Related papers: Marked metric measure spaces
Let mu(r) be the Bernoulli measure on the Cantor space given as the infinite product of two-point measures with weights r and 1-r. It is a long-standing open problem to characterize those r and s such that mu(r) and mu(s) are topologically…
Let $\mu$ be a self-similar measure generated by iterated function system of four maps of equal contraction ratio $0<\rho<1$. We study when $\mu$ is a spectral measure which means that it admits an exponential orthonormal basis $\{e^{2\pi i…
The group-theoretic method for constructing symmetric isometric embeddings is used to describe all possible four-dimensional surfaces in flat $(1,9)$-dimensional space, whose induced metric is static and spherically symmetric. For such…
Calder\'on-Zygmund theory has been traditionally developed on metric measure spaces satisfying additional regularity properties. In the lack of good metrics, we introduce a new approach for general measure spaces which admit a Markov…
The space of measured laminations $\mathcal{ML}(\Sigma)$ associated to a topological surface $\Sigma$ of genus $g$ with $n$ punctures is an integral piecewise linear manifold of real dimension $6g-6+2n$. There is also a natural symplectic…
We consider a system of weak* closed sets of finite-dimensional distributions. We show that a corresponding system of random variables can be defined on a probability space with a probability measure determined up to some set of measures,…
We introduce three metrics on the set of quantum probability measures over a compact Hausdorff space and characterize them in terms of the completely bounded norm of the corresponding unital completely positive maps. We extend the existing…
We study isometric $G$-spaces and the question of when their maximal equivariant compactification is the Gromov compactification (meaning that it coincides with the compactification generated by the distance functions to points). Answering…
In this paper, we defined three kinds of measures depending on the given finite directed graphs. For the given finite directed graph, we can construct the free semigroupoid, the diagram set and the reduced diagram set, as algebraic…
In this paper, we show that the metric space Q is a positively-curved space (PC-space) in the sense of Alexandrov. We also discuss some issues like metric tangent cone and exponential map of Q. Then we give a stratification of this metric…
We consider a family of measures on a $q$-homogeneous tree that decrease exponentially with respect to the distance from the origin. Such measures are doubling with respect to the Gromov distance. We define atomic Hardy and BMO spaces for…
To give characterizations of monotonically countably paracompact spaces with set-valued maps, Yamazaki [22] introduced the notion of strictly increasing closed cover of a topological space with which the boundedness of a set-valued map was…
A Hilbert space embedding for probability measures has recently been proposed, with applications including dimensionality reduction, homogeneity testing, and independence testing. This embedding represents any probability measure as a mean…
Finite metric spaces arise in many different contexts. Enormous bodies of data, scientific, commercial and others can often be viewed as large metric spaces. It turns out that the metric of graphs reveals a lot of interesting information.…
Ergodicity of random dynamical systems with a periodic measure is obtained on a Polish space. In the Markovian case, the idea of Poincar\'e sections is introduced. It is proved that if the periodic measure is PS-ergodic, then it is ergodic.…
We introduce and investigate in this short report the new notion of uniform measure (distribution) on the arbitrary compact metric space. We consider also some possible applications of these measures in the theory of imbedding theorems and…
We study a class of Markov chains that model the evolution of a quantum system subject to repeated measurements. Each Markov chain in this class is defined by a measure on the space of matrices. It is then given by a random product of…
Here we study what we call bounded rough Riemannian metrics $(M,g)$, which are positive definite, symmetric tensors on each tangent space, $T_pM$, which are bounded and measurable as functions in coordinates. This is enough structure to…
We generalize subgraph densities, arising in dense graph limit theory, to Markov spaces (symmetric measures on the square of a standard Borel space). More generally, we define an analogue of the set of homomorphisms in the form of a measure…
We investigate the geometry of the family $\cal M$ of isometry classes of compact metric spaces, endowed with the Gromov-Hausdorff metric. We show that sufficiently small neighborhoods of generic finite spaces in the subspace of all finite…