harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left<F\circ T,R\,\right> =\int_{\mathbb{H}}F\circ T(a,x)\,R(a,x)\,\,d\sigma(x)\frac{da}{a}&=\int_{\mathbb{H}}\int_{\mathbb{H}}\frac{1}{b^n}f\!\left(\frac{a}{b},\frac{\angle(x,y)}{b}\right)T(b,y)\,R(a,x)\,\,d\sigma(y)\frac{db}{b}\,\,d\sigma(x)\frac{da}{a}.
\mathop{S}_{\r T}^{\d = 2}\mathrm{\LARGE \s l ~ S}_{\n r T}^{\d = 2}={\frac{1}{\lambda}}\int d^{3}x \, \varepsilon^{\mu \nu \rho}\, E_{\mu}^{\underline{{{\alpha}}}}\, T_{\nu \rho}^{\underline{{{\beta}}}}\, \eta_{\alpha \beta}.
\Lambda_{N,1}:=-\log(2)(n_{1}2^{n_{1}}+n_{2}2^{n_{2}}+\cdots+n_{\tau(N)}2^{n_{\tau(N)}}), \Lambda_{N,2}:=-\log(2)(2^{n_{2}+1}+2\cdot 2^{n_{3}+1}+\cdots+(\tau(N)-1) 2^{n_{\tau(N)}+1}).
S ={\frac{1}{4 \kappa}}\left\{L^{2}\left( R_{1}^{2}+ R_{2}^{2}\right) + R_{1}^{2}R_{2}^{2}( \sin \theta )^{2}+ a_{1}^{2}R_{2}^{2}+ a_{2}^{2}R_{1}^{2}- 2 a_{1}a_{2}R_{1}R_{2}\cos \theta \right\}.
\begin{array}{r c c c c c c}{}&{{\cal G}^{( 1 )}}&{\oplus}&{{\cal G}^{( 1 )}}&{\supset}&{{\cal G}^{( 1 )}}&{}\\{\mathrm{l e v e l}}&{\ell - s}&{}&{s}&{}&{\ell}&{,}\\ \end{array}
{\cal L}^{(2)}_{GF}~=~ \frac{\kappa}{2 \pi}\epsilon^{mn}A_n{\dot A}_m + \pi_a{\dot z}_a + \pi^*_a \dot{z^*}_a + \Omega^{(0)}_1 \dot{\alpha}+ \Omega^{(1)}_2 \dot{\beta}+ \Omega^{(1)}_3 \dot{\gamma}+ \Omega^{(2)}_4 \dot{\sigma}-{\cal H}^{(2)}
= \; 2 e \sum_{I = 1}^{N}\left( \varepsilon_{\mu \nu}\partial_{\mu}A_{\nu}^{( I )}, K_{I I}\sum_{b = 1}^{N}\sum_{j = 1}^{n_{b}}U_{I b}\Big [ \delta_{n}( x_{j}^{( b )}) - \delta_{n}( y_{j}^{( b )}) \Big ] \right) \; .
\simeq \frac{- \lambda_{1}( a^{+}\Gamma_{1}\Lambda_{3}) + i \lambda_{2}( \Lambda_{3}\Gamma_{2}\Gamma_{3}a^{+})}{\sqrt{2}}u_{R}( \lambda_{1}, \lambda_{2}) \mid 0 \, \rangle \mid \psi ( R , \Lambda_{3}) \, \rangle .
L_1 = -\nu_1 + \partial_x^2 +U_1 + 2 a_1 \phi_1 \phi_1^* + a_1 \phi_2 \phi_2^* \, L_2 = -\nu_2 + \partial_x^2 +U_2 + a_2 \phi_1 \phi_1^* + 2 a_2 \phi_2 \phi_2^* \,
\chi ({\rm \mathcal{CSR}}(m,n)) \geq \chi({\rm \mathcal{SR}}(m,n-1)) \geq \frac{r-\lambda_{min}}{-\lambda_{min}}\geq \frac{n(m-1)+\binom{m}{2}}{\binom{m}{2}}> \frac{\binom{m}{2}(m-1)+\binom{m}{2}}{\binom{m}{2}}=m.
\tilde{t}^2-\tilde{w}^2-\tilde{x}^2-\tilde{y}^2-\tilde{z}^2=\left(\frac{\tilde{t}-\tilde{w}}{\lambda}\right)k_1\left[ h^{-1}\left(\frac{\tilde{t}-\tilde{w}}{\lambda}\right)\right] .
\frac{d}{dt}p_k^f(t)= -p_k^f(t)f(\mu k)+ \sum_{j=k+1}^{n_0}p_j^f(t) \int_0^\infty \binom{j}{k}(1-e^{- \mu s})^{j-k}e^{-\mu k s}\nu (ds) .
\frac{\left|v_1^{\sigma}a_{\sigma}e^{2t_n}x+v_2^{\sigma}\right|}{\|v^{\sigma}\|}= \left|a_{\sigma}(\rho(\alpha\beta)^n)^{-1}x\frac{v_1^\sigma}{\|v^\sigma\|}+\frac{v_2^{\sigma}}{\|v^{\sigma}\|}\right|,
(\boldsymbol{S}^{(k)}_e \boldsymbol{u}_{B_e}^{(k)},\boldsymbol{v}_{B_e}^{(k)})_{l^2}= \langle S^{(k)}_e u_{B_e}^{(k)},u_{B_e}^{(k)}\rangle = s^{(k)}_e(u_{B_e}^{(k)},u_{B_e}^{(k)}), \forall u_{B_e}^{(k)},u_{B_e}^{(k)}\in W^{(k)}.
y(x)=b-\frac{3b}{2(b^{2}+1)}x^{2}-\frac{21b(3b^{2}-1)}{24(b^{2}+1)^{3}}x^{4}-\frac{495(13b^{5}-10b^{3}+b)}{720(b^{2}+1)^{5}}x^{6}+\cdot\cdot\cdot
\nabla\cdot F=(e_{x}\frac{\partial}{\partial x}\cdot F)+(e_{y}\frac{\partial}{\partial y}\cdot F)+(e_{z}\frac{\partial}{\partial z}\cdot F)=\frac{\partial F_{x}}{\partial x}+\frac{\partial F_{y}}{\partial y}+\frac{\partial F_{z}}{\partial z}
Q_{3,k}\leq C_{H,l_{0}}|I_{k}|^{2(1-H)}\int_{0}^{1}\left(\sum_{j=1}^k|I_{j}|\right)^{2H-1}\cdot\left|\int_{0}^{v_1}\frac{dv_2}{(\sum_{j=1}^k|I_{j}|)^{H-\frac{1}{2}}(v_1-v_2)^{H-\frac{1}{2}}}\right|^{2}dv_1 \leq C_{H,l_{0}}|I_{k}|^{2(1-H)}.
\lambda^{1/2}(|u_0|^p)^{1/p}_{Q_{\nu r}(z_0)}+ (|D_v u_0|^p)^{1/p}_{Q_{\nu r}(z_0)}\le N \sum_{k = 0}^{\infty}2^{-k^2/8}F_k (1), \lambda^{1/2}(|u_0|^p)^{1/p}_{Q_{r}(z_0)}+ (|D_v u_0|^p)^{1/p}_{Q_{r}(z_0)}\le N \nu^{(4d+2)/p}\sum_{k = 0}^{\infty}2^{-k^2/8}F_k (1).
\cos(z;q) :=\frac{(q;q)_\infty}{(q^{1/2};q)_\infty}(z q^{-1/2})^{1/2}J^{(3)}_{-1/2}(z(1-q)/\sqrt{q};q^2) = \sum_{n=0}^{\infty}(-1)^n \frac{q^{n^2}z^{2n}}{\Gamma_q (2n+1)}.
\Omega^{1/2}_{0}:= \Omega^{-1/2}\otimes\Omega^{1}_{G}\simeq \Omega^{1/2}_{s}\otimes\Omega^{-1/2}_{r}\otimes s^{*}\Omega_{M}\simeq \Omega^{1/2}_{r}\otimes\Omega^{-1/2}_{s}\otimes r^{*}\Omega_{M}\simeq r^{*}\Omega^{1/2}_{M}\otimes s^{*}\Omega^{1/2}_{M}.
\begin{cases}\bar u_{tt}= \bar u_{xx}, (x,t)\in(0,1)\times(0,\infty), \bar u(0,t) = 0, t\in (0,\infty) \bar u_x(1,t) = - a \bar u_t(1,t) + \frac{bd}{c}\bar u(1,t), t\in (0,\infty) \bar u(x,0) = \bar u_0(x),\: \bar u_t(x,0) = \bar u_1(x), x\in(0,1),\end{cases}
\mathbf{\tilde{Q}}_\alpha (\delta(\beta) g) = \delta(\beta)\mathbf{\tilde{Q}}_\alpha (g), \mathbf{\tilde{Q}}_\alpha (g) = \mathbf{\tilde{Q}}_\alpha (g \, \mathbf{\tilde{Q}}_\alpha (h)^{-1}h) = \mathbf{\tilde{Q}}_\alpha (g \, h\mathbf{\tilde{Q}}_\alpha (h^{-1})) \ .
a_{i,j}=a_{j,i}\quad(0\leq i<j\leq d), g(a_{i,j},a_{j,k},a_{i,k}) =0\quad(\{i,j,k\}\in N_3), h(a_{i,j},a_{i,k},a_{i,\ell},a_{j,k},a_{j,\ell},a_{k,\ell}) =0\quad(\{i,j,k,\ell\}\in N_4).
(p - 1) \| (y_1, y_2)\|_{1,p}< \lambda(\kappa_1 + \kappa_2 + 2)(\kappa_1 + \kappa_2 + 1)\int_{\Omega}\vert y_1\vert^{\kappa_1 + 1}\vert y_2\vert^{\kappa_2 + 1}dz - (q - 1)\| (\nabla y_1, \nabla y_2)\|_{q,\eta}- \nu\int_{\Omega}\left[a_1\vert y_1 \vert^{1 - \nu}+ a_2\vert y_2 \vert^{1 - \nu}\right]dz.
g_2(z,\bar z,\xi)=-\bar b_N(\bar z^N+(-\bar z-\xi)^N)+wt(N+1), \bar g_1(\bar z,z)+\frac{1}{2}g_2(z,\bar z,\xi)=\eta\bar b_N\sum\bar z^i(-\bar z-\xi)^{N-1-i}+wt(N+1), (2\bar z+\xi)\bar g_1(\bar z,z)+\bar zg_2(z,\bar z,\xi)=\bar b_N(\bar z^{N-1}+(-\bar z-\xi)^{N-1})w+wt(N+2).
{\bf M}={\bf \hat{r}}{\frac{i m}{r}}- \mathrm{\boldmath{~ \hat{\t h e t a}~}}{\frac{\partial}{\partial r}}, \quad{\bf N}={\bf \hat{r}}i k{\frac{\partial}{\partial r}}- \mathrm{\boldmath{~ \hat{\t h e t a}~}}{\frac{m k}{r}}-{\bf \hat{z}}d_{m},
\lim_{j\to\infty}\int_\Omega\left(p|u_{h_j}|^{p-2}+qa(x)|u_{h_j}|^{q-2}\right)u_{h_j}vdx=\int_\Omega\left(p|u|^{p-2}+qa(x)|u|^{q-2}\right)uvdx.
\frac{1}{2\pi}\int_{\mathbb{R}^2}e^{2u}dx = \frac{1}{2\pi}\int_{M}K d v_g=\chi(M) + \sum_{j=1}^m \left(\frac{\alpha_j}{2\pi}-1\right) + \left(\frac{\alpha_\infty}{2\pi}-1\right)
E[|Z^{a,n}(s,t) -E[P^n_t|\mathcal{F}_s]|^2] = E[|E[Y^{a,n}(t)-P^n(t)|\mathcal{F}_s]|^2], \intertext{which by Jensen's inequality,}\leq E[E[|Y^{a,n}(t)-P(t)|^2|\mathcal{F}_s]] = E[Y^{a,n}(t)-P^n(t)|^2],
\tau^*(\mu_R,\mu_T) \le \tau_U(\mu_R,\mu_T) \le \tau_U(\mu_R,1/N_T) \le \tau_U(1,1/N_T)+\frac{\tau_U(\mu_R^0,1/N_T)-\tau_U(1,1/N_T)}{1-\mu_R^0}(1-\mu_R) = \frac{\tau_U(\mu_R^0,1/N_T)}{1-\mu_R^0}(1-\mu_R).
\lim_{n\rightarrow \infty}n^{2\alpha+1}\int_0^{\tau \wedge t_1 \wedge t_2}(t_1-s)^{\alpha}(t_2-s)^{\alpha}(\sigma'(X_s))^2 \Psi^{n,1}_s ds =\kappa_3 \int_0^{\tau \wedge t_1 \wedge t_2}(t_1-s)^{\alpha}(t_2-s)^{\alpha}(\sigma'\sigma)^2(X_s)ds,
W_{ii}=\left|V_{i}\right|^2, W_{ij}^\mathrm{r}=\Re\left\{W_{ij}\right\}=\left|V_{i}\right| \left|V_{j}\right| \cos\left(\theta_{i}-\theta_{j}\right), W_{ij}^\mathrm{i}=\Im\left\{W_{ij}\right\}=\left|V_{i}\right| \left|V_{j}\right| \sin\left(\theta_{i}-\theta_{j}\right),
y_{(2k-1)m+i,n^{+}_0}= MN - x_{i,n^{+}_0}+(k-\tfrac{1}{2})(mn) +\tfrac{3}{2}\mbox{\if}i+n^{+}_0 \mbox{is odd and}y_{(2k)m+i,n^{+}_0}= x_{i,n^{+}_0}+k(mn) \mbox{\if}i+n^{+}_0 \mbox{is even.}
E_{v \rightarrow c}^{(0)}\left(\mathcal{X}_{v}[m]\right)= \begin{cases}\frac{1}{1+\exp \left(l_{\mathrm{BP}}\left(a_{v}\right)\right)}, m=0 \frac{1}{M}\left(1-\frac{1}{1+\exp \left(l_{\mathrm{BP}}\left(a_{v}\right)\right)}\right). 1 \leq m \leq M\end{cases}
\overline{C}_{\lambda,\mu,\nu}=\sum_{\lambda^{\prime},\mu^{\prime},\nu^{\prime},\alpha,\beta,\gamma}C_{\lambda^{\prime},\mu^{\prime},\nu^{\prime}}c_{\lambda^{\prime}\beta\gamma}^{\lambda}c_{\mu^{\prime}\alpha\gamma}^{\mu}c_{\nu^{\prime}\alpha\beta}^{\nu}
S = \frac{1}{16\pi}\, \int \;\left\{(-{\it R}+ \, 2\partial_{\mu}\Phi \partial^{\mu}\Phi ) - \alpha'\exp (-2 \Phi) (F_{a\mu\nu}\, F_a^{\mu\nu}- \beta{G})\right\}\, \sqrt{-g}d^4x \; ,
S_{a,b}(z,w) = \frac{z+1}{z}\frac{w+1}{w}\frac{{\mathcal X}_a(z){\mathcal X}_b(w)\eta({\mathcal X}_a(z),{\mathcal X}_b(w))}{H_{a,b}({\mathcal X}_a(z),{\mathcal X}_b(w))}
V_{0}( F , \phi ; F_{T}, t ) = - F \mathrm{l n}\left( \frac{\phi^{( N_{c}- N_{f})}\mathrm{d e t}( t )}{\Lambda^{( 3 N_{c}- N_{f})}}\right) - \phi \mathrm{T r}\left[ F_{T}t^{- 1}\right] + \mathrm{h . c .}\ .
\limsup_{k\rightarrow \infty}n!k^{-n}\dim H_{(2)}^{j}(X_c,L^k)\leq \int_{K'(j,h^L_\chi)}(-1)^j c_1(L,h_\chi^L)^n\leq \int_{X_c(j,h^L_\chi)}(-1)^j c_1(L,h_\chi^L)^n.
\lim\limits_{N \rightarrow\infty}\frac{K \left (\lambda_0-\frac{|\lambda_1|}{2\cos^2 \frac{\pi}{2N-1}}\right )}{N\cdot \left[ (2N-1)K-1-\frac{1}{2\cos^2 \frac{\pi}{2N-1}}\right ]}=\frac{3M}{4K}+\frac{1}{2}-\frac{1}{\pi^2}.
\|w_n\|_{\dot H}^2 =\|w_n-u\|_{\dot H}^2+ \|u\|_{\dot H}^2+o_n(1), \|w_n\|_{q}^q = \|w_n- u\|_{q}^q + \|u\|_{q}^q + o_n(1) \ \ \ \ 1 \leq q < \infty.
\left\vert y_{m}\left( x\right) -\overset{\ast}{y_{m}}\left( x\right)\right\vert \leq \left\vert \varepsilon \right\vert \left( \psi \left(x\right) -\psi \left( a\right) \right)^{\gamma -1}\overset{m}{\underset{j=0}{\sum}}A^{j}\frac{\left( \psi \left( x\right) -\psi \left( a\right) \right)^{\alpha j}}{\Gamma \left( \alpha j+\gamma \right)}.
(u_{\mp}-v_{\pm}-\frac{1}{2}h)(u_{\pm}-v_{\mp}+ \frac{1}{2}h)H_{i}^{\pm}(u)H_{i-1}^{\mp}(v) &=(u_{\mp}-v_{\pm}+\frac{1}{2}h)(u_{\pm}-v_{\mp}-\frac{1}{2}h)H_{i-1}^{\mp}(v)H_{i}^{\pm}(u).
\mathcal{L}_{m_{3 / 2}}= \frac{1}{2}\sum_{n}\frac{n + \omega}{R}\bar{\psi}_{\mu}^{( n )}\gamma^{\mu \nu}\psi_{\nu}^{( n )}+ \frac{W}{\pi R}\sum_{n , m}\bar{\psi}_{\mu}^{( n )}\gamma^{\mu \nu}\psi_{\nu}^{( m )}\, ,
\psi_{1 a}( x^{\mu}) = \left( \begin{matrix}{{\nu_{e}}}\\{{e^{-}}}\\{{u}}\\{{d}}\\ \end{matrix}\right) ~ , ~ ~ \psi_{2 a}( x^{\mu}) = \left( \begin{matrix}{{\nu_{\mu}}}\\{{\mu^{-}}}\\{{c}}\\{{s}}\\ \end{matrix}\right) \, ,
\frac{1}{1-x}\mathrm{Li}_{p,\lambda}(x) =\sum_{l=0}^{\infty}x^{l}\sum_{k=1}^{\infty}\frac{(-\lambda)^{k-1}(1)_{k,1/\lambda}}{(k-1)!k^{p}}x^{k}&=\sum_{n=1}^{\infty}\sum_{k=1}^{n}\frac{(-\lambda)^{k-1}(1)_{k,1/\lambda}}{k^{p}(k-1)!}x^{n}.
\Pr(|S_{2k}|\geq 2a_{k}+2t) \leq\Pr(|S_{k}|\geq a_{k}+t)+\Pr(|S_{k+1,2k}|\geq a_{k}+t) &= 2\Pr(|S_{k}|\geq a_{k}+t)\leq \frac{1}{3}.
|\nabla^{l-1}u_N(x)-\nabla^{l-1}u_N(y)| =N^{-1}|\nabla^{l-1}u(Nx-z_i)-\nabla^{l-1}u(Ny-z_i)| &\leq \|\nabla^l u\|_{\infty}|x-y| &\leq (\sqrt{n}N^{-1})^{1-\alpha}\|\nabla^l u\|_{\infty}|x-y|^\alpha,
|M(f)| = |\frac{m(fg_0)}{m(g_0)}| \leq \frac{\|f g_0\|_K}{|m(g_0)|}& = \frac{\|f g_0\|_\infty + \|f g_0\|_{D(\delta_1)}}{|m(g_0)|}\leq \frac{\|f\|_\infty \|g_0\|_\infty + \|f\|_\infty \|g_0\|_{D(\delta_1)}}{|m(g_0)|}& = \frac{\|g_0\|_K}{|m(g_0)|}\|f\|_\infty,
\mathcal{D}(\widetilde{S})\; =\;\overline{C^\infty_0(\mathbb{R}^+,\mathbb{C}^2)}^{\|\cdot\|_S}\mathcal{D}(S^*)\; =\;\{\psi\in L^2(\mathbb{R}^+,\mathbb{C}^2)\,|\,\widetilde{S}\psi \in L^2(\mathbb{R}^+,\mathbb{C}^2)\}\,,
S_{A}= - \frac{2}{\kappa_{1 1}^{2}}\int d^{1 1}x \left\{\frac{\sqrt{| g |}}{2 \cdot 2 \cdot 4 !}F^{\mu \nu \lambda \xi}F_{\mu \nu \lambda \xi}+ \frac{1}{1 2}\frac{1}{3 ! ( 4 ! )^{2}}\epsilon^{\mu_{1}\ldots \mu_{1 1}}A_{\mu_{1}\mu_{2}\mu_{3}}F_{\mu_{4}\ldots \mu_{7}}F_{\mu_{8}\ldots \mu_{1 1}}\right\}
\vec{A}_{\theta}= - \vec{e}_{\phi}\frac{1 - K ( r )}{e r}\ , \ \ \, v e c A_{\phi}= \vec{e}_{\theta}\frac{1 - K ( r )}{e r}\sin \theta \ , \ \ \ \phi = \upsilon ( H_{1}( r ) \vec{\tau}\vec{e}_{r}+ H_{2}( r ) \lambda_{8}) \ ,
|(N(u_{1})-N(u_{2}))(t,R\lambda(t))| \leq C\frac{|\overline{v}_{1}-\overline{v}_{2}|\left(\overline{v}_{1}^{2}+\overline{v}_{2}^{2}\right)}{R^{2}\lambda(t)^{2}}&+C \frac{|\overline{v}_{1}-\overline{v}_{2}|\left(|\overline{v}_{1}|+|\overline{v}_{2}|\right)}{R^{2}\lambda(t)^{2}}\left(|Q_{1}(R)|+|v_{corr}(t,R\lambda(t))|\right)
\int_{\omega_0}|v_x|^2dx \leq \int_{\omega}\xi|v_x|^2dx=\int_{\omega}\xi v_xv_xdx =-\int_{\omega}(\xi_xv_x+\xi v_{xx})vdx =\frac{1}{2}\int_{\omega}\xi_{xx}v^2dx-\int_{\omega}\xi v_{xx}vdx,
A_{2}=\frac{(\gamma^{4}-4\gamma^{3}+8\gamma^{2}-8\gamma+4)\omega^{2}h^{2}+4(\gamma-2)^{2}}{(\gamma^{2}\omega^{2}h^{2}+4)((\gamma-1)^{2}\omega^{2}h^{2}+(\gamma-2)^{2})}
\mathcal{H}^I\mathcal{H}^{\tilde{I}}=R_{L_1}S_{U_0}R_{\tilde{L}_1}S_{\tilde{U}_0}=R_{L_1}R_{\tilde{L}_1}S_{U_0}S_{\tilde{U}_0}=R_{L_1}S_{U_0}=\mathcal{H}^I.
\frac{P_{\lambda}\big(q^rt^{N-1}, t^{N-2}, \dots, t, 1; q, t\big)}{P_{\lambda}\big(t^{N-1}, \dots, t, 1; q, t\big)}= \frac{(q; q)_r}{\big(q^{\theta N}; q\big)_r}\frac{1}{2\pi\sqrt{-1}}\oint_C{\frac{y^{-(r+1)}{\rm d}y}{\prod\limits_{i=1}^N{\prod\limits_{j=0}^{\theta-1}{(1 - q^{\lambda_i + \theta(N - i) + j}y)}}}}.
\times \frac{\omega}{2 i \pi \hbar \sin \omega \Delta t}e^{- i \frac{\pi}{2}| \ell |}e^{\frac{i \omega}{2 \hbar}\frac{\cos \omega \Delta t}{\sin \omega \Delta t}( r_{f}^{2}+ r_{i}^{2})}J_{| \ell |}\left( \frac{\omega r_{f}r_{i}}{\hbar \sin \omega \Delta t}\right) \ .
K = C^2 - \sum_{i=1}^{L+1}\alpha_i\{A^i,B\}- \sum_{i=1}^L\beta\omega_i\{A^i,B\}-\beta\{A,B^2\}+\sum_{i=1}^{M+1}k_iA^i - (2\epsilon+\beta\eta) B +(\beta^2-\delta)B^2.
\left(\delta_t e_u^n,\bar{u}_h\right) + (\nabla e_u^n, \nabla \bar{u}_h) +(e_u^n{\boldsymbol \sigma}(t_n)+u^n_h e_{\boldsymbol \sigma}^n,\nabla \bar{u}_h)=(\xi_1^n,\bar{u}_h), \ \ \forall \bar{u}_h\in U_h,
\dim(W^m) \geq\dim\left(\sum_{i=0}^mU_l^{m-i}\left( W_{l+1}z^{p_{l+1}}\right)^i\right) \geq\dim\left( U_l^m+\sum_{i=1}^m(W_lz^{p_l})^{m-i}\left( W_{l+1}z^{p_{l+1}}\right)^i \right).
\frac{1}{n}AA^{\mathrm{T}}=\left( \begin{array}{ll}\beta J 2\beta^3J \left( \frac{1}{n}\sum_{i=1}^{n}\sigma_{i}\right) 2\beta^3J\left( \frac{1}{n}\sum_{i=1}^{n}\sigma_{i}\right) 2\beta^2\end{array}\right).
\int_a^b F_n(\varphi(t))\d t=\sum_k \lambda^k_n \mathcal{L}^1\big(\{t\,:\, a\leq t\leq b, \, t^k_n\leq \varphi(t)\leq t^{k+1}_n\}\big)=\sum_k \lambda^k_n\big|(G(t^{k+1}_n)\wedge b)-(G(t^k_n)\vee a)\big|.
\Pi \otimes \Pi \otimes \Pi(f, g)(x, y, z)=\sum_{k, l, m}Q_k^1 Q_l^2 P_m^3 \left( P_k^1 Q_l^2 Q_m^3 f \cdot Q_k^1 P_l^2 Q_m^3 \right)(x,y,z).
\breve{A}-\lambda \breve{B}:=\left[ \begin{array}{cc}A U V^* 0 \end{array}\right]\left[ \begin{array}{cc}x z \end{array}\right]=\lambda \, \left[ \begin{array}{cc}B 0 0 0 \end{array}\right]\left[ \begin{array}{cc}x z \end{array}\right].
\left|\partial^{\beta}\Theta(t,x,p^{1},q^{1})-\partial^{\beta}\Theta(t,x,p^{2},q^{2})\right|\leq c \left(|p^{1}-p^{2}| + \sum_{i=1}^{d}|q^{1}_{i}-q^{2}_{i}|\right).
(V(B^h;k)^n) \leq C n^{-4h_{\min}+1}\sum_{i=1}^n i^{4h_{\min}-4}\leq C\begin{cases}n^{-4h_{\min}+1}h_{\min}\in (0,3/4) \ln n\cdot n^{-2}h_{\min}= 3/4 n^{-2}h_{\min}\in (3/4,1)\end{cases}
\begin{cases}T(u) \equiv u \ \ \mbox{on}\V, ||T(u)||_{L^2(\mathbb{B}^4)}\leq C_2 ||u||_{L^2(V)}, \ \ \mbox{for some constant}\C_2 >0, ||\Delta T(u)||_{L^2(\mathbb{B}^4)}\leq ||T||||u||_{W^{2,2}(V)}\leq C_0||T|| \ ||u||_{H^2_g(U)}\end{cases}
\underline{K_{t}}\|A^{1-\nu}XB^{\nu}\|_{2}^{2}\leqslant \|(1-\nu)AX-\nu XB \|_{2}^{2}-r_{0}^{2}\|AX-XB\|_{2}^{2}& -\sum_{k=1}^{t-1}r_{k}\|A^{1-\frac{m_{k}}{2^{k}}}XB^{\frac{m_{k}}{2^{k}}}-A^{1-\frac{m_{k}+1}{2^{k}}}XB^{\frac{m_{k}+1}{2^{k}}}\|_{2}^{2}&\leqslant \overline{K_{t}}\|A^{1-\nu}XB^{\nu}\|_{2}^{2}.
\mathrm{n g h}( F_{m}Q ) = \mathrm{n g h}( Q_{1}Q ) = - n_{\varphi_{u}}- 4 n_{c_{\xi}^{1}}- 3 n_{B_{u}}+ 2 n_{c_{u}^{1}}+ 2 n_{B_{\xi}}= - n_{\varphi_{u}}- n_{B_{u}}- 2 n_{B_{\xi}}< 0 ,
C_{i+1}= C(n,p)\tilde{C}^{\frac{p}{n+p}}(2^{\frac{1}{2}-\alpha_i}RCC_i)^{\frac{n}{n+p}}\alpha_{i+1}= \frac{n}{n+p}\left(\alpha_i - \frac{1}{2}\right) \beta_{i+1}= \frac{p}{n+p}+ \beta_i \frac{n}{n+p}
[ P , R ]_{\scriptscriptstyle S N}\hat{\longrightarrow}\sum_{i = 1}^{p}( - 1 )^{i + 1}\widehat{X}_{( 1 )}\cdots{\widehat{\widehat X}}_{( i )}\cdots \widehat{X}_{( p )}\{- \{\widehat{X}_{( i )}, Q \}, \widehat{R}\}
qp_{bl}(y_n,y_m) \leq \sum \limits_{k=n}^{m-1}s^k qp_{bl}(y_k,y_{k+1}) & \leq \sum \limits_{k=n}^{m-1}(2s\lambda)^k \Big( \frac{qp_{bl}(y_0,y_1)+qp_{bl}(y_1,y_0)}{2}\Big) & \leq \frac{(2s\lambda)^n}{1-2s\lambda}\Big( \frac{qp_{bl}(y_0,y_1)+qp_{bl}(y_1,y_0)}{2}\Big).
\{\widetilde{a}^{I}( \theta ) , \widetilde{a}^{J}( \theta^{\prime}) \}= g^{I J K}\widetilde{a}^{K}( \theta ) \delta ( \theta - \theta^{\prime}) - \frac{k}{2 \pi}\, h^{I J}\, \delta^{\prime}( \theta - \theta^{\prime})
\begin{aligned}f(y)\leq f(x)+\langle \nabla f(x),y-x\rangle+\frac{L}{2}\|y-x\|^2,~\forall x,y\in\mathbb{R}^n. \Longleftrightarrow f(y)\geq f(x)+\langle \nabla f(x),y-x\rangle+\frac{1}{2L}\|\nabla f(x)-\nabla f(y)\|^2,~ \forall x,y\in\mathbb{R}^n.\end{aligned}
{}_1F_1(a,b,z) = \frac{1}{\Gamma(a)}\int_0^\infty t^{a-1}e^{-t}{}_0F_1(b,zt)\mathrm{d}t, & = \frac{2\pi^a}{\Gamma(a)}\int_0^\infty r^{2a-1}e^{-\pi r^2}{}_0F_1(b,z\pi r^2)\mathrm{d}r,
\int_Q T_{\mu, \sharp}(1_{aQ\setminus Q}b_1,1_Qb_2)^{s/2}d \mu \lesssim \int_Q \Big[\int_{aQ \setminus Q}\int_Q \frac{d\mu(z) d\mu(y)}{(|x-y|+|x-z|)^{2m}}\Big]^{s/2}d\mu(x) &\lesssim \int_Q \Big[\int_{aQ \setminus Q}\frac{d\mu(y)}{|x-y|^{m}}\Big]^{s/2}d\mu(x) & \lesssim t_0 \mu(2Q).
- ({\frac{\partial S}{\partial t}})^{2}+ D_{6}^{- 1}[ ({\frac{\partial S}{\partial r}})^{2}+{\frac{1}{r^{2}}}({\frac{\partial S}{\partial \theta}})^{2}+{\frac{1}{r^{2}}}({\frac{\partial S}{\partial \phi}}- q_{6}( 1 - \cos \theta ){\frac{\partial S}{\partial x^{1 1}}})^{2}] + D_{6}({\frac{\partial S}{\partial x^{1 1}}})^{2}= 0
({\tilde{\Gamma}}_{{\bar{i}}{\bar{j}}}^{\bar{k}}-{\tilde{\Gamma}}_{{\bar{j}}{\bar{i}}}^{\bar{k}}) \langle{\hat{\phi}}_{\bar{k}}\prod_{l = 1}^{s}\phi_{l}\rangle_{g}= \sum_{l = 1}^{s}{\cal R}_{D_{l}}+ \sum_{n o d e s}{\cal R}_{\Delta}\,
S_{\underset{k}{\underbrace{a,\ldots,a}}}\left(N\right)=\sum_{l=1}^{N}\left(\prod_{t=0}^{m-1}{N \choose \xi_{m}^{t}l}\frac{\pi\left(1-\xi_{m}^{t}\right)l}{\sin\left(\pi\xi_{m}^{t}l\right)}\right)\frac{1}{l^{mk}},
\int_{0}^1 f(x)L_{X}(x,t)\, dx = \int_{0}^t f(X_t)\, dt =\int_{0}^{T^{-1}(t)}f(s^{-1}(B_u)T'(u)\, du=\int_{0}^{T^{-1}(t)}\frac{f(s^{-1}(B_u)}{(\sigma \cdot s')^2\circ s^{-1}(B_u)}\, du = \int_{0}^\infty \frac{f(b)L_B(b,T^{-1}(t))}{(\sigma \cdot s')^2\circ s^{-1}(b)}\, db =\int_{0}^\infty \frac{f(b)L_B(s(x),T^{-1}(t))}{(\sigma(x)^2 s'(x)}\, dx
{\cal I}^{\rm het}_{\rm deg}=\frac{V^{(8)}T_2}{2^8\pi^4}t_8 F_+^{4-l}F_-^l\Delta^{\rm deg}_{F_+^{4-l}F_-^l}(T_2,U)\leftrightarrow{\cal I}^{\rm I}_{\rm 1-loop}\, .
\sum\limits_{i\in\left[ k\right]}\left\Vert \mathbf{y}^{(i)}\right\Vert^{2}=\sum\limits_{i\in\left[ k\right]}\left( \left( k-1\right)^{2}\left\vert V_{i}\right\vert +n-\left\vert V_{i}\right\vert \right)=\left( k^{2}-k\right) n.
A=\nabla (H\cdot \nabla u)= (\partial_j H \cdot \nabla u^i)_{(ij)}+(H\cdot \nabla \partial_j u^i)_{(ij)}, B=\nabla (u \cdot \nabla H)= (\partial_j u\cdot \nabla H^i)_{(ij)}+ (u\cdot \nabla \partial_j H^i)_{(ij)}, C=\nabla(H u)= \nabla H u+H \otimes \nabla u,
\widetilde{K}(x,y) = - \sum_{p \in \pi\mathbb{N}_0^3}n\widehat{\omega}_{\ell,\lambda}(p) u_p(x)u_p(y) \ , K(x,y) = - \sum_{p \in \pi\mathbb{N}_0^3 \setminus \{0\}}n\widehat{\omega}_{\ell,\lambda}(p) u_p(x)u_p(y).
\widehat c_n^{(k)}=\underbrace{\int_0^1\cdots\int_0^1}_k(-x_1\cdots x_k)_n dx_1\dots dx_k &=n!\underbrace{\int_0^1\cdots\int_0^1}_k\binom{-x_1\cdots x_k}{n}dx_1\dots dx_k\,.
\tan \mbox{\boldmath\Large $\left( \right.$}\! \! \! \phi_{l}^{(D)}(k; a)\mbox{\boldmath\Large $\left. \right)$}\! \!=\tanh \left( \frac{\pi \Theta_{l}}{2}\right)\,\frac{1 -{\mathcal T}_{l}(k;a) \, \varrho_{l}}{{\mathcal T}_{l}(k;a) + \varrho_{l}}\; ,
Z_{3}=(\begin{matrix}A_{1}-B_{1}^{2}(D_{1}+A_{2})^{-1}&-B_{1}B_{2}(D_{1}+A_{2})^{-1}\\ -B_{1}B_{2}(D_{1}+A_{2})^{-1}&D_{2}-B_{2}^{2}(D_{1}+A_{2})^{-1}\end{matrix}).
\hat{\mathbb{E}}_{(\tau\wedge\sigma)+}[XI_{\{\tau\leq \sigma\}}] =\sum_{i=1}^n\hat{\mathbb{E}}_{\tau+}[\xi_i]I_{A_i}I_{\{\tau\leq \sigma\}}&=\hat{\mathbb{E}}_{\tau+}[X]I_{\{\tau\leq \sigma\}}.
\omega \wedge \iota_{E}(\eta) + \iota_{E}(\omega \wedge \eta) = \omega \wedge \iota_{E}(\eta) + \iota_{E}(\omega) \wedge \eta - \omega \wedge \iota_{E}(\eta) = \iota_{E}(\omega) \eta = (\sum \lambda_{k}\iota_{E}(d \log f_{k}) )\eta = (\sum \lambda_{k}) \eta.
\frac{2 (\log n_i')^2}{\pi^4 n_i'}E[\chi_{n'_i}(1,1)] \to E[\alpha(A_1^1)] = \int_{A_1^1}E [ G_\beta (\beta_s, \beta_t) ] ds dt &= \frac{1}{2\pi^2}E[|\beta_1|^{-2}] \int_{A_1^1}|t - s|^{-1}dt ds = \frac{\log 2}{4 \pi^2}\,,
\lambda_{h}^{N, (n,k)}= i R_0 - h^{\frac 23}a'_n e^{-i\pi/3}+ h (2k-1)\frac{e^{-i\pi/4}}{\sqrt{2 R_0}}+ h^{\frac 43}\frac{e^{-\pi i/6}}{2R_0\, a'_n}+ \mathcal O(h^{\frac 53})\, .
W ( p^{\Lambda}, q_{\Lambda}) = -{( p \cdot q )}^{2}+ 4 \bigl ( ( p^{1}q_{1}) ( p^{2}q_{2}) + ( p^{1}q_{1}) ( p^{3}q_{3}) + ( p^{3}q_{3}) ( p^{2}q_{2}) \bigr ) \, - 4 p^{0}q_{1}q_{2}q_{3}+ 4 q_{0}p^{1}p^{2}p^{3}\ .
\begin{array}{l}{\partial_{0}x^{i}= - p^{i}, \qquad \partial_{0}p^{i}= - \partial_{1}\partial_{1}x^{i},}\\{( \partial_{0}+ \partial_{1}) \theta_{a}= 0 , \qquad ( \partial_{0}- \partial_{1}) \bar{\theta}_{\dot{a}}= 0 ,}\\ \end{array}
h_{\i}^{\underline{{{a}}}}\, \dot{\xi}_{\alpha}^{\underline{{{a}}}}- \varepsilon_{i j}\, B_{\alpha}^{\underline{{{a}}}}\, \xi_{\alpha}^{\underline{{{a}}}}\, \dot{x}^{j}-{\cal E}_{j ; \alpha}^{\underline{{{a}}}}\, \xi_{\alpha}^{\underline{{{a}}}}= 0 .
A_{\rho,k}=\left(\begin{array}{cccccc}A_{1,1}\cdots A_{1,k-1}A_{1,k+1}\cdots A_{1,n}\vdots \vdots \vdots \vdots A_{n-1,1}\cdots A_{n-1,k-1}A_{n-1,k+1}\cdots A_{n-1,n}\end{array}\right).
\tilde{\alpha}_p(u) = \inf \{\sum_{m=1}^M \Vert (\lambda_{m,i_m^1, \dots,i_m^N})_{i_m^1,\dots,i_m^N =1}^{I_m^1, \dots, I_m^N}\Vert_{p'}. \Vert (x_{m,i_m^1}^1)_{i_m^1=1}^{I_m^1}\Vert_p^w\dots \Vert (x_{m,i_m^N}^1)_{i_m^N=1}^{I_m^N}\Vert_p^w \}
\delta_{g t}| \Phi_{p h}\rangle = ( \alpha^{I}\hat{\partial}^{I}+ ( s - 1 ) \alpha^{z}) | \Lambda_{0}\rangle - \frac{\alpha_{J}^{2}}{2 s + d - 4}\Bigl ( ( \bar{\alpha}^{I}\hat{\partial}^{I}+ ( s - 1 ) \bar{\alpha}^{z}) | \Lambda_{0}\rangle + \hat{\partial}^{+}| \Lambda_{1}\rangle \Bigr ) \, .
\mu_{\tau}= \frac{2(2n+5)}{9n (n-1)}, \mu_{\hat{\rho}_s}= \frac{(n^2 - 3)}{n(n-1)(n-2)}, \mu_{D}= \frac{2(n^2 + 5n - 32)}{9n (n-1)(n-3)(n-4)}, \mu_{t^*}= \frac{8}{75}\frac{3n^2 + 5n - 18}{n(n-1)(n-2)(n-3)}.
F_{\bar{\gamma}(n)}(x) = 1- c_n ~\phi^{n-1}\frac{x^{n_T-n+1}}{\left(x+ \frac{n_T-1}{2n_T-1}\right)^{2n_T -1}}\exp\left(- \frac{x}{\phi}\right) + O\left(x^{-n_T-n+1}\exp\left(- \frac{x}{\phi}\right) \right)
\left\|\Pi_{H_2}|_{H_1}\right\|^2_{HS}=\sum_{k\geq 1}\|\Pi_{H_2}\phi_{1k}\|^2=\sum_{k\geq 1}\left\langle \Pi_{H_1}\Pi_{H_2}\Pi_{H_1}\phi_{1k},\phi_{1k}\right\rangle^2=\sum_{k\geq 1}\alpha_k.
\begin{array}{l}{{\theta_{1}= \theta^{u}( v ) - \theta^{v}( w ) + \theta^{u}( w ) - \theta^{w}( v )}}\\{{\theta_{2}= \theta^{v}( u ) - \theta^{w}( v ) + \theta^{v}( v ) - \theta^{u}( u )}}\\{{\theta_{3}= \theta^{v}( w ) - \theta^{w}( u ) + \theta^{v}( u ) - \theta^{u}( w )}}\\{{\theta_{4}= \theta^{w}( v ) - \theta^{u}( w ) + \theta^{w}( w ) - \theta^{v}( v )}}\\ \end{array}
\exp^x(p+q)-\exp^x(p)\exp^x(q)=\sum_{n \geq 2}\frac{x^n}{n!}\sum_{k=0}^n{n \choose k}p^kq^{n-k}-\sum_{n \geq 2}\sum_{k=0}^n\frac{x^k}{k!}p^k\frac{x^{n-k}}{(n-k)!}q^{n-k}=0.
\mathrm{V o l ( Q^{1 , 1 , 1})}= \oint_{Q^{1 , 1 , 1}}\left( \frac{3}{8 \Lambda}\right)^{7 / 2}8 \prod_{i = 1}^{3}\sin \theta_{i}d \theta_{i}d \phi_{i}d \psi = \frac{\pi^{4}}{8}\left( \frac{6}{\Lambda}\right)^{7 / 2}.
\Phi\bigl(2 u+\tfrac{n+1}{2}v +\ell\bigr)=2 u+\tfrac{n+1}{2}v +\ell\ ,\ \ \Phi(v+\ell)=-v-\ell\ , \ \ \Phi\bigl( u-\tfrac{n+1}{4}v\bigr)= -u+\tfrac{n+1}{4}v,
\langle hD\rangle^{-1}v = \varepsilon a_{\varepsilon}(x) \exp\left[i\varphi(x) \int_1^t\theta_{1/\tau}(x)d\tau + i\varepsilon^2|a_{\varepsilon}(x)|^2\Phi^{\Sigma}_1(x)\int_1^t\theta_{1/\tau}(x)\frac{d\tau}{\tau}\right] + t^{-\frac{1}{4}+\sigma}r(t,x) \,,
-\frac{e^{i\frac{3}{4}\pi{\rm sign}\left(\frac{H}{P}\right)}}{2\sqrt{|P|}}{\rm sign}(P)e^{-\frac{i\pi}{2K}\phi}\int_{-\infty}^{+\infty}d\beta\,e^{-\frac{i\pi}{2K}\frac{H}{P}\beta^{2}}\frac{\prod_{i=1}^{3}2i\sin\frac{\pi\beta}{p_{i}}}{2i\sin\frac{\pi\beta}{K}}.
\frac{q^{4/p}(v\sigma)^{4}}{B(\frac{1}{p},q)^{4}}(-48\lambda^{4}B(\frac{2}{p},q-\frac{1}{p})^{4}+24\lambda^{2}(1+3\lambda^{2})B(\frac{1}{p},q)B(\frac{2}{p},q-\frac{1}{p})^{2}
d s^{2}= ( 1 -{\frac{q c o s \theta}{r}})^{\frac{2}{1 + \alpha^{2}}}\lbrace d r^{2}+ r^{2}d \theta^{2}+ r^{2}s i n^{2}\theta d \varphi^{2}\rbrace -{\frac{d t^{2}}{( 1 -{\frac{q c o s \theta}{r}})^{\frac{2}{1 + \alpha^{2}}}}}\, .
\varrho_{*}\Big ( \sum_{k = 1}^{n}\, f_{k}\, d^{\prime}f_{k}^{\prime}\Big ) \: = \: \sum_{k = 1}^{n}\, \sum_{\iota \in I}\, \big ( r_{U_{\iota}\cap U}^{U}( f_{k}^{\prime}) \, d^{\prime}r_{U_{\iota}\cap U}^{U}( f_{k}) \big ) \otimes 1 \, + \, 0 ,