(φ,ξ,η,λ)‐Structures of Parabolic Type

We consider (,,,)structures of parabolic type on hypersurfaces of dual spaces and study the rank of the affinor . We consider almost contact metric structures of parabolic type of the first kind on hypersurfaces of 4dimensional dual metric space. We study the properties of these structures and give examples of normal, integrable, and Sasakian parabolic structures.     

On a Positivity Property of the Riemann ξ‐Function

We consider a quantitative version of the positivity condition of the Riemann function. J. C. Lagarias obtained that for >10. The main aim of this paper is to prove the equality for 1.     

Mathematical expectations of functions of sums of a random number of independent terms

Conditions are found which must be imposed on a function g(x) in order that M g(₁+₂+ + _v < if M g(_i) < and M g(v) < ,, ₁, ₂, , _n, ... being non-negative and independent, being integral, and {_i} being identically distributed. The result is applied to the theory of branching processes.Translated from Matematicheskie Zametki, Vol. 3, No. 4, pp. 387–394, April, 1968.     

On the rate of convergence of an unstable solution of a stochastic differential equation

We study the rate of convergence of the process(tT)/T to the processw(t)/ asT , where(t) is a solution of the stochastic differential equationd(t)=a((t))dt+((t))dw(t) Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 10, pp. 1424–1427, October, 1994.     

Feller Processes Generated by Pseudo-Differential Operators: On the Hausdorff Dimension of Their Sample Paths

Let {X _t} _t0 be a Feller process generated by a pseudo-differential operator whose symbol satisfiesÇⁿ|q(Ç,)|c(1=)()) for some fixed continuous negative definite function (). The Hausdorff dimension of the set {X _t:tE}, E [0, 1] is any analytic set, is a.s. bounded above by dim E. is the Blumenthal–Getoor upper index of the Levy Process associated with ().     

Equations de convolution et formes quadratiques

Résumé On étudie, sans hypothèse de convexité, les équations f=g, f=g et f=g.

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Summary We study, without any convexity hypothesis, equations f=g, f=g and f=g where and respectively denote infimal convolution and deconvolution. We give an explicit formulation of these results in the quadratic hilbertian frame, and we interpret them in terms of parallel addition and subtraction of non necessarily semi-definite positive operators.

     

The η-Invariant of Twisted Dirac Operators of S3/Γ

The aim of this paper is to compute the and ~-invariants for the Dirac operator of the quotient of the sphere S ³ by a finite subgroup, twisted by a representation of its fundamental group.     

Minimax filtration of linear transformations of stationary sequences

We will consider the problem of determining a linear, mean-square optimal estimate of the transformation of a stationary random sequence (k) with density f() from observations of the sequence (k) + n(k) withk0, where (k) is a stationary sequence not correlated with (k) with density g(). The least favorable spectral densities and minimax (robust) spectral characteristics of an optimal estimate A for different classes of densities are found.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 1, pp. 92–99, January, 1991.     

Compound Poisson Processes with Two-Sided Reflection

We consider a compound oscillating Poisson process with two-sided reflection. This process is defined by an upper-semicontinuous compound Poisson process (t) and its functionals, namely the first-exit time of (t) from an interval and the first-exit time of (t) across the upper and lower levels. We study the main characteristics of this oscillating process in terms of the potential and resolvent of the process (t) introduced by Korolyuk. For this purpose, we refine the Pecherskii identities and some other results for upper-semicontinuous Poisson processes.     

On a class of 3-dimensional contact metric manifolds

Let M³ be a 3-dimensional contact metric manifold with contact structure (, , , g), such that and =R(.,)) commute. Such a manifold is called 3--manifold. We prove that every 3--manifold with -parallel Weyl tensor is either flat or a Sasakian manifold with constant curvature 1.     

The problem of controlling a Wiener process when the number of switchings is bounded

We are considering the problem of controlling a one-dimensional Wiener process (t) (0)=0,E=0,D= ²t.Translated fromProblemy Ustoichivosti Stokhasticheskikh Modelei. Trudy Seminara, 1988, pp. 53–55.     

Fine Densities for Excessive Measures and the Revuz Correspondence

Suppose that U is the resolvent of a Borel right process on a Lusin space X. If is a U-excessive measure on X then we show by analytical methods that for every U-excessive measure with the Radon–Nikodym derivative d/d possesses a finely continuous version. (Fitzsimmons and Fitzsimmons and Getoor gave a probabilistic approach for this result.) We extend essentially a technique initiated by Mokobodzki and deepened by Feyel. The result allows us to establish a Revuz type formula involving the fine versions, and to study the Revuz correspondence between the -finite measures charging no set that is both -polar and -negligible (U being the potential component of ) and the strongly supermedian kernels on X. This is an analytic version of a result of Azéma, Fitzsimmons and Dellacherie, Maisonneuve and Meyer, in terms of additive functionals or homogeneous random measures. Finally we give an application to the context of the semi-Dirichlet forms, covering a recent result of Fitzsimmons.     

Confidence sets centered at C p -estimators

Suppose X _n is an observation, or average of observations, on a discretized signal _n that is measured at n time points. The random vector X _n has a N( _n , ² _n I) distribution, the mean and variance being unknown. Under squared error loss, the unbiased estimator X _n of _n can be improved by variable-selection. Consider the candidate estimator _n (A) whose i-th component equals the i-th component of X _n whenever i/(n+1) lies in A and vanishes otherwise. Allow the set A to range over a large collection of possibilities. A C _p -estimator is a candidate estimator that minimizes estimated quadratic loss over A. This paper constructs confidence sets that are centered at a C _p -estimator, have correct asymptotic coverage probabiligy for _n , and are geometrically smaller than or equal to the competing confidence balls centered at X _n . The asymptotics are locally uniform in the parameters ( _n , ² _n ). The results illustrate an approach to inference after variable-selection.     

Special directions on contact metric three-manifolds

Blair [5] has introduced special directions on a contact metric 3-manifolds with negative sectional curvature for plane sections containing the characteristic vector field and, when is Anosov, compared such directions with the Anosov directions. In this paper we introduce the notion of Anosov-like special directions on a contact metric 3-manifold. Such directions exist, on contact metric manifolds with negative -Ricci curvature, if and only if the torsion is -parallel, namely (1.1) is satisfied. If a contact metric 3-manifold M admits Anosov-like special directions, and is -parallel, where is the Berger-Ebin operator, then is Anosov and the universal covering of M is the Lie group (2,R). We note that the notion of Anosov-like special directions is related to that of conformally Anosow flow introduced in [9] and [14] (see [6]).Supported by funds of the M.U.R.S.T. and of the University of Lecce. 1991.     

A characterization of Gaussian measures on locally compact Abelian groups

Let and be independent random variables having equal variance. In order that + and – be independent, it is necessary and sufficient that and have normal distributions. This result of Bernshtein [1] is carried over in [7] to the case when and take values in a locally compact Abelian group. In the present note, a characterization of Gaussian measures on locally compact Abelian groups is given in which in place of + and –, functions of and are considered which satisfy the associativity equation.Translated from Matematicheskie Zametki, Vol. 22, No. 5, pp. 759–762, November, 1977.     

Differentiation of fractional order onp-groups,approximation properties

G- p- . [5] - (G) L _r(G) (1r<), . . , - . , , , . . , X. , . (. [1], [2] [4]).     

Inverse problem of the scattering of plane waves for a class of layered media

The direct and inverse problems of the scattering of plane waves in a layered, inhomogeneous medium are considered in the paper. In the appropriate variables the wave equation of the problem has the formu (z,)=Q(Z)u _ZZ(Z,), – < Z, <, Q(Z)|_Z<01. A special feature of the case considered, in contrast to those studied earlier, is that Q(Z)|_Z0 may change sign; because of this, the equation of the problem is, in general, an equation of mixed type. The correct formulation of the direct problem for such an equation and the study of the properties of its solution form a necessary step in the investigation. For a very broad class of media including cases of Q(z) of variable sign (Q(z) can change sign by a jump a finite number of times without vanishing anywhere) a procedure is developed for solving the corresponding inverse problem of determining Q(z) on the basis of the scattering datau(0,)|_(–,). This procedure makes it possible to recover Q(z) for all z[0,). The solution of the inverse problem is unique in this class.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 78, pp. 30–53, 1978.The author thanks his scientific supervisor A. S. Blagoveshchenskii for his constant attention and assistance in the work.     

A variant of an infinite-dimensional local limit theorem

Let _n be a sequence of independent, identically distributed random elements in a separable Banach space X, for which the CLTholds: the normalized sums (₁+...+_n)/n^1/2 converge weakly to the Gaussian random element . It is proved that, under certain conditions on the distribution of ₁ and on the measurable mappingf: X R¹, the distribution of the random variable converges in variation to the distribution of the variablef().Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 177, pp. 46–50, 1989.     

On a transformation of the wiener process in ℝ m by a functional of the local time type on a surface

A transformation of the Wiener process _t in ^m is considered. This transformation is realized by a multiplicative functional _l=u(_l/u(₀), where the functionu is constructed in a certain way by using a functional of the local time type on a surface. It is proved that this transformation is equivalent to the successive application of an absolutely continuous change of a measure and killing on the surface.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 6, pp. 863–866, June, 1993.