harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
G ( W , Z ) = 1 +{\frac{P}{[{\frac{4}{( c - 2 )^{2}}}W^{2 - c}+{\frac{4}{( d - 2 )^{2}}}Z^{2 - d}]^{\frac{c d - c - d}{( c - 2 ) ( d - 2 )}}}}, ( c \neq 2 \neq d ) ,
\begin{aligned}\lim_{n \to \infty}\mathcal{X}^n = \mathcal{X}, \lim_{n \to \infty}\mathcal{M}^n = \mathcal{M}, \lim_{n \to \infty}\mathcal{S}^n = \mathcal{S}, \lim_{n \to \infty}\tilde{\mathcal{B}}^n = \tilde{\mathcal{B}}\end{aligned}
\begin{array}{lll}\lambda = (2n-1)\alpha^2-\beta^2+\alpha+\beta+a \mu = 2(n-1)\alpha\beta+(1-2n)\alpha+\beta+b \nu = \beta^2+(1-2n)\alpha^2+(\alpha-\alpha\beta)(2n-2)-(2n+2)\beta+c\end{array}
\mathrm{U}_t=\mathrm{Y}_0^t+\boldsymbol{\xi}^\star_t\big(\dot{\mathbf{Y}}^t_{\cdot}\boldsymbol{\upsilon}^\star_\nu \mathbf{L}\boldsymbol{\upsilon}_\nu\dot{\mathbf{U}}_{\cdot}\big)\boldsymbol{\xi}_t
M(z) =\left(\begin{array}{cc}1 0 0 z(1+zD^{-1}C)^{-1}D^{-1}\end{array}\right) =\left(\begin{array}{cc}1 0 0 0\end{array}\right)+z\left(\begin{array}{cc}0 0 0 D^{-1}\end{array}\right)+\sum_{k=1}^{\infty}(-1)^{k}z^{k+1}\left(\begin{array}{cc}0 0 0 \left(D^{-1}C\right)^{k}D^{-1}\end{array}\right),
\Bigl(\frac{g(x_*,R)}{g(y,R)}\Bigr)^{\alpha/\delta}=\frac{R-d(z,x_*)}{R-d(y,z)}&\leq \frac{R-d(z,x_*)}{R-\bigl(d(z,x_*)+\beta(2K)^{1-\alpha}\rho^{\alpha}(d(z,x_*)+\rho)^{1-\alpha}\bigr)}&\leq \frac{1}{1-\frac{\beta(2K)^{1-\alpha}\rho^{\alpha}(R+\rho)^{1-\alpha}}{\beta_*\rho^{\alpha}R^{1-\alpha}}}&\leq\frac{1}{1-\frac{2\beta(2K)^{1-\alpha}}{\beta_*}}
{\hat \Pi}_{{\hat \mu}{\hat \nu}}\partial_{\hat \imath}{\hat X}^{\hat \mu}\partial_{\hat \jmath}{\hat X}^{\hat \nu}\rightarrow{\hat \Pi}_{{\hat \mu}{\hat \nu}}{\cal D}_{\hat \imath}{\hat X}^{\hat \mu}{\cal D}_{\hat \jmath}{\hat X}^{\hat \nu}\, ,
L(p_1,p_2) = -\frac{\sqrt{g(p_1)}}{4 \pi}\log \left( g_{xx}(p_1) x^2 + g_{yy}(p_2) y^2 \right) \L(p_1,p_2) = \L_{g(p_1)}\left( \frac{x}{\epsilon}, \frac{y}{\epsilon}\right)
\Vert W_{l-1}\Vert_{\cal H}\le |\bigcup\nolimits_{j}\partial T^0_{j}|_{\cal H}+ |\bigcup\nolimits^{l-1}_{i=1}\partial X_i' \setminus \bigcup\nolimits_j T^{0}_j|_{\cal H}+ \frac{1}{2}\sum^{l-1}_{i=1}|\partial X_i' \cap \bigcup\nolimits_{j}T^{0}_j|_{\cal H}.
\begin{aligned}y^{k+1}(\ell) - v[\bar y^{k+1}]_\ell = \bar C^k(y^k(\ell)-v[\bar y^k]_\ell ) +\gamma_2^k\Pi_v \xi_w^{k}(\ell) &+\Pi_v \left(g^{k+1}(\ell)- (1-\alpha^k)g^{k}(\ell)\right)\end{aligned}
\sup_{\pi \in \Pi(\mu_1, \dotsc, \mu_L)}\int_{\mathcal{X}}f d\pi = \inf_{(\phi_\ell)_{\ell \in [L]}\in \Phi_f}\left\{\sum_{\ell=1}^{L}\int_{\mathcal{X}_\ell}\phi_\ell d \mu_\ell \right\}.
I(t):= i\kappa \int_{0}^{t\wedge \tau}S(t-s)(|\tilde{z}_{R^\prime}(s)|^{2}\tilde{z}_{R^\prime}(s)-|\tilde{z}_{R}(s)|^{2}\tilde{z}_{R}(s)) \d s &-i \int_{0}^{t}S(t-s)((\tilde{z}_{R^\prime}(s)-\tilde{z}_{R}(s)) \d W(s)) &-\frac{1}{2}\int_{0}^{t}S(t-s)\left(\left(\tilde{z}_{R^\prime}(s)-\tilde{z}_{R}(s)\right) F_{\Phi}\right) \d s.
p_1 = \frac{\omega (g^3+1)^2f^2 + g(2-g^3)f + \omega^2 g^2}{fg^3}p_2 = \frac{\omega^2 (g^3+1)^2f^2 + g(2g^3-1) + \omega g^2}{fg^2}p_3 = \frac{1}{2}(p_1p_2 - \{p_2,p_1\}+9)
\mathcal{L}_{\omega}\Phi=\Phi (\mathcal{D}_{\omega}+b_3\partial_{yyy}+b_2 \partial_{yy}+b_1 \partial_y +b_0) \Pi_S^{\perp}+\mathcal{R}_{\mathit{II}}, &\mathcal{R}_{\mathit{II}}:=\{\Pi_S^{\perp}(\mathcal{A}-\mathrm{I})\Pi_S-\mathcal{R}_{\Phi}\}(b_3 \partial_{yyy}+b_2 \partial_{yy}+b_1 \partial_y +b_0)\Pi_S^{\perp}+\mathcal{R}_{\mathit{I}}.
\frac12 \frac{d}{dt}\big(\|\nabla\partial^k_{\phi}u\|^2_{L^2(B)}\big) =(-1)^k\int_B\nabla\partial^{2k}_{\phi}u\nabla u_t\;dx =(-1)^k(\partial^{2k}_{\phi}u_r,u_t)_{L^2(S^1)}=(-1)^{k+1}(\partial^{2k}_{\phi}u_r, (\varepsilon+d\pi_N(u))u_r)_{L^2(S^1)}\le-\|d\pi_N(u)\partial^k_{\phi}u_r\|^2_{L^2(S^1)}-I,
\Big\{-\Delta_{\bf y}-2\lambda_{Q}H_{Q}^{\lambda_{Q}}({\bf y})\cdot\nabla_{\bf y}& +\big[\lambda_{Q}V_{Q}^{\lambda_{Q}}({\bf y})+\lambda_{Q}^2(K_{Q}^{\lambda_{Q}}({\bf y})-E)\big] \Big\}\psi_Q^{\lambda_Q}= 0\,.
\Big(\sum_{j=1}^{n}\lambda_{j}^{d}-1\Big)\mathcal{L}(\Lambda) =\sum_{1\leq i<j\leq n}^{n}\mathcal{L}(f_{i}(\Lambda)\cap f_{j}(\Lambda))-\sum_{1\leq i<j<h\leq n}\mathcal{L}(f_{i}(\Lambda)\cap f_{j}(\Lambda)\cap f_{k}(\Lambda)) + & \cdots +(-1)^{n}\mathcal{L}(\cap_{j=1}^{n}f_{j}(\Lambda)).
h +{\rm Im}({\rm Hom}_{\mathcal{C}}(f_1, D)) = h' \circ \widehat{f_1}+{\rm Im}({\rm Hom}_{\mathcal{C}}(f_1, D)) = \Phi( [0 \longrightarrow D \longrightarrow D \coprod_{{\rm Ker}(f_0)}F_0 \longrightarrow C \longrightarrow 0] ).
\frac{\big(x q^{-L}/\xi, q^{-L}/(\beta_1 \xi ), q^{-L}/(\beta_2 \xi ), q^{-L}/(\beta_3 \xi ) ;q\big)_{\infty}}{\big(x q^{-\lambda -1 -L}/ \xi, q^{-L}/(\alpha_1 \xi ), q^{1-L}/(\alpha_2 \xi ), q^{-L}/(\alpha_3 \xi ) ;q\big)_{\infty}}=0
\sum_{k=0}^a \binom{a}{k}\binom{b}{m+k}\binom{k}{c}\binom{m+k}{d}k^2 = \binom{a+b-c-d}{m+a-d}\binom{a}{c}\binom{b}{d}\cdot \frac{m(a-c)[m(a-c-1)+2(cd-ab)+b+d]+(cd-ab)^2-bd(a-c)-ac(b-d)}{(a+b-c-d)(a+b-c-d-1)}
\langle\Psi |\int_{-\infty}^\infty{\widehat T}_{00}(t,0,0,0) b(t)\, \d t|\Psi\rangle /\langle\Psi |\Psi \rangle \geq -{\frac{3}{32\pi^2}}{\frac{\hbar c}{(ct_0)^4}}\, ,
\nu\approx\frac{(\frac{s_{1}^{2}}{N_{1}}+\frac{s_{2}^{2}}{N_{2}})^{2}}{\frac{s_{1}^{4}}{N_{1}^{2}\nu_{1}}+\frac{s_{2}^{4}}{N_{2}^{2}\nu_{2}}}.
w\big(\big\{x\in B(x_0,r):|b(x)-b_{B(x_0,r)}|>\lambda\big\}\big) &\leq C_1w\big(B(x_0,r)\big)\exp\bigg[-\bigg(1+\frac{r}{\rho(x_0)}\bigg)^{-\theta^{\ast}}\frac{C_2 \lambda}{\|b\|_{\mathrm{BMO}_{\rho,\theta}}}\bigg]\left(1+\frac{r}{\rho(x_0)}\right)^{\eta},
\frak{m}'^{(0,2)}=\bigg\{p_-(A): A=\begin{bmatrix}A_1 0 0 0 A_1 0 0 0 A_2\end{bmatrix}, A_i=\begin{bmatrix}0 Y_i^tY_i 0 \end{bmatrix}, Y_i\in M_{2\times 1}(\mathbb{C}), i=1,2\bigg\}
K_{i \Theta_{l}}( z ) \stackrel{( z \rightarrow 0 )}{\sim}- \sqrt{\frac{\pi}{\Theta_{l}\sinh \left( \pi \Theta_{l}\right)}}\, \sin \left[ \Theta_{l}\ln \left( \frac{z}{2}\right) - \delta_{\Theta_{l}}\right] \, \left[ 1 + O \left( z^{2}\right) \right] \; ,
z+ \lim\limits_{t \to \infty}{Q\big(z+(1-z)(1-p_t)\big)-(1-p_t) -zp_t \over p_t\big(1-Q'(1-p_t)\big)}=z+\lim\limits_{x \rightarrow 1-}{Q\big(z+(1-z)x\big)-\big(z+(1-z)x\big) \over (1-x)(1-Q'(x))}&=z+(1-z) ~=1
\sum\limits_{k=1}^n\tau\sum\limits_{i=0}^{m-1}h\Big[\big(b_n(v_i(k))\big)_{\bar t}\psi_i(k) + v_{ix}(k)\psi_{ix}(k)-f_{ik}\psi_i(k)\Big] -\sum\limits_{k=1}^n\tau p_k\psi_m(k) +\sum\limits_{k=1}^n\tau g_k^n\psi_0(k).
m^j_{t}=\int_{[0,\,\max\{0,\,\tau^{-1}(t)\}]}\,\delta_{\Phi_t(x,\,0)}\,dm_0^j(x)+\int_{(\max\{0,\,\varsigma^{-1}(t)\},\,t]}\delta_{\Phi_t(0,\,s)}\,dm_{x=\pi_j(0)}^j(s) m^j_{x=\pi_j(L_j)}=\int_{(\max\{0,\,\tau^{-1}(t)\},\,L_j]}\delta_{\tau(x)}\,dm_0^j(x)+\int_{[0,\,\max\{0,\,\varsigma^{-1}(t)\}]}\delta_{\varsigma(s)}\,dm_{x=\pi_j(0)}^j(s),
\partial_{t}K_{t}( x , x^{\prime}) + A \, K_{t}( x , x^{\prime}) = 0 \: , \qquad \qquad \lim_{t \to 0_{+}}K_{t}( x , x^{\prime}) = [ g ( x ) ]^{1 / 4}\delta ( x , x^{\prime}) [ g ( x^{\prime}) ]^{1 / 4}\: ,
[\begin{matrix}X\\Y\\Z\end{matrix}]^{B}=[\begin{matrix}c_{x}\\c_{y}\\c_{z}\end{matrix}]+(1+s\times10^{-6})\cdot[\begin{matrix}1&-r_{z}&r_{y}\\r_{z}&1&-r_{x}\\ -r_{y}&r_{x}&1\end{matrix}]\cdot[\begin{matrix}X\\Y\\Z\end{matrix}]^{A}
[P_0(\psi_B>t)]^{\lfloor e^{(\beta m+\delta)t}\rfloor}=\left[1-\exp\left(-\frac{[(\sqrt{2\beta m}+\varepsilon)t]^2}{2t}\right)\exp(o(t))\right]^{\lfloor e^{(\beta m+\delta)t}\rfloor}.
| f \rangle = \int \left [ \prod_{k=1}^N d^2 z_k \right ] f(z_1, z_2, \cdots, z_N ) e^{-{1 \over 2}\sum_{k=1}^N | z_k |^2}\psi^\dag_{\rm LLL}(z_1) \psi^\dag_{\rm LLL}(z_2) \cdots \psi^\dag_{\rm LLL}(z_N) | 0 \rangle.
\sum_{n=0}^{\infty}\sum_{m=-\infty}^{\infty}N_{[1,4]}(m,n)z^{m}q^{n}=\dfrac{1}{(zq,z^{-1}q;q)_{\infty}(zq^{4},z^{-1}q^{4};q^{4})_{\infty}}.
G^{\varepsilon}(\eta) =\int_{\Omega}\left[ \sqrt{g}g^{ij}\theta_{i}\right]^{\varepsilon}\eta_{j}dx G(\eta) =\int_{\Omega}\sqrt{g}g^{ij}\theta_{i}\eta_{j}dx=\int_{\Omega}\sqrt{g}g^{ij}g^{ab}u_{abi}\eta_{j}dx
\Gamma(p,P) = -e^2 \int \mbox{$ \, \frac{d^3q}{(2\pi)^3}\,$}D(p-q) \gamma_\mu S(\mbox{$ \, \frac{1}{2}\,$}P+q) \Gamma(q,P)S(-\mbox{$ \, \frac{1}{2}\,$}P+q) \gamma_\mu,
\frac{1}{2}\left< P^0[h]_{\mu \nu}(x) \, P^\infty [h]_{\rho \sigma}(y) \right>_0 + \frac{1}{2}\left< P^\infty[h]_{\mu \nu}(x) \, P^0 [h]_{\rho \sigma}(y) \right>_0
\begin{aligned}\frac{dD_{V,[l]}}{dt}\leq -\frac{\kappa_1|[l]|\mathcal{A}_{[l]}(v)(0)}{NT_M^\infty}\phi(D_{X,[l]})D_{V,[l]}+\frac{4\kappa_1(N-|[l]|)\phi(\delta)}{NT_m^\infty}=: -C_1\phi(D_{X,[l]})D_{V,[l]}+C_2,\mbox{a.e.}~t\in (0,t_0). \end{aligned}
G_0 =G_{0,0}-G_{0,0}vM_0^{-1}v^*G_{0,0}, G_1 =G_{0,1}-G_{0,1}vM_0^{-1}v^*G_{0,0}&\quad+G_{0,0}vM_0^{-1}M_1M_0^{-1}v^*G_{0,0}-G_{0,0}vM_0^{-1}v^*G_{0,1}&=(I-G_{0,0}vM_0^{-1}v^*)G_{0,1}(I-vM_0^{-1}v^*G_{0,0}).
\begin{aligned}b_{s}^+(L^f) =q^{\frac{s^2-1}{2}f}\, b_{s}^+(L) + \dfrac{(-N + s) \, f \, \{1\}}{[s]^2}\, \left.V_{s}(L)\right|_{q=\xi}, b_{s}^-(L^f) =q^{\frac{s^2-1}{2}f}\, b_{s}^-(L) + \dfrac{s\, f \, \{1\}}{[s]^2}\, \left.V_{s}(L)\right|_{q=\xi}. \end{aligned}
<\dot{\omega}\left( q \right)~\omega\left( f_{1}\right)\cdots\omega\left(f_{n}\right)>=const L^{2+\lambda}~ \left\{q_{+}<\Delta|L_{1}\omega\left(q \right)\cdots\omega\left( f_{n}\right)| >-c.c. \right\}~+~O\left(L^{\lambda}\right)
\sum_{n\ge 0}g\left(3^{2\alpha-1}n+\frac{3^{2\alpha}-1}{4}\right)q^n =\frac{E(q^{6})^5}{E(q^3)^4}\sum_{j\ge 1}d_{2\alpha-1}(j)X^j\equiv 0 \pmod{3^{2\alpha}},
\left( \begin{array}{c c}{\left\{\bar{D}_{a}\, , D^{b}\right\}}{\left\{\bar{D}_{a}\, , \bar{D}_{b}\right\}}{\left\{D^{a}\, , D^{b}\right\}}{\left\{D^{a}\, , \bar{D}_{b}\right\}}\end{array}\right) = 2 i m^{2}\Delta \, .
\frac{1}{g_{\Lambda}}- \frac{1}{4}\mathrm{L o g}\left( \frac{\Lambda^{2}}{M^{2}}\right) + \frac{1}{6}\frac{\Lambda^{2}}{M^{2}}e^{- 4 / g_{\Lambda}}+ O \left( \frac{\Lambda^{4}}{M^{4}}e^{- 8 / g_{\Lambda}}\right)
\|(u_k,v_k)\|^p=\frac{2(\alpha+\beta-q)}{p-q}\int_{\Omega}|u_{k}|^{\alpha}|v_{k}|^{\beta}dx+o_k(1), \|(u_k,v_k)\|^p=\frac{\alpha+\beta-q}{\alpha+\beta-p}\int_\Omega(\lambda|u_k|^q+\mu|v_k|^q)dx+o_k(1).
\big(\tau_{k, 0}^{(\alpha, \beta)}\big)^{2}=\tau_{k, 0}^{(\alpha, \beta + 1)}\tau_{k, 0}^{(\alpha, \beta -1)}-\tau_{k-1, 0}^{(\alpha,\beta-1)}\tau_{k + 1, 0}^{(\alpha, \beta + 1)},
\begin{array}{ccllcl}G_1\cap G_2 = \langle (2,3) \rangle, G_1\cap G_3 = \langle (3,4) \rangle G_2\cap G_3 = \langle (1,2) \rangle, G_1\cap G_4 = \langle (2,4) \rangle G_2\cap G_4 = \langle (1,3) \rangle, G_1\cap G_5 = \{1\}\end{array}
A_U:=\big\{k\geq 0: \overline{Bd}\Big(N(x, U)\cap \big(N(x, U)-k\big)\cap\dots \cap \big(N(x, U)-rk\big)\Big)>0\big\}\subseteq \big\{k\geq 0: T^{-k}U\cap \dots \cap T^{-rk}U\cap U \neq \emptyset\big\}.
\mathcal{L}= \frac 1 2 \left( \mathbf{E}^{2}- \mathbf{H}^{2}\right) - \frac{2 s ( s + 1 ) \sigma g}{s ( s + 1 ) ( g^{2}- \sigma^{2}) - 1}\left( \mathcal{G}- \mathcal{G}_{0}\right) + \frac{\epsilon \alpha^{2}( 2 s + 1 )}{9 0 m^{4}}
S=\int d^4x \{\frac{1}{4}F_{\mu\nu}^a F^{a,\mu\nu}+\frac{1}{2}Tr\{(D_\mu\phi_1)^2\}+ \frac{1}{2}Tr\{(D_\mu\phi_2)^2\}-\frac{\lambda}{4}(Tr\{\phi_1^2\}-Tr\{\phi_2^2\}-f^2)^2 \}\quad.
\|\widetilde\Phi_j\|_{\widetilde H^{k-3j}(\Sigma)}\leq c(k,b)\Bigl( \|u_0\|_{\widetilde H^k(\Sigma)}+\sum_{m=0}^{j-1}\|\partial_t^mf\big|_{t=0}\|_{\widetilde H^{k-3(m+1)}(\Sigma)}\Bigr).
\Vert u \Vert_{{\Lambda_{2,2}^{\alpha,\beta/2}}}= \Vert u \Vert_{L^2}+ \big (\iint_{K \times K}\frac{|u(\xi)-u(\eta)|^2}{|\xi-\eta|^{\alpha+\beta}}d\nu(\xi)d\nu(\eta) \big )^{1/2},
F_{QR,P}(\Omega)(\zeta_i \otimes c(\sigma(j))\alpha_{\sigma(j)}) =-(2g_C+1)\left(\int_{C-\gamma}\log(f) dx_j \wedge dz_i + 2\pi i \int_{\gamma}dx_jdz_i \right) &=-F_{QR,P}(\Omega)( c(\sigma(j))\alpha_{\sigma(j)}\otimes \zeta_i)
\mathbb{X}^{i,j}_{s,t}:= \int_0^t X^i_{r-}\d X^j_r - \int_0^s X^i_{r-}\d X^j_r - X^i_{s}X^j_{s,t}\mathbb{X}^{i,i}_{s,t}:= \frac{1}{2}(X^i_{s,t})^2,(s,t)\in \Delta_T,
\lim_{\delta \to 0}\lim_{N \to \infty}\frac{1}{N}\log \left( \int_{\overline{\mathbf{P}}_{N}\in B( \overline{\mathbf{P}}, \delta)}\Pi_{i=1}^{N}\mu_{\theta}(x_{i}) \, dx_{i}\right) \leq - \overline{{\rm Ent}}[ \overline{\mathbf{P}}| \overline{\mathbf{\Pi}}^{\mu_{\theta}}].
\sum \limits_{\substack{I^{c}_{k+l}(n)}}\tilde{b}(i_{1},i_{2},...,i_{k+l};n) & = \sum\limits_{\substack{I^{c}_{k+l}(n)}}\left ( \frac{\tilde{b}(i_{1},i_{2},...,i_{k+l};n)}{b(i_{1};n)b(i_{2};n)\cdots b(i_{k+l};n)}\cdot b(i_{1};n)b(i_{2};n)\cdots b(i_{k+l};n)\right ),
\d Y_t^{x_1,y}=f(x_1,Y_t^{x_1,y})\d t+g(x_1,Y_t^{x_1,y})\d B_t,Y_0^{x_1,y}=y, \d Y_t^{x_2,y}=f(x_2,Y_t^{x_2,y})\d t+g(x_2,Y_t^{x_2,y})\d B_t,Y_0^{x_2,y}=y.
\int_0^t\!\!\int_0^t \vert r-s\vert^{-{2p-d\over 2}}Q\Big(a\sigma^{-1}\vert r-s\vert^{-1/2}\Big)drds =2\int_0^t(t-s)s^{-{2p-d\over 2}}Q\Big(a\sigma^{-1}s^{-1/2}\Big)ds.
A^{M^{10}}(y) = \int_0^\infty{{dt}\over{2t}}{1 \over{V_6}}\left( 8 \pi^2 \alpha' t \right)^{6/2}e^{-{{y^2}\over{2 \pi \alpha'}}t}\cdot 2 Z^o_B(t)Z^o_F(t).
\Gamma_0(b_i \to b_f + \gamma) ={2 \over 3\pi}\left({M(N_i)^2 - M(N_f)^2 \over 2 \hbar M(N_i)}\right)^3 A_H(N_i) \;{2J_f+1\over2J_i+1}\;{N_f \over N_i}.
\frac{\nabla^i u\nabla^j u}{|\nabla u|^2(1+|\nabla v|^2)}(\psi'\nabla_{ij}u+\psi''\nabla_i u\nabla_ju) \ge -\frac{|\psi''||\nabla u|^2+C_0|\psi'|}{1+(\psi')^2|\nabla u|^2}&\ge -\left(C_0+\min\left\{C_0^2|\psi''|,\frac{|\psi''|}{(\psi')^2}\right\}\right),
d(lnS_{t})=(lnS_{t})^{\prime}dS_{t}+\frac{1}{2}(lnS_{t})^{\prime\prime}dS_{t}dS_{t}=\frac{dS_{t}}{S_{t}}-\frac{1}{2}\frac{1}{S_{t}^{2}}dS_{t}dS_{t}
X(z) =\exp\Big(\sum_{m\nmid n\geq1}\frac{z^{n}h_{-n}}{n}(1-\xi_m^n)c^n\Big)\exp\Big(\sum_{m\nmid n\geq1}\frac{z^{-n}h_n}{n}(1-\xi_m^n)\Big) =\sum_n X_{n}z^{-n},
e^{-t}\sum_{i=1}^{+\infty}\frac{t^i}{i!}R_i\left((1-m)L\right)=e^{-t}\sum_{i=M(t)}^{N(t)}\frac{t^i}{i!}R_i\left((1-m)L\right)+o\left(e^{-r^- t}\right), t \to +\infty,
A'_{i-1,j-2}A_{i-1,j}+A_{i-2,j-1}A'_{i,j-1}= A'_{i-1,j-1}A_{i-1,j-1}, A'_{i,j-1}A_{i,j+1}+A_{i-1,j}A'_{i+1,j}= A'_{i,j}A_{i,j}.
\mu_{i}= \sigma_{i}^{2}\sigma_{i+1}, \nu_{i}= \sigma_{3}^{-1}\sigma_{4}^{-1}\cdots\sigma_{i+1}^{-1}, \omega_{i}= \mu_{i+1}\mu_{i}\cdots\mu_{2}, \xi_{j,i}= \nu_{j}\nu_{i}\nu_{i-1}\cdots\nu_{2}.
x_{i-1}(v_i) = x(v_i) + \sum_{j=1}^{i-1}\overrightarrow{d}(v_j,v_i) \geq d^-_T(v_i) + \sum_{j=1}^{i-1}\overrightarrow{d}(v_j,v_i) = d^-(v_i) = d^+ (v_i).
\begin{aligned}(f^j,\phi) =\lim_{n\to\infty}(f_n\ast\varphi_j,\phi)=\lim_{n\to\infty}(f_n\ast\varphi_j\ast(\varphi_{j-1}+\varphi_j+\varphi_{j+1}),\phi) &=(f^j\ast(\varphi_{j-1}+\varphi_j+\varphi_{j+1}),\phi),\end{aligned}
\begin{aligned}\int_0^T\int_0^1 \left(s\Theta a(v_x)^2 + s^3 \Theta^3\frac{(x-x_0)^2}{a}v^2\right)e^{2s\varphi}dxdt \le C \int_0^T \int_\omega v^2dxdt & + C\left(\int_0^T\int_0^1 h^{2}e^{2s\varphi}dxdt+\int_0^T \int_0^{\beta_1}h^2 e^{2s \Phi_1(t,-x)}dxdt+ \int_0^T \int_{\lambda_2}^1 h^2e^{2s \Phi_2(t,x)}dxdt\right). \end{aligned}
= 2 \sum_{s = o d d}^{\infty}e^{- \pi \tau_{2}\frac{s^{2}}{2}}( \frac{s^{2}}{4}+ \frac{1}{2 \pi}\frac{d}{d \tau_{2}}) \left( \frac{1}{\sqrt{2 \tau_{2}}}e^{- \frac{\pi \tau_{1}^{2}s^{2}}{2 \tau_{2}}}\sum_{n}e^{- \frac{\pi n^{2}}{2 \tau_{2}}- \frac{n \pi \tau_{1}s}{\tau_{2}}}\right)
R_k=\frac{4}{\sqrt{5}}[(\alpha(\begin{bmatrix}0 -1\end{bmatrix},t))(\alpha(\begin{bmatrix}2 1\end{bmatrix},t))-(\alpha(\begin{bmatrix}0 1\end{bmatrix},t))(\alpha(\begin{bmatrix}2 -1\end{bmatrix},t))]=0,
\ll\frac{\lambda^2_{max}}{\varphi(M)^{1/2}}\biggl(\sum_{\substack{q<W^\prime R^2 (q,MP_f)=1}}\mu(q)^2\tau_{3k}^2(q)\frac{X}{\varphi(q)}\biggr)^{1/2}\biggl(\sum_{\substack{q<W^\prime R^2 (q,MP_f)=1}}\mu(q)^2E(X,qM)\biggr)^{1/2}.
J_2 :=-s^3\lambda^3\int_{\Gamma_T}\xi^3 \sigma (\partial_\nu^A \eta^0) \psi^2 \,\d S\,\d t + \lambda \int_{\Gamma_T}\left(s\xi +\frac{\tau}{2}\right) \left(\frac{\tau}{2}-s\alpha\right) (\partial_t \log \gamma) (\partial_\nu^A \eta^0) \psi^2 \,\d S\,\d t & \ge Cs^3\lambda^3 \int_{\Gamma_T}\xi^3 \psi^2 \,\d S\,\d t.
\begin{cases}\|\sum_{l\le j\epsilon}T_a^{j, l}\|_{L^2\to L^2}\lesssim_{\epsilon}2^{jm}, \|T_a^{j, l}\|_{L^2\to L^2}\lesssim_{\epsilon}2^{10n(m-n)(j+l)}, l>j\epsilon.\end{cases}
\pi^{*}\circ \sigma^{*}\circ \sigma_{*}(f) = \sum_{\overline{w}\in W_{G}/W_{H}}w \cdot \left [ \dfrac{\pi^{*}(f)}{\prod_{\alpha \in \Delta^{+}\setminus \Delta^{+}_{H}}c^{h}_{1}(L_{-\alpha})}\right ].
\mathcal{E}_{k,m}= \underset{\lambda \mid b}\sum a_f(a \lambda^2) \underset{(s,b)=\lambda}\sum E_{k,m,s}& = \underset{\lambda \mid b}\sum a_f(a \lambda^2) \underset{(s',b')=1}\sum E_{k,m, \lambda s'}.
( J_{0}^{-})^{1 + t}J_{- 1}^{+}( J_{0}^{-})^{1 - t}= ( J_{- 1}^{+}( J_{0}^{-})^{1 + t}- 2 ( 1 + t ) J_{- 1}^{0}( J_{0}^{-})^{t}- t ( 1 + t ) J_{- 1}^{-}( J_{0}^{-})^{t - 1}) ( J_{0}^{-})^{1 - t}
(T_sT_tT_s\cdots)_m = (T_tT_sT_t\cdots T_s)_{m+1}= T_t(T_tT_sT_t\cdots)_m=q_t (T_sT_tT_s\cdots)_{m-1}+(1-q_t)(T_tT_sT_t\cdots)_m.
E\Big[ \Big( \sum_{i=1}^m \omega_i \big(\epsilon_i^p -E[\epsilon_i^p]\big) \Big)^q \Big] \leq q^q \max\Big( \sum_{i=1}^m (2\omega_i)^q (Apq)^{pq}, \big( \sum_{i=1}^m \omega_i^2 (2Ap)^{2p}\big)^{q/2}\Big) &\leq q^q 2^q (2Ap)^{pq}m \max\big( \|\omega\|_\infty q^p, \|\omega\|_2\big)^q.
\mbox{dim}\,E^{lk}(\underline{a})=\mbox{dim}\left ( \mathcal{R}^{k}(\underline{a})+\widehat{m}_{\underline{\varphi}(\underline{a})}\right )/\widehat{m}_{\underline{\varphi}(\underline{a})}=\mbox{dim}\,\mbox{Ker}\,\mbox{ad}^{r}T^{lk}(\underline{a})\cdot S^{lk}(\underline{a}).
i{\cal D}^{-1}(p^2) = \mu^2 - (p^2 + \omega^2) + \frac{N\beta^2}{2\pi}\left(f\biggl(-\frac{p^2}{\beta^2\sigma^2}\biggr) - f\biggl(\frac{\omega^2}{\beta^2\sigma^2}\biggr)\right)
\{x^{\odot \star}\in X^{\odot \star}\ : \ \lim_{h\downarrow 0}\|U^{\odot \star}(t+h,t)x^{\odot \star}- x^{\odot \star}\| = 0\}= \overline{\mathcal{D}(A^{\odot \star}(t))}= \overline{\mathcal{D}(A_0^{\odot \star})}= X^{\odot \odot}= j(X),
\begin{array}{r c l c r c l}{{\displaystyle [ t_{a}, \, t_{b}]}}&{{=}}&{{i F_{a b c}t_{c}}}&{{,}}&{{F_{a b 0}}}&{{=}}&{{0}}\\{{\displaystyle \{t_{a}, \, t_{b}\}}}&{{=}}&{{D_{a b c}t_{c}}}&{{,}}&{{D_{a b 0}}}&{{=}}&{{\displaystyle \sqrt{\frac{2}{N}}\delta_{a b}}}\\ \end{array}
J_{o r b}( \vec{R}) = 2 \pi \kappa \frac{\sigma_{1}}{q_{1}}R^{i}\varepsilon_{i j}v_{j}^{( 2 )}( \vec{R}) - 2 \pi \kappa \frac{\sigma_{2}}{q_{2}}R^{i}\varepsilon_{i j}v_{j}^{( 1 )}( - \vec{R}) \; \; ,
\begin{cases}\mathrm{for}\; x\in \boldsymbol{S}2\int_{\mathbb{R}}d\mu_{\mathrm{eq}}(\xi)\ln\left|x-\xi\right|-V(x)=C_V \mathrm{for}\; x\;\mathrm{Lebesgue\; almost\; everywhere\; in\;}\boldsymbol{S}^c 2\int_{\mathbb{R}}d\mu_{\mathrm{eq}}(\xi)\ln\left|x-\xi\right|-V(x)<C_V.\end{cases}
W_{-,+}((y_1,y_2),u_1|u_1,u_2) = 2W_{+}((y_1,y_2),u_1|u_2) =W(y_1|u_1\oplus u_2)W(y_2|u_2) \quad= \sum_{z_1,z_2}Q(z_1|u_1\oplus u_2)P(y_1|z_1) Q(z_2|u_2)P(y_2|z_2) \quad= \sum_{z_1,z_2}Q_{-,+}((z_1,z_2),u_1|u_1,u_2)P_a(y_1,y_2|z_1,z_2),
Q_{-\sigma,\Omega}^D[u]=-\frac{2\sigma}{C_\sigma}\cdot \widetilde{\cal E}_{-\sigma}^D(w_{-\sigma}^D);(-\Delta_{\Omega})^{-\sigma}_Du(x)=\frac{2\sigma}{C_\sigma}\,w_{-\sigma}^D(x,0),\ \x\in\Omega,
\stackrel{\wedge}{\psi}^{\dagger}= \sigma_{-}= \left( \begin{array}{c c}{0}&{0}\\{1}&{0}\\ \end{array}\right) , \qquad \stackrel{\wedge}{\psi}= \sigma_{+}= \left( \begin{array}{c c}{0}&{1}\\{0}&{0}\\ \end{array}\right)
\Omega^1_Y(\log \Sigma)\simeq\mathcal{O}_Y \frac{dt_1}{t_1}\oplus (\bigoplus_{i=2}^k \mathcal{O}_Y dt_i), \Omega^1_X(\log E)\simeq(\bigoplus_{i=1}^l \mathcal{O}_X \frac{dx_i}{x_i})\oplus(\bigoplus_{i=l+1}^n \mathcal{O}_X dx_i)
\lim_{m\rightarrow\infty}\varepsilon_{m(p+1)+\ell,k-1}\,h_{m(p+1)+\ell,k}(z)=\begin{cases}\frac{z}{\sqrt{(z-b_{k-1})(z-a_{k-1})}}\mathrm{if}\ \ell<k, begin{align*}1em]\frac{1}{\sqrt{(z-b_{k-1})(z-a_{k-1})}}\mathrm{if}\k\leq \ell,\end{cases}
\lambda =\sup_{{\pi}\in \bar{\mathcal{A}}^{\mathbb{G}}}\limsup_{T\uparrow \infty}\frac{1}{T}\ln \mathbb{E}^{{\mathbb{P}}^{{\pi}}}\left[ e^{\int_{0}^{T}L^{\alpha_{s-}}(V_{s},{\pi}_{s})ds}\right],
Z_{D D}={\frac{1}{N}}\sum_{p = 0}^{N - 1}\chi_{p}^{f}( q ){\frac{1}{\eta ( q )}}\int_{- \infty}^{\infty}d x q^{x^{2}/ 2 N}\sum_{m = - \infty}^{\infty}[ | a_{m}|^{2}e^{2 \pi i m x}+ ( - 1 )^{p}| b_{m}|^{2}e^{2 \pi i ( m + 1 / 2 ) x}]
|U| =\left| \frac{1}{n}\int_{\mathbb{R}^3}\int_0^\infty p F\phi(\mathcal{I})\,d\mathcal{I}\frac{dp}{p^0}\right| =\left| \frac{1}{n}\int_{\mathbb{R}^3}\int_0^\infty p f\sqrt{F_E^0}\phi(\mathcal{I})\,d\mathcal{I}\frac{dp}{p^0}\right|\le \frac{C}{n}\|f\|_{L^2_{p,\mathcal{I}}}.
\begin{array}{lll}\delta r_+ = \delta\left(M+\sqrt{M^2-a^2-Q^2}\right) &= \delta M+{1\over2}\,{\left(M^2-a^2-Q^2\right)}^{-{1\over2}}\,2\left(M\,\delta M-a\,\delta a\right) &= \delta M+{\left(r_+-M\right)}^{-1}\left(M\,\delta M-a\,\delta a\right)\;, a\,\delta a = a\,\delta\left(J/M\right)={a\over M}\,\delta J-{a^2\over M}\,\delta M\;.\end{array}
0=\sum\limits_{j=1,\textrm{}j\neq i}^{n}\frac{m_{j}\langle Q_{j},\begin{pmatrix}0 -1 1 0 \end{pmatrix}Q_{i}\rangle_{2}}{(\sigma-\sigma(q_{i}\odot_{k}q_{j})^{2})^{\frac{3}{2}}},\textrm{}i\in\{1,...,n\},
\int_{0}^{\infty}\frac{\sin^{2}(u-Q_{\frac{1}{\lambda(t)}}+Q_{\frac{1}{\lambda(t)}})}{r^{2}}r dr \leq C\left(\int_{0}^{\infty}\frac{\sin^{2}(Q_{\frac{1}{\lambda(t)}})}{r^{2}}r dr + \int_{0}^{\infty}\frac{|u-Q_{\frac{1}{\lambda(t)}}|^{2}}{r^{2}}r dr\right)
{\cal K}^{[ j ]}= \mathrm{~ l o g ~}\| \Lambda^{[ j ]}\|^{2}\equiv \mathrm{~ l o g ~}[ < 0 |{\bar{X}}^{[ j ] 1}\cdots{\bar{X}}^{[ j ] j}{X}^{[ j ] j}\cdots{X}^{[ j ] 1}| 0 > ] .
\mathrm{S o l v}\left( \frac{\mathrm{U S p}( 3 , 3 )}{\mathrm{S U}( 3 ) \times{U}( 1 )}\right) \subset \mathrm{S o l v}\left( \frac{\mathrm{S U}( 3 , 3 )}{\mathrm{S U}( 3 ) \times \mathrm{S U}( 3 ) \times \mathrm{U}( 1 )}\right) ~ .
\sum_{n \geq 1}\frac{a_n q^{2n}}{1-q^n}= \frac{1}{(q; q)_{\infty}}\sum_{n \geq 1}\sum_{k=1}^{\lfloor n/2 \rfloor}\left( s_o(n-k, k) - s_e(n-k, k)\right) a_k \cdot q^n.
\int_0^\infty (1 - x^2) \, e^{-\frac{1}{2\sigma^2}\left(x^2-1\right)^2}\, dx \geq \frac{\sigma^2}{2}+ \frac{\sigma^2}{2}\left( e^{- \frac{1}{2 \sigma^2}\, \frac{9}{16}}- 2 e^{-\frac{1}{2 \sigma^2}}\right) - \frac{\sigma^2}{2}&= \frac{\sigma^2}{2}\, e^{- \frac{1}{2 \sigma^2}}\left( e^{\frac{1}{2 \sigma^2}\, \frac{7}{16}}- 2 \right) >0
\sup_{0<r<1}\int_{\mathbb{T}^\infty}\log(1+|(\mathcal{B}_k f)_{[r]}|)dm_\infty=\sup_{\sigma>0}\int_{\mathbb{T}^\infty}\log(1+|\mathcal{B}_k f_\sigma|)dm_\infty\leq\|f\|_0<\infty,
W \left( \left. \begin{array}{c c}{{a}}&{{b}}\\{{d}}&{{c}}\\ \end{array}\right| z \right) = \frac{1}{w ( z , w )}\overline{{{W}}}\left( \left. \begin{array}{c c}{{a}}&{{b}}\\{{d}}&{{c}}\\ \end{array}\right| z \right) ,
RE_s(G^\sigma) = 2\sum_{j=1}^{\lfloor\frac{n}{2}\rfloor}|\mu_j|\leq 2\sqrt{\sum_{j=1}^{\lfloor\frac{n}{2}\rfloor}|\mu_j|^2}\sqrt{\lfloor\frac{n}{2}\rfloor}=2 \sqrt{\lfloor\frac{n}{2}\rfloor R_{-1}(G)}.
\nabla_i\nabla_j\big(\big(1+\frac{\alpha'}{2}R\big)\Delta R\big)= \big(1+\frac{\alpha'}{2}R\big)(\Delta\nabla_i\nabla_jR-2R\nabla_i\nabla_jR+\big(R\Delta R+\frac{1}{2}|\nabla R|^2\big)g_{ij}-\nabla_iR\nabla_jR) &+\frac{\alpha'}{2}\big(\nabla_i\nabla_jR\Delta R+\nabla_jR\nabla_i\Delta R+\nabla_iR\nabla_j\Delta R\big).
F_i(z,f_1(z),\dots,f_n(z))=P_i( z,f_1(z),\dots,f_n(z), &g_1(z),g_1(f_1(z)),\dots,g_1(f_n(z)),\dots, &g_l(z),g_l(f_1(z)),\dots,g_l(f_n(z)))
a_1 \cdot ( (\alpha_0 - \beta_0) \cdot (\alpha_1 - \beta_1) ) = - \frac{1}{2^2}( \alpha_0 \cdot \beta_1 + \alpha_1 \cdot \beta_0) + \frac{1}{2^2}\langle \beta_0, \beta_1 \rangle a_1.
\alpha_{-2}:= -\frac{2t}{3}a_1 - \frac{1}{3}v_{(2,3)}+ \frac{4}{3}(a_2 + a_{-2}) \cdot v_{(1,3)}+ \frac{4}{3}(a_3 + a_{-3}) \cdot v_{(1,2)}\in M_0^{(a_1)}.
\lim_{N\rightarrow \infty}\; \langle \; O_N \; \rangle \; = \;\lim_{N\rightarrow \infty}\;\int \; \prod_{n=1}^\infty \frac{db_n}{\sqrt{2\pi}}\exp \left( \; - \; \frac{1+ \lambda}{2}\;\sum_{n=1}^N \; b_n^2 \; - \;\frac{1}{2}\;\sum_{n=N+1}^\infty \; b_n^2 \right)
-A^{2}T(At)Q_{i}=\sum\limits_{j=1,\textrm{}j\neq i}^{n}\frac{m_{j}\left(T(At)Q_{j}-\sigma(q_{i}\odot_{k}q_{j})T(At)Q_{i}\right)}{(\sigma-\sigma(q_{i}\odot_{k}q_{j})^{2})^{\frac{3}{2}}}-\sigma(\dot{q}_{i}\odot_{k}\dot{q}_{i})T(At)Q_{i},\textrm{}i\in\{1,...,n\}.
\left( L_{s}^{\rho}\omega\right) \left( s_{1},\dots,s_{q}\right) =L_{s}^{\rho}\left( \omega\left( s_{1},\dots,s_{q}\right) \right) & -{\sum\limits_{i=1}^{q}}\omega\left( s_{1},\dots,s_{i-1},\left[s,s_{i}\right]_{E},s_{i+1},\dots,s_{q}\right).
(-g)\,{\epsilon}^{{\delta}_{(1)}ba}\langle{A_{{\sf GI}\,i}^{{\delta}_{(1)}}({\bf{r}})\rangle\partial_{i}\int\frac{d{\bf x}_1}{4{\pi}|{\bf r}-{\bf x}_1|}\,j^a_{0}({{\bf x}_1}})=ig\,{\epsilon}^{{\delta}_{(1)}ba}{\phi}(r)L^{{\delta}_{(1)}}{\int\frac{d{\bf x}_1}{4{\pi}|{\bf r}-{\bf x}_1|}\,j^a_{0}({{\bf x}_1}}),
\lim_{l\to\infty}\nu_{\mathcal P_{l}}\left(\Big\{S^{k_l(n_l+M_l)}h_l-\mathbb{E}_{\mathcal{P}_{l}}(S^{k_l(n_l+M_l)}h_l)\leqslant t s_l\Big\}\right)=\mathcal{N}(t).