Symmetries of Spin Calogero Models

We investigate the symmetry algebras of integrable spin Calogero systems constructed from Dunkl operators associated to finite Coxeter groups. Based on two explicit examples, we show that the common view of associating one symmetry algebra to a given Coxeter group $W$ is wrong. More precisely, the symmetry algebra heavily depends on the representation of $W$ on the spins. We prove this by identifying two different symmetry algebras for a $B_L$ spin Calogero model and three for $G_2$ spin Calogero model. They are all related to the half-loop algebra and its twisted versions. Some of the result are extended to any finite Coxeter group.Comment: This is a contribution to the Special Issue on Dunkl Operators and Related Topics, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Spectral asymptotics of a broken delta interaction

This paper is concerned with the spectral analysis of a Hamiltonian with a $\delta$-interaction supported along a broken line with angle $\theta$. The bound states with energy slightly below the threshold of the essential spectrum are estimated in the semiclassical regime $\theta\to 0$.Comment: 20 page

A one-dimensional Keller-Segel equation with a drift issued from the boundary

We investigate in this note the dynamics of a one-dimensional Keller-Segel type model on the half-line. On the contrary to the classical configuration, the chemical production term is located on the boundary. We prove, under suitable assumptions, the following dichotomy which is reminiscent of the two-dimensional Keller-Segel system. Solutions are global if the mass is below the critical mass, they blow-up in finite time above the critical mass, and they converge to some equilibrium at the critical mass. Entropy techniques are presented which aim at providing quantitative convergence results for the subcritical case. This note is completed with a brief introduction to a more realistic model (still one-dimensional).Comment: short version, 8 page

Spectral asymptotics for the Schr\"odinger operator on the line with spreading and oscillating potentials

This study is devoted to the asymptotic spectral analysis of multiscale Schr\"odinger operators with oscillating and decaying electric potentials. Different regimes, related to scaling considerations, are distinguished. By means of a normal form filtrating the oscillations, a reduction to a non-oscillating effective Hamiltonian is performed

Generating Schemata of Resolution Proofs

Two distinct algorithms are presented to extract (schemata of) resolution proofs from closed tableaux for propositional schemata. The first one handles the most efficient version of the tableau calculus but generates very complex derivations (denoted by rather elaborate rewrite systems). The second one has the advantage that much simpler systems can be obtained, however the considered proof procedure is less efficient

Perturbative Quantum Field Theory on Random Trees

In this paper we start a systematic study of quantum field theory on random trees. Using precise probability estimates on their Galton-Watson branches and a multiscale analysis, we establish the general power counting of averaged Feynman amplitudes and check that they behave indeed as living on an effective space of dimension 4/3, the spectral dimension of random trees. In the `just renormalizable' case we prove convergence of the averaged amplitude of any completely convergent graph, and establish the basic localization and subtraction estimates required for perturbative renormalization. Possible consequences for an SYK-like model on random trees are briefly discussed.Comment: 44 page

Numerical simulation of the dynamics of molecular markers involved in cell polarisation

A cell is polarised when it has developed a main axis of organisation through the reorganisation of its cytosqueleton and its intracellular organelles. Polarisation can occur spontaneously or be triggered by external signals, like gradients of signaling molecules ... In this work, we study mathematical models for cell polarisation. These models are based on nonlinear convection-diffusion equations. The nonlinearity in the transport term expresses the positive loop between the level of protein concentration localised in a small area of the cell membrane and the number of new proteins that will be convected to the same area. We perform numerical simulations and we illustrate that these models are rich enough to describe the apparition of a polarisome.Comment: 15 page

Market Share and Price Rigidity

Survey evidence shows that the main reason why .rms keep prices stable is that they are concerned about losing customers or market share. We construct a model in which .rms care about the size of their customer base. Firms and customers form long-term relationships because consumers incur costs to switch sellers. In this environment, .rms view customers as long-lived assets. We use a general equilibrium framework where industries and .rms are buffeted by idiosyncratic marginal cost shocks. We obtain three main results. First, cost pass-through into prices is incomplete. Second, the degree of pass-through is an increasing function of the persistence of cost shocks. Third, there is a non-monotonic relationship between the size of switching costs and the rate of pass-through. In addition, we characterize the heterogenous response across industries to marginal cost shocks. The implications of our model are consistent with empirical evidence.Price Rigidity, Market Share, Customer Relations, Real Rigidities.

Robust Equilibrium Yield Curves

This paper studies the quantitative implications of the interaction between robust control and stochastic volatility for key asset pricing phenomena. We present an equilibrium term structure model with a representative agent and an output growth process that is conditionally heteroskedastic. The agent does not know the true model of the economy and chooses optimal policies that are robust to model misspecification. The choice of robust policies greatly amplifies the effect of conditional heteroskedasticity in consumption growth, improving the model’s ability to explain asset prices. In a robust control framework, stochastic volatility in consumption growth generates both a state-dependent market price of model uncertainty and a stochastic market price of risk. We estimate the model using data from the bond and equity markets, as well as consumption data. We show that the model is consistent with key empirical regularities that characterize the bond and equity markets. We also characterize empirically the set of models the robust representative agent entertains, and show that this set is ?small?. That is, it is statistically difficult to distinguish between models in this set.Yield curves, Market price of Uncertainty, Robust control.