A neural network solves, explains, and generates university math problems by program synthesis and few-shot learning at human level.

We demonstrate that a neural network pretrained on text and fine-tuned on code solves mathematics course problems, explains solutions, and generates questions at a human level. We automatically synthesize programs using few-shot learning and OpenAI's Codex transformer and execute them to solve course problems at 81% automatic accuracy. We curate a dataset of questions from Massachusetts Institute of Technology (MIT)'s largest mathematics courses (Single Variable and Multivariable Calculus, Differential Equations, Introduction to Probability and Statistics, Linear Algebra, and Mathematics for Computer Science) and Columbia University's Computational Linear Algebra.

Observations of the Outer Heliosphere, Heliosheath, and Interstellar Medium.

The Voyager spacecraft have left the heliosphere and entered the interstellar medium, making the first observations of the termination shock, heliosheath, and heliopause. New Horizons is observing the solar wind in the outer heliosphere and making the first direct observations of solar wind pickup ions. This paper reviews the observations of the solar wind plasma and magnetic fields throughout the heliosphere and in the interstellar medium.

Evidence of Large Recoil Velocity from a Black Hole Merger Signal.

The final black hole left behind after a binary black hole merger can attain a recoil velocity, or a "kick," reaching values up to 5000  km/s. This phenomenon has important implications for gravitational wave astronomy, black hole formation scenarios, testing general relativity, and galaxy evolution. We consider the gravitational wave signal from the binary black hole merger GW200129_065458 (henceforth referred to as GW200129), which has been shown to exhibit strong evidence of orbital precession.

Shocks in the Very Local Interstellar Medium.

Large-scale disturbances generated by the Sun's dynamics first propagate through the heliosphere, influence the heliosphere's outer boundaries, and then traverse and modify the very local interstellar medium (VLISM). The existence of shocks in the VLISM was initially suggested by Voyager observations of the 2-3 kHz radio emissions in the heliosphere. A couple of decades later, both Voyagers crossed the definitive edge of our heliosphere and became the first ever spacecraft to sample interstellar space.

Suppression of black-hole growth by strong outflows at redshifts 5.8-6.6.

Bright quasars, powered by accretion onto billion-solar-mass black holes, already existed at the epoch of reionization, when the Universe was 0.5-1 billion years old. How these black holes formed in such a short time is the subject of debate, particularly as they lie above the correlation between black-hole mass and galaxy dynamical mass in the local Universe.

A 62-minute orbital period black widow binary in a wide hierarchical triple.

Over a dozen millisecond pulsars are ablating low-mass companions in close binary systems. In the original 'black widow', the eight-hour orbital period eclipsing pulsar PSR J1959+2048 (PSR B1957+20), high-energy emission originating from the pulsar is irradiating and may eventually destroy a low-mass companion. These systems are not only physical laboratories that reveal the interesting results of exposing a close companion star to the relativistic energy output of a pulsar, but are also believed to harbour some of the most massive neutron stars, allowing for robust tests of the neutron star equation of state.

An exomoon survey of 70 cool giant exoplanets and the new candidate Kepler-1708 b-i.

Exomoons represent a crucial missing puzzle piece in our efforts to understand extrasolar planetary systems. To address this deficiency, we here describe an exomoon survey of 70 cool, giant transiting exoplanet candidates found by Kepler. We identify only one exhibiting a moon-like signal that passes a battery of vetting tests: Kepler-1708 b.

A repeating fast radio burst source in a globular cluster.

Fast radio bursts (FRBs) are flashes of unknown physical origin. The majority of FRBs have been seen only once, although some are known to generate multiple flashes. Many models invoke magnetically powered neutron stars (magnetars) as the source of the emission.

Hints of Spin-Orbit Resonances in the Binary Black Hole Population.

Binary black hole spin measurements from gravitational wave observations can reveal the binary's evolutionary history. In particular, the spin orientations of the component black holes within the orbital plane, ϕ_{1} and ϕ_{2}, can be used to identify binaries caught in the so-called spin-orbit resonances. In a companion paper, we demonstrate that ϕ_{1} and ϕ_{2} are best measured near the merger of the two black holes.

GJ 367b: A dense, ultrashort-period sub-Earth planet transiting a nearby red dwarf star.

Ultrashort-period (USP) exoplanets have orbital periods shorter than 1 day. Precise masses and radii of USP exoplanets could provide constraints on their unknown formation and evolution processes. We report the detection and characterization of the USP planet GJ 367b using high-precision photometry and radial velocity observations.

Bayesian Inference for Gravitational Waves from Binary Neutron Star Mergers in Third Generation Observatories.

Third generation (3G) gravitational-wave detectors will observe thousands of coalescing neutron star binaries with unprecedented fidelity. Extracting the highest precision science from these signals is expected to be challenging owing to both high signal-to-noise ratios and long-duration signals. We demonstrate that current Bayesian inference paradigms can be extended to the analysis of binary neutron star signals without breaking the computational bank.

r-Process elements from magnetorotational hypernovae.

Neutron-star mergers were recently confirmed as sites of rapid-neutron-capture (r-process) nucleosynthesis. However, in Galactic chemical evolution models, neutron-star mergers alone cannot reproduce the observed element abundance patterns of extremely metal-poor stars, which indicates the existence of other sites of r-process nucleosynthesis. These sites may be investigated by studying the element abundance patterns of chemically primitive stars in the halo of the Milky Way, because these objects retain the nucleosynthetic signatures of the earliest generation of stars.

Reverberation mapping of active galactic nuclei: from X-ray corona to dusty torus.

The central engines of active galactic nuclei (AGNs) are powered by accreting supermassive black holes, and while AGNs are known to play an important role in galaxy evolution, the key physical processes occur on scales that are too small to be resolved spatially (aside from a few exceptional cases). Reverberation mapping is a powerful technique that overcomes this limitation by using echoes of light to determine the geometry and kinematics of the central regions. Variable ionizing radiation from close to the black hole drives correlated variability in surrounding gas/dust but with a time delay due to the light travel time between the regions, allowing reverberation mapping to effectively replace spatial resolution with time resolution.

The first 5 years of gravitational-wave astrophysics.

Gravitational waves are ripples in spacetime generated by the acceleration of astrophysical objects; a direct consequence of general relativity, they were first directly observed in 2015. Here, I review the first 5 years of gravitational-wave detections. More than 50 gravitational-wave events have been found, emitted by pairs of merging compact objects such as neutron stars and black holes.

New Spin on LIGO-Virgo Binary Black Holes.

Gravitational waves from binary black holes have the potential to yield information on both of the intrinsic parameters that characterize the compact objects: their masses and spins. While the component masses are usually resolvable, the component spins have proven difficult to measure. This limitation stems in great part from our choice to inquire about the spins of the most and least massive objects in each binary, a question that becomes ill defined when the masses are equal.

Constraints on Ultralight Scalar Bosons within Black Hole Spin Measurements from the LIGO-Virgo GWTC-2.

Clouds of ultralight bosons-such as axions-can form around a rapidly spinning black hole, if the black hole radius is comparable to the bosons' wavelength. The cloud rapidly extracts angular momentum from the black hole, and reduces it to a characteristic value that depends on the boson's mass as well as on the black hole mass and spin. Therefore, a measurement of a black hole mass and spin can be used to reveal or exclude the existence of such bosons.

X-ray quasi-periodic eruptions from two previously quiescent galaxies.

Quasi-periodic eruptions (QPEs) are very-high-amplitude bursts of X-ray radiation recurring every few hours and originating near the central supermassive black holes of galactic nuclei. It is currently unknown what triggers these events, how long they last and how they are connected to the physical properties of the inner accretion flows. Previously, only two such sources were known, found either serendipitously or in archival data, with emission lines in their optical spectra classifying their nuclei as hosting an actively accreting supermassive black hole.

Enhanced x-ray emission coinciding with giant radio pulses from the Crab Pulsar.

Giant radio pulses (GRPs) are sporadic bursts emitted by some pulsars that last a few microseconds and are hundreds to thousands of times brighter than regular pulses from these sources. The only GRP-associated emission outside of radio wavelengths is from the Crab Pulsar, where optical emission is enhanced by a few percentage points during GRPs. We observed the Crab Pulsar simultaneously at x-ray and radio wavelengths, finding enhancement of the x-ray emission by 3.

A nearby transiting rocky exoplanet that is suitable for atmospheric investigation.

Spectroscopy of transiting exoplanets can be used to investigate their atmospheric properties and habitability. Combining radial velocity (RV) and transit data provides additional information on exoplanet physical properties. We detect a transiting rocky planet with an orbital period of 1.

Constraints on Axionlike Particles from a Hard X-Ray Observation of Betelgeuse.

We use the first observation of Betelgeuse in hard x rays to perform a novel search for axionlike particles (ALPs). Betelgeuse is not expected to be a standard source of x rays, but light ALPs produced in the stellar core could be converted back into photons in the Galactic magnetic field, producing a detectable flux that peaks in the hard x-ray band (E_{γ}>10  keV). Using a 50 ks observation of Betelgeuse by the NuSTAR satellite telescope, we find no significant excess of events above the expected background.

Inference of the Neutron Star Equation of State from Cosmological Distances.

Finite-size effects on the gravitational wave signal from a neutron star merger typically manifest at high frequencies where detector sensitivity decreases. Proposed sensitivity improvements can give us access both to stronger signals and to a myriad of weak signals from cosmological distances. The latter will outnumber the former and the relevant part of the signal will be redshifted towards the detector's most sensitive band.

Systematic Uncertainty of Standard Sirens from the Viewing Angle of Binary Neutron Star Inspirals.

The independent measurement of the Hubble constant with gravitational-wave standard sirens will potentially shed light on the tension between the local distance ladders and Planck experiments. Therefore, thorough understanding of the sources of systematic uncertainty for the standard siren method is crucial. In this Letter, we focus on two scenarios that will potentially dominate the systematic uncertainty of standard sirens.

Astrophysical Implications of GW190412 as a Remnant of a Previous Black-Hole Merger.

Two of the dominant channels to produce merging stellar-mass black-hole binaries are believed to be the isolated evolution of binary stars in the field and dynamical formation in star clusters. The first reported black-hole binary event from the third LIGO/Virgo observing run (GW190412) is unusual in that it has unequal masses, nonzero effective spin, and nonzero primary spin at 90% confidence interval. We show that this event should be exceedingly rare in the context of both the field and cluster formation scenarios.

A giant planet candidate transiting a white dwarf.

Astronomers have discovered thousands of planets outside the Solar System, most of which orbit stars that will eventually evolve into red giants and then into white dwarfs. During the red giant phase, any close-orbiting planets will be engulfed by the star, but more distant planets can survive this phase and remain in orbit around the white dwarf. Some white dwarfs show evidence for rocky material floating in their atmospheres, in warm debris disks or orbiting very closely, which has been interpreted as the debris of rocky planets that were scattered inwards and tidally disrupted.