Related papers: Pointwise Assouad dimension for measures
In this paper we establish concavity properties of two extensions of the classical notion of the outer parallel volume. On the one hand, we replace the Lebesgue measure by more general measures. On the other hand, we consider a functional…
In this manuscript, we show how conformal invariance can be incorporated in a classical theory of gravitation, in the context of metric measure space. Metric measure space involves a geometrical scalar $f$, dubbed as density function, which…
The concept of depth has proved very important for multivariate and functional data analysis, as it essentially acts as a surrogate for the notion a ranking of observations which is absent in more than one dimension. Motivated by the rapid…
In this paper we define a numerical shape invariant of a continuous map called shape dimension of a map, which generalizes the shape dimension of a topological space. Some basic properties and applications of this invariant are given. The…
We prove existence of (at most denumerable many) absolutely continuous invariant probability measures for random one-dimensional dynamical systems with asymptotic expansion. If the rate of expansion (Lyapunov exponents) is bounded away from…
Working on doubling metric spaces, we construct generalised dyadic cubes adapting ultrametric structure. If the space is complete, then the existence of such cubes and the mass distribution principle lead into a simple proof for the…
The hermitian analog of Aleksandrov's area measures of convex bodies is investigated. A characterization of those area measures which arise as the first variation of unitarily invariant valuations is established. General smooth area…
Bayesian inference is used to estimate continuous parameter values given measured data in many fields of science. The method relies on conditional probability densities to describe information about both data and parameters, yet the notion…
We construct measure which determines a two-variable mean in a very natural way. Using that measure we can extend the mean to infinite sets as well. E.g. we can calculate the geometric mean of any set with positive Lebesgue measure. We also…
The problem of quantization of general relativity is considered in the framework of noncommutative differential geometry. Operator analogues for interval, scalar curvature, values of the Einstein tensor are proposed. Quantum measurements of…
In this work, we are interested in characterizing typical (generic) dimensional properties of invariant measures associated with the full-shift system, $T$, in a product space whose alphabet is a countable set. More specifically, we show…
We discuss a variation of Gromov's notion of asymptotic dimension that was introduced and named Nagata dimension by Assouad. The Nagata dimension turns out to be a quasisymmetry invariant of metric spaces. The class of metric spaces with…
Projections of finite dimensional sets and their measures are investigated in infinite-dimensional power measure spaces. The starting point is the known algebraic formula, expressing \ the $y$-projection of a finite-dimensional set $a$ as a…
In this manuscript, a conformally invariant theory of gravitation in the context of metric measure space is studied. The proposed action is invariant under both diffeomorphism and conformal transformations. Using the variational method, a…
We compute explicitly traces of the Dirichlet form related to the Bessel process with respect to discrete measures as well as measures of mixed type. Then some global properties of the obtained Dirichlet forms, such as conservativeness,…
We consider suitable weak solutions of 2-dimensional Euler equations on bounded domains, and show that the class of completely random measures is infinitesimally invariant for the dynamics. Space regularity of samples of these random fields…
This article explores the generalized analysis-of-variance or ANOVA dimensional decomposition (ADD) for multivariate functions of dependent random variables. Two notable properties, stemming from weakened annihilating conditions, reveal…
This paper is devoted to study the qualitative properties of hybrid measure differential equations (HMDEs, for short). We establish several results on the existence of global solutions, including the existence of regulated, continuous,…
We consider the case with boundary of the classical Kazdan-Warner problem in dimension greater or equal than three, i.e. the prescription of scalar and boundary mean curvatures via conformal deformations of the metric. We deal in particular…
The problem is posed of further extending the axiomatic construction proposed in Part 1 for non-local point transformations mapping in each other different curved space times. The new transformations apply to curved space times when…