Non-generic Cases Research Articles

Q-boson realization of quadratic algebra A1and its representations

The non-generic central elements of the quadratic algebra A1 associated with the quantum group GL(2)q are found in the case where q is a root of unity. A q-boson realization of A1 is constructed. In terms of the q-boson realization the representations of A1 on the q-Fock space are studied in both generic and non-generic cases and the cyclic representation is obtained in the non-generic case.

The q-deformed boson realization of representations of quantum universal enveloping algebras for q a root of unity. I The case of UqSL(L)

The properties of q-deformed boson operators with non-generic q (q is a root of unity) are analysed by using the representation theory method and their finite-dimensional representations are thereby obtained. Based on this discussion, reducibilities and decompositions of q-deformed boson-realized representations of quantum universal enveloping algebra UqSL(l) are studied for non-generic cases. The explicit matrix elements of some indecomposable representations are obtained on the q-deformed Fock spaces. Necessary details are provided for UqSL(2) and UqSL(3). In particular, the Lusztig operator extension of UqSL(2) is discussed in an explicit form.

SuboptimalH∞and optimal LQG controllers for continuous-time transport-delay systems

A polynomial-based solution is presented to the continuous-time LQG and H ∞ control design problems for systems with transport delays. Polynomial solutions to these problems are not currently available in the literature. The multivariable LQG problem is first considered and solved using a Wiener type of analysis. The polynomial solution to this problem is then presented for the scalar case and this result is then used to derive the H ∞, optimal controller. In this latter case an approximation must be introduced but this does not affect the stability of the solution. The H ∞ and LQG controller structures for stable plants resemble a Smith predictor. However, the solution is also applicable to open–loop unstable systems. The solution of the H ∞ problem in both generic and non-generic cases is obtained. A detailed example is presented of the solution procedure for both of these cases. The system description and cost function are quite general.

An inverse scattering study of the radiation solution of the Korteweg–de Vries equation: the generic and nongeneric cases

Although the asymptotic (t → ∞) behaviour of radiation solutions of the Korteweg–de Vries (KdV) equation has been investigated in the literature using the inverse scattering transform method (ISTM), the complete temporal (and spatial) evolution has not been studied in detail using this method. In this paper we discuss the application of the inverse scattering expansion method, a method that we have successfully applied to a wide variety of nonlinear evolution equations of physical interest, to both the generic and nongeneric cases. Using model inputs as illustrative examples, we find that (a) unlike all other problems studied to date, the "natural" expansion parameter is not the (dimensionless) area of the input potential in the direct (eigenvalue) problem, and (b) the value of the reflection coefficient R(k) at zero eigenvalue, i.e., k = 0, plays a crucial role in the success or failure of what we will call the "standard" expansion method. The standard expansion method works for input potentials belonging to the "nongeneric" class, [Formula: see text] but breaks down beyond lowest order in the expansion for potentials belonging to the "generic" class, R(0) = −1. Re-examination of the generic problem in the vicinity of k = 0 leads to a "renormalization" in each order of the expansion, which enables the generic case to be correctly solved. Unlike the nongeneric case, the generic solution is not found to be asymptotically valid.

Generic and nongeneric bifurcation for the von Kármán equations

We consider the buckling of a rectangular plate subject to a compressive thrust and a normal load, as described by the von Kármán equations. We show how for loads with some particular symmetry properties, the generic theory given by Chow, Hale, and Mallet-Paret, [ Arch. Rational Mech. Anal. 59 (1975), 159–188] for the bifurcation near a simple characteristic value, cannot be applied. We give the bifurcation curve for this situation, and also describe the transition between the generic and the symmetric nongeneric cases.

On Some Questions Concerning Controllability and Observability Indices

This paper investigates some questions relating to the controllability and observability indices of a linear constant system, particularly with regard to the occurrence of the generic case, in which a set of indices differ by at most unity. It is argued, with the support of examples, that non-generic cases can be expected to arise in practice. It is shown that a non-generic system can be made generic by connecting to it an auxiliary system of a certain order, and the required order is evaluated; the cases of parallel, cascade and feedback connection are considered. The relationship between the indices of a continuous-time system and those of the discretetime system obtained by sampling it, is examined; it is shown that the sampled system tends to be nearer to the generic case than the original system, and it is argued that, in practice, a sampled-data system is almost certain to be generic.