Racah Polynomials and Recoupling Schemes of su(1,1)\mathfrak{su}(1,1)

The connection between the recoupling scheme of four copies of $\mathfrak{su}(1,1)$, the generic superintegrable system on the 3 sphere, and bivariate Racah polynomials is identified. The Racah polynomials are presented as connection coefficients between eigenfunctions separated in different spherical coordinate systems and equivalently as different irreducible decompositions of the tensor product representations. As a consequence of the model, an extension of the quadratic algebra ${\rm QR}(3)$ is given. It is shown that this algebra closes only with the inclusion of an additional shift operator, beyond the eigenvalue operators for the bivariate Racah polynomials, whose polynomial eigenfunctions are determined. The duality between the variables and the degrees, and hence the bispectrality of the polynomials, is interpreted in terms of expansion coefficients of the separated solutions

Models of Quadratic Algebras Generated by Superintegrable Systems in 2D

In this paper, we consider operator realizations of quadratic algebras generated by second-order superintegrable systems in 2D. At least one such realization is given for each set of St\"ackel equivalent systems for both degenerate and nondegenerate systems. In almost all cases, the models can be used to determine the quantization of energy and eigenvalues for integrals associated with separation of variables in the original system

An algebraic interpretation of the multivariate qq-Krawtchouk polynomials

The multivariate quantum $q$-Krawtchouk polynomials are shown to arise as matrix elements of "$q$-rotations" acting on the state vectors of many $q$-oscillators. The focus is put on the two-variable case. The algebraic interpretation is used to derive the main properties of the polynomials: orthogonality, duality, structure relations, difference equations and recurrence relations. The extension to an arbitrary number of variables is presentedComment: 22 pages; minor correction

Two-Variable Wilson Polynomials and the Generic Superintegrable System on the 3-Sphere

We show that the symmetry operators for the quantum superintegrable system on the 3-sphere with generic 4-parameter potential form a closed quadratic algebra with 6 linearly independent generators that closes at order 6 (as differential operators). Further there is an algebraic relation at order 8 expressing the fact that there are only 5 algebraically independent generators. We work out the details of modeling physically relevant irreducible representations of the quadratic algebra in terms of divided difference operators in two variables. We determine several ON bases for this model including spherical and cylindrical bases. These bases are expressed in terms of two variable Wilson and Racah polynomials with arbitrary parameters, as defined by Tratnik. The generators for the quadratic algebra are expressed in terms of recurrence operators for the one-variable Wilson polynomials. The quadratic algebra structure breaks the degeneracy of the space of these polynomials. In an earlier paper the authors found a similar characterization of one variable Wilson and Racah polynomials in terms of irreducible representations of the quadratic algebra for the quantum superintegrable system on the 2-sphere with generic 3-parameter potential. This indicates a general relationship between 2nd order superintegrable systems and discrete orthogonal polynomials

Algebraic calculations for spectrum of superintegrable system from exceptional orthogonal polynomials

We introduce an extended Kepler-Coulomb quantum model in spherical coordinates. The Schr\"{o}dinger equation of this Hamiltonian is solved in these coordinates and it is shown that the wave functions of the system can be expressed in terms of Laguerre, Legendre and exceptional Jacobi polynomials (of hypergeometric type). We construct ladder and shift operators based on the corresponding wave functions and obtain their recurrence formulas. These recurrence relations are used to construct higher-order, algebraically independent integrals of motion to prove superintegrability of the Hamiltonian. The integrals form a higher rank polynomial algebra. By constructing the structure functions of the associated deformed oscillator algebras we derive the degeneracy of energy spectrum of the superintegrable system.Comment: 20 page

A US hospital budget impact analysis of a skin closure system compared with standard of care in hip and knee arthroplasty.

Background: Medicare\u27s mandatory bundle for hip and knee arthroplasty necessitates provider accountability for quality and cost of care to 90 days, and wound closure may be a key area of consideration. The DERMABOND Methods: A 90-day economic model was developed assuming 500 annual hip/knee arthroplasties for a typical US hospital setting. In current practice, wound closure methods for the final skin layer were set to 50% sutures and 50% staples. In future practice, this distribution shifted to 20% sutures, 20% staples, and 60% Skin Closure System. Health care resources included materials (eg, staplers, steri-strips, and traditional/barbed sutures), standard or premium dressings, outpatient visits, and home care visits. An Expert Panel, comprised of three orthopedic physician assistants, two orthopedic surgeons, and a home health representative, was used to inform several model parameters. Other inputs were informed by national data or literature. Unit costs were based on list prices in 2016 US dollars. Uncertainty in the model was explored through one-way sensitivity and alternative scenario analyses. Results: The analysis predicted that use of Skin Closure System in the future practice could achieve cost savings of $56.70 to$79.62 per patient, when standard or premium wound dressings are used, respectively. This translated to an annual hospital budgetary savings ranging from $28,349 to$39,809 when assuming 500 arthroplasties. Dressing materials and postoperative health care visits were key model drivers. Conclusions: Use of the Skin Closure System may provide cost savings within hip and knee arthroplasties due to decreases in resource utilization in the postacute care setting