Expansion formulas for terminating balanced 4F3-series from the Biedenharn–Elliot identity for su(1,1)

AbstractIn a recent paper, George Gasper (Contemp. Math. 254 (2000) 187) proved some expansion formulas for terminating balanced hypergeometric series of type 4F3 with unit argument. In this article we show how one easily derives such expansion formulas from the Biedenharn–Elliot identity for the Lie algebra su(1,1). Furthermore, we give a rather systematic method for determining when two apparently different expansion formulas are the same up to transformation formulas. This is a rather nice application of the so-called invariance groups of hypergeometric series. The method extends to other cases; we briefly indicate how it works in the case of expansion formulas for 3F2-series. We conclude with some basic analogues and show their relation with the Askey–Wilson polynomials

Widening access in selection using situational judgement tests: evidence from the UKCAT

CONTEXT Widening access promotes student diversity and the appropriate representation of all demographic groups. This study aims to examine diversity-related benefits of the use of situational judgement tests (SJTs) in the UK Clinical Aptitude Test (UKCAT) in terms of three demographic variables: (i) socioeconomic status (SES); (ii) ethnicity, and (iii) gender. METHODS Outcomes in medical and dental school applicant cohorts for the years 2012 (n = 15 581) and 2013 (n = 15 454) were studied. Applicants' scores on cognitive tests and an SJT were linked to SES (parents' occupational status), ethnicity (White versus Black and other minority ethnic candidates), and gender. RESULTS Firstly, the effect size for SES was lower for the SJT (d = 0.13-0.20 in favour of the higher SES group) than it was for the cognitive tests (d = 0.38-0.35). Secondly, effect sizes for ethnicity of the SJT and cognitive tests were similar (d = similar to 0.50 in favour of White candidates). Thirdly, males outperformed females on cognitive tests, whereas the reverse was true for SJTs. When equal weight was given to the SJT and the cognitive tests in the admission decision and when the selection ratio was stringent, simulated scenarios showed that using an SJT in addition to cognitive tests might enable admissions boards to select more students from lower SES backgrounds and more female students. CONCLUSIONS The SJT has the potential to appropriately complement cognitive tests in the selection of doctors and dentists. It may also put candidates of lower SES backgrounds at less of a disadvantage and may potentially diversify the student intake. However, use of the SJT applied in this study did not diminish the role of ethnicity. Future research should examine these findings with other SJTs and other tests internationally and scrutinise the causes underlying the role of ethnicity

Harmonic oscillator chains as Wigner Quantum Systems: periodic and fixed wall boundary conditions in gl(1|n) solutions

We describe a quantum system consisting of a one-dimensional linear chain of n identical harmonic oscillators coupled by a nearest neighbor interaction. Two boundary conditions are taken into account: periodic boundary conditions (where the nth oscillator is coupled back to the first oscillator) and fixed wall boundary conditions (where the first oscillator and the $n$th oscillator are coupled to a fixed wall). The two systems are characterized by their Hamiltonian. For their quantization, we treat these systems as Wigner Quantum Systems (WQS), allowing more solutions than just the canonical quantization solution. In this WQS approach, one is led to certain algebraic relations for operators (which are linear combinations of position and momentum operators) that should satisfy triple relations involving commutators and anti-commutators. These triple relations have a solution in terms of the Lie superalgebra gl(1|n). We study a particular class of gl(1|n) representations V(p), the so-called ladder representations. For these representations, we determine the spectrum of the Hamiltonian and of the position operators (for both types of boundary conditions). Furthermore, we compute the eigenvectors of the position operators in terms of stationary states. This leads to explicit expressions for position probabilities of the n oscillators in the chain. An analysis of the plots of such position probability distributions gives rise to some interesting observations. In particular, the physical behavior of the system as a WQS is very much in agreement with what one would expect from the classical case, except that all physical quantities (energy, position and momentum of each oscillator) have a finite spectrum

On the eigenvalue problem for arbitrary odd elements of the Lie superalgebra gl(1|n) and applications

In a Wigner quantum mechanical model, with a solution in terms of the Lie superalgebra gl(1|n), one is faced with determining the eigenvalues and eigenvectors for an arbitrary self-adjoint odd element of gl(1|n) in any unitary irreducible representation W. We show that the eigenvalue problem can be solved by the decomposition of W with respect to the branching gl(1|n) --> gl(1|1) + gl(n-1). The eigenvector problem is much harder, since the Gel'fand-Zetlin basis of W is involved, and the explicit actions of gl(1|n) generators on this basis are fairly complicated. Using properties of the Gel'fand-Zetlin basis, we manage to present a solution for this problem as well. Our solution is illustrated for two special classes of unitary gl(1|n) representations: the so-called Fock representations and the ladder representations

The paraboson Fock space and unitary irreducible representations of the Lie superalgebra osp(1|2n)

It is known that the defining relations of the orthosymplectic Lie superalgebra osp(1|2n) are equivalent to the defining (triple) relations of n pairs of paraboson operators $b^\pm_i$. In particular, with the usual star conditions, this implies that the ``parabosons of order p'' correspond to a unitary irreducible (infinite-dimensional) lowest weight representation V(p) of osp(1|2n). Apart from the simple cases p=1 or n=1, these representations had never been constructed due to computational difficulties, despite their importance. In the present paper we give an explicit and elegant construction of these representations V(p), and we present explicit actions or matrix elements of the osp(1|2n) generators. The orthogonal basis vectors of V(p) are written in terms of Gelfand-Zetlin patterns, where the subalgebra u(n) of osp(1|2n) plays a crucial role. Our results also lead to character formulas for these infinite-dimensional osp(1|2n) representations. Furthermore, by considering the branching $osp(1|2n) \supset sp(2n) \supset u(n)$, we find explicit infinite-dimensional unitary irreducible lowest weight representations of sp(2n) and their characters.Comment: typos correcte

The Wigner function of a q-deformed harmonic oscillator model

The phase space representation for a q-deformed model of the quantum harmonic oscillator is constructed. We have found explicit expressions for both the Wigner and Husimi distribution functions for the stationary states of the $q$-oscillator model under consideration. The Wigner function is expressed as a basic hypergeometric series, related to the Al-Salam-Chihara polynomials. It is shown that, in the limit case $h \to 0$ ($q \to 1$), both the Wigner and Husimi distribution functions reduce correctly to their well-known non-relativistic analogues. Surprisingly, examination of both distribution functions in the q-deformed model shows that, when $q \ll 1$, their behaviour in the phase space is similar to the ground state of the ordinary quantum oscillator, but with a displacement towards negative values of the momentum. We have also computed the mean values of the position and momentum using the Wigner function. Unlike the ordinary case, the mean value of the momentum is not zero and it depends on $q$ and $n$. The ground-state like behaviour of the distribution functions for excited states in the q-deformed model opens quite new perspectives for further experimental measurements of quantum systems in the phase space.Comment: 16 pages, 24 EPS figures, uses IOP style LaTeX, some misprints are correctd and journal-reference is adde

Interacting universes and the cosmological constant

We study some collective phenomena that may happen in a multiverse scenario. First, it is posed an interaction scheme between universes whose evolution is dominated by a cosmological constant. As a result of the interaction, the value of the cosmological constant of one of the universes becomes very close to zero at the expense of an increasing value of the cosmological constant of the partner universe. Second, we found normal modes for a 'chain' of interacting universes. The energy spectrum of the multiverse, being this taken as a collective system, splits into a large number of levels, some of which correspond to a value of the cosmological constant very close to zero. We finally point out that the multiverse may be much more than the mere sum of its parts.Comment: 7 page

Harmonic oscillators coupled by springs: discrete solutions as a Wigner Quantum System

We consider a quantum system consisting of a one-dimensional chain of M identical harmonic oscillators with natural frequency $\omega$, coupled by means of springs. Such systems have been studied before, and appear in various models. In this paper, we approach the system as a Wigner Quantum System, not imposing the canonical commutation relations, but using instead weaker relations following from the compatibility of Hamilton's equations and the Heisenberg equations. In such a setting, the quantum system allows solutions in a finite-dimensional Hilbert space, with a discrete spectrum for all physical operators. We show that a class of solutions can be obtained using generators of the Lie superalgebra gl(1|M). Then we study the properties and spectra of the physical operators in a class of unitary representations of gl(1|M). These properties are both interesting and intriguing. In particular, we can give a complete analysis of the eigenvalues of the Hamiltonian and of the position and momentum operators (including multiplicities). We also study probability distributions of position operators when the quantum system is in a stationary state, and the effect of the position of one oscillator on the positions of the remaining oscillators in the chain

Enhanced analysis of real-time PCR data by using a variable efficiency model: FPK-PCR

Current methodology in real-time Polymerase chain reaction (PCR) analysis performs well provided PCR efficiency remains constant over reactions. Yet, small changes in efficiency can lead to large quantification errors. Particularly in biological samples, the possible presence of inhibitors forms a challenge. We present a new approach to single reaction efficiency calculation, called Full Process Kinetics-PCR (FPK-PCR). It combines a kinetically more realistic model with flexible adaptation to the full range of data. By reconstructing the entire chain of cycle efficiencies, rather than restricting the focus on a ‘window of application’, one extracts additional information and loses a level of arbitrariness. The maximal efficiency estimates returned by the model are comparable in accuracy and precision to both the golden standard of serial dilution and other single reaction efficiency methods. The cycle-to-cycle changes in efficiency, as described by the FPK-PCR procedure, stay considerably closer to the data than those from other S-shaped models. The assessment of individual cycle efficiencies returns more information than other single efficiency methods. It allows in-depth interpretation of real-time PCR data and reconstruction of the fluorescence data, providing quality control. Finally, by implementing a global efficiency model, reproducibility is improved as the selection of a window of application is avoided.JRC.I.3-Molecular Biology and Genomic