harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
var_k(P) := \sup_{\omega_{-\infty}^{-1}, \sigma_{-\infty}^{-1}\in \Omega<0, \atop \omega_{-k}^{-1}= \sigma_{-k}^{-1}}\sup_{a \in A}| P(a \mid \omega_{-1}^{-\infty}) - P(a \mid \sigma_{-1}^{-\infty}) | \rightarrow 0 k \rightarrow \infty.
[ \frac{3}{2}\times \frac{4}{3}\times \frac{5}{4}\times \cdots \times \frac{2 0 1 4}{2 0 1 3}] \times [ \frac{1}{2}\times \frac{2}{3}\times \frac{3}{4}\times \cdots \times \frac{2 0 1 2}{2 0 1 3}]
\xi \overline{\mathcal{U}}_{\mathrm{Br}}^{s}\circ \overline{\eta}_{\mathrm{BrL}}^{s}=\eta_{\mathrm{BrL}}^{s}\epsilon_{\mathrm{BrL}}^{s}\mho_{\mathrm{Br}}^{s}\circ \mathcal{U}_{\mathrm{Br}}^{s}\xi =\mho_{\mathrm{Br}}^{s}\overline{\epsilon}_{\mathrm{BrL}}^{s}.
\int_0^t \int_{\mathbb{R}}\Big( \int_{\mathbb{R}}G_{\epsilon}(v-y)G_{t-s}(x-v)|v-y|^{1/2}dv \Big)^2 f(y)^2 dy ds \leq \|f\|_{L^q(\mathbb{R})}^2 \int_0^t \Big( \int_{\mathbb{R}}G_{\epsilon}(z) |z|^{1/2}dz\Big)^2\|G_{t-s}\|_{L^{2q_1}(\mathbb{R})}^2 ds \leq C \epsilon^{1/2}.
L_{{K}}:= \sum_{\begin{subarray}N_{1}+\cdots+N_{g}=K, |\frac{\overrightarrow{\boldsymbol{N}}}{K}-\boldsymbol{f_{\star}}|<\kappa\end{subarray}}\exp((K)^{2}F^{\{-2\}}_{\frac{\overrightarrow{\boldsymbol{N}}}{K}, \beta}+(K)F^{\{-1\}}_{\frac{\overrightarrow{\boldsymbol{N}}}{K}, \beta}) .
h^{(p,q)}(x,a) =\frac{\mathbb{W}^{(p)}(x-b)}{\mathbb{W}^{(p)}(a-b)}+\frac{h^{(p,q)}(b,a)}{Z^{(p+q)}(b)}\Big( \mathcal{R}^{(p,q)}(x)-\mathcal{R}^{(p,q)}(a)\frac{\mathbb{W}^{(p)}(x-b)}{\mathbb{W}^{(p)}(a-b)}\Big).
D =(t_2t_4)^{\binom n2}q^{2\binom n3}\Gamma(t_1t_2q^{n-1},t_3t_4q^{n-1})^n\prod_{m=1}^n\prod_{\substack{1\leq j<k\leq 6 (j,k)\neq (1,2),\,(3,4)}}\Gamma(t_jt_kq^{m-1}) &\quad\times\prod_{j=1}^n(q,q^{j-n}t_1/t_2,q^{j-n}t_3/t_4)_{j-1}.
\hphantom{{}={}}\phi\circ(\varphi\circ f)(x) * =\widehat{\varphi}(f(o))+D\phi(p)^i{}_\alpha a^\alpha{}_\beta x^\beta * \hphantom{{}={}}+\frac12\left(H\phi(p)^i{}_{\alpha\beta}a^\alpha{}_ja^\beta{}_k+D\phi(p)^i{}_\alpha b^\alpha{}_{jk}\right)x^jx^k+().
\begin{aligned}\langle D^{l}e^{\frac{\alpha}{p}M_{s,t}},D^{l}e^{\frac{\alpha}{p}M_{s,t}}\rangle_{\mathcal{H}^{\otimes l}}=\left(\frac{\alpha}{p}\right)^{2l}e^{\frac{2\alpha}{p}M_{s,t}}(\langle DM_{s,t},DM_{s,t}\rangle_{\mathcal{H}})^{l}=\left(\frac{\alpha}{p}\right)^{2l}e^{\frac{2\alpha}{p}M_{s,t}}(t-s)^{2lH},\end{aligned}
\Bigl(\frac{\partial u^i}{\partial \nu}+\alpha^i u^i\frac{\partial\mathcal{V}}{\partial \nu}\Bigr)(t, x)=\sigma^i(t, x,u^i_{|_{\partial\Omega}}(t, x), \mathcal{V}_{|_{\partial\Omega}}(t, x)),(t, x)\in[0, T]\times\partial\Omega.
\begin{aligned}\frac{M}{\alpha}\leq \frac{(\alpha+1-\beta')^{2}}{\alpha}\cdot \Big\{\frac{1}{\alpha}\log d_L + \Big( \frac{G_1(\alpha; |\gamma'|)}{\alpha}+ 2 W_1(\alpha) \Big) \cdot r_1 & \qquad+ \Big(\frac{G_2(\alpha; |\gamma'|)}{\alpha}+ W_2(\alpha) \Big) \cdot 2r_2 + \frac{2}{\alpha^2}+ \frac{2}{\alpha+\alpha^2}\Big\}\end{aligned}
( \vec{\alpha}_{1}\! \cdot \vec{\alpha}_{2})^{T}\, = \, ( \vec{\alpha}_{1}\! \cdot \vec{\alpha}_{2}) \, - \,{\frac{( \vec{\alpha}_{1}\! \cdot \vec{k}) \, ( \vec{\alpha}_{2}\! \cdot \vec{k}\, )}{\vec{k}\,^{2}}}.
\frac{\partial \mathring{h}_i^j}{\partial t}=\nabla^j\left(\frac{\nabla_iH}{H^2}\right)-\frac{2}{n-1}\mathring{h}^j_i-\frac{\mathring{h}^k_i\mathring{h}^j_k}{H}+\left(\frac{1}{H}-\frac{H}{(n-1)^2}-\frac{1}{n-1}\frac{\partial H}{\partial t}\right)\delta_i^j.
\int_{0}^{1}\mathbf{1}_{[0,t]}(s)dh(s)=h(t)=\int_{0}^{t}K(t,s)\varphi(s)ds=\int_{0}^{1}\left(\mathcal{K}^{*}\mathbf{1}_{[0,t]}\right)(s)\varphi(s)ds.
V_{D}= V_{E}= V_{0}\int d^{2}z \left| \Phi_{X_{1}}( z ) \right|^{2}\left| \Phi_{X_{2}}( z ) \right|^{2}={\frac{V_{0}}{\sqrt{2 \pi}}}{\frac{1}{L_{y}}}e^{- ( X_{1}- X_{2})^{2}/ 2}.
B_{23,31}=\frac{3b_2B_{33,1}-3b_3B_{22,1}}{(b_3-b_2)^2}C_2+\frac{3b_3}{b_2-b_3}[C_{2,1}-\frac{B_{12,1}}{b_1-b_2}C_1]+B_{31,3}\frac{B_{12,1}}{b_1-b_2}, &B_{23,13}=(B_{22,1}-B_{33,1})\frac{B_{23,3}}{b_2-b_3}+(B_{11,2}-B_{33,2})\frac{B_{13,3}}{b_1-b_3},
\square \left(\Delta u -\frac{R}{4}-|\nabla u|^2 \right)=-2 \left|u_{ij}-\frac{R_{ij}}{2}\right|^2, \square \left(-\Delta u -\frac{R}{4}-|\nabla u|^2 \right)=-2 \left|u_{ij}+\frac{R_{ij}}{2}\right|^2.
\int_{0}^{2\pi}\frac{e^{iky}}{(1+r^2)+(r^2-1)\cos(y)}dy = \frac{1}{r^2-1}\int_{0}^{2\pi}\frac{e^{iky}}{\cos(y)-\frac{1+r^2}{1-r^{2}}}dy = \frac{1}{r^2-1}\frac{2}{i}\int_{|w|=1}\frac{w^{k}}{w^{2}-2\frac{1+r^2}{1-r^{2}}w+1}dw.
G_{( 3 )}^{\mathrm{( T M )}}( \tau ) = 3 6 0 \tau^{5}\int_{0}^{\Omega / 2}d \Omega^{\prime}\int_{0}^{\Omega^{\prime}}d K \, \frac{K \, \left[ K^{2}+ \Omega^{\prime}\left( \Omega - \Omega^{\prime}\right) \right]^{2}}{\sqrt{( \Omega^{2}- K^{2}) \left[ ( \Omega - \Omega^{\prime})^{2}- K^{2}\right]}}\, \left( \frac{1}{e^{\Omega^{\prime}}- 1}+ \frac{1}{e^{\Omega - \Omega^{\prime}}- 1}\right) .
\varphi_L(\mathcal{N}_L(f)([c](Z))) = \mathcal{N}_L(f)([c]([\pi_L](Z))) = \mathcal{N}_L(f)([\pi_L]([c](Z))) = \varphi_L(\mathcal{N}_L(f))([c](Z)) = \prod_{a \in LT_1}\sigma_a(f)([c](Z)) = \prod_{a \in LT_1}\sigma_{[c^{-1}](a)}(f([c](Z))) = \prod_{a \in LT_1}\sigma_a(f([c](Z))) = \varphi_L(\mathcal{N}_L(f([c](Z)))) \ .
\frac{\langle \langle \psi^{\prime}| e^{- \hat{H}\tau_{f}}\hat{O}_{1}( \tau_{1}) \hat{O}_{2}( \tau_{2}) \ldots \hat{O}_{n}( \tau_{n}) e^{\hat{H}\tau_{i}}| \psi \rangle \rangle}{\langle \langle \psi^{\prime}| e^{- \hat{H}( \tau_{f}- \tau_{i})}| \psi \rangle \rangle}\, .
Y_t=Y_{T\wedge a}+\int_t^{T\wedge a}f(r,Y_r)\,dr+\int_t^{T\wedge a}\,dV_r+\int_t^{T\wedge a}\,dK_r-\int_t^{T\wedge a}\,dM_r,t\in [0,T\wedge a],
\sum_{n = N_1}^{N_2-1}\big| m_{n+1}(x, y) - m_n(x, y) \big| =\bigg(\frac{1}{\vartheta_B(n_0)}+ \sum_{n = n_0}^{N_2-1}\frac{1}{\vartheta_B(n)}- \frac{1}{\vartheta_B(n+1)}\bigg) \prod_{j = 1}^{k''}\log y_j &=\bigg(\frac{2}{\vartheta_B(n_0)}- \frac{1}{\vartheta_B(N_2)}\bigg) \prod_{j = 1}^{k''}\log y_j.
f_{Y}(x) \oplus_{Y}f_{Y}(y) = f_{Y}(f^{-1}_{Y}(x)+f^{-1}_{Y}(y)) \; \; \; \; \; \; \; \; \; \; f_{Y}(x) \otimes_{Y}f_{Y}(y) = f_{Y}(f^{-1}_{Y}(x)\cdot f^{-1}_{Y}(y))
147J_{n+1}^{(3)}=147\left(J_{n}^{(3)}+J_{n-1}^{(3)}+2J_{n-2}^{(3)}\right) &=13K_{n}^{(3)}+48K_{n-1}^{(3)}+20K_{n-2}^{(3)}&\ \ +13K_{n-1}^{(3)}+48K_{n-2}^{(3)}+20K_{n-3}^{(3)}&\ \ +26K_{n-2}^{(3)}+96K_{n-3}^{(3)}+40K_{n-4}^{(3)}&=13K_{n+1}^{(3)}+48K_{n}^{(3)}+20K_{n-1}^{(3)}.
0=\frac{1}{2}\Delta|B|^2 =\sum_{\alpha,i,j,k}|B^{\alpha}_{ij,k}|^2+\sum_{\alpha,i,j,k,l}B^{\alpha}_{ij}B^{\alpha}_{kl}R_{kijl}&+\sum_{\alpha,i,j,k,l}B^{\alpha}_{ij}B^{\alpha}_{ik}R_{jlkl}-\sum_{\alpha,\beta,i,j,k}B^{\alpha}_{ij}B^{\beta}_{ki}R^{\bot}_{\alpha\beta jk}.
\frac{d}{ds}\int_{I(s)}|w(\xi,s)|^2\rho_d\,d\xi \le-2\int_{I(s)}|\partial_\xi w|^2\rho_d\,d\xi+d\int_{I(s)}|w(\xi,s)|^2\rho_d\,d\xi +Ce^{-\frac{\theta}{4}s}\int_{I(s)}|w(\xi,s)|^2\rho_d\,d\xi+O(e^{-2\gamma s}e^{-d\theta_*s})
F(h_1s) =F(hh^{-1}h_1s)=\Omega_{\chi}^{-1}(h,h^{-1}h_1s)\sigma_{\chi}(h) F(h^{-1}h_1s) =\Omega_{\chi}^{-1}(h,skk_1^{-1}) \sigma_{\chi}(h) F(skk_1^{-1})=F(hskk_1^{-1}),
\sigma\Big((\Phi^e_{vw})^* \Phi^e_{vw}\Big) = \sigma\Big( U^{e}_{vw}(\Psi^e_{\varphi^a(v)\varphi^b(w)})^* \Psi^e_{\varphi^a(v)\varphi^b(w)}(U^{e}_{vw})^* \Big) = \sigma\Big((\Psi^e_{\varphi^a(v)\varphi^b(w)})^* \Psi^e_{\varphi^a(v)\varphi^b(w)}\Big)
\left\| \sum_{\ell=1}^{k-1}I_\ell \right\|_{L_p(\Omega)}\le C(f) \sum_{\ell=1}^{k-1}\sum_{i=1}^\ell \| \Delta_h^k u^i \|_{L_p(\Omega)}\| u \|_{L_\infty(\Omega)}^{\ell-i}.
SP_{u}\:=\: \begin{pmatrix}S_{1}\vec{u}\vec{u}^{*}S_{1}^{*}\overline{u}_{n}S_{1}\vec{u}u_{n}( S_{1}\vec{u})^{*}|u_{n}|^{2}\end{pmatrix}+ \begin{pmatrix}\hat{S}_{0}(\vec{u}\vec{u}^{*}) - S_{1}\vec{u}\vec{u}^{*}S_{1}^{*}0 0 0 \end{pmatrix}.
F = L^d-\overline{L}^d = \left( r_1\xi^{\frac{l}{2}}x_1 + r_2\xi^{\frac{2l+k+2}{2}}x_2 + r_3\xi^{\frac{l+k+1}{2}}x_3\right)^d - \left( r_1\xi^{-\frac{l}{2}}x_1 + r_2\xi^{-\frac{2l+k+2}{2}}x_2 + r_3\xi^{-\frac{l+k+1}{2}}x_3\right)^d.
(r-1)^{n-1}\tfrac{(r-1)!}{(2r-2)![(r-1)!]^{r-1}}\tfrac{(r^{2}-1)(2r-3)![(r-2)!]^{r-1}r^{r-2}}{(r-1)!}\sum_{i=1}^{n}(-1)^{r}\tfrac{d_{i}}{[(r-1)!]^{r}}2d_{i}[(r-1)!]^{r}&=(-1)^{r}(r-1)^{n-r}(r+1)r^{r-2}\sum_{i=1}^{n}d_{i}^{2}.
\left(I-\frac{{\bf p}\cdot{\bf q}^{\rm H}}{{\bf q}^{\rm H}{\bf p}}\right)(A^{\rm H}-\bar{\theta}I)\left(I-\frac{{\bf p}\cdot{\bf q}^{\rm H}}{{\bf q}^{\rm H}{\bf p}}\right){\bf t}=-{\bf r}_p,{\bf t}\bot{\bf q}.
\int_{\mathbb R^2}\Bigl( (A^{\prime\prime}+ \frac 9 r A^{\prime})x_1^3 x_2 + 6 A x_1 x_2) \cdot x_1 \cdot \frac{x_2}{r^6}(r^2-4x_1^2) dx_1 dx_2=0, & \int_{\mathbb R^2}\Bigl( (A^{\prime\prime}+ \frac 9 r A^{\prime})x_1^3 x_2 + 6 A x_1 x_2) \cdot x_2 \cdot \frac{x_1}{r^6}(r^2-4x_2^2) dx_1 dx_2=0.
\times \int_{\Omega}\Biggl ( D_{K + N - M}{\frac{\partial^{2}f}{\partial \phi_{A}^{( I - M )}\partial \phi_{C}^{( L )}}}D_{I}{\frac{\partial h}{\partial \phi_{B}^{( J )}}}D_{J + L - N}{\frac{\partial g}{\partial \phi_{D}^{( K )}}}
R_{z_{3}}^{n,\sigma}=\sigma^{2}\mathbb{E}_{t_{n}}^{\sigma,X^{n}}\left[{\textstyle \int_{t_{n}}^{t_{n+1}}}Z_{t}^{t_{n},X^{n}}dt\right] -\sigma^{2}\Delta tZ_{t_{n}}^{t_{n},X^{n}}.
{\hat{\gamma}}_{(\lambda,\mu,\alpha)}^{r}\Rightarrow{\hat{\gamma}}_{(\lambda,\mu,\alpha)}^{r}\eta_{r}^{\lambda\mu},\quad{\hat{\gamma}}^{(\lambda,\mu,\alpha)}_{r}\Rightarrow\eta_{r}^{\lambda\mu}{\hat{\gamma}}^{(\lambda,\mu,\alpha)}_{r}={\hat{\gamma}}^{(\lambda,\mu,\alpha)}_{r}\eta_{r}^{\lambda\mu},\quad{(\eta_{r}^{\lambda\mu})}^{+}=\eta_{r}^{\lambda\mu},
A(x)-1-x= \ \frac{-x}{1-x}+\frac{x}{(1-x)^2}A(x)-\frac{2x^2}{1-x}A(x)+x\big(C(x)-1\big)-\frac{x^3(1-3x)}{(1-x)^3(1-2x)}&+xA(x)+(1-2x)C(x)-1.
T^{- 1}\, = \, \left( \begin{array}{c c}{{a^{- 1}+ a^{- 1}b d^{- 1}c a^{- 1}}}&{{- a^{- 1}b d^{- 1}}}\\{{- d^{- 1}c a^{- 1}}}&{{d^{- 1}+ d^{- 1}c a^{- 1}b d^{- 1}}}\\ \end{array}\right)
\frac{I_a^{(q,r)}(b)}{W^{(q+r)}(b)}= \frac{W^{(q,r)}_a(b)}{W^{(q+r)}(b)W^{(q)}(-a)}- r \frac{\overline{W}^{(q+r)}(b)}{W^{(q+r)}(b)}\xrightarrow{b \uparrow \infty}\frac{Z^{(q)}(-a, \Phi(q+r))}{W^{(q)}(-a)}- \frac r{\Phi(q+r)}.
\setlength{\abovedisplayskip}{5 pt}\setlength{\belowdisplayskip}{5 pt}\frac{1}{{\theta_{i,n}^S}}= \frac{1}{{\theta_{i,n}^O}}- \frac{{{{{N_h}L}\mathord{\left/{\vphantom{{{N_h}L}v}}\right. \kern-\nulldelimiterspace}v}}}{{- \log \Pr \left({D >{D_{\max}}}\right)}}.
-\Delta \varphi =-\varphi''(r)-\dfrac{n-1}{r}\varphi'(r)= =\dfrac{2\beta R}{(r^2+\alpha^2)^{\beta R+1}}\left[n-2(\beta R+1)+(\beta R+1)\dfrac{2\alpha^2}{r^2+\alpha^2}\right] &=2\beta R\varphi(r)^{\frac{1}{\beta R}+1}\left[n-2(\beta R+1)+ 2(\beta R+1)\alpha^2\varphi(r)^{\frac{1}{\beta R}}\right] &=f(\varphi(r)),
\sigma \big( D_0(b_1,\cdots,b_d) \big) = \sigma_{\textup{ess}}\big( D_0(b_1,\cdots,b_d) \big) = \big\lbrace \pm \sqrt{{\rm I}_{\textup{{\bf q}}}+ 1}:{\bf q}= (q_1,\ldots,q_d) \in \mathbb{N}^d \big\rbrace,
\widetilde{R}^{\ast}(X,Y)Z =R^{\ast}(X,Y)Z-A^{\ast}_{\sigma^{\ast}(Y,Z)}X+A^{\ast}_{\sigma^{\ast}(X,Z)}Y+(D^{\ast}_{X}\sigma^{\ast})(Y,Z) -(D^{\ast}_{Y}\sigma^{\ast})(X,Z),
\bigl[\big[X_{n1}^{(p)},\dots,X_{np}^{(p)}\big],\big[X_{m1}^{(q)},\dots,X_{mq}^{(q)}\big]\bigr]{}=\sum_{\sigma\in S}\epsilon_{\sigma}\big[X_{n\sigma(1)}^{(p)},\big[X_{n\sigma(2)}^{(p)},\dots, \big[X_{n\sigma(p)}^{(p)},\big[X_{m1}^{(q)},\dots,X_{mq}^{(q)}\big]\big],\ldots\big]\big],
Z = \frac{{\displaystyle \int}{\cal D}{\bar{\psi}}{\cal D}\psi \, \, \,{\displaystyle e}^{i S ({\bar{\psi}}, \psi , A_{-}, A_{+}, A_{-}^{5}, A_{+}^{5})}}{{\displaystyle \int}{\cal D}{\bar{\psi}}{\cal D}\psi \, \, \,{\displaystyle e}^{i S ({\bar{\psi}}, \psi )}},
|c(t)^{-1}-1| = \biggl|\int_{\mathcal{X}}\biggl(\frac{2}{1+e^{-2tg}}- 1 - tg\biggr)f\biggr| &\leq \int_{A_t}\biggl|\frac{e^{-2tg}- 1 + 2tg + tg(e^{-2tg}-1)}{1+e^{-2tg}}\biggr|f + \int_{A_t^c}(1 + t|g|)f &\leq \frac{16}{3}t^2 \int_{A_t}g^2f + 72t^2 \int_{A_t^c}g^2f \leq 72 t^2 \int_{\mathcal{X}}g^2 f.
\gamma = \frac{1}{2}c_{j k}^{i}\psi_{i}\chi^{j}\chi^{k}+ \frac{1}{2}f_{k}^{i j}\psi_{i}\psi_{j}\chi^{k}+ \frac{1}{3 !}\varphi^{i j k}\psi_{i}\psi_{j}\psi_{k}.
\liminf_{n\to\infty}\frac{1}{b_c^{(n)}}\log P(S_n(\kappa_n(x))\leq\lfloor\kappa_n(x)\rfloor-a_n) &\,\,\,\,\,\,\geq\liminf_{n\to\infty}\frac{1}{b_c^{(n)}}\log P\left(\frac{n-a_n-S_n(\kappa_n(x))}{f_2(n)}>x(1+\delta)\right) &\,\,\,\,\,\,\geq-\inf_{y\in (x(1+\delta),+\infty)}H(\ell_2 y)=-H(\ell_2 x(1+\delta)),
\sigma(\beta \alpha)_2(i) = (\beta \alpha)_2 \sigma^{-1}(i) = \alpha_2(\tau^{-1}(i')) + \beta_3(\tau^{-1}(i')) = \tau(\alpha)_2(i') + \sigma(\beta)_3 (i') = (\sigma(\beta) \tau(\alpha))_2(i).
\sum_{l = 1}^{\infty}F ( l ) \ = \ \int_{0}^{\infty}d l \F ( l ) \ \ - \ \frac 1 2 F ( 0 ) \ + \ \int_{0}^{\infty}\frac{d l}{1 - e^{2 \pi \nu}}\frac{F ( i l ) - F ( - i l )}{i}\ .
(u_{\mp}-v_{\pm}+hB_{ij})(u_{\pm}-v_{\mp}-hB_{ij})H_{i}^{\pm}(u)H_{j}^{\mp}(v) =(u_{\mp}-v_{\pm}-hB_{ij})(u_{\pm}-v_{\mp}+hB_{ij})H_{j}^{\mp}(v)H_{i}^{\pm}(u).
H_{k'}^?, H_{\ell'}^?, S_{p_{i'},q_{i'},r_{i'}}, \widehat{B}'_2, \vec{K}, C_1 = \Psi_1, \Psi_2, \Psi_3; & \vdash \Psi_1 \to H_k^?; & \vdash \Psi_2 \to H_\ell^?; & \vdash \Psi_3 \to S_{p_i,q_i,r_i}.
\lvert y+\frac{r_2}{r_1}z-(x+z)\rvert =\lvert y-x+\frac{(r_2-r_1)}{r_1}z\rvert &\leq \lvert y-x\rvert+\frac{\lvert x-y\rvert}{r_1}\lvert z\rvert\leq 2\lvert y-x\rvert\,.
\widetilde{\mathcal S}(\cos\zeta,n) = \mathcal S(\cos\zeta,n), \zeta \in \mathcal D_{-\frac\pi2+\theta_{\hat{x}}+i\infty,\zeta_o}, &\widetilde{\mathcal S}(\cos\zeta,n)=-\mathcal S(\cos\zeta,n), \zeta \in \mathcal \mathcal D_{\zeta_o,\frac\pi2+\theta_{\hat{x}}-i\infty}.
0=\nabla_{i}\widetilde{E}=b^{11}\nabla_{i}b_{11}-\tilde{d}\frac{h_{i}}{h}+2\tilde{l}\sum_{j}h_{j}h_{ji}&=b^{11}(b_{1i1})-\tilde{d}\frac{h_{i}}{h}+2\tilde{l}h_{i}h_{ii},
\aligned \int_{((X_1,\widehat{\mathcal U_1}) \times_{R\times P}(X_2,\widehat{\mathcal U_2}),(\widehat{\frak S_1}\times_{R\times P}\widehat{\frak S_2})^{\epsilon})}h_1 \wedge h_2 &=\int_{((X_2,\widehat{\mathcal U_2}),\widehat{\frak S_2})}\widehat f_2^* \widehat f_1!(h_1;(\widehat{\frak S_1})^{\epsilon})\wedge h_2.\endaligned
\begin{array}{llcl}(1~0~0~0~0~0~0~-1) \mid \mbox{gnd}> (1~0~0~0~0~0~0~-1) = \beta_i^+ \mid \mbox{1st}> (1~0~0~0~0~0~1~-1) = \mbox{SU(8)}(0~0~0~0~0~0~0~0) = \mbox{U(1)}\mid \mbox{2nd}> (0~0~0~0~0~0~1~0) = \beta_i^-. \end{array}
\phi_{- n_{2}}\partial_{- n_{4}}\phi_{- n_{3}}+ \phi_{- n_{3}}\partial_{- n_{4}}\phi_{- n_{2}}+ \phi_{- n_{4}}\partial_{- n_{3}}\phi_{- n_{2}}+ \left({\frac{\phi_{- n_{2}}\phi_{- n_{3}}\phi_{- n_{4}}}{W^{\prime}}}\right)^{\prime}
\mathbb{E}\left[\left|{\bf g}^H_k \boldsymbol{\Theta}^H_k(t){\bf f}_l\right|^2\right]=\left(\beta_k+ \left(X^{(1)}_{k,l}+X^{(2)}_{k,l}\right)\left(\frac{1-\epsilon}{N_o}+\epsilon\right)\right),
\begin{array}{ll}A = (L^{2}R \times \overline{L}R) \cdot (LR \times \overline{L}R) = O(r^{5m_{j}- 11}), B = (L^{2}R \times LR) \cdot (LR \times \overline{L}R) = O(r^{5m_{j}- 11}), C =(LR \times \overline{L}R) \cdot (LR \times \overline{L}R) = O(r^{4m_{j}- 8}).\end{array}
\Delta_n=\frac{TP_n}{(2Re\lambda_n)^2}e^{2\lambda_nT}e^{2\overline{\lambda}_nT}\prod\limits_{j=1}^{N-1}e^{2q_{j,n}T}\left(1+\bar{\xi}_n(T)+\bar{\Lambda}_n(T)+\bar{\beta}_n(T)\right).
\hat{S}= \int d^{1 0}\hat{x}\sqrt{| \hat{g}|}\left\{e^{- 2 \hat{\phi}}\left[ \hat{R}- 4 ( \partial \hat{\phi})^{2}\right] +{\textstyle \frac{( - 1 )^{7 - p}}{2 \cdot ( 8 - p ) !}}\left( \hat{G}^{( 8 - p )}\right)^{2}\right\}.
\mu(Se) = q(Se,Se\wedge m_{\mu}) \mu(Se\wedge m_{\mu}) &= q(Se,Se\wedge m_{\mu})\iota(Se\wedge m_{\mu},m_{\mu})\mu(m_{\mu}) Se\wedge m_{\mu}\le m_{\mu}&=\rho(e,ef,ef)\rho(ef,ef,f)\rho(f,f,f) m_{\mu}=\{Sf\}&=\rho(e,ef,f) &=\rho^f(Se).
H_{(X_+,D_+)}(y)=e^{\frac{t_+}{2\pi i}}\sum_{d\in\mathbb K_{+}}y^d\left(\frac{1}{\prod_{i\in M_0}\Gamma (1+\frac{\bar D_i}{2\pi i}+D_i\cdot d)}\right)\textbf{1}_{[d]}[\textbf{1}]_{(D_i\cdot d)_{i\in I_+}}.
f^{11}=-3 A^{112,1}2f^{12}=-3A^{122,1}f^{22}=-A^{222,2}g^{11}-2f^{11}{}' =2 E^{1,112}g^{12}-2 f^{12}{}' =E^{1,122}2g^{22}+f^{22}{}' =-2E^{1,222}.
i \int d x ~ e^{i k \cdot x}~ \langle 0 | T ( J_{\mu}( x ) J_{\nu}( 0 ) ) | 0 \rangle = ( k_{\mu}k_{\nu}- k^{2}g_{\mu \nu}) \left\{\Pi_{0}( k^{2}) + \Pi_{2}( k^{2}) F^{2}+ O ( F^{4}) \right\}
F_{i j}^{\alpha}\equiv \partial_{i}\bar{A}_{j}^{\alpha}( \bar{x}) - \bar{A}_{i}^{\alpha}( \bar{x}) \stackrel{\leftarrow}\partial_{j}+ g F^{\alpha \beta \gamma}\bar{A}_{i}^{\beta}( \bar{x}) \bar{A}^{\gamma}( \bar{x})
P \left(\bigcap_{i=1}^M \left \{\chi_{\frac{2n}{M}, i}^2 \in \left (1 - \frac{\delta}{\sigma_{\rm w}^2 +c}, 1 + \frac{\delta}{\sigma_{\rm w}^2 +c}\right ) \right\}\right) > 1- \frac{\epsilon}{2},
dI(z_{0};z_{1},\dots,z_{n};z_{n+1})=\sum_{i=1}^{n}d\log\left(\frac{z_{i+1}-z_{i}}{z_{i}-z_{i-1}}\right)I(z_{0};z_{1},\dots,\widehat{z_{i}},\dots,z_{n};z_{n+1}).
V \left( M \right) = - C \left( N_{c}, N_{f}\right) \left[ \frac{\Lambda_{Q C D}^{\frac{1 1}{3}N_{c}- \frac{2}{3}N_{f}}}{\mathrm{d e t}M}\right]^{\frac{1 2}{1 1 \left( N_{c}- N_{f}\right)}}\ ,
\begin{array}{lll}\alpha^{d4}_{max}< \alpha^{d6}_{max}< \dots < \alpha^{d,n-2}_{max}n = 2k, \alpha^{d3}_{max}< \alpha^{d5}_{max}< \dots < \alpha^{d,n-2}_{max}n = 2k+1. \end{array}
f_k(z)= kz^k+\sum_{n=1}^{\infty}\frac{(-1)^n}{n!}\sum_{m_j>0}ka_{m_1}(k+m_1)a_{m_2}(k+m_1+m_2)\dots &\quad\dots a_{m_n}(k+\sum_{j=1}^n m_j) z^{k+\sum_{j=1}^n m_j}.
\begin{aligned}\sup\limits_{x \in [0,L]}\Vert (u,h)\Vert_{H^m_l}\leq C \left(F_0+\int_0^x F\right)^{1/2}\left\{1-2C\vartheta_0^{-8}\left(F_0+\int_0^x F\right)^2x\right\}^{-1/4}.\end{aligned}
\! \{[0\!:\! z_2\!:\!\cdots\!:\! z_{n+1}]: z_2,\!\cdots\! z_{n+1}\!\in\!\mathbb{C}\}\setminus \!\{[0\!:\! 1\!:\! z_3\!:\!\cdots\!:\! z_{n+1}]: z_3,\!\cdots\!,\! z_{n+1}\!\in\!\mathbb{C}\}= &\{[0\!:\! 0\!:\! z_3\!:\!\cdots\!:\! z_{n+1}]: z_3,\!\cdots\!,\! z_{n+1}\!\in\!\mathbb{C}\}
B_{12}\left(\begin{array}{c}z_1 z_2 z_3 \end{array}\right)=\ \frac12\left(\begin{array}{c}iz_2 iz_1 0 \end{array}\right), B_{12}\left(\begin{array}{c}z^*_1 z^*_2 z^*_3 \end{array}\right)=\ \frac12 \left(\begin{array}{c}-z^*_2 -z^*_1 0 \end{array}\right),
\mathcal{L}_sf(z,w) = (2\pi)^{-n-1}\int_{-\infty}^\infty \Big( \sum_{k=0}^\infty (2|\lambda|)^s \frac{\Gamma(\frac{2k+n}{2}+\frac{1+s}{2})}{\Gamma(\frac{2k+n}{2}+\frac{1-s}{2})}f^\lambda*_\lambda \varphi_k^\lambda(z)\Big) e^{-i\lambda w}|\lambda|^n d\lambda.
\mathcal{W}(X,Y)Z= R(X,Y)Z+\frac{r}{(n-1)(n-2)}[g(X,Z)Y-g(Y,Z)X] + \frac{1}{n-2}[Ric(Y,Z)X-Ric(X,Z)Y+g(Y,Z)QX-g(X,Z)QY],
\int_{Q_t}|\partial_t\kappa_{\varepsilon}|^2+ \frac{1}{2}\int_{\Omega}|\nabla \kappa_{\varepsilon}(t)|^2 + \rho \| \kappa_{\varepsilon}(t) \|_{H, \varepsilon}= \frac{1}{2}\int_{\Omega}|\nabla \kappa_0|^2
\dot{z}_{1}=a_{1}z_{1}+a_{2}x_{1}\dot{z}_{2}=a_{3}z_{2}+z_{1}x_{1}\dot{x}_{1}=x_{2}+a_{4}z_{1}+a_{5}x_{1}-z_{1}z_{2}&\dot{x}_{2}=bu+a_{6}z_{1}y= x_{1}
G(\rho,\zeta)= \partial_n \mathcal W^+(\rho,\zeta)\Big |_{J_{\mathcal S(\rho)}}-\partial_n \mathcal W^-(\rho,\zeta)\Big |_{J_{\mathcal S(\rho)}}\mbox{for}\;\;\rho\in\mathcal O_{\delta_1},\;\; \zeta\in h^{4+\alpha}(\Bbb S).
\left\| \Pi_{I_0}^{F, G, H'}\otimes \Pi \right\|_{L_x^\infty L_y^\infty \times L_x^q L_y^q \to L_x^q L_y^q}= \left\| \left(\Pi_{I_0}^{F, G, H'}\otimes \Pi \right)^{*, 1}\right\|_{L_x^{q'}L_y^{q'}\times L_x^q L_y^q \to L_x^1 L_y^1}.
[\phi(C(\phi^{-1},U)):C(\phi^{-1},U)] =[\phi(C(\phi^{-1},U))\cdot U:U] &=[\phi(C_n(\phi^{-1},U))\cdot U:U] &=[\phi(C_n(\phi^{-1},U)):\phi(C_n(\phi^{-1},U))\cap U] &=[\phi(C_n(\phi^{-1},U)):C_{n+1}(\phi^{-1},U)].
L^{i n t}= - e ( \overline{{\psi}}_{1}\gamma^{\mu}\gamma^{5}\psi_{1}- \overline{{\psi}}_{2}\gamma^{\mu}\gamma^{5}\psi_{2}) C_{\mu}+ i e ( \overline{{\psi}}_{1}\gamma^{\mu}\psi_{1}+ \overline{{\psi}}_{2}\gamma^{\mu}\psi_{2}) B_{\mu}
\gamma_j^*(A,b) =\min_{i\colon a_{ij}\neq 0}b_i a_{ij}^{-1},a_{kj}\gamma_j^*(A,b)= b_k\Leftrightarrow k\in\arg\min_{i\colon a_{ij}\neq 0}b_i a_{ij}^{-1}.
\beta_{a}(I)=\beta_{a,b}(I) =\big|\big\{w\in\mathcal{L}_t^f(u):\min(u)=1\big\}\big|-\big|\big\{w\in\mathcal{L}_t^f(v)\setminus\{v\}:\max(w)=n\big\}\big| &\ge\big|\big\{u>u_{\ell}>\dots>u_{d-1}\}\big|-(d-\ell)= d+1-\ell-(d-\ell)= 1.
\omega_{1,m,n}(x) I_{1,m,n}(x) = \omega_{1,m,n}(x) \, \sqrt{\frac{L_m(x)}{6 \pi n^{4/3}}}\exp(\tfrac{3}{2}n^{2/3}L_m(x) ) \, \left(1+O_X( n^{-1/3}) \right)
I_{U_{1}}= ( - )^{i_{2}- m_{2}+ i_{3}- m_{3}}\int D U_{1}D_{i_{1}m_{1}}^{J_{1}}( U_{1}) D_{- i_{2}- m_{2}}^{J_{2}}( U_{1}) D_{- i_{3}- m_{3}}^{J_{3}}( U_{1}) ,
\epsilon^{\mu \nu \rho \sigma}\partial_{\nu}R_{\rho \sigma}= \epsilon^{\mu \nu \rho \sigma}\partial_{\nu}( \vec{\Gamma}_{\rho}\times \vec{\Gamma}_{\sigma}) \vec{T}= - \frac{1}{2}\epsilon^{\mu \nu \rho \sigma}\partial_{\nu}( \partial_{\rho}\Gamma_{\sigma}- \partial_{\sigma}\Gamma_{\rho}) = 0
\begin{array}{rl}A_{\lambda}^{(nn)}(x| a_1 , \cdots , a_n |b_1 , \cdots b_n )= \displaystyle\prod_{j=1}^{n}(x-a_j \tau )\,g_{\lambda}^{(n)}(x| b_1 , \cdots ,b_n ) & + \tau^{2(\lambda -1)}\displaystyle\prod_{i=1}^{n}(x-b_i \tau^{-1})\,g_{\lambda}^{(n)}(x| a_1 , \cdots , a_n ).\end{array}
8\|\nabla_H\bar v\|_{L^2(G)}^2 + \|v_z\|_{L^2(\Omega)}^2 + (C_3/4)\|\tilde v\|_{L^4(\Omega)}^4 \le \left( C(\|a\|_{H^1(\Omega)}^2 + \|a\|_{H^1(\Omega)}^4) + \int_0^t K_2(s)\,ds \right) e^{\int_0^t K_1(s)\,ds}&=: B_1(t, \|a\|_{H^1(\Omega)}), t\ge0,
Y_{1}X_{2}Z_{3}=[\begin{matrix}c_{1}c_{3}+s_{1}s_{2}s_{3}&c_{3}s_{1}s_{2}-c_{1}s_{3}&c_{2}s_{1}\\c_{2}s_{3}&c_{2}c_{3}&-s_{2}\\c_{1}s_{2}s_{3}-c_{3}s_{1}&c_{1}c_{3}s_{2}+s_{1}s_{3}&c_{1}c_{2}\end{matrix}]
\widetilde{\left(\overline{L}Z\right)}\, \frac{\partial \widetilde{h}^{j}}{\partial \overline{\zeta}}=\frac{\widetilde{A}}{\widetilde{C}}\,\widetilde{h}^{j}-\frac{\widetilde{B}}{\widetilde{C}}\,\overline{\widetilde{h}^{j}}+\widetilde{M},
\int_{\Omega_{\ell-\ell_0}}A\nabla \phi_\ell.\nabla\phi_\ell = \frac{2}{\eta}\int_{\omega}a_{11}W_1^2 + \frac{2\eta}{3}\int_{\omega}A_{22}\nabla W_1.\nabla W_1 = \frac{2}{\eta}\int_{\omega}a_{11}W_1^2 + \frac{2\eta\mu{}^1}{3}\,.
\frac{1}{2}\int_{\epsilon^2 R^2}^{\infty}\frac{dt}{t^{3}}\sum_{k=-\infty}^{\infty}e^{-k^2 \pi t/R^2}=R^{-4}{\cal I}(\epsilon^2,\alpha=-2)=\frac{1}{R^{4}}\left\{\frac{3}{4\pi^2}\zeta(5)+\frac{1}{5}\epsilon^{-5}\right\}
L=i\overline{\psi}_{a}\partial/\psi^{a}+\frac{\lambda}{4N}[(\overline{\psi}_{a}\psi^{b})(\overline{\psi}_{b}\psi^{a})-(\overline{\psi}_{a}\gamma^{5}\psi^{b})(\overline{\psi}_{b}\gamma^{5}\psi^{a})]
A =A\|x\|^{2}\leq\sum\limits_{i=1}^{N}a_{i}=\sum\limits_{i=1}^{m}a_{i}+\sum\limits_{i=m+1}^{N}a_{i}begin{align*}2 pt] <m\frac{A}{N}+(N-m)a_{m}<m\frac{A}{N}+(N-m)\frac{A}{N}begin{align*}2 pt] =A,
W_{3}(n) =45(9 n^2-73 n+176), W_{4}(n) =7(1215 n^3-19710 n^2+121685 n-266398), W_{5}(n) =945 \left(243 n^4-6570 n^3+74165 n^2-394878 n+805440\right), W_{6}(n) =165 \left(45927 n^5-1862595 n^4+33070275 n^3-310359581 n^2+1497391014 n-2916611728\right).
\sum_{j=0}^k\frac{\left({\mathcal Q}_j^n(\zeta)\right)^2}{j!(n+2)_j(a_{2,0}a_{3,2})^j\left(n+1+\frac{a_{2,2}}{a_{3,2}}\right)_j}=\frac{[{\mathcal Q}_{k+1}^n(\zeta)]'{\mathcal Q}_{k}^n(\zeta)-[{\mathcal Q}_{k}^{n}(\zeta)]'{\mathcal Q}_{k+1}^n(\zeta)}{k!(n+2)_k(a_{2,0}a_{3,2})^k\left(n+1+\frac{a_{2,2}}{a_{3,2}}\right)_k}.
\mathrm{BF}^{k,\ell,j}(\beta):=\begin{cases}\tilde{\Xi}^{k,\ell,j}(\beta) \textup{if $j > 0$}, \tilde{\Xi}^{k,\ell ,0}(\beta)-i_{\mathrm{cusp},*}(\xi_\beta) \textup{if $j=0$}.\end{cases}
\pi(y_{n_i+1}^{n_{i+1}},\zeta_{n_{i+1}}|\zeta_{n_i})\le\sum_{\zeta_{n_i+1}^{n_{i+1}-1}}\prod_{t=n_i+2}^{n_{i+1}-1}\pi(y_t,\zeta_t|\zeta_{t-1}).
\mu\left( \left\{x\in S:\left\vert f(x)-f_{S}\right\vert >\lambda\right\}\right) \leq\mu\left( \left\{x\in S:\left\vert f(x)-f_{S}\right\vert>N\lambda_{1}\right\}\right) & \leq\frac{1}{2^{N}}\mu\left( S\right) =e^{-N\log2}\mu(S)\leq2e^{-\left(\frac{\log2}{\lambda_{1}}\right) \lambda}\mu(S)
H_j(g) =y_j\int_{\mathbb R}\Big[{\sinh(\lambda+i\phi)\over \lambda+i\phi}\Big]^{1\over 2}\cosh(\lambda+i\phi)(\cosh \lambda)^{-n-{3\over 2}}d\lambda &\quad+ix_j\int_{\mathbb R}(\lambda+i\phi)\Big[{\sinh(\lambda+i\phi)\over \lambda+i\phi}\Big]^{3\over 2}(\cosh \lambda)^{-n-{3\over 2}}d\lambda.
- ( 2 \pi k_{f}) \int_{- \infty}^{+ \infty}\mathrm{~}\frac{d q_{1}}{2 \pi}\mathrm{~}\frac{\Lambda_{k + q_{1}/ 2}( - q_{1})}{2 \omega_{R}( q_{1}) ( \omega_{R}( q_{1}) + \omega_{k + q_{1}/ 2}( q_{1}) )^{2}( \frac{m^{3}}{q_{1}^{4}}) ( c o s h ( \lambda ( q_{1}) ) - 1 )}
|D^{\sigma}_{ij}u^1(x)-D^{\sigma}_{ij}u^1(x')| \le|\int(\delta u^1(x,y)-\delta u^1(x',y))\frac{(2-\sigma)\langle y,e_i\rangle\langle y,e_j\rangle \rho(x,y)}{|y|^{n+\sigma+2}}dy| &+|\int\delta u^1(x',y)\frac{(2-\sigma)\langle y,e_i\rangle\langle y,e_j\rangle (\rho(x,y)-\rho(x',y))}{|y|^{n+\sigma+2}}dy| &=I_1+I_2.
\mathcal{C}_{21}:= \left[\frac{2A+1}{\log q}-iq,-\frac{1}{2}-iq\right] \cup \left[-\frac{1}{2}-iq,-\frac{1}{2}+iq\right] \cup \left[-\frac{1}{2}+iq,\frac{2A+1}{\log q}+iq\right],