Related papers: On a new measure on infinite dimensional unite cub…
A new sequential approach to investigations of structure of metric spaces at infinity is proposed. Criteria for finiteness and boundedness of metric spaces at infinity are found.
The classic Thue--Morse measure is a paradigmatic example of a purely singular continuous probability measure on the unit interval. Since it has a representation as an infinite Riesz product, many aspects of this measure have been studied…
We introduce the notion of coarse metric. Every coarse metric induces a coarse structure on the underlying set. Conversely, we observe that all coarse spaces come from a particular type of coarse metric in a unique way. In the case when the…
In this contribution I review rigorous formulations of a variety of limitations of measurability in quantum mechanics. To this end I begin with a brief presentation of the conceptual tools of modern measurement theory. I will make precise…
Let (X,d) be a finite metric space. This paper first discusses the spectrum of the p-distance matrix of a finite metric space of p-negative type and then gives upper and lower bounds for the so called gap of a finite metric space of strict…
In this mainly pedagogical article, we discuss under what circumstances measurements play a special role in quantum processes. In particular we discuss the following facts which appear to be a common area of confusion: i) from a fundamental…
The paper deals with measures of nonlinearity. In state estimation, they are utilized i) to select a suitable state estimation algorithm by assessing the nonlinearity of a system model, ii) to adapt the estimation algorithm structure or…
Due to considerable recent interest in the use of density matrices for a wide variety of purposes, including quantum computation, we present a general method for their parameterizations in terms of Euler angles. We assert that this is of…
For every nonempty compact convex subset $K$ of a normed linear space a (unique) point $c_K \in K$, called the generalized Chebyshev center, is distinguished. It is shown that $c_K$ is a common fixed point for the isometry group of the…
We study some new invariant measures arising from local inverse iterates. Examples are also given.
The quantum measurement incompatibility is a distinctive feature of quantum mechanics. We investigate the incompatibility of a set of general measurements and classify the incompatibility by the hierarchy of compatibilities of its subsets.…
Let $(X,d)$ be a compact metric space. We consider the behavior of probability measures $\mu$ with the property that $$ \int_{X} d(x, y) d\mu(y) \qquad \mbox{is independent of}~x \in X.$$ It appears that such measures, when they exist,…
The quantum metric tensor was introduced for defining the distance in the parameter space of a system. However, it is also useful for other purposes, like predicting quantum phase transitions. Due to the physical information this tensor…
This is an exposition of the theory of differentiable structures on metric measures spaces, in the sense of Cheeger and Keith.
A new refinement of the triangle inequality is presented in normed linear spaces. Moreover, a simple characterization of inner product spaces is obtained by using the skew-angular distance.
We introduce a new intrinsic metric in subdomains of a metric space and give upper and lower bounds for it in terms of well-known metrics. We also prove distortion results for this metric under quasiregular maps.
A new infinite series of Einstein metrics is constructed explicitly on S^2 x S^3, and the non-trivial S^3-bundle over S^2, containing infinite numbers of inhomogeneous ones. They appear as a certain limit of a nearly extreme 5-dimensional…
We describe all boundedly finite measures which are invariant by Cartesian powers of an infinite measure preserving version of Chacon transformation. All such ergodic measures are products of so-called diagonal measures, which are measures…
The notion of $\ast$-idempotent measure is a modification of the notion of idempotent measure defined for every triangular norm $\ast$. We prove existence and uniqueness of invariant $\ast$-idempotent measures for iterated function systems…
We discuss various infinite-dimensional configuration spaces that carry measures quasiinvariant under compactly-supported diffeomorphisms of a manifold M corresponding to a physical space. Such measures allow the construction of unitary…