harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\rm ch}({\rm gr}W_{-n})(x,q^2,q^2)=\sum_{k+s\le n}q^{k(k-1)+2k+2s}x^{-n+k+2s}\left(\begin{array}{c}n k,s,n-k-s \end{array}\right)_{q^2},{\rm ch}({\rm gr}W^\sigma_{-n})(x,q,q)=\sum_{k+s\le n}q^{k^2+s}x^{-n+k+2s}\left(\begin{array}{c}n k,s,n-k-s \end{array}\right)_{q^2}.
-\left\langle \phi,f'(\phi)\phi_{k}+ \mathcal{K}*\phi_{k}- k\mathcal{K}_{1}*\phi_{k}' - \mathcal{K}_{1}*\phi' + \frac{1}{2}k\mathcal{K}_{2}*\phi'' \right\rangle = \langle\Psi_{3}^{0},(\mathcal{A}_{1}- c)\phi_{k}+ \mathcal{A}_{2}\Phi_{1}^{0}\rangle - \langle\phi,\mathcal{K}_{1}*\phi'\rangle.
0 \le \dfrac{\varphi(t)}{t-t_1}= \dfrac{1}{t-t_1}\int_{t_1}^t{\varphi'(s) \, ds}& \le \dfrac{1}{t-t_1}\int_{t_1}^t{p(s) \psi(s,\varphi(s)) \, ds}& \le \dfrac{\max_{t_1 \le s \le t}\psi(s,\varphi(s))}{t-t_1}\int_{t_1}^{t_2}{p(s) \, ds},
\tilde{L}({\rm D}({\rm reg}_{0}(w))) =\sum_{N\geq2}\zeta^{\mathfrak{m}}(N)\tilde{L}({\rm Cut}_{(\emptyset;\{1\}^{N-1})}({\rm reg}_{0}(w))) =\sum_{N\geq2}\zeta^{\mathfrak{m}}(N)\tilde{L}({\rm reg}_{0}({\rm Cut}_{(\emptyset;\{1\}^{N-1})}(w))) =\sum_{N\geq2}\zeta^{\mathfrak{m}}(N)\tilde{L}({\rm Cut}_{(\emptyset;\{1\}^{N-1})}(w)),
\gamma_{k}= \left( \begin{matrix}{{0}}&{{- i \sigma_{k}}}\\{{i \sigma_{k}}}&{{0}}\\ \end{matrix}\right) \qquad \gamma_{4}= \left( \begin{matrix}{{0}}&{{I}}\\{{I}}&{{0}}\\ \end{matrix}\right) \qquad \gamma_{5}= \left( \begin{matrix}{{I}}&{{0}}\\{{0}}&{{I}}\\ \end{matrix}\right)
\omega={\mathrm{curl}\,}\boldsymbol u={\mathrm{curl}\,}( (\boldsymbol u\cdot\boldsymbol\tau)\boldsymbol\tau + (\boldsymbol u\cdot\boldsymbol n)\boldsymbol n)=\frac{\partial(\boldsymbol u\cdot\boldsymbol n)}{\partial \boldsymbol\tau}- \frac{\partial(\boldsymbol u\cdot\boldsymbol \tau)}{\partial\boldsymbol n}=0,\quad\partial\Omega.
E_N^{(\beta)}\le\; 4\pi \mathfrak{a}_N^{(\beta)}(N-1) &-\frac{1}{2}\sum_{p\in \Lambda_+^*}\left[ p^2+\widehat V(0)-\sqrt{|p|^4+2p^2 \widehat{V}(0)}-\frac{\widehat{V}^2(0)}{2p^2}\right]+ CN^{-\alpha},
T_\lambda^* T_\lambda(f) = T_\lambda^* ( f_\lambda \cdot (f \circ \tau^{d(\lambda)})) = \frac{\chi_{D_\lambda}\cdot ( (f_\lambda \cdot (f \circ \tau^{d(\lambda)})) \circ \tau_\lambda)}{f_\lambda \circ \tau_\lambda}=\chi_{D_\lambda}f = T_v (f),
\begin{array}{l l l}{\left( E_{4 s}- 2 Q - 2 Q^{- 1}\right) A ( k_{1}, k_{2})}&{=}&{A ( k_{1}- 1 , k_{2}) + A ( k_{1}+ 1 , k_{2})}\\{}&{}&{}\\{}&{+}&{A ( k_{1}, k_{2}- 1 ) + A ( k_{1}, k_{2}+ 1 )}\\ \end{array}
\pi(\rho|r)=\frac{(1-r^{2})^{\frac{\nu-1}{2}}\cdot(1-\rho^{2})^{\frac{\nu-2}{2}}}{\pi(\nu-2)!}\partial_{\rho r}^{\nu-2}\{\frac{\theta-\frac{1}{2}sin2\theta}{sin^{3}\theta}\}
\lim_{t \rightarrow 0}{\cal K}[H,\sqrt{g},\sqrt{g}](\xi ~ \eta ~\phi ~ t; ~\xi_0 ~ \eta_0 ~ \phi_0 ~ 0) = \frac{1}{\xi \eta \left(\xi^2 + \eta^2 \right)}\delta (\xi -\xi_0)\delta(\eta -\eta_0)\delta (\phi -\phi_0)
\overline\square_B\phi = \nabla_T^*\nabla_T\phi + \sum_{a,b}\bar\omega^a\wedge \bar V_b\lrcorner R^Q(V_b,\bar V_a)\phi+\sum_a \bar\omega^a\wedge (\nabla_{\bar V_a}H^{0,1})\lrcorner\,\phi, \square_B\phi = \bar\nabla_T^*\bar\nabla_T\phi + \sum_{a,b}\omega^a\wedge V_b\lrcorner R^Q(\bar V_b ,V_a)\phi+\sum_a\omega^a\wedge (\nabla_{V_a}H^{1,0})\lrcorner\,\phi.
\tilde{L}=\tilde{L}(\dot{Q}_{(d-1)}^{i},\cdots,\dot{Q}_{(1)}^{i},\dot{Q}_{(0)}^{i};Q_{(d-1)}^{i},\cdots,Q_{(1)}^{i},Q_{(0)}^{i};\dot{\lambda}^{(\alpha)}_{i};\lambda^{(\alpha)}_{i}).
\begin{aligned}\int_0^T \int_{B_{2R}^d}\int_{\mathbb{S}^{d-1}}\mathbf{1}_{(\tau,v_{s+k+1},\omega_{k+1}) \in \mathcal{B}_{V}^+}d\omega_{k+1}dv_{s+k+1}d\tau \leq C_{d}T \eta^{d}\end{aligned}
\begin{aligned}z^{\gamma(y) \gamma(x)}= \Delta(z^{\gamma(y)}K , x K) z^{\gamma(y)}&= \Delta(\Delta(z K, y K) z K, x K) \Delta(z K, y K) z &= \Delta(z K, x K) \Delta(z K, y K) z &= \Delta(z K, x y K) z &= z^{\gamma(x y)}, \end{aligned}
0=\lim_{n \to \infty}\sqrt{\frac{2}{\log(e)}E^{\mu}[D(\pi^{\mu}_{n}\|\pi_{n}^{\nu})]}&\geq \lim_{n \to \infty}E^{\mu}\left[\sqrt{\frac{2}{\log(e)}D(\pi^{\mu}_{n}\|\pi_{n}^{\nu})}\right]
U^{-1}(z_1,\dots,z_{N_p}) = P \exp\left[-{1\over 2\pi\kappa}\int_\Gamma\sum_\alpha dz^\prime_\alpha\sum_{\beta\not=\alpha}\hat Q^a_\alpha \hat Q^a_\beta\frac{1}{z^\prime_\alpha -z^\prime_\beta}\right]
\sum_{\lambda \vdash n}\epsilon^\lambda(G)f_\lambda = \sum_{\lambda \vdash n}\eta^\lambda(G)m_\lambda = \sum_{\lambda \vdash n}\frac{\psi^\lambda(G)}{z_\lambda}p_\lambda = \sum_{\lambda \vdash n}\chi^\lambda(G)s_\lambda = \sum_{\lambda \vdash n}\phi^\lambda(G)h_\lambda = \sum_{\lambda \vdash n}\gamma^\lambda(G)e_\lambda.
\begin{array}{l l l}{{X_{i}= \textstyle{\frac{\mu}{3}}J_{i}\, ,}}&{{[ J_{i}, J_{j}] = i \epsilon_{i j k}J_{k}\, ,}}&{{A_{0}= 0 \, ,}}\\{{{}}}&{{{}}}&{{{}}}\\{{Z_{1}( t ) = e^{- i \frac{\mu}{6}t}z_{1}{1}\, ,}}&{{Z_{2}( t ) = e^{- i \frac{\mu}{6}t}z_{2}{1}\, ,}}&{{Z_{3}( t ) = e^{- i \frac{\mu}{6}t}z_{3}{1}\, ,}}\\ \end{array}
|g(t,x)-g(t,y)| =|\psi(t,u_f(t,x))-\psi(t,u_f(t,y))| &\le L |u_f(t,x)-u_f(t,y)| & \le L \sup_{t \in [s,\tau]}\|\,|\nabla_x u_f(t,\cdot)|\, \|_{L^\infty(B_R)}|x-y|,
M\big( S\big(\phi^{(0)}\prec \phi^{(1)}\prec \cdots\prec \phi^{(N-1)}\prec \phi^{(N)}\big) \big) \qquad{}=\Lambda^{N}_{N-1}\big(\phi^{(N)}, \phi^{(N-1)}\big) \cdots \Lambda^{1}_0\big(\phi^{(1)}, \phi^{(0)}\big) M_N\big(\phi^{(N)}\big),
\widehat{P}^{(n)}_{q}= ( I\otimes \widehat{P}^{(n-1)}_{q}) \widehat{R}^{(n)}_{q}= (\widehat{R}^{(n)}_{q})^* ( I\otimes \widehat{P}^{(n-1)}_{q}), n\geq 2, &\widehat{P}^{(1)}_{q}= \widehat{R}^{(1)}_{q},
\sum_{f\in \mathcal{H}^*_k(N)}\omega_f\lambda^z_{{\rm sym}^m f}(n)=\lambda_{{\rm sym}^m}^z (n)+O_m\left(k^{-5/6}N^{-1+\varepsilon}n^{m/4}\log (2n)r^z_m(n)\right),
[T; N+1, c]^+ (q^\mu T_{N+1}T_{N+2}\dots T_{N+\mu}) =(q^\mu T_{N+1}T_{N+2}\dots T_{N+\mu}) [T; N,c]^+, &[T; N-1,c]^- (q^{\mu}T_{N-1}T_{N-2}\dots T_{N-\mu}) = (q^{\mu}T_{N-1}T_{N-2}\dots T_{N-\mu}) [ T;N,c]^-.
(zQ \partial_Q)^{N+1}Q^{\frac{H}{z}}\frac{Q^d}{\prod_{r=1}^d \left( H + rz\right)^{N+1}}= (H + dz)^{N+1}Q^{\frac{H}{z}}\frac{Q^d}{\prod_{r=1}^d \left( H + rz\right)^{N+1}}= Q^{\frac{H}{z}}\frac{Q^d}{\prod_{r=1}^{d-1}\left( H + rz\right)^{N+1}}
\begin{aligned}\mathbf{\hat h}^{LoS}(\varphi) = [1,e^{j\frac{2\pi d}{\lambda}\sin\varphi},\cdots,e^{j\frac{2\pi d}{\lambda}l\sin\varphi}, &\cdots,e^{j\frac{2\pi d}{\lambda}(L-1)\sin\varphi}]^T,\end{aligned}
\mathcal{A}(a^{\dagger}_+ a_+ + a^{\dagger}_- a_-) + \mathcal{B}(a_+^{\dagger}a^{\dagger}_- + a_+ a_-) + \kappa (a^{\dagger}_+ + a_-) + \overline{\kappa}(a_+ + a_-^{\dagger}) &= \mathcal D ( b^\dagger_+ b_+ + b^\dagger_- b_- ) -\frac{1}{2}\alpha\mathcal{B}([a_+,a^{\dagger}_+] + [a_-, a^{\dagger}_-]) - \frac{2|\kappa|^2}{\mathcal{A}+\mathcal{B}},
(-1)^{\ell(\Lambda^{T_m})\binom{m}{2}}p_{\Lambda^{T_m}_1}^{\{T_m \}}(x,\theta) p_{\Lambda^{T_m}_1}^{\{T_m \}}(y,\phi)p_{\Lambda^{T_m}_2}^{\{T_m \}}(x,\theta) p_{\Lambda^{T_m}_2}^{\{T_m \}}(y,\phi)\cdots p_{\Lambda^{T_m}_{\ell(\Lambda^{T_m})}}^{\{T_m \}}(x,\theta) p_{\Lambda^{T_m}_{\ell(\Lambda^{T_m})}}^{\{T_m \}}(y,\phi).
G=\begin{bmatrix}B 0 0 0 0 B 0 0 0 0 B 0 0 0 0 B\end{bmatrix}\begin{bmatrix}H_{11}H_{12}H_{13}H_{14}H_{21}H_{22}H_{23}H_{24}H_{31}H_{32}H_{33}H_{34}H_{41}H_{42}H_{43}H_{44}\end{bmatrix}\begin{bmatrix}B^\top 0 0 0 0 B^\top 0 0 0 0 B^\top 0 0 0 0 B^\top\end{bmatrix}.
\left\|\int_0^1 T_{-1}(s)Bu(s)\,ds\right\|^{2}= \sum_{n=1}^\infty \left|\int_0^1 e^{\lambda_n s}\lambda_n (u(s))_n\, ds \right|^2 &= \sum_{m=1}^\infty \left( e^{-1/2}-e^{-1}\right)^2 &=\infty,
d s^{2}= \frac{s i n^{2}{\theta}}{\rho^{2}}\left[ \left( r^{2}+ a^{2}\right) d \phi - a \, d t \right]^{2}- \frac{D}{\rho^{2}}\left[ d t - a \sin^{2}{\theta}\, d \phi \right]^{2}+ \frac{\rho^{2}}{D}\, d r^{2}+{\rho^{2}}d \theta^{2}\; ,
\forall x\in \Theta: \mathbb{E}\sum_{t=1}^{T}\big(f(x_t) - f(x)\big) \le \sum_{t=1}^{T}\Big[ \Big( L + \frac{9L^2d^2}{4\alpha t}\Big)h_t^2 + \frac{3 d^2 \sigma^2}{4h_t^2\alpha t}+ \frac{9G^2d}{2\alpha t}\Big].
{\boldsymbol N}:= \left(\begin{array}{cc}\gamma_n 0 \smallskip 0 \gamma_n^* \end{array}\right) \widetilde{\boldsymbol N}, \widetilde{\boldsymbol N}:=\left(\begin{array}{cc}\Psi_n \Psi_n^*/w_\Delta \smallskip \Psi_{n-1}\Psi_{n-1}^*/w_\Delta \end{array}\right).
\delta(G_4) \ge D_4 \stackrel{(\ref{eq:phi0rob})}{=}D_1 - 2 \phi_0^{\rm rob}n \stackrel{(\ref{eqD1})}{=}D - (\phi_0 + 2 \phi^{\rm rob}_0 )n \ge (1 - 6 \phi^{\rm rob}_0) n/2
\begin{cases}D_{0+}^{\frac{1}{2},\frac{1}{2}}x(t)=t^{-\frac{1}{3}}[1+t(x(t))^{\frac{4}{3}}],t>{0}, &\lim_{t\to{0}}{t^{\frac{1}{4}}}x(t)=3.\end{cases}
f_\S(x) = \left(1 + xq + xq^2 + xq^3\right) f_\S\left(xq^3\right)- \left(x^2 q^6 + x^2 q^7 + x^2 q^8\right) f_\S\left(xq^6\right) & + \left(x^3 q^{15}- x^2 q^9\right) f_\S\left(xq^9\right).
2\alpha \int_{G_0^+}\frac{x_1}{|x|^{2+2\alpha}}dx = \int_0^1 \frac{dx_2}{(x_2^2+2^2)^\alpha}+ \int_0^1 \frac{2^{-\alpha}}{((x_2+1)^2+1)^\alpha}dx_2 = 2^{1-2\alpha}f \left( \frac 12 \right) + 2^{-\alpha}(f(2)-f(1)).
\lim_{t \to 0^{+}}\frac{1}{t^{n+p}}\int_A \int_A{|f(x) - f(y)|}^p \cdot J \bigg( \frac{x-y}{t}\bigg) \dxdy = K(n,p) \cdot{\big({[ f]}_{W^{1,p}(A)}\big)}^p,
\begin{array}{l c r}{\tau^{2}+ k a^{2}= t_{+}^{2}}&{\qquad \mathrm{f o r}\qquad}&{k = \pm 1}\\{a = \sqrt{2 | t_{+}| \tau}}&{\qquad \mathrm{f o r}\qquad}&{k = 0 \ .}\\ \end{array}
P_N(\gamma_1,\gamma_2) = 2 \int_{0<s_1<s_2<1}ds_1ds_2 \int_\mathbb{R}[s_1-x_1]_N^{\gamma_1}[s_2-x_1]_N^{\gamma_1}dx_1 \int_{\mathbb{R}}[s_1-x_2]_N^{\gamma_2}[s_2-x_2]_N^{\gamma_2}dx_2,
\begin{array}{rcl}\mathcal A_1 = \{A_{1,1}, A_{1,2}, \dots, A_{1,e_1}\}\mathcal A_2 = \{A_{2,1}, A_{2,2}, \dots, A_{2,e_2}\}& \vdots \mathcal A_s = \{A_{s,1}, A_{s,2}, \dots, A_{s,e_s}\}\end{array}
\varphi ( x ) = \int \frac{d^{D}\mathbf{k}}{( 2 \pi )^{D}}A ( \mathbf{k}) e^{i k x}, \pi ( x ) = \int \frac{d^{D}\mathbf{k}}{( 2 \pi )^{D}}A^{\dagger}( \mathbf{k}) e^{- i k x},
\begin{aligned}\mathcal{G}_{1}:= \{a_{20}y+b_{20}x+1, r_{11}x y+r_{20}x^2-r_{20}y^2+a_{20}x-b_{20}y, a_{20}r_{11}y^2+a_{20}r_{20}x y +b_{20}r_{20}y^2+a_{20}^2 y+b_{20}^2 y+r_{11}y+r_{20}x+a_{20}\}.\end{aligned}
\tilde{\Theta}_\pm D_aA\varphi =\tilde{\Theta}_\pm D_a\gamma^{(2n+1)}(P_{\rm F}-D_aP_{\rm F}D_a)\varphi &=-\tilde{\Theta}_\pm \gamma^{(2n+1)}D_a(P_{\rm F}-D_aP_{\rm F}D_a)\varphi &=\tilde{\Theta}_\pm AD_a\varphi=-A\tilde{\Theta}_\pm D_a\varphi.
{\sum_J}' \sum_{\alpha}(-t)^{||\alpha||+\|\beta\|+n}\frac{(-1)^{s(J)}}{\beta(\alpha, J)!\cdot a_J^{\beta+I}}\cdot \frac{\partial^{||\beta(\alpha,J)||}}{\partial z^{\beta(\alpha, J)}}\left[\frac{\Delta(t)}{z_1^{\gamma_1+1}\cdot \ldots \cdot z_n^{\gamma_n+1}}\cdot \frac{Q^{\alpha}}{q^{\alpha+I}(J)}\right]_{z=\tilde a_J}.
\begin{array}{l}[J_1^\prime,J_2^\prime]_{GPB}=-\displaystyle\displaystyle\frac{\sinh 2 \gamma J_3^\prime}{2\sinh\gamma}~, begin{align*}4mm][J_2^\prime,J_3^\prime]_{GPB}=-J_1^\prime~,{}~~~~~~[J_3^\prime,J_1^\prime]_{GPB}=-J_2^\prime~;\end{array}
\widehat x = (x_1, \ldots, x_{d+1}), \widehat v = (v_1, \ldots, v_{d+1}), \widehat z = (t, \hat x, \hat v), & \widehat P (\widehat z) = \partial_t - \sum_{i = 1}^{d+1}v_i D_{x_i}- \sum_{i, j = 1}^{d}a^{i j}(z) D_{v_i v_j}- D_{v_{d+1}v_{d+1}}.
\begin{aligned}\mathcal{S}^n_t (\varphi ) = \frac{g^\prime (0)}{2\sqrt{n}}\int_0^t \sum_{j \in \mathbb{Z}}\eta^n_j (s) \Delta^n \varphi^n_j (s) ds + \frac{1}{2\sqrt{n}}\int^t_0 \sum_{j \in \mathbb{Z}}[g_n (\eta^n_j (s) ) - g^\prime(0) \eta^n_j (s) ] \Delta^n \varphi^n_j (s) ds .\end{aligned}
a(x) = \big|\{m<0:f(m,\sigma(x))\ge -n(x)\}\big| - \big|\{n\ge 0:f(n,\sigma(x))<-n(x)\}\big| - 1 = \big|\{m<0:f(m,\sigma(x))\ge 0\}\big| - \big|\{n\ge 0:f(n,\sigma(x))<0\}\big| + n(x) - 1 = a(\sigma(x)) + n(x) - 1,
\frac{\partial}{\partial q_{t A}}= \frac{\partial r}{\partial q_{t A}}\frac{\partial}{\partial r}+ \frac{\partial e_{B}}{\partial q_{t A}}\frac{\partial}{\partial e_{B}}+ \frac{\partial E_{s}}{\partial q_{t A}}\frac{\partial}{\partial E_{s}}+ \frac{\partial y_{s B}}{\partial q_{t A}}\frac{\partial}{\partial y_{s B}}\; .
B_{0}^{o}= \left(\begin{tabular}{cc}$c$ $0$ $c_{0}$ $c$\end{tabular}\right) ,\B_{\infty}= \left(\begin{tabular}{cc}$ B^{\infty}_{11}$ $B^{\infty}_{12}$ $B^{\infty}_{21}$ $B^{\infty}_{22}$\end{tabular}\right)
{\cal B}_{\Sigma}= \displaystyle{\int}{d}^{3}x \displaystyle{s u m_{\Phi = A_{\mu}^{a}, c^{a}, \Psi_{_A}, \varphi_{i}}^{}}\left({\displaystyle{\ \frac{\delta \Sigma}{\delta \Phi^{*}}}}{\displaystyle{\frac{\delta}{\delta \Phi}}}+{\displaystyle{\frac{\delta \Sigma}{\delta \Phi}}}{\displaystyle{\ \frac{\delta}{\delta \Phi^{*}}}}\right) + b .
\Phi_{0,1}^{(3,0)}= a_{11}^{*^{2}}V^{(3,0)}\hspace{20mm}\Phi_{1,1}^{(3,0)}= ( a_{11}^{*^{2}}a_{22}^{*}- a_{12}^{*}a_{11}^{*}) V^{(3,0)}
h = \left( \begin{array}{c c}{{1}}&{{0}}\\{{0}}&{{- 1}}\\ \end{array}\right) , \x = \left( \begin{array}{c c}{{0}}&{{1}}\\{{0}}&{{0}}\\ \end{array}\right) , \y = \left( \begin{array}{c c}{{0}}&{{0}}\\{{1}}&{{0}}\\ \end{array}\right)
\lambda^*(\widetilde\vartheta^t_\Gamma)=\int_{\underline C_{k+n}/\underline C_k}\lambda^*\!\left(\widetilde\theta^t_\Gamma\right)=\widetilde\vartheta^t_\Gamma+\log(\lambda)\int_{\underline C_{k+n}/\underline C_k}\alpha^t_\Gamma,
||\partial^{\alpha}u||_{L^p(\mathcal{D})}= \sup_{\varphi \in C_c^{\infty}(\mathcal{D})\setminus\{0\}}\bigg|\int_{\mathcal{D}}u(x)\frac{\partial^{\alpha}\varphi(x)}{||\varphi||_{L^q}}dx \bigg|
C_{1,k}= c_1 \Big( e^{\frac{16w_{k+1}}{3 R}}w_{k+1}^{3+\frac{4}{R\log t_0}}- e^{\frac{16w_k}{3 R}}w_k^{3+\frac{4}{R\log t_0}}\Big) ,\C_{2,k}= c_2 \big(w_{k+1}^2 - w_{k}^2\big).
h = \frac{1}{Q^{3}}\left( \begin{array}{c}{\begin{array}{c c}{- 1 -{y^{\prime}}^{2}}{y^{\prime}{\phi}^{\prime}}{y^{\prime}{\phi}^{\prime}}{1 -{{\phi}^{\prime}}^{2}}\end{array}}\end{array}\right) , \; \;{\hat{H}}^{\dagger}={\hat{H}},
\sigma_{l}= \frac{2 \pi^{\frac{d - 1}{2}}\Gamma \left( l + d - 2 \right) \left( l + d / 2 - 1 \right)}{\omega^{d - 1}\Gamma \left( d / 2 - 1 / 2 \right) \Gamma \left( l + 1 \right)}P ( l ) \, ,
\Delta_0\frac{d}{d\epsilon}\vert_{\epsilon=0}(\overline{\chi_2^{\epsilon(\mu\oplus\nu)}})^T\chi_2^{\epsilon(\mu\oplus\nu)}= -\bar\mu \bar\partial\bar\partial\overline{\chi_+^{0(\mu\oplus\nu)}\circ\chi_1^{0(\mu\oplus\nu)}}^T &-\mu\partial\partial{\chi_+^{0(\mu\oplus\nu)}\circ\chi_1^{0(\mu\oplus\nu)}}=0
\beta_1^*=\mathcal{F}\Big(b_1-\dfrac{\partial z_{g_1}}{\partial t},l_1^{*},l_1^{**}\Big),~\beta_2^*=\mathcal{F}\Big(b_2-\dfrac{\partial z_{g_2}}{\partial t},l_2^{*},l_2^{**}\Big),~~a.s.
\frac{d\Gamma_{0}}{ds}= [1+y_{x}^{2}+z_{x}^{2}]^{-1/2}\langle 1,y'(x),z'(x)\rangle \frac{d^{2}\Gamma_{0}}{ds^{2}}= [1+y_{x}^{2}+z_{x}^{2}]^{-1/2}(\frac{d}{dx})(\frac{d\Gamma_{0}}{ds}) \frac{d^{2}\Gamma_{0}}{ds^{2}}= [1+y_{x}^{2}+z_{x}^{2}]^{-2}\{-(y''y'+z''z')\langle 1, y',z'\rangle+[1+y_{x}^{2}+z_{x}^{2}]\langle 0,y'',z''\rangle\}
u_{\beta}^{(2)}\left( 0,x\right) +\Pi_{0,x}\int_{0}^{\tau}ds\psi_{\beta}^{(2)}\left( s,\xi_{s};u_{\beta}^{(2)}\right) =\Pi_{0,x}f_{\beta}^{(2)}\left( \tau ,\xi_{\tau}\right)
n \alpha \hbar = \frac{1}{2 \pi}\oint_{C_1}\frac{p dy}{1-y^2}= \frac{1}{2\pi}\left( \oint_{\gamma_0}\frac{p dz}{1-z^2}- \oint_{\gamma_1}\frac{p dy}{1-y^2}- \oint_{\gamma_2}\frac{p dy}{1-y^2}\right).
\frac{2}{e^t+1}= \sum_{n=0}^\infty C_n \frac{1}{4^n}\big( 1- e^{2t}\big)^n &= \sum_{n=0}^\infty C_n \frac{(-1)^n}{4^n}\big( e^{2t}-1 \big)^n &= \sum_{n=0}^\infty C_n \frac{(-1)^n}{4^n}n! \sum_{m=n}^\infty S_2(m,n) \frac{2^m t^m}{m!}&= \sum_{m=0}^\infty \left( \sum_{n=0}^m C_n (-1)^n 2^{m-2n}n! S_2(m,n) \right) \frac{t^m}{m!}.
\|[\partial_0 A_0, A_j] \|_{\ell^1 L^2 \dot H^{-\frac12}}\lesssim \| \partial_0 A_0\|_{L^2 \dot H^\frac12}\| A_j\|_{\ell^2 L^\infty \dot H^1}\|[A_0, \partial_0 A_j] \|_{\ell^1 L^2 \dot H^{-\frac12}}\lesssim \| A_0\|_{L^2 \dot H^\frac32}\| \partial_0 A_j\|_{\ell^2 L^\infty L^2}
L=\limsup_{R\to \infty}\,\sup_{t\in \mathbb{R}}\,\frac{1}{R}\,\int_{\prod_{k=1}^s [t, t+N_k(R)\,\,a_k^{-1})}\,\big|Q(\xi)\big|^2\,d\nu_{\mu}(\xi) &=\limsup_{R\to \infty}\,\sup_{t\in \mathbb{R}}\,\frac{1}{R}\,\sum_{0\le k_1\le a_1 R-1}\,\dots\, \sum_{0\le k_s\le a_s R-1}\,\int_{I(t,k_1,\dots,k_s)}\,\big|Q(\xi)\big|^2\,d\nu_{\mu}(\xi).
\tau_n := \begin{cases}n^k R_n^d f(R_ne_1)^k nf(R_ne_1) \to 0\,, R_n^d nf(R_ne_1) \to \xi \in (0,\infty)\,, n^{2k-1}R_n^d f(R_ne_1)^{2k-1}nf(R_ne_1) \to \infty\,.\end{cases}
\tilde{e}_{\mu, \nu;\mu',\nu'}^{\tau,i}(t,\xi,\eta, \sigma) = \frac{{e}_{\mu, \nu;\mu',\nu'}^{\tau,i}(t,\xi,\eta, \sigma) \theta(\xi-\eta, \min\{|\sigma|,|\eta-\sigma|,|\xi|\})\psi_{\geq 0}(\xi(1+t)^{5/6+5p_0})}{\mu |\xi|^{1/2}-\nu|\xi-\eta|^{1/2}- \mu' |\eta-\sigma|^{1/2}-\nu'|\sigma|^{1/2}}.
\begin{aligned}E(A,\Pi_h\xi,\Pi_h \zeta) \leq C h^2( \|A\|_{2,\infty}\|\Pi_h \xi\|_0 \|\Pi_h\zeta\|_0 + \|A\|_{1,\infty}(\|\xi\|_1\|\Pi_h\zeta\|_0 + \|\Pi_h\xi\|_0\|\zeta\|_1) + \|A\|_0\|\xi\|_1 \|\zeta\|_1). \end{aligned}
|\mathcal{X}_E(x)-\mathcal{X}_E*\vartheta_{\tilde{\eta}}(x)| = \tilde{\eta}\left | \int_{x-E_U}^{x-E^-}\frac{1}{\pi(y^2+\tilde{\eta}^2)}dy\right | =\frac{\tilde{\eta}}{\pi}\left[\int_{x-E_U}^{x-E^-}\frac{1}{(y+\tilde{\eta})^2}+\frac{2 \tilde{\eta}y}{(y^2+\tilde{\eta}^2)(y+\tilde{\eta})^2}dy\right] & \leq C \tilde{\eta}\frac{E_U-E^-}{d_U(x)d(x)}.
I_+(f) := \lim_{y\to\infty}2\pi y f(iy) = \lim_{y\to\infty}\bigl\langle \chi_y, f \bigr\rangle= \lim_{\xi\downarrow 0}\sqrt{{2\pi}}\widehat f(\xi) \ \ \chi_y(x) = \tfrac{iy}{x+iy}.
\sum_{n=k_{tot}}^\infty \frac{\Gamma(- p-x+2m-n-1)\Gamma( q-x)}{\Gamma(-p-x+2m)\Gamma( q-x-n+k_{tot})}\frac{\Gamma(n+1)}{\Gamma(n-k_{tot}+1)}=-(-1)^{k_{tot}}k_{tot}!\frac{\Gamma (p+q-2 m)}{\Gamma (p+q-2m+k_{tot}+1)}
~~~~~~~~~~~~~~~~~~~~~~~~d\o_i=\sum_{j=1}^4\o_{ij}\wedge\o_j, ~~~1\leq i\leq 4; &d\o_{ij}-\sum_{k=1}^{4}\o_{ik}\wedge\o_{kj}+\sum_{1\leq k<l\leq4}K_{ijkl}\o_k\wedge\o_l=0,~~~1\leq i,j\leq 4,
\begin{aligned}I_1 =\int\sum_{l=0}^{k-1}C_{k-1}^l \nabla^{l}u \nabla^{k-l}n\ \nabla^{k+1}n \dx &\lesssim \sum_{l=0}^{k-1}\|\nabla^{l}u\|_{L^3}\|\nabla^{k-l}n\|_{L^6}\|\nabla^{k+1}n \|_{L^2}.\end{aligned}
\hat{R}d K_{1}^{\dagger}\hat{R}K_{1}= K_{1}\hat{R}d K_{1}^{\dagger}\hat{R}^{- 1}, \quad \hat{R}d K_{1}^{\dagger}\hat{R}d K_{1}^{\dagger}= - d K_{1}^{\dagger}\hat{R}d K_{1}^{\dagger}\hat{R}^{- 1}.
[d,[x_{1}^{+}\otimes a_{1}^{-},x_{2}^{+}\otimes a_{2}^{-}]]=[[d,x_{1}^{+}\otimes a_{1}^{-}],x\otimes a_{2}^{-}]+[x_{1}^{+}\otimes a_{1}^{-},[d,x_{2}^{+}\otimes a_{2}^{-}]]
M_i\leq \sum_{T}\binom{C(T)}{d_i}\leq\sum_T\binom{\frac{(2^{\ell}+1)n}{d_i^{\ell-1}}}{d_i}\leq \sum_T 2^{\frac{\ell}{1.88}\left((2^{\ell}+1)en\right)^{1/\ell}}\leq \sum_T 2^{3\ell n^{1/\ell}}\leq 2^{\left(3\ell+1\right)n^{1/\ell}}
n=q^{2}\frac{(1+q^{-2}\bar{f}_{1})(f_{0}+q^{-2}\bar{f}_{1})}{(1+q^{2}\bar{f}_{1})(f_{0}+q^{2}\bar{f}_{1})}+q^{-2}\frac{(1+q^{2}\bar{f}_{1})(f_{0}+q^{2}\bar{f}_{1})}{(1+q^{-2}\bar{f}_{1})(f_{0}+q^{-2}\bar{f}_{1})}\;.
w_i^n=\delta_x^2 u_i^n-\frac{h^2}{12}\delta_x^2 w_i^n,1\leq i \leq M-1,0\leq n\leq N, u_i^0=\varphi_1 (x_i),v_i^0=\varphi_2 (x_i), 0\leq i \leq M, u_0^n=u_M^n=w_0^n=w_M^n=0,0\leq n\leq N.
E(\eta ,\vartheta )=-\frac{M}{2\cos \frac{\pi}{2\lambda}}\left( \cos \left( \frac{\eta}{\lambda}\right) +\cosh \left( \frac{\vartheta}{\lambda}\right) -\frac{1}{2}\cos \left( \frac{\pi}{2\lambda}\right) +\frac{1}{2}\sin \left( \frac{\pi}{2\lambda}\right) -\frac{1}{2}\right)\,.
\frac{1}{\|u'_j\|^2}+(c^2b_1^ha_j^v)^2\!\!\!\!\sum_{i=j+1}^{m_1+m_2}\!\!\!\frac{(a_i^v)^2}{(1-c^2b_1^hA_i^v)(1-c^2b_1^hA_{i+1}^v)}=\frac{1-c^2b_1^h(b_2^v-(a_j^v)^2)}{1-c^2b_1^hb_2^v}.
\sum_{k=1}^{\infty}\left[ (\sigma_k^{\delta(m)}\wedge T) - (\tau_{k-1}^{\delta(m)}\wedge T) \right]=\int_0^T \sum_{k=1}^{\infty}1\{t \in [\tau_{k-1}^{\delta(m)},\sigma_{k}^{\delta(m)}]\}dt \leq \int_0^T 1\{Z(t) \in S \backslash S_{2 \delta (m)}\}dt,
a_l^j(f^{(d)})=\left(\prod_{\iota=0}^{j-1}\beta_{l,\iota}\right)\cdot p_{d,j}\,(\beta_{l,0}^2,\beta_{l,1}^2,\dots,\beta_{l,\frac{d-j}{2}}^2)\cdot a_l^j(f)
\psi_{M}( \rho_{M}) = c_{1}( p ) \sqrt{\rho_{M}}e^{- \pi p / 2}H_{i p}^{( 1 )}( \omega \rho_{M}) + c_{2}( p ) \sqrt{\rho_{M}}e^{\pi p / 2}H_{i p}^{( 2 )}( \omega \rho_{M}) ,
a\star_{\vec{b}}^{n}b =\frac{n-1}{n}ab+\frac{i\varepsilon_{abc}}{n}c a\star_{\vec{b}}^{n}a =\frac{n-1}{n}a^{2}+\frac{1}{n}a\star_{\vec{b}}^{n}a+b\star_{\vec{b}}^{n}b+c\star_{\vec{b}}^{n}c = \frac{n+2}{n}= \frac{j+1}{j}
\xi_{t_n}^+ = \xi_{t_n}+ K_n \begin{pmatrix}\xi_{t_n}\times d^0_n + V_n^0 \xi_{t_n}\times d^1_n + V_n^1 \end{pmatrix}= \left( I_3-K_n \begin{pmatrix}H_n^0 H_n^1 \end{pmatrix}\right) \xi_{t_n}+K_n \begin{pmatrix}V_n^0 V_n^1 \end{pmatrix}.
z^\top \tilde{E}z = \sum_{\ell=1}^k \gamma_{\ell}z^\top ( S(\mu_{\ell}) \otimes{E}'(\mu_{\ell}) ) z = \sum_{\ell=1}^k \gamma_{\ell}\sum_{i,j=1}^m \Phi_i(\mu_{\ell}) \Phi_j(\mu_{\ell}) z_i^\top{E}'(\mu_{\ell}) z_j = \sum_{\ell=1}^k \gamma_{\ell}v_{\ell}^\top{E}'(\mu_{\ell}) v_{\ell}.
\frac{3}{8 \pi^{2}}\omega_{i}^{( 4 )}+ \frac{1}{8 \pi^{2}}\sum \Omega_{i j k}^{( 2 )}\omega_{i j k}^{( 2 )}+ \frac{1}{4 \pi^{2}}\sum \Omega_{i j k l m}^{( 4 )}= \frac{1}{2},
\mathbb{E}\left[\ln\gamma_o\right]= \underbrace{\mathbb{E}\left[\ln\left({{P_t}G_\mathrm{M}^2\beta{r^{-{\alpha_o}}}}\right)\right]}_{Z_1}+ &\underbrace{\mathbb{E}\left[\ln\left(\frac{1}{\sum\nolimits_{i \in \Phi /{o}}{{P_t}{G_i}\beta{{\left|{{X_{i,o}}}\right|}^{-{\alpha_i}}}}+{N_o}}\right)\right]}_{Z_2}.
( \underbrace{( 2 + ( 2 + \dots + ( 2}_{k}+ 1 / z^{2^{k + 1}})^{\frac{1}{2}}\ldots )^{\frac{1}{2}})^{\frac{1}{2}})^{- 1}= z^{2}( ( 2 z^{2^{2}}+ ( 2 z^{2^{3}}+ \ldots ( 2 z^{2^{k + 1}}+ 1 )^{\frac{1}{2}}\ldots )^{\frac{1}{2}})^{\frac{1}{2}})^{- 1}.
\mathcal{A}:=u_{\mathbf{b}}\left( a,a^{\dag}\right) =\Xi\left(b_{1}\right) \dots\Xi\left( b_{k}\right) \mathcal{D}:=u_{\mathbf{d}}\left( a,a^{\dag}\right) =\Xi\left(d_{1}\right) \dots\Xi\left( d_{j}\right) .
\sum_{i\ge0}u^if_{i+1}=\sum_{i\ge0}u^izf_i+\sum_{i\ge0}u^izg_i, \sum_{i\ge1}u^ig_i =\sum_{i\ge1}u^izf_{i+1}+\sum_{i\ge1}u^izg_{i+1}+\sum_{i\ge1}u^izh_{i+1}, \sum_{i\ge0}u^ih_i =\sum_{i\ge0}u^izh_{i+1}+\sum_{i\ge0}u^izg_{i+1}.
Z = \lim_{\delta \to 0}\int_{0}^{2 \pi}{\frac{d \lambda}{2 \pi}}e^{- i \lambda Q}\sum_{l_{1}= 0}^{\infty}\cdots \sum_{l_{N + 1}= 0}^{\infty}\exp{\left\{{- i T \sum_{\alpha = 1}^{N + 1}c_{\alpha}l_{\alpha}+ i \left( \lambda + i \delta \right) \sum_{\alpha = 1}^{N + 1}l_{\alpha}}\right\}}\ ,
\mathbf{A}=\mbox{Prod}\left(\mbox{Prod}\left(\mathbf{Q},\mathbf{D},\mathbf{D}^{\top}\right),\mbox{Prod}\left(\mathbf{Q},\mathbf{D},\mathbf{D}^{\top}\right)^{\top^{2}},\mbox{Prod}\left(\mathbf{Q},\mathbf{D},\mathbf{D}^{\top}\right)^{\top}\right),
\hat{S}= \int d^{1 1}\hat{x}\sqrt{| \hat{g}|}\left[ \hat{R}-{\textstyle \frac{1}{2 \cdot 4 !}}\hat{G}{}^{2}-{\textstyle \frac{1}{( 1 4 4 )^{2}}}{\textstyle \frac{1}{\sqrt{| \hat{g}|}}}\, \hat{\epsilon}\hat{G}\hat{G}\hat{C}\right] \, .
g ( m_{i}^{2}) \equiv{\frac{1}{1 6 \pi^{2}}}\left[ \sum_{s}m_{s}^{D - 2}{}~ \Gamma \left( 1 - \frac D 2 , m_{s}^{2}\delta \right) + 2 \sum_{d}m_{d}^{D - 2}\Gamma \left( 1 - \frac D 2 , m_{d}^{2}\delta \right) \right] ,
( 3 n_{a}^{2}+ 3 n_{d}^{2}+ 3 n_{e}^{2}) \neq 0 , \; \Q^{l}= n_{c}^{1}( Q_{a}- 3 Q_{d}- 3 Q_{e}) - \frac{3{\tilde{\epsilon}}\beta^{2}( n_{a}^{2}+ n_{d}^{2}+ n_{e}^{2})}{2 \beta^{1}}Q_{c}
\begin{array}{lcl}\sigma_1 (z) \sigma_1 (w) = \ln(z-w), \; \sigma_2 (z) \sigma_2(w) = \ln(z-w), \bar{\sigma}_1 (z) \bar{\sigma}_1 (w) = \ln(z-w), \; \bar{\sigma}_2 (z) \bar{\sigma}_2 (w) =\ln(z-w).\end{array}
1 + 2 \sum_{m=1}^\infty \frac{B_{2m}}{(2m)!}\Omega_m(x_2, \dots, x_{2m}) u^{2m}:= \exp \left(2 \sum_{\nu=1}^\infty \frac{B_{2\nu}}{(2\nu)!}\left(2 \sinh^{-1}\left(\frac{u}{2}\right)\right)^{2\nu}x_{2\nu}\right).
\sum_{k'}\sum_{m=0}^\infty \sum_{|k_0-\ell_0|>N}\cdots \sum_{|k_m-\ell_m|>N}2^{-|k-k_0|(\varepsilon/2-|\alpha|)}2^{-|k_0-\ell_0|(\varepsilon/2-|\alpha|)}&\hskip 5cm \times 2^{-|\ell_0-k_1|(\varepsilon/2-|\alpha|)}\cdots2^{-|k_m-\ell_m|(\varepsilon/2-|\alpha|)}2^{-|\ell_m-k'-j|(\varepsilon/2-|\alpha|)}&\le C2^{-N(\varepsilon/2 -|\alpha|)}
h(\hat{X}) =\left(h^2\big(\hat{x}^{(i)}_1,h^{d-1}(\hat{X}_{-i})_1\big),\dots,h^2\big(\hat{x}^{(i)}_n,h^{d-1}(\hat{X}_{-i})_n\big)\right)^T \prec_w\left(h^2\big(\tilde{x}^{(i)}_1,h^{d-1}(\tilde{X}_{-i})_1\big),\dots,h^2\big(\tilde{x}^{(i)}_n,h^{d-1}(\tilde{X}_{-i})_n\big)\right)^T=h(\tilde{X}).
X_{n+1}^{(\alpha)}:=X_n^{(\alpha)}+ \left(\xi_{n+1}\mathbb 1_{\{X_n^{(\alpha)}\leq -1\}}+ \eta_{n+1}\mathbb 1_{\{X_n^{(\alpha)}=0\}}+ \xi_{n+1}'\mathbb 1_{\{X_n^{(\alpha)}\geq 1\}}\right),
\left({\cal H}_1 \otimes{\cal H}_2 \right)_{\mathrm f}^{0}\subset{\cal H}_1^{\mathrm{s}}\otimes{\cal H}_2^{0}\hspace*{1cm}\mbox{and}\hspace*{1cm}\left({\cal H}_1 \otimes{\cal H}_2 \right)_{\mathrm f}^{0}\subset{\cal H}_1^0 \otimes{\cal H}_2^{s}\,.