Showing papers in "Acta Applicandae Mathematicae in 1999"

Moving Coframes: II. Regularization and Theoretical Foundations

Abstract: The primary goal of this paper is to provide a rigorous theoretical justification of Cartan’s method of moving frames for arbitrary finite-dimensional Lie group actions on manifolds. The general theorems are based a new regularized version of the moving frame algorithm, which is of both theoretical and practical use. Applications include a new approach to the construction and classification of differential invariants and invariant differential operators on jet bundles, as well as equivalence, symmetry, and rigidity theorems for submanifolds under general transformation groups. The method also leads to complete classifications of generating systems of differential invariants, explicit commutation formulae for the associated invariant differential operators, and a general classification theorem for syzygies of the higher order differentiated differential invariants. A variety of illustrative examples demonstrate how the method can be directly applied to practical problems arising in geometry, invariant theory, and differential equations.

The Devil's Invention: Asymptotic, Superasymptotic and Hyperasymptotic Series

Abstract: Singular perturbation methods, such as the method of multiple scales and the method of matched asymptotic expansions, give series in a small parameter e which are asymptotic but (usually) divergent. In this survey, we use a plethora of examples to illustrate the cause of the divergence, and explain how this knowledge can be exploited to generate a 'hyperasymptotic' approximation. This adds a second asymptotic expansion, with different scaling assumptions about the size of various terms in the problem, to achieve a minimum error much smaller than the best possible with the original asymptotic series. (This rescale-and-add process can be repeated further.) Weakly nonlocal solitary waves are used as an illustration.

Ordered Random Variables

Abstract: We discuss methods, which enable us to express distributions of ordered random variables (classical order statistics, induced order statistics, records and generalized order statistics) via distributions of sums of independent terms.

Abstract Differential Geometry, Differential Algebras of Generalized Functions, and de Rham Cohomology

Abstract: differential geometry is a recent extension of classical differential geometry on smooth manifolds which, however, does no longer use any notion of Calculus. Instead of smooth functions, one starts with a sheaf of algebras, i.e., the structure sheaf, considered on an arbitrary topological space, which is the base space of all the sheaves subsequently involved. Further, one deals with a sequence of sheaves of modules, interrelated with appropriate ‘differentials’, i.e., suitable ‘Leibniz’ sheaf morphisms, which will constitute the ‘differential complex’. This abstract approach captures much of the essence of classical differential geometry, since it places a powerful apparatus at our disposal which can reproduce and, therefore, extend fundamental classical results. The aim of this paper is to give an indication of the extent to which this apparatus can go beyond the classical framework by including the largest class of singularities dealt with so far. Thus, it is shown that, instead of the classical structure sheaf of algebras of smooth functions, one can start with a significantly larger, and nonsmooth, sheaf of so-called nowhere dense differential algebras of generalized functions. These latter algebras, which contain the Schwartz distributions, also provide global solutions for arbitrary analytic nonlinear PDEs. Moreover, unlike the distributions, and as a matter of physical interest, these algebras can deal with the vastly larger class of singularities which are concentrated on arbitrary closed, nowhere dense subsets and, hence, can have an arbitrary large positive Lebesgue measure. Within the abstract differential geometric context, it is shown that, starting with these nowhere dense differential algebras as a structure sheaf, one can recapture the exactness of the corresponding de Rham complex, and also obtain the short exponential sequence. These results are the two fundamental ingredients in developing differential geometry along classical, as well as abstract lines. Although the commutative framework is used here, one can easily deal with a class of singularities which is far larger than any other one dealt with so far, including in noncommutative theories.

Classical Integrable Systems and Billiards Related to Generalized Jacobians

Abstract: We study some classical integrable systems of dynamics (the Euler top in space, the asymptotic geodesic motion on an ellipsoid) which are linearized on unramified coverings of generalized Jacobian varieties. We find explicit expressions for so called root functions living on such coverings which enable us to solve the problems in terms of generalized theta-functions. In addition, general and asymptotic solutions for ellipsoidal billiards and the billiard in an ellipsoidal layer are obtained.

Affine Invariant Detection: Edge Maps, Anisotropic Diffusion, and Active Contours

Abstract: In this paper we undertake a systematic investigation of affine invariant object detection and image denoising Edge detection is first presented from the point of view of the affine invariant scale-space obtained by curvature based motion of the image level-sets In this case, affine invariant maps are derived as a weighted difference of images at different scales We then introduce the affine gradient as an affine invariant differential function of lowest possible order with qualitative behavior similar to the Euclidean gradient magnitude These edge detectors are the basis for the extension of the affine invariant scale-space to a complete affine flow for image denoising and simplification, and to define affine invariant active contours for object detection and edge integration The active contours are obtained as a gradient flow in a conformally Euclidean space defined by the image on which the object is to be detected That is, we show that objects can be segmented in an affine invariant manner by computing a path of minimal weighted affine distance, the weight being given by functions of the affine edge detectors The gradient path is computed via an algorithm which allows to simultaneously detect any number of objects independently of the initial curve topology Based on the same theory of affine invariant gradient flows we show that the affine geometric heat flow is minimizing, in an affine invariant form, the area enclosed by the curve

Wavelets in Banach Spaces

Abstract: We describe a construction of wavelets (coherent states) in Banach spaces generated by ‘admissible’ group representations. Our main targets are applications in pure mathematics while connections with quantum mechanics are mentioned. As an example, we consider operator-valued Segal–Bargmann-type spaces and the Weyl functional calculus.

Almost Periodic Solutions of Abstract Retarded Functional Differential Equations with Unbounded Delay

Abstract: We establish the existence of bounded, almost periodic and asymptotically almost periodic mild solutions for first- and second-order abstract-retarded functional differential equations with unbounded delay.

On the Dido Problem and Plane Isoperimetric Problems

Abstract: This paper is a continuation of a series of papers, dealing with contact sub-Riemannian metrics on R3 We study the special case of contact metrics that correspond to isoperimetric problems on the plane The purpose is to understand the nature of the corresponding optimal synthesis, at least locally It is equivalent to studying the associated sub-Riemannian spheres of small radius It appears that the case of generic isoperimetric problems falls down in the category of generic sub-Riemannian metrics that we studied in our previous papers (although, there is a certain symmetry) Thanks to the classification of spheres, conjugate-loci and cut-loci, done in those papers, we conclude immediately On the contrary, for the Dido problem on a 2-d Riemannian manifold (ie the problem of minimizing length, for a prescribed area), these results do not apply Therefore, we study in details this special case, for which we solve the problem generically (again, for generic cases, we compute the conjugate loci, cut loci, and the shape of small sub-Riemannian spheres, with their singularities) In an addendum, we say a few words about: (1) the singularities that can appear in general for the Dido problem, and (2) the motion of particles in a nonvanishing constant magnetic field

The Inverse Problem of the Calculus of Variations: Separable Systems

Abstract: This paper deals with the inverse problem of the calculus of variations for systems of second-order ordinary differential equations. The case of the problem which Douglas, in his classification of pairs of such equations, called the ‘separated case’ is generalized to arbitrary dimension. After identifying the conditions which should specify such a case for n equations in a coordinate-free way, two proofs of its variationality are presented. The first one follows the line of approach introduced by some of the authors in previous work, and is close in spirit, though being coordinate independent, to the Riquier analysis applied by Douglas for n = 2. The second proof is more direct and leads to the discovery that belonging to the ‘separated case’ has an intrinsic meaning for the given second-order differential equations: the system is separable in the sense that it can be decoupled into n pairs of first-order equations.

On Stochastic Integration and Differentiation

Abstract: In a standard integration scheme for a measurable/integrable modification existence, a certain criterion is suggested. It is also shown, how a stochastic differential can be determined for a given stochastic function.

Co)Bordism Groups in PDEs

Abstract: We introduce a geometric theory of PDEs, by obtaining existence theorems of smooth and singular solutions. Within this framework, following our previous results on (co)bordisms in PDEs, we give characterizations of quantum and integral (co)bordism groups and relate them to the formal integrability of PDEs. An explicit proof that the usual Thom–Pontryagin construction in (co)bordism theory can be generalized also to a singular integral (co)bordism on the category of differential equations is given. In fact, we prove the existence of a spectrum that characterizes the singular integral (co)bordism groups in PDEs. Moreover, a general method that associates, in a natural way, Hopf algebras (full p-Hopf algebras, 0 ≤ p ≤ n − 1), to any PDE, recently introduced, is further studied. Applications to particular important classes of PDEs are considered. In particular, we carefully consider the Navier–Stokes equation (NS) and explicitly calculate their quantum and integral bordism groups. An existence theorem of solutions of (NS) with a change in sectional topology is obtained. Relations between integral bordism groups and causal integral manifolds, causal tunnel effects, and the full p-Hopf algebras, 0 ≤ p ≤ 3, for the Navier–Stokes equation are determined.

Transfer Operators and State Spaces for Discrete Multidimensional Linear Systems

Abstract: This paper treats multidimensional discrete input-output systems from the constructive point of view. We adapt and improve recursive algorithms, derived earlier by E. Zerz and the second author from standard Grobner basis algorithms, for the solution of the canonical Cauchy problem for linear systems of partial difference equations with constant coefficients on the lattices N = ℕr1 × ℤr2. These recursive algorithms, in turn, furnish four other solution methods for the initial value problem, namely by transfer operators, by canonical Kalman global state equations, by parametrizations of controllable systems and, for systems with proper transfer matrix and left bounded input signals, by convolution with the transfer matrix. In the 2D-case N = ℤ2 the last method was studied by S. Zampieri. Minimally embedded systems are studied and give rise to especially simple Kalman equations. The latter also imply a useful characterization of the characteristic or polar variety of the system by eigenvalue spectra. For N = ℕr we define reachability of a system and prove that controllability implies reachability, but not conversely. Moreover we solve, in full generality, the modelling problem which was introduced and partially solved by F. Pauer and S. Zampieri. Various algorithms have been implemented by the first author in axiom, and examples are demonstrated by means of computer generated pictures. Related work on state space representations has been done by the Padovian and Groningian system theory schools.

On the Estimation of the Parameters of Multivariate Stable Distributions

Abstract: In the paper, the asymptotic normality for a new estimator for the spectral measure of a multivariate stable distribution is proved. Also an estimator for the density of a multivariate stable distribution is proposed, its properties are investigated. The dependence of a stable density on exponent α and the spectral measure is investigated.

Markov Control Processes with the Expected Total Cost Criterion: Optimality, Stability, and Transient Models *

Abstract: This paper studies the expected total cost (ETC) criterion for discrete-time Markov control processes on Borel spaces, and possibly unbounded cost-per-stage functions. It presents optimality results which include conditions for a control policy to be ETC-optimal and for the ETC-value function to be a solution of the dynamic programming equation. Conditions are also given for the ETC-value function to be the limit of the α-discounted cost value function as α ↑ 1, and for the Markov control process to be `stable" in the sense of Lagrange and almost surely. In addition, transient control models are fully analized. The paper thus provides a fairly complete, up-dated, survey-like presentation of the ETC criterion for Markov control processes on Borel spaces.

Pre-limit Theorems and Their Applications

Abstract: There exists a considerable debate in the literature about the applicability of α-stable distributions as they appear in Levy"s central limit theorems. A serious drawback of Levy"s approach is that, in practice, one can never know whether the underlying distribution is heavy tailed, or just has a long but truncated tail. Limit theorems for stable laws are not robust with respect to truncation of the tail or with respect to any change from 'light" to 'heavy" tail, or conversely. In this talk we provide a new 'pre-limiting" approach that helps overcome this drawback of Levy-type central limit theorems.

Lagrangian Theory of Tensor Fields over Spaces with Contravariant and Covariant Affine Connections and Metrics and Its Application to Einstein's Theory of Gravitation in V 4 Spaces

Abstract: The Lagrangian formalism for tensor fields over differentiable manifolds with contravariant and covariant affine connections (whose components differ not only by sign) and a metric \([(\bar L_n ,g) - spaces]\) is considered. The functional, the Lie, the covariant, and the total variations of a Lagrangian density, depending on components of tensor fields (with finite rank) and their first and second covariant derivatives, are established. A variation operator is determined and the corollaries of its commutation relations with the covariant and the Lie differential operators are found. The canonical (common) method of Lagrangians with partial derivatives (MLPD) and the method of Lagrangians with covariant derivatives (MLCD) are outlined. They differ each other by the commutation relations the variation operator has to obey with the covariant and the Lie differential operator. The covariant Euler–Lagrange equations are found on the basis of the MLCD. The energy-momentum tensors are considered on the basis of the Lie variation and the covariant Noether identities.

Integrability Criterion of Geodesical Equivalence. Hierarchies

Abstract: We prove that the Riemannian metrics g and \(\bar g\) (given in `general position") are geodesically equivalent if and only if some canonically given functions are pairwise commuting integrals of the geodesic flow of the metric g. This theorem is a multidimensional generalization of the well-known Dini theorem proved in the two-dimensional case. A hierarchy of completely integrable Riemannian metrics is assigned to any pair of geodesically equivalent Riemannian metrics. We show that the metrics of the standard ellipsoid and the Poisson sphere lie in such an hierarchy.

Non-Archimedean Analogues of Orthogonal and Symmetric Operators and p-adic Quantization

Abstract: We study orthogonal and symmetric operators in non-Archimedean Hilbert spaces in the connection with p-adic quantization. This quantization describes measurements with finite precision. Symmetric (bounded) operators in the p-adic Hilbert spaces represent physical observables. We study spectral properties of one of the most important quantum operators, namely, the operator of the position (which is represented in the p-adic Hilbert L2-space with respect to the p-adic Gaussian measure). Orthogonal isometric isomorphisms of p-adic Hilbert spaces preserve precisions of measurements. We study properties of orthogonal operators. It is proved that each orthogonal operator in the non-Archimedean Hilbert space is continuous. However, there exist discontinuous operators with the dense domain of definition which preserve the inner product. There also exist nonisometric orthogonal operators. We describe some classes of orthogonal isometric operators and we study some general questions of the theory of non-Archimedean Hilbert spaces (in particular, general connections between topology, norm and inner product).

Second-Order Subelliptic Operators on Lie Groups I: Complex Uniformly Continuous Principal Coefficients

Abstract: We consider second-order subelliptic operators with complex coefficients over a connected Lie group G. If the principal coefficients are right uniformly continuous then we prove that the operators generate strongly continuous holomorphic semigroups with kernels K satisfying Gaussian bounds. Moreover, the kernels are Holder continuous and for each ν ∈〈0, 1〉 and κ > 0 one has estimates $$\left| {K_z \left( {k^{ - 1} g;l^{ - 1} h} \right) - K_z \left( {g;h} \right)} \right| \leqslant a\left| z \right|^{ - D'/2_e {\omega }\left| z \right|} \left( {\frac{{\left| k \right|^\prime + \left| l \right|^\prime }}{{\left| z \right|^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern- ulldelimiterspace} 2}} + \left| {gh^{ - 1} } \right|^\prime }}} \right)^v {e - b}\left( {\left| {gh^{ - 1} } \right|^\prime } \right)^2 \left| z \right|^{ - 1} $$ for g, h, k, l ∈ G and all z in a subsector of the sector of holomorphy with $$\left| k \right|^\prime + \left| l \right|^\prime \leqslant \kappa \left| z \right|^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern- ulldelimiterspace} 2}} + 2^{ - 1} \left| {gh^{ - 1} } \right|^\prime$$ where $$\left| {\; \cdot \;} \right|^\prime $$ denotes the canonical subelliptic modulus and D " the local dimension. These results are established by a blend of elliptic and parabolic techniques in which De Giorgi estimates and Morrey–Campanato spaces play an important role.

Kinematics of Mechanisms to the Second Order - Application to the Closed Mechanisms

Abstract: The degree of freedom of a closed mechanism is the dimension of a subset M of R n , M being the inverse image of the unity by the closure function f : (q 1, ..., q n ) ↦ f(q 1, ..., q n ), where q 1, ..., q n are the articular coordinates. We first study the regular points for the mapping f from R n into the Lie group of displacements and, second, study the singularities of the mapping f. The classical theory of mechanisms considers, often implicitly, that f is a subimmersion. Here, the calculations are made in a larger case, up to second order, and the results are then slightly different. The case of such classical mechanisms as Bennett, Bricard, and Goldberg mechanisms, justify the considerations of this more general framework and the example of a Bricard mechanism is chosen as an application of the method.

Approximation of Laws of Multinomial Parameters by Mixtures of Dirichlet Distributions with Applications to Bayesian Inference

Abstract: Within the framework of Bayesian inference, when observations are exchangeable and take values in a finite space X, a prior P is approximated (in the Prokhorov metric) with any precision by explicitly constructed mixtures of Dirichlet distributions. Likewise, the posteriors are approximated with some precision by the posteriors of these mixtures of Dirichlet distributions. Approximations in the uniform metric for distribution functions are also given. These results are applied to obtain a method for eliciting prior beliefs and to approximate both the predictive distribution (in the variational metric) and the posterior distribution function of ∫ψd $$\tilde p$$ (in the Levy metric), when $$\tilde p$$ is a random probability having distribution P.

Contact Transformations and Local Reducibility of ODE to the Form y‴=0

Abstract: Orbits of the ODE y‴=0 in corresponding jet bundles are investigated. Explicit relations for the right-hand side of an arbitrary 3-order ODE necessary and sufficient for the existence of a contact transformation reducing this equation locally to the form y‴=0 are obtained.

Numerical Applications of the Scaling Concept

Abstract: We present a survey of recent developments in the applications of the scaling concept to numerical analysis. In addition, we report on some relevant topics not covered in existing surveys. Therefore, the present work updates and complements the existing surveys on the subject concerned. Applications of the scaling concept are useful in the numerical treatment of both ordinary and par- tial differential problems. Applications to boundary-value problems governed by ordinary differential equations are mainly related to their transformation into initial-value problems. Within this context, special emphasis is placed on systems of governing equations, eigenvalue, and free boundary-value problems. An error analysis for a truncated boundary formulation of the Blasius problem is also reported. As far as initial-value problems governed by ordinary differential equations are concerned, we discuss the development of adaptive mesh methods. Applications to partial differential problems considered herein are related to the construction of finite-difference schemes for conservations laws, the solution structure of the Riemann problem, rescaling schemes and adaptive schemes for blow-up problems. In writing this paper, our aim was to promote further and more important numerical applications of the scaling concept. Meanwhile, the pertinent bibliography is highlighted and is available on inter- net as the BIB file sc-gita.bib from the anonymousftp area at the URL ftp://dipmat.unime.it/

Symmetries of Linear and Linearizable Systems of Differential Equations

Abstract: Symmetries (point, classical, and higher) of linear and linearizable differential equations and systems are studied, including the cases of ordinary equations, equations of the Burgers type, and Lax pairs.

One-Term Edgeworth Expansion for Finite Population U-Statistics of Degree Two

Abstract: By means of Hoeffding"s decomposition, we represent a finite population U-statistic of degree two by the sum of a linear and a quadratic part. Assuming that the linear part is nondegenerate, we prove the validity of one-term Edgeworth expansion for the distribution function of the statistic under the optimal (minimal) conditions on the linear part and 2 + δ moment condition on the quadratic part. No condition is imposed on the ratio N / n, where N, respectively n, denotes the sample size respectively the population size.

Quantum Mechanics on Manifolds

Abstract: In this paper, we investigate the relationships between quantum mechanics and the theory of partial differential equations. We closely follow the De Broglie and Schrodinger picture. Namely, we consider the well-known wave-particle duality as a relation between solutions of partial differential equations, describing waves, and singularities of solutions, that is particles. Our analysis of these relations shows that the necessary ingredients of any quantum mechanical picture are two connections. The first one is a connection in the tangent bundle of the configuration manifold and the second one is a connection in the trivial linear bundle. We also consider mechanical systems equipped with an inner structure and show that quantization of these systems requires a linear connection in the corresponding vector bundle. These are gravity and electromagnetic fields, or Yang–Mills fields if the configuration space is the Minkowski space. In the case of general mechanical systems, they should be considered as natural generalizations of these fields. Explicit formulas for quantizations of some mechanical systems and the corresponding star-products are given.

Poisson Broken Lines Process and its Application to Bernoulli First Passage Percolation

Abstract: In this note we introduce a process, which we call 'the Poisson broken lines process", and we compute the intensity of a point process which is obtained by intersecting the Poisson broken lines process with an abscissa axis. In the second part we apply this result to compute an explicit lower bound for the time constant of a planar Bernoulli first passage percolation model with the parameter p < pc.

Asymptotic Expansions in the Compound Poisson Limit Theorem

Abstract: A triangular array of independent infinitesimal integer-valued random variables is considered. Asymptotic expansions for the probability distributions of sums of these variables are investigated in the case of the limiting compound Poisson laws.

Bounds for the Concentration of Asymptotically Normal Statistics

Abstract: Let X1,..., X n be independent, not necessarily identically distributed random variables. An optimal bound is derived for the concentration function of an arbitrary real-valued statistic T = T (X 1,...,X n ) for which ET2 < ∞. Applications are given for Wilcoxon"s rank-sum statistic, U-statistics, Student"s statistic, the two-sample Student statistic and linear regression.