Choquet-type integral representation of polysupermedian measures

Some aspects of multi-parameter potential theory are developed: we give a Choquet-type integral representation for measures which are supermedian for a countable family of submarkovian resolvents of commuting kernels on a Radon measurable space. For the subclass of polysupermedian measures we prove a Riesz-type decomposition, and we show that there is a unique integral representation by minimal polysupermedian measures. The setting covers a variety of very different examples like random fields, measures on product spaces which are supermedian for resolvents on the factor spaces, and completely supermedian measures.     

Generalized Sobolev's phi function for the resolvent of a Milne-type integral equation with a degenerate kernel

It is well known in the field of radiative transfer that Sobolev was the first to introduce the resolvent into Milne's integral equation with a displacement kernel. Thereafter it was shown that the resolvent plays an important role in the theory of formation of spectral lines. In the theory of line-transfer problems, the kernel representation in Milne's integral equation has been used to provide an approximate solution in a manner similar to that given by the discrete ordinales method.In this paper, by means of invariant imbedding we show how to determine an exact solution of a Milne-type integral equation with a degenerate kernel, whose form is more general than the Pincherle-Gourast kernel. A Cauchy system for the resolvent is expressed in terms of generalized Sobolev's Φ- and Ψ-functions, which are computed by solving a system of differential equations for auxiliary functions. Furthermore, these functions are expressed in terms of components of the kernel representation.     

Integral formulas for differential forms of type (p,q) on complex Finsler manifolds

Using the invariant integral kernel introduced by Demailly and Laurent-Thiebaut, complex Finsler metric and nonlinear connection associating with Chern-Finsler connection, we research the integral representation theory on complex Finsler manifolds. The Koppelman and Koppelman-Leray formulas are obtained, and the -equations are solved.     

The Koppelman-Leray formula on complex Finsler manifolds

By means of the invariant integral kernel (the Berndtsson kernel), the complex Finsler metric and the non-linear connection associated with the Chern-Finsler connection to research into the integral representation theory on complex Finsler manifolds, the Koppelman and Koppelman-Leray formulas are obtained, and the (?)--equations are solved.     

Averaging of the Cauchy kernels and integral realization of the local residue

We consider a family of kernels of integral representations associated with toric varieties. These kernels generalizes, in particular, the Bochner-Martinelli form. We show that the integral representation formulas can be derived by averaging of the Cauchy kernels on some positive measures. We apply then the obtained result to get an integral realization of the local residue corresponding to each kernel of integral representation.     

Coercive estimates and integral representation formulas on carnot groups

We establish Sobolev-type integral representation formulas for differential functions on Carnot groups. We prove coercive estimates for homogeneous differential operators with constant coefficients and finite-dimensional kernel. These integral representation formulas are new in the Euclidean spaces as well.     

On a Conformally Invariant Integral Equation Involving Poisson Kernel

We study a prescribing functions problem of a conformally invariant integral equation involving Poisson kernel on the unit ball. This integral equation is not the dual of any standard type of PDE. As in Nirenberg problem, there exists a Kazdan–Warner type obstruction to existence of solutions. We prove existence in the antipodal symmetry functions class.     

Multivariate fractionally integrated CARMA processes

A multivariate analogue of the fractionally integrated continuous time autoregressive moving average (FICARMA) process defined by Brockwell [Representations of continuous-time ARMA processes, J. Appl. Probab. 41 (A) (2004) 375-382] is introduced. We show that the multivariate FICARMA process has two kernel representations: as an integral over the fractionally integrated CARMA kernel with respect to a Lévy process and as an integral over the original (not fractionally integrated) CARMA kernel with respect to the corresponding fractional Lévy process (FLP). In order to obtain the latter representation we extend FLPs to the multivariate setting. In particular we give a spectral representation of FLPs and consequently, derive a spectral representation for FICARMA processes. Moreover, various probabilistic properties of the multivariate FICARMA process are discussed. As an example we consider multivariate fractionally integrated Ornstein-Uhlenbeck processes.     

超Cartan域上的积分表示

目的是研究第一类超Cartan域{(w,z) ||w|2 相似文献   

On numerical optimization theory of infinite kernel learning

In Machine Learning algorithms, one of the crucial issues is the representation of the data. As the given data source become heterogeneous and the data are large-scale, multiple kernel methods help to classify “nonlinear data”. Nevertheless, the finite combinations of kernels are limited up to a finite choice. In order to overcome this discrepancy, a novel method of  “infinite”  kernel combinations is proposed with the help of infinite and semi-infinite programming regarding all elements in kernel space. Looking at all infinitesimally fine convex combinations of the kernels from the infinite kernel set, the margin is maximized subject to an infinite number of constraints with a compact index set and an additional (Riemann–Stieltjes) integral constraint due to the combinations. After a parametrization in the space of probability measures, it becomes semi-infinite. We adapt well-known numerical methods to our infinite kernel learning model and analyze the existence of solutions and convergence for the given algorithms. We implement our new algorithm called “infinite” kernel learning (IKL) on heterogenous data sets by using exchange method and conceptual reduction method, which are well known numerical techniques from solve semi-infinite programming. The results show that our IKL approach improves the classifaction accuracy efficiently on heterogeneous data compared to classical one-kernel approaches.     

Dilated Fractional Stable Motions

Dilated fractional stable motions are stable, self-similar, stationary increments random processes which are associated with dissipative flows. Self-similarity implies that their finite-dimensional distributions are invariant under scaling. In the Gaussian case, when the stability exponent equals 2, dilated fractional stable motions reduce to fractional Brownian motion. We suppose here that the stability exponent is less than 2. This implies that the dilated fractional stable motions have infinite variance and hence they cannot be characterised by a covariance function. These dilated fractional stable motions are defined through an integral representation involving a nonrandom kernel. This kernel plays a fundamental role. In this work, we study the space of kernels for which the dilated processes are well-defined, indicate connections to Sobolev spaces, discuss uniqueness questions and relate dilated fractional stable motions to other self-similar processes. We show that a number of processes that have been obtained in the literature, are in fact dilated fractional stable motions, for example, the telecom process obtained as limit of renewal reward processes, the Takenaka processes and the so-called random wavelet expansion processes.     

Invariant tests for symmetry about an unspecified point based on the empirical characteristic function

This paper considers a flexible class of omnibus affine invariant tests for the hypothesis that a multivariate distribution is symmetric about an unspecified point. The test statistics are weighted integrals involving the imaginary part of the empirical characteristic function of suitably standardized given data, and they have an alternative representation in terms of an L²-distance of nonparametric kernel density estimators. Moreover, there is a connection with two measures of multivariate skewness. The tests are performed via a permutational procedure that conditions on the data.     

Hook length products and Cayley operators of classical invariant theory

We show how hook length products of frames associated with integral partitions are involved in the evaluation of the Cayley operators of classical invariant theory. In this setting, hook length products play a more fundamental role than they do in the representation theory of the symmetric group. We give an elementary combinatorial argument which derives the result of Frame, Robinson and Thrall [2] concerning the dimensions of the irreducible representations of the symmetric group from these same general results of invariant theory.     

Ergodic Type Bellman Equations of First Order with Quadratic Hamiltonian

We consider Bellman equations of ergodic type in first order. The Hamiltonian is quadratic on the first derivative of the solution. We study the structure of viscosity solutions and show that there exists a critical value among the solutions. It is proved that the critical value has the representation by the long time average of the kernel of the max-plus Schrödinger type semigroup. We also characterize the critical value in terms of an invariant density in max-plus sense, which can be understood as a counterpart of the characterization of the principal eigenvalue of the Schrödinger operator by an invariant measure.     

$C^1$ error estimation on the boundary for an exterior Neumann problem in $\mathbb R^3$

Summary.   In this paper we establish a error estimation on the boundary for the solution of an exterior Neumann problem in . To solve this problem we consider an integral representation which depends from the solution of a boundary integral equation. We use a full piecewise linear discretisation which on one hand leads to a simple numerical algorithm but on the other hand the error analysis becomes more difficult due to the singularity of the integral kernel. We construct a particular approximation for the solution of the boundary integral equation, for the solution of the Neumann problem and its gradient on the boundary and estimate their error. Received May 11, 1998 / Revised version received July 7, 1999 / Published online August 24, 2000     

The spectra and singular values of multidimensional integral operators with bihomogeneous kernels

We consider the multidimensional integral operators with bihomogeneous kernel invariant under all rotations. For truncated operators of the type we describe the limit behavior of the set of singular values and in the case when these operators are selfadjoint we describe the limit behavior of their spectra.     

Local bases for refinable spaces

We provide a new representation of a refinable shift invariant space with a compactly supported generator, in terms of functions with a special property of homogeneity. In particular, these functions include all the homogeneous polynomials that are reproducible by the generator, which links this representation to the accuracy of the space. We completely characterize the class of homogeneous functions in the space and show that they can reproduce the generator. As a result we conclude that the homogeneous functions can be constructed from the vectors associated to the spectrum of the scale matrix (a finite square matrix with entries from the mask of the generator). Furthermore, we prove that the kernel of the transition operator has the same dimension as the kernel of this finite matrix. This relation provides an easy test for the linear independence of the integer translates of the generator. This could be potentially useful in applications to approximation theory, wavelet theory and sampling.

     

Gibbs <Emphasis Type="Italic">u</Emphasis>-states for the foliated geodesic flow and transverse invariant measures

This paper is devoted to the study of Gibbs u-states for the geodesic flow tangent to a foliation F of a manifold M having negatively curved leaves. By definition, they are the probability measures on the unit tangent bundle to the foliation that are invariant under the foliated geodesic flow and have Lebesgue disintegration in the unstable manifolds of this flow. p]On the one hand we give sufficient conditions for the existence of transverse invariant measures. In particular we prove that when the foliated geodesic flow has a Gibbs su-state, i.e. an invariant measure with Lebesgue disintegration both in the stable and unstable manifolds, then this measure has to be obtained by combining a transverse invariant measure and the Liouville measure on the leaves. p]On the other hand we exhibit a bijective correspondence between the set of Gibbs u-states and a set of probability measure on M that we call φ ^u -harmonic. Such measures have Lebesgue disintegration in the leaves and their local densities have a very specific form: they possess an integral representation analogue to the Poisson representation of harmonic functions.     

Integral Kernel Operators on Fock Space – Generalizations and Applications to Quantum Dynamics

A general theory of operators on Boson Fock space is discussed in terms of the white noise distribution theory on Gaussian space (white noise calculus). An integral kernel operator is generalized from two aspects: (i) The use of an operator-valued distribution as an integral kernel leads us to the Fubini type theorem which allows an iterated integration in an integral kernel operator. As an application a white noise approach to quantum stochastic integrals is discussed and a quantum Hitsuda–Skorokhod integral is introduced. (ii) The use of pointwise derivatives of annihilation and creation operators assures the partial integration in an integral kernel operator. In particular, the particle flux density becomes a distribution with values in continuous operators on white noise functions and yields a representation of a Lie algebra of vector fields by means of such operators.     

Asymptotical representation of singular integral with the Hilbert kernel near a point of weak continuity of density

We derive an asymptotical representation for singular integral with the Hilbert kernel near a fixed point where its density vanishes as a negative power of module of logarithm of distance from this point.