Convergence of U-Statistics for Interacting Particle Systems

The convergence of U-statistics has been intensively studied for estimators based on families of i.i.d. random variables and variants of them. In most cases, the independence assumption is crucial (Lee in Statistics: Textbooks and Monographs, vol. 10, Dekker, New York, 1990; de la Peña and Giné in Decoupling. Probability and Its Application, Springer, New York, 1999). When dealing with Feynman?CKac and other interacting particle systems of Monte Carlo type, one faces a new type of problem. Namely, in a sample of N particles obtained through the corresponding algorithms, the distributions of the particles are correlated??although any finite number of them is asymptotically independent with respect to the total number N of particles. In the present article, exploiting the fine asymptotics of particle systems, we prove convergence theorems for U-statistics in this framework.     

Tanaka Formula and Local Time for a Class of Interacting Branching Measure-valued Diffusions

We construct superprocesses with dependent spatial motion(SDSMs) in Euclidean spaces R^d with d≥1 and show that,even when they start at some unbounded initial positive Radon measure such as Lebesgue measure on R^d,their local times exist when d≤3.A Tanaka formula of the local time is also derived.     

On a Relation between Intrinsic and Extrinsic Dirichlet Forms for Interacting Particle Systems

In this paper we extend the result obtained in [AKR98] (see also [AKR96a]) on the representation of the intrinsic pre–Dirichlet form ℰ^Γ of the Poisson measure π_σ in terms of the extrinsic one ℰ^P. More precisely, replacing π_σ by a Gibbs measure μ on the configuration space Γ_X we derive a relation between the intrinsic prend–Dirichlet form ℰ^Γ_μ of the measure μ and the extrinsic one ℰ^P. As a consequence we prove the closability of ℰ^Γ_μ on L²(Γ_X, μ) under very general assumptions on the interaction potential of the Gibbs measures μ.     

Hamiltonian Dynamics of a Particle Interacting with a Wave Field

We study the Hamiltonian equations of motion of a heavy tracer particle interacting with a dense weakly interacting Bose-Einstein condensate in the classical (mean-field) limit. Solutions describing ballistic subsonic motion of the particle through the condensate are constructed. We establish asymptotic stability of ballistic subsonic motion.     

An Indefinite Metric Model for Interacting Quantum Fields on Globally Hyperbolic Space-Times

We consider a relativistic ansatz for the vacuum expectation values (VEVs) of a quantum field on a globally hyperbolic space-time which is motivated by certain Euclidean field theories. The Yang-Feldman asymptotic condition w.r.t. an in-field in a quasi-free representation of the canonic commutation relations (CCR) leads to a solution of this ansatz for the VEVs. A GNS-like construction on a non-degenerate inner product space then gives local, covariant quantum fields with indefinite metric on a globally hyperbolic space-time. The non-trivial scattering behavior of quantum fields is analyzed by construction of the out-fields and calculation of the scattering matrix. A new combined effect of non-trivial quantum scattering and non-stationary gravitational forces is described for this model, as quasi-free in- fields are scattered to out-fields which form a non-quasi-free representations of the CCR. The asymptotic condition, on which the construction is based, is verified for the concrete example of de Sitter space-time. Communicated by Klaus Fredenhagen submitted 03/05/02, accepted: 18/07/03     

Funk-Hecke Formula for Orthogonal Polynomials on Spheres and on Balls

Analogues of the Funk–Hecke formula for spherical harmonicsare proved for Dunkl's h-harmonics associated to the reflectiongroups, and for orthogonal polynomials related to h-harmonicson the unit ball. In particular, an analogue and its applicationare discussed for the weight function (1–|x|²)^µ–1/2on the unit ball in R^d. 2000 Mathematics Subject Classification33C50, 33C55, 42C10.     

The Green–Kubo Formula for Locally Interacting Fermionic Open Systems

We consider a model describing finitely many free Fermi gas reservoirs coupled by local interactions and prove the Green–Kubo formulas and the Onsager reciprocity relations for heat and charge fluxes generated by temperature and chemical potential differentials. Submitted: September 2, 2006. Accepted: November 21, 2006.     

An Explicit Formula for a Strong Connection

An explicit formula for a strong connection form in a principal extension by a coseparable coalgebra is given.      

An Asymptotic Formula for the Iterates of a Function

Let IK be either IR or ? and D an open set of IK containing 0 and starlike with respect to 0 (i.e. an open interval containig 0 in the case IK = IR). If f: D » IK is a continuous function with fixed point 0, then under certain conditions stated below we can prove for the kn- th iterates of f the following asymptotic formula: 1 $$f^{(kn)}\bigg({x \over n}\bigg )=\sum_{i-1}^r{1\over (nk)^i}\ f_i(kx)+o \bigg({1\over n^r}\bigg),$$ for n » ∞, k, n and r beeing positive integers and x close enough to 0. The functions f _i are continuous and uniquely determined by f. In particular (1) holds for any function holomorphic on a neighbourhood of zero, having a convergent power series expansion of the form $$f(z)=z+a_2z^2+\cdots=\sum_{j=1}^\infty\ a_jz^j,\ a_j\in {\cal C},a_1=1,$$ and for any integers k, r with r > 0.     

An Optimal Reduced-Order Model by Orthogonal State Variables

A zero-input linear time-invariant system is considered. A setof state variables are formulated. The location of zeros ofsuch state variables are used to demonstrate some concepts incontrol theory. Applications to reduced-order models are discussed.The concept of orthogonal state variables—first introducedby the author—is shown, in a particular case, to be superiorto some other concepts in model-reduction analysis.     

Generalized Trace Formula for Polynomials Orthogonal in Continuous-Discrete Sobolev Spaces

Functional Analysis and Its Applications - In continuous-discrete Sobolev spaces a generalized trace formula for orthogonal polynomials $$\{\widehat{q}_n\}_{n=0}^\infty$$ is obtained. The proof of...     

Emerging Patterns in a Hyperbolic Model for Locally Interacting Cell Systems

Summary. Morphogenetic processes such as neurulation and gastrulation involve co- ordinated movements of cells. It is assumed that these processes happen due to long-range signaling, although the detailed mechanisms are not completely understood. Therefore, one is interested in biological ``model-systems' where self-organization of cells and in particular the mechanisms of signaling can be analyzed in greater detail. A major question is whether or not short-range signaling or local interaction of cells can also be the cause of coordinated movement and morphogenetic processes. As a model problem we analyze ripple formation of myxobacteria due to purely local interaction, a hypothesis which is discussed in the biological literature. These ripples can be observed before the final aggregation of the bacteria and fruiting body formation take place. Our basic mathematical model is a one-dimensional hyperbolic system of Goldstein-Kac type with density-dependent coefficients. Conditions for the existence of traveling waves are discussed by means of linear analysis and the construction of invariant domains.     

Bireflectionality of the Orthogonal Group over a Principal Ideal Domain

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具有年龄结构和约束的群落系统的最优收获

研究一类有年龄结构和相互作用的两种群构成的群落系统的最优收获问题,要求控制过程结束时的种群状态充分接近预先指定的年龄分布.证明了最优控制的存在性,运用Dubovitskii-Milyutin理论导出了最优性条件.这种处理方法为研究连续年龄分布下种群收获问题提供了统一框架.     

An Optimal Reduced-Order Model by Discrete Orthogonal State Variables

A linear discrete autonomous (i.e. unforced and time-invariant)systems is considered, and a set of discrete state variablesformulated. The location of zeros of such state variables isused to demonstrate some concepts in control theory. Applicationsto reduced-order models are discussed. The concept of discreteorthogonal state variables is shown, in a particular case, tobe superior to some other concepts in model-reduction analysis.