I heard about this problem more than a year ago, I couldn’t solve it then and I stopped thinking about it, but I refused to look up a solution because I thought I could do it.

I have started looking into quant interview questions again and I went down a rabbit hole with one I found on quantable.io. It goes like this: Suppose you draw points $p = (x,y)$ uniformly at random from the unit square.

A basic python package featuring one module and some data files. proj/ ├─ package/ │ ├─ data/ │ │ ├─ __init__.py │ │ ├─ data_file.csv │ │ ├─ custom_style.mplstyle │ ├─ __init__.

Useful link: https://kb.rice.edu/page.php?id=108237 Connect to NOTS via SSH. If on campus, or at RGA/RVA, connect to “Rice Owls” network. (You can’t connect to NOTS if you’re on “Rice Visitor”.) If off-campus, follow these instructions to set up the Rice VPN.

Installing WSL2: https://www.omgubuntu.co.uk/how-to-install-wsl2-on-windows-10 This context menu is super useful: https://github.com/lextm/windowsterminal-shell.git

The cores of dense stars are a powerful laboratory for studying feebly-coupled particles such as axions. Some of the strongest constraints on axionlike particles and their couplings to ordinary matter derive from considerations of stellar axion emission. In this work we study the radiation of axionlike particles from degenerate neutron star matter via a lepton-flavor-violating (LFV) coupling that leads to muon-electron conversion when an axion is emitted. We calculate the axion emission rate per unit volume (emissivity) and by comparing with the rate of neutrino emission, we infer upper limits on the LFV coupling that are at the level of $|g_{ae\mu}| \lesssim 10^{-6}$. For the hotter environment of a supernova, such as SN 1987A, the axion emission rate is enhanced and the limit is stronger, at the level of $|g_{ae\mu}| \lesssim 10^{-11}$, competitive with laboratory limits. Interestingly, our derivation of the axion emissivity reveals that axion emission via the LFV coupling is suppressed relative to the familiar lepton-flavor-preserving channels by a factor of $T^2 E_{F,e}^2 / (m_\mu^2 - m_e^2)^2 \sim T^2/m_\mu^2$, which is responsible for the relatively weaker limits.

The presence of axion strings in the Universe after recombination can leave an imprint on the polarization pattern of the cosmic microwave background radiation through the phenomenon of axion-string-induced birefringence via the hyperlight axion-like particle’s coupling to electromagnetism. Across the sky, the polarization rotation angle is expected to display a patchwork of uniform regions with sharp boundaries that arise as the `shadow' of axion string loops. The statistics of such a birefringence sky map are therefore necessarily non-Gaussian. In this article we quantify the non-Gaussianity in axion-string-induced birefringence using two techniques, kurtosis and bispectrum, which correspond to 4- and 3-point correlation functions. If anisotropic birefringence were detected in the future, a measurement of its non-Gaussian properties would facilitate a discrimination across different new physics sources generally, and in the context of axion strings specifically, it would help to break degeneracies between the axion-photon coupling and properties of the string network.

Axion-like particles (ALPs) can form a network of cosmic strings and domain walls that survives after recombination and leads to anisotropic birefringence of the cosmic microwave background (CMB). In addition to studying cosmic strings, we clarify and emphasize how the formation of ALP-field domain walls impacts the cosmic birefringence signal; these observations provide a unique way of probing ALPs with masses in the range $3H_0\lesssim ma \lesssim 3H_\text{cmb}$. Using measurements of CMB birefringence from several telescopes, we find no evidence for axion-defect-induced anisotropic birefringence of the CMB. We extract constraints on the model parameters that include the ALP mass ma, ALP-photon coupling $A\propto g_{a\gamma\gamma f_a}$, the domain wall number $N_\mathrm{dw}$, and parameters characterizing the abundance and size of defects in the string-wall network. Considering also recent evidence for isotropic CMB birefringence, we find it difficult to accommodate this with the non-detection of anisotropic birefringence under the assumption that the signal is generated by an ALP defect network.

A new analysis is presented of the angular correlation function C(Θ) of cosmic microwave background temperature at large angular separation, based on published maps derived from Wilkinson Microwave Anisotropy Probe and Planck satellite data, using different models of astrophysical foregrounds. It is found that using a common analysis, the results from the two satellites are very similar. In particular, it is found that previously published differences between measured values of C(Θ) near Θ = 90° arise mainly from different choices of masks in regions of largest Galactic emissions, and that demonstrated measurement biases are reduced by eliminating masks altogether. Maps from both satellites are shown to agree with C(90°) = 0 to within estimated statistical and systematic errors, consistent with an exact symmetry predicted in a new holographic quantum model of inflation.