1,025 research outputs found
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 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 . 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 , 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
-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 (), 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 , 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
and . 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 , 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
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