The multivariate Faa di Bruno formula and multivariate Taylor expansions with explicit integral remainder term

Copyright © Australian Mathematical Society This paper is made available with the permission of the Australian Mathematical Society Inc.The Faà di Bruno formulae for higher-order derivatives of a composite function are important in analysis for a variety of applications. There is a substantial literature on the univariate case, but despite significant applications the multivariate case has until recently received limited study. We present a succinct result which is a natural generalization of the univariate version. The derivation makes use of an explicit integral form of the remainder term for multivariate Taylor expansions.Roy B. Leipnik and Charles E. M. Pearc

Importance of the Internet in University Curriculums: A Case Study at Sam Houston State University

The advances in technology today have made the use of the Internet important in almost every discipline. Educators, business people, scientists and those in the criminal justice field all rely on the Internet to help them perform their jobs to the fullest. The Internet is immense and has many uses that can assist student in each discipline. Knowledge of the Internet and the full extent of its capabilities are important to anyone entering the workforce in today’s technologically advanced environment. In order to keep their graduates competitive in this environment, it is important that universities offer courses which not only cover the basics of Internet use but also show how the Internet can help them advance and excel no matter which field they enter. The current lack of a course for all students which covers in-depth internet use opposed to the number of fields that utilize the Internet, and the extent to which they use it, exposes the need for a course to better prepare students for the changing environment they will enter after graduation

Universal geometric approach to uncertainty, entropy and information

It is shown that for any ensemble, whether classical or quantum, continuous or discrete, there is only one measure of the "volume" of the ensemble that is compatible with several basic geometric postulates. This volume measure is thus a preferred and universal choice for characterising the inherent spread, dispersion, localisation, etc, of the ensemble. Remarkably, this unique "ensemble volume" is a simple function of the ensemble entropy, and hence provides a new geometric characterisation of the latter quantity. Applications include unified, volume-based derivations of the Holevo and Shannon bounds in quantum and classical information theory; a precise geometric interpretation of thermodynamic entropy for equilibrium ensembles; a geometric derivation of semi-classical uncertainty relations; a new means for defining classical and quantum localization for arbitrary evolution processes; a geometric interpretation of relative entropy; and a new proposed definition for the spot-size of an optical beam. Advantages of the ensemble volume over other measures of localization (root-mean-square deviation, Renyi entropies, and inverse participation ratio) are discussed.Comment: Latex, 38 pages + 2 figures; p(\alpha)->1/|T| in Eq. (72) [Eq. (A10) of published version

Field induced stationary state for an accelerated tracer in a bath

Our interest goes to the behavior of a tracer particle, accelerated by a constant and uniform external field, when the energy injected by the field is redistributed through collision to a bath of unaccelerated particles. A non equilibrium steady state is thereby reached. Solutions of a generalized Boltzmann-Lorentz equation are analyzed analytically, in a versatile framework that embeds the majority of tracer-bath interactions discussed in the literature. These results --mostly derived for a one dimensional system-- are successfully confronted to those of three independent numerical simulation methods: a direct iterative solution, Gillespie algorithm, and the Direct Simulation Monte Carlo technique. We work out the diffusion properties as well as the velocity tails: large v, and either large -v, or v in the vicinity of its lower cutoff whenever the velocity distribution is bounded from below. Particular emphasis is put on the cold bath limit, with scatterers at rest, which plays a special role in our model.Comment: 20 pages, 6 figures v3:minor corrections in sec.III and added reference

Strange Attractors in Dissipative Nambu Mechanics : Classical and Quantum Aspects

We extend the framework of Nambu-Hamiltonian Mechanics to include dissipation in $R^{3}$ phase space. We demonstrate that it accommodates the phase space dynamics of low dimensional dissipative systems such as the much studied Lorenz and R\"{o}ssler Strange attractors, as well as the more recent constructions of Chen and Leipnik-Newton. The rotational, volume preserving part of the flow preserves in time a family of two intersecting surfaces, the so called {\em Nambu Hamiltonians}. They foliate the entire phase space and are, in turn, deformed in time by Dissipation which represents their irrotational part of the flow. It is given by the gradient of a scalar function and is responsible for the emergence of the Strange Attractors. Based on our recent work on Quantum Nambu Mechanics, we provide an explicit quantization of the Lorenz attractor through the introduction of Non-commutative phase space coordinates as Hermitian $N \times N$ matrices in $R^{3}$. They satisfy the commutation relations induced by one of the two Nambu Hamiltonians, the second one generating a unique time evolution. Dissipation is incorporated quantum mechanically in a self-consistent way having the correct classical limit without the introduction of external degrees of freedom. Due to its volume phase space contraction it violates the quantum commutation relations. We demonstrate that the Heisenberg-Nambu evolution equations for the Quantum Lorenz system give rise to an attracting ellipsoid in the $3 N^{2}$ dimensional phase space.Comment: 35 pages, 4 figures, LaTe

Implementing large-system, value-based healthcare initiatives: a realist study protocol for seven natural experiments

IntroductionValue-based healthcare delivery models have emerged to address the unprecedented pressure on long-term health system performance and sustainability and to respond to the changing needs and expectations of patients. Implementing and scaling the benefits from these care delivery models to achieve large-system transformation are challenging and require consideration of complexity and context. Realist studies enable researchers to explore factors beyond ‘what works’ towards more nuanced understanding of ‘what tends to work for whom under which circumstances’. This research proposes a realist study of the implementation approach for seven large-system, value-based healthcare initiatives in New South Wales, Australia, to elucidate how different implementation strategies and processes stimulate the uptake, adoption, fidelity and adherence of initiatives to achieve sustainable impacts across a variety of contexts.Methods and analysisThis exploratory, sequential, mixed methods realist study followed RAMESES II (Realist And Meta-narrative Evidence Syntheses: Evolving Standards) reporting standards for realist studies. Stage 1 will formulate initial programme theories from review of existing literature, analysis of programme documents and qualitative interviews with programme designers, implementation support staff and evaluators. Stage 2 envisages testing and refining these hypothesised programme theories through qualitative interviews with local hospital network staff running initiatives, and analyses of quantitative data from the programme evaluation, hospital administrative systems and an implementation outcome survey. Stage 3 proposes to produce generalisable middle-range theories by synthesising data from context–mechanism–outcome configurations across initiatives. Qualitative data will be analysed retroductively and quantitative data will be analysed to identify relationships between the implementation strategies and processes, and implementation and programme outcomes. Mixed methods triangulation will be performed.Ethics and disseminationEthical approval has been granted by Macquarie University (Project ID 23816) and Hunter New England (Project ID 2020/ETH02186) Human Research Ethics Committees. The findings will be published in peer-reviewed journals. Results will be fed back to partner organisations and roundtable discussions with other health jurisdictions will be held, to share learnings.</jats:sec

Diversity sensitivity and multimodal Bayesian statistical analysis by relative entropy

© Australian Mathematical SocietyA list of recognised social diversities is assembled, including those used in social action programmes in the USA. Responses to diversity are discussed and diversity sensitivity defined as the derivative of response with respect to a defining parameter of a diversity distribution. Rewards (or penalties) for diversity are listed also; sensitivities to the responses to the rewards for diversity are called diversity sensitivities of the second kind. The statistics of bimodal and multimodal distributions are discussed, including the parametric estimation of such distributions by mixtures of multivariate normal distributions. An example is considered in detail.Roy B. Leipnik and C. E. M. Pearc

Thermodynamics, mnemonic matrices and generalized inverses

© Australian Mathematical Society 2007We present an alternative matrix mnemonic for the basic equations of simple thermodynamics. When normalized, this permits an explicit generalized inverse, allowing inversion of the mechanical and chemical thermodynamic equations. As an application, the natural variables S, V, P and T are derived from the four energies E (internal), F (free), G (Gibbs) and H (enthalpy).R.B. Leipnik and C.E.M. Pearc

The decoupling & solution of logistic & classical two-species lotka-volterra dynamics with variable production rates

Charles E. M. Pearce and Roy B. Leipnikhttp://trove.nla.gov.au/work/3545617