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.     

On the concentration of stresses due to a small elliptic inclusion on the neutral axis of a deep beam under constant bending moment

Zusammenfassung Es wird die Spannungsverteilung untersucht, die sich in einem breiten Balken mit konstanter Höhe unter einem konstanten Biegemoment ausbildet, wenn er eine kleine elliptische Einschliessung mit Zentrum auf der Neutralachse enthält. Insbesondere werden die Fälle eines sehr starren Einschlusses sowie eines elliptischen Loches im Detail diskutiert.

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Nomenclature x, y Cartesian coordinates - , elliptic coordinates - u, v (u ,u )=components of displacements - , unit elongations in -and -directions - shearing strain - , normal stress components in elliptical coordinates - shearing stress in elliptic coordinates - _x , _y normal stress in Cartesian coordinates - _xy shearing stress in Cartesian coordinates - E Young's modulus for the beam - v Poisson's ratio for the beam - 1/h ₁, 1/h ₂ stretch ratios - e _x + _y dilatation - 2 rotation - M bending moment     

On the structure of stationary flat processes. II

Summary The present paper continues the work by Davidson, Krickeberg, Papangelou, and the author on proving, under weakest possible assumptions, that a stationary random measure or a simple point process on the space of k-flats in R ^d is a.s. invariant or a Cox process respectively. The problems for and are related by the fact that is Cox whenever the Papangelou conditional intensity measure of (a thinning of) is a.s. invariant. In particular, is shown to be a.s. invariant, whenever it is absolutely continuous with respect to some fixed measure and has no (so called) outer degeneracies. When k=d–22, no absolute continuity is needed, provided that the first moments exist and that has no inner degeneracies either. Under a certain regularity condition on , it is further shown that and are simultaneously non-degenerate in either sense.     

Spline polynomials with a prescribed sequence of extrema

In the present note a theorem about strong suitability of the space of algebraic polynomials of degree n in C[a,b] (Theorem A in [1]) is generalized to the space of spline polynomials _{[a, b} ^{]n, k} (n2, 0) in C[a, b]. Namely, it is shown that the following theorem is valid: for arbitrary numbers ₀, ₁, ..., _n+k, satisfying the conditions (_i–_i–1) (_{i+1{ i}< 0(i=1, ..., n +k–1), there is a unique polynomials _n,k (t) _{[a, b} ^]/n,k and pointsa=₀,<₁<...< _n+k– 1< _n+k = b (₁1 <_n, ..., _kk<_n+k–1), such that s_n,k(_i) = _i(i=0, ..., n + k), s_n,k(_i)=0 (i=1, ..., n + k–1).Translated from Matematicheskii Zametki, Vol. 11, No. 3, pp. 251–258, March, 1972.     

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.     

On 3-dimensional contact metric manifolds with ▽ξτ=0

We give the classification of the 3-dimensional contact metric manifolds satisfying =0, which they have: harmonic curvature, or -parralel Ricci tensor or cyclic -parallel Ricci tensor.     

Stochastic models of homogeneity

The probabilityp()=P{–<) is used as a characteristic of homogeneity of the system (, ) of independent identically distributed random variables. We consider some estimation methods of this characteristic and a method of classification of observations minimizing the between-class dissimilarity measure.Translated from Statisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 120–123, 1986.     

Classification of (fξηρ)-structures

The classification of (f)-structures on differentiable manifolds of even and odd dimensions is given.Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 14, pp. 57–72, 1983.     

Two classes of conformally flat contact metric 3-manifolds

We give a classification of 3—dimensional conformally flat contact metric manifolds satisfying: =0(=L g) orR(Y, Z)=k[(Z)Y–(Y)Z]+[(Z)hY]–(Y)hZ] wherek and are functions. It is proved that they are flat (the non-Sasakian case) or of constant curvature 1 (the Sasakian case).     

Non-Uniformly Elliptic Equations with Natural Growth in the Gradient

We prove the existence of bounded solutions for a class of nonlinear elliptic problems of type–div(a(x,u,Du))=H(x,u,Du)+f, uW ^1,p ₀()L (),where a(x,,)b(||)|| ^p , b is a continuous monotone decreasing function and |H(x,,)| k()|| ^p , k is a continuous monotone increasing function.     

On minimax filtration of vector processes

We study the problem of optimal linear estimation of the transformation of a stationary random process (t) with values in a Hilbert space by observations of the process (t) + (t) fort0. We obtain relations for computing the error and the spectral characteristic of the optimal linear estimate of the transformationA for given spectral densities of the processes (t) and (t). The minimax spectral characteristics and the least favorable spectral densities are obtained for various classes of densities.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 3, pp. 389–397, March, 1993.     

Asymptotic behavior of the stationary distributions in the GI/PH/c queue with heterogeneous servers

Summary This paper deals with the stablec-server queue with renewal input. The service time distributions may be different for the various servers. They are however all probability distributions of phase type. It is shown that the stationary distribution of the queue length at arrivals has an exact geometric tail of rate , 0<<1. It is further shown that the stationary waiting time distribution at arrivals has an exact exponential tail of decay parameter >0. The quantities and may be evaluated together by an elementary algorithm. For both distributions, the multiplicative constants which arise in the asymptotic forms may be fully characterized. These constants are however difficult to compute in general.This research was supported by the National Science Foundation under Grant Nr. ENG-7908351 and by the Air Force Office of Scientific Research under Grant Nr. AFOSR-77-3236.     

Reversible subsets of certain linear transformations

Summary Suppose U is a set,F is a field of subsets of U, p_AB is the set of all real-valued bounded finitely additive functions defined onF, and for each in p_AB, A()={: in p_AB, absolutely continuous with respect to }. SupposeM is a linear subspace of p_AB such that . A generalisation of a previously discussed collection of linear transformations (see J. London Math. Soc., vol. 44 (1969), pp. 385–396) is treated by letting C_M denote the set to which T belongs iff T is a linear transformation from M into p_AB such that for some K inR and all in M and V inF, . Certain theorems of the aforementioned reference are generalized, as well as one of Trans. Amer. Math. Soc., vol. 199, (1974), pp. 131–140. The principal result of the present paper is the following generalisation of a reversibility characterisation in the first mentioned reference: Theorem: If T is in C_M, then (, T()): in M A(T()) is the only reversible subset T⁰ if T such that: i) the domain M⁰ of T⁰ is a linear subspace of M and , and ii) the range of T⁰ is the range of T.     

On the approximate calculation of the coefficient of correlation between two functions of dependent random variates

A general formula is derived for the calculation of the coefficient of correlation between functions of random variates. A formula is presented for the approximate calculation of the correlation coefficient with error estimation. The case of the calculation of the coefficient of correlation between ln and ln with and connected by gamma correlation is considered in detail.Translated from Matematicheskie Zametki, Vol. 7, No. 5, pp. 617–625, May, 1970.     

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.

     

An explicit solution of the linear filtering problem for certain classes of nonrational spectral densities

We present an explicit solution of the problem of optimal linear filtering: the recovery of the useful signal(s) at the instantt+, (>0,<0, or=0) from known values of the received signal(s)=(s)+(s) in the past, i.e., at the instantt–s, s0. In doing so we assume the random processes(s) and /gr(s) are stationary and jointly stationary, while the stationary process of noise (s) with zero mean is assumed to be mutually correlated and jointly stationary with the process(s) under the assumption that there exists a common spectral densityf() for these processes.Translated fromTeoriya Sluchaínykh Protsessov, Vol. 14, pp. 83–91, 1986.     

Extrapolation of transformations of random processes perturbed by white noise

We consider the problem of linear mean square optimal estimation of transformation of a stationary random process (t) in observations of process (t) + n(t) for t < – 0, where (t) is white noise uncorrelated with (t). We find least favorable spectral densities f₀() D and minimax (robust) spectral characteristics of an optimal estimator of transformation A for various classesD of densities.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 2, pp. 216–223, February, 1991.     

Distribution of the supremum of sums of independent variables with negative drift

Let {_n} be a sequence of identically distributed independent random variables,M₁=<0,M ₁ ² <;S ₀=0,S _n =₁+₂,+...+ _{n, n}1;¯ S=sup {S _n n=0.} The asymptotic behavior ofP(¯ St) as t is studied. If _t ^P (₁x dx=0((t)), thenP(¯ St)– 1/¦¦ _t P (₁x dx=0((t)) (t) is a positive function, having regular behavior at infinity.Translated from Matematicheskie Zametki, Vol. 22, No. 5, pp. 763–770, November, 1977.The author thanks B. A. Rogozin for the formulation of the problem and valuable remarks.     

Asymptotic behavior of the dwell time distribution for a random walk on a positive semi-axis

Let ₁, ₂, ... be a sequence of independent identically distributed random variables with zero means. We consider the functional _n = _k=o ⁿ (S _k ) where S₁=0, S_k= _i=1 ^k _i (k1) and(x)=1 for x0,(x) = 0 for x<0. It is readily seen that _n is the time spent by the random walk S_n, n0, on the positive semi-axis after n steps. For the simplest walk the asymptotics of the distribution P (_n = k) for n and k, as well as for k = O(n) and k/n<1, was studied in [1]. In this paper we obtain the asymptotic expansions in powers of n^–1 of the probabilities P(h_n = nx) and P(nx_{1 n} nx₂) for 0<₁, x = k/n ₂<1, 0<₁x₁2₂<1.Translated from Matematicheskie Zametki, Vol. 15, No. 4, pp. 613–620, April, 1974.The author wishes to thank B. A. Rogozin for valuable discussions in the course of his work.     

Contact metric manifolds

In this paper we study contact metric manifoldsM ²ⁿ⁺¹(, , ,g) with characteristic vector field belonging to thek-nullity distribution. Moreover we prove that there exist i) nonK-contact, contact metric manifolds of dimension greater than 3 with Ricci operator commuting with and ii) 3-dimensional contact metric manifolds with non-zero constant -sectional curvature.