harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Theta= \omega^0\wedge dm-\omega^1\wedge dt-H dt\wedge dm \equiv \pi_x^t dx\wedge d\tilde{m}_0+\pi^t_S dS\wedge d\tilde{m}_0+ \pi_x^m dx\wedge d\tilde{m}_1-H dt\wedge dm, \equiv (udx+rdS)\wedge dm-pdx\wedge dt-HdV.
\mathcal{X}_\mathcal{Q}:= \{x \in \mathcal{X}: \Delta u_q^T l_q(x) - \Delta u_k^T l_k(x) \geq 0 \ \forall q \in \mathcal{Q}, \ \Delta u_q^T l_q(x) - \Delta u_k^T l_k(x) \leq 0 \ \forall q \in [m] \setminus \mathcal{Q}\}.
\int_\Omega\big(\mathrm{div}\,V\big)^2 \,dx=\int_\Omega\mathrm{tr}\big((\nabla\,V)^2\big)dx+\int_{\partial\Omega}\Big((\mathrm{div}_T V)\,V\cdot\nu- \nabla_T V\, V_T \cdot \nu \Big)d\mathcal{H}^{n-1}.
\phi_k^D(u) = \bar p_1 \phi_k^R(u) + \bar p_2 \sum_{j}u_k u_j (G_{k,k}-G_{j,k}+ G_{k,j}-G_{j,j}) & -(1/\kappa) p(v_1|v_2) \sum_{j}u_k u_j ( G_{k,j}- G_{j,k}).
H_{3,j}T_j^* = u_{3,j}v_{j}^*- b v_{j}u_{1,j}^* -(I-bT_j)H_{1,j}, H_{4,j}T_j^* = u_{4,j}v_{j}^* + a v_{j}u_{1,j}^* + (I-aT_j)H_{1,j}.
\underline{\Delta}\varphi =\frac{\psi^{'}\underline\Delta d}{\rho}+\frac{\psi^{''}\left\|\overline{\nabla}d\right\|^{2}}{\rho}\ge \frac{c_{1}}{\rho}(n-1)\sqrt{\rho_{2}}\;\coth(\sqrt{\rho_{2}}\rho)-\frac{c_{2}}{\rho^{2}},
h_{\epsilon}^{\ast}(1-\delta) = \begin{cases}\bar{\epsilon}H_b(\delta) - (1-\delta)\bar{\epsilon}\frac{H_b(p_\epsilon)}{p_\epsilon+\frac{1}{\bar{\epsilon}}}\mbox{if}1- \delta \leq p_\epsilon \frac{H_b(p_\epsilon)}{p_\epsilon+\frac{1}{\bar{\epsilon}}}\mbox{if}1-\delta \geq p_\epsilon. \end{cases}
e^{-\left(\tfrac{1}{\alpha}-\tfrac{1}{\beta}\right)z}\int_{(0,\infty)^2}\,\tfrac{x}{1+\alpha x}\,e^{-z(x+v)}\P_X(dx)\P_V(dv)=\tfrac{\beta c(0)}{\alpha a(0)}\,\int_{(0,\infty)^2}\,\tfrac{v}{1+\beta v}\,e^{-z(x+v)}\,\P_X(dx)\P_V(dv).
c_{1}( \alpha ) = \frac 1 6 \left( \left( 2 \pi \alpha^{- 1}\right)^{2}- 1 \right) ~ ~ , ~ ~ c_{2}( \alpha ) ={\frac{1}{1 5}}c_{1}( \alpha ) \left( \left( 2 \pi \alpha^{- 1}\right)^{2}+ 1 1 \right) .
-\langle R^N (V,d\phi(e_j))d\phi(e_j) ,\bar\Delta^2 V\rangle-\langle R^N(\bar\Delta^2V,d\phi(e_j))d\phi(e_j),V\rangle &=-2\langle R^N(V,d\phi(e_j))d\phi(e_j) ,\bar\Delta^2 V\rangle
\begin{aligned}[b] |\hat{S}_1|-|\hat{S}_2|-\frac{2E_{\hat{S}_1}+E_{\hat{S}_1,\hat{S}_3}+E_{\hat{S}_1,V_1}-E_{\hat{S}_2,V_1}}{\tau-1}& <|S_1^*|-|S_2^*|-\frac{2E_{S_1^*}+E_{S_1^*,S_3^*}+E_{S_1^*,V_1}-E_{S_2^*,V_1}}{\tau-1}.\end{aligned}
\gamma D_\alpha^{-1}C u_2 + \gamma D_\alpha^{-1}v_1 =u_1 - D_\alpha^{-1}D_{-\beta}u_1, \gamma A_\beta^{-1}B u_1 + \gamma A_\beta^{-1}v_2 =u_2 - A_\beta^{-1}A_{-\alpha}u_2.
i \frac{\partial}{\partial t}\langle \vec{q}\,^{\prime}, t \mid \vec{q}, 0 \rangle = - \frac{1}{2}\sum_{i = 1}^{3}\frac{\partial^{2}}{\partial q_{i}^{2}}\langle \vec{q}\,^{\prime}, t \mid \vec{q}, 0 \rangle + V ( \vec{q}\,^{\prime}) \langle \vec{q}\,^{\prime}, t \mid \vec{q}, 0 \rangle
\rho \leq\begin{cases}\tilde{c}_{1}\left( \frac{B_{n}^{2}\log^{7}(pn)}{n}\right)^{1/6}+ \tilde{c}_{1}\left( \frac{B_{n}^{2}\log^{3}(pn)}{n^{1-2/q}}\right)^{1/3}, \tilde{c}_{1}\left( \frac{B_{n}^{2}\log^{7}(pn)}{n}\right)^{1/6}.\end{cases}
r_1\sum_{\tau\in T(r_1, \ldots,r_t)}a_{\tau(1)}a_{\tau(2)}\ldots a_{\tau(p)}=\sum_{\tau\in T(r_1, \ldots, r_t),\tau(1)=1}[[\cdots[[a_{\tau(1)},a_{\tau(2)}],a_{\tau(3)}],\ldots],a_{\tau(p)}].
(\lambda_1,\cdots,\lambda_n,\lambda_{n+1}, \cdots,\lambda_m)(z_1,\cdots, z_n, w_1,\cdots, w_n, u_1,\cdots, u_{|S|}) &=(\lambda_1z_1,\cdots,\lambda_nz_n,\lambda_1^{-1}w_1,\cdots,\lambda_n^{-1}w_n, \lambda_{n+1}u_1, \cdots,\lambda_mu_{|S|});
H(W_{d_1\sim d_{s_1}},W_{d_{l+1}\sim d_{l+s_2}}|Y_{1\sim l},U_{l+1\sim N_T},V_{1\sim s_1}, V_{l+1\sim l+s_2},W_{d_{s_1+1}\sim d_{l}},W_{d_{l+s_2+1}\sim d_{N_R}},W_{1\sim L}\setminus W_\mathbf{d}) = F\varepsilon_F+T\varepsilon_P\log P,
[ x^{I}, p^{J}] = i \delta^{I J}~ , [ \alpha_{n}^{i}, \alpha_{m}^{j}] = \omega_{n}\delta^{i j}\delta_{n + m , 0}~ , [ \alpha_{n}^{i^{\prime}}, \alpha_{m}^{j^{\prime}}] = \omega_{n}^{\prime}\delta^{i^{\prime}j^{\prime}}\delta_{n + m , 0}~ .
\chi(s) \ = \ \prod_{(p,M) = 1}\sum_{t= 0}^{\infty}\frac{R(1,p^t)R(m^2,p^t)}{\varphi(p^t)p^{ts}}= \ \prod_{(p,M) = 1}\begin{cases}\left(1 + \frac{1}{(p-1)p^s}\right), (p,m) = 1, \left(1 - \frac{1}{p^s}\right) (p,m) > 1, \end{cases}
\Xi_{\tilde{\bf i}}[\lambda] := \{\widehat{S}_{j_k}\cdots \widehat{S}_{j_1}a_{j_0}\mid k \ge 0\{\rm and}\j_0, \ldots, j_k \ge 1\}&\cup \{\widehat{S}_{j_k}\cdots \widehat{S}_{j_1}\lambda^{(i)}({\bf a}) \mid k \ge 0,\i \in I,\{\rm and}\j_1, \ldots, j_k \ge 1\}.
U_{2n,x}= U_{2n,2N-x}= \frac{1}{\sqrt{2}}\tilde R_n(\lambda(x);\gamma,\delta,N), x=0,\ldots,N-1, U_{2n+1,x}= -U_{2n+1,2N-x}= -\frac{1}{\sqrt{2}}\tilde R_n(\lambda(x);\gamma,\delta,N-1), x=0,\ldots,N-1, U_{2n,N}= \tilde R_n(\lambda(N);\gamma,\delta,N),U_{2n+1,N}=0,
\limsup_{\epsilon\to 0}\limsup_{L\to\infty}\frac 1{L^d}\sum_{i\in\mathbb T_L^d}\sum_{k=1}^d \sum_{\eta\in\Omega_L}\mu^L_{[0,T]}(\eta)\;\!\Bigl|\hat\chi_{i,i+e_k}^{\lfloor \epsilon L\rfloor}(\delta_\eta) - \hat\chi_{i,i+e_k}(\nu_{\eta^{\lfloor \epsilon L\rfloor}(i)})\Bigr|= 0,
\begin{aligned}I_1:= ~{}\iint_{D_t}\varphi(\underline{u})|\underline{L}\tilde{\Lambda}||Q_0(\ln \alpha, \tilde{\Lambda})| \lesssim ~{}\iint_{D_t}\varphi(\underline{u})|\underline{L}\tilde{\Lambda}|[|L(\ln \alpha)||\underline{L}\tilde{\Lambda}|+|\underline{L}(\ln \alpha)||L\tilde{\Lambda}|] =: I_{1,1}+ I_{1,2}.\end{aligned}
\widetilde{d}(\widehat{\Xi}(\lambda_0, \ldots, \lambda_n ; J, q)) = \sum_{i =1}^{m}d_{J_{(i)}}(\lambda_{i-1}) e_{J_{(i)}}+ \sum_{i =m+1}^{n-1}d_{J_{(i)}}(\lambda_{i})e_{J_{(i)}}
\int_D p(\theta)d\theta \leq\sqrt{\int_D d\theta}\cdot\sqrt{\int_D p(\theta)^2d\theta}\leq\sqrt{|D|}\cdot\sqrt{\int_{-\pi}^{\pi}p(\theta)^2d\theta}\leq &\leq \sqrt{(1-\eta)^2\frac{2\pi}{2n+1}}\cdot\sqrt{\frac{2n+1}{2\pi}}\int_{-\pi}^{\pi}p(\theta)d\theta=(1-\eta)\int_{-\pi}^{\pi}p(\theta)d\theta.
\langle \nabla \times{\bf G},\varphi\rangle=\left(\langle G_2,\varphi_{x_3}\rangle-\langle G_3,\varphi_{x_2}\rangle\right){\bf i}-\left(\langle G_1,\varphi_{x_3}\rangle-\langle G_3,\varphi_{x_1}\rangle\right){\bf j}+\left(\langle G_1,\varphi_{x_2}\rangle-\langle G_2,\varphi_{x_1}\rangle\right){\bf k}.
R_{\textrm{IPLA}}^* ={\mathrm{argmax}\atop{\scriptstyle{R \geq 0}}}~ \sum_{i=1}^{N_{max}}\frac{R}{i\cdot\pi}\cdot &\int_{0}^{\infty}\bigg[ \mathcal{I}m \Big\{\Psi_{\textrm{IPLA}}(t;i, R) - \Psi_{\textrm{IPLA}}(t;i-1,R) \Big\}\bigg] dt.
\mathcal R_V^{\pm}(\lambda) = \mathcal R_0^{\pm}(\lambda)- \mathcal R_0^{\pm}(\lambda) V\mathcal R_0^{\pm}(\lambda) +\mathcal R_0^{\pm}(\lambda) V\mathcal R_V^{\pm}(\lambda) V \mathcal R_0^{\pm}(\lambda).
g^{00}=x^4, g^{0i}=x^4\tilde{g}^{0i}, g^{i0}=x^4\tilde{g}^{i0}, 1 \leq i \leq n-1, & g^{ij}=x^2\tilde{g}^{ij}, 1 \leq i,j \leq n-1,
\nabla' p (\cdot,y_d) = - (\nabla' \nabla' (-\Delta')^{-\frac12}) \cdot P (y_d) \gamma \partial_{y_d}u' , y_d>0, \partial_{y_d}p (\cdot,y_d) = \nabla' \cdot P (y_d) \gamma \partial_{y_d}u' , y_d>0.
d s^{2}= 2 ( k - 2 ) [ d R^{2}- \frac{( 1 + \rho^{2}) d t^{2}}{( 1 + \rho^{2}) \coth^{2}R - \rho^{2}- 2 / k}+ \frac{\rho^{2}\coth^{2}R \; d x^{2}}{( \rho^{2}+ 1 - 2 / k ) \coth^{2}R - \rho^{2}}] ,
0 = \|\nabla (\eta h)\|^2_{L^2}- \||d\eta | h\|^2_{L^2}+ \langle c(F)\eta h,\eta h\rangle_{L^2}\geq \|\nabla_{\hat e_1}(\eta h)\|^2_{L^2}+\|\frac{\eta \nabla_{\hat e_1}h}{\sqrt{2+r^{-1}}}\|^2_{L^2}- \||d\eta | h\|^2_{L^2}&-\int O(r^{-3}\eta^2|h|^2)dv-\int O(\eta^2r^{1- N})dv,
|\int_{|x|<\frac{1}{|y|}}(e^{2\pi ix\cdot y}-1)K_1(x)\,dx| \leq C|y|\int_{|x|<\frac{1}{|y|}}|x|\frac{1}{|x|^{n-\beta}}\,dx &\leq \frac{1}{\beta+1}\beta^{-\beta}.
\varrho (x) = \frac 1N \sum_{j=1}^N \varphi (x-x_j)\,, \varrho u (x) = \frac 1N \sum_{j=1}^N \varphi (x-x_j)v_j\,, \varrho(|u|^2 +3T)(x) = \frac 1N \sum_{j=1}^N \varphi (x-x_j)v_j^2\,,
S(t)=\frac{w_q}{qL}\sum_{\substack{s=0 s \in W^{q}(Y)}}^{q-1}e^{2\pi \textnormal{i}h_d(s)a/q}\int_0^Mh_d'(x)e^{2\pi\textnormal{i}h_d(x)\beta}dx+ O_h\left(e^{-c\frac{\log\left(\frac{M}{q}\right)}{\log Y}}Z^{3k}\right),
|-4 \int_{t}^{\infty}\frac{e'''(s) ds}{(1+s-t)}\left(\frac{1}{\log(\lambda_{0,0}(s))}-\frac{1}{\log(\lambda_{0,0}(t))}\right)| \leq \frac{C}{t (\log(\log(t)))^{2}\log(t)}\int_{t}^{\infty}|e'''(s)| ds &\leq \frac{C \sup_{x \geq t}\left(|e'''(x)| x^{3/2}\right)}{t^{3/2}(\log(\log(t)))^{2}\log(t)}
\int_{|x|>1}u(\dot{x})|x|^{-d-\gamma}e^{-i(x,k)}dx=\sum_{0\neq z\in Z^d}e^{-i(z,k)}\sum_{s=0}^{n-1}v_s(\dot{z})|z|^{-d-\gamma-s}+q(k), q\in C^m.
D_{112}\ll \left( t_3^{-\sigma_3}+ 1 \right) \begin{cases}t_3^{-\sigma_1-\sigma_3-1}(\sigma_2>\frac{5}{2}) t_3^{-\sigma_1-\sigma_3-1}\log t_3 (\sigma_2=\frac{5}{2}) t_3^{\frac{3}{2}-\sigma_1-\sigma_2-\sigma_3}(\sigma_2<\frac{5}{2}). \end{cases}
{}_2 F_1\bigg(1,\frac{1}{2};\frac{d}{2};\cos^2\bigg(\frac{\zeta}{2}\bigg)\bigg) + \frac{d-1}{d}\, \frac{\cos^2(\zeta/2)}{1-\cos^2(\zeta/2)}\,{}_2 F_1\bigg(1,\frac{1}{2};\frac{d}{2}+1;\cos^2\bigg(\frac{\zeta}{2}\bigg)\bigg) = \sin^{-2}\bigg(\frac{\zeta}{2}\bigg),
{\cal L}= - \sqrt{- G}\sqrt{1 +{\frac{\tilde{Y}M \tilde{Y}}{G ( \partial a )^{2}}}+ \left({\frac{\tilde{Y}L \tilde{Y}}{2 G ( \partial a )^{2}}}\right)^{2}}-{\frac{\tilde{Y}L \partial Y \cdot \partial a}{2 ( \partial a )^{2}}},
\left |\frac{b_i(\tau)}{\tau}-2p_i \right | = \left |\frac{b_i(\tau_0)-2p_i \tau_0+\int_{\tau_0}^\tau \left( \frac{db_i(s)}{ds}-2p_i \right) ds}{\tau}\right | \leq \frac{\left | b_i(\tau_0)-2p_i \tau_0 \right |}{\tau}+ \frac{\int_{\tau_0}^\tau \left | \frac{db_i(s)}{ds}-2p_i \right | ds}{\tau}\leq \varepsilon + \varepsilon.
dq^\prime_t= (\tilde\gamma^T)^{-1}(t^*,q^\prime_t)\left(-\partial_t\psi(t^*,q^\prime_t)-\nabla_qV(t^*,q_t^\prime)+ \tilde F(t^*,q^\prime_t,\psi(t^*,q^\prime_t))\right) dt & +\tilde S(t^*,q_t^\prime)dt+ \left(S^\prime(t^*,q^\prime_t)-S(t^*,q_t^\prime)\right)dt +\tilde\gamma^{-1}(t^*,q^\prime_t)\sigma(t^*,q^\prime_t)\circ d\tilde W_t,
K_{\mu}^{i}= \frac 1 2 ~ i ~ \varepsilon^{i a b}\overline{{\psi}}^{a}\gamma_{\mu}\psi^{b}- i ~{\frac{\varphi^{a}\overline{{\psi}}^{a}\gamma_{\mu}\psi^{i}}{\sqrt{1 -{\varphi}^{2}}}}~ .
\sum_{n\leq x}(n, u_{n})^{k}=\sum_{n\leq x}\sum_{\substack{d\mid (n, u_n)}}J_{k}(d)=\sum_{d\leq x}J_{k}(d)\sum_{\substack{n\leq x d\mid (n, u_n)}}1=\sum_{\substack{d\leq x (d,a_2)=1}}J_{k}(d)\sum_{\substack{n\leq x \ell_u(d)\mid n}}1,
a_h^+(r_h^+ u - r_h u, r_h^+ u - r_h u) = \; a_h^+\left(r_h^+ u - r_h u, r_h^+ u - r_h u - (e_h + e_h^+)\right) &+ a_h^+(r_h^+ u - r_h u, e_h + e_h^+).
\begin{aligned}g(t) = \zeta E_{\alpha}\big[-\gamma t^{\alpha}\big]g(-2) &+ \frac{\alpha c_{0}\zeta}{B(\alpha)}\int_{-2}^{t}\bigg( E_{\alpha,\alpha}\big[-\gamma(t-s)^{\alpha}\big] + \frac{(1-\alpha)}{\alpha}\gamma^{-2\alpha}E_{\alpha,\alpha}\big[-\gamma^{-2\alpha}(t-s)^{\alpha}\big]\bigg)(t-s)^{\alpha-1}h(s)ds .\end{aligned}
g_i(1/i)=\lambda^{-\frac{\rho_1}{(1-m)\beta}}i^{\frac{\alpha}{\beta}}\quad\mbox{and}g_i'(1/i)=-\frac{\alpha}{\beta}\lambda^{-\frac{\rho_1}{(1-m)\beta}}i^{\frac{\alpha}{\beta}+1}.
\sup_{\{f=r\}}f^l|h_a-\lambda_m\operatorname{Ric}(g)| \leq \max\{r_m^l \sup_{\{f=r_m\}}|h_a-\lambda_m\operatorname{Ric}(g)|,\rho_0^l \sup_{\{f=\rho_0\}}|h_a-\lambda_m\operatorname{Ric}(g)|\}\leq \rho_0^l \sup_{\{f=\rho_0\}}|h_a-\lambda_m\operatorname{Ric}(g)|, ~r\in[\rho_0,r_m].
\aligned\int_{\Omega}(x\cdot\nabla u(x))(\int_{\Omega}\frac{|u(y)|^{2_{\mu}^{\ast}}}{|x-y|^{\mu}}dy)|u(x)|^{2_{\mu}^{\ast}-1}dx=\frac{\mu-2N}{22_{\mu}^{\ast}}\int_{\Omega}\int_{\Omega}\frac{|u(x)|^{2_{\mu}^{\ast}}|u(y)|^{2_{\mu}^{\ast}}}{|x-y|^{\mu}}dxdy.\endaligned
\frac{d\pi_{n-}^{\mu}}{d\pi_{n-}^{\nu}}(x) =\frac{E^{\nu}[\frac{d\mu}{d\nu}(X_{0})|Y_{[0,n-1]},X_{n}=x]}{E^{\nu}[\frac{d\mu}{d\nu}(X_{0})|Y_{[0,n-1]}]}~~~~~P^{\mu}~a.s.
\partial_{\omega}F^{z\bar{z}}_{ij}+\mathbf{i}\ \Gamma_K((\Omega_i-\Omega_j)F^{z\bar{z}}_{ij})=\Gamma_KR^{z\bar{z}}_{ij},\Gamma_KF^{z\bar{z}}_{ij}=F^{z\bar{z}}_{ij};
[U,A]\alpha =(\nabla_U A-\nabla_A U)\alpha=-\frac{1}{\alpha-\gamma}\left(\frac{\alpha b (\alpha -\gamma )}{a}+\frac{b c (\alpha -\gamma )}{4 a (\beta -\alpha )}+\frac{3 a b c}{4}\right)U\gamma, UA\alpha =\frac{b(c-4\alpha(\alpha-\beta))}{4a(\alpha-\beta)}U\gamma, \qquad AU\alpha=0.
\dim R^1\psi_u(j_!{\cal F},f)-{\rm rank}\{\cal F}\cdot\dim R^1\psi_u (j_!\Lambda_U,f) &=\sum_{x\in (D,f)\times_S{\bar \eta}}{\rm dim\tot}_x(j_!{\cal F}_0|_{(U,f)\times_S{\bar \eta}})-{\rm dim\tot}_u(j_!{\cal F}_0|_{(U,f)\times_S{\bar s}})
a(j+1,k)=\begin{cases}\sum_{i=1}^{\infty}a(j,i)m(6i,i+k) \textrm{if}~j~\textrm{is~odd}, \cr \sum_{i=1}^{\infty}a(j,i)m(6i+2,i+k) \textrm{if}~j~\textrm{is~even}.\end{cases}
C:=c \left\{\left( \frac{\sigma}{\sigma-\rho}\right)^{p^+}\left(\frac{\sigma^{p^-}}{(k-h)^{p^+}}\right)^{\beta-\frac{1}{t}}+ \frac{2+k^{p^+_{B_\sigma}}}{\sigma^{\frac{n}{t}}}\left( \frac{\sigma^{p^-}}{(k-h)^{p^+}}\right)^{1+\beta-\frac{1}{t}}\right\}.
{{\left\langle T\left( A{{B}_{1}}\ldots{{B}_{n}}\right) \right\rangle}_{0}}=\sum\limits_{1\leq i\leq n}{{{\left\langle T\left( \overbracket[0.5pt]{AB_{1}\ldots B_{i}}\ldots{{B}_{n}}\right) \right\rangle}_{0}}}\,\,.
\begin{array}{l}F_{+}(z)=\left(\begin{array}{cc}-ix_{+}{\cdot\tilde{\sigma}}\,\tau_{+}2\bar{\theta}_{b}\,p_{+}2\theta^{a}\,\tau_{+}\delta^{a}_{~b}\,p_{+}\end{array}\right){}F_{-}(z)=\left(\begin{array}{cc}1 0 0 -\bar{{\cal E}}\end{array}\right)F_{+}(-z)^{t}\left(\begin{array}{cc}1 0 0 -{\cal E}\end{array}\right)\end{array}
s_8 =512 b^3 p^8-2688 b^3 p^7+5760 b^3 p^6+3456 b^2 p^7-6912 b^3 p^5-12672 b^2 p^6+5184 b^3 p^4 +16416 b^2 p^5+5184 b p^6-1944 b^3 p^3-11664 b^2 p^4-10368 b p^5+7776 b^2 p^3+1728 p^5-2916 b^2 p^2 &+11664 b p^3-11664 b p^2-7776 p^3+4374 b p+17496 p^2-17496 p+6561
\frac{1}{T}\int_{0}^{T}F(Z_N(s+i \tau, \alpha)) \,d \tau= \int_{\mathcal{T}^k}(F \circ \psi_N) \,d \mu_{\alpha,T}, \mathbf{E}\left[ F(Z_N(s, \mathbb{X}_\alpha)) \right]= \int_{\mathcal{T}^k}(F \circ \psi_N) \,d \nu_{\alpha}.
{S_L}[\varphi,\hat{g}]={{26-d}\over{48\pi}}\int{d^2}\xi\sqrt{\hat{g}}\left({1\over{2}}{\hat{g}^{ab}}{\partial_a}\varphi{\partial_b}\varphi+\hat{R}\varphi\right)+{{26-d}\over{24\pi}}\oint{d}\hat{s}{k_{\hat{g}}}\varphi.
{\widehat P}_{ab}(s)= [n+2-\min(a,b)]\;s_{a-1}\;s_{b -1}- \sum_{i=\max(1,a+b-n-2)}^{\min(a,b)}\,(a+ b - 2i)\;s_{i -1}\;s_{a+b - 1-i}\,,\forall\,a,b=1,\ldots,n+1\,,
R|_H =R^h-(\sin{2t})^2\pi_1-(\cos{2t})^2\pi_2+\sin{2t}\cos{2t}\pi_3 begin{align*}4pt] =\left\{\frac1{a^2+b^2}(a\cos{2t}+b\sin{2t})-(\sin{2t})^2\right\}\pi_1 begin{align*}4pt] -\left\{\frac1{a^2+b^2}(a\cos{2t}+b\sin{2t})+(\cos{2t})^2\right\}\pi_2 begin{align*}4pt] -\left\{\frac1{a^2+b^2}(b\cos{2t}-a\sin{2t})-\sin{2t}\cos{2t}\right\}\pi_3.
F(n_1,n_2)= \left( \frac{\partial}{\partial t_1}\right)^{n_1}\left( \frac{\partial}{\partial t_2}\right)^{n_2}\det\left( g_{i+j+1}(x,t_1,t_2) \right)_{i,j=0,\ldots,k-1}\Bigg|_{t_1=t_2=0}.
f_{m,r,s}(\sigma^{-1}\bar \mu_{n})=\sum_{\substack{p+q=n+1 2\leq p,q\leq r}}\mu_{p}\{\mu_{q}\}={}\sum_{\substack{p+q=n+1 2\leq p<q\leq r}}[\mu_{p},\mu_{q}] &+\mu_{\frac{n+1}{2}}\{\mu_{\frac{n+1}{2}}\}.
\frac{[x^{n-\Sigma_{\mathrm{k}}}]S_{\neq \mathrm{k}}(x)}{[x^n]S(x)}= \frac{[x^{n-\Sigma_{\mathrm{k}}}]S(x)}{[x^n]S(x)}+ \sum_{1\le u\le n}\frac{[x^{n-\Sigma_{\mathrm{k}}- u}]S(x)}{[x^n]S(x)}[x^u]T_1(x).
\sum_{i=1}^{3}(\beta_{i}-\beta_{i+1})(\beta_{i}-\delta_{1})^{3}(\beta_{i+1}-\delta_{1})^{3}- \sum_{i=1}^{3}(\beta_{i}-\beta_{i+1})(\beta_{i}-\delta_{2})^{3}(\beta_{i+1}-\delta_{2})^{3},
\left\|A_QA_R^{-1}\right\|\lesssim\begin{cases}\displaystyle\left[\frac{\ell(R)}{\ell(Q)}\right]^{d/p}Q\subset R, \displaystyle\left[\frac{\ell(Q)}{\ell(R)}\right]^{\widetilde d/p'}R\subset Q.\end{cases}
X_{F}=[\Pi_{X_1,X_2},F]=[X_1\wedge X_2 ,F]=[F,X_1\wedge X_2] &=[F,X_1]X_2 - [F,X_2]X_1 = [X_2, F]X_1 - [X_1,F]X_2 &=(\mathcal{L}_{X_2}F)X_1 - (\mathcal{L}_{X_1}F)X_2.
1+(r\!+\!1)(n\!+\!1)f^\sigma + \sum_{i=1}^{r+1}(r\!+\!1\!-\!i)\!\cdot\! f^{{}^i\sigma}- \sum_{i=1}^{r+1}(r\!+2\!-\!i)\!\cdot\! f^{\sigma^i},
\Delta ( \hat{u}_2(x,k)- \hat{u}_1(x,k))+\frac{k^2}{c_2^2(x)}\hat{u}_2(x,k)-\frac{k^2}{c_1^2(x)}\hat{u}_1(x,k)=-\frac{i k}{2\pi}\left( \frac{f_2(x)}{c_2^{2}(x)}-\frac{f_1(x)}{c_1^{2}(x)}\right),
\|V_t\|(\phi(\cdot,t))\Big|_{t=t_1}^{t_2}\leq \int_{t_1}^{t_2}\int_{U}(-\phi h(V_t,\cdot)+\nabla\phi)\cdot h(V_t,\cdot)+\frac{\partial\phi}{\partial t}\, d\| V_t\|dt.
\begin{array}{c}\int_{M}\langle P\varphi +\overline{P}\varphi ,d_{b}\varphi \rangle d\mu +\frac{1}{4}\int_{M}(P_{0}\varphi )\varphi d\mu =\frac{1}{2}iC_{n}\int_{\Sigma}\varphi \left( P_{n}\varphi -P_{\overline{n}}\varphi\right) d\Sigma_{p}.\end{array}
R_{\ell,1}= I(V,X;Z|A)+I(X;A,V,Y)-I(V,X;Y|A) &R_{\ell,2}= I(V,X;Z|A,U)+I(X;A,V,Y) &\quad\qquad-I(V,X;Y|A,U) &R_{\ell,3}=I(X;A,U,Z).
J_{\mu \nu}= 2 i \big \{e_{[ \mu}{}^{\alpha \dot{\alpha}}e_{\nu ]}{}^{\beta}{}_{\dot{\alpha}}\widehat A_{\alpha}^{( 1 )}\star \widehat A_{\beta}^{( 1 )}+ \mathrm{h . c .}\big \}_{z = 0}+ L_{\mu \nu}+ \mathrm{W - t e r m s},
\delta_{ijkl}a_0 = \beta_0(i,j,k,l) - \beta_0(i+1,j,k,l) &{}+ \beta_0(j,i,k,l) - \beta_0(j+1,i,k,l) &\qquad{}+ \alpha_0(i,j,k,l) - \alpha_0(i,j,k+1,l).
Z_r-Z_{t_i}= (D_r^B \xi -D_{t_i}^B \xi) + \int_r^{t_i}D_r^B Y_s{{W}}(ds,B_s) +\int_r^{t_i}Y_s \nabla_x{{W}}(ds, B_s) -\int_r^{t_i}D_r^B Z_s dB_s,\ 0\leq t_i\leq r\leq T =\tilde Z_1+\tilde Z_2+\tilde Z_3+\tilde Z_4\,.
{\cal G}_{H H}= | z |^{4}\langle e^{\varepsilon X^{0}( z )}X^{4}( z ) \partial_{\tau}X^{5}( z ) e^{\varepsilon X^{0}( 0 )}X^{4}( 0 ) \partial_{\tau}X^{5}( 0 ) \rangle \sim e^{4 \varepsilon^{2}\mathrm{l n}| L / a |^{2}}\mathrm{l n}| L / a |^{2}
H={{\phi}_{0}}\left( h\left( t \right)\frac{\partial}{\partial t}+r\frac{\partial}{\partial r}\right)+\left({{c}_{1}}\sin \varphi -{{c}_{2}}\cos \varphi \right)\frac{\partial}{\partial \vartheta}+\left( \cot \vartheta \left({{c}_{1}}\cos \varphi +{{c}_{2}}\sin \varphi \right)+{{c}_{3}}\right)\frac{\partial}{\partial \varphi}.
\frac{d^k\tilde{F}}{dv^k}=P_{n-k+2}-\frac{v^{n+2-k}}{6}\sum_{l=1}^{n+3-k}\frac{(-1)^l(n+1-l)(n+2-l)}{(l-1)!(n+3-k-l)!}\hat{I}_{n,l-1}.
\lim_{n \to +\infty}\frac{a_n^\alpha}{n L^*(a_n)}\log \P\left[ \frac{S_j}{a_n}\in \left[f\left(j/n\right), g\left(j/n\right)\right], 0 \leq j \leq n \right] = - C_* \int_0^1 \frac{ds}{(g(s)-f(s))^\alpha},
\rho ( E_{i}^{( n )}) = \rho^{2}( E_{i}^{( n - 1 )}) - \frac{2 \cos \alpha \sin ( n - 1 ) \alpha}{\sin n \alpha}\rho^{2}( E_{i}^{( n - 1 )}) \rho ( E_{j}) \rho^{2}( E_{i}^{( n - 1 )})
|n_{k_{1}},n_{k_{2}}...\rangle=\frac{(b_{k_{1}}^{\dagger})^{n_{k_{1}}}}{\sqrt{n_{k_{1}}!}}\frac{(b_{k_{2}}^{\dagger})^{n_{k_{2}}}}{\sqrt{n_{k_{2}}!}}...|0_{k_{1}},0_{k_{2}},...\rangle
[\hat{\cal H}_{0}^{\pm}(n), \hat{\cal H}_{0}^{\pm}(m)]_{-}=\pm \frac{2\pi}{\rm L}(n-m) \hat{\cal H}_{0}^{\pm}(n+m)\pm \frac{1}{3}(\frac{2\pi}{\rm L})^2 n(n^2-1) \delta_{n,-m},
\mu=\mu(\phi,{\mathfrak M})=\sum_{\{n_1,n_2,\ldots,n_\nu\}\in{\mathfrak M}}\frac{1}{C_{n_1}C_{n_2}\ldots C_{n_\nu}}\left|{\textbf E}[\phi_{n_1}\phi_{n_2}\ldots \phi_{n_\nu}]\right|.
- \int_{0}^{\sigma}ds\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\frac{s}{(4\pi)^{\frac{D-1}{2}}}(\omega^{2}+s^{2})^{\frac{D-3}{2}}\Gamma(-\frac{D-3}{2}) \mbox{tr}\mbox{\boldmath$1$}.
i_{\alpha\beta\gamma}(t)=Ti_{abc}(t)=\sqrt{\frac{2}{3}}[\begin{matrix}1&-\frac{1}{2}&-\frac{1}{2}\\ 0&\frac{\sqrt{3}}{2}&-\frac{\sqrt{3}}{2}\\ \frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\ \end{matrix}][\begin{matrix}i_{a}(t)\\i_{b}(t)\\i_{c}(t)\end{matrix}],
X_n(x) =\begin{cases}\ \ \frac{T_{N+1}(x)-T_{N-1}(x)}{2(x^2-1)}\ (n>0) &-\frac{T_{N+1}(x)-T_{N-1}(x)}{2(x^2-1)}\ (n<0). \end{cases}X'_n(x) =T_{N}(x).
d_3 = -{q (p^2 + q^2)\over 2 \pi^3}, \: \: d_4 ={(p^2+q^2)(1-p^2-5q^2) \over 8 \pi^4}, \: \: d_5 ={q(p^2+q^2)(5-3p^2-7 q^2) \over 8 \pi^5}.
\int_0^{+\infty}\!\!\!iu_t\bar{u}_xdx =-\int_0^{+\infty}\!\!\!u_t\bar{u}_xdx &=-\int_0^{+\infty}\frac{d}{dt}(u\bar{u}_x)dx+\int_0^{+\infty}\!\!\!u\bar{u}_{xt}dx &=-\int_0^{+\infty}\frac{d}{dt}(u\bar{u}_x)dx-\int_0^{+\infty}\!\!\!u_x\bar{u}_tdx-(u(0,t)\bar{u}_t(0,t)).
E_1'\times_{\mathbb{P}^1}E_2' \colon\begin{cases}y_1^2= (x^2-1)\left(x^2-\left(\frac{1-\mu\sqrt{\lambda}}{1+\mu\sqrt{\lambda}}\right)^2\right) y_2^2= (x^2-1)\left(x^2-\left(\frac{1-\sqrt{\lambda}}{1+\sqrt{\lambda}}\right)^2\right)\end{cases}.
\eta_n(r) =\sum_{\substack{k\le nt b_{nk}^{-1}J_{nk}\le r}}b_{nk}^{-1}J_{nk}\le\sum_{\substack{k\le nt b_{nk}^{-1}J_{nk}\le r}}\left(b_{nk}^{-1}J_{nk}\right)^{\beta(kn^{-1})/\beta^*}&=a_n^{-1}\sum_{\substack{k\le nt b_{nk}^{-1}J_{nk}\le r}}J_{nk}^{\beta(kn^{-1})/\beta^*},
f_{\mathrm{NML}}({\bf x}^n)\buildrel \rm def \over =\frac{f({\bf x}^n ; \hat{\mu}({\bf x}^n),\hat{\Sigma}({\bf x}^n))}{\int_{Y(R, \epsilon_1,\epsilon_2)}f({\bf y}^n ; \hat{\mu}({\bf y}^n),\hat{\Sigma}({\bf y}^n)){\rm d}{\bf y}^n}
\left|\int_{[0,1]^d}f(\mathbf{x})~d\mathbf{x}- \frac{1}{n}\sum_{i=1}^n f(\mathbf{x}_i) \right| \leq \left(\textup{Var}_{HK}f \right) D_n^* (\mathbf{x}_1, \dots, \mathbf{x}_n).
\sum_{\mathcal{S}_1}\frac{\mathbb{E}_{Z=1}\left(|\Delta_jf(W)-\tilde{\Delta}_jf(W)|\mathbf{1}_{(A,B,j,k) \in \mathcal{S}_{1,2}}\right)}{\binom{2n}{|A|}(2n-|A|)\binom{2n}{|B|}(2n-|B|)}\leq C n^{1 + \alpha}.
\frac{P(a_j\>\big |\>{\cal C})}{P(a_i\>\big |\>{\cal C})}\>=\>\frac{\frac{P(a_j)}{P({\cal C})}}{\frac{P(a_i)}{P({\cal C})}}\>=\>\frac{P(a_j)}{P(a_i)}\>=\>\frac{p_j}{p_i}\>=\>r_{ij}.
=-i\frac{3\hbar c}{2}\int d^3\vec{r}[\epsilon_0 (\varphi^{\dagger}\varphi+\chi^{\dagger}\chi)+\varphi^{\dagger}(\vec{\sigma}\cdot\vec{\epsilon})\chi+\chi^{\dagger}(\vec{\sigma}\cdot\vec{\epsilon})\varphi]\,{.}
{{P}_{}}=\int\limits_{\mathbf{I}}{{{f}_{\mathbf{I}}}}\left( \mathbf{I}\right)Q\left( \frac{1}{MN}\sqrt{\frac{{\bar{\gamma}}}{2}\sum\limits_{n=1}^{N}{{{\left( \sum\limits_{m=1}^{M}{{{I}_{mn}}}\right)}^{2}}}}\right)d\mathbf{I}
\|(B+L_n)^{-1/2}A(B+L_n)^{-1/2}\|_\infty = \|(B+L_n)^{-1/2}B^{1/2}WB^{1/2}(B+L_n)^{-1/2}\|_\infty &\le c\|(B+L_n)^{-1/2}B(B+L_n)^{-1/2}\|_\infty &=c\|B^{1/2}(B+L_n)^{-1}B^{1/2}\|_\infty\le c,
\sum_{n= 1}^\infty n^2\Big( |C_n|^2+|d_nD_{n}|^2 \Big) \asymp \|z_{1}^{0}\|^2_{H^1_0}+\|z_{1}^{1}\|^2_{L^2}+\|z_{2}^{0}\|^2_{H^1_0}+\|z_{2}^{1}\|_{L^2}^2 \,.
A_{\mu}dx^\mu = \frac{1}{2e}\left( \begin{array}{cccc}3\cos\theta d\phi \omega_1 \Theta 0 0 \omega_1 \bar \Theta \cos\theta d\phi \omega_2 \Theta 0 0 \omega_2 \bar \Theta -\cos\theta d\phi \omega_3 \Theta 0 0 \omega_3 \bar \Theta -3\cos\theta d\phi\ \end{array}\right)\ ,
\phi_{SS}/2=\tan^{-1}\left( u\frac{\sinh (8m\gamma x)}{\cosh(8m\gamma ut)}\right) ,\,\,\,\phi_{S\overline{S}}/2=\tan^{-1}\left( \frac{1}{u}\frac{\sinh (8m\gamma ut)}{\cosh (8m\gamma x)}\right) ,
\frac{1}{m!}\sum_{\sigma \in{\rm Sym}(I)}\prod_{j\in I}\cos(2\pi y_{\sigma^{-1}(j)}\xi_j)=\frac{1}{{m \choose l}}\sum_{\substack{J\subseteq I |J|=l}}\prod_{j\in J}\cos(2\pi \xi_j).
R_{T}=-\Phi_{\left\Vert \eta_{1}\right\Vert ,\left\Vert x\right\Vert}+\Phi_{\left\Vert \eta_{1}\right\Vert ,\left\Vert y\right\Vert}+\Phi_{\left\Vert \eta_{2}\right\Vert ,\left\Vert y\right\Vert}+\Phi_{\left\Vert \eta_{2}\right\Vert ,\left\Vert z\right\Vert}+\Phi_{\left\Vert \eta_{3}\right\Vert ,\left\Vert x\right\Vert}+\Phi_{\left\Vert \eta_{3}\right\Vert ,\left\Vert z\right\Vert};
\langle\Psi_{n}|\Psi_{n}\rangle_{\nu}=\sum_{i_{1},...i_{n},j_{1},...j_{n}}a_{i_{1},...,i_{n}}^{*}a_{j_{1},...,j_{n}}\langle\psi_{i_{1}}|\psi_{j_{1}}\rangle\cdot\cdot\cdot\langle\psi_{i_{n}}|\psi_{j_{n}}\rangle
\| f(G)- f(\widehat G) \|_{\infty}\leq \min_{r \in \{1, \dots, d \}}L^{-1}\left( B\left(\lambda_1 \right)+ \sqrt{4r B\left(\lambda_1 \right)^2 + 2\sum_{i=r+1}^{d}\lambda_i^2}\; \right),
\mathcal{F}[p]:=\int_\Omega\int_{\mathbb{S}_+^1}\frac{\gamma-1}{2\gamma-1}c_0^{\frac{2}{\gamma-1}}|\theta\cdot\nabla p|^{\frac{2\gamma-1}{\gamma-1}}\, d\theta \, dx-\int_\Omega pS\, dx
\frac{2 \, J ( r )^{2}K ( r )^{2}}{\mathcal{L}( r )^{2}}= \frac{\left[ M ( r )^{4}\mathcal{L}( r )^{2}J^{\prime}( r ) \right]^{\prime}}{M ( r )^{4}\mathcal{L}( r )^{2}}J ( r ) - \alpha J ( r )^{2}\left( J ( r )^{2}- 1 \right) \, .
\int\limits_{\mathbb{R}^{N}}\left\vert v\right\vert^{q}\frac{dx}{\left\vert x\right\vert^{s}}=d^{1+\frac{1}{2}q}\int\limits_{\mathbb{R}^{N}}\frac{\left\vert u\right\vert^{q}}{\left\vert x\right\vert^{N+sd-Nd}}dx.
\sum_{j=t+1}^{2t-l}d_{i,j}(x_{j,1}^3+x_{j,2}^3-x_{j,3}^3-x_{j,4}^3) = 0 (1 \le i \le t-l), d_{i,i}(x_{i,1}^3 - x_{i,2}^3) + \sum_{j=t+1}^{2t-l}d_{i,j}(x_{j,1}^3+x_{j,2}^3-x_{j,3}^3-x_{j,4}^3) = 0 (t-l+1 \le i \le t).
{\cal L}= \frac{1}{1 6 \pi \alpha^{\prime}}\sqrt{- g}\e^{\mathrm{~ - \p h i ~}}\Biggl \{R + \nabla \phi \cdot \nabla \phi -{\frac{1}{3}}H^{2}- \frac{1}{4}\mathrm{T r}F^{2}+ \cdots \Biggr \}
H_{1\pm}=\left( \begin{array}{ll}1 \mp \sqrt{2(1-72s)}\mp \sqrt{2(1-72s)}0\end{array}\right) H_{2\pm}=\left( \begin{array}{ll}\pm 6\sqrt{2s}0 0 2\left( 1\mp 6\sqrt{2s}\right)\end{array}\right)