Singularity and L2-dimension of self-similar measures

We study self-similar measures defined by non-uniformly contractive iterated function systems of similitudes with overlaps. In the case the contraction ratios of the similitudes are exponentially commensurable, we describe a method to compute the L²-dimension of the associated self-similar measures. Our result allows us to determine the singularity of some of such measures.     

Invariant measures and maximal L2 regularity for nonautonomous Ornstein-Uhlenbeck equations

We characterize the domain of the realizations of a linear parabolicoperator defined in L² spaces with respect to a suitable measurethat is invariant for the associated evolution semigroup. Asa byproduct, we obtain optimal L² regularity results for evolutionequations with time-dependant Ornstein–Uhlenbeck operators.     

Extensions of spaces with cylindrical measures and supports of measures determined by the Lévy Laplacian

Extensions of noncountably additive (cylindrical) measures are described, and examples of Hilbert supports of the Lévy-Gauss measure are given.Translated fromMatematicheskie Zametki, Vol. 64, No. 4, pp. 483–492, October, 1998.L. Accardi acknowledges partial support of the Russian Foundation for Basic Research under grant No. 96-01-00030.     

Quantifying the value of reconnaissance using Lanchesterian type equations

This paper presents a method to quantify the value of reconnaissance for both direct and indirect fire weapons for the defense in sector battle scenario. The Lanchester area fire model and the Helmbold equations were modified to allow the lethality of the defending blue force to be increased as they gained more combat intelligence about the attacking red force, thus modeling intelligence as a true combat multiplier. By adjustments made to parameters in the model, the lethality of the blue's direct and indirect fire weapons could be adjusted based on the quantity and quality of their intelligence assets. With information from a computer database, and the COMAN model, maximum likelihood attrition rate estimates were calculated for both red and blue forces for ten heavy defensive battles conducted at the Army's National Training Center. In each battle, the red force attrition rate was fit to a curve which represented a percentage of blue's full potential, represented here by the square law. Using this model in a combat simulation, and with some preliminary work with comparable systems, one could implement a change in blue's intelligence assets and then provide a quantitative measure of the effect that this had on the outcome of a battle     

Elliptic operators with unbounded diffusion coefficients in L 2 spaces with respect to invariant measures

We study the self-adjoint and dissipative realization A of a second order elliptic differential operator with unbounded regular coefficients in , where μ(dx) = ρ (x)dx is the associated invariant measure. We prove a maximal regularity result under suitable assumptions, that generalize the well known conditions in the case of constant diffusion part. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

About the projection of kinematical quantities within the FE2-approach

Within the FE²-approach strain–like quantities are projected to the boundary of the attached microstructure yielding Dirichlet boundary conditions for the microscopic boundary value problem. By use of the divergence theorem one can show that the projection of gradient information to the boundary of the attached microstructure fulfills the condition that the volumetrical average of the strain–like quantities in the inner of the microstructure is exactly the same as the projected quantity, provided that the microstructure is continous. However, within the Cosserat continuum theory [1] the strain tensor depends directly on the extra rotation and not on the gradient of it. Thus, the projection can not be carried out in the same way as for gradient information. The present approach shows how to circumvent this problem by giving the extra–rotation within the Cosserat continuum theory a strong kinematical interpretation in the form of a displacement field, which shows the same effects as the extra–rotation itself. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Jones polynomial of knots formed by repeated tangle replacement operations

In this paper, we prove that the Jones polynomial of a link diagram obtained through repeated tangle replacement operations can be computed by a sequence of suitable variable substitutions in simpler polynomials. For the case that all the tangles involved in the construction of the link diagram have at most k crossings (where k is a constant independent of the total number n of crossings in the link diagram), we show that the computation time needed to calculate the Jones polynomial of the link diagram is bounded above by O(nk). In particular, we show that the Jones polynomial of any Conway algebraic link diagram with n crossings can be computed in O(n²) time. A consequence of this result is that the Jones polynomial of any Montesinos link and two bridge knot or link of n crossings can be computed in O(n²) time.     

Characterization of finite fuzzy measures using Markoff-kernels

Generalizing the definitions of L. A. Zadeh (J. Math. Anal. Appl. 23 (1968), 421–427) and E. P. Klement, R. Lowen, and W. Schwyhla (Fuzzy Sets and Systems, in press) a larger class of finite fuzzy measures is defined. It is shown that these fuzzy measures can be characterized in a unique way by a finite (classical) measure and a Markoff-kernel.     

Zeros of the Jones Polynomial for Torus Knots and 2-bridge Knots

We study zeros of the Jones polynomial and their distributions for torus knots and 2-bridge knots. We prove that e(2m+1)πi/2and e(2m+1)πi/4(m is a positive integer)can not be the zeros of Jones polynomial for torus knots T p,q by the knowledge of the trigonometric function. We elicit the normal form of Jones polynomials of the 2-bridge knot C(-2, 2, · · ·,(-1)r2) by the recursive form and discuss the distribution of their zeros.     

A note on the Poisson structures of the space of probability measures

The space of probability measures on a Riemannian manifold is endowed with the Fisher information metric. In [4] T. Friedrich showed that this space admits also Poisson structures {, }. In this note, we give directly another proof for the structure {, } being Poisson. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Minimal relative entropy for equivalent martingale measures by low-discrepancy sequence in Lévy process

In this paper, we focused on computing the minimal relative entropy between the original probability and all of the equivalent martin gale measure for the Lévy process. For this purpose, the quasiMonte Carlo method is used. The probability with minimal relative entropy has many suitable properties. This probability has the minimal Kullback-Leibler distance to the original probability. Also, by using the minimal relative entropy the exponential utility indifference price can be found. In this paper, the Monte Carlo and quasi-Monte Carlo methods have been applied. In the quasi-Monte Carlo method, two types of widely used lowdiscrepancy sequences, Halton sequence and Sobol sequence, are used. These methods have been used for exponential Lévy process such as variance gamma and CGMY process. In these two processes, the minimal relative entropy has been computed by Monte Carlo and quasi-Monte Carlo, and compared their results. The results show that quasi-Monte Carlo with Sobol sequence performs better in terms of fast convergence and less error. Finally, this method by fitting the variance gamma model and parameters estimation for the model has been implemented for financial data and the corresponding minimal relative entropy has been computed.     

Convergence of integrals with respect to L0-valued measures

Translated from Matematicheskie Zametki, Vol. 53, No. 5, pp. 102–106, May, 1993.