harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
(-\Delta_p)^s \widetilde\Gamma (x) =2\,\int_{B_1}\frac{J_p(\Gamma(x)-1)-J_p(\Gamma(x)-\Gamma(y))}{|x-y|^{N+s\,p}}\, dy &=2\,\int_{B_1}\frac{J_p(\Gamma(y)-\Gamma(x))-J_p(1-\Gamma(x))}{|x-y|^{N+s\,p}}\, dy.
\mathcal{A}_{i}=\{X_{j}:{\textstyle\sum_{x\in X_{j}}}f(x)=j,\j=1,2,3\}\mathcal{A}_{4}=\{X_{j}:{\textstyle\sum_{x\in X_{j}}}f(x)\geq4\}\mathcal{A}={\textstyle\bigcup_{i=1}^{4}}\mathcal{A}_{i}.
\Re(C_E) = O+\frac{\beta(l_j)}{2}\Re(\bar{z}_1z_2\mu_{\beta}) & + \tanh(\beta(a))\Re(\bar{z}_1z_2\frac{-\beta(l_j)}{4}(\tanh(\beta(a))+\coth(\beta(a)))\mu_{\beta}) & + \tanh(-\beta(a))\Re(z_1\bar{z}_2\frac{3\beta(l_j)}{4}(\coth(\beta(a))-\tanh(\beta(a)))\theta(\mu_{\beta})) & = O+\beta(l_j)(1-\tanh^2(\beta(a)))\Re(\bar{z}_1z_2\mu_{\beta})
\langle g L, g L_w \rangle = \langle g L, \chi(g)^{-1}L_w \rangle = \overline{\chi(g)}^{-1}\langle gL, L_{gw}\rangle =\overline{\chi(g)}^{-1}(gL )(gw) = \overline{\chi(g)}^{-1}L(w) =\overline{\chi(g)}^{-1}\langle L, L_w \rangle,
\mathcal A_{\omega_1}([\gamma_+,w_+]) - \mathcal A_{\omega_0}([\gamma_-,w_-]) = \int_{-\infty}^{\infty}\frac{d}{ds}\mathcal A_{\omega_s}([u(s, \cdot), w_- \# u((-\infty, s] \times S^1)]) ds.
{\mathcal{R}_{mi}}(A\ast B)\simeq\bigoplus_{t=a}^{b}\mathbf{1}_{pt}^{f_{m,t}(v)}\boxtimes({\mathcal{R}_{ti}}(A)\ast{\mathcal{R}_{(m-t)i}}(B))[-P'_t](-\frac{P'_t}{2}),
\begin{array}{l}\nabla_B\nabla_Bh(X,\nabla_BX)=g_{B}(\nabla_X\nabla_Bh,\nabla_Bh)=\frac{1}{2}X(|\nabla_Bh|^2), \noalign{\smallskip}\nabla_B\nabla_Bh(X,\nabla_BX)=\left( |\nabla_B h|^2-c\right)h^{-1}X(h), \end{array}
\sum_{n \in D}\frac{1}{q(n)}\geq \sum_{k>d}\frac{|D \cap \{2^{(k-1)d},...,2^{kd}-1\}|}{q(2^{kd})}\geq \sum_{k>d}\frac{2^{-d}\cdot 2^{kd-1}}{q(2^{kd})}= \infty.
- \left( 1 - \frac{w \kappa}{4}e^{2\phi}\right) \partial^2 \phi + 2(\partial \phi)^2 + \frac{\kappa}{4}e^{2\phi}\partial^2 Z +e^{2\rho+2\phi}\left( \Lambda e^{-2\phi}- 1 \right) = 0;
P \bigg( \big|Z_j^{S}(\tau_{j+1})\big| > \frac{e^{-b}}{2}e^{sq_j (\tau_j' - \tau_j)}\bigg|{\cal F}_{\tau_j'}\bigg) \leq \frac{C X'_j(\tau_j')}{sq_j e^{-2b}}e^{-2sq_j(\tau_j' - \tau_j)}= C X_j'(\tau_j') (sq_j)^5 e^{2b}.
{\cal H}_{i n t}( x ) = - e \overline{{{\psi}}}^{( 0 )}( x ) \beta^{\mu}A_{\mu}^{( 0 )}( x ) [ I + \frac{e}{m}( I - \beta_{0}^{2}) \beta^{\nu}A_{\nu}^{( 0 )}( x ) ] \psi^{( 0 )}( x ) \, \, ,
\begin{aligned}\rho_{n+1}(x,y)={}\rho(x,y,n\tau +\tau) =\rho(x,y,n\tau)+\tau\frac{\partial}{\partial t}\rho(x,y,n\tau)+\frac{1}{2}\tau^{2}\frac{\partial^{2}}{\partial t^{2}}\rho(x,y,n\tau)\end{aligned}
\i=1 \ldots t, \ \left| \frac{P_i(v)}{Q_i(v)}\right| = \left| \frac{P_i(y)}{Q_i(y)}\right|. \i=t+1\ldots n, \ \widetilde{\frac{P_i(v)}{Q_i(v)}}=\widetilde{\frac{P_i(y)}{Q_i(y)}}.
\left\{-\frac{\gamma}{8N^2}\int d\theta \rho(\theta)\left\{\left(\frac{\partial}{\partial\theta}\frac{\delta}{\delta\rho(\theta)}\right)^2-{\cal V}^2(\theta)\right\}+V[\rho]\right\}~\tilde\psi[\rho]~=~E~\tilde\psi[\rho] \; .
\|\pi(\sigma_{\check F}(f))\xi\|^2 =\int_{r(W)}D(r^{-1}(x))^{-1}|f(r^{-1}(x))|^2\|\xi_{s(r^{-1}(x))}\|^2d\nu(x) &=\int_{s(W)}|f(s^{-1}(x))|^2\|\xi_x\|^2d\nu(x),
c_i(\lambda) = \frac{1}{3(f^i(\lambda))^2}\left[ Q^{ii}(\lambda)- \lambda^i Q^{ii}_1(\lambda) + \sum_{k \neq i}\frac{(P_2^{ki}(\lambda)-\lambda^i P_1^{ki}(\lambda))^2}{f^k(\lambda)(\lambda^k - \lambda^i)}\right].
\mathcal{T}_{S}(x) :=\big\{w\in\mathbb{X}\ |\ \exists\, \tau_k\downarrow 0\ \{\rm s.t.}\ \{\rm dist}(x+\tau_kw,S)=o(\tau_k)\big\}, \mathcal{T}_{S}^{i}(x) :=\big\{w\in\mathbb{X}\ |\{\rm dist}(x+\tau w,S)=o(\tau)\ \ \forall\tau\ge 0\big\}.
\widetilde{\psi}\left( \mathrm{i}\alpha\right) =\psi\left(\mathrm{i}\alpha\right) -\int K\left( \alpha,s\right) \mathrm{d}\mu\left(s\right) & =\psi\left( \mathrm{i}\alpha\right) -\int K\left( \alpha,s\right)y\left( s\right) \mathrm{d}\sigma\left( s\right) & =\psi-\mathbb{K}y=y.
P(b) =\left(\frac{4\pi Vc}{b}\right)^N\prod_{i=1}^N \bigl({m_i}^2K_2(m_ibc^2)\bigr)\,, \rho_b =\frac{1}{P(b)}\exp\left(-bc\sum_{i=1}^N\sqrt{{p_i}^2+{m_i}^2c^2}\right)\,.
x_1 \begin{pmatrix}1 0 0 -1 \end{pmatrix}+ y_1 \begin{pmatrix}0 1 0 0 \end{pmatrix}, &x_2 \begin{pmatrix}1 0 0 -1 \end{pmatrix}+ y_2 \begin{pmatrix}0 1 1 0 \end{pmatrix}, &x_3 \begin{pmatrix}1 0 0 -1 \end{pmatrix}+ y_3 \begin{pmatrix}0 \epsilon 1 0 \end{pmatrix},
A'(t)\geq \max \left( (\kappa_2+\frac 1C) \sum_{j=1}^{J_0}\theta_j \beta_j^2(t) \ , \ (\frac{1}{\kappa_2}+\frac 1C) \sum_{j=1}^{J_0}\theta_j (\lambda_j'(t))^2 \right).
L^0 \,=\,\eta_{ij}\Big( \epsilon P_0^i \partial_\tau X^j_0,+ \,P_1^i \partial_\tau X^j_1\, + ...+ P_n^i \partial_\tau X^j_n \,+\,...+P_{N-1}^i \partial_\tau X^j_{N-1}\,+\,\epsilon P_N^i \partial_\tau X^j_N \,\Big)\,+ V
\partial_k^*(a^*) \circ b^* = a^* \circ \eth_k^*(b^*) + \partial_k^*(a^* \circ b^*), a^* \circ \partial_k^*(b^*) = \eth_k^*(a^*) \circ b^* + \partial_k^*(a^* \circ b^*), \forall k=1, \dots, m,
f_{\beta,\gamma}= \frac{zb_jf_{j+1}+\overline\beta(\beta\alpha_j+\gamma\rho_j)b_j +\overline\gamma(\beta\rho_j-\gamma\overline\alpha_j)f_{j+1}}{1+z\gamma(\overline\beta\rho_j-\overline\gamma\alpha_j)b_j +\beta(\overline\beta\overline\alpha_j+\overline\gamma\rho_j)zf_{j+1}}.
\|F\|_{L_\omega^q L_t^\infty(C_u)}^q = \int_{{\mathbb S}^2}\sup_{\Upsilon_\omega}|F|^q d\mu_{{\Bbb S}^2}\mbox{and}\|F\|_{L_x^q L_t^\infty(C_u)}^q=\int_{{\Bbb S}^2}\sup_{t\in \Upsilon_\omega}(v_{t}|F|^q) d\mu_{{\Bbb S}^2}
F_{B}= L \int_{0}^{1}d \alpha \, \left[ 2 x_{B}- \frac{1}{L}\sum_{j = - L / 2 + 1}^{L / 2}\frac{2 \cos ( 2 \pi j / L )}{1 + \alpha - 2 \alpha \cos ( 2 \pi j / L ) + \alpha g x_{B}}+ \frac{g x_{B}^{2}}{2}- x_{B}\right]
-{2 \over 3}\sum_i{\rm tr}F_{9i}{\rm tr}F_{9i}^3-{2 \over 3}\sum_i{\rm tr}F_{5i}{\rm tr}F_{5i}^3 -{1 \over 6}\sum_{ij}w_{ij}{\rm tr}F_{5i}^3{\rm tr}F_{9j}-{1 \over 6}\sum_{ij}w_{ij}{\rm tr}F_{5i}{\rm tr}F_{9j}^3.
\frac{(2n+|x|_{1})!}{n! \, (n+|x|_{1})!}\leq 2^{2n+|x|_{1}}\mbox{and}\frac{n!}{\prod_{i=1}^{d}q_{j}!}\leq \frac{n!}{\left(\lfloor{\frac{n}{d}\rfloor}\right)!^{d}}\leq C_{d}\frac{d^{n}}{n^{\frac{d-1}{2}}},
|E_{s}| = 2 \cdot b_{k-2}\cdot 2^{\mathfrak z(\lfloor b_{k-3}\cdots b_0\rfloor_2)}&=b_{k-2}\cdot 2^{\mathfrak z(s)}=\begin{cases}2^{\mathfrak z(s)}b_{k-2}=1, 0 .\end{cases}
\widetilde{\Gamma}_{ac}^d \widetilde{\Gamma}^c_{db}+\frac{\theta}{2}\partial_1\Theta_{ab}\chi -\theta (\partial_c\ln |\theta|+ \widetilde{\Gamma}^d_{dc})\widetilde{\Gamma}^c_{ab}= \frac{\widetilde{R}^{(n-1)}-\Theta^{cd}\widetilde{T}_{cd}}{n-1}\Theta_{ab}+\widetilde{T}_{ab};\; a,b,c,d,e,f=2,...,n;
\begin{gathered}\tau_1+\tau_2= a_{ijk}e^i\otimes e^{jk}\in V^*\otimes \Lambda^2V^*, \tau_3+\tau_4= c_{ijk}e^i\otimes e_{jk}\in V^*\otimes \Lambda^2V, \tau_5+\tau_6= b_{ijk}e^{n+i}\otimes e_{jk}\in V\otimes \Lambda^2V, \tau_7+\tau_8= d_{ijk}e^{n+i}\otimes e^{jk}\in V\otimes \Lambda^2V^*.\end{gathered}
u_{\underline{{{m}}}}^{i}= G_{{\cal L}}^{i k}H_{R}^{k j}( - r_{\underline{{{m}}}}^{j}+ e^{2 W}l^{( -- ) \underline{{{n}}}}r_{\underline{{{n}}}}^{j}r_{\underline{{{m}}}}^{( + + )}) ~ .
\sum_{\ell\geq \beta m}{d-k+m\choose m-\ell}=\sum_{\ell\leq (1-\beta) m}{d-k+m\choose \ell}\leq\left(\frac{e(d-k+m)}{\beta_2 m}\right)^{\beta_2 m}\leq \left(\frac{e d}{\beta_2 m}\right)^{\beta_2 m}.
\begin{cases}\displaystyle HS_{q}(gr^{i,j}(J_{\infty}(B_{i.j-1})))=HS_{q}(J_{\infty}(B_{i.j-1})) \mbox{if}j-1\geq 1 HS_{q}(gr^{i.j}(J_{\infty}(B_{i-1.i-1})))=HS_{q}(J_{\infty}(B_{i-1.i-1})) \mbox{if}j=1 \end{cases}.
f^{1}( u ) \equiv f^{( + 1 , 0 )}( u ) = f^{i}u_{i}^{1}+ g^{( i j ) k}u_{i}^{1}u_{j}^{1}u_{k}^{4}+ h^{i j k}u_{1}^{1}u_{j}^{2}u_{k}^{3}+ \ldots \; .
\mathbb{E}\left[\|v^{k+1}(\ell) - [y^{k+1}]_\ell\mathbf{1}\|\right] \le \left(\sqrt{\lambda}\right)^{k+1}\mathbb{E}\left[\|v^0(\ell) - [y^{0}]_\ell\mathbf{1}\|\right]+ \sqrt{2}C\alpha_{\max}\sum_{s=0}^k(\sqrt{\lambda})^s\qquad\hbox{for all}k\ge0.
D_{\;\;ij}^{kl}=\frac 12\left( \delta_{\;\;i}^{\left[ k\right.}\delta_{\;\;j}^{\left. l\right]}-\frac 1{\triangle}\delta_{\;\;m}^{\left[ l\right.}\partial_{}^{\left. k\right]}\delta_{\;\;\left[ j\right.}^m\partial_{\left. i\right]}^{}\right) .
(\begin{matrix}B^{+}\\B^{-}\end{matrix})_{n}=\frac{1}{2\gamma_{n}}(\begin{matrix}exp(-u_{n}L_{n})&0\\ 0&exp(u_{n}L_{n})\end{matrix})(\begin{matrix}\gamma_{n}+\gamma_{n+1}&\gamma_{n}-\gamma_{n+1}\\ \gamma_{n}-\gamma_{n+1}&\gamma_{n}+\gamma_{n+1}\end{matrix})(\begin{matrix}B^{+}\\B^{-}\end{matrix})_{n+1}
\left\vert u\right\vert_{b,2}:=\left( \int_{\Theta}b(x)u^{2}\right)^{1/2},\qquad\left\vert u\right\vert_{c,2^{\ast}}:=\left( \int_{\Theta}c(x)\left\vert u\right\vert^{2^{\ast}}\right)^{1/2^{\ast}},
u\otimes u=uu^{T}=[\begin{matrix}u_{x}^{2}&u_{x}u_{y}&u_{x}u_{z}\\u_{x}u_{y}&u_{y}^{2}&u_{y}u_{z}\\u_{x}u_{z}&u_{y}u_{z}&u_{z}^{2}\end{matrix}],[u]_{\times}=[\begin{matrix}0&-u_{z}&u_{y}\\u_{z}&0&-u_{x}\\ -u_{y}&u_{x}&0\end{matrix}].
F_{n+\frac{1}{2}}= F \left( t_{n+\frac{1}{2}}, u\left(t_{n+\frac{1}{2}}\right) -\frac{\Delta t^2}{8}F' +O(\Delta t^3) \right) = F \left( t_{n+\frac{1}{2}}, u\left(t_{n+\frac{1}{2}}\right) \right) - \frac{\Delta t^2}{8}\frac{\partial F}{\partial u}F' + O(\Delta t^3),
\Sigma' \le \sum_{\substack{q_2\sim Q_2 (q_2,10)=1}}\sum_{\substack{a< d_1d_2q_1q_2 (a,d_1d_2q_1q_2)=1}}\sup_{|\eta|\ll 1/R}F_{R}\Bigl(\frac{a}{d_1d_2q_1q_2}+\eta\Bigr)^2 &\ll d_1d_2Q_1Q_2^2\int_0^1 F_{R}(t)^2 d t+\frac{d_1d_2Q_1Q_2^2}{R}\int_0^1|F_{R}'(t)|F_{R}(t) d t.
\left(I_{0+}^{\alpha,\alpha',\beta,\beta',\gamma}t^{\rho-1}\right)(x)=\frac{\Gamma(\rho)\Gamma(-\alpha'+\beta'+\rho)\Gamma(-\alpha-\alpha'-\beta+\gamma+\rho)}{\Gamma(\beta'+\rho)\Gamma(-\alpha-\alpha'+\gamma+\rho)\Gamma(-\alpha'-\beta+\gamma+\rho)}x^{-\alpha-\alpha'+\gamma+\rho-1}.
\dim ev^{-1}(x,y) = \dim \overline{\mathcal{M}_{0,2}}(X, l_1+l_2) - 2 \dim X &= \dim X + \int_{l_1 + l_2}c_1(T_X) -1 - 2 \dim X &= \int_{l_1+l_2}((n-1)[h_1] + (n-1)[h_2]) - \dim X -1 &= 0.
\Big| \frac{1}{2 \varepsilon n^2}\sum_{x}\Phi_n(s, \tfrac{x}{n}) \sum_{r=1}^{\varepsilon n}\int_{\Omega_1(x)}\overleftarrow{\eta}^{\ell}(x) [\eta(x+1)-\eta(x+1+r) ] [f(\eta) - f( \eta_{1,x,r}) ] d\nu_{b}\Big|,
\bar h^{1^{\ast}}_{11}\bar h^{1^{\ast}}_{11k}+3\bar h^{1^{\ast}}_{12}\bar h^{1^{\ast}}_{12k}+3\bar h^{2^{\ast}}_{12}\bar h^{2^{\ast}}_{12k}+\bar h^{2^{\ast}}_{22}\bar h^{2^{\ast}}_{22k}=0, \ \ \k=1, 2.
J ={\frac{i g^{2}f_{a c d}f_{b c d}}{( 2 \pi )^{6}}}{\frac{( 1^{-}2^{-}) ( \theta_{3 1})^{2}( \theta_{3 2})^{2}}{\hat{x}_{1 2}^{2}\check{x}_{3 1 ; 2}^{2}\check{x}_{3 2 ; 1}^{2}}}.
\mathcal{L}[\beta](k, 2\pi i y) = \int_{-\infty}^{\infty}\beta(k,t)H(t) e^{-2\pi i y t}dt &=\widehat{\beta}^t \ast \widehat{H}^t (y) &=\frac{1}{2}\widehat{\beta}^t(k,y) - \frac{i}{2\pi}\int_{0}^{\infty}\frac{\widehat{\beta}^t(k,y-\tau)-\widehat{\beta}^t(k,y+\tau)}{\tau}d\tau.
\exp \left({\sum_{j = 1}^{m + 1}\sum_{n = 1}^{\infty}E_{j j}^{( n )}u_{j}^{( n )}}\right) = \sum_{j = 1}^{m + 1}E_{j j}e^{\sum_{n = 1}^{\infty}\lambda^{n}u_{j}^{( n )}}= D ({\bf u}, \lambda )
(\Box_{b,m}u)\otimes s=e^{-im\theta}e^{-m\varphi}(\Box_{B,m}e^{m\varphi}(\tilde u\otimes s))\quad\mbox{for\,\,\,$u=e^{-im\theta}\tilde u\in \Omega^{0,q}_m(D, E)$},
\sigma_{i}e_{k_{1}k_{2}\ldots k_{n}}= \sum_{l_{i}, l_{i + 1}\in{\cal I}}{\hat{\cal R}}\left( k_{i}, k_{i + 1}; l_{i}, l_{i + 1}\right) e_{k_{1}k_{2}\ldots k_{i - 1}l_{i}l_{i + 1}k_{i + 2}\ldots k_{n}}\; .
\begin{array}{l l}{{A_{4}( \hat{S}_{3}) A_{1}( \hat{S}_{3}) = i [ [ \hat{S}_{3}] ] ~ , ~}}&{{A_{2}( \hat{S}_{3}) = 2 i ( A_{1}( \hat{S}_{3}) - A_{1}( \hat{S}_{3}- 1 ) )}}\\{{A_{3}( \hat{S}_{3}) = 2 i ( A_{1}( \hat{S}_{3}) - A_{1}( \hat{S}_{3}+ 1 ) ) ~ ,}}&{{A_{5}= 4 ( 2 A_{1}( \hat{S}_{3}) - A_{1}( \hat{S}_{3}- 1 ) - A_{1}( \hat{S}_{3}+ 1 ) ) ~ .}}\\ \end{array}
s-\frac{\epsilon'}{1+\epsilon}\ =\ \frac{y'}{1+y}- \frac{\epsilon'}{1+\epsilon}\ =\ \frac{y'(1+\epsilon)-\epsilon'(1+y)}{(1+y)(1+\epsilon)}\ =\ \frac{(1+\epsilon)(y-\epsilon)' - \epsilon'(y-\epsilon)}{(1+y)(1+\epsilon)}
\langle x,y|n_{12},\mu_{12},\epsilon_{12}\rangle = \displaystyle\sum\limits_{n1,n2}\langle x | n_1,\mu_1\rangle \langle y | n_2,\mu_2\rangle \langle n_1, n_2, \mu_1, \mu_2 |n_{12},\mu_{12},\epsilon_{12}\rangle.
\sum_{\bar{\nu}^\flat \in{\mathcal Wt}(\bar{\lambda}^\flat)^\tau}{\rm tr}(\tau \, | \, V_{\bar{\lambda}^\flat, 1}(\bar{\nu}^\flat)) \, e^{\bar{\nu}^\flat}= \sum_{w \in W_0}w \Big( \prod_{\alpha \in \widetilde{\Sigma}_0^+}\frac{1}{1 - e^{-\alpha^\vee}}\, \Big) \cdot e^{w\bar{\lambda}^\flat}.
\tilde{H}= \int \mathrm{d}x ~ \left[ | \tilde{\Pi}_{\alpha}+ \frac{\Theta}{8 \pi^{2}}\tilde{\Pi}_{\alpha}^{H}|^{2}+ | \partial_{i}\tilde{Z}_{\alpha}|^{2}- ( \tilde{Z}_{\alpha}^{*}\partial_{i}\tilde{Z}_{\alpha}) ( \tilde{Z}_{\beta}\partial_{i}\tilde{Z}_{\beta}^{*}) \right] .
\langle\prod\limits_{i=1}^{N}e^{\alpha_i\Phi (Z_i)}\rangle =\left(\frac{\mu}{2\pi}\right)^s\frac{\Gamma (-s)}{b}\langle\left (\int\hat E e^{b\Phi '}\right )^s\prod\limits_{i=1}^{N}e^{\alpha_i\Phi ' (Z_i)}\rangle_{S_{SL}'}
\textstyle \hat{\sigma}( \eta ) = \sum_{\alpha , z}f_{\alpha}^{z}\dots f_{\alpha}^{1}f_{\alpha}^{0}\otimes [ \hat{\pi}( a_{\alpha}^{z}) , [ \dots [ \hat{\pi}( a_{\alpha}^{1}) , [ \hat{Y}^{2}, \hat{\pi}( a_{\alpha}^{0}) ] ] \dots ] ] ~ .
f(t)=E_{\nu, 1+\frac{\gamma}{\nu}, \frac{\gamma}{\nu}}\left(\lambda t^{\nu+\gamma}\right) = 1+ \sum_{k=1}^{\infty}\left(\lambda t^{\nu+\gamma}\right)^k\prod_{j=0}^{k-1}\frac{\Gamma(\nu j+\gamma j +\gamma+1)}{\Gamma(\nu j+\gamma j+\gamma+\nu+1)},
z-1=\begin{cases}\left|z-1\right|e^{i\pi/2}e^{i\tau/2}z=e^{i\tau},\;0<\tau<\pi/4, \left|z-1\right|e^{i3\pi/2}e^{i\tau/2}z=e^{i\tau},\;7\pi/4<\tau<2\pi.\end{cases}
l \leq \sum_{p+f\leq s}\binom{|T|}{p}\binom{|T|-p}{f}+\sum_{q+g\leq s}\binom{|T|}{q}\binom{|T|-q}{g}&= 2\sum_{p+f\leq s}\binom{|T|}{p}\binom{|T|-p}{f}&\leq 2 \sum_{p+f\leq s}|T|^s\leq 2 s^2 |T|^s.
\begin{aligned}\delta( x_1^{a_1}\cdots x_n^{a_n}dx_1 \wedge \cdots \wedge \widehat{dx_i}\wedge \cdots \wedge dx_n) ={}(-1)^{i-1}a_i x_1^{a_i}\cdots x_i^{a_i-1}\cdots x_n^{a_n}\Omega &{}+ (-1)^{i-1}\hbar^{-1}r_i x_1^{a_1}\cdots x_i^{r_i + a_i - 1}\cdots x_n^{a_n}\Omega\end{aligned}
f_N\circ\cdots\circ f_1 \,- \,g_N\circ\cdots\circ g_1 = \sum_{i=1}^N \Big(g_N\circ\cdots \circ g_{N-i+1}\circ f_{N-i}\, -\, g_N\circ\cdots \circ \,g_{N-i+1}\circ\, g_{N-i}\Big)\circ f_{N-i-1}\circ\cdots\circ f_1,
\lim_{\ell\rightarrow\infty}a_\ell =\lim_{\ell\rightarrow\infty}(b_\ell+c_\ell) &=\lim_{\ell\rightarrow\infty}b_\ell &=\sum_{i=0}^{\infty}\frac{1}{(i!)^2}&=I_0(2) \mbox{(by (\ref{eq:I_02}))}.
m_{1}^{2}\; - \; \frac{1}{N - 1}\sum_{I = 2}^{N}\; m_{I}^{2}\; \; = \; \; \frac{e^{2}N}{\pi}\; \; \stackrel{!}{=}\; \; \frac{4 N}{( f_{1}^{0})^{2}}\; P^{0}( 0 ) \; .
\sum_{j=1}^{p-1}\frac{\partial}{\partial x_{j}}\frac{\partial}{\partial x_{2}}\frac{\partial^{2}}{\partial x_{3}^{2}}\dots\frac{\partial^{p-2}}{\partial x_{p-1}^{p-2}}f \biggr|_{x_1=x_2=\dots=x_{p-1}}=0.
\tilde{\mathrm{E}}_{0}( x_{0}, y_{0}; b ) \simeq \frac{( e q )^{2}}{m_{0}}( 1 - e^{- m_{0}| x_{0}- y_{0}|}) + \hbar q \frac{e^{2}}{2 \mathrm{L}}[ \frac{e b \mathrm{L}}{2 \pi}] ( x_{0}^{2}- y_{0}^{2}) + \hbar q^{2}\frac{e^{4}}{8 \pi m_{0}}( x_{0}- y_{0})^{2}.
(\hat{g}_{IJ}) = 6\, \left( \begin{array}{cccccccc}2 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 2 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0\end{array}\right).
\begin{aligned}P_0/Q_0 \,{\it is \, a \, locally \, free\,}R{-\it module \, of \, rank\,}n P_i = Q_i \,\,{\it if}\, r_i = 0 \, ,\, \, I_R \cdot P_i = Q_i \,\,{\it if}\, r_i = n \, . \end{aligned}
\binom{pn+pk-1}{pk}\equiv\prod_{i\ge0}\binom{(pn+pk-1)_i}{(pk)_i}\equiv \binom{p-1}{0}\prod_{i\ge1}\binom{(pn+pk-1)_i}{(pk)_i}&\equiv \prod_{i\ge0}\binom{(n+k-1)_i}{k_i}\equiv \binom{n+k-1}{k}\pmod{p},
g^{\beta \gamma}\xi_{,\beta \gamma}^{\alpha}+\xi^{\gamma}\Gamma_{,\gamma}^{\alpha}-\xi_{,\gamma}^{\alpha}\Gamma^{\gamma}+\left(a-\lambda \right) \Gamma^{\alpha}=g^{\beta \gamma}\left( L_{\xi^{\alpha}}\Gamma_{\beta \gamma}^{\alpha}\right)
(G^{\vee^{*}_{1}})^{\vee^{*}_{E'}}=R p^{\prime}_{E',1*}\, p^{\prime\, !}_{E',2}\,Rp^{\prime}_{1,1*}\,p^{\prime\,!}_{1,2}G= R p^{\prime}_{E',1*}\, Rp^{\prime}_{1,1*}\, p^{\prime\, !}_{E',2}\,p^{\prime\,!}_{1,2}G = Rp'_{1*}p^{\prime \,!}_{2}G.
c_k(x_1,\ldots,x_k) := f_k^F(x_1,\ldots,x_k) &+{k!}\sum_{j=k+1}^{+\infty}\frac{(-1)^{j-k}}{j!}\frac{j!}{{k!}(j-k)!}\int_{([0,T]\times[0,M])^{j-k}}f_j^F(x_1,\ldots,x_k,x_{k+1},\ldots,x_{j}) dx_{k+1}\cdots dx_j.
W_{\rm eff}=S \left\{\ln\left[\frac{S^{N_c-N_f}{\rm det}\left(T\right) \left(1- \frac{BT^{N_f-N_c}\bar{B}}{{\rm det}(T)}\right)^\rho}{\Lambda^{3N_c-N_f}}\right] +N_f-N_c\right\},
\bigl\| \sum_i (a_i\otimes1) V x_i \bigr\|^2 \ = \ \sum_{i,j}\langle x_i,\, V^* ((a_i^*a_j) \otimes 1) V x_j\rangle \ = \ \sum_{i,j}\langle x_i,\, W^* ((a_i^*a_j) \otimes 1) W x_j\rangle \ = \ \bigl\| \sum_i (a_i\otimes1) W x_i \bigr\|^2.
\| A\varepsilon^\dagger\|_F^2 = \sum_{u=1}^m\sum_{k=1}^K(\sum_{t=1}^n [A]_{ut}[\varepsilon]_{kt})^2 =\sum_{u=1}^m\sum_{k=1}^K(\sum_{t\in T_{(u)}}[A]_{ut}[\varepsilon]_{kt})^2\cr \le \sum_{u=1}^m\sum_{k=1}^K(\sum_{t\in T_{(u)}}[A]_{ut}^2)(\sum_{t\in T_{(u)}}[\varepsilon]_{kt}^2) = O(\delta^2mn \|\varepsilon \|_F^2),
\begin{gathered}Ai(s):=\frac{1}{2\pi}\int e^{i(st+t^3/3)}dt,A_-(s):=Ai(e^{2\pi i/3}s),\Phi_-(s):=A_-'(s)/A_-(s), 0>\zeta_1>\zeta_2>\dots Ai(s).\end{gathered}
\nu\left(\tilde x_m^-(x^-_{n_1}\cdots x^-_{n_k})v_{\lambda}\right) =\nu\left( x_m^-(x^-_{n_1}\cdots x^-_{n_k})v_{\lambda}\right) &= x_m^-(x^-_{n_1}\cdots x^-_{n_k})\nu\left( v_{\lambda}\right) &=\tilde x_m^-\left(x^-_{n_1}\cdots x^-_{n_k}\nu(v_{\lambda})\right) &=\tilde x_m^-\nu\left(x^-_{n_1}\cdots x^-_{n_k}v_{\lambda}\right) .
(^{c}_{0}{D}^{\alpha}_{t}u, v ) \, + \, a\big(l(u)\big) \, (\nabla u, \nabla v) \, = \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \big(f, v\big), \forall v \in H^1_0(\Omega). u(\cdot, 0) = \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! u_0 (\cdot), \: \mbox{in}\; \, \Omega. \quad
d \leq n - k + 2 - \sum_{j=1}^{s^*-1}\left\lceil \frac{n_j}{r_j + 1}\right\rceil - \left\lceil \frac{k - \sum_{j=1}^{s^*-1}\left\lceil \frac{n_j}{r_j + 1}\right\rceil r_j}{r_{s^*}}\right\rceil
\nabla_q (D_p P_{ij \nu \nu \nu})= D^2_{q,p}P_{ij \nu \nu \nu}- h_{qp}D_\nu P_{ij \nu\nu\nu}-h_{qi}D_p P_{\nu j \nu\nu\nu}-h_{qj}D_p P_{i \nu\nu\nu\nu}& +h_{qk}(D_p P_{ijk \nu \nu}+ D_p P_{ij\nu k\nu}+D_p P_{ij \nu \nu k}),
\overline{\zeta}_q'(0)= 4\xi_q'(-1)+2(n+1-q)(\xi_q(0)-\xi_q'(0))+\sum_{j=2}^{2n}(n+1-q)^j w_j\gamma_{n,q}(j-1) &+\sum_{j=2n+1}^\infty (n+1-q)^j \frac{j-2}{j(j-1)}\xi_q(j-1),
\Phi :=\{e_i \pm e_j , 1\leq i \neq j \leq 3\}\cup \{\pm 2e_i, 1\leq i \leq 3 \}\Phi^+ :=\{e_i \pm e_j , 1\leq i< j \leq 3\}\cup \{2e_i, 1\leq i \leq 3 \}\Delta :=\{\alpha_1, \alpha_2,\alpha_3\}
dn(x;k)=\frac{k^{2}-\{zn(\frac{1}{2}x;k)+zn[K(k)-\frac{1}{2}x;k]\}^{2}}{k^{2}+\{zn(\frac{1}{2}x;k)+zn[K(k)-\frac{1}{2}x;k]\}^{2}}
\psi_{\rm AB}(r,\theta)={i\over 4\pi}e^{(N+{1\over2})\theta}\int_{\cal C}dt\;e^{-ikr\cos t}\left[{e^{i\gamma t}\over\sin{1\over2}(t-\theta)}+{e^{-i\gamma t}\over\sin{1\over2}(t+\theta)}\right].
\scriptsize{{\rm SINR}_{sr_{p}}=\frac{\sum\limits_{k=1}^K \textbf{w}_{s_kr_p}^H \rho_{s_kr_p}\textbf{w}_{s_kr_p}}{\sum\limits_{k=1}^K \sum\limits_{\substack{l=1 l\neq p}}^L \textbf{w}_{s_kr_l}^H \rho_{s_kr_l}\textbf{w}_{s_kr_l}+ \sigma^2\textbf{w}_{s_kr_p}^H \textbf{w}_{s_kr_p}},}
( e_{a}^{( 1 )}g_{b}^{( 1 )}- e_{b}^{( 1 )}g_{a}^{( 1 )}) + ( e_{a}^{( 2 )}g_{b}^{( 2 )}- e_{b}^{( 2 )}g_{a}^{( 2 )}) = 2 \pi \hbar n , n \in Z .
h(\sqrt{b}X_2^n+Z_2^n) \geq \frac{n}{2}\log\left[ 2\pi eb 2^{2R_3}+2\pi e(1-b)\right] =\frac{n}{2}\log\left[1+b(2^{2R_3}-1)\right]+\frac{n}{2}\log 2\pi e=n\xi(b)+\frac{n}{2}\log 2\pi e .
\lambda_{k+1}\int_{A_{3,4}(p)}|\langle\nabla f(z), V^{k+1}\rangle|^2\,dv_g(z)= \lambda_{k+1}\sum_{j=1}^{k+1}\int_{A_{3,4}(p)}|\langle\nabla f(z), v_j\rangle|^2\,dv_g(z)\leq C \int W_0(x)\, d\mu(x)\, .
\hat{A}\Psi_0^{(3)}= \Biggl[\frac{\partial}{\partial a^*}+ \sum_{k = 0}^{3}b_k^{(3)}a^{* (3 - k)}\Bigl(\frac{\partial}{\partial a^*}\Bigr)^k\Biggr] \Psi^{(3)}_0 = 0.
({\eta}_{\mu \nu}) = \left( \begin{array}{c c c c c}{{1}}&{{0}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{1}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{1}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{0}}&{{- 1}}\\{{0}}&{{0}}&{{0}}&{{- 1}}&{{0}}\\ \end{array}\right) .
\frac{dP_\theta^n}{dP_{\theta_0}^n}= \prod_{i=1}^N \frac{1 + \sum_{j=1}^m \theta_j \psi_j(X_i)}{1 +\sum_{j=1}^m \theta_j^0 \psi_j(X_i)}= \prod_{j=1}^m \prod_{i=1}^{N_j}\frac{1 + \theta_j \psi_j(X_i^{(j)})}{1 + \theta_j^0 \psi_j(X_i^{(j)})}=: \prod_{j=1}^m P_j(\theta_j)
\eta_{i}^{(k)}\colon x_{i}^{(k+4)}x_{i}^{(k)}=v^{m_k}\bigg[ e_{i,i+2}e_{i,i+1}e_{i+1,i+2}\bigg(\prod_{j=i}^{k-1}(e_{j+2,j+3})^{2}\bigg)y_{i+k+2}^{(3)}\bigg]+v^{n_k}(x_{i+2}^{(k+2)})^2,
\left\|\frac{LT(p)}{LT(f_1)}f_1\right\|^2 = \left|\frac{b_t}{a_{1j_1}}\right|^2 \sum |a_{1j}|^2 \frac{(\frac{n-n_2(t+j-j_1)}{n_1})!(t+j-j_1)!}{(\frac{n+ (n_1 - n_2)(t+j-j_1)}{n_1})!}.
\prod_{u = 1}^{N}d v_{u}^{l}d v_{u}^{l *}= \prod_{k = 1}^{l - 1}d \xi_{k}^{l}d \xi_{k}^{l *}\cdot 2^{N - l - 1}( \xi_{l}^{l})^{2 N - 2 l + 1}d \xi_{l}^{l}d^{2 N - 2 l + 1}\hat{\Omega}_{l}\ .
(T^2_DT^2_C-T^2_C)T^2_CT_{U_{2,4}}^2=T^2_DT^2_{U_{2,3}}-(T^2_{U_{2,3}}T^2_C-T^2_D)T_L^2T^2_{U_{2,3}}+(T^2_{U_{2,3}}T^2_C-T^2_D)T^2_{U_{1,3}}.
\left(\frac{\langle\mathbf{A}(Q_{1r}-Q_{2r}),Q_{1r}-Q_{2r}\rangle}{\|Q_{1r}-Q_{2r}\|^{2}}\right)\geq(m_{1}+m_{2})f(\|Q_{1r}-Q_{2r}\|)+0 +\sum\limits_{j=l+1}^{n}m_{j}f(\|Q_{1r}-Q_{jr}\|)+\sum\limits_{j=l+1}^{n}m_{j}\langle Q_{jr}-Q_{2r},\frac{Q_{1r}-Q_{2r}}{\|Q_{1r}-Q_{2r}\|}\rangle B_{jr}.
( 2 \pi )^{3}\lambda_{\bf g}^{3 / 2}k^{- 3 / 2}\tau ( S^{3}, A_{f}\! = \! 0 , \langle \cdot , \cdot \rangle_{H^{0}( A_{f}= 0 )})^{1 / 2}V ( S U ( 2 ) )^{- 1}
\Delta(z^2){\cal F}(z)={\rm const.}\int_0^\infty \prod_i dy_i ~y_i^{n-1}\prod_{i>j}(y_i-y_j)~e^{-\alpha\sum_iz_i^2/4y_i-\beta\sum_iy_i-\gamma\sum_i\ln y_i}.
J(B_0, \sigma) = \left(B_0\, R^2\right)^{\sigma+\frac 12}\left( 1 + \sum_{k=1}^\infty \left( \left(\frac{B_0\, R^2}{2}\right)^{k+1}\frac{1}{k!}+ \frac{1}{2\sqrt{2\pi k}}\right)^{\sigma+\frac 12}\, \right)\, .
u_{\beta_{1},\beta_{2},\beta_{3}}\left(t,x\right) =u\left(t,x\right)-\beta_{1}\alpha\left(t,x\right)-\beta_{2}-\frac{\beta_{3}}{T-t}, v_{\beta_{1},\beta_{2}}\left(t,x\right) =v\left(t,x\right)+\beta_{1}\alpha \left(t,x\right)+\beta_{2}.
C_{\operatorname*{reg},A_{\scriptscriptstyle 0}}:=\frac{1}{\beta_{\scriptscriptstyle0}}\frac{C_{P}^{\eta\left( p,P\right)}}{1-C_{P}^{\eta\left( p,P\right)}\left( 1-\frac{\alpha_{\scriptscriptstyle0}}{\beta_{\scriptscriptstyle0}}\right)}
\rho_{2}\approx\frac{ne\sqrt{\pi}G(i\lambda_{D1})e^{\frac{-t}{\tau_{0}}-\lambda_{D1}z}}{3\sqrt{3}\epsilon_{0}\epsilon_{r}\lambda_{D1}}(1-e^{1-\sqrt{\frac{2ne^{2}t^{2}}{m\epsilon_{0}\epsilon_{r}}}})
{^{(+)}\xi}^i = \left(-\beta \partial_\phi{^{(+)}\xi}^\phi(x^+),{^{(+)}\xi}^\phi(x^+) \right),\qquad{^{(-)}\xi}^i = \left(-\beta \partial_\phi{^{(-)}\xi}^\phi(x^-),{^{(-)}\xi}^\phi(x^-) \right)
\Theta_{i}\xi- \Theta_{j}\xi =\begin{pmatrix}(1 + i m)\xi_2 n \xi_2\end{pmatrix}- \begin{pmatrix}(1 + j m)\xi_2 n \xi_2\end{pmatrix}=\begin{pmatrix}(i - j)m \xi_2 0\end{pmatrix},
M_{T_1}\, N_{T_2}(t) = \sum_{{v_0 \in \Delta^0 (t) , \varepsilon\in \Delta^1 (t) \atop v_0 \in R_{\varepsilon}(t) ,P_{\varepsilon}(t) = T_1}}\ \delta_{R_{\varepsilon}(t) \,\cup_{v_0}\, T_2}\, .
\mathrm{hofib}(\mathcal N^{\geq i}A\Omega\{i\}/p^n\xrightarrow{\varphi_i-1}A\Omega\{i\}/p^n) = \tau^{\leq i}R\nu_\ast \mathbb Z/p^n\mathbb Z(i) = \tau^{\leq i}R\psi_\ast \mathbb Z/p^n\mathbb Z(i)\ ,
\exp(X)\exp(Y)\exp(Z) =\exp\bigg\{{(\alpha -\beta)u\over e^{-\beta u}- e^{-\alpha u}}\bigg[X+\bigg(e^{-{\alpha u\over2}}s_\alpha(u)+ e^{-{\beta u\over 2}}s_{\beta}(u)\bigg) Y+Z + \cr +{1\over \beta u}\bigg(e^{-{\alpha u\over2}}s(\alpha u)- e^{-{u\over2}}s((\alpha-\beta)u)\bigg) \tilde c I\bigg] \bigg\}\ .
\hat{h}= \frac{\mu}{\mu ( q )}\hat{h}^{( 0 )}+ \frac{C_{2}- C_{1}}{\mu c^{2}}\frac{\alpha^{2}}{q^{2}}+ \frac{\hbar^{2}}{2}\frac{\alpha}{\mu^{2}c^{2}}C_{1}\left( \frac{{\bf q}}{q^{3}}\frac{\partial}{\partial{\bf q}}+ 4 \pi \lambda \delta ({\bf q}) \right) \, ,
|U_1V_1| = \frac{|z_1| M^2}{DD_1^2s_0r_2}, |U_2V_2| = \frac{|z_2| M^2}{DD_2^2r_0s_1}, \Upsilon_1 = \frac{r_1s_1D_2}{e^2|z_1|}, \Upsilon_2 = \frac{r_2s_2D_1}{e^2|z_2|}.
L_{k}=[\begin{matrix}\frac{\sigma_{2}^{2}P_{k}}{\sigma_{2}^{2}P_{k}+\sigma_{1}^{2}P_{k}+\sigma_{1}^{2}\sigma_{2}^{2}}&\frac{\sigma_{1}^{2}P_{k}}{\sigma_{2}^{2}P_{k}+\sigma_{1}^{2}P_{k}+\sigma_{1}^{2}\sigma_{2}^{2}}\end{matrix}].
\begin{aligned}\dot{x}= -y - \frac{16}{3}x^2 - 2 k_1 x y - \frac{4}{3}y^2 + \frac{k_1}{3}x \left(16 x^2 - 3 k_1 x y - 4 y^2 \right), \dot{y}= x + k_1 x^2 + \dfrac{8}{3}x y - k_1 y^2 + \frac{k_1}{3}y \left(16 x^2 - 3 k_1 x y - 4 y^2 \right),\end{aligned}