Transfer of a Passive Vector Admixture by a Two-Dimensional Turbulent Flow

We consider a model of a passive vector field transfer by a random two-dimensional transverse velocity field that is uncorrelated in time and has Gaussian spatial statistics given by a powerlike correlator. We use the renormalization group and the operator product expansion techniques to show that the asymptotic approximation of the structure functions of a vector field in the inertial range is determined by the energy dissipation fluctuations. The dependence of the asymptotic approximation on the external scale of turbulence is essential and has a powerlike form (the case of an anomalous scaling). The corresponding exponents are calculated in the one-loop approximation for structure functions of an arbitrary order.     

Series representation for operator semistable laws and domains of normal attraction

The goal of this paper is to extend the series representation to operator semistable laws and to give necessary and sufficient conditions for a vector X to belong to the generalized domain of normal attraction of some operator semistable law having no Gaussian component. Proceedings of the Seminar on Stability Problems for Stochastic Models. Hajdúszoboszló. Hungary 1997, Part II.     

Anomalous scaling in the model of turbulent advection of a vector field

We consider the model of turbulent advection of a passive vector field ϕ by a two-dimensional random velocity field uncorrelated in time and having Gaussian statistics with a powerlike correlator. The renormalization group and operator product expansion methods show that the asymptotic form of the structure functions of the ϕ field in the inertial range is determined by the fluctuations of the energy dissipation rate. The dependence of the asymptotic form on the external turbulence scale is essential and has a powerlike form (anomalous scaling). The corresponding exponents are determined by the spectrum of the anomalous dimension matrices of operator families consisting of gradients of ϕ. We find a basis constructed from powers of the dissipation and enstrophy operators in which these matrices have a triangular form in all orders of the perturbation theory. In the two-loop approximation, we evaluate the anomalous-scaling exponents for the structure functions of an arbitrary order. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 3, pp. 467–487, March, 2006.     

Mittag-Leffler vector random fields with Mittag-Leffler direct and cross covariance functions

In terms of the two-parameter Mittag-Leffler function with specified parameters, this paper introduces the Mittag-Leffler vector random field through its finite-dimensional characteristic functions, which is essentially an elliptically contoured one and reduces to a Gaussian one when the two parameters of the Mittag-Leffler function equal 1. Having second-order moments, a Mittag-Leffler vector random field is characterized by its mean function and its covariance matrix function, just like a Gaussian one. In particular, we construct direct and cross covariances of Mittag-Leffler type for such vector random fields.     

The information operator and the asymptotic behavior of the toeplitz determinant

An asymptotic representation of the information operator, corresponding to a stationary Gaussian vector sequence, is given.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 166, pp. 179–185, 1988.     

Student's t Vector Random Fields with Power-Law and Log-Law Decaying Direct and Cross Covariances

This article deals with the Student's t vector random field, which is formulated as a scale mixture of Gaussian vector random fields, and whose finite-dimensional distributions decay in power-law and have heavy tails. There are two classes of Student's t vector random fields, one with second-order moments, and the other without a second-order moment. A Cauchy vector random field is an example of Student's t vector random fields without a first-order moment, and is also an example of Stable vector random fields. A second-order Student's t vector random field allows for any given correlation structure, just as a Gaussian vector random field does. We propose four types of covariance matrix structures for second-order Student's t vector random fields, which decay in power-law or log-law.     

The Gaussian free field and Hadamard’s variational formula

We relate the Gaussian free field on a planar domain to the variational formula of Hadamard which explains the change of the Green function under a perturbation of the domain. This is accomplished by means of a natural integral operator—called the Hadamard operator—associated with a given flow of growing domains.     

A note on quadratic forms of stationary functional time series under mild conditions

We study distributional properties of a quadratic form of a stationary functional time series under mild moment conditions. As an important application, we obtain consistency rates of estimators of spectral density operators and prove joint weak convergence to a vector of complex Gaussian random operators. Weak convergence is established based on an approximation of the form via transforms of Hilbert-valued martingale difference sequences. As a side-result, the distributional properties of the long-run covariance operator are established.     

Ricci solitons and odd-dimensional spheres

As a real hypersurface in a complex space, we prove two criterion inequalities for an odd-dimensional sphere in terms of the shape operator, the Reeb vector field and its associated 1-form. Also, we determine a real hypersurface in a complex space which admits a Ricci soliton with the Reeb vector field the potential vector field.     

On the Lie algebra structures connected with Hamiltonian dynamical systems

We construct the hierarchies of master symmetries constituting Virasoro-type algebras for the Hamiltonian vector fields preserving a recursion operator. Similarly, repeatedly contracting a Hamiltonian vector field with the corresponding recursion operator, we define an Abelian Lie algebra of the thus obtained hierarchy of vector fields. The approach is shown to be applicable for the Volterra and Toda lattices. Published in Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 5, pp. 699–705, May, 1997.     

Biharmonic submanifolds with parallel mean curvature vector field in spheres

We present some results on the boundedness of the mean curvature of proper biharmonic submanifolds in spheres. A partial classification result for proper biharmonic submanifolds with parallel mean curvature vector field in spheres is obtained. Then, we completely classify the proper biharmonic submanifolds in spheres with parallel mean curvature vector field and parallel Weingarten operator associated to the mean curvature vector field.     

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.     

Multivariate Normal Approximation on the Wiener Space: New Bounds in the Convex Distance

Journal of Theoretical Probability - We establish explicit bounds on the convex distance between the distribution of a vector of smooth functionals of a Gaussian field and that of a normal vector...     

On the semidiscrete differential geometry of A-surfaces and K-surfaces

In the category of semidiscrete surfaces with one discrete and one smooth parameter we discuss the asymptotic parametrizations, their Lelieuvre vector fields, and especially the case of constant negative Gaussian curvature. In many aspects these considerations are analogous to the well known purely smooth and purely discrete cases, while in other aspects the semidiscrete case exhibits a different behaviour. One particular example is the derived T-surface, the possibility to define Gaussian curvature via the Lelieuvre normal vector field, and the use of the T-surface??s regression curves in the proof that constant Gaussian curvature is characterized by the Chebyshev property. We further identify an integral of curvatures which satisfies a semidiscrete Hirota equation.     

Some Applications of the Hodge-de Rham Decomposition to Ricci Solitons

The aim of this paper is to present a link between the Perelman potential for a compact Ricci soliton M ⁿ and the Hodge-de Rham decomposition theorem, we shall use this result to present an integral formula which enables us to establish conditions under which the Ricci soliton is trivial. Moreover, given a Ricci soliton such that its associated vector field X is a conformal vector field we show that in the compact case X is a Killing vector field, while for the non-compact case, either the soliton is Gaussian or X is a Killing vector field.     

Maslov complex germ and high-frequency Gaussian beams for cold plasma in a toroidal domain

Asymptotic vector solutions describing, in the linear approximation, the passage of high-frequency Gaussian beams through an electroneutral plasma occupying a toroidal domain T (modeling a tokamak chamber) are constructed in a fairly effective form by using the Maslov complex germ theory. The particle density and the magnetic field in T are assumed to be given. Based on Radon transforms, the reconstruction of the particle density and the magnetic field from measurements of the characteristics of Gaussian beams after their passage through T is discussed.     

Operator-decomposability of Gaussian measures on separable Banach spaces

An operator-decomposable Gaussian measure on a separable Banach space can be factorized into a convolution product of a strongly operator-decomposable Gaussian measure and an operator-invariant Gaussian measure (with respect to the same operator). An example for this very factorization is discussed in some detail. In particular it is shown that a strongly operator-decomposable Gaussian measure need not necessarily be supported by the contraction subspace of the operator involved. Finally, the decomposability semigroup of a Gaussian measure turns out to be convex; and the corresponding invariance semigroup belongs to its extreme boundary.     

A Haar-Fisz Algorithm for Poisson Intensity Estimation

This article introduces a new method for the estimation of the intensity of an inhomogeneous one-dimensional Poisson process. The Haar-Fisz transformation transforms a vector of binned Poisson counts to approximate normality with variance one. Hence we can use any suitable Gaussian wavelet shrinkage method to estimate the Poisson intensity. Since the Haar-Fisz operator does not commute with the shift operator we can dramatically improve accuracy by always cycle spinning before the Haar-Fisz transform as well as optionally after. Extensive simulations show that our approach usually significantly outperformed state-of-the-art competitors but was occasionally comparable. Our method is fast, simple, automatic, and easy to code. Our technique is applied to the estimation of the intensity of earthquakes in northern California. We show that our technique gives visually similar results to the current state-of-the-art.     

Self-duality operators on odd dimensional manifolds

In this paper we construct a new elliptic operator associated to any nowhere zero vector field on an odd-dimensional manifold and study its index theory. It turns out this operator has several geometric applications to conformal vector fields, self-dual vector fields, locally free -actions and transversal hypersurfaces of these vector fields in an odd-dimensional manifold. In particular, we reveal a non-stable phenomena about the existence of conformal vector fields and self-dual vector fields in odd dimensions above 3. This is in sharp contrast to the stable phenomena about the existence of nowhere zero vector fields in odd dimensions. Besides these applications, the index formula of this new operator also gives the formulas for the dimensions of self-duality cohomology groups and for the virtual dimensions of the moduli spaces of anti-self-dual connections on 5-cobordisms, which are introduced in author's previous papers.

     

Integral formulae on foliated symmetric spaces

In this article, we generalize known integral formulae (due to Brito–Langevin–Rosenberg, Ranjan and the second author) for foliations of codimension 1 or unit vector fields and obtain an infinite series of such formulae involving invariants of the Weingarten operator of a unit vector field, of the Jacobi operator in its direction, and their products. We write several such formulae explicitly, on locally symmetric spaces as well as on arbitrary Riemannian manifolds where they involve also covariant derivatives of the Jacobi operator. We work also with foliations of codimension 1 (or vector fields) which admit “good” (in a sense) singularities.