{k_1}^{\alpha_{k_1}},\dots,X_{k_{n-3}}^{\alpha_{k_{n-3}}})S+(X_{j_1}^{\beta_1},X_{j_2}^{\beta_2},\dots,X_{j_{n-2}}^{\beta_{n-2}})S\right]
\int_{\underline{n}\in\Delta}^{E_1-Alg(Ch_{\mathbb{K}})}K^n(C)\otimes \overline{N^*}(\Delta^n)=\int_{\underline{n}\in\Delta}^{Ch_{\mathbb{K}}}K^n(C)\otimes \overline{N^*}(\Delta^n)\stackrel{\cong}{\rightarrow}\overline{N^*}K^{\bullet}(C)
\prod_{j=1}^{n}{{{np+j-1 \choose (n-j+1)p}}\over{{(n-j+1)p+n-j \choose n-j}}}={\prod_{j=2}^{n}{{np+j-1 \choose (p+1)(j-1)}\over{j(p+1)-1 \choose j-1}}=\prod_{j=1}^{n-1}{j ! (n(p+1) -j -1) ! \over ((n-j)(p+1)) ! ((n-j)(p+1)-1) !}}:=A_{n}^{(p)},
\mathbb P\{\biggl|\frac{1}{n-1}\sum_{k=2}^n\left[\frac{2c_k(\alpha)}{k^{\frac{1}{\alpha}}}\right]^{\alpha(\varphi^{n-k}\omega) - \alpha}- \frac{1}{n-1}\sum_{k=2}^n ( p_1 + p_2\left[\frac{2c_k(\alpha)}{k^{\frac{1}{\alpha}}}\right]^{\beta-\alpha})\biggr|\geq t_n\}\leq \exp(-2(n-1)t_n^2).
g_{1 1}( r ) \equiv - \mu ( r ) = - [ \eta ( r ) + i \zeta ( r ) ] , g_{2 2}( r ) \equiv s_{2 2}( r ) = - r^{2}, g_{3 3}( r ) \equiv s_{3 3}( r ) = - r^{2}\sin^{2}\theta , g_{0 0}( r ) \equiv \gamma ( r ) = \alpha ( r ) + i \beta ( r ) .
r_{+}= R_{0}\sqrt{1 + \frac{1}{3 ( 1 - b ) \omega}}\, , \qquad r_{-}= R_{0}\left( - \frac{1}{6}\kappa_{5}^{2}\, \rho R_{0}\sqrt{\frac{3 \omega}{( 1 - b )}}\right)^{1 / ( 1 - b )}.
\left\Vert f\right\Vert_{\widetilde{B}_{p(\cdot ),q(\cdot )}^{\alpha (\cdot),p(\cdot )}}:=\sup_{P\in \mathcal{Q}}\left\Vert \left( \frac{2^{v\alpha\left( \cdot \right)}\varphi_{v}\ast f}{|P|^{1/p(\cdot )}}\chi_{P}\right)_{v\geq v_{P}^{+}}\right\Vert_{\ell^{q(\cdot )}(L^{p(\cdot )})}<\infty ,
V = \frac{1}{\mu c^2}C_1\frac{\alpha}{q}\hat h^{(0)}+\frac{C_2-C_1}{\mu c^2}\frac{\alpha^2}{q^2}+ \frac{\hbar^2}{2}\frac{\alpha}{\mu^2c^2}C_1\left(\frac{{\bf q}}{q^3}\frac{\partial}{\partial{\bf q}}+ 4\pi\lambda\delta({\bf q})\right)\,,
M_{2\gamma \to \gamma}= -\frac{4}{15\sqrt{\pi}}\biggl( \frac{e^4}{3 m^4}\sin \theta\left[\frac{7}{4}(\phi_1^2 + \phi_2^2) - \phi_{12}^2\right] +\frac{Qg (Q^2 - g^2)}{8 M^4}\cos \theta\,\phi_{12}^2 \biggr) \omega \omega_1 \omega_2 H \cos 2 \beta
2 i H_{A}^{C}\sigma_{C \dot{B}}^{\m}+ 2 i H_{\dot{B}}^{\dot{C}}\sigma_{A \dot{C}}^{\m}- 2 i H_{A \dot{B}}^{\m}- \nabla_{A}H_{\dot{B}}^{\m}- \nabla_{\dot{B}}H_{A}^{\m}= 0
\sum_{k=0}^{b-m}\binom{a}{k}\binom{b}{m+k}\binom{k}{c}\binom{m+k}{d}H_{a-k}={}\binom{a+b-c-d}{m+a-d}\binom{a}{c}\binom{b}{d}(H_{m+a-d}-H_{a+b-c-d}+H_{a-c})
S = \frac{1}{2 \pi}\int_{M}d^{2}x \sqrt{- g}\left[ e^{- 2 \phi}[ R + 4 ( \nabla \phi )^{2}+ 4 \lambda^{2}] - \frac{1}{2}( \nabla f )^{2}+ 4 \mu^{2}( \cos f - 1 ) e^{- 2 \phi}\right] ,
S = \int L = \lambda_{u}^{2}( u ) \lambda_{v}^{2}( u ) + \lambda_{u}^{2}( u ) \lambda_{v}^{2}( v ) + \lambda_{u}^{2}( v ) \lambda_{v}^{2}( w ) + \lambda_{u}^{2}( w ) \lambda_{v}^{2}( u ) + \lambda_{u}^{2}( w ) \lambda_{v}^{2}( w ) + \lambda_{u}^{2}( v ) \lambda_{v}^{2}( v ) \qquad
\left| \sum_{k = k_i}^{k_{i+1}- 1}\log \frac{|T'(x_k)|}{|T'(y_k)|}\right| \le \left| \log \frac{| T'(x_{k_i}) |}{| T'(y_{k_i}) |}\right| + \left| \log \frac{(T^p)' (x_{k_i +1})}{(T^p)' (y_{k_i +1})}\right|,
2E'x =\theta x-(-1)^{p+q}x\theta=\xi_ia\otimes \xi^i\alpha-(-1)^{p+q}a\xi_i\otimes \alpha\xi^i= &=\xi_ia\otimes \xi^i\alpha-(-1)^p a\xi_i\otimes \xi^i\alpha=\xi_i\vdash a\otimes \xi^i\alpha=\eta_{ij}\frac{\partial}{\partial \xi_j}a\otimes \xi^i\alpha.
\mathcal{R}=\langle u\cdot\nabla\theta,\Lambda^{2r}e^{2\tau\Lambda^{\frac{1}{s}}}\theta\rangle-\langle u\cdot\nabla \Lambda^r e^{\tau\Lambda^{\frac{1}{s}}}\theta,\Lambda^r e^{\tau\Lambda^{\frac{1}{s}}}\theta\rangle.
I_k[u_i^{p_i}](x) \le C\begin{cases}m_i^{\frac{n-2k+4}{n-4}+o(1)}, \mbox{if}|x|<r_i, m_i^{-1+o(1)}|x|^{k-n}, \mbox{if}r_i \le |x|<1, \end{cases}
O((n + 2d\lambda)^{2n}\lambda^2 a \log p) = O((5dna(4d+4)^n)^{2n}(na(4d+4)^n)^2 a \log p) = O(2^{4n^2+9n}n^{2n+2}(d+1)^{2n^2+4n}a^{2n+3}\log p).
\sum\limits_{0<|u|\le U}\sum\limits_{\substack{x_1=1 x_1^2\equiv -\overline{3}u \bmod{|\tilde{m}|}}}^{|\tilde{m}|}\left(u,hd|m^{\ast}|^2|\tilde{m}|\right) |u|^{\alpha}\ll \frac{\xi^{2\tilde{\alpha}}}{X^{8\tilde{\alpha}}}\cdot X^{\varepsilon}.
{{\cal G}^{- 1}}_{\mu \nu}^{\alpha \beta}\eta_{\alpha \beta}^{\mathrm{L L T}}= \left( k^{2}- \Lambda_{l}+ \frac{1}{2 r^{2}}( N - 1 ) - \frac{N + d}{N + d - 2}\Lambda \right) \eta_{\mu \nu}^{\mathrm{L L T}}.
Y_i^{(m)}=X_i^{(m)}-\langle{\bf e}_{n+1}+X_{i-1}^{(m)},X_i^{(m)}-X_{i-1}^{(m)}\rangle ({\bf e}_{n+1}+X_{i-1}^{(m)}) Z_{i-1}^{(m)}=\langle{\bf e}_{n+1}+X_{i-1}^{(m)},X_{i-1}^{(m)}\rangle ({\bf e}_{n+1}+X_{i-1}^{(m)})
g_{2/3, 3/4}^2 (w_2) \geq \frac{\phi_{\ell}}{4}\left[\log \left( 1 + \frac{w_2 P'}{3}\right) - \frac{1}{2}\log \left( 1+ \frac{2 w_2 P'}{3}\right) \right] &- \frac{3 \ell}{4 k_{\ell}}H_2 (\bar{w}/ \ell) &\geq \epsilon' - \frac{\epsilon'}{2}& = \frac{\epsilon'}{2}.
\widehat{S}_{(\pm 1)}p^{( 0)}=\widehat{S}_{(\pm1)}p^{(\widetilde{0})}=0 ,~~S_{(1)}^2 p^{(0)}=S_{(1)}^2p^{(\widetilde{0})}=0 ,~~S_{(0)}^2 p^{(1)}=S_{(0)}^2p^{(\widetilde{1})}=0 ,
\delta_{a}( l ) = - \int \frac{k^{\prime}}{2 l}\, d l - \frac{1}{1 6 \pi^{2}}( C_{G}^{a}- C_{M}^{a}) k ( l ) - \frac{2}{1 6 \pi^{2}}\sum_{A}C_{A}^{a}\ln ( 1 - p_{A}l ) \, .
{\rm Term}_3 \,=\, \langle z, \delta\rangle_{L^2(V';I)}\,=\, \langle R_V^{-1}z, R_V^{-1}\delta\rangle_{L^2(V;I)}\,=\, \big\langle R_V^{-1}z,\delta \big\rangle_{L^2(V;I),L^2(V';I)}.
(j-1)+\cdots+2+1+1+2+\cdots+(r_i -j) = \frac{j(j-1)}{2}+\frac{(r_i -j)(r_i -j+1)}{2}= \frac{1}{2}[2j^2-2j-2jr_i+r_i(r_i+1)] = j^2-j-jr_i+\frac{r_i(r_i+1)}{2}
S [ q ] = \lim_{T \rightarrow \infty}\left\{\int_{- T}^{T}d t \; \frac 1 2 m \dot{q}^{2}- \frac 1 4 m \omega_{0}^{2}\ell_{0}^{2}T \left({\frac{1}{T}}\int_{- T}^{T}d t \;{\frac{q^{2}}{\ell_{0}^{2}}}\right)^{2}\right\}\; ,
\int_{- \infty}^{+ \infty}d y \frac{e^{i k y}}{\cosh^{\mu}u}= \frac{2^{\mu - 1}}{\Gamma ( \mu )}\Gamma \left( \frac{\mu + i k}{2}\right) \Gamma \left( \frac{\mu - i k}{2}\right) \mathrm{f o r}\mathrm{R e}\mu > 0 ,
S_{C_{1}\,\otimes\, C_{2}}\,(\,f \,\otimes\, g\,) & = \sum\limits_{i,\, j \,=\, 1}^{\,\infty}\left<\,f \otimes g,\, f_{\,i}\otimes g_{\,j}\,|\, a_{\,2}\otimes b_{\,2}, \,\cdots,\, a_{\,n}\otimes b_{\,n}\,\right>\,\left(\,C_{1}\,\otimes\, C_{2}\,\right)\left(\,f_{\,i}\otimes g_{\,j}\,\right)
N_h = x_h \sqrt{\frac{z B}{F_{k,1}(B)}}\prod_{i=2}^{h}F_{k,i}(B)^{-1}\prod_{i=1}^{h-1}G_{k,i}(B), N_{k+1-h}= x_{k+1-h}z \prod_{i=1}^{h}F_{k,i}(B) \prod_{i=1}^h G_{k,i}(B)^{-1}.
S_f(d;\alpha) \ll \begin{cases}(\log\log K)^{\O(1)}\|f\|_2^2, \mbox{if}\,\alpha =1, \exp(C(\alpha)(\log K)^{1-\alpha})(\log\log K)^{-\alpha})\|f\|_2^2, \mbox{if}\,\frac{1}{2}<\alpha<1,\end{cases}
g(V)=\sum_{v\in V}g(v) =2\sum_{v\in V}p(v) + \sum_{v\in V}\sum_{y\in N^o(v)}g_y(v)p(y) &=2+\sum_{y\in V}p(y)\sum_{v\in N^o(y)}g_y(v) &\le 2+\sum_{y\in V}p(y)(2\alpha(G)-2) &=2\alpha(G).
\iota^*\Gamma_1(a,b,i,j,a',b',f,0) = \Gamma_1(\iota_*a,\iota_*b,\iota_*i,\iota_*j,\iota_*a',\iota_*b',\iota_*f,0) = (\sqrt{-1}a,\sqrt{-1}b,\sqrt{-1}i,\sqrt{-1}j,\sqrt{-1}a',\sqrt{-1}b',\sqrt{-1}f,0)
\mu(\{E\}) = \ \prod_{p}\mu_{p}(\{E\})=\prod_{p}\frac{[p,n]}{[p,n-m]}= \prod_{p}\prod_{i=n-m+1}^n \left( 1-p^{-i}\right) =\prod_{i=n-m+1}^n \prod_{p}\left( 1-p^{-i}\right) = \ \frac{1}{\prod_{i=n-m+1}^n \zeta(i)}\,,
{\rm div}(\widetilde{J}_{{\Bbb A}_{1}^{+}}^{*}\theta_{a_{0},b_{0}}^{8})|_{U}=c\,H_{d}+\sum_{\lambda\in L^{\lor}/\pm1,\,\lambda^{2}=-9/2,\,|\langle\lambda,d\rangle|=1,\,[\lambda]={\bf 1}_{L}}c_{a_{0},b_{0}}(\lambda)\,H_{\lambda}
I_{\mathrm{e x a c t}}( D = 4 ) = - V T \lim_{s \rightarrow 0}{\frac{F_{+}F_{-}}{8 \pi^{2}}}\int_{0}^{\infty}d \tau \ \tau^{s - 1}e^{- \tau m^{2}}\coth \tau F_{+}\coth \tau F_{-}\ ,
\|\boldsymbol{\theta}^{*}-\boldsymbol{\theta}_0\|+d/\sqrt{n}=R2^{-\sum_{l=1}^d\alpha_lJ_{n,l}(\boldsymbol{\alpha})\left(\frac{1}{d}+\frac{1}{2\alpha^{*}}\right)}+d/\sqrt{n}\lesssim\sqrt{\log{n}/n}.
\lim_{n \rightarrow \infty}\frac{1}{t_{n}}\int_{0}^{t_{n}}\left\{\sum_{k=0}^{n-1}\varphi\left(X_{t_{k}}\right) \mathbf{1}_{\left[t_{k}, t_{k+1}\right)}(s)\right\}\mathrm{d}s=\mathbb{E}(\varphi(\bar{X})),
\displaystyle \int_{\mathbb{R}^{N}}\phi(|\nabla (t_{2}u)|)|\nabla(t_{2}u)|^{2}dx + \mu \displaystyle\int_{\mathbb{R}^{N}}\phi(|t_{2}u|)|t_{2}u|^{2}dx = \displaystyle\int_{[u>0]}f(t_{2}u)t_{2}udx.
\lim_{v \to x_{i}}\left[ ( \partial_{x}+ \partial_{u})^{2}f ( x , x_{i}, u , v ) \right] = - ( 2 \pi )^{2}\frac{( x - u )^{2}}{( x - x_{i})^{2}( u - x_{i})^{2}}\: g ( x , x_{i}, u ) \; .
B_{i j}= \frac{\varepsilon}{2 \pi \alpha^{\prime}}\left( \begin{array}{c c c c c}{{0}}&{{b_{1}}}&{{}}&{{}}&{{}}\\{{{}- b_{1}}}&{{0}}&{{}}&{{}}&{{}}\\{{}}&{{}}&{{0}}&{{b_{2}}}&{{}}\\{{}}&{{}}&{{- b_{2}}}&{{0}}&{{}}\\{{}}&{{}}&{{}}&{{}}&{{\ddots}}\\ \end{array}\right) \quad ( i , j = 1 , \ldots , p^{\prime}) ~ , \quad \mathrm{o t h e r w i s e ~ B_{\m u \n u}= 0 ~}~ .
g(\bar x) \leq \lambda f(x) + r(x) +(1-\lambda) (f(y) + r(y)) &\qquad+\lambda\tau\Phi(x)+(1-\lambda)\tau\Phi(y)+\lambda\rho\Phi(x)+(1-\lambda)\rho\Phi(y) &=\lambda[f(x)+ r(x) +(\tau+\rho)\Phi(x)]+(1-\lambda)[f(y) + r(y)+(\tau+\rho)\Phi(y)] &\leq \lambda g(x)+(1-\lambda) g(y).
M\left(\frac{\beta_1}{\alpha_1}, \ldots, \frac{\beta_{n-1}}{\alpha_{n-1}}, \frac{\beta_n}{\alpha_n}\right),\ \textrm{where}\ \frac{\beta_i}{\alpha_i}= \frac{5}{3}\ (1 \le i \le n-1),\ \textrm{and}\ \frac{\beta_n}{\alpha_n}= \frac{17}{6}.
g_{h o m , 2 n}^{\mu , \l}= k^{n - n^{2}}{\prod_{i = 1}^{2 n - 1}( i ! )^{- 1}}\times \det \left\{\frac{1}{\sqrt{k}}h_{\mu , \l}^{( 2 j + 2 k - 3 )}( 0 ) \right\},
\tilde{J}_{\varepsilon}(\xi)=\left( \frac{p-2}{2p}\int_{\mathbb{R}^{n}}U^{p}dz\right) \frac{c(\xi)^{\frac{n}{2}}a(\xi)^{\frac{p}{p-2}-\frac{n}{2}}}{b(\xi)^{\frac{2}{p-2}}}+O(\varepsilon),
-2T_0^3T_2 + 43T_0^2T_1T_2 + 480T_0^2T_2^2 + 2T_0T_1^3 - 36T_0T_1^2T_2 - 2460T_0T_1T_2^2 + 8700T_0T_2^3{}+ 2T_1^4 - 189T_1^3T_2 + 5170T_1^2T_2^2 - 31255T_1T_2^3 + 40250T_2^4 = 0
_{m}\overset{}=\frac{\left(\frac{1}{(\kappa^{2}_{R}m+N_{0}/p)}\mathcal{Y}\right)}{\left(\frac{(\kappa^{2}_{T}(\omega+1))+\omega}{(\kappa^{2}_{R}m+N_{0}/p)}\mathcal{Y}+1\right)},
\begin{aligned}\left(\frac{\partial e_u}{\partial t},\phi_h\right) + a_h(u,\phi_h) - a_h(u_h,\phi_h) + \sqrt{\epsilon}\,b_h(e_w,\phi_h) &+ \left(\rho u - \rho_h u_h, \phi_h\right) = \left(\rho V - \rho_h V_h, \phi_h\right) \forall \,\, \phi_h \in X_h,\end{aligned}
S_{10}=\int \left(\frac{1}{4}\,\hat{R}-\frac{1}{2}\,\partial_{M}\hat{\phi}\,\partial^{M}\hat{\phi}-\frac{1}{12}\,e^{-2\hat{\phi}}\,\hat{H}_{MNP}\,\hat{H}^{MNP}\right)\sqrt{-\hat{g}}\,d^{10}\hat{x},
\begin{aligned}x_1 =(4m+3)(8m-3), x_2 =-2(2m+3)(12m-1), x_3 =-4(3m+1)(4m-9), x_4 =6(4m+1)(8m+1), x_5 =3(4m+1)(16m-3), x_6 =8(4m+3)(m-1), x_7 =-4(16m+9)(2m-1), y_1 =8m+1)(4m-9), y_2 =(16m+9)(12m-1), y_3 =4(2m+3)(4m+3), y_4 =8(3m+1)(8m-3), y_5 =-6(4m+1)(2m-1), y_6 =-2(4m+1)(16m-3), y_7 =-12(4m+3)(m-1).\end{aligned}
Si(x)\sim\frac{\pi}{2}-\frac{cosx}{x}(1-\frac{2!}{x^{2}}+\frac{4!}{x^{4}}-\frac{6!}{x^{6}}\cdot\cdot\cdot)-\frac{sinx}{x}(\frac{1}{x}-\frac{3!}{x^{3}}+\frac{5!}{x^{5}}-\frac{7!}{x^{7}}\cdot\cdot\cdot)
\widehat{R}^{(\epsilon)}_{\,\,\epsilon^\prime}(\tau, z_1, z_2, t) = (-1)^{2\epsilon (1-2\epsilon^\prime)}\, i e^{2\pi i t}\frac{\eta(\tau)^3 \vartheta_{11}(\tau, z_1+z_2)}{\vartheta_{1-2\epsilon^\prime,\, 1-2\epsilon}(\tau, z_1) \vartheta_{1-2\epsilon^\prime,\, 1-2\epsilon}(\tau, z_2)},
\mathcal{L}_{\sigma p}(\mathbb{R}^{n}):=\left\{u\in L^{p-1}_{loc}(\mathbb{R}^{n})\,\Big|\;\int_{\mathbb{R}^{n}}\frac{|u(x)|^{p-1}}{1+|x|^{n+\sigma p}}dx<\infty \right\}.
u(x+z,\tau+(2\rho)^\theta)\ge\int_{B(z,\rho)}G(x+z-y,(2\rho)^\theta)u(y,\tau)\,dy \qquad\ge \int_{B(z,\rho)}G(2x,(2\rho)^\theta)u(y,\tau)\,dy=2^{-N}G(x,\rho^\theta)\int_{B(z,\rho)}u(y,\tau)\,dy
a_1 \star_{n-1}\cdots \star_{n-1}a_p = \sum_{i=1}^p \lambda_i b_i + \sum_{i=1}^p (1_c + 1_{d_{i-1}}+ 1_{e_{i+1}}) - \sum_{i=1}^{p-1}(\lambda_i 1_{t(b_i)}+ 1_c + 1_{d_{i-1}}+ 1_{e_{i+1}}).
\Delta=1-{a^4\cos^2\theta \over u^4}\ ,\ \ \ \ \tilde{\Delta}=1-{a^4\over u^4}\ ,\ \ \ \A\equiv{u_0^4\over u_H^4-\frac{1}{3}a^4}\ , \ \ \ \u_H^6-a^4 u_H^2-u_0^6=0\ .
w_{p\bar{a}\bar{b}\bar{c}}=w_{\bar{p}\bar{a}\bar{b}\bar{c}}+\varepsilon_{p(ad}h_{d\bar{b}\bar{c})}+5(21)^{-1}\eta_{p(a}w_{\bar{b}\bar{c})}-2(21)^{-1}w_{p(\bar{a}}\eta_{bc)},
(\begin{matrix}a\\b\end{matrix})=(\begin{matrix}q_{0}&1\\ 1&0\end{matrix})(\begin{matrix}b\\r_{0}\end{matrix})=(\begin{matrix}q_{0}&1\\ 1&0\end{matrix})(\begin{matrix}q_{1}&1\\ 1&0\end{matrix})(\begin{matrix}r_{0}\\r_{1}\end{matrix})=\cdot\cdot\cdot=\prod_{i=0}^{N}(\begin{matrix}q_{i}&1\\ 1&0\end{matrix})(\begin{matrix}r_{N-1}\\ 0\end{matrix}).
( Z^{\Lambda})^{\prime \prime}- \left({\frac{( Z N q )}{( Z N Z )}}+ q^{0}\right){\frac{( ( Z^{\Lambda})^{\prime})^{2}}{Z^{\Lambda}q^{0}- q^{\Lambda}}}+{\frac{1}{2}}\left( \ln{\frac{( Z N Z ) ( \bar{Z}N q ) ( \bar{Z}N Z )}{( \bar{Z}N \bar{Z}) ( Z N q )}}\right)^{\prime}( Z^{\Lambda})^{\prime}= 0 \ .
I_{\epsilon}( z ) = \sum_{n = 0}^{\infty}f^{( n )}( 0 ) \frac{z^{n + 1}}{( n + 1 ) !}\Gamma \left[ 1 + ( n + 1 ) \epsilon \right] \cos \left[ ( n + 1 ) \epsilon \frac{\pi}{2}\right] .
z_{t}\le\frac{1}{2C_{1}\left(R_{1}+8AC_{1}\right)}z_{k}\ge\frac{1}{2\eta_{k}\lambda_{k}\left(R_{1}+8AC_{1}\right)+16A\eta_{k}^{2}\lambda_{k}^{2}}=\frac{1}{2C_{1}\left(R_{1}+16AC_{1}\right)}
0 =\overline{F_1}(\bar{z}_1,\bar{z}_2,\bar{\alpha}_3\bar{z}_2,\ldots,\bar{\alpha}_n\bar{z}_2,0)F_2(0,\ldots,0,\bar{z}_1^2) +{\bigl( \overline{F_1}(\bar{z}_1,\bar{z}_2,\bar{\alpha}_3\bar{z}_2,\ldots,\bar{\alpha}_n\bar{z}_2,0) \bigr)}^2 .
\begin{pmatrix}x_n x_{n+1}\end{pmatrix}=\begin{pmatrix}\varpi_n^\circ \varpi_{n+1}^\circ\end{pmatrix}M,\begin{pmatrix}\varpi_n \varpi_{n+1}\end{pmatrix}=\begin{pmatrix}\varpi_n^\circ \varpi_{n+1}^\circ\end{pmatrix}Q.
\Phi(\sigma) \leq \frac{1}{\gamma}\left(\frac{c\delta^{-(1-s)p}}{(\tau-\sigma)^{n+sp}}\right)^{\frac{n+p}{p}}+\frac{\Phi(\tau)}{2}begin{align*}3mm] =\frac{\Phi(\tau)}{2}\left[1+\frac{3(n+p)}{p}\frac{\log \left(\dfrac{c\delta^{-(1-s)p}}{(\tau-\sigma)^{n+sp}}\right)}{\log\left(\dfrac{(\Phi(\tau)/2)^{p-1}}{\overline{C}}\right)}\right].