Exact and approximate energy spectrum for the finite square well and related potentials

We investigate the problem of a quantum particle in a one-dimensional finite square well. In the standard approach the allowed energies are determined implicitly as the solutions to a transcendental equation. We obtain the spectrum analytically as the solution to a pair of parametric equations and algebraically using a remarkably accurate approximation to the cosine function. The approach is also applied to a variety of other quantum wells

Does error control suppress spuriosity?

In the numerical solution of initial value ordinary differential equations, to what extent does local error control confer global properties? This work concentrates on global steady states or fixed points. It is shown that, for systems of equations, spurious fixed points generally cease to exist when local error control is used. For scalar problems, on the other hand, locally adaptive algorithms generally avoid spurious fixed points by an indirect method---the stepsize selection process causes spurious fixed points to be unstable. However, problem classes exist where, for arbitrarily small tolerances, stable spurious fixed points persist with significant basins of attraction. A technique is derived for generating such examples

Classicality of quantum information processing

The ultimate goal of the classicality programme is to quantify the amount of quantumness of certain processes. Here, classicality is studied for a restricted type of process: quantum information processing (QIP). Under special conditions, one can force some qubits of a quantum computer into a classical state without affecting the outcome of the computation. The minimal set of conditions is described and its structure is studied. Some implications of this formalism are the increase of noise robustness, a proof of the quantumness of mixed state quantum computing and a step forward in understanding the very foundation of QIP.Comment: Minor changes, published in Phys. Rev. A 65, 42319 (2002

Algebraic-geometrical formulation of two-dimensional quantum gravity

We find a volume form on moduli space of double punctured Riemann surfaces whose integral satisfies the Painlev\'e I recursion relations of the genus expansion of the specific heat of 2D gravity. This allows us to express the asymptotic expansion of the specific heat as an integral on an infinite dimensional moduli space in the spirit of Friedan-Shenker approach. We outline a conjectural derivation of such recursion relations using the Duistermaat-Heckman theorem.Comment: 10 pages, Latex fil

Estimating Error and Bias in Offline Evaluation Results

Offline evaluations of recommender systems attempt to estimate users’ satisfaction with recommendations using static data from prior user interactions. These evaluations provide researchers and developers with first approximations of the likely performance of a new system and help weed out bad ideas before presenting them to users. However, offline evaluation cannot accurately assess novel, relevant recommendations, because the most novel items were previously unknown to the user, so they are missing from the historical data and cannot be judged as relevant. We present a simulation study to estimate the error that such missing data causes in commonly-used evaluation metrics in order to assess its prevalence and impact. We find that missing data in the rating or observation process causes the evaluation protocol to systematically mis-estimate metric values, and in some cases erroneously determine that a popularity-based recommender outperforms even a perfect personalized recommender. Substantial breakthroughs in recommendation quality, therefore, will be difficult to assess with existing offline techniques

Conformally invariant bending energy for hypersurfaces

The most general conformally invariant bending energy of a closed four-dimensional surface, polynomial in the extrinsic curvature and its derivatives, is constructed. This invariance manifests itself as a set of constraints on the corresponding stress tensor. If the topology is fixed, there are three independent polynomial invariants: two of these are the straighforward quartic analogues of the quadratic Willmore energy for a two-dimensional surface; one is intrinsic (the Weyl invariant), the other extrinsic; the third invariant involves a sum of a quadratic in gradients of the extrinsic curvature -- which is not itself invariant -- and a quartic in the curvature. The four-dimensional energy quadratic in extrinsic curvature plays a central role in this construction.Comment: 16 page