harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\tilde{\pi}: \Lambda^{3}E^{*}\rightarrow E^{*}\otimes \Lambda^{1,1}_{\mathbb{H}}E^{*},\ (\tilde{\pi}\alpha ) (u, v, w):= \alpha ( u, v, w) +\sum_{i=1}^{3}\alpha (u,\mathcal J_{i}v, \mathcal J_{i}w).
\lim_{h \searrow 0}\frac{f_{\bf \tilde v}(t,x,x) -f_{{\bf v}_h}(t,x,x)}{h}= -{\bf A}^vf_{\bf \tilde v}^x(t,x), \lim_{h \searrow 0}\frac{G \diamond g_{\bf \tilde v}(t,x)-G \diamond g_{{\bf v}_h}(t,x)}{h}= -{\bf H}^vg_{\bf \tilde v}(t,x).
Q ( s ) = \sum_{n}\frac{1}{2 \Gamma ( s )}\sqrt{\frac{b}{a}}\int_{0}^{\infty}d t \; e^{- t \left( c - \frac{b^{2}}{8 a}\right)}\; t^{s - 1}\; K_{\frac{1}{4}}\left( \frac{b^{2}t}{8 a}\right)
d s_{1 0}^{2}= \frac{1}{\sqrt{f_{0}}}( - d t^{2}+ d x_{1 0 - p}^{2}+ \ \cdots \ + d x_{9}^{2}) + \sqrt{f_{0}}( d x_{1}^{2}+ \ \cdots \ + d x_{9 - p}^{2}) ,
J^{*}[\psi^{(5)}]_{T,t}=\hbox{\vtop{\offinterlineskip\halign{\hfil#\hfil\cr{\rm l.i.m.}\cr$\stackrel{}{{}_{p\to \infty}}$\cr}}}\sum\limits_{j_1,\ldots,j_5=0}^{p}C_{j_5 \ldots j_1}\zeta_{j_1}^{(i_1)}\ldots \zeta_{j_5}^{(i_5)},
\begin{aligned}\int_{\Sigma_s}(\partial_t \psi) (\partial_i \phi) + (\partial_i \psi) (\partial_t \phi) d x + \int_0^s \int_{\Sigma_t}(\Box \psi) (\partial_i \phi) + (\Box \phi) (\partial_i \psi) d x d t = \int_{\Sigma_0}(\partial_t \psi) (\partial_i \phi) + (\partial_i \psi) (\partial_t \phi) d x. \end{aligned}
-2\Re (A_L u \mid u) = \Omega\bigl((u,-A_Lu),(u,-A_Lu)\bigr) = \bigl\langle \bigl(u(0-),u'(0-),u''(0-)\bigr), \bigl(u(0-),u'(0-),u''(0-)\bigr)\bigr\rangle_- - \bigl\langle L\bigl( u(0-),u'(0-),u''(0-)\bigr), L\bigl(u(0-),u'(0-),u''(0-)\bigr)\bigr\rangle_+.
\dim_{H}\left\{x\in X \mid \overline{\lim}\left\lvert \frac{1}{N}\sum_{n=0}^{N-1}f\left(u_{p(n)}.x\right) - \frac{1}{N}\int_{0}^{N}f(u_{p(t)}.x)dt \right\rvert > 0 \right\}\leq 3-\frac{\delta}{d},
\Phi_{a}( z_{1}, \eta_{1}) \Phi_{b}( z_{2}, \eta_{2}) \sim ( z_{1}- z_{2})^{\Delta_{c}- \Delta_{b}- \Delta_{a}+ \bar{\eta}_{1}\eta_{2}+ \bar{\eta}_{2}\eta_{1}}C_{a b c}\Phi_{c}( z_{1}, \eta_{3})
g_j(x)=\sum_{k=\pm1}\frac{1}{v_j(\theta)}{\rm e}^{\theta x}\left[\mathbf{1}(k\neq j)\Lambda_{j,k}\overline{F}_{j,k}(x)+\mathbf{1}(k=j)\overline{\Upsilon}_{j}(x)\right],x\geq 0,
\tilde{m}_1=x_1, \tilde{m}_2=x_2-x_1, \tilde{\gamma}_0=-2N-2\nu_1-2\nu_2-2\nu_3-2\nu_4+1, \tilde{\gamma}_1=-2N-2\nu_2-2\nu_3-2\nu_4+1, \tilde{\gamma}_2=-2N-2\nu_3-2\nu_4+1, \tilde{\gamma}_3=-2N-2\nu_4+1,
\lambda^\mu \left( t\right) = \bar{X}^\mu +\xi^\mu t -\frac{1}{2}\Gamma^{\mu}_{\nu\rho}\xi^\nu\xi^\rho t^2 -\frac{1}{3!}\Gamma^\mu_{\nu\rho\kappa}\xi^\nu\xi^\rho\xi^\kappa t^3 +\ldots ,
\begin{array}{l}{( m , - m , 1 , p ) \longleftrightarrow \tilde{\beta}^{\frac{2 k^{\prime}}{\delta}m - \frac{k}{\delta}( p - k^{\prime})},}\\{( m , - m , 0 , p ) \longleftrightarrow \tilde{\beta}^{\frac{2 k^{\prime}}{\delta}m + \frac{k}{\delta}p}\tilde{\gamma}.}\\ \end{array}
\nu_{i}(x) = V^{-\frac{1}{2}}\sum_{{\bf k},r}\left[u^{r}_{{\bf k},i}e^{-i\omega_{k,i}t}\alpha^{r}_{{\bf k},i}\:+ v^{r}_{-{\bf k},i}e^{i\omega_{k,i}t}\beta^{r\dag}_{-{\bf k},i}\right] e^{i{\bf k}\cdot{\bf x}}, \; \; \; ~ i=1,2 \;.
y_q(u)= \sum_{\mathcal{R}:\mathcal{R}\ni q,|\mathcal{R}|=r+1}\sum_{i=1}^\rho\left[\sum_{p\in [N_T]}h_{qp}(u)v_{{\mathcal{R}},{[N_T]},p}^i(u)\right]x_{{\mathcal{R}},{[N_T]}}^i &+\sum_{\mathcal{R},\bar{\mathcal{R}}_i:\mathcal{R}\cup\bar{\mathcal{R}}_i\not\ni q}\left[\sum_{p\in [N_T]}h_{qp}(u)v_{{\mathcal{R}},{[N_T]},p}^i(u)\right]x_{{\mathcal{R}},{[N_T]}}^i,
\begin{cases}Lu= a^{ij}(x)D_{ij}u +b^i(x)D_iu+c(x)u=f(x) x\in \Omega, u\in\W^2_{p,\varphi}(\Omega,w) \cap \overset{\circ}W{}^1_p(\Omega,w), \p\in(1,\infty) \end{cases}
y'_n(z) = \sum_{m=0}^{n-1}(-1)^{n-m-1}(m+1) e_{n-m-1}z^m, y''_n(z) = \sum_{m=0}^{n-2}(-1)^{n-m-2}(m+2)(m+1) e_{n-m-2}z^m.
\begin{array}{ccc}(\gamma^0)^{\dagger}=\gamma^0, (\gamma^0)^T=-\gamma^0, (\gamma^0)^*=-\gamma^0, [2mm](\gamma^1)^{\dagger}=-\gamma^1, (\gamma^1)^T=\gamma^1, (\gamma^1)^*=-\gamma^1, [2mm](\gamma^2)^{\dagger}=-\gamma^2, (\gamma^2)^T=\gamma^2, (\gamma^2)^*=-\gamma^2.\end{array}
\binom{\nu_{\ell}-1}{h_{\ell}}\prod_{r\in J\setminus\{\ell\}}\binom{\nu_r}{h_r}=\prod_{r\in J}\binom{\nu_r-\delta_{\ell,r}}{h_r},\qquad\delta_{\ell,r}:=1_{\{r=\ell\}}.
\begin{array}[c]{l}Y_{t}^{1,v}=\mathbb{E}\left[ \int_{t}^{t+\delta}F_{1}(s,X_{s}^{v},Y_{s}^{1,v},Z_{s}^{1,v},v_{s})ds\right] , Y_{t}^{2,v}=\mathbb{E}\left[ \int_{t}^{t+\delta}F_{1}(s,x_{0},0,0,v_{s})ds\right] ,\end{array}
(q_n +1) (q_n +2) b_{q+ \overline{n},\kappa}+ \sum_{i \neq n}(q_i + 1) (q_i + 2) b_{q+ 2 \overline{i}- \overline n,\kappa}- (\kappa+1) b_{q- \overline{n}, \kappa+1}= 0.
y =\frac{\big( \sum m \tilde{u}_i e_i +(1+m\beta)\vec{v}\big)^T A \big( \sum m \tilde{u}_i e_i +(1+m\beta)\vec{v}\big)-N}{m^2Q^2}&=\frac{\sum_{1}^{d-1}\tilde{u_i}^2m^2\mu_i+(1+m\beta)^2N-N}{m^2Q^2}&=\sum_{1}^{d-1}x_i^2\mu_i+ x_d(2+x_d\frac{m^2Q^2}{N}).
\prod U \equiv U_{2}U_{4}U_{3}^{- 1}U_{3}U_{5}U_{1}^{- 1}U_{1}U_{6}U_{2}^{- 1}U_{2}U_{6}^{- 1}U_{5}^{- 1}U_{4}^{- 1}U_{2}^{- 1}= 1 .
Y^{b \beta}{}_{a \alpha}= ( ( \textstyle{\frac{4}{{\cal N}}}- 1 ) \kappa - \eta ) \delta^{b}{}_{a}\delta^{\beta}{}_{\alpha}+ \delta^{b}{}_{a}s_{\alpha}{}^{\beta}+ 2 s^{b}{}_{a}\delta^{\beta}{}_{\alpha}\, .
W=-\,\frac{1}{16}\,n\:\Gamma\left(\frac{7-p}{2}\right)\, \left[2\sum_{i}\omega_i^4-\left(\sum_{i}\omega_i^2\right)^2\right] \prod_{i}\omega_i^{-1}\left(\frac{2}{b}\right)^{7-p}+O\left(\frac{1}{b^{9-p}}\right).
\begin{array}{c}{C^{- 1}\left( \begin{array}{c}{\gamma_{4}}\\{\gamma_{5}}\\ \end{array}\right) C = R_{C}\left( \begin{array}{c}{\gamma_{4}^{T}}\\{\gamma_{5}^{T}}\\ \end{array}\right) ,}\\{\Pi^{- 1}\left( \begin{array}{c}{\gamma_{4}}\\{\gamma_{5}}\\ \end{array}\right) \Pi = R_{P}\left( \begin{array}{c}{\gamma_{4}}\\{\gamma_{5}}\\ \end{array}\right) ,}\\ \end{array}
E_{k}^{(2)}=\sum_{k^{\prime}\ne k}\frac{\langle\psi_{k}^{0}|V_{int}|\psi_{k^{\prime}}^{0}\rangle\langle\psi_{k^{\prime}}^{0}|V_{int}|\psi_{k}^{0}\rangle}{E_{k}^{(0)}-E_{k^{\prime}}^{(0)}}\equiv-\frac{1}{2}\sum_{i,j=1}^{3}\alpha_{ij}F_{i}F_{j}
\Phi_{1}^{( 1 )}= \pi_{e}, \quad \Phi_{2}^{( 1 )}= \pi_{\chi}, \quad \Phi_{3 \mu}^{( 1 )}= \pi_{\mu}- \frac{i}{2}\xi_{\mu}, \quad \Phi_{4 \mu}^{( 1 )}= \pi_{\mu}^{b}
\sin d_1 \geq |\langle p_i,[0,0,\sin d_1,-\cos d_1] \rangle| =|\sin d_1 \cos(\zeta(p_i))\cosh(r(p_i))-\cos d_1 \sin(\zeta(p_i))\cosh(r(p_i))| & =|\sin(d_1-\zeta(p_i))|\cosh(r(p_i))\,.
e^{\mu\nu\sigma\tau}e_{\mu\nu\lambda\rho}=-2\left\vert\begin{array}[c]{cc}\delta_{\lambda}^{\sigma}\delta_{\rho}^{\sigma}\delta_{\lambda}^{\tau}\delta_{\rho}^{\tau}\end{array}\right\vert =-2\left( \delta_{\lambda}^{\sigma}\delta_{\rho}^{\tau}-\delta_{\rho}^{\sigma}\delta_{\lambda}^{\tau}\right) ,
b(z_0\otimes\dots\otimes z_n) = z_0z_1\otimes\dots\otimes z_n &+\sum_{0}^{n-1}(-1)^i z_0\otimes\dots \otimes z_iz_{i+1}\otimes \dots\ \otimes z_n &+ (-1)^nz_n z_0\otimes\dots\otimes z_{n-1}
E (x,y)=1+\sum_{n\geq1}\sum_{\ell=1}^ne_{n,\ell}x^ny^\ell &=1+\frac{xy((2x^5-5x^4+4x^3-x^2)y^2+(2x^5-6x^4+11x^3-8x^2+2x)y-x^4+4x^3-6x^2+4x-1)}{(1-x)^2(1-2x)(1-xy)^2(xy+x-1)}.
~w(e\cup f)^{p-(r_1+r_2)}\cdot\prod_{v\in e\,\cup f}B(v,e\cup f) = ~\big(w_1(e)w_2(f)\big)^{p-(r_1+r_2)}\cdot\prod_{v\in e}B(v,e\cup f)\prod_{u\in f}B(u,e\cup f) = ~w_1(e)^{p-r_1}\prod_{v\in e}B_1(v,e)\cdot w_2(f)^{p-r_2}\prod_{u\in f}B_2(u,f) = ~\alpha_1\alpha_2.
\begin{aligned}\int_{T_1}^t \left< \varphi,\mathcal{T}(t-\tau) C_2 f_N^{(2)}(\tau) \right> d\tau = \int_{T_1}^t \left< \mathcal{T}(-(t-\tau)) \varphi,C_2 f_N^{(2)}(\tau) \right> d\tau & = \int_{T_1}^t \left<\Phi (\tau),C_2 f_N^{(2)}(\tau) \right> d\tau\end{aligned}
{\frac{\partial}{\partial p_{1}}}={\frac{1}{2}}\biggl ({\frac{\partial}{\partial x^{\prime \prime}}}-{\frac{\partial}{\partial y^{\prime \prime}}}\biggr ) , \quad{\frac{\partial}{\partial p_{2}}}={\frac{\partial}{\partial x^{\prime \prime}}}+{\frac{\partial}{\partial y^{\prime \prime}}},
\frac{\eta_n}{n}=\frac{\eta_{3N+2}}{3N+2}=\frac{N}{3N+2}\ \frac{1}{N}\sum_{i=1}^N(Z_{3i}-1) &+\frac{N+1}{3N+2}\Biggl(\frac{1}{N+1}\sum_{i=1}^{N+1}(Z_{3i-2}-1)+\frac{1}{N+1}\sum_{i=1}^{N+1}(Z_{3i-1}-1) \Biggr).
[T^0_g, Jx_m J]= \sum_nE_n T^{i(n)}_g [E_n, Jx_mJ]+\sum_n [E_n, Jx_m J] T^{i(n)}_g E_n & +\sum_n E_n [T^{i(n)}_g, Jx_mJ]E_n.
e = z^3 (\alpha^2 x^{4}+ \alpha^2 x^{3}+ x^{2}+ 1) + z^2 (\alpha x^{4}+ x^{2}+ \alpha^2 x) &+ z (\alpha x^{4}+ x^{3}+ \alpha x^{2}+ 1) + (\alpha^2 x^{4}+ \alpha x^{3}+ \alpha x^{2}+ \alpha^2 x + 1).
F_{1}({\bf{k}}) = 1 - \sum_{i ,{\bf{q}}}\frac{1 - n_{F}({\bf{k}}+{\bf{q}})}{({\tilde{\omega}}_{i}( -{\bf{q}}) + \frac{{\bf{k . q}}}{m}+ \epsilon_{{\bf{q}}})^{2}}g_{i}^{2}( -{\bf{q}})
(\hat a_j, \hat b_j, \hat c_j) = \begin{cases}(120, \ 119, \ 169) j=1 (2h_j-1,2h_j^2-2h_j, 2h_j^2-2h_j+1 ) h_j := 2^{j-2}+2 j =2, \dots, \nu,\end{cases}
g_{\mu\nu}=[\begin{matrix}-(1-\frac{2GM}{rc^{2}})&0&0&0\\ 0&(1-\frac{2GM}{rc^{2}})^{-1}&0&0\\ 0&0&r^{2}&0\\ 0&0&0&r^{2}sin^{2}\theta\end{matrix}].
\frac{z\;{}_{a}\mathtt{B}'_{2a-1, a \nu-a+1,1}(a^{a/2}z)}{{}_{a}\mathtt{B}_{2a-1, a \nu-a+1,1}(a^{a/2}z)}= \frac{a^{a/2}z}{a}\frac{{}_{a}\mathtt{B}_{2a-1, a \nu-a,1}(a^{a/2}z)}{{}_{a}\mathtt{B}_{2a-1, a \nu-a+1,1}(a^{a/2}z)}-(2-a)\nu+1-a &=\frac{z\mathtt{J}_{\nu-1}\left(z\right)}{\mathtt{J}_{\nu}\left(z\right)}-(2-a)\nu+1-a,
C^2_1\leq \frac{\widetilde{C}T^{p-1}_*}{p L^{2(p-1)}_*}+\frac{{C}T^{p-2}_*}{(p-1) L^{2(p-1)}_*}\textrm{and}C^2_2=\frac{2^{p-1}\Gamma(\frac{2p-1}{2})}{p\sqrt{\pi}}\frac{T^{p-1}_*}{L^{2(p-1)}_*}\,,
\bigcup_{k\ge k_0}\bigcup_{m\in \varphi^{-1}(n)\cap J_k}\Big\{s\in [1,b_{m+1}-b_m);\; \prod_{i=b_{m+1}-s}^{b_{m+1}-1}|w_{i}|= 2^{\delta^{(k)}}\Big\}
L_{k i n , L i o u}= \partial_{+}X^{\mu}\partial_{-}X^{\nu}G_{\mu \nu}- \frac{2 7 - 2 6}{4 8 \pi}( \frac{1}{2}\partial_{+}\phi \partial_{-}\phi - Q \hat{R}\phi + \mu_{t o t}e^{\alpha \phi}) ,
\mathrm{I_2}=\limsup_{\lambda\to0^+}\lambda^q\left\|\left[\int_{B}\left|\cdot-y\right|^{\gamma-n}\,dy\right]^{\frac{1}{q}}\mathbf{1}_{(2B)^\complement}(\cdot)\right\|_{X}&\leq2^{\frac{n-\gamma}{q}}\limsup_{\lambda\to0^+}\lambda^q|B|^{\frac{1}{q}}\left\||\cdot|^{\frac{\gamma-n}{q}}\mathbf{1}_{(2B)^\complement}(\cdot)\right\|_{X}=0.
h_{1;k}=(k+1)!\frac{k^{k}}{k!}\int_{\overline{M}_{1,1}}\frac{1-\lambda_{1}}{1-k\psi_{1}}=(k+1)k^{k}\{[\int_{\overline{M}_{1,1}}\psi_{1}]k-[\int_{\overline{M}_{1,1}}\lambda_{1}]\}.
({\rm Id}_4\otimes P)g(X^0)({\rm Id}_4\otimes P) =\begin{pmatrix}P(X^0_1 X^0_1-X^0_2X^0_2)P 0 0 P(X^0_1X^0_2+X^0_2X^0_1)P\end{pmatrix}\oplus \begin{pmatrix}0 P(X^0_1X^0_2-X^0_2X^0_1)P P(X^0_2X^0_1-X^0_1X^0_2)P 0\end{pmatrix}.
( \sum_{n = 1}^{1 5}( - 1 )^{n}- 1 ) ( \frac{g^{2}}{3 2 \pi^{2}}) t r \epsilon^{\mu \nu \alpha \beta}F_{\mu \nu}F_{\alpha \beta}= - 2 ( \frac{g^{2}}{3 2 \pi^{2}}) t r \epsilon^{\mu \nu \alpha \beta}F_{\mu \nu}F_{\alpha \beta}
\begin{cases}\partial_z^2\phi_2 = - \dfrac{\zeta}{h_b}\big( 1 + \dfrac{h}{h_b}\big) \Delta_X \psi \ \ \mathrm{in}\ \ \mathcal{S}_b, \phi_2|_{z=0}= 0, \ \ \partial_z \phi_2|_{z=-1 + \beta b}= 0. \end{cases}
\nabla\mathbf{u}^-(X)(A_i-X)=((A_i-X)^T\otimes I_2) \textrm{Vec}(\nabla\mathbf{u}^-(X)), &\nabla\mathbf{u}^-(\widetilde{Y}_i)(A_i-\widetilde{Y}_i)=((A_i-\widetilde{Y}_i)^T\otimes I_2) \textrm{Vec}(\nabla\mathbf{u}^-(\widetilde{Y}_i)).
\widetilde{( \delta_1 \circ \delta_2 + \delta_2 \circ \delta_1)(h)}&= (-1)^{n}~ [\mu_1, \widetilde{\delta_2 h}] ~+~ (-1)^{n}~ [\mu_2, \widetilde{\delta_1 h}] &= - [\mu_1, [\mu_2, \widetilde{h}] ] - [\mu_2, [\mu_1, \widetilde{h}] &= - [[\mu_1, \mu_2], \widetilde{h}] = 0 ~~~~ ( ~[\mu_1, \mu_2] = 0).
(D_t, U_t) =\left(\frac{N-|\mathcal{S}_r^{(t)}|}{S_r^{}-|\mathcal{S}_r^{(t)}|}, ~\frac{N-|\mathcal{S}_w^{(t)}|}{S_w^{}-|\mathcal{S}_w^{(t)}|}\right), \eta=\frac{K_c}{N},
{\cal H}_{C \omega}={\frac{1}{2}}p^{2}+{\frac{1}{2}}\omega^{2}q^{2}+{\frac{1}{2}}\sum_{\rho \in \Delta_{+}}{g_{| \rho |}^{2}}{\frac{| \rho |^{2}}{( \rho \cdot q )^{2}}}.
P_{\sigma}^{b b + 1}( a ) = \frac{S ( a ) \chi_{\mu_{a}/ 2}( q ) \chi_{a , s}^{( L + 2 )}( q )}{\sum_{a = 1}^{L}S ( a ) \chi_{\mu_{a}/ 2}( q ) \chi_{a , s}^{( L + 2 )}( q )}
MSE=\frac{1}{N}+\frac{3(N+1)}{N(N-1)}+f_{3}^{2}\frac{5(N+1)(N+2)}{N((N-1)(N-2)}+\rho_{3}^{2}[\frac{(N+1)(N+2)}{6}]^{2}(1-f_{3})^{2}
\begin{aligned}W_af =\lim_{t\to\infty}e^{itH}P_a^{{\varepsilon}_a}(t)J_ae^{-itH_a}P^a_Mf &=\lim_{t\to\infty}e^{itH}J_aP_a^{{\varepsilon}_a}(t)e^{-itH_a}P^a_Mf.\end{aligned}
C+C' \textstyle =\Big(\bigoplus_{\epsilon\in E_1}FH\epsilon\Big) +\Big(\bigoplus_{\epsilon\in E_0}FH\epsilon\Big) +\Big(\bigoplus_{\epsilon\in E_2}FH\epsilon\Big) +\Big(\bigoplus_{\epsilon\in E_0}FH\epsilon\Big) &\textstyle =\bigoplus_{\epsilon\in E_0\cup E_1\cup E_2}FH\epsilon =\bigoplus_{\epsilon\in E_C\cup E_{C'}}FH\epsilon.
\frac{d}{dt}\bigl(\mathcal{E}(\overline{D}_s)-\mathcal{E}(\overline{D}_0)\bigr)_{|_{t=s^+}}=\int_X(\log\frac{\|\cdot\|_0}{\|\cdot\|_1})c_1(L,\|\cdot\|_s)^{\wedge n}.
0 = \int_{M}|\nabla W|^{2}+ \frac{R}{2}\int_{M}|W|^{2}-3\int_{M\setminus M_{\varepsilon}}W_{ijkl}W_{ijpq}W_{klpq}-3\int_{M_{\varepsilon}}W_{ijkl}W_{ijpq}W_{klpq}&\geq \int_{M}|\nabla W|^{2}+ \frac{R}{2}\int_{M}|W|^{2}-3\Big(\sup_{M}|W|^{3}\Big)\operatorname{Vol}(M_{\varepsilon}) \,,
\phi_u(u, v)= \phi_u(t, x)= \frac{1}{2}\left(\phi_t+ \phi_x\right)(t, x), \; \; \phi_v(t, x)= \frac{1}{2}\left(-\phi_x+ \phi_t\right)(t, x), \phi_u\cdot \phi_v= \frac{1}{4}(-|\phi_x|^2+ |\phi_t|^2), \; \; \phi_{uv}= \frac{1}{4}(\phi_{tt}- \phi_{xx}).
\sum_{j=1}^\infty \frac{A_j^2}{Z_j^p}=\frac{\cos^2(Z_1)}{Z_1^{p-1}\left[Z_1 + \sin(Z_1)\cos(Z_1) \right]}+ \sum_{j=2}^\infty \frac{\cos^2(Z_j)}{Z_j^{p-1}\left[Z_j + \sin(Z_j)\cos(Z_j) \right]}.
\frac{\lambda_1^{(n)}}{\lambda_1^{(n)}-\lambda_{2}^{(n)}}=1+\frac{\lambda_{2}^{(n)}}{\lambda_1^{(n)}-\lambda_{2}^{(n)}}\leq 1+\sum_{k> 1}\frac{\lambda_k^{(n)}}{\lambda_1^{(n)}-\lambda_k^{(n)}},
tr_{M_n(N)}(Q_1) = (1/n)\sum_{i\in I}tr(q_1(i,i)) = (1/n)\sum_{i\in I}tr[E_N\{\alpha_0(\lambda_i{\lambda_j}^*)\}] = (1/n)\sum_{i\in I}tr (\lambda_i{\lambda_j}^*) ~~{\rm (by ~Lemma~ (\ref{kod})}) = tr_{M_n(N)}Q.
K(x) = 1 + k(x)\,\Big(b_0 + x \,b_0^+ + \frac1x \,b_0^-\Big), \mbox{with}k(x) = \frac{\left(x^2 - 1\right) \left(\alpha + \gamma\right)}{\left(\gamma x + \alpha\right)\left((\alpha + \gamma)(x - 1) + (q - 1)x \right)}\,.
K_{\nu}( x ) = \sqrt{\frac{\pi}{2 x}}\; e^{- x}\; \left[ 1 +{\frac{4 \nu^{2}- 1}{8 x}}+{\frac{( 4 \nu^{2}- 1 ) ( 4 \nu^{2}- 3^{2})}{2 ! \; ( 8 x )^{2}}}+ \dots \right] \; ,
\Bigl\lVert\frac{x_j^r}{1+\epsilon x^{2m}}(t)D_j^k(t)\psi\Bigl\lVert =\Bigl\langle\psi,D_j^k(t)\frac{x_j^{2r}}{(1+\epsilon x^{2m})^2}(t)D_j^k(t)\psi\Bigr\rangle^{1/2}&\leq\sum_{p=0}^k \binom{k}{p}\Bigl\lvert\Bigl\langle\psi_t,D_j^p\frac{x_j^{2r}}{(1+\epsilon x^{2m})^2}D_j^{2k-p}\psi_t\Bigr\rangle\Bigr\rvert^{1/2}
\sqrt{1-\tfrac{4\gamma\sigma}{1+2\gamma\sigma+\gamma^2\sigma\beta}}= \sqrt{1-\tfrac{2\sqrt{\sigma/\beta}}{1+\sqrt{\sigma}{\beta}}}=\sqrt{\tfrac{1-\sqrt{\sigma/\beta}}{1+\sqrt{\sigma/\beta}}}=\sqrt{\tfrac{\sqrt{\beta/\sigma}-1}{\sqrt{\beta/\sigma}+1}}.
\rho_{y_ku_k}= \frac{\sum_{l=1}^L (r_la_{l})^2 \sigma_x^2}{\sqrt{\sum_{l=1}^L r_l^2 \sigma_x^2 +\sigma_z^2}\sqrt{\sum_{l=1}^L a_{l}^2x_{l,i}}}= \frac{\sigma_x^2(\mathbf r^{T}\mathbf a)^2}{\sqrt{\sigma_x^2 \|\mathbf r\|^2 +\sigma_z^2}\sqrt{\sigma_x^2 \| \mathbf a\|^2}}.
\begin{array}{cll}E_{a}{\circ_{1,Z}}E_{b}=(Ze_{a})(Z^{\dagger}(Ze_{b})) = (Ze_{a})e_{b}& = -(Ze_{b})e_{a}\mbox{See{\bf [1]}}=-(Ze_{b})(Z^{\dagger}(Ze_{a})) =-E_{b}{\circ_{1,Z}}E_{a}. \end{array}
t_{\mu \nu}= \frac{1}{8 \pi{\cal G}}\left[ R_{\mu \nu}- \frac{1}{2}g_{\mu \nu}R_{\alpha}^{\alpha}- R_{\; \mu \nu}^{( 1 )}+ \frac{1}{2}\eta_{\mu \nu}R_{\; \alpha}^{( 1 ) \, \alpha}\right] \, .
\partial_{t}(\left\|\overline{\nabla}f \right\|^{2}) =2Ric'_{ij}\overline{\nabla}_{i}f\overline{\nabla}_{j}f+2g'(\overline{\nabla}{f},\overline{\nabla}\,\underline{\Delta}{f})+2g'(\overline{\nabla}{f},\overline{\nabla}\left\|\overline{\nabla}{f}\right\|^{2}) &-2g'(\overline{\nabla}{f},\overline{\nabla}\,{Scal'}),
\begin{pmatrix}C D \end{pmatrix}= \begin{pmatrix}a_{11}a_{12}a_{13}a_{14}a_{21}a_{22}a_{23}a_{24}\end{pmatrix}, J_{jk}:= \det \begin{pmatrix}a_{1j}a_{1k}a_{2j}a_{2k}\end{pmatrix}, j,k\in\{1,\ldots,4\}.
d s_{\mathrm{E}}^{2}= h^{- 1 / 4}H^{- 1 / 2}[ - f d x_{0}^{2}+ d x_{1}^{2}+ h ( d x_{2}^{2}+ d x_{3}^{2}) ] + h^{- 1 / 4}H^{1 / 2}[ f^{- 1}d r^{2}+ r^{2}d \Omega_{5}^{2}] .
= \frac{1}{3}+ \frac{1}{3}\times \frac{1}{4}+ \frac{1}{3}\times \frac{3}{4}\times \frac{1}{2}+ \frac{1}{3}\times \frac{3}{4}\times \frac{4}{5}\times \frac{1}{6}+ \frac{1}{3}\times \frac{3}{4}\times \frac{4}{5}\times \frac{5}{6}
\hat{\mathbf x}_1(\theta_\xi )=\begin{bmatrix}0 \sin \theta_\xi \cos \theta_\xi\end{bmatrix},\hat{\mathbf x}_2(\theta_\xi )=\begin{bmatrix}\sin \theta_\xi 0 \cos \theta_\xi\end{bmatrix},\hat{\mathbf x}_3(\theta_\xi )=\begin{bmatrix}\cos \theta_\xi \sin \theta_\xi 0\end{bmatrix}
T(p)=\frac{p^{4}-K_{4}p^{3}+80\pi^{2}p^{2}-K_{4}.4\pi^{2}p+9\pi^{4}}{p^{4}+K_{4}p^{3}+80.\pi^{2}p^{2}+K_{4}\cdot4\pi^{2}p+9\pi^{4}}whereK_{4}=\frac{9\pi^{2}}{8}
[|D_0|,|D_0|^{-\frac13}a^2|D|^{-\frac13}] = |D_0|^{-\frac13}\delta_0(a^2)|D_0|^{-\frac13}= |D_0|^{-\frac13}\delta_0(a)a|D_0|^{-\frac13}+|D_0|^{-\frac13}a\cdot\delta_0(a)|D_0|^{-\frac13}.
\omega_{\hat M_q}(0)( \theta_q (v_1 \wedge \dots \wedge v_n)) = \omega_{\hat M_q}(0)(v_1 \wedge \ldots \wedge v_n) = 1 \neq 1+ \kappa(q)^2 = \omega_M(q)(v_1\wedge \dots \wedge v_n).
S_0 = \frac{1}{4 e^2}\mbox{Re}\int d^4x\,d^2\theta\,W_a C^{ab}W_b+ \frac{1}{4}\int d^4x\, d^4\theta\,\Big(\phi^* e^{2V}\phi +\tilde\phi^*e^{-2V}\tilde\phi\Big).
\frac 1{N_k}\sum_{\lambda\in \Lambda_k}\int_M|\lambda-x|^2 \frac{|K_k(\lambda,x)|^2}{B_k(\lambda)}\,d\mu_k(x)\le \frac 1{N_k}\iint_{M\times M}|y-x|^2 |K_k(y,x)|^2\, d\mu_k(x)d\mu_k(y).
\rho_{t}=\frac{h h_{t}+\sum h_{k}h_{kt}}{\rho}, \rho_{i}=\frac{hh_{i}+\sum h_{k}h_{ki}}{\rho}=\frac{h_{i}w_{ii}}{\rho}, \rho_{ij}=\frac{hh_{ij}+h_{i}h_{j}+\sum h_{k}h_{kij}+\sum h_{ki}h_{kj}}{\rho}-\frac{h_{i}h_{j}w_{ii}w_{jj}}{\rho^{3}}.
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})}\; .
\phi(x) = \sum_{i\in F}r_{i}\phi(1_{B_{i}}) = \sum_{i\in F}\sum_{k\in F_{i}}r_{i}1_{B_{ik}}p_{\delta_{ik}}= \sum_{\delta\in \Delta}\sum_{i\in F_{\delta}}r_{i}1_{B_{i, k(\delta)}}p_{\delta}= 0.
\gamma(s,\tau)=\left( r_*\cos\left( \frac{-s-a_1\tau}{\sqrt{k^2+r_*^2}}\right), r_*\sin\left( \frac{-s-a_1\tau}{\sqrt{k^2+r_*^2}}\right), \frac{ks-b_1\tau}{\sqrt{k^2+r_*^2}}\right)^t,
\begin{aligned}\delta_n(\alpha ,\beta ) = \frac{1}{\pi}\Big[\arccos \frac{\cos \alpha}{\sqrt{[n +{\delta_n}(\alpha ,\beta )]^2 \sin^2\alpha +\cos^2\alpha}}- \arccos \frac{\cos \beta}{\sqrt{[n +{\delta_n}(\alpha ,\beta )]^2 \sin^2 \beta + \cos^2\beta}}\Big] \end{aligned}
\begin{aligned}h_4(t) =e^{8 t}(t^3-6 t^2+18 t-24)+12 e^{7 t}t-4 e^{6 t}(t^3-6 t^2+18 t-24) &\quad+4 e^{5 t}(4 t^3+6 t^2+6 t-9) t+6 e^{4 t}(t^3-6 t^2+18 t-24) &\quad+4 e^{3 t}(16 t^3-12 t+9) t-4 e^{2 t}(t^3-6 t^2+18 t-24) &\quad+4 e^t (4 t^3-6 t^2+6 t-3) t+t^3-6 t^2+18 t-24\end{aligned}
\frac{\partial}{\partial t}\int_{V}\langle\phi\rangle dV=\oint K_{0}\nabla\langle\phi\rangle\cdot\vec{n}dA-\oint\langle\phi^{\prime}\vec{u}^{\prime}\rangle\cdot\vec{n}dA-\oint\langle\phi\rangle\langle\vec{u}\rangle\cdot\vec{n}dA
{\cal L}^{(3+1)spin}_{1/m^2}=\frac{\alpha}{720 \pi m^2}\left[18 F^{\nu \lambda}F_{\nu\lambda,\mu}^{~~~~\mu}+7 F^{\nu \lambda,\mu}F_{\nu\lambda,\mu}-2F^{\nu \lambda,}_{~~~\lambda}F_{\nu\mu,}^{~~~\mu}\right],
X =2\binom m3+\binom m2-\frac 14|N|\big(|M|^2+|MN|+2|N|-2\big) &\quad+\frac 12\big\{m\left(m+M-1\right)+|M|(|M|+1-m-n)-|N|\big\}n &\quad+\frac 12\left(\binom{|M|+|N|+n}3-\binom n3\right) &\quad+\frac m2\left(\left[\frac{(|M|+|N|+n-1)^2}4\right]+\left[\frac{(n-1)^2}4\right]\right).
Z ={\cal N}\int \prod_{x}\{\mathrm{d}v_{\mu}\mathrm{d}j^{\mu}\mathrm{d}j_{5}^{\mu}\mathrm{d}| h |^{2}\mathrm{d}\alpha \}e^{\mathrm{i}\int \mathrm{d}^{4}x{\cal L}[ v_{\mu}, j^{\mu}, j_{5}^{\mu}, | h | , \alpha ]},
\zeta_{g,h}\colon\{\cal L}_{Q_g,w_g}\otimes \big(g^{-1}\big)^*{\cal L}_{Q_h,w_h}\cong{\cal L}_{Q_g+g^{-t}Q_hg^{-1},w_g+w_h}={\cal L}_{Q_{gh},w_{gh}}.
\mathfrak{A}_0=\mathbb{R}x_{\varphi_{_1}}\oplus\cdots\oplus\mathbb{R}x_{\varphi_{_r}},\quad\quad\mathfrak{A}_c=\mathbb{R}y_{\varphi_{_1}}\oplus\cdots\oplus\mathbb{R}y_{\varphi_{_r}},
c_{0}= \frac{2 ( t r 1 ) \Lambda^{D - 2}}{( 4 \pi )^{D / 2}\Gamma ( D / 2 ) ( D - 2 )}\ , \c_{1}= \frac{2 ( t r 1 ) \Gamma ( 2 - D / 2 ) ( D - 1 )}{( 4 \pi )^{D / 2}( D - 2 )}\ .
\mathrm{R}_{\mathrm{D}}(\lambda, \delta) = \langle\lambda, \delta\rangle+ \sup_{x^{\prime}\in \mathbb{R}^d}\left[ \sum_{1\leq \ell \leq 2}\varphi_\ell(x^{\prime}, \lambda_\ell)\right] \mathrm{R}_{\mathrm{D}}(\lambda, \delta) = \langle\lambda, \delta\rangle+ \sum_{1\leq \ell \leq 2}\sup_{x^{\prime}\in \mathbb{R}^d}\varphi_\ell(x^{\prime}, \lambda_\ell).
f_{t,\ell,u}(s) := \begin{cases}f_{t}(s) \ell\leq s\leq u f_{t}(u)+f_{t}'(u)(s-u)+\frac{1}{2}f_{t}''(u)(s-u)^{2}s\geq u f_{t}(\ell)+f_{t}'(\ell)(s-\ell)+\frac{1}{2}f_{t}''(\ell)(s-\ell)^{2}\end{cases}.
H(z,t) =-\int_{0}^{1}\sigma\left( T(S_{t}\dot{z}_{t})\dot{S}_{t}S_{t}^{-1}(\lambda z),z\right) d\lambda & =-\int_{0}^{1}\sigma\left( \dot{S}_{t}S_{t}^{-1}(\lambda z),z\right)d\lambda-\int_{0}^{1}\sigma\left( S_{t}\dot{z}_{t},z\right) d\lambda
\zeta_{0,p^{\prime}}(s,c):=\sum_{m,n\neq0}\frac{p^{\prime2}+\lambda_{n}^{\prime2}}{p^{\prime}\left(p^{\prime}+\frac{1}{\pi}\right)+\lambda_{n}^{\prime2}}\,\frac{1}{\left(\left(m-\frac{1}{2}\right)^{2}+c\,\lambda_{n}^{\prime2}\right)^{s}},\,\,\,\,(s)>1.
Z(\kappa) = \int\! [d\phi(x)]\, \exp\Biggl[ -\int\! d^{2}x\, \Bigl({1\over 2}\, \partial_{a}\phi\partial_{a}\phi +{\Lambda^{2}\over 2\kappa}:\!\phi^{2}\! :+ \Lambda^{2}\sqrt{2\pi\over\kappa}\, :\!\phi^{3}\!: +\pi\Lambda^{2}:\!\phi^{4}\! :\Bigr)\Biggr]\, .
\sum_{\rho}\hat{g}(\rho)= -g(1)\log|d_F|+\int_{W_F}g(|w|)\chi(w)(|w|^\frac12+|w|^{-\frac12})dw &-\sum_v pv_0\int_{W_v}g(|w|)\chi_v(w)\frac{|w|^\frac12}{|1-w|}dw,
\partial_{t}\left(\int_{0}^{\frac{1}{2}}d\xi \left(\chi_{\leq \frac{1}{4}}(\xi)-1\right) \frac{\sin(t\xi)}{t^{2}}\left(\frac{(b-1)}{\xi \log^{b}(\frac{1}{\xi})}+\frac{b(b-1)}{\xi \log^{b+1}(\frac{1}{\xi})}\right)\right) &=\frac{- \sin(\frac{t}{2})}{t^{3}}\left(\frac{b-1}{\log^{b}(2)}+\frac{b(b-1)}{\log^{b+1}(2)}\right) +
\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}
\begin{array}{ll}\displaystyle \max_{\{A_{k}\}_{k =1}^{K},\{w_{k}\}_{k =1}^{K}}\displaystyle \sum_{k = 1}^{K}\frac{1}{A_{k}^{-1}+ \rho(\tau_{k}, \alpha)}\mbox{subject to:}\displaystyle \sum_{k = 1}^{K}A_{k}= 1, \sum_{k = 1}^{K}w_{k}= 1 & A_{k}\geq 0, w_{k}\geq 0 \; \; k = 1,\cdots K. \end{array}
(I_{+}+ I_{-}) \wedge \frac{d \chi_{\gamma_e}(\zeta)}{\chi_{\gamma_e}(\zeta)}= \frac{iR}{4}(L^{+}_1 + L^{-}_1 + L^{+}_2 + L^{-}_2) \wedge \frac{d \chi_{\gamma_e}(\zeta)}{\chi_{\gamma_e}(\zeta)}
\tilde{\phi}(v) =\tilde{\phi}(v_1)e^{-\int_{v_1}^v\left(\frac{\eta}{r}\right)(v')dv'}&\leq\tilde{\phi}(v_1)\left(\frac{v^\ast-v}{v^\ast-v_1}\right)^{\frac{\underline{\eta}}{\frac{1}{2}\tilde{\phi}(v_1)-\tilde{C}}}.
\begin{array}{lcl}K_{m+\alpha}((m+\alpha)z')\ \sim\ \sqrt{\frac{\pi}{2(m+\alpha)}}\frac{e^{-(m+\alpha)\eta}}{(1+\left(z\frac{m}{m+\alpha}\right)^2)^{1/4}}\left\{1+\sum_{j=1}^{\infty}(-1)^j \frac{u_j([1+\left(z\frac{m}{m+\alpha}\right)^2]^{-1/2})}{m^j \left(1+\frac{\alpha}{m}\right)^j}\right\}\end{array}
p_1(L) = \{g\in G\ ;\ \exists h\in H,\ (g,h)\in L\}, p_2(L) = \{h\in H\ ;\ \exists g\in G,\ (g,h)\in L\}, k_1(L) = \{g\in G\ ;\ (g,1)\in L\}, k_2(L) = \{h\in H\ ;\ (1,h)\in L\}.
\textbf{R}(z|t_1 t_2\dots t_{m-1})= (z-t_1)^{\varrho_1-1}(t_1-t_2)^{\varrho_2-1}\dots(t_{m-2}-t_{m-1})^{\varrho_{m-1}-1}(t_{m-1}^{\varrho_m-1})\times &(1-z)^{\omega_1}(1-t_1)^{\omega_2-\omega_1-\varrho_1}\dots(1-t_{m-2})^{\omega_{m-1}-\omega_{m-2}-\varrho_{m-2}}(1-t_{m-1})^{\omega_{m}-\omega_{m-1}-\varrho_{m-1}}).
\lim_{(x_{1}, x_{2}) \rightarrow (x_{0}, 0)}\gamma^{a}(x_{1}, x_{2}) =\ a_{-}\int_{-\infty}^{x_{0}}H(y_1)[H(y_1)]^T dy_{1}& + a_{+}\int_{x_{0}}^{+\infty}H(y_1)[H(y_1)]^T dy_{1}.
\partial_{w_{aa}}\left(\frac{1}{I_{q'^{2}}+R\left(W'\right)}\cdot G\left(W,Z\right)\right)=\frac{1}{I_{q'^{2}}+R\left(W'\right)}\cdot\partial_{w_{aa}}\left( G\left(W,Z\right)\right)+\partial_{w_{aa}}\left(\frac{1}{I_{q'^{2}}+R\left(W'\right)}\right)\cdot G\left(W,Z\right),
\underset{x_1\in\mathbb{R}^{n_1},\ldots,x_L\in\mathbb{R}^{n_L}}{\mbox{minimize}}f(x_1,\ldots,x_L) ~ T_{K_l,m_l,p_l}(D_lx_l-c_l)=0, l=1,\ldots,L.