Negative Longitudinal Magnetoresistance in the Density Wave Phase of Y_{2}Ir_{2}O_{7}.

The ground state of nanowires of single-crystalline pyrochlore Y_{2}Ir_{2}O_{7} is a density wave. The application of a transverse magnetic field increases the threshold electric field for the collective depinning of the density wave state at a low temperature, leading to colossal magnetoresistance for voltages around the depinning threshold. This is in striking contrast to the case where even a vanishingly small longitudinal magnetic field sharply reduces the depinning threshold voltage, resulting in negative magnetoresistance. Ruling out several other possibilities, we argue that this phenomenon is likely to be a consequence of the chiral anomaly in the gapped out Weyl semimetal phase in Y_{2}Ir_{2}O_{7}.

A gate-tunable graphene Josephson parametric amplifier.

With a large portfolio of elemental quantum components, superconducting quantum circuits have contributed to advances in microwave quantum optics1. Of these elements, quantum-limited parametric amplifiers2-4 are essential for low noise readout of quantum systems whose energy range is intrinsically low (tens of µeV)5,6. They are also used to generate non-classical states of light that can be a resource for quantum enhanced detection7. Superconducting parametric amplifiers, such as quantum bits, typically use a Josephson junction as a source of magnetically tunable and dissipation-free non-linearity. In recent years, efforts have been made to introduce semiconductor weak links as electrically tunable non-linear elements, with demonstrations of microwave resonators and quantum bits using semiconductor nanowires8,9, a two-dimensional electron gas10, carbon nanotubes11 and graphene12,13. However, given the challenge of balancing non-linearity, dissipation, participation and energy scale, parametric amplifiers have not yet been implemented with a semiconductor weak link. Here, we demonstrate a parametric amplifier leveraging a graphene Josephson junction and show that its working frequency is widely tunable with a gate voltage. We report gain exceeding 20 dB and noise performance close to the standard quantum limit. Our results expand the toolset for electrically tunable superconducting quantum circuits. They also offer opportunities for the development of quantum technologies such as quantum computing, quantum sensing and for fundamental science14.

ON THE FAMILY OF ELLIPTIC CURVES X + 1/X + Y + 1/Y + t = 0.

We study various properties of the family of elliptic curves x+1/x+y+1/y+t = 0, which is isomorphic to the Weierstrass curve E t : Y 2 = X ( X 2 + ( t 2 4 - 2 ) X + 1 ) . . This equation arises from the study of the Mahler measure of polynomials. We show that the rank of E t ( Q ¯ ( t ) ) is 0 and the torsion subgroup of E t ( Q ( t ) ) is isomorphic to Z ∕ 4 Z . Over the rational field Q we obtain infinite subfamilies of ranks (at least) one and two, and find specific instances of Et with rank 5 and 6. We also determine all possible torsion subgroups of E t ( Q ) and conclude with some results regarding integral points in arithmetic progression on Et .

ELLIPTIC CURVES ARISING FROM THE TRIANGULAR NUMBERS.

We study the Legendre family of elliptic curves Et : y 2 = x(x - 1)(x - Δ t ), parametrized by triangular numbers Δ t = t(t + 1)/2. We prove that the rank of Et over the function field Q ‒ ( t ) is 1, while the rank is 0 over Q ( t ) . We also produce some infinite subfamilies whose Mordell-Weil rank is positive, and find high rank curves from within these families.

Integral Isosceles Triangle-Parallelogram and Heron triangle-Rhombus Pairs with a Common Area and Common Perimeter.

In this paper we show that there are infinitely many pairs of integer isosceles triangles and integer parallelograms with a common (integral) area and common perimeter. We also show that there are infinitely many Heron triangles and integer rhombuses with common area and common perimeter. As a corollary, we show there does not exist any Heron triangle and integer square which have a common area and common perimeter.