已完成今日 A+B Problem 大学习

内容来自 Missile。

题目描述

给定两个整数

a, b

，求下列式子的值。

\begin{aligned}\displaystyle &\mathfrak{S}\left( \bigoplus_{\kappa=1}^{\aleph_1} \mathfrak{G}_{\kappa} \otimes \bigwedge_{\lambda=1}^{\beth_2} \mathfrak{H}_{\lambda} \right) \times \coprod_{\mu \in \mathfrak{M}} \left[ \varprojlim_{\nu \to \infty} \left( \bigsqcup_{\xi=1}^{\beth_\omega} \mathfrak{J}_{\xi,\mu} \right) \right] \oplus \left( \bigvee_{\pi \in \Pi} \mathfrak{K}_\pi \right) \otimes \left( \bigcap_{\rho=1}^{\mathfrak{d}} \mathfrak{L}_\rho \right) \\ &\times \exp\left[ \sum_{n=1}^{\infty} \dfrac{(-1)^{n(n+1)/2}}{n!} \oint_{\Gamma_n} \dfrac{ \prod_{k=1}^n \left( \zeta(s_k) \Gamma(s_k) \text{Li}_{s_k}(e^{2\pi i/k}) \right) }{ \sum_{m=1}^n \left( s_m^2 + \dfrac{1}{s_m^3} \right) \prod_{p \neq m} (s_p - s_m) } ds_1 \cdots ds_n \right] \\ &\times \text{Tr}\left( \bigoplus_{r=1}^{1000} \begin{bmatrix} \mathscr{D}^{(r)}_{11} & \mathscr{D}^{(r)}_{12} & \cdots & \mathscr{D}^{(r)}_{1r} \\ \mathscr{D}^{(r)}_{21} & \mathscr{D}^{(r)}_{22} & \cdots & \mathscr{D}^{(r)}_{2r} \\ \vdots & \vdots & \ddots & \vdots \\ \mathscr{D}^{(r)}_{r1} & \mathscr{D}^{(r)}_{r2} & \cdots & \mathscr{D}^{(r)}_{rr} \end{bmatrix}^{\otimes r} \right) \times \prod_{\substack{p \text{ prime} \\ p \equiv 1 \ (\text{mod } 4)}} \left( 1 + \dfrac{1}{p^2} + \dfrac{1}{p^4} + \cdots + \dfrac{1}{p^{2\lfloor \log_2 p \rfloor}} \right) \\ &+ \lim_{\substack{\epsilon \to 0^+ \\ \delta \to 0^- \\ \eta \to \infty}} \dfrac{ \partial^{100}}{\partial \alpha_1 \cdots \partial \alpha_{100}} \left[ \int_{\mathbb{R}^{100}} e^{i\langle \mathbf{k}, \mathbf{x} \rangle} \dfrac{ J_1(\|\mathbf{k}\|) Y_2(\|\mathbf{k}\|) K_3(\|\mathbf{k}\|) I_4(\|\mathbf{k}\|) }{ \|\mathbf{k}\|^{100} + \epsilon \|\mathbf{k}\|^{98} + \delta \|\mathbf{k}\|^{96} + \eta \|\mathbf{k}\|^{94} + 1 } d^{100}\mathbf{k} \right] \\ &\quad \times \sum_{n_1,n_2,\ldots,n_{50}=0}^\infty \dfrac{ \Gamma(n_1+n_2+\cdots+n_{50}+1) }{ n_1!n_2!\cdots n_{50}! \cdot \zeta(2n_1+3n_2+5n_3+\cdots+229n_{50}+1) } \\ &\times \left( \bigcap_{\ell=1}^{\infty} \left\{ \mathbf{z} \in \mathbb{C}^\ell : \left| \sum_{j=1}^\ell z_j^j \right| < e^{-\ell} \right\} \right) \cup \left( \bigcup_{d=1}^\infty \left\{ \mathbf{w} \in \mathbb{H}^d : \|\mathbf{w}\|^2 = \zeta(2d) \right\} \right) \\ &+ \dfrac{1}{(2\pi i)^{100}} \oint_{|z_1|=R_1} \cdots \oint_{|z_{100}|=R_{100}} \dfrac{ \prod_{j=1}^{100} \left( \sin(z_j) \cos(z_j^{-1}) \tan(e^{z_j}) \cot(\ln(1+z_j)) \right) }{ \prod_{1 \leq j < k \leq 100} (z_j - z_k) \cdot \prod_{m=1}^{100} (z_m^{1000} + z_m^{500} + 1) } dz_1 \cdots dz_{100} \\ &\times \dim_{\mathbb{Q}} \left( \bigoplus_{n=1}^\infty \mathbb{Q}(\sqrt[n]{2}) \otimes \mathbb{Q}(\zeta_n) \right) \times \text{rank} \left( \bigotimes_{p \text{ prime}} \mathbb{Z}_p \right) \\ &+ \int_{\mathscr{M}_{g,n}} \left( \prod_{i=1}^n \psi_i^{k_i} \right) \times \exp\left( \sum_{h=0}^\infty \sum_{m=0}^\infty \dfrac{\lambda_h \kappa_m}{h!m!} \right) \times \det\left( \dfrac{\partial^2 \mathscr{F}}{\partial t_i \partial t_j} \right)_{i,j=1}^{1000} \\ &+ \dfrac{ \mathscr{L}\left\{ t^{\alpha-1} E_{\beta,\gamma}(\omega t^\delta) \right\}(s) \times \mathscr{F}\left\{ e^{-t^2} H_n(t) \right\}(\omega) \times \mathscr{Z}\left\{ \dfrac{1}{1-q^s} \right\}(t) }{ \mathscr{H}\left\{ \dfrac{d^n}{dt^n} \left( t^\mu e^{-\lambda t} \right) \right\}(s) \times \mathscr{M}\left\{ f(t) t^{s-1} \right\}(z) } \\ &+ \sum_{\sigma \in S_{1000}} \text{sgn}(\sigma) \prod_{j=1}^{1000} \left( \int_0^1 \dfrac{x^{\sigma(j)}}{1+x^{\sigma(j)^2}} dx \right) \times \chi_{\text{irr}}(\sigma) \times \dim V_\sigma \\ &\quad + \iiint\limits_{\mathbb{R}^{3}} \nabla \times \left( \dfrac{\mathbf{r}}{r^3} \right) \cdot d\mathbf{S} \times \oint_{\partial \Omega} \mathbf{F} \cdot d\mathbf{r} \times \iint_{\Sigma} \mathbf{G} \cdot d\mathbf{A} \\ &\times \left[ \dfrac{\partial}{\partial t} \left( \dfrac{\partial^2 u}{\partial x^2} + \dfrac{\partial^2 u}{\partial y^2} + \dfrac{\partial^2 u}{\partial z^2} \right) + \nabla \cdot (u \nabla u) + \dfrac{1}{c^2} \dfrac{\partial^2 u}{\partial t^2} \right]_{u=e^{i(kx-\omega t)}} \\ &+ \text{Res}_{s=1} \zeta(s) \times \text{Res}_{z=0} \Gamma(z) \times \text{Res}_{w=\infty} \text{Li}_s(w) \times \prod_{\text{singularities}} \left( 1 - \dfrac{1}{z_\nu} \right) \\ &+ \dim_{\mathbb{C}} H^0(X, \mathscr{O}(D)) - \dim_{\mathbb{C}} H^1(X, \mathscr{O}(D)) + \dim_{\mathbb{C}} H^2(X, \mathscr{O}(D)) - \cdots \\ &+ \dfrac{1}{\# G} \sum_{g \in G} \chi(g) \times \dfrac{1}{\text{Vol}(M)} \int_M \text{Ric}(g) dV_g \times \prod_{i=1}^n \left( 1 - \dfrac{1}{p_i^{s_i}} \right)^{-1} \\ &+ \mathscr{P}\left( \bigcap_{n=1}^\infty \bigcup_{m=n}^\infty A_m \right) \times \mathbb{E}\left[ e^{itX} \right] \times \text{Cov}(X,Y) \times \text{Corr}(Z,W) \\ &+ \dfrac{d}{dt} \left\langle \psi \left| \hat{H} \right| \psi \right\rangle \times \text{Tr} \left( e^{-\beta \hat{H}} \right) \times \prod_{n=1}^\infty \left( 1 + e^{-\beta \epsilon_n} \right) \\ &+ \int \mathscr{D}[\phi] e^{-S[\phi]} \mathcal{O}[\phi] \times \det\left( \dfrac{\delta^2 S}{\delta \phi \delta \phi} \right)^{-1/2} \times \exp\left( -\dfrac{1}{2} \phi \cdot \Delta^{-1} \cdot \phi \right) \\ &+ \sum_{\text{diagrams}} \dfrac{(-i\lambda)^n}{n!} \int d^4x_1 \cdots d^4x_n \langle 0 | T \left( \phi(x_1) \cdots \phi(x_n) \right) | 0 \rangle + \text{Index}(D) \end{aligned}

输入格式

一行两个整数

a, b

。

输出格式

一行一个整数，表示上述式子的值。

样例输入

CPP

114514 1919810

样例输出

CPP

数据范围限制

对于

100\%

的数据，

a, b

满足

\begin{aligned} &\forall \delta \in \left(0, 10^{-200}\right),\ \exists \alpha, \beta \in \mathbb{Q} \cap (0,1),\ \exists \gamma, \sigma \in \mathbb{Z}_{\geq 3}, \\ &\text{使得}\ \forall k, l \in \mathbb{N},\ \forall m, n \in \mathbb{Z},\ \forall f \in C^\infty\left(\mathbb{R}^4\right), \\ &\begin{cases} \text{1. 对}\ a:\ \text{拓扑空间}\ (X_a, \tau_a)\ \text{是紧的}, \\\displaystyle \quad \text{其中}\ X_a = \left\{ (x_1,x_2,x_3,x_4) \in \mathbb{R}^4 \mid \sum_{i=1}^4 x_i^2 \leq \left( \dfrac{|a|}{10^9} + \delta \right)^2 \right\},\ \tau_a \text{为欧氏拓扑}; \\ \quad \text{对}\ b:\ \text{拓扑空间}\ (X_b, \tau_b)\ \text{是紧的}, \\\displaystyle \quad \text{其中}\ X_b = \left\{ (y_1,y_2,y_3,y_4) \in \mathbb{R}^4 \mid \sum_{i=1}^4 y_i^2 \leq \left( \dfrac{|b|}{10^9} + \delta \right)^2 \right\},\ \tau_b \text{为欧氏拓扑}; \\ \\ \text{2. 对}\ a:\displaystyle\ \text{群同态}\ \phi_a: \mathbb{Z}^\gamma \to S_{\lfloor |a| + 10^9 \rfloor}\ \text{存在且非平凡}; \\ \quad \text{对}\ b:\ \text{群同态}\ \phi_b: \mathbb{Z}^\sigma \to S_{\lfloor |b| + 10^9 \rfloor}\ \text{存在且非平凡}, \\ \quad \text{其中}\ S_n \text{为}n\text{元对称群}; \\ \\ \text{3. 对}\displaystyle\ a:\ \text{复函数}\ g_a(z) = \sum_{n=0}^\infty \dfrac{\left( z - \dfrac{a}{10^9} \right)^n}{\Gamma(n + \alpha) \cdot \zeta(n + 3) \cdot \beta^n}\ \text{在}\ \mathbb{D}(0,2) \subseteq \mathbb{C}\ \text{内解析}; \\ \quad \text{对}\displaystyle\ b:\ \text{复函数}\ g_b(z) = \sum_{n=0}^\infty \dfrac{\left( z - \dfrac{b}{10^9} \right)^n}{\Gamma(n + \beta) \cdot \zeta(n + 3) \cdot \alpha^n}\ \text{在}\ \mathbb{D}(0,2) \subseteq \mathbb{C}\ \text{内解析}; \\ \\ \text{4. 对}\ a:\displaystyle\ \text{线性算子}\ T_a: L^2(\mathbb{R}) \to L^2(\mathbb{R}), \\ \quad T_a f(x) = \dfrac{a}{10^9} \cdot \int_{\mathbb{R}} e^{-|x - t|^2} f(t) dt\ \text{的范数}\ \| T_a \| \leq 1; \\ \quad \text{对}\ b:\displaystyle\ \text{线性算子}\ T_b: L^2(\mathbb{R}) \to L^2(\mathbb{R}), \\ \quad T_b f(x) = \dfrac{b}{10^9} \cdot \int_{\mathbb{R}} e^{-|x - t|^2} f(t) dt\ \text{的范数}\ \| T_b \| \leq 1; \\ \\ \text{5. 对}\displaystyle\ a:\ \text{同余式}\ \left( \left\lfloor \dfrac{|a|^3 + 10^{27}}{10^{27}} \right\rfloor + m \right) \equiv 0 \mod \gamma \\ \quad \text{对所有}\ m \in \mathbb{Z}\ \text{成立}; \\ \quad \text{对}\ b:\displaystyle\ \text{同余式}\ \left( \left\lfloor \dfrac{|b|^3 + 10^{27}}{10^{27}} \right\rfloor + n \right) \equiv 0 \mod \sigma \\ \quad \text{对所有}\ n \in \mathbb{Z}\ \text{成立}; \\ \\ \text{6. 对}\ a:\displaystyle\ \text{级数}\ \sum_{n=1}^\infty \dfrac{(-1)^n \cdot \sin\left( \dfrac{n a}{10^9} \right) \cdot \cos\left( \dfrac{n}{|a| + 1} \right)}{n^3 + \lfloor \sqrt{n} \rfloor! \cdot \gamma^n} \\ \quad \text{绝对收敛且和小于}\ \delta; \\ \quad \text{对}\ b:\displaystyle\ \text{级数}\ \sum_{n=1}^\infty \dfrac{(-1)^n \cdot \sin\left( \dfrac{n b}{10^9} \right) \cdot \cos\left( \dfrac{n}{|b| + 1} \right)}{n^3 + \lfloor \sqrt{n} \rfloor! \cdot \sigma^n} \\ \quad \text{绝对收敛且和小于}\ \delta; \\ \\ \text{7. 对}\ a:\displaystyle\ \text{分形集}\ F_a = \left\{ \sum_{n=1}^\infty \dfrac{\text{sign}(a) \cdot \chi_{[-\vert a \vert, \vert a \vert]}(n)}{10^{9n} \cdot \alpha^n} \mid n \in \mathbb{N} \right\} \\ \quad \text{的 Hausdorff 维度}\ \dim_H(F_a) \leq 0.5; \\ \quad \text{对}\ b:\displaystyle\ \text{分形集}\ F_b = \left\{ \sum_{n=1}^\infty \dfrac{\text{sign}(b) \cdot \chi_{[-\vert b \vert, \vert b \vert]}(n)}{10^{9n} \cdot \beta^n} \mid n \in \mathbb{N} \right\} \\ \quad \text{的 Hausdorff 维度}\ \dim_H(F_b) \leq 0.5; \\ \\ \text{8. 对}\ a:\displaystyle\ \text{多重积分}\ \int_{[0,1]^k} \int_{[0,1]^k} \max\left( \left( \dfrac{a}{10^9} \right)^3 - (x_1 y_1 + \cdots + x_k y_k), 0 \right) \\ \quad dx_1 \cdots dx_k dy_1 \cdots dy_k < \delta; \\ \quad \text{对}\ b:\displaystyle\ \text{多重积分}\ \int_{[0,1]^l} \int_{[0,1]^l} \max\left( \left( \dfrac{b}{10^9} \right)^3 - (x_1 y_1 + \cdots + x_l y_l), 0 \right) \\ \quad dx_1 \cdots dx_l dy_1 \cdots dy_l < \delta \end{cases} \end{aligned}

已完成今日 A+B Problem 大学习

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题目描述

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已完成今日 A+B Problem 大学习