Periodic Integrable Systems with Delta-Potentials

In this paper we study root system generalizations of the quantum Bose-gas on the circle with pair-wise delta-function interactions. The underlying symmetry structures are shown to be governed by the associated graded algebra of Cherednik's (suitably filtered) degenerate double affine Hecke algebra, acting by Dunkl-type differential-reflection operators. We use Gutkin's generalization of the equivalence between the impenetrable Bose-gas and the free Fermi-gas to derive the Bethe ansatz equations and the Bethe ansatz eigenfunctions.     

On type basic hypergeometric orthogonal polynomials

The five parameter family of Koornwinder's multivariable analogues of the Askey-Wilson polynomials is studied with four parameters generically complex. The Koornwinder polynomials form an orthogonal system with respect to an explicit (in general complex) measure. A partly discrete orthogonality measure is obtained by shifting the contour to the torus while picking up residues. A parameter domain is given for which the partly discrete orthogonality measure is positive. The orthogonality relations and norm evaluations for multivariable -Racah polynomials and multivariable big and little -Jacobi polynomials are proved by taking suitable limits in the orthogonality relations for the Koornwinder polynomials. In particular new proofs of several well-known -analogues of the Selberg integral are obtained.

     

Properties of Generalized Univariate Hypergeometric Functions

Based on Spiridonov’s analysis of elliptic generalizations of the Gauss hypergeometric function, we develop a common framework for 7-parameter families of generalized elliptic, hyperbolic and trigonometric univariate hypergeometric functions. In each case we derive the symmetries of the generalized hypergeometric function under the Weyl group of type E ₇ (elliptic, hyperbolic) and of type E ₆ (trigonometric) using the appropriate versions of the Nassrallah-Rahman beta integral, and we derive contiguous relations using fundamental addition formulas for theta and sine functions. The top level degenerations of the hyperbolic and trigonometric hypergeometric functions are identified with Ruijsenaars’ relativistic hypergeometric function and the Askey-Wilson function, respectively. We show that the degeneration process yields various new and known identities for hyperbolic and trigonometric special functions. We also describe an intimate connection between the hyperbolic and trigonometric theory, which yields an expression of the hyperbolic hypergeometric function as an explicit bilinear sum in trigonometric hypergeometric functions.     

Strategic control and interests,its effects on decision outcomes

Application of the model to artificial data shows that actors with strong preferences in the center have more possibilities to realize good outcomes than other actors. On the basis of an empirical application it is shown that a Nash equilibrium does not always arise after a large number of iterations unless actors have learning capabilities or are severely restricted in their strategic behavior.

In political systems and large organizations, ultimate decision makers are usually just a small subset of all actors in the social system. To arrive at acceptable decisions, decision makers have to take into account the preferences of other actors in the system. Typically preferences of more interested and more powerful actors are weighted heavier than those of less interested and powerful actors. This implies that the total leverage of an actor on the decision is determined by the combination of his power (his potential) and his interest (his willingness to mobilize his power). As the exact level of an actor's leverage is difficult to estimate for the other actors in the system, an actor is able to optimize his effects on outcomes of decisions by providing strategic informatioa

In this paper, first an analytic solution is presented for the optimization of strategic leverage in collective decision making by one single actor. In this solution, the actor makes assumptions about the leverage other actors will show in decision making. Subsequently, the actor optimizes the outcomes of decisions by manipulating the distribution of his leverage over a set of issues.

The analytic solution can be theoretically interpreted by decomposing the solution into three terms, the expected external leverage of the other actors on the issue, the evaluation of the deviance of the expected from the preferred outcome of the issue, and the restrictions on the distribution of leverage over the issues. The higher the expectation of the leverages the other actors will allocate to the issue, the less an actor is inclined to allocate leverage to the issue. The higher the evaluation of the deviance, the more an actor is inclined to allocate leverage to the issue. This is restricted, however, by the required distribution of leverages over the issues. The researcher is able to manipulate these restrictions to investigate its consequences for the outcomes.

In the next step, we investigate whether we can find a Nash equilibrium if all actors optimize their leverage simultaneously. Under certain conditions, a Nash equilibrium can be found by an iterative process in which actors update their estimates oh each other's leverages on the basis of what the other actors have shown in previous iterations.     

Macdonald difference operators and Harish-Chandra series

We analyse the centralizer of the Macdonald difference operatorin an appropriate algebra of Weyl group invariant differenceoperators. We show that it coincides with Cherednik's commutingalgebra of difference operators via an analog of the Harish-Chandraisomorphism. Analogs of Harish-Chandra series are defined andrealized as solutions to the system of basic hypergeometricdifference equations associated to the centralizer algebra.These Harish-Chandra series are then related to both Macdonaldpolynomials and Chalykh's Baker–Akhiezer functions.     

An Expansion Formula for the Askey–Wilson Function

The Askey–Wilson function transform is a q-analogue of the Jacobi function transform with kernel given by an explicit non-polynomial eigenfunction of the Askey–Wilson second order q-difference operator. The kernel is called the Askey–Wilson function. In this paper an explicit expansion formula for the Askey–Wilson function in terms of Askey–Wilson polynomials is proven. With this expansion formula at hand, the image under the Askey–Wilson function transform of an Askey–Wilson polynomial multiplied by an analogue of the Gaussian is computed explicitly. As a special case of these formulas a q-analogue (in one variable) of the Macdonald–Mehta integral is obtained, for which also two alternative, direct proofs are presented.