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exact_match.csv

@@ -50,7 +50,7 @@ A material perfect clock is in uniform circular motion with angular speed ω and
 "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. 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}.
+"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)? Derive this proportionality from the cosmic over-density field"," 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}"