harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
R_{\nu \mu}^{\pm}= \delta_{\mu}^{\pm}\partial_{\nu}H^{\pm}- \delta_{\nu}^{\pm}\partial_{\mu}H^{\pm}+ \delta_{\mu}^{\pm}[ \omega_{\nu}, H^{\pm}] - \delta_{\nu}^{\pm}[ \omega_{\mu}, H^{\pm}] ,
P ( u , x ) = \lim_{N \rightarrow \infty}\left. P_{2 n}( \frac{x}{N}) \right|_{n = N t / 2}= \cos u x = \Gamma ( \frac 1 2 ) \frac{J_{- \frac 1 2}\left( u x \right)}{\left( u x / 2 \right)^{- \frac 1 2}}.
D(\Lambda)=G_{\mathbb{R}}(\Lambda)\cdot B/B\simeq G_{\mathbb{R}}(\Lambda)/G_{\mathbb{R}}(\Lambda)\cap V, &S(\Lambda)\cap (N_+B/B)=(G_{\mathbb{C}}(\Lambda)\cap N_+)\cdot B/B\simeq G_{\mathbb{C}}(\Lambda)\cap N_+.
K^{(0)}_\alpha(x) = \frac{\delta}{\delta \xi^\alpha(x)}\int{\rm d}^2 y{\cal H}_{0}(y) = \left( \begin{array}{c}-2n^0(x)n^{a}(x) -\frac{1}{f}\vec\nabla^{2}n^a(x) f\pi^a(x) -n^a(x)n^a(x) + 1 \end{array}\right).
\lambda_L^{(3)}= \bar\lambda_L \biggr[1+ \bar\lambda_L \frac{2}{\pi}\sum_{i=0}^L\frac{k^{2(L-i)}\Lambda^{2i+1}}{2i+1}+\bar\lambda_L^2 \biggr( \frac{2}{\pi}\sum_{i=0}^L\frac{k^{2(L-i)}\Lambda^{2i+1}}{2i+1}\biggr)^2 \biggr]
I_2 \; := \; \frac{1}{N}\left[\ln (2 L) \; + \; \ln \left( \frac{e}{2}\sqrt{\frac{N}{\pi}}\right)\; + \; \mbox{K}_0 \left( 2L e \sqrt{\frac{N}{\pi}}\right) \; + \;\gamma \right] \; .
p_{kx\rho}=\,\frac{\partial \tilde{x}^\mu_k}{\partial x^\rho_k}\,\tilde{p}_{kx\mu}+\,\frac{\partial^2 \,\tilde{x}^\mu_k}{\partial x^\rho_k\,\partial x^\sigma_k}\,\tilde{p}_{ky\rho}(x)\,y^\rho_k, & p_{ky\rho}=\,\frac{\partial \tilde{x}^\mu_k}{\partial x^\rho_k}\,\tilde{p}_{ky\mu}.
Q_{S^n}^6= \tfrac{n(n^4-20 n^2+64)}{32}, P_{S^n}^6= -\Delta_{S^n}^3-\tfrac{-3n^2+6n+32}{4}\Delta_{S^n}^2-\tfrac{3n^4-12n^3-52n^2+128n+192}{16}\Delta_{S^n}+\tfrac{n-6}{2}Q_{S^n}^6 = \Big(-\Delta_{S^n}+\tfrac{(n-6)(n+4)}{4}\Big)\Big(-\Delta_{S^n}+\tfrac{(n-4)(n+2)}{4}\Big)\Big(-\Delta_{S^n}+\tfrac{n(n-2)}{4}\Big).
(\delta\omega_{ij\mu})e_{k}^{\mu}=-\frac{1}{2}\{(\nabla_{j}\delta e_{ik}-\nabla_{k}\delta e_{ij})+(\nabla_{k}\delta e_{ji}-\nabla_{i}\delta e_{jk})-(\nabla_{i}\delta e_{kj}-\nabla_{j}\delta e_{ki})\},
\lambda^{2}\left(\Phi_{2}\left(\frac{\left\Vert x\right\Vert +\mu}{c_{2}}\right)-\Phi_{2}\left(\frac{\mu}{c_{1}}\right)\right) =\Phi_{2}\left(\left\Vert x\right\Vert +\mu\right)-\Phi_{2}\left(\mu\right).
(B^+,B^-)(t) := \begin{cases}( B_1(\tau_1 + t) - B_1(\tau_1), B_2(\tau_1 + t) - B_1(\tau_1)) B_1(\tau_1 + 1) > B_2(\tau_1 + t) ( B_2(\tau_1 + t) - B_1(\tau_1), B_1(\tau_1 + t) - B_1(\tau_1)) B_2(\tau_1 + 1) > B_1(\tau_1 + t).\end{cases}
\det\left[\begin{array}{cccccc}f(x) f(x+h) \cdots f(x+nh) f(x+h) f(x+2h) \cdots f(x+(n+1)h) \vdots \vdots \ \ddots \vdots f(x+nh) f(x+(n+1)h) \cdots f(x+2nh) \end{array}\right] =0 x,h\in\mathbb{R}^d.
s(F^{[k]}) = \frac{1}{k}\left(\sum_{i=0}^{2k}\sum_{j=0}^{\min\{i,2\}}(-1)^{i-j}\binom{r-1+i}{r-1+j}q_1^{(i-j)}(s_j(F))\right)(s(F^{[k-1]})).
r(\widetilde{X}^{m}\otimes\widetilde{X}^{n})=r^{ij}(X_i\otimes X_j)(\widetilde{X}^{m}\otimes\widetilde{X}^{n})=r^{ij}\widetilde{X}^{m}(X_i)\widetilde{X}^{n}(X_j)=r^{mn},
\bigg(\int_{0}^{t}p_{n_{k_l}}(s,X)W_s\bigg)(\omega) = \sum_{m=0}^{j_{n_{k_l}}-1}\Phi_m^{n_{k_l}}(X(\omega)) J^{-1}\big(\bar{W}_{t^{n_{k_l}}_{m+1}\wedge t}(w)-\bar{W}_{t^{n_{k_l}}_{m}\wedge t}(w)\big).
\begin{array}{lll}\alpha_0^{(1)}=H_{q_1}+H_{p_1}-E_{1,2,3,4}, \alpha_1^{(1)}=H_{p_1}-E_{5,6}, \alpha_2^{(1)}=H_{q_1}-E_{7,8}, \alpha_0^{(2)}=H_{p_2}+H_{q_2}-E_{9,10,11,12}, \alpha_1^{(2)}=H_{p_2}-E_{13,14}, \alpha_2^{(2)}=H_{q_2}-E_{15,16}\end{array}
y\left[ i \right] = \sum\limits_{l' = 1}^{L'}{\sum\limits_{l \in{{\bar{\cal L}}_{l'}}}{{\bf{h}}_l^H{\bf{\bar H}}_{l'}^\bot{\bf{\bar X}}_{l'}{\bf{d}}\left[{i -{n_l}-{\kappa_{l'}}}\right]}}+ z\left[ i \right],
d s^{2}= \phi_{a b}d y^{a}d y^{b}+ ( \gamma_{\mu \nu}+ \phi_{a b}A_{\mu}^{\a}A_{\nu}^{\b}) d x^{\mu}d x^{\nu}+ 2 \phi_{a b}A_{\mu}^{\b}d x^{\mu}d y^{a},
\theta^1 :=du-u_tdt-u_xdx-u_ydy-u_zdz, \quad\theta^2:=du_t-u_{tt}dt-u_{tx}dx-u_{ty}dy-u_{tz}dz, \theta^3 :=du_x-u_{tx}dt-u_{xx}dx-u_{xy}dy-u_{xz}dz,\theta^4:=du_y-u_{ty}dt-u_{xy}d x-u_{yy}dy-u_{yz}dz \theta^5 :=du_z-u_{tz}dt-u_{xz}d x-u_{yz}dy-Fdz.
\mathbf{Y}_{nm}=\begin{cases}\mathbf{I}_Q+\mathcal{E}_{\mathrm{s}}\mathbf{C}_n\mathbf{C}_n^{\dag},\ \n=m\in\mathbb{N}, \mathcal{E}_{\mathrm{s}}\mathbf{C}_n\mathbf{C}_m^{\dag},\ \n\neq m\in\mathbb{N}.\end{cases}
\begin{array}{ll}\frac{\partial w^1}{\partial t}-\sum_{ij}a^{w^1}_{ij}\frac{\partial w^1}{\partial x_i\partial x_j}+\sum_i b^{w^1}_i(x)\frac{\partial w^1}{\partial x_i}=0,~ \frac{\partial w^2}{\partial t}-\sum_{ij}a^{w^2}_{ij}\frac{\partial w^2}{\partial x_i\partial x_j}+\sum_ib^{w^2}_i(x)\frac{\partial w^2}{\partial x_i}=0,\end{array}
\Delta(K)=K \otimes K, \Delta(E)=E \otimes K^t + K^s \otimes E, \Delta(F)=F \otimes K^{-s}+ K^{-t}\otimes F, \varepsilon(K) = 1, \varepsilon(E) = \varepsilon(F) = 0, S(K) = K^{-1}, S(E) = -K^{-s}E K^{-t}, S(F) = -K^t F K^s,
\begin{array}{c}{\displaystyle{\sum_{\nu = 3 / 2}^{\infty}\nu^{- ( s - k - 1 )}}\ \Re \left\{\displaystyle{\frac{- e^{\displaystyle{- \imath \frac{\pi}{2}( s + 1 )}}}{\pi}}\displaystyle{\int_{- \imath z}^{\infty - \imath z}}t^{- s}\displaystyle{D_{T E}^{( k )}( t )}\d t \right\},}\\ \end{array}
y^{(m)}_{r_1, \ldots, r_k}:= \left(\prod_{i = 1}^k \mu(r_i)g(r_i)\right) \sum_{\substack{d_1, \ldots, d_k r_i \mid d_i \forall i d_m = 1}}\frac{\lambda_{d_1, \ldots, d_k}}{\prod_{i = 1}^k \Phi(d_i)}.
\check{R}_i(u)= \sum_{klmp}S^{km}_{lp}(u) \cdot I^{(1)}\otimes I^{(2)}\otimes \cdots \otimes E^{(i)}_{pk}\otimes E^{(i+1)}_{ml}\otimes I^{(i+2)}\otimes \cdots \otimes I^{(n)}
\sigma' = \left( \underbrace{(e_4 e_2)^{-1}e_1}_{\partial f_1}e_1^{-1}\rhd f_4 \right) f_1 f_3 f_2 = f_1 (e_1^{-1}\rhd f_4 ) f_3 f_2 = f_1 \sigma f_1^{-1}= \sigma,
\sum_{z \in P_\nu}L_m\chi_y(z) \overline{L_m\chi_{y'}(z)}= q^{2|S_\mu(m-1)|}\sum\chi\left(\sum_{s \in S_\mu}\sum_{t \in T_\nu}\left(Y_{s,t}- Y'_{s,t}\right)Z_{s,t}\right),
\sum_{k \geq 2}\frac{I(q^{-k})}{k}\,(w^k-1) = \sum_{k \geq 2}\sum_{l=1}^k \frac{I(q^{-k})}{k}\,\binom{k}{l}\,(w-1)^l &=\left(\sum_{k\geq 2}I(q^{-k})\right)(w-1) + \sum_{l=2}^\infty \left(\sum_{k \geq l}\binom{k}{l}\,\frac{I(q^{-k})}{k}\right)(w-1)^l.
\sum_{m=1}^{\infty}|a_{m+k}|^{2}\lesssim \sum_{m=1}^{\infty}\left(\sum_{j=1}^{\infty}|\tilde{c}_{j+k}|^{2}\lambda_{-}^{|m-j|}\right)= \sum_{j=1}^{\infty}|\tilde{c}_{j+k}|^{2}\sum_{m=1}^{\infty}\lambda_{-}^{|m-j|}\lesssim \sum_{j=1}^{\infty}|\tilde{c}_{j+k}|^{2}< \infty,
\int_X\alpha\wedge\overline{*}\omega =\int_{\partial B_{2k-1}}\alpha\wedge\overline{*}(\gamma^{\partial B_{2k-2}}(\nu^{\partial B_{2k-1}})Q^{\partial B_{2k-2}}\zeta^{\partial B_{2k-1}}\omega^{2k-1})= &=\int_{\partial B_{2k-1}}i_{\partial B_1}^*(\alpha),
s_{k}^{2}=1+1-2\cdot1\cdot1\cos(\theta_{k}-\theta_{k-1})=2\left[1-\cos(\theta_{k}-\theta_{k-1})\right] & =4\,\frac{1-\cos(\theta_{k}-\theta_{k-1})}{2}=4\sin^{2}\left(\dfrac{\theta_{k}-\theta_{k-1}}{2}\right) .
\gamma^*(G) =\sum_{i}g(v_i), &=\sum_{1\le i\le r}g(v_i)+\sum_{r\le i\le 2r-1}g(v_i)-g(v_r) &=\sum_{v_i\in N^-[v_r]}g(v_i)+\sum_{v_i\in N^-[v_{2r-1}]}g(v_i)-g(v_r) &> 2-\varepsilon.
\int_{\partial D_r}e^{-\frac{i}{2}\theta_{\mathbf e}}(i\nabla+A_{\mathbf e})\Psi_k\cdot\nu\, \psi_k\, ds=i\sqrt\pi r^{k/2+1}\xi'(r)=i\sqrt\pi r^{k/2+1}\bigg(k\sqrt{\pi}r^{k/2-1}-\frac k2\,\frac{\xi(r)}{r}\bigg),
\int_{\mathbb{R}^{n-1}}fd\nu_{\tau,T}=\int_{\mathbb{R}^{nd}}f\left(Q\left(v_{1}\right)-Q\left(v_{2}\right),\dots,Q\left(v_{n-1}\right)-Q\left(v_{n}\right)\right)w_{\pm\tau,T}\left(v\right)dv.
\|\hat{V}\hat{O}- V\|_{\mathrm{F}}= 2^{1/2}\|\sin \Theta(\hat{V}, V)\|_F = (2d)^{1/2}\leq (2d)^{1/2}(1+\epsilon) = \frac{2^{3/2}d^{1/2}\|\hat{\Sigma}- \Sigma\|_{\mathrm{op}}}{\lambda_d - \lambda_{d+1}}.
V[g] =\omega \cdot V[J] =\omega \cdot \tilde{G}(V)\cdot \omega = g \cdot J \cdot \tilde{G}(V)\cdot g \cdot J = - g \cdot J \cdot (G(V) + \bar{G}(V)) \cdot J \cdot g = - i^2 g \cdot G(V) \cdot g - ( -i)^2 g \cdot \bar{G}(V) \cdot g = g \cdot \tilde{G}(V) \cdot g.
\begin{array}{rcl}G = 4 \partial C +6 F^{1\, m}B_{m}\, , & & H_{m}= 3\partial B_{m}+3\epsilon_{mnp}F^{1\, n}A^{2\, p}\, , & & F^{2\, m}= 2\partial A^{2\, m}\, , \end{array}
s = \langle \mbox{\boldmath $n$}\cdot \mbox{\boldmath $j$}_{0}\rangle = -\frac{1}{2}\left.\frac{\partial \mbox{$\cal L$}_{\rm eff}}{\partial V_{0}^{3}}\right|_{V_\mu^3 = 0}=\frac{eB}{4 \pi}.
\sum_{k\in R_{k,j}^c,k<j, k \in D}P(A'_{k,j})=\sum_{k \in R_{k,j}^c,k<j,k \in D}V(\frac{e^k\alpha(t_k)}{e^j\alpha(t_j)-e^k\alpha(t_k)}) &\le\sum_{k=1}^{\infty}V(\frac{2^k}{e^k-2^k})\le C\sum_{k=1}^{\infty}{(\frac{e}{2})}^{-k}\le C',
g^0(\lambda)\big|A_N\bigl(e^{i\lambda}\bigr)\bigl(1-e^{i\lambda\mu}\bigr)^{n}f^0(\lambda)-\lambda^{2n}C^{\mu,0}_{N}\bigl(e^{i\lambda}\bigr)\big| &=\alpha_2\big|1-e^{i\lambda\mu}\big|^{n}\bigl(f^0(\lambda)+\lambda^{2n}g^0(\lambda)\bigr),
\nu^{S_{k-1}}_j = \nu_{\pi_{k-2}}+ \frac{\displaystyle c_j - \nu_{\pi_1}\, w^{S_1}_j - \sum_{l=2}^{k-2}\left(\nu_{\pi_{l}}- \nu_{\pi_{l-1}}\right) \, w^{S_{l}}_j}{\displaystyle w^{S_{k-1}}_j}, j \in S_{k-1},
h_1=0,h_3=h_{\overline{3}}=\frac{2}{9},h_8=\frac{1}{2},h_6=h_{\overline{6}}=\frac{5}{9},h_{10}=h_{\overline{10}}=1,h_{15}=h_{\overline{15}}=\frac{8}{9}\ \ .
\mathbf P_{so}^{\,i}\!=\!1\!-\!\!\!\!\!\sum_{\mathbf D_u\in 2^{\mathcal D}}\!\!\!\underbrace{\mathbb E\!\!\left[\frac{e^{-\frac{(2^{\frac{nr_s}{\beta}}-1) (I_B+1)}{\gamma_{A_i,B}}}}{\frac{\gamma_{A_i,E}}{\gamma_{A_i,B}}\frac{I_B+1}{I_E+1}2^{\frac{nr_s}{\beta}}\!+\!1}\Bigg|\mathbf D_u\right]\!\mathbb P\!\!\left(\mathbf D_u\right)}_{T},
\chi(M^{-1})=\chi(\mathcal{O}_{Y^{\prime}})+\frac{1}{2}((C^{\prime})^2+K_{Y^{\prime}}\cdot C^{\prime})= \chi(\mathcal{O}_{Y^{\prime}})+\frac{1}{2}C^{\prime}\cdot (K_{Y^{\prime}}+C^{\prime})=\chi(\mathcal{O}_{Y^{\prime}})+\frac{1}{4}K_{X^{\prime}}\cdot \Delta^{\prime}
2\pi \int_0^\infty e^{-2\pi t\rho}J_k(2\pi r\rho)\,d\rho= \int_0^\infty e^{-t\rho}J_k(r\rho)\,d\rho=\frac{(\sqrt{t^2+r^2}-t)^k}{r^k\sqrt{t^2+r^2}}
\begin{array}{c}{{d s^{2}= \frac{V ( r ) [ d \chi - ( 2 n \cos \theta - a \sin^{2}\theta ) d \phi ]^{2}+ \sin^{2}\theta \lbrack a d \chi - ( r^{2}- n^{2}- a^{2}) d \phi ]^{2}}{[ r^{2}- ( n + a \cos \theta )^{2}]}}}\\{{+ [ r^{2}- ( n + a \cos \theta )^{2}] ( \frac{d r^{2}}{V ( r )}+ d \theta^{2})}}\\ \end{array}
\theta_{\rho^\varepsilon}(t,z_l) = F(z_l) + \rho^\varepsilon(t,z_l)\nu_\Sigma(z_l) = (0,0) + \rho^\varepsilon(t,z_l)(-1,0) = \big( -\rho^\varepsilon(t,z_l), 0 \big), \theta_{\rho^\varepsilon}(t,z_r) = F(z_r) + \rho^\varepsilon(t,z_r)\nu_\Sigma(z_r) = (0,0) + \rho^\varepsilon(t,z_r)(+1,0) = \big( \rho^\varepsilon(t,z_r), 0 \big)
\int_{S}\max\left( \vec p \cdot \hat n(\vec u),0\right) e^{-\beta H_N(\vec u, \vec p)}d\tilde{\sigma}(\vec p, \vec u) \sim \frac{2^{N-2}N^{N}(N-2)! S_{2N-3}}{\beta^{N}\prod_{j=2}^N\sqrt{\lambda^s_j}}e^{-\beta E_N^s}
E ( x ) = c o n s t \, e x p ( - \frac{i m \tau}{2}) f ( r ) \left\| \begin{array}{c}{{- e x p ( i ( \varphi - \tau ) ) r_{\bot}/ \sqrt{R^{2}+ r^{2}}}}\\{{i e x p ( i ( \varphi - \tau ) ) r_{\bot}/ \sqrt{R^{2}+ r^{2}}}}\\{{1}}\\{{i}}\\{{0}}\\ \end{array}\right\|
\begin{array}{c}\phi^{\kappa}(\lambda)=q^{\kappa}(\lambda)+\pi^{\nu}(\lambda) \phi^{\nu}(\lambda)=q^{\nu}(\lambda)+\pi^{\kappa}(\lambda) \left[ \pi^{\kappa}(\lambda),q^{\kappa}(\mu) \right] =\ln h(\mu,\lambda) \left[ \pi^{\nu}(\lambda),q^{\nu}(\mu) \right]=\ln h(\lambda,\mu)\end{array}
\textrm{sign}\left(\sin\left(\frac{\phi}{2}\pm A\right)\right) =\textrm{sign}\left(\sin\left(\pi-B\pm A\right)\right) &=\textrm{sign}\left(\sin\left(B\pm A\right)\right)=\textrm{sign}\left(\cos (A)\right) &=\textrm{sign}\left((-1)^k\sin(\phi)\right),
\begin{array}[c]{l}H(t,x,v,p,A,u) =\frac{1}{2}\mathrm{tr}[\sigma\sigma^{\intercal}(t,x,v,V(t,x,v,p,u),u)A]+p^{\intercal}b(t,x,v,V(t,x,v,p,u),u) \ \ +g(t,x,v,V(t,x,v,p,u),u), (t,x,v,p,A,u)\in\lbrack0,T]\times\mathbb{R}^{n}\times\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{S}^{n}\times U,\end{array}
(\tau_p^t)^2 = (\sigma_p^t)^2 + \sum_{u\neq q}(\sigma_u^{\theta(t)})^2 + \sigma_W^2,(\sigma_p^t)^2 = \frac{1}{\delta_p}\mathbb{E}\left[\left(f_t(\tau_p^{t-1}Z,X)\right)^2 \right],
\int_{K_r}D^2_{11}\phi dz = \int_{z\in H\cap B_{r/2}}\int_{-r}^{r}D^2_{11}\phi (z+ t \eta_1) dt dz \ge \int_{z\in H\cap B_{r/2}}\frac{1}{r^2}(r - 2|z| ) M_r dz & = C_0 r^{d-2}M_r
{\bf \nabla}W = i W \int_{0}^{\beta}d t^{\prime}[ \nabla A_{0}( t^{\prime},{\bf x}) - \partial_{t^{\prime}}{\bf A}( t^{\prime},{\bf x}) ] = - i W \int_{0}^{\beta}~{\bf E}( t^{\prime},{\bf x}) ~ d t^{\prime}, \; \; E_{i}\equiv F_{0 i}
\begin{aligned}I_3 =\int_{\alpha_1}^{\beta_1}\biggl[\Phi(x,\alpha_2)P_1(\alpha_2) - \Phi(x,\beta_2)P_1(\beta_2) &+ \frac{\partial \Phi(x,\alpha_2)}{\partial x}P_1(x)P_1(\alpha_2)-\frac{\partial \Phi(x,\beta_{2})}{\partial x}P_1(x)P_1(\beta_2)\biggl] dx ,\end{aligned}
\frac{3}{4\lambda^2\sinh^2(\rho/\lambda)}-\frac{3}{4\rho^2}=\frac{3}{4\rho^2}\left(\frac{(\rho\beta)^2-\sinh^2(\rho\beta)}{\sinh^2(\rho\beta)}\right)\leq-\frac{c}{\rho^2},
\sum_{\substack{0\le k \le n-1 begin{align*}3pt]k\not= (n-1)/2}}b_{n,k}=-q(1-q^n)\sum_{\substack{0\le k \le n-1 begin{align*}3pt]k\not= (n-1)/2}}\frac{(-1)^{k}}{1-q^{2k+1}}.
\langle\chi_T(k)|\chi_T(k')\rangle=\lim_{n\to\infty, n'\to\infty}n^{2k^2}{n'}^{2{k'}^2}\langle e^{ik'.X\left((n+n'-2)\frac{\pi}{4}\right)}e^{ik.X(0)}\rangle_{C_{n+n'-2}}.
S ( t_{j}^{i}) = \overline{{t}}{}_{j}^{i}, \S ( \overline{{t}}{}_{j}^{i}) = \Omega_{m}^{i}t_{n}^{m}\overline{{\Omega}}{}_{j}^{n}, \S ( E_{i}) = - E_{j}\overline{{t}}{}_{i}^{j}
\int_{-\infty}^{\infty}\frac{dy}{\sqrt{x^{2}+y^{2}}}=\int_{-\infty}^{\infty}\frac{dt}{\sqrt{(x/x_{0})^{2}+t^{2}}}=\int_{0}^{\infty}\frac{vol(S^{1})dr}{\sqrt{(x/x_{0})^{2}+r^{2}}}
\Upsilon_{a\alpha}(\vec{k}) = \left[ \begin{array}{c}\Upsilon_{11}\Upsilon_{21}\Upsilon_{12}\Upsilon_{22}\end{array}\right] = \left[ \begin{array}{c}c_{k\uparrow}^{\vphantom\dagger}c_{-k\downarrow}^{\dagger}c_{k\downarrow}^{\vphantom\dagger}-c_{-k\uparrow}^{\dagger}\end{array}\right].
\begin{array}{c}{\nabla{\bf K}= \frac 1{\sqrt{2 \epsilon}}\{\left( - \frac 1{\sqrt{2 \epsilon}}{\bf K}+ \frac 1{2 \epsilon}{\bf M}_{1}\right) \left( \vec{R}- i \frac 1 2 \vec{\Gamma}\right) + \frac 1{2 \Gamma}{\bf M}_{2}\vec{\Gamma}}\\{+ i \frac 1{\Gamma \left( \epsilon^{2}- \eta^{2}\right)}\frac{\xi^{*}}{\left| \xi \right|}M_{3}\left( \vec{\Gamma}\times \vec{R}\right) \}}\\ \end{array}
t_{n_{k+1}}= (-1)^{{\tfrac{(n_{k+1}- n_{k}-1)(n_{k+1}- n_{k})}{2}}}\left(s_{n_{k+1}+ n_{k}+1}- s_{n_{k+1}+ n_{k}+1}^{(n_{k}+1)}\right)^{n_{k+1}- n_{k}}t_{n_{k}}\ .
D_- = r\left( \frac{X_{Z_1}-1}{n}\right) - r\left(\frac{X_{Z_1}}{n}\right) D_+ = \begin{cases}r\left(\frac{X_{Z_1'+1}}{n}\right) - r\left(\frac{X_{Z_1'}}{n}\right) Z_1 \neq Z_1' r\left(\frac{X_{Z_1'}}{n}\right) - r\left(\frac{X_{Z_1'-1}}{n}\right) Z_1 = Z_1' \end{cases}
\frac{1}{24}\Big( R g_{ij}- 6 E_{ij}\Big) \nabla^{2}_{ij}R \,=\, |\nabla E|^{2}- \frac{1}{24}|\nabla R|^{2}+ 4 E_{ip}E_{jp}E_{ij}+\frac{5}{12}R |E|^{2}\,.
\frac{W(z)-W(v_1)}{z-v_1}= \frac{W(z)-W(v_2)}{z-v_2}\left(1-\frac{v_2-v_1}{z-v_1}\right)+\frac{W(v_2)-W(v_1)}{v_2-v_1}\frac{v_2-v_1}{z-v_1}
\tilde{F}^{(i)}(\frac{z_1}{z_2 q^{2n}}) ={\rm tr}_{V(\Lambda_i)}(q^{-2 \rho}\tilde{\Phi}^{\Lambda_i V^*}_{\Lambda_{i-1}}(z_1) \tilde{\Phi}^{\Lambda_{i-1}V^*}_{\Lambda_i}(z_2 q^{2n}))
\frac{\beta(1+(1-2\alpha)r^m)}{(1-r^m)^{2(1-\alpha)+1}}+\frac{(1-\beta)r^m}{(1-r^m)^{2(1-\alpha)}}+\sum_{n=1}^{\infty}\prod_{k=0}^{2(n-1)}\frac{k+2(1-\alpha)}{k+1}r^{2n}=\frac{1}{4^{1-\alpha}}.
E[\psi(W,\gamma,\alpha_{0},\theta_{0})] =E[m(W,\gamma)]-\theta_{0}+E[\alpha_{0}(X)\{Y-\gamma(X)\}] & =E[\alpha_{0}(X)\{\gamma(X)-\gamma_{0}(X)\}]+E[\alpha_{0}(X)\{\gamma_{0}(X)-\gamma(X)\}]=0.Q.E.D.
\mbox{var}[\overline{\mathbf{W}}(j,j)] = \mbox{var}\left[\sum^K_{i=1}\frac{a_i}{p_i}\mathbf{W}_i(j,j)\right] = \sum^K_{i=1}\frac{a_i^2}{p_i^2}p_i \sigma^4 = \sigma^4 \sum^K_{i=1}\frac{a_i^2}{p_i}.
\tilde{h}(i,\boldsymbol{x}) = C_lx_{j_l}^{\tau-\zeta_i-1}+\dots +C_2x_{j_2}^{\tau-\zeta_i-l+1}x_{j_3}\cdots x_{j_l}+C_1x_{j_1}^{\tau-\zeta_i-l}x_{j_2}\cdots x_{j_l}& =: C_l\tilde{h}_l(i,\boldsymbol{x})+ C_{l-1}\tilde{h}_{l-1}(i,\boldsymbol{x})+\dots + C_1\tilde{h}_1(i,\boldsymbol{x}).
\sum_{j=0}^{\alpha}\frac{\mathcal{C}(p,j,s_1,s_2)}{\mathcal{C}(p,0,s_1,s_2)}= 1 + \frac{\frac{k \ell}{p^{s_1+s_2+1}}+ O(p^{-1.5})}{1+ O(p^{-1.8})}+ \frac{\sum_{j=2}^{\alpha}p^{-0.7j}}{1+ O(p^{-1.8})}= 1 + \frac{k \ell}{p^{s_1+s_2+1}}+ O(p^{-1.4}) (s_1,s_2) \in D_1 \times D_2.
H(\rho,\tau)=\lim_{p\rightarrow 1}\frac{1-\left \|\rho \right \|_{L^p(M,\tau)}}{p-1}=\lim_{p\rightarrow 1}\frac{p}{1-p}\log(\left \| \rho \right\|_{L^p(M,\tau)})=\lim_{p\rightarrow 1}h_p(\rho,\tau) .
L ( \lambda ) = \sum \sb{i = 1}\sp N \left( p \sb i + \sum \sb{\alpha = 1}\sp M{\frac{f \sb{i i}\sp \alpha}{\lambda - \epsilon \sb \alpha}}\right) e \sb{i i}+ \sum \sb{\stackrel{i , j = 1}{i \neq j}}\sp N \left({\frac{f \sb{i j}}{q \sb i - q \sb j}}+ \sum \sb{\alpha = 1}\sp M{\frac{f \sb{i j}\sp \alpha}{\lambda - \epsilon \sb \alpha}}\right) e \sb{i j},
|I| \leq \sum_k \sum_{k_1+k_2+k_3=k}\int_{\eta_1+\eta_2+\eta_3=\eta}d\eta_1\, d\eta_2\,d\eta_3\,d\eta \, \frac{|c(k,\eta)|}{\left(1+|\eta+k^2|\right)^{1-b}}&\times |\tilde{v}_\delta(k_1,\eta_1)||\tilde{v}_\delta(-k_2,-\eta_2)||\tilde{v}_\delta(k_3,\eta_3)|.
\partial{}_{\mu}\overline{{{F}}}^{\prime a \mu \nu}- \overline{{{f}}}^{a}{}_{c b}\overline{{{F}}}^{\prime c \mu \nu}\overline{{{A}}}^{\prime b}{}_{\mu}{}+ \overline{{{A}}}^{\prime b}{}_{\mu}\overline{{{R}}}^{\prime a}{}_{b}{}^{\mu \nu}- \overline{{{C}}}^{a}{}_{d \mu}\overline{{{F}}}^{\prime d \mu \nu}= 0 ,
\overline{X}_t^k = \langle \overline{v}^k_t, \phi \rangle - \int_{0}^{t}\int_{0}^{\pi}v^k(s,x) \phi''(x) \, \textrm{d}s \, \textrm{d}x, \overline{X}_t = \langle \overline{v}_t, \phi \rangle - \int_{0}^{t}\int_{0}^{\pi}v(s,x) \phi''(x) \, \textrm{d}s \, \textrm{d}x
H_0^-(x) =\frac{1}{(b-a)^{2N-1}}\sum_{0\le\ell\le N-1 \atop 0\le k\le N-1-\ell}\binom{N-1-\ell}{k}\!\binom{\ell+N-1}{\ell}(x-a)^{k+\ell}(b-x)^{2N-1-k-\ell}&=\frac{1}{(b-a)^{2N-1}}\sum_{m=0}^{N-1}\left[\,\sum_{\ell=0}^m\binom{\ell+N-1}{\ell}\!\binom{N-1-\ell}{m-\ell}\!\right] (x-a)^m (b-x)^{2N-1-m}.
\epsilon_{n}=\sqrt{\langle x^{2}\rangle\langle(\gamma\beta_{x})^{2}\rangle-\langle x\cdot\gamma\beta_{x}\rangle^{2}}=\sqrt{|\begin{matrix}\langle x\cdot x\rangle&\langle x\cdot\gamma\beta_{x}\rangle\\ \langle x\cdot\gamma\beta_{x}\rangle&\langle\gamma\beta_{x}\cdot\gamma\beta_{x}\rangle\end{matrix}|}
\left( \begin{array}{c c}{n_{1}^{\prime}}&{n_{2}^{\prime}}\\{l_{1}^{\prime}}&{l_{2}^{\prime}}\\ \end{array}\right) = \left( \begin{array}{c c}{a}&{c / 2}\\{2 b}&{d}\\ \end{array}\right) \left( \begin{array}{c c}{n_{1}}&{n_{2}}\\{l_{1}}&{l_{2}}\\ \end{array}\right)
[\begin{matrix}\frac{e^{\eta_{1}}}{1+\sum_{i=1}^{k-1}e^{\eta_{i}}}\\ \vdots\\ \frac{e^{\eta_{k-1}}}{1+\sum_{i=1}^{k-1}e^{\eta_{i}}}\\ \frac{1}{1+\sum_{i=1}^{k-1}e^{\eta_{i}}}\end{matrix}]
0<1-\lambda_h(j)\kappa_{m}^2 h_{\max}^{2{m}}\le 1-\kappa_{m}^2 h_{\max}^{2m}\Vert w_{\mathrm{pw}}\Vert_{L^2(\Omega)}\le \Vert v_{\mathrm{pw}}- \kappa_{m}^2h_{\mathcal{T}}^{2m}w_{\mathrm{pw}}\Vert_{L^2(\Omega)}=\Vert v_{\mathrm{nc}}\Vert_{L^2(\Omega)}.
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) .
\frac{1}{nd_n}\left\{\frac{1}{n}\sum\limits_{t=1}^{k}S_t^2(1, k)+\frac{1}{n}\sum\limits_{t=k+1}^n S_t^2(k+1,n)\right\}^\frac{1}{2}&=\left\{\int_0^{\lambda}\left(W_n(r)-\frac{c_n(r)}{c_n(\lambda)}W_n(\lambda)\right)^2dr+\int_{\lambda}^1\left(W_n(r)-\frac{1-c_n(r)}{1-c_n(\lambda)}W_n(\lambda)\right)^2dr\right\}^{\frac{1}{2}},
E^2T_d=E^2T_{2,i_1}T_{3,i_2-1}T_{2,j_2-1}T_{1,j_1}=-E^2T_2^{-1}T_1^{-1}T_{4,i_1}T_{3,i_2-1}T_{2,j_2-1}T_{1,j_1}.
\frac{\partial^k}{\partial x_1^j \partial x_2^{k-j}}g(x) = \frac{(m-1)!}{(m-1-k)!}(-1)^k a_1^j a_2^{k-j}( 1 - a_1 x_1 - a_2 x_2)_+^{(m-1-k)}
-\Big(A^{2,0}a_0\Big)_l = \sum_{d=m}^{l+2m}\Big((-1)^{d+m-1}n^d 2^d P_{d-m}(m) \sum_{|J|=d}\alpha_p(1)^{j_1}\cdots \alpha_p(l+m)^{j_{l+m}}\ \frac{d!}{j_1! j_2! \cdots j_{l+2m}!}\Big).
G_{\mathcal R, \zeta_o, 2}(x,y) := &\frac{i}{2\pi}\int_{\mathcal D_{\zeta_o,\frac\pi2+\theta_{\hat{x}}-i\infty}}\frac{2i\sin\zeta\mathcal S(\cos\zeta,n)}{n^2-1}{e^{ik_{+}\left(-\vert y \vert\cos(\zeta+\theta_{\hat y})+ \vert x \vert\cos(\zeta-\theta_{\hat x})\right)}}d\zeta.
S_E=\frac{1}{|E^{\times}|}\sum_{\Lambda}\prod_{i}g(\psi_E,\Lambda\chi_{1,i}) \prod_{i}g(\psi_E,\Lambda\overline{\chi_{2,i}}) \prod_{j}g(\psi_E,\Lambda\rho_{1,j}) \prod_{j}g(\psi_E,\Lambda\overline{\rho_{2,j}})
< p_{1}^{\prime}|{\cal T}^{( E )}| p_{1}> = \int \frac{d^{3}x}{( 2 \pi )^{3}}e^{i ( p_{1}- p_{1}^{\prime}) \cdot x}\, i \frac{d}{d \alpha}\, \exp \left[ - \textstyle{\frac{i}{E}}\int_{\alpha E}^{\infty}d \tau V ( x + 2 \textstyle{\frac{p}{E}}\tau , p ) \right] \big |_{\alpha = 0}\, .
\Delta=\langle \alpha_1, \beta_1, ..., \alpha_{\gamma}, \beta_{\gamma}, x_1, ..., x_r: x_1^{m_1}=...=x_r^{m_r}= \prod_{j=1}^{\gamma}[\alpha_{j}, \beta_{j}]\prod_{i=1}^r\, x_i=1\rangle.
Vol(\partial T_{\delta}) \geq\int_{\partial T_{\delta}\bigcap C(S)}\lambda(t,\theta)=\int_{S}t^{2n-1}(\theta)\lambda(t,\theta)d\theta &\geq\int_{S}\lambda(L')\frac{\lambda_{k}(t(\theta))}{\lambda_{k}(L')}t^{2n-1}(\theta)d\theta\geq C\int_{S}\lambda(L')L'^{2n-1}d\theta,
\left(I+E(g)\right)^{-1}=\sum_{k=0}^{\infty}(-1)^k\left(E(g)\right)^{k}\left\|\left(I+E(g)\right)^{-1}\right\|_{\mathcal B_3}\le \frac{1}{1-\left\|E(g)\right\|_{\mathcal B_3}}.
\mathbf{v}^{0}=\mathbf{f}=[e^{-j\frac{2\pi}{N}/\sqrt{N}}, e^{-j\frac{2\pi}{N}2/\sqrt{N}}, \cdots, e^{-j\frac{2\pi}{N}N/\sqrt{N}}]^{T}&=[e^{-jA}, e^{-j2A}, \cdots, e^{-jNA}]^{T}.
A=(\begin{matrix}0&1&0\\ -1&0&0\\ 0&0&0\end{matrix}),B=(\begin{matrix}0&0&0\\ 0&0&1\\ 0&-1&0\end{matrix}),C=(\begin{matrix}0&0&1\\ 0&0&0\\ -1&0&0\end{matrix}).
P(\mathcal{B}_n,\mathbf{s}_0,\mathbf{s},w,i,\mathbf{u},\mathbf{v})=\lambda(\mathcal{B}_n)P^{(\mathcal{B}_n)}(\mathbf{s}_0,\mathbf{s},w,i,\mathbf{u},\mathbf{v}).
\phi^{-1}_1(\beta_k)-\phi^{-1}_2(\beta_k)=\phi^{-1}_1(\beta_{k'})-\phi^{-1}_2(\beta_{k'}) &\phi^{-1}_1(\beta_k)+\phi^{-1}_2(\beta_k)=\phi^{-1}_1(\beta_{k'})+\phi^{-1}_2(\beta_{k'}).
\begin{cases}dX_{t}^{i,N}= F_{A}\bigg( \frac{1}{N}\displaystyle\sum_{k=1}^{N}(K \ast V^{N})(X_{t}^{i,N}-X_{t}^{k,N})\bigg)\; dt + \sqrt{2}\; dW_{t}^{i}, t\leq T,~ 1\leq i \leq N, X_{0}^{i,N},~ 1\leq i \leq N, \{W^i,~1\leq i \leq N\},\end{cases}
l_{0}(x)=\frac{x-x_{1}}{x_{0}-x_{1}}\cdot\frac{x-x_{2}}{x_{0}-x_{2}}=\frac{x-4}{2-4}\cdot\frac{x-5}{2-5}=\frac{1}{6}x^{2}-\frac{3}{2}x+\frac{10}{3}
\tfrac{1}{4}\rho\mu(dp)^\Delta|X_i||V_i| = \tfrac{1}{8}\rho\mu(dp)^\Delta|X_i||V_i|+\tfrac{1}{8}\rho\mu(dp)^\Delta|X_i||V_i| > \frac{\mu d^\Delta}{32\rho}p^\Delta|P||D| +\frac{\mu d^\Delta}{16 \kappa r_1}p^\Delta n|D| > 7p^\Delta|P||D|+\rho p^\Delta n|D|/r_1\,,
S_p(x,u) = \frac{x(1-u^2)}{u(1-u-u^2)}S_p(x,u)-\frac{1-x-u-u^2+xu^2}{u(1-x)(1-u-u^2)}S_p(x,0) & \qquad + \frac{xu^2}{1-u-u^2}D_p(x,u) + xD_p(x,0) + \frac{x}{(1-x)(1-u-u^2)}begin{align*}10pt]D_p(x,u) = \frac{x}{1-u-u^2}S_p(x,u) + \frac{x^2}{(1-x)(1-u-u^2)}S_p(x,0) & \qquad + \frac{x(1-2u^2)}{u(1-u-u^2)}D_p(x,u) - \frac{x}{u}D_p(x,0) + \frac{x}{(1-x)(1-u-u^2)}
\Bigg\Vert\prod_{i=0}^{j}\big(\chi(g_{k_i})-1\big)\chi-\prod_{i=0}^{j}\big(\varphi(g_{k_i})-1\big)\varphi\Bigg\Vert \leqslant d_j(\chi,\mathbf{1})\Vert \chi- \varphi\Vert+ d_j(\chi,\varphi)\Vert \varphi\Vert
\lim_{\epsilon \rightarrow 0}\left[{\frac{\hbar^{2}}{2}}\int_{0}^{1}d s \,{\frac{\delta^{2}}{\delta \sigma^{\mu \nu}( s - \epsilon / 2 ) \delta \sigma_{\mu \nu}( s + \epsilon / 2 )}}- m^{4}\right] \Psi [ C ] = 0 \ ,
\sum_{j=1}^np_{n-j}\sum_{k=1}^ja_{j-k}{\delta_t\omega^k}=\sum_{k=1}^n\delta_t\omega^k\sum_{j=k}^np_{n-j}a_{j-k}=\sum_{k=1}^n\delta_t\omega^k=\omega^n-\omega^0, n\geq1,
p^{\varepsilon,\alpha}(t) = S(T-t)'\zeta^{\varepsilon}+ \int_t^T S(s-t)'\big[T_{\alpha}(s)^*p^{\varepsilon,\alpha}(s) +f^{\varepsilon}(s)\big]ds +\int_t^T S(s-t)' q^{\varepsilon,\alpha}(s)dW(s).
\dot{\epsilon}_{\delta\sigma}=\epsilon_{\delta\sigma}\left[3{\dot{a}\over a}(1+w)+2{\dot{c}\over c}{{1+\epsilon_{\delta\sigma}}\over{1+\epsilon}}+\left({\dot{G}_4\over G_4}-4{\dot{c}\over c}\right)(1+\epsilon_{\delta\sigma})+{{3Cc\dot{c}}\over{4\pi G_4a^4_0\varrho_m}}\right].
\mathcal{D}_i(c')=\left\{2^{-nR_2}\sum_{j\in\mathcal{J}_n}g_\mathbf{w}\big(\mathbf{U}(i),\mathbf{V}(i,j)\big)\geq c'\cdot 2^{n\left(I(U;W)+\epsilon^{(1)}_{\alpha,\delta_1}\right)}\right\},