Differential Calculus on a three-parameter oscillator algebra

Two differential calculi are developped on an algebra generalizing the usual q-oscillator algebra and involving three generators and three parameters. They are shown to be invariant under the same quantum group that is extended to a ten-generator Hopf algebra. We discuss the special case where it reduces to a deformation of the invariance group of the Weyl-Heisenberg algebra for which we prove the existence of a constraint between the values of the parameters.Comment: 10 pages, Late

Unitary representations of the quantum algebra su_q(2) on a real two-dimensional sphere for q∈R+q \in R^+ or generic q∈S1q \in S^1

Some time ago, Rideau and Winternitz introduced a realization of the quantum algebra su_q(2) on a real two-dimensional sphere, or a real plane, and constructed a basis for its representations in terms of q-special functions, which can be expressed in terms of q-Vilenkin functions, and are related to little q-Jacobi functions, q-spherical functions, and q-Legendre polynomials. In their study, the values of q were implicitly restricted to $q \in R^+$. In the present paper, we extend their work to the case of generic values of $q \in S^1$ (i.e., q values different from a root of unity). In addition, we unitarize the representations for both types of q values, $q \in R^+$ and generic $q \in S^1$, by determining some appropriate scalar products. From the latter, we deduce the orthonormality relations satisfied by the q-Vilenkin functions.Comment: 24 pages, LaTeX, uses amssym, 1 PostScript figur

Short-Run Assessment of French Economic Activity Using OPTIM.

This paper describes a short-term projection model for French economic activity, OPTIM, the aim of which is twofold. First it gives an early estimate of real GDP growth for the previous quarter, when no figure has yet been released by Insee, the French National Statistical Institute, along with flash estimates for main GDP components (consumption, investment, inventories and external trade) together with a breakdown by sectors (services, manufacturing, construction, equipment, agri-food). This appears particularly useful for the short-run analysis. In this respect OPTIM may be considered as a traditional bridge equation model since it links a particular indicator available generally ahead of the release of the quarterly national accounts with a quarterly aggregate like GDP, consumption…. Second, this tool supplies also estimates for GDP growth and its main components for the current quarter and for the next quarter (i.e two and three quarters respectively following the latest reference period of Insee's GDP data release). A pool of (mainly) monthly variables is used, which are, sometimes, directly introduced in the specification but, more often, summarised by the implementation of a principal component analysis (PCA). The largest part of the set of indicators comprises survey data together with monthly traditional indicators (industrial production, consumption in manufactured goods…). But other data (in particular financial data) are also introduced. The outcomes of OPTIM rely on a relatively complex procedure involving about twenty equations and mixing two alternative approaches: a supply approach consisting in a direct modelling of GDP and a demand approach where GDP is the sum of consumption, investment, changes in stocks and net trade (exports minus imports). The discrepancy between these two estimates is distributed according to an original method, yielding a unique GDP estimation. The paper is organised as follows. Section 1 presents the main features of OPTIM. Section 2 deals with data description while section 3 addresses the data assessment's issue. In section 4, the main equations are described. Section 5 presents a general assessment of OPTIM in terms of forecasting record. Finally section 6 concludes and proposes some avenues for further developments.

Invariance quantum groups of the deformed oscillator algebra

A differential calculus is set up on a deformation of the oscillator algebra. It is uniquely determined by the requirement of invariance under a seven-dimensional quantum group. The quantum space and its associated differential calculus are also shown to be invariant under a nine generator quantum group containing the previous one.Comment: 13 pages, Late

Innovation and Advertising: Theory and Evidence.

Advertising and innovation are two engines for firms to escape competition through a better attraction power toward consumers or quality advantage. We propose a model that encompasses both the static and dynamic interactions between R&D, advertising and competitive environment. This model provides two main predictions. First, for a given competitive environment, quality leaders spend more in advertising in order to extract maximal rents; thus, lower costs of ads may favor R&D. Second, more competition pushes Neck and Neck firms to advertise more to attract a larger share of consumers on their products or services. Empirical evidence from a large panel of 59,000 French firms over 1990-2004 supports these two properties.Advertising, Innovation, Competition, Lerner.