harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\big((A+UV^{\ast})\widehat{A}\big)^{\ast}=\big(I+(A^{\dagger})^{\ast}VE_{S_{A}}V^{\ast}A^{\dagger}\big)^{-1}\big(AA^{\dagger}+(UE_{S_{A}}V^{\ast}A^{\dagger})^{\ast}\big), \big(\widehat{A}(A+UV^{\ast})\big)^{\ast}=\big(A^{\dagger}A+(A^{\dagger}UF_{S_{A}}V^{\ast})^{\ast}\big)\big(I+A^{\dagger}UF_{S_{A}}U^{\ast}(A^{\dagger})^{\ast}\big)^{-1}.
\begin{array}{rl}x\in\partial \mathcal{E}_1\cap \partial \mathcal{E}_2 \leftrightarrow~x^TQ_1x=1, x^TQ_2x=1 x\in\partial \mathcal{E}_1\cap \mathcal{E}_2^o \leftrightarrow~ x^TQ_1x=1, x^TQ_2x<1 x\in \mathcal{E}_1^o\cap \partial \mathcal{E}_2 \leftrightarrow~ x^TQ_1x<1, x^TQ_2x=1.\end{array}
E_{1}(\omega)=\frac{2iq}{\epsilon(\omega)(2\pi)^{3/2}}\int d^{3}ke^{ibk_{2}}(\frac{\omega\epsilon(\omega)v}{c^{2}}-k_{1})\frac{\delta(\omega-vk_{1})}{k^{2}-\frac{\omega^{2}}{c^{2}}\epsilon(\omega)}
\sum^{\infty}_{n=1}b_n \left (\sum^{n}_{k=1}\frac{\lambda_kx_k}{\Lambda_n}\right )^p \leq U''_p\left (\sum^{\infty}_{n=1}b_nx^p_n \right )^p\left (\sum^{\infty}_{k=1}b_k \left (\sum^{k}_{i=1}\frac{\lambda_ix_i}{\Lambda_k}\right )^p\right )^{1/q},
p'(x) = \exp \left[-2 \left(-\frac{3 \zeta^4}{4}+\frac{\zeta^3 \lambda}{3}+\frac{3 \zeta^2}{2}-\zeta \lambda +\frac{3 x^4}{4}-\frac{\lambda x^3}{3}-\frac{3 x^2}{2}+\lambda x\right)\right].
\hat R_{L,T}^d = \sum_{t=1}^T \ell(\hat y_t,y_t) - \inf_{f \in \mathcal{L}_L^d}\sum_{t=1}^T \ell(f(y_{t-d}^{t-1}),y_t) & \leq t_d + \sqrt{T+1}\log D_{T+1}+ M(3+L) \Bigl(\sqrt{T}+ 2(3d)^{\frac{d}{2(d+2)}}T^{\frac{d+1}{d+2}}\Bigr)\,.
M_0(t)=\sum_{k}\int_{\mathbb{R}}A_k(t,\xi)e^{\tau\lambda_0(t)(|k- k_*|+|\xi- \eta_*|)}\big|\widehat{g_1'}(t,k,\xi)\big|^2\,\mathrm{d}\xi+\sum_{j=2}^3\int_{\mathbb{R}}A_*(t,\xi)e^{\tau\lambda_0(t)(|k_*|+|\xi- \eta_*|)}\big|\widehat{g_j'}(t,0,\xi)\big|^2\,\mathrm{d}\xi.
\frac{1}{r^{2}}\frac{\partial}{\partial r}(r^{2}\frac{\partial f}{\partial r})+\frac{1}{r^{2}sin\theta}\frac{\partial}{\partial\theta}(sin\theta\frac{\partial f}{\partial\theta})+\frac{1}{r^{2}sin^{2}\theta}\frac{\partial^{2}f}{\partial\varphi^{2}}
H_\sigma \colon V_\sigma \times \Delta^r_e\times \Delta^1_e \longrightarrow U_\sigma \times \Delta^r_e~,\bigl(x,(t_i)_{i=1}^r,(s_0,s_1)\bigr)\mapsto\bigl(x~,~(s_0\phi_{\sigma(i)}(x)+s_1t_i)_{i=1}^{r}\bigr)~;
\left\Vert \widehat{w^{\varphi}}q_{1}^{\varphi}\Psi\right\Vert^2_2 =\left\langle\Psi,q_{1}^{\varphi}\left(\widehat{w^{\varphi}}\right)^2q_{1}^{\varphi}\Psi\right\rangle=N^{-1}\sum_{k=1}^N\left\langle\Psi,q_{k}^{\varphi}\left(\widehat{w^{\varphi}}\right)^2q_{k}^{\varphi}\Psi\right\rangle\;.
I_1(x,t)=\dfrac{1}{t^{d/\ell+J}}\,{H}_{J0}\left(\dfrac{x}{t^{1/\ell}}\right)+\sum\limits_{k=M}^K\dfrac{1}{t^{(k+d)/\ell+J}}\,{H}_{Jk}\left(\dfrac{x}{t^{1/\ell}}\right)+O\left(\dfrac{1}{t^{(K+d+1)/\ell+J}}\right)
X_{t_{i+1}}^{j}= \Pi \left(X_{t_{i}}^{j}+ h \B(X_{t_i}^{j}, (u^{j-1}_i,v^{j-1}_i)(X_{t_{i}}^{j}), \mu^{j-1}_i, (u^{j-1}_i,v^{j-1}_i)\sharp\mu^{j-1}_i ) + \sigma \Delta W_{i}\right) [X_{t_{i}}^{j}] = \mu^j_i(\cdot) \mu^j_{i+1}(\cdot) = [X_{t_{i+1}}^{j}].
F\hat{R}(v_{1}\otimes v_{0}) =F(v_{0}\otimes v_{1})+(q^{2}-q^{-2})F(v_{1}\otimes v_{0}) =[2]^{1/2}((q^{2}-q^{-2})v_{1}\otimes v_{-1}+(q^{2}+1-q^{-2})v_{0}\otimes v_{0}+q^{-2}v_{-1}\otimes v_{1}).
\sum_{k=0}^{p-1}\frac{(\frac{d+r}{d})_k^{d-r}(\frac{r}{d})_k^{r-1}(\frac{r-d}{d})_k}{k!^d}\equiv \tfrac{d-r}{d}\big(\tfrac{r}{d}\big)^{r}\Gamma_p \big(-\tfrac{r}{d}\big)^d\pmod{p^2}.
{\mathbb P}(\bar Y(T_\beta)>u \,|\,S_0,S_1,\ldots,{S}_{N_{\beta,\omega}}) ={\mathbb P}(\bar Y(T_{\beta+\omega})+W_{N_{\beta,\omega}}>u \,|\,S_0,S_1,\ldots,{S}_{N_{\beta,\omega}}),
\Sigma_{\bar{X}^j}= \begin{pmatrix}\mbox{cov}(x_1^j,x_1^j) \mbox{cov}(x_1^j,x_2^j) ... \mbox{cov}(x_1^j,x_n^j) \mbox{cov}(x_2^j,x_1^j) \mbox{cov}(x_2^j,x_2^j) ... \mbox{cov}(x_2^j,x_n^j) ... ... ... ... \mbox{cov}(x_n^j,x_1^j) \mbox{cov}(x_n^j,x_2^j) ... \mbox{cov}(x_n^j,x_n^j) \end{pmatrix},
\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}|\mathcal{A}(u_k(x)-u_{k}(y))(\varphi(x)-\varphi(y))K(x,y)\,dy\,dx \longrightarrow \int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}|\mathcal{A}(u(x)-u(y))(\varphi(x)-\varphi(y))K(x,y)\,dy\,dx
_{2}\phi_{4}\left(a,aq;cq,cq^{2},az,azq;q^{2},c^{2}z^{2}q^{3}\right)=\frac{\left(z;q\right)_{\infty}}{\left(az;q\right)_{\infty}}\sum_{n=0}^{\infty}\frac{\left(a;q\right)_{n}S_{n}\left(c;q\right)z^{n}}{\left(cq;q\right)_{n}}.
C(k_1,k_2,j;\psi,\eta){=}\frac{1}{4^{k_1{+}k_2}k_1!(k_1{+}2j)!k_2!(k_2{+}j)!}\left(\frac{\mu^2_1\sigma^4_{2}{+}\mu^2_2\sigma^4_{1}}{\sigma^4_1\sigma^4_{2}}\right)^{k_{1}}\left(\frac{(\sigma^2_2{-}\sigma^2_1)^2}{16\sigma^4_1\sigma^4_{2}}\right)^{k_2}.
\inf_{x\in [1,C_1]}f_{w_1,\alpha_1,H}(x) \geq \inf_{x\in [1,C_1]}w_1\alpha_1 C_1^{-(\alpha_1+1)}& \geq (w_0-\eta)(\alpha_0-\eta)C_1^{-(\alpha_0+\eta+1)}:=C_2.
\lim_{R \to \infty}\lim_{\delta \to 0}\lim_{i \to \infty}E_n(u_i; B_{\delta r_{2,i}}(x^j) \setminus B_{Rr_{1,i}}(x^j)) =\lim_{R \to \infty}\lim_{\delta \to 0}\lim_{i \to \infty}E_n(\tilde u_{2,i}; B_{\delta}(0) \setminus B_{\frac{Rr_{1,i}}{r_{2,i}}}(0)).
\prod_{j=1}^{kd_1}\prod_{i=1}^{d_1}\begin{bmatrix}1 1\end{bmatrix}_{1,\frac{2kd_1 + 1 + k - 2 j - 2k i}{kd_1}}=\prod_{i=1}^{d_1}\begin{bmatrix}1 1\end{bmatrix}_{\frac{1}{kd_1},\frac{d_1+1-2i}{d_1}}.
\left|\int_{I(s)^c}\hat{w}\xi^{d-1}\,d\xi\right|\le\left(\int_{I(s)^c}\hat{w}^2\rho_d\,d\xi\right)^{1/2}\left(\int_{I(s)^c}\xi^{d-1}e^{-\frac{\xi^2}{4}}\,d\xi\right)^{1/2}=O(e^{-d\theta_*s})
\mathbb{P}(w_n = K) = \sum_{k=1}^{n}\mathbb{P}(w_n = K, w_{n-k}< K, w_{n-j}= K, j = 1, \ldots , k-1) = \sum_{k=1}^{n}\int_{0^-}^{K-0}\left[ 1 - G(b_k(w)) \right]dW_{n-k}.
|G(x^{3^n})|\prod_{k=0}^{n-1}(x^{3^{k+1}}(x^{3^k}-1)) = \left|g_3(x)-\frac{p_t(x^{3^{n+1}})\prod_{k=0}^{n-1}(x^{3^{k+1}}(x^{3^k}-1))}{q_t(x^{3^{n+1}})}\right|
f_2(x,y,t) = f(x+t,y+t)f(x+t,y-t)f(x-t,y+t) &+f(x+t,y+t)f(x+t,y-t)f(x-t,y-t) &+f(x+t,y+t)f(x-t,y+t)f(x-t,y-t) &+f(x+t,y-t)f(x-t,y+t)f(x-t,y-t)\neq 0.
\mathcal{L}_{I_U}(s) = \exp \Big(-2 \pi \sum_{t \in \mathcal{K}}\lambda_{t}\int_{0}^{\infty}\frac{y}{1 + \frac{y^{\alpha}}{s P_U}}dy \Big) & \overset{(a)}{=}\exp\Big(\frac{2}{\alpha}\pi (s P_U)^{2 / \alpha}\mathcal{B}(\frac{2}{\alpha}, 1 - \frac{2}{\alpha})\sum_{t \in \mathcal{K}}\lambda_t\Big),
z^{-\alpha}{}_2F_1\bigg[\begin{matrix}\alpha \alpha &1\end{matrix}\bigg|\,\frac1z\bigg]=(z-1)^{-\alpha}{}_2F_1\bigg[\begin{matrix}\frac12\alpha \frac12-\frac12\alpha &1\end{matrix}\bigg|\,-\frac{4z}{(1-z)^2}\bigg].
\sum_{t \rhd r, t \lhd s}w_{i+1}^r w_i^t w_{i+1}^s = \sum_{t \rhd r, t \lhd s}w_{i+1}^s w_i^t w_{i+1}^r, \sum_{t \lhd r, t \rhd s}w_{i+1}^r w_i^t w_{i+1}^s = \sum_{t \lhd r, t \rhd s}w_{i+1}^s w_i^t w_{i+1}^r.
+ I^{\lambda \rho \sigma \tau}\left( p_{2}^{\nu}- p_{1}^{\nu}\right) \left( p_{1}^{\mu}- p_{3}^{\mu}\right) + I^{\lambda \rho \mu \nu}\left( p_{1}^{\sigma}- p_{3}^{\sigma}\right) \left( p_{2}^{\tau}- p_{1}^{\tau}\right) + I^{\lambda \rho \sigma \nu}\left( p_{1}^{\mu}- p_{3}^{\mu}\right) \left( p_{2}^{\tau}- p_{1}^{\tau}\right)
\mathbf{I(t)}=K_{1}(t)X_{2}(t)+\frac{1}{2}K_{2}(t)X_{1}^{2}(t)+(1-p(t)A(t))^{-1}p(t)(\sigma_{y}(t)\hat{Y}(t)+A(t)\hat{Z}(t))+P(t)\delta\sigma(t,\Delta)X_{1}(t)I_{E_{\epsilon}}(t) & \;+(1-p(t)A(t))^{-1}p(t)\left[ \delta\sigma_{x}(t,\Delta)X_{1}(t)+\delta\sigma_{y}(t,\Delta)p(t)X_{1}(t)\right] I_{E_{\epsilon}}(t).
w^\beta_+ w^\gamma_- = (\theta^\beta_+ \theta^\gamma_- b_{\beta,\gamma}+ b_{\alpha+\beta, \alpha+\gamma})e^{\beta+\gamma}+ (\theta^\beta_+ b_{\beta, \alpha + \gamma}+ \theta^\gamma_- b_{\alpha+\beta, \gamma})e^{\alpha+\beta+\gamma}
a_{n}=\frac{a_{2}a_{n-1}+\sum_{j=2}^{n-1}(-1)^{j+1}\zeta(j)a_{n-j}}{n-1}=\frac{\gamma a_{n-1}-\zeta(2)a_{n-2}+\zeta(3)a_{n-3}-\cdot\cdot\cdot}{n-1}
N^l_{{\scriptscriptstyle + \atop (-)},m}= \underbrace{\sum_{n_1 \ge 0}\sum_{n_2 \ge 0}\cdots\cdots\cdots \sum_{n_{13}\ge 0}}_{n_1+n_2+\dots\dots\dots\dots\dots+n_{12}+n_{13}=l \atop{n_1+\dots+n_5-l=0 \, mod \; 2 \;\; (=1 \, mod \; 2) \atop -2(n_1+n_9)-n_2-n_6-n_{10}+n_4+n_8+n_{12}+2(n_5+n_{13})=m}}\!\!\!\!\!\!\!\!\!\! M_0(n_1,\dots,n_{13}).
\frac{d}{dx}\left(x \frac{\theta_2'(x)}{\theta_2(x)}\right) =\frac{d}{dx}\left( - \tfrac{1}{x}\frac{\theta_4' \left( \tfrac{1}{x}\right)}{\theta_4 \left( \tfrac{1}{x}\right)}-\frac{1}{2}\right) < 0.
\log |\Pi(z)| \le \sum_{j=1}^{+\infty}\log \big( 1+ r^2/|z_j|^2| \big) & = \int_1^{+\infty}\log \big( 1+ r^2/t^2| \big)\, dn(t) & = \log \big( 1+ r^2/t^2 \big) n(t)\Big|_1^{+\infty}+ 2r^2 \int_1^{+\infty}\frac{n(t)}{t^3(1+ r^2/t^2)}\, dt \,.
x_{-1}\ = \int_{[-T,0]}\frac{1}{T}d^-\eta(x) - \frac{1}{T}\eta(-T), x_i \ = \ \int_{[-T,0]}\bigg (\tilde e_i(0)-\tilde e_i(x)- \frac{1}{T}\int_{-T}^0 xe_i(x)dx\bigg )d^-\eta(x) + \frac{1}{T}\int_{-T}^0 xe_i(x)dx\,\eta(-T),
\Psi_{\mathrm{bv}}=\left(\psi_1(0),\dots,\psi_{|\mathcal{E}|}(0),\psi_1(l_1),\dots, \psi_{|\mathcal{E}|}(l_{|\mathcal{E}|})\right), \Psi_{\mathrm{bv}}' =\left(\psi_1'(0),\dots,\psi_{|\mathcal{E}|}'(0),-\psi_1'(l_1),\dots, -\psi_{|\mathcal{E}|}'(l_{|\mathcal{E}|})\right),
\lambda_{\rm atom}(K) = \inf \bigg\{\frac{h(k)}{\sum_{\rho \in \mathcal{R}}(1-\cos(\rho k))^2+ \sin^2(\rho k)}\colon k \in [0,2\pi)^d\backslash \{0\}\bigg\}, \tilde{\lambda}_{\rm LH}(K) = \lim\limits_{s \to 0^+}\inf \bigg\{\frac{h(k)}{\sum_{\rho \in \mathcal{R}}(1-\cos(\rho k))^2+ \sin^2(\rho k)}\colon k \in (-s,s)^d\backslash \{0 \}\bigg\},
\tilde{\phi}_{1}(\textbf{z})= I \left\lbrace \textbf{z}\in \mathcal{C}_{1}\cap \mathcal{C}_{0}, p(\textbf{z},1) \dfrac{p}{\mu{(\mathcal{C}_{1})}}> p(\textbf{z},0) \dfrac{1-p}{\mu{(\mathcal{C}_{0})}}\right\rbrace + I \left\lbrace \textbf{z}\in \mathcal{C}_{1}- \mathcal{C}_{0}\right\rbrace
E\left[S_{s,t}^{anom}\right] = \int_s^tE\left[\left( \beta^{-3}\nabla_q\beta\cdot \left( \frac{3n+2}{6}- \int_0^\infty Tr[\gamma e^{-2y \gamma}] e^{-y \gamma}dy\right)\gamma^{-1}\nabla_q\beta\right)(r,q_r) \right]dr.
\left\langle(J_U\circ f^t)(\varphi),\delta(x)\right\rangle =\left\langle J_U(\varphi\circ f),\delta(x)\right\rangle =\varphi(f(x))=\left\langle\varphi,T_f(\delta(x))\right\rangle &=\left\langle (T_f)^*(\varphi),\delta(x)\right\rangle
\hat{H}_{g h}= ( \hat{p}_{z}^{\dagger}, \hat{p}_{z}) + ( \hat{p}_{\eta}^{\dagger}, \hat{p}_{\eta}) + ( \hat{z}^{\dagger}, \Omega ( \hat{x}) \hat{z}) + ( \hat{\eta}^{\dagger}, \Omega ( \hat{x}) \hat{\eta})
\left ( \begin{array}{c}\tilde{\varphi}_{0}\vdots \tilde{\varphi}_{p}\end{array}\right ) = D^{p}\left ( \begin{array}{c}\tilde{f}_{0}\vdots \tilde{f}_{p}\end{array}\right ) = D^{p}\mathcal{F}^{-1}\Big [ \prod_{j = 1}^{\infty}2^{-p}\tilde{\mathcal{A}}( e^{- i 2^{-j}\xi}) v \Big ] .
\lim_{N \rightarrow \infty}D_{bil}(N,p)N^{-\gamma_{bil}(p)-\epsilon}=0,\; \mathrm{for}\; \mathrm{each}\; \epsilon>0, \limsup_{N \rightarrow \infty}D_{bil}(N,p)N^{-\gamma_{bil}(p)+\epsilon}=\infty,\; \mathrm{for}\; \mathrm{each}\; \epsilon>0.
~ h_{i,j}+ (-1)^{m-1}h'_{i+1, j}+ (-1)^{(m-1) + (n - 1)}h''_{i+1, j+1}= ~ (\delta_\alpha h)( a^{m+n-2}_{1, i}, ~ f^{n-1}, ~ a^{m+n-2}_{i+m+1, j}, ~ g^{m-1},~ a^{m+n-2}_{j+n+1, m+n+p -1}) = 0 ,
c_g: =k(0+)+k(0-) &=\lim_{r\to 0+}\int_{[0,t]}l(r/g(s))\lambda(ds)+\lim_{r\to 0-}\int_{[0,t]}l(r/g(s))g(s)\lambda(ds) &=\int_{[0,t]}l(0+)\lambda(ds)+\int_{[0,t]}l(0-)\lambda(ds) &=(l(0+)+l(0-))t,
E (n')^{-1}C_{Mn}^{(3)}(\nu, \mu) \le c_{24}\frac{1}{n'}\sum_{k=m'+1}^{m'+n'}\sum_{l=0}^\infty \left( \sup_{1 \le \nu}| f_{l,0}^{(n)}( \nu ) | \right)^2 & \le c_{24}\sqrt{\frac{1}{n'}\sum_{k=m'+1}^{m'+n'}\sum_{l=0}^\infty \left( \sup_{1 \le \nu}| f_{l,0}^{(n)}( \nu ) | \right)^4}& \stackrel{n',m'}{\ll}(n')^{-\theta},
y_{i+1}=y_{i}+hf\left( x_{i},y_{i}\right) +h\int\limits_{x_{0}}^{x_{i}}Kdt & =y_{i}+hf\left( x_{i},y_{i}\right) +\frac{h^{2}}{2}\left(\sum\limits_{j=0}^{j=i}2K_{j}-\left( K_{0}+K_{i}\right) \right) .
\widetilde{t}_{j,k}^{\left[\widetilde{p},q,L1b\right]}=\widetilde{t}_{j,k}^{\left[\widetilde{p},q,L1a\right]}+{q \over \widetilde{p}-1}\,\widetilde{t}_{j,k}^{\left[\widetilde{p}-1,q-1,L1a\right]}\left(t_{j,k+\widetilde{p}}-t_{j,k+\widetilde{p}-1}\right).
|I_{2}(x)| \leq C\|f\|_{L^{p, \phi}(\mathbb{G})}\sum_{k=0}^{\infty}\frac{(2^{k}R)^{\alpha}\phi(2^{k}R)}{(2^{k}R)^{\alpha-Q}}\frac{\left(\int_{2^{k}R\leq|xy^{-1}|<2^{k+1}R}K_{\alpha,\gamma}^{p_{2}}(xy^{-1})dy\right)^{1/p_{2}}}{(2^{k}R)^{Q/p_{2}}}.
\big(\omega^{S}_{\xi,\lambda}\phi\big)(f) =\int_{G}\Big(\phi(x),\psi_{0}(x) f(x)\Big)\,dx &=\int_{M_{S}}\int_{A_{S}}\int_{N_{S}}a^{2\rho_{S}}\psi_{0}(man)\,dn\,da\,dm\int_{K}\Big(\phi(k),f(k)\Big)\,dk.
h_g(f_0,g_0)=\frac{\left|A(e^{i\theta})f_0(\theta)-\sum\limits_{j=0}^{\infty}((\bold{B}^0)^{-1}\bold{R}^0\bold{a})_je^{ij\theta}\right|^2}{(f_0(\theta)+g_0(\theta))^2}
\mu '(\mathbf{x}; \mathbf{d})= \langle{\mathbf{g}(\mathbf{x}), \mathbf{d}}\rangle + h'(\mathbf{x};\mathbf{d}), \ \mbox{with}\h'(\mathbf{x}; \mathbf{d})=\sum\limits^{n}_{i=1}\max_{\mathbf{b}_{i}\in\partial h_{i}(\mathbf{x}_{i})}\langle{\mathbf{b}_{i},\mathbf{d}_{i}}\rangle ;
{\it L}_{g h}= \overline{c}^{\mu}\biggl ( g_{\mu \nu}\nabla^{2}+ R_{\mu \nu}- \frac{2 \xi \varphi}{\alpha ( \varphi )}( \nabla_{\nu}\varphi ) \nabla_{\mu}- \frac{2 \xi \varphi}{\alpha ( \varphi )}( \nabla_{\mu}\nabla_{\nu}\varphi ) \biggr ) c^{\nu}
e^D = \sum_{\Gamma \in G_{g,n}}\frac{1}{|{\rm Aut}(\Gamma)|}{\xi_{\Gamma}}_* \left( \prod_{i=1}^n e^{a_i \psi_i}\prod_{v\in V(\Gamma)}e^{c\lambda}\prod_{e=(h,h')\in E(\Gamma)}\frac{1-e^{b(\psi_h + \psi_{h'})}}{\psi_h + \psi_{h'}}\right).
X_{a}^{i}= \left( \begin{array}{c c}{{X_{a , 1 1}^{i}}}&{{X_{a , 1 2}^{i}\nonumber}}\\{{X_{a , 2 1}^{i}}}&{{X_{a , 2 2}^{i}}}\\ \end{array}\right) = \left( \begin{array}{c c}{{C_{a}^{i}+{{\cal O}( t )}}}&{{{{\cal O}( t^{\gamma})}\nonumber}}\\{{{{\cal O}( t^{\gamma})}}}&{{{\frac{T_{a}^{i}}{s - \mu_{a}}}}}\\ \end{array}\right) .
k>\ell \Rightarrow x_{ik}x_{i\ell}=qx_{i\ell}x_{ik}i>j \Rightarrow x_{ik}x_{jk}=qx_{jk}x_{ik}k>\ell i>j \Rightarrow x_{i\ell}x_{jk}=x_{jk}x_{i\ell}k>\ell i>j \Rightarrow x_{ik}x_{j\ell}-x_{j\ell}x_{ik}=(q-q^{-1})x_{i\ell}x_{jk}.
\textbf{err}^{(1)}= \mathcal{O}\Bigg(\delta^{7}\sqrt{\mathcal{K}^{(\mathrm{T_0})}_{\mathrm{r}}}\frac{1}{|\xi-z|^{\frac{7}{2}-2\mathrm{r}}}\Big\Vert|\mathrm{E}|^{2}\Big\Vert_{\mathrm{L}^2(\Omega)}\Bigg)
\rho(\lambda) = \frac{p_0'(\lambda)}{2\pi}+ \sum_i \int_{\lambda_{iL}}^{\lambda_{iR}}\frac{d\nu}{2\pi}K(\lambda-\nu)\rho(\nu) + \frac{1}{24 L^2}\sum_{ia}\frac{s_aK'(\lambda - \lambda_{ia})}{2\pi \rho(\lambda_{ia})}
\rho_{\circ}^+ = (c, a_{j_1+1}, \dots ,a_{i_1}) , \rho_k^+ = (a_{i_k+1}, \dots , a_{i_{k+1}}) k=1, \dots, n-1, \rho_k^- = (a_{j_{k+1}+1}, \dots, a_{j_k}) k=1, \dots, n -1, \rho_n^- = (a_{1}, \dots, a_{j_n}).
[y_1, y_1]=y_4+\theta_1y_5, [y_1, y_2]=\frac{\alpha_3}{\gamma_1}y_4+\theta_2y_5, [y_2, y_1]=\frac{\alpha_1\alpha_6}{\gamma_1\gamma_6}y_5, [y_1, y_3]=\frac{\beta_4}{\gamma_6}y_5, [y_3, y_1]=\frac{\beta_6}{\gamma_6}y_5, [y_2, y_3]=y_4+\theta_3y_5, [y_3, y_2]=-y_4, [y_3, y_3]=y_5.
\sum_{i=1}^3 \rho^+(\hat{E}^{0i})\rho^+(\hat{E}^{0i}) = -\xi_{\rho^+}I_{\rho^+}, \sum_{i=1}^3 \rho^-(\hat{E}^{\tau(i)})\rho^-(\hat{E}^{\tau(i)}) = -\xi_{\rho^-}I_{\rho^-},
H \; = \; H_{e}+ H_{p}\; = \; \frac{\alpha_{f}\hbar c}{r_{0}}\int_{0}^{\infty}{d \rho}\left[ \frac{\sin^{4}\alpha}{2 \rho^{2}}+ ( \partial_{\rho}\cos \alpha )^{2}+ \rho^{2}\cos^{2 m}\alpha \right] .
\chi(S^2) - \bigg{(}1-\frac{1}{a}\bigg{)}- \bigg{(}1-\frac{1}{b}\bigg{)}- \bigg{(}1-\frac{1}{c}\bigg{)}= d \bigg{(}\chi(S^2) - \bigg{(}1-\frac{1}{a'}\bigg{)}- \bigg{(}1-\frac{1}{b'}\bigg{)}- \bigg{(}1-\frac{1}{c'}\bigg{)}\bigg{)}.
\vert\mathcal J_{1}\vert\le \varepsilon I_{1,n}(t)+\frac{C_2}{\varepsilon}\left\{\Vert{\mathbf F}\Vert_{L^2(\Omega_t)}^2+\Vert\varphi(t)\Vert_{L^2(\mathbb R)}^2+\int_0^t\Vert\varphi(s)\Vert^2_{L^2(\mathbb R)}ds\right\}&+\frac{C_3}{\varepsilon}\int_0^t (I_{1,\ast}+I_{1,n})(s)ds\,.
\begin{aligned}p_2 = \omega p_1,p_3 = \omega^2 p_1, p'_1 =p'_2=p'_3 p_5 = \omega p_4,p_6 = \omega^2 p_4, p'_4 =p'_5=p'_6 p_8 = \omega p_7,p_9 = \omega^2 p_7, p'_7 =p'_8=p'_9\end{aligned}
\hat q_n=\frac{(7n)!\,(9n)!\,(10n)!}{n!\,(2n)!\,(4n)!\,(5n)!\,(6n)!\,(8n)!}\,{}_5F_4\biggl(\begin{matrix}-6n, \, -6n, \, 10n+1, \, \tfrac92n+\tfrac12, \, \tfrac92n+1 begin{align*}2.5pt]2n+1, \, n+1, \, 2n+\tfrac12, \, 2n+1 \end{matrix}\biggm|1\biggr).
A(p,u) =\begin{cases}\frac{p}{p-1}(\varrho_{\ell}(p,u)-1)-1 u\equiv 0(\bmod p) \frac{p}{p-1}(\varrho_{\ell}(p,u)-1)+\frac{1}{p-1}u\not\equiv 0(\bmod p)\end{cases}
\|f_2-f_1\|_{C^{l-3}}= \|\tilde{g}_2-\tilde{g}_1\|_{C^{l-3}}\lesssim \|\psi -Id\|_{C^{l-2}}\lesssim \|f_1\|_{C^{l-1}}, \mbox{for all}\,\, 3\leq l\leq k.
\|\varphi_1\|_{W^{2,p_0}}=\|(-\alpha \Delta +2\omega +i\mathbf{c}\cdot \nabla )^{-1}\{(\nabla \cdot \varphi_3)\varphi_2\}\|_{W^{2,p_0}}\lesssim \|(\nabla \cdot \varphi_3)\varphi_2\|_{L^{p_0}}<\infty
J_2= \int_{|z'|\geq\left(\frac{1}{3M_0}\right)^{\frac{1}{1+\alpha}}}dz' \int_{-M_0\left(\frac{h}{\rho_0}\right)^\alpha|z'|^{1+\alpha}}^{M_0\left(\frac{h}{\rho_0}\right)^\alpha|z'|^{1+\alpha}}\frac{1}{D(z)}dz_3.
\|\theta-\theta_0\|^2 \geq \tilde{C}(\alpha,\beta) \varepsilon^2 \sqrt{\sum_{j=1}^{D \wedge M_1}b_j^{-4}}+ (7+4\sqrt{x_{\alpha/2}}) \left[\sigma^2 \ln^{3/2}(1/\sigma) \vee a^{-2}_{D \wedge M_0}\right],
\|fg\|_{L^q_{z}L^p_{xy}}\le \|f\|_{L^{q_1}_{z}L^{p_1}_{xy}}\|g\|_{L^{q_2}_{z}L^{p_2}_{xy}}, 1/p=1/p_1+1/p_2,\; 1/q=1/q_1+1/q_2.
\nabla Q_\mu\cdot y =(y\cdot\nabla\phi_{Q}+\mu y\cdot\nabla\phi_{T_1}+y\cdot\nabla\sigma-\mu|y|^2)Q_\mu &=(-\frac{4|y|^2}{1+|y|^2}+\mu y\cdot\nabla\phi_{T_1}+y\cdot\nabla\sigma-\mu|y|^2)Q_\mu,
V(P(z))< \prod_{2<p\leq z}\left(1-\frac{1}{p}\right) \prod_{\substack{p\leq z, p\equiv 7, 13\pmod{24}}}\left(1-\frac{1}{p}\right)^2 < \frac{2e^{-\gamma}}{(0.95442)^2\log^\frac{3}{2}z}\left(1+\frac{1}{5\log^2 z}\right) < \frac{1.23274}{\log^\frac{3}{2}z}z>e^{40}.
\sigma_i \coloneqq \sigma_i(n, \overline{\alpha})=(-1)^i\sum_{\substack{i_1+\ldots+i_k=i 0 \leq i_j \leq n}}\binom{n}{i_i}\binom{n}{i_2}\cdots \binom{n}{i_k}\alpha_1^{n-i_i}\alpha_2^{n-i_2}\cdots \alpha_k^{n-i_k}.
\begin{array}{l}\langle :\Phi\Phi^{*}:(x) :\Phi\Phi^{*}:(y)\rangle\cr=\int d\theta drr r^{-d-(1-\gamma)(D-d)}F(\theta,r^{-\frac{1}{2}}({\bf x}_{F}-{\bf y}_{F}),r^{-\frac{1}{2}+\frac{\gamma}{2}}({\bf x}_{G}-{\bf y}_{G}))\end{array}
S = \int d^{6}x \left[{\frac 1 6}F_{{\mu}{\nu}{\sigma}}( A ) F^{{\mu}{\nu}{\sigma}}( A ) +{\frac 1 2}{\lambda}_{{\mu}{\nu}}{\cal F}^{{\mu}{\rho}{\sigma}}( A ){{\cal F}^{\nu}}_{{\rho}{\sigma}}( A ) \right] ,
|x_i -(\Gamma x^{(N)}+\eta) |_Q^2=\ |x_i^\dag-(\Gamma \bar m+\eta)|_Q^2+|(x_i-x_i^\dag)+\Gamma(\bar m-x^{(N)})|_Q^2 &+2 [x_i^\dag -(\Gamma \bar m+\eta)]^TQ [(x_i-x_i^\dag)+\Gamma(\bar m-x^{(N)})].
\int_{E}\widetilde{\mathfrak{S}}_{t}f(x)\rho({\rm d}x) =\int_{E}e^{\lambda_{1}t}\mathfrak{P}_{t}(fh)(x)\widehat{h}(x)m({\rm d}x) &=\int_{E}e^{\lambda_{1}t}T_{t}(fh)(x)\widehat{h}(x)m({\rm d}x) &=\int_{E}e^{\lambda_{1}t}f(x)h(x)\widehat{T}_{t}\widehat{h}(x)m({\rm d}x) &=\int_{E}f(x)\rho({\rm d}x).
\kappa_m \left( \sum_{i=1}^n c_i X_\rho(t_i) \right)=\begin{cases}\rho^{m/2}(m-1)! \left[\sum_{i,j=1}^n c_ic_j~ \frac{1}{2}\left( |t_i| +|t_j| -|t_i-t_j| \right)\right]^{m/2}m even; 0 m odd.\end{cases}
\begin{array}{l}u_{xx}=f_\alpha \mbox{\ \in}(0,1/4), u(0)=0, u_x(1/4)=0; u(s)=const. \mbox{\ \for}s\in [1/4,c]; u_{xx}=f_\alpha \mbox{\ \in}(c,e), u_x(c)=u_x(e)=0, u(s)=const. \mbox{\ \for}s\in [e,1/2]. \end{array}
h_t =h_{ss}+2h_su_s+4e^{2u}h+2(a-b)u_{ss}^2 +(6a+2b-6c)e^{2u}u_s^2-2ce^{4u}-4\varphi e^{2u}-\varphi_{ss}+\varphi_t-2\varphi_su_s
\tilde{\Phi}(x) = \alpha_1x_1 B_{\frac{1}{8}}(P_1), \tilde{\Phi}(x) = \alpha_2x_1 B_{\frac{1}{8}}(P_2) \tilde{\Phi}(x) = \alpha_3x_1 B_{\frac{1}{8}}(P_3).
\tilde{R}_{ab}\equiv\partial_{c}\tilde{\Gamma}_{ab}^{c}-\partial_{b}\tilde{\Gamma}_{ca}^{c}+\tilde{\Gamma}_{cd}^{c}\tilde{\Gamma}_{ab}^{d}-\tilde{\Gamma}_{bd}^{c}\tilde{\Gamma}_{ac}^{d}
\partial_t G(t,s) = \sum_{k=1}^\infty k s^{k-1}\left( qR_{k+1}(t) +\varphi(t) R_k(t) \right) = \partial_s \left(\sum_{k=1}^\infty s^k \left( qR_{k+1}(t) + \varphi(t)R_k(t) \right)\right) = \partial_s \left( q G(t,s) + \varphi(t)s G(t,s) \right).
\tilde{\phi}_{1}(\textbf{z}) = I \left\lbrace \textbf{z}\in \mathcal{C}_{1}\cap \mathcal{C}_{0}, p(\textbf{z},1)P\{Y=1\vert \textbf{Z}= \textbf{z}\}> p(\textbf{z},0) P\{Y=0 \vert \textbf{Z}= \textbf{z}\}\right\rbrace &+ I \left\lbrace \textbf{z}\in \mathcal{C}_{1}- \mathcal{C}_{0}, p(\textbf{z}, 1)>0 \right\rbrace.
\int_{0}^{1}d x_{3}\int_{0}^{x_{3}}d x_{2}x_{3}^{2 \alpha^{\prime}\alpha_{1 3}}( 1 - x_{3})^{2 \alpha^{\prime}\alpha_{3 4}}{x_{2}}^{2 \alpha^{\prime}\alpha_{1 2}}( 1 - x_{2})^{2 \alpha^{\prime}\alpha_{2 4}}( x_{3}- x_{2})^{2 \alpha^{\prime}\alpha_{2 3}}\cdot \left\{{\kappa ( x_{2}, x_{3}) \atop \lambda ( x_{2}, x_{3})}\right\},
H_{\nu}(t,z,w)=N_t^{(n,\nu)}(z,w)\sum\limits_{m=0}^{+\infty}(2m+2\nu+n)\frac{\Gamma (m+2\nu +n)}{\Gamma (m+2\nu +1)}e^{-4t(m+\nu +\frac{n}{2})^2}P_{m}^{(n-1,2\nu )}\left( \cos 2d_{FS}(z,w)\right)
{\hat{F}}[ P^{3}] ~ = ~{\alpha}^{- 1}~ ~ \sum_{m , \ell , R}{\frac{( - )^{6 \epsilon_{\ell}- 3 \epsilon_{R}}{S_{0 \Lambda_{\ell}}~ S_{0 \Lambda_{R}}~ S_{\Lambda_{\ell}\Lambda_{m}}~ S_{\Lambda_{\ell}\Lambda_{m}}~ ~ S_{\Lambda_{R}\Lambda_{m}}^{*}~ q^{- 5 C_{\ell}+{\frac{3}{2}}C_{R}}}}{S_{0 0}~ S_{0 \Lambda_{m}}}}~ .
\begin{aligned}F_{j}= -\nabla \chi_{1,loc}(x-x_{j})\frac{dx_{j}}{dt}+\Delta \chi_{1,loc}(x-x_{j}) u(t,x)+2\nabla \chi_{1,loc}(x-x_{j})\nabla u &+(\chi^{3}_{1,loc}-\chi_{1,loc})(x-x_{j})|u|^{2}u .\end{aligned}
(\tan \vartheta)^{2}= \frac{m_{2}+n_2 \pm \sqrt{(m_{2}+n_2)^2-4n_2(m_{1}+m_{2})}}{2n_2}&= \frac{m_{2}+n_2 \pm \sqrt{(m_{2}-n_2)^2-4n_2m_{1}}}{2n_2}.
\frac{\big\langle \psi_N, \sum_{j=1}^N -\Delta_{x_j}\psi_N\big\rangle}{\|\psi_N\|^2}\le \frac{E_\mathrm{kin}(\phi_N)+E_\mathrm{pot}(\phi_N)}{\|\psi_N\|^2}+\mathcal{E},
U(B_{\theta,\phi=\frac{\pi}{2}})=[\begin{matrix}t&r\\r&t\end{matrix}]=[\begin{matrix}cos\theta&-isin\theta\\ -isin\theta&cos\theta\end{matrix}]=cos\theta\hat{I}-isin\theta\hat{\sigma}_{x}=e^{-i\theta\hat{\sigma}_{x}}
W_{q}(\pi_{k}(\mu,y_{0:k},\cdot),\pi_{k}(\nu,y_{0:k},\cdot))\leq\exp\left[-\sum_{j=1}^{k}\int_{0}^{\Delta}\lambda(j,y_{j},t)\mathrm{d}t\right]W_{q}(\mu_{0,k},\nu_{0,k}),
\lim_{n\to \infty}\lim_{m\to\infty}\int 1_{F(t,n)}\d\mu_{(z,m)}= \lim_{n\to\infty}\mu_j(F(t,n)) = \frac{1}{\sigma_j\sqrt{2\pi}}\int_{-\infty}^t e^{-\frac{s^2}{2\sigma_j^2}}\ds
\eta_A^\prime(v_i \otimes w_j)=\eta_A(f^{-1}(v_i)\otimes f^\vee(w_j))=\eta_A( \sum_r f_{ri}v_r \otimes \sum_s g_{sj}w_s)=\prod_{r,s}\eta_A(v_r\otimes w_s)^{f_{ri}g_{sj}}
C_{1}(d,p):=\begin{cases}C(d,p) 1<p\leq\frac{3}{2}C^{\frac{3}{p}(2-p)}\left( d,\frac{3}{2}\right) \frac{3}{2}<p\leq2 C^{\frac{3}{p^{\prime}}(2-p^{\prime})}(d,\frac{3}{2}) 2\leq p< 3 C(d,p^{\prime}) 3\leq p<\infty\end{cases}
\frac{1}{G}= \frac{2^{[ \frac{D + 1}{2}]}N}{( 4 \pi )^{D / 2}}\int_{1 / \Lambda^{2}}^{\infty}\frac{d s}{s^{D / 2 - [ ( D - 1 ) / 2 ]}}e^{- s \rho^{2}}\prod_{k = 1}^{[ ( D - 1 ) / 2 ]}e H_{k}\coth ( e H_{k}s ) ~ .
S_{fi}\equiv \left\langle{{p_3,p_4;out}}\mathrel{\left |{\vphantom{{out\,p_3,p_4}{p_1,p_2\,in}}}\right. \kern-\nulldelimiterspace}{{p_1,p_2;in}}\right\rangle =\delta_{fi}+{{i{\cal M}_{fi}}\over{\prod\limits_i{\sqrt{(2\pi)^32\omega_i}}}}(2\pi )^4\delta^{(4)}(p_1+p_2-p_3-p_4)
\begin{aligned}\mathcal{Z}_r + \frac{\alpha + \frac{1}{2}}{2 \alpha r}\xi \mathcal{Z}_\xi = \mathcal{F}_\xi [ \mathcal{Z}, \mathcal{U}_\xi ] \mathcal{U}_r + \frac{\alpha + \frac{1}{2}}{2 \alpha r}( \xi \mathcal{U}_\xi - 1 ) = \mathcal{E}_\xi [ \mathcal{Z}, \mathcal{U}]\end{aligned}
\begin{array}{r c l}{{}}&{{}}&{{{^1 \Phi}_{h \frac{n - 1}{2}}= \displaystyle \sum_{i}a_{i}^{h}~{^1 C}_{i}\mid 1 \rangle ~ ,}}\\{{}}&{{}}&{{\left( [ 2 ] -{^1 E}_{h}\right) a_{i}^{h}+ a_{i + 1}^{h}+ a_{i - 1}^{h}= 0 ~ ,}}\\{{}}&{{}}&{{a_{0}^{h}= a_{n + 1}^{h}= 0 ~ .}}\\ \end{array}
\arraycolsep=2.8pt C=\begin{pmatrix}2 0 -1 0 0 0 0 2 -1 0 0 0 -1 -1 2 -1 0 0 0 0 -1 2 -1 -1 0 0 0 -1 2 0 0 0 0 -1 0 2 \end{pmatrix}.
\psi( x^\prime,x_n,0,\xi_n) &=x_n\psi^\prime_{x_n}(x^\prime,0,0,\xi_n)+x_n^2\int_0^1(1-s)\psi^{\prime\prime}_{x_n x_n}(x^\prime,sx_n,0,\xi_n)ds &=\left[\pm x_n\psi^\prime_{x_n}(x^\prime,0,0,\pm1)\pm x_n^2\int_0^1(1-s)\psi^{\prime\prime}_{x_n x_n}(x^\prime,sx_n,0,\pm1)ds\right]\xi_n,\; \xi_n\gtrless 0.
+\left(\frac{F[p,q](i\mu_{0}+1)w_{1}[p,q](i\mu_{0}+1)w_{2}[p,q](i\mu_{0}+1)}{w_{3}[p,q](i\mu_{0}+1)}\right)^{-\frac{1}{2}}\gamma^{3}\partial_{\psi}u_{n}(\mu)\mid_{\mu=\mu_{0}-i}
(\widehat{\psi}^\ast \widetilde{\sigma})_e(x,y) =\widetilde{\sigma}_{\widehat{\psi}(e)}(\widehat{\psi}_{\ast,e}(x),\widehat{\psi}_{\ast,e}(y)) &=\widetilde{\sigma}_e(\psi(x),\psi(y)) &=\sigma(\psi(x),\psi(y)) &=(\psi^\ast\sigma)(x,y) &=\beta(x,y).