167,957 research outputs found
Twisted Poincare duality for some quadratic Poisson algebras
No full textWe exhibit a Poisson module restoring a twisted Poincaré duality between Poisson homology and cohomology for the polynomial algebra R=C[X1Xn] endowed with Poisson bracket arising from a uniparametrised quantum affine space. This Poisson module is obtained as the semiclassical limit of the dualising bimodule for Hochschild homology of the corresponding quantum affine space. As a corollary we compute the Poisson cohomology of R, and so retrieve a result obtained by direct methods (so completely different from ours) by Monnier
Polynomial Poisson algebras: Gel'fand-Kirillov problem and Poisson spectra
No full textWe study the fields of fractions and the Poisson spectra of polynomial Poisson algebras.
First we investigate a Poisson birational equivalence problem for polynomial Poisson algebras over a field of arbitrary characteristic. Namely, the quadratic Poisson Gel'fand-Kirillov problem asks whether the field of fractions of a Poisson algebra is isomorphic to the field of fractions of a Poisson affine space, i.e. a polynomial algebra such that the Poisson bracket of two generators is equal to their product (up to a scalar). We answer positively the quadratic Poisson Gel'fand-Kirillov problem for a large class of Poisson algebras arising as semiclassical limits of quantised coordinate rings, as well as for their quotients by Poisson prime ideals that are invariant under the action of a torus. In particular, we show that coordinate rings of determinantal Poisson varieties satisfy the quadratic Poisson Gel'fand-Kirillov problem. Our proof relies on the so-called characteristic-free Poisson deleting derivation homomorphism. Essentially this homomorphism allows us to simplify Poisson brackets of a given polynomial Poisson algebra by localising at a generator.
Next we develop a method, the characteristic-free Poisson deleting derivations algorithm, to study the Poisson spectrum of a polynomial Poisson algebra. It is a Poisson version of the deleting derivations algorithm introduced by Cauchon [8] in order to study spectra of some noncommutative noetherian algebras. This algorithm allows us to define a partition of the Poisson spectrum of certain polynomial Poisson algebras, and to prove the Poisson Dixmier-Moeglin equivalence for those Poisson algebras when the base field is of characteristic zero. Finally, using both Cauchon's and our algorithm, we compare combinatorially spectra and Poisson spectra in the framework of (algebraic) deformation theory. In particular we compare spectra of quantum matrices with Poisson spectra of matrix Poisson varieties
A quadratic Poisson Gel'fand-Kirillov problem in prime characteristic
No full textThe quadratic Poisson Gel’fand-Kirillov problem asks whether the field of fractions of a Poisson algebra is Poisson birationally equivalent to a Poisson affine space, i.e. to a polyno-mial algebra K[X1,..., Xn] with Poisson bracket defined by {Xi, Xj} = λijXiXj for some skew-symmetric matrix (λij) ∈Mn(K). This problem was studied in [9] over a field of charac-teristic 0 by using a Poisson version of the deleting derivation homomorphism of Cauchon. In this paper, we study the quadratic Poisson Gel’fand-Kirillov problem over a field of arbitrary characteristic. In particular, we prove that the quadratic Poisson Gel’fand-Kirillov problem is satisfied for a large class of Poisson algebras arising as semiclassical limits of quantised co-ordinate rings. For, we introduce the concept of higher Poisson derivation which allows us to extend the Poisson version of the deleting derivation homomorphism from the characteristic 0 case to the case of arbitrary characteristic. When a torus is acting rationally by Poisson automorphisms on a Poisson polynomial algebra arising as the semiclassical limit of a quantised coordinate ring, we prove (under some technical assumptions) that quotients by Poisson prime torus-invariant ideals also satisfy the quadratic Poisson Gel’fand-Kirillov problem. In particular, we show that coordinate rings of determinantal varieties satisfy the quadratic Poisson Gel’fand-Kirillov problem
Lattice permutations and Poisson-Dirichlet distribution of cycle lengths
No full textWe study random spatial permutations on ℤ3 where each jump x↦π(x) is penalized by a factor e−T∥x−π(x)∥2 . The system is known to exhibit a phase transition for low enough T where macroscopic cycles appear. We observe that the lengths of such cycles are distributed according to Poisson-Dirichlet. This can be explained heuristically using a stochastic coagulation-fragmentation process for long cycles, which is supported by numerical data
Congestion probabilities in CDMA-based networks supporting batched Poisson traffic
No full textWe propose a new multirate teletraffic loss model for the calculation of time and call\ud
congestion probabilities in CDMA-based networks that accommodate calls of different serviceclasses\ud
whose arrival follows a batched Poisson process. The latter is more “peaked” and\ud
“bursty” than the ordinary Poisson process. The acceptance of calls in the system is based on the\ud
partial batch blocking discipline. This policy accepts a part of the batch (one or more calls) and\ud
discards the rest if the available resources are not enough to accept the whole batch. The\ud
proposed model takes into account the multiple access interference, the notion of local (soft)\ud
blocking, user’s activity and the interference cancellation. Although the analysis of the model\ud
does not lead to a product form solution of the steady state probabilities, we show that the\ud
calculation of the call-level performance metrics, time and call congestion probabilities, can be\ud
based on approximate but recursive formulas. The accuracy of the proposed formulas are\ud
verified through simulation and found to be quite satisfactory
Poisson algebras via model theory and differential-algebraic geometry
No full textBrown and Gordon asked whether the Poisson Dixmier–Moeglin equivalence holds for any complex affine Poisson algebra, that is, whether the sets of Poisson rational ideals, Poisson primitive ideals, and Poisson locally closed ideals coincide. In this article a complete answer is given to this question using techniques from differential-algebraic geometry and model theory. In particular, it is shown that while the sets of Poisson rational and Poisson primitive ideals do coincide, in every Krull dimension at least four there are complex affine Poisson algebras with Poisson rational ideals that are not Poisson locally closed. These counterexamples also give rise to counterexamples to the classical (noncommutative) Dixmier–Moeglin equivalence in finite GK dimension. A weaker version of the Poisson Dixmier–Moeglin equivalence is proven for all complex affine Poisson algebras, from which it follows that the full equivalence holds in Krull dimension three or less. Finally, it is shown that everything, except possibly that rationality implies primitivity, can be done over an arbitrary base field of characteristic zero
Bivariate Poisson and Diagonal Inflated Bivariate Poisson Regression Models in R
No full textIn this paper we present an R package called bivpois for maximum likelihood estimation of the parameters of bivariate and diagonal inflated bivariate Poisson regression models. An Expectation-Maximization (EM) algorithm is implemented. Inflated models allow for modelling both over-dispersion (or under-dispersion) and negative correlation and thus they are appropriate for a wide range of applications. Extensions of the algorithms for several other models are also discussed. Detailed guidance and implementation on simulated and real data sets using bivpois package is provided.
Mixture of bivariate Poisson regression models with an application to insurance
No full textIn a recent paper Bermúdez [2009] used bivariate Poisson regression models for ratemaking in car insurance, and included zero-inflated models to account for the excess of zeros and the overdispersion in the data set. In the present paper, we revisit this model in order to consider alternatives. We propose a 2-finite mixture of bivariate Poisson regression models to demonstrate that the overdispersion in the data requires more structure if it is to be taken into account, and that a simple zero-inflated bivariate Poisson model does not suffice. At the same time, we show that a finite mixture of bivariate Poisson regression models embraces zero-inflated bivariate Poisson regression models as a special case. Additionally, we describe a model in which the mixing proportions are dependent on covariates when modelling the way in which each individual belongs to a separate cluster. Finally, an EM algorithm is provided in order to ensure the models’ ease-of-fit. These models are applied to the same automobile insurance claims data set as used in Bermúdez [2009] and it is shown that the modelling of the data set can be improved considerably.Zero-inflation, Overdispersion, EM algorithm, Automobile insurance, A priori ratemaking.
Intensity estimation of non-homogeneous Poisson processes from shifted trajectories
No full textIn this paper, we consider the problem of estimating nonparametrically a mean pattern intensity λ from the observation of n independent and non-homogeneous Poisson processes N1,…,Nn on the interval [0,1]. This problem arises when data (counts) are collected independently from n individuals according to similar Poisson processes. We show that estimating this intensity is a deconvolution problem for which the density of the random shifts plays the role of the convolution operator. In an asymptotic setting where the number n of observed trajectories tends to infinity, we derive upper and lower bounds for the minimax quadratic risk over Besov balls. Non-linear thresholding in a Meyer wavelet basis is used to derive an adaptive estimator of the intensity. The proposed estimator is shown to achieve a near-minimax rate of convergence. This rate depends both on the smoothness of the intensity function and the density of the random shifts, which makes a connection between the classical deconvolution problem in nonparametric statistics and the estimation of a mean intensity from the observations of independent Poisson processes
Consistency and asymptotic normality of the maximum likelihood estimator in a zero-inflated generalized Poisson regression
No full textPoisson regression models for count variables have been utilized in many applications. However, in many problems overdispersion and zero-inflation occur. We study in this paper regression models based on the generalized Poisson distribution (Consul (1989)). These regression models which have been used for about 15 years do not belong to the class of generalized linear models considered by (McCullagh and Nelder (1989)) for which an established asymptotic theory is available. Therefore we prove consistency and asymptotic normality of a solution to the maximum likelihood equations for zero-inflated generalized Poisson regression models. Further the accuracy of the asymptotic normality approximation is investigated through a simulation study. This allows to construct asymptotic confidence intervals and likelihood ratio tests
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