harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
P(\ell):=\; \mbox{all the prime factors of $ \ell $}, P_{D}(\ell):=\; \mbox{all the prime factors of $ \ell $ in $ \mathbb{P}(D) $}, P_{m}(\ell_{1},\ell_{2}):=\; P_{-4\ell_{2}}(\ell_{1})\backslash P(m-2), G_{m}(\ell_{1},\ell_{2}):=\; P(\gcd(\ell_{1},\ell_{2}))\backslash P(2(m-2)).
\begin{aligned}\nu_i\geq 0,F_i(\bar{x})\leq 0, \nu_i F_i(\bar{x})=0,i \in N, F_0^\prime (\bar{x}) + \nu_1 F_1^\prime (\bar{x}) + \dots + \nu_n F_n^\prime (\bar{x}) = 0.\end{aligned}
- \left( \kappa + \mathrm{i}\omega \right) A + g \S = 0 , - \left( \gamma + \mathrm{i}\omega \right) \S + g \A \D = 0 , - 4 g \ \Re \left( A \ \overline{S}\right) - 2 \gamma \left( D - d \right) = 0 .
\psi_h^*i(X)\Omega_h = (\rho_2^r \circ \psi)^*i(X)\Omega_h= \psi^*((\rho_2^r)^*i(X)\Omega_h) = \psi^*i(Y)(\rho_2^r)^*\Omega_h &= \psi^*i(Y)(h \circ \rho_2^r)^*\Omega_1^s = \psi^*i(Y)(\rho_2 \circ \hat{h})^*\Omega_1^s= \psi^*i(Y)\Omega_r \, .
\begin{aligned}-16L\mathcal Q(t) \geq ~{}\frac12\tilde \sigma \int |\varphi'| f^2 + 3\tilde\sigma \int |\varphi'| f_x^2 +2 \tilde\sigma \int |\varphi'| f_{xx}^2 + \frac14\tilde \sigma\int |\varphi'| f_{xxx}^2 , \end{aligned}
T_{\mu}^{\nu}= \left( \begin{array}{c c c c}{{- \frac{\alpha \beta}{\kappa}\frac{e^{- \beta r}}{r^{2}}}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{- \frac{\alpha \beta}{\kappa}\frac{e^{- \beta r}}{r^{2}}}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{\frac{\alpha \beta^{2}}{2 \kappa}\frac{e^{- \beta r}}{r}}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{\frac{\alpha \beta^{2}}{2 \kappa}\frac{e^{- \beta r}}{r}}}\\ \end{array}\right)
\mathrm{Perf}^{\mathrm{gr}}(\mathcal{X}\times \mathbb{C}) =\langle \Gamma_G(\chi) \otimes \mathcal{O}_{X \times \mathbb{C}}(i) \mid \chi \in M^+, i \in \mathbb{Z}\rangle, \mathrm{Perf}^{\mathrm{gr}}(\mathcal{X}) =\langle \Gamma_G(\chi) \otimes \mathcal{O}_{X}(i) \mid \chi \in M^+, i \in \mathbb{Z}\rangle.
\frac{1}{g_{Y M}^{3}}= \frac{\tau_{5}( 2 \pi \alpha^{\prime})^{2}}{2}\int_{{\cal{C}}_{2}}{e}^{- \phi}B_{2}= \frac{1}{4 \pi g_{s}( 2 \pi \sqrt{\alpha^{\prime}})^{2}}\int_{{\cal{C}}_{2}}{e}^{- \phi}B_{2}
\frac{2\sum_{n=0}^{k}\left( -1\right)^{n}\chi\left( n\right) \cosh\left(nt\right)}{\cosh(kt)+1}=\frac{2\sum_{n=1}^{k-1}\left( -1\right)^{n}\chi\left( n\right) \sinh\left( nt\right)}{\sinh(kt)}& =\sum\limits_{j=0}^{\infty}E_{2j,\overline{\chi}}\left( 0\right)\frac{t^{2j}}{(2j)!}
\int_{Y}i_Y^*(\omega\wedge\eta)=\sum_i\int_{S_i\times T_i}\pi_1^*(\omega)\wedge\pi_2^*(\eta)=\sum_{(\dim(S_i),\dim( T_i))=(k,l)}\int_{S_i}i_{S_i}^*(\omega)\int_{T_i}i_{T_i}^*(\eta).
D_\textrm{hr}(k=2) \approx 2\frac{C(2,2,G_{2})}{N}\left( \int_{\mathbb{R}^2}\sqrt{\frac{\mathrm{e}^{-\frac{x^2+y^2}{\sigma^2}}}{\pi \sigma^2}}\, \mathrm{d}x \, \mathrm{d}y\right)^2 &=\frac{2\pi\sigma^2}{3N}.
\uppercase\expandafter{\romannumeral2}\leq \exp\left({-\frac{1}{8}\tau\delta (k + 1)N_2}\right)\exp\left((l-N_2)\tau(\int_{\Omega}\log X dP -\frac{\delta}{2})\right) = \exp\left({-\frac{1}{8}\tau\delta n}\right)\exp\left((l-N_2)\tau(\int_{\Omega}\log X dP -\frac{3}{8}\delta)\right)
\begin{aligned}\bigg|(D^\gamma (h\partial_x h)\cdot D^\alpha u)|_{y=0}\bigg| \leq \frac{\delta_1}{3}\Vert \partial_y (u,h) \Vert_{H^m_0}^2+\frac{\nu}{14}\Vert \partial_y D^\alpha u\Vert_{L^2_y}^2+C\delta_1^{-1}E_{u,h}^4+CE_{u,h}^2,\end{aligned}
\Phi_2:=\exp (\varepsilon^2 A_2)=\mathrm{I}_{H_S^{\perp}}+\varepsilon^2 A_2+\varepsilon^4 \hat{A}_2, \hat{A}_2:=\sum_{k\geq 2}\frac{\varepsilon^{2(k-2)}}{k!}\,A_2^k
A_{i,k}:= \left( [a, a + 3 i \right] \cup [b, b + 1+3i] \cup [2a, 2a+3i]) + \{2 i b \}, B_{i,k}:= \left( [a, a+1+3i] \cup [b, b+3+3i] \cup [2a, 2a+1+3i] \right) + \{(2 i + 1 ) b \},
e^{-t\widetilde{\Box}^{\pm}_{b,m}}(x,y)=P^{\pm}_{m,0}(x,y)+\sum_{\nu\in\mathrm{Spec\,}\widetilde{\Box}^{\pm}_{b,m},\nu>0}e^{-\nu t}P^{\pm}_{m,\nu}(x,y)
\Psi_{a,b}[n]^c(x^{\tilde m}) = \exp\Biggl(\, c \sum_{j=1}^{\infty}\frac{(-1)^{j+1}}{j[ja]_q}q^{jb}X_{jn}\Biggr)(x^{\tilde m}) &=x^{\tilde m}\exp\Biggl( c \sum_{j=1}^{\infty}\frac{q^{2j\langle n,\tilde m \rangle_1}-1}{q^{2ja}-1}\frac{(-1)^{j+1}}{j}q^{ja}q^{jb}x^{j p_1^*(n)}\Biggr).
\Lambda_{-1,1}=\lambda^{-1}(e_{1,2p}-e_{2,2p+1}) +(e_{2,1}-e_{2p+1,2p})+2\sum_{k=1}^{p-1}e_{2+k,1+k}-2\sum_{k=1}^{p-1}e_{p+1+k,p+k},
\Lambda' (R, m) &\triangleq \left\{\alpha \in \Omega^T \colon \alpha = \alpha^1 + \alpha^2 + \tfrac{\sin(\pi \theta)}{\pi}\alpha^3,\ \alpha^1 \in \Lambda^\star (m), \alpha^2 \in \Lambda (R, 1, m), \alpha^3 \in \Lambda (R, \theta, m) \right\}.
\begin{array}{ll}(\xi^2 + |v|^2 + \phi^2)_t + 2k_2\xi\nabla\cdot v + 2k_2 v\cdot\nabla\xi + 2a |v|^2 begin{align*}6pt]= -2 \gamma k_1\xi^2\nabla\cdot v - 2 k_1 \xi v\cdot\nabla \xi - 2k_1 v\cdot\nabla v\cdot v+ \frac{2}{k_1}(\frac{1}{\bar{\varrho}}- \frac{1}{\varrho})v\cdot\nabla\xi -2 k_1\phi v\cdot\nabla\phi.\end{array}
\lambda'(t) =\psi^{(m+1)}_{p,q}(s+t) - \psi^{(m+1)}_{p,q}(t) &= (\ln q)^{m+2}\sum_{n=1}^{p}\left[ \frac{n^{m+1}q^{n(s+t)}}{1-q^{n}}- \frac{n^{m+1}q^{nt}}{1-q^{n}}\right] &= (\ln q)^{m+2}\sum_{n=1}^{p}\left[ \frac{n^{m+1}q^{nt}(q^{ns}-1)}{1-q^{n}}\right]\leq0. \,\,
G_k = P_1(G_k) = \langle x,y \rangle \; P_2(G_k) G_k/P_2(G_k) \cong C_p \times C_p, P_2(G_k) = \langle x^p, y^p, [y,x] \rangle \; P_3(G_k) P_2(G_k)/P_3(G_k) \cong C_p \times C_p \times C_p,
p'_{1}\circ (p_{12}\circ f\circ \iota_{12})=p_{1}\circ f\circ \iota_{12}=p_{1}\circ f\circ l_{1}\circ \iota_{12}=p_{1}\circ f\circ \iota_{12}\circ l'_{1}=p'_{1}\circ (p_{12}\circ f\circ \iota_{12})\circ l'_{1},
{\cal L}= \frac{f_{\pi}^{2}}{4}\mathrm{T r}\partial_{\mu}U^{\dagger}\partial^{\mu}U + \frac{1}{3 2 e^{2}}\mathrm{T r}\left( \left[ U^{\dagger}\partial_{\mu}U , U^{\dagger}\partial_{\nu}U \right]^{2}\right) ,
\delta \Gamma^\sigma = \frac{i 36 g^2}{M^2}\int \frac{d^4 k}{(2 \pi)^4}\int_0^1 dx \;dy\; \delta(x+y-1) \Biggl\{\frac{N_1^\sigma + \epsilon N_2^\sigma}{\left[k^2 - \Delta_1 \right]^2}- \frac{N_3^\sigma}{\left[k^2 - \Delta_2 \right]^2}\Biggr\},
F = V_{p + 1}\, ( 8 \pi^{2}\alpha^{\prime})^{- \frac{p + 1}{2}}\int_{0}^{\infty}\frac{d \tau}{\tau}\tau^{- \frac{p + 1}{2}}~ e^{- \frac{y^{2}\tau}{2 \pi \alpha^{\prime}}}~ \left( f_{1}( e^{- \pi \tau}) \right)^{- 2 4}=
R_{s}=\frac{1}{2}[(\frac{cosi-\sqrt{m^{2}-sin^{2}i}}{cosi+\sqrt{m^{2}-sin^{2}i}})^{2}+(\frac{m^{2}cosi-\sqrt{m^{2}-sin^{2}i}}{m^{2}cosi+\sqrt{m^{2}-sin^{2}i}})^{2}]
P_{{\bf X}_i| Y^{i-1}}({\bf x}_i| y^{t-1}, y_t^{i-1}) = P_{{\bf X}_i| Y^{i-1}}({\bf x}_i|{\bf a}_{t-1}, y_t^{i-1}) & = P_{{\bf X}_i| Y^{i-1}}({\bf x}_i| \tilde{{\bf a}}_{t-1}, y_t^{i-1}) & =P_{{\bf X}_i| Y^{i-1}}({\bf x}_i| \tilde{y}^{t-1}, y_t^{i-1}).
= \exp \left\{- \frac{g^{2}}{2}\left[ \frac{M^{2}}{2}\int_{\Sigma}^{}d \sigma_{\mu \nu}({\bf x}) \int_{\Sigma}^{}d \sigma_{\mu \nu}({\bf y}) + \oint_{{\cal C}}^{}d x_{\mu}\oint_{{\cal C}}^{}d y_{\mu}\right] D_{M}({\bf x}-{\bf y}) \right\}.
Y_t \ = \g(\hat X) + \int_t^T f(\hat X,\hat I_s)\,ds + K_T - K_t - \int_t^T\int_\Lambda R_s(b)\,\hat \theta(ds\,db) &\ -\int_t^T Z_s\,d\hat W_s - \int_t^T\int_U L_s(z)\,(\hat\pi(ds\,dz)-\lambda_\pi(dz)\,ds),0 \leq t \leq T,\,\,\hat\P R_t(b) \ \leq \ 0, dt\otimes d\hat\P\otimes\lambda_0(db)\hat\Omega\times[0,T]\times \Lambda.
f_Af_{25}f_5f_{31}\widehat{b}= f_Af_{28}f_5f_{31}\widehat{b}f_Af_{25}f_5f_{29}\widetilde{c}= f_Af_{28}f_5f_{29}\widetilde{c}f_Af_{25}\widetilde{c}= f_Af_{28}\widetilde{c}.
\sum_{1\leq i,i',j,j'\leq n}= \sum_{|i-i'|>g,|j-j'|> g}+\sum_{|i-i'|>g,|j-j'|\leq g}+\sum_{|i-i'|\leq g,|j-j'|> g}+\sum_{|i-i'|\leq g,|j-j'|\leq g}=: I+II+III+IV.
\| (h_0(u,X_u) -h_0(u,X_u^{\epsilon}))\lambda+h_0(u,X_u^{\epsilon})\lambda_{\varepsilon}\|^p_{L^p(\Omega,L^2(\nu))}&\leq 2^{p-1}\Big(L\| X_u-X_u^{\epsilon}\|^p_{S^p(\Omega;\mathcal{D})}\|\lambda\|^p_{L^2(\nu)}+K(1+\| X_u^{\epsilon}\|^p_{S^p(\Omega;\mathcal{D})})\|\lambda_{\varepsilon}\|^p_{L^2(\nu)}\Big),
\begin{array}{ll}U^{t_{0}}_{,s}+\sqrt{1-\Delta t^2}^3v\nabla_x U^{t_0}+\frac{\sqrt{1-\Delta t^2}^3}{1+t}Q(t,(1+t )U^{t_{0}}(s,.),(1+t )U^{t_{0}}(s,.))=-\frac{\sqrt{1-\Delta t^2}^3}{1+t}U^{t_{0}}, ~U^{t_{0}}(0,.)=F(t_{0},.).\end{array}
\frac{\partial\Pi(\xi,t)}{\partial t}=\sum_{j}(\sum_{ik}-S_{ij}\frac{\partial f_{j}}{\partial\phi_{k}}\frac{\partial(\xi_{k}\Pi(\xi,t))}{\partial\xi_{i}}+\frac{1}{2}f_{j}\sum_{ik}S_{ij}S_{kj}\frac{\partial^{2}\Pi(\xi,t)}{\partial\xi_{i}\partial\xi_{k}}),
\begin{array}{l l}{{}}&{{R_{2}^{\mathrm{i n}}= e^{- i \omega v_{*}}\, ,}}\\{{}}&{{R_{2}^{\mathrm{o u t}}= e^{- i \omega v_{*}}e^{2 [ i ( \omega - \omega_{0}) - C ] r_{*}}\, , ~ ( r > r_{H}) \, .}}\\ \end{array}
{\bf\Phi}^{(n_\rho,m)}_1={\bf\Phi}^{(n_\rho,m)}_1(\rho,k)=\left(\begin{array}{c}\phi_1^{(n_\rho,m)}(\rho,k) \phi_2^{(n_\rho,m)}(\rho,k)\end{array}\right)
{\cal L}= - \frac{1}{4}F_{\mu \nu}F^{\mu \nu}+ \frac{1}{2}m^{2}( A_{\mu}+ \partial_{\mu}\theta ) ( A^{\mu}+ \partial^{\mu}\theta ) + A^{\mu}\partial_{\mu}B - \partial_{\mu}\bar{\cal C}\partial^{\mu}{\cal C}- \frac{1}{2}\alpha B^{2}.
\frac{\left( e_{j}^{( \alpha )}\cdot e_{l}^{( \alpha )}\right)_{C}}{| e_{j}^{( \alpha )}|_{C}\; | e_{l}^{( \alpha )}|_{C}}= \frac{C_{j l}}{\sqrt{C_{j j}\; C_{l l}}}= c_{j l}\equiv \operatorname{c o s}\tau_{j l}.
f_1 * f_2 = \sum_{u \in \mathcal{O}_1}\sum_{v\in \mathcal{O}_2}\delta_{(u,\, z_1)\cdot (v,\, z_2)}begin{align*}5 pt] = \sum_{u\in \mathcal{O}_1}\sum_{v\in \mathcal{O}_2}\delta_{\left(u+v,\, z_1z_2[u,v]^{1/2}\right)}.
\sum_{M\geq1}g(M)\,x^M =\frac{x-x^3}{(1-x-x^2)^2}, \sum_{M\geq1}h(M)\,x^M=\frac{x-x^2}{(1-x-x^2)^2}, \sum_{M\geq1}g_1(M)\,x^M =\frac{x^2}{(1-x-x^2)^2}=\sum_{M\geq1}h_1(M)\,x^M.
c_{1}=c_{2}=\cdots=c_{k_{i_{1}}}=i_{1}, \c_{k_{i_{1}}+1}=c_{k_{i_{1}}+2}=\cdots=c_{k_{i_{1}}+k_{i_{2}}}=i_{2}, \ \ldots, &\ldots, \c_{k_{i_{1}}+k_{i_{2}}+\cdots+k_{i_{r-1}}+1}=c_{k_{i_{1}}+k_{i_{2}}+\cdots+k_{i_{r-1}}+2}=\cdots=c_{k_{i_{1}}+k_{i_{2}}+\cdots+k_{i_{r-1}}+k_{i_{r}}}=i_{r}.
g(\zeta) = \cot^{d-3}\bigg(\frac{\zeta}{2}\bigg)\, \frac{d}{d\zeta}\int_0^\zeta \frac{Q(\theta)\, \sin^{d-3}(\theta/2) \sin\theta \, d\theta}{\sqrt{\cos\theta - \cos \zeta}}, 0 \leq \zeta \leq \alpha.
q'_j = \sum_{i \in A_j}q_i = \sum_{i \in [k]\setminus [m]: \tilde{\delta}_{i-m}\in (\nu_{j+1}, \nu_{j}]}q_i = \sum_{i \in [k]\setminus [m]: \tilde{\delta}_{i-m}\in (\nu_{j+1}, \nu_{j}]}0.5 \tilde{q}_{i-m}= 0.5 \P\{\tilde{X}\in (\nu_{j+1}, \nu_{j}]\}.
T(\rho_n, \vec{a}) = e^{-i[\vec{a}\cdot (\hat{H}(\rho_n)\hat{f}_0 + \hat{P}(\rho_n)\hat{f}_1)]}, T(\rho_n, \vec{a})^{-1}= e^{i[\vec{a}\cdot (\hat{H}(\rho_n)\hat{f}_0 + \hat{P}(\rho_n)\hat{f}_1)]},
\sum_{j=1}^m\sum_{h=1}^2C_{j,h}\int_{\Omega}\frac{\partial}{\partial z_{j,h}}(-\delta^2(K(z_j)\nabla V_{\delta, Z,j}))\frac{\partial V_{\delta, Z,i}}{\partial z_{i,\hbar}}=\int_{\Omega}u \frac{\partial V_{\delta, Z,i}}{\partial z_{i,\hbar}},\ \ \forall i=1,\cdots,m,\ \hbar=1,2.
x.s=xs+sx^{t}x\in sl_{n},\,s\in S, x.\lambda=x\lambda+\lambda x^{t}x\in sl_{n},\,\lambda\in\Lambda, x.s'=-s'x-x^{t}s'x\in sl_{n},\,s'\in S', x.\lambda'=-\lambda'x-x^{t}\lambda'x\in sl_{n},\,\lambda'\in\Lambda'.
\varkappa(m,p_1,p_2):=\sum_{s=0}^{n}\sum_{\nu\in I_s}k^{\frac{sr}{6}(\frac{1}{p_1}-\frac{1}{p_2}-2)}\cdot (\prod_{\nu_j>k}\nu_j)^{\frac{r}{6}(\frac{1}{p_1}-\frac{1}{p_2}-2)}|m(\nu)|^r<\infty,
T_{s_1,\ldots,s_k}=\begin{vmatrix}\alpha_{1,s_1}\alpha_{1,s_2}\cdots \alpha_{1,s_k}\alpha_{2,s_1}\alpha_{2,s_2}\cdots \alpha_{2,s_k}\vdots \vdots \vdots \alpha_{k,s_1}\alpha_{k,s_2}\cdots \alpha_{k,s_k}\end{vmatrix}.
\left\Vert \frac{d\tau(y)^{-1}(g\nu_{P})^{y}}{d\bar{m}_{K\cap Q}}\right\Vert_{\infty}=\left\Vert \frac{d(g\nu_{P})^{y}}{d\left(\tau(y).\bar{m}_{K}\right)^{y}}\right\Vert_{\infty}\le\left\Vert \frac{d(g\nu_{P})}{d\left(\tau(y).\bar{m}_{K}\right)}\right\Vert_{\infty}\left\Vert \frac{d\left(\tau(y).\bar{m}_{K}\right)}{d\left(g\nu_{P}\right)}\right\Vert_{\infty}.
F_\alpha(f)(vv_1)=\sum_{u\in J(H_{\Delta,0}/t_\alpha H_{\Delta,0}t_\alpha^{-1})}(t_\alpha f)(vv_1u)= =\sum_{u\in J(H_{\Delta,0}/t_\alpha H_{\Delta,0}t_\alpha^{-1})}\chi^w(t_\alpha)f(t_\alpha^{-1}vv_1ut_\alpha)=\sum_{u\in J(H_{\Delta,0}/t_\alpha H_{\Delta,0}t_\alpha^{-1})}\chi^w(t_\alpha)f(vt_\alpha^{-1}v_1ut_\alpha)
\sum_{n}\mathbb E_{\mathcal F_{\frac{n}{2^m}}}\bigl((|F|^2 \star \nu)_{\frac{n+1}{2^m}}- (|F|^2 \star \nu)_{\frac{n}{2^m}}\bigr) \to (|F|^2\star \nu)_T = (|F|^2\star \nu)_{\infty}
Q_{n}(x)- Q_{n}(x+1) = \frac{n(n+\alpha+\beta+1)}{N(\alpha+1)}\hat Q_{n-1}(x), -(x+1)(N-x+\beta) \hat Q_{n-1}(x)+(N-x-1)(x+\alpha+2) \hat Q_{n-1}(x+1) \qquad{}= N(\alpha+1)Q_{n}(x+1).
\sum_{i = 0}^{\max\left\{n, 0 \right\}}\frac{\left(q^{-n}\right)_{i}}{(q)_{i}}(x q^{n})^{i}= (x)_{\max\left\{n, 0 \right\}}, \sum_{i = 0}^{-\min\left\{n, 0 \right\}}\frac{\left(q^{n}\right)_{i}}{(q)_{i}}x^{i}= (x q^{\min\left\{n, 0 \right\}})_{- \min\left\{n, 0 \right\}}.
W(c_{1}) =\displaystyle \sum_{i}\displaystyle \sum_{s \in L(\sigma(b))}(s\rho(bwa))^{\ast}\left(\hat{\Delta}(\rho(c_{i})) \Diamond t_{i}z(\rho(c_{1}))r_{i}\right)s &=\displaystyle \sum_{i}\displaystyle \sum_{s \in L(\sigma(b))}(s\rho(bwa))^{\ast}\left(\rho(c_{i})t_{i}z(\rho(c_{1}))r_{i}\right)s=0.
|\frac{16}{\lambda(t)^{2}}\int_{0}^{1}dw \lambda''(w+t) \left(\partial_{22}K_{1}(w,\lambda(t))\lambda'(t)-\frac{\lambda'(t)}{2(1+w)}\right)\lambda'(t)| &\leq \frac{C}{t^{3}\log^{2}(t)}\int_{0}^{1}dw \left(w+\frac{1}{1+w}\right)\frac{1}{t \log^{b+1}(t)}\leq \frac{C}{t^{4}\log^{b+3}(t)}
X_{L}( a , b , c ; q ) = \sum_{w \in W}( - 1 )^{l ( w )}q^{{\frac{1}{2}}| \xi + \rho - w ( a + \rho ) |^{2}+ c_{L}}{\frac{[ L ] !}{\prod_{i = 0}^{n - 1}[ \alpha_{i}] !}},
\| x_j\|_{\mathcal{CG}_q^\omega}\ \stackrel{\eqref{n6}}{\gtrsim}\ \left(\sum_{n=1}^{\eta_j}\frac{1}{n}\left(\omega(n)h_r(k_j)\right)^q\right)^{1/q}&\ =\h_r(k_j)(\widetilde{\zeta}(\eta_j))^{1/q},
\lambda_1(a,b,c) =a^2b-ac+b^\ell-a^{\ell+3}, \lambda_2(a,b,c) =a^\ell b^\ell-c^\ell+ab^2+bc-a^{2\ell+3}, \lambda_3(a,b,c) =ac^\ell-a^{\ell+1}b^\ell+a^{\ell+3}b+a^2b^2-b^{\ell+1}-c^2+a^{2\ell+4}.
\begin{aligned}2\nu (l+k) \int_0^\infty \langle y \rangle^{2(l+k)-2}y\partial_y D^\alpha u\cdot D^\alpha u\mathrm{d}y & \leq \frac{\nu}{14}\Vert \langle y\rangle^{l+k}\partial_y D^\alpha u\Vert_{L^2_y}^2+14\nu (l+k)^2\Vert \langle y\rangle^{l+k}D^\alpha u\Vert_{L^2_y}^2,\end{aligned}
\mathcal{K}_s = \left\{Z_s = \left(X_s,V_s\right) \in \overline{\mathcal{D}_s}\left|\psi_s^{-\tau}Z_s = \left( X_s-V_s \tau,V_s \right) \; \forall\; \tau > 0 \right. \right\}\subset \mathbb{R}^{2ds}
P_{A_0,\ldots, A_n}(A)= P(\psi(X_0)\in A_0, \ldots,\psi(X_n)\in A_n, ,\psi(X_{n+1})\in A) = \int_D K_l(x_0,\psi^{-1}(A))\> d(P_{A_0,\ldots, A_{n-1}}|_{A_n}*\mu_1)(l) =(P_{A_0,\ldots, A_{n-1}}|_{A_n}*\mu_1)(A).
\psi^{(j)}_n(b)=\frac{\psi^{(j+1)}_{n-1}(b)-\psi^{(j)}_{n-1}(b)}{D^{(j)}_n}, \ \D^{(j)}_n=\frac{G^{(j)}_{n+1}G^{(j+1)}_{n-1}}{G^{(j)}_{n}G^{(j+1)}_{n}}.
\int_{\partial D_1}\dfrac{\partial \psi_k}{\partial \nu}\,w_k\,ds - 2\int_0^1 \dfrac{\partial_+ \psi_k}{\partial x_2}(x_1,0)\,w_k (x_1,0)\,dx_1 = \int_{D_1}\nabla w_k \cdot \nabla \psi_k\,dx.
\left[{\frac{d^{2}}{d t^{2}}}+{\frac{1}{2}}\sum_{k}{\frac{1}{t - a_{k}}}\;{\frac{d}{d t}}-{\frac{1}{4 \hbar^{2}}}\sum_{k}{\frac{f_{k}}{t - a_{k}}}\right] \, \Psi ( t ) = 0
F(0) =(4 (\cosh 1 -1) (\eta \cosh 1 + \gamma )^2+ k (\eta \sinh 2 + 2 \gamma \sinh 1 ))^2 & -16 ((s+1)^2-1)^2 (\eta \cosh 1 + \gamma )^4 \intertext{and}F\left(\frac{\pi}{2}\right) = (4 (\cos 1-1) (\eta \cos 1 + \gamma )^2 - k(\eta \sin 2 + 2 \gamma \sin 1 ))^2 & -16 ((s+1)^2-1)^2 (\eta \cos 1 + \gamma )^4.
\mathcal{F}(r) =\frac{2\frac{r}{1-r}}{\left(1-\left(\frac{r}{1-r}\right)^2\right)^2}\left(\frac{r}{(1-r)^3}+\frac{r}{(1-r)^4}\right)+\frac{\left(\frac{r}{1-r}\right)^2}{1-2r}&\quad+2\frac{\left(\frac{r}{1-r}\right)^2}{1-\left(\frac{r}{1-r}\right)^2}\frac{1}{(1-r)^3}+\frac{r}{1-r}+\frac{r}{(1-r)^2}.
\beta_{1}=\underbrace{\left(p_{1}^{r_{1}}\cdots p_{m}^{r_{m}}\right)}_{\sigma_{1}}\cdot \left(q_{1}^{u_{1}}\cdots q_{l}^{u_{l}}\right)\cdot \left(q_{l+1}^{u_{l+1}}\cdots q_{k}^{u_{k}}\right)\cdot d=\sigma_{1}\cdot \beta '_{1},
\phi_{0}^{( 1 )}( y ) \equiv \frac{1}{\sqrt{2}}( \phi_{- 1}( y ) + \phi_{0}( y ) ) , \quad \phi_{0}^{( 2 )}( y ) \equiv \frac{1}{\sqrt{2}}( \phi_{- 1}( y ) - \phi_{0}( y ) ) ,
(f_1(x)+2f_2(x)+2uf_{14}(x))^* = f_1^*(x)+2x^if_2^*(x)+2ux^jf_{14}^*(x) = f_1(x)+2x^if_2^*(x)+2ux^jf_{14}^*(x) & = (f_1(x)+2f_2(x)+2uf_{14}(x))a(x) +u (f_3(x)+2 f_4(x))b_1(x) & = (f_1(x)+2f_2(x)+2uf_{14}(x))a(x) +u f_4(x)b(x),
\Pr[E(x_1,v_1,\ldots,x_{\ell},v_{\ell},X_1,\ldots,X_{\ell})] &= \prod_{i=1}^{\ell}\Pr[v_i,v_{i+1}\in \Gamma(x_i)] \cdot \prod_{i=1}^{\ell}\prod_{x \in X_i}\Pr[v_i \in \Gamma(x)] &=\left( \frac{{t \choose 2}}{{m \choose 2}}\right)^{\ell}\cdot \left( \frac{t}{m}\right)^{(6t-2)\ell}\le \left( \frac{t}{m}\right)^{6 t \ell}.
\sum_{n}\lambda(n)e\left(\frac{bn}{c}\right)V\left(\frac{n}{N}\right)=\frac{N}{c}\sum_{n}\lambda(n)e\left(-\frac{\bar{b}n}{c}\right)\cdot 2\pi i^\kappa\int_{0}^{\infty}V(x)J_{k-1}\left(\frac{4\pi\sqrt{nNx}}{c}\right)\mathrm{d}x
p({\|{x}\|}) = \int_{\mathbb{R}_+}\widehat{p}(t)e^{- t^2\|{x}\|^2}\,\mathrm{d}t \approx \sum_{k=-M}^{M}p_k e^{- t_k^2\|{x}\|^2}= \sum_{k=-M}^{M}p_k \prod_{\ell=1}^3 e^{-t_k^2 x_\ell^2},
\left(\mathcal{A}f\right)(x)=(a-bx)\frac{\partial f(x)}{\partial x}+\frac{1}{2}\sigma^{2}x\frac{\partial^{2}f(x)}{\partial x^{2}}+\int_{0}^{\infty}\left(f(x+z)-f(x)\right)\nu(\mathrm{d}z),
W_{-,+}((y_1,y_2),u_1|u_1,u_2) = \frac{1}{2}W(y_1|u_1\oplus u_2)W(y_2|u_2) = \sum_{z_1,z_2}\frac{1}{2}Q(z_1|u_1\oplus u_2)P(y_1|z_1) Q(z_2|u_2)P(y_2|z_2) = \sum_{z_1,z_2}Q_{-,+}((z_1,z_2),u_1|u_1,u_2)P_a(y_1,y_2|z_1,z_2),
\int_{[0,1]^{d-1}}f(x) e^{- 2 \pi i \langle n,x \rangle}\, \textrm{d}x = \sum_{j=1}^m \frac{\langle a_j, n \rangle}{2 \pi i |n|^2}\int_{A_j}e^{-2 \pi i \langle n,x \rangle}\, \textrm{d}x.
\delta^{( 4 )}( \dot{-}k_{3}\dot{+}( \dot{-}p ) \dot{+}k_{3}\dot{+}p^{\prime}) \sim \delta ( p_{0}- p_{0}^{\prime}) \delta^{( 3 )}( e^{\lambda p_{0}}\vec{k_{3}}- \vec{p}+ \vec{k_{3}}+ e^{\lambda k_{3 , 0}}\vec{p^{\prime}}) .
\int_{-\infty}^\infty |f(x_1, x')|^2\,dx_1 = x_1|f(x_1, x')|^2 \bigr|_{x_1=-\infty}^\infty -\int_{-\infty}^\infty x_1 \frac{\partial\,|f(x_1, x')|^2}{\partial x_1}\,dx_1.
\varepsilon_{\mu \nu \lambda \rho}\partial_{\lambda}\partial_{\rho}\theta_{a}^{\mathrm{s i n g}}( x ) = 2 \pi \Sigma_{\mu \nu}^{a}( x ) \equiv 2 \pi \int_{\Sigma_{a}}^{}d \sigma_{\mu \nu}\left( x^{( a )}( \xi ) \right) \delta \left( x - x^{( a )}( \xi ) \right) .
b_{{\bf{k + q / 2}}}^{\dagger}b_{{\bf{k - q / 2}}}= N_{0}\delta_{{\bf{k}}, 0}\delta_{{\bf{q}}, 0}+ [ \delta_{{\bf{k + q / 2}}, 0}( \sqrt{N_{0}}) d_{{\bf{k}}}( -{\bf{q}}) + \delta_{{\bf{k - q / 2}}, 0}d_{{\bf{k}}}^{\dagger}({\bf{q}}) ( \sqrt{N_{0}}) ]
\langle a| T(z) |b\rangle =\lim_{w\rightarrow z}\langle a| \left\{-{1\over2}\partial\varphi^i(w) g_{ij}\partial\varphi^j(z) -{1\over(w-z)^2}\right\}|b\rangle =C_a^b{1\over8}z^{-2},
Ric(\Sigma_l,\Sigma_l) =(n-2)(\frac1{g^2}-\frac{g_t^2}{g^2})-\frac{g_{tt}}g-m\frac{u_t}u\frac{g_t}g \ge(n-2)\left(\frac1{\alpha^{\gamma}t^{\gamma}}-\frac{4(1+\frac r6)^2\gamma^2}{t^2}\right) -\frac{12\gamma(1+2r)(1+\frac{r\gamma}3)}{rt^2}\quad-m\frac{(1+3c)(1+\frac r6)\gamma}{t^2\cos\Delta}>0,
\nu=\frac{\hbar}{2m}\, , \; \; \; \; \; \; \; \; \;\rho(x,t)=|\Psi(x,t)|^{2}\, , \; \; \; \; \; \; \; \; \;v(x,t) = \frac{1}{m}\frac{\partial S(x,t)}{\partial x}\, ,
\ddot{\chi}+ \dot{\alpha}_{0}\left( 1 - \frac{\ddot{\alpha}_{0}}{\alpha_{1}^{2}}\right) \dot{\chi}- \frac{1}{3}\left( 1 + \frac{\ddot{\alpha}_{0}}{\alpha_{1}^{2}}\right) e^{- 2 \alpha_{0}}\nabla^{2}\chi = 0 .
\frac{\delta \hat{A}}{\delta\tau}= \rho\varepsilon_\rho-\varepsilon-\lambda\equiv (p-\lambda)=0, \frac{\delta \hat{A}}{\delta\lambda}=(x_m-\tau)=0, \frac{\delta\hat{A}}{\delta u}= (u-\nu)=0,\frac{\delta\hat{A}}{\delta\nu}=(x_t-u)=0, \frac{\delta\hat{A}}{\delta S}= -\left(r_t+T\right)=0,\frac{\delta\hat{A}}{\delta r}=S_t=0.
H(\tilde\gamma(t_j)) =H(\tilde\gamma(t_{j-1}))+\int_{\tilde\gamma|_{I_j}}p^*\omega &=H(\tilde\gamma(t_{j-1}))+\int_{p_*\tilde\gamma|_{I_j}}\omega &=H(\tilde\gamma(t_{j-1}))+\int_{\gamma}\widetilde{d}G_{\Omega}(\cdot,p_*\tilde\gamma(t_j))-\int_{\gamma}\widetilde{d}G_{\Omega}(\cdot,p_*\tilde\gamma(t_{j-1}))+2k_j\pi,
( B )_{j , j - 1}= ( B )_{j - 1 , j}= \frac{j ( ( 2 j_{0}+ 1 )^{2}- j^{2})}{4 ( j_{0}+ 1 ) \sqrt{4 j^{2}- 1}}, j = 1 , 2 , \ldots , 2 j_{0}.
\begin{aligned}W_{i,j}W_{j,t}=\binom{t-i}{j-i}W_{i,t}, W_{i,j}\overline W_{j,t}=\binom{n-t-i}{j-i}\overline W_{i,t}W_{t,k}=\sum_{i=0}^t(-1)^iW_{i,t}^T\overline W_{i,k}, \qquad \overline W_{t,k}=\sum_{i=0}^t(-1)^iW_{i,t}^TW_{i,k}.\end{aligned}
\sum_{j=0}^k (-1)^j \binom{k+1}{j}B_j x^{k+1-j}=(1+x)\left(x^k+\sum_{j=1}^{k-1}(-1)^j\left(\sum_{i=0}^j \binom{k+1}{i}B_i \right)x^{k-j}\right) (k \geq 0),
{\cal A}_{\mathrm{e f f}}^{\mathrm{m f}}= \frac{R \beta M^{2}}{2}\sqrt{\rho_{0}\rho_{1}}\left[ \left( \frac{\lambda_{0}+ 1}{\rho_{0}}+ \frac{\lambda_{1}+ 1}{\rho_{1}}\right) - \left( \lambda_{0}+ \lambda_{1}\right) \right] ,
\mathbb{P}\bigg( \sup_{t\in [0,T]}Z_t \geq \frac{a}{2}\bigg) = \mathbb{P}\bigg( t \in [0,T] \alpha_t+x \geq \frac{a}{2}\exp\big(bt\big) \bigg) =\mathbb{P}\bigg( t \in \big[0,\bar{\eta}_T\big] \upsilon_t+x \geq \frac{a}{2}\exp\big(b\bar{\iota}_t\big) \bigg).
\left( \begin{array}{l}{s}\\{v}\\{t}\\ \end{array}\right) ( 4 , 2 ; 3 , 1 ) = - \frac{1}{4}\left[ \begin{array}{c c c}{1}&{1}&{1}\\{5}&{- 3}&{1}\\{1 0}&{2}&{- 2}\\ \end{array}\right] \left( \begin{array}{l}{s}\\{v}\\{t}\\ \end{array}\right) ( 4 , 1 ; 3 , 2 )
f^{\#}(x,A) := \limsup_{\substack{x' \to x A' \to A t \to \infty}}\,\frac{f(x',tA')}{t}, f_\#(x,A) := \liminf_{\substack{x' \to x A' \to A t \to \infty}}\, \frac{f(x',tA')}{t},
h_1(T)W_{T,1}(f_1)-h_1(0)W_{0,1}(f_1)-\int_0^T\partial_sh_1(s)W_{s,1}(f_1)ds &=-\int_0^Th_1(s)W_{s,1}\bigotimes W_{s,3}(\lambda(\cdot, \ast)f_1(\cdot))ds.
\bar{\Pi}_{i j}^{g}= \frac{\partial{\cal L}}{\partial ( \partial_{0}g_{i j})}\, = \, \frac{1}{\lambda^{2}}\partial^{0}g_{j i}^{- 1}\, - \, \frac{n}{4 \pi}A_{j i}( g ) \, + \, \frac{i \alpha}{4 \lambda^{2}}\bar{\psi}_{j k}\gamma^{1}\psi_{k m}g_{m i}^{- 1}
S_{I}(N,z)=\sum_{k=N}^{\infty}\; f(k) \, (-z)^k \equiv \frac{(-1)^N}{2\pi i}\int_{\scriptstyle{c-i\infty}\atop{N-1 < c=\Re s<N}}^{c+i\infty}z^s f(s) & \times \;\; \Gamma(1+s-N) \, \Gamma(N-s) \;ds \;\;.
\textrm{B1}= \frac{2I-1}{K-2I+1}\frac{2I-3}{K-2I+3}\ldots \frac{1}{K-1}\begin{cases}\frac{K-1}{K}\frac{K-3}{K-2}\ldots \frac{1}{2}\ \pi \phantom{2}K \textrm{even}, \frac{K-1}{K}\frac{K-3}{K-2}\ldots \frac{2}{3}\ 2 \phantom{\pi}K \textrm{odd}.\end{cases}
\nabla_g \mathcal{U}(g) := g\mathbb{P}((I_4-g^{-1})\mathbb{A}) , &\Psi_{\mathcal{U}}(g) := \{I_4\}{\textstyle \bigcup}\left\{g=\mathcal{T}(R,p): R=\mathcal{R}_{a}(\pi,v) , \right. &~~~~~~~~~~~~~~~~~~~ \left. p= (I_3 - \mathcal{R}_{a}(\pi,v))bd^{-1}, v \in \mathcal{E}(Q)\right\}.
D_{\zeta}^{2s+\tau ,2t+\tau^{\prime}}(x,x_{1},\ldots,x_{n})=(-i)^{2s+\tau +1}\sigma_{2s+\tau}^{+}\sum_{p=0}^{t}x^{2s-2p+\tau+1-\zeta}\hat{\sigma}_{2p+\zeta}^{-}\,\,.
\mathbb{E}[f(\Delta_{i,n}^{(1)}X^{toy,(1)},\Delta_{j,n}^{(2)}X^{toy,(2)})|\mathcal{S}]=(\sigma^{(1)})^{p_1}(\sigma^{(2)})^{p_2}\mathbb{E}[f(Z,Z')]|\mathcal{I}_{i,n}^{(1)}|^{p_1/2}|\mathcal{I}_{j,n}^{(2)}|^{p_2/2}
A := \big\{[T(\cdot)-T(0)]\mathbf{1}_+(\cdot), \,\, -R_+(-\cdot)\mathbf{1}_-(\cdot), \,\, R_-(\cdot)\mathbf{1}_+(\cdot),-[T(-\cdot)-T(0)]\mathbf{1}_-(\cdot), \,\, 1, \,\, -1 \big\}.
\Sigma^0_{div}={Ce^2 \over \beta}\int{|{\bf k}| d|{\bf k}| \over (2 \pi)}{{\cal M}({\bf k},p_{0f}) \over{\bf k}^2+p_{0f}^2+{\cal M}^2({\bf k},p_{0f})}{1 \over |{\bf k}^2-{\bf p}^2|}
\|S_{\xi\chi}f\| =\sup_{\|g\|=1}\bigg|\langle S_{\xi\chi}f,g\rangle \bigg| &\leq \sup_{\|g\|=1}\sqrt{D_{2}}\|g\|\bigg(\int_{\Theta}v^{2}(w)\|\chi_{w}\pi_{F(w)}f\|^{2}d\mu(w)\bigg)^{\frac{1}{2}}&\leq \sqrt{D_{2}}\bigg(\int_{\Theta}v^{2}(w)\|\chi_{w}\pi_{F(w)}f\|^{2}d\mu(w)\bigg)^{\frac{1}{2}}
\P\Big(\|\hat P_j- P_j\|_2>2-2\delta\Big) \geq \P\Big(\Big\{\|\tilde P_j- P_j\|_2>2-2\delta+\|\tilde P_j- \hat P_j\|_2\Big\}\cap \tilde{\mathcal{E}}_{z_1,z_2}\Big) \geq \P\Big(\Big\{\|\tilde P_1- P_1\|_2>2-\delta\Big\}\cap \tilde{\mathcal{E}}_{z_1,z_2}\Big)-e^{-t}= 1-e^{-t}-\P(\tilde{\mathcal{E}}_{z_1,z_2}^c).
\bgroup\begin{pmatrix}{x}_{1}0{-{y}_{1}^{2}}0{-{y}_{1}{y}_{2}}0 0{-{y}_{2}^{2}}0{-{x}_{2}}{-{y}_{1}^{2}}0{-{y}_{1}{y}_{2}}0 0{-{y}_{2}^{2}}0 0 0 0 0 0 0 0{x}_{1}{x}_{2}{y}_{1}0 0 0{x}_{1}{x}_{2}{y}_{1}0 0{-{y}_{2}}0{x}_{1}{x}_{2}0 0{-{y}_{2}}0 0 0 \end{pmatrix}\egroup
d_{\; \; \alpha_{0}}^{\alpha_{1}}= \left( \delta_{\; \; \beta_{1}}^{\alpha_{1}}+ m_{\beta_{1}}^{\alpha_{1}\beta_{0}}\chi_{\beta_{0}}\right) \bar{D}_{\; \; \gamma_{1}}^{\beta_{1}}A_{\alpha_{0}}^{\; \; \gamma_{1}},