harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
|\frac{16}{\lambda(t)^{2}}\int_{t}^{\infty}dx \lambda'''(x) K(x-t,\lambda(t))| \leq C \sup_{x \geq t}|\lambda'''(x)| &\leq \frac{C}{t^{3}\log^{b+1}(t)}+ C \sup_{x \geq t}|e'''(x)|
\gamma_j = 2 ((\imath_{e_k}\ast \varphi) \wedge (\imath_{e_j}\nabla_{e_k}\ast \varphi) + (\imath_{e_j}\imath_{e_k}\ast \varphi) \wedge e^l \wedge (\imath_{e_k}\nabla_{e_l}\ast \varphi)).
\begin{aligned}\tau:{}X\mapsto X^{-1}, \Y \mapsto Y^{-1}, \varphi_X:{}X \mapsto b(Y)X, \Y \mapsto Y,b(Y) \in k(Y)^{\times}, \varphi_Y:{}X \mapsto X,\Y \mapsto a(X)Y,a(X) \in k(X)^{\times}. \end{aligned}
\langle (\mathcal{M}^{(2)}+2I)^{-1}\partial_c\phi',\partial_c\psi' \rangle =\frac{1}{2}\langle \partial_c\phi',\partial_c\psi' \rangle-\frac{1}{2}\langle \mathcal{M}^{(2)}(\mathcal{M}^{(2)}+2I)^{-1}\partial_c\phi',\partial_c\psi' \rangle.
d((\log_{e}|F(z)|^2) \overline{g(z)}d\overline{z})=\frac{\partial_z F(z) \overline{F(z)}}{F(z) \overline{F(z)}}\overline{g(z)}dz d\overline{z}=\frac{\partial_z F(z)}{F(z)}\overline{g(z)}(-2i)dxdy.
S_{H,L}(\Omega):=\displaystyle\inf\limits_{u\in D_{0}^{1,2}(\Omega)\backslash\{{0}\}}\ \ \frac{\displaystyle\int_{\Omega}|\nabla u|^{2}dx}{\left(\displaystyle\int_{\Omega}\int_{\Omega}\frac{|u(x)|^{2_{\mu}^{\ast}}|u(y)|^{2_{\mu}^{\ast}}}{|x-y|^{\mu}}dxdy\right)^{\frac{N-2}{2N-\mu}}}=S_{H,L},
A'(x,v) =\frac{x}{1-v}\left(A'(x,v)-\frac{1}{v}A'(vx,1)\right)+\frac{x(1+v)}{v}(A(xv,1)-1)-x^2A(xv,1),\textrm{and}A''(x,v) =xA''(x,v)+(1-x)x\big(g(x)-1\big)+B(x,v)-B(x,0)+(1-x)(B'(x,v)-B'(x,0))\,.
\begin{array}{c}{{\epsilon_{c}= k^{a b}M_{a b}+ b^{a}P_{a}+ f^{a}K_{a}+ f D ;}}\\{{M_{a b}= x_{a}p_{b}- x_{b}p_{a}\; ; \; P_{a}= p_{a}\; ; \; K_{a}= x^{2}p_{a}- 2 ( x p ) x_{a}\; ; \; D = ( x p ) .}}\\ \end{array}
A=\sum\epsilon(i_1 \dots i_{r-1}n i_{r+1}\dots i_{n-1})\,\epsilon(i_1\dots i_{n-1})\,\kappa_{i_1}\dots\kappa_{i_{r-1}}R_{i_r n i_rn}=(-1)^{n-r-1}\sum\kappa_{i_1}\dots\kappa_{i_{r-1}}K_{i_rn},
\left({\cal H}_{1}\otimes{\cal H}_{2}\right)_{\mathrm f}^{0}\subset{\cal H}_{1}^{\mathrm{s}}\otimes{\cal H}_{2}^{0}\hspace *{1 c m}\mathrm{a n d}\hspace *{1 c m}\left({\cal H}_{1}\otimes{\cal H}_{2}\right)_{\mathrm f}^{0}\subset{\cal H}_{1}^{0}\otimes{\cal H}_{2}^{s}.
\dot{S}_{1 2}( \theta_{1 2}) \, \dot{S}_{1 3}( \theta_{1 3}) \, \dot{S}_{2 3}( \theta_{2 3}) = \dot{S}_{2 3}( \theta_{2 3}) \, \dot{S}_{1 3}( \theta_{1 3}) \, \dot{S}
e^{4iR\lambda_j}C_{12}(\lambda_j)w_j - w_j = 0, & \big|\lambda_j-\lambda_0\big| < \epsilon^{1/2}\big\lVert v \big\rVert_Y^{-1/2}, \big\lVert v_j - w_j \big\rVert_Y < \epsilon^{1/2}\big\lVert v \big\rVert_Y^{1/2}.
S_1(q;\beta)=1, S_2(q;\beta)=1+q+\beta q, S_3(q;\beta)=1+2 q+q^2+q^3+\beta \big(q+2q^2+2q^3\big )+\beta^2 q^3, S_4(q;\beta)=1+3q+3q^2+3q^3+2q^4+q^5+q^6+\beta\big(q+3q^2+5q^3+6q^4+3q^5+3q^6\big) \hphantom{S_4(q;\beta)=}{}+ \beta^2\big(q^3+2q^4+3q^5+3q^6\big)+\beta^3 q^6.
\left\{1\right\}\bigcup\left\{2^{2j+1}\sum_{k=0}^{m-1-j}\left(\binom{n+1}{k}\right)\binom{n-2j-1}{m-1-j-k}x^{n}\right\}_{n\geq 1,\,0\leq j\leq\lfloor\frac{n-1}{2}\rfloor}
\begin{aligned}\partial_t \rho_1 = \partial_x \left( \partial_x \rho_1+ \rho_1 \partial_x \phi \right) , \partial_t \rho_2 = \partial_x \left( \partial_x \rho_2- \rho_2 \partial_x \phi \right) , -\partial_x^2 \phi = \rho_1-\rho_2, \end{aligned}
\Phi \, ( \log \frac{p}{\Lambda}) = 1 + \frac{a_{1}}{\log \frac{p}{\Lambda}}+ \frac{a_{2}+ a_{3}\log ( \log \frac{p}{\Lambda})}{\log^{2}\frac{p}{\Lambda}}+ \mathcal{O}\, ( \frac{1}{\log^{3}\frac{p}{\Lambda}}) \, .
T_{\mathcal{P}_{\infty}}=\langle T_i, t; ty^{-j}xy^jt^{-1}=\phi(y^{-j}xy^j), \forall~j\in\mathbb{Z}\rangle T_{\mathcal{P}_k}=\langle T_i, t; ty^kt^{-1}=\phi(y^k), ty^{-j}xy^jt^{-1}=\phi(y^{-j}xy^j), \forall~0\leq j<k\rangle
(r_{f_1}-r_{f_2})r_{u_{35}}=r_gr_{e_2}r_{u_{35}}/r_{u_{25}}-r_{g_1}r_{e_1}r_{u_{35}}/r_{u_{15}}=r_{g_2}r_{u_5}r_{u_{35}}/r_{u_4}-r_{g_1}r_{u_5}r_{u_{35}}/r_{u_4}=(r_{g_2}-r_{g_1})r_{e_3}
\sum_{p=1}^{s}\xi_{p}=ln\prod_{p=1}^{s}\frac{\delta_{p}}{(1-\delta_{p+1})x_{p}x_{p-1}}=ln\prod_{p=1}^{s}\frac{\delta_{p}}{(1-\delta_{p})x_{p-1}^{2}}+ln\frac{x_{0}}{x_{s}}+ln\frac{1-\delta_{1}}{1-\delta_{s+1}}
E_j=\left\langle\frac{i}{2}\epsilon^{jkl}\gamma_{kl}\psi,\psi\right\rangle ,\ \G=\left\langle\gamma_0\psi,\psi \right\rangle F=-\left\langle\gamma_{123}\psi,\psi \right\rangle ,\ \H=-\left\langle i\gamma_{0123}\psi,\psi \right\rangle.
H=-\frac{Gm_{1}m_{2}}{|r_{1}-r_{2}|}-\frac{Gm_{2}m_{3}}{|r_{3}-r_{2}|}-\frac{Gm_{3}m_{1}}{|r_{3}-r_{1}|}+\frac{{p_{1}}^{2}}{2m_{1}}+\frac{{p_{2}}^{2}}{2m_{2}}+\frac{{p_{3}}^{2}}{2m_{3}}.
\begin{aligned}||D^2 w^{s,0}_{k}||_{L^{p}(Q_k)}\leq C\left(d_k^{-2}|| w^{s,0}_{k}||_{L^{p}(\widetilde Q_k)}+ ||f\chi_{\mathcal{F}^{s,0}_{Q_k}}||_{L^{p}(\widetilde Q_k)}\right) &\leq C\left(2^{-\frac{2m}{np'}}||f||_{L^p(\mathcal{F}^{s,0}_{Q_k})}+||f||_{L^p(\mathcal{F}^{s,0}_{Q_k}\cap \widetilde Q_k)}\right).\end{aligned}
L^{{\rm cutoff}}_\psi = - \frac{1}{2}\int \frac{d\nu}{2\pi}\sum_{i=1}^2 \frac{\omega^3}{\nu^2}\frac{v_i \bar\gamma_i^2}{\omega^2 \Sigma((-1)^i \bar R_i)}\left( G^{ii}_\psi(-i\nu) - G^{ii}_\psi(0) \right) \,.
\langle p_\Lambda, p_\Omega \rangle_{q,t}{:=}\delta_{\Lambda \Omega}(-1)^{\binom{m}{2}}z_\Lambda(q,t),z_\Lambda(q,t){=}z_{\Lambda^s}q^{|\Lambda^a|}\prod\limits_i \frac{1-q^{\Lambda^s_i}}{1-t^{\Lambda^s_i}},
H^s_p(0,2\pi)=\left\{v=\sum_{k\in\mathbb{Z}}\widehat{v}_ke^{ikx}\in L^2(0,2\pi)\, \left| \,\sum_{k\in\mathbb{Z}}|\widehat{v}_k|^2(1+k^2)^s <\infty\right.\right\}
\bar{K}(s|\tau, \mbox{\boldmath $x$}; \tau', \mbox{\boldmath $x$}')= \frac{1}{(4\pi s)^{1/2}}\exp ({-\frac{(\tau -{\tau}')^2}{4s}})\bar{K}^{(3)}(s|\mbox{\boldmath $x$}; \mbox{\boldmath $x$}'),
\sum_{g \geq 2}(y^{1/2}+ y^{-1/2})^{2g+2}\varphi_g(q) =\left( q \frac{d}{dq}\right)^2 \frac{K(z,\tau)^4}{4}= \sum_{m \geq 2}u^{2m}f_m(q) \,,
\lambda B-A = \left[\begin{array}{cccc}(\lambda - \rho_1) I - J_{s_1}0 \cdots 0 0 (\lambda - \rho_2) I - J_{s_2}\cdots 0 \vdots \vdots \ddots \vdots 0 0 \cdots (\lambda - \rho_t) I - J_{s_t}\end{array}\right],
\begin{array}{rcl}dA+A^2+\hat{\psi}^2-\psi^2+\hat{A}= 0, d\hat{\psi}+[A,\hat{\psi}]-\{\psi,\hat{A}\}= 0, d\psi+\{A,\psi\}+[\hat{A},\hat{\psi}] = 0, d\hat{A}+[A,\hat{A}]+\{\hat{\psi},\psi\}= 0. \end{array}
\bar{a}\nabla P_{L}=\mathcal{Q}f-\mathcal{Q}\partial_{t}u_{L}+\bar{a}\bar{b}\mathcal{Q}\Delta u_{L}+\bar{a}\bar{b}\nabla\operatorname{div}R & =\mathcal{Q}f-\mathcal{Q}\partial_{t}R+2\bar{a}\bar{b}\nabla\operatorname{div}R
< \phi_{\mathcal{B}}{}_{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}}^{N_{1}N_{2}N_{3}N_{4}}\ldots \phi_{\mathcal{B}}{}_{\alpha_{1 0}^{\prime}\alpha_{8}^{\prime}\alpha_{5}^{\prime}\alpha_{1}^{\prime}}^{N_{1 0}N_{8}N_{5}N_{1}}> \ \ = \ \ \lambda \ \ \delta^{\alpha_{1}\alpha_{1}^{\prime}}\ldots \delta^{\alpha_{1 0}\alpha_{1 0}^{\prime}}\ \{\mathcal{B}}_{N_{1}\ldots N_{1 0}}~ ~ ,
\prod_{\substack{l,m,n \geq 1 l,m \leq n ; \, \gcd(l,m,n)=1}}\left( 1-z^n \right)^{\frac{l m^2}{n^4}}= \sqrt[3]{\left( 1-z \right)}\; \exp\left\{- \frac{1}{12}\left( Li_2(z) + \frac{z(7-5z)}{(1-z)^2}\right) \right\}.
F ( t ) \equiv \sum_{n = 1}^{\infty}{\frac{1}{n^{4}}}\cos n t = -{\frac{1}{4 8}}t^{2}( t - 2 \pi )^{2}+{\frac{\pi^{4}}{9 0}}\quad \mathrm{f o r}\quad 0 \le t \le 2 \pi .
\int_{B_\rho}|\nabla\Delta\varphi|_g^2d\mu_g =\int_{B_\rho}g^{ij}(\Delta\varphi)_{,i}(\Delta\varphi)_{,j}d\mu_g &= \int_{B_\rho}(\delta^{ij}+O(r^2))(\Delta_0\varphi+O(r^{N-1})\varphi')_{,i}(\Delta_0\varphi+O(r^{N-1})\varphi')_{,j}(1+O(r^N))dx &= \int_{B_\rho}|(\nabla\Delta)_0\varphi|^2dx+\int_{B_\rho}(\Delta_0\varphi)'(O(r^{N-2})\varphi'+O(r^{N-1})\varphi'')dx
g_s =g_0+s\delta+O(s^2), \Sigma_s =\{(t,s\eta(t)+O(s^2))\}, \psi_s(t,z) =\psi(t,z-s\eta+O(s^2))+s\phi_s=\left(\begin{array}{c}a^+(t)\sqrt{z-s\eta}a^-(t)\sqrt{\bar{z}-s\bar{\eta}}\end{array}\right)+O_{L^2_1}(s)+s\phi_s
K_{n}( r^{\prime}, r ; t ) = \int_{r}^{r^{\prime}}[ r d r ] B_{n}[ r ] \exp \left[ i \int_{0}^{t}d s \left\{\frac 1 2 \dot{r}^{2}- \frac{n^{2}}{2 r^{2}}- V ( r ) \right\}\right] ,
q_n^x(1, y) = \int_{-1}^1 q_n(x, y)\, dx= \iint p_n(t, y) \varrho_\varepsilon(x - t)\, dt\, dx &=\int_{-1}^1 p_n(t, y) \int_{-1}^1 \varrho_\varepsilon(x - t)\, dx\, dt= \int_{-1}^1 p_n(t, y)\, dt = 0.
1 - \gamma_m^{-1}\gamma_{j, \ell}= 1 + O_g(q^{-1/2 + 1/2k}), \frac{1}{1 - \gamma_{j, \ell}^{-1}}= 1 + O_g(q^{-1/2k}), \frac{1}{1 - q^{-1}\gamma_{j, \ell}}= 1 + O_g(q^{-1 + 1/2k}).
|\langle f,g \rangle| =\bigg|\int_{\Bbb R^n}\sum_{k\in \Bbb Z}D_k T^{-1}_N(f) D^N_k(g) d\mu \bigg| &\le \sum_{k\in \Bbb Z}\|D_k T^{-1}_N(f)\|_{L^p_\mu}\|D^N_k(g)\|_{L^{p'}_\mu}&\le \bigg\{\sum_{k\in \Bbb Z}2^{k\alpha q}\|D_k T^{-1}_N(f)\|_{L^p_\mu}^q \bigg\}^{1/q}\bigg\{\sum_{k\in \Bbb Z}2^{-k\alpha q'}\|D^N_k(g)\|_{L^{p'}_\mu}^{q'}\bigg\}^{1/q'}.
R_{k}&=\frac{\frac{2}{k-1}(3^{k-1}-1)+\frac{1}{k}(1-3^{k})}{\frac{2}{k+1}(3^{k+1}-1)+\frac{1}{k+2}(1-3^{k+2})}&=\frac{1}{9}\times\frac{\frac{2}{k-1}(3^{k-1}-1)+\frac{1}{k}(1-3^{k})}{\frac{2}{k+1}(3^{k-1}-\frac{1}{9})+\frac{1}{k+2}(\frac{1}{9}-3^{k})}.
F_{k}(\zeta) = I+\frac{c_k}{\zeta^k}\begin{bmatrix}0 1 0 0\end{bmatrix},c_k=\frac{1}{2\mathrm{i}\pi}\int_{\cal L}\frac{s^{k-1}e^{s}}{s^{c}}ds=\frac{\sin(c\pi)\,\Gamma(k-c)}{\pi(-1)^{k-1}}.
\Omega_t:= \phi_t-s(\phi_t)_s-z(\phi_t)_z=\frac{1-t}{\sqrt{1+z^2}}+t\Omega>0, \Lambda_t:= \Omega_t(\phi_t)_{zz}+(r^2-s^2)\left((\phi_t)_{ss}(\phi_t)_{zz}-(\phi_t)^2_{sz}\right) = \frac{(1-t)^2}{(1+z^2)^2}+\frac{t(1-t)}{\sqrt{1+z^2}}\phi_{zz}+\frac{t(1-t)}{(1+z^2)^{3/2}}\left(\Omega+(r^2-s^2)\phi_{ss}\right)+t^2\Lambda>0.
(\mu^{\nabla^{L_1}})^{-1}(0)/G \cong (\mu^{\nabla^{L_2}})^{-1}(0)/G =: M_0; \bigl(L_1|_{(\mu^{\nabla^{L_1}})^{-1}(0)}\bigr)/G \cong \bigl(L_2|_{(\mu^{\nabla^{L_2}})^{-1}(0)}\bigr)/G =: L_0; \bigl(E_1|_{(\mu^{\nabla^{L_1}})^{-1}(0)}\bigr)/G \cong \bigl(E_2|_{(\mu^{\nabla^{L_2}})^{-1}(0)}\bigr)/G =: E_0.
\int_Q f v \,dx dt + \int_{\Sigma_0}u_0 v \,ds = \int_Q \partial_t u v \,dx dt + \int_Q \nabla_x u \cdot \nabla_x u \,dx dt - \int_{\partial Q}n_t u v \,ds = \int_Q \partial_t u v \,dx dt + \int_Q \nabla_x u \cdot \nabla_x u \,dx dt + \int_{\Sigma_0}u v\, ds \forall v \in H^{1,1}_{0,{\overline 0}}(Q).
\big(S^{\wedge, (l)}\:V\: S^{\bowtie, (r)}\big)(x,y) = - \sum_{n=0}^\infty \frac{1}{n!}\:S^{\bowtie, (n+l+r+1)}(x,y) & \times \int_{-\infty}^0 \alpha^{l}\:(1-\alpha)^r\:(\alpha - \alpha^2)^n \: (\Box^n V) \big|_{\alpha y + (1-\alpha) x}\:d\alpha \:.
P ( x , y ) \; = \; \int \frac{d^{4}k}{( 2 \pi )^{4}}\: \hat{P}( k ) \: e^{- i k ( x - y )}\; \; , \; \; Q ( x , y ) \; = \; \int \frac{d^{4}k}{( 2 \pi )^{4}}\: \hat{Q}( k ) \: e^{- i k ( x - y )}\; ,
d_{\pm}= \frac{i}{k_{z}r_{0}}\left[ a_{\pm}- \left( \omega \pm \frac 1 2 \right) \right] , \quad a_{\pm}= \textrm{s g}\left( \omega \pm \frac 1 2 \right) \sqrt{\left( k_{z}r_{0}\right)^{2}+ \left( \omega \pm \frac 1 2 \right)^{2}}
[\begin{matrix}X_{1}\\X_{2}\\X_{3}\\X_{4}\\X_{5}\\X_{6}\end{matrix}]=[\begin{matrix}0&0&d_{31}\\ 0&0&d_{32}\\ 0&0&d_{33}\\ 0&d_{15}&0\\d_{15}&0&0\\ 0&0&0\end{matrix}][\begin{matrix}E_{1}\\E_{2}\\E_{3}\end{matrix}]
\overline{M}\equiv M_{[\bar{\rho},\bar{u},\bar{\theta}]}(t,x,v):=\frac{\bar{\rho}(t,x)}{\sqrt{(2\pi R\overline{\theta}(t,x))^{3}}}\exp\big\{-\frac{|v-\overline{u}(t,x)|^2}{2R\overline{\theta}(t,x)}\big\},
(\varphi(t),\psi(t),x'(t)) \in \bigcap_{\delta>0}\overline{\mathrm{co}}\,\tilde{\Gamma}\left(t,B^\delta_\mathit{w}\left( \int_0^t\psi(s)ds,x(t) \right) \right) & =\tilde{\Gamma}\left(t,\int_0^t\psi(s)ds,x(t) \right) \quad.
\beta_1=\frac{x+\sqrt{ad}}{2},\beta_2=\frac{y+\sqrt{bd}}{2},\beta_3=\frac{z+\sqrt{cd}}{2}&\beta_4=\frac{\sqrt{c}(\varepsilon \sqrt{a}+\sqrt{d})}{\sqrt{a}(\varepsilon \sqrt{c}+\sqrt{d})},\beta_5=\frac{\sqrt{c}(\varepsilon \sqrt{b}+\sqrt{d})}{\sqrt{b}(\varepsilon \sqrt{c}+\sqrt{d})},
h_{k,N}(u,v) =u^N P^{(-N-2\nu_1-1,-N-2\nu_2)}_k\left(1+2\frac{v}{u}\right), g_{k,N}^{+}(u,v) =\frac{2N+2\nu_{12}-k}{N+2\nu_1-k}\u^N P^{(-N-2\nu_1,-N-2\nu_2)}_k\left(1+2\frac{v}{u}\right), g_{k,N}^{-}(u,v) =- u^N P^{(-N-2\nu_1,-N-2\nu_2)}_{k-1}\left(1+2\frac{v}{u}\right).
\begin{array}{c}{V ( z_{1 2}^{\prime}) = U^{- 1}( z_{1}; g ) V ( z_{1 2}) U ( z_{2}; g ) ,}\\{{}}\\{V ( z_{2 1}^{\prime}) = U^{- 1}( z_{2}; g ) V ( z_{2 1}) U ( z_{1}; g ) .}\\ \end{array}
e_7 =1344+ (168 b+9072) p+ (-12 b^2-4716 \mu^2+1824 b-44340 \mu-3024) p^2 + (624 b\mu^2-276 b^2+51252 \mu^2-504 b+19764 \mu-144) p^3 & + (-24 b^3+1152 b^2\mu-8760 b\mu^2-1200 \mu^3+48 b^2-26568 \mu^2-1584 \mu+48) p^4 + (-96 b^2\mu+1008 b\mu^2+7072 \mu^3+3600 \mu^2-48 \mu) p^5+ (96 b\mu^2-1536 \mu^3-96 \mu^2) p^6
Z_{i_0}= \left(1-\prod_{\substack{i=1 i\neq i_0}}^{{{s}}}(\zeta_1^{q^{i_0-1}}-\zeta_1^{q^{i-1}})\right) &= \left(1-\prod_{i=2}^{{{s}}}(\zeta_1^{q^{i_0-1}}-\zeta_1^{q^{i+i_0-2}}) \right) & = \left(1-\prod_{i=2}^{{{s}}}(\zeta_1-\zeta_1^{q^{i-1}}) \right)^{q^{i_0-1}}= Z_1^{q^{i_0-1}},
\begin{aligned}c(x',y,t) ={}\sum_{x\in T^{-1}(x')}\frac{1}{\vert T'(x)\vert}\bigg\{c(x,y,t)-x\frac{\partial}{\partial y}b(x,y,t) & +h(x)\left[\frac{\partial}{\partial y}\int dx \xb(x,y,t)+\frac{1}{2}(x^{2}-\langle x^{2}\rangle)\frac{\partial^{2}}{\partial y^{2}}\alpha(y,t)\right]\bigg\}\end{aligned}
-\int_{0}^{M}(u_{n},\partial_{t}\varphi )dt-\int_{0}^{M}\sum_{i=1}^{2}(u_{n}^{i}u_{n},\partial_{i}\varphi )dt+r\!\int_{0}^{M}(u_{n},\varphi )dt=\int_{0}^{M}\!(g,\varphi )dt
\mathbf{J\left(\boldsymbol{\theta}\right)=\mathbb{E}}\left\{\left[\frac{\partial\ln p\left(\boldsymbol{y}\mid\boldsymbol{\theta}\right)}{\partial\boldsymbol{\theta}}\right]\left[\frac{\partial\ln p\left(\boldsymbol{y}\mid\boldsymbol{\theta}\right)}{\partial\boldsymbol{\theta}}\right]^{T}\right\},
A_{ijk}(x,y^\perp) \, =\, \sum_r \xi_0^{ir}(x,y)\big(\bar{\xi}_1^{rj}(x,y^\perp)y_k+\bar{\xi}_1^{rk}(x,y^\perp)y_j\big), \bar{\xi}_1^{rj}(x,y^\perp)\, =\, \frac{1}{\gamma(x)}\Big(\beta(x,y)\xi_1^{rj}(x,y)+\beta(x,-y)\xi_1^{rj}(x,-y)\Big), H_j(x,y^\perp) \, =\, \sum_iA_{iji}(x,y^\perp) \,=\, \sum_{i,r}\xi_0^{ir}(x,y)\bar{\xi}_1^{ri}(x,y^\perp)y_j,
\mathcal{M}\left( \mathcal{G}^{1},\mathbf{V}^{2}\right) =\mathcal{M}\left( \mathcal{G}^{2}\right) \mathcal{M}\left( \mathcal{G}^{2},\mathbf{V}^{3}\right) =\mathcal{M}\left( \mathcal{G}^{3}\right) \Rightarrow \mathcal{M}\left( \mathcal{G}^{1},\mathbf{V}^{3}\right) =\mathcal{M}\left( \mathcal{G}^{3}\right)
Q_R = \frac{1}{\sqrt{2 T_2 U_2}}(n_2{\bar U}\tau+ n_1 \tau -{\bar U}l_1 +l_2),\;\;\;{\bar Q}_R = \frac{1}{\sqrt{2 T_2 U_2}}(n_2 U \tau + n_1 \tau - U l_1 +l_2).
\int_X\langle [P_yG(x,y)]\psi(x),\varphi(y)\rangle\, d_{y\in X}=\int_X\langle \delta(y-x)\psi(x),\varphi(y)\rangle\, d_{y\in X}=\langle\psi(x),\varphi(x)\rangle_x.
\mathbb{P}\left(\Psi(x,\gamma,Z)=y\right)= \mathbb{P}\left(( \mathbf{1}_{[0, \frac{3}{4}]}- \mathbf{1}_{(\frac{3}{4},1]})(Z)=-1\right)=\mathbb{P}\left(Z \in \left(\frac{3}{4}, 1\right]\right)=\frac{1}{4}.
\begin{array}{l}\vphantom{\Big)}\displaystyle{(H_1)_{\alpha,\beta}(\partial)=[e_\beta,e_\alpha]-(\alpha|\beta)e_\alpha\partial^{-1}\circ e_\beta\,\, \beta\neq-\alpha\,,}\vphantom{\Big)}\displaystyle{(H_1)_{\alpha,-\alpha}(\partial)=\partial+(\alpha|\alpha)e_\alpha\partial^{-1}\circ e_{-\alpha}\,.}\end{array}
Li_{n}(z)=\sum_{k=1}^{\infty}\frac{z^{k}}{k^{n}}\Rightarrow Li_{n}(e^{i\theta})=\sum_{k=1}^{\infty}\frac{(e^{i\theta})^{k}}{k^{n}}=\sum_{k=1}^{\infty}\frac{e^{ik\theta}}{k^{n}}
\int_{0}^{1}{\rm d}\rho_{1}\ldots\int_{0}^{1}{\rm d}\rho_{N}\delta(\sum\rho_{\mu}-1)\prod_{\nu=1}^{N}\rho_{\nu}^{n_{\nu}}=\frac{\prod_{\nu=1}^{N}n_{\nu}!}{\left(\sum_{\nu=1}^{N}n_{\nu}+N-1\right)!}\:,
\delta_1^{(j)}= \frac{1}{(2^{R^{th}}+ C)^{\ell-1}}, \mbox{and}\delta_{i}^{(j)}= \frac{2^{R^{th}}-1 + C}{(2^{R^{th}}+ C)^{\ell-i+1}}, i = 2,\cdots, \ell,
A_w =\begin{bmatrix}0 0 \cdots 0 0 -a_{1,2}+a_{1,n}0 \cdots a_{2,n-1}-a_{1,n-2}a_{2,n}-a_{1,n-1}\vdots \vdots \ddots \vdots \vdots a_{1,3}-a_{1,n-1}a_{1,4}-a_{2,n-1}\cdots 0 a_{n-1,n}-a_{1,2}a_{1,2}-a_{1,n}a_{1,3}-a_{2,n}\cdots a_{1,n}-a_{n-1,n}0\end{bmatrix}\in W.
\sum_{t=1}^T\sum_{i=1}^n\frac{q_{i,t+1}^\top g_{i,t}(x_{i,t})}{\gamma_t}\le h_T+\sum_{t=1}^T\frac{nF}{\gamma_{t}}+\sum_{t=1}^T\frac{\tilde{\Delta}_t}{\gamma_{t}}+\sum_{t=1}^T\sum_{i=1}^n\frac{\Delta_{i,t}(y)}{\gamma_{t}}.
F_{n}(a,q;0)=(-1)^{n}q^{-n(n-1)/2}\,\frac{1-a^{n+1}}{1-a},\G_{n}(a,q;0)=(-1)^{n+1}q^{-n(n-1)/2}\,\frac{1-a^{n}}{1-a},
\begin{aligned}\phi_\lambda (v, s^{\prime}) =f (s^{\prime})-\lambda_1 \boldsymbol{d}_{\mathcal{S}_1}(s_1, s_1^{\prime})^{p_1}-\lambda_2 \boldsymbol{d}_{\mathcal{S}_2}(s_2, s_2^{\prime})^{p_2}& \geq B + (B - \lambda_1) \boldsymbol{d}_{\mathcal{S}_1}(s_1, s_1^\prime )^{p_1}+ (B - \lambda_2) \boldsymbol{d}_{\mathcal{S}_2}(s_2, s_2^\prime )^{p_2}\geq B.\end{aligned}
S ={\frac{1}{g^{2}}}\int d^{p + 1}y \; d^{2}x \left[ -{\frac{1}{4}}F_{A B}* F^{A B}+ D_{A}\phi^{+}* D^{A}\phi -{\frac{1}{2}}( \phi^{+}* \phi - v^{2})^{2}\right]
0 = \int_{\Omega}\partial_{z_i}\tilde u \partial_{z_i}\tilde \varphi + 4|z|^2 \bigg(\partial_{x_i}\tilde u \partial_{x_i}\tilde \varphi + h b^{ij}\partial_{y_i}\tilde u \partial_{y_j}\tilde \varphi + \hat \varepsilon^{-2}h (|\tilde u|^2 - 1) \tilde u \tilde \varphi\bigg) \, dz dx.
\bar R(X,Y)Z= c\{\langle Y, Z\rangle X-\langle X, Z\rangle Y-\langle Y,\xi\rangle\langle Z,\xi\rangle X+\langle X,\xi\rangle\langle Z,\xi\rangle Y +\langle X,Z\rangle\langle Y,\xi\rangle\xi-\langle Y,Z\rangle\langle X,\xi\rangle\xi\},
b(x, y) = \sum_{r=1}^m \frac{y^r w_r^{-r+1}\alpha_r(w_r x)^3}{r!}+ \sum_{r=2}^m \frac{y^r x^{-1}\alpha_r'(x)^3}{r!}\ll \sum_{r=1}^m e^{O(r)}\delta^{r+2}x^{r-1}y^r + \sum_{r=2}^m r^{O(1)}(e^{-1/2}r^{1/2}x^{3/2}y)^r x^{-1}.
G^{\mathrm{QAM}}_\mathrm{\tilde{\rho}}(\tilde{\rho}) &=1-\exp\left(-d^\alpha\tilde{\rho}N_{\mathrm{F}}\sigma^2_N \right)\exp\left(-\lambda d^2\tilde{\rho}^{\frac{2}{\alpha}}\frac{2\pi^2}{\alpha\sin(2\pi/\alpha)}\right).
\begin{array}{l}P=(x-1-c_2^2\chi_{G_1}(B))I_n+\dfrac{A(G)}{2r+n_1}-\frac{c_1^2}{x-1-c_3^2\chi_{G_2}(C)}MM^T \;\;\;\;=(x-1-c_2^2\chi_{G_1}(B))I_n+\dfrac{A(G)}{2r+n_1}-\frac{c_1^2}{x-1-c_3^2\chi_{G_2}(C)}(rI_n+A(G)).\end{array}
\tilde{\Lambda}(x) + \tilde{M}_{0,\varphi}(x) = \left[I + \tilde{N}_{\varphi}^{-}(x) (\tilde{\Lambda}^{+}(x))^{-1}\right] \tilde{\Lambda}(x) \left[I + \tilde{N}_{\varphi}^{+}(x) (\tilde{\Lambda}^{-}(x))^{-1}\right],
\Gamma [ \sigma ] = \int d^{D}\! x \, \sqrt{- g}( - \frac{\sigma^{2}}{2 \lambda_{0}}) - i \mathrm{T r}\mathrm{L n}[ - \sqrt{- g}( \gamma^{\mu}\nabla_{\mu}+ \sigma ) ] +{\cal O}( \frac{1}{N}) .
M_{X}(a,b) = &\left(\begin{array}{lll}u_1 \mu_1(a), \ldots, u_k \mu_k(a) \end{array}\right)\left(\begin{array}{lll}\hdots \hdots \hdots \hdots u_{ij}\hdots \hdots \hdots \hdots\end{array}\right)\left(\begin{array}{l}\mu_1(b) \vdots \mu_k(b) \end{array}\right).
\frac{\Gamma_{1}\vdash\Delta_{1}|...|\Gamma_{n}\vdash\Delta_{n}|\Omega\vdash A\Sigma_{1}\vdash\Pi_{1}|...|\Sigma_{m}\vdash\Pi_{m}|\Theta\vdash B}{\Gamma_{1}\vdash\Delta_{1}|...|\Gamma_{n}\vdash\Delta_{n}|\Sigma_{1}\vdash\Pi_{1}|...|\Sigma_{m}\vdash\Pi_{m}|\Omega\vdash B|\Theta\vdash A}
\int\limits_{[k,\Lambda]}\frac{d^4p}{(2\pi)^4}\, \frac{p_\lambda p_\kappa p_\mu p_\nu}{p^8}=\frac{1}{3}\frac{1}{64\pi^2}\ln \frac{\Lambda}{k}\,\bigl( \delta_{\lambda\kappa}\delta_{\mu\nu}+\delta_{\lambda\mu}\delta_{\kappa\nu}+\delta_{\lambda\nu}\delta_{\kappa\mu}\bigr).
\mathcal{R}_{1}^{\frac{a_{1}}{\nu_{1}}}(|\mathcal{R}_{1}^{\frac{a_{1}}{\nu_{1}}}u|^{p-2}\mathcal{R}_{1}^{\frac{a_{1}}{\nu_{1}}}u)+\mathcal{R}_{2}^{\frac{a_{2}}{\nu_{2}}}(|\mathcal{R}_{2}^{\frac{a_{2}}{\nu_{2}}}u|^{p-2}\mathcal{R}_{2}^{\frac{a_{2}}{\nu_{2}}}u)=|u|^{q-2}u.
\big\langle\omega_i,\omega_{1,\infty}\big\rangle_{Z_{1,R_i}}=\big\langle\omega_i,\omega_{1,\infty}\big\rangle_{Z_{1,0}}+\big\langle\omega_i^\mathrm{nz},\omega_{1,\infty}^\mathrm{nz}\big\rangle_{Y_{[0,R_i]}}.
H_N(\alpha,\beta) = \sum_{\substack{\lambda \vdash d \ell(\lambda) \leq N}}\frac{(\dim \mathsf{V}^\lambda)^2}{d!}\omega_\alpha(\lambda)\Omega_{\frac{q}{N}}^{-1}(\lambda) \Omega_{\frac{1}{N}}^{-1}(\lambda) \omega_\beta(\lambda)
n^d_t\le C_{d,T}(t)+ 2000\big( C_2^4 t^3+3( C_8+C_{5}^2) \big)\exp\left( C_{\tau,1}C_{s,1}^4\sigma^{-4}\tau^4 \right) \eta^4 d^4 t^3 \sum_{k=0}^{t-1}C_{d,T}(k)\le C_{\tau,2}C_{d,T}(t).
\varrho_* \Big( \sum \limits_{k = 1}^{n}\, f_k \, d' f'_k \Big) \: = \: \sum \limits_{k = 1}^{n}\, \sum \limits_{\iota \in I}\, \big( r_{U_{\iota}\cap U}^U (f'_k) \, d' r_{U_{\iota}\cap U}^U (f_k) \big) \otimes 1 \, + \, 0,
\frac{1}{n}\sum_{k=1}^n \alpha_{k,n}= \frac{1}{n}\sum_{k=1}^n \Re(\alpha_{k,n}) \ge \frac{1}{n}\sum_{k=1}^n |\alpha_{k,n}|\cos\gamma \ge \left(\frac{|a_0|}{|a_n|}\right)^{1/n}\cos\gamma \ge \frac{\cos\gamma}{|a_n|^{1/n}}.
\widetilde{w}_{(i,1)}\sigma_{1}\widetilde{w}_{(i,1)}\sigma_{1}=\sigma_{1}\widetilde{w}_{(i,1)}\sigma_{1}\widetilde{w}_{(i,1)}\Longleftrightarrow(\widetilde{w}_{1}\widetilde{w}_{(i,1)}\widetilde{w}_{1^{*}}\widetilde{w}_{(i,1)})^{3}=1,
\langle u(t), \phi\rangle = \langle h, \phi\rangle +\frac{1}{2}\int_0^t \langle u(s), \phi'' \rangle ds + \int_0^t \langle b(u(s)), \phi \rangle ds & + \int_0^t\int_0^1 \sigma(u(s,x))\phi(x)W(dsdx) +\int_0^t\int_0^1 \phi(x)\eta(dsdx), \t\geq 0 \a.s.,
({\mathfrak A}^{2,n})'=\Big(\Delta_{s,n}^{l,r}:=T_{1s}^r\prod_{r=s+1}^np^{-1}C(T_{1r})\mid 1\leq l\leq2,\,\, 3\leq s\leq n,\,r\in{\mathbb F}_p\setminus\{0\}\Big)''.
\frac{1+t_{+}^{0, n}t}{t-t_{+}^{0, n}}:=\begin{cases}1/t, n<0 -t, n>0 \end{cases}, \frac{1-t_{+}^{0, n}t}{t-t_{+}^{0, n}}:=\begin{cases}1/t, n<0 t, n>0\,. \end{cases}
{\cal{A}}_{l o w e n e r g y}^{r e l a t i v}\, = \, \int^{\infty}d^{3}k I \left( k^{0}, \vec{k}, w ( p ) , \vec{p}\right) = \int_{0}^{\Lambda}d^{2}k \int_{- \infty}^{\infty}d k^{0}I + \int_{\Lambda}^{\infty}d^{2}k \int_{- \infty}^{\infty}d k^{0}I
\left.\begin{array}{l l}\mathcal{H}\!\!\! = \lambda_{x_E}u_{x_E}+\lambda_{y_E}u_{y_E}+\lambda_{z_E}u_{z_E}~~ + \sum_{i=1}^3 ( \lambda_{x_i}u_{x_i}+ \lambda_{y_i}u_{y_i}+\lambda_{z_i}u_{z_i}) \end{array}\right.
\|\hat\Delta_H^{1/4}\hat\Delta_K^{-1/4}\|_{T,1}=\sum_{n=0}^\infty \|\Delta_H^{1/4}\Delta_K^{-1/4}\|_1^n=\frac{1}{1-\|\Delta_H^{1/4}\Delta_K^{-1/4}\|_1}<\infty.
I + f(u)E_{i}+f(u+v)E_{i+1}+f(u)f(u+v)E_{i}E_{i+1}+f(v)E_{i}+ f(u)f(v)E_{i}^2+f(u+v)f(v)E_{i+1}E_{i}+f(u)f(u+v)f(v)E_{i}E_{i+1}E_{i}=I + f(v)E_{i+1}+f(u+v)E_{i}+f(v)f(u+v)E_{i+1}E_{i}+f(u)E_{i+1}+f(v)f(u)E_{i+1}^2+f(u+v)f(u)E_{i}E_{i+1}+f(v)f(u+v)f(u)E_{i+1}E_{i}E_{i+1}
~- \partial_y (K(\chi) \partial_y \chi) - \frac{K'(\chi)}{2}(\partial_y \chi)^2 + V'(\chi) + v_i'(\chi) \delta(y - y_i) = \sum_{n>0}\frac{1}{M^n}\frac{\delta S^{(n)}_{h.d.}}{\delta \chi}.
|c_{i,j}^{(k-2s-2,l)}| \le \dfrac{1}{2i}\left(|c_{i-1,j}^{(k-2s-1,l)}| + |c_{i+1,j}^{(k-2s-1,l)}|\right) \leq \dfrac{1}{2i}\frac{4V_{k,l}}{\pi^2j}\left(\Gamma_{1,-2}[s](i)+\Gamma_{1,0}[s](i)\right)= \dfrac{4V_{k,l}}{\pi^2j}\Gamma_{0,-2}[s](i).
E(\lambda n_s^{-1}) = q_\lambda^{1/2}q_s^{-1}\theta(\lambda)T_{n_s^{-1}}& = q_\lambda q_s^{-1}T_{n_s^{-1}}\theta(n_s(\lambda)) - \sum_{k = 1}^{\langle\nu(\lambda),\alpha\rangle}q_\lambda^{1/2}q_s^{-1}q_{\lambda\mu_{n_s^{-1}}(-k)}^{-1/2}E(\lambda \mu_{n_s^{-1}}(-k))c_{s,-k}.
\int dx \left(\,{1\over 2}\, \tilde{g}^{ij}{\delta^2 \Psi \over \delta \phi^i \delta \phi^j}\, +\,{1\over 2}\, g_{ij}\partial_x \phi^i \partial_x \phi^j\Psi \, \right)\, =\, E \Psi\, .
- \left(2g+1-\sum b_j \right)! \sum_{c_2, \ldots, c_n}(2k-1)!! \cdot (2c_i+1)!! \cdot \psi_1^k(\psi')^{c_i}& \cdot \prod_{j \neq i}\frac{\psi_j^{c_j}}{(2c_j)!!(b_j - 2c_j)!}\cdot \cdot \sum_{d=0}^{2c_i+2}{4g-1+n - \sum b_j \choose 2g+1 - \sum b_j - d}\sum_{e_{n+1}=0}^1 \frac{1}{b_i!}{b_i \choose 2ci_i + 2 - d- e_{n+1}}.
A' = \left( \bigoplus_{j}\left((\mathbb{F}_{(n_j)}\oplus \mathbb{F}_{(n_j + 1)}\right) \right) \oplus \left( \bigoplus_{k}\left((\mathbb{F}_{(n_k)}\oplus \mathbb{F}_{(n_k + 2)}\right) \right)
1+b = \dfrac{r_p(r-p)_q + r_p j-i}{r_p(r-p)_q}\ge \dfrac{r_p(r-p)_q -(r_p -2)}{r_p(r-p)_q}= \dfrac{r_p(r-p-q)_q + 2}{r_p(r-p)_q}> 0, 1-b = \dfrac{r_p(r-p)_q - r_p j+i}{r_p(r-p)_q}\ge \dfrac{r_p(r-p)_q -r_p (r-p-q)_q}{r_p(r-p)_q}= \dfrac{1}{(r-p)_q}> 0,
E_n^{etd}= \Delta t^2 \varphi_2(\Delta t L)F^{(1)}(t_n,u_n) + \Delta t^3 \varphi_3(\Delta t L)F^{(2)}(t_n,u_n) + O(\Delta t^4) = \frac{\Delta t^2}{2!}F^{(1)}+ \frac{\Delta t^3 L}{3!}F^{(1)}+ \frac{\Delta t^3}{3!}F^{(2)}+ O(\Delta t^4).
A_1 =\left ({{\omega - m \Omega_H}\over 4 \pi T_H}\right )^2,~~B_1 = -{\rho_-^2 \over \rho_+^2}\left ({{\omega - m \Omega_H \rho_+^2 / \rho_-^2}\over4 \pi T_H^{\rm BTZ}}\right )^2.
\quad\psi\left(z+\frac{1+x}{2}\right)-\psi\left(z+\frac{1-x}{2}\right) =\frac{x}{z}+\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{(n+1)z^{n+1}}\left(B_{n+1}\left(\tfrac{1-x}{2}\right)-B_{n+1}\left(\tfrac{1+x}{2}\right)\right)=2\sum_{n=0}^{\infty}\frac{B_{2n+1}\left(\frac{1+x}{2}\right)}{(2n+1)z^{2n+1}},