harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
R=d\omega+\frac{1}{2}\left[ \omega,\omega\right] \propto J_{ab}, T=de+\left[ \omega,e\right] \propto P_{a}, \left[ e,e\right] =\frac{1}{\ell^{2}}e^{a}e^{b}\left[ P_{a},P_{b}\right] \propto Z_{ab}.
{\rm Tr\,}(P_\lambda)={\rm Tr\,}(\sum_{i=1}^k \lambda_{u_i}e_B\lambda_{u_i}^{-1})=\sum_{i=1}^k{\rm Tr\,}(\lambda_{u_i}e_B\lambda_{u_i}^{-1})=\sum_{i=1}^k{\rm Tr\,}(e_B)=\sum_{i=1}^k 1 =k\, .
\bold{J}_{\Gamma\backslash C, v_1, v_2}:= \bold{M}_{\Gamma\backslash C, v_1, v_2}\cap \bold{J}_{\Gamma\backslash C, v_1+v_2}= \{x \in \bold{M}_{\Gamma\backslash C, v_1, v_2}\mid{\partial W}/{\partial a}(x)=0, \forall a\in C\}.
S^{j+1}(m) = S\left( 2^{v-2j}3^j w + 1\right) = \frac{2^{v-2j}3^{j+1}w + 4}{4}& = 2^{v-2(j+1)}3^{j+1}w +1 = \left(\frac{3}{4}\right)^{j+1}(2^v w)+1 & = \left(\frac{3}{4}\right)^{j+1}(m-1)+1.
\sum_{n=1}^{\infty}\, f\left( n\right) =-\frac{1}{2}f\left( 0\right) +\int_{0}^{\infty}\, dx\, f\left( x\right) +i\int_{0}^{\infty}\, dx\, \frac{f\left( ix\right) -f\left( -ix\right)}{e^{2\pi x}-1}.
\bar{\Delta}_\alpha{\rm exp}\{(i/\hbar) W_m \}= 0\quad\Longrightarrow\quad\int d^2 \theta \, \theta^2 \frac{\partial^2 \delta W_m}{\partial \theta^2 \delta \Phi^A(\theta)}\, \Phi^B(\theta) (\sigma_\alpha)_B^{~~\!A}+ V_\alpha W_m = 0,
J\eta =-X-W+(\|X\|^2+\|W\|^2)B &=\left(\frac{\|W\|^2}{1+\|X\|^2}(B+X)-W\right)+\frac{1+\|X\|^2+\|W\|^2}{1+\|X\|^2}(\|X\|^2B-X),
\mathcal{X}_h = \left\{X \in{\rm St}(m,n) \mid X ={\rm BlkDiag}(P_1,\ldots,P_{n_A}) \cdot E(h) \cdot{\rm BlkDiag}(Q_1,\ldots,Q_{n_B}) \right. &\quad\left. \mbox{for some}\, P_i \in \mathcal{O}^{s_i-s_{i-1}}\, (i=1,\ldots,n_A), \, Q_j \in \mathcal{O}^{t_j-t_{j-1}}\, (j=1,\ldots,n_B) \right\}.
\tilde{f}= \sum_{\{(p, \ell, \alpha_j): 2p+|\ell|-\alpha_j \leq 2\}}\sum_{i=1}^{N_j}a_{p,\ell,i}^jr^{2p-\alpha_j}x^{\ell}\phi^j_{i}+ \tilde{f}_{>2}
\sum_{\ell \in \L_{0}}\chi_{\ell}(y; x) = 1, \left| \frac{\partial^{k}\chi_{\ell}(y;x)}{\partial y(j_{1})\cdots \partial y(j_{k})}\right| \leq C e^{- \gamma|x - y(\ell)|}\prod_{1 \leq i \leq k}e^{- \gamma|x - y(j_{i})|},
\begin{aligned}\mathbb{E}^{u_1,u_2}[\tilde{H}_{1{v_1}}(t,x,y,z,z_1,z_2,u_1,u_2;q_1,k_1,k_{11},k_{21},p_1,Q_{11},Q_{21})(v_1-u_1(t))|\mathcal{F}_t^1]\geq 0, \mathbb{E}^{u_1,u_2}[\tilde{H}_{1{v_2}}(t,x,y,z,z_1,z_2,u_1,u_2;q_1,k_1,k_{11},k_{21},p_1,Q_{11},Q_{21})(v_2-u_2(t))|\mathcal{F}_t^2]\leq 0,\end{aligned}
\Bigl( y(Z,t)\le s \Bigr) =\Bigl( \alpha_{m}(Z^{t_{m}})\le \frac{s-y(t_{m})}{t-t_{m}}\Bigr) =\Bigl( \alpha_{m}(Z^{t_{m}})\le \frac{s-y(t_{m})}{t-t_{m}}\Bigr)\cap (y(Z,t_{m})\le s)
2\lambda mT_{m,0}-2\lambda^{2m+1}mT_{-m,0}+2\lambda^mT_{1,0}-2\lambda^{m+2}T_{-1,0}&=(m+1)T_{m+1,0}+(m-1)T_{m-1,0}\lambda^2+(-m+1)\lambda^{2m}T_{-m+1,0}+(-m-1)T_{-m-1,0}\lambda^{2m+2}.
\begin{cases}\overline u_t-\overline u_{rr}-\frac{N-1}{r}\overline u_r\geq f(\overline u) \mbox{for}r\in [\tilde c_{k-1}t, ct], \overline u(\tilde c_{k-1}t,t)\geq u(\tilde c_{k-1}t, t) \mbox{for}t\geq T, \overline u(ct,t)\geq u(c t, t) \mbox{for}t\geq T, \overline u(r, T)\geq u(r, T) \mbox{for}r\in [\tilde c_{k-1}T, cT].\end{cases}
\frac{\mathrm{d}Z(t)}{Z(t)}= \sum_{i}{\it \Pi}_i(t,\mathfrak{X}) \frac{\mathrm{d}X_i(t)}{X_i(t)}={\it \Pi}'(t,\mathfrak{X})\sigma(t,\mathfrak{X}) \left[\vartheta(t,\mathfrak{X})\, \mathrm{d}t+ \mathrm{d}W(t) \right] \,,\mathrm{by}\ \eqref{eq:SDE}\,.
\nabla_{0}\nabla_{0}X_{\mu}= q \cos{( \mathrm{I m}\Phi / 2 )}\in^{i j}t^{j}n_{\mu}^{i}\ , \ \, n a b l a_{0}\nabla_{1}X_{\mu}= q \sin{( \mathrm{I m}\Phi / 2 )}t^{i}n_{\mu}^{i}\ ,
\overline{\Pi}_{0,2}^{\sharp}-\overline{\Pi}_{0,2}^{\sharp}\circ B^{\sharp}\circ\Pi_{0,2}^{\sharp}=\Pi_{0,2}^{\sharp}, \overline{\Pi}_{0,2}^{\sharp}\circ B^{\sharp}\circ\Pi_{2,0}^{\sharp}=-\bar{p}_{V}\circ\Pi_{2,0}^{\sharp}.
W ( \Phi ) = \sum_{i = 0}^{N}M_{i}W ( \imath e_{i}) + \frac{1}{2}\alpha \sum_{l = 0}^{N}s_{N - l}\sum_{m = 0}^{l}\mathrm{T r}( \Psi \Phi_{0}^{m}\Psi \Phi_{0}^{2 l - m}) +{\cal O}( \Psi^{3}) .
[K_5, L_3 ] = K_5^2+\{K_3, K_5\}+d_3 K_3 +e_{31}, [L_3, K_3 ] = K_3^2+\{K_3, K_5\}+d_3 K_4 +e_{32}, d_{3}=\frac12\big(Q-2K_1+Q^{(3)}+Q^{(4)}\big), e_{31}=-\frac{1}{4}\big(Q^{(3)}-Q^{(4)}\big)(Q+2K_1), e_{32}=\frac{1}{4}\big(Q-Q^{(4)}\big)\big(2K_1+Q^{(3)}\big).
l(T^*(n)) \leq n-hc(T^*(n)) begin{align*}5pt] \leq n- \left\lceil \frac{out(T^*)+\sum_{v\in V_{in}}ex(v)}{2}\right\rceil begin{align*}5pt] = n- \left \lfloor \frac{n-2}{2r-1}\right\rfloor-1 begin{align*}5pt] = \left \lceil \frac{(2r-2)n+2}{2r-1}\right\rceil-1= \left \lfloor \frac{(2r-2)n+2}{2r-1}\right\rfloor.
|N_0(t)-N(B(t),C(t))| =\left|\int_{\mathbb{R}}(N(B(t)+y,C(t))-N(B(t),C(t))p_{t,0}(y)dy\right| &\leq\int_{\mathbb{R}}|N(B(t)+y,C(t))-N(B(t),C(t)|p_{t,0}(y)dy.
\tfrac{1}{3}\Phi_{abc}^{(ijk)}\{x_{i},x_{j},x_{k}\}= \Phi_{abc}^{ijk}\{x_{i},x_{j},x_{k}\}, \tfrac{1}{15}\overline{\Phi}_{abc}^{(ij(klm))}\{x_{i},x_{j},x_{k},x_l,x_m\}= \Phi_{abc}^{ijklm}\{x_{i},x_{j},x_{k},x_l,x_m\},
\bar{q}_{\beta}(n,t)=\sum_{k=1}^{n}\sum_{\Lambda_{n}^{k}}k!\prod_{j=1}^{n-k+1}\frac{\left(\rho^{j}\binom{r+j-1}{j}\right)^{x_{j}}}{x_{j}!}\left(\frac{\lambda t^{\beta}(1-\rho)^{r}}{1-(1-\rho)^{r}}\right)^{k}E_{\beta,k\beta+1}^{k+1}(-\lambda t^{\beta}),
-\varphi_t -(-\Delta)^s\varphi + \alpha_{n, s}(n-2s)\mu^{-(n-2s)-1}_j\frac{|y_j|^2}{\left(1+|y_j|^2\right)^{\frac{n-2s}{2}+2}}\dot{\xi}_j\cdot y_j = 0 \mathbb{R}^n\times (t_0, +\infty).
\det\left(I_{N}+H\right)^{-1}\exp\left(Z^{*}\left(I_{N}+H\right)^{-1}Z\right) =\sum_{K}\prod_{1\le m,n\le N}\frac{\left(h_{m,n}\right)^{k_{m,n}}}{k_{m,n}!}\psi_{r_1,c_1}\left(\overline{w_1},w_1\right)\dotsm\psi_{r_N,c_N}\left(\overline{w_N},w_N\right),
f_{\epsilon,t}(x-1,x) = 1+ \frac{\epsilon}{M(ut)}\sum_{y\in (-\epsilon t,x)}G^\omega_{(-\epsilon t,x)}(x-1,y)\mu(y) f_{\epsilon,t}(y,x) &\le 1+ \frac{\epsilon}{M(ut)}\sum_{y\in (-\epsilon t,x)}G^\omega_{(-\epsilon t,x)}(x-1,y)\mu(y) (1+\eta(\epsilon))^{x-y}.
g (g \varphi^{j}_{s}+ \lambda \psi^{j}_{s}) + \lambda (g \varphi^{j}_{t}+ \lambda \psi^{j}_{t}) = - \eta^{t}\Lambda \begin{pmatrix}\varphi^{j}\psi^{j}\end{pmatrix}+ g^{2}F^{j}+ g \lambda G^{j}+ \lambda^{2}H^{j},
F_{r_1,\ldots,r_k}^{(m)}=\int_0^1F\bigl(\frac{\log r_1}{\log R},\ldots,\frac{\log r_{m-1}}{\log R},t_m,\frac{\log r_{m+1}}{\log R},\ldots,\frac{\log r_k}{\log R}\bigr)dt_m.
\mathcal{L}_\Omega: 0 \to \underset{|\alpha| = 0}{\oplus}\mathbb{C}_{V_\Omega^{(\alpha)}\times K}\to\underset{|\alpha| = 1}{\oplus}\mathbb{C}_{V_\Omega^{(\alpha)}\times K}\to\cdots \to\underset{|\alpha| = n}{\oplus}\mathbb{C}_{V_\Omega^{(\alpha)}\times K}\to 0,
\frac{d}{dt}\langle F(x)\rangle_t\sim\frac{1}{Z_t}\int_{D_t}dx\,\frac{\partial}{\partial x}e^{S(x)}F'(x)=\left.\frac{1}{Z_t}e^{S(x)}F'(x)\right|^{x_+(t)}_{x_-(t)}.
\left( 1-{r_0\over r}\right){r\over r_0}{d\over dr}\left( r^2{df_\ell\over dr}\right)+ \left( 2\ell + (2\ell+1+2ikr_0\cosh\gamma){r_0\over r}\right){r^2\over r_0}\;{df_\ell\over dr}+ \ell (-\ell + 2ikr_0\cosh\gamma) \; f_\ell = 0
\begin{aligned}\sum\limits_{i=B_{1}+1}^{B_{2}}\sum\limits_{j=1}^{K}d_{i,j}=B_2-B_1, &\sum\limits_{i=B_{1}+1}^{B_{2}}d_{i,j}=1, ~\forall j\in\left[B_{1}+1:B_{2}\right].\end{aligned}
\langle R^N(\bar\nabla_{e_j}\tau(\phi),R^N(V,d\phi(e_i))d\phi(e_i))d\phi(e_j),V\rangle-\langle R^N(R^N(V,d\phi(e_j))\bar\nabla_{e_j}\tau(\phi),d\phi(e_k))d\phi(e_k),V\rangle &=2\langle R^N(d\phi(e_j),V)\bar\nabla_{e_j}\tau(\phi),R^N(V,d\phi(e_k))d\phi(e_k)\rangle.
Z(\eta) = \int{\cal D}a{\cal D}A_\alpha^a\ \exp \left[i \int -{1\over2}(i \partial_\mu a + \eta_\mu)^2 +{g^2 a \over 32 \pi^2 f_0}\;F_{\mu\nu}^a \, \tilde F^{a \mu\nu}+{\cal L}_{YM}\right] \ .
f_{D,\lambda}=\arg\min{}_{f\in\mathcal{H}}\lambda\|f\|_{\mathcal{H}}^{2}+\frac{1}{n}{\displaystyle \sum_{i=1}^{n}}\left[\frac{(1-\Delta_{i})2(C_{i}-f(Z_{i}))}{\hat{g}(C_{i}|Z_{i})}+(f(Z_{i}))^{2}\right]
\begin{cases}-\Delta\eta ' =(k^2) ' \eta +k^2 \eta ' \,\,\,\,\Omega begin{align*}0.5cm] \frac{\partial \eta'}{\partial n}=\left(-\frac{\partial^2 \eta}{\partial n^2}\right) V\cdot n+\nabla \eta\cdot\nabla_\Gamma(V\cdot n)\,\,\,\,\partial\Omega begin{align*}0.5cm] \int_{\partial\Omega}\eta^2(V\cdot n)d\sigma+2\int_\Omega \eta\eta'dx=0.\end{cases}
\mathcal{F}(g)(\xi) = \int_{0}^{\infty}\phi(r, \xi) g(r) dr = \int_{0}^{\infty}\frac{\sqrt{r}\tilde{\phi}_{\sqrt{\xi}}(r)}{f(\sqrt{\xi})}g(r) dr = \frac{1}{f(\sqrt{\xi})}\mathcal{F}_{H}(\frac{g(\cdot)}{\sqrt{\cdot}})(\sqrt{\xi})
\begin{array}{rl}Tr W(w,z_2)W(z_1,w)W(w,z_1)W(z_2,w) = Tr RW(z_1,w)W(w,z_2)R^{-1}RW(z_2,w)W(w,z_1)R^{-1}= Tr W(z_1,z_2)W(z_2,z_1) = Tr W(z_1,z_1) \end{array}
F_{2 s , 2 t}^{\Theta}= \sigma_{1}( x_{1}, \ldots , x_{n}) \sigma_{1}( x_{1}^{- 1}, \ldots , x_{n}^{- 1}) F_{2 s , 2 t}^{{\cal O}_{2 , 2}^{1 , 1}}\, \, .
\frac{d\Phi}{d\epsilon}|_{\epsilon=0}=\int_{a}^{b}[\eta(x)\frac{\partial L}{\partial f}(x,f(x),f^{\prime}(x))+\eta^{\prime}(x)\frac{\partial L}{\partial f^{\prime}}(x,f(x),f^{\prime}(x))]dx=0.
{\cal Z}=\int{\cal D}A^\mu{\cal D}B^{\mu\nu}{\cal D}\theta{\cal D}Q^i{\cal D}\Phi^i{\cal D}\Phi^{ij}{\cal D}Q^0\delta(Q^0)\delta(\Gamma_\beta)det\mid\{\widetilde{\varphi}_\alpha, \Gamma_\beta \}\mid e^{iS_T},
2^{n_1}(1+o(1))\left\lfloor\frac{C_{t-k}}{\binom{n_1-w+k}{k}}\right\rfloor \ge \frac{2^n}{n_2^{2(t-k)}\binom{n_1/2+\sqrt{n_1\ln n_1}+k}{k}}(1+o(1)) &=\frac{2^{n+k}k!}{n_1^kn_2^{2(t-k)}}(1+o(1))
x_0.\left(e^{\frac{\pi}{Y_0}}-1\right)+x_0.\left(e^{\frac{2.\pi}{Y_0}}-e^{\frac{\pi}{Y_0}}\right)+x_0.\left(e^{\frac{3.\pi}{Y_0}}-e^{2.\frac{\pi}{Y_0}}\right)+ &\ldots +x_0.\left(e^{\frac{j.\pi}{Y_0}}-e^{\frac{(j-1).\pi}{Y_0}}\right)+\ldots
\Box \tilde \phi_1= \left(|\tilde \phi_{1, t}|^2- |\tilde \phi_{1, x}|^2\right) \tilde \phi_{1}+ \mathbf{1}_{\omega}f_{0}^{\tilde \phi^{\perp}_1}+ e_0, \; \tilde \phi_1[0]= (a, b)
\sum_{k=0}^N I_n(x+k)= \sum_{k=0}^N \sum_{m=0}^k \binom{k}{m}m! S(n,m;x) = \sum_{m=0}^N m! S(n,m;x) \sum_{k=m}^N \binom{k}{m}=\sum_{m=0}^{n\wedge N}\binom{N+1}{m+1}m! S(n,m;x),
H_n^{(JW)}=\frac{1}{\sqrt{\mathcal{C}}}\sum_{j=1}^{n-1}\sum_{a,b=1}^2{\alpha}_{a,b,j}\sigma_{j}^{(a)}\sigma_{j +1}^{(b)}+\frac{1}{\sqrt{\mathcal{C}}}\sum_{j=1}^{n}{\alpha}_{3,0,j}\sigma_j^{(3)}
S(x) \log x = \sum_{n \leq x}g(n) \log x &= \sum_{n \leq x}g(n) \log \left( \frac{x}{n}\right) + \sum_{n \leq x}g(n) \sum_{p||n}\log p + \sum_{n \leq x}g(n) \sum_{\alpha \geq 2 \atop p^{\alpha}||n}\log p^{\alpha}&= S_1 + S_2 + S_3.
\tilde{Q}_{Y_p}\cap [K,N_{Y_p}/2] =\{N_{Y_p}-x \mid x\in \tilde{S}_{Y_p}\cap[N_{Y_p}/2, N_{Y_p}-K]\}= \{N_{Y_p}- x \mid x \in S(r_-,r_+) \cap [N_{Y_p}/2,N_{Y_p}-K], x \neq 2\xi r_+\}.
\cdot \{\langle m ; A ; \pm | \frac{\delta : \hat{\mathrm{H}}_{\pm}:}{\delta A_{1}( y , t )}| \overline{{{m}}}; A ; \pm \rangle \langle \overline{{{m}}}; A ; \pm | \frac{\delta : \hat{\mathrm{H}}_{\pm}:}{\delta A_{1}( x , t )}| m ; A ; \pm \rangle - ( x \longleftrightarrow y ) \}. \,
\int_{\Omega}(1-\sigma)D^2 v_{j,\mu}:D^2\varphi+\sigma\Delta v_{j,\mu}\Delta\varphi dx=\lambda_j(\mu)\left(v_{j,\mu},\varphi\right)_{\partial\Omega}\,,\ \ \ \forall\varphi\in \mathcal{H}^2_{0,D}(\Omega),
\int_{\Omega_t}(\partial_1\mathcal B_1+\partial_2\mathcal B_2-\partial_t\mathcal B_0-\mathcal B_3){\mathbf V}_{s}\cdot{\mathbf V}_{s}d{\bf x}ds\le C_1\Vert{\mathbf V}_{s}\Vert_{L^2(\Omega_t)}^2=C_1\int_0^t I_0(s)ds\,,
\prod_{i=1}^N\prod_{j=1}^mt^{-j\cdot \kappa^{(i)}_{m-j+1}}=\prod_{i=1}^N\prod_{j=1}^{m}t^{-(m-j+1)\cdot \kappa_{j}^{(i)}}=\prod_{i=1}^Nt^{-m\sum_{j=1}^m\kappa_{j}^{(i)}}\prod_{j=1}^{m}t^{(j-1)\cdot \kappa_{j}^{(i)}}= &=t^{-m(n_1+\cdots n_N)}\prod_{i=1}^N\prod_{j=1}^{m}t^{(j-1)\cdot \kappa_{j}^{(i)}}
c_{1}(m)\Lambda \geq \theta(x,\overline{\sigma}) \geq \sum_{i=0}^{M}\theta(x, 2^{-i}\overline{\sigma}) - \theta(x,2^{-(i+1)}\overline{\sigma}) \geq M \frac{2c_{1}(m) \Lambda}{- \log_{2}(2\tau)}\geq \frac{3}{2}c_{1}(m) \Lambda ,
\Big| \frac{1}{2m \ell n}\sum_{x}\Phi_{n}(s,\tfrac{x}{n}) \sum_{j=1}^{m-1}\sum_{z = x-\ell}^{x-1}\int_{\Omega - \Omega_2^j(x)}[ \eta(z) - \eta(z - j \ell) ] \overrightarrow{\eta}^{\varepsilon n}(x+1) [f(\eta) - f(\eta^{z,z- j \ell}) ] d\nu_{b}\Big|
\mathcal{L}_i(\widetilde{g})=\Psi_{0,i}\cdot\nabla \widetilde{g_i}+\Phi_{1,i}(\widetilde{g}),\mathcal{N}_{p,i}(g^*,\cdots,g^*)=\Psi_{p-1,i}(g^*,\cdots,g^*)\cdot\nabla g_i+\Phi_{p,i}(g^*,\cdot,g^*),
\frac{d\rho_{2}(t)}{dt}=\frac{A}{V}\rho_{2}(t)\sqrt{2(\frac{P_{2}(0)}{\rho_{2}(0)}-\frac{P_{1}(0)}{\rho_{1}(0)}\frac{\rho_{1}(0)+\rho_{2}(0)-\rho_{2}(t)}{\rho_{2}(t)})}
\tilde{B}_{2}= \prod_{i=0}^{r_{1}}\tilde{B}_{1}1^{e_{1}+i}= \Big{(}\prod_{i=0}^{r_{1}-1}\tilde{B}_{1}1^{e_{1}+i}\Big{)}\tilde{B}_{1}1^{e_{1}+r_{1}}= \Big{(}\prod_{i=0}^{r_{1}-1}B_{1}1^{c_{1}+i}\Big{)}B_{1}1^{e_{1}+r_{1}}
\varrho_\varepsilon \rightharpoonup \varrho,\varrho_\varepsilon\mathbf u_\varepsilon\rightharpoonup\overline{\varrho\mathbf u}L^1(\Omega\times (0,T)), \mathbf{u}_\varepsilon \rightharpoonup \mathbf{u}\text{weaklyin~}L^2(0,T; W^{1,2}_0(\Omega)),
\mathcal{D}(\mathbf{x}_k,{\mathbf{u}}_k)_{\{i,j\}}={\partial h_i(\boldsymbol{f}_O^{d_i-1}\left(\boldsymbol{f}(\mathbf{x}_k,{\mathbf{u}}_k)\right))}\big/{\partial{u}_{j,k}}.
( \kappa ) ={\frac{1}{\sqrt{1 + \vert \nu_{B}\vert^{2}}}}\, \left( \begin{array}{c c}{{\kappa_{B}}}&{{\kappa_{B}\, \nu_{B}}}\\{{\nu_{B}\, \kappa_{B}}}&{{\nu_{B}^{c}\, \kappa_{B}\, \nu_{B}}}\\ \end{array}\right) \,{\frac{1}{\sqrt{1 + \vert \nu_{B}\vert^{2}}}}\; ,
\sqrt{\frac{1}{\left( L\frac{1}{t}\right)^{\frac{\varepsilon}{2}}}}\sqrt{\frac{L2}{2}}\sum_{j=0}^{\infty}2^{-j/2}\sqrt{\frac{1}{2L2}+\frac{1+j}{L_{2}2^{5}}}\leq \frac{3.59}{\sqrt{\left( L\frac{1}{t}\right)^{\frac{\varepsilon}{2}}}}.
a_n^{q_n(\vec x, \vec y)}={a_{n-1}}^{-q_{n-1}(\vec x, \vec y)}\cdots a_1^{-q_1(\vec x, \vec y)}\cdot a_1^{x_1}\cdots a_n^{x_n}\cdot a_1^{y_1}\cdots a_n^{y_n}.
\rho\; F[\phi ]\; =\; N^2\;\det \{X\}^{-1/2}\;\exp\{-\textstyle{\frac{1}{4}}\phi c^{\ast}X^{-1}c\phi\}\;F[\frac{\delta}{\delta (\phi c^{\ast})}+\frac{\delta}{\delta (c\phi )}]\;\exp\{+\textstyle{\frac{1}{4}}\phi c^{\ast}X^{-1}c\phi\}\;\;,
f(w; \lambda) = C \cdot d^w \cdot \dfrac{\prod_{i=0}^{r-1}\varGamma\left(w+\frac{i}{r}\right)}{\prod_{i=1}^{r}\varGamma\left(w+v_i\right)}\mapsto f(w; \lambda') = C' \cdot d^w \cdot \dfrac{\prod_{i=0}^{r-1}\varGamma\left(w+\frac{i}{r}\right)}{\prod_{i=1}^{r}\varGamma\left(w+v_i'\right)},
\big|d|\psi|^2\big|^2= 4\sum_{j=1}^2|\operatorname{Re}f_j|^2|\psi|^4, |\nabla\psi|^2= \sum_{j=1}^2|f_j|^2|\psi|^2, |D\psi|^2= \big((\operatorname{Re}f_1+\operatorname{Im}f_2)^2+(\operatorname{Re}f_2+\operatorname{Im}f_1)^2\big)|\psi|^2.
A\Omega_A^a=L\eta_\mu R\Gamma(\mathrm{Spf}(A)_C,\mathbb A_{\inf})^a\to \varprojlim (L\eta_\xi\varphi_\ast)^{\circ r}R\Gamma(\mathrm{Spf}(A)_C,\mathbb A_{\inf})^a
{\rm Hilb}\big(3T_6^{(0)}(K_{2,2,2})^{ab},t\big)=(1,12,58,137,154,64)=11t^4(1+t)+25t^3(1+t)^2 \hphantom{{\rm Hilb}\big(3T_6^{(0)}(K_{2,2,2})^{ab},t\big)=}{}+20 t^2(1+t)^3+7 t (1+t)^4+(1+t)^5,
\sum_{\mathbf{d},\mathbf{e}\in\mathcal{D}_k}|\lambda_{\mathbf{d}}\lambda_{\mathbf{e}}|E^{(1)}_q \ll (\log{x})^{2k}\sum_{q<R^2W, (q,B)=1}\mu^2(q)\tau_{3k}(q)E^{(1)}_q &\ll (\log{x})^{2k}\Bigl(\sum_{q<R^2W, (q,B)=1}\mu^2(q)\tau_{3k}(q)^2E_q^{(1)}\Bigr)^{1/2}\Bigl(\sum_{q<R^2W, (q,B)=1}\mu^2(q)E_q^{(1)}\Bigr)^{1/2}.
\langle\psi\rangle_{\mathcal{Y}^{t,p}_s(F;E)}:=\sup_{0<|h|<\frac{1}{2}\,d(F,E)}\left(\int_{F}\mathrm{Snail}\left(\frac{\psi(\cdot+h)-\psi}{|h|^t};x,E\right)^{p}\,dx\right)^\frac{1}{p},
\int_{\mathbb{R}^{d}}\frac{\left\vert \rho_{\mathbb{C}}(\tau,z-x)\right\vert^{2}}{\rho_{s}(x)^{2}}\rho_{s}(x)\,dx=\left( \frac{\pi s}{\sqrt{\alpha}}\right)^{d}\exp\left( \frac{t/2}{2\alpha}|\xi|^{2}+\frac{s-t/2}{2\alpha}|\eta|^{2}+\frac{u}{2\alpha}\xi\cdot\eta\right)
\langle\mathcal{M}^N_{g,2}, \hat{\Lambda}^N_{F, G, H}\rangle_t= \int_0^t\mu_{s,1}^N\bigotimes\mu_{s,3}^N\left(\lambda(\cdot, \ast)g(\cdot)\left(e^{-F_s(\cdot)+G_s(\cdot)}-1\right)\right)ds &-\int_0^t\mu_{s,2}^N\left(\psi(\cdot)g(\cdot)\left(e^{-G_s(\cdot)+H_s(\cdot)}-1\right)\right)ds
\mathrm{ch}(a_0\otimes a_1\otimes\cdots\otimes a_p) := \Gamma F[F,a_0][F,a_1]\cdots[F,a_p], \Omega(a_0\otimes a_1\otimes\cdots\otimes a_p) := \Gamma a_0\partial(a_1)\partial(a_2)\cdots \partial(a_p)
||(\widetilde{m_w})^*|| \ge \mathbb{E}||\sum_{g\in G}w(g)\lambda_g r_g||_{(C^*_r(G))^*}& \ge C \left(\sum_{g\in G}||w(g)\lambda_g ||^2_{(C^*_r(G))^*}\right)^{\frac{1}{2}}= C\cdot ||w||_{\ell^2(G)}
E ( \vec{n}) = \frac{e^{2}L}{4}\left[ \sum_{\alpha = 1}^{N - 1}\left( N n_{\alpha}+ m \right)^{2}- \frac{1}{N}\left( \sum_{\alpha = 1}^{N - 1}\left( N n_{\alpha}+ m \right) \right)^{2}\right]
\vert \nu_{A}\rangle ={\frac{1}{\sqrt{1 + \vert \nu_{A}\vert^{2}}}}\left( \begin{array}{c}{{- \nu_{A}}}\\{{1}}\\ \end{array}\right) \quad , \quad \langle \nu_{A}\vert ={\frac{1}{\sqrt{1 + \vert \nu_{A}\vert^{2}}}}\Bigl ( - \nu_{A}^{*}\; \quad \; 1 \Bigr ) \; .
=\sum_{j=1}^{d_1}\Big(\int_{S_1}\langle (1-\Pi_{V_2})f_2,(1-\Pi_{V_2})(r(x_1,\cdot))\rangle_{L^2(\mathbb{P}^{X_2})}\phi_j(x_1)d\mathbb{P}^{X_1}(x_1) \Big)^2 &\leq\int_{S_1}\langle (1-\Pi_{V_2})f_2,(1-\Pi_{V_2})(r(x_1,\cdot))\rangle_{L^2(\mathbb{P}^{X_2})}^2d\mathbb{P}^{X_1}(x_1) &\leq \Vert h_1\Vert^2(\psi_\Pi(V_2)\phi(V_2))^2,
\mbox{\boldmath $ z$}^*:=(z^*,z^*_{\Gamma}) \in \partial \varphi (\mbox{\boldmath $ z$}) \hbox{is in \mbox{\boldmath $ H$}if and~only~if}&(z^*,z^*_{\Gamma}) = \bigl( -\Delta z+\beta (z), \partial_\nu z-\Delta_{\Gamma}z_\Gamma + \beta_{\Gamma}(z_\Gamma ) \bigr).
\lambda\!=\!\min\left\{1, \min_{\substack{x,y: \gamma_1(x,y)>0}}\frac{\gamma(x,y)}{\gamma_1(x,y)}, \min_{\substack{x,y: \mu_1(x,y)>0}}\frac{\mu(x,y)}{\mu_1(x,y)}\right\}.
\big[u_{ij}^{\pm},u_{ik}^{\pm}+ u_{jk}^{\pm}\big]+\big[u_{ik}^{\pm},u_{jk}^{\pm}\big] +\bigg[u_{ik}^{\pm}, \sum_{j < i}\delta_{x_i,x_j}\bigg] =0 \ \i < j < k.
\int_0^t \left\langle f(\tau, \cdot), \partial_{\tau}\varphi(\tau,\cdot)\right\rangle \,\mathrm{d}\tau = \lim_{h\to 0}\int_0^t \left\langle f(\tau, \cdot) + f(\tau+h,\cdot), \frac{\varphi(\tau+h,\cdot)-\varphi(\tau,\cdot)}{2h}\right\rangle\,\mathrm{d}\tau,
G = \langle g_1,...,g_{13}\mid [g_{10},g_6]=g_{11}, [g_{10},g_7] = g_{12}, &\;\;\; [g_2,g_1]=[g_4,g_3]=[g_6,g_5]=[g_8,g_7]=[g_{10},g_9]=g_{13}, p\rangle.
\left\|\frac{1}{\sqrt{p}}\nabla_p s_{f}\right\|_{p-Sl}^2 = \frac{1}{p}\inf_{g\in L^2[0,1]:\nabla_p s_{f}=\nabla_p s_{g}}\|g\|^2_{L^2[0,1]}&=\frac{1}{2p(3p-1)}(2s_{f}(p)+\delta(1-p))^2+\frac{1}{2p}\|(f(t)-f(t-p))_{t\in [p,1]}\|^2_{L^2[p,1]},
S^*_{n}(qp^d) = \sum_{\substack{k=1, (k,p)=1}}^{p^dq}k^{n}= \sum_{j=0}^{q-1}\sum_{\substack{k=1, (k,p)=1}}^{p^d}(p^d j+k)^{n}\equiv q \sum_{\substack{k=1, (k,p)=1}}^{p^d}k^{n}\equiv qS^*_{n}(p^d)\pmod{p^d}.
t \cdot (f(x_0,x_1),s_0,s_1,s_\infty) = \left(t^{2g}f(x_0/t,x_1/t), t^{-1}s_0, t^{-1}s_1, t^{-1}s_\infty \right) = \left(t^{-2}f(x_0,x_1), t^{-1}s_0, t^{-1}s_1, t^{-1}s_\infty \right).
G(m,k) =-\frac{(1+q^{m+2k-1})(q;q^2)_{m}(q^{2k+1};q^2)_{m-1}^2 q^{-{m\choose 2}-(2m-1)k}}{(1-q)(q;q)_{m-1}^3 (-q;q)_{m-1}}\equiv 0\pmod{[m]\Phi_m(q)^2},
K=\frac{\overline{P}-\overline{P}_{e}}{1-\overline{P}_{e}}where\overline{P}=\frac{1}{N}\sum_{u=1}^{N}\sum_{c}\frac{n_{cu}(n_{cu}-1)}{m(m-1)}=\sum_{c}\frac{o_{cc}}{mN},and\overline{P}_{e}=\sum_{c}\frac{n_{c}^{2}}{(mN)^{2}}
(\varphi_2^{-1})^\ast(A_1 - A_2)\ =\ \left(\frac{1}{z}\right)^\ast\!\! A\, - A \ =\i\,\left(\frac{|z|}{z}\right)^N d\left(\frac{z}{|z|}\right)^N \, =\i\,(\varphi_2^{-1})^\ast (g_{12}\, dg_{21})\ ,
\delta_{H}= 2 \times 1 0^{- 5}\alpha_{0}^{- 1}\, ( \Omega_{\mathrm{N R}})^{c_{1}+ c_{2}\ln \Omega_{\mathrm{N R}}}\exp{[ c_{3}( n - 1 ) + c_{4}( n - 1 )^{2}]}
\hspace{1 . 4 i n}\times e^{- \overline{{{\Lambda}}}_{1}}( ( - 1 )^{n - 1}q z )^{- \partial_{\overline{{{\Lambda_{1}}}}}+ \frac{n - i - 1}{n}}q^{- i}( - 1 )^{i n + \frac{1}{2}i ( i + 1 )}
{\cal L}= \frac{1}{2}\partial_{\mu}\varphi \partial^{\mu}\varphi - \frac{1}{2}m^{2}\varphi^{2}- \frac{\lambda}{4 !}\varphi^{4}+ \frac{1}{2}\delta Z \partial_{\mu}\varphi \partial^{\mu}\varphi - \frac{1}{2}\delta m^{2}\varphi^{2}- \frac{1}{4 !}\delta \lambda \varphi^{4}\,
\sum_{f=1}^{F}\sum_{k=1}^{K}\mathbb{E}\{\mid\tilde{\boldsymbol{H}}(f,k)\mid^2\}=\sum_{f=1}^{F}\mathbb{E}\{\sum_{k=1}^{K}\mid\tilde{\boldsymbol{H}}(f,k)\mid^2\}&=\sum_{f=1}^{F}\mathbb{E}\{a_{f:M}\},
\frac{d}{dx}I_{1}^{n,k}(x) =-\left(1-\frac{k}{n}\right)p^{n,k}(x)f_{n,1}(x;x)+nI_{1}^{n,k}(x) &=n(I_{1}^{n,k}(x)-I_{0}^{n,k}(x)).
ds^2 ={1\over \sqrt{H_{p+r}H_{p+s}}}ds^2_{p+1}- \sqrt{{H_{p+r}\over H_{p+s}}}ds^2_s - \sqrt{{H_{p+s}\over H_{p+r}}}ds^2_r - \sqrt{H_{p+r}H_{p+s}}ds^2_{9-p-r-s}\ .
< \bar{\psi}\psi ( x ) ~ \bar{\psi}\psi ( 0 ) >_{T \gg \mu}\ = \ 2 T^{2}\ \left[ \sum_{n = 0}^{\infty}\exp \{- \pi T ( 2 n + 1 ) x \}\right]^{2}= \frac{T^{2}}{2 \sinh^{2}( \pi T x )}
\sum_{k=1}^{n}\frac{q^k}{1-q^k}-\sum_{k=1}^{n-1}\frac{xq^k}{1-xq^k}=\frac{x}{1-x}-\frac{1}{(x)_n}\sum_{k=1}^{n}\left[\begin{matrix}n k\end{matrix}\right]\frac{(q/x)_k(x)_{n-k}x^k}{1-q^k}.
F_{n}(a,q;x)=(-1)^{n}q^{-n(n-1)/2}\sum_{j=0}^{n}\frac{(-1)^{j}q^{j(j-1)/2}}{(q;q)_{j}^{\,2}}\!\left(\,\sum_{k=0}^{n-j}(q^{k+1};q)_{j}(q^{n-j-k+1};q)_{j}\, a^{k}\!\right)\! x^{j}.
\frac{\Gamma(\vec a;K-k-\sigma)\Gamma(k+1+2\sigma)^2\Gamma(k+1)^2}{\Gamma(\vec a;K-k-1-\sigma)\Gamma(k+2+2\sigma)^2\Gamma(k+2)^2}&\quad=\frac{P(\vec{a};K-k-1-\sigma)}{(k+1+2\sigma)^2(k+1)^2}&=\frac{(-1)^{p}\prod_{i=1}^{p}(k+1+\sigma-K-a_i)}{(k+1+2\sigma)^2(k+1)^2}\ ,
H/\hbar=g_{4}(e^{i(\omega_{b}-\omega_{a})t}ab^{\dagger}+e^{i(\omega_{a}-\omega_{b})t}a^{\dagger}b)(c^{\dagger}+\xi_{1}^{*}e^{i\omega_{1}t}+\xi_{2}^{*}e^{i\omega_{2}t})(c+\xi_{1}e^{-i\omega_{1}t}+\xi_{2}e^{-i\omega_{2}t})
P= P(x,l,m,n) =: \frac{\Gamma\left(\frac 12(x+l+m+n+1)\right)\Gamma\left(\frac 12(x+l-m-n+1)\right)\Gamma\left(\frac 12(x-l+m-n+1)\right)\Gamma\left(\frac 12(x-l-m+n+1)\right)}{\Gamma\left(\frac 12(x-l-m-n+1)\right)\Gamma\left(\frac 12(x-l+m+n+1)\right)\Gamma\left(\frac 12(x+l-m+n+1)\right)\Gamma\left(\frac 12(x+l+m-n+1)\right)}.
f(x + y) = f((x_+ + y_+) - (x_- + y_-)) = g(x_+ + y_+) - g(x_- + y_-) = g(x_+) + g(y_+) - g(x_-) - g(y_-) = (g(x_+) - g(x_-)) + (g(y_+) - g(y_-)) = f(x) + f(y).
\begin{matrix}z_{1,i}&=&l_{1,1}F_{1,i}&+&l_{1,2}F_{2,i}&+&\epsilon_{1,i}\\ \vdots&&\vdots&&\vdots&&\vdots\\z_{10,i}&=&l_{10,1}F_{1,i}&+&l_{10,2}F_{2,i}&+&\epsilon_{10,i}\end{matrix}
\Omega_1\mid_{A_0}=\frac{dy_3\wedge dy_2}{f_{y_0}},~\Omega_1\mid_{A_2}=\frac{dy_0\wedge dy_3}{f_{y_2}},~\Omega_1\mid_{A_3}=\frac{dy_2\wedge dy_0}{f_{y_3}},
\int_{[0,1]^{n/m}}\sqrt{\mathbb{P}_{\beta_k|T^{(i)}=t}^{(i)}\big(X^{(i)}_{[(\ell_{k-1}+1):\ell_k]}\notin B_k(t)\big)}dt &\leq \int_{\mathcal{B}}\sqrt{\mathbb{P}_{\beta_k|T^{(i)}=t}^{(i)}\big(X^{(i)}_{[(\ell_{k-1}+1):\ell_k]}\notin B_k(t)\big)}dt+\mathbb{P}_T(T^{(i)}\notin \mathcal{B}) &\leq \sqrt{2}(md)^{-2}+ 2(md)^{-4}\leq 2 (md)^{-2},
S = d + i \theta \cdot \sigma^{\mu}\cdot \bar{\theta}\, \partial_{\mu}d + \frac{1}{4}\theta^{2}\bar{\theta}^{2}\partial^{2}d + \sqrt{2}\theta \cdot \delta - \frac{i}{\sqrt{2}}\theta^{2}\partial_{\mu}\delta \cdot \sigma^{\mu}\cdot \bar{\theta}+ \theta^{2}Z
2\int\nabla\xi:u\nabla^2 G_h*u\,dx =2\int \nabla \xi \colon G_{h/2}\ast u\,\nabla^2 G_{h/2}\ast u\,dx+ 2\int \left[ G_{h/2}\ast,\nabla \xi \right] u \, \nabla^2 G_{h/2}\ast u\,dx,
\frac{d^2\bar{x}}{d\bar{t}^2}=\frac{1}{h}\left[\frac{d}{dx}\left(\frac{1}{h}\frac{d\varphi}{dx}\right)v^2+\left(\frac{1}{h}\frac{d\varphi}{dx}\right)\frac{d^2x}{dt^2}\right].
\begin{cases}\frac{\partial \varphi_{\epsilon}}{\partial t}=\log \frac{\omega_{\varphi_{\epsilon}}^n}{\omega_0^n}+F_0+\gamma (k\chi+\varphi_{\epsilon})+\log(\prod_{i=1}^d (\epsilon^2+|s_i|_{H_i}^2))^{(1-\beta)\tau_i}+\theta_X(\omega_{\varphi_{\epsilon}}) \varphi_{\epsilon}|_{t=0}=c_{\epsilon 0}.\end{cases}
\left.\begin{array}{ll}\mathcal{S}(u,v) \displaystyle=\dfrac{N-4}{2N}\left(\dfrac{a_2}{m_1}\int_{\mathbb{R}^N}|\nabla u|^2dx+\dfrac{a_1}{2m_2}\int_{\mathbb{R}^N}|\nabla v|^2dx\right) begin{align*}0.5cm] ~~\displaystyle+a_2\omega_1\int_{\mathbb{R}^N}|u|^2dx+\dfrac{a_1}{2}\omega_2\int_{\mathbb{R}^N}|v|^2dx.\end{array}\right.
K_{m+1}(t) \le C_1\biggl( \lVert a_0\rVert_{L^\infty_H L^p_z}+ K_m(t)H_m(t)+ K_m(t)^2+ R t^{1/2}K_m(t)\biggr), H_{m+1}(t) \le C_1\biggl( \lVert a_0\rVert_{L^\infty_H L^p_z}+ K_m(t)H_m(t)+ R t^{1/2}H_m(t)+ R t^{1/2}K_m(t)\biggr).
2N+O(\operatorname{nz}(2N+1)) < \frac{1}{2}\left( \operatorname{nz}_0(N)^2+\operatorname{nz}_1(N)^2\right)+2\operatorname{nz}_0(N)\operatorname{nz}_1(N) =\frac{1}{2}\operatorname{nz}(2N+1)^2-\operatorname{nz}_0(N)^2+\operatorname{nz}(2N+1)\cdot \operatorname{nz}_0(N).
S[\Phi] \equiv \int \!d^4x d^8\theta\; \Phi{\bar\Phi}+ 24 \int \!d^4x d^4\theta \;\Phi (m^2-\hspace{0.2ex}\raisebox{.5ex}{\fbox{}}\hspace{0.3ex}) \Phi+ 24 \int \!d^4x d^4{\bar\theta}\;{\bar\Phi}(m^2 - \hspace{0.2ex}\raisebox{.5ex}{\fbox{}}\hspace{0.3ex}){\bar \Phi}.