Poisson sigma models and symplectic groupoids

We consider the Poisson sigma model associated to a Poisson manifold. The perturbative quantization of this model yields the Kontsevich star product formula. We study here the classical model in the Hamiltonian formalism. The phase space is the space of leaves of a Hamiltonian foliation and has a natural groupoid structure. If it is a manifold then it is a symplectic groupoid for the given Poisson manifold. We study various families of examples. In particular, a global symplectic groupoid for a general class of two-dimensional Poisson domains is constructed.Comment: 34 page

An introduction to simulation of risk processes

A typical model for insurance risk, the so-called collective risk model, has two main components: one characterizing the frequency (or incidence) of events and another describing the severity (or size or amount) of gain or loss resulting from the occurrence of an event. Here we focus on simulating the point process N(t) of the incidence of events. We discuss five prominent examples of N(t), namely the classical (homogeneous) Poisson process, the non-homogeneous Poisson process, the mixed Poisson process, the Cox process (also called the doubly stochastic Poisson process) and the renewal process.Collective risk model; Poisson process; Non-homogeneous Poisson process; Mixed Poisson process; Cox process; Renewal process;

Integrability vs Supersymmetry: Poisson Structures of The Pohlmeyer Reduction

We construct recursively an infinite number of Poisson structures for the supersymmetric integrable hierarchy governing the Pohlmeyer reduction of superstring sigma models on the target spaces AdS_{n}\times S^n, n=2,3,5. These Poisson structures are all non-local and not relativistic except one, which is the canonical Poisson structure of the semi-symmetric space sine-Gordon model (SSSSG). We verify that the superposition of the first three Poisson structures corresponds to the canonical Poisson structure of the reduced sigma model. Using the recursion relations we construct commuting charges on the reduced sigma model out of those of the SSSSG model and in the process we explain the integrable origin of the Zukhovsky map and the twisted inner product used in the sigma model side. Then, we compute the complete Poisson superalgebra for the conserved Drinfeld-Sokolov supercharges associated to an exotic kind of extended non-local rigid 2d supersymmetry recently introduced in the SSSSG context. The superalgebra has a kink central charge which turns out to be a generalization to the SSSSG models of the well-known central extensions of the N=1 sine-Gordon and N=2 complex sine-Gordon model Poisson superalgebras computed from 2d superspace. The computation is done in two different ways concluding the proof of the existence of 2d supersymmetry in the reduced sigma model phase space under the boost invariant SSSSG Poisson structure.Comment: 33 pages, Published versio

Two sufficient conditions for Poisson approximations in the ferromagnetic Ising model

A $d$-dimensional ferromagnetic Ising model on a lattice torus is considered. As the size of the lattice tends to infinity, two conditions ensuring a Poisson approximation for the distribution of the number of occurrences in the lattice of any given local configuration are suggested. The proof builds on the Stein--Chen method. The rate of the Poisson approximation and the speed of convergence to it are defined and make sense for the model. Thus, the two sufficient conditions are traduced in terms of the magnetic field and the pair potential. In particular, the Poisson approximation holds even if both potentials diverge.Comment: Published in at http://dx.doi.org/10.1214/1214/07-AAP487 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Transition from Poisson to gaussian unitary statistics: The two-point correlation function

We consider the Rosenzweig-Porter model of random matrix which interpolates between Poisson and gaussian unitary statistics and compute exactly the two-point correlation function. Asymptotic formulas for this function are given near the Poisson and gaussian limit.Comment: 19 pages, no figure

On the Equivalence of Location Choice Models: Conditional Logit, Nested Logit and Poisson

It is well understood that the two most popular empirical models of location choice - conditional logit and Poisson - return identical coefficient estimates when the regressors are not individual specific. We show that these two models differ starkly in terms of their implied predictions. The conditional logit model represents a zero-sum world, in which one region's gain is the other regions' loss. In contrast, the Poisson model implies a positive-sum economy, in which one region's gain is no other region's loss. We also show that all intermediate cases can be represented as a nested logit model with a single outside option. The nested logit turns out to be a linear combination of the conditional logit and Poisson models. Conditional logit and Poisson elasticities mark the polar cases and can therefore serve as boundary values in applied research.firm location, residential choice, conditional logit, nested logit, Poisson count model

Algebra of Lax Connection for T-Dual Models

We study relation between T-duality and integrability. We develop the Hamiltonian formalism for principal chiral model on general group manifold and on its T-dual image. We calculate the Poisson bracket of Lax connections in T-dual model and we show that they are non-local as opposite to the Poisson brackets of Lax connection in original model. We demonstrate these calculations on two specific examples: Sigma model on S(2) and sigma model on AdS(2).Comment: 24 pages, references adde

Mean-parametrized Conway-Maxwell-Poisson regression models for dispersed counts

Conway-Maxwell-Poisson (CMP) distributions are flexible generalizations of the Poisson distribution for modelling overdispersed or underdispersed counts. The main hindrance to their wider use in practice seems to be the inability to directly model the mean of counts, making them not compatible with nor comparable to competing count regression models, such as the log-linear Poisson, negative-binomial or generalized Poisson regression models. This note illustrates how CMP distributions can be parametrized via the mean, so that simpler and more easily-interpretable mean-models can be used, such as a log-linear model. Other link functions are also available, of course. In addition to establishing attractive theoretical and asymptotic properties of the proposed model, its good finite-sample performance is exhibited through various examples and a simulation study based on real datasets. Moreover, the MATLAB routine to fit the model to data is demonstrated to be up to an order of magnitude faster than the current software to fit standard CMP models, and over two orders of magnitude faster than the recently proposed hyper-Poisson model.Comment: To appear in Statistical Modelling: An International Journa

Derivation of reduced two-dimensional fluid models via Dirac's theory of constrained Hamiltonian systems

We present a Hamiltonian derivation of a class of reduced plasma two-dimensional fluid models, an example being the Charney-Hasegawa-Mima equation. These models are obtained from the same parent Hamiltonian model, which consists of the ion momentum equation coupled to the continuity equation, by imposing dynamical constraints. It is shown that the Poisson bracket associated with these reduced models is the Dirac bracket obtained from the Poisson bracket of the parent model