harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\int_0^1(1-x)^{-X}x^{m-1}dx =B(1-X,m)=\frac{\Gamma(1-X)\Gamma(m)}{\Gamma(1-X+m)}&=\frac{(m-1)!}{(1-X)(2-X)\cdots(m-X)}&=\frac{1}{m}\prod_{a=1}^m\biggl(1-\frac{X}{a}\biggr)^{-1}=\frac{1}{m}\prod_{a=1}^m\sum_{n=0}^\infty \frac{X^n}{a^n}&=\frac{1}{m}\sum_{k=0}^\infty X^k\sum_{0<a_1\leq\cdots\leq a_k\leq m}\frac{1}{a_1\cdots a_k}
g\left((r,c)(r',c')\right) = g(rr'+cc'h,rc'+r'c) & = rr'+cc'h+rc'r_0+r'cr_0 & =r(r'+c'r_0)+cc'r_0^2+r'cr_0 & = r(r'+c'r_0)+cr_0(c'r_0+r') & = (r+cr_0)(r'+c'r_0) & = g(r,c)g(r',c')
\begin{aligned}\int_\Omega (D_\alpha u\cdot \phi_\alpha+p \psi )\,dx &=\int_{\Omega_r(x_0)}\big(D_\alpha v- (D_\alpha v)_{\Omega_r(x_0)}\big)\cdot f_\alpha+\big(\pi-(\pi)_{\Omega_r(x_0)}\big) g\,dx.\end{aligned}
A_{37,\ell}= \int_{C/\ell}^{\pi -C/\ell}\frac{a_{3,\ell}(\phi) a_{7,\ell}(\phi)}{(1-4 \alpha_2^2/\lambda_{\ell}^2)^{1/2}(1-4 \alpha_1^2/\lambda_{\ell}^2)^{1/2}}\sin \phi d \phi
f(k)\leq f(x^*)= \left(\log cen - (\ell + 1)\left(\frac{\log cen}{\ell + 1}- \frac{1}{\ln 2}\right)\right)2^{\frac{\log cen}{\ell + 1}- \frac{1}{\ln 2}}=\frac{\ell + 1}{2^{1/\ln 2}\ln 2}(cen)^{\frac{1}{\ell + 1}}.
b = \frac{- ( y , z )}{\ln \left( \mu L \left( 1 - \frac{4 s^{2}}{L^{2}}\right) \right)}, \qquad \qquad a = - b \ln \left( \frac{L}{2}\left( 1 - \frac{4 s^{2}}{L^{2}}\right) \right)
\left( \begin{array}{c}{{\partial_{\mu}\theta}}\\{{\sin \theta \hspace{1 m m}\partial_{\mu}\varphi}}\\ \end{array}\right) = L \left( \begin{array}{c}{{G_{\mu}}}\\{{H_{\mu}}}\\ \end{array}\right) , \hspace{3 m m}L = \left( \begin{array}{c c}{{\beta}}&{{\gamma}}\\{{- \gamma}}&{{\beta}}\\ \end{array}\right) ,
|A|_{i,j}=a_{i,j}-r_i^j(A^{i,j})^{-1}c_j^i:=\left|\begin{array}{cc}A^{i,j}c_j^i r_i^j \boxed{a_{i,j}}\end{array}\right|,A^{-1}=\left(|A|_{j,i}^{-1}\right)_{i,j=1}^n.
\mathcal{T'}_{12}(s, t) = \chi_{\{s \leq t\}}g_z^2 (t) , \mathcal{T'}_{21}(s, t) = \chi_{\{s \geq t\}}g_z^2 (t), \mathcal{T'}_{11}(s, t) = \mathcal{T'}_{22}(s, t) = 0.
|\mathcal{F}|\leq \begin{cases}\displaystyle\sum_{i=(n-1)/2}^n{n \choose i}\mbox{if}n\mbox{is odd,}{n-1 \choose \frac{n}{2}-2}+\displaystyle\sum_{i=\frac{n}{2}}^n{n \choose i}\mbox{if}n\mbox{is even.}\end{cases}
\Phi_{b}(\vec{y}_{1},\eta)...\Phi_{b}(\vec{y}_{n/2},\eta)\Phi_{b}(\vec{y}_{n/2+1},-\eta)...\Phi_{b}(\vec{y}_{n},\eta)=\Psi_{1}+\bar{\eta}\Psi+\bar{\Psi}\eta+\bar{\eta}\eta\Psi
cap_p(B^{g^+}_q(t))\geq (\int_{Vol(B^{g^+}_q(t))}^{+\infty}(n\tau)^{\frac{p}{1-p}}d\tau)^{1-p}=\frac{n^p}{(p-1)^{p-1}}Vol(B^{g^+}_q(t))
B_2 \le 2 \int_0^{(n-m)s/N}\frac{2s}{Nn}\d h + 2 \int_{(n-m)s/N}^{(n+m)s/N}\Bigl( \frac s N \Bigl( \frac1n + \frac1m \Bigr) - \frac h{mn}\Bigr) \d h = \frac{4s^2}{N^2}\frac{n-m}n + \frac{4s^2}{N^2}m \Bigl(\frac1n + \frac1m\Bigr) - \frac1{mn}\frac{s^2}{N^2}((n+m)^2 - (n-m)^2),
\Delta k_{\pm}(a\longrightarrow j)=\int_{0}^{\infty}(a\longrightarrow j)d\omega-\frac{\pi^{2}}{\hbar}(N_{a}-N_{j})(\frac{\alpha_{2}}{n})[|\langle a|m_{-}|j\rangle|^{2}-|\langle a|m_{+}|j\rangle|^{2}]
F(x,u;v) =1+xuK+xu\big(F(x,u;0)-1+vF(x,u;v)-v\big) &+\frac{x}{1-v}\big(F(x,u;1)-vF(x,u;v)\big)-x-ux^2K &+\frac{ux^3}{(1-x)^2}\left(\frac{F(x,u;0)+F(x,u;1)-1}{(1-ux)(1-uvx)}-1\right) &+\frac{ux^3}{(1-x)^2(1-ux)(1-vux)}\left(\frac{1}{1-x}F\Big(x,u;\frac{1}{1-x}\Big)-\frac{1}{1-x}-F(x,u;1)+1\right),
A \,\, = \,\,\begin{pmatrix}A_0 A_1 A_2 \cdots A_{r-2}A_{r-1}0 A_0 A_1 \cdots A_{r-3}A_{r-2}0 0 A_0 \cdots A_{r-4}A_{r-3}\vdots \vdots \vdots \ddots \vdots \vdots 0 0 0 \cdots A_0 A_1 0 0 0 \cdots 0 A_0 \end{pmatrix}.
2V_k(n) =\Big(\sum_{1\leq i \leq n}f_k(i)f_k(i+1)\Big)^2-\sum_{1\leq i \leq n}\big(f_k(i)f_k(i+1)\big)^2+O_\varepsilon(n^{1+\varepsilon}) &=\Big(\sum_{1\leq i \leq n}f_k(i)f_k(i+1)\Big)^2+O_\varepsilon(n^{1+\varepsilon}).
\mathcal{C}^0_{\rm{RBA}_\lambda}(A, T)= \ 0, \mathcal{C}^1_{\rm{RBA}_\lambda}(A, T)= mathcal{C}^1_{\rm{Alg}}(A), \mathcal{C}^n_{\rm{RBA}_\lambda}(A, T)= mathcal{C}^n_{\rm{Alg}}(A)\oplus \mathcal{C}^{n-1}_{\rm{RBO}_\lambda}(T),\,n\geqslant 2,
B_{2,Z}=\left\{\varphi=\sum_{j=1}^{\infty}a_{j}\varphi_{\zeta_{j}}:\sum_{j=1}^{\infty}\left\vert a_{j}\right\vert^{2}\mu\left( \zeta_{j}\right) <\infty\right\}, \left\Vert \varphi\right\Vert_{B_{2,Z}}\approx\left\Vert \left\{a_{j}\right\}_{j=1}^{\infty}\right\Vert_{\ell^{2}\left( \mu\right)}.
\big|\mathcal{F}(f\circ \theta_a^X)( \xi )\big|^2 ={}4|\xi|^{-d+1}a^2\bigg( \int_{\varphi(\omega)}^{\varphi(\beta)}f\circ \theta^H(r) \cos(2\pi(R+ar)|\xi| + \nu_d) &\times (R+ar)^{\frac{d-1}{2}}dr\bigg)^2 +O(R^{d-1}a^2|\xi|^{-d-1})
\frac{d\tilde{x}^\mu_k}{dt}=\,\gamma^\mu_{kx}(\tilde{x},\dot{\tilde{x}},t), \frac{d\dot{\tilde{x}}^\mu_k}{dt}=\,\gamma^\mu_{ky}(\tilde{x},\dot{\tilde{x}},t),{\tilde{y}}^\mu_k =\,\frac{d\tilde{x}^\mu_k}{dt}\quad\mu=1,2,3,4;\,\,k=1,...,N.
|A(z,\xi)|\leq L(1+|\xi|+|F|)^{p(z)-1}&|B(z,\xi)|\leq L(1+|\xi|+|F|)^{p(z)-1}&\langle A(z,\xi),\xi\rangle\geq \nu|\xi|^{p(z)}-|F|^{p(z)}
\left[\begin{matrix}\mathbf{b}_1 \cdots \mathbf{b}_{2g}\end{matrix}\right]=\left[\begin{matrix}\mathbf{d}_1 \cdots \mathbf{d}_{2g}\end{matrix}\right]\delta\quad\textrm{for some}~\delta\in M_{2g}(\mathbb{Z})\cap\mathrm{GL}_{2g}(\mathbb{Q}).
[ V_{\alpha}, V_{\beta}] = \epsilon_{\alpha \beta}^{\! \gamma}V_{\gamma}, [ V_{\alpha}, V_{m}^{a}] = V_{m}^{b}( \sigma_{\alpha})_{b}^{a}, \{V_{m}^{a}, V_{m}^{b}\}= - m^{2}( \sigma^{\alpha})^{a b}V_{\alpha},
\begin{aligned}- F^{ii}\nabla_{ii}\phi = F^{ii}\nabla_{ii}u = F^{ii}U_{ii}- \, F^{ii}A^{ii}= u_t + \psi - F^{ii}A^{ii}\geq \, u_t + \psi - C (1 + |\nabla u|) \sum F^{ii}\geq \, u_t - C (1 + |\nabla u|) \sum F^{ii}- C |\nabla u|^{\gamma}.\end{aligned}
s_\lambda(1,t,t^2, \ldots, t^{n-1}) = t^{n(\lambda)}\prod_{1\le i<j\le n}\frac{1-t^{\lambda_i-\lambda_j+(j-i)}}{1-t^{j-i}}= t^{n(\lambda)}\prod_{b\in \lambda}\frac{1-t^{n+c(b)}}{1-t^{h(b)}},
w(t,r,v)- w(b,r,v)\,=\,- \big[ v(t)-v(b)- \frac{v'(b)}{r'(b)}(r(t)-r(b))\big] & - \Big[ \frac{v'(t)}{r'(t)}- \frac{v'(b)}{r'(b)}\Big](r-r(b))+ \Big[ \frac{v'(t)}{r'(t)}- \frac{v'(b)}{r'(b)}\Big](r(t)-r(b)).
J_{\infty}(R) =\displaystyle\lim_{\underset{m}{\rightarrow}}R_{m}&=\frac{\mathbb{C}[x_{j,(-1-i)}\;|\;0\leq i,\; 1\leq j\leq n]}{( T^{j}f_{i}|i=1,\ldots n,\; j\in\mathbb{N})}.
\frac{\partial e_j}{\partial [\tilde{v}_i]_{j,\alpha}}=\frac{[\tilde{w}_i]_{\alpha,j}}{m|g|\lambda_j}\bigg[c_{i-1}+\frac{i}{m}|g|-[\tilde{v}_i\tilde{w}_i]_{j,j}-\sum_{s=0}^{m-1}\frac{m-s}{m}[\tilde{v}_s\tilde{w}_s]_{j,j}\bigg].
u_{1}=\frac{2\tilde{r}-1+\alpha+\beta}{2n},\tilde{r}\in \{1, 2, \ldots, n\}, u_{2}=\frac{2\tilde{p}-1-\sqrt{1+4E}}{4m},\tilde{p}\in \{1, 2, \ldots, 2m\},
\begin{array}{ll}H= \sum_{i=2}^{M}(\mbox{\cal A}c^{+}_{i}c_{i-1}+1/2B_{i}c^{+}_{i}c_{i}+\gamma c^{+}_{i}c^{+}_{i-1}) +1/2B_{1}c^{+}_{1}c_{1}-\mbox{\cal A}(-1)^{\hat N}c^{+}_{1}c_{M}-\gamma (-1)^{\hat N}c^{+}_{1}c^{+}_{M}+H.c.\end{array}
\frac{1}{2i\pi}\int_{\mathcal{C}(\sigma^2,\varepsilon)}(zI-K_v^2)^{-1}dz= \sum_{\substack{\tilde{\sigma}^2\in\, \Xi^K_v |\tilde{\sigma}^2-\sigma^2|<\varepsilon}}\pi^{\tilde{\sigma}},
\left\langle x \circ_i y, x' \circ_{i'}y' \right\rangle := \begin{cases}1 \mbox{if}x = x', y = y', \mbox{and}i = i' = 1, -1 \mbox{if}x = x', y = y', \mbox{and}i = i' = 2, 0 \mbox{otherwise}. \end{cases}
\omega \mathrm{~}C_{{\bf{k}}}({\bf{q}}) = \omega_{{\bf{k}}}({\bf{q}}) C_{{\bf{k}}}({\bf{q}}) + ( \frac{v_{{\bf{q}}}}{V}) \Lambda_{{\bf{k}}}( -{\bf{q}}) \sum_{{\bf{k}}^{'}}[ \Lambda_{{\bf{k}}^{'}}( -{\bf{q}}) C_{{\bf{k}}^{'}}({\bf{q}}) + \Lambda_{{\bf{k}}^{'}}({\bf{q}}) D_{{\bf{k}}^{'}}^{*}( -{\bf{q}}) ] + \Lambda_{{\bf{k}}}( -{\bf{q}})
\Theta_h = \left( \frac{1}{2}\left(p^{(2,0)}\right)^2 - p^1u_1 - u_1^3 \right) d x \wedge d t+ p^1 d u \wedge d t &\quad{}- \frac{1}{2}u_1 d u \wedge d x + p^{(2,0)}d u_1 \wedge d t \, .
f_{0}(z)=\gamma_{0}+\frac{1-|\gamma_{0}|^{2}}{\overline{\gamma_{0}}+\frac{1}{z\gamma_{1}+\frac{z(1-|\gamma_{1}|^{2})}{\overline{\gamma_{1}}+\frac{1}{z\gamma_{2}+\cdot\cdot\cdot}}}}
-\dot u_0(Tx)=\sum_{k=0}\sp \infty\frac{\lambda_k(T\sp{-k}x)}{\eta_k(T\sp{-k}x)}=\sum_{k=1}\sp \infty\frac{\lambda_k(T\sp{-k}x)}{\eta_k(T\sp{-k}x)}+\frac{\lambda(x)}{\eta(x)}>\frac{\lambda(x)}{\eta(x)},
\mathrm{Res}_{t=0}\left(\frac{\Gamma(k+t)^2}{(2\pi)^{2t}}\frac{X^t\zeta(2t+1)}{t}\right) = \lim_{t \to 0}\left(\frac{d}{dt}\left(\frac{t^2\Gamma(k+t)^2}{(2\pi)^{2t}}\frac{X^t\zeta(2t+1)}{t}\right)\right) =\frac{\left((k-1)!\right)^2}{2}\log X+c_0;
C_0 =\frac{\Gamma((n+2)/2)^2}{\Gamma((n+1)/2) \Gamma((n+3)/2)}&= \Big( \frac \pi 2 \Big)^{2 ( n \thinspace \mathrm{mod}\thinspace 2) - 1}\frac{1}{(n+1)}\Big( \frac{n!!}{(n-1)!!}\Big)^2
\tilde{\mu}^{*}\tilde{\mu}= \frac{1 + \frac{j ( j + 1 )}{k - 2}}{2 j + k + 1 + \frac{j ( j + 1 )}{k - 2}}> 1 ; j = - N - \frac{k}{2}( N ={1 , 2 , . . .})
V_{n s}({\tilde{x}}) = \left( \frac{{\tilde{m}}N^{2}}{1 6 \pi}\right) N g^{2 n s - 1}{\tilde{x}}^{2}\left\{{\tilde{x}}^{2}- \frac{2{\tilde{b}}}{3}{\tilde{x}}+ \frac{{\tilde{b}}^{2}}{3}- ( - 1 )^{n s}{\tilde{c}}\right\}.
\left(K_{n}^{(3)}\right)^{2}+\left(K_{n+1}^{(3)}\right)^{2}+\left(K_{n+2}^{(3)}\right)^{2}=21\cdot 2^{2n}+2^{n+1}\left(M_{n+1}^{(2)}+3M_{n+2}^{(2)}\right)+6,
{\bar{t}\over \bar{t}_*}= \left({9-4r^2 \over 12}\right)\left[{2 \over 3-2r}\left|{\eta\over\eta_*}\right|^{(3/2)-r}+{2 \over 3+2r}\left|{\eta\over\eta_*}\right|^{(3/2)+r}\right]
gu_ig^{-1}u_i^{-1}= \exp((at_{i+1}+b)s_i Z_{i+1}+ Y_i) \in G' \cap G_{i+1}, hu_ih^{-1}u_i^{-1}= \exp((ct_{i+1}+d)s_i Z_{i+1}+ Y_i') \in G' \cap G_{i+1},
G_p =\frac{\kappa_g}{16\pi^2}\ln(r^{-2}) + \frac{\kappa_g}{16\pi^2}S_0 + \frac{\kappa_g}{16\pi^2}a_ix^i + \frac{\kappa_g}{16\pi^2}b_{ij}x^ix^j + o_4(r^{2}).
0 =F^{(n+1)}(x)+F^{(n+1)}(x+F(x))(1+F^{\prime}(x))^{n+1}&\qquad+nF^{(n)}(x+F(x))(1+F^{\prime}(x))^{n-1}F^{\prime \prime}(x) &\qquad+\sum_{j=1}^{n-1}F^{(j+1)}(x+F(x))(1+F^{\prime}(x))G_{j}^{n}(x)+\sum_{j=1}^{n-1}F^{(j)}(x+F(x))(G_{j}^{n})^{\prime}(x)
\begin{aligned}1- L(1+\kappa +{\rm i}v,{\rm sym}^m f)M_x(1+\kappa+{\rm i}v,{\rm sym}^m f) \noalign{\vskip 2mm}\ll_\varepsilon(kN)^\varepsilon \Biggl|\sum_{\substack{x<n\leq X (n,P(z))=1}}\frac{\mu(n)\lambda_{{\rm sym}^m f}(n)}{n^{1+\kappa+{\rm i}v}}\Biggr|+(kN)^\varepsilon \sum_{n>X}\frac{d_{m+1}(n)}{n^{1+\kappa}},\end{aligned}
a^{h_s}|_k = h_s|_{k}^{-1}h_s|_{k+1}= (a^s\prod_{i = 0}^{k-1}g|_i)^{-1}a^s\prod_{i = 0}^{k}g|_i &= \begin{cases}g|_k, k \neq p-1, (\prod_{i = 0}^{p-2}g|_i)^{-1}= g|_{p-1}, k = p-1, \end{cases}
\displaystyle{\frac{\phi^{( 0 )}( a , b + 1 , c | u , v , w + 1 / 4 )}{\phi^{( 0 )}( a , b - 1 , c | u , v , w + 1 / 4 )}}= \displaystyle{\frac{W ( a , b + 1 | A_{0}) \overline{{W}}( b + 1 , c | B_{0})}{W ( a , b - 1 | A_{0}) \overline{{W}}( b - 1 , c | B_{0})}}.
\|\mathcal W_\Psi f\|^2 &=\int_{\mathcal S^n}\!\int_{\mathcal S^n}\!\int_0^\infty \frac{\sum_{l,k}\bigl|a_l^k(\Psi_\rho)\bigr|^2}{N(n,l)}\,\mathcal K_l^\lambda(x\cdot y)\,\alpha(\rho)\,d\rho\,\overline{f(x)}\,d\sigma_n(x)\,f(y)\,d\sigma_n(y).
\dfrac{\partial I_{\varepsilon_\alpha}}{\partial t}(t_\alpha,\xi_\alpha)=\sum_{i=0}^n \lambda_{i,\alpha}\left\langle Z_{i,\alpha},\dfrac{\partial}{\partial t}(-W_{\delta_{\varepsilon_\alpha}(t_\alpha),\xi_\alpha}+\phi_{\delta_{\varepsilon_\alpha}(t_\alpha),\xi_\alpha}) \right\rangle_{P_g}.
\psi (t)=e^{\varepsilon^{-2}t\mathcal{A}}\psi (0)+\varepsilon^{-\rho}\int_{0}^{t}e^{\varepsilon^{-2}(t-\tau )\mathcal{A}_{s}}\mathcal{F}^{s}(a+\varepsilon \psi )d\tau +\mathcal{Z}^{s}(t).
h= \sum \limits_{j\in \mathbb{J}}\langle h, x_j\rangle \tau_j \iff \sum\limits_{j\in\mathbb{J}}(2-c_j)\langle h, x_j\rangle \langle \tau_j, h\rangle = \|h\|^2 \iff \sum\limits_{j\in\mathbb{J}}c_j^2|\langle h, x_j\rangle |^2 = \|h\|^2.
\dot{x}^j =\mu_j\cos\left(\mu_0 t\right)-\mu_{j+n}\sin\left(\mu_0 t\right) \dot{x}^{j+n}=\mu_{j+n}\cos\left(\mu_0 t\right)+\mu_{j}\sin\left(\mu_0 t\right) \dot{x}^{2n+1}=\frac{1}{2}\sum_{j=1}^n(\dot{x}^{j+n}x^j-\dot{x}^jx^{j+n}).
\mathbb{E}\left[\sum_{A\in \mathcal{Z}_\ell}d_Gk|\mathcal{T}_{A}\cap [0,t]|\right] \leq d_G k | \mathcal{Z}_\ell | \mathbb{E}\left [|\mathcal{T}_{A}\cap [0,t]| \right ]= d_G k t| \mathcal{Z}_\ell | \leq d_G k t M_{k,G}\ell^{d-1}.
E(\zeta_\chi)=\zeta_\chi\mbox{a.e. in}\Omega_a^b\!\!\!\!\!\!\!-\ , \|E(\zeta_\chi)\|_{{k+2},\mathbb{R}^3}\leqslant\tilde{c}\|\zeta_\chi\|_{k+2,\Omega_a^b\!\!\!\!\!\!\!-\},
\tilde{B}_f (x, a) = f (x+a) + f (x) + f (a) + f(0) = (x+a) (x^{2^{i_2}}+ a^{2^{i_2}}) + x^{1 + 2^{i_2}}+ a^{1 + 2^{i_2}}&= a x^{2^{i_2}}+ a^{2^{i_2}}x.
\left[X_j,\overline{X_k}\right]=\frac{\partial^2 \Phi}{\partial z_j\partial \overline{z_k}}\left( \frac{\partial}{\partial \overline{z_0}}-\frac{\partial}{\partial z_0}\right)=2i\frac{\partial^2 \Phi}{\partial x_j\partial x_k}\frac{\partial}{\partial y_0}\quad\forall\, j,k\in\{1,\ldots, n\},
{\mathbb E}\,{\rm e}^{-\alpha (S_{N_{\beta,\omega}}-\bar S_{N_{\beta,\omega}})- \gamma (U_{N_{\beta,\omega}}-G_{N_{\beta,\omega}})}&= \exp\left[-\int_0^\infty \int_{(-\infty,0)}\frac{1}{t}{\rm e}^{-\beta t}(1-{\rm e}^{-\omega t})(1-{\rm e}^{- \gamma t-\alpha x})\,{\mathbb P}(Y(t)\in{\rm d}x)\,{\rm d}t\right],
\mathcal{E}^{\prime}\left( t\right) =4h^{\prime}\left( t\right) h\left(t\right) \left( \Phi\left( x\left( t\right) \right) -\Phi^{\ast}\right)-2h\left( t\right) \left\langle \nabla\Phi\left( x\left( t\right)\right) ,x\left( t\right) -x^{\ast}\right\rangle .
\lim_n\P(Y^{(n)}_n=1) =\lim_n\sum_{i\ge0}\P(Y^{(n)}_n=1|r_{n}(\boldsymbol\xi)=i)\P(r_{n}(\boldsymbol\xi)=i) &=\sum_{i\ge0}\lim_n\P(Y^{(n)}_n=1|r_{n}(\boldsymbol\xi)=i)\lim_n\P(r_{n}(\boldsymbol\xi)=i).
T_{\ell}(f) + p^{j-1-\nu_p(n)}\left(T_{\ell}(gE_{p-1}^n) - \lambda_{\ell}(g)E_{p-1}^n \right) = \lambda(\ell)f,\mbox{\ \ \i.e.,}T_{\ell}(f + p^{j-1-\nu_p(n)}gE_{p-1}^n) = \lambda(\ell)(f + p^{j-1-\nu_p(n)}gE_{p-1}^n),
\frac{\partial (\L^{-1})}{\partial \sigma}(\sigma,\tau) = (\gamma^{(1)})' (\sigma_0+\sigma)+ \tau (n^{(1)})^\prime (\sigma_0+\sigma)\,,\frac{\partial (\L^{-1})}{\partial \tau}(\sigma,\tau) = n^{(1)}(\sigma_0+\sigma)\,,
V=\left( \begin{array}{cc}e^{-D^- D^+}e^{-\frac{1}{2}D^- D^+}\left( \frac{I- e^{-D^- D^+}}{D^- D^+}\right) D^- e^{-\frac{1}{2}D^+ D^-}D^+ I- e^{-D^+ D^-}\end{array}\right)
\delta_Y(y)=\frac{1}{2\pi}\int_{\mathbb{R}}\exp^{\diamond}\big[(e^{ix}-1)\tilde{N}(T_0)+i x \beta B(T_0)+\lambda T_0 (e^{ix}-1-ix)-\frac{1}{2}x^2 \beta^2 T_0-ixy\big]dx
\int_{\mathbb{R}^N}\left(\left|\nabla v_n\right|^2+(V(\epsilon x)+1)\left|v_n\right|^2\right) d x+\int_{\mathbb{R}^N}F'_1(v_n)v_nd x-\lambda_\epsilon \int_{\mathbb{R}^N}\left|v_n\right|^2 d x=\int_{\mathbb{R}^N}F'_2(v_n)v_n d x+o_n(1).
B^{\rho_{\pm}}_{n}(u):=\frac{L^{\rho_{\pm}}(\gamma^2n+\beta_1 u n^{2/3},n)-(L^{\rho_{\pm}}(\gamma^2 n,n)+\frac{1}{1-\rho_\pm}\beta_1 u n^{2/3})}{\beta_2 n^{1/3}}.
a_{14}^{1423}=a_{14}^{1432}=a_{14}^{1243}=a_{14}^{1234}=a_{14}^{1342}=a_{14}^{1324}= a_{14}^{2143}=a_{14}^{2134}=a_{14}^{2314}=a_{14}^{3142}=a_{14}^{3124}=a_{14}^{3214}
u(\rho)=\left(C_1F\left(-\frac{e^2+1}{4},-\frac{1}{2},\rho^2\right) +C_2\rho^3F\left(-\frac{e^2-5}{4},\frac{5}{2},\rho^2\right)\right)e^{-\frac{\rho^2}{2}}
\Vert l_{jk}\Vert_{0\le j,k\le2}= \left( \begin{array}{ccc}- \frac{1}{3}\frac{2}{3}\frac{1}{3}\frac{2}{3}- \frac{1}{3}- \frac{2}{3}\frac{1}{3}- \frac{2}{3}- \frac{1}{3}\end{array}\right) .
\begin{aligned}\left(1+\frac{k}{q-k+1}\right)^{q-k+1}= \left(1+\frac{k}{q-k+1}\right)^{\left(\frac{q-k+1}{k}+1/2\right)k}\left(1+\frac{k}{q-k+1}\right)^{-k/2}< e^k\cdot \left(1+ \frac{k^2}{12(q-k+1)^2}\right)^k \left(\frac{q-k+1}{q+1}\right)^{k/2}.\end{aligned}
\mathfrak{M}(q\,|\tau, 0, 0)=\frac{(2\pi)^q}{\Gamma^q\bigl(1-\frac{1}{\tau}\bigr)}\Gamma\bigl(1-\frac{q}{\tau}\bigr), M_{(\tau, 0, 0)}= \frac{2\pi \tau^{1/\tau}}{\Gamma(1-1/\tau)}\beta_{1,0}^{-1}(\tau, b_0=\tau).
det(A)=a_{1j}C_{1j}+a_{2j}C_{2j}+a_{3j}C_{3j}+\cdot\cdot\cdot+a_{nj}C_{nj}=\sum_{i=1}^{n}a_{ij}C_{ij}=\sum_{i=1}^{n}a_{ij}(-1)^{i+j}M_{ij}
\Bbb{P}\left[\sum_{i=1}^n\log\left(\frac{dQ}{dP}(X_i)\right)\ge y\right]\le\exp\left[-\frac{y}{2}\right]\rho^n(P,Q)\quad\mbox{with}\quad\rho(P,Q)=\int\sqrt{\frac{dP}{d\lambda}\frac{dQ}{d\lambda}}\,d\lambda.
\lim_{N\to \infty}\iint f(x,y)\nu_x^{(N)}(dy)\mu(dx) = \int \lim_{N\to \infty}\left( \int f(x,y)\nu_x^{(N)}(dy)\right)\mu(dx) = \int \left( \int f(x,y)\nu_x(dy)\right)\mu(dx) = \iint f(x,y)\nu_x(dy)\mu(dx).
\begin{array}{c}{{l i m}}\\{{\varepsilon \rightarrow 0}}\\ \end{array}\int D \phi \exp \{i \int d^{4}x [ \frac{1}{2}\phi ( \alpha \partial_{0}^{2}- \nabla^{2}- m^{2}+ i \varepsilon ) \phi + \frac{\lambda}{4}\phi^{4}] + i \int d^{4}x J \phi \}\bullet \left( \exp [ i \delta \alpha \int d^{4}x \phi \partial_{0}^{2}\phi ] - 1 \right)
\Phi_{\ell}^{(m)}(r, \varphi) = \mathrm{e}^{\mathrm i m \varphi}J_m( r \varkappa \ell) = \frac{\ell^{m}\varkappa^{m}}{2^{m}}\; r^{m}\; \mathrm{e}^{\mathrm{i}m \varphi}+ \mathcal O(\ell^{m+1}) & = \frac{\ell^{m}\varkappa^{m}}{2^{m}}\cdot \Psi_m (r, \varphi) + \mathcal O(\ell^{m+1}) ,
(1/f)^{\prime\prime\prime}(x)=-\frac{f^{\prime\prime\prime}(x)}{f(x)^{2}}+6\frac{f^{\prime}(x)f^{\prime\prime}(x)}{f(x)^{3}}-6\frac{f^{\prime}(x)^{3}}{f(x)^{4}}
E[e^{-2\alpha \overline{X(t)}}(e^{\alpha X(t)}-e^{\alpha X^n(t)})^2] \leq e^{-\alpha X(0)}E[(e^{\alpha X(t)}-e^{\alpha X^n(t)})^2] &= e^{-\alpha X(0)}E[e^{2\alpha X(t)}+ e^{2\alpha X^n(t)}- 2 e^{\alpha X^n(t)}e^{\alpha X(t)}].
g^{*}h_{\alpha \beta \gamma \delta}\: = \: \left( h_{\alpha \beta \gamma \delta}\right) \left( \nu_{\beta \gamma \delta}^{g}\right) \left( \nu_{\alpha \gamma \delta}^{g}\right)^{- 1}\left( \nu_{\alpha \beta \delta}^{g}\right) \left( \nu_{\alpha \beta \gamma}^{g}\right)^{- 1}
- \mathrm{T r}\left( P_{-}\Gamma_{7}\frac{\mu^{2}+ \hat{\phi}^{i}\hat{\phi}^{i}}{- ( \Gamma \cdot \Delta )^{2}+ \mu^{2}}\right) = - \int d^{3}x \ \mathrm{t r}\left( P_{-}\Gamma_{7}\Gamma_{\mu}{\partial}_{\mu}\, \langle x | ( \Gamma \cdot \Delta + \mu )^{- 1}| x \rangle \right)
\P =\left\{x\in \mathcal{B}: x \mbox{is nonnegative, nondecreasing, and}\right. &\phantom{==}\left.(y-T_1)^kx(w)\le (w-T_1)^kx(y)\mbox{for all}y,w\in[T_1,T_2]\mbox{with}y\le w \right\}.
L(f, 3) = \frac{4\pi^{3}}{9\sqrt{3}}\int_{0}^{\infty}c^{3}(e^{-2\pi u}) \sum_{k, r=1}^{\infty}\frac{\chi_{-3}(kr)}{k}\left(e^{-\frac{2\pi kr}{9u}}- e^{-\frac{2\pi kr}{3u}}\right) udu .
T \left( \vartheta | \vartheta_{1}, \vartheta_{2}, \ldots , \vartheta_{N}\right)_{a_{1}a_{2}\ldots a_{N}}^{b_{1}b_{2}\ldots b_{N}}= \prod_{j = 1}^{N}S_{b_{j}a_{j}}^{a_{j + 1}b_{j + 1}}\left( \vartheta - \vartheta_{j}\right)
k *_C A = kA + (k-1)C = kA + (k-1)(C + \{g \}) + k \{-g \}+ \{g \}= k (A + \{-g \}) + (k - 1)(C + \{g \}) + \{g \}= k *_{C + \{g \}}(A + \{-g \}) + \{g \},
\begin{gathered}\left (\eta(t) \cdot (\nabla \nabla g_{\nu (t-s)}) * \sigma (s) - (\nabla \nabla g_{\nu (t-s)}) * (\eta(t) \sigma (s) ) \right ) (x) =\int_{\mathbb{R}^d}\nabla \nabla g_{\nu (t-s)}(z) z \cdot \left ( \int_0^1 \nabla \eta ( x - (1-\lambda) z, t) d\lambda \right ) \sigma(x-z, s) dz\end{gathered}
\sum_{n=0}^m \frac{(-1)^{m-n}q^{(m-n)(m-n-1)/2}}{q^{n(n-1)/2}[m-n]_q! [n]_q!}= \frac{(-1)^m q^{m^2/2}}{[m]_q !}\sum_{n=0}^m (-1)^n q^{(m-1)n}\begin{bmatrix}a-m n\end{bmatrix}_q = &= \prod_{n=0}^{m-1}( 1- q^{2(n+1 -m)}),
Li_{s}(e^{\mu})=\Gamma(1-s)[(-2\pi i)^{s-1}\sum_{k=0}^{\infty}(k+\frac{\mu}{2\pi i})^{s-1}+(2\pi i)^{s-1}\sum_{k=0}^{\infty}(k+1-\frac{\mu}{2\pi i})^{s-1}],
{\phi}^{self}(x^{+}) =\int_0^L dy^{-}\int_0^L dz^{-}\int_0^L d{\eta}^{-}{\Phi}^{\star,++}(z^{-},x^{+}|y^{-},x^{+})i{\epsilon}(y^{-}-{\eta}^{-}){\Phi}^{++}({\eta}^{-},x^{+}|z^{-},x^{+})
B'(I_d - \tilde{C}_{(k-1)}\tilde{C}_{(k-1)}')w_k \quad=x - B'\tilde{C}_{(k-1)}\tilde{C}_{(k-1)}'w_k &\quad=x - B'\tilde{C}_{(k-1)}(t_{1,k},\ldots,t_{k-1,k})'.
\|u_{i_0}\|_{\lambda_{d}}^2 \leq \|u_{i_0}\|_{\lambda_{i_0}}^2= \mu_{i_0}|u_{i_0}|_4^4+b\sum_{{\substack{i=1 i\neq i_{0}}}}^{d-1}|u_{i}u_{i_0}|_2^2<b\sum_{i=1}^{d-1}|u_{i}u_{i_0}|_2^2
A^{\pm}=-i\sqrt{2}\cdot \alpha \left( x\right) \cdot e^{\pm \xi\left( x\right)}\left[ \sinh \rho_{\frac{s}{2}}\left( x\right)\cosh \frac{i}{2}\frac{d}{dx}\mp \cosh \rho_{\frac{s}{2}}\left( x\right)\sinh \frac{i}{2}\frac{d}{dx}\right]
\mathcal{R}_{T}^c(\alpha)=\left\{T_{np}\ge \frac{p(p-1)}{2(n-4)}\left(\sqrt{\frac{n-3}{n-6}}-1\right)+\chi^2_{\alpha}(p(p-1)/2)\sqrt{\frac{(n-3)}{(n-4)^2(n-6)}}\right\}.
\frac{\partial \mathcal{S}}{\partial T}= \frac{2 l \pi^{2}\Xi^{2}r_{+}^{5}( \Xi^{2}+ 1 ) ( r_{+}^{4}+ q^{2}l^{4})}{3 [ 4 r_{+}^{8}( \Xi^{2}- 1 ) + \Xi^{2}( q^{2}l^{4}- r_{+}^{4})^{2}+ 4 q^{2}r_{+}^{4}l^{4}]}.
\Delta^2 S_1 = \Delta^2 \sum\limits_{i=0}^{\frac{t_B}{\Delta}- 2}\left( \frac{t_A}{\Delta}+ i \right) = \left( t_B - \Delta \right) t_A + \Delta^2 \sum\limits_{i=1}^{\frac{t_B}{\Delta}- 2}i &= \left( t_B - \Delta \right) t_A + \frac{1}{2}\left( t_B - \Delta \right) \left( t_B - 2\Delta \right).
|Y_{\scriptscriptstyle{W}}^2(kr) - J_{\scriptscriptstyle{W}}^2(kr)|\,\sin^2 \Delta_{\scriptscriptstyle{l}}\leq\left(e^{a ||B||/ \sqrt \pi}+ 1\right)^2\,\frac{(M a/2)^{2W}}{\Gamma^2(W+1)}\,(1 + \mbox{O}( (Ma)^2,(Ma)^{2W}) )\,,
\int_M -\psi dd^c_B(\phi T)\wedge (\omega^T)^{n-p-1}\wedge \eta= \int_M dd^c_B(-\psi) \wedge \phi T\wedge (\omega^T)^{n-p-1}\wedge \eta \leq \int_M \phi T\wedge (\omega^T)^{n-p}\wedge \eta \leq \sup_M \phi \int_M T\wedge (\omega^T)^{n-p}\wedge \eta=(\sup_M\phi) \|T\|.
R_{\frac{1}{2}\frac{1}{2}}^{0}( \theta ) = ( 1 + A \sinh ( \theta / 2 ) ) M ( \theta ) \quad ; \qquad R_{\frac{1}{2}\frac{1}{2}}^{1}( \theta ) = ( 1 - A \sinh ( \theta / 2 ) ) M ( \theta )
\xi^{\prime}\equiv{\frac{2}{\hbar}}( x - x^{\prime}) , \quad \xi \equiv \tau -{\frac{2}{\hbar}}( x - x^{\prime}) , \quad \eta^{\prime}\equiv{\frac{2}{\hbar}}( p^{\prime}- p ) , \quad \eta \equiv \sigma -{\frac{2}{\hbar}}( p^{\prime}- p ) ,
\rho_{s}^{( N_{f})}( z , m ) = \frac{z}{2}\frac{B^{( N_{f})}( z , m ) \sum_{f = 0}^{N_{f}}\widetilde{D}_{f}^{( N_{f})}( z , m ) - D^{( N_{f})}( z , m ) \sum_{f = 0}^{N_{f}}\widetilde{B}_{f}^{( N_{f})}( z , m )}{[ C^{( N_{f})}( m ) ]^{2}\prod_{f = 1}^{N_{f}}( z^{2}+ m_{f}^{2})}\: ,
\hat{B}_{\mathrm{x},i}(x)\!=\!\begin{cases}-\ln(z_i(x))+\ln(d_{\mathrm{x}}^i)-\frac{{C_{\mathrm{x}}^i}x}{d_{\mathrm{x}}^i}z_i(x) > \delta begin{align*}0.15cm] \, \beta(z_i(x);\delta) +\ln(d_{\mathrm{x}}^i)-\frac{{C_{\mathrm{x}}^i}x}{d_{\mathrm{x}}^i}z_i(x) \leq \delta \, \end{cases}
\frac{d}{dt}|\overline{u}|^2_2 +|\phi_1 \nabla \overline{u}|^2_2\leq C|\nabla v|_\infty |\overline{u}|^2_2+C|\overline{u}|_2 |\nabla \phi_1|_\infty |\phi_1 \nabla \overline{u}|_2 \leq \frac{1}{10}|\phi_1 \nabla \overline{u}|^2_2+C(|\nabla v|_\infty +|\nabla \phi_1|^2_\infty)|\overline{u}|^2_2.
T_{s_1s_0}T_{(s_1s_0)^n}\star\varphi_m =T_{s_1s_0}\star q^{2n}\varphi_{m+n}+(q-1)\sum_{k=1}^{2n}q^{2n-k}\psi_{m+n-k}&=q^{2n+2}\varphi_{m+n+1}+(q-1)q^{2n}\left(q\psi_{m+n}+\psi_{m+n-1}\right)+(q-1)\sum_{k=1}^{2n}q^{2n-k}\psi_{m+n-1-k}&=q^{2n+2}\varphi_{m+n+1}+(q-1)\left(q^{2n+1}\psi_{m+n}+q^{2n}\psi_{m+n-1}+\sum_{k=3}^{2n+2}q^{2n+2-k}\psi_{m+n+1-k}\right),
y^{(1)}_{i}(z,\lambda) =1+\sum_{k=1}^{\infty}a_{ik}(z-z_{i})^{k},i=1,2,3, y^{(1)}_{4}(z,\lambda) =\frac{1}{z}+\sum_{k=2}^{\infty}\frac{a_{4k}}{z^{k}}\intertext{and}y^{(2)}_{i}(z,\lambda) =y^{(1)}_{i}(z,\lambda)\log(z-z_{i}) + \tilde{y}_{i}(z,\lambda),i=1,2,3, y^{(2)}_{4}(z,\lambda) =y^{(1)}_{4}(z,\lambda)\log\frac{1}{z}+\tilde{y}_{4}(z,\lambda),
|\Psi\rangle_{\nu}=|\Psi_{0}\rangle_{\nu}\oplus|\Psi_{1}\rangle_{\nu}\oplus|\Psi_{2}\rangle_{\nu}\oplus\cdot\cdot\cdot=a|0\rangle\oplus\sum_{i}a_{i}|\psi_{i}\rangle\oplus\sum_{ij}a_{ij}|\psi_{i},\psi_{j}\rangle_{\nu}\oplus\cdot\cdot\cdot,
\|\hat{Q}_r-Q_r\|_2^2=2\langle\hat Q_r,I-Q_r\rangle =\sum_{\substack{s< r_0: s\neq r}}2\langle\hat Q_r,Q_s\rangle+2\langle\hat Q_r,Q_{\geq r_0}\rangle &=\sum_{\substack{s< r_0: s\neq r}}2\|\hat{Q}_rQ_s\|_2^2+2\|\hat{Q}_rQ_{\geq r_0}\|_2^2.
W[\phi,\pi] ={\cal N}\exp\left( -\frac{1}{\hbar}\int dx\,\left( \left( \phi\left( x\right) \,\sqrt{m^{2}-\nabla_{x}^{2}}\,\phi\left( x\right)\right) + \left({\pi}\left( x\right) \,\left(\sqrt{m^{2}-\nabla_{x}^{2}}\right)^{-1}\!{\pi}\left( x\right) \right) \right) \right) ,
v_1 :\; s_1s_3s_7=-1 ~,~~ v_4 :\; s_7=-1 \ , v_2 :\; s_1s_2s_9=1 ~,~~ v_5 :\;s_{11}=1 \ , v_3 :\; s_2s_3s_{11}=1~,~~ v_6 :\;s_{9}=1 \ .
s(x)=s^{(1)}(x)\footnote{In $s^{(1)}(x), \ (1)$ indicates the denominator is $p(x|t_1).$}=\left ( \frac{p(x|t_2)}{p(x|t_1)}, \frac{p(x|t_3)}{p(x|t_1)}, \ldots, \frac{p(x|t_k)}{p(x|t_1)}\right )
S_{\delta}= S_{\delta}(H, M, N):= \sum_{h\sim H}\sum_{m\sim M}\sum_{n\sim N}a_{h, m}b_n{\rm e}\bigg(X\frac{M^{\beta}N^{\gamma}}{H^{\alpha}}\frac{h^{\alpha}}{m^{\beta}n^{\gamma}+\delta}\bigg),
L ( t ) : = \sum_{l = 1}^{\infty}\frac{1}{2}\frac{1}{1 + \cosh l t}\, \frac{1}{l}, \quad \widetilde{K}( s_{1}) : = s_{1}^{3 N^{2}/ 2 - 1}\, K_{3 N^{2}- 4}( 2 \sqrt{s_{1}}) ,