harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left| \int_{\gamma_i}G \right| \leq \int_{\gamma_i}|G| = \int_{\gamma_i}\exp\left(-W(z) \right) = \int_{\gamma_i}\exp\left(-\sum \frac{c}{\pi}a_k \log |z-(x_k,0)| \right) & = \int_{\gamma_i}\prod |z-(x_k,0)|^{-\frac{c}{\pi}a_k}.
z_1\equiv\frac{{\rm tr}(\tilde{G}\tilde{{\cal H}}\tilde{G}\tilde{{\cal H}})}{2(-G)(1+C_1^2)}=\frac{\tilde{{\cal H}}^{\mu\nu}\tilde{{\cal H}}_{\nu\mu}-2y^{\mu}y_{\mu}}{2(1+C_1^2)},
0\, \,= \,\,\frac{\partial f}{\partial \lambda^j_\alpha}\,\,=\,\, \frac{\partial f'}{\partial \lambda^j_\alpha}\,\, = \! \sum_{c_\beta \in C_{d-e+1}}\!\!\! c_\beta \cdot f_{\alpha+\beta}d-e+2 \leq j \leq d+1 \alpha \in M_{d_j,d_j}.
(\mathcal{S},\Lambda)=(\mathcal{V}\times \mathbb{P}_\mathbb{C}^1\times \mathbb{C}, \left( t\xi+ c_0\xi^2\right)\frac{\partial}{\partial \xi}\wedge \frac{\partial}{\partial u})\,\,\,\bigcup \,\,\, (\mathcal{V}_1\times \mathbb{P}_\mathbb{C}^1\times \mathbb{C}, \left( t\xi_1+ c_0u_1 \xi_1^2\right)\frac{\partial}{\partial \xi_1}\wedge \frac{\partial}{\partial u_1})
\mathcal{L}^{-1}\left( \left( 2\pi\right)^{3/2}e^{\frac{1}{2}t^{2}}m_{\Sigma}\left( t/\kappa\right) \right)(\sqrt{2x}) = h(\sqrt{2x}) = e^{-x}{\displaystyle\iint\limits_{\sqrt{2x}T_{N}}}e^{-\frac{1}{2}\left( y^{2}+z^{2}\right)}\, d y \, d z,
C_5(y_0) =\sup_{t\ge y_0}\frac{\exp(C_2(y_0)R(t))-1}{R(t)}=\frac{\exp(C_2(y_0)R(y_0))-1}{R(y_0)}, C_6(y_0) =\sup_{t\ge y_0}\frac{1-\exp(-C_1(y_0)R(t))}{R(t)}=C_1(y_0).
\begin{aligned}\mathcal{L}(t) = \mathcal{T}(t) - \mathcal{V}(t) &= \frac{1}{2}\int_{-\infty}^\infty u_t^2 - u_x^2 - (u^2-1)^2 - \frac{1}{2}\Omega^2 x^2u^2 \, dx.\end{aligned}
\tau^a ={n_1 \choose d_1}{n_2 \choose d_2+2}, \tau^b = (d_2-d_1+1)!{d_1+1 \choose d_2-d_1+1}{d_2+1 \choose d_2-d_1+1}{n_1 \choose d_1+1}{n_2 \choose d_2+1}& + \sum_{2r_0+r_1=2d_1-d_2}{d_1+1 \choose r_0}{d_1+1-r_0 \choose r_1}\frac{(d_2+2)!}{2^{r_0+d_2-d_1+1}}{n_1 \choose d_1+1}{n_2 \choose d_2+2}.
{\frac{d^{2}t}{d \tilde{\lambda}^{2}}}+{\frac{\dot{n}}{n}}\left({\frac{d t}{d \tilde{\lambda}}}\right)^{2}+{\frac{a \dot{a}}{n^{2}}}\sum_{j}\left({\frac{d x^{j}}{d \tilde{\lambda}}}\right)^{2}= \left[ n n^{\prime}\left({\frac{d t}{d \tilde{\lambda}}}\right)^{2}- a a^{\prime}\sum_{j}\left({\frac{d x^{j}}{d \tilde{\lambda}}}\right)^{2}- 2{\frac{n^{\prime}}{n}}\right]{\frac{d t}{d \tilde{\lambda}}},
f_2 = Y^{p^s}- \tfrac{b_1}{a_0}Y^{p^{s-1}}X^{p^s-p^{s-1}}-\tfrac{b_2}{a_0}\Delta^{p^{s-2}}X^{p^s-2p^{s-2}}\ldots -\tfrac{b_{r-1}}{a_0}\Delta X^{p^s-2}-\tfrac{b_r}{a_0}YX^{p^s-1}.
\frac{(\log(i+1))^{\beta-1}[g(0)+\cdots+g(i)]}{(i+1)g(i)}\begin{cases}\le\frac{1}{\alpha\beta}+\frac{i_0(\log(i+1))^{\beta-1}}{i+1}, \ge\frac{A_{i,j}}{\alpha\beta+(\log(j+1))^{1-\beta}}.\end{cases}
\tilde{W}_{\pm}( \tilde{\phi}; ( \tilde{\phi}_{0}, \tilde{W}_{0}) ) = \tilde{W}_{\pm}( \tilde{\phi}+ \pi ; ( \tilde{\phi}_{0}+ \pi , \tilde{W}_{0}) ) = - \tilde{W}_{\pm}( - \tilde{\phi}; ( - \tilde{\phi}_{0}, - \tilde{W}_{0}) ) ,
\nabla_{Z}Z =- \eta(Y) Z + \phi (\nabla_{Z}Y) + f Y + Z(f) \xi, \nabla_{Y}Y =- \phi \nabla_{Y}(\phi Y) + \eta (\nabla_{Y}Y) \xi - \eta(Y) \phi (Y), \nabla_{Z}Z + \nabla_{Y}Y =\phi ([Z, Y]) -2 \eta(Y) \phi (Y) +2 f Y + (Z(f)+ \eta(\nabla_{Y}Y) -2 f \eta (Y)) \xi,
\frac{d^{n-1}z}{dv^{n-1}}=\bar{P}_1+nvI_{n,0}-(n-1)I_{n,1}, \frac{d^{n-2}z}{dv^{n-2}}=\bar{P}_2+\frac{n}{2}v^2I_{n,0}-(n-1)vI_{n,1}+\frac{n-2}{2}I_{n,2}, \frac{d^{n-k}z}{dv^{n-k}}=P_{k-1}:3\leq k\leq n-2, \frac{dz}{dv}=-1+P_{n-2,1},
\big(T_j(PQ)\big)_i =(PQ)_i(u)(PQ)_j(u-1)(PQ)_j(u) &=\big((P)_i(u)(Q)_i(u)\big)\big((P)_j(u-1)(Q)_j(u-1)\big)\big((P)_j(u)(Q)_j(u)\big) &=\big((P)_i(u)(P)_j(u-1)(P)_j(u)\big)\big((Q)_i(u)(Q)_j(u-1)(Q)_j(u)\big) &=\big(T_j(P)T_j(Q)\big)_i.
2L\left[ xP_{n}P_{n+1}\right] =2h_{n+1},L\left[ \left(x^{2}-z^{2}\right) P_{n+1}\partial_{x}P_{n}\right] =nh_{n+1}, L\left[ -z^{2}P_{n}\partial_{x}P_{n+1}\right] =-z^{2}\left( n+1\right)h_{n}.
g_{1}(\nabla^{^{M_1}}_{U}V,X) =-\cos^{2}\theta g_{1}(V,\nabla^{^{M_1}}_{U}X)+g_{1}(\pi_*(\omega\phi V),\pi_*(\mathcal{H}\nabla^{^{M_1}}_{U}X))-g_{1}(\omega V,\mathcal{T}_{U}\mathcal{B}X) &-g_{1}(\pi_*(\omega V),\pi_*(\mathcal{H}\nabla^{^{M_1}}_{U}\mathcal{C}X))-g_{1}(V,\phi U)\eta (X)
|m+j|^{-4}\ge (|m|+|j|)^{-4}= |m|^{-4}(1+|j|/|m|)^{-4}&\ge |m|^{-4}(1-4|j|/|m|) \ge |m|^{-4}(1-2^{2+r}|m|^{r-1}).
W^{(\nu)}(y) =(1-y)^{\nu-1/2}y^{\nu-1/2}\, \Psi_0(y) \,T^{(\nu)}\, \Psi_0(y)^*, D^{(\nu)}=y(1-y)\partial_y^2+\partial_y(C+\nu-1-y(U+2\nu+1))-(V+(\nu-1)U+(\nu-1)(\nu-2)),
{\sf A}_{n}= \frac{( - 1 )^{n}}{2}\sum_{s = 0}^{\left[ \frac{n - 2}{2}\right]}\frac{( n - s - 1 ) !}{( n - 2 s - 2 ) ! ( s + 1 ) !}L^{2 s}\mathrm{f o r}n{\geq}2 ,
J^{*}[\psi^{(4)}]_{T,t}=\hbox{\vtop{\offinterlineskip\halign{\hfil#\hfil\cr{\rm l.i.m.}\cr$\stackrel{}{{}_{p\to \infty}}$\cr}}}\sum\limits_{j_1, j_2, j_3,j_4=0}^{p}C_{j_4j_3 j_2 j_1}\zeta_{j_1}^{(i_1)}\zeta_{j_2}^{(i_2)}\zeta_{j_3}^{(i_3)}\zeta_{j_4}^{(i_4)},
C_\nu :=\min\bigg\{\eta_\nu^{1/2}\bigg(1 -\frac{V_{-1/2}(0)}{V_{-1/2}(\mathrm i\beta)}\bigg), \, \eta_\nu^{1/2}\bigg(1 -\frac{V_{1/2}(0)}{V_{1/2}(\mathrm i\beta)}\bigg), \Big(1 -\nu\big(3(16 +\nu^2)^{1/2}-5\nu\big)/8\Big)^{1/2}\bigg\}.
A_{K}^{N S N S}={\frac{1}{2}}\int_{0}^{\infty}{\frac{d t}{2 t}}( 4 \pi^{2}t )^{- 5}{\frac{f_{4}( e^{- 2 \pi t})^{8}}{f_{1}( e^{- 2 \pi t})^{8}}}\, ,
F_{B}(\beta)=F_{B}(2^{N}\beta)+\sum_{k=1}^{N}\int_{\cal S}\frac{d^{2}\tau}{\tau_{2}^{2}}\tau_{2}^{-\frac{d-2}{2}}\left[\frac{\bar{\theta}_{4}^{4}}{\bar{\eta}^{12}}(z_{v}-z_{o})-\frac{\bar{\theta}_{3}^{4}}{\bar{\eta}^{12}}(z_{v}+z_{o})\right]\theta_{2}\left(0\left|\frac{i(2^{k}\beta)^{2}}{2\pi^{2}\tau_{2}}\right.\right)
\mathcal{I}^2 \ll \frac{1}{T^2}\int_{|y| > M^{\varepsilon}}\Big|\widehat{w_2}\left( \frac{y}{T}\right) \Big |^2 \Big | \Sigma(M^{\frac \eta 4},y) \Big|^8 \Big|\Sigma(M^{1-\frac \eta 4},y) \Big|^8dy.
E = - \frac{3 \xi^{2}}{6 4 a}\left[ \sum_{l = 1}^{\infty}\nu^{0}- \frac{9}{1 2 8}\sum_{l = 1}^{\infty}\frac{1}{\nu^{2}}+ \frac{4 2 3}{1 6 3 8 4}\sum_{l = 1}^{\infty}\frac{1}{\nu^{4}}+{\cal O}\left( \frac{1}{\nu^{6}}\right) \right]{.}
\zeta_{\bar{\cal V}}(s) = \frac{m L}{\sqrt{8 \pi}}\frac{1}{\Gamma(s)}\int_0^\infty d\beta \, \beta^{s-\frac{3}{2}}e^{-4 \beta}=\frac{m L}{\sqrt{8 \pi}}\frac{1}{4^{s-\frac{1}{2}}}\frac{\Gamma(s-\frac{1}{2})}{\Gamma(s)}
\begin{gathered}B^{(m)}_{n-m}=A^{(m-1)}_{n-m+1}A^{(m)}_{n-m}=B^{(m-1)}_{n-m+1}+a_{n-m+1}A^{(m-1)}_{n-m+1}B^{(0)}_{n}=0 B^{(1)}_{n-1}=1=A^{(0)}_{n}\end{gathered}
\Psi_+(z)=\Psi_-(z)\begin{pmatrix}1 0 -e^{\pi i\alpha}1\end{pmatrix}, z\in\Sigma_1, &\Psi_+(z)=\Psi_-(z)\begin{pmatrix}0 -1 1 0\end{pmatrix}, z\in \Sigma_2, &\Psi_+(z)=\Psi_-(z)\begin{pmatrix}1 0 -e^{-\pi i\alpha}1\end{pmatrix}, z\in\Sigma_3.
\frac{\partial}{\partial X^{\mu_0}}j_{b}^{\mu_{0}\mu_{1}\ldots\mu_{p}}(X)=- g_{b}\sum_{i=0}^{p}\int d\xi^0_{b}d\xi^1_{b}\ldots\hat{d\xi^{i}_{b}}\ldots d\xi^{p}_{b}\left[K_{bi}^{\mu_{1}\mu_{2}\ldots\mu_{p}}\delta^{(D)}(X-X_b)\right]_{\xi_b^i=0}^{\xi_b^i=l_b^i},
G(A{\mathbin{\rm\vphantom{X}\hskip 1.1pt\hskip -0.5pt\_\hskip -0.3em\_\vrule width 0.5pt\hskip 1.7pt}}B,C) = G(B,\tilde{A}\wedge C) , \, \, \, \, \, \, \forall C \in \bigwedge(V) ,
\frac{\displaystyle \prod_{i=0}^{t-1}(q^m - q^{a+i}) \prod_{i=0}^{b-t-1}(q^{a}- q^i)}{\displaystyle \prod_{i= 0}^{b-1}(q^m - q^i)}\begin{bmatrix}b t \end{bmatrix}_q.
( \kappa ) ={\frac{1}{\sqrt{1 + \vert \nu_{B}\vert^{2}}}}\left( \begin{array}{c c}{\kappa_{B}}&{\kappa_{B}\nu_{B}}\\{\nu_{B}\kappa_{B}}&{\nu_{B}^{c}\kappa_{B}\nu_{B}}\\ \end{array}\right){\frac{1}{\sqrt{1 + \vert \nu_{B}\vert^{2}}}},
\hat{S}\ [\hat{X}^{\hat{\mu}},\gamma] =-{\textstyle\frac{p}{2}}\int d\xi \sqrt{|\gamma|}\ \gamma^{-1}partial_{\xi}\hat{X}^{\hat{\mu}}\partial_{\xi}\hat{X}^{\hat{\nu}}\hat{g}_{\hat{\mu}\hat{\nu}}\, ,
\sum_{h=2}^{j-1}(-1)^{h}S_{h-1}^{(1)}H_{k,Y_{l-h,j-h}}=\sum_{h=2}^{j-1}(-1)^{h}(\alpha+k-1+l-j+h) H_{k,Y_{l-1,j-h}}+H_{k,Y_{l-1,j-1}}\sum_{h=2}^{j-1}(\alpha+k-j+h).
\o_{X_{Y^1}/Y^1}:= \o_{X_{Y^1}}\otimes{f_{Y^1}}^*\o_{Y^1}^{-1}\cong \o_{X_{Y^1}}\otimes{f_{Y^1}}^*\o_{Y^1}^{p-1}\otimes{f_{Y^1}}^*{F_Y}^*\o_Y^{-1}\cong{(F_Y)_X}^!\O_X\otimes{(F_Y)_X}^*\o_X\otimes ({f_{Y^1}}^*{F_Y}^!\O_Y)^{-1}\otimes{(F_Y)_X}^*f^*\o_Y^{-1}\cong{(F_Y)_X}^*\o_{X/Y}=(\o_{X/Y})_{Y^1}
Z[Y,\psi,\eta]=\exp\left\{-{S_c}[{X_c},\tilde{g}]\right\}\int{\mathcal{D}_{\tilde{g}}}\varphi{{\left(Det'\hat{\Delta}\right)}^{-d/2}}{{\sqrt{Det'{\hat{P}^{\dagger}}\hat{P}}}\over{Vol(CKV)}}\exp\left\{-{S_L}[\varphi,\hat{g}]\right\},
K_{l + \nu}( r^{\prime \prime}, r^{\prime}; T ) = \int_{r ( 0 ) = r^{\prime}}^{r ( T ) = r^{\prime \prime}}{\cal D}r ( t ) \, \; \mu_{l + \nu}[ r^{2}] \exp \left( \frac{i}{\hbar}\int_{0}^{T}d t \left\{\frac{M}{2}\, \left[ \stackrel{.}{r}( t ) \right]^{2}- V ( r ( t ) ) \right\}\right) \;
\sum_{i=0}^k\frac{\lambda_i^\rho}{S_i^{\rho-1}}=\frac{1}{2^{2-\rho}}\sum_{i=0}^k\frac{i+1}{(i+2)^{\rho-1}}\leq \frac{1}{2^{2-\rho}}\sum_{i=0}^k(i+1)^{2-\rho}\leq \frac{1}{2^{2-\rho}(3-\rho)}(k+2)^{3-\rho}
\P(L[(S(0) , \dots , S({\tau_D}))^-]=\omega^-) =\prod_{j=0}^{n-1}p^*(\omega_{j+1}, \omega_{j})G_{p^*}(\omega_{j}; D \setminus \{\omega_{n-1}, \dots , \omega_{j+1}\}) &=\prod_{j=0}^{n-1}p(\omega_{j}, \omega_{j+1})G_{p^*}(\omega_{j}; D \setminus \{\omega_{j+1}, \dots , \omega_{n-1}\}).
\begin{dcases}\|\xi_{\lambda^p_k}\|_{L^2(0,1)}\leq Ck^{-1}, \|\xi_{\lambda^p_k}\|_{H^{-s}_{per}(0,1)}\leq Ck^{-s-1}, \ \ 0<s<1, \|\xi_{\lambda^p_k}\|_{H^{-s}_{per}(0,1)}\leq Ck^{-2}, \ \s\geq 1, \|\eta_{\lambda^p_k}\|_{L^2(0,1)}\leq C.\end{dcases}
Z=\int \big[{(\Pi_{i=0}^2{\rm det}\Delta_{0,i}^{(-1)^i})^2(\Pi_{i=0}^2{\rm det}\Delta_{i,0}^{(-1)^i})^2\over(\Pi_{i=0}^2{\rm det}\Delta_{i,1}^{(-1)^i})^{-1}(\Pi_{i=0}^2{\rm det}\Delta_{1,i}^{(-1)^i})^{-2}(\Pi_{i=0}^2{\rm det}\Delta_{0,i}^{(-1)^i})^2}\big]^{1\over 4}.
\| g \|^2_{s+\gamma}\le \|f \|_s^2 \sum_{\ell \ge 0}|\widehat u(\ell) |^2 \sum_{p,n \ge 0,p + n = \ell}\frac{\langle n \rangle^{2(s+\gamma)}}{\langle p \rangle^{2s}}\, .
\begin{cases}\partial_tu(x,t)+\triangle_{p(x,t)}^Nu(x,t)+\sum_{i=1}^n\mu_i\frac{\partial u}{\partial x_i}(x,t)=ru(x,t)~~ \mathbb R^n\times (0,T), u(x,T)=g(x)~~ \mathbb R^n\end{cases}
\langle\tilde\psi,R^P(\tilde\psi,\tilde\psi)\tilde\psi\rangle_{SM\otimes\phi^\ast TP}=R_{IJKL}\langle\tilde\psi^I,\tilde\psi^K\rangle_{SM}\langle\tilde\psi^J,\tilde\psi^L\rangle_{SM},
\psi = \psi \left( \varepsilon^{a b}\right) = \left( 1 , 0 , \sqrt{{\frac{1}{2}}}\varepsilon^{a b}, 0 , \sqrt{{\frac{3 . 1}{4 . 2}}}\varepsilon^{\left( a b \right.}\varepsilon^{\left. c d \right)}, 0 , . . . \right) .
O_0 f=\frac{1}{A_1(x,0)}\cal{M}_0 f-\frac{A_2(x,0)}{A_1(x,0)}\int_0^x \exp\left( -\int_\xi^x\frac{A_2(t,0)}{A_1(t,0)}dt\right)\frac{(\cal{M}_0f)(\xi)}{A_1(\xi,0)}d\xi
\left[ 2i{\left(-1\right)}^{n-1}{\epsilon}_1 \cdot p_1 P^{n-1}\left( p_1^2,p^2 \right) \right]\frac{{\left(-1\right)}^n i}{p^{2n}-{\left( -m^2 \right)}^n}\left[ 2i{\left(-1\right)}^{n-1}{\epsilon}_2 \cdot p_2 P^{n-1}\left( p^2,p_2^2 \right) \right]+
\mathcal{M}_\psi\left[\left(\Psi_n^{(s)}\ast\Psi_m^{(r)}\right);\alpha+s/2\right]=(-1)^n\mathcal{M}_\psi\left[\left(\Psi_n^{(s)}\star\Psi_m^{(r)}\right);\alpha+s/2\right]\;,
s = a{\theta_3}^4 + b{\theta_3^\prime\theta_3^3 \over \eta^{12}}\ , \ \ -t= a{\theta_4}^4 + b{\theta_4^\prime\theta_4^3 \over \eta^{12}}\ , \ \ -u = a{\theta_2}^4+ b{\theta_2^\prime\theta_2^3 \over \eta^{12}}\ ,
\varphi(K_1^+(w), K_1^-(z)) = 1 = \varphi(\phi^+(w), \phi^-(z)), & \varphi(E^+(w), F^-(z)) = \frac{(q-q^{-1})z}{z-w}, \varphi(\phi^+(w), K_1^-(z)) = \frac{z-w}{qz-q^{-1}w}, & \varphi(F^+(w), E^-(z)) = \frac{(q^{-1}-q)w}{z-w}, \varphi(K_1^+(w), \phi^-(z)) = \frac{q^{-1}z - qw}{z-w}.
\begin{array}{cccccc}(n_m + 0)_{m=1}^\infty (n_m + 1)_{m=1}^\infty (n_m + 2)_{m=2}^\infty (n_m + 3)_{m=3}^\infty (n_m + 4)_{m=4}^\infty \dots 1 2 3 4 5 6 7 8 9 10 11 12 13 14 \vdots \vdots \vdots \vdots \vdots \ddots \end{array}
\sum_{\substack{q\le x^{\theta}(q,B)=1}}\max_{(L(a),q)=1}\Bigl|\#\mathcal{P}_{L,\mathcal{A}}(x;q,a)-\frac{\#\mathcal{P}_{L,\mathcal{A}}(x)}{\phi_L(q)}\Bigr|\ll \frac{\#\mathcal{P}_{L,\mathcal{A}}(x)}{(\log{x})^{100k^2}}.
\gamma_{(lr-r)}[X,Y]:=(\mathrm{ad}_{X}^{(l)}\otimes 1)\gamma_{(lr-r)}(Y)+(\mathrm{ad}_{Y}^{(r)}\otimes 1)\gamma_{(lr-r)}(X), & \gamma_{(l-lr)}[X,Y]:=(1\otimes \mathrm{ad}_{X}^{(l)}+ \mathrm{ad}_{X}^{(l)}\otimes 1)\gamma_{(l-lr)}(Y), &\gamma_{(lr-l)}[X,Y]:=(1\otimes \mathrm{ad}_{X}^{(l)})\gamma_{(lr-l)}(Y)+(1\otimes \mathrm{ad}_{Y}^{(r)})\gamma_{(lr-l)}(X).
\left\langle \frac{\partial^2 V}{\partial\left({\rm Re}F\right)^2}\right\rangle = -\left\langle \frac{\partial^2 V}{\partial\left({\rm ImF}\right)^2}\right\rangle =\frac{2}{\Lambda_{YM}^4}\left(\frac{11 N_c}{12}\right)^2 \ .
\kappa_1: (\kappa_1+1)\tilde{q}_1 = (\kappa_0+1)p^*, \kappa_2: (\kappa_2+1)\tilde{q}_1 = (\kappa_1+1) p^*, \kappa_3: (\kappa_3+1)\tilde{q}_1 = (\kappa_2+1) p^*, \vdots \vdots \, .
( k + 2 ) [ ( k + 1 )^{2}( k + 2 ) - 8 ( 2 k + 5 ) ] G_{2 k + 6}= 1 2 ( k + 2 )^{2}G_{2 ( k + 1 )}+ c_{2 ( k + 1 )}^{( 1 )}+ G_{k + 1}c_{k - 1}^{( 1 )}+ c_{2 k}^{( 0 )}
\frac{\partial}{\partial t}\frac{\delta \bar\ell}{\delta\bar{\sigma}^1}+\operatorname{div}\frac{\delta \bar\ell}{\delta\bar{\sigma}^2}=-\operatorname{ad}^*_{\bar\sigma^1}\frac{\delta \bar\ell}{\delta\bar{\sigma}^1}-\operatorname{Tr}\left(\operatorname{ad}^*_{\bar\sigma^2}\frac{\delta \bar\ell}{\delta\bar{\sigma}^2}\right),
\int_{\omega_k}|A_k-Dg_k(x)|dx \leq \sum_{\ell=1}^{N_k}\int_{\omega_{k,\ell}}\left(|A_k-M_{k,\ell}|+|M_{k,\ell}-Dg_k(x)|\right)dx &\lesssim \sum_{\ell=1}^{N_k}\int_{\omega_{k,\ell}}|A_k-M_{k,\ell}|dx=\sum_{\ell=1}^{N_k}\lambda_{k,\ell}|A_k-M_{k,\ell}||\omega_{k}|\lesssim j^{-2}|\omega_{k}|,
\eta(A^{\not\ge\lambda}M_\lambda\otimes_{B_\lambda}N_\lambda) =A^{\not\ge\lambda}A^{\le\lambda}+A^{<\lambda}=A^{<\lambda}, \eta(M_\lambda\otimes_{B_\lambda}N_\lambda A^{\not\ge\lambda}) =A^{\le\lambda}A^{\not\ge\lambda}+A^{<\lambda}=A^{<\lambda}.
U([(1,0,0), (1,1,1)]) = U^\prime_{\bar{0}}([(1,0,0),(1,1,1)]) & = U^\prime_{\bar{0}}((1-1), -(1-0), (1-0)) & = U^\prime_{\bar{0}}((1-1), -(1-0), (1-0)) & = U^\prime_{\bar{0}}(0,-1,1) = 0
g(m^2_i)\equiv{1 \over 16\pi^2}\left[\sum_s m_s^{D-2}{}~\Gamma\left(1-\frac D2,m_s^2\delta\right)+2\sum_d m_d^{D-2}\Gamma\left(1-\frac D2,m_d^2\delta\right)\right]~~~,
c_{-i-1}= (-1)^{i+1}\,\,{^{\mathbf{x}_{-i-1}}\Delta_{y}^{-i-1}}\left. p(y) \right|_{y=3}= \sum_{d_{i+1}=3}^{2i+3}\sum_{d_{i}= d_{i+1}}^{2i+1}\cdots \sum_{d_2 = d_3}^5 \sum_{d_1=d_2}^3 p(d_1).
\left|\Box^*_{g_{1,t}}v_{2,t}\right| = \left|\Box^*_{g_{1,t}}v_{2,t}- \Box^*_{g_{2,t}}v_{2,t}\right| \le \left|\Delta_{g_{1,t}}v_{2,t}- \Delta_{g_{2,t}}v_{2,t}\right| + |R_{g_{1,t}}-R_{g_{2,t}}|v_{2,t}\le C \varepsilon \left( |\nabla^2 v_{2,t}|_{g_{2,t}}+|\nabla v_{2,t}|_{g_{2,t}}+ v_{2,t}\right).
\alpha = \alpha_{0}[ 1 - \frac{\alpha_{0}}{3 \pi}l o g ( \frac{m^{2}}{p_{0}^{2}}) ]^{- 1}= \alpha_{0}[ 1 - \frac{\alpha_{0}}{3 \pi}l o g ( 2 / n_{0}) ]^{- 1}
\mathcal{A}_1 =r_1h_2+(r_1+r_2)h_3+\cdots (r_1+r_2+\cdots +r_{n-2})h_{n-1}+(r_1+r_2+\cdots +r_{n-1})h_n &=r_1(h_2+h_3+\cdots h_n)+r_2(h_3+h_4+\cdots +h_n)+\cdots +r_{n-2}(h_{n-1}+h_n)+r_{n-1}h_n &=\sum \limits_{j=1}^{n-1}r_j\sum \limits_{k=1}^{n-j}h_{j+k}.
\left\{\widehat{A}(TX){\rm exp}(\frac{c}{2}){\rm ch}(\widetilde{T_CM}-\widetilde{L_C}-16+W_i+W_j)\right\}^{(14)}&=-24\left\{\widehat{A}(TX){\rm exp}(\frac{c}{2})\right\}^{(14)}.
\delta_{0}{\cal L}_{0}= - \Omega [ ( d + 2 ) \chi^{\mu , \alpha \beta}( D R )_{\nu , \alpha \beta}+ 1 0 \chi^{\alpha , \beta \nu}D_{\nu}R_{\alpha \beta}+ 1 0 \chi^{\alpha}( D R )_{\alpha}+ ( d - 8 ) \chi^{\alpha}D_{\alpha}R ]
Spec\left(C_{\alpha}\right)=\left\lbrace\left[\alpha^2+2\alpha\right]^1,\left[\alpha\right]^{\frac{\alpha^3+2\alpha^2-\alpha-2}{2}},\left[-\alpha\right]^{\frac{\alpha^3+2\alpha^2+\alpha+2}{2}}\right\rbrace.
\sum_{\beta < \gamma}a_{\beta \gamma}^{\alpha}\theta^{\beta}\wedge \theta^{\gamma}= - \sum_{\beta <\gamma}\omega_{\gamma}^{\alpha}(X_{\beta}) \theta^{\beta}\wedge \theta^{\gamma}+ \sum_{\beta <\gamma}\omega_{\beta}^{\alpha}(X_{\gamma})\theta^{\beta}\wedge \theta^{\gamma}&= \omega_{\beta}^{\alpha}(X_{\gamma}) \theta^{\beta}\wedge \theta^{\gamma}.
\Delta( \tau_i^{\pm})=\tau_i^{\pm}\otimes \tau_i^{\pm},\ \Delta( \tau_{21})=\tau_1^{+}\otimes \tau_{21}+ \tau_{21}) \otimes \tau^+_{2}, \ \Delta( \tau_{12})= \tau_2^{-}\otimes \tau_{12}+ \tau_{12}) \otimes \tau^+_{1},
N_{t}^{\max}= \max_{f\in\mathcal{F}_0}\big| \{\tilde{f}\in\mathcal{F}_0:\, d(f,\tilde{f})\leq t\}\big|, N_{t}^{\min}= \min_{f\in\mathcal{F}_0}\big|\{\tilde{f}\in\mathcal{F}_0:\, d(f,\tilde{f})\leq t\}\big|.
(\begin{matrix}f_{11}&f_{12}&\cdot\cdot\cdot&f_{1n}\\f_{21}&f_{22}&\cdot\cdot\cdot&f_{2n}\\ \vdots&\vdots&\cdot\cdot\cdot&\vdots\\f_{m1}&f_{m2}&\cdot\cdot\cdot&f_{mn}\end{matrix})
\sum_{m\in Z}\oint \frac{d z}{2 \pi i z}\prod_{i=1}^3{\mathcal B}[ a_i z, n_i+m; b_i z^{-1}, m_i-m] ={\mathcal B}[a_1 b_2, n_1+m_2; a_3 b_1; n_3+m_1] \;{\mathcal B}[a_2 b_1, n_2+m_1; a_3 b_2, n_3+m_2] \;.
(1+\varepsilon)\vert B\vert^d \geq (1+\varepsilon)\mathcal{H}^{d}_{\infty}(B )\geq (1+\varepsilon)\mathcal{H}^{d}_{\infty}(E)\geq\sum_{n\geq 0}\vert L_n \vert^d & \geq \sum_{n\geq 0}\mathcal{L}^d (L_n)\geq \mathcal{L}^d (E)=\mathcal{L}^d (B)=\vert B\vert^d.
p_{k,0}={k+3 \choose 3}- 4{k -N+2 \choose 3}-{k+2 \choose 2}+ 3{k-N+1 \choose 2}-{2N-k+2 \choose 3}&= \frac{1}{6}(k-N) \left(-2 k^2+4 k N-3 k+4 N^2+15 N+5\right),
A_{0}<\frac{A_{2}}{B_{2}}<\frac{A_{4}}{B_{4}}<\cdot\cdot\cdot<\frac{A_{2n}}{B_{2n}}<x<\frac{A_{2n+1}}{B_{2n+1}}<\cdot\cdot\cdot<\frac{A_{5}}{B_{5}}<\frac{A_{3}}{B_{3}}<\frac{A_{1}}{B_{1}}.
\gamma_{\kappa,k}= \gamma_{k,\kappa}= d_k - \sum_{k'\leq \kappa-1}\gamma_{k,k'}\,\,\mbox{for}k\leq \kappa-1, \gamma_{\kappa,\kappa}= d_{\kappa}- \sum_{k\leq \kappa}d_k + \sum_{k,k'\leq \kappa-1}\gamma_{k,k'}.
\gamma^{\epsilon,L}(d\varphi)\propto \exp\left[-\frac{1}{2}\sum_{x\in \Lambda_{\epsilon,L}}\left(\varphi_x(-\Delta^1+\epsilon^2 m)\varphi_x\right)\right]d\varphi=\exp\left[-\frac{1}{2}(\varphi,A_{\epsilon}\varphi)\right]d\varphi,
\zeta_p(q)=\Lambda_q^- \sum_{\gamma \in \sum_p^q}\prod_{i=1}^{|\gamma|}\frac{\psi(r_i)-\psi(r_{i-1})}{\psi(q)-\psi(r_{i-1})}\frac{\Theta(r_{i-1},r_i)}{w(\overrightarrow{r_{i-1}\,r_i})},
\operatorname{Pr}\Big(|\langle x,\varphi\rangle|\geq\xi\Big)=\operatorname{Pr}\Big(|Z|\geq\xi/\|\Sigma^{1/2}x\|_2\Big)\leq\operatorname{Pr}\Big(|Z|\geq\xi/\sigma_\mathrm{max}\Big)\leq e^{-\xi^2/2\sigma_\mathrm{max}^2}
\begin{matrix}A\\B\\C\\D\\ \end{matrix}[\begin{matrix}1&2&3&4\\ 2&1&4&3\\ 3&4&1&2\\ 4&3&2&1\\ \end{matrix}]\begin{matrix}E\\F\\G\\H\\ \end{matrix}[\begin{matrix}1&3&4&2\\ 2&4&3&1\\ 3&1&2&4\\ 4&2&1&3\\ \end{matrix}]\begin{matrix}I\\J\\K\\L\\ \end{matrix}[\begin{matrix}1&4&2&3\\ 2&3&1&4\\ 3&2&4&1\\ 4&1&3&2\\ \end{matrix}]
\left|\prod_{p=0}^{s-1}(1-\zeta_q^{b^{p}m})\right| =\prod_{p\in S_1}|(1-\zeta_q^{b^{p}m})(1-\zeta_q^{b^{p+1}m})|\prod_{p\in S_3}|(1-\zeta_q^{b^{p}m})| &<\left(2\sin\frac{\pi l}{2l+1}\right)^s.
\left( I - \, k^{2}\, \eta \, a^{2}N \right) \left( \overset{1}{\mathbb{P}}\left(\tilde{W}\right)\right) (x) + (1 + \eta) \; \overset{2}{\mathbb{P}}\left(\tilde{W}\right)(x) + \left( I + \eta \, \nabla M \right) \left( \overset{3}{\mathbb{P}}\left(\tilde{W}\right) \right)(x) = a \, \mathcal{P}(x,0) - T(x),
\beta_{\nu}^{( 1 )}= \varepsilon^{\mu , [ \mu \nu ]}+ \varepsilon^{[ \mu \nu ] , \mu}, \beta_{\nu}^{( \widetilde{1})}= \frac 1 2 e_{\mu \nu \rho \omega}\left( \varepsilon^{\widetilde{\mu}, [ \rho \omega ]}+ \varepsilon^{[ \rho \omega ] , \widetilde{\mu}}\right) ,
\partial_{\mu}\Big ( \exp ( \frac{\phi}{f_{\phi}}) F^{\mu \nu}{}_{i}\Big ) + q \exp ( \frac{\phi}{f_{\phi}}) A_{\mu}{}^{j}f_{i j}{}^{k}F^{\mu \nu}{}_{k}= - q \overline{{\psi}}\gamma^{\nu}X_{i}\psi ,
\mathbf{P}\Big(|\Delta X_{s}|>c\,(t-s)^{\lambda}s<t\Big)\,\leq\,\mathbf{P}\Bigl(\sum_{n=0}^{\infty}Y_{n}\geq1\Bigr)\,\leq\,\sum_{n=0}^{\infty}\mathbf{E}Y_{n\,},
\Psi_{E}^{( - )}= \left( \begin{array}{l}{{e^{i \frac{\theta}{2}}\sum f_{n}e^{i \left( n + \frac{1}{2}\right) \theta}}}\\{{e^{- i \frac{\theta}{2}}\sum g_{n}e^{i \left( n + \frac{1}{2}\right) \theta}}}\\ \end{array}\right)
\times\!\bigg(\!\!\left(t_{C_{u-1}}\right)_{R_ur_uS_us_u}\!\!\left(t_B\right)_{V_uv_uW_uw_u}\!\left(\!\left(t_{E_u\alpha_u}\right)_{S_us_uV_uv_u}\delta_{W_uR_u}\!\!\delta_{w_ur_u}\!\!-\!\delta_{S_uV_u}\delta_{s_uv_u}\!\left(t_{E_u\alpha_u}\!\right)_{W_uw_uR_ur_u}\!\right)\!\!\bigg)\!\times
\lim_{n\longrightarrow \infty}(A_n, B_n, C_n, N_n, K_{W_n})=(A, B, C, N, K_{W}), \: K_W\succ 0, \lim_{n\longrightarrow \infty}(F_n,G_n, \Gamma_n, D_n, K_{Z_n})=(F,G, \Gamma, D, K_{Z}), \: K_Z \succeq 0
c_{s}^{(n)}(\chi_\mathbf{z}) = c_{s}^{(n)}(\mathbf{z}) = \frac{1-q^{-1}\mathbf{z}^{n_\alpha \alpha^\vee}}{1-\mathbf{z}^{n_\alpha \alpha^\vee}}= \frac{1-q^{-1}(\mathbf{z}')^{\alpha'}}{1-(\mathbf{z}')^{\alpha'}}.
J \epsilon_{a b 3}\epsilon_{i j}\int_{\tau_{0}}^{\tau_{0}- i \beta}d z^{0}\left[ S_{c}( z^{0}- x^{0}, \vec{p}) \right]_{\rho \mu}\left( \gamma_{5}C \right)_{\rho \sigma}\left[ S_{c}( z^{0}- y^{0}, - \vec{p}) \right]_{\sigma \nu}.
\begin{array}{l}{\mathrm{H}_{2 j}^{\left( k \right)}= \varepsilon^{j}\left( 1 - \varepsilon \right)^{j}}\\{\mathrm{H}_{2 j + 1}^{\left( k \right)}= \varepsilon^{j}\left( 1 - \varepsilon \right)^{j + 1}}\\ \end{array}\forall 0 \leq j \leq \frac{k - 1}2
{}_2F_1\left(2,\tfrac{1}{2}+\nu;\tfrac{3}{2};-x^2\right) = (1+x^2)^{-2}\,{}_2F_1\left(2,1-\nu;\tfrac{3}{2};\tfrac{x^2}{1+x^2}\right) = (1+x^2)^{-(\frac{1}{2}+\nu)}\,{}_2F_1\left( \tfrac{1}{2}+\nu,-\tfrac{1}{2};\tfrac{3}{2};\tfrac{x^2}{1+x^2}\right).
\sum_{n=1}^{\infty}\frac{1}{n^{2}}=lim_{n\rightarrow\infty}(\frac{1}{1^{2}}+\frac{1}{2^{2}}+\frac{1}{3^{2}}+\cdot\cdot\cdot+\frac{1}{n^{2}})=\frac{\pi^{2}}{6}.
\int_0^1\log(\mu(u,e^{T_0}e^{S_0}))\,du =\log(\Delta_\tau(e^{T_0}e^{S_0})) &=\log(\Delta_\tau(e^{T_0}))+\log(\Delta_\tau(e^{S_0}))=\tau(T_0)+\tau(S_0).
-2\langle \nabla a_{3,j},\nabla u_j\rangle =-a_{3,j,z_j}+(p-2)\left( \frac{1}{2}-\frac{\langle \nabla z_j, \nabla u_j \rangle^2}{z_j |\nabla z_j|^2}\right)\frac{|\nabla z_j|^2}{z_j}.
A_{s}^{( L )}\simeq \frac{\lambda^{2}}{3 2 \pi m}\left\{- \left( 1 - \frac{\vec{p}^{2}}{2 m^{2}}\right) \left[ \ln \left( \frac{\Lambda^{2}}{\vec{p}^{2}}\right) + i \pi \right] + \frac{\vec{p}^{2}}{\Lambda^{2}}+ \frac{\vec{p}^{4}}{2 \Lambda^{4}}+ \frac{\Lambda^{2}}{2 m^{2}}- \frac{3 \Lambda^{4}}{1 6 m^{4}}\right\}.
\vartheta - h d t , \qquad \mathrm{w i t h}\qquad \vartheta = ( p_{i}- \frac{1}{2}{B_{c}}\varepsilon_{i j}x_{j}) d x_{i}+ \frac{1}{2}\theta \varepsilon_{i j}p_{i}d p_{j}= P_{i}d Q_{i},
d s^{2}= - \left( r^{2}/ l^{2}- \frac{\mu}{r^{d - 3}}- 1 \right) d t^{2}+ \frac{d r^{2}}{\left( r^{2}/ l^{2}- \frac{\mu}{r^{d - 3}}- 1 \right)}+ r^{2}d \sigma_{- 1},
f_\psi(k) - \phi_k \leq f_\psi(k) - \frac{2}{d\gamma_d}\Phi(C_4 \gamma_d^k k^n) &= f_\psi(k) - f_\psi\left(k + \log_{\gamma_d}(C_4k^n)\right) &\leq \frac{2}{k^3}\log_{\gamma_d}(C_4k^n)\asymp \frac{\log(k)}{k^3}\cdot
|k^3 -\alpha k_1^3 - \alpha k_2^3 | = 3\alpha |k(k_1-c_1k)(k_1-c_2k)| \geq 3\alpha M_{\epsilon_0}|k|^{-\nu_{c_1}-\epsilon_0}\Big((c_1-c_2)|k| - \frac12\Big) &\gtrsim |k|^{1-\nu_{c_1}- \epsilon_0}.
\mathcal ({\mathcal{P}}_{n}f)(t)=\frac{\sum_{i=0}^{n}\frac{\beta_{i}}{t-t_{i}}f(t_{i})}{\sum_{i=0}^{n}\frac{\beta_{i}}{t-t_{i}}}= \sum_{i=0}^{n}f(t_{i})\mathcal{L}_{i}(t),
a(S) =C_{S}\,\det(S)^{k/2}\prod\limits_{p|N_1}B_p^{S'}\Big(h\Big(v_p(L),v_p\Big(\frac{N_1}{L}\Big)\Big)\Big) =C_{S}\,\det(S)^{k/2}\prod\limits_{p|N_1}B_p^{S'}(h(b_p,a_p-b_p)).
\overline{\mathbf{d}_B \mathbf{d}_C}=\{(d_1,d_2):d_1=\tfrac{\tau_1}{\gamma(P_1)},\tfrac{\tau_1}{\gamma(P_1)}\leq d_2\leq \tfrac{\tau_1}{\gamma(P_1)}+\tfrac{\tau_2}{\gamma(P_2)}\}
Q\big(f \cdot \frac{\rho_1}{\rho_2}\big) (KnH) =\int_K\int_H f(k^{-1}n h) \frac{\rho_1(k^{-1}nh)}{\rho_2(k^{-1}nh)}dh dk &= \frac{\rho_1(n)}{\rho_2(n)}\int_K \int_H f(k^{-1}n h) dh dk,
{i\over 4\pi}\delta_{ab}\epsilon^{\mu\nu\rho}{(x-y)_\rho\over |x-y|^3}, \,\,\,\,\,\,\,\,-i g f_{abc}\epsilon_{\mu\nu\rho}\int d^3x , \,\,\,\,\,\,\,\,g (T^a)_i^j.
\|u\|^{\tau}_{L^{\tau}(X)}= C \int_{X}|u(x)|^{a\tau}|u(x)|^{(1-a)\tau}dx & \leq \left(\int_{X}|u(x)|^{q}dx\right)^{\frac{a\tau}{q}}\left(\int_{X}|u(x)|^{\nu}dx\right)^{\frac{(1-a)\tau}{\nu}}& \stackrel{\eqref{Sobolevin}}\leq C\|u\|^{a\tau}_{H^{\sigma,p}(X)}\|u\|^{(1-a)\tau}_{L^{\nu}(X)},
\begin{array}{rll}\partial_tv + v\cdot\nabla_H v + w\partial_zv - \Delta v + \nabla_H\pi = f \Omega\times(0,T), w = \int_z^0 \mathrm{div}_H\,v\,d\zeta \Omega\times(0,T), \mathrm{div}_H\,\bar v = 0 G\times(0,T), v(0) = a, \end{array}
\frak{m}^{(0,2)}=\bigg\{p_-(A): A=\begin{bmatrix}0 0 A_1 0 0 A_2{}^tA_1{}^tA_2 0\end{bmatrix}, A_i=\begin{bmatrix}x_i 0 0 y_i I_2\end{bmatrix}, x_i, y_i\in\mathbb{C}, i=1,2\bigg\}