harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
T \triangleq \mathrm{Tr}(M) =e^{t(1-q)^2}qs + 2 e^{-tq(1-q)}q(1-s) + e^{tq^2}(1-2q+qs) D \triangleq \det(M) =\left(e^{t(1-q)^2+tq^2}-e^{-2tq(1-q)}\right)q(s-q) \, .
\alpha_\lambda : = \frac{1}{4}\begin{pmatrix}i\,u_y -e^{u/2}- \lambda^{-1}\,e^{-u/2}e^{u/2}+\lambda\,e^{-u/2}-i\,u_y \end{pmatrix}\mathrm{d}x & \qquad+ \frac{i}{4}\begin{pmatrix}-u_x e^{u/2}-\lambda^{-1}\,e^{-u/2}e^{u/2}-\lambda\,e^{-u/2}u_x \end{pmatrix}\mathrm{d}y \; .
W \left(\begin{array}{cc}(d_1,d_2) (c_1,c_2) (a_1,a_2) (b_1,b_2)\end{array}\right) =W \left(\begin{array}{cc}(\tilde{d}_1,\tilde{d}_2) (\tilde{c}_1,\tilde{c}_2) (\tilde{a}_1,\tilde{a}_2) (\tilde{b}_1,\tilde{b}_2)\end{array}\right).
\nu_1 D_s^i u_1(s_0) D_t^j u_2(t_0) + \nu_2 D_s^i u_1(s_0) D_t^j v_2(t_0) & + \nu_3 D_s^i v_1(s_0) D_t^j u_2(t_0) + \nu_4 D_s^i v_1(s_0) D_t^j v_2(t_0) = D_s^i D_t^j f(s_0,t_0),
\Phi^{\ell}( \tau , \bar{\tau}, \phi ) = \sum_{n = 0}^{2 \ell}a_{n}t_{0 n}^{\ell}( \tau , \bar{\tau}, \phi ) = \frac{e^{- i \ell \phi}}{( 1 + | \tau |^{2})^{\ell}}\sum_{n = 0}^{2 \ell}b_{n}\tau^{n}\equiv e^{- i \ell \phi}\varphi ( \tau , \bar{\tau}) \, .
\beta \ll_{c,C,\varepsilon}\Big\vert\frac{1}{N^{h-m}}\int\limits_{\mathbf{x}\in \mathbb{R}^{h-m}}F_1(\mathbf{x}) \prod\limits_{j=1}^d g_j(\xi_j(\Phi(\mathbf{x}))+a_j) \, d\mathbf{x}\Big\vert,
F(n_1,n_2) = \sum_{\substack{\sum_{j=0}^{k-1}h_{1j}=n_1 \sum_{j=0}^{k-1}h_{2j}=n_2}}\binom{n_1}{h_{11},\ldots,h_{1k}}\binom{n_2}{h_{21},\ldots,h_{2k}}\det\Big( I_{2h_{1j}+3h_{2j}+i+j+1}(2\sqrt{x}) \Big)_{i,j=0,\ldots,k-1}.
\psi(z)=lnz-\frac{1}{2z}-\sum_{n=1}^{N}\frac{B_{2n}}{2nz^{2n}}+(-1)^{N+1}\frac{2}{z^{2N}}\int_{0}^{\infty}\frac{t^{2N+1}dt}{(t^{2}+z^{2})(e^{2\pi t}-1)}.
\begin{aligned}\int_{B_{t}}(|\nabla u|^{2})^{\mu}dx \le\theta\int_{B_{s}}(|\nabla u|^{2})^{\mu}dx +C\left\{\frac{1}{(s-t)^{n}}\int_{B_{s}}d^{n}(u,p_{0})dx+\int_{B_{R}}gdx\right\}\end{aligned}
{}\Lambda_r(n) = \sum_{d|n, d\leq R}\mu(d) \left(\mathrm{log}\left(\frac{n}{d}\right)\right)^r - \sum_{lm|n, m \leq R, l \leq Q}\mu(m)\Lambda_r(l) + \sum_{lm|n, l \leq Q}\mu(m)\Lambda_r(l) + \sum_{lm|n, m > R, l>Q}\mu(m)\Lambda_r(l).
\Biggl(\prod_{i=1}^k p_{s_i-s_{i-1}}(y_{i-1},y_i)\Biggr) p_{T-s_k}(y_k,z) begin{align*}-2pt] - \Biggl( \prod_{i=1}^kp_{\eta_n(s_i)-\eta_n(s_{i-1})}(y_{i-1},y_i) \Biggr) p_{T-\eta_n(s_k)}(y_k,z)
T_{+ +}= -{\frac{N}{2 4}}\left( e^{2 \rho}\, \partial_{+}^{\, 2}e^{- 2 \rho}-{\frac{1}{2}}\, e^{4 \rho}{\left( \partial_{+}e^{- 2 \rho}\right)}^{2}- 2 t_{+}\right) \; .
|E\Delta F| \le |{\rm cov}(E)\setminus E|+|{\rm cov}(E)\setminus F|= 2 |{\rm cov}(E)\setminus E| &\le 2(\sqrt\pi)^{-1}|{\rm cov}(E)|^{\frac 1 2}(P(E)- P(F)) \leq 2\sqrt{2}(P(E)- P(F)).
\tilde{P}\colon \mathbb{Z}^n_{\geq 0}\times 2^{\binom{[n]}{2}}\times [n]^2 \rightarrow \mathbb{R}, \tilde{Y}\colon \mathbb{Z}^n_{\geq 0}\times 2^{\binom{[n]}{2}}\times [n]^3 \rightarrow \mathbb{R}.
R_{a b c}^{d}- \delta_{b}^{d}g_{a c}+ \delta_{c}^{d}g_{a b}+ D_{b}T_{a c}^{d}- D_{c}T_{a b}^{d}+ T_{e b}^{d}T_{a c}^{e}- T_{e c}^{d}T_{a b}^{e}= 0 .
\sum_{i=0}^\infty M^{l i}I_i \leq \sum_{i=0}^\infty M^{l i}\nu\big(E_g(Q_{2 R}, c_0 \lambda_0 M^i)\big) +\Big[\sum_{i=0}^\infty M^{\frac{l}{\hat p}i}\hat\nu\big(E_{\hat g}(Q_{2 R}, c_0 \lambda_0 M^i)\big)\Big]^{\hat p}.
\begin{cases}\omega < \delta^{(1)}, \Omega_2 e \geq D_{A\Omega_1}e, \omega < \delta^{(2)}, \frac{1}{2}(\vert M_{\Omega_2}\vert + \vert N_{\Omega_2}\vert-\langle A\Omega_1\rangle)Ve < \Omega_2 Ve < D_{M_{\Omega_2}}Ve, \end{cases}
{\check{c}}_{\gamma , \sigma \dot{\beta}}^{\enspace \enspace \enspace \sigma \dot{\sigma}}= \bar{c_{a}}^{\sigma \dot{\sigma}}E_{\gamma}{c_{\sigma \dot{\beta}}}^{a}+{c_{\sigma \dot{\beta}}}^{a}\bar{c_{b}}^{\sigma \dot{\sigma}}{c_{\gamma a}}^{b}-{c_{\gamma \dot{\beta}}}^{\dot{\sigma}}
\beta_{1 0}^{2}= \frac{4}{3 5}\frac{( r - 5 L ) ( 5 r^{7}+ 2 5 r^{6}L + 9 7 r^{5}L^{2}+ 4 8 5 r^{4}L^{3}+ 2 4 9 5 r^{3}L^{4}+ 1 2 4 7 5 r^{2}L^{5}+ 6 2 2 3 5 L^{6}+ 7 L^{7})}{( r^{2}- L^{2})^{4}}.
\Gamma (s_1) \Gamma (s_2) \zeta_2 (s_1,s_2 \,; a) = &\int_0^\infty \!\!\! \int_0^\infty \frac{y^{s_2-1}}{e^y-1}H(a,x+y)x^{s_1-1}dx dy +\int_0^\infty \!\!\! \int_0^\infty \frac{x^{s_1-1}}{x+y}H(1,y)y^{s_2-1}dx dy,
\frac{\delta n\lambda}{16\sigma^2}\|\theta\|_{w^*,1}\!+\!\frac{\delta n}{2\sigma^2}\sum_{j=1}^p\frac{w_j^{2}}{w_j^*}|\theta_j|\!+\!\frac{\log 4p}{\beta\delta}\|\theta\|_{w^*,1}\!+\!\frac{\log2}{\beta}.
\sum_{\substack{\tau\in D(\ell) \mathrm{sign}(\tau)=1}}\prod_{j=1}^{\ell}\frac{1}{x_{\tau(j)}-x_j}=\sum_{\substack{X=\bigsqcup_{i=1}^{s}X_i |X_i|\geqslant 2 s\}}\left(\prod_{j=1}^sf(X_i)\right).
(\deg(x) - \lambda) f_1(x) = \sum_{y \in S_2}b(x,y) f_1(y) = \sum_{y \in S_2}b(x,y) s_2 f_2(y) = s_2(\deg(x) - \lambda) f_2(x) = \frac{s_2}{s_1}(\deg(x) - \lambda) f_1(x)
S_1 \leq \frac{Q}{2}\sum_{k= 1}^I \Delta x_k |n_k- \hat{n}_k| \sum_{j=\alpha_{0,k}}^{\alpha_{I,k}-1}(N_j+\hat{N}_j)+ \frac{Q}{2}\sum_{k=1}^{I}(N_k+ \hat{N}_k) \sum_{j=\alpha_{0,k}}^{\alpha_{I,k}-1}\Delta x_j |n_j-\hat{n}_j|.
\int_{([0,T]\times[0,M])^k}\frac{1}{k!}c_k(x_1,\ldots,x_k) N(dx_1) \cdots N(dx_k) &= \int_{\mathbb X^k}\frac{1}{k!}c_k(x_1,\ldots,x_k) N(dx_1) \cdots N(dx_k), \; \textrm{on}\Omega^M
\lim_{x\to 1^-}(1-x)^{k+4}Z^{(k+1)}(x) =3\sum_{l=0}^k \binom{k}{l}\lim_{x\to 1^-}\left((1-x)^{l+1}f^{(l)}(x)\right)\lim_{x\to 1^-}\left((1-x)^{k-l+3}Z^{(k-l)}(x)\right), & =3\frac{2p-1}{8}\sum_{l=0}^k \binom{k}{l}l!(k+2-l)!, &=3\frac{2p-1}{8}k!\sum_{l=0}^k (k+2-l)(k+1-l), &=3\frac{2p-1}{8}k!\sum_{l=1}^{k+1}l(l+1)=\frac{2p-1}{8}(k+3)!.
\begin{bmatrix}x_{1, k+1}x_{2,k+1}\end{bmatrix}=\hat A\begin{bmatrix}x_{1,k}x_{2,k}\end{bmatrix},\|\hat A z\|_{\hat{\mathcal{X}}}\leq \|z\|_{\hat{\mathcal{X}}},\quad \|z\|_{\hat{\mathcal{X}}}:=\left\| \begin{bmatrix}\mathcal{X}_1 0 0 \mathcal{X}_2\end{bmatrix}z\right\|.
- 2 \sum_{k=1}^d \sum_{i=1}^{k-1}\sum_{j=k+1}^{d}\frac{1}{((x_k-x_i)+\epsilon^2)((x_j-x_k)+\epsilon^2)}= -2 \sum_{k=1}^d \sum_{i=k+1}^{d}\sum_{j=i+1}^{d}\frac{1}{((x_i-x_k)+\epsilon^2)((x_j-x_i)+\epsilon^2)}.
\mathrm{d}Y_\alpha(t) = \frac{1}{2}(\omega_\alpha-\omega Y_\alpha) \mathrm{d}t+ \sum_{\beta=1}^{N-1}\sqrt{Y_\alpha(\delta_{\alpha\beta}-Y_\beta)}\mathrm{d}W_{\alpha\beta}(t), \alpha = 1,\dots,N-1,
0\le \langle Ag, g\rangle_{X}= \int_D \int_X\int_X g(x)g(y) \> K_h(x,dy) \> d\omega_X(x) \> \cdot\> f(h)\> d\omega_D(h) = \int_{W_e}\int_{U_z}\int_{U_z}g(x)g(y) \> K_h(x,dy) \> d\omega_X(x) \> \cdot\> f(h)\> d\omega_D(h) =: \int_{W_e}\phi(x)\> f(x)\> d\omega_D(x)
f_i(X)=\begin{cases}1\quad\mbox{if}|X-(x_i,0)|\le r_i, 2-\frac{|X-(x_i,0)|}{r_i}\quad\mbox{if}r_i\le |X-(x_i,0)|\le 2r_i, 0\quad\mbox{if}|X-(x_i,0)|\ge 2r_i.\end{cases}
\frac{1}{2^9}\frac{1}{\pi^4}\big{[}\Omega_5\big{]}^{\frac{1}{2}}\sum_{N=1}^{\infty}\frac{(N + 1)(N + 2)}{N(N + 3)}C_N^{\frac{3}{2}}(cos{\theta_4^0})\big{[}a^{-2}C_N^{\frac{3}{2}}(cos{\theta_4})\big{]}(a^{-2})a^3K \frac{1}{\epsilon}+ F. T.
\|C_\phi(f)\|_{2,\omega}^2 = |f(\phi(0))|^2 + \|f_+\circ\phi\|_{2,\omega}^2 + \|f_-\circ\phi\|_{2,\omega}^2 & \le |f(\phi(0))|^2 + \Big(\frac{1+r}{1-r}\Big)^{2\Lambda}\Big( \|f_+\|_{2,\omega}^2+\|f_-\|_{2,\omega}^2\Big).
\log \det \big ( R^{(1)}(x) \big ) - \log \det \big (R^{(0)}(x) \big ) =g(1,x)-g(0,x) &=g_t'(s,x)=\sum_{i,j=1}^n \big (R^{(s)}(x) \big )^{-1}_{ji}\big (R^{(1)}_{ij}(x) - R^{(0)}_{ij}(x) \big ),
\phi_N(\xi) = E\left[\zeta^{\,-\frac{(1 - \theta) S(\theta) - N \ln N}{N}}\right]= \zeta^{\,\ln N}E\left[\left(\zeta^{\frac{(1 - \theta)}{N}}\right)^{-S(\theta)}\right]= \zeta^{\,\ln N}G\left(\zeta^{\frac{(1 - \theta)}{N}}\right).
\|h_0\|-(A'+B') = (\|a_1'\|+n_1\|g_1\|+\|b_1'\|) - (\|a_1'\|+\|a_2'\|) - (\|b_1'\|+\|b_2'\|) &= n_1\|g_1\|-\|a_2'\|-\|b_2'\|\ge n_1\|g_1\|-\alpha_1\|g_1\|-\beta_1\|g_1\| &=(n_1-\alpha_1-\beta_1)\|g_1\|\ge (2V+1) \|g_1\|.
{\bf W}_\alpha= \begin{bmatrix}b_0 0 0 0 0 \cdots 0 b_1 b_0 0 0 0 \cdots 0 b_2 b_1 b_0 0 0 \cdots 0 b_3 b_2 b_1 b_0 0 \cdots 0 \vdots \vdots \vdots \vdots \vdots \cdots \vdots b_{P-1}b_{P-2}w_{P-3}b_{P-4}\cdots b_1 b_0\end{bmatrix},
\begin{cases}\partial_t^2\partial_{\alpha}^k u+A\bold{n}\partial_{\alpha}^k \frac{\partial_{\alpha}}{z_{\alpha}}u=g_k, A=a|z_{\alpha}| \bold{n}=\frac{iz_{\alpha}}{|z_{\alpha}|}=\frac{\bar{u}_t+i}{|\bar{u}_t+i|}g_k=\partial_{\alpha}^k g-\sum_{m=1}^k c_{m,k}\partial_{\alpha}^m (A\bold{n}) \partial_{\alpha}^{k-m}Du.\end{cases}
u(x) \,=\, v(x) \, = \, L[{\rm div}\, \mathcal{F}] + L[g] & \, = \, \big( \tilde c [\mathcal{F}] + c[g] \big) \frac{x^{\bot}}{4\pi |x|^2}+ \mathcal{R}(x) \,, |x|\geq 4 R_0\,,
\#\Bigl\{(a_1,a_2)\in\mathcal{C}^2:\exists (v_1,v_2,v_3,v_4)\in[-V,V]^4\backslash\{\mathbf{0}\}v_1a_1+v_2a_2+v_3 X+v_4=0\Bigr\}\ll X^{o(1)}\Bigl(\#\mathcal{C}^{5/4}V^2+\frac{\#\mathcal{C}^{3/2}V^{3}}{X^{1/2}}\Bigr).
\dfrac{M_{2}(\mathcal{NC}(G))}{|e(\mathcal{NC}(G))|}= \dfrac{(p^{2}n-n)(p^{2}n-pn)(p^{4}n^{2}-2p^{3}n^{2}+p^{2}n^{2})}{(p^{2}n-n)(p^{2}n-pn)}.
(A_1,\dots,A_t)=\begin{cases}(1,\dots,1,x,1,\dots,1) x\in H_j\cap\tilde{A}_{n_j}, (S_1,\dots,S_{j-1},x,1,\dots,1) x\in H_j\backslash\tilde{A}_{n_j}.\end{cases}
1 ={\frac{Q ( \infty , s )}{2 \pi N}}\equiv{\frac{2}{( 2 \pi )^{2}N}}\left[ \left( 1 -{\frac{( 0 . 1 2 5 )^{2}}{( 0 . 1 2 5 )^{2}+ s^{2}}}\right){\frac{\pi}{2 s}}-{\frac{0 . 1 2 5}{( 0 . 1 2 5 )^{2}+ s^{2}}}\ln \left({\frac{s}{0 . 1 2 5}}\right) \right] .
\begin{aligned}\ge \frac{n(n-1)\tau^2}{\Im(\rho)^2}\min\left\{\frac{-1+(2e)^2}{2\left(\frac{1}{2e}+2e\right)^2},\lim_{n \to \infty}\left(\frac{-1+(ne)^2}{2\left(\frac{1}{ne}+ne\right)^2}-\frac{0.184(n-2)}{ne}\right)\right\}>\frac{0.432n(n-1)\tau^2}{\Im(\rho)^2}.\end{aligned}
[{\bf T}_{\bf m},{\bf T}_{\bf n}] = - 2 i \sin \left({\frac{2 \pi}{N}}\epsilon^{a b}m_{a}n_{b}\right){\bf T}_{\bf m + n}. ( \epsilon^{1 2}= - \epsilon^{2 1}= 1 , a , b = 1 , 2 )
v_{\{0,1,2\}}= \textstyle\frac{1}{6}v_{\{0,1\}}+ \frac{1}{6}v_{\{0,2\}}+ \frac{1}{6}v_{\{0,3\}}+ \frac{1}{2}v_{\{0,1,2,3\}}& = \textstyle\frac{1}{3}v_{\{0\}}+ \frac{2}{3}v_{\{0,1,2,3\}}.
\widehat{h}_{5}=h_{5}^{[KP,SG]}\left( r,\theta ,\chi \right) =\eta_{5}^{[KP,SG]}\left( r,\theta ,\chi \right) \eta_{5}\left( r,\theta \right)h_{5}(r),\qquad h_{5}(r)=-r_{0}^{2}e^{2\psi (r)},
\bar{t}_k = \widetilde{t}_k, \bar{\lambda}_{u_1}= \widetilde{\lambda}_{u_1}- \sum_{i \neq u_1}\widetilde{\lambda}_i \widetilde{\underline{t}}^{\,\underline{a}_i - \underline{a}_{u_1}}, \bar{\lambda}_{u_2}= \widetilde{\lambda}_{u_1}\widetilde{\underline{t}}^{\,\underline{a}_{u_1}- \underline{a}_{u_2}}, \bar{\lambda}_i = \widetilde{\lambda}_i
\nabla I P_{> M}O((u_{> \frac{M}{8}})(u_{\leq \frac{M}{8}})^{2}) = O((\nabla I P_{> \frac{M}{8}}u)(P_{\leq \frac{M}{8}}u)^{2}) + O((I P_{> \frac{M}{8}}u)(\nabla u_{\leq \frac{M}{8}})(u_{\leq \frac{M}{8}})).
\begin{aligned}\le \varepsilon^{-2N}h^{j+l+k-\delta j}\mathcal{O}(|X|^{4N-j-2l-k+|\gamma^1|})\mathcal{O}(|X|^2) &\le \varepsilon^{-2N}\mathcal{O}(h h^{\delta(4N+1)}h^{(1-2\delta)(j+l+k-1)})\le \varepsilon^{-2N}\mathcal{O}(h h^{\delta(4N+1)})\le \mathcal{O}(h).\end{aligned}
A_\alpha(p_\alpha f)(x) = d p_\alpha(x) f(x) + \langle x ; p_\alpha(x) \nabla(f)(x) \rangle + \langle x ; f(x) \nabla(p_\alpha)(x) \rangle + \sum_{k=1}^d \partial_k D_k^{\alpha-1}(p_\alpha f)(x).
\begin{array}{l}\displaystyle\int_{{\mathcal W}^k(x^{(k+1)})}\Lambda^{(k)}(x^{(k+1)},x^{(k)})\,J_\kappa(x^{(k)};\theta)\,\mathrm{d}x^{(k)}\displaystyle=\,J_\kappa(x^{(k+1)};\theta)\,\frac{\Gamma((k+1)\,\theta)}{\Gamma(\theta)}\prod_{i=1}^k \,\frac{\Gamma((k+1-i)\,\theta+\kappa_i\big)}{\Gamma((k+2-i)\,\theta+\kappa_i)},\end{array}
\{\eta_{M N}\}= \left( \begin{array}{c c}{g_{\alpha \beta}+ g_{\rho \sigma}A_{\alpha}^{\rho}A_{\beta}^{\sigma}}&{g_{\mu \rho}A_{\beta}^{\rho}}\\{A_{\alpha}^{\rho}g_{\rho \nu}}&{g_{\mu \nu}}\\ \end{array}\right) ,
c_0 :K_0^{1/8}(S)\to K_0(A),~~~[p,m]\mapsto [\kappa(p)]-[1_m] c_1 :K_1^{1/8}(S)\to K_1(A),~~~[u]\mapsto [\kappa(u)] c :=c_0\oplus c_1 :K_*^{1/8}(S)\to K_*(A).
\frac{1}{(q+\xi_2)^2 - E - i \eta}- \frac{1}{(q+\xi_1)^2 - E - i \eta}= \frac{2 q (\xi_1 - \xi_2) + \xi_1^2 - \xi_2^2}{[(q+\xi_2)^2 - E - i \eta ][ (q+\xi_1)^2 - E - i \eta ]}\leq \frac{\tau}{[(q+\xi_2)^2 - E - i \eta ][ (q+\xi_1)^2 - E - i \eta ]}
Q_1=\tilde{q}_1+\frac{1}{8}\left( \frac{\tilde{p}_1}{\varepsilon^6}-\frac{3\tilde{p}_1{}^2}{\varepsilon^2}-\frac{3\tilde{p}_2}{\varepsilon^2}-\frac{2\tilde{t}_1}{\varepsilon^2}-\frac{3}{\varepsilon^{10}}\right),Q_2=\tilde{q}_2-\frac{3\tilde{p}_1}{8\varepsilon^2}+\frac{1}{8\varepsilon^6}.
S_\gamma^1 := \{\Phi^t(X) \colon X \in \gamma t = 0 \}, S_\gamma^2 := \{\Phi^t(X) \colon X \in \gamma t = \tau(X) \}, S_\gamma^3 := \{\Phi^t(X) \colon X=X_1 0 \leq t \leq \tau(X) \}, S_\gamma^4 := \{\Phi^t(X) \colon X=X_2 0 \leq t \leq \tau(X) \},
\frac{\zeta(2)\zeta(3)}{\zeta(6)}=\prod_{p}\left(1+\frac{1}{p^2}+ \frac{1}{p^3}+ \cdots \right)=\lim_{n\to \infty}\prod_{p}\left(1+\frac{1}{p^2}+ \frac{1}{p^3}+ \cdots + \frac{1}{p^n}\right).
e^{4 i p^{\prime}L}= \frac{( \sigma - \gamma )^{2}- \lambda^{2}}{( \sigma + \gamma )^{2}- \lambda^{2}}= \frac{( \kappa - i p^{\prime})^{2}+ W^{2}}{( \kappa + i p^{\prime})^{2}+ W^{2}}
\big | \widehat f_{2n_*}\big(\tfrac in \big) - f_0\big(\tfrac in \big) \big | \leq C_3 L_{n_*}^{\beta/(\beta+1)}+ C_3 \big( f_0\big(\tfrac in \big) L_{n_*}\big)^{\beta/(2\beta+1)}\ \ \i=1,\ldots,n,
\begin{cases}\frac{1}{2}r_2(b^2-4c+1) + \left|\mathcal F_{[c,b]}^{(1)}\right| begin{align*}0.75em]\frac{1}{4}r_2(b^2-4c+1) + \left|\mathcal F_{[c,b]}^{(1)}\right| \end{cases}
\ddot{h}(t) + \frac{\alpha}{t}\dot{h}(t) = \| \dot{x}(t) \|^2 + \langle x(t) - x^{*}, \ddot{x}(t) + \frac{\alpha}{t}\dot{x}(t) \rangle, = \| \dot{x}(t) \|^2 + \langle x(t) - x^{*}, -\nabla \Phi (x(t)) + g(t)\rangle .
\left\{R_{\pm 0 A ( u )}, R_{\pm 0 ( v )}^{B}\right\}= 0 , \qquad \left\{R_{\pm 0 A ( u )}, R_{\mp 0 ( v )}^{B}\right\}= \delta_{u v}\frac{1}{2}( 1 \mp i \Omega \Pi )_{A}{}^{B},
\frac{d}{dt}\left( \begin{array}{c}\omega_1 \eta_1 \end{array}\right) = \left( \begin{array}{ccc}- \frac{1}{12}\frac{d \ln\Delta}{dt}\frac{3\,\delta}{2\,\Delta}- \frac{g_2 \, \delta}{8 \, \Delta}\frac{1}{12}\frac{d \ln \Delta}{dt}\end{array}\right) \cdot \left( \begin{array}{c}\omega_1 \eta_1 \end{array}\right) \;,
T^{i,j}f(y) = \sum_{\substack{R_1 \in \mathcal{D}_1 R_2 \in \mathcal{D}_2}}\sum_{\substack{P_1 \in (R_1)_{i_1}P_2 \in (R_2)_{i_2}}}\sum_{\substack{Q_1 \in (R_1)_{j_1}Q_2 \in (R_2)_{j_2}}}a_{PQR}\cdot \hat{f}(P) h_{Q}(y)
\omega_t \left( \prod_{j=1}^m C(f_j, g_j) \right) = \omega \left( \prod_{j=1}^m \tau_t (C(f_j,g_j)) \right) =\omega \left( \prod_{j=1}^m C( (f_j)_t, (g_j)_t) \right) ={\rm pf}[ \mathcal{C}_t]
\begin{aligned}f =\lim_{m\to\infty}e^{it_mH}P_a^{{\varepsilon}}(t_m)J_ae^{-it_mH_a}P^a_{M^m}e^{it_mH_a}J_a^*P_a^{{\varepsilon}}(t_m)e^{-it_mH}f &= W_a\Omega_af\in \mathcal{R}(W_a).\end{aligned}
\sum_{1\leq i_1<\cdots<i_k\leq n}\lambda_{i_1}^l\cdots\lambda_{i_k}^l\otimes(t_{i_1}+\cdots+t_{i_k}) = (-1)^{k}\sum_{p=1}^n(-1)^{p}\sum_{\substack{0\leq k_r\leq l-1 k_1+\cdots+k_n=lk-p}}\lambda_1^{k_1}\cdots\lambda_n^{k_n}p_G^*\delta_G(\bigwedge\nolimits^{p}\sigma_n).
\frac{1}{s_c}\times \alpha (d,m) = \max \left( 1-\frac{1}{d}, \frac{1}{2}, 1-\frac{(m-1)d}{(d+1) ((m-1)d-2)}\right) \rightarrow 1, & s_c - \alpha (d,m) = \min \left( \frac{(m-1)d-2}{2(m-1)d}, \frac{(m-1)d-2}{4(m-1)}, \frac{d}{2(d+1)}\right) \rightarrow \frac{1}{2}
\int_{B_{R}(0^n)\setminus\Lambda}(-\triangle)^{\gamma}(e^{\frac{n-2\gamma}{2}w})=\lim\limits_{\lambda\rightarrow+\infty}\int_{\mathcal{O}_{\lambda}}(-\triangle)^{\gamma}(e^{\frac{n-2\gamma}{2}w})\leq C(n,\gamma,g)<+\infty.
S_g(f+g'+h) = S(g)^{-1}S(f+(g+g')+h) &= S(g)^{-1}S(f+(g+g'))S(g+g')^{-1}S((g+g')+h) &= S_g(f+g')S_g(g')^{-1}\underbrace{S(g)^{-1}S(g)}_{=1}S_g(g'+h)\ .\ \ \ \square
R^{'\mathsf{proper}}_j(\eta) =\log\left(1+\frac{\eta{\rm Tr}({\bf G}_{jl}{\bf C}_{x_{l}})}{\eta\left({\rm Tr}({\bf G}_{jB}{\bf C}_{x_{B}})+\sum_{\substack{i=1 i\neq j,l}}^{J}{\rm Tr}({\bf H}_{ji}{\bf C}_{x_{i}})+\kappa{\rm Tr}\left({\bf G}_{jj}{\rm diag}({\bf C}_{x_{j}})\right)+\sigma^{2}_w\right)+\sigma_n^{2}}\right),
\langle (\xi_1,\eta_1,\zeta_1),(\xi_2,\eta_2,\zeta_2) \rangle_{(A_r,B_r,C_r)}:={\rm tr}( A_r^{-1}\xi_1 A_r^{-1}\xi_2 ) +{\rm tr}(\eta_1^T\eta_2) +{\rm tr}(\zeta_1^T \zeta_2)
\int_0^x tf(t)^2f^{(1)}(t) \;dt =\left[\frac{tf(t)^3}{3}\right]_0^x-\frac{1}{3}\int_0^x f(t)^3 \;dt=\frac{xf(x)^3}{3}-\frac{1}{3}\int_0^x f(t)^3 \;dt.
\big|\|\varphi_{n,t}(b)\|^2-\|b\|^2\big| =\big|\|\varphi_{n,t}(b)^*\varphi_{n,t}(b)\|-\|b^*b\|\big| &\leq\big|\|\varphi_{n,t^{-1}}(b^*)\varphi_{n,t}(b)\|-\|\varphi_{n,e}(b^*b)\|\big|+\big|\|\varphi_{n,e}(b^*b)\|-\|b^*b\|\big| &\leq\big|\|\varphi_{n,t^{-1}}(b^*)\varphi_{n,t}(b)-\varphi_{n,e}(b^*b)\|\big|+\big|\|\varphi_{n,e}(b^*b)\|-\|b^*b\|\big|
|f(z)-f(w)|\; \leq\; C_1 \left(\int_{2D}|Df|^2\,\right)^{\frac{1}{2}}&\leq\; C_2 \left(m^{2p}\mathrm{Area}(2D)\right)^{\frac{1}{2}}\;=\; 2C_2\sqrt{\pi}m^p|z-w|\ .
\tilde{\mathbf{b}}^{[i,j]}_s = \mathbf{f}_{i}^{[i,j]}-\sum_{s'=0}^{s-1}\frac{{\mathbf{b}_{s'}^{[i]}}^{\mathsf{H}}\mathbf{f}_{i}^{[i,j]}}{\|\mathbf{b}_{s'}^{[i]}\|^2}\mathbf{b}_{s'}^{[i]}
|\tau(M^{a}F_x(r)y_r)| =\Big|\frac{1}{r^a}\frac{1}{\Gamma(a)}\int^r_0(r-s)^{a-1}\tau(F_x(s)y_r)ds\Big| &\leq\frac{1}{r^{Re(a)}}\frac{C_ae^{\pi|Im(a)|}}{{\Gamma(Re(a))}}\int^r_0(r-s)^{Re(a)-1}\tau(F_x(s)y_r)ds &\leq{C_ae^{\pi|Im(a)|}}\frac{1}{r}\int^r_0\tau(F_x(s)y_r)ds={C_ae^{\pi|Im(a)|}}\tau(\alpha(u_r)x\cdot y_r).
P(S_n(n-x f_2(n))+a_n>\lfloor n-\varepsilon f_2(n)\rfloor) =P(n-a_n-S_n(n-x f_2(n))<\lceil\varepsilon f_2(n)\rceil) &=P\left(\frac{n-a_n-S_n(n-x f_2(n))}{f_2(n)}\leq\varepsilon\right).
\limsup_{M \rightarrow \infty}\limsup_{n \rightarrow \infty}\frac{1}{n}\sum_{|x| \geq M n}\sup_{s \in [0,T]}| [ -(-\Delta)^{\gamma/2}G_s](\tfrac{x-1}{n}) - [ -(-\Delta)^{\gamma/2}G_s](\tfrac{x}{n}) |=0.
a_{3}= 1 / 2 \ \ , \qquad a_{n}={\frac{1}{2}}{\frac{n ( n - 2 )}{( n - 1 ) ( n - 3 )}}\sum_{k = 1}^{n - 3}a_{k + 2}a_{n - k}\ \ , \qquad n \ge 4 \ \ .
Z^{( 1 )}[ \eta ] = \mathrm{S d e t}^{- \frac 1 2}{\sf Y}[ \eta ] = \mathrm{S d e t}^{- \frac 1 2}\left( \begin{array}{c c}{{p^{2}}}&{{\lambda \bar{\eta}}}\\{{\lambda \eta}}&{{i p \llap /}}\\ \end{array}\right) .
{\cal Z}=\sum_{k=0}^{+\infty}(-1)^k\exp(-\frac{\pi k}{g_{nc}^2\theta})\frac{1}{2\pi i}\int_{{\cal C}_0}\frac{dz}{z^{k+1}}\exp\left[z\Phi(-z;\frac{3}{2};1)\frac{A_{nc}}{\sqrt{2\pi g_{nc}^2\theta^3}}\,\,\right],
-6\sum_{\substack{a,b\in \mathbb{Z}ab=4n+1 12|3a-b-2}}d\left(N,\tilde{N}, \frac{1}{6}, \frac{1}{3}\right) =\sum_{\substack{a,b\in\mathbb{Z}^+ ab=4n+1 1\leq b\leq 3a-2 12|3a-b-2 \ \ 12|3a-b+2}}b - \sum_{\substack{a,b\in\mathbb{Z}^+ ab=4n+1 b\geq 3a+2 12|3a-b-2\ \ 12|3a-b+2}}3a & =:S_1(n)-3S_2(n).
|\partial_t \phi(t,x)| \lesssim \Bigl|\int_{{\mathbb{R}}^n}\partial_t g_t(x-y)1\!{\rm l}_{B(0,R_0)}(y)\,{\rm d}y\Bigr| &= \Bigl|\int_{{\mathbb{R}}^n}\partial_t g_t(x-y)1\!{\rm l}_{B(0,R_{0})^c}(y)\,{\rm d}y\Bigr| \le C\int_{{\mathbb{R}}^n}t^{-n/2-1}e^{-\frac{R^2}{4t}}e^{-\frac{c|x-y|^2}{4t}}\,{\rm d}y\lesssim R^{-2}.
\displaystyle\beta_k=\left(\begin{smallmatrix}k-\frac{1}{2}\!~~k\end{smallmatrix}\right)\displaystyle\int_0^{+\infty}\!\!\!\!\!\!\!\!\frac{dx}{(1+x^4)^{k+1/2}}\bigg\slash \left(\displaystyle\int_0^{+\infty}\!\!\!\!\!\frac{dx}{(1+x^4)^{1/2}}\right)^{4k+1}\!\!\!\!.
\frac{1}{\lambda_N^2}\prod\limits_{k=1}^{\infty}\Bigg( \frac{\lambda_N^2 + \mu_{k}^{2}}{\lambda_N^2 +\lambda_{k}^{2}}\Bigg)=\sum\limits_{k=1}^{\infty}\frac{a_k}{\lambda_k^2(\lambda_N^2 +\lambda_{k}^{2})}.
(\frac{\partial}{\partial t}+iP)U(t,\gamma) = \int e^{i(\langle \gamma, \xi\rangle - h(t,\xi))}\frac{\partial u}{\partial t}(t,\xi)d\xi + \int e^{i(\langle \gamma, \xi\rangle - h(t,\xi))}i(p(\gamma,\xi)- h'_{t}(t,\xi))u(t,\xi) d\xi.
\left( T_{1}+T_{2}\right)^{2m}\left( T_{1}-T_{2}\right) h_{2m+1}^{-}& =2\left( N\kappa+m+1\right) \left( T_{1}+T_{2}\right)^{2m}h_{2m}^{+}& =2^{m+1}\left( N\kappa+m+1\right) \left( N\kappa-\kappa+1\right)_{m}\left( N\kappa+1\right)_{m}& =2^{m+1}\left( N\kappa-\kappa+1\right)_{m}\left( N\kappa+1\right)_{m+1}.
\rho=\begin{cases}-2\min(f,g) 2\mid u_2 2\mid\frac{cu_2+2}{N/c}, 1-\min(f,g)-\min(f+1,g-1) 2\mid u_2 2\nmid\frac{cu_2+2}{N/c}, 1-\min(f,g)-\min(f-1,g+1) 2\nmid u_2 2\mid \frac{cu_2+2}{N/c}, 2-2\min(f,g) 2\nmid u_2 2\nmid \frac{cu_2+2}{N/c}.\end{cases}
A_1 \cap A_2 = A_1 \cap A_4 = A_3 \cap A_4 = \{(1,1)\}, A_1 \cap A_3 = \{(1,1), \, (1,-1)\}, A_2 \cap A_3 = \{(1,1), \, (\xi,\xi), \, (\xi^2,\xi^2)\}, A_2 \cap A_4 = \{(1,1), \, (-1,1)\},
\frac{1}{g_{a}^{2}( p^{2})}= \frac{k_{a}}{g_{M_{s t r i n g}}^{2}}+ \frac{b_{a}}{1 6 \pi^{2}}\ln \frac{M_{s t r i n g}^{2}}{p^{2}}+ \frac{1}{1 6 \pi^{2}}\triangle_{a},
L_{q}= \frac{1}{2 e}\left( \dot{x}_{\mu}- \frac{i}{2}v \varepsilon_{\mu \nu \lambda}\xi_{a}^{\nu}\xi_{a}^{\lambda}\right)^{2}- \frac{1}{2}e m^{2}- i q m v \xi_{1}^{\mu}\xi_{2 \mu}+ \frac{i}{2}\xi_{a}^{\mu}\dot{\xi}_{a \mu},
\sum_{k=r_m}^{r_{m+1}-1}\frac{\log^{2\ell}p_k}{p_k}\lesssim \sum_{k=\lfloor e^{\log^2m}\rfloor}^{\lfloor e^{\log^2(m+1)}\rfloor}\frac{\log^{2\ell}k}{k\log k}\lesssim \log m^{4\ell-2}\big( \log^2(m+1)-\log^2m \big)\lesssim \log m^{4\ell-1}m^{-1}\lesssim m^{-1/2},
K^p[H_1,H_2] =\mathring{K}^p[H_1,H_2]+Err^p[H_1,H_2], \mathring{K}^p[H_1,H_2] =\epsilon_3\mathring{K}_2[H_1,H_2]+\mathring{K}_3[H_1,H_2]+\epsilon_2 \mathring{K}^p_4[H_1,H_2]+\mathring{K}_5[H_1,H_2], Err^p[H_1,H_2] =\epsilon_3 Err_2[H_1,H_2]+Err_3[H_1,H_2]+\epsilon_2 Err^p_4[H_1,H_2]+Err_5[H_1,H_2].
I_1 =d^{f(p)}\big(H_0(x),\exp_{f(p)}^{-1}\circ f(p)\big)=d^{f(p)}\big(H_0(x),Df(p)\circ \exp_p^{-1}(p)\big) &=d^{f(p)}\big(Df(p)\circ \exp_p^{-1}(x),Df(p)\circ \exp_p^{-1}(p)\big) &\geq \frac{\lambda}{2}d^{p}\big(\exp_p^{-1}(x),\exp_p^{-1}(p)\big)\geq \frac{\lambda}{2L}d(x,p).
\omega_{1,1}(z)=2\sum_{a}e^{\hat{t}_{a,0}}(\frac{1}{24}d\xi_{a,1}(z)+\frac{\hat{t}_{a,1}}{24}d\xi_{a,0}(z)+\frac{1}{2}B_{a,0;a,0}d\xi_{a,0}(z)).
\lim \limits_{n \to \infty}\frac{1}{n}\sum_{j=0}^{n-1}\Big|\mu^A\Big(S^A_j(Q_1) \cap Q_2\Big) - \mu^A (Q_1) \mu^A(Q_2)\Big| =0,\forall Q_1\ne Q_2 \in \mathcal{F}^A
\begin{array}{l}{{< \chi , \chi^{\prime}> = \displaystyle \prod_{r = 1}^{N}\delta_{n_{r}n_{r}^{\prime}}\cdot \displaystyle \prod_{r = 1}^{M}\delta_{m_{r}m_{r}^{\prime}}\cdot \displaystyle \prod_{r = 1}^{Q}\delta_{q_{r}q_{r}^{\prime}}\cdot \displaystyle \prod_{r = 1}^{T}\delta_{t_{r}t_{r}^{\prime}}\cdot \displaystyle \prod_{r = 1}^{4}\delta_{\nu_{r}\nu_{r}^{\prime}}.}}\\ \end{array}
P_0 =\frac{1}{2}\big((f_6)_{\xi^6}-(f_5)_{\xi^5}+(f_4)_{\xi^4}-(f_3)_{\xi^3}+(f_2)_{\xi^2}-(f_1)_{\xi}\big)+f_0, P_1 =-3(f_6)_{\xi^4}+\frac{5}{2}(f_5)_{\xi^3}-2(f_4)_{\xi^2}+\frac{3}{2}(f_3)_{\xi}-f_2, P_2 =\frac{9}{2}(f_6)_{\xi^2}-\frac{5}{2}(f_5)_{\xi}+f_4, P_3 =-f_6.
\int_{0}^{T}\bigg[\mathrm{i}\big(\Psi,\,\frac{\partial \widetilde{\Psi}}{\partial t}\big)+\frac{1}{2}\big(\left(\mathrm{i}\nabla +\mathbf{A}\right)\Psi,\,\left(\mathrm{i}\nabla +\mathbf{A}\right) \widetilde\Psi\big) + \big(V\Psi, \widetilde\Psi \big)+\big(\phi\Psi, \widetilde\Psi\big)\bigg]\mathrm{d}t = 0,
\begin{cases}\Gamma_1(u,v):=\psi_T S_{\lambda}(t)u_{0,\lambda}-i\psi_T\int_{0}^{t}S_{\lambda}(t-t')(\alpha \lambda uv +\beta \lambda^{-3}|u|^2u) dt', \Gamma_2(u,v):=\psi_T W_{\lambda}(t)v_{0,\lambda}+\psi_T\int_{0}^{t}W_{\lambda}(t-t')(-v\partial_{x}v+\varepsilon\partial_{x}|u|^2) dt'. \end{cases}
h = \frac{8 \pi}{1 5}l_{p}^{\prime 6}\sum_{n = 1}^{N}\sum_{n_{0}= - \infty}^{\infty}\frac{Q_{n}}{\left( | \vec{r}- \vec{r}_{n}|^{2}+ ( x_{1 1}- x_{n 1 1}+ n_{0}R_{1 1}^{\prime})^{2}\right)^{3}},
\left\|\sum_{i=0}^{n-1}g_i'(0)\right\|_U^2 = \sum_{\substack{i=0,\dots,n-1 j=0,\dots,n-1}}\langle g_i'(0), g_j'(0)\rangle = \sum_{i=0}^{n-1}\|g_i'(0)\|_U^2 + 2 \sum_{0\le i< j< n}\langle g_i'(0), g_j'(0)\rangle.
\vec{M}\equiv ( M ,{\cal Q}_{d},{\cal Q}_{e}^{1},{\cal Q}_{e}^{2}) \, , \hspace{. 3 c m}\vec{N}\equiv ( N ,{\cal Q}_{a},{\cal Q}_{m}^{1},{\cal Q}_{m}^{2}) \, , \hspace{. 3 c m}\vec{J}\equiv ( J ,{\cal F},{\cal P}_{m}^{1},{\cal P}_{m}^{2}) \, ,
( \mathcal{L}_1,\mathcal{L}_2 , \mathcal{L}_3, \phi) \otimes \mathcal{L}= ( \mathcal{L}_1 \otimes \mathcal{L}, \mathcal{L}_2 \otimes \mathcal{L}, \mathcal{L}_3 \otimes f_3^* \mathcal{L}, \phi \otimes \mathrm{id}),
d\check{X}_t = -\mathbf{L}\check{X}_tdt-\alpha(t)\left(\nabla U_v(X(t))-\mathbf{1}\otimes\frac{1}{m}\sum_{i=1}^m \nabla u_i(X^{(i)}(t))\right)dt+\sqrt{2\sigma\alpha(t)}\left(dB_t - \mathbf{1}\otimes d\bar{B}_t\right)
\begin{aligned}\delta^{u}_{j}(T) \leq C \tilde{C}_{T}\Big[\delta^{u}_{j-1}(T)(k^{q}_{j}(T)+k^{q}_{j-1}(T)) + \delta^{\nabla y}_{j-1}(T)(k^{q}_{j}(T)+k^{q}_{j-1}(T)) \Big].\end{aligned}
\mathbf{H}= \left[ \mathbf{h}_{1,1}\ldots \mathbf{h}_{1,N_t}\ldots \ldots \mathbf{h}_{k,1}\ldots \mathbf{h}_{k,j}\ldots\mathbf{h}_{k,N_t}\ldots \mathbf{h}_{N,N_t}\right].