harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\int_{0}^{2 \pi}\frac{d \theta}{2 \pi}e^{- | z |^{2}}\, \prod_{\pm}\left\{e^{x e^{\pm i \theta}| z_{2 M + 1}^{\pm}|^{2}}\, \prod_{\mu = 1}^{M}e^{x e^{\pm i \theta}\left[ e^{i \omega_{\mu}}| z_{\mu +}^{\pm}|^{2}+ e^{- i \omega_{\mu}}| z_{\mu -}^{\pm}|^{2}\right]}\right\}\ \ \ ,
g_{i j}( x , t ) = h_{i j}( x ) e^{2 \lambda ( x , t )}= \frac{\cosh^{2}t}{2 \Lambda}\frac{\partial \phi_{\xi}^{k}( x )}{\partial x^{i}}\frac{\partial \phi_{\xi}^{l}( x )}{\partial x^{j}}h_{k l}\left( \phi_{\xi}( x ) \right) ,
r(y_1,y_2) := 1+\frac{1}{4}(\cos(2\pi y_1)-2\sin(2\pi y_1))\sin(2\pi y_2) = r_1(y_1+y_2) + r_2(y_1 - y_2), a(y_1,y_2) := 1-\frac{1}{2}\sin(2\pi y_1)\sin(2\pi y_2)
S=\int dx^D\sqrt{g_{D}}\left[-\frac{M^{D-2}_*}{16\pi}(R+2\Lambda)-\frac12 g^{AB}\nabla_A{\bf \Phi}\cdot \nabla_B{\bf \Phi}- \frac{\lambda}{4}({\bf \Phi}^2 -v^2)^2 \right],
\Psi_{\lambda_{i}}( Z_{i}, \tau ) = \frac{e x p \left[ i ( \frac{\dot{\sigma}}{2 \hbar \sigma_{i}}Z_{i}^{2}+ \alpha_{\lambda{i}}( \tau ) + \frac{\lambda_{i}Z_{i}}{\sigma_{i}}) \right]}{\sigma_{i}^{1 / 2}}
\begin{aligned}\operatorname{tr}(\psi(h_1))^2-4\det(\psi(h_1)) =\left(2+2^{2n_1}+O(2^{3n_1})\right)^2-4 = 2^{2n_1+2}\left( 1+ O\left(2^{n_1}\right) \right)\end{aligned}
\P_{s,x}(X_{s+ u_{0}+ \gamma}\in \cdot | \tau_{h}>s+ u_{0}+ \gamma) = \phi_{s+u_0+\gamma, s}(\delta_x) &= \phi_{s+u_0+\gamma,s+g(s)}\circ \phi_{s+g(s),s}(\delta_x) &= \P_{s+g(s), \phi_{s+g(s),s}(\delta_x)}(X_{s+u_0+\gamma}\in \cdot | \tau_h > s+ u_0 + \gamma).
\int_{a}^{b}f(x)dx=A_{0}(\frac{h}{2^{n}})+a_{0}(\frac{h}{2^{n}})^{2}+a_{1}(\frac{h}{2^{n}})^{4}+a_{2}(\frac{h}{2^{n}})^{6}+\cdot\cdot\cdot
\mathbb{E}(X_{p}Y_p) =1 + \sum_{i=1}^{\alpha}(\tau_k(p^i)\tau_{\ell}(p^i)-\tau_k(p^{i-1})\tau_{\ell}(p^{i-1}))X^i + \sum_{i=\alpha+1}^{\infty}(\tau_k(p^{\alpha}) \tau_{\ell-1}(p^i) + \tau_{\ell}(p^{\alpha}) \tau_{k-1}(p^i))X^i.
|p_{(a_1,\cdots,a_k)}|^2-|p_{(a_1,\cdots,a_k,a_{k+1})}|^2-|p_{(a_1,\cdots,a_k,a_{k+2})}|^2+|p_{(a_1,\cdots,a_k,a_{k+1},a_{k+2})}|^2=0.
\frac{t_0+f_2(\mathbf{S},\mathbf{\Phi}')}{\bar{f}_1(\mathbf{S},\mathbf{\Phi}')}=\left(t_0+f_2(\mathbf{S},\mathbf{\Phi}')\right)\prod_{k=1}^{K_1}\left(\frac{g_{1k}}{\gamma_{1k}}\right)^{-\gamma_{1k}}\leq 1
\begin{pmatrix}a\phi_{d_0,d_1}b\phi_{d_0,d_2}\phi_{d_3,d_1}\phi_{d_3,d_2}\end{pmatrix}\! \colon\{\cal E}_{d_1}\times{\cal E}_{d_2}\to{\cal E}_{d_0}\times{\cal E}_{d_3}
ch_{h}(\iota^{*}_{D}B_{i,m,n})=ch_{h}(\iota^{*}_{D}L^{m}_{i})ch_{h}(\wedge^{i}\iota^{*}_{D}W^{-}_{i})ch_{h}(\iota^{*}_{D}S^{q_{i}-(i)}U_{i}).
\begin{array}{rl}G(x_1,x_2,x_3,x_4) = \displaystyle\int d^4x_5\int d^4x_6<0|T\left(\pi_1\pi_2\chi_3\chi_4(\pi\star\sigma\star\pi)_5(\chi\star\sigma\star\chi)_6\right)|0>,\end{array}
g_{{\rm SO}(M)}\big(\Pi^P(X^{h'},Y^{h'}),\alpha^{+}\big) =\frac{1}{2}{\bf B}((\nabla_X\xi)_Y+(\nabla_Y\xi)_X-\xi_{R_{\xi_X}(Y)+R_{\xi_Y}(X)},\alpha), g_{{\rm SO}(M)}\big(\Pi^P(X^{h'},\gamma^{\ast}),\alpha^{+}\big) =\frac{1}{2}{\bf B}([\xi_X,\gamma]_{\mathfrak{m}}-\xi_{R_{\gamma}(X)},\alpha), g_{{\rm SO}(M)}\big(\Pi^P(\beta^{\ast},\gamma^{\ast}),\alpha^{+}\big)=0.
y_{i,j}+ y_{mr+1-i,n+1-j}= \tfrac{1}{2}mnr +\tfrac{3}{2}, \mbox{\if}i+j \mbox{is even, and}y_{i,j}+ y_{mr+1-i,n+1-j}= 2MN -\tfrac{1}{2}mnr +\tfrac{3}{2}, \mbox{\if}i+j \mbox{is odd.}
Z_{1 / R}( 1 , 0^{1 5}) \to \int \frac{d^{2}\tau}{\tau_{2}}\tau_{2}^{- 5}| \eta ( \tau ) |^{- 1 6}\left( Z_{0}^{0}( \tau )^{1 6}{Z_{0}^{0}( \tau )^{*}}^{4}- Z_{1}^{0}( \tau )^{1 6}{Z_{1}^{0}( \tau )^{*}}^{4}- Z_{0}^{1}( \tau )^{1 6}{Z_{0}^{1}( \tau )^{*}}^{4}\right) \ ,
\begin{array}{lll}F^{ij}D_\xi w_{ij}= F^{ij}\left[D_{ij}u_\xi-D_\xi A_{ij}-(D_{p_k}A_{ij})D_ku_\xi\right] = D_\xi\tilde{B}+(D_{p_k}\tilde{B})D_ku_\xi. \end{array}
\|m_{Q^{(-\frac{1}{2}+\theta)(1+\mathrm i t)}a^*Q^{(\frac{1}{2}-\theta)(1+\mathrm i t)}}^R\|_{B(L^p(\mathbb G))}=\|\sigma_{(-\frac{1}{2}+\theta)t}\circ m_{Q^{-\frac{1}{2}+\theta}a^*Q^{\frac{1}{2}-\theta}}^R \circ \sigma_{(\frac{1}{2}-\theta)t}\|_{B(L^p(\mathbb G))}&=\|m_{Q^{-\frac{1}{2}+\theta}a^*Q^{\frac{1}{2}-\theta}}^R\|_{B(L^p(\mathbb G))}.
C= C_1 \oplus C_2 = \begin{pmatrix}0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 \end{pmatrix}V = V_1 \cup V_2 = \{(0,1,0,0),(1,0,0,0), (0,0,0,2),(0,0,1,1),(0,0,2,0)\}.
\begin{array}{c}\displaystyle j_s^*x^i = x^i ; j_s^*u^\alpha = u^\alpha ; j_s^*u_i^\alpha = u_i^\alpha; j_s^*p_\alpha^i = p_\alpha^i \, , begin{align*}10pt]\displaystyle j_s^*p_\alpha^{ij}= \frac{1}{n(ij)}\, p_\alpha^{1_i+1_j}\, , \mbox{where}n(ij) = \begin{cases}1 \, , \mbox{if}i=j 2 \, , \mbox{if}i \neq j \end{cases}\end{array}
A_j = \left(\begin{array}{cccc}\Phi (\boldsymbol{x}^j_1,\boldsymbol{x}^j_1 ) \cdots \Phi(\boldsymbol{x}^j_1,\boldsymbol{x}_{N_j}^j ) \vdots \ddots \vdots \Phi (\boldsymbol{x}^j_{N_j},\boldsymbol{x}^j_1) \cdots \Phi (\boldsymbol{x}^j_{N_j},\boldsymbol{x}^j_{N_j}) \end{array}\right).
\begin{array}{l l l}{{\mathrm{r e g}\, \, \varepsilon}}&{{=}}&{{\mathrm{r e g}\, \, p / D = \varepsilon_{c}^{( 1 )}}}\\{{\mathrm{r e g \,}p_{\perp}}}&{{=}}&{{- \mathrm{r e g}\, \, \varepsilon ,}}\\ \end{array}
\frac{- 1}{r}\frac{\partial ( r V_{\theta}^{( 2 )})}{\partial r}= q_{1}\gamma F^{2}( H^{2}- 1 ) \; \; , \; \; \gamma = \frac{2 q_{1}q_{2}c_{1}^{2}c_{2}^{2}}{\kappa^{2}}\; \; .
{\frac{m}{M_{P l}}}={\frac{( 3 \pi P_{k}^{S})^{1 / 2}}{2 L}}\left[{\frac{( 1 + u^{2})^{3 / 4}}{( 1 + 3 u^{2})^{1 / 4}}}\right] \approx 2 . 6 \times 1 0^{- 7}\left( 1 0^{1 0}P_{k}^{S}\right)^{1 / 2}\left({\frac{6 0}{L}}\right) \left[{\frac{( 1 + u^{2})^{3 / 4}}{( 1 + 3 u^{2})^{1 / 4}}}\right] ;
\sum_{k = 1}^{r - 1}{}_{r - 1}C_{k}\hat{h}_{i}^{( k + 1 )}\hat{H}_{i}^{( r - k )}= \hat{h}_{i}^{( r )}\hat{H}_{i}^{( 1 )}+ \sum_{k = 2}^{r - 1}{}_{r - 1}C_{k - 1}\hat{h}_{i}^{( k )}\hat{H}_{i}^{( r + 1 - k )}.
\sum_{n=0}^{\infty}\textup{PDO}_\textup{t}(2n+1)q^n =\dfrac{f_2^4f_6^4}{f_4f_{12}}\cdot\dfrac{1}{f_1^4}&=\dfrac{f_2^4f_6^4}{f_4f_{12}}\cdot\left(\dfrac{f_4^{14}}{f_2^{14}f_8^4}+4q\dfrac{f_4^2f_8^4}{f_2^{10}}\right),
\overbrace{[x+a, y+b]}= (x+a)\star (y+b) -(y+b)\star (x+a) = x\ast y+ \mathfrak{l}(x)b+ \mathfrak{r}(y)a - y\ast x -\mathfrak{l}(y)a - \mathfrak{r}(x)b = [x, y] + (\mathfrak{l}-\mathfrak{r})(x)b - (\mathfrak{l}-\mathfrak{r})(y)a.
\partial_t u_{i}^{(n)}= \tilde{L}_k u_{i}^{(n)}+ \frac{\delta_i}{Z_{k,i}}\sum_{r \in \nu_k^{-1}(i)}\delta_r\Bigl( \prod_{j=1}^q (u_j^{(n-1)})^{\alpha_j^{(r)}}\, u_{r}^{(n)}-\prod_{j=1}^q (u_j^{(n-1)})^{\beta_{j}^{(i)}}\, u_{i}^{(n)}\Bigr).
(\eta_R-\eta_L)\left( (v_1^A)^n/\pi^{a(n)}\right) = \binom{n}{b(n)}(v_1^A)^{n-b(n)}\pi^{b(n)-a(n)}(t_1^A)^{b(n)}\in (v_1^A)^{-1}V^A/\pi \otimes_{V^A}V^AT,
\left( \begin{array}{c}{{z^{\prime}}}\\{{\bar{z}^{\prime}}}\\{{\bf 1}}\\ \end{array}\right) ={\bf M}\left( \begin{array}{c}{{z}}\\{{\bar{z}}}\\{{\bf 1}}\\ \end{array}\right) \equiv \left( \begin{array}{c c c}{{u}}&{{0}}&{{t}}\\{{0}}&{{\bar{u}}}&{{\bar{t}}}\\{{0}}&{{0}}&{{{\bf 1}}}\\ \end{array}\right) \left( \begin{array}{c}{{z}}\\{{\bar{z}}}\\{{\bf 1}}\\ \end{array}\right) \ .
SO_{\alpha}(G^*)-SO_{\alpha}(G^{*'})= \left(d_{G^*}^2(v_{i-1})+9\right)^\alpha-\left(d_{G^*}^2(v_{i-1})+4\right)^\alpha+2\cdot13^\alpha+ 2\cdot8^\alpha- 3\cdot 20^\alpha -17^\alpha < 10^\alpha-5^\alpha + 2 \cdot 13^\alpha+ 2\cdot8^\alpha-3\cdot 20^\alpha-17^\alpha<0.
y_{1}^{3}-y_{1}=r(T), y_{1}=y^{9}+y^{3}+y. y_{2}^{3}-y_{2}=\omega r(T), y_{2}=(\omega y)^{9}+(\omega y)^{3}+\omega y. y_{3}^{3}-y_{3}=\omega^{2}r(T), y_{3}=(\omega^{2}y)^{9}+(\omega^{2}y)^{3}+\omega^{2}y.
s_{(2,1,1)}(x_{1},x_{2},x_{3})=\frac{1}{\Delta}det[\begin{matrix}x_{1}^{4}&x_{2}^{4}&x_{3}^{4}\\x_{1}^{2}&x_{2}^{2}&x_{3}^{2}\\x_{1}&x_{2}&x_{3}\end{matrix}]=x_{1}x_{2}x_{3}(x_{1}+x_{2}+x_{3})
\lambda\left( A_{\alpha}\right) =\max_{\left\Vert \mathbf{x}\right\Vert_{2}=1}\left\langle A_{\alpha}\mathbf{x},\mathbf{x}\right\rangle \text{\ \and \ \ \}\lambda_{\min}\left( A_{\alpha}\right) =\min_{\left\Vert\mathbf{x}\right\Vert_{2}=1}\left\langle A_{\alpha}\mathbf{x},\mathbf{x}\right\rangle .
\lambda (\lambda' v) = x_{\alpha}x_{-\alpha, \lambda}x_{\alpha, \lambda'}x_{-\alpha}v & = x_{\alpha}h_{\alpha, \lambda \lambda'}x_{-\alpha}v & = - 2 x_{\alpha, \lambda \lambda'}x_{-\alpha}v + h_{\alpha, \lambda \lambda'}x_{\alpha}x_{-\alpha}v & = 2 h_{\alpha, \lambda \lambda'}v - h_{\alpha, \lambda \lambda'}v &= (\lambda \lambda') v
T_{n}(q) =\sup_{0<F_0(t)<1}\frac{\sqrt{n}|\mathbb{F}_{n}(t)-F_{0}(t)|}{q(F_{0}(t))},T_{n}^{+}(q) =\sup_{0<F_0(t)<1}\frac{\sqrt{n}\left( \mathbb{F}_{n}(t)-F_{0}(t) \right )}{q(F_{0}(t))},
\gamma_1 =\{(u, 0)|u\ge i_0\}, \gamma_2 =(i_0, 0), \gamma_3 =\{(u, v)|j_0u+i_0v=i_0j_0, 0\le u\le i_0, 0\le v\le j_0\}, \gamma_4 =(0, j_0), \gamma_5 =\{(0, v)|v\ge j_0\}
\Gamma(t):=e^{\sum_{i=1}^N\left[\int_0^tB_i(s)d\beta_i+\beta_i\theta_i-\frac{1}{2}\int_0^tB_i^2(s)ds-\frac{1}{2}\theta_i^2t-\theta_i\int_0^tB_i(s)ds\right]}
\begin{array}{l}{{c_{2 a}f_{-}( u_{+}){\bf D}_{a + 1}{\bf D}_{a + 1}^{'}+ c_{1 a}f_{+}( u_{-}){\bf D}_{a + 1}{\bf U}_{a - 1}^{'}+ f_{-}{\bf X}_{a , a + 2}^{a + 1}{\bf X}_{a + 2 , a}^{' a + 1}}}\\{{\mathrm{}= c_{1 a}f_{-}( u_{+}){\bf U}_{a - 1}^{'}{\bf U}_{a - 1}+ c_{2 a}f_{+}( u_{-}){\bf D}_{a + 1}^{'}{\bf U}_{a + 1}+ f_{-}{\bf X}_{a , a - 2}^{' a - 1}{\bf X}_{a - 2 , a}^{a - 1}}}\\ \end{array}
\varphi^{(p,q)}(\zeta)=\textstyle\frac{(-1)^p}{q(p+q-1)!}\,\displaystyle(1-\zeta\bar{\zeta})\frac{\partial^{\,p+q}}{\partial\bar{\zeta}^p\partial\zeta^q}(1-\zeta\bar{\zeta})^{p+q-1}.
\tilde\delta_n=\tilde\kappa_g'\tilde\kappa_t-\tilde\kappa_g\tilde\kappa_t',\tilde\delta_n'=\tilde\kappa_\nu'(\tilde\kappa_g^2+\tilde\kappa_t^2)+\tilde\kappa_g''\tilde\kappa_t-\tilde\kappa_g\tilde\kappa_t''u=0
a^L_i := (\-a^L_i \vee \bigvee_{s \in B_i \cap S}\-a^L_{s,i}) \wedge \bigwedge_{s \in S \setminus B_i}\-a^L_{s,i}, a^R_j := (\-a^R_j \wedge \bigwedge_{s \in C_j \cap S}\-a^R_{s,j}) \vee \bigvee_{s \in S \setminus C_j}\-a^R_{s,j}.
\begin{aligned}\frac{\phi_N^{(s)}(t,Z_s^*)}{N-s+1}+ \phi_N^{(s-1)}(t,(Z_s^*)^{(i)})+ \phi_N^{(s-1)}(t,(Z_s^*)^{(j)}) = & = \frac{\phi_N^{(s)}(t,Z_s)}{N-s+1}+ \phi_N^{(s-1)}(t,Z_s^{(i)}) + \phi_N^{(s-1)}(t,Z_s^{(j)})\end{aligned}
x^{k+1}= x^k - \lambda \left( \nabla f_{i_k}(x^k) - \nabla f_{i_k}(\widetilde{x}^k) + \frac{1}{N}\sum_{i=1}^N \nabla f_i(\widetilde{x}^k)\right); \widetilde{x}^{k+1}= \begin{cases}x^{k - t_k}; \widetilde{x}^{k+1}\end{cases}
\Bigl\|P_{\mathcal H_0}T\sum_{n\geq 0}a_ne_{2n+1}\Bigr\|^2 =\sum_{n\geq 0}|a_nc_n(\lambda_{2n+1}-\lambda_{2n})|^2 &\leq(\sup_{n\geq 0}c_n|\lambda_{2n}-\lambda_{2n+1}|)^2\sum_{n\geq 0}|a_n|^2.
r^{\prime}( t ) = r ( t ) \ \ , \ \ \theta^{\prime}( t ) = \theta + \alpha ( t ) \ \ , \ \z^{\prime}( t ) = z ( t ) + \frac{1}{g}\alpha ( t ) \ \ , \ \ \xi^{\prime}( t ) = \xi ( t ) + \frac{1}{g}\dot{\alpha}( t ) \ \ .
p_{a}(D^{n}\varphi):= \max_{\alpha\in\mathcal{A}}\sup_{x\in(l_\alpha,r_\alpha)}\max\{|D^{n+1}\varphi(x)(x-l_\alpha)^{1+a}|,|D^{n+1}\varphi(x)(r_\alpha-x)^{1+a}|\}
I \leq t^{1-\eta-\theta}\left[h^{-1}(1/t)\right]^{\gamma+\beta}\int_0^1 s^{\gamma/2-\eta}(1-s)^{\beta/2-\theta}\, ds &= B(\beta/2+1-\theta, \gamma/2+1-\eta)t^{1-\eta-\theta}\left[h^{-1}(1/t)\right]^{\gamma+\beta}\, .
f_{T}(\mathbf{x}|K_{n}) =\sum_{S}(\prod_{1\leq i<j\leq n}\left( \rho_{ij}\right)^{s_{ij}})\prod_{m=1}^{n}T_{\sigma_{m}}(x_{m}), f_{U}(\mathbf{x}|K_{n}) =\sum_{S}(\prod_{1\leq i<j\leq n}\left( \rho_{ij}\right)^{s_{ij}})\prod_{m=1}^{n}U_{\sigma_{m}}(x_{m}),
\times \prod_{b = 1}^{N}\prod_{j = 1}^{m_{b}}\exp \left( - 2 e \sum_{I = 1}^{N}\Big ( \varepsilon_{\mu \nu}\partial_{\mu}A_{\nu}^{( I )}, K_{I I}U_{I b}\; \delta ( y_{j}^{( b )}) \Big ) \right)
\frac{1}{n}\int_{-1}^{1}\frac{1}{T(a_nt)}\frac{a_ntQ'(a_nt)}{\pi\sqrt{1-t^2}}\, dt=\frac{1+o(1)}{\alpha}\frac{1}{n}\int_{-1}^{1}\frac{a_ntQ'(a_nt)}{\pi\sqrt{1-t^2}}\, dt=\frac{1+o(1)}{\alpha}.
{\rm ind}_0^{D=4}({\rm G}_2)=\frac{1}{12}~\frac{1}{E^2}~\oint \frac{d x_1}{2 \pi i}\frac{d x_2}{2 \pi i}\frac{(x_1-x_2)^2 (x_1+x_2)^2 x_1^2 x_2^2}{\Big[(x_1-x_2)^2-E^2 \Big]\Big[ (x_1+x_2)^2-E^2 \Big] \Big[ x_1^2-E^2 \Big] \Big[ x_2^2-E^2 \Big]}\times \makebox[5cm]{}
(L\chi)(t,q,z)= \frac{1}{2}\Sigma_{kl}(t,q)(\partial_{z_k}\partial_{z_l}\chi)(t,q,z)-\tilde\gamma_{kl}(t,q)\delta^{li}z_i(\partial_{z_k}\chi)(t,q,z), (L^*h)(t,q,z)= \partial_{z_k}\bigg(\frac{1}{2}\Sigma_{kl}(t,q)\partial_{z_l}h(t,q,z)+\tilde\gamma_{kl}(t,x)\delta^{li}z_ih(t,q,z)\bigg).
\sum_{i_1=N+1}^{N+p-1}\bigg(\binom{N+p-1}{i_1}x_1^{i_1}\sum_{i=0}^{p-2}\frac{(-1)^i(\sum_{j=2}^{p-i}x_j)^{N-i_1+p-1}}{\prod_{j=2}^{p-i}(\sum_{k=j}^{p-i}x_k)\cdot \prod_{j=p-i+1}^{p}(\sum_{k=p-i+1}^j x_k)}\bigg)=\frac{(-1)^{p-2}x_1^{N+p-1}}{\prod_{j=2}^{p}(\sum_{k=1}^j x_k)}.
\omega_{( - )}\partial_{-}\omega_{( - )}^{- 1}= \left( g_{( 0 )}^{- 1}y_{( 0 ) +}\right)^{- 1}L_{( - 1 )}g_{( 0 )}^{- 1}y_{( 0 ) +}= y_{( 0 ) +}^{- 1}N_{( - 1 )}^{- 1}\partial_{-}N_{( - 1 )}y_{( 0 ) +}.
{\frac{1}{g_{1}^{2}( q^{2})}}-{\frac{1}{g_{2}^{2}( q^{2})}}\simeq \mathrm{b d . t e r m s}+{\frac{1}{2 4 \pi^{2}}}\ln \left({\frac{\sqrt{q^{2}}}{k}}\right) .
\bar{c}_i = \min \{c \in \mathbb{Z}^+ \colon c n_i d_i\}= d_i / d_{i+1}= a_i, c_i^* = \min \{c \in \mathbb{Z}^+ \colon c n_i \in \langle n_1, \ldots, n_{i-1}\rangle \}= a_i.
W_m^*(r)-\frac{r}{\lambda}W_m^*(\lambda) =\int J(x)dF(x)\left\{\left(Z(r)-rZ(1)\right)-\frac{r}{\lambda}\left(Z(\lambda)-\lambda Z(1)\right)\right\}&=\int J(x)dF(x)\left\{Z(r)-\frac{r}{\lambda}Z(\lambda)\right\}
\begin{aligned}\phi_q^{1111}= 0, \phi_q^{211}= 0, \phi_q^{22}= 0, \phi_q^{31}= q^2 + q^3, \phi_q^4 = 1 + 3q + 4q^2 + 4q^3 + 3q^4 + q^5. \end{aligned}
\begin{aligned}\int_K \int_{K_2}|x-y|^{-(\alpha+d-2)}|x|^{\alpha-d}|y|^{\alpha-d}dydx &\leq \left(\int_K\int_{K_2}\frac{|x|^{2(\alpha-d)}}{|x-y|^{\alpha+d-2}}dydx\right)^{1/2}\cdot\left(\int_K\int_{K_2}\frac{|y|^{2(\alpha-d)}}{|x-y|^{\alpha+d-2}}dydx\right)^{1/2}&\leq \int_K|x|^{2(\alpha-d)}dx\int_{|y|\leq\varepsilon}|y|^{-(\alpha+d-2)}dy<\infty.\end{aligned}
I_{0}^{(2)}\left(-\frac{31}{19}\right)=\frac{19}{9\cdot{31^{4}}}\left(-2268214\cdot{e^{-\frac{31}{19}}}+80048\cdot{e^{\frac{31}{19}}}\right)<0, &I_{1}^{(2)}\left(-\frac{31}{19}\right)=\frac{62}{855}>0.
\begin{aligned}\frac{d}{dt}\star_t\varphi (t) =\frac{A\sqrt{6}\big(y(t)^3 z(t)^2-y(t) z(t)^4\big)-1}{3\,y(t)^2 z(t)^4}(e^{1234}+ e^{1256}) -\frac{2A\sqrt{6}\,y(t) z(t)^4+1}{3\,y(t)^2 z(t)^4}e^{3456},\end{aligned}
= (\gamma^{-1})^{ij}(t,q)\left(-\nabla_q V(t,q)+\tilde F(t,q)\right)_j &+ \beta^{-1}\partial_{q^k}(\gamma^{-1})^{ik}&-(\gamma^{-1})^{ij}\partial_{q^j}\beta^{-1}&-\beta^{-1}\partial_{q^j}(\gamma^{-1})^{ij}= (\gamma^{-1})^{ij}(t,q)\left(-\nabla_q V(t,q)+\tilde F(t,q)\right)_j &-(\gamma^{-1})^{ij}\partial_{q^j}\beta^{-1}
(\mathbf{C}/(\mathbf{Z}+ \frac{-1}{pz}\cdot \mathbf{Z}), <\frac{1}{p}>, (\frac{-1}{2pz}, \frac{1}{2})) = (\mathbf{C}/(z\cdot \mathbf{Z}+ \frac{1}{p}\cdot \mathbf{Z}), <\frac{z}{p}>, (\frac{-1}{2p}, \frac{z}{2})) &=\epsilon(w_p(\mathbf{C}/(\mathbf{Z}+z\cdot \mathbf{Z}), <\frac{1}{p}>, (\frac{z}{2}, \frac{1}{2})))
\langle \eta_t^\varepsilon(x_1), A_{1}^{}\rangle =E^{}+E^{}+E^{}+E^{}+\sum_{i=1,2}\tilde{\Theta}^{\varepsilon_1,\mathbf{e}_i}_{1,r}\star\left[\operatorname{div}(\mathbf{B}_t)\right](x_i),
S_{E}^{( 5 )}= - \sum_{j = 0}^{1}( - 1 )^{j}\left[ \frac{z}{\tilde{\rho}_{0}\sinh^{2}\left( z / \tilde{\rho}_{0}\right)}- \coth \left( z / \tilde{\rho}_{0}\right) + \frac{4}{1 5}\cosh \left( z / \tilde{\rho}_{0}\right) \sinh \left( z / \tilde{\rho}_{0}\right) \right] S_{E}^{( 5 ) j},
\cos_q z :=\frac{(q^2;q^2)_\infty}{(q;q^2)_\infty}(z)^{1/2}J^{(1)}_{-1/2}(2 z;q^2) = \sum_{n=0}^{\infty}(-1)^n \frac{z^{2n}}{[2n]_q!}.
|\psi(x)\rangle= \,\frac{1}{\sqrt{N}}\,\sum^N_{k=1}\,\int_{\varphi^{-1}_{k*}(T_xM_4)}\,d^4z_k\,e^{\imath\,\vartheta_{k\Psi}(\varphi^{-1}_{k}(x),z_k)}& \,n_{k\Psi}(\varphi^{-1}_{k}(x),z_k)\,|\varphi^{-1}_{k}(x),z_k\rangle.
\xi_{2}=\frac{1}{rR^{3}}\left(\left(r_{3}\tau-\frac{p r_{2}^{2}r_{3}^{2}}{r}\right)x^{i},\left(r_{3}\tau-\frac{p r_{2}^{2}r_{3}^{2}}{r}\right)y^{i},\left(r \tau -\sqrt{\Delta}r_{2}^{2}r_{3}-\frac{rR^{2}\sigma}{r_{3}^{2}}\right)z^{j}\right),
\|X\omega u\|^2 \leq 2(\|\omega\|_\infty^2\|g u\|^2+\|g\|_\infty^2\|P_-\overline\theta\omega u\|^2)+\|X_0\|^2\|P_-\overline\theta\omega u\|^2 &=2\|\omega\|_\infty^2\|g u\|^2+(2\|g\|_\infty^2+\|X_0\|^2)\|P_-\overline\theta\omega u\|^2.
x_{\sigma(\iota)}= w_{j_k,i_{\iota}}\bigl(1+\frac{\ell(i_{\iota})}{\mu_{i_{\iota}}}\bigr)\Biggl(\frac{y_{\sigma(\iota)}}{w_{j_k,i_{\iota}}\bigl(1+\frac{\lambda_{\ell(i_{\iota})}}{\mu_{i_{\iota}}}\bigr)}-\frac{y_{\sigma(\iota_{i^*_k})}}{w_{j_k,i^*_k}\bigl(1+\frac{\lambda_{\ell(i^*_k)}}{\mu_{i^*_k}}\bigr)}\Biggr);
R_{x}^{\alpha}\psi_{n}(x)\ =-\frac{1}{2\Gamma(2-\alpha)\cos\alpha\pi/2}\left[\frac{d^{2}}{dx^{2}}\int_{-a}^{a}(x-x^{\prime})^{-\alpha+1}\psi_{n}(x^{\prime})dx^{\prime}\right] ,x\geq a.
\begin{aligned}\widetilde{V}_k = \bigcup \{\psi_{\boldsymbol{i}}(V_0) : \boldsymbol{i}\in \Lambda_k \}, \widetilde{E}_k = \bigcup \{\psi_{\boldsymbol{i}}(E_0 ) : \boldsymbol{i}\in \Lambda_k \}, \widetilde{G}_k = (\widetilde{V}_k, \widetilde{E}_k).\end{aligned}
\frac{\partial}{\partial t}R_{ijkl}=\frac{1}{2}(\nabla_{j}\nabla_{k}v_{il}+\nabla_{i}\nabla_{l}v_{jk}-\nabla_{i}\nabla_{k}v_{jl}-\nabla_{j}\nabla_{l}v_{ik})+\frac{1}{2}{R_{ijk}}^{p}v_{pl}-\frac{1}{2}{R_{ijl}}^{p}v_{pk}
\prod_{u = 1}^V{N_u^{(N)}\choose n_u}\left(\rho_u^{(N)}p_u^{(N)}\right)^{n_u}\left(1-\rho_u^{(N)}p_u^{(N)}\right)^{N_u^{(N)}- n_u}&=\ \prod_{u = 1}^V \frac{N_u^{(N)}\cdots (N_u^{(N)}- n_u + 1)}{n_u!}\left(\rho_u^{(N)}p_u^{(N)}\right)^{n_u}
\bf{U}= \left( \begin{array}{cc}0 x^2 - u(t)^2 \frac{\frac{l^2 - \frac{1}{4}}{x^2}+ \omega^2 x^2 -\frac{l^2}{u(t)^2}-\omega^2 u(t)^2 - \dot{u}(t)^2- \frac{1 - 2iu(t) \dot{u}(t)}{x^2 - u(t)^2}}{x^2 - u(t)^2}0 \end{array}\right) ,
\mathcal{I}_1:=\int_{B_R}\int_{B_R}\frac{\Big(J_p(u_h(x)-u_h(y))-J_p(u(x)-u(y))\Big)}{|h|^{\gamma+t}}\,\Big(\delta_h u(x)\,\eta(x)^p-\delta_h u(y)\,\eta(y)^p\Big)\,d\mu(x,y),
C_0:=\sum_{i=1}^{k-1}h_i^2+\sum_{i=1}^{w}x_ix^*_i+\sum_{i=1}^{w}x^*_ix_i+\sum_{i=1}^{\ell}y_iy^*_i-\sum_{i=1}^{\ell}y^*_iy_i
S\times gU\to H,(s,gu)\mapsto f(s)(gu) =f(s)(g)f(s)(u) &=f(s)\bigl(\exp_G(X)\bigr)f(s)\bigl(\exp_G(\exp_G^{-1}(u))\bigr) &=\exp_H\bigl(\mathrm Lf(s)X\bigr)\exp_H\bigl(\mathrm Lf(s)\exp_G^{-1}(u)\bigr) &=\exp_H\bigl(\lambda(s)X\bigr)\exp_H\bigl(\lambda(s)\exp_G^{-1}(u)\bigr)
(\sum_{k=1}^{\nu}Y_{k}X_{k}^{j+\nu})+(\sum_{k=1}^{\nu}\Lambda_{1}Y_{k}X_{k}^{j+\nu-1})+(\sum_{k=1}^{\nu}\Lambda_{2}Y_{k}X_{k}^{j+\nu-2})+\cdot\cdot\cdot+(\sum_{k=1}^{\nu}\Lambda_{\nu}Y_{k}X_{k}^{j})=0
Q'_2(0)=\sum\limits_{r=1}^{\min(n_1,\Delta n)}\sum\limits_{s=0}^{n_1}\sum\limits_{t=0}^{\Delta n-1}\sum\limits_{\substack{\mathbf{c}\in [1,n_1]_<^r \mathbf{c'}\in [1,n_1]_<^s}}\sum\limits_{\substack{\mathbf{d}\in [n_1+2,n_2]_<^{r-1}\mathbf{d'}\in [n_1+2,n_2]_<^t}}\det U(\mathbf{c},\mathbf{c'},\mathbf{d},\mathbf{d'}).
W ={\frac{m^{3}N^{2}}{2 4 g_{\mathrm{Y M}}^{2}}}\left[ E_{2}( \tau ) -{\frac{q}{p}}E_{2}\left({\frac{q}{p}}\tau \right) \right] \to{\frac{m^{3}N^{2}E_{2}( \tau )}{9 6 \pi g}}-{\frac{m^{3}N^{3}}{9 6 \pi g p^{2}}}\ .
z_{2}~ = ~ \frac{3 \mu^{2}( \mu - 1 ) \Gamma ( \mu ) ( \eta_{1}^{\mathrm{o}})^{2}}{2 ( \mu - 2 )^{2}( 2 \mu - 1 )^{2}}\left[ \hat{\Theta}- \frac{1}{( \mu - 1 )^{2}}\right]
\Vert u^{\infty}\Vert_{\mathcal{F}}^2 := \sum_{(\ell_1,m_1)\in\Gamma}\sum_{(\ell_2,m_2)\in\Gamma}\Big( \frac{2\ell_1+1}{eka}\Big)^{2\ell_1}\Big(\frac{2\ell_2+1}{eka}\Big)^{2\ell_2}\left\vert \mu_{(\ell_1, m_1; \ell_2, m_2)}\right\vert^2.
\delta \Gamma_{r e g}^{( k )}[ A ] = - \int_{0}^{\infty}\delta ( \mathrm{T r}e^{- t D D^{\dagger}}) \left( 1 + \sum_{i = 1}^{r}e_{i}e^{- t M_{i}^{2}L_{1}^{2}}\right) \frac{d t}{t}.
\Phi = \left( \begin{array}{c c c c}{0}&{\phi_{1}\otimes M_{e}+ \phi_{2}\otimes M_{\nu}}&{0}&{0}\\{\phi_{1}^{*}\otimes M_{e}^{*}+ \phi_{2}^{*}\otimes M_{\nu}^{*}}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}\\ \end{array}\right)
a^{2}H^{2}\delta^{\prime \prime}+ ( 3 H^{2}+ a H H^{\prime}) a \delta^{\prime}= \frac{3}{2}\frac{H_{0}^{2}\Omega_{\mathrm{N R}}}{a^{3}}\delta ( 1 + \delta ) + \frac{4}{3}\frac{a^{2}H^{2}}{( 1 + \delta )}\delta^{' 2}.
\int_{r>t}r^{1-2m}[|\nabla v_m|^2-q_0v_m^2]e^{-2\rho}dx= -\frac{1}{2}\frac{d}{dt}(t^{1-2m}\int_{S_t}v_m^2e^{-2\rho}dx)-\frac{1}{2}\int_{S_t}r^{-2m}(2m-1)v_m^2e^{-2\rho}dx
\{x\}_{y}^{\pm}:=\prod\limits_{k=0}^{\infty}\left( 1\pm q^{x+ky}\right)\,,\quad \left\{x_{1};\ldots ;x_{n}\right\}_{y}^{\pm}:=\prod\limits_{a=1}^{n}\{x_{a}\}_{y}^{\pm}\,.
\sum_{k=0}^n(-1)^{k+n}\frac{2n}{n+k}\binom{n+k}{n-k}m_k(t)[4(1-\epsilon)]^k = 2\int_0^{\pi}e^{in\zeta}\nu_t(d(\cos^2(\zeta/2)/(1-\epsilon)):= M_n(t,\epsilon)
w^{S_1}_j = \lambda, 1 \leq j \leq n-1 w^{S_2}_0 = \frac{1}{1 + \rho}\, \lambda w^{S_{j+1}}_{j-1}= \frac{1}{1 + \rho}\, \frac{w^{S_j}_{j-2}}{a_j}, 2 \leq j \leq n.
|\alpha_x \rangle = \frac{1}{\sqrt{C_R(x)^2 + \pi^2 |g(x)|^4}}\left(C_R(x) \left(\begin{array}{c}0 |\varepsilon_x \rangle\end{array}\right) + \overline{g}(x) \left( \begin{array}{c}1 \frac{\mathcal{P}}{x \Omega}|g\rangle \end{array}\right)\right).
D_f(g)= \sum_{\alpha=0}^{l-1}(\mathrm{ad}x)^k(\mathrm{ad}y)^{\alpha}(\big[ [y,[f_1,\ldots,[f_i,f_{i+1}]]],(\mathrm{ad}y)^{l-1-\alpha}([x,y])\big]) &+(\mathrm{ad}x)^k(\mathrm{ad}y)^l(\mathrm{ad}x)(\mathrm{ad}y)([f_1,\ldots,[f_{i-1},f_i]]).
\|\varphi\|=\lim\limits_{n\rightarrow \infty}\limsup\limits_{t\rightarrow \infty}\|P_{\delta}(W_{n,m_0;\textrm{sur}})e^{-itH_0}\varphi+P_{\delta}(W_{n,m_0;\textrm{far}})e^{-itH_0}\varphi\|< \varepsilon
{\cal{S}}= \frac{1}{1 6 \pi}\int d^{4}x \sqrt{- \tilde{g}}[ \tilde{\Phi}\tilde{R}- \frac{\omega ( \tilde{\Phi})}{\tilde{\Phi}}\tilde{g}^{\mu \nu}\partial_{\mu}\tilde{\Phi}\partial_{\nu}\tilde{\Phi}] +{\cal{S}}_{m}[ \psi_{m}, \tilde{g}_{\mu \nu}]
\begin{array}{c}{{< O_{k}O_{l}P >_{c}= ( \frac{1}{\rho^{2}}\frac{\partial \rho}{\partial t_{l}}+ \frac{1}{\rho^{2}}\frac{\partial \rho}{\partial \mu}\frac{\partial \mu}{\partial t_{l}}) < O_{k}P >}}\\{{- \frac{1}{\rho}( < O_{k}O_{l}P > + \frac{\partial \mu}{\partial t_{l}}< O_{k}P P > ) .}}\\ \end{array}
{\rm Im}\,{\bf w}^\ell=A_\ell\cdot\left( \begin{array}{c}{\sf t}^\ell_1 \vdots{\sf t}^\ell_{k_\ell}\end{array}\right), \ \ \ \ \ \ \ \{\scriptstyle (\ell\,=\,2\,,\,\ldots\,,\,\rho)}.
\sqrt[\leftroot{-4}\uproot{5}p]{pabK_p}=\begin{cases}2^d p^e r' s \sqrt[\leftroot{-4}\uproot{5}p]{K_p}, \textnormal{for FLT2 Case I}, 2^d p^e r_0 s' \sqrt[\leftroot{-4}\uproot{5}p]{K_p}, \textnormal{for FLT2 Case II}.\end{cases}
(\rho^{k}\otimes^{k}\rho)\Delta(m_{1}rm_{2}) =(\rho^{k}\otimes (^{k}\rho))(\Delta(m_{1})rm_{2}+m_{1}\Delta(rm_{2})) &=(\rho^{k}\otimes^{k}\rho)(1 \otimes m_{1}rm_{2}+m_{1}\otimes rm_{2}) &=\bar{e_{k}}\otimes \rho(m_{1}rm_{2}) + m_{1}\bar{e_{k}}\otimes (^{k}\rho(rm_{2})) &=\bar{e_{k}}\otimes \rho(m_{1}rm_{2})
d_{\mathfrak{m}}= \dim_{E}A(\sigma_{1},\sigma_{2}) \otimes_{\mathfrak{Z}_{\Omega}}\kappa(\mathfrak{m}) = \dim_{E}\mathrm{Hom}_{G}(I_{\sigma_{1}}\otimes_{\mathfrak{Z}_{\Omega}}\kappa(\mathfrak{m}), I_{\sigma_{2}}\otimes_{\mathfrak{Z}_{\Omega}}\kappa(\mathfrak{m})) = m_{\sigma_{1}}m_{\sigma_{2}}.
\delta_{v_i^c}= \delta_{k_{i+1}-1}(\delta_{k_{i+1}-2}\delta_{k_{i+1}-1}) \cdots (\delta_{k_i+2}\cdots \delta_{k_{i+1}-2}\delta_{k_{i+1}-1})(\delta_{k_i+1}\delta_{k_i+2}\cdots \delta_{k_{i+1}-2}\delta_{k_{i+1}-1}).
\{( \left( \begin{array}{cccc}\sigma(t) \sigma(t) \sigma(t)^{-1}\sigma(t)^{-1}\end{array}\right) )_{\sigma \in I_F}\ \vert t=\gamma_1^{p_1}\dots \gamma_{d-1}^{p_{d-1}}, p_1, \dots, p_{d-1}\in \mathbb{Z}\}\hookrightarrow Q_0/W_0(L).
\int_{A}^{\infty}\frac{1}{x\sqrt{2\pi\sigma^{2}}}e^{-\frac{(lnx-\mu)^{2}}{2\sigma^{2}}}x^{2}dx=\frac{1}{2}e^{2(\mu+\sigma^{2})}(1+erf(\frac{\mu+2\sigma^{2}-lnA}{\sqrt{2}\sigma}))
\begin{gathered}\Delta g_{\nu (t-s)}* \left ( \nabla \cdot \eta(t) \Delta_2 \tau (s,t) \right ) (x) = \int_{\mathbb{R}^d}( K(x,z,t,s) \left ( \nabla \cdot \eta \right ) (X(z,t), t) + \Delta g_{\nu (t-s)}(x - X(z,t) ) \left ( \left (\nabla \cdot \eta \right ) (X(z,s), t) - \left (\nabla \cdot \eta \right ) (X(z,t), t) \right ) ) dz,\end{gathered}
E^{\nu}\left[\left.\frac{d\mu}{d\nu}(X_{0})\right|\bigcap_{n \geq 1}\mathcal{F}_{0,\infty}^{\mathcal{Y}}\vee \mathcal{F}_{n,\infty}^{\mathcal{X}}\right] =E^{\nu}\left[\left.\frac{d\mu}{d\nu}(X_{0})\right|{F}_{0,\infty}^{\mathcal{Y}}\right]~P^{\nu}~a.s.
g_K^{(j)}(z) = \int_0^{K-1}h_{j,z}(x)\,dx + \sum_{\ell=1}^L \frac{B_{2\ell}}{(2\ell) !}(h_{j,z}^{(2\ell-1)}(K-1)-h_{j,z}^{(2\ell-1)}(0)) &+ \frac{1}{2}(h_{j,z}(K-1)+ h_{j,z}(0))+\mathcal{E}_{K,j,z,L},