harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
(\phi_2^{\mathbb I}, \phi_0^{\mathbb I})=c_1(\Phi_2,\, \Phi_0),(-\phi_2^{\mathbb R}, \phi_0^{\mathbb R})=c_2(\Phi_2,\, \Phi_0),\phi_1= i (c_3\, U^+, c_4\,U^-),
\boldsymbol{T}_{k}\, \left( E_{j_{0}}^{T}\right)(x) + \eta \, \sum_{j \neq j_{0}}^{\aleph}\left( \underset{x}{\nabla}M^{k}- k^{2}\, N^{k}\right) (E_{j}^{T})(x) = E^{Inc}_{j_{0}}(x).
= \left( 1-yz \right)^{\frac{y}{1-y}}\exp\left\{\frac{y (y^2 +4y +1)}{(1-y)^4}(Li_4(z)-Li_4(yz)) + \frac{3y(1+y)}{(1-y)^3}Li_3(yz) + \frac{3y}{(1-y)^2}Li_2(yz) \right\};
\textrm{N}_{\vec d, I}= \bigoplus_{i\in I,\,\,j\in [N],\,\, i\ne j}\pi_*\left(\mathcal{K}_i^{\vee}\right)-\bigoplus_{i,j\in I,\,\,i\ne j}\pi_*\left(\mathcal{K}^{\vee}_i \otimes\mathcal{K}_j\right) .
\chi(\tilde{B},X,\Lambda) = \left\{X_{i}^{\pm 1}\;|\; i \in [n+1,m] \, \right\}\cup \bigcup_{(\tilde{B}',X',\Lambda') \sim (\tilde{B},X,\Lambda)}\left \{X_i' \;|\; i\in [n] \,\right \}.
\textrm{Cov}_B[X,Y] = \textrm{Var}_B[X] a^T & =k(B) \textrm{Var}[X]a^T + (\textrm{Var}_B[Y] -k(B) \textrm{Var}[Y]) \beta &= k(B)(\textrm{Cov}[X,Y] - \textrm{Var}[Y]\beta ) +\textrm{Var}_B[Y] \beta & = \textrm{Var}_B[Y] \beta.
\big(\tfrac{\lambda_{I}}{\lambda_{I_0}}\big)^{h}\exp\big\{C_{h}|I_0| \log(\tfrac{en}{|I_0|})-D_{h}|I| \log(\tfrac{en}{|I|})\big\}&=(\tfrac{ne}{|I|})^{-c_1 |I|}\exp\big\{(\varkappa h +C_h)|I_0|\log(\tfrac{en}{|I_0|}) - A_h|I| \log(\tfrac{en}{|I|})\big\}.
(u_{0}^{n}, u_{1}^{n}) = \sum_{j = 1}^{N}(\lambda_{n}^{j}\phi_{0}^{j}(\lambda_{n}^{j}x), (\lambda_{n}^{j})^{2}\phi_{1}^{j}(\lambda_{n}^{j}x)) + (R_{0,n}^{N}, R_{1,n}^{N}),
\begin{array}{c}{{~ ~ e_{i}e_{j}= - \delta_{i j}+ c_{i j k}\, e_{k}\, , ~{i , j , k = 1 , 2 , \cdots , 7}}}\\{{{}}}\\{{1 = c_{1 2 7}= c_{1 6 3}= c_{2 4 6}= c_{3 4 7}= c_{5 6 7}= c_{2 5 3}= c_{1 5 4}\mathrm{( o t h e r s ~ z e r o )}\, .}}\\ \end{array}
\| u^{(n+1)}\|_{\ell^1 X^{s}[0,T]}\lesssim \e^{C(M) L}( \| u_0\|_{\ell^1 H^{s}}+ \| G(u^{(n)}) \|_{l^1Y^s}) \lesssim \e^{C(M) L}( M_s + T^\delta C(M) \|u^{(n)}\|_{\ell^1 X^{s}[0,T]}) .
\partial_{tr}v_{2}(t,r) = \frac{c_{b}}{2}\int_{0}^{\infty}\cos(t\xi) \xi^{2}(J_{0}(r\xi)-J_{2}(r\xi)) \frac{\chi_{\leq \frac{1}{4}}(\xi) d\xi}{\log^{b-1}(\frac{1}{\xi})}
x^w = ((x\pi_1)(x\pi_2))^{(u^{-1}\Pi_1)(u^{-1}\Pi_2)u}= \left((x\pi_1)^{(u^{-1}\Pi_1)}(x\pi_2)^{(u^{-1}\Pi_2)}\right)^u = \left((x^{u^{-1}}) \pi_1 (x^{u^{-1}}) \pi_2 \right)^u = x.
\ll\frac{y_{max}^2X}{V}\biggl(\sum_{\substack{u<R (u,VP_f)=1}}\frac{\mu(u_i)^2}{\varphi(u_i)}\biggr)^k\biggl(\sum_{s_{i,j}>D_0}\frac{\mu(s_{i,j})^2}{\varphi(s_{i,j})^2}\biggr)\biggl(\sum_{s>1}\frac{\mu(s)^2}{\varphi(s)^2}\biggr)^{k^2-k-1}&\ll\frac{y_{max}^2\varphi(VP_f)^kX(\log X)^k}{V(VP_f)^kD_0}.
\big|\int_0^t \Delta f_k(\omega(s))ds\big| \leq \int_0^t |\Delta f_k|(\omega(s))ds &\leq c_2 \int_0^t | f|(\omega(s))ds+\int_0^t |\Delta f|(\omega(s))ds+c_2 \int_0^t |d f|_P(\omega(s))ds.
(\tilde{U}_{1}^{i},\tilde{U}_{2}^{i},...,\tilde{U}_{d}^{i})=(F_{1}^{n}(X_{1}^{i}),F_{2}^{n}(X_{2}^{i}),...,F_{d}^{n}(X_{d}^{i})),i=1,...,n.
\,\, k[V]^{g^*}= k[x_1, x_2, x_3, x_4][y_1^{m_1}y_2^{m_2}y_3^{m_3}y_4^{m_4}| \sum_{i=1}^{4}m_i \equiv 0 \, (\rm{mod}\, 6)]\,\, .
A(z)=\frac{1}{4}\left(X^2(z)-X(z)-\frac{3}{4}\right),B(z)=\frac{1}{4}\left(Y^2(z)-Y(z)-\frac{3}{4}\right), C(z)=\frac{1}{4}\left(Z^2(z)-Z(z)-\frac{3}{4}\right),
\eta_t =\min\left(\frac{\mathfrak{y}}{d},\, d^{-\frac{2(\beta-1)}{2\beta-1}}T^{-\frac{\beta}{2\beta-1}}\right) \qquad h_t = \mathfrak{h}\cdot T^{-\frac{1}{2(2\beta-1)}}\enspace.
\sum_{\imath=1}^{\infty}\frac{1}{\imath^\alpha}\bigg(\sum_{\jmath=1}^{\imath}\varrho(\jmath)\bigg)^p \leq \c{C}(\alpha,p)\sum_{\imath=1}^{\infty}\frac{1}{\imath^{\alpha-p}}\varrho^p(\imath);
g_\lambda(N_{k,q}) =-\lambda\frac{q+1}{\pi}\mu_{k+2,q+1}^\lambda, g_\lambda(B_{k,q}) =\mu_{k,q}^\lambda,k\neq 2q g_\lambda(B_{2q,q}) =\mu_{2q,q}^\lambda-\lambda\frac{q+1}{\pi}\mu_{2q+2,q+1}^\lambda.
\lim_{\lambda\rightarrow\infty}a_{\lambda p(p+1)+\rho}=a^{(\rho)}=(1-\widetilde{F}_{0}^{(\rho)}(z)) \prod_{i=\rho-p}^{\rho-1}\widetilde{F}_{0}^{(i)}(z),z\in\mathbb{C}\setminus[a_{0},b_{0}].
S_{\mathrm{Y M}}= - \frac{1}{4 g^{2}}\int d^{4}x F_{\alpha \beta}^{a}F_{a}^{\alpha \beta}+ \frac{\theta_{\mathrm{Y M}}}{3 2 \pi^{2}}\int d^{4}x F_{\alpha \beta}^{a}{\widetilde{F}}_{a}^{\alpha \beta}
\begin{array}{r c l}{{}}&{{}}&{{t t^{- 1}= 1 = t^{- 1}t ~ ,}}\\{{}}&{{}}&{{t e t^{- 1}= q^{2}e ~ ,}}\\{{}}&{{}}&{{\Delta ( e ) = e \otimes 1 + t \otimes e ~ , ~ \Delta ( t ) = t \otimes t ~ ,}}\\{{}}&{{}}&{{\epsilon ( e ) = 0 ~ , ~ \epsilon ( t ) = 1 ~ ,}}\\{{}}&{{}}&{{S ( e ) = - t^{- 1}e ~ , ~ S ( t ) = t^{- 1}~ .}}\\ \end{array}
\mathfrak{J}_\mathfrak{m}^1=\begin{bmatrix}\tilde{\mathfrak{J}}^1 \tilde{\mathfrak{J}}^0 \varpi_E^{-1}\tilde{\mathfrak{H}}^1 \tilde{\mathfrak{H}}^1 \tilde{\mathfrak{J}}^1 \tilde{\mathfrak{J}}^0 \varpi_E\tilde{\mathfrak{J}}^0 \tilde{\mathfrak{H}}^1 \tilde{\mathfrak{J}}^1\end{bmatrix}\cap A_W,
\tilde{\mathsf{d}}(t+v_1 , s+v_2) = \mathsf{d}_t (t+v_1, t) + \mathsf{d}_s (s+v_2,s) + \delta &= \mathsf{d}_0 (v_1, 0) + \mathsf{d}_0 (v_2,0) + \delta &= \mathsf{d}_0 (-v_1, 0) + \mathsf{d}_0 (-v_2,0) + \delta &= \tilde{\mathsf{d}}(-t-v_1 , -s-v_2),
\begin{aligned}x(t) =x(0)+B(t)-\min\big\{0,\inf_{0\leq s \leq t}(x(0)+B(s)-f(s))\big\}&= \max\big\{x(0)+B(t),\sup_{0\leq s \leq t}(f(s)+B(t)-B(s))\big\}.\end{aligned}
\Pr\left[G_n > \frac{n+4}{4}\right] = \Pr\left[\frac{n}{2}- \frac{C_{w_1,n}}{2}+ 1 > \frac{n+4}{4}\right] = 1-\Pr\left[C_{w_1,n}\geq \frac{n}{2}\right] = 1 - O\left(\frac{n}{2^{n/2}}\right)
\mathcal{B}=*B^A\left[\begin{array}{c}v_A \overline{v}_A\end{array}\right]-iB^A\left[\begin{array}{c}v_A -\overline{v}_A\end{array}\right]+b^{\alpha}\left[\begin{array}{c}e_{\alpha}\overline{e}_{\alpha}\end{array}\right]+i*b^{\alpha}\left[\begin{array}{c}e_{\alpha}-\overline{e}_{\alpha}\end{array}\right]
\bigl|\varpi\circ W^{t}\left(z\right)-\varpi\left(z^{*}\right)\bigr|\leq\sum_{i=1}^{n+\nu}\bigl|x_{i}^{\:\left(t\right)}-x_{i}^{*}\bigr|=\left\Vert W^{t}\left(z\right)-z^{*}\right\Vert ,
|E_{2}(t,r)| \leq C\left(\frac{1}{\sqrt{r}\sqrt{|t-r|}\log^{b}(|t-r|)}\right)+\frac{C}{\sqrt{r}\sqrt{t+r}\log^{b-1}(t+r)}+\frac{C}{r \log^{b-1}(r)}, r>\frac{t}{2}, |t-r| > 5
\mathsf u\sum_{|\vec{d}|=d}\left[h_1^{d_1}\ldots h_N^{d_N}\right] \bigg\{\prod_{i}\bigg(\frac{z_i(\alpha_i+y)}{\alpha_i(z_i+y)}\bigg)^{b_i}\bigg(\frac{z_i+y}{z_i}\bigg)^{d_i}\bigg(\frac{h_i}{R(z_i)}\bigg)^{d_i+1}z_i^{d+1}\bigg(\frac{R(z_i)}{\prod_{j}(z_j-\alpha_i)}\bigg)^{b_i}\cdot \prod_{i<j}(z_i-z_j)^2\bigg\}\bigg{|}_{\epsilon=0}.
\nu_{s,t}(z):=\begin{cases}\kappa_t\delta_1+(1-\kappa_t)\delta_0, s<t; \delta_1, s=t; \left(1-\frac{1-\kappa_t}{1-\kappa_t(z_{0,...,t})}\right)\delta_1+\frac{1-\kappa_t}{1-\kappa_t(z_{0,...,t})}\delta_0, s>t.\end{cases}
\lVert AB-A_0 B_0 \rVert^2 \sim \sum_{i=1}^{M-1}\left[ x_i^2 + \sum_{j=2}^{N}\{a_i(b_j-b_1)-a^0_i(b^0_j-b^0_1)\}^2 \right] &=\sum_{i=1}^{M-1}\left\{x_i^2 + \sum_{j=2}^{N}(a_i b_j - a^0_i b^0_j)^2 \right\}&=\sum_{i=1}^{M-1}x_i^2 +\sum_{i=1}^{M-1}\sum_{j=2}^N (a_i b_j - a^0_i b^0_j)^2
\frac{(u_{+}-v_{-})^{2}}{(u_{+}-v_{-})^{2}-h^{2}}\bar R_{21}(u_{+}-v_{-})L_{2}^{-}(u)L_{1}^{+}(v)=\frac{(u_{-}-v_{+})^{2}}{(u_{-}-v_{+})^{2}-h^{2}}L_{1}^{+}(v) &L_{2}^{-}(u)\bar R_{21}(u_{-}-v_{+})
p^i_{n}(x,y,z)= \frac{(a_i+b_i)^n b_i^n \left(\left(\frac{a_i}{b_i}\right)^n |b|^{n+1}+|a|^{n+1}-\frac{a_i^n}{(a_i+b_i)^n}\right)}{|a|^{n+1}|b|^{n+1}}.
\langle b, \Pi_{j}(G, \tilde G)\rangle =\langle b, G\, R_j (\tilde G)-\tilde G\, R_j^t( G)\rangle= \langle G,\, b\, R_j(\tilde G)- R_j(b\tilde G)\rangle &= \langle G, [b, R_j](\tilde G)\rangle,
\psi_{k}^{+}( x ) = \left( \begin{matrix}{{\eta_{k}^{S}}}\\{{\tilde{\eta}_{k}^{A}}}\\ \end{matrix}\right) \mathrm{~ a n d ~}\psi_{k}^{-}( x ) = \left( \begin{matrix}{{\eta_{k}^{A}}}\\{{\tilde{\eta}_{k}^{S}}}\\ \end{matrix}\right)
\begin{array}{c}{\langle e_{a_{k}. . . a_{1}}\mid = ( \mid e_{a_{1}. . . a_{k}}\rangle )^{+}}{\langle e_{a_{k}. . . a_{1}}\mid e_{b_{1}. . . b_{k}}\rangle = \delta_{a_{1}b_{1}}. . . \delta_{a_{k}b_{k}}}\end{array}
\Omega_k=\{x\in\Omega\, :\, |x|<k\}\mbox{and}\delta_k=\left(\frac{\displaystyle\int_{\Omega_k}|\phi_+|^{q-1}\,dx}{\displaystyle\int_{\Omega_k}|\phi_-|^{q-1}\,dx}\right)^{1/(q-1)}.
\begin{aligned}\lambda_{\mathcal W}(\mathcal M)^2 =\rho\left(\left(\frac{1}{N}\sum_{i=1}^N W_i \otimes A_i\right)^{2}\right) &=\rho \left(\frac{1}{N^2}{\sum_{i, j = 1}^N (W_i W_{\!j}) \otimes (A_i A_j)}\right) &=\lambda_{\mathcal W^2}(\mathcal M^2).\end{aligned}
M_1:=\begin{pmatrix}1 2 1 1 2 2 1 1 1 \end{pmatrix}, \M_2:=\begin{pmatrix}0 2 1 2 1 1 0 1 2 \end{pmatrix}, \M_3:=\begin{pmatrix}2 2 0 1 1 1 0 1 2 \end{pmatrix}, \M_4:=\begin{pmatrix}0 2 2 0 0 0 2 0 1 \end{pmatrix}.
\lambda_{1}= \frac{1}{2d}\left(c-\sqrt{c^2-4ds}\right), \lambda_{3}=\frac{1}{2}\left(c+\sqrt{c^{2}-4f(1) p'(1)}\right), begin{align*}0.2cm]\lambda_{2}= \frac{1}{2d}\left(c+\sqrt{c^2-4ds}\right), \lambda_{4}=\frac{1}{2}\left(c-\sqrt{c^{2}-4f(1) p'(1)}\right).
\lim\limits_{N\rightarrow\infty}\frac{|gF_N\Delta F_N|}{|F_N|}\le\lim\limits_{N\rightarrow\infty}\frac{|F_{N-1}|+|gF_{N-1}|+|gH_{n_N}\Delta H_{n_N}|}{|H_{n_N}|}&\le\lim\limits_{N\rightarrow\infty}\frac{2\sum_{i=1}^{N-1}|H_{n_i}|+|gH_{n_N}\Delta H_{n_N}|}{|H_{n_N}|}&=0
\psi_{L}=\frac{1}{2}(1-\gamma^{5})\psi=(\begin{matrix}I_{2}&0\\ 0&0\end{matrix})\psi,\psi_{R}=\frac{1}{2}(1+\gamma^{5})\psi=(\begin{matrix}0&0\\ 0&I_{2}\end{matrix})\psi.
j=\left(\frac{a(z_1)z_0+b(z_1)}{c(z_1)z_0+d(z_1)},\frac{\alpha z_1+\beta}{\gamma z_1+\delta}\right)=\left(z_0,\frac{\alpha z_1+\beta}{\gamma z_1+\delta}\right)\circ\left(\frac{a(z_1)z_0+b(z_1)}{c(z_1)z_0+d(z_1)},z_1\right)
\bigr(\mathcal{R}(F\cdot{G})\bigr)(x)=(\mathcal{R}F)(x)\cdot(\mathcal{R}G)(x)-\bigr(\mathcal{R}\left((\mathcal{R}F)\cdot{G}\right)\bigr)(x)-\bigr(\mathcal{R}\left(F\cdot(\mathcal{R}G)\right)\bigr)(x)
V_{e f}\left( \phi_{c}\right) = \frac{2 \alpha^{2}}{4 !}\phi_{c}^{4}+ \frac{\hbar}{2 4 \pi a}\alpha^{3}\phi_{c}^{3}- \frac{\hbar}{1 6 \pi^{2}a^{4}}\sum_{N = 1}^{m}\left( \frac{1}{N}\right)^{4}+ \frac{\hbar}{1 6 \pi^{2}a^{2}}\alpha^{2}\phi_{c}^{2}\sum_{N = 1}^{m}\left( \frac{1}{N}\right)^{2}.
\frac{\rho^2 (\xi_1 \omega_2 - \xi_2 \omega_1 )^2}{(\tau^2-|\xi|^2)^2}\frac{\partial \tau}{\partial \rho}= \frac{\rho^2 (\xi_1 \omega_2 - \xi_2 \omega_1 )^2 [2 (\tau - \xi \cdot \omega)^2]}{(\tau^2-|\xi|^2)^2 (\tau^2 - 2 \tau \xi \cdot \omega + |\xi|^2)}= \frac{(\xi_1 \omega_2 - \xi_2 \omega_1 )^2}{2(\tau^2 - 2 \tau \xi \cdot \omega + |\xi|^2)}
\left( nh_{n}\right)^{-(d+1)/5d}\leq \left( nh_{n}\right)^{-3/10}=\left(h_{n}^{2/5}\right)^{3/4}\left( \left( nh_{n}^{2}\right)^{-6/5}\right)^{1/4}\leq h_{n}^{2/5}+\left( nh_{n}^{2}\right)^{-6/5}.
a_{\delta}\geq \frac{1-c_1\delta^{\frac{p}{n-1+p}}(1+s_1(\delta))}{1-c_3\delta^{\alpha \frac{p'}{n-1+p'}}(1+s_3(\delta'))}=1+c_3\delta^{\alpha \frac{p'}{n-1+p'}}(1+o_{\delta}(1)+o_{\delta'}(1)) \geq 1+c_2\delta^{\frac{2}{n+1}}(1+o_{\delta}(1)+o_{\delta'}(1))\quad.
\theta^{(n)}_d\left(q\to 1\ \mathrm{limit\of}\ \frac{\prod_{j=1}^k x_{i-1,j}}{\prod_{j=1}^k x_{i,j}}F_{0,k}\right)= \mu_k A_i(d^k)\otimes Z^k\ \mathrm{for\any}\ 1\leq i \leq n.
I(A:B|MZ)_{\hat{\sigma}_{ABMZ}}= \int_{\mathbb{R}^{2n}}I(A:B|M)_{\hat{\sigma}_{ABM|Z=\mathbf{z}}}\,\mathrm{d}p_Z(\mathbf{z}) &= \int_{\mathbb{R}^{2n}}I(A:B|M)_{\hat{\rho}_{ABM}}\,\mathrm{d}p_Z(\mathbf{z}) = 0\;.
k+h^\vee = \frac{u}{v}\, \in \, \mathbb Q_{>0}, \ \ (u, v)=1, \ \u, v >0 \ \ \ \ \begin{cases}u\geq h^\vee, \ \v \geq h \ \ (v, r^\vee) =1 u\geq h, \ \v \geq r^\vee h^\vee \ \ (v, r^\vee) =r^\vee \end{cases}
\big\|C \star \big(S^3 - C^3\big) \star S\|_{L^\infty}\leq c\|E\|_{L^\infty}\Big(\frac{1}{m^4_t}{\bf 1}_{d=2}+ \frac{1}{m^2_t}{\bf 1}_{d=3}\Big)+ c\|E\|_{L^\infty}^2\Big(\frac{1}{m^4_t}{\bf 1}_{d=2}+ \frac{1}{m^3_t}{\bf 1}_{d=3}\Big) &+ \|E\|_{L^\infty}^3\Big(\frac{4}{m^4_t}+ \frac{\|E\|_{L^1}}{m^2_t}\Big).
H = \left( \begin{array}{c c c c}{{r_{5}}}&{{0}}&{{r_{3}+ i r_{4}}}&{{r_{1}+ i r_{2}}}\\{{0}}&{{r_{5}}}&{{- r_{1}+ i r_{2}}}&{{r_{3}- i r_{4}}}\\{{r_{3}- i r_{4}}}&{{- r_{1}- i r_{2}}}&{{- r_{5}}}&{{0}}\\{{r_{1}- i r_{2}}}&{{r_{3}+ i r_{4}}}&{{0}}&{{- r_{5}}}\\ \end{array}\right) .
\mathcal{F}_{t_1}= \cap_{n=1}^{\infty}\mathcal{F}_{t_1+1/n}^0, \mathcal{F}_{t_1}^0 =\sigma \{W_s(A): A \in \mathcal{B}_b(S), 0 < s \leq t_1 \}\vee \mathcal{N},
\mathrm{ch}(\mathrm{ind}\{D_B^+\}) =\langle e^{\Omega},[T^n]\rangle=\langle \frac{\Omega^n}{n!},[T^n]\rangle= &=\pm\langle x_1\cup\cdots \cup x_n\otimes y_1\cup \cdots \cup y_n,[T^n]\rangle=\pm y_1\cup\cdots\cup y_n\neq 0
-\frac{Q_{i}'(0)}{Q_{i}''(0)}= \frac{\frac{r_{i}^{2}\vartheta_{i}}{\lambda_{i,2}^{2}}-\frac{r_{i}}{\lambda_{i,2}^{2}}}{\frac{r_{i}^{2}\vartheta_{i}}{\lambda_{i,2}^{4}}}= \lambda_{i,2}^{2}\frac{\vartheta_{i}r_{i}-1}{r_{i}(\vartheta_i-1)}, i\in\mathbb{N}\cap[1,\eta],
\P(S_n\geq x)=\P(T_x\leq n) = \P(T_x\leq n,\: |X_n|\geq x-b_0\sqrt{n}) +\P(T_x\leq n, \: |X_n|< x-b_0\sqrt{n})\leq \P(|X_n|\geq x-b_0\sqrt{n}) +\P(T_x\leq n, \: |X_n-X_{T_x}|>b_0\sqrt{n}).
\delta(A)\sum_{j_2,\dots, j_\ell\geq 0}\prod_{r=2}^\ell \delta_{j_r}(A)=\delta(A)\prod_{r=2}^\ell\sum_{j_r=0}^m \delta_{j_r}(A)=\delta(A)\prod_{r=2}^\ell 1=\delta(A)
P_{x,r}^U\left(\bigcup_{k=1}^n A_k^c \right)\leq n \cdot c_1 \exp\left( -\frac{c_2}{2}\epsilon^{2/3}x^{1/3}\right)\leq \frac{1}{2}\epsilon^{2/3}x^{1/3}\cdot c_1 \exp\left( -\frac{c_2}{2}\epsilon^{2/3}x^{1/3}\right).
c^k_{\mathrm{b2s},1}\left( N_{\mathrm{sc}}\right) = \mathsf{E}\left\lbrace B N_{\mathrm{sc}}\log_2\left[ \frac{P_{\mathrm{bs}}}{BN_{\mathrm{bs}}\sigma^2_{\mathrm{sc}}}\right. \right. \left. \left. \,\,\, \times \left(\max \{\|\mathcal{H}^{\dagger}_{\mathrm{b2s},i}\| \}_{i=1}^{N_{\mathrm{bs}}}\right)^{-2}+1 \right] \right\rbrace,
\left|\Delta(\l)\right| \geqslant C\left|\omega_0(z_k) + \frac{\omega_1(z_k)}{\l}\right|e^{\Re(i \tau_k \l)}\geqslant C_1 \frac{e^{\Re(i \tau_k \l)}}{|\l|}, |\l|>R,\ \arg \l = \arg z_k,
\sqrt{5}M_n (1) = \sum_{m=-\infty}^\infty I_{n,n}(\omega + 2\pi im), 2 M_n (1) + 5 M_n' (1) = \sum_{m=-\infty}^\infty I_{n+1,1}(\omega + 2\pi im), 2\sqrt{5}M_n (1) + 11\sqrt{5}M_n' (1) + 5\sqrt{5}M_n'' (1) = \sum_{m=-\infty}^\infty I_{n+2,n}(\omega + 2\pi im),
G_{\alpha_{1}\ldots \alpha_{n}}( p_{1}, \ldots , p_{n}) \; \prod_{i = 1}^{n}\; J_{\alpha_{i}}^{i}( p_{i}) = \hat{G}_{\alpha_{1}\ldots \alpha_{n}}( p_{1}, \ldots , p_{n}) \; \prod_{i = 1}^{n}\; \hat{J}_{\alpha_{i}}^{i}( p_{i}) \ .
\rho(y) L_i \rho(u_{i,j}) = L_i \rho(y) \rho(u_{i,j})=\sum_{k\in [n_i]}L_i \rho(u_{i, k}y_{i, k, j})=\sum_{k\in [n_i]}V_{i,k}\rho(y_{i, k, j}).
A_\alpha\left(Y_\alpha\!=\!0,Y_{\beta\ne\alpha},t\right) \ge 0 \mathrm{and}B_{\alpha\beta}\left(Y_\alpha\!=\!0,Y_{\beta\ne\alpha},t\right) = 0, A_\alpha\left(\sum_{\alpha=1}^{N-1}Y_\alpha\!=\!1,t\right) \le 0 \mathrm{and}B_{\alpha\beta}\left(\sum_{\alpha=1}^{N-1}Y_\alpha\!=\!1,t\right) = 0, &\qquad\qquad\qquad\qquad\qquad\alpha,\beta=1,\dots,N-1.
\begin{cases}H_\alpha(\beta,\beta) = \Delta_\alpha(\beta) \mu(0) \ \ \beta\ge 1, begin{align*}0.1cm]H_\alpha(\beta,k) = \Delta_\alpha( k) \mu(\beta-k)+\sum_{l=1}^{\beta-k}H_\alpha(\beta-k,l) \Delta_l(k) \ \ \beta\ge 2,\,1\le k\le \beta-1. \end{cases}
-\tau((i,j)) B_{\gamma}^{(j,i)}+B_{\gamma}^{(i,j)}\tau((i,j)) +\tau((i,j)) B_{\delta}^{(j,i)}+B_{(i,j) \delta}^{(i,j)}\tau( (i,j)) \qquad{}=-\tau((i,j)) B_{\gamma}^{(j,i)}+B_{\gamma}^{(i,j)}\tau((i,j)) .
P\{T_1 = T_{\min}\}= & K \int_0^{\infty}\cdots \int_0^{\infty}e^{-(p_g t_g + \cdots + p_2 t_2)}\left(1 - e^{-p_g t_g}\right)^{M_g - 1}\cdots\left(1 - e^{-p_2 t_2}\right)^{M_2 - 1}\left[1 - e^{-p_1 (t_2 \wedge \cdots \wedge t_g)}\right]^{M_1}\, dt_2 \cdots dt_g,
\kappa_{0,\infty}:=\sum_{k\geq 0}\|Qe_k\|^2_{L^{\infty}_x}<\infty,\kappa_{1,\infty}:=\sum_{k\geq 0, \,1\leq i\leq N}\|\partial_{x_i}Qe_k\|^2_{L^{\infty}_x}<\infty.
| J_{a}= \epsilon , \; D_{b}= \lambda_{b}, \; D_{c}= \lambda_{c}, \; D_{d}= \lambda_{d}\rangle \equiv{\frac{1}{4}}| a_{\epsilon}9 \rangle -{\frac{i \lambda_{b}}{4 \sqrt{3}}}| a_{\epsilon}b \rangle -{\frac{i \lambda_{c}}{4 \sqrt{3}}}| a_{\epsilon}c \rangle -{\frac{i \lambda_{d}}{4 \sqrt{3}}}| a_{\epsilon}d \rangle
\Delta = \delta^* \overline{\delta}, \ \ \ \ \ \zeta_\alpha = \left( \frac{\alpha}{\alpha + \Delta}\right)^{1/2}, \ \ \ \ \ \tilde \delta_\alpha = \frac{1}{\sqrt{\alpha}}\overline \delta \circ \zeta_\alpha, \ \ \ \ \ \tilde \Delta_\alpha = \frac{1}{\sqrt{\alpha}}\Delta^{1/2}\circ \zeta_\alpha, \ \ \ \ \ \theta_\alpha = 1 - \tilde \Delta_\alpha.
R^{\rm boundary}_{\rm massive}(r) = A_{\rm out}\frac{1}{r^\beta}\left(1-\frac{i}{\pi}\left(\frac{2r_{H}}{r}\right)^{1-2\beta}\right)+ A_{\rm in}\frac{1}{r^\beta}\left(1+\frac{i}{\pi}\left(\frac{2r_{H}}{r}\right)^{1-2\beta}\right),
=C_{k}h^{k-i}y_{0}^{-(k-i)}\left\langle D_{y}^{m}u\right\rangle_{B}^{(\gamma)}\leq C_{k}\varepsilon^{(k-i)}\left\langle D_{y}^{m}u\right\rangle_{B}^{(\gamma)},m<i\leq k-1.
\mathbf{a}^\top\mathbf{R}_{\boldsymbol{\epsilon}}\mathbf{a}\geq\mbox{ZZB}=\int_{0}^{\infty}\max_{\boldsymbol{\delta}:\mathbf{\mathbf{a}}^\top\boldsymbol{\delta}=h}A(\boldsymbol{\delta})\mathbb{P}_e(\boldsymbol{\delta})h ~ dh.
L_{s}= \; \frac{q^{2}}{1 + q^{2}}\; e^{- 1}\; ( \dot{x}_{\mu})^{2}+ \; \frac{e}{1 + q^{2}}\; m^{2}+ b \; \dot{e}+ \frac{b^{2}}{1 + q^{2}}+ \dot{\bar{c}}\; \dot{c}.
3x=3y =2z=2w 3\bigg(\frac{A}{BE}\bigg)=3\bigg(\frac{A}{CD}\bigg) =2\bigg(\frac{B}{CE}\bigg)=2\bigg(\frac{C}{DE}\bigg). \intertext{Plugging in $t=0$ gives the initial requirements for this set of solutions:}\lambda_2\lambda_5=\lambda_3\lambda_4, \lambda_2\lambda_4 =\lambda_3^2,3\lambda_1\lambda_5=2\lambda_3^2.
\mathbf{a}_{k}(l) = \langle \lambda_k+\cdots+\lambda_r,\alpha_{u_{k,l}}^{\vee}+ \cdots+\alpha_{u_{k,m_k}}^{\vee}\rangle 1 \leq l \leq m_k, & \mathbf a_{k}(m_k+1) = 0.
\dfrac{M_{1}(\mathcal{NC}(G))}{|v( \mathcal{NC}(G) )|}= \dfrac{2^{9k}-5\cdot2^{7k}-2^{6k}+9\cdot2^{5k}-5\cdot2^{3k}-3\cdot2^{2k}+3\cdot2^{k}}{2^{3k}-2^{k}-1}
\frac{\partial E}{\partial z^j_{\eta}}\frac{\partial^2\varphi^j}{\partial x^{\epsilon}\partial x^{\eta}}= 2b^2f(x^1)\Big[\delta_{\epsilon 1}f' \left( x^1\right) \left\lbrace f(x^1)f''\left( x^1\right)+f'^2(x^1)+\sin^2\left( x^2\right)\right\rbrace +\delta_{\epsilon 2}f(x^1) \sin\left( x^2\right) \cos\left( x^2\right) \Big].
{\frac{d \operatorname{l o g}{y_{\mathrm{c}}}}{d \operatorname{l o g}{k}}}={\frac{m_{\mathrm{P l}}^{2}}{4 \pi}}\left[{\frac{H^{\prime}\left( \phi \right)}{H \left( \phi \right)}}\right]^{2}= \epsilon .
\sum_{q\in S}\frac{(-1)^k (2k+\gamma +\delta +1) (\gamma+1)_k (-N)_k N!}{(k+\gamma +\delta +1)_{N+1}(\delta+1)_k k!}(1+\delta_{q,0}) P_n(q) P_{n'}(q){}= \left[ \binom{\gamma+\lfloor n/2\rfloor}{\lfloor n/2\rfloor}\binom{\delta+N-\lfloor n/2\rfloor}{N-\lfloor n/2\rfloor}\right]^{-1}\delta_{n,n'}
f_{\mathrm{NML}}({\bf x}^n;\mathcal{M})\buildrel \rm def \over =\frac{f({\bf x}^n ; \hat{\mu}({\bf x}^n),\hat{\Sigma}({\bf x}^n))}{\int_{Y}f({\bf y}^n ; \hat{\mu}({\bf y}^n),\hat{\Sigma}({\bf y}^n)){\rm d}{\bf y}^n}.
\Sigma^t = \begin{pmatrix}\sigma^{k_1}(\alpha) \sigma^{k_2}(\alpha) \cdots \sigma^{k_\nu}(\alpha) \sigma^{k_1+1}(\alpha) \sigma^{k_2+1}(\alpha) \cdots \sigma^{k_\nu+1}(\alpha) \vdots \vdots \ddots \vdots \sigma^{k_1+t}(\alpha) \sigma^{k_2+t}(\alpha) \cdots \sigma^{k_\nu+t}(\alpha) \end{pmatrix}_{(t+1) \times \nu}.
\Big\lfloor \frac{m}{r^{t+1}}\Big\rfloor r^{t+1}e_{t+1}+\sum_{i=0}^{t}\alpha_i r^i e_i &= \sum_{i=0}^{t+1}\Big\lfloor \frac{m}{r^{t+1}}\Big\rfloor r^{i}+\sum_{i=0}^{t}\alpha_i r^i e_i
\left(n\left(i+\frac q2\right)-m(j+q)\right)d^1_{r,s}(m+n,i+j)+2q\cdot d^0_{r,s}(n,j) =0, 2q\cdot d^0_{r,s}(m+n,i+j)-\left(m(j+s+q)-(n+r)\left(i+\frac q2\right)\right)d^1_{r,s}(n,j) =0.
N(\star_{i=1}^{m+k}H,\alpha^{mN+kN+1}) = N(\star_{i=1}^{m+k}H, (\star_{i=1}^{m}\beta) \star (\star_{i=m+1}^{m+k}\beta) ) = & N( \star_{i=1}^{m+k}H,(\star_{i=1}^{m}\beta) \star (\star_{i=1}^{k}\beta) ) \le N(\star_{i=1}^{m}H,\star_{i=1}^{m}\beta) N( \star_{i=1}^{k}H,\star_{i=1}^{k}\beta),
S_i=-\frac{k}{2\pi}\int_{\cal M}d^2x\left[ \frac{1}{2y_{(i)}^2}\partial_\mu y_{(i)}\partial^\mu y_{(i)}+\frac{1}{4y_{(i)}^2}f_{(i)\mu\nu}f_{(i)}^{\mu\nu}\right] + (SU(2)\mbox{~WZW with level~}k )_i
\begin{array}{ll}[h_i,h_j]=0, [e_i,f_j]=\delta_{ij}h_i, \mbox{$[$}h_{i},e_{j}\mbox{$]$}= A_{ij}e_{j}, \mbox{$[$}h_{i},f_{j}\mbox{$]$}=-A_{ij}f_{j},\end{array}
c_{1}({\cal N}) = n \left({\frac{1}{2}}+ \lambda \right) \sigma + \left({\frac{1}{2}}- \lambda \right) \pi^{*}\eta + \left({\frac{1}{2}}+ n \lambda \right) \pi^{*}\left( c_{1}( B ) \right) + \sum_{i}\kappa_{i}N_{i}
\sum_{a<n\le b}\varphi(n)=\int_{a}^{b}\varphi(x)dx+\int_{a}^{b}(\{x\}-\frac{1}{2})\varphi^{\prime}(x)dx+(\{a\}-\frac{1}{2})\varphi(a)-(\{b\}-\frac{1}{2})\varphi(b).
\tau = -{\frac{1}{\left( a H \right) \left( 1 - \epsilon \right)}}\left[ 1 +{\frac{2 \epsilon \left( \epsilon - \eta \right)}{\left( 1 - \epsilon \right)^{2}}}+ \mathrm{O}\left( \epsilon \eta^{2}\right) + \cdots \right] .
\begin{aligned}M(x,y) :=\{((\tilde x,g(\tilde x)),(0,\tilde y))\,|\,\tilde x=x,\,\tilde y\in -D, y=g(\tilde x)+\tilde y\}&=\begin{cases}\{((x,g(x)),(0,y-g(x)))\}g(x)-y\in D, \varnothing \end{cases}\end{aligned}
\lim_{m \rightarrow 0}I m T_{1}( t ) = - \lim_{m \rightarrow 0}\frac{m^{2}}{4 t^{2}}\left( 1 - \frac{4 m^{2}}{t}\right)^{\frac{1}{2}}\theta ( t - 4 m^{2}) = - \frac{1}{2 4}\delta ( t ) .
\frac{d \rho}{d t}< \frac{\rho^{7 / 2}}{d_{0}^{1 / 2}e}\frac{( 1 - \frac{\rho_{0}^{7}}{\rho^{7}})}{( 1 + \frac{\rho_{0}^{7}}{\rho^{7}})^{2 / 7}}.
\begin{array}{l}{{\displaystyle{\frac{A_{n}}{B_{n}}}\sim r_{0}^{2 \left( k + \alpha - n \right)}\qquad n - k + \frac{1}{2}- \alpha \leq 0}}\\{{\displaystyle{\frac{B_{n}}{A_{n}}}\sim r_{0}^{2 \left( n + 1 - k - \alpha \right)}\qquad n - k + \frac{1}{2}- \alpha > 0}}\\ \end{array}
c^\prime_{j}\left((x^*)^{-1}\right)=\left<\tau_{j}\left(\tilde{w}x^T\tilde{w}\right)\xi_{\tau_{j}},\check{\xi}_{\tilde{\tau}_{j}}\right>=\left<\xi_{\tau_{j}},\tilde{\tau}_{j}\left(x^*\right)\check{\xi}_{\tilde{\tau}_{j}}\right>.
B_2=\begin{pmatrix}a_{1,4}a_{1,6}a_{1,7}a_{1,8}\cdots a_{1,n}a_{2,4}a_{2,6}a_{2,7}a_{2,8}\cdots a_{2,n}a_{3,4}a_{3,6}a_{3,7}a_{3,8}\cdots a_{3,n}\end{pmatrix},
\varepsilon \sum_{i,j=1}^d g^\varepsilon b_i \Lambda^\varepsilon b_j D_i D_j e^{-\mathcal{A}_N^0t}= \varepsilon \sum_{i,j=1}^d g^\varepsilon b_i \Lambda^\varepsilon b_j D_i D_j e^{-\mathcal{A}_N^0t}{\mathcal P}
\bar{a}_{ij}(t, x) := \tfrac{1}{\gamma}\hat{a}\big(\bar{z}(t, x)\big) \delta_{ij}\bar{f}(t, x) := \tfrac{1}{\gamma}\big((1 - K) f(\bar{z}, \nabla \bar{z})\big)(t, x) + \tfrac{1}{\gamma}\big(K \big(\hat{a}(\bar{z}) A \bar{z}\big)\big)(t, x) - \tfrac{\alpha}{\gamma}\big((1 - K) A \theta\big)(t, x)
F(t)=-(-1)^{\frac{k}{2}}(it)^{\frac{2-k}{2}+m}(k-1)\sum_{0\leq j\equiv \frac{k-2}{2}+m\;({\rm mod}\;2)}\binom{k}{j}\sum_{s\geq 0}C(j-2-2s)(it)^{j-2-2s}.
W^{\Omega_{z_1}, \wedge^m (\Omega_{z_2})}(v^{(d,a)}\otimes v^{(d_m , d)})=\sum_{a_m}v^{(d_m, a_m)}\otimes v^{(a_m , a)}W^{(1,m)}\left( \left. \begin{array}{cc}d_m d a_m a \end{array}\right| z_1 -z_2 \right),
\mu^{k+1}+A^T \nu^{k+1}= -\rho(\tilde x^{k}+ x^{k+1/2}- x^{k+1}+A^T ( \tilde y^{k}+ y^{k+1/2}- y^{k+1})) &= -\rho(\tilde x^{k}+ A^T\tilde y^k + x^{k+1/2}+A^Ty^{k+1/2}- (I+A^TA)x^{k+1}),
\overline{{{( \Delta n )^{2}}}}= \frac{n \, d \, ( n + d )}{\left( d - n \, \displaystyle{\frac{\partial d}{\partial n}}\right)^{2}- n \, d \, ( n + d ) \, \displaystyle{\frac{\partial^{2}d}{\partial n^{2}}}\, \log \frac{n + d}{d}}\ \ .
\operatorname{Ass}_A\left( \operatorname{Tor}_{2i}^A(M, N) \right) = \phi \quad\mbox{and}& \operatorname{Ass}_A\left( \operatorname{Tor}_{2i-1}^A(M, N) \right) = \operatorname{Ass}_A(k) = \{(u,x)/(ux) \}.
\|\lambda(\varphi_i)D\lambda(\varphi_i).v\|_s \le c_s\left[\|g^{}\|_s\|T\varphi_i\|_{s_0}^2+\|g^{}\|_{s_0}\|T\varphi_i\|_{s}^2\right]\cdot &\quad\cdot 2c_s\left[\|g^{}\|_s\|Tv\|_{s_0}\|T\varphi_i\|_{s_0}+\|g^{}\|_{s_0}\|Tv\|_{s}\|T\varphi_i\|_{s}\right].
q_{\beta}(n,t)=\sum_{k=1}^{n}\underset{m_j\in\mathbb{N}}{\underset{m_{1}+m_{2}+\dots+m_{k}=n}{\sum}}\prod_{j=1}^{k}\frac{\theta^{m_{j}}}{m_{j}!}\left(\alpha t^{\beta}\right)^{k}E_{\beta,k\beta+1}^{k+1}\left(-\alpha(e^{\theta}-1)t^{\beta}\right).
S=-\sum{\hbox{Tr}({{U}_{\mu}{U}_{\nu}{U}^{\dagger}_{\mu}{U}^{\dagger}_{\nu}})}+\frac{i}{2}\hbox{Tr}{{\bar \psi}{\Gamma^\mu}(U_\mu \psi U_\mu^\dagger-U_\mu^\dagger \psi U_\mu)},
Q = -\frac{12}{25}(\mu^3 - 7\lambda\mu^2 + 33\lambda^2\mu - 9\lambda^3) , L_1 = 12(\mu+\lambda)^2(\mu-3\lambda) , L_2 = \frac{6}{25}(\mu+\lambda)^2(7\mu-33\lambda) , L_3 = \frac{39}{25}(\mu+\lambda)^3 .
\min_{x}\ (\max_{k=1}^{m}w_{k}(\max_{i,j=1}^{n}a_{ij}^{(k)}x_{j}/x_{i}))=\min_{x}\ (\max_{i,j=1}^{n}\max_{k=1}^{m}(w_{k}a_{ij}^{(k)})x_{j}/x_{i}).