DeepSeek 著。

第一份测试

$$ \begin{aligned} &\mathcal{L}_{GQCD}(\psi, \bar{\psi}, A_\mu^a, G_{\mu\nu}^a, \theta, \xi, J_\mu^a, \eta, \bar{\eta}) = \\ &\quad -\frac{1}{4} G_{\mu\nu}^a G_a^{\mu\nu} + \bar{\psi}_i \left( i \gamma^\mu D_\mu - m_i \right) \psi_i + \frac{\theta g^2}{32\pi^2} G_{\mu\nu}^a \tilde{G}_a^{\mu\nu} \\ &\quad - \frac{1}{2\xi} (\partial^\mu A_\mu^a)^2 + \bar{c}^a \partial^\mu D_\mu^{ab} c^b + J_\mu^a A_a^\mu + \bar{\eta}_i \psi_i + \bar{\psi}_i \eta_i \\ &\quad + \frac{g}{2} f^{abc} (\partial_\mu A_\nu^a - \partial_\nu A_\mu^a) A_b^\mu A_c^\nu + \frac{g^2}{4} f^{abc} f^{ade} A_\mu^b A_\nu^c A_d^\mu A_e^\nu \\ &\quad + g \bar{\psi}_i \gamma^\mu T_{ij}^a \psi_j A_\mu^a - g f^{abc} \bar{c}^a c^b A_\mu^c + \frac{g}{2} f^{abc} (\partial^\mu \bar{c}^a) c^b A_\mu^c \\ &\quad + \frac{\kappa_1}{M_P} R_{\mu\nu\rho\sigma} G_a^{\mu\nu} G_a^{\rho\sigma} + \frac{\kappa_2}{M_P^2} \bar{\psi}_i \gamma^\mu \overset{\leftrightarrow}{D}_\mu \psi_i R \\ &\quad + \frac{\kappa_3}{M_P} \bar{\psi}_i \sigma^{\mu\nu} T_{ij}^a \psi_j F_{\mu\nu}^a + \frac{\kappa_4}{M_P^2} (\bar{\psi}_i \psi_i)^2 R \\ &\quad + \sum_{n=5}^\infty \frac{\lambda_n}{M_P^{n-4}} \mathcal{O}_n + \frac{\zeta_1}{\Lambda^2} (\bar{\psi}_i D_\mu \psi_i)(\bar{\psi}_j D^\mu \psi_j) \\ &\quad + \frac{\zeta_2}{\Lambda^2} G_{\mu\nu}^a D_\rho D^\rho G_a^{\mu\nu} + \frac{\zeta_3}{\Lambda^2} f^{abc} f^{ade} G_{\mu\nu}^b G_c^{\nu\rho} G_{\rho\mu}^{d,e} \\ &\quad + \frac{\zeta_4}{\Lambda^4} (\bar{\psi}_i \psi_i)^3 + \frac{\zeta_5}{\Lambda^4} (\bar{\psi}_i \gamma_\mu \psi_i)^4 + \frac{\zeta_6}{\Lambda^6} (\bar{\psi}_i \psi_i)^4 \\ &\quad + \epsilon^{\mu\nu\rho\sigma} \left[ \frac{\alpha_1}{\Lambda} \bar{\psi}_i \gamma_\mu D_\nu \psi_i G_{\rho\sigma}^a + \frac{\alpha_2}{\Lambda^2} G_{\mu\nu}^a D_\rho G_{\sigma\lambda}^a A^\lambda \right] \\ &\quad + \beta_1 \frac{\det(\bar{\psi}_i \psi_j)}{\Lambda^{2N_f-3}} + \beta_2 \frac{(\bar{\psi}_i \psi_i)^{N_f}}{\Lambda^{3N_f-4}} + \beta_3 \frac{G_{\mu\nu}^a G_a^{\mu\nu} \bar{\psi}\psi}{\Lambda^2} \\ &\quad + \gamma_1 e^{-\frac{8\pi^2}{g^2}} \cos\theta + \gamma_2 e^{-\frac{16\pi^2}{g^2}} \cos 2\theta + \gamma_3 e^{-\frac{8\pi^2}{g^2}} \frac{\bar{\psi}\psi}{\Lambda} \\ &\quad + \delta_1 \frac{\nabla_\mu G_a^{\mu\nu} \nabla_\rho G_a^{\rho\nu}}{\Lambda^4} + \delta_2 \frac{D_\mu \bar{\psi} D_\nu \psi G_a^{\mu\nu}}{\Lambda^3} + \delta_3 \frac{\bar{\psi} \sigma_{\mu\nu} \psi \bar{\psi} \sigma^{\mu\nu} \psi}{\Lambda^2} \\ &\quad + \epsilon_1 \frac{R_{\mu\nu\rho\sigma} R^{\mu\nu\rho\sigma} G_{\alpha\beta}^a G_a^{\alpha\beta}}{M_P^4} + \epsilon_2 \frac{R \bar{\psi} i \gamma^\mu D_\mu \psi}{M_P^2} \\ &\quad + \epsilon_3 \frac{R_{\mu\nu} \bar{\psi} \gamma^\mu D^\nu \psi}{M_P^2} + \epsilon_4 \frac{R_{\mu\nu\rho\sigma} \bar{\psi} \sigma^{\rho\sigma} D^\mu \psi}{M_P^2} \\ &\quad + \frac{\eta_1}{\Lambda^8} (\bar{\psi} \psi)^2 (\bar{\psi} \gamma_\mu \psi)^2 G_{\alpha\beta}^a G_a^{\alpha\beta} + \frac{\eta_2}{\Lambda^{10}} (\bar{\psi} \psi)^5 \\ &\quad + \frac{\eta_3}{\Lambda^6} f^{abc} f^{ade} f^{afg} G_{\mu\nu}^b G_c^{\nu\rho} G_{\rho\sigma}^d G_e^{\sigma\mu} \bar{\psi} \psi \\ &\quad + \frac{\eta_4}{\Lambda^8} \epsilon^{\mu\nu\rho\sigma} \bar{\psi} \gamma_\mu D_\nu \psi \bar{\psi} \gamma_\rho D_\sigma \psi G_{\alpha\beta}^a G_a^{\alpha\beta} \\ &\quad + \frac{\eta_5}{\Lambda^{12}} (\bar{\psi} \psi)^6 + \frac{\eta_6}{\Lambda^7} D_\mu G_a^{\mu\nu} D_\rho G_a^{\rho\sigma} D_\lambda G_\sigma^\lambda \\ &\quad + \zeta_7 \frac{\det(\bar{\psi}_i (1-\gamma_5) \psi_j)}{\Lambda^{2N_f-3}} + \zeta_8 \frac{\det(\bar{\psi}_i (1+\gamma_5) \psi_j)}{\Lambda^{2N_f-3}} \\ &\quad + \theta_1 \frac{\bar{\psi} \sigma_{\mu\nu} T^a \psi \bar{\psi} \sigma^{\mu\nu} T^a \psi}{\Lambda^2} + \theta_2 \frac{\bar{\psi} \gamma_\mu D_\nu \psi \bar{\psi} \gamma^\mu D^\nu \psi}{\Lambda^4} \\ &\quad + \theta_3 \frac{G_{\mu\nu}^a D_\rho G_a^{\rho\sigma} D_\lambda G_\sigma^\lambda}{\Lambda^6} + \theta_4 \frac{\bar{\psi} \psi \bar{\psi} \gamma_\mu \psi \bar{\psi} \gamma^\mu \psi}{\Lambda^4} \\ &\quad + \kappa_5 \frac{R_{\mu\nu} G_a^{\mu\rho} G_{\rho}^{a\nu}}{M_P^2} + \kappa_6 \frac{R \bar{\psi} \psi}{M_P} + \kappa_7 \frac{R_{\mu\nu} \bar{\psi} \sigma^{\mu\nu} \psi}{M_P} \\ &\quad + \lambda_1 \frac{(\bar{\psi} D_\mu \psi)(\bar{\psi} D^\mu \psi)}{\Lambda^4} + \lambda_2 \frac{(D_\mu \bar{\psi} \psi)(D^\mu \bar{\psi} \psi)}{\Lambda^4} \\ &\quad + \lambda_3 \frac{G_{\mu\nu}^a G_a^{\nu\rho} G_{\rho\sigma}^b G_b^{\sigma\mu}}{\Lambda^4} + \lambda_4 \frac{G_{\mu\nu}^a G_b^{\mu\nu} G_{\rho\sigma}^a G_b^{\rho\sigma}}{\Lambda^4} \\ &\quad + \mu_1 \frac{\epsilon^{\mu\nu\rho\sigma} G_{\mu\nu}^a G_{\rho\sigma}^b G_{\alpha\beta}^a G_b^{\alpha\beta}}{\Lambda^6} + \mu_2 \frac{\epsilon^{\mu\nu\rho\sigma} G_{\mu\alpha}^a G_{\nu\beta}^a G_{\rho\gamma}^b G_{\sigma\delta}^b g^{\alpha\beta} g^{\gamma\delta}}{\Lambda^6} \\ &\quad + \nu_1 \frac{\bar{\psi} \gamma_\mu D_\nu \psi \bar{\psi} \gamma^\mu D^\nu \psi \bar{\psi} \psi}{\Lambda^6} + \nu_2 \frac{\bar{\psi} \sigma_{\mu\nu} \psi \bar{\psi} \sigma^{\mu\nu} \psi \bar{\psi} \psi}{\Lambda^6} \\ &\quad + \xi_1 \frac{D_\mu G_a^{\mu\nu} D_\rho G_a^{\rho\sigma} D_\lambda G_\sigma^\lambda}{\Lambda^9} + \xi_2 \frac{D_\mu \bar{\psi} D_\nu \psi D_\rho \bar{\psi} D^\rho \psi}{\Lambda^8} \\ &\quad + \xi_3 \frac{G_{\mu\nu}^a G_a^{\mu\nu} \bar{\psi} \psi \bar{\psi} \psi}{\Lambda^4} + \xi_4 \frac{G_{\mu\nu}^a G_a^{\rho\sigma} \bar{\psi} \sigma_{\mu\nu} \psi \bar{\psi} \sigma_{\rho\sigma} \psi}{\Lambda^4} \\ &\quad + \omega_1 \frac{\det(\bar{\psi}_i \gamma_\mu D_\nu \psi_j)}{\Lambda^{4N_f-4}} + \omega_2 \frac{\det(\bar{\psi}_i \sigma_{\mu\nu} \psi_j)}{\Lambda^{4N_f-4}} \\ &\quad + \chi_1 \frac{\epsilon^{\mu\nu\rho\sigma} \bar{\psi} \gamma_\mu \psi \bar{\psi} \gamma_\nu \psi \bar{\psi} \gamma_\rho \psi \bar{\psi} \gamma_\sigma \psi}{\Lambda^{12}} \\ &\quad + \chi_2 \frac{\epsilon^{\mu\nu\rho\sigma} G_{\mu\nu}^a G_{\rho\sigma}^a \bar{\psi} \psi \bar{\psi} \psi}{\Lambda^6} + \chi_3 \frac{\epsilon^{\mu\nu\rho\sigma} G_{\mu\alpha}^a G_{\nu\beta}^a \bar{\psi} \gamma^\alpha \psi \bar{\psi} \gamma^\beta \psi}{\Lambda^6} \\ &\quad + \phi_1 \frac{R_{\mu\nu\rho\sigma} R^{\mu\nu\rho\sigma} \bar{\psi} \psi}{M_P^4} + \phi_2 \frac{R_{\mu\nu} R^{\mu\nu} \bar{\psi} \psi}{M_P^4} \\ &\quad + \phi_3 \frac{R^2 \bar{\psi} \psi}{M_P^4} + \phi_4 \frac{R_{\mu\nu\rho\sigma} \bar{\psi} \sigma^{\mu\nu} \sigma^{\rho\sigma} \psi}{M_P^2} \\ &\quad + \psi_1 \frac{(\bar{\psi} D_\mu D_\nu \psi)(\bar{\psi} D^\mu D^\nu \psi)}{\Lambda^6} + \psi_2 \frac{(D_\mu D_\nu \bar{\psi} \psi)(D^\mu D^\nu \bar{\psi} \psi)}{\Lambda^6} \\ &\quad + \psi_3 \frac{G_{\mu\nu}^a D_\rho D^\rho G_a^{\mu\nu} \bar{\psi} \psi}{\Lambda^4} + \psi_4 \frac{D_\mu G_a^{\mu\nu} D_\rho G_a^{\rho\sigma} \bar{\psi} \gamma_\nu \gamma_\sigma \psi}{\Lambda^5} \\ &\quad + \tau_1 \frac{\epsilon^{\mu\nu\rho\sigma} D_\mu G_{\nu\alpha}^a D_\rho G_{\sigma\beta}^a \bar{\psi} \gamma^\alpha \gamma^\beta \psi}{\Lambda^7} + \tau_2 \frac{\epsilon^{\mu\nu\rho\sigma} D_\mu G_{\nu\alpha}^a D_\rho G_{\sigma\beta}^a \bar{\psi} \sigma^{\alpha\beta} \psi}{\Lambda^7} \\ &\quad + \tau_3 \frac{\det(\bar{\psi}_i D_\mu \psi_j)}{\Lambda^{3N_f-3}} + \tau_4 \frac{\det(\bar{\psi}_i \gamma_\mu D_\nu \psi_j)}{\Lambda^{4N_f-4}} \\ &\quad + \rho_1 \frac{(\bar{\psi} \psi)^{N_f+1}}{\Lambda^{3N_f-1}} + \rho_2 \frac{(\bar{\psi} \gamma_\mu \psi)^{2N_f}}{\Lambda^{4N_f-4}} + \rho_3 \frac{(\bar{\psi} \sigma_{\mu\nu} \psi)^{N_f}}{\Lambda^{2N_f-2}} \\ &\quad + \sigma_1 \frac{G_{\mu\nu}^a G_a^{\mu\nu} G_{\rho\sigma}^b G_b^{\rho\sigma} G_{\alpha\beta}^c G_c^{\alpha\beta}}{\Lambda^8} \\ &\quad + \sigma_2 \frac{G_{\mu\nu}^a G_a^{\nu\rho} G_{\rho\sigma}^b G_b^{\sigma\alpha} G_{\alpha\beta}^c G_c^{\beta\mu}}{\Lambda^8} \\ &\quad + \sigma_3 \frac{\epsilon^{\mu\nu\rho\sigma} G_{\mu\alpha}^a G_{\nu\beta}^a G_{\rho\gamma}^b G_{\sigma\delta}^b G_{\epsilon\zeta}^c G_c^{\epsilon\zeta} g^{\alpha\beta} g^{\gamma\delta}}{\Lambda^{10}} \\ &\quad + \sigma_4 \frac{\epsilon^{\mu\nu\rho\sigma} G_{\mu\alpha}^a G_{\nu\beta}^a G_{\rho\gamma}^b G_{\sigma\delta}^b G_{\epsilon\zeta}^c G_c^{\eta\theta} g^{\alpha\beta} g^{\gamma\delta} g_{\eta\zeta} g^{\epsilon\theta}}{\Lambda^{10}} \\ &\quad + \mathcal{A} + \mathcal{B} + \mathcal{C} + \mathcal{D} + \mathcal{E} + \mathcal{F} + \mathcal{G} + \mathcal{H} + \mathcal{I} + \mathcal{J} \end{aligned} $$

其中：

$$ \begin{aligned} \mathcal{A} &= \int d^4x \, e^{-S_{\text{inst}}} \prod_{f=1}^{N_f} \det(D_\mu \gamma^\mu + m_f) \\ \mathcal{B} &= \sum_{k=1}^\infty \frac{1}{k!} \left( \frac{8\pi^2}{g^2} \right)^k e^{-k\frac{8\pi^2}{g^2}} \cos(k\theta) \prod_{i=1}^k \int d^4x_i \, \rho_i^4 \\ \mathcal{C} &= \text{Tr} \left[ \ln\left(1 - \frac{g A_\mu^a T^a \gamma^\mu}{i \partial_\mu \gamma^\mu - m} \right) \right] \\ \mathcal{D} &= \int \mathcal{D}A \mathcal{D}\bar{\psi} \mathcal{D}\psi \, e^{i \int d^4x \, \mathcal{L}_{\text{QCD}}} \prod_{f=1}^{N_f} \det(i D_\mu \gamma^\mu - m_f) \\ \mathcal{E} &= \sum_{n=1}^\infty \frac{(-1)^n}{n} \text{Tr} \left[ \left( \frac{1}{i \partial_\mu \gamma^\mu - m} g A_\mu^a T^a \gamma^\mu \right)^n \right] \\ \mathcal{F} &= \int dU \, e^{-S_{\text{eff}}[U]} \prod_{f=1}^{N_f} \det(M U + m_f) \\ \mathcal{G} &= \sum_{q=-\infty}^\infty e^{iq\theta} \int \mathcal{D}A_q \, e^{-S_{\text{YM}}[A_q] + i \frac{\theta}{32\pi^2} \int G\tilde{G}} \\ \mathcal{H} &= \text{P} \exp \left[ i g \int_0^1 ds \, A_\mu^a(x(s)) T^a \frac{dx^\mu}{ds} \right] \\ \mathcal{I} &= \int \mathcal{D}G \mathcal{D}q \mathcal{D}\bar{q} \, e^{i \int d^4x \, (\mathcal{L}_{\text{free}} + \mathcal{L}_{\text{int}})} \prod_{\text{colors}} \delta(G_a^\mu A_\mu^a - J^\mu) \\ \mathcal{J} &= \sum_{\text{top sectors}} e^{-S_{\text{top}}} \prod_{\text{zero modes}} \int d\xi \, d\bar{\xi} \, e^{-\bar{\xi} M \xi} \end{aligned} $$

第二份测试

$$ \begin{aligned} \mathscr{F}(\mathbf{X}, \mathbf{Y}, \mathbf{Z}) = &\min_{\mathbf{U} \in \mathbb{C}^{m \times n}} \max_{\mathbf{V} \in \mathbb{C}^{p \times q}} \left\{ \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \text{Tr} \left[ \mathbf{U}^\dagger \mathbf{V} \mathbf{X} \mathbf{Y}^\top \mathbf{Z} \right] e^{-\frac{1}{2}\|\mathbf{W}\|_F^2} d\mathbf{W} \right. \\ &+ \sum_{k=1}^{\infty} \frac{(-1)^k}{k!} \oint_{\gamma} \frac{\det(\mathbf{A} - z\mathbf{I})}{\prod_{j=1}^m (z - \lambda_j(\mathbf{B}))} dz \\ &+ \lim_{\epsilon \to 0^+} \frac{1}{\pi} \int_{\mathbb{R}^{n}} \frac{\Im\left\langle \psi \left| \mathbf{H} \right| \psi \right\rangle}{\omega - E_0 + i\epsilon} d\omega \\ &+ \sup_{\mathbf{P} \succeq 0} \inf_{\mathbf{Q} \succ 0} \left\{ \log \frac{\det(\mathbf{P} + \mathbf{Q})}{\det(\mathbf{P})\det(\mathbf{Q})} + \text{Tr} \left[ \mathbf{P}^{-1} \mathbf{Q} \mathbf{R} \mathbf{S}^\top \right] \right\} \\ &+ \prod_{i=1}^n \prod_{j=1}^m \left( \sum_{k=1}^p \mathbf{X}_{ijk} \mathbf{Y}_{kji} + \oint_{\partial \Omega} \mathbf{F}(\mathbf{r}) \cdot d\mathbf{r} \right) \\ &+ \bigcap_{\alpha \in \mathcal{A}} \bigcup_{\beta \in \mathcal{B}} \left\{ \mathbf{x} \in \mathbb{R}^n : \|\mathbf{A}\mathbf{x} - \mathbf{b}\|_2 \leq \epsilon \right\} \\ &+ \sum_{\substack{S \subseteq \{1,\dots,n\} \\ |S| = k}} \left( \prod_{i \in S} \lambda_i(\mathbf{C}) \right) \exp\left( -\frac{1}{2} \mathbf{x}_S^\top \mathbf{\Sigma}^{-1} \mathbf{x}_S \right) \\ &+ \int_{\mathcal{M}} \left\langle \nabla f, \nabla g \right\rangle dV + \oint_{\partial \mathcal{M}} f \frac{\partial g}{\partial n} dS \\ &+ \min_{\mathbf{\Theta} \in \text{SO}(n)} \max_{\mathbf{\Phi} \in \text{U}(m)} \left\| \mathbf{\Theta} \mathbf{X} \mathbf{\Phi}^\dagger - \mathbf{Y} \right\|_{\text{F}}^2 \\ &+ \sum_{d=0}^{\infty} \int_{\overline{\mathcal{M}}_{g,n}} \psi_1^{d_1} \cdots \psi_n^{d_n} \cap [\overline{\mathcal{M}}_{g,n}]^{\text{vir}} \\ &+ \sup_{f \in \mathcal{F}} \inf_{g \in \mathcal{G}} \left\{ \mathbb{E}_{(x,y) \sim \mathcal{D}} \left[ \ell(f(x), g(y)) \right] + \lambda \|f\|_{\mathcal{H}}^2 \right\} \\ &+ \prod_{v \in V} \sum_{c \in C} \prod_{(u,v) \in E} \mathbf{1}_{c(u) \neq c(v)} \exp\left( -\beta H(\mathbf{c}) \right) \\ &+ \lim_{N \to \infty} \frac{1}{N} \log \int e^{-\beta H_N(\sigma)} d\sigma \\ &+ \sum_{\pi \in S_n} \text{sgn}(\pi) \prod_{i=1}^n \mathbf{A}_{i,\pi(i)} \oint_{\gamma} \frac{f(z)}{(z-a)^{n+1}} dz \\ &+ \min_{\mathbf{W}_1, \mathbf{W}_2} \max_{\mathbf{D}} \left\{ \mathbb{E}[\log D(x)] + \mathbb{E}[\log(1-D(G(z)))] \right\} \\ &+ \int_{\text{Conf}_N(\mathbb{R}^2)} \prod_{i<j} |z_i - z_j|^\beta e^{-\frac{\beta}{4} \sum |z_i|^2} d^2 z_1 \cdots d^2 z_N \\ &+ \sum_{k=1}^K \int_{\Theta_k} p(x|\theta_k) \pi(\theta_k) d\theta_k \prod_{j\neq k} \mathbb{P}(\theta_j \in \Theta_j) \\ &+ \sup_{\mathbf{M} \in \mathcal{C}} \inf_{\mathbf{N} \in \mathcal{D}} \left\{ \text{Tr}[\mathbf{M}\mathbf{N}] + \log \det(\mathbf{I} + \mathbf{M} + \mathbf{N}) \right\} \\ &+ \prod_{i=1}^\infty \left( 1 + \frac{z}{i} \right) e^{-\frac{z}{i}} \Gamma(1+z) \zeta(1+z) \\ &+ \sum_{n=1}^\infty \frac{\chi(n)}{n^s} \prod_{p \mid n} \left(1 - \frac{1}{p^s}\right) L(s, \chi) \\ &+ \min_{\mathbf{U}} \max_{\mathbf{V}} \left\{ \left\| \mathbf{U} \mathbf{V}^\top - \mathbf{X} \right\|_* + \lambda \|\mathbf{U}\|_{2,1} + \mu \|\mathbf{V}\|_{1,2} \right\} \\ &+ \int_{\text{Gr}(k,n)} \sigma_\lambda \cdot \sigma_\mu \cup \sigma_\nu \cap [\text{Gr}(k,n)] \\ &+ \sum_{\mathbf{m} \in \mathbb{Z}^n} e^{-\mathbf{m}^\top \mathbf{A} \mathbf{m} + \mathbf{b}^\top \mathbf{m}} \prod_{i=1}^n \theta_3(z_i, q_i) \\ &+ \limsup_{n \to \infty} \frac{1}{n} \log \mathbb{P} \left( \frac{1}{n} \sum_{i=1}^n X_i \in A \right) \\ &+ \min_{\rho \in \mathcal{D}(\mathcal{H})} \max_{M \in \text{POVM}} I(X;Y) + \lambda S(\rho) \\ &+ \int_{\mathcal{A}/\mathcal{G}} e^{-S_{\text{YM}}[A]} \mathcal{D}A \prod_{x \in M} \delta(G(A)(x)) \\ &+ \sum_{\Gamma} \frac{1}{|\text{Aut}(\Gamma)|} \int_{\overline{\mathcal{M}}_{g,n}} \omega_\Gamma \cap [\overline{\mathcal{M}}_{g,n}] \\ &+ \sup_{f \in BL_1} \left\{ \int f d\mu - \int f d\nu \right\} + \lambda W_2^2(\mu, \nu) \\ &+ \prod_{p \in \text{Spec}(\mathbf{D})} \left( 1 - \frac{s}{p} \right)^{-1} \exp\left( \sum_{m=1}^\infty \frac{N_p^m}{m} p^{-ms} \right) \\ &+ \min_{\mathbf{T}} \max_{\mathbf{S}} \left\{ \text{Tr}[\mathbf{T}^\top \mathbf{S}] + \|\mathbf{T} - \mathbf{S}\|_{\text{HS}}^2 \right\} \\ &+ \int_{\text{Lag}(M)} \mathcal{F}(\mathcal{L}) \wedge \star \mathcal{F}(\mathcal{L}) \exp\left( -\int_M R \right) \\ &+ \sum_{\chi \in \widehat{G}} \dim V_\chi \oint_{\gamma} \frac{\chi(g)}{\det(1 - g)} dg \\ &+ \lim_{\Lambda \to \infty} \int_{|p|<\Lambda} \frac{d^4p}{(2\pi)^4} \frac{\text{Tr}[\gamma^\mu \gamma^\nu \gamma^\rho \gamma^\sigma]}{p^2 - m^2} \\ &+ \min_{\pi} \max_{s \in \mathcal{S}} \left\{ R(s, \pi(s)) + \gamma \sum_{s'} P(s'|s,\pi(s)) V(s') \right\} \\ &+ \int_{\mathcal{X}} \left( \sup_{f \in \mathcal{F}} \frac{1}{n} \sum_{i=1}^n f(X_i) - \mathbb{E}[f(X)] \right) d\mathbb{P} \\ &+ \sum_{n=1}^\infty \frac{\tau(n)}{n^s} \prod_{p} \left( 1 - \frac{\tau(p)}{p^s} + \frac{p^{11}}{p^{2s}} \right)^{-1} \\ &+ \min_{\mathbf{W}} \max_{\mathbf{H}} \left\{ \text{Tr}[\mathbf{W}^\top \mathbf{H} \mathbf{W}] + \lambda \|\mathbf{W}\|_{2,1} + \mu \|\mathbf{H}\|_* \right\} \\ &+ \oint_{\gamma} \frac{\prod_{i=1}^n (z - a_i)^{m_i}}{\prod_{j=1}^p (z - b_j)^{n_j}} dz \times \frac{1}{2\pi i} \int_{\Gamma} f(w) dw \\ &+ \sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)!} z^{2k+1} \prod_{j=1}^k \frac{1}{(2j-1)!!} \\ &+ \lim_{t \to \infty} \frac{1}{t} \log \mathbb{E} \left[ \exp\left( \int_0^t V(X_s) ds \right) \right] \\ &+ \min_{\mathbf{P}} \max_{\mathbf{Q}} \left\{ \text{Tr}[\mathbf{P} \mathbf{Q}] + \log \det(\mathbf{I} + \mathbf{P} + \mathbf{Q}) \right\} \\ &+ \int_{\text{Bun}_G(X)} \mathcal{F}(\mathcal{E}) \wedge \star \mathcal{F}(\mathcal{E}) \exp\left( -\int_X \text{ch}(\mathcal{E}) \right) \\ &+ \sum_{\rho \in \widehat{G}} \dim \rho \oint_{\gamma} \frac{\chi_\rho(g)}{\det(1 - \text{Ad}(g))} dg \\ &+ \lim_{\epsilon \to 0} \frac{1}{\epsilon} \left( \int_{\mathbb{R}^n} f(x) e^{-\frac{\|x\|^2}{2\epsilon}} dx - (2\pi\epsilon)^{n/2} f(0) \right) \\ &+ \min_{\theta \in \Theta} \max_{x \in \mathcal{X}} \left\{ \ell(\theta, x) + \lambda R(\theta) + \mu \Omega(x) \right\} \\ &+ \int_{\mathcal{M}} \text{Tr}[\mathcal{R} \wedge \mathcal{R}] \wedge \star \text{Tr}[\mathcal{F} \wedge \mathcal{F}] \\ &+ \sum_{n=1}^\infty \frac{a_n}{n^s} \prod_{p} \left( 1 - \frac{a_p}{p^s} + \frac{\psi(p)}{p^{2s}} \right)^{-1} L(s, \psi) \\ &+ \sup_{f \in \mathcal{F}} \inf_{g \in \mathcal{G}} \left\{ \mathbb{E}[\ell(f,g)] + \lambda \|f\|_{\mathcal{H}}^2 + \mu \|g\|_{\mathcal{H}}^2 \right\} \\ &+ \prod_{v \in V} \sum_{\sigma_v \in \Sigma} \prod_{(u,v) \in E} \exp\left( -\beta J_{uv} \sigma_u \sigma_v - h_v \sigma_v \right) \\ &+ \lim_{N \to \infty} \frac{1}{N} \log \int \exp\left( -\beta H_N(\sigma) + h \cdot \sigma \right) d\sigma \\ &+ \sum_{\pi \in S_n} \varepsilon(\pi) \prod_{i=1}^n \mathbf{A}_{i,\pi(i)} \oint_{\gamma} \frac{f(z)}{\prod_{j=1}^m (z - a_j)^{k_j}} dz \\ &+ \min_{\mathbf{W}} \max_{\mathbf{D}} \left\{ \mathbb{E}[\log D(x)] - \mathbb{E}[\log(1-D(G(z)))] \right\} \\ &+ \int_{\text{Sym}^N(M)} \prod_{i<j} d(z_i, z_j)^\beta e^{-\beta U(\mathbf{z})} d\text{vol}(\mathbf{z}) \\ &+ \sum_{k=1}^K \int_{\Theta_k} p(x|\theta_k) \pi(\theta_k) d\theta_k \prod_{j\neq k} F(\theta_j) d\theta_j \\ &+ \sup_{\mathbf{M}} \inf_{\mathbf{N}} \left\{ \text{Tr}[\mathbf{M}\mathbf{N}] + \log \det(\mathbf{I} + \mathbf{M}\mathbf{N}) \right\} \\ &+ \prod_{n=1}^\infty (1 - q^n)^{24} \eta(\tau)^{24} \Delta(\tau) \\ &+ \sum_{\chi} \frac{L(s, \chi)}{L(2s, \chi^2)} \prod_{p} \left( 1 - \frac{\chi(p)}{p^s} \right)^{-1} \\ &+ \min_{\mathbf{U}} \max_{\mathbf{V}} \left\{ \|\mathbf{U} \mathbf{V}^\top - \mathbf{X}\|_F^2 + \lambda \|\mathbf{U}\|_* + \mu \|\mathbf{V}\|_* \right\} \\ &+ \int_{\mathcal{M}_{g,n}} \psi_1^{a_1} \cdots \psi_n^{a_n} \kappa_1^{b_1} \cdots \kappa_m^{b_m} \cap [\mathcal{M}_{g,n}]^{\text{vir}} \\ &+ \sum_{\mathbf{m} \in \mathbb{Z}^n} e^{-\mathbf{m}^\top \mathbf{\Sigma} \mathbf{m} + \mathbf{\mu}^\top \mathbf{m}} \prod_{i=1}^n \theta(z_i | \tau_i) \\ &+ \liminf_{n \to \infty} \frac{1}{n} \log \mathbb{P} \left( \bigcap_{i=1}^n \{ X_i \in A_i \} \right) \\ &+ \min_{\rho} \max_{\Lambda} \left\{ S(\rho \| \sigma) + \lambda \text{Tr}[\rho \log \rho] + \mu \text{Tr}[\sigma \log \sigma] \right\} \\ &+ \int_{\mathcal{A}} e^{-S_{\text{CS}}[A]} \mathcal{D}A \prod_{x} \delta(F_A(x)) \\ &+ \sum_{\Gamma} \frac{1}{|\text{Aut}(\Gamma)|} \int_{\mathcal{M}_{g,n}} \omega_\Gamma \wedge \star \omega_\Gamma \\ &+ \sup_{f} \left\{ \int f d\mu - \int f d\nu \right\} + \lambda \text{TV}(\mu, \nu) + \mu \text{KL}(\mu \| \nu) \\ &+ \prod_{p} \left( 1 - \frac{\alpha_p}{p^s} \right)^{-1} \left( 1 - \frac{\beta_p}{p^s} \right)^{-1} L(s, f) L(s, g) \\ &+ \min_{\mathbf{T}} \max_{\mathbf{S}} \left\{ \text{Tr}[\mathbf{T}^\top \mathbf{S} \mathbf{T}] + \|\mathbf{T} - \mathbf{S}\|_F^2 + \lambda \|\mathbf{T}\|_* \right\} \\ &+ \int_{\text{Loc}(M)} \mathcal{F}(\nabla) \wedge \star \mathcal{F}(\nabla) \exp\left( -\int_M \text{scal} \right) \\ &+ \sum_{\chi} \dim \rho_\chi \oint_{\gamma} \frac{\chi(g)}{\det(1 - \rho(g))} dg \\ &+ \lim_{\Lambda \to \infty} \int \frac{d^d p}{(2\pi)^d} \frac{\text{Tr}[\Gamma^{\mu_1} \cdots \Gamma^{\mu_n}]}{(p^2 - m^2)^k} \\ &+ \min_{\pi} \max_{s} \left\{ Q(s, \pi(s)) + \gamma \mathbb{E}[V(s')] + \lambda H(\pi(\cdot|s)) \right\} \\ &+ \int_{\mathcal{X}} \left( \mathbb{E}[f] - \frac{1}{n} \sum f(X_i) \right)^2 d\mathbb{P} + \lambda \|f\|_{\mathcal{H}}^2 \\ &+ \sum_{n=1}^\infty \frac{\sigma_k(n)}{n^s} \prod_{p} \left( 1 - \frac{\sigma_k(p)}{p^s} + \frac{p^{2k}}{p^{2s}} \right)^{-1} \\ &+ \min_{\mathbf{W}} \max_{\mathbf{H}} \left\{ \text{Tr}[\mathbf{W}^\top \mathbf{H} \mathbf{W}] + \lambda \|\mathbf{W}\|_{2,1} + \mu \|\mathbf{H}\|_F^2 \right\} \\ &+ \oint_{\gamma} \frac{\prod (z - a_i)^{m_i}}{\prod (z - b_j)^{n_j}} \frac{f(z)}{g(z)} dz \times \frac{1}{2\pi i} \int_{\Gamma} h(w) dw \\ &+ \sum_{k=0}^\infty \frac{(-1)^k}{(2k)!} z^{2k} \prod_{j=1}^k \frac{1}{(2j)!!} B_{2k} \\ &+ \lim_{t \to 0} \frac{1}{t} \left( \mathbb{E}[f(X_t)] - f(X_0) \right) \\ &+ \min_{\mathbf{P}} \max_{\mathbf{Q}} \left\{ \text{Tr}[\mathbf{P} \mathbf{Q} \mathbf{P}^\top] + \log \det(\mathbf{I} + \mathbf{P} \mathbf{Q}) \right\} \\ &+ \int_{\text{Coh}(X)} \text{ch}(\mathcal{E}) \wedge \text{td}(X) \exp\left( -\int_X c_1(\mathcal{E}) \right) \\ &+ \sum_{\rho} \dim \rho \oint_{\gamma} \frac{\chi_\rho(g)}{\det(1 - \text{Ad}^*(g))} dg \\ &+ \lim_{\epsilon \to 0} \frac{1}{\epsilon^2} \left( \int f(x) e^{-\frac{\|x\|^2}{2\epsilon}} dx - (2\pi\epsilon)^{n/2} \sum_{|\alpha| \leq k} \frac{D^\alpha f(0)}{\alpha!} \right) \\ &+ \min_{\theta} \max_{x} \left\{ L(\theta, x) + \lambda \|\theta\|_1 + \mu \|x\|_\infty + \nu R(\theta, x) \right\} \\ &+ \int_{\mathcal{M}} \text{Tr}[\mathcal{R} \wedge \star \mathcal{R}] \wedge \text{Tr}[\mathcal{F} \wedge \star \mathcal{F}] \\ &+ \sum_{n=1}^\infty \frac{\lambda_f(n)}{n^s} \prod_{p} \left( 1 - \frac{\lambda_f(p)}{p^s} + \frac{\psi(p)}{p^{2s}} \right)^{-1} \\ &+ \sup_{f} \inf_{g} \left\{ \mathbb{E}[\ell(f,g)] + \lambda \|f\|^2 + \mu \|g\|^2 + \nu \text{Cov}(f,g) \right\} \\ &+ \prod_{v} \sum_{\sigma_v} \prod_{(u,v)} \exp\left( -\beta J_{uv} \sigma_u \sigma_v - h_u \sigma_u - h_v \sigma_v \right) \\ &+ \lim_{N \to \infty} \frac{1}{N} \log \int \exp\left( -\beta H_N(\sigma) + \mathbf{h} \cdot \sigma + \mathbf{J} \cdot \sigma \sigma^\top \right) d\sigma \\ &+ \sum_{\pi} \varepsilon(\pi) \prod_{i} \mathbf{A}_{i,\pi(i)} \oint_{\gamma} \frac{f(z)}{\prod (z - a_j)} \frac{g(z)}{\prod (z - b_k)} dz \\ &+ \min_{\mathbf{W}} \max_{\mathbf{D}} \left\{ \mathbb{E}[\log D] + \mathbb{E}[\log(1-D)] + \lambda \|\mathbf{W}\|^2 + \mu \|\mathbf{D}\|^2 \right\} \\ &+ \int_{\text{Conf}_N} \prod_{i<j} |z_i - z_j|^{2\beta} e^{-\beta \sum V(z_i)} d^2 z_1 \cdots d^2 z_N \\ &+ \sum_{k=1}^K \int_{\Theta_k} p(x|\theta_k) \pi(\theta_k) d\theta_k \prod_{j} \mathbb{P}(\theta_j \in A_j) F_j(\theta_j) \\ &+ \sup_{\mathbf{M}} \inf_{\mathbf{N}} \left\{ \text{Tr}[\mathbf{M}\mathbf{N}\mathbf{M}^\top] + \log \det(\mathbf{I} + \mathbf{M}\mathbf{N}) + \lambda \|\mathbf{M}\|_F^2 \right\} \\ &+ \prod_{n=1}^\infty (1 - q^n)^{24} (1 - q^{2n})^{24} \eta(\tau)^{48} \Delta(\tau)^2 \\ &+ \sum_{\chi} \frac{L(s, \chi) L(s, \chi^2)}{L(2s, \chi^4)} \prod_{p} \left( 1 - \frac{\chi(p)^2}{p^s} \right)^{-1} \\ &+ \min_{\mathbf{U}} \max_{\mathbf{V}} \left\{ \|\mathbf{U} \mathbf{V}^\top - \mathbf{X}\|^2 + \lambda \|\mathbf{U}\|_{2,1} + \mu \|\mathbf{V}\|_{1,2} + \nu \|\mathbf{U} \mathbf{V}^\top\|_* \right\} \\ &+ \int_{\overline{\mathcal{M}}_{g,n}} \psi_1^{d_1} \cdots \psi_n^{d_n} \kappa_1^{e_1} \cdots \kappa_m^{e_m} \lambda_1^{f_1} \cdots \lambda_k^{f_k} \cap [\overline{\mathcal{M}}_{g,n}] \\ &+ \sum_{\mathbf{m} \in \mathbb{Z}^n} e^{-\mathbf{m}^\top \mathbf{\Sigma}^{-1} \mathbf{m} + \mathbf{\mu}^\top \mathbf{m}} \prod_{i=1}^n \theta(z_i | \tau) \vartheta(z_i | \tau) \\ &+ \lim_{n \to \infty} \frac{1}{n} \log \mathbb{P} \left( \bigcap_{i=1}^n \bigcup_{j=1}^m \{ X_{ij} \in A_{ij} \} \right) \\ &+ \min_{\rho} \max_{\Lambda} \left\{ I(X;Y) + \lambda S(\rho) + \mu \text{Tr}[\rho \log \rho] + \nu \text{Tr}[\sigma \log \sigma] \right\} \\ &+ \int_{\mathcal{A}} e^{-S[A]} \mathcal{D}A \prod_{x} \delta(\mathcal{G}(A)(x)) \det(\mathcal{M}(A)) \\ &+ \sum_{\Gamma} \frac{1}{|\text{Aut}(\Gamma)|} \int_{\mathcal{M}_{g,n}} \omega_\Gamma \wedge \overline{\omega_\Gamma} \wedge \Phi_\Gamma \\ &+ \sup_{f} \left\{ \int f d\mu - \int f d\nu \right\} + \sum_{k=1}^\infty \lambda_k W_k^k(\mu, \nu) + \text{KL}(\mu \| \nu) \\ &+ \prod_{p} \left( 1 - \frac{\alpha_p}{p^s} \right)^{-1} \left( 1 - \frac{\beta_p}{p^s} \right)^{-1} \left( 1 - \frac{\gamma_p}{p^s} \right)^{-1} L(s,f) L(s,g) L(s,h) \\ &+ \min_{\mathbf{T}} \max_{\mathbf{S}} \left\{ \text{Tr}[\mathbf{T}^\top \mathbf{S} \mathbf{T} \mathbf{S}^\top] + \|\mathbf{T} - \mathbf{S}\|_F^2 + \lambda \|\mathbf{T}\|_* + \mu \|\mathbf{S}\|_* \right\} \end{aligned} $$

第三份测试

$$ \begin{aligned} \mathbf{J} &= \begin{bmatrix} \frac{\partial}{\partial t}\min_{\theta\in\Theta}\max_{\phi\in\Phi}\mathcal{L}(\theta,\phi) & \int_{-\infty}^{\infty}\frac{\sum_{k=1}^\infty e^{-k\lambda}\lambda^k}{k!}f_k(x)dx & \nabla_\mathbf{x}\left[\prod_{i=1}^n\sigma(w_i^\top\mathbf{x}+b_i)\right] \\ \lim_{N\to\infty}\frac{1}{N}\sum_{i=1}^N\nabla_\theta\log\pi_\theta(a_i|s_i) & \oint_\gamma\frac{\prod_{j=1}^m(z-\alpha_j)^{\beta_j}}{\sum_{k=0}^n c_kz^k}dz & \frac{d}{dt}\arg\min_{\mathbf{u}\in\mathcal{U}}\left\|\mathbf{A}\mathbf{u}-\mathbf{b}\right\|_2^2 \\ \mathbb{E}_{q_\phi(z|x)}\left[\log\frac{p_\theta(x|z)p(z)}{q_\phi(z|x)}\right] & \max_{\|\mathbf{v}\|=1}\min_{\|\mathbf{w}\|=1}\mathbf{v}^\top\mathbf{A}\mathbf{w} & \int_{\mathcal{M}}\text{Tr}(\mathbf{R}\wedge\mathbf{R})d\text{vol} \\ \frac{\partial^2}{\partial x_i\partial x_j}\sup_{f\in\mathcal{F}}\mathbb{E}[f(X)] & \sum_{T\in\mathcal{T}}\prod_{(u,v)\in E_T}w_{uv}\det\mathbf{L}_T & \nabla_\theta\left[\min_{\mathbf{D}}\max_{\mathbf{G}}V(\mathbf{G},\mathbf{D})\right] \\ \oint_{\partial\Omega}\mathbf{F}\cdot d\mathbf{S} = \iiint_\Omega\nabla\cdot\mathbf{F}dV & \lim_{h\to0}\frac{f(x+h)-f(x-h)}{2h} & \frac{\delta}{\delta\rho(x)}\mathcal{F}[\rho] \\ \min_{\pi}\max_{s\in\mathcal{S}}\left[R(s,\pi(s))+\gamma\sum_{s'}P(s'|s,\pi(s))V(s')\right] & \int_0^\infty e^{-st}\mathcal{L}\{f(t)\}dt & \det\left(\lambda\mathbf{I}-\mathbf{A}^\top\mathbf{\Sigma}^{-1}\mathbf{A}\right) \\ \sum_{k=1}^\infty\frac{(-1)^{k-1}}{k}\text{Tr}(\mathbf{A}^k) & \inf_{\mathbf{P}\succ0}\log\det(\mathbf{P}^{-1})\text{s.t.}\mathbf{A}^\top\mathbf{P}+\mathbf{P}\mathbf{A}\prec0 & \frac{\partial}{\partial\beta}\log\int e^{-\beta H(\mathbf{x})}d\mathbf{x} \\ \nabla_\mathbf{W}\left[\min_{\mathbf{b}}\max_{\|\mathbf{\xi}\|\leq\epsilon}\mathcal{L}(\mathbf{W},\mathbf{b},\mathbf{\xi})\right] & \oint_C\frac{f(z)}{(z-a)^{n+1}}dz = \frac{2\pi i}{n!}f^{(n)}(a) & \mathbb{E}_{p(x)}[\nabla_\theta\log p_\theta(x)] \\ \frac{d}{d\lambda}\det(\mathbf{A}+\lambda\mathbf{B})\big|_{\lambda=0} = \text{Tr}(\mathbf{A}^{-1}\mathbf{B})\det\mathbf{A} & \sup_{f\in\mathcal{F}}\left|\mathbb{E}_P[f]-\mathbb{E}_Q[f]\right| & \min_{\mathbf{U},\mathbf{V}}\|\mathbf{X}-\mathbf{U}\mathbf{V}^\top\|_F^2 + \lambda\|\mathbf{U}\|_1 \\ \int_{\mathbb{R}^n}e^{-\frac{1}{2}\mathbf{x}^\top\mathbf{\Sigma}^{-1}\mathbf{x}+\mathbf{j}^\top\mathbf{x}}d\mathbf{x} & \max_{\mathbf{X}\succeq0}\min_{\mathbf{Y}\succ0}\text{Tr}(\mathbf{X}\mathbf{Y}) & \frac{\partial\mathcal{L}}{\partial\mathbf{W}^{(l)}} = \delta^{(l)}(\mathbf{a}^{(l-1)})^\top \\ \sum_{\sigma\in S_n}\text{sgn}(\sigma)\prod_{i=1}^n a_{i,\sigma(i)} & \lim_{n\to\infty}\frac{1}{n}\log\mathbb{P}\left(\frac{1}{n}\sum X_i\in A\right) & \nabla_\theta\text{KL}(q_\phi(z|x)\|p_\theta(z|x)) \\ \oint_\gamma\frac{dz}{\sqrt{(z-a)(z-b)(z-c)}} & \min_{\mathbf{\Pi}\geq0}\langle\mathbf{C},\mathbf{\Pi}\rangle + \lambda H(\mathbf{\Pi}) & \frac{d}{dt}\mathbf{U}(t) = -i\mathbf{H}(t)\mathbf{U}(t) \\ \mathbb{E}_{z\sim p(z)}[\nabla_\theta f_\theta(g_\phi(z))] & \max_{\|\mathbf{x}\|=1}\mathbf{x}^\top\mathbf{A}\mathbf{x} = \lambda_{\text{max}}(\mathbf{A}) & \int_{\mathcal{M}}e^{-\mathcal{S}[\phi]}\mathcal{D}\phi \\ \frac{\delta^2\mathcal{F}}{\delta\rho(x)\delta\rho(y)} & \sum_{k=0}^\infty\frac{(-1)^k}{(2k+1)!}z^{2k+1} & \min_{\mathbf{P}}\|\mathbf{A}\mathbf{P}-\mathbf{B}\|_F^2 + \lambda\|\mathbf{P}\|_* \\ \oint\frac{P(z)}{Q(z)}dz = 2\pi i\sum\text{Res}(f,z_k) & \lim_{\epsilon\to0}\frac{1}{\epsilon}\left(\int f(x)e^{-\frac{\|x\|^2}{2\epsilon}}dx - (2\pi\epsilon)^{n/2}f(0)\right) & \nabla_\mathbf{x}\left[\max_{\|\mathbf{\delta}\|_\infty\leq\epsilon}\mathcal{L}(\mathbf{x}+\mathbf{\delta})\right] \\ \frac{\partial}{\partial t}u(t,x) = \Delta u(t,x) + f(u) & \inf_{\mathbf{M}}\sup_{\mathbf{N}}\text{Tr}(\mathbf{M}\mathbf{N}) & \int_0^1 B_n(x)f^{(n)}(x)dx \\ \sum_{d=0}^\infty\int_{\overline{\mathcal{M}}_{g,n}}\psi_1^{d_1}\cdots\psi_n^{d_n} & \min_{\theta}\max_{x\in\mathcal{X}}\ell(\theta,x) + \lambda R(\theta) & \frac{d}{d\epsilon}\det(\mathbf{A}+\epsilon\mathbf{B})\big|_{\epsilon=0} \\ \nabla_\theta\mathbb{E}_{q_\phi(z|x)}[f_\theta(z)] & \oint_\gamma\frac{e^z}{z^2+1}dz & \max_{\mathbf{v}\in\mathcal{V}}\min_{\mathbf{w}\in\mathcal{W}}\mathbf{v}^\top\mathbf{A}\mathbf{w} \\ \frac{\partial}{\partial\Sigma_{ij}}\log\det(\mathbf{\Sigma}) = (\mathbf{\Sigma}^{-1})_{ji} & \lim_{N\to\infty}\frac{1}{N}\log\int e^{-\beta H_N(\sigma)}d\sigma & \min_{\mathbf{U}}\|\mathbf{X}-\mathbf{U}\|_F^2 + \lambda\|\mathbf{U}\|_1 \\ \int_{\text{SO}(n)}f(R)d\mu(R) & \sup_{f\in BL_1}\left|\int fd\mu-\int fd\nu\right| & \nabla_\mathbf{x}\left[\prod_{i=1}^m(1+\exp(-y_i\mathbf{w}_i^\top\mathbf{x}))^{-1}\right] \\ \frac{d}{dt}\arg\max_{\mathbf{p}\in\Delta^n}\mathbf{p}^\top\mathbf{r} - H(\mathbf{p}) & \sum_{n=1}^\infty\frac{\mu(n)}{n^s} = \frac{1}{\zeta(s)} & \min_{\mathbf{P}\in\Pi(\mu,\nu)}\int c(x,y)d\mathbf{P}(x,y) \\ \oint\frac{\sin z}{z^3}dz = -\frac{\pi i}{2} & \lim_{t\to\infty}\frac{1}{t}\log\mathbb{E}[e^{\int_0^t V(X_s)ds}] & \nabla_\theta\left[\min_{\mathbf{D}}\mathbb{E}[\log D(x)] + \mathbb{E}[\log(1-D(G_\theta(z)))]\right] \\ \frac{\partial^2 f}{\partial z_i\partial\bar{z}_j} & \max_{\pi}\mathbb{E}[\sum_{t=0}^\infty\gamma^t r(s_t,a_t)] & \int_{\mathbb{C}^n}e^{-\frac{1}{2}\|z\|^2}dz \\ \sum_{k=0}^n\binom{n}{k}a^kb^{n-k} = (a+b)^n & \inf_{\mathbf{Q}\succ0}\log\det(\mathbf{Q}^{-1}) & \frac{d}{d\lambda}\sigma(\mathbf{A}+\lambda\mathbf{B}) \\ \oint_\gamma\frac{\log z}{z^2+1}dz & \min_{\mathbf{W}}\max_{\mathbf{D}}V(\mathbf{W},\mathbf{D}) + \lambda\|\mathbf{W}\|_2^2 & \nabla_\mathbf{x}\left[\sum_{i=1}^n x_i\log x_i\right] \\ \frac{\partial}{\partial\theta_j}\log\sum_{i=1}^N e^{\theta_i} & \lim_{\beta\to\infty}\frac{1}{\beta}\log\int e^{-\beta f(x)}dx & \min_{\mathbf{U},\mathbf{V}}\|\mathbf{X}-\mathbf{U}\mathbf{V}^\top\|_F^2 + \lambda(\|\mathbf{U}\|_F^2 + \|\mathbf{V}\|_F^2) \\ \int_{\mathbb{R}^n}\frac{e^{i\mathbf{k}\cdot\mathbf{x}}}{(2\pi)^n}d\mathbf{k} = \delta(\mathbf{x}) & \sup_{f\in\mathcal{F}}\mathbb{E}[f(X)] - \mathbb{E}[f(Y)] & \frac{d}{dt}\mathbf{P}(t) = \mathbf{A}\mathbf{P}(t) + \mathbf{P}(t)\mathbf{A}^\top \\ \sum_{n=1}^\infty\frac{\Lambda(n)}{n^s} = -\frac{\zeta'(s)}{\zeta(s)} & \min_{\mathbf{\Theta}}\|\mathbf{Y}-\mathbf{X}\mathbf{\Theta}\|_F^2 + \lambda\|\mathbf{\Theta}\|_{2,1} & \nabla_\theta\text{Var}_{p_\theta(x)}[f(x)] \\ \oint\frac{dz}{z^n+1} = \frac{2\pi i}{n} & \lim_{\epsilon\to0}\frac{f(x+\epsilon)-f(x)}{\epsilon} & \max_{\mathbf{v}\in\mathbb{S}^{n-1}}\mathbf{v}^\top\mathbf{\Sigma}\mathbf{v} \\ \frac{\partial}{\partial g_{ij}}\mathcal{R}(g) & \inf_{\mathbf{P}}\|\mathbf{X}-\mathbf{P}\|_\infty & \int_{\mathcal{M}}R_g d\text{vol}_g \\ \mathbb{E}_{p(x)}[\nabla_\phi\log q_\phi(z|x)] & \sum_{k=0}^\infty\frac{z^k}{k!} = e^z & \min_{\mathbf{C}}\|\mathbf{X}-\mathbf{C}\|_1 + \lambda\text{Tr}(\mathbf{C}) \\ \oint_\gamma\frac{e^{-z^2}}{z-a}dz & \max_{\mathbf{U}^\top\mathbf{U}=\mathbf{I}}\text{Tr}(\mathbf{U}^\top\mathbf{A}\mathbf{U}) & \nabla_\mathbf{x}\left[\log\sum_{i=1}^n e^{x_i}\right] \\ \frac{d}{d\alpha}I_\alpha(\mu,\nu) & \lim_{n\to\infty}\left(1+\frac{z}{n}\right)^n = e^z & \min_{\mathbf{B}}\|\mathbf{A}-\mathbf{B}\|_2 + \lambda\|\mathbf{B}\|_* \end{bmatrix} \end{aligned} $$

测试结束。

注意到 有些网络剪贴板 的 Markdown 无法渲染。我们的 $\KaTeX$ 还是很强大的。