harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sum_{\substack{\rho'=\beta'+i\gamma', \zeta'\left(\rho'\right)=0,\, 0<\gamma'\leq T}}\left(\beta'-\frac{1}{2}\right) = \frac{T}{2\pi}\log{\log{\frac{T}{2\pi}}}+ \frac{1}{2\pi}\left(\frac{1}{2}\log{2}- \log{\log{2}}\right)T &\quad\quad- \operatorname{Li}\left(\frac{T}{2\pi}\right) + O((\log{\log{T})^2}),
H^{0}( \mathcal{K}_{\S}) \cong H^{0}( \pi_{*}( \O_{\S})^{*}\otimes \mathcal{K}_{0}) = H^{0}( \oplus_{i = 1}^{r}\mathcal{K}_{0}^{\otimes i}) = \mathcal{T}S p e c .
=\int(\frac{1}{e^{i\cdot x}-1}-\frac{1}{e^{i\cdot x}+1})\cdot d(e^{i\cdot x})=\int(\frac{d(e^{i\cdot x}-1)}{e^{i\cdot x}-1}-\frac{d(e^{i\cdot x}+1)}{e^{i\cdot x}+1})
R_0 := C_{-1}^{10}(3 C_{-1}^9 - 36 C_{-1}^2 F_{-5}^2 + 18 C_{-1}^6 F_{-3}-96 F_{-3}^3 - 6 C_{-1}^6 \lambda - 48 C_{-1}^3 F_{-3}\lambda - 96 F_{-3}^2 \lambda) & - (C_{-1}^6 F_{-5}F_{-4}^2 (-48 C_{-1}^3 - 96 F_{-3}) + F_{-4}^4 (16 C_{-1}^6+ 64 C_{-1}^3 F_{-3}+ 64 F_{-3}^2))=0,
Y_1 = \begin{cases}h_{11}X_{1}+h_{12}X_2 S_1=1, h_{21}X_{1}+h_{22}X_2 S_1=2,\end{cases}Y_2 = \begin{cases}h_{11}X_{1}+h_{12}X_2 S_2=1, h_{21}X_{1}+h_{22}X_2 S_2= 2,\end{cases}
\sqrt[\leftroot{-4}\uproot{5}3]{3abK_3}=\begin{cases}2^d 3^e r' s \sqrt[\leftroot{-2}\uproot{3}3]{K_3}, \textnormal{for FLT2 Case I}, 2^d 3^e r_0 s' \sqrt[\leftroot{-2}\uproot{3}3]{K_3}, \textnormal{for FLT2 Case II}.\end{cases}
p^*\sum_{t=k_0}^k e^TA_{S\bar{S}}(t)e \leq\left(\pi_{\bar{S}}^T(k_0)-\pi_{\bar{S}}^T(k)\right)e+\sum_{t=k_0}^ke^TA_{\bar{S}S}(t)e\cr \leq 1 +\sum_{t=k_0}^ke^TA_{\bar{S}S}(t)e
\frac{1}{2}\int d t \, d^{3}x h_{\mu}^{\nu}T_{\nu}^{\mu}= \frac{1}{2}\int d t \, d^{3}x \left( H_{t t}^{0}T^{t t}+ \frac{1}{2}H_{a a}^{0}T^{a a}+ H_{z z}^{0}T^{z z}+ 2 H_{t z}^{0}T^{t z}\right) \, .
\varphi_{n,m}(h\chi) =\sum_{g\in G_n}[\pi^m]((h\chi)(g))g &=\sum_{g\in G_n}[\pi^m](\chi(h^{-1}g))g &=\sum_{g\in G_n}[\pi^m](\chi(g))(hg) &=h\bigg(\sum_{g\in G_n}[\pi^m](\chi(g))g\bigg)=h\varphi_{n,m}(\chi)
S(p^t)=\begin{cases}0, \beta< t-1, -\frac{p^{\min(\alpha+t,2t)}p^{t/2}}{p^{3t}}p^{t-1}, \beta=t-1, \frac{p^{\min(\alpha+t,2t)}p^{t/2}}{p^{3t}}\phi(p^t), \beta\geq t.\end{cases}
\left( M_{5}\right)_{a_{2}b_{| 2}}q^{b_{| 2}\left[ N_{b_{| 2}}+ n_{2}\right]}+ K_{a_{2}}^{\left( 2 \right)}\left( \cdots q^{b \left[ N_{b}+ n_{2}- 1 \right]}\right) = 0 \, .
B = \frac{\Gamma \left( 1 + n - m \alpha_{-}^{2}\right)}{\Gamma \left( n + ( 1 - m ) \alpha_{-}^{2}\right) \Gamma \left( 1 - \alpha_{-}^{2}\right)}C_{n m , 1 2}^{n , m - 1}A_{n , m - 1}^{\alpha}
\sum_{k \ge 0}\frac{(\pi q_2 + q_1^{k-1}) (\pi q_2 + q_1^{(k-2)}) \dotsm (\pi q_2 + 1)}{(1-q_1^k) (1-q_1^{k-1}) \dotsm (1-q_1)}z^k = \prod_{n \ge 0}\frac{1}{1 - \pi q_2 q_1^nz}\prod_{n \ge 0}(1+q_1^nz).
\sup_{x \in [-1+h-a,-1+h+a]}f(x) = \max\{f(-1+h-a),f(-1+h+a)\}\stackrel{(\ref{eq:f-endpoints-max-left})}{=}f(-1+h-a) = 2 r_x \left( \frac{1}{r_x}+ h\left(1 - \frac{1}{r_x}\right)\right).
{\mathcal{F}}(0)\delta_{{\bf{k}}, 0}+ \sum_{n=1}^{\infty}\frac{{\mathcal{F}}^{(n)}(0)}{n!}\mbox{}\sum_{\{{\bf{q}}_{i}\}}{\tilde{f}}_{{\bf{q}}_{1}}{\tilde{f}}_{{\bf{q}}_{2}}...{\tilde{f}}_{{\bf{q}}_{n}}\delta_{({\bf{k}}- \sum_{i=1}^{n}{\bf{q}}_{i}),{\bf{0}}}
I_2(\beta, N, T) = E^{\mathbb{P}_T^\lambda}\left[\sum_{k=1}^{N}\left\{\int_{0}^{T}\lambda_k(t)\cdot dB^{(k)}_t -\frac{1}{2}\int_{0}^{T}|\lambda_k(t)|^2dt\right\}\right] &= -\frac{N}{2}\kappa^2(\alpha+1)^2E\left[\int_{0}^{T}t^{2\alpha}dt\right] &= -\frac{\kappa^2(\alpha+1)^2}{2}N\cdot\frac{T^{2\alpha+1}}{2\alpha+1}.
\frac{1}{Q_1}\sum_{\substack{q\sim Q_1 (q,10)=1 q>1}}\sum_{\substack{a\le d q (a,d q)=1}}F_X\Bigl(\frac{a}{d q}\Bigr)\ll Q_1\sup_{\substack{(a,q)=1 1<q\le Q_1 (q,10)=1 d|10}}F_X\Bigl(\frac{a}{d q}\Bigr)\ll_A\frac{Q_1}{(\log{X})^{100(A+1)}}.
\int_{\Omega_{2}}\!\!|\nabla\phi(x)|^2dx = \sum_{n\in J_{k}}|c_{n}|^2\int_{\Omega_{2}}\!\!|\nabla\phi_{n}(x)|^2dx & + \sum_{n,m \in J_{k}\atop n\neq m}c_{n}\overline{c_{m}}\int_{\Omega_{2}}\!\! \nabla\phi_{n}(x)\cdot\nabla\phi_{m}(x)\, dx.
\langle I'(u_k)-I'(u),u_k-u\rangle &=\langle I'(u_k),u_k-u\rangle -\langle I'(u),u_k-u\rangle &\leq\|I'(u_k)\|_{(E_0^{\alpha,p})^*}\|u_k-u\|_{E^{\alpha,p}}-\langle I'(u),u_k-u\rangle &\rightarrow0\ \ \mbox{as}\k\rightarrow\infty.
W =\sum_{j\in J}\Bigl(\frac{1}{3}(z_{j_1,1}^2-z_{{j_2},1}^2)(z_{j_1,2}^2-z_{{j_2},2}^2)-(z_{j_1,1}^2-z_{{j_2},2}^2)(z_{j_1,2}^2-z_{{j_2},1}^2)\Bigr),
24310=(\begin{matrix}24310\\ 1\end{matrix})=(\begin{matrix}24310\\ 24309\end{matrix})=(\begin{matrix}221\\ 2\end{matrix})=(\begin{matrix}221\\ 219\end{matrix})=(\begin{matrix}17\\ 8\end{matrix})=(\begin{matrix}17\\ 9\end{matrix})
\delta_{\mu_{1}...\mu_{n}}^{\nu_{1}...\nu_{n}}\delta_{\nu_{1}...\nu_{p}}^{\mu_{1}...\mu_{p}}=n!\frac{(d-p+n)!}{(d-p)!}\delta_{\nu_{n+1}...\nu_{p}}^{\mu_{n+1}...\mu_{p}}.
\sum_{i,j\geq0}\Big(G_i(\alpha(x),G_j(y,z))-G_i(G_j(x,y),\alpha(z))-(-1)^{|x||y|}G_i(\alpha(y),G_j(x,z)) &+(-1)^{|x||y|}G_i(G_j(y,x),\alpha(z))\Big)=0, &\sum_{i,j\geq0}\Big(G_i(G_j(x,y),\alpha(z))-(-1)^{|y||z|}G_i(G_j(x,z),\alpha(y))\Big)=0.
\alpha=\frac{1}{1-p}s_f(1-p), a(t)={\bf 1}_{[0,1-p]}(t)(f(t)-\alpha), &\beta=\frac{1}{2p-1}(s_f(p)-s_f(1-p)), b(t)={\bf 1}_{(1-p,p)}(t)(f(t)-\beta), &\gamma=\frac{1}{1-p}(s_f(1)-s_f(p)), c(t)={\bf 1}_{[p,1]}(t)(f(t)-\gamma).
\prod_{\substack{p p \neq 2 p \nmid h}}\Big( 1 + \frac{\big(\frac{h (-1)^{k - \frac{1}{2}}}{p}\big)}{p^{2w - \frac{1}{2}}}\Big) = \frac{L^{(2)}(2w - \frac{1}{2}, \chi_{k,h})}{\zeta^{(2h)}(4w-1)}.
B_{W}(z,w) =2m\alpha^{\tfrac{3-2m+n}{2m}}\sum_{k=0}^{\infty}\alpha^{k/m}\frac{(z\bar{w})^k}{\Gamma\left(\frac{2k+2+n}{2m}\right)}=2m\alpha^{\tfrac{3-2m+n}{2m}}\sum_{k=0}^{\infty}\frac{\big(\alpha^{1/m}(z\bar{w})\big)^k}{\Gamma\left(\tfrac{k}{m}+\tfrac{2+n}{2m}\right)}.
V( B[f_1,f_2]_c^{\textsf{com}}) \underset{\ne}{\subset}V(B[f_1, f_2]_{f_2}^{\textsf{com}}) , | V( B[f_1, f_2]_{f_2}^{\textsf{com}}) | = | V( B[f_1]_{f_2}^{\textsf{com}}) | < | V( B[f_1]_{f_1}^{\textsf{com}}) | = | V( B_{f_1}^{\textsf{com}}) | .
{\cal P}_{\Lambda}^{0}\, = \, e^{\cal K}\, \left( \, F_{\Delta}\, f_{\phantom{\Delta}\Lambda \Sigma}^{\Delta}\,{\bar{X}}^{\Sigma}\, + \,{\bar{F}}_{\Delta}\, f_{\phantom{\Delta}\Lambda \Sigma}^{\Delta}\,{X}^{\Sigma}\right)
\prescript L{}{}\ell{}^R (x,\alpha) + \prescript L{}{}\ell{}^R(x,-\alpha)= \langle \mu,\alpha\rangle + \langle \mu,-\alpha\rangle + \underbrace{\chi_R(\alpha) + \chi_R(-\alpha)}_{\in\{0,1\}}&\qquad-\underbrace{(\chi_L(w\alpha) + \chi_L(-w\alpha))}_{\in\{0,1\}}\in \{-1,0,1\}.
\Phi(t) = B\mu(t)+ B_1r_1 \Bigl(\frac{t}{T}\Bigr)^a +B_2r_2 \Bigl(\frac{t}{T}\Bigr)^a + B_2Lr_1 \Bigl(\frac{t}{T}\Bigr)^a \log\Bigl(\frac{T}{t}\Bigr).
\partial_{x_i}M=M\big\{\frac{\partial_{x_i}\rho}{\rho}+\frac{(v-u)\cdot\partial_{x_i}u}{R\theta}+(\frac{|v-u|^{2}}{2R\theta}-\frac{3}{2})\frac{\partial_{x_i}\theta}{\theta}\big\}.
\Gamma_2(z\,|\,1,\tau) = (2\pi)^{\frac{z}{2}}\tau^{\frac{(z-z^2)}{2\tau}- \frac{z}{2}}e^{\gamma\frac{(z^2-z)}{2\tau}}\Gamma(z) \prod\limits_{m=1}^\infty(m\tau)^{1-z}e^{\frac{(z-z^2)}{2m\tau}}\frac{\Gamma(z+m\tau)}{\Gamma(1+m\tau)}.
q^{-m-k}(\gamma(s,\pi,\eta,\psi)-\gamma(s,\pi',\eta,\psi)) = \int_{F^\times}( W_{v_k}(t(a)w)-W_{v_k'}(t(a)w))(\omega_{\psi^{-1}}(w)\phi^m)(a)\chi(a)\eta^{-1}(a)|a|^{-s}da.
\left(n(i+q)-m\left(j+\frac q2\right)\right)(2d^1_{0,0}(m+n,i+j)-d^1_{0,0}(n,j)-d^0_{0,0}(0,0)) =0, d^1_{0,0}(m,i)+d^1_{0,0}(n,j)=2d^0_{0,0}(0,0).
\frac{d}{d\tau}M(\tau) = \int (\frac{\cosh s}{\sinh s}) (\frac{s}{\sinh s})^{2}\tilde{w}^{4}s^{2}ds + \int \frac{x}{|x|}\cdot (\nabla \tilde{w}) (\tilde{u}^{3}- \tilde{w}^{3}) s^{2}ds d\tau.
g= (g_{ij})= \begin{pmatrix}\frac{\left\lbrace a^2 (y^1)^4+3\tilde{b}^2(y^2)^4\right\rbrace}{(y^1)^4}\frac{-4\tilde{b}^2(y^2)^3}{(y^1)^3}\frac{-4\tilde{b}^2(y^2)^3}{(y^1)^3}\frac{2\tilde{b}\left\lbrace a (y^1)^2+3\tilde{b}^2(y^2)^2\right\rbrace}{(y^1)^2}\end{pmatrix},
\frac{1}{2}\begin{bmatrix}\alpha \mathrm{I}+ C B^\mathsf{T}- B \beta \mathrm{I}\end{bmatrix}\boldsymbol{z}^{(i+1)}= \frac{1}{2}\begin{bmatrix}\alpha \mathrm{I}- B^\mathsf{T}B \beta \mathrm{I}\end{bmatrix}\boldsymbol{z}^{(i)}+ \boldsymbol{d},i = 1, 2, \dots, \ell ~ \mbox{or}~ \ell_k
\mathfrak{H}_m(|g|^2) = \sum_{\l = 0}^m \begin{pmatrix}m \l\end{pmatrix}H_{2\l}(g_1; \tfrac 12) H_{2m - 2\l}(g_2 ; \tfrac 12), \mathfrak{L}_m(|g|^2) = (-1)^m m!\cdot L_m(|g|^2).
\sum_{\substack{X^{\theta_2-\theta_1}\le p_1\le \dots\le p_\ell X^{1-\theta_2}\le\prod_{i\in\mathcal{I}}p_j\le X^{1-\theta_1}p_1\cdots p_\ell \le X/p_j}}^*S_{p_1\cdots p_\ell}(p_j)=o_{\mathcal{L}}\Bigl(\frac{\#\mathcal{A}}{\log{X}}\Bigr),
\begin{array}[c]{l}\left\langle J_{ab},J_{cd},Z_{e}\right\rangle =\alpha_{0}\,\varepsilon_{abcde}, \left\langle J_{ab},Z_{cd},P_{e}\right\rangle =-\alpha_{0}\,\varepsilon_{abcde}, \left\langle Z_{ab},Z_{cd},P_{e}\right\rangle =-\alpha_{0}\,\varepsilon_{abcde}.\end{array}
Q_{m,\Omega}^N[u]=Q_{\sigma,\Omega}^N[(-\Delta)^ku]=c_2(n,\sigma)\!\!\inf\limits_{\scriptstyle w|_{x\in\partial\Omega}=0\atop \scriptstyle w|_{y=0}=(-\Delta)^ku}\,\int\limits_0^\infty\int\limits_{\Omega}y^{1-2\sigma}|\nabla w|^2\,dxdy.
Z_{q_0}(\Delta,b ,\sigma):= \begin{vmatrix}\sigma_{2,3}\sigma_{1,2}\sigma_{1,3}\sigma_{0,1}\sigma_{0,3}\sigma_{0,2}\end{vmatrix}:=(w_{\sigma_{0,2}}w_{\sigma_{1,3}})^{-1}\begin{Bmatrix}\sigma_{2,3}\sigma_{1,2}\sigma_{1,3}\sigma_{0,1}\sigma_{0,3}\sigma_{0,2}\end{Bmatrix}.
H_{2,j-1}= -ab H_{1,j-1}+ (a+b) \widetilde H_{1,j-1}- L_{2,j}^*\widetilde H_{1,j}L_{1,j}= b H_{4,j-1}- \widetilde H_{4,j-1}= -a H_{3,j-1}+ \widetilde H_{3,j-1}.
\varpi(x_{2i+1}) =x,\varpi(x_{2i+2})=x^{-1},i=0,...,(l-3)/2, \varpi(x_{l-2}) =x,\varpi(x_{l-1})=x,\varpi(x_{l})=x^{-2}\varpi(x_{l+1}) =y^{2},\varpi(d)=y^{-1}ll>1
-\frac{i}{2}\sum_n g_n \oint_{\gamma_n}(\phi_1+\alpha_1\phi_B) \frac{dz}{z-z_n}-\frac{i}{2}g_\infty \oint_{\gamma_\infty}(\phi_1+\alpha_1\phi_B)\frac{dz}{z}-\alpha_1\int\phi_1\partial_z\partial_{\bar z}\phi_B \frac{idz\wedge d\bar z}{2}.
\int_{0}^{1}|h(s)|\,\d s = \int_{0}^{t_{1}}|h(s)|\,\d s+ \sum_{k=1}^{n-1}\int_{t_{k}}^{t_{k+1}}|h(s)|\,\d s+ \int_{t_{n}}^{1}|h(s)|\,\d s & \ge |Vh(t_{1})|+ \sum_{k=1}^{n-1}|Vh(t_{k+1}) - Vh(t_{k})|+ |Vh(t_{n})| & = 1+2(n-1)+1=2n.
\tau=f(l,\sigma) =f(l,\alpha l_0\beta)=f(l,\alpha l_0)\beta=ll_0\alpha\beta f(k,\tau) =f(k,ll_0\alpha\beta)=f(k,ll_0\alpha)\beta=f(k,l)l_0\alpha\beta=kll_0\alpha\beta f(k,\sigma) =f(k,\alpha l_0\beta)=f(k,\alpha l_0)\beta=kl_0\alpha\beta.
\varphi'^{(m)}_{s}(\bar{a})=\begin{cases}\overline{-F^{s-1}da}\otimes \overline{1}((m,p)\neq (2,2)), \overline{-F^{s-1}da}\otimes \overline{1}+\overline{d\pi/\pi^{2}}\otimes\sqrt{\overline{a_{0}\pi^{2}}}((m,p)= (2,2)) \end{cases}
\underbrace{\sum_{i=1}^{K-1}\!\left[\!\frac{h(\textbf{Y}_i^n|\Omega^n,W_{i+1},\ldots,W_K)}{r_i}-\!\!\right. \left. \frac{h(\textbf{Y}_{i+1}^n|\Omega^n,W_{i+1},\ldots,W_K)}{r_{i+1}}\!\!\right]}_{\leq 0}\!\!.
\mathcal{G}(\mathcal{Q}h) =\int_0^T(g(t),\mathcal{Q}h(t))_{L^2_{x,v}}dt &=(g_0,h(0))_{L^2_{x,v}}- \int_0^T (\mathcal{L}_2 f(t)\,, h(t))_{L^2_{x,v}}dt\,.
\kappa{{T}_{00}}=0,\kappa{{T}_{11}}=\frac{{{\nu}'}}{r},\kappa{{T}_{22}}=\frac{1}{4}r\left({\nu}'\left( r{\nu}'+2 \right)+2r{\nu}'' \right), \kappa{{T}_{33}}=\kappa{{T}_{22}}{{\sin}^{2}}\theta ,\kappa T=-\frac{{\nu}'\left( r{\nu}'+4 \right)}{2r}-{\nu}''.
\operatorname{Cov}\bigl( Z^H_s,Z^H_u \bigr)\operatorname{Cov}\bigl( Z^H_t, Z^H_t \bigr)= \operatorname{Cov}\bigl( Z^H_s,Z^H_t \bigr)\operatorname{Cov}\bigl( Z^H_t, Z^H_u \bigr).
\begin{gathered}F_{[j,j+1]}= \frac{1}{1 - g_L g_R - \alpha g_L + \overline\alpha g_R}\begin{pmatrix}1 + g_L g_R + \alpha g_L + \overline\alpha g_R 2\rho g_L 2\rho g_R 1 + g_L g_R - \alpha g_L - \overline\alpha g_R \end{pmatrix}, \noalign{\vskip5pt}\alpha=\alpha_j, \rho=\rho_j, g_L(z)=zb_j(z), g_R(z)=zf_{j+1}(z).\end{gathered}
||v_{6}(t)||_{\dot{H}^{1}_{e}}= ||v_{6}(t,\cdot \lambda(t))||_{\dot{H}^{1}_{e}}\leq C \left(||v_{6}(t,\cdot \lambda(t))||_{L^{2}(R dR)}+||L(v_{6}(t,\cdot \lambda(t)))||_{L^{2}(R dR)}\right)
\Vert J_{44}^{\left( 9\right)}\Vert_{p}=\mathcal{O}\left( 1\right)(v(|z_{1}|)+(v(|z_{2}|))\left( \frac{\omega (h_{1})}{v(h_{1})}+\frac{\omega(h_{2})}{v(h_{2})}\right) .
\mathrm{I}_\mathrm{SD}\approx \sum^2_{q=1}\left(1 - \frac{1}{N_\mathrm{s}}\sum_{n=1}^{N_\mathrm{s}}\log_2 \left[ 1+\exp{\left( (-1)^{c_q^{(n)}}\lambda_q^{(n)}\right)}\right]\right)
\sigma(a_1)\circ T(\sigma(a_2)) =B_1(\sigma(a_1))S(B_1(\sigma(a_3)))B_2(\sigma(a_4))S(B_2(\sigma(a_2))) &=B_1(a_1)S(B_1(a_2))B_2(a_3)S(B_2(a_4)) &=\epsilon(B_1(a_1))\epsilon(B_2(a_2))1=\epsilon(a)1.
S \left[ Z \left( \zeta \right) \right] = \int d^{3}\zeta \left[ - \sqrt{- G \left( Z \right)}-{\frac{1}{6}}\epsilon^{a b c}\Pi_{a}^{A}\Pi_{b}^{B}\Pi_{c}^{C}B_{C B A}\left( Z \left( \zeta \right) \right) \right]
G_m(x;q,c)=\frac{(\ln q)^{m+1}}{c^2}\sum^{\infty}_{n=2}q^{-nx}H(m,n,q,c)+(1-\delta_{1m})(\ln q)^{m+1}q^{-x}\frac{(m-2)(1-q^{-c})}{(1-q^{-1})c},
S(E_{\xi_{i}})^{*}S^{-1}(E_{\xi_{i}}) - q^{-2}S^{-1}(E_{\xi_{i}}) S(E_{\xi_{i}})^{*}\sim - q^{-1}(q - q^{-1}) \sum_{k = i + 1}^{N}q^{3(i - k)}S^{-1}(E_{\xi_{k}}) S(E_{\xi_{k}})^{*}.
T ( z ) = \frac{1}{8 \pi}\partial_{z}\varphi \partial_{z}\varphi = \sum_{k = - \infty}^{\infty}L_{k}z^{- k - 2}\quad , \quad \bar{T}( \bar{z}) = \frac{1}{8 \pi}\partial_{\bar{z}}\varphi \partial_{\bar{z}}\varphi = \sum_{k = - \infty}^{\infty}\bar{L}_{k}\bar{z}^{- k - 2}
(C\otimes C)_n \cong \begin{cases}(k \cong k)\otimes_k k \mbox{if}n=0 (k\otimes_k (k\oplus k)) \oplus ((k\oplus k)\otimes k) \cong k^{\oplus 4}\mbox{if}n=1 (k \otimes_k (k\oplus k)) \oplus ((k\oplus k)\otimes_k (k\oplus k)) \oplus ((k\oplus k)\otimes_k k) \cong k^8 \mbox{if}n=2 \end{cases}
{\cal M}^{2}\sim \tilde{g}^{2}g_{i j}( n^{i}+ A^{i k l}n_{k l}) ( n^{j}+ A^{j r s}n_{r s}) + \left( \frac{2 \pi}{\tilde{g}}\right)^{2}g^{i j}g^{k l}n_{i k}n_{j l},
{M\over M_{Pl}}={(P_k^S)^{1/4}\over 2\sqrt{L}u^{1/2}[(1+u^2)(1+3u^2)]^{1/8}}\approx{2.0\times 10^{-4}\left(10^{10}P_k^S\right)^{1/4}\left({60/L}\right)^{1/2}\over u^{1/2}[(1+u^2)(1+3u^2)]^{1/8}};
I'=\langle E_{1,2}^{2},E_{1,3}^{2},E_{1,4}^{2},E_{2,3}^{2},E_{2,4}^{2},E_{3,4}^{2}, E_{1,2}E_{1,3}, E_{1,2}E_{1,4},E_{1,3}E_{1,4}, E_{2,3}E_{2,4}, E_{1,3}E_{2,4}, E_{1,3}E_{2,3},E_{1,4}E_{2,4},E_{1,4}E_{3,4},E_{2,4}E_{3,4}\rangle,
\Theta^{\rho \sigma}\mbox{}_{\mu \nu}\Theta^{-1}_{\rho \sigma ,\kappa \lambda}=(I)_{\mu \nu ,\kappa\lambda}=\left(P^{(2)}+P^{(1)}_m+P^{(0)}_s+P^{(0)}_{\omega}\right)_{\mu \nu ,\kappa \lambda}\ .
\int_{\mathbb{R}^{2n}}e^{-\pi \langle x,\mathcal{Q}x\rangle}\prod_{j=1,2}f_j(x_j)^{c_j}\, dx=\big(\frac{(2\pi)^{\frac12(c_1+c_2)-1}}{\sqrt{1-e^{-2s}}}\big)^{-n}\int_{\mathbb{R}^n}P_s \big[\big( \frac{f_1}{\gamma}\big)^\frac1p \big] \big( \frac{f_2}{\gamma}\big)^{\frac1{q'}}\, d\gamma.
\begin{array}{l l}{{\displaystyle{\sum_{s = 1}^{D / 2}\, [ Z_{s}, \bar{Z}_{s}] = \sum_{s = 1}^{D / 2}\, \frac{1}{\, \theta_{s}}}\, ,}}&{{~ [ Z_{s}, Z_{r}] = 0 \, ,}}\\ \end{array}
I_{n,B}=\frac1{(4\pi^2)^2}\int_0^\Lambda\, |\varphi_k\dot\varphi_{k'}-\dot\varphi_k\varphi_{k'}|^2 \frac{k^2k'^2k^{2n}}{k'^2-k^2}\coth\frac{\beta W_k}2 \coth\frac{\beta W_{k'}}2\, dkdk'\ .
d,\rho ,s ; \Vert g\Vert_{L_\infty}, \Vert g^{-1}\Vert_{L_\infty}, \Vert \mathbf{A}\Vert_{L_\rho (\Omega )},\Vert v\Vert_{L_s(\Omega)},\Vert \mathcal{V}\Vert_{L_s(\Omega )}, &\Vert Q_0\Vert_{L_\infty}, \Vert Q_0^{-1}\Vert_{L_\infty};\ \, \Gamma .
Q_{N}\left( i,n\right) :=\mathcal{S}_{i}^{t}\cap{\displaystyle\bigcap\limits_{j=2,..,d}}\left\{z\in\mathbb{R}^{d}:\left\langle v_{j},z\right\rangle \in\left[n_{j}N,\left( n_{j}+1\right) N\right] \right\}\ ,
[\nabla_{\beta_1}\nabla_{\beta_2}\cdots\nabla_{\beta_{l+1}}, \nabla_{\gamma_1}]W = \nabla_{\beta_1}([\nabla_{\beta^{\prime}}, \nabla_{\gamma_{1}}]) + [\nabla_{\beta_1}, \nabla_{\gamma_1}](\nabla)_{\beta^{\prime}}W,
F ( w ) = C_{1}( e^{A ( w )})^{\cdot}+ C_{2}( e^{A ( w )})^{\cdot}\int_{0}^{w}d w^{\prime}\frac{\ddot{A}( w^{\prime}) + \dot{A}( w^{\prime})^{2}}{\dot{A}( w^{\prime})^{2}}e^{- 3 A ( w^{\prime})}.
S_{P}[{\bar{x}};{\hat{g}}] ={\frac{1}{4 \pi \alpha^{\prime}}}\sum_{n = - \infty}^{\infty}{\frac{2 n \pi}{\mathrm{s i n h}( 2 n \pi / l )}}\left[ ( | x_{n}^{i}|^{2}+ | x_{n}^{f}|^{2}) \mathrm{C o s h}( 2 n \pi / l ) - 2 \mathrm{R e}( x_{n}^{i}\cdot x_{n}^{f *}) \right] \quad .
\frac{s_3}{s_1+s_3-1}\sum_{n \leq at_3}\frac{1}{n^{s_2-1}}\int_{at_3}^\infty \frac{du}{u^{s_1}(u+n)^{s_3+1}}&= \frac{s_3}{2\pi i (s_1+s_3-1) \Gamma(s_3+1)}\sum_{n \leq at_3}\int_{at_3}^\infty \int_{(c)}\frac{\Gamma(s_3+1+z)\Gamma(-z)}{u^{s_1+s_3+1+z}n^{s_2-1-z}}dz du,
\sum_{m\ll \delta^{-1}}\Bigl|\sideset{}{'}\sum_{p_1,\dots,p_\ell}\Bigl(V_m(\mathcal{A}{}_d;d)-\frac{\kappa_\mathcal{A}\#\mathcal{A}{}}{\#\mathcal{B}{}}V_m(\mathcal{B}_d;d)\Bigr)\Bigr|=o_{\delta,\mathcal{L}}\Bigl(\frac{\#\mathcal{A}}{\log{X}}\Bigr).
\textstyle{\binom{e_u+1+n-2}{n-2}}= \textstyle{\binom{e_u+n-1}{n-2}}& = \textstyle{\binom{e_u+n}{n-1}-\binom{e_u+n-1}{n-1}>\binom{e_v+n-1}{n-1}-\binom{e_v-1+n-1}{n-1}=\binom{e_v-1+n-1}{n-2}=\binom{e_v+n-2}{n-2},}
sp_\lambda(\mathbf{x}^{\pm};z)=\sum_{\substack{\varepsilon_i\in\{0,1\}1\leq i\leq n+1}}(-z^{-1})^{|\varepsilon|}sp_{\lambda_1-\varepsilon_1,\dots,\lambda_{n+1}-\varepsilon_{n+1}}(\mathbf{\bar{x}}^{\pm}).
\lim_{z\rightarrow1}\frac{A^{(1)}(z,\xi)}{D(z)}= \frac{\left(1-\frac{\lambda_1}{\mu_1}\right)C^{(1)}}{k_1\left(1-\frac{\lambda_1}{\mu_1}\right)-k_2\frac{\lambda_1}{\mu_2}},
||\mathcal{E}_{\alpha}||_2 \leq \alpha\big((1+\sqrt{2}\alpha n)^2+[2(1+\sqrt{2}\alpha n)^2+4\sqrt{2}n(1+\sqrt{2}\alpha n)] &\quad~+[8\sqrt{2}n(1+\sqrt{2}\alpha n)+8n^2]+16n^2\big) &=3\alpha[\sqrt{2}n(\alpha+2)+1]^2.
\sum_{Q\in \Gamma\backslash \mathcal Q_D}\chi_d(Q) \int_{C_Q}\varphi(z) y^{s-1}\, |dz| = \pi^{-s-\frac 12}\sqrt{d}\, \Gamma(\tfrac s2+\tfrac{ir}2+\tfrac 14)\Gamma(\tfrac s2-\tfrac{ir}2+\tfrac 14) L(s+\tfrac 12,\varphi\times\chi_d).
\mathrm{d}f:=\left(\frac{\partial f}{\partial z_{1}},\cdots,\frac{\partial f}{\partial z_{n}}\right),\overline{\mathrm{d}}f:=\left(\frac{\partial f}{\partial\overline{z}_{1}},\cdots,\frac{\partial f}{\partial\overline{z}_{n}}\right)
\tilde{\mu}_{k}(u) =\underline{\overline{u}+\overline{\rho(u)}}=u_{1}: \overline{N}\rightarrow \overline{N^{1}}\tilde{\mu}_{k}^{2}(u)=\tilde{\mu}_{k}(u_{1}) =\underline{\overline{u_{1}}+\overline{\rho(u_{1})}}=u_{2}: \overline{\overline{N}}\rightarrow \overline{\overline{N^{1}}}
\overline{{{\lambda_{\uparrow L -}}}}\: \Lambda + \lambda_{\uparrow L -}\: \overline{{{\Lambda}}}\; = \; 2 \: I_{\uparrow}\: \hat{Y}_{L}\: I_{\downarrow}\: \hat{Y}_{R}\: I_{\uparrow}\: \left( T_{[ 1 ]}^{( 0 )}\: T_{[ 1 ]}^{( 0 )}\: \overline{{{T_{[ 0 ]}^{( - 1 )}\: T_{[ 0 ]}^{( 0 )}}}}\: + \: T_{[ 0 ]}^{( - 1 )}\: T_{[ 0 ]}^{( 0 )}\: \overline{{{T_{[ 1 ]}^{( 0 )}\: T_{[ 1 ]}^{( 0 )}}}}\right) \; .
\delta((a, b), x) = \delta((t_1, t_2) + (p 3^n, q 3^n), x) = w + p 3^n \alpha + q 3^n \beta = (a - t_1) \alpha + (b - t_2) \beta + w = a \alpha + b \beta - t_1 \alpha - t_2 \beta + w.
\renewcommand{\arraystretch}{2}\begin{array}{lcl}\displaystyle \Omega_1 = -\frac{1}{4}(F_2+F_3+F'_4)+\cdots, ~ \displaystyle \Omega_2 = -\frac{1}{4}(-F_1+F_3+F'_4)+\cdots, \displaystyle \Omega_3 = -\frac{1}{4}(-F_1-F_2+F'_4)+\cdots, ~ \displaystyle \Omega_4 = -\frac{1}{4}(-F_1-F_2-F_3)+\cdots. \end{array}
Y_{0}^{(a,b)}=Y_0, Y_{1}^{(a,b)}=\frac{rY_0+bX_0}{2},Y_{h'+2}^{(a,b)}=rY_{h'+1}^{(a,b)}-Y_{h'}^{(a,b)}, X_{0}^{(a,b)}=X_0, X_{1}^{(a,b)}=\frac{rX_0+aY_0}{2},X_{h'+2}^{(a,b)}=rX_{h'+1}^{(a,b)}-X_{h'}^{(a,b)},
\sigma(B)-\sigma(0)=-\frac{e^{2}}{2\pi^{2}\hbar}[\psi(\frac{1}{2}+\frac{1}{\tau a})-\psi(\frac{1}{2}+\frac{1}{\tau_{1}a})+\frac{1}{2}\psi(\frac{1}{2}+\frac{1}{\tau_{2}a})-\frac{1}{2}\psi(\frac{1}{2}+\frac{1}{\tau_{3}a})]
\begin{array}{l l l}{{q}}&{{=}}&{{( + \frac{1}{2}, \, \, \vartheta_{1}- \frac{1}{2}, \vartheta_{2}- \frac{1}{2}, \vartheta_{3}- \frac{1}{2})}}\\{{q}}&{{=}}&{{( \pm \frac{1}{2}, \pm \frac{1}{2}, \, \, \vartheta_{2}- \frac{1}{2}, \vartheta_{3}- \frac{1}{2})}}\\{{q}}&{{=}}&{{( \pm \frac{1}{2}, \pm \frac{1}{2}, \pm \frac{1}{2}, \, \, \vartheta_{3}- \frac{1}{2})}}\\ \end{array}
Q(y)=\operatorname{sgn}(y)\begin{dcases}1 |y|{<}\tau_{0}i \tau_{i}{<}|y|{<}\tau_{i+1}, 0{<}i{<}2^{w-1}{-}2 2^{w-1}|y|{<}\tau_{2^{w-1}-2}\end{dcases}.
L_1= \ -\frac{4H}{H^2-4K}(\frac{1}{t}-\frac{1}{t^2}) +\frac{2H}{H^2-4K}(1-\frac1q)\frac{(\lambda_2+\frac{\lambda_1}{t})^2}{K}&+\left[-\frac{4K}{H^2-4K}(1-\frac1q)-\alpha^2-\frac{\alpha}q\right]\frac{\lambda_2(\lambda_2+\frac{\lambda_1}{t})^2}{K^2}+\frac{2\lambda_2(1+\frac 1 t)^2}{H^2-4K}
\widetilde{\varepsilon_5}(m):=\min\Bigl\{4 F_m, \frac{(\pi^2 B_1a)^{m/3}}{316\,A(\sigma)}\Bigl(\frac{m!}{\Gamma(m/3+2)}\Bigr)^{1/2}\,\varepsilon_4\Bigr\}.
x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2 = 0 \, , x_1^2 + \varepsilon x_2^2 + \varepsilon^2 x_3^2 + \varepsilon^3 x_4^2 + \varepsilon^4 x_5^2 = 0 \, , (\varepsilon^5=1.) \varepsilon^4 x_1^2 + \varepsilon^3 x_2^2 + \varepsilon^2 x_3^2 + \varepsilon x_4^2 + x_5^2 =0 \, .
{\rm Tr}e^{-\beta \, P \bar{\cal K}}= \left[ \frac{mL}{\sqrt{4\pi \beta}}+\frac{1}{\sqrt{\pi}}\sum_{n=1}^\infty \frac{2^n\,\beta^{n-\frac{1}{2}}}{1\cdot 3\cdot 5 \cdot ... \cdot (2n-1)}\right] e^{-\beta}-1
(g^{-1}k)^{-1}h & = \underbrace{\theta (n (\theta (g^{-1}) k))^{-1}}_{\in \overline{N}}\underbrace{a (\theta (g^{-1}) k)}_{\in A}\underbrace{\theta (m (\theta (g^{-1})k))^{-1}}_{\in M}\underbrace{\kappa (\theta (g^{-1}) k)^{-1}h}_{\in K}.
\vec{\psi}_{\mp}( x ) \left( \frac{d \vec{\phi}}{d x}\pm \vec{\nabla}W \right) | \vec{\phi}( x ) , \vec{\psi}( x ) \rangle = 0 \qquad , \, \vec{\psi}_{\pm}( x ) = \vec{\psi}_{1}( x ) \pm \vec{\psi}_{2}( x )
\sum_{n=0}^\infty\sum_{|m|\leq n/2}(-1)^mq^{n^2-2m^2}(1-q^{2n+1}) = qg_{1,3,1}(q^6,q^6;q^4) + f_{1,3,1}(q^2,q^2;q^4) &+q^4f_{1,3,1}(q^{10},q^{10};q^4)-q^9g_{1,3,1}(q^{14},q^{14};q^4) &=f_{1,3,1}(q^2,q^2;q^4) =j(q^2;q^4).
\frac{n_0 \alpha'_{\tilde{m}(i)}}{i}= \frac{n_0( \lfloor\gamma\tilde{m}(i) \rfloor+ 1 ) \alpha}{i(n + \lfloor\gamma\tilde{m}(i) \rfloor+ 1 - \tilde{m}(i))}\le\frac{n_0 \alpha}{n + i -\tilde{m}(i)}\le\alpha.
\sum_{g\in S}\varphi(g)\|a_g\|_2^2\leq|F|^2\sum_{g\in S}\|a_g\|_2^2\leq |F|^2\sum_{i=1}^m\|E_{A\rtimes\Sigma_i}(u_{b_i}^*uu_{c_i}^*)\|_2^2\leq\frac{\kappa}{2}.
\begin{aligned}\mathbf{R}_{\mathrm{c},k}=\mathbf{I}+\sum_{j=1}^K\mathbf{H}_k^H\mathbf{P}_{j}\mathbf{P}_{j}^H\mathbf{H}_k, \mathbf{R}_{k}=\mathbf{I}+\sum_{j=1,j\neq k}^K\mathbf{H}_k^H\mathbf{P}_{j}\mathbf{P}_{j}^H\mathbf{H}_k.\end{aligned}
W_{e f f}= m_{i j}\Phi_{i}\Phi_{j}f \left( \frac{\lambda_{a}\Phi_{a}}{m_{i j}\Phi_{i}\Phi_{j}}; \frac{g_{a b c}\Phi_{a}\Phi_{b}\Phi_{c}}{m_{i j}\Phi_{i}\Phi_{j}}\right) .
B^{(\ell)}=\underline{B}^{(\ell)}+HB^{(\ell-1)}+\sum_{t=1}^{\ell-1}\underline{B}^{(t-1)}H^{(2)}B^{(\ell-t-1)}+\underline{B}^{(\ell-1)}H^{(1)}-\sum_{t=0}^{\ell}R_t^{(\ell)},
u = -\hat{s}\bigl((\hat{s}+q_{\widehat{x}}(s))^\wedge \hat{x}\bigr) = s^{-1}\bigl((s^{-1}- \pi(x)^{-1}s\pi(x)^{-1})^{-1}\pi(x)^{-1}(e-w)\bigr) = \bigl( \pi(x)^{-1}- s \pi(x)^{-1}s \bigr)^{-1}(e-w) = (\pi(x)+s)^{-1}\pi(x)(\pi(x)-s)^{-1}(e-w).
\left| L_n(w, \theta_1) - L_n(w, \theta_2) \right|\leq \frac1{2n}\sum\limits_{i = 1}^n \frac{| D_{\theta_1}(X_i) - D_{\theta_2}(X_i) |}{D_{\min}}+ \frac1{2n}\sum\limits_{i = 1}^n \frac{| D_{\theta_1}(X_i) - D_{\theta_2}(X_i) |}{1 - D_{\max}}.
\Bigr(\mathrm{Tr\,}b^+_0(z,z)-\mathrm{Tr\,}b^-_0(z,z)\Bigr)dv_X(z,\theta) &=\left[\mathrm{Td_b\,}(\nabla^{T^{1,0}X},T^{1,0}X)\wedge\mathrm{ch_b\,}(\nabla^{E},E)\wedge e^{-m\frac{d\omega_0}{2\pi}}\wedge\omega_0\right]_{2n+1}(z,\theta),\ \ \forall(z,\theta)\in D.
s_0 (t)=\frac{1}{\sqrt{\sinh1}}\sinh\left(t-\frac{1}{2}\right)\,, c_0 (t) =1\,, s_k (t)=\sqrt{\frac{2}{1+4\pi^2 k^2}}\sin(2\pi k t)\,,k\in\mathbb{N}\,, c_k (t)=\sqrt{\frac{2}{1+4\pi^2 k^2}}\cos(2\pi k t)\,,k\in\mathbb{N}\,,
\sum_{n=0}^{\infty}\textup{PDO}_\textup{t}(2n)q^n =2q{f_2f_4f_6f_{12}}\cdot\dfrac{f_3}{f_1^3}&=2q{f_2f_4f_6f_{12}}\left(\dfrac{f_4^6f_6^3}{f_2^9f_{12}^2}+3q\dfrac{f_4^2f_6f_{12}^2}{f_2^7}\right).
\sum\limits_{k = 0}^n f(k) = \int_0^n f(x)dx + \frac{1}{2}[f(n)+f(0)] + \sum\limits_{k = 2}^\infty \frac{B_{k}}{k!}\left[f^{(k - 1)}(n) - f^{(k - 1)}(0) \right]
I(\tilde{A}(\varphi_1) \cdots \tilde{A}(\varphi_n)\Omega) = I(A(\varphi_1) \cdots A(\varphi_n)\Omega) &= \langle \cdot, \varphi_1 \rangle \cdots \langle \cdot ,\varphi_n \rangle\in L^2(\Phi', \mu)
\lim_{t\rightarrow\infty}\ell\left( t\right) =\alpha\gamma_{1}^{\alpha-1}\left( \mu_{\alpha}^{\left( 2\right)}\left( \rho_{1}\right)-\mu_{\alpha}^{\left( 1\right)}\mu_{1}^{\left( 2\right)}\left( \rho_{1}\right) \right) .
\varpi_{\{3,4,5\}}= \frac{1}{6}\varpi_3\varpi_4\varpi_5 = e_3(x_1,x_2,x_3,x_4,x_5), \varpi_{\{2,4,5\}}= \frac{1}{2}\varpi_2\varpi_4\varpi_5 = e_1(x_1,x_2)e_2(x_1,x_2,x_3,x_4,x_5).