Upload exact_match.csv

exact_match.csv

@@ -1,13 +1,13 @@
 Question,Answer,Domain,Difficulty,Type,Task,Reference,Explanation (Optional),,,
-"Consider a tripartite state <Math>\rho_{ABC}<\Math> consisting of d-dimensional qudits. Let A, B, C contain NA, NB, and NC qubits, respectively. Suppose there exist a channel R acting on the Hilbert space and outputing to the Hilbert space of AB such that <Math> || R[\rho_{BC}] - \rho_{ABC} ||_1 = \epsilon<\Math>. One can show that the conditional mutual information <Math>I(A:C|B) \le c*\sqrt{\epsilon}<\Math>. How does the prefactor c depend on d, NA, NB, and NC? You can throw away all the constants",<Math>d^NC<\Math>,Math/Physics,Easy,Advanced Reasoning,Quantum Information,Yifan Zhang,,,Quantum Information,15
-"Given a TFIM Hamiltonian <Math>H_L = \sum_{i=1}^{L-1} X_i X_{i+1} + \sum_{i=1}^L Z_i</Math> of size L, rewrite <Math>H_{L+1}</Math> as a function of <Math>H_L</Math>",H_{L+1} = H_L \otimes I + X_L X_{L+1} + Z_{L+1},Math/Physics,Easy,Advanced Reasoning,Condensed Matter Physics,Roger,,,Condensed Matter Physics,6
-"Given a TFIM Hamiltonian <Math>H_L = \sum_{i=1}^{L-1} X_i X_{i+1} + \sum_{i=1}^L Z_i</Math> of size L, rewrite <Math>H_{2L+2}</Math> as a function of <Math>H_L</Math> such that the expression splits the total system into a bipartite system connected by 2 sites, one is <Math>H_L^{sys}</Math> represents the system, the other is <Math>H_L^{env)</Math> represents the environment",H_{L+1} = H_L \otimes I_{L+2} + I_{L+2}\otimes H_L + X_L X_{L+1} + X_{L+1} X_{L+2} + X_{L+2}X_{L+3} + Z_{L+1} + Z{L+2},Math/Physics,Medium,Advanced Reasoning,Condensed Matter Physics,Roger,,,Probability/Statistics,2
-"We can expand the power of the adjoint of the sum of operators <Math>sum_{i=1}^n A_i</Math> with an operator B into a power form, in short <Math>ad^k_{sum_{i=1}^n A_i} B = (?)^k</Math>",(sum_i ad_{A_i})^k (B),Math/Physics,Medium,Advanced Reasoning,Quantum Information,Roger,,,Astrophysics/Cosmology,5
+"Consider a tripartite state <Math>\rho_{ABC}<\Math> consisting of d-dimensional qudits. Let A, B, C contain NA, NB, and NC qubits, respectively. Suppose there exist a channel R acting on the Hilbert space and outputing to the Hilbert space of AB such that <Math> || R[\rho_{BC}] - \rho_{ABC} ||_1 = \epsilon<\Math>. One can show that the conditional mutual information <Math>I(A:C|B) \le c*\sqrt{\epsilon}<\Math>. How does the prefactor c depend on d, NA, NB, and NC? You can throw away all the constants",d^NC,Math/Physics,Easy,Advanced Reasoning,Quantum Information,Yifan Zhang,,,Quantum Information,14
+"Given a TFIM Hamiltonian <Math>H_L = \sum_{i=1}^{L-1} X_i X_{i+1} + \sum_{i=1}^L Z_i</Math> of size L, rewrite <Math>H_{L+1}</Math> as a function of <Math>H_L</Math>",H_{L+1} = H_L \otimes I + X_L X_{L+1} + Z_{L+1},Math/Physics,Easy,Advanced Reasoning,Condensed Matter Physics,Roger,,,Condensed Matter Physics,5
+"Given a TFIM Hamiltonian <Math>H_L = \sum_{i=1}^{L-1} X_i X_{i+1} + \sum_{i=1}^L Z_i</Math> of size L, rewrite <Math>H_{2L+2}</Math> as a function of <Math>H_L</Math> such that the expression splits the total system into a bipartite system connected by 2 sites, one is <Math>H_L^{sys}</Math> represents the system, the other is <Math>H_L^{env)</Math> represents the environment",H_{L+1} = H_L \otimes I_{L+2} + I_{L+2}\otimes H_L + X_L X_{L+1} + X_{L+1} X_{L+2} + X_{L+2}X_{L+3} + Z_{L+1} + Z{L+2},Math/Physics,Medium,Advanced Reasoning,Condensed Matter Physics,Roger,,,Astrophysics/Cosmology,8
+"We can expand the power of the adjoint of the sum of operators <Math>sum_{i=1}^n A_i</Math> with an operator B into a power form, in short <Math>ad^k_{sum_{i=1}^n A_i} B = (?)^k</Math>",(sum_i ad_{A_i})^k (B),Math/Physics,Medium,Advanced Reasoning,Quantum Information,Roger,,,Probability/Statistics,3
 "Given a fermionic model <Math>H_L = \sum_{i=1}^{L-1} [a^{\dagger}_i a_{i+1} + a_i a^{\dagger}_{i+1} +n_i n{i+1}] </Math>, write it into a qubit format with Jordan Wigner transformation","H_L=\sum_{i=1}^{L-1}[\tfrac12\bigl(\sigma_i^{x}\sigma_{\,i+1}^{x}+\sigma_i^{y}\sigma_{\,i+1}^{y})+\tfrac14\bigl(\sigma_i^{z}\sigma_{\,i+1}^{z}+\sigma_i^{z}+\sigma_{\,i+1}^{z}+1)]",Math/Physics,Medium,Advanced Reasoning,Condensed Matter Physics,Yuxuan Zhang,,,,0
-"Consider a bipartite state <Math>\rho_{AB}</Math> consisting of d-dimensional qudits. Let A, B contain NA, NB qubits, respectively. Find a bound on <Math>||\rho_{AB} - \rho_A \otimes I_B ||_1<\Math> that is a second moment quantity of <Math>\rho_{AB}<\Math>. I_B is the maximally mixed state on B.",<Math>d^{(NA+NB)/2} ||\rho_{AB} - \rho_A \otimes I_B ||_2<\Math>,Math/Physics,Medium,Advanced Reasoning,Quantum Information,Yifan Zhang,,,,0
-"Consider a bipartite state <Math>\rho_{AB}</Math> consisting of d-dimensional qudits. Let A, B contain NA, NB qubits, respectively. Find a bound on <Math>||\rho_{AB} - \rho_A \\rho_B ||_1<\Math> that is a fourth moment quantity of <Math>\rho_{AB}<\Math>. ",<Math>d^{3(NA+NB)/4} 4 ||\rho_{AB} - \rho_A \otimes \rho_B ||_2^2<\Math>,Math/Physics,Hard,Advanced Reasoning,Quantum Information,Yifan Zhang,,,,
-"Using combinatorics, write down the size of the Fock (sub-)spaces for m different modes. Consider the following 4 cases: Bosons, Spinless Fermions, Spinful Fermions, qubits? Output the latex file for final answers separated by commas only.","Infinite, 2^m, 4^m, 2^m",Math/Physics,Easy,Basic Knowledge,Condensed Matter Physics,Yuxuan Zhang,,,,
-"Using combinatorics, write down the size of the Fock (sub-)spaces for m different modes and n different particles. Consider the following 4 cases: Bosons, Spinless Fermions, Spinful Fermions, qubits? Output the latex file for final answers separated by commas only.","<Math>\binom{n+m-1}{\,n},\ \binom{m}{\,n},\ \binom{2m}{\,n},\ \binom{m}{\,n}<\Math>",Math/Physics,Easy,Basic Knowledge,Condensed Matter Physics,Yuxuan Zhang,,,,
+"Consider a bipartite state <Math>\rho_{AB}</Math> consisting of d-dimensional qudits. Let A, B contain NA, NB qubits, respectively. Find a bound on <Math>||\rho_{AB} - \rho_A \otimes I_B ||_1<\Math> that is a second moment quantity of <Math>\rho_{AB} - \rho_A \otimes I_B<\Math>. I_B is the maximally mixed state on B. Express your answer in Scahtten norms if it is possible.",d^{(NA+NB)/2} ||\rho_{AB} - \rho_A \otimes I_B ||_2,Math/Physics,Medium,Advanced Reasoning,Quantum Information,Yifan Zhang,,,,0
+"Consider a bipartite state <Math>\rho_{AB}</Math> consisting of d-dimensional qudits. Let A, B contain NA, NB qubits, respectively. Find a bound on <Math>||\rho_{AB} - \rho_A \otimes \rho_B ||_1<\Math> that is a fourth moment quantity of <Math>\rho_{AB} - \rho_A \otimes I\rho_B<\Math>. Express your answer in Scahtten norms if it is possible.",d^{3(NA+NB)/4} 4 ||\rho_{AB} - \rho_A \otimes \rho_B ||_2^2,Math/Physics,Hard,Advanced Reasoning,Quantum Information,Yifan Zhang,,,,
+"Using combinatorics, write down the size of the total Fock space for m different modes. Consider the following cases: Spinless Fermions. Output the LaTeX for final answers only without specifying the variable.",2^m,Math/Physics,Easy,Basic Knowledge,Condensed Matter Physics,Yuxuan Zhang,,,,
+"Using combinatorics, write down the size of each Fock sub-spaces for m different modes and n different particles. Consider the following cases: Spinless Fermions. Output the LaTeX for final answers only without specifying the variable.","\binom{m}{\,n}",Math/Physics,Easy,Basic Knowledge,Condensed Matter Physics,Yuxuan Zhang,,,,
 "Consider the following Hamiltonian: <Math>
 \hat{H} = -\frac{1}{2} \sum_{n=1}^N \left\{
 \left[J_0 + (-1)^{n-1} J_1 \right]
@@ -21,29 +21,8 @@ Question,Answer,Domain,Difficulty,Type,Task,Reference,Explanation (Optional),,,
 \sigma_{n-1}^x \sigma_n^z \sigma_{n+1}^x +
 \sigma_{n-1}^y \sigma_n^z \sigma_{n+1}^y
 \right)
-<\Math> Can it be written as a free fermion system? Answer with 'Yes' or 'No'. If 'Yes', write out its fermionic representation.","Yes. <Math>\hat{H} = 
-- \frac{1}{2} \sum_{n=1}^{N}
-\left[
-c^\dagger_{n,1} c_{n,2} 
-+ \gamma\, c^\dagger_{n,1} c^\dagger_{n,2} 
-+ \text{h.c.}
-\right]
-- \frac{1}{2} \sum_{n=1}^{N'}
-\left[
-\alpha\left(c^\dagger_{n,2} c_{n+1,1} 
-+ \gamma\, c^\dagger_{n,2} c^\dagger_{n+1,1} \right)
-+ \text{h.c.}
-\right]
-- \frac{1}{2} \sum_{n=1}^{N'}
-h\left(
-c^\dagger_{n,1} c_{n,1} + c^\dagger_{n,2} c_{n,2}
-\right)
-+ \frac{1}{2} \sum_{n=1}^{N'}
-\Omega\left(
-c^\dagger_{n,1} c_{n+1,1} + c^\dagger_{n,2} c_{n+1,2}
-+ \text{h.c.}
-\right)<\Math>",Math/Physics,Hard,Advanced Reasoning,Quantum Information,Yuxuan Zhang,,,,
-"You are given three gates. <Math> Z_1(\theta_1) = exp(i \theta_1 |0><0|_1) <\Math> is a phase gate on qubit 1. <Math> Z_2(\theta_2) = exp(i \theta_2 |0><0|_2) <\Math> is a phase gate on qubit 2. <Math> R = exp(i \theta_r |11><11|) <\Math> is a phase gate acting on qubit 1 and 2. Let <Math> U = Z_1 R Z_2 <\Math>. How to choose <Math> \theta_1, \theta_2, \theta_r <\Math> such that the diagonal elements of U is 1 on <Math> |00>, |11> <\Math> and is -i on <Math> |01>, |10> <\Math>. Answer in the format like <Math>\theta_1=\pi, \theta_2=8\pi/7, theta_r=-6\pi/5<\Math>. ",,Math/Physics,Medium,Advanced Reasoning,Condensed Matter Physics,,,,,
+<\Math> Can it be written as a free fermion system? Answer with 'Yes' or 'No'.",Yes,Math/Physics,Hard,Advanced Reasoning,Quantum Information,Yuxuan Zhang,,,,
+"You are given three gates. <Math> Z_1(\theta_1) = exp(i \theta_1 |0><0|_1) <\Math> is a phase gate on qubit 1. <Math> Z_2(\theta_2) = exp(i \theta_2 |0><0|_2) <\Math> is a phase gate on qubit 2. <Math> R = exp(i \theta_r |11><11|) <\Math> is a phase gate acting on qubit 1 and 2. Let <Math> U = Z_1 R Z_2 <\Math>. How to choose <Math> \theta_1, \theta_2, \theta_r <\Math> such that the diagonal elements of U is 1 on <Math> |00>, |11> <\Math> and is -i on <Math> |01>, |10> <\Math>. Answer in the format like \theta_1=\pi, \theta_2=8\pi/7, theta_r=6\pi/5. Make all angles non-negative.","\theta_1=3\pi/2, \theta_2=3\pi/2, theta_r=\pi",Math/Physics,Medium,Advanced Reasoning,Quantum Information,Yifan Zhang,,,,
 "Rényi-$\alpha$ negativity is defined as:
 <Math>
 \mathcal{N}_\alpha(\rho) = -\log \left( \operatorname{Tr} \left[ \left( \rho^{T_A} \right)^\alpha \right] / \operatorname{Tr} \left[ \rho ^\alpha \right]\right)
@@ -57,41 +36,28 @@ First, consider a two-level system with lower state |g⟩and excited state |e⟩
 separation ω_eg . Let the dipole moment for this transition be d_eg. Find the AC Stark shift for the state |g⟩in terms of the laser intensity,
 transition dipole moment, and relevant frequencies. You can assume that we are
 far from the atomic resonance, but do not make the rotating wave approximation!
-What is the AC Stark shift for the state |e⟩? ",<Math>\Delta E_g=I \frac{\left|d_{e g}\right|^2}{\hbar \epsilon_0 c} \frac{\omega_{e g}}{\omega^2-\omega_{e g}^2}<\Math> \n <Math>\Delta E_e=-I \frac{\left|d_{e g}\right|^2}{\hbar \epsilon_0 c} \frac{\omega_{e g}}{\omega^2-\omega_{e g}^2}<\Math>,Math/Physics,Hard,Advanced Reasoning,Quantum Information,,,,,
-What is the change in the transition frequency ωeg due to the presence of the trapping light?,<Math>\Delta \omega_{e g}=\frac{\Delta E_e-\Delta E_g}{\hbar}=-\frac{\left|d_{e g}\right|^2}{\hbar^2 \epsilon_0 c} \frac{2 I \omega_{e g}}{\omega^2-\omega_{e g}^2}<\Math>,Math/Physics,Hard,Advanced Reasoning,Quantum Information,,,,,
+What is the AC Stark shift for the state |e⟩? ",\Delta E_g=I \frac{\left|d_{e g}\right|^2}{\hbar \epsilon_0 c} \frac{\omega_{e g}}{\omega^2-\omega_{e g}^2}<\Math> \n <Math>\Delta E_e=-I \frac{\left|d_{e g}\right|^2}{\hbar \epsilon_0 c} \frac{\omega_{e g}}{\omega^2-\omega_{e g}^2},Math/Physics,Hard,Advanced Reasoning,Quantum Information,Xiang Zou,,,,
+What is the change in the transition frequency ωeg due to the presence of the trapping light?,\Delta \omega_{e g}=\frac{\Delta E_e-\Delta E_g}{\hbar}=-\frac{\left|d_{e g}\right|^2}{\hbar^2 \epsilon_0 c} \frac{2 I \omega_{e g}}{\omega^2-\omega_{e g}^2},Math/Physics,Hard,Advanced Reasoning,Quantum Information,Xiang Zou,,,,
 "Now consider a three-level atom, where the excited state |e⟩is additionally coupled to
 a higher-energy level |f⟩with transition frequency ω_fe and dipole moment d_fe. Find the AC Stark shift for the state |e⟩taking into account both
-states |f⟩and |g⟩.",<Math>\Delta E_e=\frac{I}{\hbar \epsilon_0 c}\left(\frac{\omega_{f e}\left|d_{f e}\right|^2}{\omega^2-\omega_{f e}^2}-\frac{\omega_{e g}\left|d_{e g}\right|^2}{\omega^2-\omega_{e g}^2}\right)<\Math>,Math/Physics,Hard,Advanced Reasoning,Quantum Information,,,,,
+states |f⟩and |g⟩.",\Delta E_e=\frac{I}{\hbar \epsilon_0 c}\left(\frac{\omega_{f e}\left|d_{f e}\right|^2}{\omega^2-\omega_{f e}^2}-\frac{\omega_{e g}\left|d_{e g}\right|^2}{\omega^2-\omega_{e g}^2}\right),Math/Physics,Hard,Advanced Reasoning,Quantum Information,Xiang Zou,,,,
 "At what laser frequency ω_m does the transition frequency ω_eg become
-independent of the trap laser intensity I?",<Math>\omega_m=\sqrt{\frac{\left(2 \omega_{f e}\left|d_{e g}\right|^2-\omega_{e g}\left|d_{f e}\right|^2\right) \omega_{e g} \omega_{f e}}{2 \omega_{e g}\left|d_{e g}\right|^2-\omega_{f e}\left|d_{f e}\right|^2}}<\Math>,Math/Physics,Hard,Advanced Reasoning,Quantum Information,,,,,
-"1) The states |ψ1⟩, |ψ2⟩,· · ·, |ψN ⟩ are linearly independent and are not all
-orthogonal.
-Is it possible to build a device which, if given as input an unknown choice |ψi⟩, (where 1 ≤ i ≤ N ), always returns as output the value of i? Justify your answer","No. Since {|ψi⟩} does not form a set of orthogonal basis, it is impossible to completely distinguish them.",Math/Physics,Hard,Advanced Reasoning,Quantum Information,,,,,
-"Is it possible to build a device, which, if given as input an unknown choice |ψi⟩, (where 1 ≤ i ≤ N ), either (with some probability pi > 0) returns as output the value of i, or (with probabilty (1− pi)) returns as output the declaration ”state not identified”. Justify your answer.","Yes. Since the original set of vectors are linearly independent, the span of the
-original set of vectors has dimension N . Now, any any fixed but arbitrary vector
-|ψi⟩ in the set, we can consider the span, Si of the complementary sets
-|ψ1⟩, |ψ2⟩, · · · |ψi−1⟩, |ψi+1⟩,· · ·, |ψN ⟩.
-[The span of a set means the vector space generated by a set.] Such a span Si
-has only dimension N− 1. This implies the existence of a vector |φi⟩ that is
-orthogonal to the Si.
-The device can work as follows. It randomly chooses a value i and performs a
-measurement into either i) the span Si or ii) the complementary vector |φi⟩.
-Notice that, from the linear independence of the vector, we must have ⟨φi|ψi⟩ ̸= 0,
-for each i. Notice further than the state |φi⟩ is orthogonal to all the original states
-|ψj ⟩ for j ̸= i. Therefore, a successful projection into case ii) |φi⟩ will identify
-the original state as |ψi⟩.",Math/Physics,Hard,Advanced Reasoning,Quantum Information,,,,,
-"How many different quantum states are there in a k-design? Assume the Hilbert space size is $d$, output your answer in combinatorics",<Math>\binom{d + k - 1}{k}<\Math>,Math/Physics,Easy,Basic Knowledge,Quantum Information,Yuxuan Zhang,,,,
+independent of the trap laser intensity I?",\omega_m=\sqrt{\frac{\left(2 \omega_{f e}\left|d_{e g}\right|^2-\omega_{e g}\left|d_{f e}\right|^2\right) \omega_{e g} \omega_{f e}}{2 \omega_{e g}\left|d_{e g}\right|^2-\omega_{f e}\left|d_{f e}\right|^2}},Math/Physics,Hard,Advanced Reasoning,Quantum Information,Xiang Zou,,,,
+A material perfect clock is in uniform circular motion with angular speed ω and radius R in flat spacetime. Is there any physical restriction on \omega and R?,\omega*R <1,Math/Physics,Medium,Advanced Reasoning,Astrophysics/Cosmology,Xiang Zou,,,,
+"After one full circle, how much will the reading of the perfect clock advance?",\frac{2\pi}{\omega}\sqrt{1-\omega^2R^2},Math/Physics,Hard,Advanced Reasoning,Astrophysics/Cosmology,Xiang Zou,,,,
+"Some student said that the following invariant interval for some physical spacetime is problematic, ds^2 =du^2 +2dudv+dv^2 +dy^2 +dz^2. He is correct, why can you infer the determinant of the metric tensor represented by this invariant interval?",0,Math/Physics,Medium,Advanced Reasoning,Astrophysics/Cosmology,Xiang Zou,,,,
+"How many different quantum states are there in a k-design? Assume the Hilbert space size is $d$, output your answer in combinatorics only",\binom{d + k - 1}{k},Math/Physics,Easy,Basic Knowledge,Quantum Information,Yuxuan Zhang,,,,
 "For a spherical 3-design with N states, sample each quantum state according to the probability distribution of each state. What is the probability that a sampled state has a probability weight higher than 1/N? Hint: recall Paley–Zygmund inequality; you may assume the Hilbert space size d ->inf in the end",1/6,Math/Physics,Hard,Advanced Reasoning,Quantum Information,Yuxuan Zhang,,,,
-"T_n = \sum_i R_i 1(X_0>0), where R_i is the rank of X_i in the sorted order |X_1|\leq |X_2|.... Decompose T_n into a linear combination of U-statistic of order 1 and a U-statistic of order 2. (Hint: Express the result as aU_2+bU_2, where the coefficients a and b depend on n.)",\sum_{i<j} 1(X_1+X_j>0)+\sum_{i=i}^n 1(X_i>0) = \binom{n}{2}U_2 + nU_1,Math/Physics,Hard,Advanced Reasoning,Probability/Statistics,Shunan Zheng,,,,
-"During the supernovae explosion, it will eject a large amount of material into the surrounding environment (chracterized by number density n_H), with mass given by M_{ej}, and release amount of energy E_{SN}. Assume that during its evolution phase, the ionizing radiation has lower the ambient density of supernovae such that the explosion bubble could freely expand. Derive a rough estimate for the timescale of this exploding phase for a star with mass M_{\star}, assume all mass has been ejected away, using the quantities mentioned above.",t_{free expansion} ~  r_{free expansion} /(2E_{SN} / M_{ej})^{1/2} ~ M_{\star}^{5/6} (1/(4 pi m_p n_H))^{1/3}(1/ 2E_{SN})^{1/2},Math/Physics,Medium,Advanced Reasoning,Astrophysics/Cosmology,Saiyang Zhang,"Assume the expansion halts when M_ej is equal to the mass swept away by the ejected mass: M_ej ~ M_swept = 4/3 pi r_{free expansion}^3 mu n_H m_p, where mu is the molecular weight of the universe. We derive the sweeping radius as r_{free expansion} ~(3M_{ej}/4 pi m_p n_H)^{1/3}. By the conservation of energy, the time scale for the expansion phase is t_{free expansion} ~  r_{free expansion} /(2E_{SN} / M_{ej})^{1/2} ~ M_{\star}^{5/6} (1/(4 pi m_p n_H))^{1/3}(1/ 2E_{SN})^{1/2}",,,
-"If we are living in a universe with primordial black holes as the only dark matter candidate (i.e. \Omega_{DM} = \Omega_{PBH}), and all of them are centered around some mass M_{PBH}, with number density given by n_{PBH}. What is the dominant change (Poisson fluctuation) induced by PBHs to the linear power spectrum P(k) related to the number density of PBHs n_{PBH} (only show the proportional relationship)?"," P(k) \propto 1/n_{PBH}.
+"T_n = \sum_i R_i 1(X_0>0), where R_i is the rank of X_i in the sorted order |X_1|\leq |X_2|.... Decompose T_n into a linear combination of U-statistic of order 1 and a U-statistic of order 2. (Hint: Express the result as aU_2+bU_2, where the coefficients a and b depend on n. U_2 = \frac{\sum_{i<j} 1(X_1+X_j>0))}{\binom{n}{2}} and U_1 = \frac{\sum_{i=1}^n 1(X_i>0)}{n}). Don't need to show the expression for U_1 and U_2 again.)",T_n = \binom{n}{2}U_2 + nU_1,Math/Physics,Medium,Advanced Reasoning,Probability/Statistics,Shunan Zheng,,,,
+"During the supernovae explosion, it will eject a large amount of material into the surrounding environment (chracterized by number density n_H), with mass given by M_{ej}, and release amount of energy E_{SN}. Assume that during its evolution phase, the ionizing radiation has lower the ambient density of supernovae such that the explosion bubble could freely expand. Using the quantities mentioned above, derive proportionality relation with the timescale of this exploding phase for a star with mass M_{\star}, assuming all mass has been ejected away.",t_{free expansion} ~  r_{free expansion} /(2E_{SN} / M_{ej})^{1/2} ~ M_{\star}^{5/6} n_H^{-1/3}E_{SN}^{-1/2},Math/Physics,Medium,Advanced Reasoning,Astrophysics/Cosmology,Saiyang Zhang,"Assume the expansion halts when M_ej is equal to the mass swept away by the ejected mass: M_ej ~ M_swept = 4/3 pi r_{free expansion}^3 mu n_H m_p, where mu is the molecular weight of the universe. We derive the sweeping radius as r_{free expansion} ~(3M_{ej}/4 pi m_p n_H)^{1/3}. By the conservation of energy, the time scale for the expansion phase is t_{free expansion} ~  r_{free expansion} /(2E_{SN} / M_{ej})^{1/2} ~ M_{\star}^{5/6} (1/(4 pi m_p n_H))^{1/3}(1/ 2E_{SN})^{1/2}",,,
+," P(k) \propto 1/n_{PBH}.
 ",Math/Physics,Hard,Advanced Reasoning,Astrophysics/Cosmology,Saiyang Zhang,"Assume PBHs are located at spatial coordinate \mathbf{x}, the overdense field at spatial point is given by: \delta_{iso}(\mathbf{x})\equiv \frac{\delta \rho_{PBH}}{\Bar{\rho}_{ DM}}=\frac{1}{n_{PBH}} \sum_i \delta_D\left(\mathbf{x}-\mathbf{x}_i\right)-1. Taking the two point correlation of the density fluctuation, we have \left\langle\delta_{PBH}(\mathbf{x}) \delta_{PBH}(\mathbf{0})\right\rangle =\frac{1}{n_{PBH}} \delta_D(\mathbf{x})+\xi_{PBH}(x). Taking the fourier transform of the twopoint correlation function, we have the power spectrum taking the form of P(k) = D_{0}^{2}/n_{ PBH} +.... where D_{0} is the growth factor. Therefore, we have P(k) \propto 1/n_{PBH}.
 ",,,
 "Work out the temperature profile (T(r)) of the black hole of mass M_{BH} accretion disk with radial denpendence (r), assume that black hole accretes at a rate \dot{M}, and assume that half of the graviational potential energy is converted into thermal heating energy, radiates away like black body. Express the answer in the quantities mentioned above.",T(t) = (G M_{BH}\dot{M} / 8 \pi \sigma_{SB} r^3)^{1/4},Math/Physics,Medium,Advanced Reasoning,Astrophysics/Cosmology,Saiyang Zhang,"For a mass element M move from r+\delta r to r, the change in potential energy can be written as \delta E = GM_{BH}M \delta r / r^2. If half of the energy is converted to luminosity, we have \delta L = 2G M_{BH}\dot{M} / 2r^2 \delta r. If each segment of the accretion disk emits like a black body, the luminosity can be written as \delta L = 4 \pi r \delta r \sigma_{SB} T(r)^4. Solving for T(r), we have T(t) = (G M_{BH}\dot{M} / 8 \pi \sigma_{SB} r^3)^{1/4}",,,
-"In a standard LCDM universe, if a Black Hole seed is born in an ideal environment and always growing at an Eddington rate, with radiation efficiency of 0.1, what will be the mass of the BH at z =10 if it is born with a mass of 1000 Solar mass at z=20? (Give order of magnitude estimate)",4 \times 10^5 solar mass.,Math/Physics,Medium,Advanced Reasoning,Astrophysics/Cosmology,Saiyang Zhang,"The characteristic Salpter time(e-fold growing time) scale for BH growth is 5\times 10^7 yr, where M(t) = M_{ini} exp(t/t_{Salpter}). Therefore, the time difference between redshift 20 and redshift 10 is about 3\times 10^8 yrs. Therefore, the BH mass estimated at z = 10 is about 4 \times 10^5 solar mass.",,,
+"In a standard LCDM universe, if a Black Hole seed is born in an ideal environment and always growing at an Eddington rate, with radiation efficiency \epsilon of 0.1, what will be the mass of the BH at z =10 if it is born with a mass of 1000 Solar mass at z=20? (Round to one significant figure)",6 \times 10^5 M_\odot ,Math/Physics,Medium,Advanced Reasoning,Astrophysics/Cosmology,Saiyang Zhang,"The characteristic Salpter time(e-fold growing time) scale for BH growth is 5\times 10^7 yr, where M(t) = M_{ini} exp(t/t_{Salpter}). Therefore, the time difference between redshift 20 and redshift 10 is about 3\times 10^8 yrs. Therefore, the BH mass estimated at z = 10 is about 4 \times 10^5 solar mass.",,,
 "Consider a dark matter halo at redshift $z$, capable of cooling through molecular hydrogen (H$\_2\$). The halo is characterized by its virial temperature $T_{\rm vir}$, molecular hydrogen cooling function $\Lambda(T, n_{\rm H})$, and hydrogen number density $n_{\rm H}$. To determine the minimum mass of such a halo capable of cooling within a local Hubble time, we impose the condition that the cooling time $\tau_{\rm cool}(M,z)$ be less than or equal to the local Hubble time $t_H(z)$. Using the equations provided below, derive the minimum halo mass $M_{t_H-\rm cool}(z)$ as a function of redshift z.
 
-\textbf{Given Equations related to cooling of hydrogen:}
+Given Equations related to cooling of hydrogen:
 \tau_{\rm cool}(M,z) = \frac{3 k_B T_{\rm vir}(M,z)}{2 \Lambda(T_{\rm vir}, n_{\rm H}) f_{\rm H_2}}, 
 T_{\rm vir}(M,z) &\approx 2554,\mathrm{K} \left(\frac{M}{10^6 M_\odot}\right)^{2/3} \left(\frac{1+z}{31}\right), 
 \Lambda(T,n_{\rm H}) \approx 10^{-31.6}\left(\frac{T}{100,\mathrm{K}}\right)^{3.4}\left(\frac{n_{\rm H}}{10^{-4},\mathrm{cm}^{-3}}\right),\mathrm{erg,s^{-1}}, 
@@ -99,36 +65,27 @@ n_{\rm H} \approx 1.0\left(\frac{1+z}{31}\right)^3,\mathrm{cm}^{-3},
 t_H(z) \approx 3.216 \times 10^{15},\mathrm{s}\left(\frac{1+z}{31}\right)^{-3/2}, 
 f_{\rm H_2,max}(T) \approx 3.5 \times 10^{-4}\left(\frac{T}{1000,\mathrm{K}}\right)^{1.52}.
 ", M_{t_H-\rm cool} \approx 1.54 \times 10^{5} M_\odot \left(\frac{1+z}{31}\right)^{-2.074}.,Math/Physics,Hard,Advanced Reasoning,Astrophysics/Cosmology,Saiyang Zhang,"To find the minimum mass, we first equate the cooling time $\tau_{\rm cool}$ to the local Hubble time $t_H(z)$: \frac{3 k_B T_{\rm vir}(M,z)}{2 \Lambda(T_{\rm vir}, n_{\rm H}) f_{\rm H_2}} = t_H(z). Plugging in the expressions for $T_{\rm vir}$, $\Lambda$, and $n_{\rm H}$, we obtain: \tau_{\rm cool} \approx 3.46 \times 10^{10}\,\mathrm{s} \left(\frac{M}{10^6 M_\odot}\right)^{-1.6}\left(\frac{1+z}{31}\right)^{-5.4}f_{\rm H_2}^{-1}. Equating this to $t_H(z)$, we have: 3.46 \times 10^{10}\,\mathrm{s} \left(\frac{M}{10^6 M_\odot}\right)^{-1.6}\left(\frac{1+z}{31}\right)^{-5.4}f_{\rm H_2}^{-1} = 3.216 \times 10^{15}\,\mathrm{s}\left(\frac{1+z}{31}\right)^{-3/2}. Solving for the required molecular hydrogen fraction: f_{\rm H_2} \approx 1.09 \times 10^{-5}\left(\frac{M}{10^6 M_\odot}\right)^{-1.6}\left(\frac{1+z}{31}\right)^{-3.9}. Next, we equate this required fraction to the maximum fraction that can form in the halo, $f_{\rm H_2,max}$, assuming $T=T_{\rm vir}$: 1.09 \times 10^{-5}\left(\frac{M}{10^6 M_\odot}\right)^{-1.6}\left(\frac{1+z}{31}\right)^{-3.9} = 3.5 \times 10^{-4}\left(\frac{T_{\rm vir}}{1000\,\mathrm{K}}\right)^{1.52}. Replacing $T_{\rm vir}$ with its expression: T_{\rm vir}=2554\,\mathrm{K}\left(\frac{M}{10^6 M_\odot}\right)^{2/3}\left(\frac{1+z}{31}\right), we get: 1.09 \times 10^{-5}\left(\frac{M}{10^6 M_\odot}\right)^{-1.6}\left(\frac{1+z}{31}\right)^{-3.9}=3.5\times 10^{-4}\left[2.554\left(\frac{M}{10^6 M_\odot}\right)^{2/3}\left(\frac{1+z}{31}\right)\right]^{1.52}. Solving for $M$, we obtain the minimum halo mass: M_{t_H-\rm cool} \approx 1.54 \times 10^{5} M_\odot \left(\frac{1+z}{31}\right)^{-2.074}.",,,
-"The lifetime of a lightbulb is an exp(\lambda) random variable. Replacement happens every fixed T time unit, starting at time 0. What is the expected time when the lightbulb is discovered to have failed. Denote the time the lightbulb has failed with D.","
+"The lifetime of a lightbulb is an exp(\lambda) random variable. Replacement happens every fixed T time unit, starting at time 0. What is the expected time when the lightbulb is discovered to have failed. Let D be the time when the machine is discovered to be down.","
 E[D] = \frac{T}{1-e^{-\lambda T}}
 ",Math/Physics,Medium,Advanced Reasoning,Probability/Statistics,Shunan Zheng,,,,
-"The lifetime of a lightbulb follows exp(1/\theta), where the expected lifetime \theta is unknown. In one experiment, test N lightbulbs and record their exact lifetime y_1,...,y_N. In another experiment, test M lightbulbs.  At some time t, record if the lightbulbs are still burning with indicators Z_1,...,Z_M. Z_i = 1 if the lightbulb i is still burning; Z_i = 0 if the lightbulb i is out. To estimate the unknown \theta, implement the EM algorithm. What’s the corresponding E-step and M-step? (Hint: Denote the observed data with:  $y= (y_1, \ldots, y_N, Z_1, \ldots, Z_M)$. Let $Z = \sum_{i=1}^M Z_i $. Denote the unobserved data with: $X = (X_1, \ldots, X_M)$.)","
-The E-step:
-
-Q(\theta | \theta_t) 
-= E [l^c(\theta; y, X)] = -N ( \ln \theta + \bar{y}/\theta) - \sum_{i=1}^{M} ( \ln \theta + E(X_i | E_i)/\theta) 
-= -(N + M) \ln \theta - \frac{1}{\theta} ( N \bar{y} + Z(t + \theta_t) + (M - Z) ( \theta_t - \frac{t \exp(-t/\theta_t)}{1 - \exp(-t/\theta_t)} ) ).
-
-
-The M-step:
-
-\theta_{t+1} 
-=\arg\max_\theta Q(\theta \mid \theta_t) 
-= \frac{1}{N + M} ( \bar{y} + Z(t + \theta_t) + (M - Z) ( \theta_t - \frac{t \exp(-t/\theta_t)}{1 - \exp(-t/\theta_t)} ) ).
+"The lifetime of a lightbulb follows exp(1/\theta), where the expected lifetime \theta is unknown. In one experiment, test N lightbulbs and record their exact lifetime y_1,...,y_N. In another experiment, test M lightbulbs.  At some time t, record if the lightbulbs are still burning with indicators Z_1,...,Z_M. Z_i = 1 if the lightbulb i is still burning; Z_i = 0 if the lightbulb i is out. To estimate the unknown \theta, implement the EM algorithm. What’s the corresponding expression Q(\theta | \theta_t) for E-step? (Hint: Denote the observed data with:  $y= (y_1, \ldots, y_N, Z_1, \ldots, Z_M)$. Let $Z = \sum_{i=1}^M Z_i $. Denote the unobserved data with: $X = (X_1, \ldots, X_M)$.)","Q(\theta | \theta_t) = -(N + M) \ln \theta - \frac{1}{\theta} ( N \bar{y} + Z(t + \theta_t) + (M - Z) ( \theta_t - \frac{t \exp(-t/\theta_t)}{1 - \exp(-t/\theta_t)} ) )
 
 ",Math/Physics,Hard,Advanced Reasoning,Probability/Statistics,Shunan Zheng,,,,
 "Let ${N(t), t \ geq 0}$ be an RP with integer valued inter-renewal times with pmf 
-$P(X_n = 0) = 1-\alpha$, $P(X_n = 1) = \alpha$, $n \geq 1$, where $0<\alpha<1$. Compute $P(N(t) = k)$ for k = 0,1,2,....
+$P(X_n = 0) = 1-\alpha$, $P(X_n = 1) = \alpha$, $n \geq 1$, where $0<\alpha<1$. Compute $P(N(t) = k)$ for k = 0,1,2,....
 ",Math/Physics,Medium,Advanced Reasoning,Probability/Statistics,Shunan Zheng,,,,
-"Assume the distribution of the population G_i  is:  $G_i(m = 1) \sim N(\mu^{(i)}, \sigma_i^2), \quad (i = 1, 2)$. According to the distance-based classification rule (assume $\mu^{(1)} > \mu^{(2)}$, $\sigma_1 < \sigma_2$):
-\[
-\begin{cases}
-x \in G_1, & \text{if } \mu^* < x < \mu^{(1)} \\
-x \in G_2, & \text{if } x \leq \mu^* \text{ or } x \geq \mu^{(1)}
-\end{cases}
-\]
-where  $\mu^* = \frac{\sigma_1 \mu^{(2)} + \sigma_2 \mu^{(1)}}{\sigma_1 + \sigma_2}$.
-Find the misclassification probabilities P(2|1).
-
-
-
+"Assume the distribution of the population G_i  is:  $G_i(m = 1) \sim N(\mu^{(i)}, \sigma_i^2), \quad (i = 1, 2)$. According to the distance-based classification rule (assume $\mu^{(1)} > \mu^{(2)}$, $\sigma_1 < \sigma_2$):
+\[
+\begin{cases}
+x \in G_1, & \text{if } \mu^* < x < \mu^{(1)} \\
+x \in G_2, & \text{if } x \leq \mu^* \text{ or } x \geq \mu^{(1)}
+\end{cases}
+\]
+where  $\mu^* = \frac{\sigma_1 \mu^{(2)} + \sigma_2 \mu^{(1)}}{\sigma_1 + \sigma_2}$.
+Find the misclassification probabilities P(2|1).
+
+
+
+",P(2\mid 1) = \Phi(\frac{\mu^{(2)}-mu^{(1)}}{\sigma_1+\sigma_2}) +\Phi(\frac{\mu^{(2)}-mu^{(1)}}{\sigma_2-\sigma_1}),Math/Physics,Medium,Advanced Reasoning,Probability/Statistics,Shunan Zheng,,,,
+"how many elements are in the group described by the following relation: \begin{aligned} & {\left[(-1)^F\right]^2=1, \quad \mathrm{C}^2=1, \quad \mathrm{R}^2=1, \quad(-1)^F \mathrm{C}=\mathrm{C}(-1)^F} \\ & \mathrm{R}(-1)^F=(-1)^F \mathrm{R}, \quad \mathrm{RC}=\mathrm{C}(-1)^F \mathrm{R} \\ & \mathrm{T}^2=1, \quad \mathrm{~T}(-1)^F=(-1)^F \mathrm{T}, \quad(\mathrm{TR})^2=1, \quad \mathrm{TC}=\mathrm{C}(-1)^F \mathrm{T} \end{aligned}",16,Math/Physics,Medium,Advanced Reasoning,Mathematical Physics,Wucheng Zhang,,,,
+"What is the group described by the following relation: \begin{aligned} & {\left[(-1)^F\right]^2=1, \quad \mathrm{C}^2=1, \quad \mathrm{R}^2=1, \quad(-1)^F \mathrm{C}=\mathrm{C}(-1)^F} \\ & \mathrm{R}(-1)^F=(-1)^F \mathrm{R}, \quad \mathrm{RC}=\mathrm{C}(-1)^F \mathrm{R} \\ & \mathrm{T}^2=1, \quad \mathrm{~T}(-1)^F=(-1)^F \mathrm{T}, \quad(\mathrm{TR})^2=1, \quad \mathrm{TC}=\mathrm{C}(-1)^F \mathrm{T} \end{aligned}",D_4 \times \mathbb{Z}_2,Math/Physics,Medium,Advanced Reasoning,Mathematical Physics,Wucheng Zhang,,,,