Nonlocal Hamiltonian operators of hydrodynamic type with flat metrics, integrable hierarchies, and the associativity equations

We solve the problem of describing all nonlocal Hamiltonian operators of hydrodynamic type with flat metrics. This problem is equivalent to describing all flat submanifolds with flat normal bundle in a pseudo-Euclidean space. We prove that every such Hamiltonian operator (or the corresponding submanifold) specifies a pencil of compatible Poisson brackets, generates bihamiltonian integrable hierarchies of hydrodynamic type, and also defines a family of integrals in involution. We prove that there is a natural special class of such Hamiltonian operators (submanifolds) exactly described by the associativity equations of two-dimensional topological quantum field theory (the Witten-Dijkgraaf-Verlinde-Verlinde and Dubrovin equations). We show that each N-dimensional Frobenius manifold can locally be represented by a special flat N-dimensional submanifold with flat normal bundle in a 2N-dimensional pseudo-Euclidean space. This submanifold is uniquely determined up to motions.     

Differential invariants of generic hyperbolic Monge-Ampère equations

In this paper basic differential invariants of generic hyperbolic Monge-Ampère equations with respect to contact transformations are constructed and the equivalence problem for these equations is solved.      

On the existence of solutions to the Navier–Stokes–Poisson equations of a two‐dimensional compressible flow

In this paper, we consider the Navier–Stokes–Poisson equations for compressible, barotropic flow in two space dimensions. We introduce useful tools from the theory of Orlicz spaces. Then we prove the existence of globally defined finite energy weak solutions for the pressure satisfying p(?)=a?log^d (?) for large ?. Here d>1 and a>0. Copyright © 2006 John Wiley & Sons, Ltd.     

Approximate Simulation of Hawkes Processes

Hawkes processes are important in point process theory and its applications, and simulation of such processes are often needed for various statistical purposes. This article concerns a simulation algorithm for unmarked and marked Hawkes processes, exploiting that the process can be constructed as a Poisson cluster process. The algorithm suffers from edge effects but is much faster than the perfect simulation algorithm introduced in our previous work Møller and Rasmussen (2004). We derive various useful measures for the error committed when using the algorithm, and we discuss various empirical results for the algorithm compared with perfect simulations. Extensions of the algorithm and the results to more general types of marked point processes are also discussed.     

A Remark on the Equivalence between Poisson and Gaussian Stochastic Partial Differential Equations

We discuss the connection between Gaussian and Poisson noise Wick-type stochastic partial differential equations.     

Limiting distributions of randomly accelerated motions

In this paper, the process {X(t); t>0}, representing the position of a uniformly accelerated particle (with Poisson-paced) changes of its acceleration, is studied. It is shown that the distribution ofX(t) (suitably normalized), conditionally on the numbern of changes of acceleration, tends in distribution to a normal variate asn goes to infinity. The asymptotic normality of the unconditional distribution ofX(t) for large values oft is also shown. The study of these limiting distributions is motivated by the difficulty of evaluating exactly the conditional and unconditional probability laws ofX(t). In fact, the results obtained in this paper permit us to give useful approximations of the probability distributions of the position of the particle. Dipartmento di Statistica, Probabilità Statistiche Applicate University of Rome “La Sapienza,” Italy. Published in Lietuvos Matematikos Rinkinys, Vol. 37, No. 3, pp. 295–308, July–September, 1997.     

Irreducible Representations of Quantum 3 × 3 Matrices

The irreducible representations of quantum 2×2 and 3×3 matrices at the roots of unity are classified.     

A Vlasov–Poisson plasma model with non‐L1 data

It is considered the Vlasov–Poisson equation for a plasma confined in an unbounded cylinder and it is proven an existence and uniqueness result for non‐L₁ (but almost L₁) initial charge distribution. Copyright © 2004 John Wiley & Sons, Ltd.     

Geometric norm equality related to the harmonicity of the Poisson kernel for homogeneous Siegel domains

In this paper, we present a geometric norm equality involving an admissible linear form ω for the Shilov boundary of a homogeneous Siegel domain D. We prove that the validity of this norm equality is equivalent to the symmetry of D and the reduction of ω essentially to the Koszul form. This, in particular, reveals a geometric reason that the Poisson kernel is annihilated by the Laplace-Beltrami operator if and only if D is symmetric, a theorem due to Hua, Look, Korányi and Xu.     

Affine bracket algebra theory and algorithms and their applications in mechanical theorem proving

This paper discusses two problems:one is some important theories and algorithms of affine bracket algebra;the other is about their applications in mechanical theorem proving.First we give some efficient algorithms including the boundary expanding algorithm which is a key feature in application.We analyze the characteristics of the boundary operator and this is the base for the implementation of the system.We also give some new theories or methods about the exact division,the representations and structure of affine geometry and so on.In practice,we implement the mechanical auto-proving system in Maple 10 based on the above algorithms and theories.Also we test about more than 100 examples and compare the results with the methods before.