A note on occupation times of random walks

In this note I show that some asymptotic results on the average occupancy time of an interval derived for lattice random walks with negative exponential transition probabilities are true for all random walks whose transition probabilities have a finite variance. The proof is based on the continuum limit.     

On a first-passage problem for a cumulative process with exponential decay

A first-passage problem for a cumulative process is investigated. The cumulative process is assumed to be generated by a Poisson process, and the amplitude generated by an event is assumed to decay exponentially. An integral equation for the probability density of the first-passage time until the total amplitude exceeds a pre-specified threshold level is derived. The Laplace transform of the probability density of the first-passage time is obtained explicity when each amplitude generated by an event is distributed exponentially. The mean first-passage times are given in a closed form and plotted versus the threshold level.     

Reply to “Comment on ‘Calculation of Two-center Nuclear Attraction Integrals over Integer and Noninteger N-slater-type Orbitals in Nonlined-up Coordinate Systems” ’

The comments of Guseinov on our recent paper (T. Özdo?an, S. Gümü? and M. Kara, J. Math. Chem. 33 (2003) 181) are critically analyzed. It is proved that the expansion formula for the product of two normalized associated Legendre functions and the expressions for two-center nuclear attraction integrals over Slater type orbitals have been obtained independently, by the use of basic mathematical rules.     

Lack of natural weighted estimates for some singular integral operators

We show that the classical Hörmander condition, or analogously the -Hörmander condition, for singular integral operators is not sufficient to derive Coifman's inequality

where , is the Hardy-Littlewood maximal operator, is any weight and is a constant depending upon and the constant of . This estimate is well known to hold when is a Calderón-Zygmund operator.

As a consequence we deduce that the following estimate does not hold:

where and where is an arbitrary weight. However, by a recent result due to A. Lerner, this inequality is satisfied whenever is a Calderón-Zygmund operator.

One of the main ingredients of the proof is a very general extrapolation theorem for weights.

     

Some Processes Associated with Fractional Bessel Processes

Let be a d-dimensional fractional Brownian motion with Hurst parameter H and let be the fractional Bessel process. Itôs formula for the fractional Brownian motion leads to the equation . In the Brownian motion case is a Brownian motion. In this paper it is shown that X_t is not an -fractional Brownian motion if H 1/2. We will study some other properties of this stochastic process as well.     

Some Categorical Algebraic Properties: Counter-Examples for Functor Categories

This work is a complement to the authors earlier papers, where it is shown that a functor category inherits from such properties as amalgamation, transferability and congruence extension if has either products or certain pushouts. A general scheme is given for constructing counter-examples which show that the latter condition on is essential. In particular, it is shown that the functor categories , , ( resp.) do not satisfy the amalgamation (congruence extension resp.) property in general. Moreover, one class of categories is described, where the condition of the existence of certain pushouts is not only sufficient, but also necessary for to preserve the considered properties of .Mathematics Subject Classifications (2000) 18A25, 18A32, 18B99, 08B26.Dali Zangurashvili: The support rendered by INTAS Grant 97 31961 is gratefully acknowledged.     

WEAK TYPE（H^1 ,L^1） ESTIMATE FOR COMMUTATOR OF MARCINKIEWICZ INTEGRAL

Let μΩ,b be the commutator generalized by the n-dimensional Marcinkiewicz integral μΩ and a function b∈ BMO（R^n）. It is proved that μΩ,bis bounded from the Hardy space H^1 （R^n） into the weak L^1（R^n） space.     

A spectral characterization of the -torus by the first stability eigenvalue

Let be a compact hypersurface with constant mean curvature immersed into the unit Euclidean sphere . In this paper we derive a sharp upper bound for the first eigenvalue of the stability operator of in terms of the mean curvature and the length of the total umbilicity tensor of the hypersurface. Moreover, we prove that this bound is achieved only for the so-called -tori in , with . This extends to the case of constant mean curvature hypersurfaces previous results given by Wu (1993) and Perdomo (2002) for minimal hypersurfaces.

     

-additive families and the invariance of Borel classes

We prove that any -additive family of sets in an absolutely Souslin metric space has a -discrete refinement provided every partial selector set for is -discrete. As a corollary we obtain that every mapping of a metric space onto an absolutely Souslin metric space, which maps -sets to -sets and has complete fibers, admits a section of the first class. The invariance of Borel and Souslin sets under mappings with complete fibers, which preserves -sets, is shown as an application of the previous result.