The discretised harmonic oscillator: Mathieu functions and a new class of generalised Hermite polynomials

We present a general, asymptotical solution for the discretised harmonic oscillator. The corresponding Schr\"odinger equation is canonically conjugate to the Mathieu differential equation, the Schr\"odinger equation of the quantum pendulum. Thus, in addition to giving an explicit solution for the Hamiltonian of an isolated Josephon junction or a superconducting single-electron transistor (SSET), we obtain an asymptotical representation of Mathieu functions. We solve the discretised harmonic oscillator by transforming the infinite-dimensional matrix-eigenvalue problem into an infinite set of algebraic equations which are later shown to be satisfied by the obtained solution. The proposed ansatz defines a new class of generalised Hermite polynomials which are explicit functions of the coupling parameter and tend to ordinary Hermite polynomials in the limit of vanishing coupling constant. The polynomials become orthogonal as parts of the eigenvectors of a Hermitian matrix and, consequently, the exponential part of the solution can not be excluded. We have conjectured the general structure of the solution, both with respect to the quantum number and the order of the expansion. An explicit proof is given for the three leading orders of the asymptotical solution and we sketch a proof for the asymptotical convergence of eigenvectors with respect to norm. From a more practical point of view, we can estimate the required effort for improving the known solution and the accuracy of the eigenvectors. The applied method can be generalised in order to accommodate several variables.Comment: 18 pages, ReVTeX, the final version with rather general expression

Replication, Communication, and the Population Dynamics of Scientific Discovery

Many published research results are false, and controversy continues over the roles of replication and publication policy in improving the reliability of research. Addressing these problems is frustrated by the lack of a formal framework that jointly represents hypothesis formation, replication, publication bias, and variation in research quality. We develop a mathematical model of scientific discovery that combines all of these elements. This model provides both a dynamic model of research as well as a formal framework for reasoning about the normative structure of science. We show that replication may serve as a ratchet that gradually separates true hypotheses from false, but the same factors that make initial findings unreliable also make replications unreliable. The most important factors in improving the reliability of research are the rate of false positives and the base rate of true hypotheses, and we offer suggestions for addressing each. Our results also bring clarity to verbal debates about the communication of research. Surprisingly, publication bias is not always an obstacle, but instead may have positive impacts---suppression of negative novel findings is often beneficial. We also find that communication of negative replications may aid true discovery even when attempts to replicate have diminished power. The model speaks constructively to ongoing debates about the design and conduct of science, focusing analysis and discussion on precise, internally consistent models, as well as highlighting the importance of population dynamics

Improving Science That Uses Code

As code is now an inextricable part of science it should be supported by competent Software Engineering, analogously to statistical claims being properly supported by competent statistics.If and when code avoids adequate scrutiny, science becomes unreliable and unverifiable because results — text, data, graphs, images, etc — depend on untrustworthy code.Currently, scientists rarely assure the quality of the code they rely on, and rarely make it accessible for scrutiny. Even when available, scientists rarely provide adequate documentation to understand or use it reliably.This paper proposes and justifies ways to improve science using code:1. Professional Software Engineers can help, particularly in critical fields such as public health, climate change and energy.2. ‘Software Engineering Boards,’ analogous to Ethics or Institutional Review Boards, should be instigated and used.3. The Reproducible Analytic Pipeline (RAP) methodology can be generalized to cover code and Software Engineering methodologies, in a generalization this paper introduces called RAP+. RAP+ (or comparable interventions) could be supported and or even required in journal, conference and funding body policies.The paper’s Supplemental Material provides a summary of Software Engineering best practice relevant to scientific research, including further suggestions for RAP+ workflows.‘Science is what we understand well enough to explain to a computer.’ Donald E. Knuth in A=B [ 1]‘I have to write to discover what I am doing.’ Flannery O’Connor, quoted in Write for your life [ 2]‘Criticism is the mother of methodology.’ Robert P. Abelson in Statistics as Principled Argument [ 3]‘From its earliest times, science has operated by being open and transparent about methods and evidence, regardless of which technology has been in vogue.’ Editorial in Nature [4