harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
A_1(t)=\frac{(1+t)(1+5t-5t^2)}{24},\;\; A_2(t)=\frac{(1 + t)^2 (-143+298t+231t^2-770t^3+385t^4)}{1152}, A_3 (t)=\frac{(1+t)^3(-6187-240549t+750468t^2-334565t^3-1021020t^4+1276275t^5-425425t^6)}{414720}.
w(z)=(\frac{z-a}{z-b})^{\alpha}(\frac{z-c}{z-b})_{2}^{\gamma}F_{1}(\alpha+\beta+\gamma,\alpha+\beta^{\prime}+\gamma;1+\alpha-\alpha^{\prime};\frac{(z-a)(c-b)}{(z-b)(c-a)})
Z = \int{\cal D}[f(\xi),g(\xi),X] \,\delta \left( g_{zz}^{f}\right)\delta \left( g_{\bar{z}\bar{z}}^{f}\right) \,{\rm det}\left( \frac{g_{zz}^{f}}{\delta f}\right){\rm det}\left( \frac{g_{\bar{z}\bar{z}}^{f}}{\delta f}\right) \, e^{-S[g^{f},X]}.
(gw,w^{-1}t_1w,\dots,w^{-1}t_nw) =((gw)w^{-1}t_1w(gw)^{-1},\dots,(gw)w^{-1}t_nw(gw)^{-1}) & =(gt_1 g^{-1},\dots,gt_n g^{-1}).
S_K^{\frac{N(r+1)}{N+prs}}< C_2 \varepsilon_W (\ell^{-1}(C_3 \varepsilon_W)^{q_1}+ \ell^{-1}(C_3 \varepsilon_W)^{q_2})= O(\varepsilon_W^{\frac{p}{p-\min\{q_1,q_2\}}})
\int_{|y|^{\alpha}t^{-\beta}}^{1}\xi^{-d/\alpha}t^{-\beta d/\alpha}\xi^{-1-1/\beta +1+1/\beta}d\xi \le \int_{|y|^{\alpha}t^{-\beta}}^{1}\xi^{-d/\alpha}t^{-\beta d/\alpha}d\xi = C t^{-\beta d/\alpha}(1 - |y|^{\alpha}t^{-\beta}).
{\cal L}_{0}= - \frac{1}{4}\partial_{\mu}A_{\nu}\partial^{\mu}A^{\nu}- \frac{1}{2}\partial_{\mu}\phi_{\alpha}\partial^{\mu}\phi^{\alpha}- D_{( 0 ) \mu}T \overline{{{\: D_{( 0 )}^{\mu}T}}}- m^{2}T \overline{{{T}}}
a_k = \frac1{\tau}\int\limits_{t_{k-1}}^{t_k}a(t)\,dt,\qquad\Phi_i = \frac 1h\int\limits_{x_i}^{x_{i+1}}\Phi(x)\,dx,~~ \Phi_m=\Phi(\ell), f_{ik}=\frac{1}{\tau h}\int\limits_{t_{k-1}}^{t_k}\int\limits_{x_i}^{x_{i+1}}f(x,t)\,dxdt,k=\overline{1,n},i=\overline{0,m-1},
\lim\limits_{r\to+\infty}\frac{\bar u(r,\theta)}{u_0(r)}=\lim\limits_{r\to+\infty}\sum_{k=0}^{\infty}\frac{\bar f_k}{u_0}\varphi_k=\sum_{k=0}^{\infty}\lim\limits_{r\to + \infty}\frac{\bar f_k}{u_0}\varphi_k=g(\theta).
|z^* (1)|^2_M=(2(7 + t_2(4 + (4 - 5 t_1)t_1 - 4t_2 + t_1(-3 + (t_1-1 )t_1^2)t_2 + (t_1^2-1)(1 + 2t_1)t_2^2 + (1 + t_1)^2 t_2^3)))/(3\exp(2)) .
\frac{1}{\tau}=\frac{ne^{4}}{2\pi\sqrt{2m^{*}(E_{c}-E_{db})}\hbar^{2}\epsilon\epsilon_{r}}(\frac{1}{q_{s}^{2}}-\frac{1}{q_{s}^{2}+\frac{8m^{*}(E_{c}-E_{db})}{\hbar^{2}}})(11)
{\Phi}^{(I)}= -\frac{s\,e^{2}F(1-F)}{12{\pi}^{2}|m|}\int_{1}^{\infty}\frac{dv}{v{\sqrt{v-1}}}\frac{(1+F)(1 + A{v}^{F}) - (2-F)(1 +A^{-1}v^{1-F})}{A{v}^{F}+ 2 + A^{-1}v^{1-F}}.
\Phi(t,x) = \Phi(t,r(t))+ D\Phi(t,r(t))(x-r(t))+D^2\Phi(t,z)(x-r(t))\cdot(x-r(t)) &=\tilde\Phi(t,x)+D^2\Phi(t,z)(x-r(t))\cdot(x-r(t))\,.
\delta x^{\mu}={\dot{x}}^{\mu}\xi \; , \; \; \delta e = \frac{d}{d \tau}\left( e \xi \right) \; , \; \; \delta b^{\mu}= \frac{d}{d \tau}\left( b^{\mu}\xi \right) \; , \; \; \delta \psi^{\mu}= \dot{\psi}^{\mu}\xi \; , \; \; \delta \chi = \frac{d}{d \tau}\left( \chi \xi \right) \; ,
d\beta_t = \mathfrak{M}(u_t,\beta_t)^{-1}\left [\left (\big\langle F(u_t),\Phi'(\beta_t) \big\rangle_{\beta_t}+\epsilon \kappa(u_t,\beta_t)\right )dt + \sqrt{\epsilon}\bigg\langle G(u_t)dW_t ,\Phi'(\beta_t) \bigg\rangle_{\beta_t}\right ],
\sigma_{K,m,p}(D)\coloneqq\begin{cases}\qquad\qquad\qquad\qquad\qquad\sigma_{\min}(D), \mbox{$D$ is surjective}, \displaystyle\min\{\sigma_{\min}((D)_\Lambda)\mid\Lambda\subset[m], |\Lambda|=m-K, (D)_\Lambda\neq 0\}, \mbox{$D$ is not surjective}.\end{cases}
8\pi(1-{\rm gen}(M^2))= 2\int_{M}KdV_{M}=\int_{M}(|H|^{2}-|h|^{2})dV_{M}= \int_{M}(|\xi|^{2}+4-(|h|^{2}+|H-\xi|^{2}))dV_{M}\geq 0,
[\begin{matrix}1&a&c\\ 0&I_{n}&b\\ 0&0&1\end{matrix}]\cdot[\begin{matrix}1&-a&-c+a\cdot b\\ 0&I_{n}&-b\\ 0&0&1\end{matrix}]=[\begin{matrix}1&0&0\\ 0&I_{n}&0\\ 0&0&1\end{matrix}].
y=C{}_{2}F_{1}(\alpha,\beta;1;x)+D\sum_{r=0}^{\infty}\frac{(\alpha)_{r}(\beta)_{r}}{(1)_{r}^{2}}(ln(x)+\sum_{k=0}^{r-1}(\frac{1}{\alpha+k}+\frac{1}{\beta+k}-\frac{2}{1+k}))x^{r}
\nabla^{2}u-\frac{1}{c_{0}^{2}}\frac{\partial^{2}u}{\partial t^{2}}+\tau_{\sigma}^{\alpha}\frac{\partial^{\alpha}}{\partial t^{\alpha}}\nabla^{2}u-\frac{\tau_{\epsilon}^{\beta}}{c_{0}^{2}}\frac{\partial^{\beta+2}u}{\partial t^{\beta+2}}=0.
\Sigma' \le \frac{c(\delta_2/\delta_1)^d(\delta_1/\delta)^d}{(\delta_1^{-1}\rho(x,y))^\sigma}= \frac{c(\delta_1/\delta)^d}{(\delta_2^{-1}\rho(x,y))^d(\delta_1^{-1}\rho(x,y))^{\sigma-d}}&\le \frac{c(\delta_1/\delta)^d}{(\delta_2^{-1}\rho(x,y))^\sigma}\le \frac{c2^\sigma(\delta_1/\delta)^d}{(1+\delta_2^{-1}\rho(x,y))^\sigma}.
\rho_1 =\iota_0\rho_1 \iota_1 \rho_2 =\iota_1\rho_2 \iota_0 \rho_3 =\iota_0\rho_3 \iota_1 \rho_{12}=\iota_0\rho_{12}\iota_0 \rho_{23}=\iota_1\rho_{23}\iota_1 \rho_{123}=\iota_0\rho_{123}\iota_1.
V_l{}^{ij}= e^{2U}\delta_{l\, [i}\partial_j\left ( \ln \chi^{(1)}\chi^{(2)}\right ) \qquad V_l{}^{i4}={1\over 2}e^{2U}\epsilon_{lim}\partial_m \left ( \ln{\chi^{(1)}\over\chi^{(2)}}\right )
Y(\mathcal{A})_{\omega\beta}^{iv}=\left(Y(\mathcal{A})_{\alpha\kappa_{2}}^{gv}\cdot Y(\mathcal{A})_{\kappa_{2}\beta}^{gv}\right)\cdot\left(Y(\mathcal{A})_{\beta\kappa_{2}}^{gi}\cdot Y(\mathcal{A})_{\kappa_{2}\alpha}^{gi}\right)\cdot Y(\mathcal{A})_{\omega\alpha}^{iv}
P(G_i^c) =1-P(G_i)=1-P(G_{i,2}|G_{i,1})P(G_{i,1})=1-P(G_{i,2}|G_{i,1})\bigl(1-P(G_{i,1}^c)\bigr) &\leq 1-P(G_{i,2}|G_{i,1})+P(G_{i,1}^c)
\frac{V}{N}= - \frac{\sigma}{g_{0}}+ \frac{1}{4 \pi \rho^{2}}\sum_{l = 0}^{I}( 2 l + 1 ) \ln ( 1 + \frac{\sigma}{\frac{l ( l + 1 )}{\rho^{2}}+ \frac{2 \xi}{\rho^{2}}}) ,
r=\frac{n(\sum_{i=1}^{n}(X_{i}-\overline{X})(Y_{i}-\overline{Y}))}{n[\sqrt{\sum_{i=1}^{n}(X_{i}-\overline{X})^{2}}]\times\sqrt{\sum_{i=1}^{n}(Y_{i}-\overline{Y})^{2}}]}
\langle P_h^k(\cdot, s), V \rangle = \int \int_{\Gamma_0(4) \backslash \mathcal{H}}P_h(z,s) \overline{V(z)}\frac{dx dy}{y^2}= \int_0^\infty \int_0^1 y^{s-1}e^{2\pi i h z}\overline{V(z)}dx \frac{dy}{y}.
\tilde{v}(a): =-\frac{1}{2}[\hat{g}(a_+),\hat{g}(1)], \tilde{w}(a): =\frac{1}{2}[\hat{f}(1),\hat{f}(a_+)], \tilde{f}(a): =-\frac{1}{2}(-1)^{|a|}[\hat{g}(\bar{a}),\hat{w}(1)], \tilde{g}(a): =\frac{1}{2}(-1)^{|a|}[\hat{v}(1),\hat{f}(\bar{a})], \tilde{\pi}_1(a): =\frac{1}{2}[\hat{v}(1),\hat{g}(a)], \tilde{\pi}_2(a): =\frac{1}{2}[\hat{f}(a),\hat{w}(1)],
\|\pi_n u-\pi_nv\|_{\L^p}= \bigg\|u-\frac{n}{\|v\|_{\L^p}}v\bigg\|_{\L^p}= \frac{1}{\|v\|_{\L^p}}\|(u-v)\|v\|_{\L^p}+v(\|v\|_{\L^p}-n)\|_{\L^p}& \leq \|u-v\|_{\L^p}+ \|v\|_{\L^p}-n \leq \|u-v\|_{\L^p}+( \|v\|_{\L^p}-\|u\|_{\L^p}) & \leq 2\|u-v\|_{\L^p},
|x|^{-n/2}\, \left( \int_{B ( x, |x|/4)}|\nabla \bar e |^2 \, dy \right)^{1/2}\, \leq c \, |x|^{-1}\max_{B ( x, |x|/2)}\, \bar e = o\left( |x|^{\frac{1-n}{p-1}}\right)\, \, \mbox{as}\, \, x \to \infty,
E[X_j^+(\tau_j' - \tau_j)|{\cal F}_{\tau_j}] = \int_0^{\tau_j' - \tau_j}\phi_j^+(u) e^{(\lambda_j^+ - \nu_j^+)(\tau_j' - \tau_j - u)}\: du &= (1 + \delta) s \int_0^{\xi_j - \tau_j}e^{s(q_j + C_4) u}e^{s(q_j + C_4 + 1)(\tau_j' - \tau_j - u)}\: du &= (1 + \delta) s e^{s(q_j + C_4 + 1)(\tau_j' - \tau_j)}\int_0^{\xi_j - \tau_j}e^{-su}\: du.
\ell_{n_s}=\lfloor\sqrt[3]{n_s-d_s+d_s}\rfloor=\left\lfloor \sqrt[3]{n_s-d_s}\,\sqrt[3]{1+\frac{d_s}{n_s-d_s}}\right\rfloor &=\left\lfloor s\,\sqrt[3]{1+\frac{d_s}{s^3}}\right\rfloor\le\left\lfloor s\,\Big(1+\frac{1}{3}\frac{d_s}s^3\Big)\right\rfloor=s
\mathbf{PW}^{k, p}_{int}(\widetilde{\Omega}) = \{\mathbf{u}\,:\, \mathbf{u}\in \mathbf{W}^{k,p}(\widetilde{\Omega}^s),~s=\pm; ~ [\mathbf{u}]_{\Gamma}=\mathbf{0},~\textrm{and}~ [\sigma(\mathbf{u})\mathbf{n}]_{\Gamma}=\mathbf{0}\},
J_{83}= -2\int \sum_{i,j}\overline{\phi}^{k+1}\phi^{k+1}a^{ij}_k\partial_i \overline{u}^{k+1,j}\leq C| \overline{\phi}^{k+1}|_2 |\phi^{k+1}\nabla\overline{u}^{k+1}|_2|\nabla u^k|_\infty \leq C|\nabla u^k|^2_\infty| \overline{\phi}^{k+1}|^2_2+\frac{\alpha}{20}|\phi^{k+1}\nabla\overline{u}^{k+1}|^2_2.
H=\left[\begin{array}{cccc}H_{00}H_{01}\cdots H_{0(n-1)}H_{10}H_{11}\cdots H_{1(n-1)}\vdots \vdots \ddots \vdots H_{(m-1)0}H_{(m-1)1}\cdots H_{(m-1)(n-1)}\end{array}\right],
\zeta = \frac{a \hat{\zeta}+ b}{- \overline{{{b}}}\hat{\zeta}+ \overline{{{a}}}}, \qquad \eta = \frac{\hat{\eta}}{( - \overline{{{b}}}\hat{\zeta}+ \overline{{{a}}})^{2}}, \qquad | a |^{2}+ | b |^{2}= 1 ,
\int_{-5}^{5}dz\int_{0}^{2\pi}d\varphi\int_{0}^{3}(\rho^{3}+\rho z)d\rho=2\pi\int_{-5}^{5}[\frac{\rho^{4}}{4}+\frac{\rho^{2}z}{2}]_{0}^{3}dz=2\pi\int_{-5}^{5}(\frac{81}{4}+\frac{9}{2}z)dz=\cdot\cdot\cdot=405\pi.
\dot{\mathrm{B}}^{s_0}_{p_0,q_0}(\Omega)\cap \dot{\mathrm{B}}^{s_1}_{p_1,q_1}(\Omega) = [\dot{\mathrm{B}}^{s_0}_{p_0,q_0}\cap \dot{\mathrm{B}}^{s_1}_{p_1,q_1}](\Omega)
\begin{cases}x([a]Q_{n_k}) = \frac{x(\Psi_{a}^\prime)^2 - \Psi_{a+1}^\prime\Psi_{a-1}^\prime}{(\Psi_{a}^\prime)^2}; y([a]Q_{n_k}) = \frac{\Psi_{a+2}^\prime (\Psi_{a-1}^\prime)^2 - \Psi_{a-2}^\prime (\Psi_{a+1}^\prime)^2}{4 y (\Psi_{a}^\prime)^3},\end{cases}
\mathrm{Leb}(\mathcal W^+_n)\leq\sum_{j=0}^n(M_j+1)\varepsilon_j<2M_0\varepsilon_0+\frac{8}{s}\cdot\sum_{j=1}^{n}\alpha^{-\frac{M_{j-1}}{4p}}<\frac{1}{2(p^2+2)},
dX_t = -\varepsilon^{-1}\nabla_x V(X_t, Y_t) dt +{\varepsilon}^{-1/2}\sqrt{2{\beta_X^{-1}}}dB^X_t dY_t = b_Y(X_t, Y_t) dt + \sqrt{2 \beta_Y^{-1}}dB^Y_t
H ( x , y ) : = \int{\frac{d^{4}p}{( 2 \pi )^{4}}}\left({\frac{\chi}{p^{2}+ M_{1}^{2}}}-{\frac{\chi}{p^{2}+ M_{2}^{2}}}\right){\frac{1}{p^{2}}}{\frac{{\frac{- 1}{\kappa}}+{\frac{p^{2}}{M^{2}}}\left( 1 +{\frac{p^{2}}{\tilde{M}^{2}}}\right)}{1 +{\frac{p^{2}}{M^{\prime}{}^{2}}}}}e^{i p ( x - y )}.
\mathrm{P}_{m,k}^1 = \mathrm{P}(_{m,k}<R_0)=\mathrm{P}\left(\bar{z}_{m,k}<\frac{{\epsilon}_0}{\rho \zeta_1}\right) = \mathcal{E}_{L\left(||y_{m,k}+x_m||\right)}\left\{1- e^{-L\left(||y_{m,k}+x_m||\right) \frac{{\epsilon}_0}{\rho \zeta_0}}\right\},
\int\limits_0^{\pi}Cf(0,e_n,\psi)\sin\psi d\psi &=\frac{n-2}{|S^{n-3}|}\int\limits_0^{\pi/2}\int\limits_{S^{n-2}}Rf(((\cos\psi_0)\omega,\sin \psi_0),0)d\omega \sin\psi_0(\cos\psi_0)^{n-2}d\psi_0 &=\frac{n-2}{|S^{n-3}|}\int\limits_0^{\pi/2}\int\limits_{S^{n-2}}Rf(((\sin \phi) \omega,\cos \phi),0) \cos\phi(\sin\phi)^{n-2}d\omega d\phi,
\lambda_1 :=2[\mathcal{O}_X]-[\mathcal{O}_{\Sigma}]-2[\mathcal{O}_L]+[\mathcal{O}_P] \lambda_2 :=4[\mathcal{O}_X]-2[\mathcal{O}_H]-[\mathcal{O}_S]-5[\mathcal{O}_L]+5[\mathcal{O}_P].
\;\;\; = \frac{1}{4}{\eta}_{\mu \nu}{\eta}_{\rho \sigma}\sum\limits_{n=1}^{m-1}\sum\limits_{s=1}^{m-1}\langle 0 \mid \left({\alpha}_{m-n}^{\mu}\left[{\alpha}_n^{\nu},{\alpha}_{s-m}^{\rho}\right]{\alpha}_{-s}^{\sigma}+ \right.
(l+1)\,C_{l+1}^\lambda(t)=2(\lambda+l)\,t\,C_l^\lambda(t)-(2\lambda+l-1)\,C_{l-1}^\lambda(t) &(l+1)\,C_{l+1}^{\lambda-1}(t)=2\,(\lambda-1)\left[t\,C_l^\lambda(t)-C_{l-1}^\lambda(t)\right] &l\,C_l^\lambda(t)=(2\lambda+l-1)\,t\,C_{l-1}^\lambda(t)-2\lambda\,(1-t^2)\,C_{l-2}^{\lambda+1}(t) &\frac{d}{dt}\,C_l^\lambda(t)=2\lambda\,C_{l-1}^{\lambda+1}(t)
\mathcal{H}^{\prime}(t) = \lambda(t)\mathcal{E}^{\prime}(t) +\lambda^{\prime}(t) \mathcal{E}(t) + \mu \alpha^{\prime}(t) \mathcal{E}^r(t)\langle P(u_t), u\rangle + \mu\alpha(t) \mathcal{E}^r(t) \frac{d}{dt}\langle P(u_t), u\rangle &+\mu r\alpha(t) \mathcal{E}^{r-1}(t)\mathcal{E}^{\prime}(t)\langle P(u_t), u\rangle.
\overline{}_\infty=Ch_\infty^p\left(\frac{\int_{0}^\infty |k^+(u)||u|^p\, du}{\int_{0}^\infty k^+(u)\, du}+\frac{\int_{-\infty}^0 |k^-(u)||u|^p\, du}{\int_{-\infty}^0 k^-(u)\, du}\right)
F^{ij}\nabla_i\nabla_j K =q(q-1)u^{q-2}F^{ij}\nabla_iu\nabla_ju+qu^{q-1}F^{ij}\nabla_i\nabla_ju &=qu^{q-1}\left[\langle\nabla K, X\rangle+2K-uF^{ij}h_{im}h_{mj}+(q-1)\frac{F^{ij}\nabla_iu\nabla_ju}{u}\right] &=qu^{q-1}\langle\nabla K, X\rangle+2qu^{q-1}K-qK^2H+(1-\frac1q)\frac{F^{ij}\nabla_i K \nabla_j K}{K}.
T_{k+1,i}= T^2_{k,i-1}+D_kT_{k,i}\cr =S^2_{k-2,k-2-(i-1)}(D_2,D_3,\ldots,D_{k-1})+D_kS_{k-2,k-2-i}(D_2,D_3,\ldots,D_{k-1})\cr =S^2_{k-2,k-1-i}(D_2,D_3,\ldots,D_{k-1})+D_kS_{k-2,k-1-i-1}(D_2,D_3,\ldots,D_{k-1})\cr =S_{k-1,k-1-i}(D_2,D_3,\ldots,D_{k}).
\tilde{p}_{\mathrm{min}}= \max\Big\{0, \frac{C_{2}}{\alpha|w_{\bar{k}1}|^2}\Big\}, \tilde{p}_{\mathrm{max}}= \min\Big\{\frac{B_k}{\alpha T_f}+ E_k, \frac{C_{1}}{\alpha|w_{\bar{k}2}|^2}\Big\},
\rho_\kappa(x)=\begin{cases}0, 0\leq x \leq 1-\kappa; \frac{1}{2\delta\gamma}(x-(1-\kappa))^2, 1-\kappa<x<1-\delta; \frac{1}{\delta}(x-1)+1+\frac{\gamma}{2\delta}, 1-\delta\leq x\leq 1.\end{cases}
\begin{multlined}\widehat{[\rho,x]}_{\vec a}:= [\rho;\mathrm{id}_{a_{x^{-1}(\delta(1))}},\dots,\mathrm{id}_{a_{x^{-1}(\delta(n))}};x] \mkern100mu:a_1\dots a_m\to a_{x^{-1}(\delta(1))}\dots a_{x^{-1}(\delta(n))}\end{multlined}
\stackrel{(0)}{S}=S_{0}\left[ A_{\alpha \beta}^{\;\;\;(\sigma )}\right]+\int d^{D}x\left( A_{\;\;\;\;(\sigma )}^{*\alpha \beta}\partial_{\left[\alpha \right.}\eta_{\left. \beta \right]}^{\;\;\;(\sigma )}+\eta_{\;\;\;(\sigma )}^{*\alpha}\partial_{\alpha}C^{(\sigma )}\right) ,
\|x+y\|^4+ \|x-y\|^4 \leq2(\|x\|^4+\|y\|^4)+12\|x\|^2y\|^2 =2((\|x\|^2+\|y\|^2)^2+4\|x\|^2\|y\|^2) ,~\forall x ,y \in \mathcal{X}.
H3(U,\overline{U},-\widehat{\overline{U}},\widehat{U};\alpha,\beta;0;0)=0 ~\Leftrightarrow~ \cfrac{\widehat{\overline{U}}}{U}=\cfrac{\alpha\overline{U}-\beta\widehat{U}}{\alpha\widehat{U}-\beta\overline{U}}\,;
Y(z)=\exp\left(\sum^\infty_{n=1}\frac{a_{-n}}{n}z^n\right)\exp\left(-\sum^\infty_{n=1}\frac{a_n}{n}(z^{-n}+z^n)\right)=\sum_{n\in \mathbb{Z}}Y_nz^{-n}, &Y^*(z)=(1-z^2)\exp\left(-\sum^\infty_{n=1}\frac{a_{-n}}{n}z^n\right)\exp\left(\sum^\infty_{n=1}\frac{a_n}{n}(z^{-n}+z^n)\right)=\sum_{n\in \mathbb{Z}}Y^*_nz^n,
A = \left(\begin{array}{c|c|c}\alpha_{t_1}^{[\sigma]}\dots \alpha_{t_m}^{[\sigma]}\end{array}\right) = \begin{pmatrix}\alpha_{t_1}\cdots \alpha_{t_{m}}\sigma(\alpha_{t_1}) \cdots \sigma(\alpha_{t_{m}}) \vdots \ddots \vdots \sigma^{n-1}(\alpha_{t_1}) \cdots \sigma^{n-1}(\alpha_{t_{m}})\end{pmatrix}.
{\lambda_{nk}}^\pm = \ln ({n^2}) - \ln \left|{\frac{\pi}{2}+ 2\pi k}\right| + \frac{\pi}{{2\pi k}}+ o\left({\frac{1}{k}}\right) \pm i\left[{\frac{\pi}{2}+ 2\pi k + O\left({\frac{{\ln k}}{k}}\right)}\right].
(\inf_{z\in L^0(Z)}\Phi^\ast(\cdot,z))^\ast(x) =\sup_{y\in L^0(Y)}\{\langle x,y\rangle - \inf_{z\in L^0(Z)}\Phi^\ast(y,z)\}&= \sup_{y\in L^0(Y), \, z\in L^0(Z)}\{\langle x,y\rangle - \Phi^\ast(y,z)\}&= \sup_{y\in L^0(Y), \, z\in L^0(Z)}\{\langle x,y\rangle + \langle 0,z\rangle - \Phi^\ast(y,z)\}&= \Phi(x,0).
\left\vert \mathbf{K}_{2}\right\vert^{2}=\mathbf{K}_{2}^{\ast}\mathbf{K}_{2}=\boldsymbol{\mu}+\frac{1}{2}(\mathbf{K}^{\ast}\mathbf{K}-\mathbf{I}_{B})\geq \mathbf{K}^{\ast}\mathbf{K}.
I ( d , a , c ) ={\frac{1}{2 ( a^{2}+ c^{2}- d^{2})}}\ln \left[{\frac{c^{2}}{( d + a )^{2}}}\right] +{\frac{a}{( c^{2}- d^{2})^{1 / 2}( a^{2}+ c^{2}- d^{2})}}\operatorname{a r c t a n}\left[ \frac{( c^{2}- d^{2})^{1 / 2}}{d}\right] .
2 \big\langle \varrho^j, E^{j-1}\big\rangle = 2\big\langle P_h^{j-1}\varrho^j, E^{j-1}\big\rangle \le \frac{1}{h}\big\| P_h^{j-1}\varrho^j \big\|^2 + h \big\| E^{j-1}\big\|^2.
c_{n}^{\ast\ast}=\frac{(m+n-2)(m+n-4)\Gamma(n/2)}{(2\pi)^{n/2}m^{2}}\left[\int_{0}^{\infty}t^{n/2-1}\left(1+\frac{2t}{m}\right)^{-(m+n-4)/2}\mathrm{d}t\right]^{-1}=\frac{(m+n-2)(m+n-4)\Gamma(n/2)}{(m\pi)^{n/2}m^{2}B(\frac{n}{2},~\frac{m-4}{2})},~if~m>4,
\int^{1}_0 G\left(\frac{1}{8},\tau\right)d\tau =2\int^{1/2}_0 G\left(\frac{1}{8},\tau\right)d\tau =2\int^{1/8}_0\left(-\frac{7}{48}\tau^3+\frac{35}{1024}\tau\right)d\tau &\phantom{=}+ 2\int^{1/2}_{1/8}\left(\frac{1}{48}\tau^3-\frac{1}{16}\tau^2+\frac{43}{1024}\tau-\frac{1}{3072}\right)d\tau &=\frac{277}{49152}.
\Delta(z, t) = \frac{1}{2^t}{t \choose (t+z+1)/2}- \frac{1}{2^t}{t \choose (t+z-1)/2}&= \frac{1}{2^t}{t \choose (t+z-1)/2}\biggl(\frac{(t-z -1)/2}{(t+z+1)/2}- 1 \biggr) &= -\frac{1}{2^t}{t \choose (t-z+1)/2}\frac{z}{(t+z+1)/2}.
0 = \sum_{p,m}\, \left( a_{ip}'(0) \, \delta_{pm}\, \delta_{jm}+ \delta_{ip}\, J_{pm}'(0) \, \delta_{jm}+ \delta_{ip}\, \delta_{pm}\, a_{jm}'(0) \right) & = a_{ij}'(0) + (1 - 2 \, \lambda_i ) \, \delta_{ij}+ a_{ji}'(0) \, .
e^{SO_{red}}(T)\le \frac{2n+2}{3}f(1,4)+\frac{n-5}{3}f(4,4)+rM_{12}+xM_{24}+yM_{24}+sM_{12}= \frac{2n+2}{3}f(1,4)+\frac{n-5}{3}f(4,4)+2M_{12}+(t-1)M_{24}\le \frac{2n+2}{3}f(1,4)+\frac{n-5}{3}f(4,4)+2M_{12}+2M_{24}< \frac{2n+2}{3}f(1,4)+\frac{n-5}{3}f(4,4)+2M_{13}+M_{34}= 2f(1,3)+f(3,4)+\frac{2n-5}{3}f(1,4)+\frac{n-7}{3}f(4,4).
\Big\| \sum_{n=1}^{N}a_{n}n^{-s}\Big\|_{\mathfrak{X}_{1}(X)}\leq C_{\sigma}\, N^{\sigma}\, \Big\| \sum_{n=1}^{N}a_{n}n^{-s}\Big\|_{\mathfrak{X}_{2}(X)}\,.
\phi(t) = jC_{D,E}(t) - jC_{\tilde E,\tilde D}(t) + jC_{D,E}(t) -jC_{\tilde E,\tilde D}(t) = 0, \psi(t) = -iC_{\tilde D,D}(t) - C_{E,\tilde E}(t) + iC_{\tilde D,D}(t) = -C_{E,\tilde E}(t) .
f(\gamma_\infty(s)) = \lim_{i\to \infty}f(\gamma_{s_i}(s)) &\le \lim_{i\to\infty}\left(\frac{s_i-s}{s_i}f(x_0) + \frac{s}{s_i}f(x_{s_i})\right) &= f(x_0) - s\cdot a,
\Phi_{c}(p,J)=\frac{1}{i}(2\pi)^{\,4}\delta_{J(-p)}\,W_{c}^{\,\alpha,\,\alpha_{0}}(J)=\tilde C_{\alpha}^{\,\alpha_{0}}(p)\,\Bigl \{\,(2\pi)^{\,4}\,\delta_{\Phi(-p)}\,i\,L^{\,\alpha,\,\alpha_{0}}(\Phi)\,\vert_{\,\Phi=i\tilde C_{\alpha}^{\,\alpha_{0}}\!J}\,+i\,J(p)\,\Bigr \}\ .
B^{(\ell)}= \underline{B}^{(\ell)}+HB^{(\ell-1)}+\sum_{t=1}^{\ell-1}\sum_{k=1}^r\nu_k^2\underline{B}^{(t-1)}\chi_k\check{\chi}_k^* B^{(\ell-t-1)}+\sum_{t=1}^{\ell-1}\underline{B}^{(t-1)}\tilde{H}B^{(\ell-t-1)}+\underline{B}^{(\ell-1)}H^{(1)}-\sum_{t=0}^{\ell}R_t^{(\ell)}.
u_1 (x_1, t_1 )\rho (x_1 ) \ge \frac{{\psi (u)(x_0, t_0)}}{{\psi (u(x_1, t_1))}}|D u|(x_0, t_0)\rho (x_0)\ge \sqrt{\frac{1}{2}}\cdot 32 \sqrt{2}\frac{{M_0}}{\mu}\cdot \mu^2 = 2 \cdot 8 M_0 \cdot 2\mu \ge 2\frac{\psi}{{\psi '}}|\rho_1 |,
A_{[g]}=\left[{}^{[g]}\nabla-\left({}^g\nabla+g\otimes X_g\right)\right]_0 &=\left[{}^g\nabla+g\otimes \beta^{\sharp}-\iota(\beta)-{}^g\nabla-g\otimes \beta^{\sharp}\right]_0=\left[-\iota(\beta)\right]_0=0.
\phi_{i}\left(t,x\right):=u_{i}^{\varepsilon}\left(t,x\right)-\left(u_{i,0}^{\varepsilon}\left(t,x\right)+\varepsilon m^{\varepsilon}\left(x\right)\bar{u}_{i}\left(x,\frac{x}{\varepsilon}\right)\cdot\nabla_{x}u_{i}^{0}\left(t,x\right)\right),
\nabla \phi-\operatorname{Id}=\nabla (\psi \circ \widetilde{\phi}\circ \psi^{-1})-\operatorname{Id}= \nabla \psi (\widetilde{\phi}\circ \psi^{-1})\cdot (\nabla \widetilde{\phi}(\psi^{-1})-\operatorname{Id})\cdot \nabla \psi^{-1}&+(\nabla \psi (\widetilde{\phi}\circ \psi^{-1})-\nabla \psi (\psi^{-1}))\cdot \nabla \psi^{-1}
f_{\lambda_1, 1}(s_1, s_2) = \sup_{y_1' \in \mathcal{Y}_1}\{f(y_1', y_2, x_2) - \lambda_1 c_1((y_1, x_1), (y_1', x_2))\}f_{\lambda_2, 2}(s_1, s_2) = \sup_{y_2' \in \mathcal{Y}_2}\{f(y_1, y_2', x_1) - \lambda_2 c_2((y_2, x_2), (y_2', x_1)\},
\det \left(z_j^{i-1}\right)_{1\le i,j\le n_2}=(-1)^{\frac{n_2(n_2-1)}2}\det \left(z_j^{n_2-i}\right)_{1\le i,j\le n_2}=(-1)^{\frac{n_2(n_2-1)}2}\left(\prod\limits_{j=1}^{n_2}z_j^{n_2}\right)\det \left(\frac 1{z_j^i}\right)_{1\le i,j\le n_2}.
s = l + 2 , \; s^{2}= \nu^{2}+ \frac{h^{4}}{2 ( \nu^{2}- 1 )}+ \frac{( 5 \nu^{2}+ 7 ) h^{8}}{3 2 ( \nu^{2}- 1 )^{3}( \nu^{2}- 4 )}+ \cdot \cdot \cdot
J^{*}[\psi^{(5)}]_{T,t}={\int\limits_t^{*}}^T\psi_5(t_5)\ldots{\int\limits_t^{*}}^{t_2}\psi_1(t_1)d{\bf w}_{t_1}^{(i_1)}\ldots d{\bf w}_{t_5}^{(i_5)}
L_{\rm eff}={\mu \over 2}\dot r^2 +{\kappa\over 4}\left[{(\dot b+ \dot r)^2 \over b+r}+{(\dot b - \dot r)^2 \over b-r}\right] -{b{\bf J}^2\over 2 r^2(\kappa + \mu b)}-{b Q^2\over 2\kappa(b^2 -r^2)}\, \,.
{\cal Z}_{\delta}( x , z ; p , \epsilon ) \equiv \langle \alpha | \delta \rangle \theta_{\delta}( 0 )^{- 1}\langle \psi ( x_{1}) \! \cdot \! \psi ( x_{2}) \prod_{i = 1}^{4}p_{i}\! \cdot \! \psi ( z_{i}) \epsilon_{i}\! \cdot \! \psi ( z_{i}) \rangle_{\delta}.
\begin{array}{cc}\tan \left( k_{2}a/2\right) =\mu_{1}^{\left( -\right)}/\mu_{2}, \phi textrm{even}\;\;\;(B=B^{^{\prime}},\;C=+A^{^{\prime}}) & -\cot \left( k_{2}a/2\right) =\mu_{1}^{\left( -\right)}/\mu_{2}, \phi textrm{odd}\;\;\;(B=-B^{^{\prime}},\;C=-A^{^{\prime}})\end{array}
\sum_m(X_{m_1}^{J_1})_{l m}(X_{m_2}^{J_2})_{m n}= \sum_{J, M}\left ( \begin{array}{ccc}J_1 J_2 J m_1 m_2 M\end{array}\right ) \left \{\begin{array}{ccc}J_1 J_2 J S S S\end{array}\right \}(X_M^J)_{l n},
\delta_{0}\left( \frac{R^{2}}{4 \phi^{2}}\eta_{i j}d \phi^{i}d \phi^{j}\right) = \frac{R^{2}}{4 \phi^{2}}\phi^{i}\left( d \alpha d \phi^{j}+ d \phi^{j}d \alpha \right) \eta_{i j}
\lim_{n \to +\infty}\frac{4\pi}{f(z(t_n))}=(4 \pi )^{\frac{3-2p}{3-p}}, \lim_{n\to +\infty}\frac{f(4 \pi )-f(z(t_n))}{4\pi-z(t_n)}= f'(4 \pi ) = \frac{p}{3-p}(4\pi)^{\frac{2p-3}{3-p}},
\int_{- \infty}^{\infty}d q_{-}\, e^{i \omega q_{-}}\delta \epsilon_{+}( q_{-}) = ( 2 \overline{{\mu}})^{i \omega}\int_{- \infty}^{\infty}d q_{+}\, e^{- i \omega q_{+}}\sum_{k = 1}^{\infty}\frac{\overline{{\mu}}^{1 - k}}{k !}\frac{\Gamma ( 1 + i \omega )}{\Gamma ( 2 - k + i \omega )}\delta \epsilon_{+}( q_{+})^{k}.
\nonumber \bar{\delta}_{s}a^{i}= \omega^{k i}\bar{c}_{k}, \quad \bar{\delta}_{s}c^{i}= i \omega^{i k}q_{k}+ \beta \omega^{i k}\partial_{k}H , \quad \bar{\delta}_{s}\bar{c}_{i}= 0 , \quad \bar{\delta}_{s}q_{i}= i \beta \omega^{j k}\bar{c}_{j}\partial_{i}\partial_{k}H .
P(\lambda_1,\dots,\lambda_i +\partial,\dots,\lambda_{n-1})_{\to}*_{n-i}f &=\sum\sum^{k_i}_{j=0}\binom{k_i}{j}a_1\otimes \cdots\otimes a_i(\partial^jf)\otimes \cdots\otimes a_n\,\lambda^{k_1}_1\cdots\lambda^{k_i-j}_i\cdots\lambda^{k_{n-1}}_{n-1}.
\delta L_{TG}=\varepsilon^{\mu \nu\rho}(R_{\beta\mu \nu}^{\alpha}\delta \Gamma_{\alpha\rho}^{\beta}-\bar{R}_{\beta\mu\nu}^{\alpha}\delta\bar{\Gamma}_{\alpha \rho}^{\beta})-\partial_{\mu}\,[\varepsilon^{\mu\nu\rho}N_{\beta\nu}^{\alpha}(\delta\Gamma_{\alpha\rho}^{\beta}+\delta\bar{\Gamma}_{\alpha\rho}^{\beta})]
|\int_{t}^{\infty}ds \lambda''(s) \partial_{2}K(s-t,\lambda(t)) \lambda'(t)| \leq \frac{C}{t^{3}\log^{2b+2}(t)}\int_{t}^{\infty}ds |\partial_{2}K(s-t,\lambda(t))| &\leq \frac{C}{t^{3}\log^{3b+2}(t)}
D^4(f)-E_2 D^3(f)+(3D(E_2)+\alpha_1(s) E_4)D^2(f) &-\Bigl(D^2(E_2)+\frac{\alpha_1(s)}{2}D(E_4)-\alpha_2(s) E_6\Bigr)D(f)+\alpha_3(s) E_8 f=0,
\sum_{j}q_{a}^{j}\varphi_{j}^{+}\overline{{{\varphi}}}_{j}^{+}= \zeta_{a}^{+}\overline{{{\zeta}}}_{a}^{+}= u_{( \alpha}^{+}u_{\beta )}^{+}\zeta_{a}^{( \alpha}\overline{{{\zeta}}}_{a}^{\beta )}.
A^2 =\frac{\lambda_1^2\lambda_2^2\lambda_4^2}{B^2D^2}&>\frac{\lambda_1^2\lambda_2^2\lambda_4^2}{\lambda_2^2K_1^2}\Bigg(1+\frac{3\lambda_1}{\lambda_2^2}t\Bigg)^{-2/3}A >\frac{\lambda_1\lambda_4}{K_1}\Bigg(1+\frac{3\lambda_1}{\lambda_2^2}t\Bigg)^{-1/3}.
(\tau \cdot v_1^*) (v_1) = v_1^* (\tau^{-1}\cdot v_1) = v_1^* (t_1^{-1}v_1) = t_1^{-1}, (\tau \cdot v_1^*) (v_2) = v_1^* (\tau^{-1}\cdot v_2) = v_1^* (t_2^{-1}v_2) = 0.
L = -(e_3e_2^{(2)}) \otimes (f_1 \wedge f_2 \wedge f_3) \otimes (e_1 \otimes f_3) - (e_3e_2^{(2)}) \otimes (e_1 \otimes f_1 \wedge f_3) \otimes (f_2 \wedge f_3) &+ (e_3e_2^{(2)}) \otimes (e_1 \otimes f_2 \wedge f_3) \otimes (f_1 \wedge f_3),
\xi^{+}_{f|W_N}(j;u,v)=\frac{j! (N_{\overline{S}}N'_S)^{j+1-\frac{k}{2}}}{(-2\pi i)^{j+1}\phi(N')}\sum_{\alpha}\alpha(n')\overline{\alpha^2_S}(B) \overline{\alpha^2_{\overline{S}}}\left(\frac{A}{N_S N'_S}\right)L\left(S_\alpha(f)|W_S^{NN'},j+1\right).
A_0 =\mathbb{C}|_{M},A_1=-V_\mathbb{C},A_2=T_\mathbb{C}M\oplus \Lambda^2 (V_\mathbb{C}), B_0 =\mathbb{C}|_{M},B_1=-E_\mathbb{C},B_2=\Lambda^2 (E_\mathbb{C}).
\frac{T_{p+2}}{\kappa}\int_{\mathcal{M}_{p+3}}C_{p+3}=\frac{T_{p+2}}{\kappa}\int_{\mathcal{M}_{p+3}}A_{p+1}\wedge\omega_2 =\frac{T_{p}}{2\kappa}\int_{\mathcal{M}_{p+1}}\frac{A_{p+1}}{2\pi^2\alpha'}\,,
\sum_{t = 0}^\infty f_\rho(t) = \sum_{t = 1}^\infty \frac{\mu_\rho f_\rho^\ast(t)}{t}= \mu_\rho\left(\sum_{t = 1}^{K_\delta}\frac{f_\rho^\ast(t)}{t}+ \sum_{t = K_\delta + 1}^\infty \frac{f^\ast(t)}{t}\right) = \mu_\rho\left(\sum_{t = 1}^{K_\delta}\frac{f_\rho^\ast(t)}{t}+ \mu(1 - F(K_\delta))\right) = 1,
\begin{array}{c c}{{R_{P}= \left( \begin{array}{c c}{{- \cos 2 \phi_{P}}}&{{- \sin 2 \phi_{P}}}\\{{- \sin 2 \phi_{P}}}&{{\cos 2 \phi_{P}}}\\ \end{array}\right) ,}}&{{R_{C}= \left( \begin{array}{c c}{{- \cos 2 \phi_{C}}}&{{\sin 2 \phi_{C}}}\\{{\sin 2 \phi_{C}}}&{{\cos 2 \phi_{C}}}\\ \end{array}\right) .}}\\ \end{array}
\mathcal{L}(v )=\div \Big(e^{\frac{|\Lambda+\nabla_y\widetilde{w}|^2}{2}+\lambda V (x,y )}\big( (\Lambda+\nabla_y\widetilde{w})^T\nabla_yv (\Lambda+\nabla_y\widetilde{w})+\nabla_y v\big) \Big).
\sum_{w'\in W_J}\sum_{w\in{}^JW}(-1)^{\ell(w w')}w^{-1}c_{W_J w W_I}w^{\prime{-}1}= \sum_{w\in W^J}\sum_{w'\in W_J}(-1)^{\ell(w w')}w c_{W_J w^{-1}W_I}w'
\frac{|(V_1^H \times V_2^H)\cup (W_1^H \times W_2^H)|}{|(V_1 \times V_2) \cup (W_1 \times W_2)|}= \frac{|V_1^H||V_2^H|+|W_1^H||W_2^H|}{|V_1|| V_2|+ |W_1|| W_2|}\leq \frac{\alpha|V_1|\alpha|V_2|+\alpha|W_1|\alpha|W_2|}{|V_1|| V_2|+ |W_1|| W_2|}= \alpha^2,