Common eigenfunctions of commuting differential operators of rank 2

In this paper we find coomon eigenfunctions of commuting differential operators of rank 2 with polynomial coefficients in some partial cases.Comment: 6 page

Commuting differential operators of rank 2 with polynomial coefficients

In this paper we study self-adjoint commuting ordinary differential operators with polynomial coefficients. These operators define commutative subalgebras of the first Weyl algebra. We find new examples of commuting operators of rank 2.Comment: 12 page

AKNS hierarchy and finite-gap Schrodinger potentials

In this paper we study AKNS hierarchy. We find explicit necessary conditions for functions $p$ and $q$ to be solution of some equation of AKNS hierarchy. Then we construct finite-gap Schrodinger potential using functions $p$ and $q$.Comment: 16 page

Explicit characterization of some commuting differential operators of rank 2

In this paper we consider differential opeartor L=d^4_x + u(x). We find the commutativity condition for operator L with a differential operator M of order 4g+2, where L and M are operators of rank 2. Some examples are constructed. These examples don't commute with differential opeartors of odd order

Products and connected sums of spheres as monotone Lagrangian submanifolds

We obtain new restrictions on Maslov classes of monotone Lagrangian submanifolds of $\mathbb{C}^n$. We also construct families of new examples of monotone Lagrangian submanifolds, which show that the restrictions on Maslov classes are sharp in certain cases.Comment: 18 pages, clarified and simplified. arXiv admin note: text overlap with arXiv:1812.0500

Noise-tolerant quantum speedups in quantum annealing without fine tuning

Quantum annealing is a powerful alternative model for quantum computing, which can succeed in the presence of environmental noise even without error correction. However, despite great effort, no conclusive proof of a quantum speedup (relative to state of the art classical algorithms) has been shown for these systems, and rigorous theoretical proofs of a quantum advantage generally rely on exponential precision in at least some aspects of the system, an unphysical resource guaranteed to be scrambled by random noise. In this work, we propose a new variant of quantum annealing, called RFQA, which can maintain a scalable quantum speedup in the face of noise and modest control precision. Specifically, we consider a modification of flux qubit-based quantum annealing which includes random, but coherent, low-frequency oscillations in the directions of the transverse field terms as the system evolves. We show that this method produces a quantum speedup for finding ground states in the Grover problem and quantum random energy model, and thus should be widely applicable to other hard optimization problems which can be formulated as quantum spin glasses. Further, we show that this speedup should be resilient to two realistic noise channels ($1/f$-like local potential fluctuations and local heating from interaction with a finite temperature bath), and that another noise channel, bath-assisted quantum phase transitions, actually accelerates the algorithm and may outweigh the negative effects of the others. The modifications we consider have a straightforward experimental implementation and could be explored with current technology.Comment: 21 pages, 7 figure

Localization of interacting fermions at high temperature

We suggest that if a localized phase at nonzero temperature $T>0$ exists for strongly disordered and weakly interacting electrons, as recently argued, it will also occur when both disorder and interactions are strong and $T$ is very high. We show that in this high-$T$ regime the localization transition may be studied numerically through exact diagonalization of small systems. We obtain spectra for one-dimensional lattice models of interacting spinless fermions in a random potential. As expected, the spectral statistics of finite-size samples cross over from those of orthogonal random matrices in the diffusive regime at weak random potential to Poisson statistics in the localized regime at strong randomness. However, these data show deviations from simple one-parameter finite-size scaling: the apparent mobility edge ``drifts'' as the system's size is increased. Based on spectral statistics alone, we have thus been unable to make a strong numerical case for the presence of a many-body localized phase at nonzero $T$

The Ideal Judge: How Implicit Bias Shapes Assessment of State Judges

Judicial Performance Evaluation (JPE) is generally seen as an important part of the merit system, which often suffers from a lack of relevant voter information. Utah’s JPE system has undergone significant change in recent years. Using data from the two most recent JPE surveys, we provide a preliminary look at the operation of this new system. Our results suggest that the survey component has difficulty distinguishing among the judges on the basis of relevant criteria. The question prompts intended to measure performance on different ABA categories are also indistinguishable. We find evidence that, on some measures, female judges do disproportionately worse than male judges. We suggest that the free response comments and the new Court Observation Program results may improve the ability of the commission to make meaningful distinctions among the judges on the basis of appropriate criteria