Related papers: Abstract Action Spaces and their topological and d…
The aim of this note is to introduce a notion of dynamical entropy, which we call infinite-product entropy, for probability measures on (countable) infinite cartesian product of any measurable space with itself. The idea behind the…
We construct an example of a real-valued continuous non-constant function $f$ defined on a connected complete metric space $X$ such that every point of $X$ is a point of local minimum or local maximum for $f$. The space $X$ is connected but…
The notion that the geometry of our space-time is not only a static background but can be physically dynamic is well established in general relativity. Geometry can be described as shaped by the presence of matter, where such shaping…
We present results on simplifying an acting group preserving properties of actions: transitivity, being a coset space and preserving a fixed equiuniformity in case of a $G$-Tychonoff space.
We present a new partial order reduction method for reachability analysis of nondeterministic labeled transition systems over metric spaces. Nondeterminism arises from both the choice of the initial state and the choice of actions, and the…
In this paper we present a new characterization of free group actions (in classical differential geometry), involving dynamical systems and representations of the corresponding transformation groups. In fact, given a dynamical system, we…
We focus on autonomously generating robot motion for day to day physical tasks that is expressive of a certain style or emotion. Because we seek generalization across task instances and task types, we propose to capture style via cost…
If $N \subset \R$ is a separable II$_1$-factor, the space $\Hom(N,\R)$ of unitary equivalence classes of unital *-homomorphisms $N \to \R$ is shown to have a surprisingly rich structure. If $N$ is not hyperfinite, $\Hom(N,\R)$ is an…
Spontaneous directed motion, a hallmark of cell biology, is unusual in classical statistical physics. Here we study, using both numerical and analytical methods, organized motion in models of the cytoskeleton in which constituents are…
In a space-time $M$ with a Killing vector field $\xi^a$ which is either everywhere timelike or everywhere spacelike, the collection of all trajectories of $\xi^a$ gives a 3-dimension space $S$. Besides the symmetry-reduced action from that…
Active matter encompasses different nonequilibrium systems in which individual constituents convert energy into non-conservative forces or motion at the microscale. This review provides an elementary introduction to the role of topology in…
Motivated by persistent homology and topological data analysis, we consider formal sums on a metric space with a distinguished subset. These formal sums, which we call persistence diagrams, have a canonical 1-parameter family of metrics…
Classical finite association schemes lead to a finite-dimensional algebras which are generated by finitely many stochastic matrices. Moreover, there exist associated finite hypergroups. The notion of classical discrete association schemes…
Let $(X, d)$ be a compact metric space and let $\mathcal{M}(X)$ denote the space of all finite signed Borel measures on $X$. Define $I \colon \mathcal{M}(X) \to \R$ by \[I(\mu) = \int_X \int_X d(x,y) d\mu(x) d\mu(y),\] and set $M(X) = \sup…
A general framework for investigating topological actions of $Z^d$ on compact metric spaces was proposed by Boyle and Lind in terms of expansive behavior along lower-dimensional subspaces of $R^d$. Here we completely describe this expansive…
It is proposed that gravity may arise in the low energy limit of a model of matter fields defined on a special kind of a dynamical random lattice. Time is discretized into regular intervals, whereas the discretization of space is random and…
We give an explicit description of all finite Borel measures on Noetherian topological spaces X, and characterize them as objects dual to a space of functions on X. We use these results to study the asymptotic behavior of continuous…
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.…
What defines an action like "kicking ball"? We argue that the true meaning of an action lies in the change or transformation an action brings to the environment. In this paper, we propose a novel representation for actions by modeling an…
We wish to investigate some elementary problems concerning topological dynamics revolving around our proposed definition of escaping set. We also discuss the notion of escaping set in the induced dynamics of the hyperspace. Moreover, we…