Time evolution of coupled multimode and multiresonator optomechanical systems

We study the time evolution of bosonic systems where multiple driven bosonic modes of light interact with multiple mechanical resonators through arbitrary, time-dependent, optomechanical-like interactions. We find the analytical expression for the full time evolution of the system and compute the expectation value of relevant quantities of interest. Among the most interesting ones, we are able to compute the first-order quantum bipartite coherence between pairs of subsystems, and the analytical expression for the mixedness induced by the nonlinear interaction in the reduced state of the mechanical oscillators. Finally, we also compare our results with a linearised version of the system, and we find a regime where there are qualitative and quantitative differences in the behavior of some measurable quantities. Our results can therefore be used to describe the full time-evolution of the system, to characterise its nonlinear character and explore the validity of the linearisation approach.Comment: 23 pages, 1 figure

Perspectives on Competition Law: Problems and Solutions

This essay suggests that it does not follow that competition between jurisdictions is good merely because competition between economic operators in pursuit of economic goals is a good thing. The result, as the discussion on television indicated, may simply be a jurisdictional mess, as Dr. Markus Wagemann put it. You end up with all sorts of people seeking to pursue their own values: cultural values, regional values, and linguistic values; and the economic operator simply does not know where he or she stands in this jurisdictional competition. This point can perhaps be completed by simply mentioning a remark made to me by Klaus Dieter Ehlermann, the former Director General of Competition and of the legal service of the Commission: ‘It is the lawyers who make the good distinctions; politicians only make the distinctions that are convenient for them.‘ It is important to begin by making sure what we are talking about. My second point arises out of Professor Dr. Friedl Weiss\u27s paper. From the vantage point of a judge, we are increasingly faced, not with a hierarchy of norms, but a competition between norms of apparently equal value. This essay then reflects that this is not the first time we have experienced a world in which there was the fullest competition between lower level jurisdictions. Third, this essay considers the importance of taking state aid into account. This essay concludes that greater jurisdictional autonomy leads to greater barriers to trade

Baxter's Relations and Spectra of Quantum Integrable Models

Generalized Baxter's relations on the transfer-matrices (also known as Baxter's TQ relations) are constructed and proved for an arbitrary untwisted quantum affine algebra. Moreover, we interpret them as relations in the Grothendieck ring of the category O introduced by Jimbo and the second author in arXiv:1104.1891 involving infinite-dimensional representations constructed in arXiv:1104.1891, which we call here "prefundamental". We define the transfer-matrices associated to the prefundamental representations and prove that their eigenvalues on any finite-dimensional representation are polynomials up to a universal factor. These polynomials are the analogues of the celebrated Baxter polynomials. Combining these two results, we express the spectra of the transfer-matrices in the general quantum integrable systems associated to an arbitrary untwisted quantum affine algebra in terms of our generalized Baxter polynomials. This proves a conjecture of Reshetikhin and the first author formulated in 1998 (arXiv:math/9810055). We also obtain generalized Bethe Ansatz equations for all untwisted quantum affine algebras.Comment: 41 pages (v3: New Section 5.6 added in which Bethe Ansatz equations are written explicitly for all untwisted quantum affine algebras. New examples, references, and historical comments added plus some minor edits. v4: References added.

Airline Price Competition: A Time Series Analysis of 'Low-Cost' Carriers.

This paper, after providing an introduction to the operating context of low cost carriers in Europe, examines the competitive pricing behaviour of airlines. Data is collected by route for cases where more than one airline is in direct competition. Data on fares is obtained from the internet for two airlines with competing services to Alicante, Prague and Malaga, departing from Nottingham East Midlands Airport in the UK, for the six working weeks up to and including the actual departure. These destinations represent leisure traffic. Two domestic business destinations were also selected to illustrate price competition on business demand where departure times were within a maximum of 20 minutes of each other and a further examination of competing services from London Gatwick (LGW) was made. Cross Correlation Analysis is used to examine whether, subject to a variety of lags, the prices offered by one airline can be seen to be both correlated with the other price series and to lead it. This provides some insight into the pricing strategy adopted by the competitors. Autocorrelation Functions (ACFs) and Partial Autocorrelation Functions (PACFs) can also be produced on the prices offered by each airline. These suggest the nature of the ARIMA model that can be fitted to the series and these models can show the degree to which series values are correlated with their own past values and whether a reasonable model could be based on an ARIMA approach. The relative strength of these two relationships is examined; are prices more closely explained by the competitor's actions or the airlines own past price setting?

Spectral Curves, Opers and Integrable Systems

We establish a general link between integrable systems in algebraic geometry (expressed as Jacobian flows on spectral curves) and soliton equations (expressed as evolution equations on flat connections). Our main result is a natural isomorphism between a moduli space of spectral data and a moduli space of differential data, each equipped with an infinite collection of commuting flows. The spectral data are principal G-bundles on an algebraic curve, equipped with an abelian reduction near one point. The flows on the spectral side come from the action of a Heisenberg subgroup of the loop group. The differential data are flat connections known as opers. The flows on the differential side come from a generalized Drinfeld-Sokolov hierarchy. Our isomorphism between the two sides provides a geometric description of the entire phase space of the Drinfeld-Sokolov hierarchy. It extends the Krichever construction of special algebro-geometric solutions of the n-th KdV hierarchy, corresponding to G=SL(n). An interesting feature is the appearance of formal spectral curves, replacing the projective spectral curves of the classical approach. The geometry of these (usually singular) curves reflects the fine structure of loop groups, in particular the detailed classification of their Cartan subgroups. To each such curve corresponds a homogeneous space of the loop group and a soliton system. Moreover the flows of the system have interpretations in terms of Jacobians of formal curves.Comment: 64 pages, Latex, final version to appear in Publications IHE