deep-principle/science_physics · Datasets at Hugging Face

Question stringlengths 78 1.28k	Answer stringlengths 1 232	Domain stringclasses 1 value	Difficulty stringclasses 3 values	Type stringclasses 2 values	Task stringclasses 4 values	Reference stringclasses 8 values	Explanation (Optional) stringclasses 5 values
Consider a tripartite state \boxed{\rho_{ABC}} 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 \boxed{I(A:C\|B) \le c*\sqrt{\epsilon}}. 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	null
Given a TFIM Hamiltonian \boxed{H_L = \sum_{i=1}^{L-1} X_i X_{i+1} + \sum_{i=1}^L Z_i} 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	null
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	null
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	null
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	null
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. If you get <Math>\|\|\rho_{AB} - \rho_A \otimes I_B \|\|_1 \le K <\Math> where K is the bound you obtain, answer "K" directly. Don't answer "K=..."	d^{(NA+NB)/2} \|\|\rho_{AB} - \rho_A \otimes I_B \|\|_2	Math/Physics	Medium	Advanced Reasoning	Quantum Information	Yifan Zhang	null
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} <\Math>. I am expecting an answer that is a function of. <Math>\rho_{AB} - \rho_A \otimes I\rho_B<\Math>. Express your answer in Scahtten norms if it is possible. If you get <Math>\|\|\rho_{AB} - \rho_A \otimes \rho_B \|\|_1 \le K <\Math> where K is the bound you obtain, answer K directly.	d^{3(NA+NB)/4} 4 \|\|\rho_{AB} - \rho_A \otimes \rho_B \|\|_2^2	Math/Physics	Hard	Advanced Reasoning	Quantum Information	Yifan Zhang	null
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	null
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	null
Consider the following Hamiltonian: <Math> \hat{H} = -\frac{1}{2} \sum_{n=1}^N \left\{ \left[J_0 + (-1)^{n-1} J_1 \right] \left[ \frac{1+\gamma}{2} \, \sigma_n^x \sigma_{n+1}^x + \frac{1-\gamma}{2} \, \sigma_n^y \sigma_{n+1}^y \right] + h \, \sigma_n^z \right\} - \frac{\Omega}{2} \sum_{n=1}^N \left( \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'.	Yes	Math/Physics	Hard	Advanced Reasoning	Quantum Information	Yuxuan Zhang	null
You are given three gates. <Math> Z_1(\theta_1) = exp(i \theta_1 \|1><1\|_1) <\Math> is a phase gate on qubit 1. <Math> Z_2(\theta_2) = exp(i \theta_2 \|1><1\|_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	Easy	Advanced Reasoning	Quantum Information	Yifan Zhang	null
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) <\Math> Given two-qubit density matrix: <Math> \rho = \begin{bmatrix} 0.5782 & 0.2853 - 0.1241i & 0.0008 - 0.0034i & 0.3634 + 0.1226i \\ 0.2853 + 0.1241i & 0.1674 & 0.0011 - 0.0015i & 0.1529 + 0.1385i \\ 0.0008 + 0.0034i & 0.0011 + 0.0015i & 0.00002 & -0.0002 + 0.0023i \\ 0.3634 - 0.1226i & 0.1529 - 0.1385i & -0.0002 - 0.0023i & 0.2544 \end{bmatrix} <\Math> What is the third Renyi negativity of this state (with partial transpose on the first qubit)? Round to the 3rd digit after the decimal point.	0.586	Math/Physics	Medium	Advanced Reasoning	Quantum Information	Yuxuan Zhang	null
the power of an adjoint of tensor product <Math>ad^k_{A\otimes B}</Math> can be expand into what form of tensor product of adjoint operator <Math> (?)^k </Math>, use <Math>\sigma</Math> to annotate the signs of commutator, e.g <Math>ad_{A,\sigma}</Math> where <Math>ad_{A,+}</Math> represents anti-commutator, <Math>ad_{A,-}</Math> represents commutator.	\frac{1}{2^k} (\sum_{\sigma_i\in{+,-} ad_{A,\sigma_i} \otimes ad_{B,-\sigma_i})^k	Math/Physics	Hard	Advanced Reasoning	Quantum Information	Xiu-Zhe Luo	null
One of the key conceptual hurdles that was overcome in the development of neutral atom-based optical clocks is the need to trap and confine the atoms (using an optical lattice). However, the accuracy and precision of such a clock degrades significantly in the presence of differential AC Stark shift between the two levels which comprise the clock superposition. Suppose a laser of frequency ω and intensity I creates a dipole trap for the atom. First, consider a two-level system with lower state \|g⟩and excited state \|e⟩ with energy 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⟩?	\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	null
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	null
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⟩.	\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	null
At what laser frequency ω_m does the transition frequency ω_eg become 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	null
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	null
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	null
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	null
How many different quantum states are there in a k-design? Assume the Hilbert space size is $d$, output your answer in an equation using combinatorics only	\binom{d + k - 1}{k}	Math/Physics	Easy	Basic Knowledge	Quantum Information	Yuxuan Zhang	null
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? Output a number or expression only.	1/6	Math/Physics	Hard	Advanced Reasoning	Quantum Information	Yuxuan Zhang	null
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	null
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} ~ 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}
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, and the Stefan-Boltzmann Constant \sigma_{SB}.	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 \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 (express in M_\odot ) 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. 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}}, 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. 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	null
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	null
Let ${N(t), t \ geq 0}$ be an RP with integer valued inter-renewal times with pmf $P(X_n = 0) = 1-\alpha$, $P(X_n = 1) = \alpha$, for $n \geq 1$, where $0<\alpha<1$. Compute $P(N(t) = k)$ for k = 0,1,2,....	P(N(t) = k) = \binom{t}{k} (1-\alpha)^{k-t}\alpha^{t+1}	Math/Physics	Medium	Advanced Reasoning	Probability/Statistics	Shunan Zheng	null
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	null
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	Condensed Matter Physics	Wucheng Zhang	null
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	Condensed Matter Physics	Wucheng Zhang	null

Consider a tripartite state \boxed{\rho_{ABC}} 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 \boxed{I(A:C|B) \le c*\sqrt{\epsilon}}. 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

null

Given a TFIM Hamiltonian \boxed{H_L = \sum_{i=1}^{L-1} X_i X_{i+1} + \sum_{i=1}^L Z_i} 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

null

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

null

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

null

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

null

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. If you get <Math>||\rho_{AB} - \rho_A \otimes I_B ||_1 \le K <\Math> where K is the bound you obtain, answer "K" directly. Don't answer "K=..."

d^{(NA+NB)/2} ||\rho_{AB} - \rho_A \otimes I_B ||_2

Math/Physics

Medium

Advanced Reasoning

Quantum Information

Yifan Zhang

null

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} <\Math>. I am expecting an answer that is a function of. <Math>\rho_{AB} - \rho_A \otimes I\rho_B<\Math>. Express your answer in Scahtten norms if it is possible. If you get <Math>||\rho_{AB} - \rho_A \otimes \rho_B ||_1 \le K <\Math> where K is the bound you obtain, answer K directly.

d^{3(NA+NB)/4} 4 ||\rho_{AB} - \rho_A \otimes \rho_B ||_2^2

Math/Physics

Hard

Advanced Reasoning

Quantum Information

Yifan Zhang

null

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

null

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

null

Consider the following Hamiltonian: <Math> \hat{H} = -\frac{1}{2} \sum_{n=1}^N \left\{ \left[J_0 + (-1)^{n-1} J_1 \right] \left[ \frac{1+\gamma}{2} \, \sigma_n^x \sigma_{n+1}^x + \frac{1-\gamma}{2} \, \sigma_n^y \sigma_{n+1}^y \right] + h \, \sigma_n^z \right\} - \frac{\Omega}{2} \sum_{n=1}^N \left( \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'.

Yes

Math/Physics

Hard

Advanced Reasoning

Quantum Information

Yuxuan Zhang

null

You are given three gates. <Math> Z_1(\theta_1) = exp(i \theta_1 |1><1|_1) <\Math> is a phase gate on qubit 1. <Math> Z_2(\theta_2) = exp(i \theta_2 |1><1|_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

Easy

Advanced Reasoning

Quantum Information

Yifan Zhang

null

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) <\Math> Given two-qubit density matrix: <Math> \rho = \begin{bmatrix} 0.5782 & 0.2853 - 0.1241i & 0.0008 - 0.0034i & 0.3634 + 0.1226i \\ 0.2853 + 0.1241i & 0.1674 & 0.0011 - 0.0015i & 0.1529 + 0.1385i \\ 0.0008 + 0.0034i & 0.0011 + 0.0015i & 0.00002 & -0.0002 + 0.0023i \\ 0.3634 - 0.1226i & 0.1529 - 0.1385i & -0.0002 - 0.0023i & 0.2544 \end{bmatrix} <\Math> What is the third Renyi negativity of this state (with partial transpose on the **first qubit**)? Round to the 3rd digit after the decimal point.

0.586

Math/Physics

Medium

Advanced Reasoning

Quantum Information

Yuxuan Zhang

null

the power of an adjoint of tensor product <Math>ad^k_{A\otimes B}</Math> can be expand into what form of tensor product of adjoint operator <Math> (?)^k </Math>, use <Math>\sigma</Math> to annotate the signs of commutator, e.g <Math>ad_{A,\sigma}</Math> where <Math>ad_{A,+}</Math> represents anti-commutator, <Math>ad_{A,-}</Math> represents commutator.

\frac{1}{2^k} (\sum_{\sigma_i\in{+,-} ad_{A,\sigma_i} \otimes ad_{B,-\sigma_i})^k

Math/Physics

Hard

Advanced Reasoning

Quantum Information

Xiu-Zhe Luo

null

One of the key conceptual hurdles that was overcome in the development of neutral atom-based optical clocks is the need to trap and confine the atoms (using an optical lattice). However, the accuracy and precision of such a clock degrades significantly in the presence of differential AC Stark shift between the two levels which comprise the clock superposition. Suppose a laser of frequency ω and intensity I creates a dipole trap for the atom. First, consider a two-level system with lower state |g⟩and excited state |e⟩ with energy 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⟩?

\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

null

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

null

\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

null

At what laser frequency ω_m does the transition frequency ω_eg become 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

null

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

null

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

null

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

null

How many different quantum states are there in a k-design? Assume the Hilbert space size is $d$, output your answer in an equation using combinatorics only

\binom{d + k - 1}{k}

Math/Physics

Easy

Basic Knowledge

Quantum Information

Yuxuan Zhang

null

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? Output a number or expression only.

1/6

Math/Physics

Hard

Advanced Reasoning

Quantum Information

Yuxuan Zhang

null

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

null

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} ~ 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}

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, and the Stefan-Boltzmann Constant \sigma_{SB}.

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 \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 (express in M_\odot ) 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. 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}}, 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. 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

null

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

null

Let ${N(t), t \ geq 0}$ be an RP with integer valued inter-renewal times with pmf $P(X_n = 0) = 1-\alpha$, $P(X_n = 1) = \alpha$, for $n \geq 1$, where $0<\alpha<1$. Compute $P(N(t) = k)$ for k = 0,1,2,....

P(N(t) = k) = \binom{t}{k} (1-\alpha)^{k-t}\alpha^{t+1}

Math/Physics

Medium

Advanced Reasoning

Probability/Statistics

Shunan Zheng

null

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

null

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

Condensed Matter Physics

Wucheng Zhang

null

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

Condensed Matter Physics

Wucheng Zhang

null