Showing papers in "Acta Applicandae Mathematicae in 2003"

Invariant Euler–Lagrange Equations and the Invariant Variational Bicomplex

Abstract: In this paper, we derive an explicit group-invariant formula for the Euler- Lagrange equations associated with an invariant variational problem. The method relies on a group-invariant version of the variational bicomplex induced by a general equivariant moving frame construction, and is of independent interest.

Linearisable third-order ordinary differential equations and generalised Sundman transformations : The Case X′′′=0

Abstract: We calculate in detail the conditions which allow the most general third-order ordinary differential equation to be linearised in X ′′′(T)=0 under the transformation X(T)=F(x,t), dT=G(x,t) dt.

Transition Semigroups of Banach Space Valued Ornstein-Uhlenbeck Processes

Abstract: We investigate the transition semigroup of the solution to a stochastic evolution equation dX(t)=AX(t) dt+dW H (t), t≥0, where A is the generator of a C 0-semigroup S on a separable real Banach space E and W H (t) t≥0 is cylindrical white noise with values in a real Hilbert space H which is continuously embedded in E. Various properties of these semigroups, such as the strong Feller property, the spectral gap property, and analyticity, are characterized in terms of the behaviour of S in H. In particular we investigate the interplay between analyticity of the transition semigroup, S-invariance of H, and analyticity of the restricted semigroup S H .

Existence Theory for Finite-Dimensional Pseudomonotone Equilibrium Problems

Abstract: This article was originally written to be delivered during a short course, but because of its finite-dimensional setting, it can also be addressed to nonspecialists and those only possessing a basic background on real analysis and mathematical programming. Thus, it should be conceived as an introduction to the existence theory for equilibrium (general optimization) problems including minimization and variational inequality under the assumption of no compactness and possibly having an unbounded solution set. Nevertheless, some of the results that are established here have not appeared elsewhere. Our approach is based on the asymptotic description of the functions and constraint set. In particular, this allows us to give various characterizations of the nonemptiness (and, in another case, boundedness) of the solution set. Several applications to convex problems in mathematical programming are given, along with applications to vector equilibrium problems. A guide to historical references is also provided.

The Universality of Zeta-Functions

Abstract: The first part of the paper contains a survey on the universality of zeta-functions. Zeta-functions with Euler's product as well as zeta-functions without Euler's product are discussed. Also, the joint universality theorems are considered. In the second part of the paper the universality of zeta-functions of finite Abelian groups of rank ≤3 is proved.

On Navier–Stokes Equations with Slip Boundary Conditions in an Infinite Pipe

Abstract: The paper examines steady Navier–Stokes equations in a two-dimensional infinite pipe with slip boundary conditions. At both inlet and outlet, the velocity of flow is assumed to be constant. The main results show the existence of weak and regular solutions with no restrictions of smallness of the flux vector, also simply connectedness of the domain is not required.

Continuous-Time Controlled Markov Chains with Discounted Rewards

Abstract: This paper studies denumerable state continuous-time controlled Markov chains with the discounted reward criterion and a Borel action space. The reward and transition rates are unbounded, and the reward rates are allowed to take positive or negative values. First, we present new conditions for a nonhomogeneous Q(t)-process to be regular. Then, using these conditions, we give a new set of mild hypotheses that ensure the existence of ∈-optimal (∈≥0) stationary policies. We also present a ‘martingale characterization’ of an optimal stationary policy. Our results are illustrated with controlled birth and death processes.

Univalent Functions in Two-Dimensional Free Boundary Problems

Abstract: The main goal of the paper is to bring together methods of the classical theory of univalent functions and some problems of fluid mechanics. Our interest centers on free boundary problems. We study the time evolution of the free boundary of a viscous fluid in the zero- and nonzero-surface-tension models for planar flows in Hele-Shaw cells either with an extending to infinity free boundary or with a bounded free boundary. We consider special classes of univalent functions that admit an explicit geometric interpretation to characterize the shape of the free interface. Another model is two-dimensional solidification/melting of a nucleus in a forced flow.

Special Classes of Snarks

Abstract: We report the most relevant results on the classification, up to isomorphism, of nontrivial simple uncolorable (i.e., the chromatic index equals 4) cubic graphs, called snarks in the literature. Then we study many classes of snarks satisfying certain additional conditions, and investigate the relationships among them. Finally, we discuss connections between the snark family and some significant conjectures of graph theory, and list some problems and open questions which arise naturally in this research.

A New Estimator for a Tail Index

Abstract: We investigate properties of a new estimator for a tail index introduced by Davydov and co-workers. The main advantage of this estimator is the simplicity of the statistic used for the estimator. We provide results of simulation by comparing plots of our's and Hill's estimators.

On Bayesian adaptation

Abstract: We show that Bayes estimators of an unknown density can adapt to unknown smoothness of the density. We combine prior distributions on each element of a list of log spline density models of different levels of regularity with a prior on the regularity levels to obtain a prior on the union of the models in the list. If the true density of the observations belongs to the model with a given regularity, then the posterior distribution concentrates near this true density at the rate corresponding to this regularity.

Monodromy of Variations of Hodge Structure

Abstract: We present a survey of the properties of the monodromy of local systems on quasi-projective varieties which underlie a variation of Hodge structure In the last section, a less widely known version of a Noether–Lefschetz-type theorem is discussed

The Problem of Aliasing in Identifying Finite Parameter Continuous Time Stochastic Models

Abstract: This note exposits the problem of aliasing in identifying finite parameter continuous time stochastic models, including econometric models, on the basis of discrete data. The identification problem for continuous time vector autoregressive models is characterised as an inverse problem involving a certain block triangular matrix, facilitating the derivation of an improved sufficient condition for the restrictions the parameters must satisfy in order that they be identified on the basis of equispaced discrete data. Sufficient conditions already exist in the literature but these conditions are not sharp and rule out plausible time series behaviour.

Characterizing Specific Riemannian Manifolds by Differential Equations

Abstract: Some characterizations of certain rank-one symmetric Riemannian manifolds by the existence of nontrivial solutions to certain partial differential equations on Riemannian manifolds are surveyed.

Motion Control Algorithms for Simple Mechanical Systems with Symmetry

Abstract: We treat underactuated mechanical control systems with symmetry, taking the viewpoint of the affine connection formalism. We first review the appropriate notions and tests of controllability associated with these systems, including that of fiber controllability. Secondly, we present a series expansion describing the evolution of the trajectories of general mechanical control systems starting from nonzero velocity. This series is then used to investigate the behavior of the system under small-amplitude periodic forcing. On this basis, motion control algorithms are designed for systems with symmetry to solve the tasks of point-to-point reconfiguration, static interpolation and stabilization problems. Several examples are given and the performance of the algorithms is illustrated in the blimp system.

The Continuous Wavelet Transform and Symmetric Spaces

Abstract: The continuous wavelet transform has become a widely used tool in applied science during the last decade. In this article we discuss some generalizations coming from actions of closed subgroups of GL(n, R) acting on R n . In particular, we propose a way to invert the wavelet transform in the case where the stabilizer of a generic point in R n is not compact, but a symmetric subgroup, a case that has not previously been discussed in the literature. Introduction, Wavelets on ax +b-group The continuous wavelet transform has become a widely used tool in applied science during the last decade. The best known example deals with wavelets on the real line. Each wavelet by taking "matrix coefficients", i.e., inner products with translations and dilations of the wavelet, defines a wavelet transform which can be used to reconstruct the function from the dilations and translations of the wavelet. This process helps compensate for the local-nonlocal behavior of the Fourier transform. Translations and dilations form the so-called ax + b-group of transformations of the real line. These act in a natural manner as unitary operators on L 2 (R). In that way, wavelet transforms are simply a part of the representation theory of the ax + b-group. This observation is the basis for the generalization of the continuous wavelet transform to higher dimensions and more general settings. There have been further attempts to generalize these ideas to arbitrary groups; see (4) and the references therein. In this article we review some of the basic ideas for the general wavelet transform for groups acting on R n . We start by reviewing the classical one dimensional wavelet transform. In this simple setting, the usual definition of a wavelet is equivalent to the corresponding matrix coefficients being square integrable on theax+b-group. A natural generalization of this to an arbitrary representation (�, H) of a topological group G is to say, that a vector u ∈ H \ {0} is a wavelet if a 7→ (v | �(a)u) is square integrable for all v ∈ H. The generalized wavelet transform is then Wv(a) := (v | �(a)v). The basic facts for this transform and, in particular, the inversion formula are presented in section 1. In section 2 these notions are applied to topological groups H acting on R n by a representation �. Let G = H ×� R n be the semi-direct product of H and R n . Then G acts on R n and L 2 (R n ). When the action of H on R n is sufficiently well behaved, the decomposition of L 2 (R n ) under G can be described in terms of the orbits in R n under the transpose (contragredient) representation � ' where � ' (a) = �(a −1 ) t . In general, one

Linear Functional Equations in Banach Modules over a C *-Algebra

Abstract: The paper is a survey on the Hyers–Ulam–Rassias stability of linear functional equations in Banach modules over a C*-algebra. Its contents is divided into the following sections: 1. Introduction; 2. Stability of the Cauchy functional equation in Banach modules; 3. Stability of the Jensen functional equation in Banach modules; 4. Stability of the Trif functional equation in Banach modules; 5. Stability of cyclic functional equations in Banach modules over a C*-algebra; 6. Stability of cyclic functional equations in Banach algebras and approximate algebra homomorphisms; 7. Stability of algebra *-homomorphisms between Banach *-algebras and applications.

Geometric Completeness of Distribution Spaces

Abstract: This paper introduces a notion of geometric completeness for spaces of distributions, modelled after the notion of a complete variety in algebraic geometry. It is related to the following elimination problem for systems of PDE: consider the set of homogeneous solutions of a system of PDE in some space of distributions. When is the projection of this set onto some of its coordinates also the set of homogeneous solutions of a system of PDE?

The Hilbert-Caratheodory form for parametric multiple integral problems in the calculus of variations

Abstract: We define, and discuss some properties of, a form which bears a relation to the Caratheodory form for multiple integral problems in the calculus of variations similar to the relation that the Hilbert 1-form of Finsler geometry bears to the Poincare–Cartan form for one-dimensional variational problems.

Convergence Dynamics of Delayed Hopfield-Type Neural Networks Under Almost Periodic Stimuli

Abstract: Convergence dynamics of Hopfield-type neural networks subjected to almost periodic external stimuli are investigated. In this article, we assume that the network parameters vary almost periodically with time and we incorporate variable delays in the processing part of the network architectures. By employing Halanay inequalities, we obtain delay independent sufficient conditions for the networks to converge exponentially toward encoded patterns associated with the external stimuli. The networks are guaranteed to have exponentially hetero-associative stable encoding of the external stimuli.

Forecasting for Stationary Binary Time Series

Abstract: The forecasting problem for a stationary and ergodic binary time series {X n }n=0∞ is to estimate the probability that Xn+1=1 based on the observations X i , 0≤i≤n without prior knowledge of the distribution of the process {X n }. It is known that this is not possible if one estimates at all values of n. We present a simple procedure which will attempt to make such a prediction infinitely often at carefully selected stopping times chosen by the algorithm. We show that the proposed procedure is consistent under certain conditions, and we estimate the growth rate of the stopping times.

Bias Optimality versus Strong 0-Discount Optimality in Markov Control Processes with Unbounded Costs

Abstract: This paper deals with expected average cost (EAC) and discount-sensitive criteria for discrete-time Markov control processes on Borel spaces, with possibly unbounded costs. Conditions are given under which (a) EAC optimality and strong −1-discount optimality are equivalent; (b) strong 0-discount optimality implies bias optimality; and, conversely, under an additional hypothesis, (c) bias optimality implies strong 0-discount optimality. Thus, in particular, as the class of bias optimal policies is nonempty, (c) gives the existence of a strong 0-discount optimal policy, whereas from (b) and (c) we get conditions for bias optimality and strong 0-discount optimality to be equivalent. A detailed example illustrates our results.

Geometric aspects of the maximum principle and lifts over a bundle map.

Abstract: A coordinate-free proof of the maximum principle is provided in the specific case of an optimal control problem with fixed time. Our treatment heavily relies on a special notion of variation of curves that consist of a concatenation of integral curves of time-dependent vector fields with unit time component, and on the use of a concept of lift over a bundle map. We further derive necessary and sufficient conditions for the existence of so-called abnormal extremals.

On the Degeneration, Regeneration and Braid Monodromy of T×T

Abstract: This paper is the first in a series of three papers concerning the surface T×T. Here we study the degeneration of T×T and the regeneration of its degenerated object. We also study the braid monodromy and its regeneration.

Polynomial Stability to Three-Dimensional Magnetoelastic Waves

Abstract: In this work we show that the energy associated to the linear three-dimensional magneto-elastic system decays polynomially to zero as time goes to infinity, provided the initial data is smooth enough.

Holomorphic Symplectic Manifolds and Lagrangian Fibrations

Abstract: We introduce the geometrical nature of fibre space structures of an irreducible symplectic manifold and holomorphic Lagrangian fibrations.

Finite-Type Invariants of Cubic Complexes

Abstract: The paper is for a general audience and may serve as a preliminary introduction to the theory of finite-type invariants.

Asymptotically Efficient Nonparametric Estimation of Nonlinear Spectral Functionals

Abstract: The paper considers a problem of construction of asymptotically efficient estimators for functionals defined on a class of spectral densities. We define the concepts of H0- and IK-efficiency of estimators, based on the variants of Hajek–Ibragimov–Khas'minskii convolution theorem and Hajek–Le Cam local asymptotic minimax theorem, respectively. We prove that \(\Phi (\hat \theta _T ),{\text{ where }}\hat \theta _T \) is a suitable sequence of T1/2-consistent estimators of unknown spectral density θ(λ), is H0- and IK-asymptotically efficient estimator for a nonlinear smooth functional Φ(θ).

On Poisson–Malcev Structures

Abstract: Malcev–Poisson structure on a manifold is analogous to a Poisson structure with the Lie identity replaced by a slightly more general Malcev identity. Examples of such structures arise naturally. In the second part of the paper we study Malcev bialgebras. A theorem of characterization is proved.

Stochastic Algebraic de Rham Complexes

Abstract: We define completion of the algebraic de Rham complex associated to the algebras of functionals smooth in the Chen–Souriau sense or in the Nualart–Pardoux sense over the loop space. We show that the stochastic algebraic de Rham cohomology groups are equal to the deterministic cohomology groups of the loop space.