On the product formula on non-compact Grassmannians

We study the absolute continuity of the convolution $\delta_{e^X}^\natural \star \delta_{e^Y}^\natural$ of two orbital measures on the symmetric space SO_0(p,q)/SO(p)\timesSO(q), $q>p$. We prove sharp conditions on $X$, Y\in\a for the existence of the density of the convolution measure. This measure intervenes in the product formula for the spherical functions. We show that the sharp criterion developed for \SO_0(p,q)/\SO(p)\times\SO(q) will also serve for the spaces SU(p,q)/S(U(p)\timesU(q)) and Sp(p,q)/Sp(p)\timesSp(q), $q>p$. We also apply our results to the study of absolute continuity of convolution powers of an orbital measure $\delta_{e^X}^\natural$

Duality and spatial inhomogeneity

Within the framework on non-extensive thermostatistics we revisit the recently advanced q-duality concept. We focus our attention here on a modified q-entropic measure of the spatial inhomogeneity for binary patterns. At a fixed length-scale this measure exhibits a generalised duality that links appropriate pairs of q and q' values. The simplest q q' invariant function, without any free parameters, is deduced here. Within an adequate interval q < qo < q', in which the function reaches its maximum value at qo, this invariant function accurately approximates the investigated q-measure, nitidly evidencing the duality phenomenon. In the close vicinity of qo, the approximate meaningful relation q + q' = 2qo holds.Comment: Contribution to International School and Conference on "Non Extensive Thermodynamics and physical applications", Villasimius-Capo Boi (Cagliari), Italy, 23-30 May 2001, 6 pages, 2 figures, replaced with published versio

Correlation Exponent and Anomalously Localized States at the Critical Point of the Anderson Transition

We study the box-measure correlation function of quantum states at the Anderson transition point with taking care of anomalously localized states (ALS). By eliminating ALS from the ensemble of critical wavefunctions, we confirm, for the first time, the scaling relation z(q)=d+2tau(q)-tau(2q) for a wide range of q, where q is the order of box-measure moments and z(q) and tau(q) are the correlation and the mass exponents, respectively. The influence of ALS to the calculation of z(q) is also discussed.Comment: 6 pages, 3 figure

Necessary and sufficient conditions for the existence of the q-optimal measure

This paper presents the general form and essential properties of the q-optimal measure following the approach of Delbaen and Schachermayer (1996) and proves its existence under mild conditions. Most importantly, it states a necessary and sufficient condition for a candidate measure to be the q-optimal measure in the case even of signed measures. Finally, an updated characterization of the q-optimal measure for continuous asset price processes is presented in the light of the counterexample appearing in Cerny and Kallsen (2006) concerning Hobson's (2004) approach

Algorithmic Randomness and Capacity of Closed Sets

We investigate the connection between measure, capacity and algorithmic randomness for the space of closed sets. For any computable measure m, a computable capacity T may be defined by letting T(Q) be the measure of the family of closed sets K which have nonempty intersection with Q. We prove an effective version of Choquet's capacity theorem by showing that every computable capacity may be obtained from a computable measure in this way. We establish conditions on the measure m that characterize when the capacity of an m-random closed set equals zero. This includes new results in classical probability theory as well as results for algorithmic randomness. For certain computable measures, we construct effectively closed sets with positive capacity and with Lebesgue measure zero. We show that for computable measures, a real q is upper semi-computable if and only if there is an effectively closed set with capacity q