harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
Ric'|_{S_h^{2n}}=\frac{2(n-1)}{a^2+b^2}\bigl(ah'|_{S_h^{2n}}+b\widetilde{h}'|_{S_h^{2n}}\bigr), Scal'|_{S_h^{2n}}=\frac{4n(n-1)a}{a^2+b^2}.
{\mathcal L}_i(f):=\frac{d^{m_i}z^{q+m_i+k}f_{i}}{dz^{m_i}}(z) - CG\left(z,\frac{d^lz^{m_j+d+k}f_{j}}{dz^l}(z),\;1\leq j\leq r,\; 0\leq l\leq m_j\right).
L(n_1,n_2,k) = \sum_{n_1 \ll p_1Cr}\, \frac{1}{n^{k-2/3}_1}\,\frac{1}{n^{2/3}_1}\, \left(\sum_{n^2_1n_{2}\ll N_0}\frac{|A_{\pi}(n_{2},n_{1})|^2}{n_2^{2/3}}\right)^{1/2}.
\begin{array}{l}{z^{3}( \tau ) ={\displaystyle \frac{p_{1}}{p_{3}}}\tau + z^{3}( 0 ) \quad ,}\\{}\\{z^{2}( \tau ) = ( z^{2}( 0 ) -{\displaystyle \frac{p_{4}( 0 )}{p_{3}}}\tau ) \exp ( i \gamma p_{1}\tau ) = ( z^{4}( \tau ) )^{*}\quad .}\\ \end{array}
\sum_{v\in X}\deg_G(v) =\sum_{v\in X}|N(v)\cap X|+\sum_{v\in X}|N(v)\setminus X| =\left(\frac{1}{2}\sum_{v\in X}|N(v)\cap X|+\sum_{v\in X}|N(v)\setminus X|\right)+\frac{1}{2}\sum_{v\in X}|N(v)\cap X| &= M+\frac{1}{2}\sum_{v\in X}|N(v)\cap X| &>\frac{t|X|}{2}+\frac{|X|\delta(G[X])}{2}&\geq\frac{3t|X|}{4}.
\dim_{\rm B}C =-\left(\frac{\log 2}{\log q}\right)\limsup_{k\in\mathbb{N}}\frac{\sum_{j=1}^{k}a_{2j-1}}{\sum_{j=1}^{2k-1}a_{i}}\dim_{\rm B}D =-\left(\frac{\log 2}{\log q}\right)\limsup_{k\in\mathbb{N}}\frac{\sum_{j=1}^{k}a_{2j}}{\sum_{i=1}^{2k}a_{i}}
F\Gamma = \langle r^d\rangle \langle r^e\rangle \cup \langle r^d\rangle \langle r^e\rangle s \cup \langle r^d\rangle r^ks \langle r^e\rangle\cup \langle r^d\rangle r^ks \langle r^e\rangle s &= \langle r^2\rangle \cup \langle r^2\rangle s \cup \langle r^2\rangle r^ks \cup \langle r^2\rangle r^{k}= D_n k
\partial_\mu^2 E(\lambda,\mu) = -\lim_{T\to \infty}\frac 1T \partial_\mu^2( \ln Z^{(W)}_T(\lambda,\mu) + 1) & = - \lim_{T\to \infty}\frac 1T \frac{\partial_\mu^2Z^{(W)}_T(\lambda,\mu)}{Z^{(W)}_T(\lambda,\mu)}.
\sum_{j=0}^r \Re \langle \sigma+c\gamma,[C_j] \rangle+\sum_{i=1}^{\ell}\Re\langle\sigma+c\gamma,[F_i] \rangle\leq \sum_{j=0}^r\Re \langle \sigma,[C_j] \rangle+ \sum_{i=1}^{\ell}\Re(c)\langle \gamma,[F_i]\rangle
\begin{aligned}\chi(S) = \chi(S\cap U) + \chi\bigg(S\cap \bigg(\bigcup_i e_i\bigg)\bigg) - \chi\bigg(S\cap U \cap \bigg(\bigcup_i e_i\bigg)\bigg) &=\chi(S\cap U) + \sum_i \chi(S\cap e_i) - \sum_i \chi(S\cap U \cap e_i) \ .\end{aligned}
\tilde{t}=t, \tilde{x}=\frac{\sqrt{2}}{A}\ln x+\omega_1 t+\omega_0, u=\nu_0\exp[(K+\frac{\omega_1^2}{4})t]x^{-\frac{\mathcal{D}}{A^2}+\frac{\omega_1}{\sqrt{2}A}}\tilde{u}.
\chi^K = \prod_i(f^g)^{\chi_i^{K_i}}= \prod_i(\chi_i^{-1})^{K_i}\circ g^{-1}\circ f \circ g \circ \chi_i^{K_i}&= \prod_i{g^{-1}}\circ ((\chi_i^{-1})^{K_i})^{g^{-1}}\circ f \circ (\chi_i^{K_i})^{g^{-1}}\circ g &= (\prod_i (\chi_i^{-1})^{K_i g^{-1}}\circ f \circ \chi_i^{K_i g^{-1}})^g.
\frac{\partial w}{\partial \overline{z}}=-w\left( n{{\overline{z}}^{n-1}}+{{a}_{1}}\left( n-1 \right){{\overline{z}}^{n-2}}+{{a}_{2}}\left( n-2 \right){{\overline{z}}^{n-3}}+\cdots +2{{a}_{n-2}}\overline{z}+{{a}_{n-1}}\right)
I_{n n^{\prime \prime}}( \hat{q}_{1}, \hat{q}_{2}) \equiv \int_{- \infty}^{\infty}d \eta \mathrm{e}^{i \eta \mathrm{s g n}( e H ) \hat{q}_{1}}D_{n}( \eta - \hat{q}_{2}) D_{n^{\prime \prime}}( \eta + \hat{q}_{2}) ,
\rho \to \frac{a \rho + b}{c \rho + d}, \; \; \rho \equiv \frac{{\tilde{V}}_{2}}{( 2 \pi )^{2}\alpha^{\prime}}\left( B_{p - 1 , p}+ i \sqrt{G_{( p - 1 ) ( p - 1 )}G_{p p}}\right) ,
f'_{(x,y)}(t) = -(k-1)_2(\deg_W(x)+\deg_W(y)-2d)e^{-t(k-1)_2}&\;- ((k)_2-2)(W(x,y)-\deg_W(x)-\deg_W(y)+d)e^{-t((k)_2-2)}&= \left( (k)_2 - 2 \right) (d - f_{(x,y)}(t)) + 2(k-2)(\deg_W(x) + \deg_W(y) - 2d)e^{-t(k-1)_2}&= (2k-4)(g_x(t) + g_y(t)) + (k-2)_2 d - \left( (k)_2 - 2 \right) f_{(x,y)}(t)\;.
L_{Y u k}( \overline{{{\Psi^{\prime}}}}\Psi^{\prime}\varphi \, ; \, \phi ) = - \frac{h}{\sqrt{2}}\overline{{{\Psi}}}^{\prime}\Psi^{\prime}\varphi \frac{e^{\gamma \phi / M_{p}}}{\chi^{3 / 2}}= - \frac{h}{4}\overline{{{\Psi^{\prime}}}}\Psi^{\prime}\varphi \left[ \frac{M^{4}+ V_{1}( \phi )}{V_{2}( \phi )}\right]^{3 / 2}e^{\gamma \phi / M_{p}}.
I_4(t,\theta,x):= \Big\| \| J'_g(t,r,\xi) - J_g(t,r,\xi) \|_{\Pi_x^{(\phi)};\frac{1}{\gamma_0 - 1}}\Big\|_{\mathbf 1_{0\leq r\leq \theta}dr;\frac{1}{\gamma_0 - 1}}.
\frac{R^{-1}+{\left ( R^{-1}\right )}^T}{2}= \omega^{-\frac{1}{2}}C_1 D_2 C_1^{*}\omega^{-\frac{1}{2}}, \frac{R^{-1}-{\left ( R^{-1}\right )}^T}{2}= \omega^{-\frac{1}{2}}C_1 D_3 C_1^{*}\omega^{-\frac{1}{2}},
\mathbb{P}^y\Bigg\{\sup_{0\le s< \lambda}\sup_{x\in \mathbb{R}}\frac{|U^k(s,x) - \ell^x(s)|^2}{2^{-k}log(2^k)}> b \Bigg \}= \mathbb{P}^y\Bigg\{\sup_{0\le s < \lambda}\sup_{x\in \mathbb{R}}\frac{|\frac{1}{\sqrt{\lambda}}U^k(s\lambda,\sqrt{\lambda}x) - \frac{1}{\sqrt{\lambda}}\ell^{\sqrt{\lambda}x}(s\lambda)|^2}{2^{-k}log(2^k)}> b \Bigg \}.
P(Z(\lambda)) = \sum_{\zeta \in r(\lambda) \Lambda^m}\sum_{(\rho,\xi) \in \Lambda^{\operatorname{min}}(\lambda, \zeta)}P(Z(\lambda \rho)) = \sum_{\eta \in s(\lambda) \Lambda^{m-d(\lambda)}}P(Z(\lambda \eta)) = \sum_{i=1}^p \sum_{\alpha_{ij}\in s(\eta_i) \Lambda^{m-d(\eta_i)}}P(Z(\eta_i \alpha_{ij})) = \sum_{i=1}^p P(Z(\eta_i)),
{\cal L}= - 2 f_{\phi}^{2}\ln ( S + S^{+}) |_{\theta^{2}\bar{\theta^{2}}}+ \frac{1}{2}S W_{j}^{\alpha}W_{\alpha}^{j}|_{\theta^{2}}+ \frac{1}{2}S^{+}\overline{{W}}{}_{j}^{\dot{\alpha}}\overline{{W}}{}_{\dot{\alpha}}^{j}|_{\bar{\theta}^{2}}
\sum_{i}|\nabla\sqrt{\tilde{\varphi}}|^2\left(\lambda^{-1}\left|x_i-x\right|\right)= \sum_i |\nabla\sqrt{\tilde{\varphi}}|^2\left(\lambda^{-1}\left|x_i-x\right|\right)\varphi\left((2\lambda)^{-1}\left|x_i-x\right|\right) \le C_{\varphi}(2\lambda)^d\|\nabla \sqrt{\tilde{\varphi}}\|_\infty^2 \rho_{2\lambda}(\chi, x) .
U({y},\mathbb{I}) \tilde{a}( \mathbf{x}) U({y},\mathbb{I})^{-1}=U(y^{0},\mathbb{I})\,U(\mathbf{y},\mathbb{I}) \,\tilde{a}( \mathbf{x}) \, U(\mathbf{y},\mathbb{I})^{-1}U(y^{0},\mathbb{I})^{-1}.
\begin{aligned}\tilde{L}_n ( \eta_j^2 - \eta_j ) = 2 ( W_{j -1}- W_j ) \eta_j + 2 W_j , & \tilde{L}_n \big( \eta_j^3 + \frac{3}{2}\eta_j^2 + \frac{5}{2}\eta_j \big) = 3 (W_{j -1}- W_j ) \eta_j^2 + 6 W_{j - 1}\eta_j + 3 W_j \end{aligned}
\begin{pmatrix}A_c 0 0 0 A_u 0 0 0 A_s \end{pmatrix}\begin{pmatrix}\operatorname{Id}+ k_c k_u k_s \end{pmatrix}+ \begin{pmatrix}g_c \circ K{}{}g_u \circ K{}{}g_s \circ K{}{}\end{pmatrix}- \begin{pmatrix}A_c + r + k_c \circ (A_c + r) k_u \circ (A_c + r) k_s \circ (A_c +r) \end{pmatrix}= 0 .
{\cal W}= \epsilon_{i j k}\Phi_{i}\Phi_{j}\Phi_{k}+ \epsilon_{i j k}\Phi_{i}^{l}\Phi_{j}^{l}\Phi_{k}^{l}+ \epsilon_{i j k}\Phi_{k}^{i}Q^{i j}Q^{i j}+ \Phi_{i}Q^{i}Q^{i}+ Q^{i j}Q^{j k}Q^{k i}+ Q^{i j}Q^{i}Q^{j}~ .
\zeta_{\bar{{\cal V}}}( s ) = \frac{m L}{\Gamma ( s ) \sqrt{4 \pi}}\int_{0}^{\infty}\! \! d \beta \beta^{s - \frac{3}{2}}e^{- \beta}= \frac{m L}{\sqrt{4 \pi}}\frac{\Gamma ( s - \frac{1}{2})}{\Gamma ( s )}.
\sum_{n\geq 1}\frac{(-1)^n q^{2n}}{1-q^{4n}}\frac{(-q;q^2)_{n-1}}{(q^2;q^2)_{n-1}}q^{n-1}= \frac{1}{1-q}\sum_{n\geq 0}(-1)^n q^{n^2}- \frac{1}{1-q^2}.
M_\pi(z;X) = \prod_{T\in{\cal T}^\pi}\frac{1}{1-z\,X^T}= \sum_{r\geq0}z^r\, s_{(r)}[s_\pi](X) \,; \cr L_\pi(z;X) = \prod_{T\in{\cal T}^\pi}(1-z\,X^T) = \sum_{r\geq0}(-1)^r\, z^r\, s_{(1^r)}[s_\pi](X) \,,
ds^2 = \frac{1-\left( \frac{t_-}{t}\right)^2}{-k+\left(\frac{t_+}{t}\right)^2}dy^2 - \frac{dt^2}{\left(-k+\left(\frac{t_+}{t}\right)^2\right)\left(1 -\left(\frac{t_-}{t}\right)^2\right)}+t^2 d\Omega_k^2
F_{\tiny \mathrm{d i v}}^{C}[ g , \beta , D ] = - \eta \,{\frac{\Gamma \left( 1 - \frac D 2 \right)}{( 4 \pi )^{D / 2}}}\, \,{\frac{\pi^{2}m^{D - 4}}{3 \kappa \beta^{2}}}\, \, \int_{\Sigma}\left[ f_{1}m^{2}- \left( p_{1}{\frac{4 \pi^{2}}{\kappa^{2}\beta^{2}}}{\cal P}+ p_{2}R + p_{3}{\cal Q}) \right) \right] .
e_3 =\frac16\left[(\operatorname{tr}S)^3-3(\operatorname{tr}S)\operatorname{tr}(S^2)+2\operatorname{tr}(S^3)\right], e_4 =\frac1{24}\left[(\operatorname{tr}S)^4-6(\operatorname{tr}S)^2\operatorname{tr}(S^2)+3(\operatorname{tr}(S^2))^2+8(\operatorname{tr}S)\operatorname{tr}(S^3)-6\operatorname{tr}(S^4)\right].
|h_{n+2}| \leq 2\kappa E^{c}|h_{n+3}| = |h_k^{n_1}h_{n+2}^{n_2}| = |n_1||h_k| + |n_2| |h_{n+2}| &\leq (EN)(mnN^{2})+(EN)(2\kappa E^{c}) &\leq 2mn\kappa E^{c+1}N^{3}.
\int\limits_{0}^tsgn(X_u)dX_u =\int\limits_0^t sgn(Y_u)\frac{1}{\sigma(Y_u)}\left(-\frac{\sigma^2\left(e^{Y_u}\right)}{2}du+\sigma\left(e^{Y_u}\right)dW_u\right) =\int\limits_0^t g'(Y_u)dY_u-g'(0)\int\limits_0^t 1_{\{Y_u=0\}}dY_u=\int\limits_0^t g'(Y_u)dY_u
Y^{\mu,t+s}(b,z) =Y^\mu(\prod_{n=1}^{\infty}e^{\frac{2(-t-s+\sqrt{-1}\Im(\mu)}{n}(-z)^{-n}a_n}b,z) &=Y^{\mu,t}(\prod_{n=1}^{\infty}e^{\frac{-2s}{n}(-z)^{-n}a_n}b,z),
-\dfrac{1}{2\pi}\sum_{k_{1}=1}^{+\infty}\dfrac{\cosh(\pi k_{1}x_{2}/a)\cos (\pi k_{1}x_{1}/a)}{k_{1}\sinh (\pi k_{1})}\doteq\dfrac{1}{2\pi}\log \left| \vartheta_{4}\left( \left. \frac{\pi}{2a}(x_{1}+ix_{2}) \right| i \right)\right|
R_{\ell,i,s}(z) =\displaystyle P_{\ell}(z)\psi_{i,s}\left(\dfrac{1}{z-t}\right)-P_{\ell,i,s}(z) =\displaystyle\psi_{i,s}\left(\dfrac{P_{\ell}(t)}{z-t}\right) =\sum_{k=0}^{\infty}\dfrac{\psi_{i,s}(t^kP_{\ell}(t))}{z^{k+1}}\enspace.
\mathbb{R}_{12}(u-v|g_1,g_2)\,\mathrm{L}_1(u|g_1)\,\mathrm{L}_2(v|g_2) = =\mathrm{L}_2(v|g_2)\,\mathrm{L}_1(u|g_1)\,\mathbb{R}_{12}(u-v|g_1,g_2).
F_{v_{1,1},g_1}= F_{v_{2,n},g_n}= F_{v_{2,i},g_i}= F_{v_{i+1,1},g_i}= f_3, F_{v_{1,j}, f_{1,j}}= N_{Z/Y_i}\otimes^\mathbb{L}f_2, &F_{v_{2,j},f_{2,j}}= F_{v_{1,j},f_{2,j}}= f_1, F_{v_{2,j}, f_{1,j}}= f_2.
[M:N]^{-2}\Psi_i\circ\Psi\circ T_i(x)=\sum_{j=1}^J[M:N]^{-2}\Psi_i\circ\Psi\circ T_i(z_jx)=\sum_{j=1}^J\mu_i^jz_jx=\sigma_ix.
[a_{(1\,l)}, a_{(3\,l)}]_c=[[a_{(1\,l)},a_{l-1}]_c,a_l]_c+\chi_{l-1}(g_{(1\,l)})a_{l-1}[a_{(1\,l)},a_l]_c & -\chi_l(g_{l-1})[a_{(1\,l)},a_l]_ca_{l-1}=-\lambda_{l-1l-1l}(1-\xi^2)[a_{(1\,l-2)},a_l]_cg_{l-1l-1l}& -\lambda_{l-1ll}(1-\xi)a_{(1\,l-1)}g_{l-1ll}+3\lambda_{l-1ll}\chi_{l-1}(g_{(1\,l-2)})a_{l-1}a_{(1\,l-2)}g_{l-1ll}.
{\cal H}={\frac{1}{2}}\rho v_{i}v_{i}+ R ( \rho ) ~ , ~ ~ \{v_{i}( x ) , \rho ( y ) \}={\frac{\partial}{\partial x^{i}}}\delta ( x - y ) ~ , ~ ~ \{v_{i}( x ) , v_{j}( y ) \}= -{\frac{\partial_{i}v_{j}- \partial_{j}v_{i}}{\rho}}\delta ( x - y ) .
\Delta \xi(\tau) = \Big\langle \Big(\sum_{a<b}^{0,u}y^a y^b\Big)^2\Big\rangle_\tau-\Big\langle\sum_{a<b}^{0,u}y^a y^b\Big\rangle_\tau^2=\sum_{a<b}^{0,u}\sum_{c<d}^{0,u}\big(\langle y^a y^b y^cy^d\rangle_\tau-\langle y^a y^b\rangle_\tau \langle y^cy^d\rangle_\tau\big)\ge 0,
\sum_{n=1}^{\infty}|\partial^2_x[(h_{\alpha}\circ F_{\alpha, n}^{-1}) (\hat \Psi_{\alpha}\circ F_{\alpha,n}^{-1}) G_{\alpha, n}]| \leq C\sum_{n=1}^{\infty}(n+1)n^{-\left(1+\frac{1}{\alpha}\right)}< \infty,
\begin{aligned}\begin{pmatrix}R_h 0 0 P_h \end{pmatrix}\partial_t U(0) = \begin{pmatrix}R_h 0 0 P_h \end{pmatrix}(- \mathcal A U(0) + F(0)) begin{align*}1ex] = \begin{pmatrix}R_h 0 0 P_h \end{pmatrix}\begin{pmatrix}u_1 - A u_0 + f(0)\end{pmatrix}= \begin{pmatrix}R_h u_1 - P_h A u_0 + P_h f(0)\end{pmatrix}\,.\end{aligned}
\sup_{r\leq t}|C^{\varepsilon ,\varphi}(r)| \leq 2a(\varepsilon)\sup_{r\leq t}\Vert \bar{Y}^{\varepsilon ,\varphi}(r)\Vert_{-q}^{2}\int_{\mathbb{X}\times \lbrack 0,t]}R_{G}(y)|\psi |d\nu_{T}+2\int_{\mathbb{X}\times \lbrack 0,t]}\Vert \bar{Y}^{\varepsilon ,\varphi}(s)\Vert_{-q}R_{G}(y)|\psi |d\nu_{T}& =T_{1}+T_{2}.
\left\langle{\mathcal L}\right\rangle = - \frac{1}{4}\left\langle \mathcal{F^{A}}_{\mu \nu}\mathcal F^{\mathcal A \mu \nu}\right\rangle = - \frac{1}{4}f_{\mu \nu}f^{\mu \nu}+ \left( D_{\mu}\varphi^{*}\right) \left( D^{\mu}\varphi \right) - \frac{8 g^{2}}{2 5}\left| \varphi \right|^{4}
\|\omega a\|_\tau^2=\tau(\langle\omega a, \omega a\rangle_{\mathcal A})=\tau(a^*\langle\omega,\omega\rangle_{\mathcal A}a)=\tau(a^*a\langle\omega,\omega\rangle_{\mathcal A}) \leq \|a^*a\|\|\omega\|_\tau^2.
T_{0}= \Gamma_{1 1}, \quad T_{2}={\frac{1}{2}}\psi^{2}+{\cal F}\Gamma_{1 1}, \quad T_{4}={\frac{1}{2 4}}\Gamma_{1 1}\psi^{4}+{\frac{1}{2}}{\cal F}\psi^{2}+{\frac{1}{2}}{\cal F}^{2}\Gamma_{1 1}, \quad \mathrm{e t c .}
\| u - u_h \|_{N^{st + 1 - t}_2(\mathbb{R}^d)}= \| u - u_h \|_{(H^1(\mathbb{R}^d), N^s_2(\mathbb{R}^d))_{t,\infty}}\leq C_{t,s}|h|^{(1-s)t}|h|^{1-t}\| u \|_{(H^2(\mathbb{R}^d), H^1(\mathbb{R}^d))_{t,\infty}}= C_{t,s}|h|^{1-st}\| u \|_{N^{2-t}_2(\Omega)},
\sum_{n=2}^x \frac{r(n)-r(n-1)}{\log n}= \sum_{n=2}^x r(n)\left(\frac{1}{\ln n}- \frac{1}{\ln(n+1)}\right) + \frac{r(\left\lfloor x\right\rfloor)}{\ln(\left\lfloor x\right\rfloor +1)}&=O\left(\sum_{n=2}^x \frac{1/n}{(\ln n)^2}\right) + O(1) = O(1).
(g^{i_1}, g^{i_2}, g^{i_3})\rightharpoonup c = b_{(g^{i_1}, g^{i_2}, g^{i_3})}c b^{-1}_{(g^{i_{\sigma(1)}}, g^{i_{\sigma(2)}}g^{i_{\sigma(3)}})}\shortintertext{and}(g^{i_1}, g^{i_2}, g^{i_3})x\rightharpoonup c = b_{(g^{i_1}, g^{i_2}, g^{i_3})}\bigl(b_x c - \alpha(c) b_x\bigr) b^{-1}_{(g^{i_{\sigma(1)}+1}, g^{i_{\sigma(2)}+1}, g^{i_{\sigma(3)}+1})}
\widetilde{\mathcal{H}}_n=\,\Delta x\sum_m \left\{\tfrac{1}2\left[(\delta_m^+u_{m,n})^2+(\delta_m^-u_{m,n})^2+(\delta_m^+v_{m,n})^2+(\delta_m^-v_{m,n})^2\right]-\tfrac{1}2(u_{m,n}^2+v_{m,n}^2)^2\right\}.
\begin{aligned}d_0a(b) = ba - ab, d_nf(a_1, \dots, a_{n+1}) = a_1f(a_2, \dots, a_{n+1}) + \sum_{i =1}^n (-1)^i f(a_1, \dots, a_ia_{i+1}, \dots, a_{n+1}) & + (-1)^{n+1}f(a_1, \dots,a_n)a_{n+1}.\end{aligned}
i\,[H,K_{i}]=\left(p_{i}-\frac{\kappa}{2R^{2}}\,\epsilon_{ij}x_{j}\right)\cosh\frac{t}{R}+\left(\frac{2}{R}\,\tilde{p}_{i}-\frac{\mu}{R}\,x_{i}\right)\sinh\frac{t}{R}=-\frac{\partial K_{i}}{\partial t}.
\mathcal{L}_{\Pi^{\sharp}\alpha}\Pi = [\Pi^{\sharp}\alpha, \Pi] = (-1)^{n-k}[X_k, X_1 \wedge \cdots \wedge X_n] & = (-1)^{n-k}\sum_{i=1}^nX_1 \wedge \cdots \wedge[X_k, X_i]\wedge \cdots \wedge X_n =0,
P^n (X_t \not \in C(t, \epsilon)) = P^n(X_{t \wedge \tau_{m^o-1}}\not \in C(t, \epsilon), \tau_{m^o-1}> t) + P^n(X_t \not \in C(t, \epsilon), \tau_{m^o-1}\leq t) &\leq P^n (X_{t \wedge \tau_{m^o-1}}\not \in C(t, \epsilon)) + P^{n, m^o}(\tau_{m^o-1}\leq t).
\int_{-1}^1p_x'(t)w(t)dt\leq-w(x)+p_x(1)w(1)-\sum_{i=1}^L\left(w(s_i+)-w(s_i-)\right)\left(p_x-\chi_{[-1,x]}\right)(s_i)+\int_{-1}^1 q_x(t)\cdot\bigl\lvert w'(t)\bigr\rvert dt.
\sum_{i=1}^t m_i u_i = \sum_{i=1}^{j-1}m_in_i-\sum_{i=1}^{j-1}m_i\alpha_i+\sum_{i=j}^t m_iu_i \leq\tilde{K}-m_j\delta-m_j\sum_{i=1}^{j-1}\alpha_i+m_j\sum_{i=j}^tu_i =\tilde{K}-m_j \left(\sum_{i=j}^tn_i-\sum_{i=1}^t \alpha_i+\sum_{i=1}^{j-1}\alpha_i-\sum_{i=j}^tu_i \right)=\tilde{K}.
\rho(z) \;=\;{z\over \pi}\sqrt{b-z\over z-a}\left[{1\over 4\beta}-{d-2\over 2}\int_a^b{d y\over y+z}\sqrt{y+a\over y+b}{\rho(y)\over y}\right]\;\;\;\;\;\;\;\;\;\;{\rm for}\;\;\;\; \beta < \beta_c\;.
-\frac{\mathfrak{N}(z)}{z-\gamma}+ \frac{\vartheta^2}{z-\gamma}= &=a z + b + \sum_{\substack{k \in \mathcal{M}k \ne0,k_0}}A_{k}\left(\frac{1}{\nu_k - z}-\frac{1}{\nu_k}\right) + \frac{A_0}{\nu_0-z}+ (\mathfrak{N}(\gamma)-\vartheta^2)\left(\frac{1}{\gamma - z}-\frac{1}{\gamma}\right)\,.
\left[ \begin{array}{c | | c c c c}{{7}}&{{2}}&{{2}}&{{2}}&{{2}}\\ \end{array}\right]_{- 1 2 8}^{1 , 6 5}\leftarrow \left[ \begin{array}{c | | c c c c c}{{7}}&{{1}}&{{1}}&{{2}}&{{2}}&{{2}}\\{{1}}&{{1}}&{{1}}&{{0}}&{{0}}&{{0}}\\ \end{array}\right]_{- 1 1 2}^{2 , 5 8}\; ,
\widehat{Y}(qx) = \biggl(F_{\infty}+\frac{F_1}{1- \alpha_1 x}+\frac{F_2}{1- \alpha_2 x}\biggr) \widehat{Y}(x), \widehat{Y}(x) = \begin{pmatrix}\widehat{y}_{0}(x) \widehat{y}_{1}(x) \widehat{y}_{2}(x) \end{pmatrix}\! .
\mathbb{E}_{t}[\langle \zeta_{t+1}, h \rangle f(\langle \zeta_{t+1}, h \rangle) ]=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}h_{i}\{\mathbb{E}_{t}[X_{i,t+1}f(\langle \zeta_{t+1}, h \rangle) ]-p_{i,t+1}\mathbb{E}_{t}f(\langle \zeta_{t+1}, h \rangle) \},
S_{N}={\textstyle\sum\limits_{n\geq0}}P_{N=ne}S_{n}={\textstyle\sum\limits_{n\geq0}}P_{N=ne}{\textstyle\sum\limits_{k=1}^{n}}x_{k}& ={\textstyle\sum\limits_{k\geq1}}\left({\textstyle\sum_{n\geq k}}P_{N=ne}\right) x_{k}={\textstyle\sum\limits_{k\geq1}}P_{N\geq ke}x_{k}.
\Lambda^{( 1 )}( g_{1}g_{2})^{\alpha \beta}\: + \: \Lambda^{( 3 )}( g_{1}, g_{2})^{\alpha}\: - \: \Lambda^{( 3 )}( g_{1}, g_{2})^{\beta}\: + \: d \log \lambda_{\alpha \beta}^{g_{1}, g_{2}}\: = \: \Lambda^{( 1 )}( g_{2})^{\alpha \beta}\: + \: g_{2}^{*}\Lambda^{( 1 )}( g_{1})^{\alpha \beta}
\zeta_{N}^{c}( 2 , x^{2}) = \frac{\alpha \, R}{8 \, x^{3}}+ \frac{1 5 \, \alpha}{5 1 2 \, R \, x^{5}}+ \frac{4 0 3 5}{2^{1 7}}\, \frac{\alpha}{R^{3}\, x^{7}}+{\cal O}( x^{- 9}) \,{.}
[A(s-t)tr(\lambda^{a_2}\lambda^{a_1}\lambda^{a_3}\lambda^{a_4})+A(u-t)tr(\lambda^{a_2}\lambda^{a_3}\lambda^{a_1}\lambda^{a_4})+A(s-u)tr(\lambda^{a_1}\lambda^{a_2}\lambda^{a_3}\lambda^{a_4})+3\leftrightarrow 4]
F(x)=\prod_{i=0}^k \left(\sum_{b\ge 0}x^b\frac{(a_i+b)!}{b!}\right)=\prod_{i=0}^k\frac{a_i!}{(1-x)^{1+a_i}}=\frac{a_0!\cdots a_k!}{(1-x)^{1+k+A}}\; ,
\bigl|\partial_{\alpha}(G_{,A}(\Bar{\Xi}))(\Phi_{,i\alpha}^A(F)-\Phi_{,i\alpha}^A(\Bar{F}))(V_i-\bar{V}_i)\bigr|\leqslant C \big((1+|F|^2)|F-\bar{F}|^2+|V-\bar{V}|^2 \big).
\gamma(s,(r_0 \circ \sigma) \otimes \eta,\psi) = \gamma(s,(r_0 \circ \sigma') \otimes \eta',\psi'), L(s,(r_0 \circ \sigma) \otimes \eta) = L(s,(r_0 \circ \sigma') \otimes \eta') \varepsilon(s,(r_0 \circ \sigma) \otimes \eta,\psi) = \varepsilon(s,(r_0 \circ \sigma') \otimes \eta',\psi').
\sum_{n=0}^{2^{k_i}- 1}1_{[-B,-B+b] \cup [B-b,B]}\left( t + \sum_{j=0}^{n-1}f(T^j x) \right)\le2\eta \sum_{n=0}^{2^{k_i}- 1}1_{[-B,B]}\left( t + \sum_{j=0}^{n-1}f(T^j x) \right)
\tilde{\mathcal{H}}= \{(x,y) \in \mathcal{H}, y-x > 1\}, \mathcal{H}_1 = \{(x,y) \in \mathcal{H}, 0 \leq x \textrm{and}y \leq 1 \}, \mathcal{H}_+ = \{(x,y) \in \mathcal{H}, y-x \leq 1 \textrm{and}1 < y \}, \mathcal{H}_- = \{(x,y) \in \mathcal{H}, y-x \leq 1 \textrm{and}x < 0 \}.
A(\tau) :=-\frac{1}{2\pi i}\cdot\frac{\partial}{\partial\tau}F(\tau)-\frac 16 E_2(\tau)F(\tau)+\frac{F(\tau)E_6(\tau)(7j(\tau)-6912)}{6E_4(\tau)(j(\tau)-1728)}, B(\tau) :=\frac{F(\tau)E_6(\tau)j(\tau)}{E_4(\tau)},
\omega(e_{k \ell})=\sum_{\alpha=1}^p u_{\alpha,k}^-\frac{\partial}{\partial u_{\alpha,\ell}^-}-\sum_{\mu=p+1}^{p+q}u_{\mu,\ell}^-\frac{\partial}{\partial u_{\mu,k}^-}+\frac{1}{2}\delta_{k\ell}(p-q)
\gamma (t,x) = \max_{a_i(0,x) \ne a_j(0,x)}\frac{|(a_i(t,x)- a_i (0,x)) - ( a_j(t,x) -a_j (0,x))|}{| a_i (0,x)-a_j (0,x) |}= \max_{a_i(0,x) \ne a_j(0,x)}| \frac{a_i(t,x) - a_j(t,x)}{a_i (0,x)-a_j (0,x)}- 1|.
\Z = H^{- 1}\left( \begin{array}{c c c}{{( 1 - z ) ( i \cal E \mathrm{+ 1 ) - \mathrm{\Phi}^{2}}}}&{{\hspace{5 m m}}}&{{i \cal E \it + z}}\\{{i \cal E \it + z}}&{{\hspace{5 m m}}}&{{( 1 + z ) ( i \cal E \mathrm{- 1 ) + \mathrm{\Phi}^{2}}}}\\ \end{array}\right) ,
\partial_t\bigg(\int_{[-t,0]}\varphi_i(x+t)d^-\eta(x)\bigg) \ = \ \partial_t\bigg(\eta(0)\varphi(t) - \int_{-t}^0\eta(x)\dot\varphi_i(x+t)dx\bigg) &= \ \eta(0)\dot\varphi(t) - \eta(-t)\dot\varphi_i(0^+) - \int_{-t}^0 \eta(x)\ddot\varphi_i(x+t) dx.
\int_{0}^{\infty}\frac{(\eta_k)^jz^{j-1}}{(j-1)!}e^{-\eta_k z}V_q(b+z)dz=\sum_{i=1}^{N}\frac{P_i(\eta_k)^j}{(\eta_k+\gamma_{i,q})^j}+\sum_{i=1}^{M}\frac{H_i(\eta_k)^j}{(\eta_k-\beta_{i,q})^j}e^{\beta_{i,q}(b-y)},
\theta_0(2z + 3\sigma + 1/2; 4\sigma) + e^{-2\pi i z}\theta_0(2z + \sigma + 1/2; 4\sigma) = \frac{2\theta_0(3\sigma + 1/2; 4 \sigma) e^{-2\pi i z}\theta_0(z + 1/2; \sigma)}{\theta_0(1/2; \sigma)}.
\mathbb E\left[e^{-q \tau_{x}^+}\textbf{1}_{\{X_{\tau_{x}^+}-x \in dy\}}\right]=C^q_0(x)\delta_0(dy)+\sum_{k=1}^{m^+}\sum_{j=1}^{m_k}C^q_{kj}(x)\frac{(\eta_k)^jy^{j-1}}{(j-1)!}e^{-\eta_k y}dy,
\dot{V}_{k,v}(x) :=\limsup_{h\to+0}\frac{1}{h}\Big( V_k(\phi(h,x,v)) - V_k(x) \Big) \le \limsup_{h\to+0}\frac{1}{h}(e^h - 1)V_k(x) + \lim_{h\to0}\frac{e^h-1}{kh}= V_k(x)+\frac{1}{k}.
\Psi^{(k)}_0(x) \sim \left(I + \frac{m_1}{x}+ \cdots \right) e^{\left(\frac{x^3}{6}-\frac{xt}{2}\right)\sigma_3},x \to \infty, \frac{\pi}{2}+\frac{k-2}{3}\pi < \arg x < \frac{\pi}{2}+\frac{k}{3}\pi, k = 1, 2, \dots,7.
\mu_{\lambda \rho}(C_{\bf 0}\in\Lambda (0, k_\alpha, k_\beta)\mid ~ \Gamma_0 )= \lambda^{|{\bf k}| -1}|{\bf k}|\frac{p_\alpha^{k_\alpha}p_\beta^{k_\beta}}{(k_\alpha)!k_\beta !}F_\lambda (0, k_\alpha, k_\beta),
(\frac{u^{n+1}_{h}- u^{n}_{h}}{\Delta t},v_{h}) = -b(u^{n}_{h},u^{n+1}_{h},v_{h}) - \nu (u^{n+1}_{h}, v_{h}) + (f^{n+1},v_{h}) \; \; \forall v_{h}\in V_{h}.
\mathcal{D}^\prime = \left\{(s_1,s_2,s_3) \mid \sigma_1 \geq 0, \sigma_2 \geq 0, \sigma_3 > 0, t_1 \geq 0, t_2 \geq 0, 2 \leq t_3 \leq T, \right. &\left. \sigma_1+\sigma_3 \leq 1, \sigma_2+\sigma_3 \leq 1, \sigma_1+\sigma_2+\sigma_3> 3/2, \right. &\left. s_1+s_3 \neq 1, s_2+s_3 \neq 1, s_1+s_2+s_3 \neq 2 \right\}.
0 \leq \sum \limits_{k=0}^N \gamma x_k \leq \frac{x_N}{1-x_N}\cdot (\frac45)^N \sum \limits_{k=0}^N (\frac54)^k \leq \frac{x_N}{1-x_N}\cdot 5 < \frac5{99}< \frac1{10}
\Phi_{p,q}(e_{2j-1}) = \tau_{2j-1}\cdot E \otimes...\otimes E \otimes U \otimes \underbrace{D \otimes...\otimes D}_{(j-1) \times}, \Phi_{p,q}(e_{2j}) = \tau_{2j}\cdot E \otimes...\otimes E \otimes V \otimes \underbrace{D \otimes...\otimes D}_{(j-1) \times}.
\int_0^\infty y^{s + \frac{k-1}{2}}\sum_{m \in \mathbb{Z}}r_{2k+1}(m^2 + h) e^{-2\pi y(2m^2 + h)}\frac{dy}{y}= \sum_{m \in \mathbb{Z}}\frac{r_{2k+1}(m^2 + h)}{(2m^2 + h)^{s + \frac{k-1}{2}}}\frac{\Gamma(s + \frac{k-1}{2})}{(2\pi)^{s + \frac{k-1}{2}}}.
\mathcal{T}^\nu_{2,\mp}(\mathcal{T}^\nu_{1,\mp}f)(\xi) = \mathcal{C}_{{\mp}i{\psi_2}}(\mathcal{C}_{{\mp}i{\psi_1}}f)(\xi) &= \mathcal{C}_{{\mp}i{\psi_1}}f \left(\frac{\mp i \xi}{\sqrt 2}, \frac{{\mp}\xi}{\sqrt 2}\right) &= f(-\xi)
M_{n_{1}, n_{2}}^{d}( w_{1}, w_{2}) = k^{2 ( n_{1}+ n_{2}- 1 ) - d}\int_{0}^{\infty}d x x^{\frac{d}{2}}\frac{\partial}{\partial{t}}\left\{\dot{P}^{2}( P + w_{1})^{- n_{1}}( \tilde{P}+ w_{2})^{- n_{2}}\right\}
\sum_{0 \le s \le (k-1)d/n}\binom{d}{s}(-1)^{(\frac{n}{d}+1)s}\sum_{j=0}^{n-1}(-j) (-\omega_n^{ij})^{k-\frac{n}{d}s}= \omega_n^{i\binom{k}{2}}\Big( 1_{n \mid dk}\cdot \binom{k}{2}\binom{d}{dk/n}+ \omega_n^i{n \brack k}^{'}_{\omega_n^i}\Big).
Res_{\mu=-\frac n2-\lambda-N-1}\big(M_{\tilde{\Phi}^*\bar{\theta}}(\mu;r)(\phi)\big) =Res_{\mu=-\frac n2-\lambda-N-1}\int_X r^\mu<\tilde{\Phi}^*(\bar{\theta}),\phi>Vol(r^2\Phi^* g_+)
\sigma_n^*(x) =\frac{2\sqrt{1-x^2}}{\pi^2}\left(\int_0^s+\int_s^1\right)\frac{a_ntQ'(a_nt)-a_nxQ'(a_nx)}{n(t^2-x^2)}\frac{dt}{\sqrt{1-t^2}}&=:I_1+I_2.
\int_{B_R(0)}\frac{|\nabla u|^{p-2}}{|x|^{ap}}\nabla u \nabla(x\cdot \nabla u)dx-\int_{\partial B_{R}(0)}\frac{|\nabla u|^{p-2}}{|x|^{ap}}|\partial_{\nu}u|^2ds=\int_{B_{R}(0)}\frac{u^{q}(x\cdot \nabla u)}{|x|^{b(q+1)}}dx.
g_1 \in \Gamma \vert{i}{I}{J}\ \ \mbox{,}g_2 \in \Gamma \vert{j}{J}{K}\ \ \mbox{,}\ \g_3 \in \Gamma \vert{k}{K}{L}\ \ \mbox{,}g_4 \in \Gamma \vert{l}{L}{I}
\hat{T}_{00}=\frac{1}{2}\left[\dot{\phi}^2+\left(\vec{\nabla}\phi\right)^2+m^2\phi^2+\sigma\phi^2\right]=\frac{1}{2}\int d^dy \phi(y)\left(\hat{T}^{(0)}_x+\hat{T}^{(1)}_x\right) \phi(y)
(1+(\xi/l-t)^2)^{1/4}(1+(\xi/(l\pm 1)-t)^2)^{-1/4}-1 = (1+(\xi/(l\pm 1)-t)^2)^{-1/4}((1+(\xi/l-t)^2)^{1/4}- (1+(\xi/(l\pm 1)-t)^2)^{1/4}).
[\mathcal{A},\mathcal{B}]=\sum_{\tau=i,j,k}\left(\dfrac{1}{3}[\mathcal{A}_1,\mathcal{B}_1]_\tau+[\mathcal{A}_1,\mathcal{B}_\tau]_\tau+[\mathcal{A}_\tau,\mathcal{B}_1+\mathcal{B}_\tau]_\tau\right),\mbox{~~for all ~~}\mathcal{A},\mathcal{B}\in\mathfrak{h}_{24}.
{\cal L}_{gw}=\frac{1}{g^2}\left(\frac{1}{2L^2}S^{ab}_{a'b'}\theta_{I}^{a'}\theta_{J}^{b'}A^{I}_{a}A^{J}_{b}+\frac{1}{4}T^{ab}_{a'b'}\theta_{I}^{a'}\theta_{J}^{b'}F^{I}_{ac}F^{Jc}_{b}\right)+\frac{1}{gL\sqrt{32\pi G}}U^{Kabc}_{\;a'b'c'}\theta_{I}^{a'}\theta_{J}^{b'}\theta_{K}^{c'}A^{I}_{a}F^{J}_{bc}.
I \coloneqq \frac1{2\pi i}\oint\limits_{|z|=2-\epsilon}\frac{\eta(z) (\log y)^z}{z^{k+1}}\, \mathrm{d}z = \eta_o(2) \frac{\log^2 y}{2^k}- \frac1{2\pi i}\oint\limits_{|z|=2+\epsilon_1}\frac{\eta(z) (\log y)^z}{z^{k+1}}\, \mathrm{d}z
\begin{array}{rl}p(x_a,x_b,x_c) = \displaystyle \frac{(x_a+1)(x_b+1)(x_c+1)-(x_a-1)(x_b-1)(x_c-1)}{8}&= \displaystyle \frac{x_ax_b+x_ax_c+x_bx_c+1}{4}\end{array}
\tilde{\Gamma}_{ij}^{k}=\Gamma_{ij}^{k}+\frac{\partial\varphi}{\partial x^{i}}\delta_{j}^{k}+\frac{\partial\varphi}{\partial x^{j}}\delta_{i}^{k}-\frac{\partial\varphi}{\partial x^{l}}g^{lk}g_{ij}
R ( x ) ={\frac{2 a}{( 1 + \beta^{2}) ( 1 - a x )^{2}}}\left[{\cal Z}_{1}- 2 c ( 1 - a x ) ( 1 - x )^{\frac{\beta^{2}- 1}{\beta^{2}+ 1}}+{\frac{1 - a x}{1 - x}}\right] .
\begin{aligned}\mathsf{I}\{Z(\tau)=0\}= \mathsf{I}\{Z(\tau-)=1\}\mathsf{I}\{\mathrm{M}_{1}(\tau)-\mathrm{M}_{1}(\tau-)=1\}&+\mathsf{I}\{Z(\tau-)=0\}\mathsf{I}\{\Lambda_{0}(\tau)-\Lambda_{0}(\tau-)=0\}.\end{aligned}
a_{non-relativistic}=\frac{v^{2}}{r}\rightarrow a_{relativistic}=\frac{1}{m}\frac{dp}{d\tau}=\frac{1}{m}\gamma\frac{d(\gamma mv)}{dt}=\gamma^{2}\frac{dv}{dt}=\gamma^{2}\frac{v^{2}}{r}